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http://clay6.com/qa/4454/prove-that-angle-in-a-semi-circle-is-a-right-angle-	Browse Questions # Prove that angle in a semi-circle is a right angle. Toolbox: • To prove angle between any two lines is rightangle, prove that the dot product of the two vectors =0 • $\large(\overrightarrow a+\overrightarrow b).(\overrightarrow a-\overrightarrow b)=\|\overrightarrow a\|^2-\|\overrightarrow b\|^2$ Let O be origin, A and A' be points at equal distance from O and are on positive and negative x-axis. Let AA' be diametre of semi circle. Let P be a point on the semicircle. $\Rightarrow \|\overrightarrow {OA}\|=\|\overrightarrow {OA'}\|=\|\overrightarrow {OP}\|=$radius of the circle 'r' and $\large\overrightarrow {OA}=-\overrightarrow {OA'}$ To prove that angle in a semicicle is right angle, We have to prove that angle between AP and A'P is 90$^\circ$ We know from triangle OPA, using triangular law of addition. $\large\overrightarrow {OP}+\overrightarrow {PA}=\overrightarrow {OA}$ $\Rightarrow \large\overrightarrow {PA}=\overrightarrow {OA}-\overrightarrow {OP}$ Similarly from triangle OPA' we get $\large\overrightarrow {OP}+\overrightarrow {PA'}=\overrightarrow {OA'}$ $\Rightarrow \large\overrightarrow {PA'}=\overrightarrow {OA'}-\overrightarrow {OP}$ But since $\large\overrightarrow {OA}=-\overrightarrow {OA'}$ $\Rightarrow \large\overrightarrow {PA'}=-\overrightarrow {OA}-\overrightarrow {OP}$ $\large\overrightarrow {PA}.\overrightarrow {PA'}=\big(\overrightarrow {OA}-\overrightarrow {OP}\big).-(\overrightarrow {OA}+\overrightarrow {OP}\big)$ We know that $\large(\overrightarrow a+\overrightarrow b).(\overrightarrow a-\overrightarrow b)=\|\overrightarrow a\|^2-\|\overrightarrow b\|^2$ $=\large\|\overrightarrow {OA}\|^2-\|\overrightarrow {OP}\|^2=0$ Because we know that $\|\overrightarrow {OA}\|=\|\overrightarrow {OP}\|$ If dot product of two vectors =0 then the angle between them is $90^{\circ}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9934786558151245, "perplexity": 1138.3393528097306}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501174163.72/warc/CC-MAIN-20170219104614-00630-ip-10-171-10-108.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/pendulum-with-a-twist.163886/	# Pendulum with a twist 1. Apr 3, 2007 ### ataglance05 1. The problem statement, all variables and given/known data A pendulum 2 meters long with a mass of 1kg is mounted on a circular platform on the earth's surface that's spinning at constant angular velocity of .12 rads/sec. The pendulum is mounted on a pole that's perpendicular to the platform at a distance of 5 meters from the center of rotation. If it's displaced for its equilibrium position, what will be the period of the pendulum? P.S. My physics teacher said that the diagrams below explains what's happening: FROM A BIRD'S EYE VIEW- FROM A NORMAL VIEW: Now, he mentioned how we must get the force from the equation F=mv2/r and then get the acceleration from a=F/m, but first I need to get r and I have no clue how to do that since theta isn't given. Then I have to replace the acceleration I get for g in the period equation. 2. Relevant equations a=F/m a=v2/r F=mv2/r 3. The attempt at a solution No clue!!!!!!!!?!?!?! Last edited by a moderator: May 2, 2017 2. Apr 3, 2007 ### Mentz114 Do you know how to work out the period of a pendulum that's not on a turntable ? One of the numbers in the equation is the 'return force' on the bob. You'll need to replace that with a different force for the rotating pendulum. Similar Discussions: Pendulum with a twist	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9136819243431091, "perplexity": 626.4761628971925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424910.80/warc/CC-MAIN-20170724202315-20170724222315-00656.warc.gz"}
https://ora.ox.ac.uk/objects/uuid:9ea529d1-d4e4-4c40-b1cc-83927dde47ad	Journal article ### Strong unitary and overlap uncertainty relations: Theory and experiment Abstract: We derive and experimentally investigate a strong uncertainty relation valid for any n unitary operators, which implies the standard uncertainty relation and others as special cases, and which can be written in terms of geometric phases. It is saturated by every pure state of any n-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any n þ 1 pure states, and gives an upper bound for the out-of-time-order correlation function. We test... Publication status: Published Peer review status: Peer reviewed ### Authors More by this author Institution: University of Oxford Division: MPLS Division Department: Physics Subgroup: Atomic & Laser Physics Role: Author ORCID: 0000-0001-8627-1298 Publisher: American Physical Society Publisher's website Journal: Physical Review Letters Journal website Volume: 120 Issue: 23 Pages: Article: 230402 Publication date: 2018-06-05 Acceptance date: 2018-05-01 DOI: EISSN: 1079-7114 ISSN: 0031-9007 Pubs id: pubs:911078 URN: UUID: Local pid: pubs:911078 Language: English Keywords:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9752351641654968, "perplexity": 4226.577887828357}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153803.69/warc/CC-MAIN-20210728220634-20210729010634-00203.warc.gz"}
https://portalrecerca.uab.cat/en/publications/polyakov-loop-and-the-hadron-resonance-gas-model	# Polyakov loop and the hadron resonance gas model E. Megías, E. Ruiz Arriola, L. L. Salcedo Research output: Contribution to journalArticleResearchpeer-review 43 Citations (Scopus) ## Abstract The Polyakov loop has been used repeatedly as an order parameter in the deconfinement phase transition in QCD. We argue that, in the confined phase, its expectation value can be represented in terms of hadronic states, similarly to the hadron resonance gas model for the pressure. Specifically, L(T) αg αe -Δ α/T, where g α are the degeneracies and Δ α are the masses of hadrons with exactly one heavy quark (the mass of the heavy quark itself being subtracted). We show that this approximate sum rule gives a fair description of available lattice data with N f=2+1 for temperatures in the range 150MeV<T<190MeV with conventional meson and baryon states from two different models. For temperatures below 150 MeV different lattice results disagree. One set of data can be described if exotic hadrons are present in the QCD spectrum while other sets do not require such states. © 2012 American Physical Society. Original language English 151601 Physical Review Letters 109 https://doi.org/10.1103/PhysRevLett.109.151601 Published - 9 Oct 2012	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8134950995445251, "perplexity": 2329.084339570382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141204453.65/warc/CC-MAIN-20201130004748-20201130034748-00556.warc.gz"}
http://mathoverflow.net/questions/44919/length-of-shortest-geodesic-and-cheegers-isoperimetric-constant-for-a-special-g	# Length of shortest geodesic and Cheeger's isoperimetric constant for a special genus 2 surface Let us take two copies of $Y$-pieces [ or pair of pants ] with each boundary geodesic of length $l$, and glue them together without any twisting to obtain a genus 2 closed orientable hyperbolic surface $M$. I want to find out the a) length of the shortest simple closed geodesic and b) Cheeger's isoperimetric constant for this surface. FYI : Cheeger's constant for a closed surface $M$ is defined as infimum of $\frac{l(A)}{minimum ( \area of B and \area of B')}$ where B and B' are the two components of $M \backslash A$, where infimum is taken over all 1 dimensional geodesic submanifolds A, i.e. union of simple closed geodesics A separating $M$ and boundary of A is a part of both boundary of B and that of B'. Any help ? I believe length of the shortest geodesic should be minimum over some multiples of l, coming from measuring the length of geodesics in $M$ which "we first intuitively see" while drawing a diagram of a genus 2 surface, i.e.the ones surrounding just one "hole", and the ones surrounding "both holes" in $M$, the ones surrounding the "handles".I also beleive the geodesics in the isotopy classe of curves "winding around" a collar of $M$ should have more length.But I want to make it rigorous. The reason I asked this question is if we can show that Cheeger's constant is say $\geq \frac{l}{100}$, then it gives us a concrete way to construct a cosed hyperbolic surface with arbitrarily large eigenvalue, by Cheeger's inequality. - The 3 geodesics "surrounding the handles" have length l by construction, the ones "running back and forth from one handle'surounding geodesic to the other is definitely $\geq \frac{l}{100}$ if we use the basic hyperbolic triangle identities and the same lower bound hold for the geodesics "joining the 1st "handle-surrounding" geodesic to the 3rd "handle surrounding" geodesic, then also we get a lower bound as well. – Analysis Now Nov 5 '10 at 4:53 As $l$ grows, the collars shrink, and there are geodesics crossing the collars that are short. – Sam Nead Nov 5 '10 at 9:51 Yes, but for this particular metric ( like any other hyperbolic metric ) on genus 2 surface, there would exist a shortest closed geodesic, I was interested in finding its length explicily or a lower bound. – Analysis Now Nov 5 '10 at 14:50 Any closed hyperbolic surface of genus $g$ admits some pants decomposition where each curve has length at most $B(g)$, the Bers constant. Thus, if I understand your question correctly, there is no way to achieve what you want in fixed genus. Google "Bers constant" for many papers on this topic. The standard reference is Buser's book "Geometry and spectra of compact Riemann surfaces". - Thanks, yes I read the book partly, I just remembered that for genus 2, $B_g$ is at most $21(g-1)$ . So, I cannot have a linear lower bound for the systoles for the family of the surfaces indexed by l, because then it will blow up. But honestly, I cannot see this very short geodesic in my special genus 2 surface ? – Analysis Now Nov 5 '10 at 14:54 Since you glued without twisting, the very short geodesic is exactly perpendicular to the boundary curve of the pants. – Sam Nead Nov 5 '10 at 15:06 Sam, if you have meant the ones perpendicular to the boundary geodesics , say of length 2y of the pants, then I guess we can explicitly find out their lengths. Considering the pants to be a union of isometric right-angled hexagons of 3 sides length l2 and other 3 sides, and decomposing the hexagons into two right-angled pentagons by dropping a perpendicular, we get, from pentagon identities: $cosh(0.25l)=sinh(0.5l)sinh(y)$.Which gives us : $sinh(y)=0.5(\frac{1}{sinh(0.25l)},$which is small if l is big, but which is big if l is small. – Analysis Now Nov 5 '10 at 15:59 by which I mean y or equivalently 2y, length of perpendiculsr geodesics. – Analysis Now Nov 5 '10 at 16:00 There is a universal constant $C$ such that for any hyperbolic surface $\Sigma$, $\lambda_1(\Sigma)\leq C$. This follows from Margulis' lemma and eigenvalue estimates based on the minimax principle. If $\epsilon$ is Margulis' constant for hyperbolic surfaces, then there is a disk $B$ of radius $\epsilon/2$ embedded in $\Sigma$. So one may estimate $\lambda_1(\Sigma)\leq \lambda_1^D(B)$ (the Dirichlet eigenvalue for the disk $B$). - Thank you very much . – Analysis Now Nov 23 '10 at 15:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9217937588691711, "perplexity": 421.3249605977451}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645371566.90/warc/CC-MAIN-20150827031611-00115-ip-10-171-96-226.ec2.internal.warc.gz"}
https://keio.pure.elsevier.com/en/publications/speed-limit-for-open-quantum-systems	# Speed limit for open quantum systems Ken Funo, Naoto Shiraishi, Keiji Saito Research output: Contribution to journalArticlepeer-review 48 Citations (Scopus) ## Abstract We study the quantum speed limit for open quantum systems described by the Lindblad master equation. The obtained inequality shows a trade-off relation between the operation time and the physical quantities such as the energy fluctuation and the entropy production. We further identify a quantity characterizing the speed of the state transformation, which appears only when we consider the open system dynamics in the quantum regime. When the thermal relaxation is dominant compared to the unitary dynamics of the system, we show that this quantity is approximated by the energy fluctuation of the counter-diabatic Hamiltonian which is used as a control field in the shortcuts to adiabaticity protocol. We discuss the physical meaning of the obtained quantum speed limit and try to give better intuition about the speed in open quantum systems. Original language English 013006 New Journal of Physics 21 1 https://doi.org/10.1088/1367-2630/aaf9f5 Published - 2019 Jan 9 ## Keywords • quantum speed limits	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9745460152626038, "perplexity": 661.8635401228448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711232.54/warc/CC-MAIN-20221208014204-20221208044204-00088.warc.gz"}
https://classes.areteem.org/mod/forum/discuss.php?d=946	## Online Course Discussion Forum ### MC I-A Counting and Probability Homework 4.20 MC I-A Counting and Probability Homework 4.20 Hi, Could you explain how to solve homework 4.20 in MC I-A Counting and Probability? "How many total ways can you draw red cards and black cards from a standard deck of cards so that theblack cards appear consecutively in the card draw?" Re: MC I-A Counting and Probability Homework 4.20 There are four ways to grab the $5$ cards so that the $2$ black are consecutive: $BBRRR$, $RBBRR$, $RRBBR$, and $RRRBB$. There are $26 \times 25 \times 26 \times 25 \times 24 = 10140000$ ways to grab the cards precisely in $BBRRR$ order. You can find the number of ways to grab the cards in the other three cases in a similar way.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9370518326759338, "perplexity": 715.0823062673331}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00046.warc.gz"}
https://quantumcomputing.stackexchange.com/questions/1789/bernstein-vazirani-problem-in-book-as-exercise?noredirect=1	# Bernstein–Vazirani problem in book as exercise I´ve solved the Exercise 7.1.1 (Bernstein–Vazirani problem) of the book "An introduction to quantum computing" (Mosca et altri). The problem is the following: Show how to find $a \in Z_2^n$ given one application of a black box that maps $\|x\rangle\|b\rangle \to \|x\rangle \|b \oplus x · a\rangle$ for some $b\in \{0, 1\}$. I´d say we can do it like this: • First i go from $\|0\|0\rangle \to \sum_{i \in \{0,1\}^n}\|i\rangle\| + \rangle$ using QFT and Hadamard • Then I apply the oracle: $$\sum_{i \in \{0,1\}^n}(-1)^{(i,a)} \|i\rangle\| + \rangle$$ • Then I read the pase with an Hadamard (since we are in $Z_2^n$ our QFT is an Hadamard) $$\|a\rangle \|+ \rangle$$ I think is correct. Do you agree? This is not correct: you need to use the state $\|-\rangle=(\|0\rangle-\|1\rangle)/\sqrt{2}$ instead of $\|+\rangle$. The important thing is that you've missed showing how the black box map that you've stated gives the oracle output that you've stated. To see this, apply the map on $$\|x\rangle\|+\rangle\mapsto\|x\rangle(\|0\oplus x\cdot a\rangle+\|1\oplus x\cdot a\rangle)/\sqrt{2}=\|x\rangle(\|0\rangle+\|1\rangle)/\sqrt{2}.$$ When the $\|+\rangle$ state is there, you get no phase. Meanwhile, with the $\|-\rangle$ state, $$\|x\rangle\|-\rangle\mapsto\|x\rangle(\|0\oplus x\cdot a\rangle-\|1\oplus x\cdot a\rangle)/\sqrt{2}=\left\{\begin{array}{cc} \|x\rangle(\|0\rangle-\|1\rangle)/\sqrt{2} & x\cdot a=0 \\ \|x\rangle(\|1\rangle-\|0\rangle)/\sqrt{2} & x\cdot a=1\end{array}\right..$$ This can simply be written as $(-1)^{x\cdot a}\|x\rangle\|-\rangle$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8790776133537292, "perplexity": 518.8740837287792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00597.warc.gz"}
https://answerstoprayers.info/romans_7_7-13/	## God’s Law Reveals Our Sin Well then, am I suggesting that the law of God is sinful? Of course not! In fact, it was the law that showed me my sin. I would never have known that coveting is wrong if the law had not said, “You must not covet.” But sin used this command to arouse all kinds of covetous desires within me! If there were no law, sin would not have that power. At one time I lived without understanding the law. But when I learned the command not to covet, for instance, the power of sin came to life, and I died. So I discovered that the law’s commands, which were supposed to bring life, brought spiritual death instead. Sin took advantage of those commands and deceived me; it used the commands to kill me. But still, the law itself is holy, and its commands are holy and right and good. But how can that be? Did the law, which is good, cause my death? Of course not! Sin used what was good to bring about my condemnation to death. So we can see how terrible sin really is. It uses God’s good commands for its own evil purposes.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9500773549079895, "perplexity": 2826.983049189503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488550571.96/warc/CC-MAIN-20210624015641-20210624045641-00549.warc.gz"}
https://primethinker.com/calculus/study-questions-list	# Study Questions List Make it easier to study later on by summarizing your notes. You will be taught and tested on the concepts highlighted by these questions. Creating FLASHCARDS with an answer and an example on each card will support you as concepts build up in this course. Also, just go though and answer each question to study in general. Functions 1. What is a function? Draw a graph that represents a function and one that does not represent a function. Explain why. 2. What is domain? When you want to find the domain of a function what should you think about? 3. What is an even or odd function? How do you verify if a function is even or odd? 4. How do you find the points of intersection of the graphs of two functions? 5. How do you find x and y intercepts? 6. What would you do if you have to find roots of functions, zeroes of functions or solutions of functions? 7. What is the point slope form of the equation of the line? 8. How do the slopes of parallel and perpendicular lines compare? 9. What is the vertical line test and how is it used? 10. How do you evaluate f+g, f-g, f*g, f/g, and f( g(x))? 11. How do we sketch the graphs of functions using transformations? Limits 1. What is the definition of the limit? Give an example on how to read/estimate the limit of a function from a graph or a table. 2. How can a limit fail to exist? 3. Note some examples of how different techniques are used to evaluate limits - Special Trig limits, rationalizing technique, factoring technique. 4. How do you find limits of piece-wise defined functions? 5. Create a table of basic trig function values ( 0o, 30o 45o….) Remember these. 6. What does it mean for a function to be continuous? Conditions? 7. What is a removable/ non-removable discontinuity? 8. How can you check the continuity of a function at a point? Find the value of c that makes a piecewise defined function continuous. 9. How do you evaluate one sided limits? Give an example. 10. How do you find the equations of vertical asymptotes? Using limits? 11. How do you find horizontal asymptotes of a function? What limits do you evaluate? Derivatives 1. What is a difference quotient? 2. How can the secant lines be used to approximate the slope of the tangent line graphically? 3. How do you find the derivative using the limiting process? 4. What is the relationship between continuity and differentiability? 5. How do you find the derivative at a given point (x,y)? 6. How do you find the equation of the tangent line? 7. What is a normal line and how do you find its equation? 8. How can you find the x-values of horizontal tangents of a function? 9. How can you find the x values of vertical tangent of a function? 10. Note down the derivative rules. 11. How can you show if a piecewise defined function is differentiable where it splits? 12. What are the derivatives of trigonometric functions? Know the basic trig values from the table. Applications of derivatives 1. What do you think about when drawing the graph of f ‘ given the graph of f ? 2. What does the graph of f’ tell about f? Give an example on how to read the graph. 3. What does the graph of f’’ tell about f? Give an example on how to read the graph. 4. How do you find critical values of f(x)? 5. How to you find the intervals where f(x) is increasing or decreasing? 6. What does the first derivative test say about finding relative extrema? 7. What does concave up and concave down mean graphically? How do you find concavity on a given interval? 8. How do you find points of inflection? 9. What does the second derivative test say? What do you do when it fails? 4 Theorems 1. What is the Intermediate values theorem? Explain it graphically. 2. What is the Extreme Value Theorem? How is it used to find absolute extrema? 3. What is the Rolle’s Theorem? Can you explain it graphically too. 4. What does the Mean Value Theorem say? Explain it graphically too. Velocity Acceleration 1. How do we find velocity and acceleration if the position function is given? 2. How do we find the average rate of change of a function? Average velocity? 3. What do we mean by instantaneous rate of change? What does it have to do with the derivative? Instantaneous velocity? 4. How are speed and velocity related? 5. How can you determine when an object is moving forward or backwards or standing still? 6. How do you find the velocity or acceleration of an object given the graph of the position function? 7. Learn how to analyze the position, velocity and acceleration graphs. 8. What can we say about the speed of an object if velocity and acceleration have the same signs/ opposite signs? 9. Can you determine the total distance traveled by a particle from the position graph? The Integral 1. What is an antiderivative? (indefinite Integral) 2. Does a function have a unique antiderivative? Explain. 3. Learn the basic integration rules. See page 250. 4. How can you check the result of a indefinite integral? 5. How are areas under graphs approximated by sums? 6. What is an upper sum? Lower sum? Riemann Sum? 7. What is the difference between a definite integral and an indefinite integral? 8. Why do we say that the definite integral represents “signed area” and NOT geometric area? 9. What does the first fundamental theorem of calculus say in your own words? (And how do you use it?) 10. What does the Mean Value Theorem for Integrals say? 11. How can you find the average value of a function over an interval? 12. What does the Second Fundamental Theorem say? 13. How do you use the u-substitution method to evaluate integrals? 14. How does knowing whether a function is even or odd help us in integrating it? 15. What is the trapezoidal sum? Simpson’s rule (BC)? Logarithmic, Exponential, and other transcendental functions – Derivatives and Integrals 1. How is the natural logarithmic function defined using the integral? 2. What are the properties of the natural log function? 3. What is the definition of the number e? 4. How do you find the derivative of the natural log function? 5. What is logarithmic differentiation? 6. What is the log rule for integration? 7. How do you compute the derivative of the inverse function? 8. How do you integrate and differentiate exponential functions and logarithmic function with other bases? Differential Equations 1. Describe the difference between a general solution and a particular solution? 2. What is a slope field and how do you interpret it? 3. How is the Euler’s method used to approximate the solution of a differential equation? (BC) 4. Describe how to recognize and solve differential equations that can be solved by separation of variables? 5. How do you model exponential growth and decay with differential equations? 6. What situations are modeled by the logistic differential equation? (BC) Applications of Integration 1. How do you use integrals to find the area bounded by two curves? 2. How do you use integrals to find the volumes of solids? 3. Describe how Disk Method, Washer Method, and Shell Method works? 4. How can you find the length of a curve? (BC) More methods for Integration 1. How do you find the antiderivative using integration by parts? 2. What is partial fraction decomposition and how is it used to find antiderivatives? 3. What are improper integrals? 4. Define the terms converges and diverges when working with improper integrals? 5. When and how do you use the L’hopital’s rule? Sequences and Series 1. What is the difference between a sequence and a series? 2. What does converges or diverges mean? 3. How do you check if a sequence converges or diverges? 4. What are the tests by which you can check the convergence or divergence of a series? (know the conditions of each test (page 644)) 5. What is the alternating series remainder? 6. How can you find the radius and interval of convergence using ratio and root tests? 7. What is a taylor polynomial? Taylor series/ Maclauren series? 8. How can you compute the Langrange error bound or the remainder for Pn taylor polynomial? 9. What is a power series? How do you express a function by a power series?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8877345323562622, "perplexity": 452.40768133547965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145747.6/warc/CC-MAIN-20200223062700-20200223092700-00539.warc.gz"}
http://komal.elte.hu/feladat?a=honap&h=202011&t=mat&l=en	Mathematical and Physical Journal for High Schools Issued by the MATFUND Foundation # KöMaL Problems in Mathematics, November 2020 Please read the rules of the competition. Show/hide problems of signs: ## Problems with sign 'K' The deadline is: December 10, 2020 24:00 (UTC+01:00). K. 669. Let us consider the set of 3-digit positive integers containing all the digits 1, 2, 3 exactly once. Find the smallest positive integer that contains each number from the previous set as consecutive digits. (6 pont) This problem is for grade 9 students only. K. 670. Grandma bought two candles: the red candle was 2 cm longer than the blue one. On All Saints' Day she lit the red candle at 5:30 p.m. then she lit the blue candle at 7 p.m. and let them burn all the way down. The two candles were equal in length at 9:30 p.m. The blue candle burned out at 11 p.m and the red one finished at 11:30 p.m. What was the initial length of the red candle? (6 pont) This problem is for grade 9 students only. K. 671. We know that the first five terms of an increasing arithmetic sequence are all positive primes. Find the smallest prime at the $\displaystyle 5^\text{th}$ position. (6 pont) This problem is for grade 9 students only. K. 672. A garden is divided into 16 patches as shown in the figure. In each patch, either roses or tulips or daisies or gerberas are grown: only one type of flower in each, but every row, every column, and both diagonals contain every type of flower. In how many different ways is it possible to arrange the flowers in this way? (Two arrangements are considered different if there exists a patch that contains a different kind of flower.) (6 pont) This problem is for grade 9 students only. K. 673. The students in a class (we do not know how many of them there are) decided that everyone would buy some small present to everyone else for Christmas, and they would also buy some present together for each of their 11 teachers. Unfortunately, the Christmas party was cancelled. Then they decided to divide the presents equally among all the siblings of the students. (Each sibling gets the same present.) Was that possible if the total number of siblings was 15? (6 pont) This problem is for grade 9 students only. ## Problems with sign 'C' The deadline is: December 10, 2020 24:00 (UTC+01:00). C. 1630. The numbers 1 to 32 are written in the white fields of a chessboard, using each number once. Then the sum of the numbers in the adjacent fields is entered in each black field. What are the smallest and largest possible values of the sum of the numbers in the black fields? (5 pont) This problem is for grades 1–10 students only. C. 1631. Let $\displaystyle AB$ be a chord in a unit circle. Triangle $\displaystyle ABC$ is right-angled at $\displaystyle B$, and vertex $\displaystyle C$ lies on the circle. Triangle $\displaystyle ABD$ is isosceles right-angled, and $\displaystyle AB$ is the hypotenuse. How long is the chord $\displaystyle AB$ if the two triangles have equal areas? What is this area? (5 pont) This problem is for grades 1–10 students only. C. 1632. How many different infinite arithmetic sequences of positive integer terms are there in which 24, 744 and 2844 all occur? Two arithmetic sequences are considered different if they have different first terms or different common differences. (5 pont) C. 1633. Let $\displaystyle P$ be an interior point of one side of a unit square. Consider all parallelograms with one vertex at $\displaystyle P$, and one on each side of the square. Prove that if $\displaystyle P$ is not the midpoint of the side then $\displaystyle (i)$ there are exactly two rectangles among these parallelograms, and $\displaystyle (ii)$ the sum of the areas of these two rectangles is 1. (5 pont) C. 1634. Prove the following inequality: $\displaystyle \frac{1}{4} +\frac{1}{28} +\frac{1}{70} +\cdots +\frac{1}{(3k-2)(3k+1)} +\cdots +\frac{1}{2017\cdot 2020} < \frac{1}{3}.$ (5 pont) C. 1635. Given two intersecting circles, construct\footnote[1]with straight edge and compasses on paper, or with appropriate geometric construction software a secant through one of the intersection points such that the segment bounded by the two circles is divided $\displaystyle 1:2$ by the intersection point. Write down and explain the steps of the construction. (Elementary steps like bisecting an angle or reflecting a point in a line do not need to be described in detail.) (5 pont) This problem is for grades 11–12 students only. C. 1636. The Hungarian poet Dezső Kosztolányi spent a few weeks in Paris when he was a student. When he was given for change a ten-centime coin not in circulation any more, he wanted to give it away. He did not succeed, which he explained to himself by the expression on his face revealing his intentions. Therefore he decided to get 9 valid ten-centime coins, mix them with the worthless coin in his pocket, and by not looking at them he pays with one of them in a shop. He continued doing so until he had a single coin in his pocket: the coin out of circulation. What is the probability of this? (5 pont) This problem is for grades 11–12 students only. ## Problems with sign 'B' The deadline is: December 10, 2020 24:00 (UTC+01:00). B. 5126. Prove that if $\displaystyle n\ge 3$, then there exist $\displaystyle n$ distinct positive integers such that the sum of their reciprocals is 1. (3 pont) B. 5127. Given a convex angle and a line segment of length $\displaystyle k$, determine the locus of those points inside the angle through which there exists a line cutting off a triangle of perimeter $\displaystyle k$ from the angle. (4 pont) B. 5128. Find all pairs of relatively prime integers $\displaystyle (x,y)$ such that $\displaystyle x^2 + x = y^3 + y^2$. Proposed by L. Surányi, Budapest (4 pont) B. 5129. Two players are taking turns in selecting one of the coefficients $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ of the polynomial $\displaystyle x^3+ax^2+bx+c$, and giving it some integer value of their choice. Prove that the starting player can achieve that (after the three steps) all three roots of the polynomial should be integers (i.e. that the polynomial can be expressed as a product of three polynomials of integer coefficients). (3 pont) B. 5130. There are $\displaystyle n$ points in the plane. We know that for any $\displaystyle k$ points ($\displaystyle k\ge 2$), it is possible to select two of them with distance at most 1. Show that the points can be covered with $\displaystyle k-1$ disks of unit radius. (5 pont) B. 5131. Let $\displaystyle H$ be an equilateral triangle of unit area, let $\displaystyle O$ be a fixed point, and for any point $\displaystyle P$ let $\displaystyle H_P$ denote the triangle obtained from triangle $\displaystyle H$ by a parallel shift with vector $\displaystyle \overrightarrow{OP}$. Consider the set $\displaystyle N$ of points $\displaystyle P$ for which the area of the intersection $\displaystyle H\cap H_P$ is at least $\displaystyle 4/9$. What is the area of $\displaystyle N$? Based on the idea of V. Vígh, Székkutas (5 pont) B. 5132. How many different strings of 2021 letters can be made of letters A, B and C such that the number of A's is even and the number of B's is of the form $\displaystyle 3k+2$? (6 pont) B. 5133. Given six points in the space, no four of which are coplanar, prove that they can be divided into two sets of three such that the two triangular plates spanned by the two sets of three points should intersect each other. (6 pont) ## Problems with sign 'A' The deadline is: December 10, 2020 24:00 (UTC+01:00). A. 786. In a convex set $\displaystyle S$ that contains the origin it is possible to draw $\displaystyle n$ disjoint unit circles such that viewing from the origin non of the unit circles blocks out a part of another (or a complete) unit circle. Prove that the area of $\displaystyle S$ is at least $\displaystyle n^2/100$. Submitted by: Dömötör Pálvölgyi, Budapest (7 pont) A. 787. Let $\displaystyle p_n$ denote the $\displaystyle n^{\text{th}}$ prime number and define $\displaystyle a_n=\lfloor p_n \nu \rfloor$, where $\displaystyle \nu$ is a positive irrational number. Is it possible that there exist only finitely many $\displaystyle k$ such that $\displaystyle \binom{2a_k}{a_k}$ is divisible by $\displaystyle p_i^{10}$ for all $\displaystyle i=1,2,\ldots, 2020$? Submitted by: Abhishek Jha, Delhi, India and Ayan Nath, Tezpur, India (7 pont) A. 788. Solve the following system of equations: $\displaystyle x+\frac1{x^3}=2y, \qquad y+\frac1{y^3}=2z, \qquad z+\frac1{z^3}=2w, \qquad w+\frac1{w^3}=2x.$ (7 pont) ### Upload your solutions above or send them to the following address: KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112, Pf. 32. 1518, Hungary	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.800505518913269, "perplexity": 466.86178886384283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141723602.80/warc/CC-MAIN-20201203062440-20201203092440-00463.warc.gz"}
https://www.physicsforums.com/threads/ev-unit-vector-and-projection-of-a-vector-in-a-dot-product.556371/	# Ev(Unit Vector) and projection of a vector in a dot product 1. Dec 3, 2011 ### Usernam So my book says Lets suppose, We have two vector v and u w=projection of u ev= unit vector θ=angle between the two w=(u.ev)ev or w=( (u.v)/(v.v) )v Now, the second equation is fairly easy to understand if we understand the first one because ev= v / \|v\| What is bothering me is I have no idea why w=(u.ev)ev. It would be more reasonable, as per me, if w=(u.ev) But that is not the case. Why is that extra ev lingering around in the equation. ANY IDEAS? 2. Dec 3, 2011 ### mathman w is a vector parallel to ev. (u.ev) is a scalar (± length of w), so you need the entire expression to define the vector w.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9522036910057068, "perplexity": 3159.7241822169126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583509960.34/warc/CC-MAIN-20181016010149-20181016031649-00116.warc.gz"}
https://en.wikipedia.org/wiki/Bartlett%27s_bisection_theorem	Bartlett's bisection theorem Bartlett's Bisection Theorem is an electrical theorem in network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical two-port network can be transformed into a lattice network.[1] The theorem often appears in filter theory where the lattice network is sometimes known as a filter X-section following the common filter theory practice of naming sections after alphabetic letters to which they bear a resemblance. The theorem as originally stated by Bartlett required the two halves of the network to be topologically symmetrical. The theorem was later extended by Wilhelm Cauer to apply to all networks which were electrically symmetrical. That is, the physical implementation of the network is not of any relevance. It is only required that its response in both halves are symmetrical.[2] Applications Lattice topology filters are not very common. The reason for this is that they require more components (especially inductors) than other designs. Ladder topology is much more popular. However, they do have the property of being intrinsically balanced and a balanced version of another topology, such as T-sections, may actually end up using more inductors. One application is for all-pass phase correction filters on balanced telecommunication lines. The theorem also makes an appearance in the design of crystal filters at RF frequencies. Here ladder topologies have some undesirable properties, but a common design strategy is to start from a ladder implementation because of its simplicity. Bartlett's theorem is then used to transform the design to an intermediate stage as a step towards the final implementation (using a transformer to produce an unbalanced version of the lattice topology).[3] Definition and proof Definition Start with a two-port network, N, with a plane of symmetry between the two ports. Next cut N through its plane of symmetry to form two new identical two-ports, ½N. Connect two identical voltage generators to the two ports of N. It is clear from the symmetry that no current is going to flow through any branch passing through the plane of symmetry. The impedance measured into a port of N under these circumstances will be the same as the impedance measured if all the branches passing through the plane of symmetry were open circuit. It is therefore the same impedance as the open circuit impedance of ½N. Let us call that impedance ${\displaystyle Z_{oc}}$. Now consider the network N with two identical voltage generators connected to the ports but with opposite polarity. Just as superposition of currents through the branches at the plane of symmetry must be zero in the previous case, by analogy and applying the principle of duality, superposition of voltages between nodes at the plane of symmetry must likewise be zero in this case. The input impedance is thus the same as the short circuit impedance of ½N. Let us call that impedance ${\displaystyle Z_{sc}}$. Bartlett's bisection theorem states that the network N is equivalent to a lattice network with series branches of ${\displaystyle Z_{sc}}$ and cross branches of ${\displaystyle Z_{oc}}$.[4] Proof Consider the lattice network shown with identical generators, E, connected to each port. It is clear from symmetry and superposition that no current is flowing in the series branches ${\displaystyle Z_{sc}}$. Those branches can thus be removed and left open circuit without any effect on the rest of the circuit. This leaves a circuit loop with a voltage of 2E and an impedance of ${\displaystyle 2Z_{oc}}$ giving a current in the loop of; ${\displaystyle I={\frac {2E}{2Z_{oc}}}}$ and an input impedance of; ${\displaystyle {\frac {E}{I}}=Z_{oc}}$ as it is required to be for equivalence to the original two-port. Similarly, reversing one of the generators results, by an identical argument, in a loop with an impedance of ${\displaystyle 2Z_{sc}}$ and an input impedance of; ${\displaystyle {\frac {E}{I}}=Z_{sc}}$ Recalling that these generator configurations are the precise way in which ${\displaystyle Z_{oc}}$ and ${\displaystyle Z_{sc}}$ were defined in the original two-port it is proved that the lattice is equivalent for those two cases. It is proved that this is so for all cases by considering that all other input and output conditions can be expressed as a linear superposition of the two cases already proved. Examples Lattice equivalent of a T-section high-pass filter Lattice equivalent of a Zobel bridge-T low-pass filter It is possible to use the Bartlett transformation in reverse; that is, to transform a symmetrical lattice network into some other symmetrical topology. The examples shown above could just as equally have been shown in reverse. However, unlike the examples above, the result is not always physically realisable with linear passive components. This is because there is a possibility the reverse transform will generate components with negative values. Negative quantities can only be physically realised with active components present in the network. Extension of the theorem Example of impedance and frequency scaling using a Π-section low-pass filter prototype. In the first transformation, the prototype is bisected and the cut-off frequency is rescaled from 1 rad/s to 105 rad/s (15.9 kHz). In the second transformation, the bisected network is rescaled on the left side to operate at 600 Ω and on the right side to operate at 50 Ω. There is an extension to Bartlett's theorem that allows a symmetrical filter network operating between equal input and output impedance terminations to be modified for unequal source and load impedances. This is an example of impedance scaling of a prototype filter. The symmetrical network is bisected along its plane of symmetry. One half is impedance-scaled to the input impedance and the other is scaled to the output impedance. The response shape of the filter remains the same. This does not amount to an impedance matching network, the impedances looking in to the network ports bear no relationship to the termination impedances. This means that a network designed by Bartlett's theorem, while having exactly the filter response predicted, also adds a constant attenuation in addition to the filter response. In impedance matching networks, a usual design criteria is to maximise power transfer. The output response is "the same shape" relative to the voltage of the theoretical ideal generator driving the input. It is not the same relative to the actual input voltage which is delivered by the theoretical ideal generator via its load impedance.[5][6] The constant gain due to the difference in input and output impedances is given by; ${\displaystyle A={\frac {V_{2}}{E}}={\frac {2R_{2}}{R_{1}+R_{2}}}}$ Note that it is possible for this to be greater than unity, that is, a voltage gain is possible, but power is always lost. References 1. ^ Bartlett, AC, "An extension of a property of artificial lines", Phil. Mag., vol 4, p902, November 1927. 2. ^ Belevitch, V, "Summary of the History of Circuit Theory", Proceedings of the IRE, vol 50, pp850, May, 1962. 3. ^ Vizmuller, P, RF Design Guide: Systems, Circuits, and Equations, pp 82–84, Artech House, 1995 ISBN 0-89006-754-6. 4. ^ Farago, PS, An Introduction to Linear Network Analysis, pp117-121, The English Universities Press Ltd, 1961. 5. ^ Guillemin, EA, Synthesis of Passive Networks: Theory and Methods Appropriate to the Realization and Approximation Problems, p207, Krieger Publishing, 1977, ISBN 0-88275-481-5 6. ^ Williams, AB, Taylor, FJ, Electronic Filter Design Handbook, 2nd ed. McGraw-Hill, New York, 1988.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 13, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8592326641082764, "perplexity": 826.3146360176047}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720962.53/warc/CC-MAIN-20161020183840-00070-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/loop-quantum-gravity-and-qm-possibly-in-danger-after-discovery-at-esas-integral.513609/	# Loop Quantum Gravity and QM possibly in danger after discovery at ESA’s Integral? 1. Jul 12, 2011 ### eiyaz http://www.esa.int/esaCP/SEM5B34TBPG_index_0.html "ESA’s Integral gamma-ray observatory has provided results that will dramatically affect the search for physics beyond Einstein. It has shown that any underlying quantum ‘graininess’ of space must be at much smaller scales than previously predicted. Einstein’s General Theory of Relativity describes the properties of gravity and assumes that space is a smooth, continuous fabric. Yet quantum theory suggests that space should be grainy at the smallest scales, like sand on a beach." According to the article since QM implies that space-time is actually distorted at small lengths especially near the Planck length of 10^-35, the lack of distortion could hinger many QM theories. I am going to make a bold statement, please correct me if I am wrong. The article states that no distortion was found down to 10^-48. Since Planck's length is only 10^-35 how is this even possible? Does this mean Planck length is not the smallest length, thereby suggesting that a large part of QM is incorrect? GR states that space-time is essentially smooth while QM suggests that space is distorted at the micro level. If the distortion is less than planck's length, would this not mean there is no distortion, thus disproving everything we know about QM? I hope I'm not looking at this correctly! 2. Jul 12, 2011 ### marcus 3. Jul 12, 2011 ### eiyaz Share this great discussion with others via Reddit, Google+, Twitter, or Facebook	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8889700174331665, "perplexity": 1326.0839032754704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221219109.94/warc/CC-MAIN-20180821210655-20180821230655-00639.warc.gz"}
https://www.nature.com/articles/s41467-018-03864-y?error=cookies_not_supported&code=8ee60f77-0e06-4133-a3c3-15cf205bf0ee	Article \| Open \| Published: # Dissociation of two-dimensional excitons in monolayer WSe2 ## Abstract Two-dimensional (2D) semiconducting materials are promising building blocks for optoelectronic applications, many of which require efficient dissociation of excitons into free electrons and holes. However, the strongly bound excitons arising from the enhanced Coulomb interaction in these monolayers suppresses the creation of free carriers. Here, we identify the main exciton dissociation mechanism through time and spectrally resolved photocurrent measurements in a monolayer WSe2 pn junction. We find that under static in-plane electric field, excitons dissociate at a rate corresponding to the one predicted for tunnel ionization of 2D Wannier–Mott excitons. This study is essential for understanding the photoresponse of 2D semiconductors and offers design rules for the realization of efficient photodetectors, valley dependent optoelectronics, and novel quantum coherent phases. ## Introduction As Johan Stark first observed in hydrogen atoms1, applying an electric field on Coulomb-bound particles shifts their energy levels and eventually leads to their dissociation (Fig. 1a). In condensed matter physics, Wannier–Mott excitons display features analogous to those of hydrogen2, but with the crucial difference that they recombine if they are not dissociated. Thermal energy is usually sufficient to ionize excitons in 3D semiconductors owing to their small binding energy EB (typically a few meV). In contrast, quantum confinement effects and reduced Coulomb screening in low-dimensional materials give rise to large exciton binding energy (EB > 100 meV), which prevents thermal or spontaneous dissociation even at elevated temperatures and exciton densities. In particular, monolayer transition metal dichalcogenides (TMDs) have aroused tremendous interest due to their unique optical properties governed by prominent excitonic features3,4,5,6 and spin- and valley dependent effects7,8,9,10,11. These 2D semiconductors provide an exciting testbed for probing the physics arising from many-body Coulomb interactions6,12. Recently, all-optical experiments have revealed a wealth of physical phenomena such as exciton13,14, trion15,16, and biexciton17 formation, bandgap renormalization18, exciton–exciton annihilation19,20,21,22,23,24,25, and optical Stark effect7,11. Exciton dissociation, on the other hand, can in principle be assessed through photocurrent measurements since photocurrent directly stems from the conversion of excitons into free carriers. A large number of studies have investigated photodetection performances of 2D TMDs26,27,28,29 and demonstrated their potential as photodetectors and solar cells. However, it is still unclear which dissociation process can overcome the large exciton binding energy and lead to efficient photocurrent generation in these devices. Theoretical studies suggest that strong electric fields may provide the energy required to dissociate the excitons30,31,32, but the precise mechanism governing exciton dissociation in 2D TMDs remains to be experimentally investigated. Here, we address this important issue by monitoring the exciton dissociation and subsequent transport of free carriers in a monolayer TMD pn junction through spectrally and temporally resolved photocurrent measurements. Combining these two approaches allow us to assess and correlate two essential excitonic properties under static electric field, namely the Stark shift and the dissociation time. Further, we make use of the extreme thinness of 2D materials and their contamination-free assembly into heterostructures to reliably control the potential landscape experienced by the excitons. By placing the monolayer TMD in close proximity to metallic split gates, we can generate high in-plane electric fields and drive a photocurrent (PC). We find that at low field the photoresponse time of our device is limited by the rate at which excitons tunnel into the continuum through the potential barrier created by their binding energy, a process known as tunnel ionization (Fig. 1a). Tuning the electric field inside the pn junction further allows us to disentangle various dynamical processes of excitons and free carriers and to identify the kinetic bottlenecks that govern the performance of TMD-based optoelectronic devices. ## Results ### Device structure and characterization Figure 1b, c presents a schematic and optical micrograph of our lateral pn junction device made by assembling exfoliated flakes on metallic split gates (VG1 and VG2) separated by 200 nm (see “Methods”). Few-layer graphite flakes placed on both ends of a monolayer WSe2 flake serve as ambipolar electrical contacts33 that we use to apply a bias voltage VB and collect the photogenerated charges. The lateral graphite-WSe2-graphite assembly is fully encapsulated in hexagonal boron nitride, typically 20 nm thick, which provides a clean and flat substrate. Three devices were measured (see Supplementary Note 1 and Supplementary Figs. 13), but unless otherwise specified, all measurements presented in the main text are obtained at room temperature from the device shown in Fig. 1c. Tuning of bias and gate voltages allows us to finely control the in-plane electric field F. Finite-element and analytical calculations of the electric field distribution in our device (see Supplementary Note 2 and Supplementary Figs. 47) provide us with a precise estimate of F and the electrostatic doping inside the WSe2 (Fig. 1d). Applying gate voltages of opposite polarity (Vasym = VG1 = −VG2 = −10 V) leads to the formation of a sharp pn junction (Fig. 1e) with an in-plane electric field reaching 21 V µm−1 (Fig. 1d). The photoresponse that we observed at the junction (Fig. 1c) follows a photodiode-like behavior: PC is only generated in the pn or np configuration (see Supplementary Fig. 1c) and can be increased by applying a reverse bias voltage (Fig. 1f). ### Spectral response We probe the absorption spectrum in the photoactive region by measuring the PC as a function of photon energy at a constant laser power P and in-plane electric field F. Figure 2a shows the responsivity (PC/P) spectra of a device similar to the one presented in Fig. 1c, measured at various VB and at low temperature (T = 30 K) in order to reduce thermal broadening. We observe a pronounced peak at a photon energy = 1.73 eV, corresponding to the A exciton, and a step-like increase around 1.87 eV. For increasing electric field, this step-like feature broadens and an additional shoulder appears at 1.83 eV. To identify the various spectral features, we compare the experimental spectra with first-principles calculations for a monolayer WSe2 embedded in hBN (see Supplementary Note 3 and Supplementary Fig. 8). By including the electronic screening from the hBN layers in the many-body G0W0 and Bethe–Salpeter Equation (BSE) frameworks34 we obtain a bandgap of 1.85 eV and a lowest bound exciton at 1.67 eV in good agreement with the experimental spectra. To account for the effect of a constant in-plane electric field we use a model based on the 2D Wannier equation (see Supplementary Note 4 and Supplementary Fig. 9). In these model calculations, screening by the TMD itself as well as the surrounding dielectric materials is described via the Keldysh potential for the electron–hole interaction. Figure 2b shows calculated absorption spectra for different in-plane fields F. Excellent agreement between experiment and calculations is found assuming a bandgap of 1.9 eV, which yields a binding energy of EB = 170 meV for the A excitons consistent with the first-principles calculations. The unbroadened spectrum calculated at zero field (Fig. 2b, solid black line) confirms the presence of multiple overlapping excited excitonic peaks below the bandgap. The calculated spectra for higher field reproduce remarkably well the field-induced increase of the sub-bandgap absorption observed experimentally. This is a manifestation of the Franz–Keldysh effect, which results from the leakage of the free electron and hole wave functions into the bandgap (inset of Fig. 2b). We note that our experimental value of EB agrees well with the one estimated from the diamagnetic shift of a monolayer WSe2 encapsulated between silica and hBN35. Larger EB has been observed in SiO2-supported WSe2 samples36,37,38, underlining the role of the dielectric environment on the excitonic properties39. ### Excitonic Stark effect Turning our attention to the A exciton photocurrent peak, we observe a pronounced red-shift as VB (Fig. 2c) and Vasym increase. We attribute this to the DC Stark effect. In first approximation, the Stark shift of a 1s exciton (without dipole moment) is given by $${\mathrm{\Delta }}E = - \frac{1}{2}\alpha F^2$$, where $$\alpha$$ is the in-plane polarizability. As shown in Fig. 2d, the A exciton energy shows a quadratic dependence with the maximum in-plane electric field FM calculated for different values of Vasym and VB (Fig. 2e), yielding a polarizability of $$\alpha = \left( {1 \pm 0.2} \right) \times 10^{ - 6}$$ Dm/V. This shift matches well with the predicted polarizability of $$\alpha = 9.4 \times 10^{ - 7}$$ Dm/V for EB = 170 meV, thus supporting our previous spectral analysis. Interestingly, we note that the measured in-plane polarizability is two order of magnitude larger than the out-of-plane value recently obtained in PL experiments40. This strong anisotropy confirms the 2D nature of the A exciton and demonstrates the advantage of using in-plane electric fields for controlling the optical properties of TMDs31. ### Photoresponse dynamics Along with the Stark shift, the application of a large in-plane electric field shortens the lifetime of excitons, which eventually decay into free electrons and holes (Fig. 1a). We probe these decay dynamics by assessing the photoresponse time $$\tau$$ of the device with time-resolved photocurrent measurements (TRPC), banking on the nonlinear photoresponse of the WSe2. Figure 3a, b shows the strong sublinear power dependence of the photocurrent (and the corresponding responsivity) under resonant pulsed optical excitation ( = 1.65 eV, see “Methods”). Many physical processes may be responsible for or contribute to the observed sublinearity, including phase space filling41 and dynamic screening effects (e.g., bandgap renormalization18). These many-body effects become intricate as the exciton gas approaches the Mott transition42. However, recent time-resolved spectroscopy19,22 and photoluminescence20,23 experiments indicate that in this exciton density regime (1011 $$\lesssim$$ N $$\lesssim$$ 1013 cm−2), exciton–exciton annihilation (EEA, or exciton Auger recombination) is the dominant decay process for excitons in TMDs24. To account for EEA in the rate equation governing the photocurrent we add a loss term that scales quadratically with the exciton density ($$\gamma N^2$$, where $$\gamma$$ is the EEA rate). Assuming that each pulse generates an initial exciton population $$N_0$$, this model yields $${\rm{PC}} \propto {\mathrm{ln}}\left( {1 + \gamma \tau N_0} \right)$$, which reproduces well the observed sublinear photoresponse (black lines in Fig. 3a, b, see Supplementary Note 5). Moreover, the fits capture adequately the variation of the sublinear photoresponse with bias (Fig. 3a, b) and gate (Supplementary Fig. 10a) voltages, from which we extract the values of $$1/\gamma \tau$$ (Fig. 3c). Hence, these nonlinear measurements already offer an indirect way to probe the photoresponse time. In order to directly extract $$\tau ,$$we resonantly excite A excitons in the pn junction with a pair of 200 fs-long laser pulses separated by a variable time delay Δt, for various values of Vasym (Fig. 3d, e). Due to the sublinear power dependence, the photocurrent displays a symmetric dip when the two pulses coincide in time (Δt = 0). By extending our nonlinear photocurrent model to the case of two time-delayed pulses (see Supplementary Note 5 and Supplementary Fig. 10), we can show that the time dependence of this dip is dominated by an exponential time constant corresponding to the intrinsic photoresponse time τ of the device. The photoresponse rate $$\Gamma = \frac{1}{\tau }$$ is extracted from TRPC measurements at various values of Vasym (Fig. 3d, e) and VB (see Supplementary Fig. 10d) and presented in Fig. 3c. We observe that Γ increases markedly with gate and bias voltages, and remarkably follows the same trend as the values of $$1/\gamma \tau$$ obtained from the power dependence measurements. Comparing these two results, we obtain an EEA rate of $$\gamma$$ = 0.05 cm2/s, which is similar to those found in WSe219,23, MoS221,22, and WS220,25. We also note that the shortest response time we measure, $$\tau$$ = 10.3 ± 0.4 ps, translates into a bandwidth of f = 0.55/$$\tau$$ ~ 50 GHz, which compares with the fastest responses measured in TMD-based photodetectors43,44. ## Discussion To directly address the exciton dissociation caused by the in-plane electric field FM, we examine the dependence of the photoresponse rate Γ on FM at the pn junction (Fig. 4a). Clearly, two regimes can be distinguished. The rapid increase of Γ with FM is attributed to dissociation by tunnel ionization. We verify this by comparing the measured Γ to the calculated tunnel ionization rate $${\mathrm{\Gamma }}_{{\mathrm{diss}}}$$, obtained by introducing the complex scaling formalism in the 2D Wannier–Mott exciton model (see Supplementary Note 4 and Supplementary Table 1). According to this model, $${\mathrm{\Gamma }}_{{\mathrm{diss}}}$$ can be evaluated in first approximation by the product of the “attempt frequency”45, which scales with $$E_{\mathrm{B}}/h$$, and the exponential tunneling term $${\mathrm{exp}}( - E_{\mathrm{B}}/e_0dF_{\mathrm{M}})$$, where $$e_0$$ is the elementary charge, d is the exciton diameter, and $$h$$ is the Plank constant. We find that the dependence of Γ at low field (FM < 15 V µm−1) coincides well with the calculated dissociation rate of excitons with EB = 170 meV, in agreement with our photocurrent spectroscopy analysis. More importantly, this shows that in the low-field regime the exciton dissociation process is the rate-limiting step governing the generation of photocurrent. We note that in multilayer TMDs, where EB ~ 50 meV, the ionization rate is two orders of magnitude larger than in the monolayer case46, and hence this process was not found to limit the photoresponse rate of multilayer devices44. At high electric field (FM > 20 V µm−1), the photoresponse rate deviates from the dissociation rate-limited model and enters a new regime characterized by a more moderate increase of Γ with FM. The observed linear scaling of Γ(FM) suggests that, in this regime, the photoresponse rate is limited by the drift-diffusive transport of free carriers out of the pn junction. By considering a carrier drift velocity $$v_{{\mathrm{drift}}} = \mu F$$, we estimate that carriers generated in the center of the junction of length L = 200 nm escape the junction at a rate $${\mathrm{\Gamma }}_{{\mathrm{drift}}} = 2\mu F/L$$. Comparing this simple expression (dotted line in Fig. 4a) to the measured Γ at high field, we find $$\mu = 4 \pm 1$$ cm2 V−1 s–1, which is very similar to the room temperature field-effect mobility that we measure in our sample (μFE ~ 3 cm2 V−1 s–1, see Supplementary Note 1). A complete photocurrent model is achieved by introducing competing loss mechanisms caused by the radiative and non-radiative recombination of excitons (see Supplementary Note 6). Good agreement with the experimental data is obtained by considering the finite lifetime of excitons ($$\tau _{{\mathrm{r}},N} = 1/{\mathrm{\Gamma }}_{{\mathrm{r}},N}$$ ~ 1 ns20,23, see Supplementary Note 1) and free carriers ($$\tau _{{\mathrm{r}},n} = 1/{\mathrm{\Gamma }}_{{\mathrm{r}},n}$$~ 30 ps41) at zero electric field. This comprehensive picture of the dynamical processes (Fig. 4b) offers valuable insights into the internal quantum efficiency (IQE) of the photocurrent generation mechanism in this device. Indeed, the efficiency $$\eta$$ of each photocurrent step depends on the competition between the PC-generating ($$\tau _{{\mathrm{drift}}}$$, $$\tau _{{\mathrm{diss}}}$$) and the loss ($$\tau _{{\mathrm{r}},N/n}$$) pathways, such that $$\eta _{{\mathrm{diss}}/{\mathrm{drift}}} = {\mathrm{\tau }}_{{\mathrm{r}},N/n}/({\mathrm{\tau }}_{{\mathrm{r}},N/n} + {\mathrm{\tau }}_{{\mathrm{diss}}/{\mathrm{drift}}})$$. In the inset of Fig. 4a, we compare the IQE measured at low power as a function of VB with the total extraction efficiency $$\eta _{{\mathrm{extract}}} = \eta _{{\mathrm{drift}}}\,\eta _{{\mathrm{diss}}}$$ derived from the kinetic model shown in Fig. 4b. We find that $$\eta _{\rm{extract}}$$ captures very well the bias dependence of the IQE, indicating that we correctly identified the relevant PC-generating processes. The field-independent discrepancy of 30% is attributed to the collection efficiency $$\eta _{{\mathrm{coll}}}$$, which we define as the ratio between the number of excitons reaching the pn junction and the number of absorbed photons. This value coincides with our analysis of the measured photocurrent profile and with the prediction of our exciton diffusion model (see Supplementary Note 7 and Supplementary Fig. 11). In summary, our study offers a global understanding of the fundamental mechanisms governing the exciton dynamics and associated photoresponse in monolayer TMDs under in-plane electric field. We demonstrate that despite their large binding energy, photogenerated excitons can rapidly dissociate into free carriers via tunnel ionization, thereby outcompeting recombination processes. Importantly, this knowledge allows us to identify the main material properties that limit photocurrent generation in TMDs such as carrier mobility, exciton binding energy, and lifetime. This provides guidelines in terms of device design, material quality improvement, and Coulomb engineering of the van der Waals heterostructure to further improve the performances of TMD-based optoelectronics devices and develop their applications in valleytronics. We finally note that the observed Stark and Franz–Keldysh effects open up exciting opportunities for modulating light with 2D materials47. ## Methods ### Device fabrication Exfoliated layers are assembled in a van der Waals heterostructure using the same technique as described in ref. 48. The monolayer of WSe2 is identified by photoluminescence measurement (see Supplementary Note 1). The heterostructure is deposited onto metallic split gates (15 nm palladium) defined by electron-beam lithography on a degenerately doped silicon substrate covered with a 285-nm-thick SiO2 layer. The two graphite flakes are electrically connected by one-dimensional contacts48 made of Ti/Au (2/100 nm). ### Photocurrent measurements Photocurrent measurements are performed using a photocurrent scanning microscope setup, where a laser beam is focused by a microscope objective (Olympus LUCPlanFLN × 40) onto the device placed on a piezoelectric stage (Attocube ANC300). Photocurrent is measured with a preamplifier and a lock-in amplifier synchronized with a mechanical chopper. A supercontinuum laser (NKT Photonics SuperK Extreme), with a pulse duration of 40 ps, repetition rate of 40 MHz and tunable wavelength (from 500 to 1500 nm) is employed to characterize the devices, perform photocurrent spectroscopy, and measure the photocurrent power dependence. 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One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013). ## Acknowledgements T.G.P. and K.S.T. acknowledge support for CNG by the Danish National Research Foundation, project DNRF103. T.P.G. also acknowledges support for the VKR center of excellence QUSCOPE by the Villum foundation. M.M. thanks the Natural Sciences and Engineering Research Council of Canada (PGSD3-426325-2012). P.S. acknowledges financial support by a scholarship from the “la Caixa” Banking Foundation. F.V. acknowledges financial support from Marie-Curie International Fellowship COFUND and ICFOnest program. F.H.L.K. acknowledges financial support from the Government of Catalonia trough the SGR grant (2014-SGR-1535), and from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0522), support by Fundacio Cellex Barcelona, CERCA Programme/Generalitat de Catalunya and the Mineco grants Ramón y Cajal (RYC-2012-12281) and Plan Nacional (FIS2013-47161-P and FIS2014-59639-JIN). Furthermore, the research leading to these results has received funding from the European Union Seventh Framework Programme under grant agreement no. 696656 Graphene Flagship and the ERC starting grant (307806, CarbonLight). ## Author information ### Affiliations 1. #### ICFO–Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, 08860, Spain • Mathieu Massicotte • , Fabien Vialla • , Peter Schmidt • , Mark B. Lundeberg • , Diana Davydovskaya • & Frank H. L. Koppens 2. #### CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs, Lyngby, Denmark • Simone Latini • , Sten Haastrup • & Kristian S. Thygesen 3. #### Center for Nanostructured Graphene (CNG), Technical University of Denmark, Kongens, Lyngby, 2800, Denmark • Simone Latini • & Kristian S. Thygesen 4. #### National Graphene Institute, University of Manchester, Booth St E, Manchester, M13 9PL, UK • Mark Danovich 5. #### National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan • Kenji Watanabe • & Takashi Taniguchi 6. #### Department of Physics and Nanotechnology, Aalborg University, DK-9220, Aalborg East, Denmark • Thomas G. Pedersen 7. #### Center for Nanostructured Graphene (CNG), DK-9220, Aalborg Øst, Denmark • Thomas G. Pedersen 8. #### ICREA – Institució Catalana de Recerça i Estudis Avancats, 08010, Barcelona, Spain • Frank H. L. Koppens ### Contributions M.M. conceived and designed the experiments under the supervision of F.H.L.K., M.M., D.D., and F.V. fabricated the samples. M.M. and F.V. carried out the experiments. M.M. performed the data analysis and discussed the results with F.H.L.K., F.V., and P.S. T.G.P. developed the Wannier–Mott exciton model. T.P.G, M.B.L., M.D., and V.I.F. performed the electrostatic calculations, and S.H., S.L., and K.S.T. performed the ab-initio calculations. K.W. and T.T. provided hBN crystals. M.M., F.V., P.S., and F.H.L.K. co-wrote the manuscript, with the participation of T.G.P. and K.S.T. ### Competing interests The authors declare no competing interests. ### Corresponding author Correspondence to Frank H. L. Koppens. ## Electronic supplementary material ### DOI https://doi.org/10.1038/s41467-018-03864-y	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8445746898651123, "perplexity": 4685.425803460964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247508363.74/warc/CC-MAIN-20190221193026-20190221215026-00317.warc.gz"}
https://www.physicsforums.com/threads/to-take-diff-eq-vs-pde-sequence.841529/	# To take DiFF EQ vs. PDE (sequence) 1. Nov 5, 2015 ### RJLiberator Greetings all, I am registering for spring 2016 courses and have one question. I can pick up a math course and I have the option between two courses: 430 Formal Logic vs. 481 Applied Partial Differential Equations. I am a math and physics double major. Course list and description: http://catalog.uic.edu/ucat/course-descriptions/math/ I have taking the intro to diff equations course 220. I think taking further differential equation courses would be a very good idea for me and my future. However, due to time constraints of other courses I cannot take class 480: Applied Differential equations this semester. So my question is: Is there a reason why I should wait and take 480 Applied Differential Equations before I take class 481 Partial Differential Equations? Will I be unprepared for 481 if I do not take 480 first? Or are they completely separate. If so, is there any reason to consider taking Formal Logic over 481 knowing that I am interested in mathematical physics? I would think PDE would help me more. 2. Nov 5, 2015 ### micromass Staff Emeritus Don't worry, 220 will be enough preparation for 481. There is no reason to take Formal logic for physics unless logic is of a significant interest to you. 3. Nov 5, 2015 ### RJLiberator Thank you for the response! That helps. Similar Discussions: To take DiFF EQ vs. PDE (sequence)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8699355721473694, "perplexity": 1630.6842784092871}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806620.63/warc/CC-MAIN-20171122175744-20171122195744-00106.warc.gz"}
https://math.stackexchange.com/questions/1721175/prove-that-for-a-power-series-function-that-is-constantly-zero-that-the-coeffic	# Prove that for a power series function that is constantly zero, that the coefficients are zero Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots$ is constantly zero on a bounded non-empty open interval $I$, which may or may not contain $0$. Prove that $a_j = 0$ for every $j$, so that the power series is constantly zero on its whole domain (which is of course centered at $0$).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.989885151386261, "perplexity": 91.61462410473293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232258453.85/warc/CC-MAIN-20190525224929-20190526010929-00213.warc.gz"}
http://www.physicsforums.com/showthread.php?t=516434	# Abundance of Isotopes by FeDeX_LaTeX Tags: abundance, isotopes P: 427 1. The problem statement, all variables and given/known data 1) "The relative atomic mass of antimony is 121.8. Antimony exists as two isotopes; antimony-121 and antimony-123. Calculate the relative abundances of the two isotopes." 2) "The relative atomic mass of rubidium is 85.5. Rubidium consists of two isotopes, rubidium-85 and rubidium-87. Calculate the percentage of rubidium-87 in naturally occurring rubidium." 2. Relevant equations N/A 3. The attempt at a solution I thought these questions should be relatively easy but the solutions at the back of the book don't seem to agree with my answers. For question 1, I did [121x + 123(100-x)]/100 = 121.8 and solved for x. Likewise for question 2. For question 1, I get that antimony-121 is 60% and antimony-123 is 40%, but the textbook says the answers should be 37.5% and 62.5% respectively. For question 2, I get Rb-85 occurring 75% and Rb-87 occurring 25%, yet the textbook says Rb-87 occurs 23.5%. However I checked my answers and I do get the average relative atomic masses that they state. Where have I gone wrong? Thanks. HW Helper Thanks P: 4,958 Quote by FeDeX_LaTeX 1. The problem statement, all variables and given/known data 1) "The relative atomic mass of antimony is 121.8. Antimony exists as two isotopes; antimony-121 and antimony-123. Calculate the relative abundances of the two isotopes." 2) "The relative atomic mass of rubidium is 85.5. Rubidium consists of two isotopes, rubidium-85 and rubidium-87. Calculate the percentage of rubidium-87 in naturally occurring rubidium." 2. Relevant equations N/A 3. The attempt at a solution I thought these questions should be relatively easy but the solutions at the back of the book don't seem to agree with my answers. For question 1, I did [121x + 123(100-x)] = 121.8 and solved for x. Likewise for question 2. For question 1, I get that antimony-121 is 60% and antimony-123 is 40%, but the textbook says the answers should be 37.5% and 62.5% respectively. For question 2, I get Rb-85 occurring 75% and Rb-87 occurring 25%, yet the textbook says Rb-87 occurs 23.5%. However I checked my answers and I do get the average relative atomic masses that they state. Where have I gone wrong? Thanks. According to your type of calculation the average of the numbers 5 and 5 should be [5x + 5(100-x)] = 500. Do you see anything wrong with this calculation? RGV Emeritus Sci Advisor HW Helper PF Gold P: 7,800 You need to know the atomic mass of each of the isotopes: antimony-121, antimony-123, rubidium-85, and rubidium-87. You can't assume that they're 121, 123, 85, and 87. Wikipedia page on isotopes of Antimony. Wikipedia page on isotopes of Rubidium. P: 427 Abundance of Isotopes Hello, Thank you for your responses. I'm just a little confused and would appreciate if someone could explain to me why my method is incorrect though. For example, for chlorine, Cl-35 occurs 75% of the time and Cl-37 occurs 25% of the time. So the average, according to the textbook, is; (350.75)+(370.25) = 35.5 Which is why 35.5 amu is the relative atomic mass of chlorine. In the questions I did, I followed the same method, but since the percentages are unknown I let the percentage be x, and the other percentage be (100-x), since percentages must total to 100. I don't understand why the atomic mass Ar of Rb-85 wouldn't just be 85? All questions are supposed to be doable using only the textbook. The question before this I got correct using this method, and the method working they've shown looks identical to mine... Sci Advisor HW Helper Thanks P: 4,958 In my first reply I asked you a question---for a good reason. So far you have not answered it, but had you done so you would see right away where your problem lies. RGV P: 427 Hello, I saw your post but I still don't see the error, solving for x I get x = 50% (or 0.5) which is how often each of your numbers (5 and 5) occur... I don't understand. I'm not sure how your example is relevant because it doesn't involve weighted averages... For chlorine, the approximate relative atomic mass is 35.5, because the textbook explains that Cl-35 occurs 75% of the time and Cl-37 25% of the time, so by weighted averages, 350.75 + 370.25 = 35.5 This is exactly what I have done for my questions, so why are they incorrect? HW Helper Thanks P: 10,380 Find and use more accurate data for the average atomic masses and isotope masses. The links SammyS gave you are excellent. ehild Emeritus Sci Advisor HW Helper PF Gold P: 7,800 I believe that the 60% you calculated is the percent mass of Sb-121 in a sample of antimony. Since each atom of Sb-121 has less mass than each atom of Sb-123, the number of atoms of Sb-121 is more than 60% of the total number of antimony atoms in the sample. Sci Advisor HW Helper Thanks P: 4,958 I was responding to the unedited post and had not noticed that you had edited it. That said, you got good advice from Sammy S. The actual weight of Antimony 121 is not exactly 121 times the weight of one nucleon, because of binding energy, etc. Of course, there is the possibility that the book's answer is wrong; that happens frequently. RGV RGV HW Helper Thanks P: 10,380 Yes, the book mixed the abundances, that of Sb121 is greater. You can find accurate abundances here: http://presolar.wustl.edu/ref/IsotopeAbundanceTable.pdf ehild Related Discussions Biology, Chemistry & Other Homework 3 Advanced Physics Homework 0 Biology, Chemistry & Other Homework 3 Chemistry 2 Nuclear Engineering 48	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9137182235717773, "perplexity": 1232.892894826662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510268533.31/warc/CC-MAIN-20140728011748-00233-ip-10-146-231-18.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/73847/evaluating-partial-sums	# Evaluating Partial Sums I need some help with the following question from my homework, I do not exactly understand what to do. Question at Hand Text is: Evaluate the partial sums of the infinite series $\displaystyle \sum_{n=1}^\infty \frac1{n(n+2)}$, and then evaluate the infinite series. The trouble I am having is understanding exactly what is asked of me to do. - Well, you try to find an expression for the partial sums first. Do you know partial fraction decomposition? – Ｊ. Ｍ. Oct 19 '11 at 1:30 The question is asking you to find a formula for $S(M) = \sum_{n \leq M} \frac{1}{n(n+2)}$. – JavaMan Oct 19 '11 at 1:31 The m-th partial sum of $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n(n+2) }$ is the sum truncated to the m-th term. In other words, it first wants you to find the finite sum $$s_m = \sum_{n=1}^m \frac{1}{n(n+2)}$$ for all $m\in \mathbb{N}$. Then it wants you to find the original infinite sum by recalling the definition that $$\sum_{n=1}^{\infty} \frac{1}{n(n+2) } = \lim_{m\to \infty} s_m .$$ @Steve: And to find $s_m$, you’ll want to use partial fractions. After you’ve found the partial fraction decomposition, you may also find it very helpful to write out the first few partial sums in full, paying close attention to the algebraic signs of the terms. – Brian M. Scott Oct 19 '11 at 1:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9193180799484253, "perplexity": 170.16503603345197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422121833101.33/warc/CC-MAIN-20150124175033-00016-ip-10-180-212-252.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/permeability-of-light.65331/	# Permeability Of Light 1. Feb 28, 2005 ### mustaffarel What does a substance's permeability of light depend on? 2. Feb 28, 2005 ### Integral Staff Emeritus The permeability of free space,or the permeability of a substance, is the parameter that determines how the substance effects a magnetic field. It does not just apply to light but to all magnetic fields 3. Feb 28, 2005 ### mustaffarel Is it because of visible light is a kind of electromagnetic wave? 4. Feb 28, 2005 ### kirovman Visible light is only a small part of the Electromagnetic wave spectrum, ranging from radio waves, up to X-rays and Gamma Rays, with visible light somewhere in the middle of the scale. The parameter $$\mu_o$$ is the permeability of free space. $$\mu_r$$ is the relative permeability (analogous to $$\epsilon_r$$) It can be defined as $$B=\mu_0 (H+M)$$ Where B is the magentic induction field, H is the magnetic field strength and M is the magnetisation across the sample. or with some working: $$B = \mu_0 \mu_r H$$ Since $$M = \chi_m H$$ and $$\mu_r = (1+\chi_m)$$ ($$\chi_m$$ is magnetic susceptibility) relative permeability in a material is just some kind of a measure of how much of an applied magnetic field will be "used up" due to magnetisation in the material. Also it is said that B is what you pay for, H is what you get. Last edited: Feb 28, 2005 Similar Discussions: Permeability Of Light	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9319729804992676, "perplexity": 1168.8495058701585}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463609404.11/warc/CC-MAIN-20170528004908-20170528024908-00099.warc.gz"}
http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume23/cayrol05a-html/node22.html	# Computation of tupled values We propose an algorithm for computing the tupled values for an arbitrary graph (cyclic or acyclic, the cycles may be isolated or not). This algorithm uses a principle of propagation of values: an argument is evaluated when the values of its direct attackers are known. We must consider the cycles as meta-arguments which are evaluated when all the direct attackers of the cycle'' (i.e. the direct attackers of one of the elements of the cycle which do not belong to the cycle) are evaluated. The beginning of the process is as follows: we consider that all the arguments have the initial value , and only the leaves of the graph are marked'' as having their final values. Thus, we have the following partition of the graph : • : the part of the graph already evaluated (at the beginning, this part contains only the leaves of the graph), • : the part of the graph which is not evaluated (at the beginning, this part contains all the arguments of the graph except the leaves). The algorithm also relies on a special data structure denoted by giving the list of the cycles in the graph and their main characteristics: • list of the arguments which belong to this cycle, • list of the arguments which belong to this cycle and which have direct attackers outside the cycle (these arguments are called inputs of the cycle; those which will be used in order to propagate the values across the cycle in the case of a non isolated cycle); this list will be empty in the case of an isolated cycle. Remark: For the sake of efficiency, the interconnected cycles (see Definition 1) will be considered as a whole'' by the algorithm and will be used like a meta-cycle''. For example, the two cycles and which do not have any direct attacker outside of the cycles, will be described in the data structure as only one meta-cycle'' with the following lists: • , , , • nothing (because it is an isolated meta-cycle''). In order to avoid some ambiguity, these meta-cycles'' are defined as mcycles: Definition 21 (mcycle) Let be an attack graph. Let be the set of all the cycles of . Let and be a set of cycles. Let be the set: . If satisfies the following properties: • , a path from to such that each element (arguments or edges between arguments) of the path belongs to cycles of , • and , such that is interconnected with . Then the union of the belonging to is a mcycle. Thus, we make a partition of using the notion of interconnection between cycles, each element of the partition being a different mcycle. See on the following example: In this graph, there are 6 cycles: • , • , • , • , • , • . and 3 mcycles: • , • , • . Algorithm 2 is the main algorithm used for computing the tupled values. The function ADD-NODE (respectively REMOVE-NODE) whose parameters are a subgraph of the attack graph and a node , adds (resp. removes) in (resp. of) . The other functions are described in [5]. Algorithm 2 has been applied on an example after the step of rewriting (see Figure 1). Note that in order to make the understanding of the results easier, we do not have created new arguments (as in Definitions 11 and 12), but of course, it would be necessary for a rigorous formalization. The previous argumentation graph can be rewritten as follows: The results of the valuation obtained after one propagation step are: , , , , , , , Marie-Christine Lagasquie 2005-02-04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9195408821105957, "perplexity": 696.8421240225296}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378232.65/warc/CC-MAIN-20141119123258-00177-ip-10-235-23-156.ec2.internal.warc.gz"}
https://aar.pausd.org/project/martinezpanamalraj	# Optimal Environmental Conditions for Biohydrogen Production by Chlamydomonas Moewusii by Cory P. & Jessica A. ## Summary What are the optimal conditions for algal biohydrogen production by Chlamydomonas moewusii? For hundreds of years, mankind has utilized fossil fuels, such as petroleum, coal, and natural gas, to power many facets of its society. For example, most power plants run on coal or natural gas, and most large trucks and airplanes run off kerosene-based fuels or petroleum diesel. However, concerns regarding greenhouse gas emissions and consequent global warming have led scientists and engineers to research and/or develop new ways of obtaining clean energy. One promising solution is the use of hydrogen fuel. Hydrogen fuel (H2) is seen as a potential alternative to fossil fuel because of its high specific energy/energy density and clean combustion, which can be shown with its simple reaction equation: 2H2 + O2 → 2H2O A reason hydrogen fuel is not currently mainstream is that hydrogen fuel, by itself, is not a renewable energy source. It does not hide in pockets underground, and cannot be found in large amounts on Earth’s surface. The production of hydrogen fuel determines the cleanliness of its use; hydrogen fuel produced by burning fossil fuels is not considered clean energy, whereas hydrogen fuel produced by renewable energy sources, such as solar, wind, or nuclear energy, is considered clean energy. In addition to its ability to be used in hydrogen fuel cells, hydrogen fuel can also be reacted with carbon monoxide (CO) to produce liquid hydrocarbon fuels. This research exploration focuses on biohydrogen, or hydrogen fuel produced by algae under certain environmental conditions. We are going to manipulate the conditions on algae in its optimal environment, to see the effect on hydrogen gas production. The enzyme hydrogenase is responsible for creating hydrogen gas. The optimal temperature is 37 deg C, and the optimal pH is 6.0. We will test H2 production in the presence of various substances, like acetic acid or hydrochloric acid. We will likely relate the treatment substance to something that is becoming more prevalent in the environment due to climate change. The measure of H2 gas produced would show the flexibility of and strength of the hydrogenase...	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8038987517356873, "perplexity": 1784.461156613329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00509.warc.gz"}
http://mathhelpforum.com/calculus/15469-need-help-forgotten-basics.html	# Math Help - need help...forgotten the basics 1. ## need help...forgotten the basics i am trying to solve this [t is time and T is temperature] to find the time t, 6450Pi(0.00185^2)(0.0625)836 dt = --------------------------------------------- dT (1.2^2)7.5 - 108Pi0.001850.25T between the limits 0 and 180degC i have tried & got this answer, t = 3.6236 ln (10.8 - 0.156923T) and when solving between the limits, I am getting error because of ln (10.8-0.156923(180)) which is negative. can someone help me 2. Your equation is a mess. Type it more neatly. 3. sorry, hope this is more readable dt = (3.6236)/(10.8 - 0.156923T) dT thanks 4. Originally Posted by hks sorry, hope this is more readable dt = (3.6236)/(10.8 - 0.156923T) dT thanks Ignore the numbers, Say, $a,b,c>0$ and we have: $dt = \frac{a}{b-c\tau }d\tau$ I hate the subhuman differencial notation, let me write instead, $\frac{dt}{d\tau}=t' = \frac{a}{b-c\tau}$ Now integrate both sides, $\int t' d\tau = \int \frac{a}{b-c\tau} d\tau$ $t = -\frac{a}{c}\ln \|b-c\tau\|+C$ Now just evaluate this at $\tau = 180$ and then subtract $\tau =0$. (You can ignore the constant). Once you do that you can substitute the true values for $a,b,c$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9760457277297974, "perplexity": 4460.767177613605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823963.50/warc/CC-MAIN-20160723071023-00296-ip-10-185-27-174.ec2.internal.warc.gz"}
http://www.physicsforums.com/showpost.php?p=2894773&postcount=13	View Single Post P: 315 integral of e^(-x/5) is -5e^(-x/5) but why am i integrating again? i can see there might be a little confusion where i did my integral... i skipped a step near the end... but the complete integral of the given function is -5xe(-x/5) - e(-x/5) so howcome i need to integrate the second term?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9929108619689941, "perplexity": 890.7443385185298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999655109/warc/CC-MAIN-20140305060735-00078-ip-10-183-142-35.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1996876/help-with-rules-of-identity-of-first-order-logic-with-equality	# Help with Rules of Identity of First Order Logic with Equality I'm a beginner in logic and I'm studying with textbooks. Right now I've just got to predicate logic with identity and I need to ask a few questions, so I can free my mind of doubts and sleep well at night. Do the identity rules (Id) p//x=x (reflexivity); x=y ⇔ y=x (symmetry); x=y/y=z//x=z (transitivity); Fx/x=y//Fy, Fx/¬Fy//¬(x=y) (substitution); apply to both variables and constants? I'm almost certain that they do, but there's this textbook that says they only apply to constants, and then a more recent edition of the same book says it apply to both variables and constants. So I just need to be sure. Another question: If I have 1. Raa 2. ¬Rab can I infer from both premises the line ¬(a=b) with the Id rules, or do I need some intermediate step? Or is it just wrong? One last question: when doing Existential Instantiation (EI), I know I can replace the variable with a new constant, one that did not appear in the proof in any preceding line and in the conclusion line, and then drop the quantifier; but there's a textbook that says I could instantiate with a variable, providing it's a new one that has not been used, so this mean I can do EI with both variables and constants? I was sure that I could only instantiate with a constant, and that the constant was supposed to be a "temporary name". Can anyone clear this to me? • If you have the rule : "from $Fx$ and $\lnot Fy$ derive : $\lnot (x=y)$", you can apply it with $Rax$ as $Fx$. Thus $Raa$ is $Fa$ and $\lnot Rab$ is $\lnot Fb$ and you can conclude with : $\lnot (a=b)$. – Mauro ALLEGRANZA Nov 3 '16 at 13:53 • But there is no need to have this rule, because it is a simple consequence of the preceeding one (by tautological equivalence between : $(p \land q) \to r$ and $(p \land \lnot r) \to \lnot q$). – Mauro ALLEGRANZA Nov 3 '16 at 15:32 • I see, does this mean relational predicates are no different than monadic predicates when we apply the identity rules? Even if it is an intransitive relation? And about the equivalences you mention how do they relate to the identity rules? – Iconoclasteretic Nov 5 '16 at 6:44 Then, to infer $\neg a = b$ you probably need to do a proof by contradiction: assume $a=b$, infer $Rab$ by substituting $b$ for the second $a$ in $Raa$, and that contradicts with $\neg Rab$ • Your textbooks are not contradictory; they just define the rules differently. And so what you can or cannot do all depends on how the rules are defined for the particular system you decide to use. So yes, maybe there is a system out there that can get the $\neg a = b$ in one step ... but none that I am familiar with. – Bram28 Nov 3 '16 at 0:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8282257318496704, "perplexity": 453.7672995595604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528290.72/warc/CC-MAIN-20190722221756-20190723003756-00448.warc.gz"}
https://slideplayer.com/slide/733858/	# Infinite Limits at Limits & Infinity. ## Presentation on theme: "Infinite Limits at Limits & Infinity."— Presentation transcript: Limits & Limits & Limits at Limits at Infinite Infinite Infinity Infinity Infinity Infinity Infinite Infinite Limits at Limits at Limits & Limits & Infinite Limits at Limits & Infinity Infinite Limits vs. Limits at Infinity Recall the limit notation: limxc f(x) = N means “as x is approaching c, but remains unequal to c, the corresponding value of f(x) is approaching to N.” Now, If we allow N to be  and –, e.g., limxc f(x) =  and limxc f(x) = –, we have infinite limits. If we allow c to be  and –, e.g., limx  f(x) and limx – f(x), we have limits at infinity. limxc f(x) =  limx f(x) c c limx– f(x) limxc f(x) = – Infinite Limits We know, when k  0, k /0 is undefined. However, in terms of limit, it’s really either + or –. Of course, depends on whether k is positive or negative and, also, whether 0 is “positive” or “negative” If k is positive, then 0 is “positive”, then k/0 = ____. If k is positive, then 0 is “negative”, then k/0 = ____. If k is negative, then 0 is “positive”, then k/0 = ____. If k is negative, then 0 is “negative”, then k/0 = ____. Infinite Limits (cont’d) Definition of “Positive Zero” and “Negative Zero” “Positive zero” (denoted by +0) is defined as a quantity (usually the denominator) is approaching 0 but remains slightly greater than 0 (i.e, positive). “Negative zero” (denoted by –0) is defined as a quantity (usually the denominator) is approaching 0 but remains slightly less than 0 (i.e, negative). Examples: Infinite Limits (cont’d) Example 10: Example 11: Final Notes: When we do have the denominator approaching 0 (but not the numerator), we should always consider the ___________ limits, i.e., the _____________ limit and the ______________ limit. Recall that, if the right-sided limit is the same as the left-sided limit (including  and –), then the limit is that quantity. Otherwise, the limit doesn’t exist (DNE). Limits at Infinity Q: What is k/ where k is any constant? A: ___ Examples: Q: What is /k where k is positive constant? Q: What is /? A: It’s one of those ____________ forms, i.e, a __________ Examples: Note: A limit in an indeterminate form, e.g. [0/0] and [/], only means the limit can’t be determined by simply the “plug-it-in” method, but may be determined by other means. Limits at Infinity (cont’d) Q: What is /? A: As mentioned earlier, it’s an indeterminate form, i.e., and it depends on _________ and the ___________, whichever is “larger”. Q: Can one  be “larger” or “smaller” than another ? A: Yes, of course. Here you go: i)  < 2 < 3 < 4 < 5 < ... That is, if m and n are positive, and m < n, then m < n. Here we say, the magnitude of m is smaller than the magnitude of n. ii)  < 2 < 3 < 4 < 5 < ... Although here we see an ∞ on the left is “smaller” than an ∞ on the right of the inequality above, we say , 2, 3, 4, 5, etc. have the same magnitude. In general, we say the magnitudes of two ∞’s are equal if they have the same exponents. Q: So what do you think the following should be? Summary If the numerator  has a larger magnitude than the denominator , then it’s ___; If the numerator  has a smaller magnitude than the denominator , then it’s ___; If the numerator  has the same magnitude than the denominator , then it’s a _____________, which can be obtained by _____________________. Examples: Limits at Infinity (cont’d) In the previous slide, the meaning of one  having a larger magnitude than the other if it has a larger exponent (e.g., 3 > 2, 7 > 4) and the meaning of one  having the same magnitude as the other if they have the same exponent (e.g., 53 and 43, 67 and 27). The problem with this definition is that, though it is conceivable that 3 is greater than 2, it is almost unimaginable to say 32 has the same magnitude as 22, when obviously 32 should be greater than 22. Therefore, we are going to redo and re-explain the limits a rational function as x   or x – (not mentioned previously), as such: When we have to find the limit of a rational function as x  , or x –, which term from the numerator and denominator really matters? Answer: _________________ Examples: 1. 2. 3. 4. 5. 6.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9957888126373291, "perplexity": 2662.595198869683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589980.6/warc/CC-MAIN-20180718002426-20180718022426-00435.warc.gz"}
https://because0fbeauty.wordpress.com/2013/02/28/integrals-are-not-infinite-sums/	## Integrals are not infinite sums I realized a mistake I’ve had today while sitting in my local analysis class. Realized is generous, rather it was pointed out to me. I’d generally thought of integrals such as $\int_a^b f(x) dx$ as sums of f(x) where the index ranged over [a,b]. Clearly I had never thought very hard about this. If you add up every value of f(x) and it’s not zero a lot your sum will diverge. The integral is different because of the measure it uses (I’m assuming a Lesbesgue measure on R). The measure for the uncountable sum is the counting measure on the set [a,b] in which every singleton has measure one, versus Lebesgue which gives measure zero to such a set. We can certainly write something like $\sum_{\alpha\in A} f(\alpha)=\int_A f(\alpha) d\mu$ where $\mu$ is the counting measure, but we can’t think of it as the Lebesgue measure. Which means I think it’s about time that I look into the Stieltjes integral and see how to understand the spectrum of an operator that has both discrete and continuous spectrum. Next post perhaps.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 3, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9937075972557068, "perplexity": 294.42589538071877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423839.97/warc/CC-MAIN-20170722002507-20170722022507-00105.warc.gz"}
http://math.stackexchange.com/questions/147525/sage-returning-0-for-curve-arc-length-integrals	# Sage returning 0 for curve arc length (integrals) I'm using Sage to calculate a curve arc length. Basically, the curve is given by this equation: In sage, I'm calculating it's arc length by using this formula: var('t') integral(sqrt((derivative(t - sin(t)))^2 + (derivative(1 - cos(t)))^2 + (derivative(4 * cos(t/2)))^2), t, 0, 8pi); The formula is from here: http://www.mathwords.com/a/arc_length_of_a_curve.htm Basically, it should return 32 sqrt(2) but it always returns 0. Could anyone tell me why? I probably did something wrong with the syntax, but can't figure out where. - I deleted my answer because it is indeed a branch problem and not a syntax problem. – Phira May 20 '12 at 23:13 var('t'); f = sqrt(sin(t)^2); integral(f,t,0,2*pi) <-- this throws an error, asking for an assumption on the sign of sin(t). If assume(sin(t)>0) is added [though false], the answer is 0. As much as I like open-source software, I guess the message is to look elsewhere when you need symbolic integration. – user31373 May 20 '12 at 23:18 Probably Sage did not keep track of branches, and used a discontinuous anti-derivative. Maple, also, has a discontinuous anti-derivative. But Maple manages to take the jumps into account and gets the right answer. \begin{align} &\int 2 \sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{sin} \biggl(\frac{t}{2}\biggr) d t = -4\sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{cos} \biggl(\frac{t}{2}\biggr) \\ &\int_{0}^{8 \pi} 2 \sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{sin} \biggl(\frac{t}{2}\biggr) d t = 32 \sqrt{2} \end{align}	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9556014537811279, "perplexity": 2355.056161006298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510267075.55/warc/CC-MAIN-20140728011747-00096-ip-10-146-231-18.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-equations/132615-second-order-differential-equation-problem.html	# Math Help - Second Order differential equation problem 1. ## Second Order differential equation problem y''+9y=30sin(3x) I'm told to use the method of solving y''+9y=30e^(3ix), but I'm not sure what to do after I find (D-3)(D-3)y=30e^(3ix). Could I have some help? 2. In terms of Laplace Transform the particular integral of... $y^{''} + 9\cdot y = 30 \cdot \sin 3x$ (1) ... is written as... $Y_{p} (s) = \frac{10}{(s^{2}+9)^{2}}$ (2) ... so that is... $y_{p} (x) = \mathcal {L} \{Y_{p} (s)\} = \frac {5}{27}\cdot (\sin 3x - 3x\cdot \cos 3x)$ (3) Alternatively [and is is a not confortable job ...] You can search for function of the type... $y_{p} (x) = x \cdot (\alpha\cdot \cos 3x + \beta\cdot \sin 3x)$ (4) ... that satisfies (1)... Kind regards $\chi$ $\sigma$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9211094379425049, "perplexity": 2385.610137690961}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445033.85/warc/CC-MAIN-20151124205405-00044-ip-10-71-132-137.ec2.internal.warc.gz"}
https://www.gerad.ca/fr/papers/G-2006-72	Groupe d’études et de recherche en analyse des décisions # Isoperimetric Polygons of Maximal Width ## Charles Audet, Pierre Hansen et Frédéric Messine The value $\frac{1}{2n} \cot\left( \frac{\pi}{2n}\right)$ is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd, and clipped regular Reuleaux polygons when n is even but not a power of 2. Using a global optimization algorithm, we solve the problem for n =4. The optimal width for the quadrilateral is shown to be $\frac{1}{4} \sqrt{ 3( 2\sqrt 3 -3 )} \approx 0.2949899\ldots$ We propose two mathematical programs to determine the maximal width when n =2s with $s\geq 3$ and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n =8,16 and 32. , 24 pages	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8862274289131165, "perplexity": 576.8466975715334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102307.32/warc/CC-MAIN-20170816144701-20170816164701-00150.warc.gz"}
https://en.m.wikisource.org/wiki/Talk:Electromagnetic_phenomena	# Talk:Electromagnetic phenomena Active discussions ## section 5 Near end of section 5 it says "take \varrho and the vector \varphi,\ \mathfrak{u}', such as they are in the element dS" It seems as if "as" should read "that". However, I am not sure I ought to make such a change, because as it it is maybe not strictly incorrect English (grammar or spelling), so maybe it should be considered a part of the author's style rather than a typo or error. ??? Knotwork (talk) 13:18, 11 April 2009 (UTC) ## section 7 Near start of section 7 it says "Our second special case is that of a particle having an electric moment, i.e. a small space S, with a total charge \int\varrho\ x\ dS,\ \int\varrho\ y\ dS,\ \int\varrho\ z\ dS have values differing from 0." This seems incorrect English grammar. But I am not sure whether or how to fix it. My intuition tells me it should say "has" instead of "have", but the grammar is bad enough that I am not totally sure what it is really trying to say, and afterall it is a historic document so to speak so I am hesitant to mess with it too much. Knotwork (talk) 13:42, 11 April 2009 (UTC) It goes on to say "Let x, y, z be the coordinates, taken relatively to a fixed point A" ; normally in such papers they tend to say "relative to", leading me to wonder what difference might or might not be intended by saying "relatively to" instead of the (in my reading seemingly more normal) "relative to". Is there a difference? Or is this merely style? Again I am hesitant to change it though in a Wikipedia article I certainly would change it... Knotwork (talk) 13:46, 11 April 2009 (UTC) Thank you for your edits, which are mostly correct. However, those grammatical errors you pointed in sections 5 and 7, actually appeared in the original paper of Lorentz - who was no native English speaker. So those passages should remain unchanged in the article. --D.H (talk) 14:51, 11 April 2009 (UTC) ## section 10 I changed "deflexions" to "deflections" because dictionaries returned the word deflection when consulted about deflexions, but it seems to me that if a word "deflexions" did exist it would have a meaning different from that of deflections; the word deflexions does appear in google lookups, although not always clearly intended seriously. Maybe online dictionaries are simply not complete as so highly technical terms of physics? Knotwork (talk) 15:03, 11 April 2009 (UTC)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8971911072731018, "perplexity": 1274.3317705394516}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570901.18/warc/CC-MAIN-20220809033952-20220809063952-00547.warc.gz"}
http://clay6.com/qa/11969/www.clay6.com/qa/11969/using-differentials-find-the-approximate-value-of-each-of-the-following-up-	Comment Share Q) # Using differentials, find the approximate value of each of the following up to $3$ places of decimal. $(0.999)^{\Large\frac{1}{10}}$ $\begin{array}{1 1} 0.9999 \\ 0.0999 \\ 0.9000 \\ 0.8999 \end{array}$ Comment A) Toolbox: • Let $y=f(x)$ • $\Delta x$ denote a small increment in $x$ • $\Delta y=f(x+\Delta x)-f(x)$ • $dy=\big(\large\frac{dy}{dx}\big)\Delta x$ Step 1: Let $y=x^{\Large\frac{1}{10}}$ Then $y+\Delta y=(x+\Delta x)^{\Large\frac{1}{10}}$ $\Delta y=(x+\Delta x)^{\Large\frac{1}{10}}-x^{\Large\frac{1}{10}}$ $\Delta y=\sqrt{x+\Delta x}-x^{\Large\frac{1}{10}}$ $x+\Delta x=0.999$ and $x=1$ $\therefore \Delta x=-0.001$ Hence $\Delta y=(0.999)^{\Large\frac{1}{10}}-(1)^{\Large\frac{1}{10}}$ $(0.999)^{\Large\frac{1}{10}}$$=1+\Delta y-----(1) Step 2: \Delta y\approx dy=\large\frac{dy}{dx}.$$\Delta x$ $\qquad\;\;\;\;=\large\frac{1}{10}$$x^{\Large\frac{-9}{10}}\Delta x \qquad\;\;\;\;=\large\frac{1}{10}$$1^{\Large\frac{-9}{10}}(-0.001)$ $\qquad\;\;\;\;=-0.0001$ From equ(1) we have $(0.999)^{\Large\frac{1}{10}}=1-0.0001$ $\qquad\;\;\;\;\;\;\;\;=0.9999$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9989880323410034, "perplexity": 4970.176934350085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825363.58/warc/CC-MAIN-20181214044833-20181214070333-00295.warc.gz"}
http://www.probabilaball.com/2015/09/more-pitching-stabilization-points.html	## 03 September, 2015 ### More Pitching Stabilization Points Using the beta-binomial model (notated BB) or the gamma-Poisson model (notated GP, and in this post what I call M is what in the previous post I called K - the variance parameter of the population talent distribution), I calculated the stabilization point for some more pitching statistics. I don't think the model(s) fit perfectly to the data, but they provide a good approximation that generally matches up with results I've seen elsewhere on the web. Data was acquired from fangraphs.com. I only considered starting pitchers from 2009 - 2014, splitting the same pitcher between years, and did not adjust the data in any way. All the data and code I used here may be found on my github. I make no claims to efficiency or ease of use. The "cutoff value" is the minimum number of the denominator (IP, TBF, BIP, etc.) in a year in order to be included in the data set. These numbers were chosen somewhat arbitrarily, and for some of my statistics, changing the cutoff value will change the stabilization point. I'm not sure which statistics this will happen to - I know WHIP for sure, and I suspect ER as well, whereas I think BABIP doesn't exhibit this tendency. It's a function of the change (or lack thereof) in population variance of talent levels as the cutoff value increases - if somebody wants to take a look at it, it would be neat. I wanted have a little fun and apply the model to stats where it clearly is silly to do so, such as win rate (I defined as wins per game started) and extra batters faced per inning (the total number of additional batters a pitcher faced beyond what is required by their IP). The model still produces estimates, but of course, but bad data fed into a good model doesn't magically produce good analysis. \begin{array}{\| l \| l \| c \| c \| c \| c \| c \|} \hline \textrm{Stat}&\textrm{Formula}&\hat{M}&SE(\hat{M})&\textrm{95% CI}&\textrm{Cutoff}&\textrm{Model}\\ \hline \textrm{BABIP}&\textrm{(H-HR)/n}&2006.71&484.94&(1056.22,2957.20)&300&BB\\ \textrm{GB Rate}&\textrm{GB/BIP}&65.52&3.63&(58.39,72.64)&300&BB\\ \textrm{FB Rate}&\textrm{FB/BIP}&61.96&3.42&(55.25,68.66)&300&BB\\ \textrm{LD Rate}&\textrm{LD/BIP}&768.42&94.10&(583.99,952.86)&300&BB\\ \textrm{HR/FB Rate}&\textrm{HR/FB}&505.11&93.95&(320.96,689.26)&100&BB\\ \textrm{SO Rate}&\textrm{SO/TBF}&90.94&5.04&(81.06,100.82)&400&BB\\ \textrm{HR Rate}&\textrm{HR/TBF}&931.59&107.80&(720.30,1142.88)&400&BB\\ \textrm{BB Rate}&\textrm{(BB-IBB)/(TBF-IBB)}&221.25&14.43&(192.97,249.53)&400&BB\\ \textrm{HBP Rate}&\textrm{HBP/TBF}&989.30&119.95&(754.21,1224.41)&400&BB\\ \textrm{Hit rate}&\textrm{H/TBF}&623.35&57.57&(510.51,736.18)&400&BB\\ \textrm{OBP}&\textrm{(H + BB + HBP)/TBF}&524.73&44.96&(436.62,612.84)&400&BB\\ \textrm{Win Rate}&\textrm{W/GS}&57.23&8.68&(40.21,74.24)&15&BB\\ \textrm{WHIP}&\textrm{(H + BB)/IP}&77.20&5.46&(66.50,87.90)&80&GP\\ \textrm{ER Rate}&\textrm{ER/IP}&59.55&3.94&(51.82,67.25)&80&GP\\ \textrm{Extra BF}&\textrm{(TBF - 3IP)/IP}&73.00&5.08&(63.05,82.95)&80&GP\\ \hline \end{array} I'm not exactly sure what combinations of statistics fangraphs is using for the denominator of their BABIP - it's not BIP = GB + FB + LD. I know the numerator of H - HR is correct, but the denominator was usually smaller , though sometimes larger, than BIP. I solved for what fangraphs was using and used that in my calculations - if somebody wants to let me know exactly what they're using for n, please do. ** When dividing by IP, I corrected the 0.1 and 0.2 decimal representations to 0.33 and 0.67. I've also created histograms of each observed statistic with an overlay of the estimated distribution of true talent levels. They can be found in this imgur gallery. Remember that the dashed line represents the distribution of talent levels, not of observed data, so it's not necessarily bad if it is shaped differently than the observed data. $\hat{M}$ is the estimated variance parameter of the underlying talent distribution. Under the model, it is equal to the number of plate appearances at which there is 50% shrinkage. $SE(\hat{M})$ is the standard error of the estimate $\hat{M}$. It is on the same scale as the divisor in the formula. The 95% CI is calculated as $\hat{M} \pm 1.96 SE(\hat{M})$ It represents a 95% confidence interval for the number of plate appearances at which there is 50% shrinkage. For an arbitrary stabilization level $p$, the number of required plate appearances can be estimated as $\hat{n} = \left(\dfrac{p}{1-p}\right) \hat{M}$ And a 95% confidence interval for the required number of plate appearances is given as $\left(\dfrac{p}{1-p}\right) \hat{M} \pm 1.96 \left(\dfrac{p}{1-p}\right) SE(\hat{M})$ Since the denominators are so different (as opposed to offensive statistics where PA was used for almost everything except for batting average, and AB are fairly close to PA), I don't feel as comfortable putting everything on the same plot. That being said, the stats that use TBF look like And the stats that use BIP for their denominator look like	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8537899255752563, "perplexity": 1044.3101142472253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662573189.78/warc/CC-MAIN-20220524173011-20220524203011-00639.warc.gz"}
https://www.physicsforums.com/threads/nuclear-reaction-question.147034/	Homework Help: Nuclear Reaction Question 1. Dec 6, 2006 Scottlow How much more energy is in a nuclear reaction than a chemical reaction. 2. Dec 6, 2006 Staff: Mentor What do you know about the question so far? Where is the energy for a chemical reaction stored? Where is the energy for a nuclear reaction stored? What is the difference between a nuclear fusion reaction and a nuclear fission reaction? Tell us what you know so we can point you to resources where you can figure out this homework problem of yours.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833690524101257, "perplexity": 614.643352248744}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864063.9/warc/CC-MAIN-20180521102758-20180521122758-00471.warc.gz"}
http://math.stackexchange.com/questions/478789/inverse-laplace-transform-with-symbolic-variables	# inverse laplace transform - with symbolic variables Transform: $$F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s^2-a^2)(s-2b)}$$ My steps: $$F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s+a)(s-a)(s-2b)}$$ $$= \frac{A}{s+a} + \frac{B}{s-a} + \frac{C}{s-2b} + K$$ $$K = 0$$ $$A = F(s) * (s+a)$$ at s = -a $$A = \frac{2a^2 + (a-6b)(-a) +a^2 - 4ab}{4ab+2a^2}$$ $$A = \frac{a+b}{2b+a}$$ I problems like this (that I've seen) at the step above the fraction would reduce into just a number. In this case, it doesn't and its surprising because I've never seen it happened before. Is there something I am doing wrong here? - $$\frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s^2-a^2)(s-2b)}=F= \frac{A}{s+a} + \frac{B}{s-a} + \frac{C}{s-2b}$$ then by doing boring :-) handy calculations we can find $A,B$ and $C$. We have then: $$\frac{-Asa+2Aab+As^2-2Asb+Bsa-2Bab+Bs^2-2Bsb+Cs^2-Ca^2}{(-s+2b)(-s^2+a^2)}$$ Now if we put $s=-a$ in the numerators, we get $A=\frac{a+b}{a+2b}$ and this is what you already got. With the similar approach $s=+a$ for $B$ and $s=2b$ for $C$, we get: $$B=\frac{2a-5b}{a-2b},~~C=\frac{4b^2+2ab-a^2}{a^2-4b^2}$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9640049934387207, "perplexity": 118.93664603038543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115867691.21/warc/CC-MAIN-20150124161107-00066-ip-10-180-212-252.ec2.internal.warc.gz"}
https://pt.scribd.com/doc/55390761/MODELING-AND-SIMULATION-OF-WIND-TURBINES	P. 1 MODELING AND SIMULATION OF WIND TURBINES # MODELING AND SIMULATION OF WIND TURBINES \|Views: 3.008\|Likes: Publicado porm33m00 ### More info: Published by: m33m00 on May 13, 2011 Direitos Autorais:Attribution Non-commercial ### Availability: Read on Scribd mobile: iPhone, iPad and Android. download as PDF, TXT or read online from Scribd See more See less 01/09/2013 pdf text original ## Sections • Table 1.1 World Electricity Consumption with Estimations • Table 1.2 Wind Power Installations Worldwide • Table 1.3 Wind Energy Capacity Leaders Worldwide by End 2001 • Figure 1.3 Power transfer in a wind energy converter • Figure 2.2 Nacelle • Figure 2.3 Horizontal axis wind turbines according to number of blades • Table 2.1 Number of Blades for Commercial Wind Turbine Designs • 2.2.1. GEAR BOX • Figure 2.4 A typical gear • 2.2.2. V / Hz CONVERTER • Figure 2.5 AC – AC signal conversion • 2.2.3. YAW ASSEMBLY • 2.2.4. PITCH CONTROL MECHANISM • 2.2.5. ELECTRONIC CONTROLLER • Figure 2.6 A typical wind turbine in detail (VESTAS V27 / 225 kW • 3.1. AERODYNAMICS OF WIND TURBINES • 3.1.1. AERODYNAMIC FORCES • Figure 3.2 Lift and drag forces acting on rotor blade • 3.1.1.1. DRAG FORCES • 3.1.1.2. LIFT FORCES • 3.1.2. AERO-FOILS • Figure 3.3 Components of wind power acting on rotor blade • 3.2 ENERGY AND POWER IN THE WIND • 3.2.1. POWER COEFFICIENT • Figure 3.5 Wind flow through a wind turbine • 3.2.2. TIP SPEED RATIO • Table 3.1 Speed Definitions • 3.2.3. EFFECT OF THE NUMBER OF BLADES • 3.3. GENERATOR THEORY • 3.3.1.1. THEORY: • Figure 3.8 The equivalent circuit for DC motors • 3.3.1.2. DC GENERATOR APPLICATIONS IN WIND TURBINES • 3.3.2. SYNCHRONOUS AC MACHINES (ALTERNATORS) • 3.3.2.1. THEORY • Figure 3.9 A salient six-pole rotor for a synchronous machine • Figure 3.10 A non-salient two-pole rotor for a synchronous machine • 3.3.2.2. THE ROTATION SPEED OF A SYNCHRONOUS GENERATOR • 3.3.2.3. INTERNAL VOLTAGE OF A SYNCHRONOUS GENERATOR • 3.3.2.4. THE EQUIVALENT CIRCUIT OF AN ALTERNATOR • Figure 3.12 A simple circuit for alternators • Figure 3.13 The per-phase equivalent circuit for synchronous generators • 3.3.3. ASYNCHRONOUS (INDUCTION) AC MACHINES • Figure 3.14 Cutaway diagram for a wound-rotor induction machine • Figure 3.15 Cutaway diagram for a squirrel-cage induction machine • Figure 3.16 Transformer model for an induction machine • a transformer • 3.3.3.1.1. ROTOR CIRCUIT MODEL • Figure 3.18 The rotor circuit model for induction machines • concentrated in resistor RR • 3.3.3.1.2. FINAL EQUIVALENT CIRCUIT • Figure 3.20 The per-phase equivalent circuit for induction machines • Figure 3.21 Torque-Speed curve for a MW-size induction machine • Table 3.2 Common Synchronous Speeds for Generators • 3.3.4. RECENT DEVELOPMENTS IN GENERATORS FOR WIND • 3.3.4.1. DUAL GENERATORS • 3.3.4.2. DIRECT-DRIVE GENERATORS • 3.4. GRID INTEGRATION • 3.4.1. FREQUENCY CONVERTER SYSTEMS • Figure 3.22 Electrical energy conversion by power converters • Figure 3.23 Basic wiring diagram for direct frequency converters • Figure 3.24 Indirect frequency converters • 3.4.1.1.1. SEMICONDUCTOR DIODES • 3.4.1.1.2. THYRISTORS • 3.4.1.1.3. TRANSISTORS • semiconductors; • Semiconductors • 3.4.1.2. CHARACTERISTICS OF POWER CONVERTERS • • Classification by axis of rotation • 4.1. CLASSIFICATION BY AXIS OF ROTATION • Figure 4.1 Horizontal and vertical axis wind turbines • 4.1.1. HORIZONTAL AXIS WIND TURBINES (HAWT) • Figure 4.2 Horizontal axis wind turbine configurations • 4.1.2. VERTICAL AXIS WIND TURBINES (VAWT) • Figure 4.3 Vertical axis wind turbine configurations • 4.2. CLASSIFICATION BY ROTOR SPEED • 4.2.1. VARIABLE ROTOR SPEED • 4.2.2. CONSTANT ROTOR SPEED • 4.3. CLASSIFICATION BY POWER CONTROL • Figure 4.4 Operating regions of a typical wind turbine • Table 4.1 Descriptions of Operational Regions for a Typical Wind Turbine • Figure 4.5 Rotor diameter vs. power output • Figure 4.6 Swept area by rotor blades • 4.3.1. PITCH CONTROL • Figure 4.7 Pitch Control • 4.3.2. STALL CONTROL • Table 4.2 Pitch vs. Stall Issues • Figure 4.9 Stall & Pitch controlled power schemes • 4.4. CLASSIFICATION BY LOCATION OF INSTALLATION • 4.4.1 ON-SHORE WIND TURBINES • 4.4.2 OFF-SHORE WIND TURBINES • Figure 5.2 Yaw control block • Figure 5.3 Turbine efficiency block • Figure 5.4 Turbine efficiency characteristics corresponding to wind speed • mechanism • 5.1.4 ANGULAR SPEED CALCULATION BLOCK • Figure 5.18 Tip speed ratio vs. power coefficient • Table 5.1 Modelled Wind Turbine Simulation Results • REFERENCES • APPENDICES # MODELING AND SIMULATION OF WIND TURBINES A Thesis Submitted to the Graduate School of Natural and Applied Sciences of Dokuz Eylül University In Partial Fulfillment of the Requirements for the Degree of Master of Science in Electrical & Electronics Engineering, Electrical & Electronics Engineering Program by Osman Oral KIVRAK February, 2003 IZMIR M.Sc. THESIS EXAMINATION RESULT FORM We certify that we have read this thesis and “MODELING AND SIMULATION OF WIND TURBINES” completed by OSMAN ORAL KIVRAK under supervision of PROF. DR. MUSTAFA GÜNDÜZALP and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Prof. Dr. Mustafa GÜNDÜZALP Supervisor (Committee Member) (Committee Member) Approved by the Graduate School of Natural and Applied Sciences Prof. Dr. Cahit HELVACI Director I ACKNOWLEDGMENTS I wish to thank to my supervisor Prof. Dr. Mustafa GÜNDÜZALP for his guidance and understanding throughout my project. I wish also thank to Prof. Dr. Eyüp AKPINAR for his support on critical points. I am also grateful to my family and colleagues for their advices. Osman Oral KIVRAK II ABSTRACT Increasing worldwide energy deficiency causes raising importance of development of new energy resources. It is foreseen that new energy resources should not harm environment and natural life beside meeting present and future energy demand. Accordingly, a great tendency towards renewable energy resources took place in the market. Wind energy has become the most popular resource in the last decade by its purity and sustainability. Wind energy conversion systems convert the aerodynamic power in an air stream into the electric power. Principally, a wind energy conversion system consists of blade(s), which captures the aerodynamic power in the wind, shaft, which transfers the torque created by the turning action of blade(s) and generator, which converts this torque into electric power. Unlike other energy production systems, wind, as a source of energy for wind energy conversion systems, has a structure of showing sudden changes depending on climatic conditions. These sudden changes in wind speed may cause some unwanted mechanical or electrical damages, therefore it is necessary to supervise produced power curve continuously. Several power control methods are developed for this purpose. Pitch control – opening and closing of blades along their longitudinal axes - is the most efficient and popular power control method especially for variable-speed wind turbines. In this project, status and importance of wind energy conversion systems throughout the world, the energy conversion operation in wind turbines and components of them are investigated. Then, wind turbines are classified according to different categories. At final, a megawatt size, variable-speed wind turbine is modeled and its operation is observed by using MATLAB v5.2 – SIMULINK III software. Output power curve regulation is carried out by ‘pitch control’ method. The prototype for the simulation is VESTAS V80 – 2.0 MW model wind turbine. Keywords: Wind energy, renewable, turbine, variable speed, pitch control, energy conversion, MATLAB. IV ÖZET Enerji açiginin her geçen gün arttigi dünyamizda, yeni enerji kaynaklari gelistirmenin önemi de her geçen gün artmaktadir. Olusturulacak yeni enerji kaynaklarinin, mevcut ve gelecekteki enerji ihtiyacini karsilamasi ile birlikte, çevreyi ve dogal yasami da olumsuz yönde etkilememesi öngörülmektedir. Bu dogrultuda, enerji sektöründe yenilenebilir enerji kaynaklarina yönelim artmaktadir. Rüzgar enerjisi, temizligi ve sürekliligi ile, son 10 yilda en popüler kaynak olmustur. Rüzgar enerjisi dönüsüm sistemleri, rüzgarin içinde bulundurdugu aerodinamik gücü elektriksel güce dönüstürürler. Bir rüzgar enerjisi dönüsüm sistemi, prensip olarak, rüzgardaki aerodinamik gücü yakalayan kanat(lar), kanatlarin dönme hareketi ile olusan torku ileten saft ve bu mekanik torku elektriksel güce çeviren jeneratörden olusmaktadir. Diger enerji üretim sistemlerinden farkli olarak, rüzgar enerjisi dönüsüm sistemlerinde enerji kaynagi olarak kullanilan rüzgar, iklim kosullarina bagli olarak ani degisimler gösterebilen bir yapidadir. Bu ani degisimler, sistemde mekaniki ve elektriki birçok hasara yol açabileceginden, üretilen güç egrisinin sürekli denetim altinda bulundurulmasi gerekmektedir. Bu amaçla, çesitli güç kontrol yöntemleri gelistirilmistir. Pitch kontrolü – türbin kanatlarinin kendi dikey eksenlerinde açilip kapatilmasi -, özellikle degisken hizlarda çalisan rüzgar türbinleri için en verimli ve popüler güç kontrolü yöntemidir. Bu projede, rüzgar enerjisi dönüsüm sistemlerinin önemi ve dünyadaki durumu, rüzgar türbinlerinde gerçeklesen enerji dönüsüm islemi ve türbin aksamlari incelenmistir. Daha sonra rüzgar türbinleri çesitli kategorilere göre siniflandirilmistir. Son olarak, MATLAB v5.2 – SIMULINK yazilimi kullanilarak, degisken hizlarda çalisan megawatt boyutunda bir rüzgar türbini modellenerek çalismasi gözlenmistir. V Çikis gücü ayari ‘pitch control’ yöntemiyle gerçeklestirilmistir. Modelde prototip olarak VESTAS V80 – 2.0 MW model rüzgar türbini alinmistir. Anahtar Kelimeler: Rüzgar enerjisi, yenilenebilir, türbin, degisken hizli, açi kontrolü. VI CONTENTS Page Contents………………………………………………………………………... VI List of Tables…………………………………………………………………... X List of Figures...……………………………………………………………….. XI Chapter One INTRODUCTION 1.1 Historical Background…………………...………………………………....... 4 1.2 Functional Structure of Wind Turbines….………………………………....... 6 Chapter Two COMPONENTS OF WIND TURBINES 2.1 Common Components……………………...……………………………..... 8 2.1.1 Nacelle……………………..………………………………………........ 8 2.1.2 Blade……………..……………………...…………………………........ 8 2.1.3 Low Speed Shaft………………..….………………………………........ 11 2.1.4 High Speed Shaft…………..………………………………………........ 11 2.1.5 Disc Brake……………………………...….………………………........ 11 2.1.6 Generator……….……………………….…………………………........ 12 2.1.7 Tower……………………..………..………………………………........ 12 2.2 Optional Components……………………………………………………..... 13 VII 2.2.1 Gear Box……………..…………….………………………………..... 13 2.2.2 V / Hz Converter………………………..…………………………..... 13 2.2.3 Yaw Assembly………………………………………….…………..... 14 2.2.4 Pitch Control Mechanism……………...……………………………... 14 2.2.5 Electronic Controller…………………...…………………………...... 15 Chapter Three ELECTROMECHANICAL ENERGY CONVERSION 3.1 Aerodynamics of Wind Turbines………...………………………………....... 18 3.1.1 Aerodynamic Forces………..……...………………………………........ 18 3.1.1.1 Drag Forces……………….......………………………………........ 19 3.1.1.2 Lift Forces……………………………………….……………........ 19 3.1.2 Aero-Foils…………………………..………...……………………........ 20 3.2 Energy and Power in The Wind………….………………………………....... 22 3.2.1 Power Coefficient ……………………..…………………..………........ 25 3.2.2 Tip Speed Ratio………………………………………………................ 27 3.2.3 Effect of The Number of Blades……...................................................... 28 3.3 Generator Theory………………………...………………………………....... 33 3.3.1 DC Machines……..……………………………………………….......... 33 3.3.1.1 Theory…………………………...……………………………........ 33 3.3.1.2 DC Generator Applications in Wind Turbines…………………….. 36 3.3.2 Synchronous AC Machines (Alternators)………………………………. 36 3.3.2.1 Theory…………………………………………………................... 37 3.3.2.2 The Rotation Speed of a Synchronous Generator…………………. 39 3.3.2.3 Internal Voltage of a Synchronous Generator……………………... 40 3.3.2.4 The Equivalent Circuit of an Alternator……………………………42 3.3.3 Asynchronous (Induction) AC Machines………………………………. 44 3.3.3.1 Equivalent Circuit of an Induction Machine………………………. 46 3.3.3.1.1 Rotor Circuit Model………………………………………...... 48 3.3.3.1.2 Final Equivalent Circuit………………………………………. 50 VIII 3.3.4 Recent Developments in Generators for Wind Turbines……………….. 56 3.3.4.1 Dual Generators……………………………………………………. 56 3.3.4.2 Direct-Drive Generators……………………………………………57 3.4 Grid Integration……………………………………………………………..... 58 3.4.1 Frequency Converter Systems………………………………………...... 59 3.4.1.1 Power Semiconductors for Frequency Converters…………………63 3.4.1.1.1 Semiconductor Diodes……………………………………...... 64 3.4.1.1.2 Thyristors…………………………………………………...... 65 3.4.1.1.3 Transistors…............................................................................. 65 3.4.1.2 Characteristics of Power Converters………………………………. 67 Chapter Four CLASSIFICATION OF WIND TURBINES 4.1 Classification by Axis of Rotation……………………...………………......... 69 4.1.1 Horizontal Axis Wind Turbines (HAWT)…………………………........ 70 4.1.2 Vertical Axis Wind Turbines (VAWT)……………………………........ 71 4.2 Classification by Rotor Speed……………………………………………....... 72 4.2.1 Variable Rotor Speed…………..….………………………………........ 73 4.2.2 Constant Rotor Speed.…………………..…………………………........ 74 4.3 Classification by Power Control…………………………………………...…75 4.3.1 Pitch Control……………………………………………………………. 80 4.3.2 Stall Control…………………………………………………………….. 81 4.4 Classification by Location of Installation…………………………………..... 83 4.4.1 On-Shore Wind Turbines……………………………………………….. 83 4.4.2 Off-Shore Wind Turbines………………………………………………. 84 IX Chapter Five EXPERIMENTAL WORK 5.1 Sub-Systems in The Model……………………………...………………........ 89 5.1.1 Yaw Control Block………………………...………………………........ 89 5.1.2 Turbine Efficiency Block…………….……………………………........ 90 5.1.3 Pitch Control Block…………………………………………………...... 91 5.1.4 Angular Speed Calculation Block…........................................................ 93 5.1.5 Cp – ? Selection Block………………………………………………….. 95 5.2 Simulation Results……………………………………………………………95 Chapter Six CONCLUSIONS 6.1 Future Prospects………………………………………...……………….........106 References………...………………………………………...………………....... 108 Appendices….…………………………………………………………………... 110 Appendix A – Flowchart of The Simulated System………………………..... A Appendix B – VESTAS V80 – 2.0 MW Wind Turbine…………………....... B X LIST OF TABLES Page Table 1.1 World Electricity Consumption with Estimations………...………... 2 Table 1.2 Wind Power Installations Worldwide…..…………………………... 3 Table 1.3 Wind Energy Capacity Leaders Worldwide by End 2001....……….. 4 Table 2.1 Number of Blades for Commercial Wind Turbine Designs………… 11 Table 3.1 Speed Definitions…………………………………………………… 27 Table 3.2 Common Synchronous Speeds for Generators……………………... 55 Table 3.3 Characteristics and Maximum Ratings of Switchable Power Semiconductors………………………………….………………….. 67 Table 4.1 Descriptions of Operational Regions for a Typical Wind Turbine…. 77 Table 4.2 Pitch vs. Stall Issues………………………………………………… 82 Table 5.1 Modelled Wind Turbine Simulation Results……….......................... 103 XI LIST OF FIGURES Page Figure 1.1 World electricity consumption with estimations ..……………….. 1 Figure 1.2 Wind power installations worldwide…..…………………............. 2 Figure 1.3 Power transfer in a wind energy converter…………….................. 6 Figure 2.1 Wind turbine types by rotor assemblies………………………….. 7 Figure 2.2 Nacelle………...………………………………………….............. 8 Figure 2.3 Horizontal axis wind turbines according to number of blades…… 10 Figure 2.4 A typical gear…………………………………………………….. 13 Figure 2.5 AC – AC signal conversion………………………………............. 14 Figure 2.6 A typical wind turbine in detail (VESTAS V27 / 225 kW)...……. 16 Figure 3.1 A typical wind turbine showing all components…………………. 17 Figure 3.2 Lift and drag forces acting on rotor blade…………………........... 19 Figure 3.3 Components of wind power acting on rotor blade……………….. 21 Figure 3.4 Cylindrical volume of air passing at velocity V (10 m/s) through a ring enclosing an area, ‘A’, each second……………………….. 23 Figure 3.5 Wind flow through a wind turbine……………………………….. 25 Figure 3.6 Power coefficient versus tip speed ratio for a constant speed wind turbine…………………………………………………………….. 31 Figure 3.7 Power coefficient versus tip speed ratio for a variable speed wind turbine for different pitch angles from 0 to 15 degrees by 0.5 degree increments…………….…………………………………... 32 Figure 3.8 The equivalent circuit for DC motors……………………….……. 34 Figure 3.9 A salient six-pole rotor for a synchronous machine……………… 38 Figure 3.10 A non-salient two-pole rotor for a synchronous machine………... 39 XII Figure 3.11 a. Plot of flux vs. field current for synchronous generators b. The magnetization curve for synchronous generators…………. 41 Figure 3.12 A simple circuit for alternators…………………………………… 42 Figure 3.13 The per-phase equivalent circuit for synchronous generators……. 43 Figure 3.14 Cutaway diagram for a wound-rotor induction machine…………. 45 Figure 3.15 Cutaway diagram for a squirrel-cage induction machine………… 45 Figure 3.16 Transformer model for an induction machine……………………. 47 Figure 3.17 Magnetization curve for an induction machine compared to that for a transformer………………………………………………….. 47 Figure 3.18 The rotor circuit model for induction machines………………….. 49 Figure 3.19 The rotor circuit model with all the frequency (slip) effects concentrated in resistor R R ………………………..……………... 49 Figure 3.20 The per-phase equivalent circuit for induction machines………… 51 Figure 3.21 Torque-Speed curve for a MW-size induction machine………….. 52 Figure 3.22 Electrical energy conversion by power converters……………….. 60 Figure 3.23 Basic wiring diagram for direct frequency converters…………… 62 Figure 3.24 Indirect frequency converters…………………………………….. 63 Figure 4.1 Horizontal and vertical axis wind turbines……………………….. 70 Figure 4.2 Horizontal axis wind turbine configurations……………………... 71 Figure 4.3 Vertical axis wind turbine configurations………………………... 72 Figure 4.4 Operating regions of a typical wind turbine……………………… 76 Figure 4.5 Rotor diameter vs. power output…………………………………. 78 Figure 4.6 Swept area by rotor blades……………………………………….. 79 Figure 4.7 Pitch Control……………………………………………………… 81 Figure 4.8 Stall Control………………………………………………………. 81 Figure 4.9 Stall & Pitch controlled power schemes………………………….. 83 Figure 5.1 Overview of the wind turbine simulation…...……………………. 88 Figure 5.2 Yaw control block……………………………………………....... 90 Figure 5.3 Turbine efficiency block..........………………………………….... 90 Figure 5.4 Turbine efficiency characteristics corresponding to wind speed.... 91 Figure 5.5 Graphical demonstrations for the response of pitch control mechanism....................................................................................... 92 XIII Figure 5.6 Pitch control block with 0-15 degrees adjustment interval………. 93 Figure 5.7 Angular speed calculation block..................................................... 94 Figure 5.8 Wind speed values filtered by yaw control block………………... 96 Figure 5.9 Aerodynamic power in the wind…………………………………. 96 Figure 5.10 Captured wind power by the turbine (Input power to generator)… 97 Figure 5.11 Angular speed variation of the turbine in respect of each wind speed change (Change of input torque)…………………………... 97 Figure 5.12 Angular shaft speed of the turbine………………………………... 98 Figure 5.13 Rotational speed of turbine shaft before gearbox………………… 98 Figure 5.14 Rotational speed of turbine shaft after gearbox (Rotational speed of generator rotor)………………………………………………… 99 Figure 5.15 Tip speed ratio…...……………………………………………….. 99 Figure 5.16 Blade pitch angle (a)………………...…………………………… 100 Figure 5.17 Power coefficient (C p )……………………………………………. 100 Figure 5.18 Tip speed ratio vs. power coefficient…………….........…………. 101 Figure 5.19 Turbine wind speed – power characteristics…………………....... 101 Figure 5.20 Turbine efficiency vs. wind speed………………………………... 102 1 CHAPTER ONE INTRODUCTION World electrical energy consumption gets higher as the technology being developed and the human life’s dependency on electricity is growing. Predictions say that world electrical energy demand will continue to increase in the following 20 years period as shown in Figure 1.1. So, electrical energy supplies will be insufficient to respond this demand. Therefore, new and cost-reduced energy supplies must be introduced into the market. World Electricity Consumption 0 6000 12000 18000 24000 1990 1995 2000 2005 2010 2015 2020 Years N e t E l e c t r i c a l E n e r g y C o n s u m p t i o n ( G W h ) Figure 1.1 World electricity consumption with estimations 2 Table 1.1 World Electricity Consumption with Estimations World Electricity Consumption Annual Consumption (GWh) 1990 10,549 1998 12,725 1999 12,833 2005* 15,182 2010* 17,380 2015* 19,835 2020* 22,407 * Estimated values. Wind energy offers the potential to generate substantial amounts of electricity without the pollution problems of most conventional forms of electricity generation. The scale of its development will depend critically on the car e with which wind turbines are selected and sited. (Boyle, 1996, p.267) Figure 1.2 shows that, for about 10 years, generating electricity from wind sites is one of the most popular methods to provide demanded electricity of the world. Wind Power Installation History 1991 - 2002 0 4000 8000 12000 16000 20000 24000 28000 32000 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Year I n s t a l l e d M W Annual Installation Cumulative Installation Figure 1.2 Wind power installations worldwide 3 Table 1.2 Wind Power Installations Worldwide WECS Installations Annual Installation (MW) Cumulative Installation (MW) 1991 2,223 1992 338 2,561 1993 480 3,041 1994 730 3,771 1995 1,290 5,061 1996 1,292 6,353 1997 1,568 7,921 1998 2,597 10,518 1999 3,922 14,440 2000 4,495 18,935 2001 6,824 25,759 2002* 6,000 31,759 * Estimated value. Since 1996, global wind power capacity has continued to grow at an annual cumulative rate close to 40%. Over the past decade, installations have roughly doubled every two and a half years. During 2001 alone, close to 6,800 MW of new capacity was added to the electricity grid worldwide. (EWEA, European Wind Energy Association, 2002, p.11) By the end of 2001, global wind power installed had reached a level of almost 25,000 MW. This is enough power to satisfy the needs of around 14 million households, over 35 million people. Europe accounts for around 70% of this capacity, and for two-thirds of the growth during 2001. But other regions are beginning to emerge as substantial markets for the wind industry. Over 45 countries around the world now contribute to the global total, and the number of people employed by the industry world-wide is estimated to be around 70,000. (EWEA, European Wind Energy Association, 2002, p.11) 4 Table 1.3 Wind Energy Capacity Leaders Worldwide by End 2001 COUNTRY Installed MW Germany 8,734 USA 4,245 Spain 3,550 Denmark 2,456 India 1,456 Italy 700 UK 525 China 406 Greece 358 Japan 357 Turkey 19 Others 2,121 TOTAL 24,927 1.1 HISTORICAL BACKGROUND Wind energy has been used for thousands of years for milling grain, pumping water, and other mechanical power applications. Today there are over one million windmills in operation around the world; these are used principally for water pumping. Whilst the wind will continue to be used for this purpose, it is the use of wind energy as a pollution- free means of generating electricity on a potentially significant scale that is attracting most current interest in the subject. Strictly speaking, a windmill is used for milling grain, so modern ‘windmills’ tend to be called wind turbines, partly because of their functional similarity to other types of turbines that are used to generate electricity. They are also sometimes referred to as wind energy conversion systems (WECS) and those used to generate electricity are sometimes described as wind generators or aero-generators. For utility-scale sources of wind energy, a large number of wind turbines are usually built close together to form a wind plant. 5 Attempts to generate electricity from wind energy have been made (with various degrees of success) since the end of the nineteenth century. Small wind machines for charging batteries have been manufactured since the 1940s. It is, however, only since the 1980s that the technology has become sufficiently mature. An extensive range of commercial wind turbines is currently available from over 30 manufacturers around the world. Several electricity providers today use wind plants to supply power to their customers. (Boyle, 1996, p.267) Wind turbines, like windmills, are mounted on a tower to capture the most energy. At 30 meters or more above ground, they can take the advantage of faster and less turbulent wind. Turbines catch the wind’s energy with their propeller- like blades. Usually, two or three blades are mounted on a shaft to form a rotor. A blade acts much like an airplane wing. As wind blows, a pocket of low-pressure air forms on the downwind side of the blade. The low-pressure air pocket then pulls the blade toward it, causing the rotor to turn. This is called lift. The force of the lift is actually much stronger than the wind's force against the front side of the blade, which is called drag. The combination of lift and drag causes the rotor to spin like a propeller, and the turning shaft spins a generator to make electricity. Wind turbines can be used in stand-alone applications, or they can be connected to a utility power grid or even combined with a photovoltaic (solar cell) system. Stand- alone wind turbines are typically used for water pumping or communications. However, homeowners or farmers in windy areas can also use wind turbines as a way to cut their electric bills. The cost of wind energy equipment fell steadily between the early 1980s and the early 1990s. The technology is continually being improved to make it both cheaper and more reliable, so it can be expected that wind energy will tend to become more economically competitive over the coming decades. 6 An understanding of machines that extract energy from the wind involves many fields of knowledge, including meteorology, aerodynamics, electricity and planning control, as well as structural, civil and mechanical engineering. 1.2 FUNCTIONAL STRUCTURE OF WIND TURBINES Figure 1.3 Power transfer in a wind energy converter As shown in Figure 1.3, blades of a wind turbine rotor extract some of the flow energy from air in motion, convert it into rotational energy then deliver it via a mechanical drive unit (shafts, clutches and gears) to the rotor of a generator and thence to the stator of the same by mechanical-electrical conversion. The electrical energy from the generator is fed via a system of switching and protection devices, leads and any necessary transformers to the mains, to the end user or to some means of storage. (Heier, 1998, p.21) 7 CHAPTER TWO COMPONENTS OF WIND TURBINES A wind turbine converts the kinetic energy of the wind firstly to the rotational mechanical energy then to the electrical energy. All of these duties are carried out by special components. The rotor assembly may be placed either; 1. Upwind of the tower and nacelle, so receiving wind unperturbed by the tower itself or, 2. Downwind of the tower, which enables self alignment of the rotor with the wind direction (yawing), but causes the wind to be deflected and made turbulent by the tower before arriving at the rotor (tower shadow). Figure 2.1 Wind turbine types by rotor assemblies 8 The lifetime of a rotor is related to variable loads and environmental conditions that it experiences during service. Therefore, the rotor's inherent mechanical properties and design will affect its useful service life. 2.1. COMMON COMPONENTS 2.1.1. NACELLE Nacelle contains the key components of a wind turbine, including the gearbox, and electrical generator. Service personnel may enter the nacelle from the tower of the turbine in order to make maintenances. Towards the other side of the nacelle, there is wind turbine rotor, i.e. rotor blades and the hub. Figure 2.2 Nacelle 2.1.2. BLADE Rotor blade design has advanced with knowledge from wing technology, and utilizes the aerodynamic lift forces that an airfoil experiences in a moving stream of air. The shape of the blade and its angle in relation to the relative wind direction both affect its aerodynamic performance. 9 The materials used in modern wind turbine blade construction may be grouped into three main classes; • Wood (including laminated wood composites) • Synthetic composites (a polyester or epoxy matrix reinforced by glass fibers) • Metals (predominantly steel or aluminum alloys) Rotor blades should have the optimum design in order to capture maximum amount of wind and so to provide maximum rotation of the shaft. Wind turbines can have different number of rotor blades. The principle rule is; the lower the number of rotor blades the faster turns the rotor. The measure for this is called tip speed ratio, λ, which is defined as rotor tip speed divided by the wind velocity. If λ = 1, the blade tip velocity is as high as the wind speed. Rotors of wind turbines should have rotational speeds as high as possible to reduce the masses of gearboxes and generators. So, the number of rotor blades is low and in general not more than three. Most of today’s wind turbines have blade tip speeds of less than 65 m/s. In the old prototypes of large wind turbines, designers tried to increase the blade tip speed more and more because the shaft torque reduces with increasing rotational speed, but high blade tip speeds have the disadvantage of high noise emissions and physical damages of the rotor. 3-bladed rotors are the most common ones all over the world. The main reason to use 3 blades is the constant inertia moment of the rotor for all circumferential azimuth angles in relation to operational motions around the longitudinal axis of the tower. (German Wind Energy Institute - DEWI, 1998, p.40) 2-bladed rotor offered the chance to reduce the cost for the rotor, but unfortunately the dynamic behaviour of the 2-bladed rotor caused additional efforts that increase again the overall cost. (German Wind Energy Institute - DEWI, 1998, p.41) 10 As compared to 3-bladed rotors, 1-bladed rotors have tip speed two times that of 3-bladed ones. This means a 1-bladed wind turbine is several times noisier than a 3- bladed one. Additionally, the rotor blade can be fixed to the hub by a single hinge that allows for a movement that reduces structural loads on the blade. On the other hand, 1-bladed rotors principally have an aerodynamic unbalance, which introduces additional motions, causes loads and needs complicated hub constructions to keep the movements under control. (German Wind Energy Institute - DEWI, 1998, p.41) a. One-Bladed b. Two-Bladed c. Three-Bladed Figure 2.3 Horizontal axis wind turbines according to number of blades If 1, 2 or 3 bladed rotors are designed for similar tip speeds (as they have not been in the past but would require to be in the future for European land based applications subject to current sound limits), then the blades of the 3-bladed rotor are more highly stressed than for the 2 or 1 bladed system and thus rotor blade costs will be high for the 3 bladed system. Table 2.1 illustrates the relative proportion of 1, 2 and 3 bladed designs among present commercially available wind turbines of over 30 kW rated output. If the data were presented as the proportion of operational machines the dominance of the 3- 11 bladed designs would be still more pronounced. (European Commission Directorate- General for Energy, 1997, pp.5-6) Table 2.1 Number of Blades for Commercial Wind Turbine Designs Number of Blades % of Designs 1 2 2 24 3 74 Conventional wisdom holds that three-bladed machines will deliver more energy and operate more smoothly than either one or two bladed turbines. They will also incur higher blade and transmission costs as a result. Some experiments say that rotors with three blades can capture 5% more energy than two-bladed turbines while encountering less cyclical loads than one and two bladed turbines. 2.1.3. LOW SPEED SHAFT While transferring the primary torque to the gear train from the rotor assembly, the main shaft is usually supported on journal bearings. Due to its high torque loadings, the main shaft is susceptible to fatigue failure. Thus, effective pre-service non-destructive testing procedures are advisable for this component. 2.1.4. HIGH SPEED SHAFT The high-speed shaft rotates with over 1,000 revolutions per minute (rpm) and drives the electrical generator. It is equipped with an emergency mechanical disc brake. 2.1.5. DISC BRAKE This may be situated either on the main shaft before the gearbox, or on the high- speed shaft after the gearbox. The latter arrangement requires a smaller (and cheaper) 12 brake assembly in order to supply the necessary torque to slow down the rotor. However, this arrangement does not provide the most immediate control of the rotor, and in the event of a gearbox failure, braking control of the rotor is lost. 2.1.6. GENERATOR The generator converts the mechanical energy of the input shaft to electrical energy. It must be compatible at input with the rotor and gearbox assemblies, but at output with the utility's power distribution (if connected to a grid) or to local power requirements (if the turbine is part of a stand alone system). The generator can be either DC, synchronous or induction (asynchronous). DC machines are used for stand alone systems such as battery charging which do not need to produce grid compatible electricity. Synchronous machines are generally used for high synchronous speeds, but induction machines can be used for low variable speeds. Generally for wind turbines, induction generators are used for the opportunity of controlling the system under different wind speeds. This situation is the result of unstable wind speeds. In some systems, permanent magnet generators can also be used. 2.1.7. TOWER The tower of a wind turbine carries the nacelle and the rotor. Generally, it is an advantage to have a high tower, since wind speeds increase farther away from the ground. For example, a typical modern 600 kW turbine will have a tower of 40 to 60 metres (the height of a 13-20 story building). Towers may be either of tubular or lattice types. Tubular towers are safer for the personnel that have to maintain the turbines, as they may use an inside ladder to get to the top of the turbine. The advantage of lattice towers is primarily that they are cheaper. 13 2.2. OPTIONAL COMPONENTS 2.2.1. GEAR BOX Gearboxes are used for non-direct drive designs. In general, the transmission gear is used to adapt WECS to low wind speeds in order to help the rotational speed getting close to the frequency of the grid system. But, this adaptation brings the addition of mechanical machinery parts (Large gearboxes, coupling elements etc.) to be installed. Figure 2.4 A typical gear Gearboxes are not intrinsic to wind turbines. Designers use them only because they need to increase the speed of the slow-running main shaft to the speed required by mass-produced generators. Manufacturers can produce for special purpose, slow- speed generators and drive them directly without using a transmission. For this reason, specially designed permanent- magnet alternators have revolutionized the reliability and serviceability of small wind turbines. 2.2.2. V / Hz CONVERTER The AC-AC converter includes a rectifier and an inverter to control the frequency. Its aim is to keep the generated system voltage near grid frequency (50 or 60 Hz). A controlled rectifier-inverter group converts the generated AC voltage to a DC signal and then again to an AC signal. The controlling principle is based on the controlling of the inverter elements (IGBTs, thyristors etc.). 14 Figure 2.5 AC – AC signal conversion 2.2.3. YAW ASSEMBLY It is necessary for the rotor axis to be aligned with the wind direction in order to extract as much of the wind's kinetic energy as possible. The smallest upwind machines (up to 25 kW) most commonly use tail vanes to keep the machine aligned with the wind. However, larger wind turbines with upwind rotors require active yaw control to align the machine with the wind. To enable this, when a change in wind direction occurs, sensors activate the yaw control motor, which rotates the nacelle and rotor assembly until the turbine is properly aligned. Downwind machines of all sizes may possess passive yaw control, which means that they can self-align with the wind direction without the need for or a tail vane or yaw drive. Yaw system can also be used to shut down the wind turbine in order to save it from the physical effects of very high wind speeds. 2.2.4. PITCH CONTROL MECHANISM This mechanism is used on wind turbines for active power control. At a sufficiently high level of wind, a blade pitch adjuster ensures that the turbine speed is kept roughly constant by altering the blade angle. 15 For reasons of stability and to reduce the component loading, this mechanism changes the blade pitch angle along its longitudinal axis to limit the input torque loading to turbine blades. A simple pitch control design can be achieved by using a hydraulic or mechanical centrifugal governor. 2.2.5. ELECTRONIC CONTROLLER It contains a computer, which continuous ly monitors the condition of the wind turbine and controls the pitch and yaw mechanisms. In case of any malfunction, (e.g. overheating of the gearbox or the generator), it automatically stops the wind turbine and calls the turbine operator's computer via a telephone modem link. Another important characteristic of the electronic controller is to control the AC- AC converter elements (i.e. firing angles of thyristors). At this point, electronic controller takes on the frequency synchronization duty between generated signal and grid. 16 F i g u r e 2 . 6 A t y p i c a l w i n d t u r b i n e i n d e t a i l ( V E S T A S V 2 7 / 2 2 5 k W ) 17 CHAPTER THREE ELECTROMECHANICAL ENERGY CONVERSION Electromechanical energy conversion is carried out by the full operation of wind turbine. In case of any component’s failure, either the complete energy conversion stopped or some losses must be taken into account. Figure 3.1 A typical wind turbine showing all components 18 As shown in Figure 3.1, the wind blade(s) is able to capture the wind energy and rotates itself. This rotation of the blade is transferred to the generator shaft or namely to the rotor by an optional gearbox. This box increases the rotational speed of the shaft, which provides more electrical energy production. The high- speed generator (asynchronous or synchronous) is connected to the V/Hz converter to keep the frequency of the generated voltage in the order of the grid frequency. The sequence of events in the generation and transmission of wind power can be summarized as follows: 1. A torque is produced as the wind interacts with the rotor, 2. The relatively low rotational frequency of the rotor is increased via a gearbox, 3. The gearbox output shaft turns a generator, 4. The electricity produced by the generator passes through the turbine controller and circuit breakers and is stepped up to an intermediate voltage level (generally 690 V) by the turbine transformer, 5. The site cabling system delivers the electricity to the site transformer via the site control and circuit breaker system, 6. The site transformer steps up the voltage to the grid value, 7. The grid system transmits the electricity to the locality of its end use, 8. Transformer substations reduce the voltage to domestic or industrial values, 9. Local low voltage networks transmit the electricity to homes, offices and factories. 3.1. AERODYNAMICS OF WIND TURBINES 3.1.1. AERODYNAMIC FORCES An object in an air stream experiences a force that is imparted from the air stream to that object. This force can be considered to be equivalent to two component forces, acting in perpendicular directions, known as the drag force and the lift force. 19 The magnitudes of drag and lift forces depend on the shape of the object, its orientation to the direction of the air stream, and the velocity of the air stream. Figure 3.2 Lift and drag forces acting on rotor blade 3.1.1.1. DRAG FORCES Drag forces are in line with the direction of the air stream. For example, a flat plate in an air stream experiences maximum drag forces when the direction of the air flow is perpendicular to the flat side of the plate. When the direction of the air stream is in line with the flat side of the plate, the drag forces are at a minimum. (Boyle, 1996, p.284) For wind turbine blades, the objective is to minimize drag forces. 3.1.1.2. LIFT FORCES Lift forces are perpendicular to the direction of the air stream. They are termed ‘lift’ because they are the forces that enable aero planes to lift off the ground and fly. Lift forces acting on a flat plate are smallest when the direction of the air stream is at a zero angle to the flat surface of the plate. At small angles relative to the direction of the air stream (that is, when the so called angle of attack is small), a low pressure region is created on the downstream side of the plate as a result of an increase in the air velocity on that side. In this 20 situation, there is a direct relationship between air velocity and pressure: The faster the air flow, the lower the pressure. This phenomenon is known as the Bernoulli’s Effect. The lift force thus acts as a ‘suction’ or ‘pulling’ force on the object. Lift forces are the principal that cause a modern wind turbine to operate. (Boyle, 1996, p.284) 3.1.2. AERO-FOILS The angle that an object makes with the direction of an air flow, measured against a reference line in the object, is called the angle of attack or angle of incidence. The reference line on an aero- foil section is usually referred to as the chord line. Arching or cambering a flat plate will cause it to induce higher lift forces for given angle of attack, but the use of so-called aero-foil sections is even more effective. When employed as the profile of a wing, these sections accelerate the air flow over the upper surface. The high air speed thus induced results in a large reduction in pressure over the upper surface relative to the lower surface. (Boyle, 1996, p.284) 21 Figure 3.3 Components of wind power acting on rotor blade The lift force, in a direction at right angles to the air stream, is described by the lift coefficient C L , and is defined by Equation (3.1); L 2 L A V ? L 2 C ⋅ ⋅ · (3.1) where C L : Lift coefficient ρ : Air density (kg/m 2 ) A L : Area of aero- foil in plan (m 2 ) V : Wind speed (m/s) L : Lift force (N) Similarly, the drag force is described by the drag coefficient C D by Equation (3.2); 22 D 2 D A V ? D 2 C ⋅ ⋅ · (3.2) where C D : Drag coefficient ρ : Air density (kg/m 2 ) A D : Area of aero- foil in plan (m 2 ) V : Wind speed (m/s) D : Lift force (N) Horizontal and vertical axis wind turbines both make use of the aerodynamic forces generated by aero- foils in order to extract power from the wind, but each harnesses these forces in a different way. In a fixed pitch horizontal axis wind turbine, the angle of attack at a given position on the rotor blade stays constant throughout its rotation cycle. In a vertical axis wind turbine, the angle of attack at a given position on the rotor blade is constantly varying throughout its rotation cycle. 3.2 ENERGY AND POWER IN THE WIND A wind turbine obtains its power input by converting the force of the wind into torque (turning force) that is acting on the rotor blades. The amount of energy which the wind transfers to the rotor depends on the density of the air, the rotor area, and the wind speed. Power can be defined as the rate at which energy is used or converted and it can therefore be expressed as energy per unit of time; s j 1 W 1 · (3.3) 23 The energy contained in the wind is its kinetic energy; 2 V m 2 1 E ⋅ ⋅ · (3.4) where m is the mass and V is the velocity with which this mass is moving. It can be considered that the air is passing through a circular ring (enclosing a circular area, say 100 m 2 ) at a velocity V (say 10 m/s) as shown in Figure 3.4; Figure 3.4 Cylindrical volume of air passing at velocity V (10 m/s) through a ring enclosing an area, ‘A’, each second As the air is moving at a velocity of 10 m/s, a cylinder of air with a length of 10 m will pass through the ring each second. Therefore, a volume of air equal to 100x10=1000 cubic meters will pass through the ring each second. By multiplying this volume by the air density, the mass of the air moving through the ring each second can be obtained. 24 In other words; Mass of air per second = air density x volume of air passing each second = air density x area x length of cylinder of air passing each second = air density x area x velocity V A ⋅ ⋅ · ? m (3.5) where ρ : Air density (kg/m 3 ) A : Rotor disk Area (m 2 ) V : Wind velocity (m/s) Consequently the kinetic energy formula becomes; 3 V A 2 1 E ⋅ ⋅ ⋅ · ? (3.6) However, energy per unit of time is equal to power (1 W = 1 j/s), so above formula is also the expression for the power in the wind; 3 V A 2 1 P ⋅ ⋅ ⋅ · ? (3.7) An airstream moving through a turbine rotor disc cannot give up all of its energy to the blades because some kinetic energy must be retained in order to move the airstream away from the disc area after interaction. In addition, there are frictional effects, which produce heat losses. Thus, a turbine rotor will never extract 100 % of the wind's energy. There are some new parameters to be introduced into calculations in order to express the system efficiency. 25 3.2.1. POWER COEFFICIENT The ability of a turbine rotor to extract the wind's power depends upon its "efficiency". Thus, to express the power output of the turbine, a non-dimensional power co-efficient C p is included. Also, rotors reduce the wind velocity from the undisturbed wind speed V 1 far in front of the rotor to a reduced air stream velocity V 2 behind the rotor as shown in Figure 3.5; Figure 3.5 Wind flow through a wind turbine The difference in the wind velocity is a measure for the extracted kinetic energy which turns the rotor and at the opposite end of the drive train, the connected electrical generator. By including the losses, the power theoretically extracted by the wind turbine can be described by Equation (3.8); 3 1 V A p C 2 P ⋅ ⋅ η ⋅ ⋅ · ? (3.8) 26 where ? : Air density (kg/m 3 ) p C : Non-dimensional power coefficient η : Mechanical / Electrical efficiency A : Rotor disk area (m 2 ) V 1 : Undisturbed wind velocity in front of the rotor (m/s) This describes the fraction of the wind's power per unit area extracted by the rotor, governed by the aerodynamic characteristics of the rotor and its number of blades. As the air stream interacts with the rotor disc and power is extracted, the air stream speed is reduced by an amount described by the axial interference factor, a. This is the ratio of the upstream to the downstream wind speed. Equation (3.9) expresses the power using the axial interference factor; ) a 1 ( a V A 2 P 2 3 1 − ⋅ ⋅ ⋅ ⋅ η ⋅ ⋅ · ? (3.9) where "a" is the dimensionless axial interference factor. Thus, by substitution, the power co-efficient C p may be defined as; ) a 1 ( a 4 C 2 p − ⋅ ⋅ · (3.10) By differentiating (3.10) with respect to a, the maximum value of C p occurs when a = 0.33. Thus, Cp max = 16/27 = 0.593. 27 3.2.2. TIP SPEED RATIO The speed of rotation of a wind turbine is usually given in either revolutions per minute (rpm) or radians per second (rad/s). The rotation speed in rpm is usually symbolized by n r and the angular velocity in rad/s is by ? r . Table 3.1 Speed Definitions Definition Symbol Unit Rotational Speed n r rpm Angular Speed ? r rad/s 1 rpm = 60 2 π ⋅ rad/s = 0.10472 rad/s Another measure of a wind turbine’s speed is its tip speed, U, which is the tangential velocity of the rotor at the tip of blades, measured in meters per second. It is the product of the angular velocity, ? r , of the rotor and the tip radius, r. Alternatively, it can be defined as; 60 n r 2 U r ⋅ ⋅ π ⋅ · (3.11) By dividing the tip speed, U, by the undisturbed wind velocity, V, at the upstream of the rotor, the very useful non-dimensional ratio known as the tip speed ratio, which is usually symbolized by λ is obtained. This ratio provides us with a useful measure with which to compare wind turbines of different characteristics. (Boyle, 1996, p.283) If a rotor turns very slowly, it will allow wind to pass unperturbed through the gaps between the blades. Likewise, a rotor turning very rapidly will appear as a solid wall to the wind. Therefore, it is necessary to match the angular velocity of the rotor to the wind speed in order to obtain maximum efficiency. 28 The relationship between the wind speed and the rate of rotation of the rotor is characterized by a non-dimensional factor, known as the tip speed ratio, λ, given by Equation (3.12). Note that this factor arises from the full aerodynamic theory of wind power extraction; V U V r Speed Wind Speed Tip Blade r · ⋅ ω · · λ (3.12) where r : Rotor radius measured at the blade tip (m) ? r : Angular speed of the blade tip (rad/s) U : Blade tip speed (m/s) V : Wind Speed (m/s) 3.2.3. EFFECT OF THE NUMBER OF BLADES The optimum tip speed ratio may be inferred however by relating the time taken for the disturbed wind to re-establish itself t w , to the time taken for a blade of rotational frequency omega to move into the position occupied by its predecessor t b . For an n-bladed rotor, the time period for the blade to move to its predecessor's position is given by Equation (3.13); r b n 2 t ω ⋅ π ⋅ · (3.13) where t b : Time period for the blade to move its predecessor’s position (sec) ? r : Angular speed of the blade tip (rad/s) n : Number of blades 29 If the length of the strongly disturbed airstream upwind and downwind of the rotor is d, then the time for the wind to return to normal is given by Equation (3.14); V d t w · (3.14) where t w : Time period for the wind to return to normal (sec) d : Length of disturbed air stream (m) V : Wind Velocity (m/s) Maximum power extraction occurs when these time periods are equal (If t b exceeds t w , then some wind is unaffected. If t w exceeds t b , then some wind is not allowed to move through the rotor). For this case, Equation (3.15) applies; d 2 V n r π ⋅ ω ⋅ (3.15) where ? r : Angular speed of the blade tip (rad/s) n : Number of blades d : Length of disturbed air stream (m) V : Wind velocity (m/s) Therefore, for optimum power extraction, the rotor must turn at a frequency which is related to the speed of the oncoming wind. This rotor frequency decreases as the radius of the rotor increases, and may be characterized by calculating the optimum tip speed ratio by Equation (3.16); , ` . \| π ⋅ ≈ λ d r n 2 0 (3.16) 30 where λ 0 : Optimum tip speed ratio r : Blade tip radius of rotation (m) n : Number of blades d : Length of disturbed air stream (m) If we substitute a constant k for the term (r/d), which practical results have shown to be approximately 2 for an n bladed machine, then the optimum tip speed ratio is defined by Equation (3.17); n 4 0 π ⋅ ≈ λ (3.17) Thus, for a two-bladed rotor, the maximum power extracted from the wind (at Cp max ) occurs at a tip speed ratio of about 6, and for a four-bladed machine at a tip apeed ratio of about 3. If the aerofoil is carefully designed, the optimum tip speed ratios may be about 30% above these values. (De Montfort University- http://www.iesd.dmu.ac.uk/wind_energy/m32extex.html, 1996). Most modern horizontal axis wind turbine rotors consist of two or three thin blades. These are known as "low solidity" rotors, due to the low fraction of the swept area which is solid. This arrangement gives a relatively high tip speed ratio in comparison to rotors with a high number of blades (such as those used in water pumps, which require a high starting torque), and gives an optimum match to the frequency requirements of modern electricity generators. This minimizes the size of the gearbox required and increases efficiency. Figure 3.6 shows the relationship between rotor efficiency (C p ) and the tip speed ratio for a typical wind turbine; as wind speed increases, it is necessary for the rotor to speed up in order to remain near the optimum tip speed ratio. However, this is in conflict with the requirements of most generating systems, which require a constant generator frequency in order to supply electricity of a fixed frequency. Thus, the 31 wind turbine which has a generator directly coupled to the grid operates for much of the time with a tip speed ratio which is not optimized. Figure 3.6 Power coefficient versus tip speed ratio for a constant speed wind turbine The alternative is to decouple the generator from the grid by an intermediate system which facilitates variable speed operation. Some manufactures are producing variable speed turbines (where the rotor speeds up with the wind velocity), in order to maintain a tip speed ratio near the optimum. These turbines utilize electronic inverter/rectifier based control systems to stabilize the fluctuating voltage from the turbine before feeding into the grid supply. For a variable-speed turbine, the objective is to operate near maximum efficiency, where the resulting target power can be expressed as; 3 r 3 et arg t et arg t , p et arg t r C A p C 2 P ω ⋅ , ` . \| λ ⋅ ⋅ ⋅ η ⋅ ⋅ · ? (3.18) 32 where ? : Air density (kg/m 3 ) p C target : Power coefficient target η : Mechanical / Electrical efficiency A : Rotor disk area (m 2 ) r : Rotor radius measured at the blade tip (m) ? r : Angular speed of the blade tip (rad/s) λ target : Tip speed ratio target 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 TSR C p Figure 3.7 Power coefficient versus tip speed ratio for a variable speed wind turbine for different pitch angles from 0 to 15 degrees by 0.5 degree increments Figure 3.7 illustrates the Cp-λ relationship for a variable-speed wind turbine at different pitch angles. For constant-speed turbines, only one of the curves will be valid and an attempt is made to design the rotor blades to operate near maximum efficiency (Cp max ) at wind speeds that occur most frequently at the design site. The rotor speed varies by only a few percent, but the wind speed varies over a wide range. Therefore, the operating point is rarely, and randomly, at λ for Cp max . It is apparent from Equation (3.18) and Figure 3.7 that the power at any wind speed is 33 maximized by operating near the tip-speed ratio which results in the maximum power coefficient. For a variable-speed turbine, this means that as the wind speed changes, the rotor speed should be adjusted proportionally. 3.3. GENERATOR THEORY All generators produce electricity by Faraday Law of electromagnetic induction: A magnetic field cuts a wire with a relative velocity, so inducing an electric potential difference in the wire. If this wire forms a circuit, then an electrical current is produced. The magnitude of the current is being increased with the strength of the field, the length of wire cut by the field and the relative velocity. Of the wind turbine systems currently being manufactured, their generating systems may be classed as follows; 3.3.1. D.C. GENERATORS 3.3.1.1. THEORY: DC machines convert mechanical power to dc electric power, and vice versa. Most dc machines are like ac machines in that they have ac voltages and currents within them – dc machines have a dc output only because a mechanism exists that converts the internal ac voltages to dc voltages at their terminals. Since this mechanism is called commutator, dc machinery is also known as commutating machinery. DC generators are dc machines used as generators. There is no real difference between a generator and a motor except for the direction of power flow. (Chapman, 1999, p.566) 34 Figure 3.8 The equivalent circuit for DC motors In Figure 3.8, the armature circuit is represented by an ideal voltage source E A and a resistor R A . This representation is really the Thevenin equivalent of the entire rotor structure, including rotor coils, interpoles and compensating windings, if present. The brush voltage drop is represented by a small battery V brush opposing the direction of current flow in the machine. The field coils, which produce the magnetic flux in the generator, are represented by inductor L F and resistor R F . The separate resistor R adj represents an external variable resistor used to control the amount of current in the field circuit. (Chapman, 1999, p.508) The internal generated voltage in a DC machine is given by Equation (3.19); ω ⋅ Φ ⋅ ⋅ π ⋅ · a 2 P Z E A (3.19) where ‘Z’ is the total number of conductors and ‘a’ is the number of current paths in the machine. This equation is sometimes rewritten in a simpler form that emphasizes the quantities that are variable during machine operation. This simpler form is; ω ⋅ Φ ⋅ · K E A (3.20) where K is a constant representing the construction of the machine. 35 The induced torque developed by the machine is given by; A ind I K ⋅ Φ ⋅ · Τ (3.21) Equations (3.20) and (3.21), the Kirchhoff’s Voltage Law equation of the armature circuit and the machine’s magnetization curve, are all the tools necessary to analyze the behaviour and performance of a dc motor. (Chapman, 1999, p.508) There are five major types of dc generators, classified according to the manner in which their field flux is produced: 1. Separately Excited Generator: In a separately excited generator, the field flux is derived from a separate power source independent of the generator itself. 2. Shunt Generator: In a shunt generator, the field flux is derived by connecting the field circuit directly across the terminals of the generator. 3. Series Generator: In a series generator, the field flux is produced by connecting the field circuit in series with the armature of the generator. 4. Cumulatively Compounded Generator: In a cumulatively compounded generator, both a shunt and a series field are present, and their effects are additive. 5. Differentially Compounded Generator: In a differentially compounded generator, both a shunt and a series field are present, but their effects are subtractive. These various types of dc generators differ in their terminal (voltage-current) characteristics, and therefore in the applications to which they are suited. DC generators are compared by their voltages, power ratings, efficiencies, and voltage regulations. Voltage regulation (VR) is defined by Equation (3.22); % 100 V V V VR fl fl nl × · (3.22) 36 where V nl is the no- load terminal voltage of the generator and V fl is the full- load terminal voltage of the generator. It is a rough measure of the shape of the generator's voltage-current characteristic—a positive voltage regulation means a drooping characteristic, and a negative voltage regulation means a rising characteristic. All generators are driven by a source of mechanical power, which is usually called the prime mover of the generator. A prime mover for a dc generator may be a wind or steam turbine, a diesel engine, or even an electric motor. Since the speed of the prime mover affects the output voltage of a generator, and since prime movers can vary widely in their speed characteristics, it is customary to compare the voltage regulation and output characteristics of different generators, assuming constant-speed prime movers. (Chapman, 1999, pp.566-567) 3.3.1.2. DC GENERATOR APPLICATIONS IN WIND TURBINES Small scale stand-alone wind turbines are the most commonly used to charge batteries at relatively low voltages. They use simple DC generators. In these systems, the rotating generator shaft (connected to the turbine blades either directly or through a gearbox) turns the rotor within a magnetic field produced by either the field coil windings or by an arrangement of permanent magnets on the armature. The rotation causes an electric current to be set up in the rotor windings as the coils of wire cut through the magnetic field. This current (whose magnitude depends upon the number of turns in the windings, the strength of the magnetic field and the speed of rotation) is drawn off from the commutator through graphite brushes and fed directly to the battery, sometimes via a voltage regulator which smoothes out fluctuations in the generated voltage. 3.3.2. SYNCHRONOUS AC MACHINES (ALTERNATORS) AC generators employ a rotary magnetic field, known as a rotary field. This may be obtained by the use of a rotating permanent magnet or by rotary excitation using a current fed via so-called brushes and slip-rings. In stationary conductors—the stator 37 windings of the generator—such rotary fields excite electric currents that vary with the frequency of rotation. In these synchronous generators, coils are set (spatially) at e.g. 120° intervals or an integral multiple thereof. The voltage is dependent on the construction of the generator, the speed of rotation of the rotary field, the excitation and the load characteristics, and in isolated and stand-alone operation can be regulated by varying the excitation. When connected to the public supply, both voltage and frequency are dictated by the grid. If the three-phase alternating current stator of a generator is supplied with alternating current from the grid, it also sets up a rotary field. This excites currents in the rotor windings of the generator, which vary with a frequency corresponding to the difference between the field rotation frequency and the mechanical speed of rotation. These currents cause torques on the rotor, which, in synchronous machines, have a damping effect. 3.3.2.1. THEORY A synchronous generator or alternator is a device for converting mechanical power from a prime mover to AC electric power at a specific voltage and frequency. The term synchronous refers to the fact that this machine's electrical frequency is locked in or synchronization with its mechanical rate of shaft rotation. The synchronous generator is used to produce the vast majority of electric power used throughout the world. (Chapman, 1999, p.316) In a synchronous generator, a dc current is applied to the rotor winding, which produces a rotor magnetic field. The rotor of the generator is then turned by a prime mover, producing a rotating magnetic field within the machine. This rotating magnetic field induces a three-phase set of voltages within the stator windings of the generator. Two terms commonly used to describe the windings on a machine are field windings and armature windings. In general, the term "field windings" applies to 38 the windings that produce the main magnetic field in a machine, and the term "armature windings" applies to the windings where the main voltage is induced. For synchronous machines, the field windings are on the rotor, so the terms "rotor windings" and "field windings" are used interchangeably. Similarly, the terms "stator windings" and "armature windings" are used interchangeably. The rotor of a synchronous generator is essentially a large electromagnet. The magnetic poles on the rotor can be of either salient or non-salient construction. The term salient means "protruding" or "sticking out" and a salient pole is a magnetic pole that sticks out from the surface of the rotor. On the other hand, a non-salient pole is a magnetic pole constructed flush with the surface of the rotor. Non-salient pole rotors are normally used for two- and four-pole rotors, while salient-pole rotors are normally used for rotors with four or more poles. (Chapman, 1999, pp.250-252) Figure 3.9 A salient six-pole rotor for a synchronous machine 39 Figure 3.10 A non-salient two-pole rotor for a synchronous machine A DC current must be supplied to the field circuit on the rotor. Since the rotor is rotating, a special arrangement is required to get the DC power to its field windings. There are two common approaches for supplying this DC power; 1. Supply the DC power from an external DC source to the rotor by means of slip rings and brushes. 2. Supply the DC power from a special DC power source mounted directly on the shaft of the synchronous generator. 3.3.2.2. THE ROTATION SPEED OF A SYNCHRONOUS GENERATOR Synchronous generators are by definition synchronous, meaning that the electrical frequency produced is locked in or synchronized with the mechanical rate of rotation of the generator. A synchronous generator’s rotor consists of an electromagnet to which direct current is supplied. The rotor magnetic field points in whatever direction the rotor is turned. Now, the rate of rotation of the magnetic fields in the machine is related to the stator electrical frequency by; 120 p n f m e · (3.23) 40 where f e : Electrical frequency (Hz) n m : Mechanical speed of the magnetic field (rpm) (equals the speed of the rotor for synchronous machines) p : Number of poles Since the rotor turns at the same speed as the magnetic field, this equation relates the speed of the rotor rotation to the resulting electrical frequency. (Chapman, 1999, pp.254-255) 3.3.2.3. INTERNAL VOLTAGE OF A SYNCHRONOUS GENERATOR The magnitude of the voltage induced in a given stator phase is; f N 2 E C A ⋅ Φ ⋅ ⋅ π ⋅ · (3.24) In solving problems with synchronous machines, this equation is sometimes rewritten in a simpler form that emphasizes the quantities that are variable during machine operation. This simpler form is; ω ⋅ Φ ⋅ · K E A (3.25) where K is a constant representing the construction of the machine. If ? is expressed in radians per second, then 2 p N K C · (3.26) The internal generated voltage E A is directly proportional to the flux and to the speed, but the flux itself depends on the current flowing in the rotor field circuit. The field current I F is related to the flux in the manner shown in Figure 3.11 (a). Since E A is directly proportional to the flux, the internal generated voltage E A is related to the 41 field current as shown in Figure 3.11 (b). This plot is called the magnetization curve or the open-circuit characteristic of the machine. Figure 3.11 a. Plot of flux vs. field current for synchronous generators b. The magnetization curve for synchronous generators The voltage E A is the internal generated voltage produced in one phase of a synchronous generator. However, this voltage E A is not usually the voltage that appears at the terminals of the generator. In fact, the only time the internal voltage E A is the same as the output voltage V F of a phase is when there is no armature current flowing in the machine. (Chapman, 1999, pp.255-256) There are number of factors that cause the difference between E A and V F ; 1. The distortion of the air-gap magnetic field by the current flowing in the stator, called armature reaction 2. The self inductance of armature coils 3. The resistance of armature coils 4. The effect of salient-pole rotor shapes 42 3.3.2.4. THE EQUIVALENT CIRCUIT OF AN ALTERNATOR Figure 3.12 A simple circuit for alternators The armature reaction voltage on a phase is; A A I X j E V ⋅ ⋅ − · Φ (3.27) In addition to the effects of armature reaction, the stator coils have a self inductance and resistance. If the stator self inductance is called L A (and its corresponding reactance is called X A ) while the stator resistance is called R A , then the total difference between E A and V F is given by; A A A A A A I R I X j I X j E V ⋅ − ⋅ ⋅ − ⋅ ⋅ − · Φ (3.28) The armature reaction effects and the self inductance in the machine are both represented by reactances, and it is customary to combine them into a single reactance, called the synchronous reactance of the machine; A S X X X + · (3.29) 43 Therefore, the final equation describing V F is; A A A S A I R I X j E V ⋅ − ⋅ ⋅ − · Φ (3.30) Figure 3.13 The per-phase equivalent circuit for synchronous generators The way in which a synchronous generator operates in a real power system depends on the constraints on it. When a generator operates alone, the real and reactive powers that must be supplied are determined by the load attached to it, and the governor set points and field current control the frequency and terminal voltage, respectively. When the generator is connected to an infinite bus, its frequency and voltage are fixed, so the governor set points and field current control the real and reactive power flow from the generator. In real systems containing generators of approximately equal size, the governor set points affect both frequency and power flow, and the field current affects both terminal voltage and reactive power flow. A synchronous generator's ability to produce electric power is primarily limited by heating within the machine. When the generator's windings overheat, the life of the machine can be severely shortened. Since here are two different windings (armature and field), there are two separate constraints on the generator. The maximum allowable heating in the armature windings sets the maximum kilovoltamperes allowable from the machine, and the maximum allowable heating in the field windings sets the maximum size of E A . The maximum size of E A and the maximum size of I A together set the rated power factor of the generator. (Chapman, 1999, p.316) 44 Early alternators, which produce an AC voltage, were developed as a replacement for DC generators. Alternators have a number of advantages. They are generally cheaper and more durable, due to the use of slip rings rather than commutators. A further design improvement is their incorporation of the armature windings in the stator, whilst the rotor provides the magnetic field. If permanent magnets are used, the power is drawn from the alternator through fixed contacts and wear due to the passage of high currents through moving contacts is eliminated. In excited field alternators, the magnetic field is provided by a supply of relatively low current to the field windings, via slip rings. Thus, in order to be compatible with a utility's grid supply, the machine must be driven at a constant speed by turbine rotors, to produce power which is in phase with grid supply. In practice, this may be achieved by altering the pitch of the turbine rotor blades to alter their lift coefficient as the wind speed varies. More commonly, however, the generator output is small enough in relation to that of the utility supply to allow it to "lock-on" to the grid frequency, ensuring a grid-compatible output frequency despite small variations in wind speed. 3.3.3. ASYNCHRONOUS (INDUCTION) AC MACHINES An induction generator differs from a synchronous generator in that its rotor consists in its simplest form of an iron cylinder with slots on its periphery that carry insulated copper bars. These are short-circuited by rings which are positioned on the flat faces of the cylinder. The currents that produce the magnetic field are in short- circuited loops. If positioned on the stator, the field current in these loops is induced from currents in the stator windings, and vice versa. In operational terms, power generation can only occur when the induced closed- loop field currents have been initiated and maintained. This is facilitated in one of three ways; • Reactive power is drawn from the live grid, to which the generator is connected, 45 • Capacitors connected between the output and the earth enable autonomous self- excited generation (some residual magnetism in the system is necessary), • A small synchronous generator may be run in parallel, which may (if diesel, fuelled, for example) then provide power at times of inadequate wind. Figure 3.14 Cutaway diagram for a wound-rotor induction machine Figure 3.15 Cutaway diagram for a squirrel-cage induction machine 3.3.3.1. EQUIVALENT CIRCUIT OF AN INDUCTION MACHINE An induction machine relies for its operation on the induction of voltages and currents in its rotor circuit from the stator circuit (transformer action). Because the induction of voltages and currents in the rotor circuit of an induction machine is 46 essentially a transformer operation, the equivalent circuit of an induction machine will turn out to be very similar to the equivalent circuit of a transformer. An induction machine is called a singly excited machine (as opposed to a doubly excited synchronous machine), since power is supplied to only the stator circuit. Because an induction machine does not have an independent field circuit, its model will not contain an internal voltage source such as the internal generated voltage E A in a synchronous machine. It is possible to derive the equivalent circuit of an induction machine from the knowledge of transformers and the variation of rotor frequency with speed in induction machines. (Chapman, 1999, p.365) A transformer per-phase equivalent circuit, representing the operation of an induction machine, is shown in Figure 3.16. Like any transformer, there is a certain resistance and self- inductance in the primary (stator) windings, which must be represented in the equivalent circuit of the machine. The stator resistance will be called as R 1 and the stator leakage reactance will be called as X 1 . These two components appear right at the input to the machine model. Also, like any transformer with an iron core, the flux in the machine is related to the integral of the applied voltage E 1 . The curve of magnetomotive force versus flux (magnetization curve) for this machine is compared to a similar curve for a power transformer in Figure 3.17. Notice that the slope of the induction machine's magnetomotive force- flux curve is much shallower than the curve of a good transformer. This is because there must be an air gap in an induction machine, which greatly increases the reluctance of the flux path and therefore reduces the coupling between primary and secondary windings. The higher reluctance caused by the air gap means that a higher magnetizing current is required to obtain a given flux level. Therefore, the magnetizing reactance X m in the equivalent circuit will have a much smaller value (or the susceptance B m will have a much larger value) than it would in an ordinary transformer. 47 Figure 3.16 Transformer model for an induction machine The primary internal stator voltage E 1 is coupled to the secondary E R by an ideal transformer with an effective turns ratio a eff . The voltage E R produced in the rotor in turn produces a current flow in the shorted rotor (or secondary) circuit of the machine. Figure 3.17 Magnetization curve for an induction machine compared to that for a transformer The primary impedances and the magnetization current of the induction machine are similar to the corresponding components in a transformer equivalent circuit. An induction machine equivalent circuit differs from a transformer equivalent circuit 48 primarily in the effects of varying rotor frequency on the rotor voltage E R and the rotor impedances R R and jX R . (Chapman, 1999, pp.366-367) 3.3.3.1.1. ROTOR CIRCUIT MODEL In an induction machine, when the voltage is applied to the stator windings, a voltage is induced in the rotor windings of the machine. In general, the greater the relative motion between the rotor and the stator magnetic fields, the greater the resulting rotor voltage and rotor frequency. The largest relative motion occurs when the rotor is stationary, called the locked-rotor or blocked-rotor condition, so the largest voltage and rotor frequency are induced in the rotor at that condition. The smallest voltage (0 V) and frequency (0 Hz) occur when the rotor moves at the same speed as the stator magnetic field, resulting in no relative motion. The magnitude and frequency of the voltage induced in the rotor at any speed between these extremes is directly proportional to the slip of the rotor. Therefore, if the magnitude of the induced rotor voltage at locked-rotor conditions is called E R0, the magnitude of the induced voltage at any slip will be given by Equation (3.31); 0 R R E s E ⋅ · (3.31) and the frequency of induced voltage at any slip will be given by Equation (3.32); e r f s f ⋅ · (3.32) This voltage is induced in a rotor containing both resistance and reactance. The rotor resistance R R is a constant (except for the skin effect), independent of slip, while the rotor reactance X R is affected in a more complicated way by slip. (Chapman, 1999, p.367) The reactance of an induction machine rotor depends on the inductance of the rotor and the frequency of the voltage and current in the rotor. With a rotor inductance of L R , the rotor reactance is given by; 49 R r R r R L f 2 L X ⋅ ⋅ π ⋅ · ⋅ ω · (3.33) Substituting Equation (3.32) into Equation (3.33); ( ) 0 R R R e R R e R X s X L f 2 s X L f s 2 X ⋅ · ⋅ ⋅ π ⋅ ⋅ · ⋅ ⋅ ⋅ π ⋅ · (3.34) where X R0 is the blocked-rotor rotor reactance. Figure 3.18 The rotor circuit model for induction machines Figure 3.19 The rotor circuit model with all the frequency (slip) effects concentrated in resistor R R 50 3.3.3.1.2. FINAL EQUIVALENT CIRCUIT To produce the final per-phase equivalent circuit for an induction machine, it is necessary to refer the rotor part of the model over to the stator side. The rotor circuit model that will be referred to the stator side is shown in Figure 3.19, which has all the speed variation effects concentrated in the impedance term. In an ordinary transformer, the voltages, currents and the impedances on the secondary side of the device can be referred to the primary side by means of the turns ratio of the transformer: s 2 s s s p s s p Z a Z I a 1 I I V a V V ⋅ · ⋅ · · ⋅ · · (3.35) where the prime refers to the referred values of voltage, current and impedance. Exactly the same sort of transformation can be done for the induction machine’s rotor circuit. If the effective turns ratio of an induction machine is a eff , then the transformed rotor voltage becomes; 0 R eff R 1 E a E E ⋅ · ′ · (3.36) and the rotor current becomes; eff R 2 a I I · (3.37) and the rotor impedance becomes 51 , ` . \| + ⋅ · 0 R R 2 eff 2 jX s R a Z (3.38) so 0 R 2 eff 2 R 2 eff 2 X a X R a R ⋅ · ⋅ · (3.39) Figure 3.20 The per-phase equivalent circuit for induction machines In wind energy conversion systems, depending on the speed of the wind, the generator may act either as a generator, supplying power to the grid, or as a motor (acting as a sink of power from the grid). In either case, there will be a difference in speed between the shaft speed n r and the output n s . This is known as generator slip, and may be expressed as; s r s n ) n n ( s · (3.40) where n s : Electrical speed of the magnetic field (or stator speed) (rpm) n r : Rotor mechanical speed (rpm) 52 The slip is defined as negative when the machine is acting as a generator, and positive when acting as a motor. (Chapman, 1999, pp.369-370) Figure 3.21 Torque-Speed curve for a MW-size induction machine The torque-speed characteristic curve in Figure 3.21 shows that, if an induction motor is driven at a speed greater than synchronous speed by an external effect (i.e. wind), the direction of its induced torque will reverse and it will act as a generator. As the torque applied to its shaft increases, the amount of power produced by that generator increases. There is a maximum possible induced torque in the generator mode of operation. This torque is known as the pushover torque of the generator. If a torque is applied to the shaft of the induction generator which is greater than the pushover torque, the generator will over-speed. (Chapman, 1999, p.436) 53 As a generator, an induction machine has severe limitations. Because it lacks a separate field circuit, an induction generator cannot produce reactive power. In fact, it consumes reactive power, and an external source of reactive power must be connected to it at all times to maintain its stator magnetic field. This external source of reactive power must also control the terminal voltage of the generator—with no field current, an induction generator cannot control its own output voltage. Normally, the generator's voltage is maintained by the external power system to which it is connected. The one great advantage of an induction generator is its simplicity. An induction generator does not need a separate field circuit and does not have to be driven continuously at a fixed speed. As long as the machine's speed is some value greater than synchronous speed for the power system to which it is connected, it will function as a generator. The greater the torque applied to its shaft (up to a certain point), the greater its resulting output power. The fact that no fancy regulation is required makes this generator a good choice for windmills, heat recovery systems, and similar supplementary power sources attached to an existing power system. In such applications, power- factor correction can be provided by capacitors, and the generator's terminal voltage can be controlled by the external power system. (Chapman, 1999, p.437) Wind machines driving electrical generators operate at either variable or constant speed. In variable-speed operation, rotor speed varies with wind speed. In constant- speed machines, rotor speed remains relatively constant, despite changes in wind speed. (Gipe, 1995, p.211) Small wind turbines typically operate at variable speed. This simplifies the turbine’s controls while improving aerodynamic performance. When these small wind machines drive an induction generator, both the voltage and frequency vary with wind speed. The electricity they produce is incompatible with the constant- voltage, constant- frequency alternating current (AC) produced by the utility, but can 54 be used as is for resistive heating or pumping water at variable rates, or it can be rectified to direct current (DC) for charging batteries. If a grid-connected turbine is fitted with an AC generator, this must produce power that is in phase with the utility's grid supply. Many commercial grid- connected turbines use induction AC generators, whose magnetizing current is drawn from the grid, ensuring that the generator's output frequency is locked to that of the utility and so controlling the rotor speed within limits. Synchronous generators produce electricity in synchronization with the generator's rotating shaft frequency. Thus, the rotor speed of grid-connected turbines must exactly match the utility supply frequency. To generate utility-compatible electricity, the output from a variable-speed generator must be conditioned. Although it is possible to use rotary inverters for this task, variable-speed turbines typically use a form of synchronous inverter to produce constant- voltage 50 or 60 Hz AC like that of the utility. Most of these inverters use the utility’s alternating current as a signal to trigger electronic switches that transfer the variable-frequency electricity at just the right moment to deliver 50 or 60 Hz AC at the proper voltage. Although some manufacturers of medium-sized wind turbines build variable- speed turbines, most operate the rotor at or near constant speed. These machines produce utility-compatible power directly via induction (asynchronous) generators. Induction generators have two advantages over alternators; • They are inexpensive. • They can supply utility-compatible electricity without complicated controls. For AC generators, a critical design factor, that is synchronous speed, must be considered. AC generators produce alternating current, the frequency of which varies directly with the speed of the rotor and indirectly with the number of poles in the 55 generator. For a given number of poles, frequency increases with increasing generator speed. p f 120 s n · (3.41) where n s : Synchronous or stator speed (rpm) f : Grid frequency (Hz) p : Number of poles Manufacturers should decide the number of poles of the generator (for either synchronous or asynchronous) for optimum conditions. Table 3.2 Common Synchronous Speeds for Generators Pole Number Europe (50 Hz) North America (60 Hz) 4-pole 1500 rpm 1800 rpm 6-pole 1000 rpm 1200 rpm An induction generator begins producing electricity when it is driven above its synchronous speed which is generally 1000 or 1500 rpm in Europe (1200 or 1800 rpm in North America). Induction generators are not true constant-speed machines. As torque increases, generator speed increases 2 to 5 %, or 20 to 50 rpm on a 1000- rpm generator. This increase of 1 to 3 rpm in rotor speed is imperceptible in a wind turbine operating at a nominal speed of 50 rpm. As torque increases, the magnetic field in the induction generator also increases. This continues until the generator reaches its limit, which is about 5 % greater than its synchronous speed. Induction generators are readily available in a range of sizes and are easily interconnected with the utility. Medium- sized wind turbines use induction generators almost exclusively. 56 3.3.4. RECENT DEVELOPMENTS IN GENERATORS FOR WIND TURBINES As well as applying to the basic process of energy conversion, technological development also relates to the design and size of machines used for the generation of electric power from wind energy. Whilst the induction machine is now well established as the most popular generator for reliable, efficient, low-cost power production from the wind, other designs of machines are used and there are several "drivers" for change. The 'traditional' Danish design of wind turbine is fixed-speed, using an induction generator. Variations on this theme which are now appearing include; • Multiple or dual (two speed) generators, • Induction machines with variable generator rotor resistance. 3.3.4.1. DUAL GENERATORS Generators operate inefficiently at partial loads. For example, in a 500-kW wind turbine, where the generator is designed to reach its rated capacity at a wind speed of 16 m/s, the generator operates at partial load much of the time. At a site with an average wind speed of 7 m/s, the generator will operate 97 % of the time at less than rated capacity and about half the time at less than 100 kW. (Gipe, 1995, pp.212-213) Efficiency drops off rapidly when the generator is operated at less than one-third its rated value. For example, the efficiency falls nearly 15 % (from 95 % at rated output) when a 500-kW wind turbine is operated at 100 kW. To avoid this, designers of constant-speed wind turbines often use dual generators or dual windings: One main generator and a small generator having the capacity from one- fifth to one-third of the main generator. The small generator operates at nearly full load in low to moderate winds. When the wind speed reaches the rated wind speed of the small generator, it switches off and the main generator switches on instead. Thus both 57 generators operate more efficiently then either one alone. At many sites, the small generator will operate more than 50 % of the total generating time, although it delivers less than half the total generation. The two generators may be in tandem and driven by the same shaft or they can be side by side, with the small generator driven by belts from the main generator. During the mid-1990s, most new constant-speed turbines used one generator with dual windings. The generator operates on 6 poles during light winds and uses 4 poles in higher winds. The use of dual generators permits the turbine to operate at two speeds, enables designers to drive the rotor at a higher aerodynamic efficiency over a broader range of wind speeds than with only one generator. Dual-speed wind turbines, while incapable of taking the full advantage of the optimum tip-speed ratio over the entire operating range, can capture most of the efficiency advantages of variable-speed turbines, at only a small increase in cost for the extra windings. (Gipe, 1995, p.213) The advantage of one single generator with dual windings becomes problematical as turbines grow ever more powerful. Because a generator’s power is proportional to its volume, while losses are proportional to its surface area, larger generators are also more efficient than smaller ones. This could add perceptibly to the improved performance of larger turbines over that of their smaller predecessors. (Gipe, 1995, p.214) 3.3.4.2. DIRECT-DRIVE GENERATORS In fact, the gearbox is needed for the generator frequency to catch grid frequency for grid-connected systems. As turbine size increases, the relative cost of the gearbox becomes more important. Removing the gearbox could save not only cost, but also mass, losses, acoustic noise and reliability problems. For a doubling of wind turbine diameter, rated power will quadruple, and rotor torque, which is closely related to 58 gearbox cost, will increase by a factor of eight. Another important issue is the integration of the generator into overall nacelle design. On mid-1990s, some manufacturers successfully developed gearless wind turbines. Instead of using a gear with a high transmission ratio, they use low speed multi-pole generators directly connected to the blade shaft. The large dimensions of these multi-pole generators lead to a certain transportation disadvantage especially in the megawatt class. As rotor diameter increases, rotor speed decreases. So, lower rotor speeds make the design of direct-drive generators problematic, requiring large-diameter ring generators with numerous poles. For example, an existing Darrieus type turbine uses a 162-pole synchronous generator coupled directly to the vertical axis turbine’s torque tube. Direct designs have the maintenance and operation advantage as compared to the usage of gearboxes. 3.4. GRID INTEGRATION With regard to the transfer of energy to electrical supply installations, we must differentiate between; • Systems with limited supply options, that either operate in isolation or supply weak grids, • Unlimited capacity connection with the rigid grid. Wind energy converters should give reliable operation in both operations. Due to its very high output capacity (in comparison with the nominal values of the consumers connected to it), the so-called rigid combined grid can be regarded both as an infinitely rich source of active and reactive current and, for the low- level energy 59 supply devices that wind power plants usually represent, as a sink of unlimited capacity and constant voltage and frequency. Unlike thermal power plants, wind turbines are usually installed at remote sites with limited supply options. Therefore a weak grid connection is often made using stub cables, which are sometimes long. In large wind energy converters and wind parks, supply power can reach the same order of magnitude as grid transfer power, or even approach its level, which means that mutual influences must be taken into account. (Heier, 1998, p.181) There is currently a clear trend in favor of robust single systems, mainly characterized by stall-controlled turbines with asynchronous generators and direct connection to the grid, rather than more expensive units. However, synchronous machines are also popular, often based on gearless, ring-type designs with non- controllable, controlled or machine-commutated rectifiers, direct-current intermediate circuits and grid- or self-commutated inverters. The increased cost of such systems is justified if, by adjusting the turbine speed to the prevailing wind speed, the compatibility of the plant to the environment and the grid can be improved, leading to a higher energy output and reduced drive-train loading. This type of system also requires a frequency-converter system that is capable of supplying the variable-frequency electrical energy from the turbine generator to a grid of (almost) constant frequency and voltage. (Heier, 1998, p.183) 3.4.1. FREQUENCY CONVERTER SYSTEMS Electronic power frequency converters, so-called power converters, are the most common solution for the conversion and control of electrical energy. They are also used to an increasing degree in wind energy converters to adjust the generator frequency and voltage to those of the grid, particularly in variable-speed systems. (Heier, 1998, p.183) 60 Power converters have significant advantages over the rotating transformers based on groups of mechanical components and the mechanical commutators that were common in the past, namely; • Low- loss energy conversion • Rapid engagement and high dynamic ratio • Wear-free operation • Low maintenance requirement • Low volume and weight Figure 3.22 Electrical energy conversion by power converters Rectifiers convert alternating or three-phase current into direct-current, with the electrical energy flowing from alternating or three-phase current systems into direct- current systems. Inverters convert direct-current into alternating or three-phase current. The energy flows into the alternating-current side. 61 Direct-current conversion is the conversion of direct-current with a given voltage and polarity for use in a direct-current system with a different voltage and possibly reversed polarity. In alternating-current conversion, alternating-current of a given voltage, frequency and number of phases is converted for use in an alternating-current system with a different voltage, frequency and possibly a different number of phases. The main components of current-conversion systems are the power section, with so-called power converter valves, which carries the electrical power, and an electronic signal processing unit, which performs numerous control, protective and regulating tasks. As wind power plants are almost always fitted with three-phase current generators, only three-phase current converters are relevant for power conditioning. Here, it must be differentiated that; • Direct frequency converters, • Intermediate circuit frequency converters. Direct frequency converters are used particularly for the reduction of frequency. In the case of supply from or to a 50 Hz grid, the operating range 0-25 Hz is preferred. Direct frequency converters require two complete anti-parallel power conversion bridges per phase to operate the consumer and supply systems. This results in high costs for power gates and control elements. 62 Figure 3.23 Basic wiring diagram for direct frequency converters The conversion of grid frequency f 1 into machine frequency f 2 or vice versa, in a direct frequency converter takes place by the selection of voltage sections from the three phases and by triggering the power converter such that the voltage path after smoothing has the amplitude, phase position and frequency required by the machine. (Heier, 1998, p.185) Indirect frequency converters consist of a rectifier, direct current or direct voltage intermediate circuit and an inverter. A frequency converter with a direct current intermediate circuit will be referred to as an I frequency converter, and one with a direct voltage intermediate circuit as a U frequency converter. (Heier, 1998, p.186) 63 a. I frequency converter b. U frequency converter Figure 3.24 Indirect frequency converters Particular characteristics of the intermediate circuit are; • The inductor for current smoothing in the I frequency converter, • The capacitor for voltage smoothing in the U frequency converter. Indirect frequency converters have achieved a clear dominance in energy conversion and the connection of variable speed wind power plants to the grid. Direct frequency converters were only used in individual cases to supply the rotor circuit of double-fed asynchronous generators. 3.4.1.1. POWER SEMICONDUCTORS FOR FREQUENCY CONVERTERS So-called power converter valves are the main components of the power section of frequency converters. They consist of one or more power semiconductors, and 64 conduct electrical current in one direction only. These valves generally alternate periodically between the electrically conductive and non-conductive states, and therefore function primarily as switches. As there is no need to operate any mechanical contacts, these can initiate and/or terminate current conduction very rapidly (i.e. in the microsecond range). Power converter valves can be either controllable or non-controllable. Non- controllable valves (diodes for example) conduct in the forward direction and block in the reverse direction. Controllable valves permit the selection of the moment at which conductivity in the forward direction begins. Thyristors can be switched on by their gate and block if the direction of the current is reversed. Switchable thyristors and transistors, on the other hand, can be switched on by one gate electrode and off by a second (or the same) gate. (Heier, 1998, pp.186-187) 3.4.1.1.1. SEMICONDUCTOR DIODES Diodes consist of positively (p) and negatively (n) doped semiconductor material with a barrier layer between them that ensures current can flow in one direction only. This is possible in the case of positive diode voltages. If the current direction and voltage are reversed, the diode becomes non-conducting and blocks the flow of current. Its application is thus limited to use in uncontrolled rectifiers and for protective and back-up functions, for example as a recovery diode in direct-current circuits or similar circuit elements. In addition to limit values for current and voltage in the forward and reverse directions, and thermal behaviour, another determining variable is conducting- state dynamic behaviour, particularly for protective functions. For the effective protection of semiconductor components, so-called fast-recovery diodes with low storage charges are necessary to protect power converter valves from destruction by overvoltage. (Heier, 1998, p.187) 65 3.4.1.1.2. THYRISTORS Thyristors are semiconductor components with four differently (p and n) doped layers. Conventional thyristors, GTO thyristors and MCTs are the main types used in frequency converters. Thyristors, unlike diodes, do not automatically go into a conducting state when an adjoining positive anode-cathode voltage is present. The transition from blocking to conducting state is initiated by the supply of a power impulse to the gate, and is known as the firing of the thyristor. Once triggered, thyristors behave like diodes. They remain in the conducting state as long as a current flows in the positive direction and the current does not fall below the component's minimum value, the so- called holding current. If a thyristor is in off-state, it can be fired by a new current impulse or periodic impulse sequences at the gate. However, in conventional thyristors, it is not possible to interrupt the current by intervention at the gate. Switchable thyristors do permit this. The best known type is the Gate-Turn-Off, or GTO thyristor. With these types of thyristors, uninterrupted current requires a free-wheeling arm. The metal-oxide-semiconductor controlled thyristor, abbreviated to MCT, behaves in a similar manner to the GTO thyristor. The MCT can be switched on almost without power by a negative voltage (in relation to the anode) at the gate. A positive gate voltage switches it off, and at null current it automatically switches to blocking operation. (Heier, 1998, p.187) 3.4.1.1.3. TRANSISTORS Transistors are semiconductor components with three differently (p and n) doped layers. Mainly bipolar, MOSFET and IGBT transistors are used in frequency converters. As valve components they function exclusively as switches. 66 Bipolar transistors (BPT), in their function as power semiconductors, are usually used in emitter mode. This allows a high level of power amplification to be achieved. Almost like switches, they become conductive when a control current is passed through the base electrode. When switched off, the on-state of the transistor is terminated and the flow of current blocked. In order to achieve low on-state voltage, and thus low losses, transistors are operated with a relatively high base current. The transistors therefore operate in the so-called saturation range. Much smaller control currents are needed for metal-oxide-semiconductor field effect transistors than those for bipolar transistors. These MOSFETs can be switched almost without power, by voltage control at the gate. This, however, requires that the internal capacities of the transistor to be reloaded. Increasing the switching frequency causes increased currents and thus higher losses in the drive level. MOSFETs are used in the lower-output range at high switching frequencies for combinational circuit components and frequency converters, and have advantages over bipolar transistors and IGBTs, particularly at high switching frequencies. IGBTs (insulated gate bipolar transistors) combine the advantageous characteristics of MOSFETs and bipolar power transistors. The field-effect transistor at the control input facilitates rapid switching at very low driving power. IGBTs automatically limit current increases at the output. This results in good excess current and short-circuit behaviour. Integrated free-wheeling diodes protect the transistor in the off-state direction. Different types of IGBTs are used as individual transistors or are connected together in modules of two to six transistors to form bridge connections. In more recent developments, transistors are built into modules with driver switches, protective switches and potential divisions. IGBTs can be connected in parallel. However, this requires that all transistors exhibit the same thermal behaviour. The development and availability of new power electronic semiconductor components has given a new impetus to power converter technology and its application in the field of drive and energy engineering. Particularly in the small and 67 medium output range, new components have largely pushed transistors and GTOs out of the market. (Heier, 1998, p.188) Table 3.3 shows symbols, maximum ratings and characteristics of power semiconductors; Table 3.3 Characteristics and Maximum Ratings of Switchable Power Semiconductors Component Rating BPT IGBT MOSFET MCT GTO Symbol Voltage (V) 1200 1700 (3300) 1000 3000 4500 Current (A) 800 600 (1200) 28 300 4000 Output (kVA) 480 360 14 450 4500 Turn-Off Time (µs) 15 - 25 1 – 4 0.3 - 0.5 5 – 10 10 - 25 Frequency (kHz) 0.5 – 5 2 – 20 5 – 100 1 – 3 0.2 – 1 Drive Requirement Medium Low Low Low High 3.4.1.2. CHARACTERISTICS OF POWER CONVERTERS The main components of power converters are the power converter valves and their electrical connections and trigger equipment. Also necessary are circuit elements, energy storages, auxiliary devices and devices for commutation, filtering, cooling and protection, and usually also transformers. 68 Power converters must be run at their voltage and timed according to frequency. The origin of the commutation voltage and commutation reactive power at the conductive connection to another valve is decisive for current carrying. Externally commutated power converters operate using natural commutation. They require a grid, load or machine that specifies the voltage and can supply reactive power. Self- commutated converters, on the other hand, operate with forced commutation. The required reactive power is provided by capacitors. The internal function of power converters must also be differentiated with regard to the origin of the elementary frequency. Externally clocked power converters take their control pulse from the system that they work in parallel with. Line clocking is the adjustment of the zero-crossings or phase intersections to the grid voltage. Thus the load- or machine-clocked power converter orientates itself to the load or machine voltage. Self-clocked power converters have an internal clock generator and are thus not dependent upon external frequency information. As well as the commutation voltage and elementary frequency, the so-called pulse number, the number of non-simultaneous conductive connections (commutations) from one valve to another within one cycle, is an important parameter of power converter circuits. Three and six, as well as twelve, pulse connections are normal for three-phase current systems. The pulse number is characterized by the number of sine peaks (pulses) of the unsmoothed direct-current. (Heier, 1998, p.190) Commutation, the transfer of current between the individual valves, can occur in different ways. If the live valve is turned off before the next valve is fired then the connection becomes temporarily dead. As ripples occur in direct-current, this process is known as intermittent flow. In contrast, it is possible to fire a second valve while the valve to be turned off is still live. This creates a temporary short-circuit between two alternating-current lines. The current in the valve to be turned off is quickly forced to be under its holding point. This interrupts the short circuit before the operating current is exceeded. This changeover is known as commutating operation. (Heier, 1998, p.191) 69 CHAPTER FOUR CLASSIFICATION OF WIND TURBINES Wind turbines can be classified in several ways due to there are more than one design criteria which affects turbine performance. Classification categories can be arranged as; • Classification by axis of rotation • Classification by rotor speed • Classification by power control • Classification by location of installation 4.1. CLASSIFICATION BY AXIS OF ROTATION As mentioned before, modern windmills are usually referred to as wind turbines or wind energy conversion systems to distinguish them from their traditional name. Apart from a few innovative designs, modern wind turbines come in two basic configurations: 1. Horizontal Axis Wind Turbines 2. Vertical Axis Wind Turbines The majority of modern wind turbines are electricity-generating devices. They range from small turbines that produce a few tens or hundreds of watts of power to relatively large turbines that produce 2 MW or more. (Boyle, 1996, p.280) 70 Figure 4.1 Horizontal and vertical axis wind turbines 4.1.1. HORIZONTAL AXIS WIND TURBINES (HAWT) Modern low-solidity horizontal axis wind turbines evolved from traditional windmills and are by far the most common wind turbines manufactured today. They have a clean, streamlined appearance; due to wind turbine designers’ improved understanding of aerodynamics, derived largely from developments in aircraft wing and propeller design. They are almost universally employed to generate electricity. (Boyle, 1996, p.280) They generally have either two or three blades or else a large number of blades, although only one is necessary. Wind turbines with large numbers of blades have what appears to be virtually a solid disc covered by solid blades and are described as high solidity devices. These include the multi-blade wind turbines used for water pumping on farms. In contrast, the swept area of wind turbines with few blades is largely void and only a very small fraction appears to be ‘solid’. These are referred to as low solidity devices. 71 The rotor axis of conventional wind turbines is seldom truly horizontal. Designers tilt the rotor axis slightly to provide more clearance between the blades and tower than with a truly horizontal driveline (i.e. 6°). (Gipe, 1995, p.175) Figure 4.2 Horizontal axis wind turbine configurations 4.1.2. VERTICAL AXIS WIND TURBINES (VAWT) Vertical axis wind turbines have an axis of rotation that is vertical, and so, unlike their horizontal counterparts, they can harness wind from any direction without the need to reposition the rotor when the wind direction changes. (Boyle, 1996, p.280) D.G.M. Darrieus invented the modern vertical axis wind turbine in the 1920s. The French engineer’s name has become synonymous with the “φ” or “eggbeater” 72 configuration, although he experimented with several designs, including a conventional two-bladed turbine. (Gipe, 1995, p.171) Figure 4.3 Vertical axis wind turbine configurations Vertical axis designs have an advantage of rotational symmetry that obviates any need for a yaw system. It was often a claimed advantage that all the drive train and power conversion equipment can be at ground level, but it was found that this implied a long and heavy torque tube for the main shaft and various designs compromised with gear boxes at the top of the main shaft. The overriding disadvantages, however, of the vertical axis design compared to horizontal axis are: • Inherently lower aerodynamic efficiency because the drive torque varies strongly with blade position in the rotor circle (and may even be negative in some positions) • Substantial passive support structure in the rotor system with an associated cost penalty • At the present time, VAWTs are not economically competitive with HAWTs. 4.2. CLASSIFICATION BY ROTOR SPEED Modern wind turbines have two types of electrical connections to the grid: • With the simple direct synchronization of an induction generator, the rotor operates with nearly constant speed because the strong grid keeps generator’s frequency. The only rotational speed variation is given by the slip range of the generator. 73 • With the help of an inverter system between the wind turbine generator and the grid, the turbine is decoupled from the grid frequency and is able to rotate at variable speeds. For a long period, directly grid coupled wind turbines dominated the world market due to their technical simplicity. But several positive aspects of variable speed turbines changed the current development situation. (German Wind Energy Institute, DEWI, 1998, p.48) 4.2.1. VARIABLE ROTOR SPEED The aerodynamically optimized lay out of wind turbines is based on a fixed relationship between wind and rotor tip speed, the so-called tip speed ratio. To keep the maximum efficiency, the rotor must change its rotational speed according to the wind speed, in other words, low winds with low rotor speeds, high winds with high rotor speeds. (German Wind Energy Institute, DEWI, 1998, p.48) Variable speed is attractive because it enables designer to gain greater rotor efficiencies by allowing rotor speed to vary with wind speed. There may be additional benefits as well. Slower rotor speeds in light winds lower noise emissions just when the aerodynamic noise of the blades is most noticeable. Variable-speed operation may also reduce dynamic loads on the turbine’s drive train, thus extending turbine life. When operating at variable speed, the rotor stores the energy of gusty winds as inertia as its speed increases, rather than forcing the drive train to absorb the increased torque instantaneously. Due to their ability to operate at tip speed ratios closer to the optimum value, variable speed machines can be more efficient than fixed speed systems. However, modification of both the generator and the intermediate electronic control systems are necessary in order to provide a grid-compatible supply. One of the main factors favoring this route is the requirement of some utilities for very smooth output power. 74 Variable rotor speeds normally are combined with a “pitch angle control system”. They have various operational advantages in comparison with constant rotor speed machines; • Higher energy extraction. • Very low power fluctuations during rated power operation. • Lower rotor loads due to rotor speed yielding in gusts. • Low blade pitch change rates possible. • Low rotor speed at low wind conditions reduces the noise emission considerably. High power variable speed drives are now being designed into turbines and with them a new set of engineering aspects need to be considered, including; • Fault level of network. • Voltage regulation. • Electromagnetic compatibility. • Electrical system behaviour during gusting conditions. • Power converter efficiency. For variable speed turbines, relatively complex power converter hardware is necessary. The power conversion equipment must provide low harmonics and unity power factor control of the current delivered to the network. 4.2.2. CONSTANT ROTOR SPEED Constant rotor speed is the simplest way of operating a wind turbine because the rotor speed is guided by the frequency of a strong grid. The tip speed ratio cannot be maintained constant during operation that means the efficiency reaches its optimum only with one wind speed, which is the design wind speed of the rotor blade. During all other wind velocities, the efficiency is smaller than maximum. To better adapt the rotor operation to the aerodynamic design point, the manufacturers often use two 75 speed induction generators which allow changing the rotor speed in two steps: At low wind speeds; generator operates with a low rotational speed (higher number of poles) and at high wind speeds; with a high rotational speed (lower number of poles). Constant one or two steps rotor speed operation is the simplest way of rotor speed control, because the strong grid takes over the speed guidance; • No rotor speed control system is necessary. • Simple rotor speed regulation by the strong grid. • Only rotor speed monitoring is necessary. • Low cost design. Due to stiff grid coupling, the rated power fluctuations reach higher values than variable speed designs. 4.3. CLASSIFICATION BY POWER CONTROL Wind turbines can be classified into 3 groups as “small scale”, “medium scale” and “large scale” in terms of their power output capacity. Wind turbines with power ratings lower than 100 kW are called as small scale where the turbines with power ratings between 100 and 700 kW are called as medium scale.The large scale wind turbines have the power output capacity of greater than 700 kW. 76 Figure 4.4 Operating regions of a typical wind turbine The maximum power which can be produced by a wind turbine is the rated power of it, and the wind speed at which the turbine reaches rated power output is called as the rated wind speed. Above this, there is a maximum wind speed, called as cut-out wind speed, at which the turbine is designed to shut down in order to save mechanical parts of the wind turbine from harmful effects of high wind speed. The lowest wind speed at which a wind turbine will operate is known as the cut-in wind speed. At or above the rated wind speed, the power output remains constant whatever the wind speed (below the cut-out wind speed), but below the rated wind speed the output power varies with the wind speed. (Boyle, 1996, pp.268-269) 77 Table 4.1 Descriptions of Operational Regions for a Typical Wind Turbine Operating Region Operational Description: Power Output vs. Wind Speed Wind Speed Range Region - I - Wind speeds too low to produce usable electric power. 0 to cut- in wind speed; 0 to 4 m/s. Region - II - Production of electric power increasing with wind speed. Cut- in to rated wind speed; 4 to 13 m/s. Region - III - Production of electric power at constant, rated power level. Wind turbine blades purposely made less efficient as wind speed increases. Rated wind speed to cut- out wind speed; 13 m/s to 25 m/s. Region - IV - No electric power output. Winds too energetic to justify added strength and cost for the small number of hours per year beyond cut-out wind speed. Cut-out wind speed to survival wind speed; 25 m/s to rated survival wind speed. As the blades of the wind turbine rotate through circular path, they sweep through a disc- like area which is referred to as the swept area. This value can be normally calculated by area formula for circles; 2 r A ⋅ π · (4.1) where r is the rotor radius. 78 Figure 4.5 Rotor diameter vs. power output The power that a wind turbine can extract from the wind at a given wind speed is directly proportional to its rotor’s swept area. It is extremely important that the maximum swept area is presented to the wind and this is achieved by making sure that the rotor’s axis is aligned with the direction from which the wind is blowing. As the wind does not always blow from the same direction, a mechanism of some kind is needed to realign the rotor axis in response to changes in wind direction. This aligning or slewing action, about a vertical axis that passes through the center of the tower, is known as yawing. A wind turbine blade has a distinctive curved cross-sectional shape, which is rounded at one end and sharp at the other. The shape of the blade’s cross-section is the key how modern wind turbines extract energy from the wind. This special profile is known as an aerofoil section and is already familiar as the cross-sectional shape of aeroplane wings. 79 Figure 4.6 Swept area by rotor blades Due to the aerodynamic forces on rotor blades, a wind turbine converts the kinetic energy of wind flow into rotational mechanical energy. These driving aerodynamic forces are generated along the rotor blades, which need specially shaped profiles that are very similar to those, used for wings or aeroplanes. With increasing airflow speed, the aerodynamic lift forces grow with the second power and the extracted energy of the turbine with the third power of the wind speed, a situation which needs a very effective, fast acting power control of the rotor to avoid mechanical and electrical overloading in the wind turbine’s energy transmission system. Modern wind turbines use two different aerodynamic control principles to limit the power extraction to the nominal power of the generator. The most passive one is the so-called stall control, the active one pitch control. Stall control is a traditional way and has restrictions. Pitch control is more flexible and has opportunities to influence the operation of the wind turbine. (German Wind Energy Institute, DEWI, 1998, p.44) 80 4.3.1. PITCH CONTROL Pitch control is an active control system, which normally needs an input signal from the generator power. Always when the generator’s rated power is exceeded due to increasing wind speeds, the rotor blades will be turned along their longitudinal axis (pitch axis), or in other words, change their pitch angle to reduce the angle of attack of incoming air flow. Under all wind conditions, the flow around the profiles of the rotor blade is well attached to the surface, thus producing aerodynamic lift under very small drag forces. Therefore, turbine blades reach the optimum pitch angle, at which it will produce the maximum power at that wind speed. Pitch controlled turbines are more sophisticated than fixed pitch stall controlled turbines, because they need a pitch changing system. (German Wind Energy Institute, DEWI, 1998, p.45) The advantages of the pitch controlled wind turbines are; • Allow for active power control under all wind conditions, also at partial power. • Straight power cur ve at high wind speeds. • They reach rated power even under low air density conditions (high site elevations, high temperatures). • Higher energy production under the same conditions (no efficiency reducing stall adaptation of the blade). • Simple start-up of the rotor by simple pitch change. • No need of strong brakes for emergency rotor stops. • Decreasing rotor blade loads with increasing wind above rated power. • Feathering position of rotor blades for low loads at extreme winds. • Lower rotor blade masses lead to lower turbine masses. 81 Figure 4.7 Pitch Control 4.3.2. STALL CONTROL Stall control is a passive control system, which reacts on the wind speed. The rotor blades are fixed in their pitch angle, and cannot be turned along their longitudinal axis. Their pitch angle is chosen in a way that for winds higher than rated wind speed the flow around the rotor blade profile separates from the blade surface (stall). This reduces the driving lift forces and increases the drag. Lower lift and higher rotational drag act against a further increase of rotor power. (German Wind Energy Institute, DEWI, 1998, p.44) The advantages of stall controlled wind turbines are; • No pitch control system. • Simple rotor hub structure. • Less maintenance due to fewer moving machinery parts. • High reliability of power control. Figure 4.8 Stall Control 82 In last years, a mixture of pitch and stall control is appeared, the so-called active stall. In that case the rotor blade pitch is turned in direction towards stall and not towards feathering position (lower lift) as it is done in normal pitch systems. The advantages of this system are; • Very small pitch angle changes necessary. • Power control under partial power conditions (low winds) is possible. • Feathering position of rotor blades for low loads at extreme winds. The main issues in deciding between pitch and stall control are listed in Table 4.2. Table 4.2 Pitch vs. Stall Issues Issues Pitch Stall Energy Capture Better in principle Compromised power curve Control With Fixed Speed Difficult in high wind speeds Generally satisfactory, although design uncertain Control With Variable Speed Better power quality, lower drive train loads than any stall option Requires proving Safety Complete rotor protection Needs auxiliary systems for over-speed protection Cost More cost in rotor systems Less cost in rotor, but more in braking system Large wind turbines almost exclusively use pitch or stall control. In a few instances, yawing out of wind is used as a back up safety procedure or as contributory to control. Recently, some manufacturers have used stall in conjunction with variable speed operation. The one configuration that has now been unanimously rejected is fixed speed pitch control. This combination produced very large transients in the power 83 output when controlling power. This rejection is, however, rather interesting since it was, in the early days, a popular choice. Figure 4.9 Stall & Pitch controlled power schemes As shown in Figure 4.9, pitch controlled power scheme results almost zero oscillations. Beside, stall control scheme shows some unwanted fluctuations causing power losses. 4.4. CLASSIFICATION BY LOCATION OF INSTALLATION Wind turbines are installed either on the land or on the sea level by some additional equipment. They are classified as on-shore and off-shore wind turbines. 4.4.1 ON-SHORE WIND TURBINES In order to get the best efficiency from wind turbine operation and provide sustainable electricity to consumers, wind turbines should be erected in windy areas. For this purpose, locations with continuous and fast wind should be selected. 84 Wind turbines on the land are called as on-shore wind turbines. In order to benefit from wind speed as much as possible, windy and smooth areas such as lowlands, sea coasts, large farms are selected for siting. 4.4.2 OFF-SHORE WIND TURBINES Off-shore wind turbines are installed on sea up to some depths. It is a fact that, there is a noteworthy difference of available wind speeds between on-shore and off- shore locations. It is possible to obtain higher output power levels for off-shore designs than the same turbines designed for on-shore. The next great leap for the wind energy industry will be in the area of offshore development. The potential for this technology is vast and it requires, and deserves sustained and substantial research and development support. (European Commission Directorate-General for Energy, 1997, p.10) Most turbines operate with a blade tip speed less than 65 m/s principally in order to contain sound emission within acceptable limits. It has been recognized that if off- shore wind turbines are remote from the coast and can be allowed increased sound emission, then there is considerable scope for reduction of the weight and cost of the turbines themselves. A tip speed of 100 m/s may be acceptable for offshore wind turbines. As with sound, if there is some relaxation in concern about the near field visual effect for offshore wind farms, there is added potential for cost reduction in support structures and greater tolerance of more unusual design configurations that may have economic merit. Thus the general view is that, if higher tip speeds can be exploited, the cost of the wind turbine component of the offshore system can be significantly reduced compared to land based designs. Obviously this is very desirable to help offset the increased costs of foundations and electrical transmission associated with offshore projects. 85 A key objective for the design of cost effective offshore wind turbines will be that inspection and maintenance requirements are reduced to a minimum. Design for high reliability will be an important priority with an emphasis on minimising long term operation and maintenance costs, possibly at the expense of a somewhat higher wind turbine capital cost. (European Commission Directorate-General for Energy, 1997, p.11) 86 CHAPTER FIVE EXPERIMENTAL WORK In this chapter, a wind turbine is modelled by MATLAB v5.2 - SIMULINK software. The prototype chosen for the simulation is VESTAS V80 – 2.0 MW wind turbine. The characteristics of the modelled wind turbine are; Rated Mechanical Power (P cap ) : 2 MW Rated Wind Speed : 12.5 m/s Cut- in Wind Speed : 4.5 m/s Cut-out Wind Speed : 20 m/s Power Regulation Method : Pitch Control (0-15 degrees) Rotor Diameter (2.r) : 74 m Disc Swept Area (A) : 4300.84 m 2 Air Density (?) : 1.225 kg/m 3 Moment of Inertia (J) : 1000 t.m 2 Gear Ratio : 38 Rotational Speed (n rlow ) : 20 – 28.5 rpm Generator Rotor Speed (n rhigh ) : 760 – 1083 rpm While constructing the closed- loop model, some mathematical expressions describing the power output and rotational motion of the turbine are used. System Equation Set: 3 p cap V A C 5 . 0 P ⋅ ⋅ ⋅ η ⋅ ρ ⋅ · (5.1) 87 ( ) ( ) ( ) α ⋅ − λ ⋅ − ] ] ] α ⋅ − − λ ⋅ π ⋅ α ⋅ − · 3 00184 . 0 3 . 0 15 3 Sin ) 0167 . 0 44 . 0 ( C p (5.2) V r r ω ⋅ · λ (5.3) dt d J P P ) t ( r ) t ( r ) t ( cap ) 1 t ( cap ω ⋅ ⋅ ω + · + (5.4) where P cap : Captured power by the turbine (input to the generator) (W) ? : Air density (kg/m 3 ) ? : Turbine mechanical efficiency C p : Power coefficient A : Swept area by rotor blades (m 2 ) V : Wind speed (m/s) a : Blade pitch angle (degree) ? : Tip speed ratio r : Rotor radius (m) ? r : Angular shaft speed (rad/s) J : Moment of inertia (kg.m 2 ) 88 F i g u r e 5 . 1 O v e r v i e w o f t h e w i n d t u r b i n e s i m u l a t i o n 89 The aim of the simulation is to observe system output power curve versus wind input that changes with time. The captured power is used to calculate shaft speed variation corresponding torque change. For example, when input wind power increases, input torque to the turbine increases as well. Then, acceleration on the turbine shaft will be observed. 5.1 SUB-SYSTEMS IN THE MODEL 5.1.1 YAW CONTROL BLOCK Yaw mechanism should be adapted to all wind turbines to avoid two unwanted effects; 1. Physical damage of turbine machinery parts due to extremely high wind speeds; occurs when the wind speed is as high as unacceptable over the rated value. This causes teetering effects on turbine tower and over-speed of generator rotor. Manufacturers should take into account the upper damage limit to keep turbine in service. This limit is called cut-out wind speed. 2. Motoring operation of the turbine generator due to very low wind speeds because of insufficient starting torque; a specific wind speed occurs as the lower limit to enable starting of generator mode of the machine. The specific lower limit of the wind speed is called cut-in wind speed. Another usage purpose of the yaw system is aligning the turbine in line with the wind direction in order to allow the turbine to absorb maximum energy from the wind. In the studied model, 4.5 m/s is defined as cut- in and 20 m/s as cut-out wind speeds. Any wind data outside the 4.5 – 20 m/s interval is neglected to make system efficient. 90 Figure 5.2 Yaw control block 5.1.2 TURBINE EFFICIENCY BLOCK At each wind speed, the mechanical torque input onto turbine shaft changes and mechanical efficiency also changes due to friction and heating. So, it may be stated that, turbine mechanical efficiency is directly proportional to the wind speed. Figure 5.3 Turbine efficiency block An efficiency curve is constituted for the model by using the operating values of different turbines present in the market. 91 Figure 5.4 Turbine efficiency characteristics corresponding to wind speed 5.1.3 PITCH CONTROL BLOCK Pitch control mechanism allows turbine blades to turn along their longitudinal axes. As any blade moved to increase the pitch angle, its capacity of absorbing wind power will decrease. In the studied system, when the absorbed wind power exceeds 2 MW, pitch control mechanism will be activated. After the power curve decreases below 2 MW, blade pitch angle will begin to decrease. To make power curve smooth while pitch control is activated, blade response time to any increment or decrement command is tried to be minimized. For this purpose, linear interpolation is applied to input wind speed data. By this way, present 137 wind inputs are raised to 2740 data with sample time equal to 0.05 second. 92 Figure 5.5 Graphical demonstrations for the response of pitch control mechanism As seen from Figure 5.5, when the captured power exceeds 2 MW level at time 70.57 seconds, pitch mechanism is activated at time 70.60 seconds and the power curve is corrupted at time 70.60 sec. approximately at 2.0135 MW. The corresponding pitch mechanism response time is approximately 30 milliseconds. After the blade opening command is received by pitch control mechanism, the time required for the output power curve to recover itself to 2 MW level is about 10 milliseconds as shown in Figure 5.5. 93 Figure 5.6 Pitch control block with 0-15 degrees adjustment interval 5.1.4 ANGULAR SPEED CALCULATION BLOCK This block is a key for turbine performance. By using the advantage of taken samples of captured power in narrow time intervals (sample time=0.05 sec.), shaft angular speed variation corresponding to changing input torque at each step is calculated accurately in this block. Then, obtained angular speed value is used to calculate tip speed ratio. The general mechanical rotational motion equation is used to define acceleration, deceleration or constant speed operations by wind speed changes; dt d J ) t ( r ) t ( ) 1 t ( ω ⋅ + τ · τ + (5.5) where t (t+1) : New captured mechanical torque input to the shaft (N.m) t (t ) : Existing mechanical torque on the shaft (N.m) J : Moment of inertia (kg.m 2 ) ? r(t) : Angular shaft speed (rad/s) This equation can be modified to provide system compatibility; 94 dt d J P P ) t ( r ) t ( r ) t ( cap ) 1 t ( cap ω ⋅ ⋅ ω + · + (5.6) where P cap(t+1) : New captured mechanical power input to the shaft (W) P cap(t ) : Existing mechanical power on the shaft (W) Here, derivative term states the speed variation between times (t) and (t+1). This value is added to the speed value at time (t) to find the new speed value at time (t+1); J P J P P dt d ) t ( r cap r ) t ( r ) t ( cap ) 1 t ( cap r ) t ( r ⋅ ω · ω ∆ ⇒ ⋅ ω · ω ∆ · ω + (5.7) Consequently, this speed difference (indicating acceleration, deceleration or constant speed operation) is added to the speed value at time (t); r ) t ( r ) 1 t ( r ω ∆ + ω · ω + (5.8) The resultant angular speed can be used to find tip speed ratio (?), power coefficient (C p ) and the power input to the generator (P cap ), respectively. Figure 5.7 Angular speed calculation block 95 5.1.5 Cp – ? SELECTION BLOCK After the system decides pitch angle in degrees, power coefficient (C p ) can be found by using its characteristic equation depending on tip speed ratio (?) and pitch angle (a). C p – ? selection block has two inputs (?, a), and one output (C p ). Block has a C p =f(?, a) function for each a input (Equation 5.2). Multiport selection block inside the sub-system decides the function to be used. After the output C p is found, it is fed back to power calculation block to determine the captured power of the turbine. This power is also the input mechanical power to the generator. At the end of simulation, output power graph says that pitch control is a very useful way to control system output whatever the wind power. Pitch control allows user to control the power absorbing capacity of the turbine. 5.2 SIMULATION RESULTS Simulation takes 137 seconds. Input wind data is interpolated by the system with 0.05 second sample time. Totally, simulation includes 20 x 137 = 2740 steps. Small sample time enables system to be stable and captured power to be kept around the rated value. Note from Figure 5.10 that, output power fluctuations can be kept in 200 kW tolerances. All graphical results of the simulation are shown below. 96 Figure 5.8 Wind speed values filtered by yaw control block Figure 5.9 Aerodynamic power in the wind 97 Figure 5.10 Captured wind powerby the turbine (Input power to generator) Figure 5.11 Angular speed variation of the turbine in respect of each wind speed change (Change of input torque) 98 Figure 5.12 Angular shaft speed of the turbine Figure 5.13 Rotational speed of turbine shaft before gearbox 99 Figure 5.14 Rotational speed of turbine shaft after gearbox (Rotational speed of generator rotor) Figure 5.15 Tip speed ratio 100 Figure 5.16 Blade pitch angle (a) Figure 5.17 Power coefficient (C p ) 101 Figure 5.18 Tip speed ratio vs. power coefficient Figure 5.19 Turbine wind speed – power characteristics 102 Figure 5.20 Turbine efficiency vs. wind speed In Table 5.1, variations of all parameters of the wind turbine can be observed corresponding to each available wind speed value. Note that, until wind speed (V) reaches the rated value, pitch angle (a) kept at zero by the system, and after the rated wind speed occurred, pitch angle is started to increase in order to allow keeping the output power (P cap ) around rated value At the same time, the available aerodynamic wind power (P w ) is still increasing. 103 Table 5.1 Modelled Wind Turbine Simulation Results V (m/s) P w (kW) ? a (degrees) C p P cap (kW) 5 330 15.7 0 0.21 51 6 570 13.4 0 0.36 165 7 904 11.9 0 0.42 323 8 1,345 10.8 0 0.44 514 9 1,922 10 0 0.44 753 10 2,632 9.4 0 0.43 1,031 11 3,505 9 0 0.42 1,360 12 4,551 8.6 0 0.41 1,732 13 5,788 8.3 2 0.35 1,925 14 7,225 7.1 2.5 0.30 2,020 15 8,890 6.7 4.5 0.24 1,999 16 10,790 6.35 6 0.21 2,052 17 12,938 5.9 7 0.16 1,869 18 18,466 5.6 8.5 0.14 1,847 104 CHAPTER SIX CONCLUSIONS Wind power is a deceptively simple technology. Behind the tall, slender towers and gently turning blades lie a complex interplay of lightweight materials, aerodynamic design and computerized electronic control. Although a number of variations continue to be explored, the most common configuration has become the horizontal three bladed turbine with its rotor positioned upwind on the windy side of the tower. With this broad envelope, continuing improvements are being made in the ability of the machines to capture as much energy as possible from the wind. These include more powerful rotors, larger blades, improved power electronics, better use of composite materials and taller towers. The most dramatic improvement has been in the increasing size and performance of wind turbines. From machines of just 25 kW twenty years ago, the typical size being sold today is up to 2500 kW. Today’s wind turbines include properties of modern technology. They are modular and very quick to install and commission. Advantages of using wind energy conversion systems instead of other energy production systems are; • Environmental protection (No CO 2 emission) • Low-cost. Wind can be competitive with nuclear, coal and gas • Diversity and security of supply • Rapid deployment. Modular and quick to install 105 • Fuel is abundant, free and inexhaustible • Costs are predictable and not influenced by fuel price fluctuations • Land- friendly. Agricultural / industrial activity can continue around it Power control of the studied horizontal axis, variable speed wind turbine is made by pitch angle adjustment. This seems as the most efficient method to supply 3-phase utility grids. As the number of wind speed samples increases, the pitch control mechanism works more efficiently, in other words; the oscillations around rated power line can be minimized above rated wind speeds. Moment of inertia, rotor diameter and gear ratio are three critical parameters for a variable speed wind turbine and must be selected carefully by manufacturers while designation. Moment of inertia is the rotational mass of the turbine rotating parts. The constructing material of blades and other rotating masses should be selected optimum to verify the minimum cut- in wind speed. This means minimum starting torque and maximum usage of the wind power. Rotor diameter is directly specifies the swept area and so captured power from the wind. It should be selected carefully to ensure reaching rated power output level and allowing minimum cut- in wind speed. For this purpose, long time wind speed measurements should be made and then it will be possible to investigate the optimum wind speed interval to allow maximum overall energy capturing. Gear ratio is the adjustment location of induction machine generator region. For example, in the studied system, 20-28.5 rpm operating interval of low-speed shaft is modified into 760-1083 rpm region for a 750 rpm synchronous speed asynchronous machine with the gear ratio of 38. 106 Although tip speed ratio values seem acceptable in both raising and falling regions of ?–Cp curve, allowing tip speed ratio to exceed 10 causes the over-speed of generator rotor, resulting in the physical damage of machinery parts. Figure 6.1 ?–Cp curve indicating operating regions of the generator 6.1 FUTURE PROSPECTS In the future, even larger turbines than today’s 2500 kW will be produced to service the new offshore market. Machines in a range from 3000 kW up to 5000 kW are currently under development. In 2002, the German company Enercon is scheduled to erect the first prototype of its 4500 kW turbine with a rotor diameter of 112 meters. (EWEA, European Wind Energy Association, 2002, p.13) European Wind Energy Association (EWEA) which is the international voice of the wind industry located in the center of Europe has launched an industrial blueprint including the targets to be reached by 2020. 107 The main objectives of this study are; • Supplying 12 % of global electricity demand, assuming that global demand doubles by then • Creation of 1475 million recruitments • Cumulative CO 2 savings of 11,768 million tones • 1,261,000 MW wind energy capacity installed generating 3093 TWh, equivalent to the current electricity use of all Europe This study demonstrates that there are no technical, economic or resource limitations to achieve this goal, but the political and policy changes are required in order for the wind industry to reach its full potential. 108 REFERENCES American Wind Energy Association. (2002). Global wind energy market report. URL: http://www.awea.org/pubs/documents/ Boyle, G. (1996). Renewable energy: Power for a sustainable future. Oxford University Press. Chapman, Stephen J. (1999). Electric machinery fundamentals. (3 rd ed). Melbourne: McGraw-Hill International Editions Electric Machinery Series. Chen, Z., & Spooner, E. (2001). Grid power quality with variable speed wind turbines. IEEE Transactions on Energy Conversion, 16, 148-153 Çam, E. (1999). Yeni tip kanat modeli ile rüzgardan elektrik eldesi. Bornova, Izmir. Aegean University. Danish Wind Turbine Manufacturers Association. (2001). Guided tour on wind energy. URL: http://www.windpower.org/download/ De Montfort University. (1998). Wind energy training course. URL: http://www.iesd.dmu.ac.uk/wind_energy/index.html European Commission Directorate-General for Energy. (1997). Wind energy - The facts. URL: http://www.ewea.org/doc/ 109 European Wind energy Association. (2002). Wind energy – Clean power for generations. URL: http://www.ewea.org/doc/ European Wind energy Association. (2002). Wind force 12. URL: http://www.ewea.org/doc/ European Wind Energy Association. (2002). Wind force 12, The new global challange. Wind Directions, XXI - 4, 16-19 URL: http://www.ewea.org/doc/ German Wind Energy Institute. (1998). Wind Energy Information Brochure. Gipe, J. (1995). Wind energy: Comes of age. John Wiley & Sons Inc. Heier, S. (1998). Grid integration of wind energy conversion systems. (Waddington R.). Swadlincote, UK: John Wiley & Sons Inc. (Original book published 1996). Muljadi, E., & Butterfield, C.P. (2000). Pitch-controlled variable-speed wind turbine generation. Phoenix, Arizona, USA: 1999 IEEE Industrial Applications Society Annual Meeting, October 3-7, 1999. Ramage, J. (1983). Energy – A guidebook. Oxford University Press. Shaltout, A. A. (1994). Analysis of torsional torques in starting of large squirrel cage induction motors. IEEE Transactions on Energy Conversion, 9, 135-141 Wang, Q., & Chang, L. (1999). An independent maximum power extraction strategy for wind energy conversion systems. Shaw Conference Center, Edmonton, Alberta, Canada May 9-12 1999: Proceedings of the 1999 IEEE Canadian Conference on Electrical and Computer Engineering. 110 APPENDICES A Get wind data (V) Rotor radius (r) Gear ratio 4.5 < V < 20 m/s Calculate aerodynamic wind power ( 3 w V A 5 . 0 P ⋅ ⋅ ρ ⋅ · ) Mechanical power ( p w m C P P ⋅ · ) Calculate captured power (Generator input power) ( η ⋅ · m cap P P ) Calculate angular speed (? r ) Tip speed ratio (?) Calculate turbine efficiency (?) (Look-up table) Calculate pitch angle (a) Calculate power coefficient (C p) V = 0 Yes No - FLOWCHART OF THE SIMULATED SYSTEM - B C M.Sc. THESIS EXAMINATION RESULT FORM We certify that we have read this thesis and “MODELING AND SIMULATION OF WIND TURBINES” completed by OSMAN ORAL KIVRAK under supervision of PROF. DR. MUSTAFA GÜNDÜZALP and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Prof. Dr. Mustafa GÜNDÜZALP Supervisor (Committee Member) (Committee Member) Approved by the Graduate School of Natural and Applied Sciences Prof. Dr. Cahit HELVACI Director I ACKNOWLEDGMENTS I wish to thank to my supervisor Prof. Dr. Mustafa GÜNDÜZALP for his guidance and understanding throughout my project. I wish also thank to Prof. Dr. Eyüp AKPINAR for his support on critical points. I am also grateful to my family and colleagues for their advices. Osman Oral KIVRAK a megawatt size. therefore it is necessary to supervise produced power curve continuously. wind turbines are classified according to different categories. wind. Wind energy has become the most popular resource in the last decade by its purity and sustainability. Accordingly. Unlike other energy production systems. These sudden changes in wind speed may cause some unwanted mechanical or electrical damages. Wind energy conversion systems convert the aerodynamic power in an air stream into the electric power. a wind energy conversion system consists of blade(s). which captures the aerodynamic power in the wind. In this project. Then. shaft. Pitch control – opening and closing of blades along their longitudinal axes is the most efficient and popular power control method especially for variable-speed wind turbines. It is foreseen that new energy resources should not harm environment and natural life beside meeting present and future energy demand. Principally.II ABSTRACT Increasing worldwide energy deficiency causes raising importance of development of new energy resources. variable-speed wind turbine is modeled and its operation is observed by using MATLAB v5. the energy conversion operation in wind turbines and components of them are investigated. a great tendency towards renewable energy resources took place in the market. as a source of energy for wind energy conversion systems. has a structure of showing sudden changes depending on climatic conditions. status and importance of wind energy conversion systems throughout the world. Several power control methods are developed for this purpose. which transfers the torque created by the turning action of blade(s) and generator. At final.2 – SIMULINK . which converts this torque into electric power. turbine.0 MW model wind turbine. pitch control.III software. Keywords : Wind energy. MATLAB. variable speed. Output power curve regulation is carried out by ‘pitch control’ method. renewable. energy conversion. The prototype for the simulation is VESTAS V80 – 2. . Daha sonra rüzgar türbinleri çesitli kategorilere göre siniflandirilmistir. rüzga r enerjisi dönüsüm sistemlerinde enerji kaynagi olarak kullanilan rüzgar. mevcut ve gelecekteki enerji ihtiyacini karsilamasi ile birlikte.IV ÖZET Enerji açiginin her geçen gün arttigi dünyamizda. çesitli güç kontrol yöntemleri gelistirilmistir. MATLAB v5. iklim kosullarina bagli olarak ani degisimler gösterebilen bir yapidadir. çevreyi ve dogal yasami da olumsuz yönde etkilememesi öngörülmektedir. Olusturulacak yeni enerji kaynaklarinin. degisken hizlarda çalisan megawatt boyutunda bir rüzgar türbini modellenerek çalismasi gözlenmistir. özellikle degisken hizlarda çalisan rüzgar türbinleri için en verimli ve popüler güç kontrolü yöntemidir. Bir rüzgar enerjisi dönüsüm sistemi. yeni enerji kaynaklari gelistirmenin önemi de her geçen gün artmaktadir. üretilen güç egrisinin sürekli denetim altinda bulundurulmasi gerekmektedir. kanatlarin dönme hareketi ile olusan torku ileten saft ve bu mekanik torku elektriksel güce çeviren jeneratörden olusmaktadir. sistemde mekaniki ve elektriki birçok hasara yol açabileceginden. Rüzgar enerjisi dönüsüm sistemleri. . prensip olarak. Diger enerji üretim sistemlerinden farkli olarak. enerji sektöründe yenilenebilir enerji kaynaklarina yönelim artmaktadir. Bu dogrultuda.2 – SIMULINK yazilimi kullanilarak. rüzgardaki aerodinamik gücü yakalayan kanat(lar). Rüzgar enerjisi. Son olarak. son 10 yilda en popüler kaynak olmustur. Pitch kontrolü – türbin kanatlarinin kendi dikey eksenlerinde açilip kapatilmasi -. Bu amaçla. rüzgar türbinlerinde gerçeklesen enerji dönüsüm islemi ve türbin aksamlari incelenmistir. Bu ani degisimler. rüzgar enerjisi dönüsüm sistemlerinin önemi ve dünyadaki durumu. Bu projede. rüzgarin içinde bulundurdugu aerodinamik gücü elektriksel güce dönüstürürler. temizligi ve sürekliligi ile. 0 MW model rüzgar türbini alinmistir. türbin. degisken hizli. Modelde prototip olarak VESTAS V80 – 2.V Çikis gücü ayari ‘pitch control’ yöntemiyle gerçeklestirilmistir. Anahtar Kelimeler: Rüzgar enerjisi. yenilenebilir. açi kontrolü. . ...... 11 2..1 Nacelle……………………......... X List of Figures...……………………………….. 11 2.. 8 2...4 High Speed Shaft………….. 11 2..……………………………….......……………………………………….…..1. XI Chapter One INTRODUCTION 1. VI List of Tables………………………………………………………………….. 12 2.....………………………..3 Low Speed Shaft………………..2 Functional Structure of Wind Turbines…..………………………………………........………………………….6 Generator………..1.. 12 2.....1..... 6 Chapter Two COMPONENTS OF WIND TURBINES 2.…………………………..………………………... 8 2....1.1 Common Components…………………….7 Tower…………………….1 Historical Background………………….………………………………………………………………..……………………………….1..... 4 1.1.1.....…………………….…………………………….....……….VI CONTENTS Page Contents……………………………………………………………………….………………………………. 13 ... 8 2.5 Disc Brake……………………………....2 Blade……………....2 Optional Components…………………………………………………….........…. .. 33 3......1 Rotor Circuit Model……………………………………….1 Aerodynamics of Wind Turbines……….………………………………....3. 33 3.1.3...1 Power Coefficient ……………………... 19 3......2 Energy and Power in The Wind…………..………………….5 Electronic Controller…………………..3 Internal Voltage of a Synchronous Generator……………………..2.3..... 36 3...……………..……....... 28 3... 18 3......……………. 50 ...2 V / Hz Converter……………………….1 Gear Box…………….1......1 Aerodynamic Forces……….... 2... 48 3.... 33 3..3.......…………………………...………………………………………………...3....2 DC Generator Applications in Wind Turbines……………………... 13 13 14 14 15 Chapter Three ELECTROMECHANICAL ENERGY CONVERSION 3..3.. 39 3..………………………………...3..2. 19 3... 25 3.... 20 3...1. 18 3. 2.4 Pitch Control Mechanism…………….3.3. 27 3...3. 2. 22 3..3......…………………………….... 2..... 46 3... 36 3..2...2.....………………………………..3.2 The Rotation Speed of a Synchronous Generator…………………..1..2.……………………………….………………………….4 The Equivalent Circuit of an Alternator…………………………… 42 3....……………………………........……………………………….2...3......……….2.... 37 3.2..... 40 3.......3...1 Equivalent Circuit of an Induction Machine………………………..………………………………....1 Theory…………………………....3 Generator Theory……………………….1.1..2 Synchronous AC Machines (Alternators)……………………………….....3 Effect of The Number of Blades……......……………………....2....3..1....VII 2..1 Drag Forces………………..2 Aero-Foils………………………….2 Tip Speed Ratio………………………………………………....2 Lift Forces………………………………………............. 44 3..3 Yaw Assembly…………………………………………......3 Asynchronous (Induction) AC Machines………………………………...1.2...2..1..2 Final Equivalent Circuit……………………………………….1.....………...1 Theory………………………………………………….1 DC Machines…….2...…………. .1 Classification by Axis of Rotation…………………….1.1 Horizontal Axis Wind Turbines (HAWT)………………………….....1.. 65 3..4..4 Recent Developments in Generators for Wind Turbines………………. 74 4. 65 3. 71 4. 73 4.....…..4.4.....3 Transistors….1 Frequency Converter Systems………………………………………...... 58 3....………………………………..1 Variable Rotor Speed…………............... 70 4.3 Classification by Power Control………………………………………….3. 83 4....2.2 Thyristors………………………………………………….... 59 3..VIII 3...4 Classification by Location of Installation…………………………………..1 Power Semiconductors for Frequency Converters………………… 63 3...4......... 81 4..………………………….......1...4. 56 3......1 Semiconductor Diodes……………………………………. 83 4..1 Pitch Control……………………………………………………………..3.....………………. 64 3..2 Stall Control…………………………………………………………….2 Classification by Rotor Speed……………………………………………. 84 .1.....3..3.…………………......2 Direct-Drive Generators…………………………………………… 57 3.… 75 4........1 On-Shore Wind Turbines……………………………………………….3.. 56 3..... 67 Chapter Four CLASSIFICATION OF WIND TURBINES 4....4..1.4..1...1..1.. 72 4.4.2 Characteristics of Power Converters………………………………......2 Vertical Axis Wind Turbines (VAWT)……………………………..1 Dual Generators……………………………………………………..............4.2......1.2 Off-Shore Wind Turbines……………………………………………….... 80 4.........4.2 Constant Rotor Speed. 69 4.4 Grid Integration……………………………………………………………....1. ..... 89 5.... 91 5.....3 Pitch Control Block…………………………………………………. 110 Appendix A – Flowchart of The Simulated System………………………...........1....1 Sub-Systems in The Model…………………………….... 108 Appendices….1 Yaw Control Block……………………….......1 Future Prospects……………………………………….……………………………....…………………………………………………………………...……………….1....2 Simulation Results…………………………………………………………… 95 Chapter Six CONCLUSIONS 6.106 References……….........4 Angular Speed Calculation Block….5 Cp – ? Selection Block…………………………………………………......IX Chapter Five EXPERIMENTAL WORK 5. 95 5...1...………………………..... B ......1. 90 5. 89 5............ A Appendix B – VESTAS V80 – 2...2 Turbine Efficiency Block…………….……………………………………….........……………….1...............……………….......... 93 5......0 MW Wind Turbine………………….. .... 67 Table 4.. 103 .... 77 Table 4......3 Characteristics and Maximum Ratings of Switchable Power Semiconductors…………………………………... Stall Issues………………………………………………… 82 Table 5..X LIST OF TABLES Page Table 1..…………………………..…………………..1 World Electricity Consumption with Estimations……….2 Wind Power Installations Worldwide….………. 2 3 4 Table 2......……….1 Descriptions of Operational Regions for a Typical Wind Turbine….. Table 1.1 Modelled Wind Turbine Simulation Results………..1 Number of Blades for Commercial Wind Turbine Designs………… 11 Table 3.3 Wind Energy Capacity Leaders Worldwide by End 2001......2 Pitch vs..2 Common Synchronous Speeds for Generators…………………….... Table 1..1 Speed Definitions…………………………………………………… 27 Table 3. 55 Table 3.... 1 Wind turbine types by rotor assemblies………………………….... each second……………………….....8 The equivalent circuit for DC motors……………………….. Figure 3...6 A typical wind turbine in detail (VESTAS V27 / 225 kW).5 degree increments…………….…………………...…….....3 Power transfer in a wind energy converter…………….... Figure 2...………………......1 World electricity consumption with estimations .. Figure 3.…………………………………...5 AC – AC signal conversion……………………………….………………………………………….2 Wind power installations worldwide…..……. Figure 3.... 32 Figure 3... Figure 2. Figure 3.9 A salient six-pole rotor for a synchronous machine……………… Figure 3.2 Nacelle………............1 A typical wind turbine showing all components………………….2 Lift and drag forces acting on rotor blade…………………...XI LIST OF FIGURES Page Figure 1.....6 Power coefficient versus tip speed ratio for a constant speed wind 1 2 6 7 8 10 13 14 16 17 19 21 23 25 turbine……………………………………………………………... Figure 3.....5 Wind flow through a wind turbine……………………………….. Figure 2.10 A non-salient two-pole rotor for a synchronous machine……….3 Horizontal axis wind turbines according to number of blades…… Figure 2. Figure 3. Figure 2. 34 Figure 3. Figure 1......3 Components of wind power acting on rotor blade………………...... Figure 1.. 31 Figure 3.. 38 39 .4 A typical gear…………………………………………………….4 Cylindrical volume of air passing at velocity V (10 m/s) through a ring enclosing an area........ ‘A’.... Figure 2.7 Power coefficient versus tip speed ratio for a variable speed wind turbine for different pitch angles from 0 to 15 degrees by 0. 5 Rotor diameter vs. field current for synchronous generators b.... Figure 3.………………………………….... Figure 3...13 The per-phase equivalent circuit for synchronous generators……....……………........... 90 Figure 5.15 Cutaway diagram for a squirrel-cage induction machine………… 45 Figure 3.24 Indirect frequency converters…………………………………….. power output…………………………………..16 Transformer model for an induction machine……………………..21 Torque-Speed curve for a MW-size induction machine………….1 Horizontal and vertical axis wind turbines………………………..23 Basic wiring diagram for direct frequency converters…………… Figure 3....17 Magnetization curve for an induction machine compared to that for a transformer…………………………………………………..4 Operating regions of a typical wind turbine……………………… Figure 4........18 The rotor circuit model for induction machines…………………......... 88 Figure 5..........14 Cutaway diagram for a wound-rotor induction machine………….1 Overview of the wind turbine simulation….. 49 47 49 Figure 3..9 Stall & Pitch controlled power schemes………………………….19 The rotor circuit model with all the frequency (slip) effects concentrated in resistor RR ………………………..5 Graphical demonstrations for the response of pitch control mechanism..... 62 63 70 71 72 76 78 79 Figure 4...6 Swept area by rotor blades……………………………………….. 90 Figure 5........... 45 Figure 3.…………………….... The magnetization curve for synchronous generators…………...11 a....3 Vertical axis wind turbine configurations………………………. Figure 5...8 Stall Control………………………………………………………... Figure 4...2 Horizontal axis wind turbine configurations……………………... Plot of flux vs.... 60 Figure 3. 83 Figure 5..2 Yaw control block……………………………………………. Figure 4.7 Pitch Control……………………………………………………… 81 Figure 4........ Figure 4..........4 Turbine efficiency characteristics correspond ing to wind speed.. 52 Figure 3. 81 Figure 4.. 92 91 ..XII Figure 3..12 A simple circuit for alternators…………………………………… Figure 3....3 Turbine efficiency block.... Figure 4. Figure 4..20 The per-phase equivalent circuit for induction machines………… 51 Figure 3.. Figure 3. 47 Figure 3... 41 42 43 Figure 3..22 Electrical energy conversion by power converters………………. 99 98 Figure 5...8 Wind speed values filtered by yaw control block………………..14 Rotational speed of turbine shaft after gearbox (Rotational speed of generator rotor)………………………………………………… 99 Figure 5..XIII Figure 5........ 101 Figure 5... 102 . Figure 5.. 101 Figure 5... Figure 5...7 Angular speed calculation block......…………………………… 100 Figure 5...17 Power coefficient (C p )……………………………………………...18 Tip speed ratio vs......6 Pitch control block with 0-15 degrees adjustment interval………....... 93 94 96 96 97 97 Figure 5..16 Blade pitch angle (a)………………. 98 Figure 5.15 Tip speed ratio…........11 Angular speed variation of the turbine in respect of each wind speed change (Change of input torque)…………………………..9 Aerodynamic power in the wind…………………………………. 100 Figure 5.19 Turbine wind speed – power characteristics………………….10 Captured wind power by the turbine (Input power to generator)… Figure 5...………….... wind speed………………………………... power coefficient……………. Figure 5..13 Rotational speed of turbine shaft before gearbox………………… Figure 5.....………………………………………………...... Figure 5..........12 Angular shaft speed of the turbine………………………………......20 Turbine efficiency vs.. Therefore. electrical energy supplies will be insufficient to respond this demand. World Electricity Consumption 24000 Net Electrical Energy Consumption (GWh) 18000 12000 6000 0 1990 1995 2000 2005 2010 2015 2020 Years Figure 1. Predictions say that world electrical energy demand will continue to increase in the following 20 years period as shown in Figure 1.1 World electricity consumption with estimations .1.1 CHAPTER ONE INTRODUCTION World electrical energy consumption gets higher as the technology being developed and the human life’s dependency on electricity is growing. new and cost-reduced energy supplies must be introduced into the market. So. 2 Table 1. The scale of its development will depend critically on the care with which wind turbines are selected and sited. generating electricity from wind sites is one of the most popular methods to provide demanded electricity of the world.2 shows that.407 Wind energy offers the potential to generate substantial amounts of electricity without the pollution problems of most conventional forms of electricity generation.1 World Electricity Consumption with Estimations World Electricity Consumption 1990 1998 1999 2005* 2010* 2015* 2020* * Estimated values.380 19. Wind Power Installation History 1991 .2002 32000 28000 Installed MW 24000 20000 16000 12000 8000 4000 0 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Year Annual Installation Cumulative Installation Figure 1. Annual Consumption (GWh) 10.182 17. for about 10 years. (Boyle. 1996.833 15.549 12.2 Wind power installations worldwide .725 12.267) Figure 1.835 22. p. European Wind Energy Association. 2002.800 MW of new capacity was added to the electricity grid worldwide.759 31.223 2.771 5.518 14. p.061 6. p. over 35 million people.2 Wind Power Installations Worldwide WECS Installations 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002* * Estimated value. (EWEA. During 2001 alone.000 2. Europe accounts for around 70% of this capacity.568 2.000 MW.922 4.921 10. and the number of people employed by the industry world-wide is estimated to be around 70.11) .353 7. global wind power installed had reached a level of almost 25.290 1. But other regions are beginning to emerge as substantial markets for the wind industry. installations have roughly doubled every two and a half years.824 6. This is enough power to satisfy the needs of around 14 million households.935 25. Over the past decade.440 18.759 Annual Installation (MW) Cumulative Installation (MW) By the end of 2001. Over 45 countries around the world now contribute to the global total. 2002.597 3. and for two-thirds of the growth during 2001. global wind power capacity has continued to grow at an annual cumulative rate close to 40%.000. European Wind Energy Association. close to 6.292 1.3 Table 1.11) 338 480 730 1.561 3. (EWEA. Since 1996.495 6.041 3. 4 Table 1.550 2.245 3. so modern ‘windmills’ tend to be called wind turbines. these are used principally for water pumping.free means of generating electricity on a potentially significant scale that is attracting most current interest in the subject. .927 1. pumping water. partly because of their functional similarity to other types of turbines that are used to generate electricity. and other mechanical power applications.456 1. a windmill is used for milling grain. Whilst the wind will continue to be used for this purpose.734 4.456 700 525 406 358 357 19 2. For utility-scale sources of wind energy. a large number of wind turbines are usually built close together to form a wind plant. it is the use of wind energy as a pollution.121 24. They are also sometimes referred to as wind energy conversion systems (WECS) and those used to generate electricity are sometimes described as wind generators or aero-generators.3 Wind Energy Capacity Leaders Worldwide by End 2001 COUNTRY Germany USA Spain Denmark India Italy UK China Greece Japan Turkey Others TOTAL Installed MW 8. Today there are over one million windmills in operation around the world.1 HISTORICAL BACKGROUND Wind energy has been used for thousands of years for milling grain. Strictly speaking. . only since the 1980s that the technology has become sufficiently mature. a pocket of low-pressure air forms on the downwind side of the blade. At 30 meters or more above ground.like blades. This is called lift. they can take the advantage of faster and less turbulent wind. Standalone wind turbines are typically used for water pumping or communications. The low-pressure air pocket then pulls the blade toward it. p. However.267) Wind turbines. Turbines catch the wind’s energy with their propeller. An extensive range of commercial wind turbines is currently available from over 30 manufacturers around the world. 1996. As wind blows. It is. however. causing the rotor to turn. are mounted on a tower to capture the most energy. The cost of wind energy equipment fell steadily between the early 1980s and the early 1990s. and the turning shaft spins a generator to make electricity. which is called drag. A blade acts much like an airplane wing. The technology is continually being improved to make it both cheaper and more reliable.5 Attempts to generate electricity from wind energy have been made (with various degrees of success) since the end of the nineteenth century. so it can be expected that wind energy will tend to become more economically competitive over the coming decades. The combination of lift and drag causes the rotor to spin like a propeller. Usually. homeowners or farmers in windy areas can also use wind turbines as a way to cut their electric bills. like windmills. (Boyle. The force of the lift is actually much stronger than the wind's force against the front side of the blade. Wind turbines can be used in stand-alone applications. two or three blades are mounted on a shaft to form a rotor. or they can be connected to a utility power grid or even combined with a photovoltaic (solar cell) system. Small wind machines for charging batteries have been manufactured since the 1940s. Several electricity providers today use wind plants to supply power to their customers. 3 Power transfer in a wind energy converter As shown in Figure 1.6 An understanding of machines that extract energy from the wind involves many fields of knowledge. The electrical energy from the generator is fed via a system of switching and protection devices. leads and any necessary transformers to the mains.2 FUNCTIONAL STRUCTURE OF WIND TURBINES Figure 1. clutches and gears) to the rotor of a generator and thence to the stator of the same by mechanical-electrical conversion. electricity and planning control. 1998. (Heier. to the end user or to some means of storage. convert it into rotational energy then deliver it via a mechanical drive unit (shafts. as well as structural.3. p. blades of a wind turbine rotor extract some of the flow energy from air in motion. civil and mechanical engineering. including meteorology.21) . aerodynamics. 1. so receiving wind unperturbed by the tower itself or. All of these duties are carried out by special components. Figure 2. Downwind of the tower. 1.1 Wind turbine types by rotor assemblies . but causes the wind to be deflected and made turbulent by the tower before arriving at the rotor (tower shadow). which enables self alignment of the rotor with the wind direction (yawing). 2.7 CHAPTER TWO COMPONENTS OF WIND TURBINES A wind turbine converts the kinetic energy of the wind firstly to the rotational mechanical energy then to the electrical energy. Upwind of the tower and nacelle. The rotor assembly may be placed either. and electrical generator.1. . the rotor's inherent mechanical properties and design will affect its useful service life. BLADE Rotor blade design has advanced with knowledge from wing technology. COMMON COMPONENTS 2.1.e.8 The lifetime of a rotor is related to variable loads and environmental conditions that it experiences during service. including the gearbox. Therefore. NACELLE Nacelle contains the key components of a wind turbine. Figure 2. rotor blades and the hub.2 Nacelle 2. i. 2. The shape of the blade and its angle in relation to the relative wind direction both affect its aerodynamic performance.1. and utilizes the aerodynamic lift forces that an airfoil experiences in a moving stream of air.1. Towards the other side of the nacelle.2. Service personnel may enter the nacelle from the tower of the turbine in order to make maintenances. there is wind turbine rotor. The principle rule is.9 The materials used in modern wind turbine blade construction may be grouped into three main classes. 1998.DEWI. If λ = 1. but high blade tip speeds have the disadvantage of high noise emissions and physical damages of the rotor. So. The main reason to use 3 blades is the constant inertia moment of the rotor for all circumferential azimuth angles in relation to operational motions around the longitudinal axis of the tower. but unfortunately the dynamic behaviour of the 2-bladed rotor caused additional efforts that increase again the overall cost. (German Wind Energy Institute . p. Rotors of wind turbines should have rotational speeds as high as possible to reduce the masses of gearboxes and generators. Most of today’s wind turbines have blade tip speeds of less than 65 m/s. Wind turbines can have different number of rotor blades. the number of rotor blades is low and in general not more than three. p. (German Wind Energy Institute . the blade tip velocity is as high as the wind speed. The measure for this is called tip speed ratio.DEWI. designers tried to increase the blade tip speed more and more because the shaft torque reduces with increasing rotational speed.40) 2-bladed rotor offered the chance to reduce the cost for the rotor. In the old prototypes of large wind turbines. 3-bladed rotors are the most common ones all over the world. 1998. which is defined as rotor tip speed divided by the wind velocity. • Wood (including laminated wood composites) • Synthetic composites (a polyester or epoxy matrix reinforced by glass fibers) • Metals (predominantly steel or aluminum alloys) Rotor blades should have the optimum design in order to capture maximum amount of wind and so to provide maximum rotation of the shaft. λ. the lower the number of rotor blades the faster turns the rotor.41) . causes loads and needs complicated hub constructions to keep the movements under control. p. which introduces additional motions. If the data were presented as the proportion of operational machines the dominance of the 3- . (German Wind Energy Institute . 1-bladed rotors principally have an aerodynamic unbalance.DEWI. This means a 1-bladed wind turbine is several times noisier than a 3bladed one. 1-bladed rotors have tip speed two times that of 3-bladed ones.10 As compared to 3-bladed rotors. the rotor blade can be fixed to the hub by a single hinge that allows for a movement that reduces structural loads on the blade. Two-Bladed c.1 illustrates the relative proportion of 1.3 Horizontal axis wind turbines according to number of blades If 1.41) a. 2 or 3 bladed rotors are designed for similar tip speeds (as they have not been in the past but would require to be in the future for European land based applications subject to current sound limits). Three-Bladed Figure 2. On the other hand. One-Bladed b. Table 2. 2 and 3 bladed designs among present commercially available wind turbines of over 30 kW rated output. Additionally. then the blades of the 3-bladed rotor are more highly stressed than for the 2 or 1 bladed system and thus rotor blade costs will be high for the 3 bladed system. 1998. HIGH SPEED SHAFT The high-speed shaft rotates with over 1.11 bladed designs would be still more pronounced. 2. the main shaft is usually supported on journal bearings. LOW SPEED SHAFT While transferring the primary torque to the gear train from the rotor assembly. pp. It is equipped with an emergency mechanical disc brake.1. DISC BRAKE This may be situated either on the main shaft before the gearbox.1 Number of Blades for Commercial Wind Turbine Designs Number of Blades 1 2 3 % of Designs 2 24 74 Conventional wisdom holds that three-bladed machines will deliver more energy and operate more smoothly than either one or two bladed turbines. (European Commission DirectorateGeneral for Energy.5. effective pre-service non-destructive testing procedures are advisable for this component. 2. 1997. 2. They will also incur higher blade and transmission costs as a result.000 revolutions per minute (rpm) and drives the electrical generator. Thus. the main shaft is susceptible to fatigue failure.4. The latter arrangement requires a smaller (and cheaper) .1.3. Due to its high torque loadings. Some experiments say that rotors with three blades can capture 5% more energy than two-bladed turbines while encountering less cyclical loads than one and two bladed turbines.1.5-6) Table 2. or on the highspeed shaft after the gearbox. Generally. TOWER The tower of a wind turbine carries the nacelle and the rotor. Generally for wind turbines. In some systems. it is an advantage to have a high tower. GENERATOR The generator converts the mechanical energy of the input shaft to electrical energy. as they may use an inside ladder to get to the top of the turbine. .7. The advantage of lattice towers is primarily that they are cheaper. but induction machines can be used for low variable speeds. Synchronous machines are generally used for high synchronous speeds. Towers may be either of tubular or lattice types. 2. a typical modern 600 kW turbine will have a tower of 40 to 60 metres (the height of a 13-20 story building). However.1. DC machines are used for stand alone systems such as battery charging which do not need to produce grid compatible electricity. since wind speeds increase farther away from the ground. 2. The generator can be either DC. synchronous or induction (asynchronous). this arrangement does not provide the most immediate control of the rotor. and in the event of a gearbox failure. Tubular towers are safer for the personnel that have to maintain the turbines. induction generators are used for the opportunity of controlling the system under different wind speeds.12 brake assembly in order to supply the necessary torque to slow down the rotor. For example. permanent magnet generators can also be used. braking control of the rotor is lost.6. It must be compatible at input with the rotor and gearbox assemblies. This situation is the result of unstable wind speeds.1. but at output with the utility's power distribution (if connected to a grid) or to local power requirements (if the turbine is part of a stand alone system). Its aim is to keep the generated system voltage near grid frequency (50 or 60 Hz). In general. specially designed permanent. The controlling principle is based on the controlling of the inverter elements (IGBTs. But.4 A typical gear Gearboxes are not intrinsic to wind turbines. thyristors etc. Figure 2.) to be installed. 2. For this reason.magnet alternators have revolutionized the reliability and serviceability of small wind turbines.1. OPTIONAL COMPONENTS 2. GEAR BOX Gearboxes are used for non-direct drive designs. .13 2. slowspeed generators and drive them directly without using a transmission.2.2.).2. Designers use them only because they need to increase the speed of the slow-running main shaft to the speed required by mass-produced generators. this adaptation brings the addition of mechanical machinery parts (Large gearboxes.2. Manufacturers can produce for special purpose. coupling elements etc. the transmission gear is used to adapt WECS to low wind speeds in order to help the rotational speed getting close to the frequency of the grid system. A controlled rectifier-inverter group converts the generated AC voltage to a DC signal and then again to an AC signal. V / Hz CONVERTER The AC-AC converter includes a rectifier and an inverter to control the frequency. 2.2. larger wind turbines with upwind rotors require active yaw control to align the machine with the wind. YAW ASSEMBLY It is necessary for the rotor axis to be aligned with the wind direction in order to extract as much of the wind's kinetic energy as possible. . a blade pitch adjuster ensures that the turbine speed is kept roughly constant by altering the blade angle.4. Downwind machines of all sizes may possess passive yaw control. At a sufficiently high level of wind. The smallest upwind machines (up to 25 kW) most commonly use tail vanes to keep the machine aligned with the wind. sensors activate the yaw control motor. which rotates the nacelle and rotor assembly until the turbine is properly aligned. when a change in wind direction occurs. Yaw system can also be used to shut down the wind turbine in order to save it from the physical effects of very high wind speeds. which means that they can self-align with the wind direction without the need for or a tail vane or yaw drive. PITCH CONTROL MECHANISM This mechanism is used on wind turbines for active power control. However.14 Figure 2. To enable this.5 AC – AC signal conversion 2.3.2. 5.15 For reasons of stability and to reduce the component loading.g. which continuous ly monitors the condition of the wind turbine and controls the pitch and yaw mechanisms. In case of any malfunction. electronic controller takes on the frequency synchronization duty between ge nerated signal and grid. (e. this mechanism changes the blade pitch angle along its longitudinal axis to limit the input torque loading to turbine blades.2. overheating of the gearbox or the generator). it automatically stops the wind turbine and calls the turbine operator's computer via a telephone modem link. . A simple pitch control design can be achieved by using a hydraulic or mechanical centrifugal governor. Another important characteristic of the electronic controller is to control the ACAC converter elements (i. At this point. 2. firing angles of thyristors).e. ELECTRONIC CONTROLLER It contains a computer. 16 Figure 2.6 A typical wind turbine in detail (VESTAS V27 / 225 kW) . 17 CHAPTER THREE ELECTROMECHANICAL ENERGY CONVERSION Electromechanical energy conversion is carried out by the full operation of wind turbine. In case of any component’s failure. either the complete energy conversion stopped or some losses must be taken into account.1 A typical wind turbine showing all components . Figure 3. The gearbox output shaft turns a generator. This box increases the rotational speed of the shaft. . Local low voltage networks transmit the electricity to homes. A torque is produced as the wind interacts with the rotor. The grid system transmits the electricity to the locality of its end use. 7. The site transformer steps up the voltage to the grid value. The high. 5. the wind blade(s) is able to capture the wind energy and rotates itself. 3. 6. This force can be considered to be equivalent to two component forces. 2. The electricity produced by the generator passes through the turbine controller and circuit breakers and is stepped up to an intermediate voltage level (generally 690 V) by the turbine transformer. This rotation of the blade is transferred to the generator shaft or namely to the rotor by an optional gearbox. 8. known as the drag force and the lift force. The site cabling system delivers the electricity to the site transformer via the site control and circuit breaker system. 4.1. 9. 3.1. offices and factories.speed generator (asynchronous or synchronous) is connected to the V/Hz converter to keep the frequency of the generated voltage in the order of the grid frequency.1. which provides more electrical energy production. acting in perpendicular directions. The relatively low rotational frequency of the rotor is increased via a gearbox. The sequence of events in the generation and transmission of wind power can be summarized as follows: 1.1. AERODYNAMIC FORCES An object in an air stream experiences a force that is imparted from the air stream to that object. AERODYNAMICS OF WIND TURBINES 3. Transformer substations reduce the voltage to domestic or industrial values.18 As shown in Figure 3. the drag forces are at a minimum. p.1.1. DRAG FORCES Drag forces are in line with the direction of the air stream. a low pressure region is created on the downstream side of the plate as a result of an increase in the air velocity on that side. the objective is to minimize drag forces. At small angles relative to the direction of the air stream (that is. its orientation to the direction of the air stream. and the velocity of the air stream. In this .284) For wind turbine blades. LIFT FORCES Lift forces are perpendicular to the direction of the air stream. They are termed ‘lift’ because they are the forces that enable aero planes to lift off the ground and fly. Figure 3. Lift forces acting on a flat plate are smallest when the direction of the air stream is at a zero angle to the flat surface of the plate.1.19 The magnitudes of drag and lift forces depend on the shape of the object. When the direction of the air stream is in line with the flat side of the plate. For example.2.2 Lift and drag forces acting on rotor blade 3.1. a flat plate in an air stream experiences maximum drag forces when the direction of the air flow is perpendicular to the flat side of the plate. 1996.1. (Boyle. 3. when the so called angle of attack is small). This phenomenon is known as the Bernoulli’s Effect. (Boyle.20 situatio n. the lower the pressure. When employed as the profile of a wing.284) .1. AERO-FOILS The angle that an object makes with the direction of an air flow.foil section is usually referred to as the chord line . The lift force thus acts as a ‘suction’ or ‘pulling’ force on the object. is called the angle of attack or angle of incidence. these sections accelerate the air flow over the upper surface. Lift forces are the principal that cause a modern wind turbine to operate. p. measured against a reference line in the object. The high air speed thus induced results in a large reduction in pressure over the upper surface relative to the lower surface. 1996. The reference line on an aero. 1996. p. there is a direct relationship between air velocity and pressure: The faster the air flow.284) 3. but the use of so-called aero-foil sections is even more effective. (Boyle. Arching or cambering a flat plate will cause it to induce higher lift forces for given angle of attack.2. .1). in a direction at right angles to the air stream.2).21 Figure 3.foil in plan (m2 ) V L : Wind speed (m/s) : Lift force (N) Similarly. the drag force is described by the drag coefficient CD by Equation (3. 2⋅L ? ⋅ V2 ⋅ AL CL = (3.3 Components of wind power acting on rotor blade The lift force. and is defined by Equation (3. is described by the lift coefficient CL.1) where CL : Lift coefficient ρ : Air density (kg/m2 ) AL : Area of aero. 2 ENERGY AND POWER IN THE WIND A wind turbine obtains its power input by converting the force of the wind into torque (turning force) that is acting on the rotor blades.foil in plan (m2 ) : Wind speed (m/s) : Lift force (N) Horizontal and vertical axis wind turbines both make use of the aerodynamic forces generated by aero.2) where CD ρ AD V D : Drag coefficient : Air density (kg/m2 ) : Area of aero. In a fixed pitch horizontal axis wind turbine.foils in order to extract power from the wind. the angle of attack at a given position on the rotor blade is constantly varying throughout its rotation cycle. Power can be defined as the rate at which energy is used or converted and it can therefore be expressed as energy per unit of time. 1 W =1 j s (3. 3. the angle of attack at a given position on the rotor blade stays constant throughout its rotation cycle. The amount of energy which the wind transfers to the rotor depends on the density of the air. the rotor area.3) . and the wind speed. but each harnesses these forces in a different way.22 CD = 2⋅D ? ⋅ V2 ⋅ AD (3. In a vertical axis wind turbine. ‘A’. the mass of the air moving through the ring each second can be obtained. It can be considered that the air is passing through a circular ring (enclosing a circular area. Therefore. say 100 m2 ) at a velocity V (say 10 m/s) as shown in Figure 3.4. a cylinder of air with a length of 10 m will pass through the ring each second. E = 12 ⋅ m ⋅ V 2 (3.4) where m is the mass and V is the velocity with which this mass is moving. .4 Cylindrical volume of air passing at velocity V (10 m/s) through a ring enclosing an area. each second As the air is moving at a velocity of 10 m/s.23 The energy contained in the wind is its kinetic energy. Figure 3. By multiplying this volume by the air density. a vo lume of air equal to 100x10=1000 cubic meters will pass through the ring each second. Thus. .7) An airstream moving through a turbine rotor disc cannot give up all of its energy to the blades because some kinetic energy must be retained in order to move the airstream away from the disc area after interaction. a turbine rotor will never extract 100 % of the wind's energy. there are frictional effects. In addition. There are some new parameters to be introduced into calculations in order to express the system efficiency. E = 12 ⋅ ? ⋅ A ⋅ V 3 (3. energy per unit of time is equal to power (1 W = 1 j/s). P = 12 ⋅ ? ⋅ A ⋅ V 3 (3.5) where ρ : Air density (kg/m3 ) A : Rotor disk Area (m2 ) V : Wind velocity (m/s) Consequently the kinetic energy formula becomes.6) However. Mass of air per second = air density x volume of air passing each second = air density x area x length of cylinder of air passing each second = air density x area x velocity m= ?⋅A⋅V (3.24 In other words. which produce heat losses. so above formula is also the expression for the power in the wind. 2. to express the power output of the turbine.25 3. By including the losses.5. POWER COEFFICIENT The ability of a turbine rotor to extract the wind's power depends upon its "efficiency".8) . the power theoretically extracted by the wind turbine can be described by Equation (3. a non-dimensional power co-efficient Cp is included. Figure 3.5 Wind flow through a wind turbine The difference in the wind velocity is a measure for the extracted kinetic energy which turns the rotor and at the opposite end of the drive train. the connected electrical generator.1. Thus.8). P= ? ⋅ C ⋅ η ⋅ A ⋅ V13 2 p (3. rotors reduce the wind velocity from the undisturbed wind speed V1 far in front of the rotor to a reduced air stream velocity V2 behind the rotor as shown in Figure 3. Also. As the air stream interacts with the rotor disc and power is extracted. P = 2 ⋅ ? ⋅ η ⋅ A ⋅ V13 ⋅ a ⋅ (1 − a 2 ) (3. by substitution. Thus.9) where "a" is the dimensionless axial interference factor.26 where ? Cp η : Air density (kg/m3 ) : : : : Non-dimensional power coefficient Mechanical / Electrical efficiency Rotor disk area (m2 ) Undisturbed wind velocity in front of the rotor (m/s) A V1 This describes the fraction of the wind's power per unit area extracted by the rotor. Equation (3.9) expresses the power using the axial interference factor. the power co-efficient Cp may be defined as. a.593.33. Cpmax = 16/27 = 0. This is the ratio of the upstream to the downstream wind speed. Thus.10) with respect to a. . the air stream speed is reduced by an amount described by the axial interference factor. the maximum value of Cp occurs when a = 0. C p = 4 ⋅ a ⋅ (1 − a 2 ) (3. governed by the aerodynamic characteristics of the rotor and its number of blades.10) By differentiating (3. The rotation speed in rpm is usually symbolized by nr and the angular velocity in rad/s is by ? r.1 Speed Definitions Definition Rotatio nal Speed Angular Speed 1 rpm = Symbol nr ?r 2⋅ π rad/s = 0.2. (Boyle. Table 3. which is the tangential velocity of the rotor at the tip of blades. V. r. .2. It is the product of the angular velocity. Likewise. TIP SPEED RATIO The speed of rotation of a wind turbine is usually given in either revolutions per minute (rpm) or radians per second (rad/s).11) By dividing the tip speed. it is necessary to match the angular velocity of the rotor to the wind speed in order to obtain maximum efficiency. Alternatively. which is usually symbolized by λ is obtained. U. p. 1996. U= 2⋅ π⋅ r ⋅ n r 60 (3. it can be defined as. at the upstream of the rotor. Therefore. This ratio provides us with a useful measure with which to compare wind turbines of different characteristics.27 3. measured in meters per second. of the rotor and the tip radius. it will allow wind to pass unperturbed through the gaps between the blades. a rotor turning very rapidly will appear as a solid wall to the wind. U. the very useful non-dimensional ratio known as the tip speed ratio. ? r. by the undisturbed wind velocity.10472 rad/s 60 Unit rpm rad/s Another measure of a wind turbine’s speed is its tip speed.283) If a rotor turns very slowly. λ= Blade Tip Speed ωr ⋅ r U = = Wind Speed V V (3. For an n -bladed rotor.13). Note that this factor arises from the full aerodynamic theory of wind power extraction. 2⋅π n ⋅ ωr tb = (3.13) where tb ?r n : Time period for the blade to move its predecessor’s position (sec) : Angular speed of the blade tip (rad/s) : Number of blades . known as the tip speed ratio.28 The relationship between the wind speed and the rate of rotation of the rotor is characterized by a non-dimensional factor. EFFECT OF THE NUMBER OF BLADES The optimum tip speed ratio may be inferred however by relating the time taken for the disturbed wind to re-establish itself tw. λ.12) where r ?r U V : Rotor radius measured at the blade tip (m) : Angular speed of the blade tip (rad/s) : Blade tip speed (m/s) : Wind Speed (m/s) 3. to the time taken for a blade of rotational frequency omega to move into the position occupied by its predecessor tb.3. given by Equation (3.2.12). the time period for the blade to move to its predecessor's position is given by Equation (3. then some wind is unaffected. then some wind is not allowed to move through the rotor). and may be characterized by calculating the optimum tip speed ratio by Equation (3. the rotor must turn at a frequency which is related to the speed of the oncoming wind. tw = d V (3.16). 2⋅π r  ⋅  n d λ0 ≈ (3.16) . For this case.14) where tw d V : Time period for the wind to return to normal (sec) : Length of disturbed air stream (m) : Wind Velocity (m/s) Maximum power extraction occurs when these time periods are equal (If tb exceeds tw.14). This rotor frequency decreases as the radius of the rotor increases. Equation (3. If tw exceeds tb. then the time for the wind to return to normal is given by Equation (3.29 If the length of the strongly disturbed airstream upwind and downwind of the rotor is d. n ⋅ ωr 2 ⋅ π ≈ V d where ? r : Angular speed of the blade tip (rad/s) n d V : Number of blades : Length of disturbed air stream (m) : Wind velocity (m/s) (3.15) Therefore.15) applies. for optimum power extraction. 1996). the . which require a constant generator frequency in order to supply electricity of a fixed frequency.dmu. for a two-bladed rotor.30 where λ0 r n d : Optimum tip speed ratio : Blade tip radius of rotation (m) : Number of blades : Length of disturbed air stream (m) If we substitute a constant k for the term (r/d). Most modern horizontal axis wind turbine rotors consist of two or three thin blades. Figure 3.17) Thus.iesd. This arrangement gives a relatively high tip speed ratio in comparison to rotors with a high number of blades (such as those used in water pumps.uk/wind_energy/m32extex. These are known as "low solidity" rotors. which require a high starting torque). then the optimum tip speed ratio is defined by Equation (3. (De Montfort Universityhttp://www.html. 4⋅ π n λ0 ≈ (3. which practical results have shown to be approximately 2 for an n bladed machine. However. this is in conflict with the requirements of most generating systems. Thus. If the aerofoil is carefully designed.17). the optimum tip speed ratios may be about 30% above these values.6 shows the relationship between rotor efficiency (C p ) and the tip speed ratio for a typical wind turbine. the maximum power extracted from the wind (at Cpmax ) occurs at a tip speed ratio of about 6. This minimizes the size of the gearbox required and increases efficiency. and for a four-bladed machine at a tip apeed ratio of about 3. it is necessary for the rotor to speed up in order to remain near the optimum tip speed ratio. and gives an optimum match to the frequency requirements of modern electricity generators. as wind speed increases. due to the low fraction of the swept area which is solid.ac. Figure 3. the objective is to operate near maximum efficiency. Some manufactures are producing variable speed turbines (where the rotor speeds up with the wind velocity).6 Power coefficient versus tip s peed ratio for a constant speed wind turbine The alternative is to decouple the generator from the grid by an intermediate system which facilitates variable speed operation. where the resulting target power can be expressed as. Pt arg et ? = ⋅ C p ⋅ η ⋅ A ⋅ C p . in order to maintain a tip speed ratio near the optimum.18) . These turbines utilize electronic inverter/rectifier based control systems to stabilize the fluctuating voltage from the turbine before feeding into the grid supply.t arget 2  r ⋅  λ t arg et    ⋅ ω3 r   3 (3. For a variable-speed turbine.31 wind turbine which has a generator directly coupled to the grid operates for much of the time with a tip speed ratio which is not optimized. but the wind speed varies over a wide range.7 illustrates the Cp-λ relationship for a variable-speed wind turbine at different pitch angles. the operating point is rarely.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 TSR Figure 3.25 0.18) and Figure 3.15 0. The rotor speed varies by only a few percent.45 0.32 where ? Cp η target : Air density (kg/m3 ) : Power coefficient target : Mechanical / Electrical efficiency : Rotor disk area (m2 ) : Rotor radius measured at the blade tip (m) : Angular speed of the blade tip (rad/s) : Tip speed ratio target A r ?r λtarget 0. at λ for Cpmax .20 0. It is apparent from Equation (3.40 0.10 0. and randomly.7 Power coefficient versus tip speed ratio for a variable speed wind turbine for different pitch angles from 0 to 15 degrees by 0.7 that the power at any wind speed is .5 degree increments Figure 3.35 0. For constant-speed turbines.05 0. only one of the curves will be valid and an attempt is made to design the rotor blades to operate near maximum efficiency (Cpmax ) at wind speeds that occur most frequently at the design site. Therefore.30 Cp 0. 33 maximized by operating near the tip-speed ratio which results in the maximum power coefficient.3. DC generators are dc machines used as generators. The magnitude of the current is being increased with the strength of the field.1. THEORY: DC machines convert mechanical power to dc electric power.3. (Chapman. For a variable-speed turbine. 3. If this wire forms a circuit. D. Since this mechanism is called commutator. There is no real difference between a generator and a motor except for the direction of power flow. 3.C. the rotor speed sho uld be adjusted proportionally.3. 1999. so inducing an electric potential difference in the wire. the length of wire cut by the field and the relative velocity.566) .1. dc machinery is also known as commutating machinery. p. GENERATORS 3. Most dc machines are like ac machines in that they have ac voltages and currents within them – dc machines have a dc output only because a mechanism exists that converts the internal ac voltages to dc voltages at their terminals. this means that as the wind speed changes.1. and vice versa. GENERATOR THEORY All generators produce electricity by Faraday Law of electromagnetic induction: A magnetic field cuts a wire with a relative velocity. their generating systems may be classed as follows. then an electrical current is produced. Of the wind turbine systems currently being manufactured. EA = K ⋅ Φ ⋅ ω (3.20) where K is a constant representing the construction of the machine.8 The equivalent circuit for DC motors In Figure 3. p.508) The internal generated voltage in a DC machine is given by Equation (3.19).19) where ‘Z’ is the total number of conductors and ‘a’ is the number of current paths in the machine. are represented by inductor LF and resistor RF.8. This simpler form is. 1999. The brush voltage drop is represented by a small battery Vbrush opposing the direction of current flow in the machine. . interpoles and compensating windings. This representation is really the Thevenin equivalent of the entire rotor structure. (Chapman. The separate resistor Radj represents an external variable resistor used to control the amount of current in the field circuit. which produce the magnetic flux in the generator. Z⋅P ⋅ Φ ⋅ω 2⋅π⋅ a EA = (3. This equation is sometimes rewritten in a simpler form that emphasizes the quantities that are variable during machine operation. The field coils. if present. including rotor coils.34 Figure 3. the armature circuit is represented by an ideal voltage source EA and a resistor RA. both a shunt and a series field are present. power ratings. and their effects are additive.21) Equations (3.35 The induced torque developed by the machine is given by. the field flux is derived from a separate power source independent of the generator itself. 1999. but their effects are subtractive. 3. p. Τind = K ⋅ Φ ⋅ I A (3. Series Generator: In a series generator. and therefore in the applications to which they are suited.22). DC generators are compared by their voltages. 4. both a shunt and a series field are present. efficiencies. and voltage regulations. Cumulatively Compounded Generator: In a cumulatively compounded generator.21). are all the tools necessary to analyze the behavio ur and performance of a dc motor.20) and (3.22) . the field flux is derived by connecting the field circuit directly across the terminals of the generator. 2. These various types of dc generators differ in their terminal (voltage-current) characteristics. the field flux is produced by connecting the field circuit in series with the armature of the generator. Voltage regulation (VR) is defined by Equation (3. Differentially Compounded Generator: In a differentially compounded generator. the Kirchhoff’s Voltage Law equation of the armature circuit and the machine’s magnetization curve. classified according to the manner in which their field flux is produced: 1.508) There are five major types of dc generators. Separately Excited Generator: In a separately excited generator. Shunt Generator: In a shunt generator. Vnl − Vfl × 100% Vfl VR = (3. (Chapman. 5. or even an electric motor. Since the speed of the prime mover affects the output voltage of a generator. In these systems.1. (Chapman. This may be obtained by the use of a rotating permanent magnet or by rotary excitation using a current fed via so-called brushes and slip-rings. 3. which is usually called the prime mover of the generator.load terminal voltage of the generator. sometimes via a voltage regulator which smoothes out fluctuations in the generated voltage. it is customary to compare the voltage regulation and output characteristics of different generators. This current (whose magnitude depends upon the number of turns in the windings. 1999.2.load terminal voltage of the generator and Vfl is the full. All generators are driven by a source of mechanical power. the strength of the magnetic field and the speed of rotation) is drawn off from the commutator through graphite brushes and fed directly to the battery. known as a rotary field. A prime mover for a dc generator may be a wind or steam turbine. In stationary conductors—the stator . SYNCHRONOUS AC MACHINES (ALTERNATORS) AC generators employ a rotary magnetic field. The rotation causes an electric current to be set up in the rotor windings as the coils of wire cut through the magnetic field. the rotating generator shaft (connected to the turbine blades either directly or through a gearbox) turns the rotor within a magnetic field produced by either the field coil windings or by an arrangement of permanent magnets on the armature. DC GENERATOR APPLICATIONS IN WIND TURBINES Small scale stand-alone wind turbines are the most commonly used to charge batteries at relatively low voltages. They use simple DC generators. and a negative voltage regulation means a rising characteristic.36 where Vnl is the no.3.3. assuming constant-speed prime movers.2. and since prime movers can vary widely in their speed characteristics.566-567) 3. pp. a diesel engine. It is a rough measure of the shape of the generator's voltage-current characteristic—a positive voltage regulation means a drooping characteristic. In these synchronous generators. The voltage is dependent on the construction of the generator. and in isolated and stand-alone operation can be regulated by varying the excitation.1. a dc current is applied to the rotor winding. in synchronous machines. This rotating magnetic field induces a three-phase set of voltages within the stator windings of the generator. In general.2. the speed of rotation of the rotary field. coils are set (spatially) at e. have a damping effect.3. If the three-phase alternating current stator of a generator is supplied with alternating current from the grid. The synchronous generator is used to produce the vast majority of electric power used throughout the world. The rotor of the generator is then turned by a prime mover. When connected to the public supply. Two terms commonly used to describe the windings on a machine are field windings and armature windings.g. both voltage and frequency are dictated by the grid. The term synchronous refers to the fact that this machine's electrical frequency is locked in or synchronization with its mechanical rate of shaft rotation. 3. 1999. THEORY A synchronous generator or alternator is a device for converting mechanical power from a prime mover to AC electric power at a specific voltage and frequency. 120° intervals or an integral multiple thereof.316) In a synchronous generator. the term "field windings" applies to . the excitation and the load characteristics. (Chapman. it also sets up a rotary field. This excites currents in the rotor windings of the generator. which produces a rotor magnetic field. producing a rotating magnetic field within the machine. These currents cause torques on the rotor. which. p.37 windings of the generator—such rotary fields excite electric currents that vary with the frequency of rotation. which vary with a frequency corresponding to the difference between the field rotation frequency and the mechanical speed of rotation. 9 A salient six-pole rotor for a synchronous ma chine . Similarly. the field windings are on the rotor.38 the windings that produce the main magnetic field in a machine. a non-salient pole is a magnetic pole constructed flush with the surface of the rotor. so the terms "rotor windings" and "field wind ings" are used interchangeably. pp.250-252) Figure 3. (Chapman. The term salient means "protruding" or "sticking out" and a salient pole is a magnetic pole that sticks out from the surface of the rotor. Non-salient pole rotors are normally used for two. while salient-pole rotors are normally used for rotors with four or more poles. and the term "armature windings" applies to the windings where the main voltage is induced. the terms "stator windings" and "armature windings" are used interchangeably. For synchronous machines. 1999.and four-pole rotors. The rotor of a synchronous generator is essentially a large electromagnet. The magnetic poles on the rotor can be of either salient or non-salient construction. On the other hand. THE ROTATION SPEED OF A SYNCHRONOUS GENERATOR Synchronous generators are by definition synchronous. A synchronous generator’s rotor consists of an electromagnet to which direct current is supplied. 1.2. Now.23) . 2. Supply the DC power from an external DC source to the rotor by means of slip rings and brushes.2. meaning that the electrical frequency produced is locked in or synchronized with the mechanical rate of rotation of the generator.10 A non-salient two -pole rotor for a synchronous machine A DC current must be supplied to the field circuit on the rotor. a special arrangement is required to get the DC power to its field windings.3. 3. There are two common approaches for supplying this DC power.39 Figure 3. Since the rotor is rotating. fe = nm ⋅ p 120 (3. Supply the DC power from a special DC power source mounted directly on the shaft of the synchronous generator. The rotor magnetic field points in whatever direction the rotor is turned. the rate of rotation of the magnetic fields in the machine is related to the stator electrical frequency by. but the flux itself depends on the current flowing in the rotor field circuit. (Chapman.2. If ? is expressed in radians per second.3. this equation relates the speed of the rotor rotation to the resulting electrical frequency.40 where fe nm p : Electrical frequency (Hz) : Mechanical speed of the magnetic field (rpm) (equals the speed of the rotor for synchronous machines) : Number of poles Since the rotor turns at the same speed as the magnetic field.25) where K is a constant representing the construction of the machine. EA = K ⋅ Φ ⋅ ω (3. this equation is sometimes rewritten in a simpler form that emphasizes the quantities that are variable during machine operation.11 (a).3. then NC ⋅ p 2 K= (3. INTERNAL VOLTAGE OF A SYNCHRONOUS GENERATOR The magnitude of the voltage induced in a given stator phase is. The field current IF is related to the flux in the manner shown in Figure 3. pp. This simpler form is.254-255) 3. Since EA is directly proportional to the flux.26) The internal generated voltage EA is directly proportional to the flux and to the speed.24) In solving problems with synchronous machines. E A = 2 ⋅ π ⋅ NC ⋅ Φ ⋅ f (3. 1999. the internal generated voltage EA is related to the . The distortion of the air-gap magnetic field by the current flowing in the stator.255-256) There are number of factors that cause the difference between EA and VF . Figure 3.41 field current as shown in Figure 3. In fact.11 a. 1999. called armature reaction 2. (Chapman. this voltage EA is not usually the voltage that appears at the terminals of the generator. field current for synchronous generators b. However. the only time the internal voltage EA is the same as the output voltage VF of a phase is when there is no armature current flowing in the machine. The magnetization curve for synchronous generators The voltage EA is the internal generated voltage produced in one phase of a synchronous generator. The self inductance of armature coils 3. 1. pp.11 (b). Plot of flux vs. This plot is called the magnetization curve or the open-circuit characteristic of the machine. The resistance of armature coils 4. The effect of salient-pole rotor shapes . 3. and it is customary to combine them into a single reactance.27) In addition to the effects of armature reaction.28) The armature reaction effects and the self inductance in the machine are both represented by reactances.2. If the stator self inductance is called LA (and its corresponding reactance is called XA) while the stator resistance is called RA. XS = X + XA (3.42 3. called the synchronous reactance of the machine.12 A simple circuit for alternators The armature reaction voltage on a phase is.29) . THE EQUIVALENT CIRCUIT OF AN ALTERNATOR Figure 3. VΦ = E A − j ⋅ X ⋅ IA − j ⋅ XA ⋅ I A − R A ⋅ I A (3. VΦ = E A − j ⋅ X ⋅ I A (3. the stator coils have a self inductance and resistance.4. then the total difference between EA and VF is given by. the final equation describing VF is. respectively. the real and reactive powers that must be supplied are determined by the load attached to it. 1999. and the governor set points and field current control the frequency and terminal voltage. When the generator's windings overheat. the life of the machine can be severely shortened.316) . When the generator is connected to an infinite bus. and the maximum allowable heating in the field windings sets the maximum size of EA. there are two separate constraints on the generator. A synchronous generator's ability to produce electric power is primarily limited by heating within the machine.13 The per-phase equivalent circuit for synchronous generators The way in which a synchronous generator operates in a real power system depends on the constraints on it. The maximum allowable heating in the armature windings sets the maximum kilovoltamperes allowable from the machine. The maximum size of E and the A maximum size of IA together set the rated power factor of the generator. In real systems containing generators of approximately equal size. When a generator operates alone. VΦ = EA − j ⋅ XS ⋅ I A − R A ⋅ IA (3. its frequency and voltage are fixed. p. so the governor set points and field current control the real and reactive power flow from the generator. Since here are two different windings (armature and field).43 Therefore. (Chapman.30) Figure 3. and the field current affects both terminal voltage and reactive power flow. the governor set points affect both frequency and power flow. 3. whilst the rotor provides the magnetic field. The currents that produce the magnetic field are in shortcircuited loops. Alternators have a number of advantages. ensuring a grid-compatible output frequency despite small variations in wind speed. due to the use of slip rings rather than commutators. the field current in these loops is induced from currents in the stator windings. to which the generator is connected. If positioned on the stator. In practice. . If permanent magnets are used. the power is drawn from the alternator through fixed contacts and wear due to the passage of high currents through moving contacts is eliminated. power generation can only occur when the induced closed. to produce power which is in phase with grid supply. and vice versa. however. This is facilitated in one of three ways. • Reactive power is drawn from the live grid. ASYNCHRONOUS (INDUCTION) AC MACHINES An induction generator differs from a synchronous generator in that its rotor consists in its simplest form of an iron cylinder with slots on its periphery that carry insulated copper bars. In operational terms. In excited field alternators. in order to be compatible with a utility's grid supply. 3. which produce an AC voltage. were developed as a replacement for DC generators. More commonly. via slip rings. the generator output is small enough in relation to that of the utility supply to allow it to "lock-on" to the grid frequency. the magnetic field is provided by a supply of relatively low current to the field windings.3. this may be achieved by altering the pitch of the turbine rotor blades to alter their lift coefficient as the wind speed varies.loop field currents have been initiated and maintained.44 Early alternators. Thus. the machine must be driven at a constant speed by turbine rotors. These are short-circuited by rings which are positioned on the flat faces of the cylinder. A further design improvement is their incorporation of the armature windings in the stator. They are generally cheaper and more durable. 15 Cutaway diagram for a squirrel-cage induction machine 3.1. • A small synchronous generator may be run in parallel.14 Cutaway diagram for a wound-rotor induction machine Figure 3.45 • Capacitors connected between the output and the earth enable autonomous selfexcited generation (some residual magnetism in the system is necessary). for example) then provide power at times of inadequate wind.3. fuelled. EQUIVALENT CIRCUIT OF AN INDUCTION MACHINE An induction machine relies for its operation on the induction of voltages and currents in its rotor circuit from the stator circuit (transformer action). Figure 3. Because the induction of voltages and currents in the rotor circuit of an induction machine is . which may (if diesel.3. Therefore. The curve of magnetomotive force versus flux (magnetization curve) for this machine is compared to a similar curve for a power transformer in Figure 3. Because an induction machine does not have an independent field circuit. the flux in the machine is related to the integral of the applied voltage E1 .17. which greatly increases the reluctance of the flux path and therefore reduces the coupling between primary and secondary windings. representing the operation of an induction machine. 1999.16. . which must be represented in the equivalent circuit of the machine. The higher reluctance caused by the air gap means that a higher magnetizing current is required to obtain a given flux level. An induction machine is called a singly excited machine (as opposed to a doubly excited synchronous machine).46 essentially a transformer operation. its model will not contain an internal voltage source such as the internal generated voltage EA in a synchronous machine. the equivalent circuit of an induction machine will turn out to be very similar to the equivalent circuit of a transformer.365) A transformer per-phase equivalent circuit. This is because there must be an air gap in an induction machine. Like any transformer. like any transformer with an iron core.inductance in the primary (stator) windings. Notice that the slope of the induction machine's magnetomotive forceflux curve is much shallower than the curve of a good transformer. Also. is shown in Figure 3. since power is supplied to only the stator circuit. The stator resistance will be called as R1 and the stator leakage reactance will be called as X1 . p. there is a certain resistance and self. It is possible to derive the equivalent circuit of an induction machine from the knowledge of transformers and the variation of rotor frequency with speed in induction machines. the magnetizing reactance Xm in the equivalent circuit will have a much smaller value (or the susceptance Bm will have a much larger value) than it would in an ordinary transformer. (Chapman. These two components appear right at the input to the machine model. The voltage ER produced in the rotor in turn produces a current flow in the shorted rotor (or secondary) circuit of the machine.47 Figure 3. Figure 3.16 Transformer model for an induction machine The primary internal stator voltage E1 is coupled to the secondary ER by an ideal transformer with an effective turns ratio aeff . An induction machine equivalent circuit differs from a transformer equivalent circuit .17 Magnetization curve for an induction machine compared to that for a transformer The primary impedances and the magnetization current of the induction machine are similar to the corresponding components in a transformer equivalent circuit. .31) and the frequency of induced voltage at any slip will be given by Equation (3. so the largest voltage and rotor frequency are induced in the rotor at that condition.1.32) This voltage is induced in a rotor containing both resistance and reactance. 1999. the magnitude of the induced voltage at any slip will be given by Equation (3. The smallest voltage (0 V) and frequency (0 Hz) occur when the rotor moves at the same speed as the stator magnetic field. the greater the resulting rotor voltage and rotor frequency. The magnitude and frequency of the voltage induced in the rotor at any speed between these extremes is directly proportional to the slip of the rotor. The largest relative motion occurs when the rotor is stationary.3. In general. when the voltage is applied to the stator windings. the rotor reactance is given by. resulting in no relative motion. the greater the relative motion between the rotor and the stator magnetic fields. (Chapman. With a rotor inductance of LR. Therefore. called the locked-rotor or blocked-rotor condition. (Chapman. E R = s ⋅ E R0 (3. if the magnitude of the induced rotor voltage at locked-rotor conditions is called ER0. a voltage is induced in the rotor windings of the machine.366-367) 3. ROTOR CIRCUIT MODEL In an induction machine.3.367) The reactance of an induction machine rotor depends on the inductance of the rotor and the frequency of the voltage and current in the rotor. p.32).1. pp.48 primarily in the effects of varying rotor frequency on the rotor voltage ER and the rotor impedances RR and jXR. while the rotor reactance XR is affected in a more complicated way by slip. fr = s ⋅ fe (3.31). The rotor resistance RR is a constant (except for the skin effect). 1999. independent of slip. 34) where XR0 is the blocked-rotor rotor reactance.19 The rotor circuit model with all the frequency (slip) effects concentrated in resistor RR .33).33) Substituting Equation (3. X R = 2 ⋅ π ⋅ s ⋅ f e ⋅ LR X R = s ⋅ (2 ⋅ π ⋅ f e ⋅ L R ) X R = s ⋅ X R0 (3.49 X R = ωr ⋅ LR = 2 ⋅ π ⋅ f r ⋅ LR (3.18 The rotor circuit model for induction machines Figure 3.32) into Equation (3. Figure 3. In an ordinary trans former. I2 = IR a eff (3.3. E1 = E′ = a eff ⋅ E R 0 R (3.3. Exactly the same sort of transformation can be done for the induction machine’s rotor circuit. The rotor circuit model that will be referred to the stator side is shown in Figure 3. the voltages. FINAL EQUIVALENT CIRCUIT To produce the final per-phase equivalent circuit for an induction machine. it is necessary to refer the rotor part of the model over to the stator side. which has all the speed variation effects concentrated in the impedance term.50 3.19. then the transformed rotor voltage becomes. currents and the impedances on the secondary side of the device can be referred to the primary side by means of the turns ratio of the transformer: ′ Vp = Vs = a ⋅ Vs ′ 1 I p = I s = ⋅ Is a ′ Zs = a 2 ⋅ Zs (3.35) where the prime refers to the referred values of voltage.36) and the rotor current becomes. If the effective turns ratio of an induction machine is aeff.37) and the rotor impedance becomes .2. current and impedance.1. 39) Figure 3. (n s − n r ) ns s= (3.40) where ns nr : Electrical speed of the magnetic field (or stator speed) (rpm) : Rotor mechanical speed (rpm) . depending on the speed of the wind.20 The per-phase equivalent circuit for induction machines In wind energy conversion systems. or as a motor (acting as a sink of power from the grid). In either case. and may be expressed as.38) so R 2 = a2 ⋅ R R eff 2 X 2 = a eff ⋅ X R 0 (3. supplying power to the grid. This is known as generator slip.51 R  2 Z2 = a eff ⋅  R + jX R 0   s  (3. the generator may act either as a generator. there will be a difference in speed between the shaft speed nr and the output ns. the generator will over-speed.e. This torque is known as the pushover torque of the ge nerator. the direction of its induced torque will reverse and it will act as a generator.436) . p. (Chapman. 1999. 1999. and positive when acting as a motor. There is a maximum possible induced torque in the generator mode of operation.369-370) Figure 3.52 The slip is defined as negative when the machine is acting as a generator. If a torque is applied to the shaft of the induction generator which is greater than the pushover torque. wind). As the torque applied to its shaft increases. the amount of power produced by that generator increases.21 shows that.21 Torque -Speed curve for a MW-size induction machine The torque-speed characteristic curve in Figure 3. (Chapman. if an induction motor is driven at a speed greater than synchronous speed by an external effect (i. pp. constant. an induction generator cannot control its own output voltage. In fact. (Gipe. As long as the machine's speed is some value greater than synchronous speed for the power system to which it is connected.437) Wind machines driving electrical generators operate at either variable or constant speed. power.211) Small wind turbines typically operate at variable speed. In constantspeed machines. This external source of reactive power must also control the terminal voltage of the generator—with no field current. 1995. and an external source of reactive power must be connected to it at all times to maintain its stator magnetic field. This simplifies the turbine’s controls while improving aerodynamic performance. Because it lacks a separate field circuit. The electricity they produce is incompatible with the constantvoltage. p. both the voltage and frequency vary with wind speed. heat recovery systems. the greater its resulting output power. p. (Chapman. 1999. it consumes reactive power. An induction generator does not need a separate field circuit and does not have to be driven continuously at a fixed speed. The one great advantage of an induction generator is its simplicity.factor correction can be provided by capacitors. The greater the torque applied to its shaft (up to a certain point).frequency alternating current (AC) produced by the utility. and similar supplementary power sources attached to an existing power system. and the generator's terminal voltage can be controlled by the external power system. it will function as a generator. Normally. despite changes in wind speed. The fact that no fancy regulation is required makes this generator a good choice for windmills. When these small wind machines drive an induction generator. but can . the generator's voltage is maintained by the external power system to which it is connected. an induction machine has severe limitations. rotor speed varies with wind speed. an induction generator cannot produce reactive power. rotor speed remains relatively constant. In variable-speed operation.53 As a generator. In such applications. If a grid-connected turbine is fitted with an AC generator. or it can be rectified to direct current (DC) for charging batteries. Induction generators have two advantages over alternators. ensuring that the generator's output frequency is locked to that of the utility and so controlling the rotor speed within limits. most operate the rotor at or near constant speed. the frequency of which varies directly with the speed of the rotor and indirectly with the number of poles in the . Although some manufacturers of medium-sized wind turbines build variablespeed turbines. Many commercial gridconnected turbines use induction AC ge nerators. For AC generators. a critical design factor. must be considered. Synchronous generators produce electricity in synchronization with the generator's rotating shaft frequency.voltage 50 or 60 Hz AC like that of the utility. To generate utility-compatible electricity. whose magnetizing current is drawn from the grid. the rotor speed of grid-connected turbines must exactly match the utility supply frequency. Thus.54 be used as is for resistive heating or pumping water at variable rates. • They can supply utility-compatible electricity without complicated controls. variable-speed turbines typically use a form of synchronous inverter to produce constant. • They are inexpensive. that is synchronous speed. AC generators produce alternating current. These machines produce utility-compatible power directly via induction (asynchronous) generators. Although it is possible to use rotary inverters for this task. this must produce power that is in phase with the utility's grid supply. the output from a variable-speed generator must be conditioned. Most of these inverters use the utility’s alternating current as a signal to trigger electronic switches that transfer the variable-frequency electricity at just the right moment to deliver 50 or 60 Hz AC at the proper voltage. Table 3.55 generator. frequency increases with increasing generator speed. Induction generators are readily available in a range of sizes and are easily interconnected with the utility. or 20 to 50 rpm on a 1000rpm generator.2 Common Synchronous Speeds for Generators Pole Number 4-pole 6-pole Europe (50 Hz) 1500 rpm 1000 rpm North America (60 Hz) 1800 rpm 1200 rpm An induction generator begins producing electricity when it is driven above its synchronous speed which is generally 1000 or 1500 rpm in Europe (1200 or 1800 rpm in North America). This continues until the generator reaches its limit. As torque increases.sized wind turbines use induction generators almost exclusively. which is about 5 % greater than its synchronous speed. . Medium. For a given number of poles. 120 ⋅ f p n = s (3. generator speed increases 2 to 5 %. the magnetic field in the induction generator also increases. As torque increases. This increase of 1 to 3 rpm in rotor speed is imperceptible in a wind turbine operating at a nominal speed of 50 rpm.41) where ns f p : : : Synchronous or stator speed (rpm) Grid frequency (Hz) Number of poles Manufacturers should decide the number of poles of the generator (for either synchronous or asynchronous) for optimum conditions. Induction generators are not true constant-speed machines. 3. • Multiple or dual (two speed) generators. the generator will operate 97 % of the time at less than rated capacity and about half the time at less than 100 kW. At a site with an average wind speed of 7 m/s. Whilst the induction machine is now well established as the most popular generator for reliable. low-cost power production from the wind. • Induction machines with variable generator rotor resistance.1. DUAL GENERATORS Generators operate inefficiently at partial loads. pp. the efficiency falls nearly 15 % (from 95 % at rated output) when a 500-kW wind turbine is operated at 100 kW. Thus both . it switches off and the main generator switches on instead. 3.4. The small generator operates at nearly full load in low to moderate winds.fifth to one-third of the main generator. The 'traditional' Danish design of wind turbine is fixed-speed. For example. in a 500-kW wind turbine.212-213) Efficiency drops off rapidly when the generator is operated at less than one-third its rated value. designers of constant-speed wind turbines often use dual generators or dual windings: One main generator and a small generator having the capacity from one. where the generator is designed to reach its rated capacity at a wind speed of 16 m/s. the generator operates at partial load much of the time. 1995. (Gipe. efficient. using an induction generator. RECENT DEVELOPMENTS IN GENERATORS FOR WIND TURBINES As well as applying to the basic process of energy conversion. other designs of machines are used and there are several "drivers" for change.56 3. To avoid this.3. For example. technological development also relates to the design and size of machines used for the generation of electric power from wind energy. Variations on this theme which are now appearing include.4. When the wind speed reaches the rated wind speed of the small generator. The two generators may be in tandem and driven by the same shaft or they can be side by side. 1995. rated power will quadruple. (Gipe. (Gipe. This could add perceptibly to the improved performance of larger turbines over that of their smaller predecessors. can capture most of the efficiency advantages of variable-speed turbines. Because a generator’s power is proportional to its volume.213) The advantage of one single generator with dual windings becomes problematical as turbines grow ever more powerful. During the mid-1990s. while incapable of taking the full advantage of the optimum tip-speed ratio over the entire operating range. As turbine size increases. For a doubling of wind turbine diameter. larger generators are also more efficient than smaller ones.2. p. which is closely related to . The use of dual generators permits the turbine to operate at two speeds. while losses are proportional to its surface area. At many sites. the relative cost of the gearbox becomes more important. the gearbox is needed for the generator frequency to catch grid frequency for grid-connected systems. The generator operates on 6 poles during light winds and uses 4 poles in higher winds. and rotor torque. with the small generator driven by belts from the main generator. acoustic noise and reliability problems. DIRECT-DRIVE GENERATORS In fact. Removing the gearbox could save not only cost. p. most new constant-speed turbines used one generator with dual windings. at only a small increase in cost for the extra windings.3. the small generator will operate more than 50 % of the total generating time. but also mass. enables designers to drive the rotor at a higher aerodynamic efficiency over a broader range of wind speeds than with only one generator. losses. Dual-speed wind turbines.57 generators operate more efficiently then either one alone. although it delivers less than half the total generation. 1995.4.214) 3. GRID INTEGRATION With regard to the transfer of energy to electrical supply installations. requiring large-diameter ring generators with numerous poles. On mid-1990s. lower rotor speeds make the design of direct-drive generators problematic.4.58 gearbox cost. some manufacturers successfully developed gearless wind turbines. Due to its very high output capacity (in comparison with the nominal values of the consumers connected to it). we must differentiate between. As rotor diameter increases. rotor speed decreases. the so-called rigid combined grid can be regarded both as an infinitely rich source of active and reactive current and. will increase by a factor of eight. they use low speed multi-pole generators directly connected to the blade shaft. Direct designs have the maintenance and operation advantage as compared to the usage of gearboxes. that either operate in isolation or supply weak grids. The large dimensions of these multi-pole generators lead to a certain transportation disadvantage especially in the megawatt class.level energy . 3. • Unlimited capacity connection with the rigid grid. For example. So. Another important issue is the integration of the generator into overall nacelle design. Wind energy converters should give reliable operation in both operations. Instead of using a gear with a high transmission ratio. an existing Darrieus type turbine uses a 162-pole synchronous generator coupled directly to the vertical axis turbine’s torque tube. for the low. • Systems with limited supply options. particularly in variable-speed systems. or even approach its level.or self-commutated inverters. wind turbines are usually installed at remote sites with limited supply options. 1998. (Heier. which are sometimes long. controlled or machine-commutated rectifiers. In large wind energy converters and wind parks. This type of system also requires a frequency-converter system that is capable of supplying the variable-frequency electrical energy from the turbine generator to a grid of (almost) constant frequency and voltage.59 supply devices that wind power plants usually represent. However. supply power can reach the same order of magnitude as grid transfer power. the compatibility of the plant to the environment and the grid can be improved. ring-type designs with noncontrollable. often based on gearless. leading to a higher energy output and reduced drive-train loading. p. synchronous machines are also popular. which means that mutual influences must be taken into account. (Heier.1. are the most common solution for the conversion and control of electrical energy.183) . p. FREQUENCY CONVERTER SYSTEMS Electronic power frequency converters.183) 3. 1998. mainly characterized by stall-controlled turbines with asynchronous generators and direct connection to the grid. They are also used to an increasing degree in wind energy converters to adjust the generator frequency and voltage to those of the grid. so-called power converters. The increased cost of such systems is justified if.4. rather than more expensive units. 1998. by adjusting the turbine speed to the prevailing wind speed. (Heier. direct-current intermediate circuits and grid.181) There is currently a clear trend in favor of robust single systems. as a sink of unlimited capacity and constant voltage and frequency. Unlike thermal power plants. Therefore a weak grid connection is often made using stub cables. p. 22 Electrical energy conversion by power converters Rectifiers convert alternating or three-phase current into direct-current.loss energy conversion • Rapid engagement and high dynamic ratio • Wear-free operation • Low maintenance requirement • Low volume and weight Figure 3. with the electrical energy flowing from alternating or three-phase current systems into directcurrent systems. The energy flows into the alternating-current side. Inverters convert direct-current into alternating or three-phase current. namely. . • Low.60 Power converters have significant advantages over the rotating transformers based on groups of mechanical components and the mechanical commutators that were common in the past. which carries the electrical power. frequency and number of phases is converted for use in an alternating-current system with a different voltage. As wind power plants are almost always fitted with three-phase current generators. • Intermediate circuit frequency converters. Here. protective and regulating tasks. which performs numerous control. In alternating-current conversion. The main components of current-conversion systems are the power section. with so-called power converter valves. only three-phase current converters are relevant for power conditioning. the operating range 0-25 Hz is preferred. Direct frequency converters are used particularly for the reduction of frequency. • Direct frequency converters. and an electronic signal processing unit. it must be differentiated that. Direct frequency converters require two complete anti-parallel power conversion bridges per phase to operate the consumer and supply systems. . This results in high costs for power gates and control elements. In the case of supply from or to a 50 Hz grid.61 Direct-current conversion is the conversion of direct-current with a given voltage and polarity for use in a direct-current system with a different voltage and possibly reversed polarity. frequency and possibly a different number of phases. alternating-current of a given voltage. and one with a direct voltage intermediate circuit as a U frequency converter.185) Indirect frequency converters consist of a rectifier. p. A frequency converter with a direct current intermediate circuit will be referred to as an I frequency converter.186) . (Heier.62 Figure 3. phase position and frequency required by the machine. 1998. (Heier. p. in a direct frequency converter takes place by the selection of voltage sections from the three phases and by triggering the power converter such that the voltage path after smoothing has the amplitude. direct current or direct voltage intermediate circuit and an inverter. 1998.23 Basic wiring diagram for direct frequency converters The conversion of grid frequency f 1 into machine frequency f 2 or vice versa. They consist of one or more power semiconductors. Indirect frequency converters have achieved a clear dominance in energy conversion and the connection of variable speed wind power plants to the grid.4. and .1. • The inductor for current smoothing in the I frequency converter.63 a. • The capacitor for voltage smoothing in the U frequency converter. Direct frequency converters were only used in individual cases to supply the rotor circuit of double-fed asynchronous generators. POWER SEMICONDUCTORS FOR FREQUENCY CONVERTERS So-called power converter valves are the main components of the power section of frequency converters.24 Indirect frequency converters Particular characteristics of the intermediate circuit are. U frequency converter Figure 3. 3.1. I frequency converter b. 1998.4.1. pp. Switchable thyristors and transistors.187) . Noncontrollable valves (diodes for example) conduct in the forward direction and block in the reverse direction.e. Its application is thus limited to use in uncontrolled rectifiers and for protective and back-up functions.186-187) 3. This is possible in the case of positive diode voltages. Thyristors can be switched on by their gate and block if the direction of the current is reversed. in the microsecond range).1. (Heier.1. p. so-called fast-recovery diodes with low storage charges are necessary to protect power converter valves from destruction by overvoltage. Controllable valves permit the selection of the moment at which conductivity in the forward direction begins. the diode becomes non-conducting and blocks the flow of current. another determining variable is conducting. particularly for protective functions. and therefore function primarily as switches. 1998.64 conduct electrical current in one direction only. For the effective protection of semiconductor components. Power converter valves can be either controllable or non-controllable. These valves generally alternate periodically between the electrically conductive and non-conductive states. can be switched on by one gate electrode and off by a second (or the same) gate. SEMICONDUCTOR DIODES Diodes consist of positively (p) and negatively (n) doped semiconductor material with a barrier layer between them that ensures current can flow in one direction only.state dynamic behaviour. for example as a recovery diode in direct-current circuits or similar circuit elements. If the current direction and voltage are reversed. (Heier. In addition to limit values for current and voltage in the forward and reverse directions. As there is no need to operate any mechanical contacts. on the other hand. and thermal behaviour. these can initiate and/or terminate current conduction very rapidly (i. 4.187) 3. abbreviated to MCT.65 3. GTO thyristors and MCTs are the main types used in frequency converters. in conventional thyristors. The transition from blocking to conducting state is initiated by the supply of a power impulse to the gate. p.1. Thyristors. Conventional thyristors. it is not possible to interrupt the current by intervention at the gate. and is known as the firing of the thyristor.3. Once triggered.1. it can be fired by a new current impulse or periodic impulse sequences at the gate. Mainly bipolar. Switchable thyristors do permit this.2. do not automatically go into a conducting state when an adjoining positive anode-cathode voltage is present. The best known type is the Gate-Turn-Off. However. THYRISTORS Thyristors are semiconductor components with four differently (p and n) doped layers. thyristors behave like diodes. behaves in a similar manner to the GTO thyristor. As valve components they function exclusively as switches. If a thyristor is in off-state.1. uninterrupted current requires a free-wheeling arm.1. A positive gate voltage switches it off.4. They remain in the conducting state as long as a current flows in the positive direction and the current does not fall below the component's minimum value. The MCT can be switched on almost without power by a negative voltage (in relation to the anode) at the gate. With these types of thyristors. the socalled holding current. unlike diodes. TRANSISTORS Transistors are semiconductor components with three differently (p and n) doped layers. The metal-oxide -semiconductor controlled thyristor. (Heier. and at null current it automatically switches to blocking operation. 1998. . or GTO thyristor. MOSFET and IGBT transistors are used in frequency converters. protective switches and potential divisions. they become conductive when a control current is passed through the base electrode. In order to achieve low on-state voltage. in their function as power semiconductors. particularly at high switching frequencies. IGBTs can be connected in parallel. however. MOSFETs are used in the lower-output range at high switching frequencies for combinational circuit components and frequency converters. However. The transistors therefore operate in the so-called saturation range. Different types of IGBTs are used as individual transistors or are connected together in modules of two to six transistors to form bridge connections. Much smaller control currents are needed for metal-oxide-semiconductor field effect transistors than those for bipolar transistors. and have advantages over bipolar transistors and IGBTs. IGBTs automatically limit current increases at the output. Particularly in the small and . Almost like switches. The development and availability of new power electronic semiconductor components has given a new impetus to power converter technology and its application in the field of drive and energy engineering. In more recent developments. transistors are operated with a relatively high base current. this requires that all transistors exhibit the same thermal behaviour. These MOSFETs can be switched almost without power. Integrated free-wheeling diodes protect the transistor in the off-state direction. Increasing the switching frequency causes increased currents and thus higher losses in the drive level. and thus low losses. When switched off. This. IGBTs (insulated gate bipolar transistors) combine the advantageous characteristics of MOSFETs and bipolar power transistors. the on-state of the transistor is terminated and the flow of current blocked. requires that the internal capacities of the transistor to be reloaded. This allows a high level of power amplification to be achieved. The field-effect transistor at the control input facilitates rapid switching at very low driving power. are usually used in emitter mode.66 Bipolar transistors (BPT). by voltage control at the gate. This results in good excess current and short-circuit behaviour. transistors are built into modules with driver switches. 3 Characteristics and Maximum Ratings of Switchable Power Semiconductors Rating BPT IGBT Component MOSFET MCT GTO Symbol Voltage (V) Current (A) Output (kVA) Turn-Off Time (µs) Frequency (kHz) Drive Requirement 1200 1700 (3300) 600 (1200) 360 1000 3000 4500 800 28 300 4000 480 14 450 4500 15 . cooling and protection. 1998.3 . new components have largely pushed transistors and GTOs out of the market. auxiliary devices and devices for commutation.2 – 1 Medium Low Low Low High 3.4.2.25 1–4 0.67 medium output range.5 – 5 2 – 20 5 – 100 1–3 0. CHARACTERISTICS OF POWER CONVERTERS The main components of power converters are the power converter valves and their electrical connections and trigger equipment.3 shows symbols.0. maximum ratings and characteristics of power semiconductors. energy storages.1. p. Table 3. (Heier. and usually also transformers. filtering.25 0.188) Table 3. Also necessary are circuit elements. .5 5 – 10 10 . If the live valve is turned off before the next valve is fired then the connection becomes temporarily dead. (Heier. (Heier. 1998. Thus the load. This interrupts the short circuit before the operating current is exceeded. Self-clocked power converters have an internal clock generator and are thus not dependent upon external frequency information. The pulse number is characterized by the number of sine peaks (pulses) of the unsmoothed direct-current. the transfer of current between the individual valves.68 Power converters must be run at their voltage and timed according to frequency. on the other hand.or machine-clocked power converter orientates itself to the load or machine voltage. As well as the commutation voltage and elementary frequency. In contrast. 1998. operate with forced commutation. They require a grid. it is possible to fire a second valve while the valve to be turned off is still live.191) . Externally clocked power converters take their control pulse from the system that they work in parallel with. this process is known as intermittent flow. pulse connections are normal for three-phase current systems. p. as well as twelve. the number of non-simultaneous conductive connections (commutations) from one valve to another within one cycle. Externally commutated power converters operate using natural commutation. The required reactive power is provided by capacitors. As ripples occur in direct-current. Three and six. can occur in different ways. p. This creates a temporary short-circuit between two alternating-current lines. Line clocking is the adjustment of the zero-crossings or phase intersections to the grid voltage. load or machine that specifies the voltage and can supply reactive power. This changeover is known as commutating operation. The internal function of power converters must also be differentiated with regard to the origin of the elementary frequency. The origin of the commutation voltage and commutation reactive power at the conductive connection to another valve is decisive for current carrying. The current in the valve to be turned off is quickly forced to be under its holding point. Selfcommutated converters. is an important parameter of power converter circuits.190) Commutation. the so-called pulse number. 1996. • Classification by axis of rotation • Classification by rotor speed • Classification by power control • Classification by location of installation 4. p. (Boyle. Classification categories can be arranged as.1. modern wind turbines come in two basic configurations: 1. Apart from a few innovative designs. CLASSIFICATION BY AXIS OF ROTATION As mentioned before. They range from small turbines that produce a few tens or hundreds of watts of power to relatively large turbines that produce 2 MW or more. Vertical Axis Wind Turbines The majority of modern wind turbines are electricity-generating devices.69 CHAPTER FOUR CLASSIFICATION OF WIND TURBINES Wind turbines can be classified in several ways due to there are more than one design criteria which affects turbine performance. Horizontal Axis Wind Turbines 2. modern windmills are usually referred to as wind turbines or wind energy conversion systems to distinguish them from their traditional na me.280) . due to wind turbine designers’ improved understanding of aerodynamics. These include the multi-blade wind turbines used for water pumping on farms. although only one is necessary. 1996. These are referred to as low solidity devices. (Boyle.280) They generally have either two or three blades or else a large number of blades.1 Horizontal and vertical axis wind turbines 4. derived largely from developments in aircraft wing and propeller design.70 Figure 4. They are almost universally employed to generate electricity. HORIZONTAL AXIS WIND TURBINES (HAWT) Modern low-solidity horizontal axis wind turbines evolved from traditional windmills and are by far the most common wind turbines manufactured today. They have a clean. the swept area of wind turbines with few blades is largely void and only a very small fraction appears to be ‘solid’. streamlined appearance.1. In contrast. . Wind turbines with large numbers of blades have what appears to be virtually a solid disc covered by solid blades and are described as high solidity devices. p.1. 71 The rotor axis of conventional wind turbines is seldom truly horizontal. Designers tilt the rotor axis slightly to provide more clearance between the blades and tower than with a truly horizontal driveline (i.e. 6°). (Gipe, 1995, p.175) Figure 4.2 Horizontal axis wind turbine configurations 4.1.2. VERTICAL AXIS WIND TURBINES (VAWT) Vertical axis wind turbines have an axis of rotation that is vertical, and so, unlike their horizontal counterparts, they can harness wind from any direction without the need to reposition the rotor when the wind direction changes. (Boyle, 1996, p.280) D.G.M. Darrieus invented the modern vertical axis wind turbine in the 1920s. The French engineer’s name has become synonymous with the “φ” or “eggbeater” 72 configuration, although he experimented with several designs, including a conventional two-bladed turbine. (Gipe, 1995, p.171) Figure 4.3 Vertical axis wind turbine configurations Vertical axis designs have an advantage of rotational symmetry that obviates any need for a yaw system. It was often a claimed advantage that all the drive train and power conversion equipment can be at ground level, but it was found that this implied a long and heavy torque tube for the main shaft and various designs compromised with gear boxes at the top of the main shaft. The overriding disadvantages, however, of the vertical axis design compared to horizontal axis are: • Inherently lower aerodynamic efficiency because the drive torque varies strongly with blade position in the rotor circle (and may even be negative in some positions) • Substantial passive support structure in the rotor system with an associated cost penalty • At the present time, VAWTs are not economically competitive with HAWTs. 4.2. CLASSIFICATION BY ROTOR SPEED Modern wind turbines have two types of electrical connections to the grid: • With the simple direct synchronization of an induction generator, the rotor operates with nearly constant speed because the strong grid keeps generator’s frequency. The only rotational speed variation is given by the slip range of the generator. 73 • With the help of an inverter system between the wind turbine generator and the grid, the turbine is decoupled from the grid frequency and is able to rotate at variable speeds. For a long period, directly grid coupled wind turbines dominated the world market due to their technical simplicity. But several positive aspects of variable speed turbines changed the current development situation. (German Wind Energy Institute, DEWI, 1998, p.48) 4.2.1. VARIABLE ROTOR SPEED The aerodynamically optimized lay out of wind turbines is based on a fixed relationship between wind and rotor tip speed, the so-called tip speed ratio. To keep the maximum efficiency, the rotor must change its rotational speed according to the wind speed, in other words, low winds with low rotor speeds, high winds with high rotor speeds. (German Wind Energy Institute, DEWI, 1998, p.48) Variable speed is attractive because it enables designer to gain greater rotor efficiencies by allowing rotor speed to vary with wind speed. There may be additional benefits as well. Slower rotor speeds in light winds lower noise emissions just when the aerodynamic noise of the blades is most noticeable. Variable-speed operation may also reduce dynamic loads on the turbine’s drive train, thus extending turbine life. When operating at variable speed, the rotor stores the energy of gusty winds as inertia as its speed increases, rather than forcing the drive train to absorb the increased torque instantaneously. Due to their ability to operate at tip speed ratios closer to the optimum value, variable speed machines can be more efficient than fixed speed systems. However, modification of both the generator and the intermediate electronic control systems are necessary in order to provide a grid-compatible supply. One of the main factors favoring this route is the requirement of some utilities for very smooth output power. 74 Variable rotor speeds normally are combined with a “pitch angle control system”. They have various operational advantages in comparison with constant rotor speed machines; • Higher energy extractio n. • Very low power fluctuations during rated power operation. • Lower rotor loads due to rotor speed yielding in gusts. • Low blade pitch change rates possible. • Low rotor speed at low wind conditions reduces the noise emission considerably. High power variable speed drives are now being designed into turbines and with them a new set of engineering aspects need to be considered, including; • Fault level of network. • Voltage regulation. • Electromagnetic compatibility. • Electrical system behavio ur during gusting conditions. • Power converter efficiency. For variable speed turbines, relatively complex power converter hardware is necessary. The power conversion equipment must provide low harmonics and unity power factor control of the current delivered to the network. 4.2.2. CONSTANT ROTOR SPEED Constant rotor speed is the simplest way of operating a wind turbine because the rotor speed is guided by the frequency of a strong grid. The tip speed ratio cannot be maintained constant during operation that means the efficienc y reaches its optimum only with one wind speed, which is the design wind speed of the rotor blade. During all other wind velocities, the efficiency is smaller than maximum. To better adapt the rotor operation to the aerodynamic design point, the manufacturers often use two the rated power fluctuations reach higher values than variable speed designs. Wind turbines with power ratings lower than 100 kW are called as small scale where the turbines with power ratings between 100 and 700 kW are called as medium scale. “medium scale” and “large scale” in terms of their power output capacity. with a high rotational speed (lower number of poles). • Low cost design.75 speed induction generators which allow changing the rotor speed in two steps: At low wind speeds. Due to stiff grid coupling. Constant one or two steps rotor speed operation is the simplest way of rotor speed control. . CLASSIFICATION BY POWER CONTROL Wind turbines can be classified into 3 groups as “small scale”. generator operates with a low rotational speed (higher number of poles) and at high wind speeds. 4. • Simple rotor speed regulation by the strong grid. • Only rotor speed monitoring is necessary.The large scale wind turbines have the power output capacity of greater than 700 kW. • No rotor speed control system is necessary. because the strong grid takes over the speed guidance.3. pp. (Boyle. but below the rated wind speed the output power varies with the wind speed.4 Operating regions of a typical wind turbine The maximum power which can be produced by a wind turbine is the rated power of it. called as cut-out wind speed. there is a maximum wind speed.268-269) .76 Figure 4. The lowest wind speed at which a wind turbine will operate is known as the cut-in wind speed. and the wind speed at which the turbine reaches rated power output is called as the rated wind speed. at which the turbine is designed to shut down in order to save mechanical parts of the wind turbine from harmful effects of high wind speed. At or above the rated wind speed. Above this. the power output remains constant whatever the wind speed (below the cut-out wind speed). 1996. in wind speed. 13 m/s to 25 m/s. Cut.III constant.1 Descriptions of Operational Regions for a Typical Wind Turbine Operating Region Region .IV strength and cost for the small number of hours per year beyond cut-out wind speed.II - As the blades of the wind turbine rotate through circular path. Rated wind speed to cutout wind speed. Wind Speed Wind speeds too low to produce usable electric power. This value can be normally calculated by area formula for circles. A = π ⋅ r2 where r is the rotor radius. they sweep through a disc. 0 to 4 m/s. Production of electric power at Region .in to rated wind speed.1) . (4. 25 m/s to rated survival wind speed. Wind Speed Range Region . 4 to 13 m/s.I Operational Description: Power Output vs. 0 to cut. Wind turbine blades purposely made less efficient as wind speed increases. Cut-out wind speed to survival wind speed. Winds too energetic to justify added Region . rated power level.like area which is referred to as the swept area. No electric power output.77 Table 4. Production of electric power increasing with wind speed. 5 Rotor diameter vs. As the wind does not always blow from the same direction. The shape of the blade’s cross-section is the key how modern wind turbines extract energy from the wind. This aligning or slewing action. power output The power that a wind turbine can extract from the wind at a given wind speed is directly proportional to its rotor’s swept area. It is extremely important that the maximum swept area is presented to the wind and this is achieved by making sure that the rotor’s axis is aligned with the direction from which the wind is blowing. about a vertical axis that passes through the center of the tower. . is known as yawing.78 Figure 4. a mechanism of some kind is needed to realign the rotor axis in response to changes in wind direction. which is rounded at one end and sharp at the other. A wind turbine blade has a distinctive curved cross-sectional shape. This special profile is known as an aerofoil section and is already familiar as the cross-sectional shape of aeroplane wings. With increasing airflow speed. Pitch control is more flexible and has opportunities to influence the operation of the wind turbine. which need specially shaped profiles that are very similar to those. 1998. the aerodynamic lift forces grow with the second power and the extracted energy of the turbine with the third power of the wind speed. p. Stall control is a traditional way and has restrictions. fast acting power control of the rotor to avoid mechanical and electrical overloading in the wind turbine’s energy transmission system.79 Figure 4. The most passive one is the so-called stall control.44) . (German Wind Energy Institute. the active one pitch control.6 Swept area by rotor blades Due to the aerodynamic forces on rotor blades. a situation which needs a very effective. Modern wind turbines use two different aerodynamic control principles to limit the power extraction to the nominal power of the generator. DEWI. used for wings or aeroplanes. a wind turbine converts the kinetic energy of wind flow into rotational mechanical energy. These driving aerodynamic forces are generated along the rotor blades. p.45) The advantages of the pitch controlled wind turbines are. (German Wind Energy Institute.80 4.3. 1998. the flow around the profiles of the rotor blade is well attached to the surface. • No need of strong brakes for emergency rotor stops. because they need a pitch changing system. also at partial power. turbine blades reach the optimum pitch angle. Always when the generator’s rated power is exceeded due to increasing wind speeds. • They reach rated power even under low air density conditions (high site elevations. the rotor blades will be turned along their longitudinal axis (pitch axis).1. DEWI. • Allow for active power control under all wind conditions. thus producing aerodynamic lift under very small drag forces. • Straight power cur ve at high wind speeds. PITCH CONTROL Pitch control is an active control system. • Decreasing rotor blade loads with increasing wind above rated power. . or in other words. • Simple start-up of the rotor by simple pitch change. Therefore. Under all wind conditions. high temperatures). at which it will produce the maximum power at that wind speed. • Feathering position of rotor blades for low loads at extreme winds. which normally needs an input signal from the generator power. Pitch controlled turbines are more sophisticated than fixed pitch stall controlled turbines. change their pitch angle to reduce the angle of attack of incoming air flow. • Higher energy production under the same conditions (no efficiency reducing stall adaptation of the blade). • Lower rotor blade masses lead to lo wer turbine masses. 3. p. and cannot be turned along their longitudinal axis.2. • Simple rotor hub structure. DEWI. This reduces the driving lift forces and increases the drag. The rotor blades are fixed in their pitch angle.44) The advantages of stall controlled wind turbines are. Lower lift and higher rotational drag act against a further increase of rotor power. • High reliability of power control. 1998. • Less maintenance due to fewer moving machinery parts. Their pitch angle is chosen in a way that for winds higher than rated wind speed the flow around the rotor blade profile separates from the blade surface (stall). (German Wind Energy Institute. STALL CONTROL Stall control is a passive control system.7 Pitch Control 4. • No pitch control system. Figure 4.8 Stall Control . which reacts on the wind speed.81 Figure 4. • Feathering position of rotor blades for low loads at extreme winds. yawing out of wind is used as a back up safety procedure or as contributory to control. In that case the rotor blade pitch is turned in direction towards stall and not towards feathering position (lower lift) as it is done in normal pitch systems.82 In last years. lower drive train loads than any stall option Complete rotor protection Needs auxiliary systems for over-speed protection Cost More cost in rotor systems Less cost in rotor. The one configuration that has now been unanimously rejected is fixed speed pitch control. In a few instances. • Power control under partial power conditions (low winds) is possible.2 Pitch vs. This combination produced very large transients in the power . The advantages of this system are. some manufacturers have used stall in conjunction with variable speed operation. the so-called active stall. Table 4. Stall Issues Issues Energy Capture Control With Fixed Speed Control With Variable Speed Better power quality. although design uncertain Safety Large wind turbines almost exclusively use pitch or stall control. The main issues in deciding between pitch and stall control are listed in Table 4. • Very small pitch angle changes necessary. but more in braking system Requires proving Pitch Better in principle Difficult in high wind speeds Stall Compromised power curve Generally satisfactory. Recently. a mixture of pitch and stall control is appeared.2. however. This rejection is.4. CLASSIFICATION BY LOCATION OF INSTALLATION Wind turbines are installed either on the land or on the sea level by some additional equipment. Beside. For this purpose.1 ON-SHORE WIND TURBINES In order to get the best efficiency from wind turbine operation and provide sustainable electricity to consumers. stall control scheme shows some unwanted fluctuations causing power losses. in the early days. pitch controlled power scheme results almost zero oscillations. a popular choice. 4.9 Stall & Pitch controlled power schemes As shown in Figure 4. They are classified as on-shore and off-shore wind turbines. wind turbines should be erected in windy areas. Figure 4. 4. . locations with continuous and fast wind should be selected.4. rather interesting since it was.9.83 output when controlling power. p. 4.10) Most turbines operate with a blade tip speed less than 65 m/s principally in order to contain sound emission within acceptable limits. windy and smooth areas such as lowlands. there is added potential for cost reduction in support structures and greater tolerance of more unusual design configurations that may have economic merit. large farms are selected for siting. Obviously this is very desirable to help offset the increased costs of foundations and electrical transmission associated with offshore projects.4. (European Commission Directorate-General for Energy. A tip speed of 100 m/s may be acceptable for offshore wind turbines.84 Wind turbines on the land are called as on-shore wind turbines. Thus the general view is that. if higher tip speeds can be exploited. sea coasts. It is possible to obtain higher output power levels for off-shore designs than the same turbines designed for on-shore. In order to benefit from wind speed as much as possible. . As with sound. then there is considerable scope for reduction of the weight and cost of the turbines themselves.2 OFF-SHORE WIND TURBINES Off-shore wind turbines are installed on sea up to some depths. and deserves sustained and substantial research and development support. if there is some relaxation in concern about the near field visual effect for offshore wind farms. It is a fact that. It has been recognized that if offshore wind turbines are remote from the coast and can be allowed increased sound emission. 1997. the cost of the wind turbine component of the offshore system can be significantly reduced compared to land based designs. there is a noteworthy difference of available wind speeds between on-shore and offshore locations. The potential for this technology is vast and it requires. The next great leap for the wind energy industry will be in the area of offshore development. 1997. p.11) . (European Commission Directorate-General for Energy. possibly at the expense of a somewhat higher wind turbine capital cost.85 A key objective for the design of cost effective offshore wind turbines will be that inspection and maintenance requirements are reduced to a minimum. Design for high reliability will be an important priority with an emphasis on minimising long term operation and maintenance costs. 84 m2 : 1.5 m/s : 20 m/s : Pitch Control (0-15 degrees) (2.loop model. some mathematical expressions describing the power output and rotational motion of the turbine are used.5 rpm (nrhigh ) : 760 – 1083 rpm While constructing the closed. The characteristics of the modelled wind turbine are.SIMULINK software. Rated Mechanical Power Rated Wind Speed Cut.5 m/s : 4. a wind turbine is modelled by MATLAB v5.in Wind Speed Cut-out Wind Speed Power Regulation Method Rotor Diameter Disc Swept Area Air Density Moment of Inertia Gear Ratio Rotational Speed Generator Rotor Speed (Pcap) : 2 MW : 12.5 ⋅ ρ ⋅ η ⋅ C p ⋅ A ⋅ V 3 (5. System Equation Set: Pcap = 0.m2 : 38 (nrlow) : 20 – 28.1) .225 kg/m3 : 1000 t.r) (A) (?) (J) : 74 m : 4300.2 .86 CHAPTER FIVE EXPERIMENTAL WORK In this chapter.0 MW wind turbine. The prototype chosen for the simulation is VESTAS V80 – 2. 00184 ⋅ (λ − 3) ⋅ α  15 − (0.2) λ= r ⋅ ωr V (5.m2 ) .3 ⋅ α ) (5.44 − 0.0167 ⋅ α) ⋅ Sin   − 0.3) Pcap(t +1) = Pcap(t ) + ωr (t ) ⋅ J ⋅ where dωr ( t ) dt (5.87  π ⋅ (λ − 3)  C p = (0.4) Pcap : Captured power by the turbine (input to the generator) (W) ? ? Cp A V a ? r ?r J : Air density (kg/m3 ) : Turbine mechanical efficiency : Power coefficient : Swept area by rotor blades (m2 ) : Wind speed (m/s) : Blade pitch angle (degree) : Tip speed ratio : Rotor radius (m) : Angular shaft speed (rad/s) : Moment of inertia (kg. 1 Overview of the wind turbine simulation .88 Figure 5. 5. a specific wind speed occurs as the lower limit to enable starting of generator mode of the machine. The captured power is used to calculate shaft speed variation corresponding torque change. occurs when the wind speed is as high as unacceptable over the rated value. This limit is called cut-out wind speed.5 m/s is defined as cut.1 SUB-SYSTEMS IN THE MODEL 5. . Physical damage of turbine machinery parts due to extremely high wind speeds. 1.1 YAW CONTROL BLOCK Yaw mechanism should be adapted to all wind turbines to avoid two unwanted effects. This causes teetering effects on turbine tower and over-speed of generator rotor.89 The aim of the simulation is to observe system output power curve versus wind input that changes with time. 4. In the studied model. Motoring operation of the turbine generator due to very low wind speeds because of insufficient s tarting torque. 2. when input wind power increases. Manufacturers should take into account the upper damage limit to keep turbine in service.1.5 – 20 m/s interval is neglected to make system efficient. Then. For example. Any wind data outside the 4. The specific lower limit of the wind speed is called cut-in wind speed. input torque to the turbine increases as well. Another usage purpose of the yaw system is aligning the turbine in line with the wind direction in order to allow the turbine to absorb maximum energy from the wind.in and 20 m/s as cut-out wind speeds. acceleration on the turbine shaft will be observed. 90 Figure 5.2 Yaw control block 5.1.2 TURBINE EFFICIENCY BLOCK At each wind speed, the mechanical torque input onto turbine shaft changes and mechanical efficiency also changes due to friction and heating. So, it may be stated that, turbine mechanical efficiency is directly proportional to the wind speed. Figure 5.3 Turbine efficiency block An efficiency curve is constituted for the model by using the operating values of different turbines present in the market. 91 Figure 5.4 Turbine efficiency characteristics corresponding to wind speed 5.1.3 PITCH CONTROL BLOCK Pitch control mechanism allows turbine blades to turn along their longitudinal axes. As any blade moved to increase the pitch angle, its capacity of absorbing wind power will decrease. In the studied system, when the absorbed wind power exceeds 2 MW, pitch control mechanism will be activated. After the power curve decreases below 2 MW, blade pitch angle will begin to decrease. To make power curve smooth while pitch control is activated, blade response time to any increment or decrement command is tried to be minimized. For this purpose, linear interpolation is applied to input wind speed data. By this way, present 137 wind inputs are raised to 2740 data with sample time equa l to 0.05 second. 92 Figure 5.5 Graphical demonstrations for the response of pitch control mechanism As seen from Figure 5.5, when the captured power exceeds 2 MW level at time 70.57 seconds, pitch mechanism is activated at time 70.60 seconds and the power curve is corrupted at time 70.60 sec. approximately at 2.0135 MW. The corresponding pitch mechanism response time is approximately 30 milliseconds. After the blade opening command is received by pitch control mechanism, the time required for the output power curve to recover itself to 2 MW level is about 10 milliseconds as shown in Figure 5.5. shaft angular speed variation corresponding to changing input torque at each step is calculated accurately in this block.m) Existing mechanical torque on the shaft (N.4 ANGULAR SPEED CALCULATION BLOCK This block is a key for turbine performance. Then.5) New captured mechanical torque input to the shaft (N. The general mechanical rotational motion equation is used to define acceleration.m) Moment of inertia (kg. By using the advantage of taken samples of captured power in narrow time intervals (sample time=0. .). obtained angular speed value is used to calculate tip speed ratio.1. deceleration or constant speed operations by wind speed changes.93 Figure 5. τ ( t +1) = τ( t ) + J ⋅ where t (t+1 ) t (t) J ? r(t) : : : : dω r (t ) dt (5.6 Pitch control block with 0-15 degrees adjustment interval 5.05 sec.m2 ) Angular shaft speed (rad/s) This equation can be modified to provide system compatibility. Figure 5. derivative term states the speed variation between times (t) and (t+1). respectively.94 Pcap( t +1) = Pcap(t ) + ω r( t ) ⋅ J ⋅ where Pcap(t+1) Pcap(t) dωr ( t ) dt (5.7) Consequently. power coefficient (C p ) and the power input to the generator (Pcap). ωr ( t +1) = ωr ( t ) + ∆ω r (5.7 Angular speed calculation block . this speed difference (indicating acceleration. This value is added to the speed value at time (t) to find the new speed value at time (t+1).8) The resultant angular speed can be used to find tip speed ratio (?). dωr ( t ) dt = ∆ωr = Pcap( t +1) − Pcap( t ) ω r( t ) ⋅ J ⇒ ∆ωr = ∆Pcap ω r (t ) ⋅ J (5.6) : New captured mechanical power input to the shaft (W) : Existing mechanical power on the shaft (W) Here. deceleration or constant speed operation) is added to the speed value at time (t). Note from Figure 5. Input wind data is interpolated by the system with 0.2 SIMULATION RESULTS Simulation takes 137 seconds. Multiport selection block inside the sub-system decides the function to be used. Cp – ? selection block has two inputs (?. 5. a). All graphical results of the simulation are shown below. it is fed back to power calculation block to determine the captured power of the turbine. Block has a Cp =f(?.1. simulation includes 20 x 137 = 2740 steps.10 that. Small sample time enables system to be stable and captured power to be kept around the rated value. . After the output Cp is found. output power fluctuations can be kept in 200 kW tolerances.5 Cp – ? SELECTION BLOCK After the system decides pitch angle in degrees. This power is also the input mechanical power to the generator. and one output (C p ).05 second sample time. At the end of simulation.2). power coefficient (C p ) can be found by using its characteristic equation depending on tip speed ratio (?) and pitch angle (a). output power graph says that pitch control is a very useful way to control system output whatever the wind power.95 5. Totally. Pitch control allows user to control the power absorbing capacity of the turbine. a) function for each a input (Equation 5. 96 Figure 5.8 Wind speed values filtered by yaw control block Figure 5.9 Aerodynamic power in the wind . 97 Figure 5.11 Angular speed variation of the turbine in respect of each wind speed change (Change of input torque) .10 Captured wind power by the turbine (Input power to generator) Figure 5. 13 Rotational speed of turbine shaft before gearbox .98 Figure 5.12 Angular shaft speed of the turbine Figure 5. 15 Tip speed ratio .14 Rotational speed of turbine shaft after gearbox (Rotational speed of generator rotor) Figure 5.99 Figure 5. 100 Figure 5.16 Blade pitch angle (a) Figure 5.17 Power coefficient (Cp) . 18 Tip speed ratio vs.101 Figure 5. power coefficient Figure 5.19 Turbine wind speed – power characteristics . the available aerodynamic wind power (P w) is still increasing. . wind speed In Table 5. pitch angle (a) kept at zero by the system. Note that. variations of all parameters of the wind turbine can be observed corresponding to each available wind speed value. until wind speed (V) reaches the rated value. and after the rated wind speed occurred.1.102 Figure 5.20 Turbine efficiency vs. pitch angle is started to increase in order to allow keeping the output power (Pcap) around rated value At the same time. 6 8.505 4.21 0.890 10.8 10 9.4 11.42 0.14 Pcap (kW) 51 165 323 514 753 1.16 0.103 Table 5.632 3.43 0.031 1.44 0.1 6.466 15.9 5.922 2.999 2.41 0.052 1.1 Modelled Wind Turbine Simulation Results V (m/s) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Pw (kW) 330 570 904 1.551 5.869 1.020 1.35 0.21 0.44 0.360 1.4 9 8.938 18.24 0.7 6.732 1.36 0.6 ? a (degrees) 0 0 0 0 0 0 0 0 2 2.345 1.3 7.925 2.225 8.5 Cp 0.7 13.35 5.788 7.42 0.847 .9 10.5 4.30 0.790 12.5 6 7 8. Wind can be competitive with nuclear. the most common configuration has become the horizontal three bladed turbine with its rotor positioned upwind on the windy side of the tower. larger blades. The most dramatic improvement has been in the increasing size and performance of wind turbines. continuing improvements are being made in the ability of the machines to capture as much energy as possible from the wind. These include more powerful rotors.104 CHAPTER SIX CONCLUSIONS Wind power is a deceptively simple technology. With this broad envelope. Modular and quick to install . the typical size being sold today is up to 2500 kW. They are modular and very quick to install and commission. aerodynamic design and computerized electronic control. slender towers and gently turning blades lie a complex interplay of lightweight materials. Advantages of using wind energy conversion systems instead of other energy production systems are. • Environmental protection (No CO2 emission) • Low-cost. Although a number of variations continue to be explored. improved power electronics. Behind the tall. From machines of just 25 kW twenty years ago. Today’s wind turbines include properties of modern technology. coal and gas • Diversity and security of supply • Rapid deployment. better use of composite materials and taller towers. Moment of inertia is the rotational mass of the turbine rotating parts. . It should be selected carefully to ensure reaching rated power output level and allowing minimum cut. The constructing material of blades and other rotating masses should be selected optimum to verify the minimum cut.105 • Fuel is abundant. variable speed wind turbine is made by pitch angle adjustment. in other words. free and inexhaustible • Costs are predictable and not influenced by fuel price fluctuations • Land.in wind speed. 20-28. Rotor diameter is directly specifies the swept area and so captured power from the wind. Agricultural / industrial activity can continue around it Power control of the studied horizontal axis.5 rpm operating interval of low-speed shaft is modified into 760-1083 rpm region for a 750 rpm synchronous speed asynchronous machine with the gear ratio of 38. Gear ratio is the adjustment location of induction machine generator region. the oscillations around rated power line can be minimized above rated wind speeds. in the studied system.in wind speed. As the number of wind speed samples increases. For this purpose. This seems as the most efficient method to supply 3-phase utility grids. rotor diameter and gear ratio are three critical parameters for a variable speed wind turbine and must be selected carefully by manufacturers while designation. the pitch control mechanism works more efficiently. This means minimum starting torque and maximum usage of the wind power. Moment of inertia.friendly. For example. long time wind speed measurements should be made and then it will be possible to investigate the optimum wind speed interval to allow maximum overall energy capturing. In 2002. Figure 6. the German company Enercon is scheduled to erect the first prototype of its 4500 kW turbine with a rotor diameter of 112 meters. Machines in a range from 3000 kW up to 5000 kW are currently under development. resulting in the physical damage of machinery parts. 2002.1 FUTURE PROSPECTS In the future. .13) European Wind Energy Association (EWEA) which is the international voice of the wind industry located in the center of Europe has launched an industrial blueprint including the targets to be reached by 2020.1 ?–Cp curve indicating operating regions of the generator 6.106 Although tip speed ratio values seem acceptable in both raising and falling regions of ?–Cp curve. allowing tip speed ratio to exceed 10 causes the over-speed of generator rotor. European Wind Energy Association. (EWEA. p. even larger turbines than today’s 2500 kW will be produced to service the new offshore market. 000 MW wind energy capacity installed generating 3093 TWh.107 The main objectives of this study are. . equivalent to the current electricity use of all Europe This study demonstrates that there are no technical. assuming that global demand doubles by then • Creation of 1475 million recruitments • Cumulative CO2 savings of 11. • Supplying 12 % of global electricity demand.261.768 million tones • 1. economic or resource limitations to achieve this goal. but the political and policy changes are required in order for the wind industry to reach its full potential. ewea. Chen. 16. URL: http://www. Bornova.org/download/ De Montfort University. Grid power quality with variable speed wind turbines.108 REFERENCES American Wind Energy Association. URL: http://www. Stephen J.dmu. (1997). E.ac.awea. & Spooner. (1998). Izmir. Oxford University Press. (3rd ed). Melbourne: McGraw-Hill International Editions Electric Machinery Series. Wind energy training course. URL: http://www.The facts. 148-153 Çam.org/doc/ .. Yeni tip kanat modeli ile rüzgardan elektrik eldesi. IEEE Transactions on Energy Conversion. (2001).html European Commission Directorate-General for Energy. Guided tour on wind energy.uk/wind_energy/index. Global wind energy market report. (1999).windpower. Danish Wind Turbine Manufacturers Association. Chapman. (2001). (2002). URL: http://www. (1999). (1996). Renewable energy: Power for a sustainable future.iesd.org/pubs/documents/ Boyle. Electric machinery fundamentals. Aegean University. Wind energy . G. Z. E. & Butterfield. J. E. Phoenix. J. Wind Energy Information Brochure.4. 135-141 Wang. Wind force 12.. (2002). (2000). An independent maximum power extraction strategy for wind energy conversion systems. Wind force 12.ewea. Heier. Arizona. Muljadi. Wind Directions. & Chang. XXI . Shaltout. Swadlincote. IEEE Transactions on Energy Conversion. (1999). URL: http://www. (1995). A. 16-19 URL: http://www.org/doc/ European Wind energy Association.). Wind energy: Comes of age. 1999. (1998). Q. (Waddington R. L. USA: 1999 IEEE Industrial Applications Society Annual Meeting. (Original book published 1996). C. Analysis of torsional torques in starting of large squirrel cage induction motors. Oxford University Press. Edmonton. (1983). URL: http://www.ewea. (2002). October 3-7. Shaw Conference Center. UK: John Wiley & Sons Inc. Pitch-controlled variable-speed wind turbine generation. (2002). (1998). A. S. The new global challange.ewea. Grid integration of wind energy conversion systems. 9.109 European Wind energy Association. (1994). Ramage. Alberta.org/doc/ German Wind Energy Institute. Wind energy – Clean power for generations.P.. Gipe. John Wiley & Sons Inc. Canada May 9-12 1999: Proceedings of the 1999 IEEE Canadian Conference on Electrical and Computer Engineering. . Energy – A guidebook.org/doc/ European Wind Energy Association. 110 APPENDICES . 5 ⋅ ρ ⋅ A ⋅ V3 ) Mechanical power ( Pm = Pw ⋅ Cp ) Calculate captured power (Generator input power) ( Pcap = Pm ⋅ η ) Calculate pitch angle (a) Calculate angular speed (? r) Tip speed ratio (?) Calculate power coefficient (C p) .FLOWCHART OF THE SIMULATED SYSTEM - .A Get wind data (V) Rotor radius (r) Gear ratio 4.5 < V < 20 m/s No V=0 Yes Calculate turbine efficiency (?) (Look-up table) Calculate aerodynamic wind power ( Pw = 0. B . C . scribd /********* DO NOT ALTER ANYTHING BELOW THIS LINE ! **********/ var s_code=s.t();if(s_code)document.write(s_code)//-->	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8512294292449951, "perplexity": 3909.824396522904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398471436.90/warc/CC-MAIN-20151124205431-00045-ip-10-71-132-137.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/rock-hanging-from-ceiling-swinging-uniform-circular-motion.157554/	# Rock hanging from ceiling swinging uniform circular motion 1. Feb 22, 2007 ### iamtheone 1. The problem statement, all variables and given/known data A rock of mass 5.5kg is hanging from the ceiling moving in uniform circular motion. The radius of the orbit is .8m. The angle of the rope in comparison to if it were hanging straight down is 15 degrees outward. What is the magnitude of the force of the cord on the mass? I already found out this is about 55.801 N What is the magnitude of the acceleration of the rock? What is the angular speed of the rock? How many revolutions per second does the rock make? 2. Relevant equations angular speed = v/r f = ma centripital accel = (v^2)/r ..others probably as well... 3. The attempt at a solution To get the magnitude of the acceleration of the rock: Ftotal = Frope + Fweight = 109.701 plug in to F = ma, a=19.9457 (incorrect) I'm guessing I need to include another force....or take the magnitude of both those force vectors added together? I don't know how to get the Frope force vector though. Any help with thi swould be apreciated. 2. Feb 22, 2007 You mentioned centripetal acceleration, so there is a centripetal force, too. 3. Feb 22, 2007 ### iamtheone i don't know how to calculate it. And I mentioned it merely because I THINK it applies. 4. Feb 22, 2007 Did you draw a free body diagram? What equations can you work out from the diagram? 5. Feb 22, 2007 ### iamtheone I did, but I'm still pretty much clueless. 6. Feb 22, 2007 Well, regarding the second question, what kind of an acceleration does the rock possess? 7. Feb 22, 2007 ### iamtheone centripetal, but i don't know the velocity, so I can't calculate that either... 8. Feb 22, 2007 Well, which equation did you use to obtain the force in the rope? Which equation can you use to obtain the magnitude of the centripetal force (and the velocity)? 9. Feb 22, 2007 ### iamtheone I used Fcos(20) = mg -> which is just a version of f = ma I can't find equations for the magnitude of the centripetal force or the velocity....which is very annoying. I've been looking in my book for a while now. 10. Feb 22, 2007 Can you find the magnitude of the centripetal force now? 11. Feb 22, 2007 ### iamtheone that gives the acceleration....not force. 12. Feb 22, 2007 ### arildno And force equals..?what?..acceleration? 13. Feb 22, 2007 ### iamtheone mass, but i can't calculate v - ive already tried this many times....i guess the question comes down to being able to calculate v. 14. Feb 22, 2007 ### arildno The tension is given by T=mg/(cos(15)), not by mg/(cos(20)! Now, what is the planar radial component of this tension? 15. Feb 22, 2007 ### denverdoc well if v^2/r=a and R is known, then we need another eqn to solve for the a. You have Tension which you have calculated, what if you were to resolve that into x and Y components? edit: whoops didn't mean to butt in. Similar Discussions: Rock hanging from ceiling swinging uniform circular motion	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9402986168861389, "perplexity": 1617.3841003193677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948522999.27/warc/CC-MAIN-20171213104259-20171213124259-00032.warc.gz"}
https://yoshiwarabooks.org/mfg/The-Natural-Base.html	## Section5.3The Natural Base There is another base for logarithms and exponential functions that is often used in applications. This base is an irrational number called $e\text{,}$ where \begin{equation} e \approx 2.71828182845 \end{equation} The number $e$ is essential for many advanced topics, and it is often called the natural base. ### SubsectionThe Natural Exponential Function The natural exponential function is the function $f(x) = e^x\text{.}$ Values for $e^x$ can be obtained with a calculator using the $\boxed{e^x}$ key ( 2nd LN on most calculators). For example, you can evaluate $e^1$ by pressing 2nd LN $1$ to confirm the value of $e$ given above. Because $e$ is a number between $2$ and $3\text{,}$ the graph of $f(x) = e^x$ lies between the graphs of $y = 2^x$ and $y = 3^x\text{.}$ Compare the tables of values and the graphs of the three functions below. As with other exponential functions, the domain of the natural exponential function includes all real numbers, and its range is the set of positive numbers. $x$ $y=2^x$ $y=e^x$ $y=3^x$ $-3$ $0.125$ $0.050$ $0.037$ $-2$ $0.250$ $0.135$ $0.111$ $-1$ $0.500$ $0.368$ $0.333$ $0$ $1$ $1$ $1$ $1$ $2$ $2.718$ $3$ $2$ $4$ $7.389$ $9$ $3$ $8$ $20.086$ $27$ ###### Example5.40 Graph each function. How does each graph differ from the graph of $y = e^x\text{?}$ 1. $g(x) = e^{x+2}$ 2. $h(x) = e^x + 2$ Solution If $f(x) = e^x\text{,}$ then $g(x) = f(x + 2)\text{,}$ so the graph of $g$ is shifted $2$ units to the left of $y = e^x\text{.}$ Also, $h(x) = f(x) + 2\text{,}$ so the graph of $h$ is shifted $2$ units up from $y = e^x\text{.}$ The graphs are shown above. ###### Checkpoint5.41 Use your calculator to evaluate the following powers. 1. $e^2$ 2. $e^{3.5}$ 3. $e^{-0.5}$ 1. $e^2\approx 7.389$ 2. $e^{3.5}\approx 33.115$ 3. $e^{-0.5}\approx 0.6065$ ### SubsectionThe Natural Logarithmic Function The base $e$ logarithm of a number $x\text{,}$ or $\log_ e x\text{,}$ is called the natural logarithm of $x$ and is denoted by $\ln x\text{.}$ ###### The Natural Logarithm The natural logarithm is the logarithm base $e\text{.}$ \begin{equation} \ln x = \log_{e}{x}, ~~~~ x\gt 0 \end{equation} The natural logarithm of $x$ is the exponent to which $e$ must be raised to produce $x\text{.}$ For example, the natural logarithm of $10\text{,}$ or $\ln 10\text{,}$ is the solution of the equation \begin{equation} e^y = 10 \end{equation} You can verify on your calculator that \begin{equation} e^{\alert{2.3}} ~~ \approx 10 \text{ or } ~~ \ln 10 \approx \alert{2.3} \end{equation} In general, natural logs obey the same conversion formulas that work for logs to other bases. ###### Conversion Formulas for Natural Logs \begin{equation} \blert{y = \ln x} ~~\text{ if and only if } ~~ \blert{e^y = x} \end{equation} In particular, \begin{equation} \begin{aligned}[t] \ln e \amp = 1 \text{ because } e^1 = e \\ \ln 1 \amp = 0 \text{ because } e^0 = 1 \\ \end{aligned} \end{equation} The conversion formulas tell us that the natural log function, $g(x) = \ln x\text{,}$ is the inverse function for the natural exponential function, $f(x) = e^x\text{.}$ ###### Example5.42 1. Graph $f(x) = e^x$ and $f^{-1}(x) = \ln x$ on the same grid. 2. Give the domain and range of the natural log function. Solution 1. We can make a table of values for $f^{-1}(x) = \ln x$ by interchanging the columns in the table for $f(x) = e^x\text{.}$ Plotting the points gives us the graph below. $x$ $y=\ln x$ $0.050$ $-3$ $0.135$ $-2$ $0.368$ $-1$ $1$ $0$ $2.718$ $1$ $7.389$ $2$ $20.086$ $3$ 2. The domain of the natural log function is the same as the range of $y = e^x\text{,}$ or all positive numbers. The range of $y = \ln x$ is the same as the domain of $y = e^x\text{,}$ or all real numbers. These results are confirmed by the graph of $y = \ln x\text{.}$ ###### Caution5.43 Observe that the natural log of a number greater than $1$ is positive, while the logs of fractions between $0$ and $1$ are negative. In addition, the natural logs of negative numbers and zero are undefined. ###### Checkpoint5.44 1. $\ln 100$ 2. $\ln 0.01$ 3. $\ln e^3$ 1. $\ln 100\approx 4.6052$ 2. $\ln 0.01\approx -4.6052$ 3. $\ln e^3 =3$ ### SubsectionProperties of the Natural Logarithm We use natural logarithms in the same way that we use logs to other bases. The properties of logarithms that we studied in Section 4.4 also apply to logarithms base $e\text{.}$ ###### Properties of Natural Logarithms If $x, y \gt 0\text{,}$ then 1. $\ln{(xy)} = \ln{x} + \ln{y}$ 2. $\ln\dfrac{x}{y} = \ln x - \ln y$ 3. $\ln{x^k} = k \ln x$ Because the functions $y = e^x$ and $y = \ln x$ are inverse functions, the following properties are also true. ###### The Natural log and $e^x$ \begin{equation} \ln{e^x} = x,~~\text{ for all }x, ~~~~~\text{ and }~~~e^{\ln x} = x,~~\text{ for }x \gt 0 \end{equation} ###### Example5.45 Simplify each expression. 1. $\ln e^{0.3x}$ 2. $e^{2 \ln(x+3)}$ Solution 1. The natural log is the log base $e\text{,}$ and hence the inverse of $e^x\text{.}$ Therefore, \begin{equation} \ln {e^{0.3x}} = 0.3x \end{equation} 2. First, we simplify the exponent using the third property of logs to get \begin{equation} 2 \ln(x + 3) = \ln(x + 3)^2 \end{equation} Then $e^{2 \ln(x+3)} = e^{\ln(x+3)^2} = (x + 3)^2\text{.}$ ###### Checkpoint5.46 Simplify each expression. 1. $e^{(\ln x)/2}$ 2. $\ln\left(\dfrac{1}{e^{4x}}\right)$ 1. $\sqrt{x}$ 2. $-4x$ ### SubsectionSolving Equations We use the natural logarithm to solve exponential equations with base $e\text{.}$ The techniques we've learned for solving other exponential equations also apply to equations with base $e\text{.}$ ###### Example5.47 Solve each equation for $x\text{.}$ 1. $e^x = 0.24$ 2. $\ln x = 3.5$ Solution 1. We convert the equation to logarithmic form and evaluate using a calculator. \begin{equation} x = \ln 0.24 \approx -1.427 \end{equation} 2. We convert the equation to exponential form and evaluate. \begin{equation} x = e^{3.5} \approx 33.1155 \end{equation} ###### Checkpoint5.48 1. $\ln x =-0.2$ 2. $e^x = 8$ 1. $0.8187$ 2. $2.0794$ To solve more complicated exponential equations, we isolate the power on one side of the equation before converting to logarithmic form. ###### Example5.49 Solve $140 = 20 e^{0.4x}\text{.}$ Solution First, we divide each side by $20$ to obtain \begin{equation} 7 = e^{0.4x} \end{equation} Then we convert the equation to logarithmic form. \begin{equation} \begin{aligned}[t] 0.4x \amp = \ln 7 \amp\amp \blert{\text{Divide both sides by 0.4.}}\\ x \amp= \frac{\ln 7}{0.4} \end{aligned} \end{equation} Rounded to four decimal places, $x \approx 4.8648\text{.}$ ###### Note5.50 We can also solve the equation in Example 5.49, \begin{equation} 7 = e^{0.4x} \end{equation} by taking the natural logarithm of both sides. This gives us \begin{equation} \begin{aligned}[t] \ln 7 \amp = \ln e^{0.4x}\amp\amp \blert{\text{Simplify the right side.}} \\ \ln 7 \amp = 0.4x \end{aligned} \end{equation} because $\ln{e^a} = a$ for any number $a\text{.}$ We then proceed with the solution as before. ###### Checkpoint5.51 Solve \begin{equation} 80 -16e^{-0.2x} = 70.3 \end{equation} Hint $\blert{\text{Subtract from both sides and divide by}~ -16.}$ $\blert{\text{Take the natural log of both sides.}}$ $\blert{\text{Divide by}~ -0.2.}$ $x = -5 \ln \left(\dfrac{9.6}{16} \right)\approx 2.5023$ ###### Example5.52 Solve $P = \dfrac{a}{1 + be^{-kt}}$ for $t\text{.}$ Solution We multiply both sides of the equation by the denominator, $1 + be^{-kt}\text{,}$ to get \begin{equation} P(1 + be^{-kt} ) = a \end{equation} Then we isolate the power, $e^{-kt}\text{,}$ as follows: \begin{equation} \begin{aligned}[t] 1 + be^{-kt} \amp = \frac{a}{P}\amp\amp \blert{\text{Subtract 1 from both sides.}}\\ be^{-kt} \amp = \frac{a}{P}- 1 \amp\amp \blert{\text{Rewrite the right side. }}\\ be^{-kt} \amp = \frac{a - P}{P}\amp\amp \blert{\text{Divide both sides by }b.}\\ e^{-kt} \amp= \frac{a - P}{bP} \end{aligned} \end{equation} Next, we take the natural logarithm of both sides to get \begin{equation} \ln {e^{-kt}} = \ln{\frac{a - P}{bP}} \end{equation} and recall that $\ln {e^x} = x$ to simplify the left side. \begin{equation} -kt = \ln{\frac{a - P}{bP}} \end{equation} Finally, we divide both sides by $-k$ to solve for $t\text{.}$ \begin{equation} t =\frac{-1}{k}\ln{\frac{a-P}{bP}} \end{equation} ###### Checkpoint5.53 Solve $N = Ae^{-kt}$ for $k\text{.}$ Hint $\blert{\text{Divide both sides by}~ A.}$ $\blert{\text{Take the natural log of both sides.}}$ $\blert{\text{Divide both sides by}~ -t.}$ $k=\dfrac{-\ln(N/A) }{t}$ ### SubsectionExponential Growth and Decay In Section 4.1, we considered functions of the form \begin{equation} P(t) = P_0\cdot b^t \end{equation} which describe exponential growth when $b \gt 1$ and exponential decay when $0 \lt b \lt 1\text{.}$ Exponential growth and decay can also be modeled by functions of the form \begin{equation} \blert{P(t) = P_0 \cdot e^{kt}} \end{equation} where we have substituted $e^k$ for the growth factor $b\text{,}$ so that \begin{equation} \begin{aligned}[t] P(t) \amp = P_0 \cdot b^t\\ \amp = P_0 \cdot \left(e^k\right)^t = P_0 \cdot e^{kt}\\ \end{aligned} \end{equation} We can find the value of $k$ by solving the equation $b = e^k$ for $k\text{,}$ to get $k = \ln b\text{.}$ For instance, in Example 4.1 in Section 4.1 we found that a colony of bacteria grew according to the formula \begin{equation} P(t) = 100 \cdot \alert{3^t} \end{equation} We can express this function in the form $P(t) = 100 \cdot \alert{e^{kt}}$ if we set \begin{equation} 3 = e^k ~ \text{ or } ~ k = \ln 3 \approx 1.0986 \end{equation} Thus, the growth law for the colony of bacteria can be written \begin{equation} P(t) \approx 100 \cdot e^{1.0986t} \end{equation} By graphing both functions on your calculator, you can verify that \begin{equation} P(t) = 100 \cdot 3t~~~\text{and}~~~P(t) = 100 \cdot e^{1.0986t} \end{equation} are just two ways of writing the same function. ###### Example5.54 From 1990 to 2000, the population of Clark County, Nevada, grew by $6.4\%$ per year. 1. What was the growth factor for the population of Clark County from 1990 to 2000? If the population of Clark County was $768,000$ in 1990, write a formula for the population $t$ years later. 2. Write a growth formula for Clark County using base $e\text{.}$ Solution 1. The growth factor was $b = 1 + r = 1.064\text{.}$ The population $t$ years later was \begin{equation} P(t) = 768,000 (1.064)^t \end{equation} 2. We use the formula $P(t) = P_0 \cdot e^{kt}\text{,}$ where $e^k = 1.064\text{.}$ Solving for $k\text{,}$ we find \begin{equation} k = \ln 1.064 = 0.062 \end{equation} so $P(t) = 768,000 e^{0.062t}\text{.}$ ###### Checkpoint5.55 From 1994 to 1998, the number of personal computers connected to the Internet grew according to the formula $N(t) = 2.8e^{0.85t}\text{,}$ where $t = 0$ in 1994 and $N$ is in millions. (Source: Los Angeles Times, September 6, 1999) 1. Evaluate $N(1)\text{.}$ By what percent did the number of Internet users grow in one year? 2. Express the growth law in the form $N(t) = N_0 (1 + r)^t\text{.}$ Hint $e^k = 1 + r$ 1. $N(1)\approx 6.55\text{,}$ $~134\%$ 2. $N(t)\approx 2.8 (1.3396)^t$ If $k$ is negative, then $e^k$ is a fraction less than $1\text{.}$ For example, if $k = -2\text{,}$ \begin{equation} e^{-2} = \frac{1}{e^2} \approx \frac{1}{7.3891} \approx 0.1353 \end{equation} Thus, for negative values of $k\text{,}$ the function $P(t) = P_0 e^{kt}$ describes exponential decay. ###### Exponential Growth and Decay The function \begin{equation} P(t) = P_0 e^{kt} \end{equation} describes exponential growth if $k \gt 0\text{,}$ and exponential decay if $k \lt 0\text{.}$ ###### Example5.56 Express the decay law $N(t) = 60 (0.8)^t$ in the form $N(t) = N_0 e^{kt}\text{.}$ Solution For this decay law, $N_0 = 60$ and $b = 0.8\text{.}$ We would like to find a value for $k$ so that $e^k = b = 0.8\text{,}$ that is, we must solve the equation \begin{equation} \begin{aligned}[t] e^k \amp = 0.8\amp\amp \blert{\text{Take natural log of both sides.}}\\ \ln e^k\amp = \ln 0.8\amp\amp \blert{\text{Simplify.}}\\ k\amp = \ln 0.8 \approx -0.2231 \end{aligned} \end{equation} Replacing $b$ with $e^k\text{,}$ we find that the decay law is \begin{equation} N(t) \approx 60e^{-0.2231t} \end{equation} ###### Checkpoint5.57 A scientist isolates $25$ grams of krypton-91, which decays according to the formula \begin{equation} N(t) = 25e^{-0.07t}\text{,} \end{equation} where $t$ is in seconds. 1. Complete the table of values showing the amount of krypton-91 left at $10$-second intervals over the first minute. $t$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $N(t)$ 2. Use the table to choose a suitable window and graph the function $N(t)\text{.}$ 3. Write and solve an equation to answer the question: How long does it take for 60% of the krypton-91 to decay? Hint If $60\%$ of the krypton-91 has decayed, $40\%$ of the original $25$ grams remains. 1. $t$ $0$ $10$ $20$ $30$ $40$ $50$ $60$ $N(t)$ $25$ $12.41$ $6.16$ $3.06$ $1.52$ $0.75$ $0.37$ 2. $25 e^{-0.07t} = 0.40(25)\text{;}$ $~~t=\dfrac{\ln(0.4)}{-0.07}\approx 13.09$ seconds ### SubsectionContinuous Compounding Some savings institutions offer accounts on which the interest is compounded continuously. The amount accumulated in such an account after $t$ years at interest rate $r$ is given by the function \begin{equation} A(t) = Pe^{rt} \end{equation} where $P$ is the principal invested. ###### Example5.58 Suppose you invest $$500$ in an account that pays $8\%$ interest compounded continuously. You leave the money in the account without making any additional deposits or withdrawals. 1. Write a formula that gives the value of your account $A(t)$ after $t$ years. 2. Make a table of values showing $A(t)$ for the first $5$ years. 3. Graph the function $A(t)\text{.}$ 4. How much will the account be worth after $10$ years? 5. How long will it be before the account is worth$$1000\text{?}$ Solution 1. We substitute $500$ for $P\text{,}$ and $0.08$ for $r$ to find \begin{equation} A(t) = 500e^{0.08t} \end{equation} 2. We evaluate the formula for $A(t)$ to obtain a table. $t$ $A(t)$ $0$ $500$ $1$ $541.64$ $2$ $586.76$ $3$ $635.62$ $4$ $688.56$ $5$ $745.91$ 3. The graph of $A(t)$ is shown above. 4. We evaluate $A(t)$ for $t = 10\text{.}$ \begin{equation} \begin{aligned}[t] A(10) \amp= 500 e^{0.08(10)}\\ \amp = 500 e^{0.8}\\ \amp \approx 500(2.2255) = 1112.77 \end{aligned} \end{equation} The account will be worth $1112.77$ after $10$ years. 5. We substitute $1000$ for $A(t)$ and solve the equation. \begin{equation} \begin{aligned}[t] 1000 \amp= 500 e^{0.08t}\amp\amp \blert{\text{Divide both sides by 500.}}\\ 2 \amp = e^{0.08t}\amp\amp \blert{\text{Take natural log of both sides.}}\\ \ln 2 \amp= \ln e^{0.08t} = 0.08t\amp\amp \blert{\text{Divide both sides by 0.08.}}\\ t \amp= \frac{\ln 2}{0.08}\approx 8.6643 \end{aligned} \end{equation} The account will be worth$1000$ after approximately $8.7$ years. ###### Checkpoint5.59 Zelda invested $$1000$ in an account that pays $4.5\%$ interest compounded continuously. How long will it be before the account is worth$$2000\text{?}$ About $15.4$ years ### SubsectionSection Summary #### SubsubsectionVocabulary Look up the definitions of new terms in the Glossary. • Natural exponential function • Natural logarithm • Continuous compounding #### SubsubsectionCONCEPTS 1. The natural base is an irrational number called $e\text{,}$ where \begin{equation} e\approx2.71828182845 \end{equation} 2. The natural exponential function is the function $f (x) = e^x\text{.}$ The natural log function is the function $g(x) = \ln x = \log_e x\text{.}$ 3. ###### Conversion Formulas for Natural Logs \begin{equation} \blert{y = \ln x} ~~\text{ if and only if } ~~ \blert{e^y = x} \end{equation} 4. ###### Properties of Natural Logarithms If $x, y \gt 0\text{,}$ then 1. $\ln{(xy)} = \ln{x} + \ln{y}$ 2. $\ln\dfrac{x}{y} = \ln x - \ln y$ 3. $\ln{x^k} = k \ln x$ 5. We use the natural logarithm to solve exponential equations with base $e\text{.}$ 6. ###### Exponential Growth and Decay The function \begin{equation} P(t) = P_0 e^{kt} \end{equation} describes exponential growth if $k \gt 0\text{,}$ and exponential decay if $k \lt 0\text{.}$ 7. Continuous compounding: The amount accumulated in an account after $t$ years at interest rate $r$ compounded continuously is given by \begin{equation} A(t) = Pe^{rt} \end{equation} where $P$ is the principal invested. #### SubsubsectionSTUDY QUESTIONS 1. State the value of $e$ to $3$ decimal places. Memorize this value. 2. Explain why $\ln e^x = x\text{.}$ 3. State the formula for exponential growth using base $e\text{.}$ 4. How is the formula for exponential decay in base $e$ different from the formula for exponential growth? #### SubsubsectionSKILLS Practice each skill in the Homework problems listed. 1. Graph exponential functions base $e\text{:}$ #1–4 2. Simplify expressions: #5 and 6 3. Solve exponential and log equations base $e\text{:}$ #7–10, 23–30 4. Use the properties of logs and exponents with the natural base: #19–22, 37–40 5. Use the natural exponential function in applications: #11–14, 47–58 6. Convert between $P(t) = P_0(1 + r )^t$ and $P(t) = P_0e^{kt}\text{:}$ #15–18, 41–46 ### SubsectionHomework 5.3 For Problems 1-4, use your calculator to complete the table for each function. Then choose a suitable window and graph the function. $x$ $-10$ $-5$ $0$ $5$ $10$ $15$ $20$ $f(x)$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ ###### 1 $f(x)=e^{0.2x}$ ###### 2 $f(x)=e^{0.6x}$ ###### 3 $f(x)= e^{-0.3x}$ ###### 4 $f(x)=e^{-0.1x}$ For Problems 5-6, simplify. ###### 5 1. $\ln e^2$ 2. $e^{\ln 5t}$ 3. $e^{-\ln x}$ 4. $\ln \sqrt{e}$ ###### 6 1. $\ln e^{x^4}$ 2. $e^{3 \ln x}$ 3. $e^{\ln x -\ln y}$ 4. $\ln \left(\dfrac{1}{e^{2t}} \right)$ For Problems 7-10, solve for $x\text{.}$ Round your answers to two decimal places. ###### 7 1. $e^x=1.9$ 2. $e^x=45$ 3. $e^x=0.3$ ###### 8 1. $e^x=2.1$ 2. $e^x=60$ 3. $e^x=0.9$ ###### 9 1. $\ln x=1.42$ 2. $\ln x = 0.63$ 3. $\ln x = -2.6$ ###### 10 1. $\ln x=2.03$ 2. $\ln x = 0.59$ 3. $\ln x = -3.4$ ###### 11 The number of bacteria in a culture grows according to the function \begin{equation} N(t) = N_0 e^{0.04t} \end{equation} where $N_0$ is the number of bacteria present at time $t = 0$ and $t$ is the time in hours. 1. Write a growth law for a sample in which $6000$ bacteria were present initially. 2. Make a table of values for $N(t)$ in $5$-hour intervals over the first $30$ hours. 3. Graph $N(t) \text{.}$ 4. How many bacteria were present at $t = 24$ hours? 5. How much time must elapse (to the nearest tenth of an hour) for the original $6000$ bacteria to increase to $100,000\text{?}$ ###### 12 Hope invests $\2000$ in a savings account that pays $5\frac{1}{2}\%$ annual interest compounded continuously. 1. Write a formula that gives the amount of money $A(t)$ in Hope’s account after $t$ years. 2. Make a table of values for $A(t)$ in $2$-year intervals over the first $10$ years. 3. Graph $A(t) \text{.}$ 4. How much will Hope's account be worth after $7$ years? 5. How long will it take for the account to grow to $\5000\text{?}$ ###### 13 The intensity, $I$ (in lumens), of a light beam after passing through $t$ centimeters of a filter having an absorption coefficient of $0.1$ is given by the function \begin{equation} I (t) = 1000e^{-0.1t} \end{equation} 1. Graph $I (t)\text{.}$ 2. What is the intensity (to the nearest tenth of a lumen) of a light beam that has passed through $0.6$ centimeter of the filter? 3. How many centimeters (to the nearest tenth) of the filter will reduce the illumination to $800$ lumens? ###### 14 X-rays can be absorbed by a lead plate so that \begin{equation} I (t) = I_0 e^{-1.88t} \end{equation} where $I_0$ is the X-ray count at the source and $I (t)$ is the X-ray count behind a lead plate of thickness $t$ inches. 1. Graph $I (t)\text{.}$ 2. What percent of an X-ray beam will penetrate a lead plate $\frac{1}{2}$ inch thick? 3. How thick should the lead plate be in order to screen out $70\%$ of the X-rays? For problems 15-18, express each exponential function in the form $P(t) = P_0b^t\text{.}$ Is the function increasing or decreasing? What is itsinitial value? ###### 15 $P(t) = 20e^{0.4t}$ ###### 16 $P(t)=0.8 e^{1.3t}$ ###### 17 $P(t) = 6500e^{-2.5t}$ ###### 18 $P(t)=1.7 e^{-0.02t}$ ###### 19 1. Fill in the table, rounding your answers to four decimal places. $x$ $0$ $0.5$ $1$ $1.5$ $2$ $2.5$ $e^x$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ 2. Compute the ratio of each function value to the previous one. Explain the result. ###### 20 1. Fill in the table, rounding your answers to four decimal places. $x$ $0$ $2$ $4$ $6$ $8$ $10$ $e^x$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ 2. Compute the ratio of each function value to the previous one. Explain the result. ###### 21 1. Fill in the table, rounding your answers to the nearest integer. $x$ $0$ $0.6931$ $1.3863$ $2.0794$ $2.7726$ $3.4657$ $4.1589$ $e^x$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ 2. Subtract each $x$-value from the next one. Explain the result. ###### 22 1. Fill in the table, rounding your answers to the nearest integer. $x$ $0$ $1.0986$ $2.1972$ $3.2958$ $4.3944$ $5.4931$ $6.5917$ $e^x$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ $\phantom{000}$ 2. Subtract each $x$-value from the next one. Explain the result. ###### 23 $6.21 = 2.3e^{1.2x}$ ###### 24 $22.26 = 5.3e^{0.4x}$ ###### 25 $6.4 = 20e^{0.3x} - 1.8$ ###### 26 $4.5 = 4e^{2.1x} + 3.3$ ###### 27 $46.52 = 3.1e^{1.2x} + 24.2$ ###### 28 $1.23 = 1.3e^{2.1x} - 17.1$ ###### 29 $16.24 = 0.7e^{-1.3x} - 21.7$ ###### 30 $55.68 = 0.6e^{-0.7x} + 23.1$ For Problems 31-36, solve the equation for the specified variable. ###### 31 $y = e^{kt},~~$ for $t$ ###### 32 $\dfrac{T}{R} = e^{t/2},~~$ for $t$ ###### 33 $y = k(1-e^{-t}),~~$ for $t$ ###### 34 $B - 2 = (A + 3)e^{-t/3},~~$ for $t$ ###### 35 $T = T_0 \ln(k + 10),~~$ for $k$ ###### 36 $P = P_0 + \ln 10k,~~$ for $k$ ###### 37 1. Fill in the table, rounding your answers to three decimal places. $n$ $0.39$ $3.9$ $39$ $390$ $\ln n$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ 2. Subtract each natural logarithm in your table from the next one. (For example, compute $\ln 3.9 - \ln 0.39\text{.}$) Explain the result. ###### 38 1. Fill in the table, rounding your answers to three decimal places. $n$ $0.64$ $6.4$ $64$ $640$ $\ln n$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ 2. Subtract each natural logarithm in your table from the next one. (For example, compute $\ln 6.4 - \ln 0.64\text{.}$) Explain the result. ###### 39 1. Fill in the table, rounding your answers to three decimal places. $n$ $2$ $4$ $8$ $16$ $\ln n$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ 2. Divide each natural logarithm in your table by $\ln 2\text{.}$ Explain the result. ###### 40 1. Fill in the table, rounding your answers to three decimal places. $n$ $5$ $25$ $125$ $625$ $\ln n$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ 2. Divide each natural logarithm in your table by $\ln 5\text{.}$ Explain the result. For Problems 41–46, 1. Express each growth or decay law in the form $N(t) = N_0e^{kt}\text{.}$ 2. Check your answer by graphing both forms of the function on the same axes. Do they have the same graph? ###### 41 $N(t) = 100\cdot 2^t$ ###### 42 $N(t) = 50\cdot 3^t$ ###### 43 $N(t) = 1200(0.6)^t$ ###### 44 $N(t) = 300(0.8)^t$ ###### 45 $N(t) = 10(1.15)^t$ ###### 46 $N(t) = 1000(1.04)^t$ ###### 47 The population of Citrus Valley was $20,000$ in $2000\text{.}$ In $2010\text{,}$ it was $35,000\text{.}$ 1. What is $P_0$ if $t = 0$ in $2000\text{?}$ 2. Use the population in $2010$ to find the growth factor $e^k\text{.}$ 3. Write a growth law of the form $P(t) = P_0 e^{kt}$ for the population of Citrus Valley. 4. If it continues at the same rate of growth, what will the population be in $2030\text{?}$ ###### 48 A copy of Time magazine cost $$1.50$ in $1981.~$ In $1988\text{,}$ the cover price had increased to$$2.00\text{.}$ 1. What is $P_0$ if $t = 0$ in $1981\text{?}$ 2. Use the price in $1988$ to find the growth factor $e^k\text{.}$ 3. Find a growth law of the form $P(t) = P_0e^{kt}$ for the price of Time. 4. In $1999\text{,}$ a copy of Time cost $$3.50\text{.}$ Did the price of the magazine continue to grow at the same rate from $1981$ to $1999\text{?}$ ###### 49 Cobalt-60 is a radioactive isotope used in the treatment of cancer. A $500$-milligram sample of cobalt-60 decays to $385$ milligrams after $2$ years. 1. Using $P_0 = 500\text{,}$ find the decay factor $e^k$ for cobalt-60. 2. Write a decay law $N(t) = N_0e^{kt}$ for cobalt-60. 3. How much of the original sample will be left after $10$ years? ###### 50 Weed seeds can survive for a number of years in the soil. An experiment on cultivated land found $155$ million weed seeds per acre, and in the following years the experimenters prevented the seeds from coming to maturity and producing new weeds. Four years later, there were $13.6$ million seeds per acre. (Source: Burton, 1998) 1. Find the annual decay factor $e^k$ for the number of weed seeds in the soil. 2. Write an exponential formula with base $e$ for the number of weed seeds that survived after $t$ years. Problems 51–58 are about doubling time and half-life. ###### 51 Delbert invests$$500$ in an account that pays $9.5\%$ interest compounded continuously. 1. Write a formula for $A(t)$ that gives the amount of money in Delbert's account after $t$ years. 2. How long will it take Delbert's investment to double to $$1000\text{?}$ 3. How long will it take Delbert's money to double again, to$$2000\text{?}$ 4. Graph $A(t)$ and illustrate the doubling time on your graph. 5. Choose any point $(t_1, A_1)$ on the graph, then find the point on the graph with vertical coordinate $2A_1\text{.}$ Verify that the difference in the $t$-coordinates of the two points is the doubling time. ###### 52 The growth of plant populations can be measured by the amount of pollen they produce. The pollen from a population of pine trees that lived more than $9500$ years ago in Norfolk, England, was deposited in the layers of sediment in a lake basin and dated with radiocarbon techniques. The figure shows the rate of pollen accumulation plotted against time, and the fitted curve $P(t) = 650e^{0.00932t}\text{.}$ (Source: Burton, 1998) 1. What was the annual rate of growth in pollen accumulation? 2. Find the doubling time for the pollen accumulation, that is, the time it took for the accumulation rate to double. 3. By what factor did the pollen accumulation rate increase over a period of $500$ years? ###### 53 Technetium-99m (Tc-99m) is an artificially produced radionuclide used as a tracer for producing images of internal organs such as the heart, liver, and thyroid. A solution of Tc-99m with initial radioactivity of $10,000$ becquerels (Bq) decays according to the formula \begin{equation} N(t) = 10,000e^{-0.1155t} \end{equation} where $t$ is in hours. 1. How long will it take the radioactivity to fall to half its initial value, or $5000$ Bq? 2. How long will it take the radioactivity to be halved again? 3. Graph $N(t)$ and illustrate the half-life on your graph. 4. Choose any point $(t_1, N_1)$ on the graph, then find the point on the graph with vertical coordinate $0.5N_1\text{.}$ Verify that the difference in the $t$-coordinates of the two points is the half-life. ###### 54 All living things contain a certain amount of the isotope carbon-14. When an organism dies, the carbon-14 decays according to the formula \begin{equation} N(t) = N_0e^{-0.000124t} \end{equation} where $t$ is measured in years. Scientists can estimate the age of an organic object by measuring the amount of carbon-14 remaining. 1. When the Dead Sea scrolls were discovered in 1947, they had $78.8\%$ of their original carbon-14. How old were the Dead Sea scrolls then? 2. What is the half-life of carbon-14, that is, how long does it take for half of an object's carbon-14 to decay? ###### 55 The half-life of iodine-131 is approximately $8$ days. 1. If a sample initially contains $N_0$ grams of iodine-131, how much will it contain after $8$ days? How much will it contain after $16$ days? After $32$ days? 2. Use your answers to part (a) to sketch a graph of $N(t)\text{,}$ the amount of iodine-131 remaining, versus time. (Choose an arbitrary height for $N_0$ on the vertical axis.) 3. Calculate $k\text{,}$ and hence find a decay law of the form $N(t) = N_0e^{kt}\text{,}$ where $k \lt 0\text{,}$ for iodine-131. ###### 56 The half-life of hydrogen-3 is $12.5$ years. 1. If a sample initially contains $N_0$ grams of hydrogen-3, how much will it contain after $12.5$ years? How much will it contain after $25$ years? 2. Use your answers to part (a) to sketch a graph of $N(t)\text{,}$ the amount of hydrogen-3 remaining, versus time. (Choose an arbitrary height for $N_0$ on the vertical axis.) 3. Calculate $k\text{,}$ and hence find a decay law of the form $N(t) = N_0e^{kt}\text{,}$ where $k\lt 0\text{,}$ for hydrogen-3. ###### 57 A Geiger counter measures the amount of radioactive material present in a substance. The table shows the count rate for a sample of iodine-128 as a function of time. (Source: Hunt and Sykes, 1984) Time (min) $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ Counts/sec $120$ $90$ $69$ $54$ $42$ $33$ $25$ $19$ $15$ $13$ 1. Graph the data and use your calculator's exponential regression feature to fit a curve to them. 2. Write your equation in the form $G(t) = G_0e^{kt}\text{.}$ 3. Calculate the half-life of iodine-128. ###### 58 The table shows the count rate for sodium-24 registered by a Geiger counter as a function of time. (Source: Hunt and Sykes, 1984) Time (min) $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$ $90$ Counts/sec $180$ $112$ $71$ $45$ $28$ $18$ $11$ $7$ $4$ $3$ 1. Graph the data and use your calculator's exponential regression feature to fit a curve to them. 2. Write your equation in the form $G(t) = G_0e^{kt}\text{.}$ 3. Calculate the half-life of sodium-24.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9107487797737122, "perplexity": 878.1166629632369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875141460.64/warc/CC-MAIN-20200217000519-20200217030519-00241.warc.gz"}
https://physics.codidact.com/users/54107/activity	# Activity for MissMulan‭ Type On... Excerpt Status Date Edit Post #286927 Initial revision about 1 month ago Question maxwell equation in 1d Maxwell's first law in differential form states that $$\triangledown \cdot E = \frac{\rho}{\epsilon{o}}$$ . In case of 1d can we say that $$\rho = \lambda$$ where $$\lambda$$ is the linear charge density of something? (more) Edit Post #286925 Initial revision about 1 month ago Question Double slit experiment with 2 electrons fired from a gun Suppose we perform the double slit experiment , but we fire instead 2 electrons instead of 1. In the double slit experiment performed in the 1920s a interference pattern was observed at the screen which showed the electron-wave nature of any subatomic particle.The particle would start moving as a ... (more) Edit Post #286923 Post edited: Edit Post #286923 Initial revision about 1 month ago Question SI Units of wavefunction What are the SI units of the wavefunction Ψ(x).I know that [Ψ(x)]^2 describes the probabilty of finding a quantum object at a certain quantum state but what about the wavefunction instead? (more) Edit Post #286906 Post edited: Edit Post #286906 Initial revision about 1 month ago Question Differential equation solution cannot describe what happens in reality Suppose we have a free falling object inside a planet's gravitational field with strength g.The planet's atmosphere provides a drag force which is dependant from the u^2 of the particle. Suppose the weight of the object is m1g and the drag force due to the atmosphere is -ku^2 and lets say we set k... (more) Edit Post #286898 Initial revision about 1 month ago Question Classical Uncertainty Suppose we are using a force gauge to measure gravity in a planet. We set the gauge force to the 1N range which has a resolution of .01 N. From its specs the error introduced during the measurment at the 1N range is 210^-3reading+10^-3range. Now suppose we perform 5 different measurements: ... (more) Comment Post #286873 Im looking something in the newtonian mechanics region of study which is modeled by a system of ordinary differential equations. (more) Edit Post #286869 Initial revision about 1 month ago Question System of ODEs models in physics What examples of a system can be described by a system of ordinary differential equations? (more) Edit Post #286842 Initial revision about 2 months ago Question How are charges sorted? Im designing a capacitor and I have decided to make the surface of 1 plate of the capacitor bigger than the other plate. Image alt text How are the charges sorted through A2?Are they spread out to cover all the surface area of the conductor (A2)? (more) Edit Post #286802 Initial revision 2 months ago Question Calculate inductance using laws of electromagnetism Get 2 conductors and seperate them we can use Gauss's law to calculate the capacitance created by the seperation of the 2 conductors.Can we use other laws of electromagnetism to calculate the inductance of a piece of wire just like we used Gauss's law to calculate the capacitance of the 2 seperated c... (more) 2 months ago Edit Post #286695 Question closed 3 months ago Edit Post #286695 Initial revision 3 months ago Question Meaning of complex frequency If we have a LC high pass filter the transfer function H(s) becomes: $$H(s) = \cfrac{sL}{sL + \cfrac{1}{sC}}$$ If we solve for s to find a pole of the transfer function we get: $$s = j \cfrac{1}{\sqrt{LC}}$$ In the case of a sinusoidal input signal = $s = j \omega \rightarrow \omega ... (more) 3 months ago Edit Post #285688 Initial revision 8 months ago Question Conductivity with angle of light In the lab I changed the angle the light hits a photoresistor and it doesnt obey Lambert's cosine law the conductivity of the photoresistor drops fast from +-20 to +-30 degrees angle.Why? (more) 8 months ago Edit Post #284165 Initial revision about 1 year ago Answer A: Is it plausible to desire a "universal" calendar applicable everywhere in our universe? I dont think anyone can make a universal calendar because time flows more slowly or more fast between different regions in the universe or it can even go backwards if you come close to a rotating black hole. And a planck unit of time is not the smallest amount of time ,the planck time is defined a... (more) about 1 year ago Edit Post #283743 Initial revision about 1 year ago Question Superconductivity In my notes from University the reason a material can exhibit superconducting properties is at really low temperatures electrons form Cooper pairs of which the electrons of the Cooper pairs have lower energy than if they were in the conduction band so which means that they dont collisions with the ca... (more) about 1 year ago Comment Post #283395 @#8056ts we should make this a new question. (more) about 1 year ago Comment Post #283395 @#8056ts we should make this a new question. (more) about 1 year ago Comment Post #283395 Olin photons can't accelerate like most of the objects because photons always move with one speed for all inertial frames. (more) about 1 year ago Comment Post #283393 But how can we find x(t) from F(r) since they have different arguments? (more) about 1 year ago Edit Post #283392 Initial revision about 1 year ago Answer A: Why we can't find a particle accelerating unless there's some other particle accelerating somewhere else? > But, when we are walking, running. We are accelerating. Wrong.When you are walking you're moving at a costant speed. > The third law says we will never find a particle accelerating unless there’s some other particle accelerating somewhere else The book says that because of Netwon's 3r... (more) about 1 year ago Edit Post #283386 Initial revision about 1 year ago Question Find position of a particle at a time If we have a force which changes depending on the position of a particle, how can we find the position of the particle at some time$t\$? We can find its velocity if it has travelled a given distance $$\int^{rf}{ro} F(r)dr = \frac{1}{2} \cdot mp(uf^2 - uo^2)$$ but this equation doesn't inv... (more) Edit Post #283009 Post edited: Edit Post #283251 Post edited: Edit Post #283251 Post edited: Comment Post #283251 I will edit the question then (more) Comment Post #283251 It is an exercise the gravitational field exists only for some r below a limit and a test particle enters somehow(pops into existence) at the edge of the field. (more) Comment Post #283251 It is a exercise it is not the real deal. The gravitational field only exists for some r below a limit. (more) Comment Post #283009 I am assuming for simplicity the B field becomes 0 after the particle reaches the end of the wire. (more) Edit Post #283259 Initial revision about 1 year ago Answer A: Signal modeling as only digital, only analogue, or as both If a signal has a analog component then it analog regardless of if it has a digital component. (more) Comment Post #283258 I don't think this has anything to do with physics. (more) Edit Post #283251 Post edited: Edit Post #283251 Post edited: Comment Post #283251 And i have something else in my mind lets just say the initial acceleration is 0. (more) Edit Post #283251 Post edited: Comment Post #283251 HDE 226868 you are correct i will edit (more) Edit Post #283251 Post edited: Edit Post #283251 Initial revision about 1 year ago Question Find jerk of time varying force This gravitational field we move inside has some distance L after which it becomes 0.Before L it is just like any gravitational field. Suppose we move inside that gravitational field.The acceleration we experience depends on the distance from the planet. $$a\sim\frac{1}{r^2}$$ At t=to we enter t... 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http://math.stackexchange.com/questions/148280/fx-to-y-is-a-homotopy-equivalence-if-there-exists-g-h-y-to-x-with-fg	# $f:X \to Y$ is a homotopy equivalence if there exists $g,h : Y \to X$ with $fg$ and $hf$ homotopy equivalences Let $X$ and $Y$ be topological spaces, and let $f: X \to Y$. I'd like to show that if there are maps $g,h : Y \to X$ such that $fg$ and $hf$ are homotopy equivalences, then $f$ is a homotopy equivalences. $fg$ is a homotopy equivalence means there is some map $k_1 : Y \to Y$ such that $fg k_1 \simeq \mathrm{id}_Y \simeq k_1 fg$. Similarly, there is some map $k_2 : X \to X$ with $hfk_2 \simeq \mathrm{id}_X \simeq k_2 hf$. I want to find a map $k_3 : Y \to X$, comprising of compositions of $f, g, h, k_1$ and $k_2$, such that $f k_3 \simeq \mathrm{id}_Y$ and $k_3 f \simeq \mathrm{id}_X$. I simply cannot find such a $k_3$. Any ideas? Thanks - $[g]=[hfg]=[h]$, hence $[f]$ is invertible... – Rasmus May 22 '12 at 16:20 @Rasmus Sorry, but I don't follow. Why is $[g] = [hfg]$? – Matt May 22 '12 at 16:27 Because $[hfg]=[hf][g]=[id_X][g]=[g]$. – Rasmus May 23 '12 at 8:36 Note that since $1_Y\simeq fgk_1$, we have that $k_2h\simeq k_2hfgk_1\simeq gk_1$. So take $k_3=k_2h\simeq gk_1$. Then: $$fk_3\simeq fgk_1\simeq 1_Y$$ $$k_3f=k_2hf\simeq 1_X$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9793859720230103, "perplexity": 102.67518557911211}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398449145.52/warc/CC-MAIN-20151124205409-00058-ip-10-71-132-137.ec2.internal.warc.gz"}
http://link.springer.com/article/10.1007%2Fs11075-007-9126-y	Numerical Algorithms , Volume 46, Issue 1, pp 45–58 # A second order numerical scheme for the solution of the one-dimensional Boussinesq equation Original Paper DOI: 10.1007/s11075-007-9126-y Bratsos, A.G. Numer Algor (2007) 46: 45. doi:10.1007/s11075-007-9126-y ## Abstract A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived. ### Keywords SolitonBoussinesq equationFinite-difference methodPredictor–corrector ### Mathematics Subject Classifications (2000) 35Q5135Q5365M0678M2065Y10	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8544734120368958, "perplexity": 1303.1824189290287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718278.43/warc/CC-MAIN-20161020183838-00436-ip-10-171-6-4.ec2.internal.warc.gz"}
http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html	## Tuesday, May 17, 2016 ### Dirichlet's theorem There have been some calls for an informal exposition of the recent formal proof of Dirichlet's theorem: Theorem 9.4.1 (dirith, Dirichlet's theorem): If $N\in\Bbb N$ and $(A,N)=1$, then there are infinitely many primes $p$ such that $N\mid p-A$. If you have a copy of Shapiro, "Introduction to the Theory of Numbers", open it to p. 375, Theorem 9.4.1, because the proof follows the book closely. Otherwise, I will do my best to explain the steps, as well as a few things that Shapiro glossed over. The major players in the proof are the von Mangoldt function $\Lambda(n)$, which is defined by $\Lambda(p^\alpha)=\log p$ if $p^\alpha$ is a prime power and $\Lambda(n)=0$ otherwise, the Möbius function $\mu(n)$, which is $1$ if $n$ is a squarefree number with an even number of factors, $-1$ if it is squarefree with an odd number of factors, and $0$ otherwise, and the Dirichlet characters $\chi(n)$, which are completely multiplicative functions on $\Bbb Z/N\Bbb Z$ that are nonzero on the units. Facts about these functions will be introduced as we go along. The starting point is the following theorem (vmadivsum, Equation 9.2.13): $$\sum_{n\le x}\frac{\Lambda(n)}n=\log x+O(1).$$ Proof: By definition, $\psi(x)=\sum_{n\le x}\Lambda(n)$ (this is the second Chebyshev function), and from chpo1ub $\psi(x)=O(x)$. Also (logfac2), $$\log\lfloor x\rfloor!=\sum_{n\le x}\Lambda(n)\left\lfloor\frac xn\right\rfloor.$$ We can put these together via \begin{align} \sum_{n\le x}\frac{\Lambda(n)}n &= \frac1x\sum_{n\le x}\Lambda(n)\frac xn\\ &=\frac1x\sum_{n\le x}\Lambda(n)\left\lfloor\frac xn\right\rfloor + \frac1x\sum_{n\le x}\Lambda(n)\left\{\frac xn\right\}\\ &=\frac1x\log\lfloor x\rfloor! + \frac1x O(\sum_{n\le x}\Lambda(n))\\ &=\frac1x(x\log x+O(x)) + \frac1x O(\psi(x))\\ &=\log x+O(1). \end{align} (We also used the asymptotic approximation of the factorial (logfaco1), $\log\lfloor x\rfloor!=x\log x+O(x)$ in this equation.) $$\tag{\blacksquare}$$ Our goal will be to strengthen this to talk about a specific progression $\bmod N$ (we will hold $N$ fixed throughout the proof) to achieve rpvmasum, Equation 9.4.3: $$\phi(N)\sum_{\substack{n\le x\\n\equiv A}}\frac{\Lambda(n)}n=\log x+O(1).$$ Once we have established this, the main theorem is simple: Theorem (rplogsum, Equation 9.4.4): $$\phi(N)\sum_{\substack{p\le x\\p\equiv A}}\frac{\log p}p=\log x+O(1).$$ Proof: (Note that $p$ is always a variable over primes here.) \begin{align} \sum_{\substack{n\le x\\n\equiv A}}\frac{\Lambda(n)}n-\sum_{\substack{p\le x\\p\equiv A}}\frac{\log p}p &= \sum_{\substack{p^\alpha\le x, \alpha>1\\p^\alpha\equiv A}}\frac{\log p}{p^\alpha} \le \sum_{p^\alpha\le x, \alpha>1}\frac{\log p}{p^\alpha}\\ &\le \sum_{p\in\Bbb P}\sum_{\alpha=2}^\infty\frac{\log p}{p^\alpha} = \sum_{p\in\Bbb P}\frac{\log p}{p(p-1)}\\ &\le \sum_{n=2}^\infty\frac{\log n}{n(n-1)} \le \sum_{n=2}^\infty\frac{\sqrt{n-1}}{n(n-1)}\\ &\le \sum_{n=2}^\infty\left(\frac2{\sqrt{n-1}}-\frac2{\sqrt n}\right)=2 \end{align} Thus $\phi(N)\sum_{n\le x,n\equiv A}\frac{\Lambda(n)}n=\log x+O(1)$ implies $\phi(N)\sum_{p\le x,p\equiv A}\frac{\log p}p=\log x+O(1)$. But this implies that $\sum_{p\le x,p\equiv A}\frac{\log p}p$ diverges, and since a finite sum cannot diverge, this means that $\{p\in\Bbb P\mid p\equiv A\pmod N\}$ is infinite. $$\tag{\blacksquare}$$ We are left with the task of establishing rpvmasum. At this point we turn to Dirichlet characters as a "better basis" from which to weight the sum. This works because of the relation (sum2dchr) $$\sum_{\chi}\bar\chi(A)\chi(n)=\begin{cases}\phi(N)&n\equiv A\\0&o.w.\end{cases}$$ (where the sum is taken over all Dirichlet characters $\chi$). Using this equation we can rewrite the sum as $$\sum_{\substack{n\le x\\n\equiv A}}\frac{\Lambda(n)}n=\sum_{\chi}\bar\chi(A)\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n.$$ There are two cases. The $\log x$ term comes from the principal character $\chi=\chi_0$ (written $\chi=\bf 1$ in Metamath): Theorem (rpvmasumlem): $$\sum_{n\le x}\frac{\chi_0(n)\Lambda(n)}n=\log x+O(1)$$ Proof: \begin{align} \sum_{n\le x}\frac{\chi_0(n)\Lambda(n)}n &=\sum_{\substack{n\le x\\(n,N)=1}}\frac{\Lambda(n)}n\\ &=\sum_{n\le x}\frac{\Lambda(n)}n-\sum_{\substack{n\le x\\(n,N)>1}}\frac{\Lambda(n)}n\\ &=\log x+O(1)-\sum_{\substack{p^\alpha\le x\\(p^\alpha,N)>1}}\frac{\log p}{p^\alpha} \end{align} If $(p^\alpha,N)>1$, then $p$ is a divisor of $N$, and there are only finitely many of those. Thus $$\sum_{\substack{p^\alpha\le x\\(p^\alpha,N)>1}}\frac{\log p}{p^\alpha} \le\sum_{p\mid N}\sum_{\alpha=1}^\infty\frac{\log p}{p^\alpha} =\sum_{p\mid N}\frac{\log p}{p-1}$$ so $\sum_{n\le x,(n,N)>1}\frac{\Lambda(n)}n=O(1)$ and $\sum_{n\le x}\frac{\chi_0(n)\Lambda(n)}n=\log x+O(1)$. $$\tag{\blacksquare}$$ The other case (the hard case) is when $\chi\ne\chi_0$. Fix a nonprincipal character $\chi$. Our goal is to show (dchrvmasum, Equation 9.4.8) $$\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n=O(1).$$ From this it follows that \begin{align} \sum_{\substack{n\le x\\n\equiv A}}\frac{\Lambda(n)}n &=\sum_{n\le x}\frac{\chi_0(n)\Lambda(n)}n+\sum_{\chi\ne\chi_0}\bar\chi(A)\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n\\ &=(\log x+O(1))+\sum_{\chi\ne\chi_0}\bar\chi(A)O(1)\\ &=\log x+O(1), \end{align} which will prove vmadivsum. Theorem (vmasum, Equation 9.2.4): $$\log n=\sum_{d\mid n}\Lambda(d)$$ Proof: \begin{align} \sum_{d\mid n}\Lambda(d) &=\sum_{p^\alpha\mid n}\log p =\sum_{p}\sum_{\alpha\le\nu_p(n)}\log p\\ &=\sum_{p}\nu_p(n)\log p =\log\prod_{p} p^{\nu_p(n)}=\log n \end{align} where $\nu_p(n)$ (written in Metamath as $(p\mbox{ pCnt }n)$) is the exponent of $p$ in the prime factorization of $n$. $$\tag{\blacksquare}$$ Using Möbius inversion (muinv) on this equation gives $$\Lambda(n)=\sum_{d\mid n}\mu(d)\log\frac nd,$$ hence (dchrvmasumlem1) \begin{align} \sum_{n\le x}\frac{\chi(n)\Lambda(n)}n &=\sum_{n\le x}\sum_{d\mid n}\frac{\chi(n)\mu(d)}n\log\frac nd\\ &=\sum_{d\le x}\sum_{m\le x/d}\frac{\chi(dm)\mu(d)}{dm}\log m\\ &=\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}\sum_{m\le x/d}\frac{\chi(m)}{m}\log m. \end{align} We have a similar equation for $\log x$ (dchrvmasum2lem): \begin{align} \log x&=\sum_{n\le x}\frac{\chi(n)}n\log\frac xn\sum_{d\mid n}\mu(d)\\ &=\sum_{n\le x}\sum_{d\mid n}\frac{\chi(n)\mu(d)}n\log\frac xn\\ &=\sum_{d\le x}\sum_{m\le x/d}\frac{\chi(dm)\mu(d)}{dm}\log\frac x{dm}\\ &=\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}\sum_{m\le x/d}\frac{\chi(m)}{m}\log\frac x{dm}. \end{align} The first step is an application of musum to collapse the double sum to just the $n=1$ term. Combining these two yields the equation $$\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n+\log x=\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}\sum_{m\le x/d}\frac{\chi(m)}{m}\log\frac xd.$$ Our next task is to get an asymptotic expression for the sums $\sum_{m\le z}\frac{\chi(m)}{m}$ and $\sum_{m\le z}\frac{\chi(m)}{m}\log m$ appearing here. To that end, we introduce the following lemma: Theorem (dchrisum, Theorem 9.4.1): Let $f:\Bbb R^+\to\Bbb R$ be an eventually monotonically decreasing function (that is, for some $M$, $M\le x\le y$ implies $f(x)\ge f(y)$) which converges to $0$. Then $\sum_n\chi(n)f(n)$ converges to some $L$, and for some $c$, $$x\ge M\implies\|\sum_{n\le x}\chi(n)f(n)-L\|\le cf(x).$$ Proof: It suffices to show that $\sum_{a\le n<b}\chi(n)f(n)=O(f(x))$ (for positive integers $a,b$ with $x\le a\le b$), because then the sequence is Cauchy and so converges to a limit $L$, and since the entire sequence above $x$ is inside the closed set $\{z\in\Bbb C\mid\|\sum_{n\le x}\chi(n)f(n)-z\|\le f(x)\}$, so is the limit. Let $R(k)=\sum_{n=0}^{k-1}\chi(n)$. Then $R(N)$ sums over all the equivalence classes of $\Bbb Z/N\Bbb Z$ exactly once, so $R(N)=\sum_{i\in\Bbb Z/N\Bbb Z}\chi(i)$, and since $\chi$ is nonprincipal, by dchrsum $R(N)=0$. Let $C\ge\|R(k)\|$ for all $0\le k<N$. (This is a finite set, so there is a finite supremum.) Then setting $k=aN+b$ where $b<N$, we have \begin{align} R(k)&=\sum_{i=0}^{a-1}\sum_{n=0}^{N-1}\chi(iN+n)+\sum_{n=0}^{b-1}\chi(aN+n)\\ &=\sum_{i=0}^{a-1}\sum_{n=0}^{N-1}\chi(n)+\sum_{n=0}^{b-1}\chi(n)\\ &=\sum_{i=0}^{a-1}R(N)+R(b)=R(b), \end{align} and $C\ge\|R(b)\|$; hence $C\ge\|R(k)\|$ for all $k$. Using summation by parts, \begin{align} \|\sum_{a\le n<b}\chi(n)f(n)\| &=\|\sum_{a\le n<b}(R(n+1)-R(n))f(n)\|\\ &=\|R(b)f(b)-R(a)f(a)-\sum_{a\le n<b}R(n+1)(f(n+1)-f(n))\|\\ &\le Cf(b)+Cf(a)+\sum_{a\le n<b}C\|f(n+1)-f(n)\|\\ &=Cf(b)+Cf(a)+\sum_{a\le n<b}C(f(n)-f(n+1))\\ &=Cf(b)+Cf(a)+C(f(a)-f(b))\le 2Cf(x). \end{align} $$\tag{\blacksquare}$$ With dchrisum in hand we can apply it to the decreasing functions $f(x)=\frac{\log x}x$ and $f(x)=\frac1x$; let $$L_0=\sum_{n=1}^\infty\frac{\chi(n)}n\qquad L_1=\sum_{n=1}^\infty\frac{\chi(n)\log n}n$$ be the limits of the series that result. Theorem (dchrmusum2, Lemma 9.4.2): $$L_0\sum_{n\le x}\frac{\chi(n)\mu(n)}n=O(1).$$ Proof: We have: \begin{align} 1&=\sum_{n\le x}\sum_{d\mid n}\frac{\chi(n)\mu(d)}n\\ &=\sum_{d\le x}\sum_{m\le x/d}\frac{\chi(dm)\mu(d)}{dm}\\ &=\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}\sum_{m\le x/d}\frac{\chi(m)}m \end{align} dchrisum provides that $\|\sum_{m\le z}\frac{\chi(m)}m-L_0\|\le\frac Cz$ for some $C$ and all $z$. Then: \begin{align} \|L_0\sum_{n\le x}\frac{\chi(n)\mu(n)}n\| &\le1+\|1-L_0\sum_{n\le x}\frac{\chi(n)\mu(n)}n\|\\ &=1+\|\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}(\sum_{m\le x/d}\frac{\chi(m)}m-L_0)\|\\ &\le1+\sum_{d\le x}\frac1d\|\sum_{m\le x/d}\frac{\chi(m)}m-L_0\|\\ &\le1+\sum_{d\le x}\frac Cx\le 1+C. \end{align} $$\tag{\blacksquare}$$ We will be doing a lot of case analysis on $L_0=0$, so let the notation $\left[a\atop b\right]$ denote $$\left[a\atop b\right]=\begin{cases}a&L_0=0\\b&L_0\ne0\end{cases}.$$ Theorem (dchrvmasumif, Equation 9.4.24): $$\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n=-\left[\log x\atop 0\right]+O(1).$$ Proof: Earlier, we showed that $$\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n+\left[\log x\atop 0\right]= \sum_{d\le x}\frac{\chi(d)\mu(d)}{d}\sum_{m\le x/d}\frac{\chi(m)}{m}\log\left[x/d\atop m\right].$$ We will show that this latter sum is bounded. Let $g(x)=\sum_{n\le x}\frac{\chi(n)}n\log\left[x\atop n\right]$, so that the sum in question is $\sum_{d\le x}\frac{\chi(d)\mu(d)}{d}g(x/d)$. Let us show that $$g(x)=\left[0\atop L_1\right]+O\left(\frac{\log x}x\right).$$ In the case $L_0=0$, we have $$g(x)=\sum_{n\le x}\frac{\chi(n)}n\log x=\left(L_0+O\left(\frac1x\right)\right)\log x=O\left(\frac{\log x}x\right),$$ and if $L_0\ne0$, we have $$g(x)=\sum_{n\le x}\frac{\chi(n)\log n}n=L_1+O\left(\frac{\log x}x\right).$$ Since $\left[0\atop L_1/L_0\right]L_0=\left[0\atop L_1\right]$ and $L_0\sum_{d\le x}\frac{\chi(d)\mu(d)}d=O(1)$, we also have $\left[0\atop L_1\right]\sum_{d\le x}\frac{\chi(d)\mu(d)}d=O(1)$. \begin{align} \sum_{n\le x}\frac{\chi(n)\Lambda(n)}n+\left[\log x\atop 0\right] &=\sum_{d\le x}\frac{\chi(d)\mu(d)}dg(x/d)\\ &=O(1)+\sum_{d\le x}\frac{\chi(d)\mu(d)}d\left(g(x/d)-\left[0\atop L_1\right]\right) \end{align} Now suppose the big-O constants take $x\ge A\implies \left\|g(x)-\left[0\atop L_1\right]\right\|\le C\frac{\log x}x$, with $A\ge1$, and set $D=\sum_{n<A}\frac{\log A}n$. Then $x<A$ implies \begin{align} \|g(x)\|&\le\sum_{n\le x}\frac1n\left\|\log\left[x\atop n\right]\right\|\\ &\le\sum_{n\le x}\frac1n\log A\le\sum_{n<A}\frac1n\log A=D\\ \end{align} Then: \begin{align} \left\|\sum_{d\le x}\frac{\chi(d)\mu(d)}d\left(g(x/d)-\left[0\atop L_1\right]\right)\right\| &\le\sum_{d\le x}\frac1d\left\|g(x/d)-\left[0\atop L_1\right]\right\|\\ &\le\sum_{d\le x/A}\frac Cd\frac{\log(x/d)}{x/d}+\sum_{x/A<d\le x}\frac Dd\\ &\le\frac Cx\sum_{d\le x}(\log x-\log d)+D(\log x-\log(x/A)+1)\\ &\le\frac Cx(x\log x-\log\lfloor x\rfloor!)+D(\log A+1) \end{align} which is bounded using $\log\lfloor x\rfloor!=x\log x+O(x)$. $$\tag{\blacksquare}$$ We're almost there: if we can prove $L_0\ne0$ unconditionally, dchrvmasumif will give us what we want. Replay the derivation of rpvmasum, with $A=1$ so that the $\bar\chi(A)$ factors disappear: \begin{align} \sum_{\substack{n\le x\\n\equiv 1}}\frac{\Lambda(n)}n &=\sum_{n\le x}\frac{\chi_0(n)\Lambda(n)}n+\sum_{\chi\ne\chi_0}\sum_{n\le x}\frac{\chi(n)\Lambda(n)}n\\ &=(\log x+O(1))+\sum_{\chi\ne\chi_0}(-\left[\log x\atop 0\right]+O(1))\\ &=(1-\|W\|)\log x+O(1), \end{align} where $W$ is the set of all $\chi\ne\chi_0$ such that $L_0(\chi)\ne 0$. Since $\sum_{n\le x,n\equiv 1}\frac{\Lambda(n)}n$ is nonnegative, we have a contradiction if $\|W\|>1$ (because the $(1-\|W\|)\log x+O(1)$ will eventually be negative). Thus there is at most one $\chi$ such that $L_0(\chi)=0$. Now fix $\chi$ again and assume $L_0=0$. Because $\bar\chi$ is also a Dirichlet character, and $L_0(\bar\chi)=\overline{L_0(\chi)}=0$, we must have $\bar\chi=\chi$, so $\chi$ is a real Dirichlet character (dchrisum0re). Now let $F(n)=\sum_{d\mid n}\chi(d)$. Because a divisor sum of a multiplicative function is multiplicative (fsumdvdsmul), $F$ is multiplicative (dchrisum0fmul). (Unlike $\chi$ it is not completely multiplicative.) Theorem (dchrisum0flb, Equation 9.4.29): $$F(n)\ge\begin{cases}1&\sqrt n\in\Bbb N\\0&o.w.\end{cases}.$$ Proof: We begin by establishing the theorem for prime powers $p^\alpha$, $\alpha\ge0$. In this case $\sqrt{p^\alpha}\in\Bbb N$ iff $\alpha$ is even. Now $$F(p^\alpha)=\sum_{d\mid p^\alpha}\chi(d)=\sum_{k\le\alpha}\chi(p^k)=\sum_{k\le\alpha}\chi(p)^k,$$ and since $\chi$ is a real Dirichlet character it can only take the values $-1,0,1$. If $\chi(p)=1$, then $F(p^\alpha)=\alpha+1\ge1$; if $\chi(p)=0$, then $F(p^\alpha)=1\ge1$; and if $\chi(p)=-1$ then $$F(p^\alpha)=\sum_{k\le\alpha}(-1)^k=\frac{1+(-1)^{\alpha}}2$$ which is $1$ when $\alpha$ is even and $0$ if it is odd. Thus in any case the theorem is proven for prime powers. Now to cover the general case, by complete induction. If $n=1$ then it is a prime power so the theorem is proven; otherwise let $p$ be a prime divisor of $n$, and let $n=p^\alpha k$ where $p\nmid k$, so $F(n)=F(p^\alpha)F(k)$. We have proven that $F(p^\alpha)$ satisfies the theorem, and $F(k)$ satisfies the theorem by the induction hypothesis, because $k<n$. The product of two nonnegatives is nonnegative, so $F(n)\ge0$. If $n$ is a square then $\nu_p(n)=\alpha$ is even, so $p^\alpha$ is a square, and so $k$ is a quotient of squares, which is thus also a square. Thus $F(p^\alpha),F(k)\ge1$, so $F(n)\ge1$. $$\tag{\blacksquare}$$ Therefore, $$\sum_{n\le x}\frac{F(n)}{\sqrt n}\ge\sum_{m^2\le x}\frac{F(m^2)}{m}\ge\sum_{m\le\sqrt x}\frac1m\ge\frac12\log x,$$ so $\sum_{n\le x}\frac{F(n)}{\sqrt n}$ is unbounded. Theorem (dchrisumn0, Lemma 9.4.4): $L_0\ne0$. Proof: We will show that $L_0=0$ implies $\sum_{n\le x}\frac{F(n)}{\sqrt n}$ is bounded. We have: \begin{align} \sum_{n\le x}\frac{F(n)}{\sqrt n} &=\sum_{n\le x}\sum_{d\mid n}\frac{\chi(d)}{\sqrt n}\\ &=\sum_{d\le x}\sum_{m\le x/d}\frac{\chi(d)}{\sqrt{dm}}\\ &=\sum_{d\le\sqrt x}\sum_{m\le x/d}\frac{\chi(d)}{\sqrt{dm}} +\sum_{m\le\sqrt x}\sum_{\sqrt x<d\le x/m}\frac{\chi(d)}{\sqrt{dm}}\\ &=\sum_{d\le y}\sum_{m\le y^2/d}\frac{\chi(d)}{\sqrt{dm}} +\sum_{m\le y}\sum_{y<d\le y^2/m}\frac{\chi(d)}{\sqrt{dm}} \end{align} (changing variables $y\mapsto\sqrt x$ in the last step). Using dchrisum once more, we obtain $L_2$ such that $\|\sum_{n\le x}\frac{\chi(d)}{\sqrt d}-L_2\|\le\frac C{\sqrt x}$. Then \begin{align} \|\sum_{m\le y}\sum_{y<d\le y^2/m}\frac{\chi(d)}{\sqrt{dm}}\| &\le\frac{2C}{\sqrt y}\sum_{m\le y}\frac1{\sqrt m}. \end{align} We need here the asymptotic estimate divsqrsum for $\sum_{m\le x}\frac1{\sqrt m}=2\sqrt x+c+O(x^{-1/2})$, so that we get $$\|\sum_{m\le y}\sum_{y<d\le y^2/m}\frac{\chi(d)}{\sqrt{dm}}\|\le4C+O(\frac1{\sqrt y})=O(1).$$ For the other sum, we can write \begin{align} \sum_{d\le y}\frac{\chi(d)}{\sqrt d}\sum_{m\le y^2/d}\frac1{\sqrt m} &=\sum_{d\le y}\frac{\chi(d)}{\sqrt d}\left(2\frac{y}{\sqrt d}+c+O(\frac{\sqrt d}{y})\right)\\ &=2y\sum_{d\le y}\frac{\chi(d)}{d}+c\sum_{d\le y}\frac{\chi(d)}{\sqrt d}+O(1)\\ &=2y(L_0+O(\frac1y))+c(L_2+O(\frac1{\sqrt y}))+O(1)=2yL_0+O(1)=O(1). \end{align} Thus $\sum_{n\le x}\frac{F(n)}{\sqrt n}=O(1)$, a contradiction. Thus our assumption $L_0=0$ was false, so $L_0\ne0$. $$\tag{\blacksquare}$$ Then dchrvmasumif simplifies to dchrvmasum, and we are done. $$\tag{\blacksquare}$$ #### 1 comment: 1. 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https://cs.stackexchange.com/questions/93428/identification-of-formal-language	Identification of Formal Language $$L = \{a^{m+n}b^{m+k}c^{n+k}\mid m,n,k\ge 1\}.$$ Is $L$ DCFL or not? According to me it should be DCFL since we can write $L$ as $\{a^{n}a^{m}b^{m}b^{k}c^{k}c^{n}\mid m,n,k\ge1\}$. So, now after push and pop on the stack, it will be empty. So, according to me it will be DCFL but I have doubt since DPDA can't count $n$ and $m$ after reading symbol $a$ so machine can't know how many $a$'s should have to be popped. According to this logic, one stack is not enough for this language. So, it should be CSL. Please tell me whether it should be DCFL or CSL? • Those are not the only two possibilities. It could be context-free but not deterministic (and probably is). – rici Jun 24 '18 at 6:20 • @rici , could you please explain how it is CFL , but not DCFL ? – user89917 Jun 24 '18 at 6:55 • Your construction makes it clear how to write a context-free grammar. So it's a CFL. But that grammar is not deterministic. It may be that no deterministic grammar exists in which case the language is not a DCFL. But it's still a CFL. – rici Jun 24 '18 at 6:59 • See Wikipedia: en.m.wikipedia.org/wiki/Deterministic_context-free_language which notes that DCFLs are a proper subset of CFLs. – rici Jun 24 '18 at 7:01 • Your PDA has to guess when to switch from $a^n$ to $a^m$. Deterministically, you can read $a^{m+n} b^{m+k}$ and have a representation of $n-k$ on the stack, but I'm not sure how this is helpful (consider for example the case $n=k$). – Yuval Filmus Jun 24 '18 at 12:25 $L$ is CFL because there is a CFG generating it: \begin{align} S&\rightarrow aAc\\ A&\rightarrow aAc \mid BC\\ B&\rightarrow aDb\\ C&\rightarrow bEc\\ D&\rightarrow aDb \mid \epsilon\\ E&\rightarrow bEc \mid \epsilon \end{align} $L$ is not DCFL. To prove this, we give some lemmas firstly: Lemma 1. \begin{align} L &=\{a^x b^y c^z\mid x,y\ge 2, x+y+z\text{ is even},\|x-y\|+2\le z\le x+y-2\}\\ &=\{a^x b^y c^{\|x-y\|+2+z}\mid x,y\ge 2, z\text{ is even},0\le z\le x+y-\|x-y\|-4\}. \end{align} Proof. Let $L'=\{a^x b^y c^z\mid x,y\ge 2, x+y+z\text{ is even},\|x-y\|+2\le z\le x+y-2\}$. Easy to see $L\subseteq L'$. Now let $a^xb^yc^z\in L'$, then we can choose $m=(x+y-z)/2, n=(x-y+z)/2, k=(y-x+z)/2$. Because $x+y+z$ is even, $m,n,k$ are all integers. Because $\|x-y\|+2\le z\le x+y-2$, we have $m,n,k\ge 1$, so $a^xb^yc^z\in L$. As a result, $L=L'$. Lemma 2. $M =\{a^x b^y c^{\|x-y\|+2}d^z\mid x,y\ge 2, z\text{ is even},z\le x+y-\|x-y\|-4\}$ is not context-free. Proof. Suppose it is context-free, according to Odgen's lemma, there exists some $p\ge 1$ such that $s=a^{p+2}b^{p+2}c^2d^{2p}\in M$ can be written as $s=uvwrt$, such that 1. the number of $d$s in $vwr$ is at most $p$ (i.e. we mark all $d$s), 2. there is at least one $d$ in $vr$, and 3. for all $q\ge 0$, $uv^qwr^qt\in M$. From condition 3, $v$ and $r$ each contains at most one character. Together with condition 2, as $q$ grows, the number of $d$s in $uv^qwr^qt$ grows infinitely. To satisfy the condition $z\le x+y-\|x-y\|-4=2\min\{x,y\}-4$, the number of $a$ and $b$ must both grow, i.e. $a$ and $b$ must both in $vr$. However, $vr$ contains at most two characters, while $d$ is already in it, $vr$ cannot contain both $a$ and $b$, a contradiction. Now let's come back to $L$. Suppose there is a DPDA $D$ accepting $L$, we create two copies $D_1$ and $D_2$ of $D$ and change the input character $c$ in $D_2$ to $d$. We then construct a new PDA $P$ as follows: 1. The states of $P$ are the union of states of $D_1$ and $D_2$. The start state of $P$ is the start state of $D_1$. The accepting states of $P$ are the accepting states of $D_2$. 2. For a transition in $D_1$, if the destination is an accepting state in $D_1$, change the destination to the corresponding state in $D_2$. Other transitions keep unchanged. Now run $P$ on a string $a^xb^yc^{\|x-y\|+2}d^z\in M$. According to lemma 1, after reading $a^xb^yc^{\|x-y\|+2}$, it enters a state in $D_2$ for the first time. Since $a^xb^yc^{\|x-y\|+2+z}\in L$ according to lemma 1, $P$ will finally accept. On the other hand, if a string $s$ is accepted by $P$, let $s=s_1s_2$ where after reading $s_1$, $P$ enters a state in $D_2$ for the first time. According to lemma 1, $s_1$ must have the form $a^xb^yc^{\|x-y\|+2}$ ($x,y\ge 2$). Also, $s_2$ does not contain $c$, and after changing all $d$s in $s_2$ to $c$s, $s_1s_2$ belongs to $L$. This means $s$ must have the form $a^xb^yc^{\|x-y\|+2}d^z$ where $x+y+\|x-y\|+2+z$ is even (i.e. $z$ is even) and $\|x-y\|+2+z\le x+y-2$ (i.e. $z\le x+y-\|x-y\|-4$). Hence $s\in M$. 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https://cplberry.com/category/astrophysics/	Second star to the right and straight on ’til morning—Astrophysics white papers What will be the next big thing in astronomy? One of the hard things about research is that you often don’t know what you will discover before you embark on an investigation. An idea might work out, or it might not, or along the way you might discover something unexpected which is far more interesting. As you might imagine, this can make laying definite plans difficult… However, it is important to have plans for research. While you might not be sure of the outcome, it is necessary to weigh the risks and rewards associated with the probable results before you invest your time and taxpayers’ money! To help with planning and prioritising, researchers in astrophysics often pull together white papers [bonus note]. These are sketches of ideas for future research, arguing why you think they might be interesting. These can then be discussed within the community to help shape the direction of the field. If other scientists find the paper convincing, you can build support which helps push for funding. If there are gaps in the logic, others can point these out to ave you heading the wrong way. This type of consensus building is especially important for large experiments or missions—you don’t want to spend a billion dollars on something unless you’re really sure it is a good idea and lots of people agree. I have been involved with a few white papers recently. Here are some key ideas for where research should go. Ground-based gravitational-wave detectors: The next generation We’ve done some awesome things with Advanced LIGO and Advanced Virgo. In just a couple of years we have revolutionized our understanding of binary black holes. That’s not bad. However, our current gravitational-wave observatories are limited in what they can detect. What amazing things could we achieve with a new generation of detectors? It can take decades to develop new instruments, therefore it’s important to start thinking about them early. Obviously, what we would most like is an observatory which can detect everything, but that’s not feasible. In this white paper, we pick the questions we most want answered, and see what the requirements for a new detector would be. A design which satisfies these specifications would therefore be a solid choice for future investment. Binary black holes are the perfect source for ground-based detectors. What do we most want to know about them? 1. How many mergers are there, and how does the merger rate change over the history of the Universe? We want to know how binary black holes are made. The merger rate encodes lots of information about how to make binaries, and comparing how this evolves compared with the rate at which the Universe forms stars, will give us a deeper understanding of how black holes are made. 2. What are the properties (masses and spins) of black holes? The merger rate tells us some things about how black holes form, but other properties like the masses, spins and orbital eccentricity complete the picture. We want to make precise measurements for individual systems, and also understand the population. 3. Where do supermassive black holes come from? We know that stars can collapse to produce stellar-mass black holes. We also know that the centres of galaxies contain massive black holes. Where do these massive black holes come from? Do they grow from our smaller black holes, or do they form in a different way? Looking for intermediate-mass black holes in the gap in-between will tells us whether there is a missing link in the evolution of black holes. The detection horizon (the distance to which sources can be detected) for Advanced LIGO (aLIGO), its upgrade A+, and the proposed Cosmic Explorer (CE) and Einstein Telescope (ET). The horizon is plotted for binaries with equal-mass, nonspinning components. Adapted from Hall & Evans (2019). What can we do to answer these questions? 1. Increase sensitivity! Advanced LIGO and Advanced Virgo can detect a $30 M_\odot + 30 M_\odot$ binary out to a redshift of about $z \approx 1$. The planned detector upgrade A+ will see them out to redshift $z \approx 2$. That’s pretty impressive, it means we’re covering 10 billion years of history. However, the peak in the Universe’s star formation happens at around $z \approx 2$, so we’d really like to see beyond this in order to measure how the merger rate evolves. Ideally we would see all the way back to cosmic dawn at $z \approx 20$ when the Universe was only 200 million years old and the first stars light up. 2. Increase our frequency range! Our current detectors are limited in the range of frequencies they can detect. Pushing to lower frequencies helps us to detect heavier systems. If we want to detect intermediate-mass black holes of $100 M_\odot$ we need this low frequency sensitivity. At the moment, Advanced LIGO could get down to about $10~\mathrm{Hz}$. The plot below shows the signal from a $100 M_\odot + 100 M_\odot$ binary at $z = 10$. The signal is completely undetectable at $10~\mathrm{Hz}$. The gravitational wave signal from the final stages of inspiral, merger and ringdown of a two 100 solar mass black holes at a redshift of 10. The signal chirps up in frequency. The colour coding shows parts of the signal above different frequencies. Part of Figure 2 of the Binary Black Holes White Paper. 3. Increase sensitivity and frequency range! Increasing sensitivity means that we will have higher signal-to-noise ratio detections. For these loudest sources, we will be able to make more precise measurements of the source properties. We will also have more detections overall, as we can survey a larger volume of the Universe. Increasing the frequency range means we can observe a longer stretch of the signal (for the systems we currently see). This means it is easier to measure spin precession and orbital eccentricity. We also get to measure a wider range of masses. Putting the improved sensitivity and frequency range together means that we’ll get better measurements of individual systems and a more complete picture of the population. How much do we need to improve our observatories to achieve our goals? To quantify this, lets consider the boost in sensitivity relative to A+, which I’ll call $\beta_\mathrm{A+}$. If the questions can be answered with $\beta_\mathrm{A+} = 1$, then we don’t need anything beyond the currently planned A+. If we need a slightly larger $\beta_\mathrm{A+}$, we should start investigating extra ways to improve the A+ design. If we need much larger $\beta_\mathrm{A+}$, we need to think about new facilities. The plot below shows the boost necessary to detect a binary (with equal-mass nonspinning components) out to a given redshift. With a boost of $\beta_\mathrm{A+} = 10$ (blue line) we can survey black holes around $10 M_\odot$$30 M_\odot$ across cosmic time. The boost factor (relative to A+) $\beta_\mathrm{A+}$ needed to detect a binary with a total mass $M$ out to redshift $z$. The binaries are assumed to have equal-mass, nonspinning components. The colour scale saturates at $\log_{10} \beta_\mathrm{A+} = 4.5$. The blue curve highlights the reach at a boost factor of $\beta_\mathrm{A+} = 10$. The solid and dashed white lines indicate the maximum reach of Cosmic Explorer and the Einstein Telescope, respectively. Part of Figure 1 of the Binary Black Holes White Paper. The plot above shows that to see intermediate-mass black holes, we do need to completely overhaul the low-frequency sensitivity. What do we need to detect a $100 M_\odot + 100 M_\odot$ binary at $z = 10$? If we parameterize the noise spectrum (power spectral density) of our detector as $S_n(f) = S_{10}(f/10~\mathrm{Hz})^\alpha$ with a lower cut-off frequency of $f_\mathrm{min}$, we can investigate the various possibilities. The plot below shows the possible combinations of parameters which meet of requirements. Requirements on the low-frequency noise power spectrum necessary to detect an optimally oriented intermediate-mass binary black hole system with two 100 solar mass components at a redshift of 10. Part of Figure 2 of the Binary Black Holes White Paper. To build up information about the population of black holes, we need lots of detections. Uncertainties scale inversely with the square root of the number of detections, so you would expect few percent uncertainty after 1000 detections. If we want to see how the population evolves, we need these many per redshift bin! The plot below shows the number of detections per year of observing time for different boost factors. The rate starts to saturate once we detect all the binaries in the redshift range. This is as good as you’ll ever going to get. Expected rate of binary black hole detections $R_\mathrm{det}$ per redshift bin as a function of A+ boost factor $\beta_\mathrm{A+}$ for three redshift bins. The merging binaries are assumed to be uniformly distributed with a constant merger rate roughly consistent with current observations: the solid line is about the current median, while the dashed and dotted lines are roughly the 90% bounds. Figure 3 of the Binary Black Holes White Paper. Looking at the plots above, it is clear that A+ is not going to satisfy our requirements. We need something with a boost factor of $\beta_\mathrm{A+} = 10$: a next-generation observatory. Both the Cosmic Explorer and Einstein Telescope designs do satisfy our goals. Title: Deeper, wider, sharper: Next-generation ground-based gravitational-wave observations of binary black holes arXiv: 1903.09220 [astro-ph.HE] Theme music: Daft Punk Extreme mass ratio inspirals are awesome We have seen gravitational waves from a stellar-mass black hole merging with another stellar-mass black hole, can we observe a stellar-mass black hole merging with a massive black hole? Yes, these are a perfect source for a space-based gravitational wave observatory. We call these systems extreme mass-ratio inspirals (or EMRIs, pronounced em-rees, for short) [bonus note]. Having such an extreme mass ratio, with one black hole much bigger than the other, gives EMRIs interesting properties. The number of orbits over the course of an inspiral scales with the mass ratio: the more extreme the mass ratio, the more orbits there are. Each of these gives us something to measure in the gravitational wave signal. A short section of an orbit around a spinning black hole. While inspirals last for years, this would represent only a few hours around a black hole of mass $M = 10^6 M_\odot$. The position is measured in terms of the gravitational radius $r_\mathrm{g} = GM/c^2$. The innermost stable orbit for this black hole would be about $r_\mathrm{g} = 2.3$. Part of Figure 1 of the EMRI White Paper. As EMRIs are so intricate, we can make exquisit measurements of the source properties. These will enable us to: • Measure the masses of both black holes to better than 10% precision • Reconstruct the mass distribution of massive black holes out to redshift $z \gtrsim 4$ • Measure massive black hole spins to a precision of better than 0.001, giving us an insight into how they formed • Perform precision tests of the no-hair theorem describing black holes in general relativity, and test alternative theories of gravity in the strong-field regime • Cross-correlate locations with galaxy catalogues to measure the expansion of the Universe Event rates for EMRIs are currently uncertain: there could be just one per year or thousands. From the rate we can figure out the details of what is going in in the nuclei of galaxies, and what types of objects you find there. With EMRIs you can unravel mysteries in astrophysics, fundamental physics and cosmology. Have we sold you that EMRIs are awesome? Well then, what do we need to do to observe them? There is only one currently planned mission which can enable us to study EMRIs: LISA. To maximise the science from EMRIs, we have to support LISA. As an aspiring scientist, Lisa Simpson is a strong supporter of the LISA mission. Credit: Fox Title: The unique potential of extreme mass-ratio inspirals for gravitational-wave astronomy arXiv: 1903.03686 [astro-ph.HE] Theme music: Muse Bonus notes White paper vs journal article Since white papers are proposals for future research, they aren’t as rigorous as usual academic papers. They are really attempts to figure out a good question to ask, rather than being answers. White papers are not usually peer reviewed before publication—the point is that you want everybody to comment on them, rather than just one or two anonymous referees. Whilst white papers aren’t quite the same class as journal articles, they do still contain some interesting ideas, so I thought they still merit a blog post. Recycling I have blogged about EMRIs before, so I won’t go into too much detail here. It was one of my former blog posts which inspired the LISA Science Team to get in touch to ask me to write the white paper. The O2 Catalogue—It goes up to 11 The full results of our second advanced-detector observing run (O2) have now been released—we’re pleased to announce four new gravitational wave signals: GW170729, GW170809, GW170818 and GW170823 [bonus note]. These latest observations are all of binary black hole systems. Together, they bring our total to 10 observations of binary black holes, and 1 of a binary neutron star. With more frequent detections on the horizon with our third observing run due to start early 2019, the era of gravitational wave astronomy is truly here. The population of black holes and neutron stars observed with gravitational waves and with electromagnetic astronomy. You can play with an interactive version of this plot online. The new detections are largely consistent with our previous findings. GW170809, GW170818 and GW170823 are all similar to our first detection GW150914. Their black holes have masses around 20 to 40 times the mass of our Sun. I would lump GW170104 and GW170814 into this class too. Although there were models that predicted black holes of these masses, we weren’t sure they existed until our gravitational wave observations. The family of black holes continues out of this range. GW151012, GW151226 and GW170608 fall on the lower mass side. These overlap with the population of black holes previously observed in X-ray binaries. Lower mass systems can’t be detected as far away, so we find fewer of these. On the higher end we have GW170729 [bonus note]. Its source is made up of black holes with masses $50.7^{+16.3}_{-10.2} M_\odot$ and $34.4^{+8.9}_{-10.2} M_\odot$ (where $M_\odot$ is the mass of our Sun). The larger black hole is a contender for the most massive black hole we’ve found in a binary (the other probable contender is GW170823’s source, which has a $39.5^{+10.0}_{-6.6} M_\odot$ black hole). We have a big happy family of black holes! Of the new detections, GW170729, GW170809 and GW170818 were both observed by the Virgo detector as well as the two LIGO detectors. Virgo joined O2 for an exciting August [bonus note], and we decided that the data at the time of GW170729 were good enough to use too. Unfortunately, Virgo wasn’t observing at the time of GW170823. GW170729 and GW170809 are very quiet in Virgo, you can’t confidently say there is a signal there [bonus note]. However, GW170818 is a clear detection like GW170814. Well done Virgo! Using the collection of results, we can start understand the physics of these binary systems. We will be summarising our findings in a series of papers. A huge amount of work went into these. The papers The O2 Catalogue Paper Title: GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs arXiv: 1811.12907 [astro-ph.HE] Data: Catalogue; Parameter estimation results LIGO science summary: GWTC-1: A new catalog of gravitational-wave detections The paper summarises all our observations of binaries to date. It covers our first and second observing runs (O1 and O2). This is the paper to start with if you want any information. It contains estimates of parameters for all our sources, including updates for previous events. It also contains merger rate estimates for binary neutron stars and binary black holes, and an upper limit for neutron star–black hole binaries. We’re still missing a neutron star–black hole detection to complete the set. More details: The O2 Catalogue Paper The O2 Populations Paper Title: Binary black hole population properties inferred from the first and second observing runs of Advanced LIGO and Advanced Virgo arXiv: 1811.12940 [astro-ph.HE] Using our set of ten binary black holes, we can start to make some statistical statements about the population: the distribution of masses, the distribution of spins, the distribution of mergers over cosmic time. With only ten observations, we still have a lot of uncertainty, and can’t make too many definite statements. However, if you were wondering why we don’t see any more black holes more massive than GW170729, even though we can see these out to significant distances, so are we. We infer that almost all stellar-mass black holes have masses less than $45 M_\odot$. More details: The O2 Populations Paper The O2 Catalogue Paper Synopsis: O2 Catalogue Paper Read this if: You want the most up-to-date gravitational results Favourite part: It’s out! We can tell everyone about our FOUR new detections This is a BIG paper. It covers our first two observing runs and our main searches for coalescing stellar mass binaries. There will be separate papers going into more detail on searches for other gravitational wave signals. The instruments Gravitational wave detectors are complicated machines. You don’t just take them out of the box and press go. We’ll be slowly improving the sensitivity of our detectors as we commission them over the next few years. O2 marks the best sensitivity achieved to date. The paper gives a brief overview of the detector configurations in O2 for both LIGO detectors, which did differ, and Virgo. During O2, we realised that one source of noise was beam jitter, disturbances in the shape of the laser beam. This was particularly notable in Hanford, where there was a spot on the one of the optics. Fortunately, we are able to measure the effects of this, and hence subtract out this noise. This has now been done for the whole of O2. It makes a big difference! Derek Davis and TJ Massinger won the first LIGO Laboratory Award for Excellence in Detector Characterization and Calibration™ for implementing this noise subtraction scheme (the award citation almost spilled the beans on our new detections). I’m happy that GW170104 now has an increased signal-to-noise ratio, which means smaller uncertainties on its parameters. The searches We use three search algorithms in this paper. We have two matched-filter searches (GstLAL and PyCBC). These compare a bank of templates to the data to look for matches. We also use coherent WaveBurst (cWB), which is a search for generic short signals, but here has been tuned to find the characteristic chirp of a binary. Since cWB is more flexible in the signals it can find, it’s slightly less sensitive than the matched-filter searches, but it gives us confidence that we’re not missing things. The two matched-filter searches both identify all 11 signals with the exception of GW170818, which is only found by GstLAL. This is because PyCBC only flags signals above a threshold in each detector. We’re confident it’s real though, as it is seen in all three detectors, albeit below PyCBC’s threshold in Hanford and Virgo. (PyCBC only looked at signals found in coincident Livingston and Hanford in O2, I suspect they would have found it if they were looking at all three detectors, as that would have let them lower their threshold). The search pipelines try to distinguish between signal-like features in the data and noise fluctuations. Having multiple detectors is a big help here, although we still need to be careful in checking for correlated noise sources. The background of noise falls off quickly, so there’s a rapid transition between almost-certainly noise to almost-certainly signal. Most of the signals are off the charts in terms of significance, with GW170818, GW151012 and GW170729 being the least significant. GW170729 is found with best significance by cWB, that gives reports a false alarm rate of $1/(50~\mathrm{yr})$. Cumulative histogram of results from GstLAL (top left), PyCBC (top right) and cWB (bottom). The expected background is shown as the dashed line and the shaded regions give Poisson uncertainties. The search results are shown as the solid red line and named gravitational-wave detections are shown as blue dots. More significant results are further to the right of the plot. Fig. 2 and Fig. 3 of the O2 Catalogue Paper. The false alarm rate indicates how often you would expect to find something at least as signal like if you were to analyse a stretch of data with the same statistical properties as the data considered, assuming that they is only noise in the data. The false alarm rate does not fold in the probability that there are real gravitational waves occurring at some average rate. Therefore, we need to do an extra layer of inference to work out the probability that something flagged by a search pipeline is a real signal versus is noise. The results of this calculation is given in Table IV. GW170729 has a 94% probability of being real using the cWB results, 98% using the GstLAL results, but only 52% according to PyCBC. Therefore, if you’re feeling bold, you might, say, only wager the entire economy of the UK on it being real. We also list the most marginal triggers. These all have probabilities way below being 50% of being real: if you were to add them all up you wouldn’t get a total of 1 real event. (In my professional opinion, they are garbage). However, if you want to check for what we might have missed, these may be a place to start. Some of these can be explained away as instrumental noise, say scattered light. Others show no obvious signs of disturbance, so are probably just some noise fluctuation. The source properties We give updated parameter estimates for all 11 sources. These use updated estimates of calibration uncertainty (which doesn’t make too much difference), improved estimate of the noise spectrum (which makes some difference to the less well measured parameters like the mass ratio), the cleaned data (which helps for GW170104), and our most currently complete waveform models [bonus note]. This plot shows the masses of the two binary components (you can just make out GW170817 down in the corner). We use the convention that the more massive of the two is $m_1$ and the lighter is $m_2$. We are now really filling in the mass plot! Implications for the population of black holes are discussed in the Populations Paper. Estimated masses for the two binary objects for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817 (solid), GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. The grey area is excluded from our convention on masses. Part of Fig. 4 of the O2 Catalogue Paper. The mass ratio is $q = m_2/m_1$. As well as mass, black holes have a spin. For the final black hole formed in the merger, these spins are always around 0.7, with a little more or less depending upon which way the spins of the two initial black holes were pointing. As well as being probably the most most massive, GW170729’s could have the highest final spin! It is a record breaker. It radiated a colossal $4.8^{+1.7}_{-1.7} M_\odot$ worth of energy in gravitational waves [bonus note]. Estimated final masses and spins for each of the binary black hole events in O1 and O2. From lowest chirp mass (left; red–orange) to highest (right; purple): GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. Part of Fig. 4 of the O2 Catalogue Paper. There is considerable uncertainty on the spins as there are hard to measure. The best combination to pin down is the effective inspiral spin parameter $\chi_\mathrm{eff}$. This is a mass weighted combination of the spins which has the most impact on the signal we observe. It could be zero if the spins are misaligned with each other, point in the orbital plane, or are zero. If it is non-zero, then it means that at least one black hole definitely has some spin. GW151226 and GW170729 have $\chi_\mathrm{eff} > 0$ with more than 99% probability. The rest are consistent with zero. The spin distribution for GW170104 has tightened up for GW170104 as its signal-to-noise ratio has increased, and there’s less support for negative $\chi_\mathrm{eff}$, but there’s been no move towards larger positive $\chi_\mathrm{eff}$. Estimated effective inspiral spin parameters for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817, GW170608, GW151226, GW151012, GW170104, GW170814, GW170809, GW170818, GW150914, GW170823, GW170729. Part of Fig. 5 of the O2 Catalogue Paper. For our analysis, we use two different waveform models to check for potential sources of systematic error. They agree pretty well. The spins are where they show most difference (which makes sense, as this is where they differ in terms of formulation). For GW151226, the effective precession waveform IMRPhenomPv2 gives $0.20^{+0.18}_{-0.08}$ and the full precession model gives $0.15^{+0.25}_{-0.11}$ and extends to negative $\chi_\mathrm{eff}$. I panicked a little bit when I first saw this, as GW151226 having a non-zero spin was one of our headline results when first announced. Fortunately, when I worked out the numbers, all our conclusions were safe. The probability of $\chi_\mathrm{eff} < 0$ is less than 1%. In fact, we can now say that at least one spin is greater than $0.28$ at 99% probability compared with $0.2$ previously, because the full precession model likes spins in the orbital plane a bit more. Who says data analysis can't be thrilling? Our measurement of $\chi_\mathrm{eff}$ tells us about the part of the spins aligned with the orbital angular momentum, but not in the orbital plane. In general, the in-plane components of the spin are only weakly constrained. We basically only get back the information we put in. The leading order effects of in-plane spins is summarised by the effective precession spin parameter $\chi_\mathrm{p}$. The plot below shows the inferred distributions for $\chi_\mathrm{p}$. The left half for each event shows our results, the right shows our prior after imposed the constraints on spin we get from $\chi_\mathrm{eff}$. We get the most information for GW151226 and GW170814, but even then it’s not much, and we generally cover the entire allowed range of values. Estimated effective inspiral spin parameters for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817, GW170608, GW151226, GW151012, GW170104, GW170814, GW170809, GW170818, GW150914, GW170823, GW170729. The left (coloured) part of the plot shows the posterior distribution; the right (white) shows the prior conditioned by the effective inspiral spin parameter constraints. Part of Fig. 5 of the O2 Catalogue Paper. One final measurement which we can make (albeit with considerable uncertainty) is the distance to the source. The distance influences how loud the signal is (the further away, the quieter it is). This also depends upon the inclination of the source (a binary edge-on is quieter than a binary face-on/off). Therefore, the distance is correlated with the inclination and we end up with some butterfly-like plots. GW170729 is again a record setter. It comes from a luminosity distance of $2.75^{+1.35}_{-1.32}~\mathrm{Gpc}$ away. That means it has travelled across the Universe for $3.2$$6.2$ billion years—it potentially started its journey before the Earth formed! Estimated luminosity distances and orbital inclinations for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817 (solid), GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. An inclination of zero means that we’re looking face-on along the direction of the total angular momentum, and inclination of $\pi/2$ means we’re looking edge-on perpendicular to the angular momentum. Part of Fig. 7 of the O2 Catalogue Paper. Waveform reconstructions To check our results, we reconstruct the waveforms from the data to see that they match our expectations for binary black hole waveforms (and there’s not anything extra there). To do this, we use unmodelled analyses which assume that there is a coherent signal in the detectors: we use both cWB and BayesWave. The results agree pretty well. The reconstructions beautifully match our templates when the signal is loud, but, as you might expect, can resolve the quieter details. You’ll also notice the reconstructions sometimes pick up a bit of background noise away from the signal. This gives you and idea of potential fluctuations. Time–frequency maps and reconstructed signal waveforms for the binary black holes. For each event we show the results from the detector where the signal was loudest. The left panel for each shows the time–frequency spectrogram with the upward-sweeping chip. The right show waveforms: blue the modelled waveforms used to infer parameters (LALInf; top panel); the red wavelet reconstructions (BayesWave; top panel); the black is the maximum-likelihood cWB reconstruction (bottom panel), and the green (bottom panel) shows reconstructions for simulated similar signals. I think the agreement is pretty good! All the data have been whitened as this is how we perform the statistical analysis of our data. Fig. 10 of the O2 Catalogue Paper. I still think GW170814 looks like a slug. Some people think they look like crocodiles. We’ll be doing more tests of the consistency of our signals with general relativity in a future paper. Merger rates Given all our observations now, we can set better limits on the merger rates. Going from the number of detections seen to the number merger out in the Universe depends upon what you assume about the mass distribution of the sources. Therefore, we make a few different assumptions. For binary black holes, we use (i) a power-law model for the more massive black hole similar to the initial mass function of stars, with a uniform distribution on the mass ratio, and (ii) use uniform-in-logarithmic distribution for both masses. These were designed to bracket the two extremes of potential distributions. With our observations, we’re starting to see that the true distribution is more like the power-law, so I expect we’ll be abandoning these soon. Taking the range of possible values from our calculations, the rate is in the range of $9.7$$101~\mathrm{Gpc^{-3}\,yr^{-1}}$ for black holes between $5 M_\odot$ and $50 M_\odot$ [bonus note]. For binary neutron stars, which are perhaps more interesting astronomers, we use a uniform distribution of masses between $1 M_\odot$ and $2 M_\odot$, and a Gaussian distribution to match electromagnetic observations. We find that these bracket the range $97$$4440~\mathrm{Gpc^{-3}\,yr^{-1}}$. This larger than are previous range, as we hadn’t considered the Gaussian distribution previously. 90% upper limits for neutron star–black hole binaries. Three black hole masses were tried and two spin distributions. Results are shown for the two matched-filter search algorithms. Fig. 14 of the O2 Catalogue Paper. Finally, what about neutron star–black holes? Since we don’t have any detections, we can only place an upper limit. This is a maximum of $610~\mathrm{Gpc^{-3}\,yr^{-1}}$. This is about a factor of 2 better than our O1 results, and is starting to get interesting! We are sure to discover lots more in O3… [bonus note]. The O2 Populations Paper Synopsis: O2 Populations Paper Read this if: You want the best family portrait of binary black holes Favourite part: A maximum black hole mass? Each detection is exciting. However, we can squeeze even more science out of our observations by looking at the entire population. Using all 10 of our binary black hole observations, we start to trace out the population of binary black holes. Since we still only have 10, we can’t yet be too definite in our conclusions. Our results give us some things to ponder, while we are waiting for the results of O3. I think now is a good time to start making some predictions. We look at the distribution of black hole masses, black hole spins, and the redshift (cosmological time) of the mergers. The black hole masses tell us something about how you go from a massive star to a black hole. The spins tell us something about how the binaries form. The redshift tells us something about how these processes change as the Universe evolves. Ideally, we would look at these all together allowing for mixtures of binary black holes formed through different means. Given that we only have a few observations, we stick to a few simple models. To work out the properties of the population, we perform a hierarchical analysis of our 10 binary black holes. We infer the properties of the individual systems, assuming that they come from a given population, and then see how well that population fits our data compared with a different distribution. In doing this inference, we account for selection effects. Our detectors are not equally sensitive to all sources. For example, nearby sources produce louder signals and we can’t detect signals that are too far away, so if you didn’t account for this you’d conclude that binary black holes only merged in the nearby Universe. Perhaps less obvious is that we are not equally sensitive to all source masses. More massive binaries produce louder signals, so we can detect these further way than lighter binaries (up to the point where these binaries are so high mass that the signals are too low frequency for us to easily spot). This is why we detect more binary black holes than binary neutron stars, even though there are more binary neutron stars out here in the Universe. Masses When looking at masses, we try three models of increasing complexity: • Model A is a simple power law for the mass of the more massive black hole $m_1$. There’s no real reason to expect the masses to follow a power law, but the masses of stars when they form do, and astronomers generally like power laws as they’re friendly, so its a sensible thing to try. We fit for the power-law index. The power law goes from a lower limit of $5 M_\odot$ to an upper limit which we also fit for. The mass of the lighter black hole $m_2$ is assumed to be uniformly distributed between $5 M_\odot$ and the mass of the other black hole. • Model B is the same power law, but we also allow the lower mass limit to vary from $5 M_\odot$. We don’t have much sensitivity to low masses, so this lower bound is restricted to be above $5 M_\odot$. I’d be interested in exploring lower masses in the future. Additionally, we allow the mass ratio $q = m_2/m_1$ of the black holes to vary, trying $q^{\beta_q}$ instead of Model A’s $q^0$. • Model C has the same power law, but now with some smoothing at the low-mass end, rather than a sharp turn-on. Additionally, it includes a Gaussian component towards higher masses. This was inspired by the possibility of pulsational pair-instability supernova causing a build up of black holes at certain masses: stars which undergo this lose extra mass, so you’d end up with lower mass black holes than if the stars hadn’t undergone the pulsations. The Gaussian could fit other effects too, for example if there was a secondary formation channel, or just reflect that the pure power law is a bad fit. In allowing the mass distributions to vary, we find overall rates which match pretty well those we obtain with our main power-law rates calculation included in the O2 Catalogue Paper, higher than with the main uniform-in-log distribution. The fitted mass distributions are shown in the plot below. The error bars are pretty broad, but I think the models agree on some broad features: there are more light black holes than heavy black holes; the minimum black hole mass is below about $9 M_\odot$, but we can’t place a lower bound on it; the maximum black hole mass is above about $35 M_\odot$ and below about $50 M_\odot$, and we prefer black holes to have more similar masses than different ones. The upper bound on the black hole minimum mass, and the lower bound on the black hole upper mass are set by the smallest and biggest black holes we’ve detected, respectively. Binary black hole merger rate as a function of the primary mass ($m_1$; top) and mass ratio ($q$; bottom). The solid line and dark and lighter bands show the median, 50% interval and 90% interval. The dashed line shows the posterior predictive distribution: our expectation for future observations averaging over our uncertainties. Fig. 1 of the O2 Populations Paper. That there does seem to be a drop off at higher masses is interesting. There could be something which stops stars forming black holes in this range. It has been proposed that there is a mass gap due to pair instability supernovae. These explosions completely disrupt their progenitor stars, leaving nothing behind. (I’m not sure if they are accompanied by a flash of green light). You’d expect this to kick for black holes of about $50$$60 M_\odot$. We infer that 99% of merging black holes have masses below $43.8 M_\odot$ with Model A, $42.8 M_\odot$ with Model B, and $41.8 M_\odot$ with Model C. Therefore, our results are not inconsistent with a mass gap. However, we don’t really have enough evidence to be sure. We can compare how well each of our three models fits the data by looking at their Bayes factors. These naturally incorporate the complexity of the models: models with more parameters (which can be more easily tweaked to match the data) are penalised so that you don’t need to worry about overfitting. We have a preference for Model C. It’s not strong, but I think good evidence that we can’t use a simple power law. Spins To model the spins: • For the magnitude, we assume a beta distribution. There’s no reason for this, but these are convenient distributions for things between 0 and 1, which are the limits on black hole spin (0 is nonspinning, 1 is as fast as you can spin). We assume that both spins are drawn from the same distribution. • For the spin orientations, we use a mix of an isotropic distribution and a Gaussian centred on being aligned with the orbital angular momentum. You’d expect an isotropic distribution if binaries were assembled dynamically, and perhaps something with spins generally aligned with each other if the binary evolved in isolation. We don’t get any useful information on the mixture fraction. Looking at the spin magnitudes, we have a preference towards smaller spins, but still have support for large spins. The more misaligned spins are, the larger the spin magnitudes can be: for the isotropic distribution, we have support all the way up to maximal values. Inferred spin magnitude distributions. The left shows results for the parametric distribution, assuming a mixture of almost aligned and isotropic spin, with the median (solid), 50% and 90% intervals shaded, and the posterior predictive distribution as the dashed line. The right shows a binned reconstruction of the distribution for aligned and isotropic distributions, showing the median and 90% intervals. Fig. 7 of the O2 Populations Paper. Since spins are harder to measure than masses, it is not surprising that we can’t make strong statements yet. If we were to find something with definitely negative $\chi_\mathrm{eff}$, we would be able to deduce that spins can be seriously misaligned. Redshift evolution As a simple model of evolution over cosmological time, we allow the merger rate to evolve as $(1+z)^\lambda$. That’s right, another power law! Since we’re only sensitive to relatively small redshifts for the masses we detect ($z < 1$), this gives a good approximation to a range of different evolution schemes. Evolution of the binary black hole merger rate (blue), showing median, 50% and 90% intervals. For comparison, reference non-evolving rates (from the O2 Catalogue Paper) are shown too. Fig. 5 of the O2 Populations Paper. We find that we prefer evolutions that increase with redshift. There’s an 88% probability that $\lambda > 0$, but we’re still consistent with no evolution. We might expect rate to increase as star formation was higher bach towards $z =2$. If we can measure the time delay between forming stars and black holes merging, we could figure out what happens to these systems in the meantime. The local merger rate is broadly consistent with what we infer with our non-evolving distributions, but is a little on the lower side. Bonus notes Naming Gravitational waves are named as GW-year-month-day, so our first observation from 14 September 2015 is GW150914. We realise that this convention suffers from a Y2K-style bug, but by the time we hit 2100, we’ll have so many detections we’ll need a new scheme anyway. Previously, we had a second designation for less significant potential detections. They were LIGO–Virgo Triggers (LVT), the one example being LVT151012. No-one was really happy with this designation, but it stems from us being cautious with our first announcement, and not wishing to appear over bold with claiming we’d seen two gravitational waves when the second wasn’t that certain. Now we’re a bit more confident, and we’ve decided to simplify naming by labelling everything a GW on the understanding that this now includes more uncertain events. Under the old scheme, GW170729 would have been LVT170729. The idea is that the broader community can decide which events they want to consider as real for their own studies. The current condition for being called a GW is that the probability of it being a real astrophysical signal is at least 50%. Our 11 GWs are safely above that limit. The naming change has hidden the fact that now when we used our improved search pipelines, the significance of GW151012 has increased. It would now be a GW even under the old scheme. Congratulations LVT151012, I always believed in you! Is it of extraterrestrial origin, or is it just a blurry figure? GW151012: the truth is out there!. Burning bright We are lacking nicknames for our new events. They came in so fast that we kind of lost track. Ilya Mandel has suggested that GW170729 should be the Tiger, as it happened on the International Tiger Day. Since tigers are the biggest of the big cats, this seems apt. Carl-Johan Haster argues that LIGO+tiger = Liger. Since ligers are even bigger than tigers, this seems like an excellent case to me! I’d vote for calling the bigger of the two progenitor black holes GW170729-tiger, the smaller GW170729-lion, and the final black hole GW17-729-liger. Suggestions for other nicknames are welcome, leave your ideas in the comments. August 2017—Something fishy or just Poisson statistics? The final few weeks of O2 were exhausting. I was trying to write job applications at the time, and each time I sat down to work on my research proposal, my phone went off with another alert. You may be wondering about was special about August. Some have hypothesised that it is because Aaron Zimmerman, my partner for the analysis of GW170104, was on the Parameter Estimation rota to analyse the last few weeks of O2. The legend goes that Aaron is especially lucky as he was bitten by a radioactive Leprechaun. I can neither confirm nor deny this. However, I make a point of playing any lottery numbers suggested by him. A slightly more mundane explanation is that August was when the detectors were running nice and stably. They were observing for a large fraction of the time. LIGO Livingston reached its best sensitivity at this time, although it was less happy for Hanford. We often quantify the sensitivity of our detectors using their binary neutron star range, the average distance they could see a binary neutron star system with a signal-to-noise ratio of 8. If this increases by a factor of 2, you can see twice as far, which means you survey 8 times the volume. This cubed factor means even small improvements can have a big impact. The LIGO Livingston range peak a little over $100~\mathrm{Mpc}$. We’re targeting at least $120~\mathrm{Mpc}$ for O3, so August 2017 gives an indication of what you can expect. Binary neutron star range for the instruments across O2. The break around week 3 was for the holidays (We did work Christmas 2015). The break at week 23 was to tune-up the instruments, and clean the mirrors. At week 31 there was an earthquake in Montana, and the Hanford sensitivity didn’t recover by the end of the run. Part of Fig. 1 of the O2 Catalogue Paper. Of course, in the case of GW170817, we just got lucky. Sign errors GW170809 was the first event we identified with Virgo after it joined observing. The signal in Virgo is very quiet. We actually got better results when we flipped the sign of the Virgo data. We were just starting to get paranoid when GW170814 came along and showed us that everything was set up right at Virgo. When I get some time, I’d like to investigate how often this type of confusion happens for quiet signals. SEOBNRv3 One of the waveforms, which includes the most complete prescription of the precession of the spins of the black holes, we use in our analysis goes by the technical name of SEOBNRv3. It is extremely computationally expensive. Work has been done to improve that, but this hasn’t been implemented in our reviewed codes yet. We managed to complete an analysis for the GW170104 Discovery Paper, which was a huge effort. I said then to not expect it for all future events. We did it for all the black holes, even for the lowest mass sources which have the longest signals. I was responsible for GW151226 runs (as well as GW170104) and I started these back at the start of the summer. Eve Chase put in a heroic effort to get GW170608 results, we pulled out all the stops for that. Thanksgiving I have recently enjoyed my first Thanksgiving in the US. I was lucky enough to be hosted for dinner by Shane Larson and his family (and cats). I ate so much I thought I might collapse to a black hole. Apparently, a Thanksgiving dinner can be 3000–4500 calories. That sounds like a lot, but the merger of GW170729 would have emitted about $5 \times 10^{40}$ times more energy. In conclusion, I don’t need to go on a diet. Confession We cheated a little bit in calculating the rates. Roughly speaking, the merger rate is given by $\displaystyle R = \frac{N}{\langle VT\rangle}$, where $N$ is the number of detections and $\langle VT\rangle$ is the amount of volume and time we’ve searched. You expect to detect more events if you increase the sensitivity of the detectors (and hence $V$), or observer for longer (and hence increase $T$). In our calculation, we included GW170608 in $N$, even though it was found outside of standard observing time. Really, we should increase $\langle VT\rangle$ to factor in the extra time outside of standard observing time when we could have made a detection. This is messy to calculate though, as there’s not really a good way to check this. However, it’s only a small fraction of the time (so the extra $T$ should be small), and for much of the sensitivity of the detectors will be poor (so $V$ will be small too). Therefore, we estimated any bias from neglecting this is smaller than our uncertainty from the calibration of the detectors, and not worth worrying about. New sources We saw our first binary black hole shortly after turning on the Advanced LIGO detectors. We saw our first binary neutron star shortly after turning on the Advanced Virgo detector. My money is therefore on our first neutron star–black hole binary shortly after we turn on the KAGRA detector. Because science… Accuracy of inference on the physics of binary evolution from gravitational-wave observations Gravitational-wave astronomy lets us observing binary black holes. These systems, being made up of two black holes, are pretty difficult to study by any other means. It has long been argued that with this new information we can unravel the mysteries of stellar evolution. Just as a palaeontologist can discover how long-dead animals lived from their bones, we can discover how massive stars lived by studying their black hole remnants. In this paper, we quantify how much we can really learn from this black hole palaeontology—after 1000 detections, we should pin down some of the most uncertain parameters in binary evolution to a few percent precision. Life as a binary There are many proposed ways of making a binary black hole. The current leading contender is isolated binary evolution: start with a binary star system (most stars are in binaries or higher multiples, our lonesome Sun is a little unusual), and let the stars evolve together. Only a fraction will end with black holes close enough to merge within the age of the Universe, but these would be the sources of the signals we see with LIGO and Virgo. We consider this isolated binary scenario in this work [bonus note]. Now, you might think that with stars being so fundamentally important to astronomy, and with binary stars being so common, we’d have the evolution of binaries figured out by now. It turns out it’s actually pretty messy, so there’s lots of work to do. We consider constraining four parameters which describe the bits of binary physics which we are currently most uncertain of: • Black hole natal kicks—the push black holes receive when they are born in supernova explosions. We now the neutron stars get kicks, but we’re less certain for black holes [bonus note]. • Common envelope efficiency—one of the most intricate bits of physics about binaries is how mass is transferred between stars. As they start exhausting their nuclear fuel they puff up, so material from the outer envelope of one star may be stripped onto the other. In the most extreme cases, a common envelope may form, where so much mass is piled onto the companion, that both stars live in a single fluffy envelope. Orbiting inside the envelope helps drag the two stars closer together, bringing them closer to merging. The efficiency determines how quickly the envelope becomes unbound, ending this phase. • Mass loss rates during the Wolf–Rayet (not to be confused with Wolf 359) and luminous blue variable phases–stars lose mass through out their lives, but we’re not sure how much. For stars like our Sun, mass loss is low, there is enough to gives us the aurora, but it doesn’t affect the Sun much. For bigger and hotter stars, mass loss can be significant. We consider two evolutionary phases of massive stars where mass loss is high, and currently poorly known. Mass could be lost in clumps, rather than a smooth stream, making it difficult to measure or simulate. We use parameters describing potential variations in these properties are ingredients to the COMPAS population synthesis code. This rapidly (albeit approximately) evolves a population of stellar binaries to calculate which will produce merging binary black holes. The question is now which parameters affect our gravitational-wave measurements, and how accurately we can measure those which do? Binary black hole merger rate at three different redshifts $z$ as calculated by COMPAS. We show the rate in 30 different chirp mass bins for our default population parameters. The caption gives the total rate for all masses. Figure 2 of Barrett et al. (2018) Gravitational-wave observations For our deductions, we use two pieces of information we will get from LIGO and Virgo observations: the total number of detections, and the distributions of chirp masses. The chirp mass is a combination of the two black hole masses that is often well measured—it is the most important quantity for controlling the inspiral, so it is well measured for low mass binaries which have a long inspiral, but is less well measured for higher mass systems. In reality we’ll have much more information, so these results should be the minimum we can actually do. We consider the population after 1000 detections. That sounds like a lot, but we should have collected this many detections after just 2 or 3 years observing at design sensitivity. Our default COMPAS model predicts 484 detections per year of observing time! Honestly, I’m a little scared about having this many signals… For a set of population parameters (black hole natal kick, common envelope efficiency, luminous blue variable mass loss and Wolf–Rayet mass loss), COMPAS predicts the number of detections and the fraction of detections as a function of chirp mass. Using these, we can work out the probability of getting the observed number of detections and fraction of detections within different chirp mass ranges. This is the likelihood function: if a given model is correct we are more likely to get results similar to its predictions than further away, although we expect their to be some scatter. If you like equations, the from of our likelihood is explained in this bonus note. If you don’t like equations, there’s one lurking in the paragraph below. Just remember, that it can’t see you if you don’t move. It’s OK to skip the equation. To determine how sensitive we are to each of the population parameters, we see how the likelihood changes as we vary these. The more the likelihood changes, the easier it should be to measure that parameter. We wrap this up in terms of the Fisher information matrix. This is defined as $\displaystyle F_{ij} = -\left\langle\frac{\partial^2\ln \mathcal{L}(\mathcal{D}\|\left\{\lambda\right\})}{\partial \lambda_i \partial\lambda_j}\right\rangle$, where $\mathcal{L}(\mathcal{D}\|\left\{\lambda\right\})$ is the likelihood for data $\mathcal{D}$ (the number of observations and their chirp mass distribution in our case), $\left\{\lambda\right\}$ are our parameters (natal kick, etc.), and the angular brackets indicate the average over the population parameters. In statistics terminology, this is the variance of the score, which I think sounds cool. The Fisher information matrix nicely quantifies how much information we can lean about the parameters, including the correlations between them (so we can explore degeneracies). The inverse of the Fisher information matrix gives a lower bound on the covariance matrix (the multidemensional generalisation of the variance in a normal distribution) for the parameters $\left\{\lambda\right\}$. In the limit of a large number of detections, we can use the Fisher information matrix to estimate the accuracy to which we measure the parameters [bonus note]. We simulated several populations of binary black hole signals, and then calculate measurement uncertainties for our four population uncertainties to see what we could learn from these measurements. Results Using just the rate information, we find that we can constrain a combination of the common envelope efficiency and the Wolf–Rayet mass loss rate. Increasing the common envelope efficiency ends the common envelope phase earlier, leaving the binary further apart. Wider binaries take longer to merge, so this reduces the merger rate. Similarly, increasing the Wolf–Rayet mass loss rate leads to wider binaries and smaller black holes, which take longer to merge through gravitational-wave emission. Since the two parameters have similar effects, they are anticorrelated. We can increase one and still get the same number of detections if we decrease the other. There’s a hint of a similar correlation between the common envelope efficiency and the luminous blue variable mass loss rate too, but it’s not quite significant enough for us to be certain it’s there. Fisher information matrix estimates for fractional measurement precision of the four population parameters: the black hole natal kick $\sigma_\mathrm{kick}$, the common envelope efficiency $\alpha_\mathrm{CE}$, the Wolf–Rayet mass loss rate $f_\mathrm{WR}$, and the luminous blue variable mass loss rate $f_\mathrm{LBV}$. There is an anticorrealtion between $f_\mathrm{WR}$ and $\alpha_\mathrm{CE}$, and hints at a similar anticorrelation between $f_\|mathrm{LBV}$ and $\alpha_\mathrm{CE}$. We show 1500 different realisations of the binary population to give an idea of scatter. Figure 6 of Barrett et al. (2018) Adding in the chirp mass distribution gives us more information, and improves our measurement accuracies. The fraction uncertainties are about 2% for the two mass loss rates and the common envelope efficiency, and about 5% for the black hole natal kick. We’re less sensitive to the natal kick because the most massive black holes don’t receive a kick, and so are unaffected by the kick distribution [bonus note]. In any case, these measurements are exciting! With this type of precision, we’ll really be able to learn something about the details of binary evolution. Measurement precision for the four population parameters after 1000 detections. We quantify the precision with the standard deviation estimated from the Fisher inforamtion matrix. We show results from 1500 realisations of the population to give an idea of scatter. Figure 5 of Barrett et al. (2018) The accuracy of our measurements will improve (on average) with the square root of the number of gravitational-wave detections. So we can expect 1% measurements after about 4000 observations. However, we might be able to get even more improvement by combining constraints from other types of observation. Combining different types of observation can help break degeneracies. I’m looking forward to building a concordance model of binary evolution, and figuring out exactly how massive stars live their lives. arXiv: 1711.06287 [astro-ph.HE] Journal: Monthly Notices of the Royal Astronomical Society; 477(4):4685–4695; 2018 Favourite dinosaur: Professor Science Bonus notes Channel selection In practise, we will need to worry about how binary black holes are formed, via isolated evolution or otherwise, before inferring the parameters describing binary evolution. This makes the problem more complicated. Some parameters, like mass loss rates or black hole natal kicks, might be common across multiple channels, while others are not. There are a number of ways we might be able to tell different formation mechanisms apart, such as by using spin measurements. Kick distribution We model the supernova kicks $v_\mathrm{kick}$ as following a Maxwell–Boltzmann distribution, $\displaystyle p(v_\mathrm{kick}) = \sqrt{\frac{2}{\pi}} \frac{v_\mathrm{kick}^2}{\sigma_\mathrm{kick}^3} \exp\left(\frac{-v_\mathrm{kick}^2}{2\sigma_\mathrm{kick}^2}\right)$, where $\sigma_\mathrm{kick}$ is the unknown population parameter. The natal kick received by the black hole $v^_\mathrm{kick}$ is not the same as this, however, as we assume some of the material ejected by the supernova falls back, reducing the over kick. The final natal kick is $v^_\mathrm{kick} = (1-f_\mathrm{fb})v_\mathrm{kick}$, where $f_\mathrm{fb}$ is the fraction that falls back, taken from Fryer et al. (2012). The fraction is greater for larger black holes, so the biggest black holes get no kicks. This means that the largest black holes are unaffected by the value of $\sigma_\mathrm{kick}$. The likelihood In this analysis, we have two pieces of information: the number of detections, and the chirp masses of the detections. The first is easy to summarise with a single number. The second is more complicated, and we consider the fraction of events within different chirp mass bins. Our COMPAS model predicts the merger rate $\mu$ and the probability of falling in each chirp mass bin $p_k$ (we factor measurement uncertainty into this). Our observations are the the total number of detections $N_\mathrm{obs}$ and the number in each chirp mass bin $c_k$ ($N_\mathrm{obs} = \sum_k c_k$). The likelihood is the probability of these observations given the model predictions. We can split the likelihood into two pieces, one for the rate, and one for the chirp mass distribution, $\mathcal{L} = \mathcal{L}_\mathrm{rate} \times \mathcal{L}_\mathrm{mass}$. For the rate likelihood, we need the probability of observing $N_\mathrm{obs}$ given the predicted rate $\mu$. This is given by a Poisson distribution, $\displaystyle \mathcal{L}_\mathrm{rate} = \exp(-\mu t_\mathrm{obs}) \frac{(\mu t_\mathrm{obs})^{N_\mathrm{obs}}}{N_\mathrm{obs}!}$, where $t_\mathrm{obs}$ is the total observing time. For the chirp mass likelihood, we the probability of getting a number of detections in each bin, given the predicted fractions. This is given by a multinomial distribution, $\displaystyle \mathcal{L}_\mathrm{mass} = \frac{N_\mathrm{obs}!}{\prod_k c_k!} \prod_k p_k^{c_k}$. These look a little messy, but they simplify when you take the logarithm, as we need to do for the Fisher information matrix. When we substitute in our likelihood into the expression for the Fisher information matrix, we get $\displaystyle F_{ij} = \mu t_\mathrm{obs} \left[ \frac{1}{\mu^2} \frac{\partial \mu}{\partial \lambda_i} \frac{\partial \mu}{\partial \lambda_j} + \sum_k\frac{1}{p_k} \frac{\partial p_k}{\partial \lambda_i} \frac{\partial p_k}{\partial \lambda_j} \right]$. Conveniently, although we only need to evaluate first-order derivatives, even though the Fisher information matrix is defined in terms of second derivatives. The expected number of events is $\langle N_\mathrm{obs} \rangle = \mu t_\mathrm{obs}$. Therefore, we can see that the measurement uncertainty defined by the inverse of the Fisher information matrix, scales on average as $N_\mathrm{obs}^{-1/2}$. For anyone worrying about using the likelihood rather than the posterior for these estimates, the high number of detections [bonus note] should mean that the information we’ve gained from the data overwhelms our prior, meaning that the shape of the posterior is dictated by the shape of the likelihood. Interpretation of the Fisher information matrix As an alternative way of looking at the Fisher information matrix, we can consider the shape of the likelihood close to its peak. Around the maximum likelihood point, the first-order derivatives of the likelihood with respect to the population parameters is zero (otherwise it wouldn’t be the maximum). The maximum likelihood values of $N_\mathrm{obs} = \mu t_\mathrm{obs}$ and $c_k = N_\mathrm{obs} p_k$ are the same as their expectation values. The second-order derivatives are given by the expression we have worked out for the Fisher information matrix. Therefore, in the region around the maximum likelihood point, the Fisher information matrix encodes all the relevant information about the shape of the likelihood. So long as we are working close to the maximum likelihood point, we can approximate the distribution as a multidimensional normal distribution with its covariance matrix determined by the inverse of the Fisher information matrix. Our results for the measurement uncertainties are made subject to this approximation (which we did check was OK). Approximating the likelihood this way should be safe in the limit of large $N_\mathrm{obs}$. As we get more detections, statistical uncertainties should reduce, with the peak of the distribution homing in on the maximum likelihood value, and its width narrowing. If you take the limit of $N_\mathrm{obs} \rightarrow \infty$, you’ll see that the distribution basically becomes a delta function at the maximum likelihood values. To check that our $N_\mathrm{obs} = 1000$ was large enough, we verified that higher-order derivatives were still small. Michele Vallisneri has a good paper looking at using the Fisher information matrix for gravitational wave parameter estimation (rather than our problem of binary population synthesis). There is a good discussion of its range of validity. The high signal-to-noise ratio limit for gravitational wave signals corresponds to our high number of detections limit. Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals The space-based observatory LISA will detect gravitational waves from massive black holes (giant black holes residing in the centres of galaxies). One particularly interesting signal will come from the inspiral of a regular stellar-mass black hole into a massive black hole. These are called extreme mass-ratio inspirals (or EMRIs, pronounced emries, to their friends) [bonus note]. We have never observed such a system. This means that there’s a lot we have to learn about them. In this work, we systematically investigated the prospects for observing EMRIs. We found that even though there’s a wide range in predictions for what EMRIs we will detect, they should be a safe bet for the LISA mission. Artistic impression of the spacetime for an extreme-mass-ratio inspiral, with a smaller stellar-mass black hole orbiting a massive black hole. This image is mandatory when talking about extreme-mass-ratio inspirals. Credit: NASA LISA & EMRIs My previous post discussed some of the interesting features of EMRIs. Because of the extreme difference in masses of the two black holes, it takes a long time for them to complete their inspiral. We can measure tens of thousands of orbits, which allows us to make wonderfully precise measurements of the source properties (if we can accurately pick out the signal from the data). Here, we’ll examine exactly what we could learn with LISA from EMRIs [bonus note]. First we build a model to investigate how many EMRIs there could be. There is a lot of astrophysics which we are currently uncertain about, which leads to a large spread in estimates for the number of EMRIs. Second, we look at how precisely we could measure properties from the EMRI signals. The astrophysical uncertainties are less important here—we could get a revolutionary insight into the lives of massive black holes. The number of EMRIs To build a model of how many EMRIs there are, we need a few different inputs: 1. The population of massive black holes 2. The distribution of stellar clusters around massive black holes 3. The range of orbits of EMRIs We examine each of these in turn, building a more detailed model than has previously been constructed for EMRIs. We currently know little about the population of massive black holes. This means we’ll discover lots when we start measuring signals (yay), but it’s rather inconvenient now, when we’re trying to predict how many EMRIs there are (boo). We take two different models for the mass distribution of massive black holes. One is based upon a semi-analytic model of massive black hole formation, the other is at the pessimistic end allowed by current observations. The semi-analytic model predicts massive black hole spins around 0.98, but we also consider spins being uniformly distributed between 0 and 1, and spins of 0. This gives us a picture of the bigger black hole, now we need the smaller. Observations show that the masses of massive black holes are correlated with their surrounding cluster of stars—bigger black holes have bigger clusters. We consider four different versions of this trend: Gültekin et al. (2009); Kormendy & Ho (2013); Graham & Scott (2013), and Shankar et al. (2016). The stars and black holes about a massive black hole should form a cusp, with the density of objects increasing towards the massive black hole. This is great for EMRI formation. However, the cusp is disrupted if two galaxies (and their massive black holes) merge. This tends to happen—it’s how we get bigger galaxies (and black holes). It then takes some time for the cusp to reform, during which time, we don’t expect as many EMRIs. Therefore, we factor in the amount of time for which there is a cusp for massive black holes of different masses and spins. That’s a nice galaxy you have there. It would be a shame if it were to collide with something… Hubble image of The Mice. Credit: ACS Science & Engineering Team. Given a cusp about a massive black hole, we then need to know how often an EMRI forms. Simulations give us a starting point. However, these only consider a snap-shot, and we need to consider how things evolve with time. As stellar-mass black holes inspiral, the massive black hole will grow in mass and the surrounding cluster will become depleted. Both these effects are amplified because for each inspiral, there’ll be many more stars or stellar-mass black holes which will just plunge directly into the massive black hole. We therefore need to limit the number of EMRIs so that we don’t have an unrealistically high rate. We do this by adding in a couple of feedback factors, one to cap the rate so that we don’t deplete the cusp quicker than new objects will be added to it, and one to limit the maximum amount of mass the massive black hole can grow from inspirals and plunges. This gives us an idea for the total number of inspirals. Finally, we calculate the orbits that EMRIs will be on. We again base this upon simulations, and factor in how the spin of the massive black hole effects the distribution of orbital inclinations. Putting all the pieces together, we can calculate the population of EMRIs. We now need to work out how many LISA would be able to detect. This means we need models for the gravitational-wave signal. Since we are simulating a large number, we use a computationally inexpensive analytic model. We know that this isn’t too accurate, but we consider two different options for setting the end of the inspiral (where the smaller black hole finally plunges) which should bound the true range of results. Number of EMRIs for different size massive black holes in different astrophysical models. M1 is our best estimate, the others explore variations on this. M11 and M12 are designed to be cover the extremes, being the most pessimistic and optimistic combinations. The solid and dashed lines are for two different signal models (AKK and AKS), which are designed to give an indication of potential variation. They agree where the massive black hole is not spinning (M10 and M11). The range of masses is similar for all models, as it is set by the sensitivity of LISA. We can detect higher mass systems assuming the AKK signal model as it includes extra inspiral close to highly spinning black holes: for the heaviest black holes, this is the only part of the signal at high enough frequency to be detectable. Figure 8 of Babak et al. (2017). Allowing for all the different uncertainties, we find that there should be somewhere between 1 and 4200 EMRIs detected per year. (The model we used when studying transient resonances predicted about 250 per year, albeit with a slightly different detector configuration, which is fairly typical of all the models we consider here). This range is encouraging. The lower end means that EMRIs are a pretty safe bet, we’d be unlucky not to get at least one over the course of a multi-year mission (LISA should have at least four years observing). The upper end means there could be lots—we might actually need to worry about them forming a background source of noise if we can’t individually distinguish them! EMRI measurements Having shown that EMRIs are a good LISA source, we now need to consider what we could learn by measuring them? We estimate the precision we will be able to measure parameters using the Fisher information matrix. The Fisher matrix measures how sensitive our observations are to changes in the parameters (the more sensitive we are, the better we should be able to measure that parameter). It should be a lower bound on actual measurement precision, and well approximate the uncertainty in the high signal-to-noise (loud signal) limit. The combination of our use of the Fisher matrix and our approximate signal models means our results will not be perfect estimates of real performance, but they should give an indication of the typical size of measurement uncertainties. Given that we measure a huge number of cycles from the EMRI signal, we can make really precise measurements of the the mass and spin of the massive black hole, as these parameters control the orbital frequencies. Below are plots for the typical measurement precision from our Fisher matrix analysis. The orbital eccentricity is measured to similar accuracy, as it influences the range of orbital frequencies too. We also get pretty good measurements of the the mass of the smaller black hole, as this sets how quickly the inspiral proceeds (how quickly the orbital frequencies change). EMRIs will allow us to do precision astronomy! Distribution of (one standard deviation) fractional uncertainties for measurements of the massive black hole (redshifted) mass $M_z$. Results are shown for the different astrophysical models, and for the different signal models. The astrophysical model has little impact on the uncertainties. M4 shows a slight difference as it assumes heavier stellar-mass black holes. The results with the two signal models agree when the massive black hole is not spinning (M10 and M11). Otherwise, measurements are more precise with the AKK signal model, as this includes extra signal from the end of the inspiral. Part of Figure 11 of Babak et al. (2017). Distribution of (one standard deviation) uncertainties for measurements of the massive black hole spin $a$. The results mirror those for the masses above. Part of Figure 11 of Babak et al. (2017). Now, before you get too excited that we’re going to learn everything about massive black holes, there is one confession I should make. In the plot above I show the measurement accuracy for the redshifted mass of the massive black hole. The cosmological expansion of the Universe causes gravitational waves to become stretched to lower frequencies in the same way light is (this makes visible light more red, hence the name). The measured frequency is $f_z = (1 + z)f$ where $f$ is the frequency emitted, and $z$ is the redshift ($z= 0$ for a nearby source, and is larger for further away sources). Lower frequency gravitational waves correspond to higher mass systems, so it is often convenient to work with the redshifted mass, the mass corresponding to the signal you measure if you ignore redshifting. The redshifted mass of the massive black hole is $M_z = (1+z)M$ where $M$ is the true mass. To work out the true mass, we need the redshift, which means we need to measure the distance to the source. Distribution of (one standard deviation) fractional uncertainties for measurements of the luminosity distance $D_\mathrm{L}$. The signal model is not as important here, as the uncertainty only depends on how loud the signal is. Part of Figure 12 of Babak et al. (2017). The plot above shows the fractional uncertainty on the distance. We don’t measure this too well, as it is determined from the amplitude of the signal, rather than its frequency components. The situation is much as for LIGO. The larger uncertainties on the distance will dominate the overall uncertainty on the black hole masses. We won’t be getting all these to fractions of a percent. However, that doesn’t mean we can’t still figure out what the distribution of masses looks like! One of the really exciting things we can do with EMRIs is check that the signal matches our expectations for a black hole in general relativity. Since we get such an excellent map of the spacetime of the massive black hole, it is easy to check for deviations. In general relativity, everything about the black hole is fixed by its mass and spin (often referred to as the no-hair theorem). Using the measured EMRI signal, we can check if this is the case. One convenient way of doing this is to describe the spacetime of the massive object in terms of a multipole expansion. The first (most important) terms gives the mass, and the next term the spin. The third term (the quadrupole) is set by the first two, so if we can measure it, we can check if it is consistent with the expected relation. We estimated how precisely we could measure a deviation in the quadrupole. Fortunately, for this consistency test, all factors from redshifting cancel out, so we can get really detailed results, as shown below. Using EMRIs, we’ll be able to check for really small differences from general relativity! Distribution of (one standard deviation) of uncertainties for deviations in the quadrupole moment of the massive object spacetime $\mathcal{Q}$. Results are similar to the mass and spin measurements. Figure 13 of Babak et al. (2017). In summary: EMRIS are awesome. We’re not sure how many we’ll detect with LISA, but we’re confident there will be some, perhaps a couple of hundred per year. From the signals we’ll get new insights into the masses and spins of black holes. This should tell us something about how they, and their surrounding galaxies, evolved. We’ll also be able to do some stringent tests of whether the massive objects are black holes as described by general relativity. It’s all pretty exciting, for when LISA launches, which is currently planned about 2034… One of the most valuable traits a student or soldier can have: patience. Credit: Sony/Marvel arXiv: 1703.09722 [gr-qc] Journal: Physical Review D; 477(4):4685–4695; 2018 Conference proceedings: 1704.00009 [astro-ph.GA] (from when work was still in-progress) Estimated number of Marvel films before LISA launch: 48 (starting with Ant-Man and the Wasp) Bonus notes Hyphenation Is it “extreme-mass-ratio inspiral”, “extreme mass-ratio inspiral” or “extreme mass ratio inspiral”? All are used in the literature. This is one of the advantage of using “EMRI”. The important thing is that we’re talking about inspirals that have a mass ratio which is extreme. For this paper, we used “extreme mass-ratio inspiral”, but when I first started my PhD, I was first introduced to “extreme-mass-ratio inspirals”, so they are always stuck that way in my mind. I think hyphenation is a bit of an art, and there’s no definitive answer here, just like there isn’t for superhero names, where you can have Iron Man, Spider-Man or Iceman. Science with LISA This paper is part of a series looking at what LISA could tells us about different gravitational wave sources. So far, this series covers 1. Massive black hole binaries 2. Cosmological phase transitions 3. Standard sirens (for measuring the expansion of the Universe) 4. Inflation 5. Extreme-mass-ratio inspirals You’ll notice there’s a change in the name of the mission from eLISA to LISA part-way through, as things have evolved. (Or devolved?) I think the main take-away so far is that the cosmology group is the most enthusiastic. GW170817—The papers gAfter three months (and one binary black hole detection announcement), I finally have time to write about the suite of LIGO–Virgo papers put together to accompany GW170817. The papers There are currently 9 papers in the GW170817 family. Further papers, for example looking at parameter estimation in detail, are in progress. Papers are listed below in order of arXiv posting. My favourite is the GW170817 Discovery Paper. Many of the highlights, especially from the Discovery and Multimessenger Astronomy Papers, are described in my GW170817 announcement post. Keeping up with all the accompanying observational results is a task not even Sisyphus would envy. I’m sure that the details of these will be debated for a long time to come. I’ve included references to a few below (mostly as [citation notes]), but these are not guaranteed to be complete (I’ll continue to expand these in the future). 0. The GW170817 Discovery Paper Title: GW170817: Observation of gravitational waves from a binary neutron star inspiral arXiv: 1710.05832 [gr-qc] Journal: Physical Review Letters; 119(16):161101(18); 2017 LIGO science summary: GW170817: Observation of gravitational waves from a binary neutron star inspiral This is the paper announcing the gravitational-wave detection. It gives an overview of the properties of the signal, initial estimates of the parameters of the source (see the GW170817 Properties Paper for updates) and the binary neutron star merger rate, as well as an overview of results from the other companion papers. I was disappointed that “the era of gravitational-wave multi-messenger astronomy has opened with a bang” didn’t make the conclusion of the final draft. More details: The GW170817 Discovery Paper summary −1. The Multimessenger Astronomy Paper Title: Multi-messenger observations of a binary neutron star merger arXiv: 1710.05833 [astro-ph.HE] Journal: Astrophysical Journal Letters; 848(2):L12(59); 2017 LIGO science summary: The dawn of multi-messenger astrophysics: observations of a binary neutron star merger I’ve numbered this paper as −1 as it gives an overview of all the observations—gravitational wave, electromagnetic and neutrino—accompanying GW170817. I feel a little sorry for the neutrino observers, as they’re the only ones not to make a detection. Drawing together the gravitational wave and electromagnetic observations, we can confirm that binary neutron star mergers are the progenitors of (at least some) short gamma-ray bursts and kilonovae. Do not print this paper, the author list stretches across 23 pages. More details: The Multimessenger Astronomy Paper summary 1. The GW170817 Gamma-ray Burst Paper Title: Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A arXiv: 1710.05834 [astro-ph.HE] Journal: Astrophysical Journal Letters; 848(2):L13(27); 2017 LIGO science summary: Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A Here we bring together the LIGO–Virgo observations of GW170817 and the Fermi and INTEGRAL observations of GRB 170817A. From the spatial and temporal coincidence of the gravitational waves and gamma rays, we establish that the two are associated with each other. There is a 1.7 s time delay between the merger time estimated from gravitational waves and the arrival of the gamma-rays. From this, we make some inferences about the structure of the jet which is the source of the gamma rays. We can also use this to constrain deviations from general relativity, which is cool. Finally, we estimate that there be 0.3–1.7 joint gamma ray–gravitational wave detections per year once our gravitational-wave detectors reach design sensitivity! More details: The GW170817 Gamma-ray Burst Paper summary 2. The GW170817 Hubble Constant Paper Title: A gravitational-wave standard siren measurement of the Hubble constant [bonus note] arXiv: 1710.05835 [astro-ph.CO] Journal: Nature; 551(7678):85–88; 2017 [bonus note] LIGO science summary: Measuring the expansion of the Universe with gravitational waves The Hubble constant quantifies the current rate of expansion of the Universe. If you know how far away an object is, and how fast it is moving away (due to the expansion of the Universe, not because it’s on a bus or something, that is important), you can estimate the Hubble constant. Gravitational waves give us an estimate of the distance to the source of GW170817. The observations of the optical transient AT 2017gfo allow us to identify the galaxy NGC 4993 as the host of GW170817’s source. We know the redshift of the galaxy (which indicates how fast its moving). Therefore, putting the two together we can infer the Hubble constant in a completely new way. More details: The GW170817 Hubble Constant Paper summary 3. The GW170817 Kilonova Paper Title: Estimating the contribution of dynamical ejecta in the kilonova associated with GW170817 arXiv: 1710.05836 [astro-ph.HE] Journal: Astrophysical Journal Letters; 850(2):L39(13); 2017 LIGO science summary: Predicting the aftermath of the neutron star collision that produced GW170817 During the coalescence of two neutron stars, lots of neutron-rich matter gets ejected. This undergoes rapid radioactive decay, which powers a kilonova, an optical transient. The observed signal depends upon the material ejected. Here, we try to use our gravitational-wave measurements to predict the properties of the ejecta ahead of the flurry of observational papers. More details: The GW170817 Kilonova Paper summary 4. The GW170817 Stochastic Paper Title: GW170817: Implications for the stochastic gravitational-wave background from compact binary coalescences arXiv: 1710.05837 [gr-qc] Journal: Physical Review Letters; 120(9):091101(12); 2018 LIGO science summary: The background symphony of gravitational waves from neutron star and black hole mergers We can detect signals if they are loud enough, but there will be many quieter ones that we cannot pick out from the noise. These add together to form an overlapping background of signals, a background rumbling in our detectors. We use the inferred rate of binary neutron star mergers to estimate their background. This is smaller than the background from binary black hole mergers (black holes are more massive, so they’re intrinsically louder), but they all add up. It’ll still be a few years before we could detect a background signal. More details: The GW170817 Stochastic Paper summary 5. The GW170817 Progenitor Paper Title: On the progenitor of binary neutron star merger GW170817 arXiv: 1710.05838 [astro-ph.HE] Journal: Astrophysical Journal Letters; 850(2):L40(18); 2017 LIGO science summary: Making GW170817: neutron stars, supernovae and trick shots (I’d especially recommend reading this one) We know that GW170817 came from the coalescence of two neutron stars, but where did these neutron stars come from? Here, we combine the parameters inferred from our gravitational-wave measurements, the observed position of AT 2017gfo in NGC 4993 and models for the host galaxy, to estimate properties like the kick imparted to neutron stars during the supernova explosion and how long it took the binary to merge. More details: The GW170817 Progenitor Paper summary 6. The GW170817 Neutrino Paper Title: Search for high-energy neutrinos from binary neutron star merger GW170817 with ANTARES, IceCube, and the Pierre Auger Observatory arXiv: 1710.05839 [astro-ph.HE] Journal: Astrophysical Journal Letters; 850(2):L35(18); 2017 This is the search for neutrinos from the source of GW170817. Lots of neutrinos are emitted during the collision, but not enough to be detectable on Earth. Indeed, we don’t find any neutrinos, but we combine results from three experiments to set upper limits. More details: The GW170817 Neutrino Paper summary 7. The GW170817 Post-merger Paper Title: Search for post-merger gravitational waves from the remnant of the binary neutron star merger GW170817 arXiv: 1710.09320 [astro-ph.HE] Journal: Astrophysical Journal Letters; 851(1):L16(13); 2017 LIGO science summary: Searching for the neutron star or black hole resulting from GW170817 After the two neutron stars merged, what was left? A larger neutron star or a black hole? Potentially we could detect gravitational waves from a wibbling neutron star, as it sloshes around following the collision. We don’t. It would have to be a lot closer for this to be plausible. However, this paper outlines how to search for such signals; the GW170817 Properties Paper contains a more detailed look at any potential post-merger signal. More details: The GW170817 Post-merger Paper summary 8. The GW170817 Properties Paper Title: Properties of the binary neutron star merger GW170817 arXiv: 1805.11579 [gr-qc] In the GW170817 Discovery Paper we presented initial estimates for the properties of GW170817’s source. These were the best we could do on the tight deadline for the announcement (it was a pretty good job in my opinion). Now we have had a bit more time we can present a new, improved analysis. This uses recalibrated data and a wider selection of waveform models. We also fold in our knowledge of the source location, thanks to the observation of AT 2017gfo by our astronomer partners, for our best results. if you want to know the details of GW170817’s source, this is the paper for you! If you’re looking for the most up-to-date results regarding GW170817, check out the O2 Catalogue Paper. More details: The GW170817 Properties Paper summary 9. The GW170817 Equation-of-state Paper Title: GW170817: Measurements of neutron star radii and equation of state arXiv: 1805.11581 [gr-qc] Neutron stars are made of weird stuff: nuclear density material which we cannot replicate here on Earth. Neutron star matter is often described in terms of an equation of state, a relationship that explains how the material changes at different pressures or densities. A stiffer equation of state means that the material is harder to squash, and a softer equation of state is easier to squish. This means that for a given mass, a stiffer equation of state will predict a larger, fluffier neutron star, while a softer equation of state will predict a more compact, denser neutron star. In this paper, we assume that GW170817’s source is a binary neutron star system, where both neutron stars have the same equation of state, and see what we can infer about neutron star stuff™. More details: The GW170817 Equation-of-state Paper summary The GW170817 Discovery Paper Synopsis: GW170817 Discovery Paper Read this if: You want all the details of our first gravitational-wave observation of a binary neutron star coalescence Favourite part: Look how well we measure the chirp mass! GW170817 was a remarkable gravitational-wave discovery. It is the loudest signal observed to date, and the source with the lowest mass components. I’ve written about some of the highlights of the discovery in my previous GW170817 discovery post. Binary neutron stars are one of the principal targets for LIGO and Virgo. The first observational evidence for the existence of gravitational waves came from observations of binary pulsars—a binary neutron star system where (at least one) one of the components is a pulsar. Therefore (unlike binary black holes), we knew that these sources existed before we turned on our detectors. What was less certain was how often they merge. In our first advanced-detector observing run (O1), we didn’t find any, allowing us to estimate an upper limit on the merger rate of $12600~\mathrm{Gpc^{-1}\,yr^{-1}}$. Now, we know much more about merging binary neutron stars. GW170817, as a loud and long signal, is a highly significant detection. You can see it in the data by eye. Therefore, it should have been a easy detection. As is often the case with real experiments, it wasn’t quite that simple. Data transfer from Virgo had stopped over night, and there was a glitch (a non-stationary and non-Gaussian noise feature) in the Livingston detector, which meant that this data weren’t automatically analysed. Nevertheless, GstLAL flagged something interesting in the Hanford data, and there was a mad flurry to get the other data in place so that we could analyse the signal in all three detectors. I remember being sceptical in these first few minutes until I saw the plot of Livingston data which blew me away: the chirp was clearly visible despite the glitch! Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. The Livinston data have had the glitch removed. The signal is clearly visible in the two LIGO detectors as the upward sweeping chirp; it is not visible in Virgo because of its lower sensitivity and the source’s position in the sky. Figure 1 of the GW170817 Discovery Paper. Using data from both of our LIGO detectors (as discussed for GW170814, our offline algorithms searching for coalescing binaries only use these two detectors during O2), GW170817 is an absolutely gold-plated detection. GstLAL estimates a false alarm rate (the rate at which you’d expect something at least this signal-like to appear in the detectors due to a random noise fluctuation) of less than one in 1,100,000 years, while PyCBC estimates the false alarm rate to be less than one in 80,000 years. Parameter estimation (inferring the source properties) used data from all three detectors. We present a (remarkably thorough given the available time) initial analysis in this paper (more detailed results are given in the GW170817 Properties Paper, and the most up-to-date results are in O2 Catalogue Paper). This signal is challenging to analyse because of the glitch and because binary neutron stars are made of stuff™, which can leave an imprint on the waveform. We’ll be looking at the effects of these complications in more detail in the future. Our initial results are • The source is localized to a region of about $28~\mathrm{deg^2}$ at a distance of $40^{+8}_{-14}~\mathrm{Mpc}$ (we typically quote results at the 90% credible level). This is the closest gravitational-wave source yet. • The chirp mass is measured to be $1.188_{-0.002}^{+0.004} M_\odot$, much lower than for our binary black hole detections. • The spins are not well constrained, the uncertainty from this means that we don’t get precise measurements of the individual component masses. We quote results with two choices of spin prior: the astrophysically motivated limit of 0.05, and the more agnostic and conservative upper bound of 0.89. I’ll stick to using the low-spin prior results be default. • Using the low-spin prior, the component masses are $m_1 = 1.36$$1.60 M_\odot$ and $m_2 = 1.17$$1.36 M_\odot$. We have the convention that $m_1 \geq m_2$, which is why the masses look unequal; there’s a lot of support for them being nearly equal. These masses match what you’d expect for neutron stars. As mentioned above, neutron stars are made of stuff™, and the properties of this leave an imprint on the waveform. If neutron stars are big and fluffy, they will get tidally distorted. Raising tides sucks energy and angular momentum out of the orbit, making the inspiral quicker. If neutron stars are small and dense, tides are smaller and the inspiral looks like that for tow black holes. For this initial analysis, we used waveforms which includes some tidal effects, so we get some preliminary information on the tides. We cannot exclude zero tidal deformation, meaning we cannot rule out from gravitational waves alone that the source contains at least one black hole (although this would be surprising, given the masses). However, we can place a weak upper limit on the combined dimensionless tidal deformability of $\tilde{\Lambda} \leq 900$. This isn’t too informative, in terms of working out what neutron stars are made from, but we’ll come back to this in the GW170817 Properties Paper and the GW170817 Equation-of-state Paper. Given the source masses, and all the electromagnetic observations, we’re pretty sure this is a binary neutron star system—there’s nothing to suggest otherwise. Having observed one (and one one) binary neutron star coalescence in O1 and O2, we can now put better constraints on the merger rate. As a first estimate, we assume that component masses are uniformly distributed between $1 M_\odot$ and $2 M_\odot$, and that spins are below 0.4 (in between the limits used for parameter estimation). Given this, we infer that the merger rate is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely within our previous upper limit [citation note]. There’s a lot more we can learn from GW170817, especially as we don’t just have gravitational waves as a source of information, and this is explained in the companion papers. The Multimessenger Paper Synopsis: Multimessenger Paper Read this if: Don’t. Use it too look up which other papers to read. Favourite part: The figures! It was a truly amazing observational effort to follow-up GW170817 The remarkable thing about this paper is that it exists. Bringing together such a diverse (and competitive) group was a huge effort. Alberto Vecchio was one of the editors, and each evening when leaving the office, he was convinced that the paper would have fallen apart by morning. However, it hung together—the story was too compelling. This paper explains how gravitational waves, short gamma-ray bursts, kilonovae all come from a single source [citation note]. This is the greatest collaborative effort in the history of astronomy. The paper outlines the discoveries and all of the initial set of observations. If you want to understand the observations themselves, this is not the paper to read. However, using it, you can track down the papers that you do want. A huge amount of care went in to trying to describe how discoveries were made: for example, Fermi observed GRB 170817A independently of the gravitational-wave alert, and we found GW170817 without relying on the GRB alert, however, the communication between teams meant that we took everything much seriously and pushed out alerts as quickly as possible. For more on the history of observations, I’d suggest scrolling through the GCN archive. The paper starts with an overview of the gravitational-wave observations from the inspiral, then the prompt detection of GRB 170817A, before describing how the gravitational-wave localization enabled discovery of the optical transient AT 2017gfo. This source, in nearby galaxy NGC 4993, was then the subject of follow-up across the electromagnetic spectrum. We have huge amount of photometric and spectroscopy of the source, showing general agreement with models for a kilonova. X-ray and radio afterglows were observed 9 days and 16 days after the merger, respectively [citation note]. No neutrinos were found, which isn’t surprising. The GW170817 Gamma-ray Burst Paper Synopsis: GW170817 Gamma-ray Burst Paper Read this if: You’re interested in the jets from where short gamma-ray bursts originate or in tests of general relativity Favourite part: How much science come come from a simple time delay measurement This joint LIGO–Virgo–FermiINTEGRAL paper combines our observations of GW170817 and GRB 170817A. The result is one of the most contentful of the companion papers. Detection of GW170817 and GRB 170817A. The top three panels show the gamma-ray lightcurves (first: GBM detectors 1, 2, and 5 for 10–50 keV; second: GBM data for 50–300 keV ; third: the SPI-ACS data starting approximately at 100 keV and with a high energy limit of least 80 MeV), the red line indicates the background.The bottom shows the a time–frequency representation of coherently combined gravitational-wave data from LIGO-Hanford and LIGO-Livingston. Figure 2 of the GW170817 Gamma-ray Burst Paper. The first item on the to-do list for joint gravitational-wave–gamma-ray science, is to establish that we are really looking at the same source. From the GW170817 Discovery Paper, we know that its source is consistent with being a binary neutron star system. Hence, there is matter around which can launch create the gamma-rays. The Fermi-GBM and INTEGRAL observations of GRB170817A indicate that it falls into the short class, as hypothesised as the result of a binary neutron star coalescence. Therefore, it looks like we could have the right ingredients. Now, given that it is possible that the gravitational waves and gamma rays have the same source, we can calculate the probability of the two occurring by chance. The probability of temporal coincidence is $5.0 \times 10^{-6}$, adding in spatial coincidence too, and the probability becomes $5.0 \times 10^{-8}$. It’s safe to conclude that the two are associated: merging binary neutron stars are the source of at least some short gamma-ray bursts! Testing gravity There is a $\sim1.74\pm0.05~\mathrm{s}$ delay time between the inferred merger time and the gamma-ray burst. Given that signal has travelled for about 85 million years (taking the 5% lower limit on the inferred distance), this is a really small difference: gravity and light must travel at almost exactly the same speed. To derive exact limit you need to make some assumptions about when the gamma-rays were created. We’d expect some delay as it takes time for the jet to be created, and then for the gamma-rays to blast their way out of the surrounding material. We conservatively (and arbitrarily) take a window of the delay being 0 to 10 seconds, this gives $\displaystyle -3 \times 10^{-15} \leq \frac{v_\mathrm{GW} - v_\mathrm{EM}}{v_\mathrm{EM}} \leq 7 \times 10^{-16}$. That’s pretty small! General relativity predicts that gravity and light should travel at the same speed, so I wasn’t too surprised by this result. I was surprised, however, that this result seems to have caused a flurry of activity in effectively ruling out several modified theories of gravity. I guess there’s not much point in explaining what these are now, but they are mostly theories which add in extra fields, which allow you to tweak how gravity works so you can explain some of the effects attributed to dark energy or dark matter. I’d recommend Figure 2 of Ezquiaga & Zumalacárregui (2017) for a summary of which theories pass the test and which are in trouble; Kase & Tsujikawa (2018) give a good review. Table showing viable (left) and non-viable (right) scalar–tensor theories after discovery of GW170817/GRB 170817A. The theories are grouped as Horndeski theories and (the more general) beyond Horndeski theories. General relativity is a tensor theory, so these models add in an extra scalar component. Figure 2 of Ezquiaga & Zumalacárregui (2017). We don’t discuss the theoretical implications of the relative speeds of gravity and light in this paper, but we do use the time delay to place bounds for particular on potential deviations from general relativity. 1. We look at a particular type of Lorentz invariance violation. This is similar to what we did for GW170104, where we looked at the dispersion of gravitational waves, but here it is for the case of $\alpha = 2$, which we couldn’t test. 2. We look at the Shapiro delay, which is the time difference travelling in a curved spacetime relative to a flat one. That light and gravity are effected the same way is a test of the weak equivalence principle—that everything falls the same way. The effects of the curvature can be quantified with the parameter $\gamma$, which describes the amount of curvature per unit mass. In general relativity $\gamma_\mathrm{GW} = \gamma_\mathrm{EM} = 1$. Considering the gravitational potential of the Milky Way, we find that $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ [citation note]. As you’d expect given the small time delay, these bounds are pretty tight! If you’re working on a modified theory of gravity, you have some extra checks to do now. Gamma-ray bursts and jets From our gravitational-wave and gamma-ray observations, we can also make some deductions about the engine which created the burst. The complication here, is that we’re not exactly sure what generates the gamma rays, and so deductions are model dependent. Section 5 of the paper uses the time delay between the merger and the burst, together with how quickly the burst rises and fades, to place constraints on the size of the emitting region in different models. The papers goes through the derivation in a step-by-step way, so I’ll not summarise that here: if you’re interested, check it out. Isotropic energies (left) and luminosities (right) for all gamma-ray bursts with measured distances. These isotropic quantities assume equal emission in all directions, which gives an upper bound on the true value if we are observing on-axis. The short and long gamma-ray bursts are separated by the standard $T_{90} = 2~\mathrm{s}$ duration. The green line shows an approximate detection threshold for Fermi-GBM. Figure 4 from the GW170817 Gamma-ray Burst Paper; you may have noticed that the first version of this paper contained two copies of the energy plot by mistake. GRB 170817A was unusually dim [citation note]. The plot above compares it to other gamma-ray bursts. It is definitely in the tail. Since it appears so dim, we think that we are not looking at a standard gamma-ray burst. The most obvious explanation is that we are not looking directly down the jet: we don’t expect to see many off-axis bursts, since they are dimmer. We expect that a gamma-ray burst would originate from a jet of material launched along the direction of the total angular momentum. From the gravitational waves alone, we can estimate that the misalignment angle between the orbital angular momentum axis and the line of sight is $\leq 55~\mathrm{deg}$ (adding in the identification of the host galaxy, this becomes $\leq 28~\mathrm{deg}$ using the Planck value for the Hubble constant and $36~\mathrm{deg}$ with the SH0ES value), so this is consistent with viewing the burst off-axis (updated numbers are given in the GW170817 Properties Paper). There are multiple models for such gamma-ray emission, as illustrated below. We could have a uniform top-hat jet (the simplest model) which we are viewing from slightly to the side, we could have a structured jet, which is concentrated on-axis but we are seeing from off-axis, or we could have a cocoon of material pushed out of the way by the main jet, which we are viewing emission from. Other electromagnetic observations will tell us more about the inclination and the structure of the jet [citation note]. Cartoon showing three possible viewing geometries and jet profiles which could explain the observed properties of GRB 170817A. Figure 5 of the GW170817 Gamma-ray Burst Paper. Now that we know gamma-ray bursts can be this dim, if we observe faint bursts (with unknown distances), we have to consider the possibility that they are dim-and-close in addition to the usual bright-and-far-away. The paper closes by considering how many more joint gravitational-wave–gamma-ray detections of binary neutron star coalescences we should expect in the future. In our next observing run, we could expect 0.1–1.4 joint detections per year, and when LIGO and Virgo get to design sensitivity, this could be 0.3–1.7 detections per year. The GW170817 Hubble Constant Paper Synopsis: GW170817 Hubble Constant Paper Read this if: You have an interest in cosmology Favourite part: In the future, we may be able to settle the argument between the cosmic microwave background and supernova measurements The Universe is expanding. In the nearby Universe, this can be described using the Hubble relation $v_H = H_0 D$, where $v_H$ is the expansion velocity, $H_0$ is the Hubble constant and $D$ is the distance to the source. GW170817 is sufficiently nearby for this relationship to hold. We know the distance from the gravitational-wave measurement, and we can estimate the velocity from the redshift of the host galaxy. Therefore, it should be simple to combine the two to find the Hubble constant. Of course, there are a few complications… This work is built upon the identification of the optical counterpart AT 2017gfo. This allows us to identify the galaxy NGC 4993 as the host of GW170817’s source: we calculate that there’s a $4 \times 10^{-5}$ probability that AT 2017gfo would be as close to NGC 4993 on the sky by chance. Without a counterpart, it would still be possible to infer the Hubble constant statistically by cross-referencing the inferred gravitational-wave source location with the ensemble of compatible galaxies in a catalogue (you assign a probability to the source being associated with each galaxy, instead of saying it’s definitely in this one). The identification of NGC 4993 makes things much simpler. As a first ingredient, we need the distance from gravitational waves. For this, a slightly different analysis was done than in the GW170817 Discovery Paper. We fix the sky location of the source to match that of AT 2017gfo, and we use (binary black hole) waveforms which don’t include any tidal effects. The sky position needs to be fixed, because for this analysis we are assuming that we definitely know where the source is. The tidal effects were not included (but precessing spins were) because we needed results quickly: the details of spins and tides shouldn’t make much difference to the distance. From this analysis, we find the distance is $41^{+6}_{-13}~\mathrm{Mpc}$ if we follow our usual convention of quoting the median at symmetric 90% credible interval; however, this paper primarily quotes the most probable value and minimal (not-necessarily symmmetric) 68.3% credible interval, following this convention, we write the distance as $44^{+3}_{-7}~\mathrm{Mpc}$. While NGC 4993 being close by makes the relationship for calculating the Hubble constant simple, it adds a complication for calculating the velocity. The motion of the galaxy is not only due to the expansion of the Universe, but because of how it is moving within the gravitational potentials of nearby groups and clusters. This is referred to as peculiar motion. Adding this in increases our uncertainty on the velocity. Combining results from the literature, our final estimate for the velocity is $v_H= 3017 \pm 166~\mathrm{km\,s^{-1}}$. We put together the velocity and the distance in a Bayesian analysis. This is a little more complicated than simply dividing the numbers (although that gives you a similar result). You have to be careful about writing things down, otherwise you might implicitly assume a prior that you didn’t intend (my most useful contribution to this paper is probably a whiteboard conversation with Will Farr where we tracked down a difference in prior assumptions approaching the problem two different ways). This is all explained in the Methods, it’s not easy to read, but makes sense when you work through. The result is $H_0 = 70^{+12}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (quoted as maximum a posteriori value and 68% interval, or $74^{+33}_{-12}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ in the usual median-and-90%-interval convention). An updated set of results is given in the GW170817 Properties Paper: $H_0 = 70^{+19}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (68% interval using the low-spin prior). This is nicely (and diplomatically) consistent with existing results. The distance has considerable uncertainty because there is a degeneracy between the distance and the orbital inclination (the angle of the normal to the orbital plane relative to the line of sight). If you could figure out the inclination from another observation, then you could tighten constraints on the Hubble constant, or if you’re willing to adopt one of the existing values of the Hubble constant, you can pin down the inclination. Data (updated data) to help you try this yourself are available [citation note]. Two-dimensional posterior probability distribution for the Hubble constant and orbital inclination inferred from GW170817. The contours mark 68% and 95% levels. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SH0ES). Figure 2 of the GW170817 Hubble Constant Paper. In the future we’ll be able to combine multiple events to produce a more precise gravitational-wave estimate of the Hubble constant. Chen, Fishbach & Holz (2017) is a recent study of how measurements should improve with more events: we should get to 4% precision after around 100 detections. The GW170817 Kilonova Paper Synopsis: GW170817 Kilonova Paper Read this if: You want to check our predictions for ejecta against observations Favourite part: We might be able to create all of the heavy r-process elements—including the gold used to make Nobel Prizes—from merging neutron stars When two neutron stars collide, lots of material gets ejected outwards. This neutron-rich material undergoes nuclear decay—now no longer being squeezed by the strong gravity inside the neutron star, it is unstable, and decays from the strange neutron star stuff™ to become more familiar elements (elements heavier than iron including gold and platinum). As these r-process elements are created, the nuclear reactions power a kilonova, the optical (infrared–ultraviolet) transient accompanying the merger. The properties of the kilonova depends upon how much material is ejected. In this paper, we try to estimate how much material made up the dynamical ejecta from the GW170817 collision. Dynamical ejecta is material which escapes as the two neutron stars smash into each other (either from tidal tails or material squeezed out from the collision shock). There are other sources of ejected material, such as winds from the accretion disk which forms around the remnant (whether black hole or neutron star) following the collision, so this is only part of the picture; however, we can estimate the mass of the dynamical ejecta from our gravitational-wave measurements using simulations of neutron star mergers. These estimates can then be compared with electromagnetic observations of the kilonova [citation note]. The amount of dynamical ejecta depends upon the masses of the neutron stars, how rapidly they are rotating, and the properties of the neutron star material (described by the equation of state). Here, we use the masses inferred from our gravitational-wave measurements and feed these into fitting formulae calibrated against simulations for different equations of state. These don’t include spin, and they have quite large uncertainties (we include a 72% relative uncertainty when producing our results), so these are not precision estimates. Neutron star physics is a little messy. We find that the dynamical ejecta is $10^{-3}$$10^{-2} M_\odot$ (assuming the low-spin mass results). These estimates can be feed into models for kilonovae to produce lightcurves, which we do. There is plenty of this type of modelling in the literature as observers try to understand their observations, so this is nothing special in terms of understanding this event. However, it could be useful in the future (once we have hoverboards), as we might be able to use gravitational-wave data to predict how bright a kilonova will be at different times, and so help astronomers decide upon their observing strategy. Finally, we can consider how much r-process elements we can create from the dynamical ejecta. Again, we don’t consider winds, which may also contribute to the total budget of r-process elements from binary neutron stars. Our estimate for r-process elements needs several ingredients: (i) the mass of the dynamical ejecta, (ii) the fraction of the dynamical ejecta converted to r-process elements, (iii) the merger rate of binary neutron stars, and (iv) the convolution of the star formation rate and the time delay between binary formation and merger (which we take to be $\propto t^{-1}$). Together (i) and (ii) give the mass of r-process elements per binary neutron star (assuming that GW170817 is typical); (iii) and (iv) give total density of mergers throughout the history of the Universe, and combining everything together you get the total mass of r-process elements accumulated over time. Using the estimated binary neutron star merger rate of $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, we can explain the Galactic abundance of r-process elements if more than about 10% of the dynamical ejecta is converted. Present day binary neutron star merger rate density versus dynamical ejecta mass. The grey region shows the inferred 90% range for the rate, the blue shows the approximate range of ejecta masses, and the red band shows the band where the Galactic elemental abundance can be reproduced if at least 50% of the dynamical mass gets converted. Part of Figure 5 of the GW170817 Kilonova Paper. The GW170817 Stochastic Paper Synopsis: GW170817 Stochastic Paper Read this if: You’re impatient for finding a background of gravitational waves Favourite part: The background symphony For every loud gravitational-wave signal, there are many more quieter ones. We can’t pick these out of the detector noise individually, but they are still there, in our data. They add together to form a stochastic background, which we might be able to detect by correlating the data across our detector network. Following the detection of GW150914, we considered the background due to binary black holes. This is quite loud, and might be detectable in a few years. Here, we add in binary neutron stars. This doesn’t change the picture too much, but gives a more accurate picture. Binary black holes have higher masses than binary neutron stars. This means that their gravitational-wave signals are louder, and shorter (they chirp quicker and chirp up to a lower frequency). Being louder, binary black holes dominate the overall background. Being shorter, they have a different character: binary black holes form a popcorn background of short chirps which rarely overlap, but binary neutron stars are long enough to overlap, forming a more continuous hum. The dimensionless energy density at a gravitational-wave frequency of 25 Hz from binary black holes is $1.1_{-0.7}^{+1.2} \times 10^{-9}$, and from binary neutron stars it is $0.7_{-0.6}^{+1.5} \times 10^{-9}$. There are on average $0.06_{-0.04}^{+0.06}$ binary black hole signals in detectors at a given time, and $15_{-12}^{+31}$ binary neutron star signals. Simulated time series illustrating the difference between binary black hole (green) and binary neutron star (red) signals. Each chirp increases in amplitude until the point at which the binary merges. Binary black hole signals are short, loud chirps, while the longer, quieter binary neutron star signals form an overlapping background. Figure 2 from the GW170817 Stochastic Paper. To calculate the background, we need the rate of merger. We now have an estimate for binary neutron stars, and we take the most recent estimate from the GW170104 Discovery Paper for binary black holes. We use the rates assuming the power law mass distribution for this, but the result isn’t too sensitive to this: we care about the number of signals in the detector, and the rates are derived from this, so they agree when working backwards. We evolve the merger rate density across cosmic history by factoring in the star formation rate and delay time between formation and merger. A similar thing was done in the GW170817 Kilonova Paper, here we used a slightly different star formation rate, but results are basically the same with either. The addition of binary neutron stars increases the stochastic background from compact binaries by about 60%. Detection in our next observing run, at a moderate significance, is possible, but I think unlikely. It will be a few years until detection is plausible, but the addition of binary neutron stars will bring this closer. When we do detect the background, it will give us another insight into the merger rate of binaries. The GW170817 Progenitor Paper Synopsis: GW170817 Progenitor Paper Read this if: You want to know about neutron star formation and supernovae Favourite part: The Spirography figures The identification of NGC 4993 as the host galaxy of GW170817’s binary neutron star system allows us to make some deductions about how it formed. In this paper, we simulate a large number of binaries, tracing the later stages of their evolution, to see which ones end up similar to GW170817. By doing so, we learn something about the supernova explosion which formed the second of the two neutron stars. The neutron stars started life as a pair of regular stars [bonus note]. These burned through their hydrogen fuel, and once this is exhausted, they explode as a supernova. The core of the star collapses down to become a neutron star, and the outer layers are blasted off. The more massive star evolves faster, and goes supernova first. We’ll consider the effects of the second supernova, and the kick it gives to the binary: the orbit changes both because of the rocket effect of material being blasted off, and because one of the components loses mass. From the combination of the gravitational-wave and electromagnetic observations of GW170817, we know the masses of the neutron star, the type of galaxy it is found in, and the position of the binary within the galaxy at the time of merger (we don’t know the exact position, just its projection as viewed from Earth, but that’s something). Orbital trajectories of simulated binaries which led to GW170817-like merger. The coloured lines show the 2D projection of the orbits in our model galaxy. The white lines mark the initial (projected) circular orbit of the binary pre-supernova, and the red arrows indicate the projected direction of the supernova kick. The background shading indicates the stellar density. Figure 4 of the GW170817 Progenitor Paper; animated equivalents can be found in the Science Summary. We start be simulating lots of binaries just before the second supernova explodes. These are scattered at different distances from the the centre of the galaxy, have different orbital separations, and have different masses of the pre-supernova star. We then add the effects of the supernova, adding in a kick. We fix then neutron star masses to match those we inferred from the gravitational wave measurements. If the supernova kick is too big, the binary flies apart and will never merge (boo). If the binary remains bound, we follow its evolution as it moves through the galaxy. The structure of the galaxy is simulated as a simple spherical model, a Hernquist profile for the stellar component and a Navarro–Frenk–White profile for the dark matter halo [citation note], which are pretty standard. The binary shrinks as gravitational waves are emitted, and eventually merge. If the merger happens at a position which matches our observations (yay), we know that the initial conditions could explain GW170817. Inferred progenitor properties: (second) supernova kick velocity, pre-supernova progenitor mass, pre-supernova binary separation and galactic radius at time of the supernova. The top row shows how the properties vary for different delay times between supernova and merger. The middle row compares all the binaries which survive the second supernova compared with the GW170817-like ones. The bottom row shows parameters for GW170817-like binaries with different galactic offsets than the $1.8~\mathrm{kpc}$ to $2.2~\mathrm{kpc}$ range used for GW1708017. The middle and bottom rows assume a delay time of at least $1~\mathrm{Gyr}$. Figure 5 of the GW170817 Progenitor Paper; to see correlations between parameters, check out Figure 8 of the GW170817 Progenitor Paper. The plot above shows the constraints on the progenitor’s properties. The inferred second supernova kick is $V_\mathrm{kick} \simeq 300_{-200}^{+250}~\mathrm{km\,s^{-1}}$, similar to what has been observed for neutron stars in the Milky Way; the per-supernova stellar mass is $M_\mathrm{He} \simeq 3.0_{-1.5}^{+3.5} M_\odot$ (we assume that the star is just a helium core, with the outer hydrogen layers having been stripped off, hence the subscript); the pre-supernova orbital separation was $R_\odot \simeq 3.5_{-1.5}^{+5.0} R_\odot$, and the offset from the the centre of the galaxy at the time of the supernova was $2.0_{-1.5}^{+4.0}~\mathrm{kpc}$. The main strongest constraints come from keeping the binary bound after the supernova; results are largely independent of the delay time once this gets above $1~\mathrm{Gyr}$ [citation note]. As we collect more binary neutron star detections, we’ll be able to deduce more about how they form. If you’re interested more in the how to build a binary neutron star system, the introduction to this paper is well referenced; Tauris et al. (2017) is a detailed (pre-GW170817) review. The GW170817 Neutrino Paper Synopsis: GW170817 Neutrino Paper Read this if: You want a change from gravitational wave–electromagnetic multimessenger astronomy Favourite part: There’s still something to look forward to with future detections—GW170817 hasn’t stolen all the firsts. Also this paper is not Abbot et al. This is a joint search by ANTARES, IceCube and the Pierre Auger Observatory for neutrinos coincident with GW170817. Knowing both the location and the time of the binary neutron star merger makes it easy to search for counterparts. No matching neutrinos were detected. Neutrino candidates at the time of GW170817. The map is is in equatorial coordinates. The gravitational-wave localization is indicated by the red contour, and the galaxy NGC 4993 is indicated by the black cross. Up-going and down-going regions for each detector are indicated, as detectors are more sensitive to up-going neutrinos, as the Cherenkov detectors are subject to a background from cosmic rays hitting the atmosphere. Figure 1 from the GW170817 Neutrino Paper. Using the non-detections, we can place upper limits on the neutrino flux. These are summarised in the plots below. Optimistic models for prompt emission from an on axis gamma-ray burst would lead to a detectable flux, but otherwise theoretical predictions indicate that a non-detection is expected. From electromagnetic observations, it doesn’t seem like we are on-axis, so the story all fits together. 90% confidence upper limits on neutrino spectral fluence $F$ per flavour (electron, muon and tau) as a function of energy $E$ in $\pm 500~\mathrm{s}$ window (top) about the GW170817 trigger time, and a $14~\mathrm{day}$ window following GW170817 (bottom). IceCube is also sensitive to MeV neutrinos (none were detected). Fluences are the per-flavour sum of neutrino and antineutrino fluence, assuming equal fluence in all flavours. These are compared to theoretical predictions from Kimura et al. (2017) and Fang & Metzger (2017), scaled to a distance of 40 Mpc. The angles labelling the models are viewing angles in excess of the jet opening angle. Figure 2 from the GW170817 Neutrino paper. Super-Kamiokande have done their own search for neutrinos, form $3.5~\mathrm{MeV}$ to around $100~\mathrm{PeV}$ (Abe et al. 2018). They found nothing in either the $\pm 500~\mathrm{s}$ window around the event or the $14~\mathrm{day}$ window following it. Similarly BUST looked for muon neutrinos and antineutrinos and found nothing in the $\pm 500~\mathrm{s}$ window around the event, and no excess in the $14~\mathrm{day}$ window following it (Petkov et al. 2019). The only post-detection neutrino modelling paper I’ve seen is Biehl, Heinze, &Winter (2017). They model prompt emission from the same source as the gamma-ray burst and find that neutrino fluxes would be $10^{-4}$ of current sensitivity. The GW170817 Post-merger Paper Synopsis: GW170817 Post-merger Paper Read this if: You are an optimist Favourite part: We really do check everywhere for signals Following the inspiral of two black holes, we know what happens next: the black holes merge to form a bigger black hole, which quickly settles down to its final stable state. We have a complete model of the gravitational waves from the inspiral–merger–ringdown life of coalescing binary black holes. Binary neutron stars are more complicated. The inspiral of two binary neutron stars is similar to that for black holes. As they get closer together, we might see some imprint of tidal distortions not present for black holes, but the main details are the same. It is the chirp of the inspiral which we detect. As the neutron stars merge, however, we don’t have a clear picture of what goes on. Material gets shredded and ejected from the neutron stars; the neutron stars smash together; it’s all rather messy. We don’t have a good understanding of what should happen when our neutron stars merge, the details depend upon the properties of the stuff™ neutron stars are made of—if we could measure the gravitational-wave signal from this phase, we would learn a lot. There are four plausible outcomes of a binary neutron star merger: 1. If the total mass is below the maximum mass for a (non-rotating) neutron star ($M < M^\mathrm{Static}$), we end up with a bigger, but still stable neutron star. Given our inferences from the inspiral (see the plot from the GW170817 Gamma-ray Burst Paper below), this is unlikely. 2. If the total mass is above the limit for a stable, non-rotating neutron star, but can still be supported by uniform rotation ($M^\mathrm{Static} < M < M^\mathrm{Uniform}$), we have a supramassive neutron star. The rotation will slow down due to the emission of electromagnetic and gravitational radiation, and eventually the neutron star will collapse to a black hole. The time until collapse could take something like $10$$5 \times 10^4~\mathrm{s}$; it is unclear if this is long enough for supramassive neutron stars to have a mid-life crisis. 3. If the total mass is above the limit for support from uniform rotation, but can still be supported through differential rotation and thermal gradients($M^\mathrm{Uniform} < M < M^\mathrm{Differential}$), then we have a hypermassive neutron star. The hypermassive neutron star cools quickly through neutrino emission, and its rotation slows through magnetic braking, meaning that it promptly collapses to a black hole in $\lesssim 1~\mathrm{s}$. 4. If the total mass is big enough($M^\mathrm{Differential} < M$), the merging neutron stars collapse down to a black hole. In the case of the collapse to a black hole, we get a ringdown as in the case of a binary black hole merger. The frequency is around $6~\mathrm{kHz}$, too high for us to currently measure. However, if there is a neutron star, there may be slightly lower frequency gravitational waves from the neutron star matter wibbling about. We’re not exactly sure of the form of these signals, so we perform an unmodelled search for them (knowing the position of GW170817’s source helps for this). Comparison of inferred component masses with critical mass boundaries for different equations of state. The left panel shows the maximum mass of a non-rotating neutron star compared to the initial baryonic mass (ignoring material ejected during merger and gravitational binding energy); the middle panel shows the maximum mass for a uniformly rotating neutron star; the right panel shows the maximum mass of a non-rotating neutron star compared of the gravitational mass of the heavier component neutron star. Figure 3 of the GW170817 Gamma-ray Burst Paper. Several different search algorithms were used to hunt for a post-merger signal: 1. coherent WaveBurst (cWB) was used to look for short duration ($< 1~\mathrm{s}$) bursts. This searched a $2~\mathrm{s}$ window including the merger time and covering the $1.7~\mathrm{s}$ delay to the gamma-ray burst detection, and frequencies of $1024$$4096~\mathrm{Hz}$. Only LIGO data were used, as Virgo data suffered from large noise fluctuations above $2.5~\mathrm{kHz}$. 2. cWB was used to look for intermediate duration ($< 500~\mathrm{s}$) bursts. This searched a $1000~\mathrm{s}$ window from the merger time, and frequencies $24$$2048~\mathrm{Hz}$. This used LIGO and Virgo data. 3. The Stochastic Transient Analysis Multi-detector Pipeline (STAMP) was also used to look for intermediate duration signals. This searched the merger time until the end of O2 (in $500~\mathrm{s}$ chunks), and frequencies $24$$4000~\mathrm{Hz}$. This used only LIGO data. There are two variations of STAMP: Zebragard and Lonetrack, and both are used here. Although GEO is similar to LIGO and Virgo and the searched high-frequencies, its data were not used as we have not yet studied its noise properties in enough detail. Since the LIGO detectors are the most sensitive, their data is most important for the search. No plausible candidates were found, so we set some upper limits on what could have been detected. From these, it is not surprising that nothing was found, as we would need pretty much all of the mass of the remnant to somehow be converted into gravitational waves to see something. Results are shown in the plot below. An updated analysis which puts upper limits on the post-merger signal is given in the GW170817 Properties Paper. Noise amplitude spectral density $\sqrt{S_n}$ for the four detectors, and search upper limits $h_\mathrm{rss}$ as a function of frequency. The noise amplitude spectral densities compare the sensitivities of the detectors. The search upper limits are root-sum-squared strain amplitudes at 50% detection efficiency. The colour code of the upper-limit markers indicates the search algorithm and the shape indicates the waveform injected to set the limits (the frequency is the average for this waveform). The bar mode waveform come from the rapid rotation of the supramassive neutron star leading to it becoming distorted (stretched) in a non-axisymmetric way (Lasky, Sarin & Sammut 2017); the magnetar waveform assumes that the (rapidly rotating) supramassive neutron star’s magnetic field generates significant ellipticity (Corsi & Mészáros 2009); the short-duration merger waveforms are from a selection of numerical simulations (Bauswein et al. 2013; Takami et al. 2015; Kawamura et al. 2016; Ciolfi et al. 2017). The open squares are merger waveforms scaled to the distance and orientation inferred from the inspiral of GW170817. The dashed black lines show strain amplitudes for a narrow-band signal with fixed energy content: the top line is the maximum possible value for GW170817. Figure 1 of the GW170817 Post-merger Paper. We can’t tell the fate of GW170817’s neutron stars from gravitational waves alone [citation note]. As high-frequency sensitivity is improved in the future, we might be able to see something from a really close by binary neutron star merger. The GW170817 Properties Paper Synopsis: GW170817 Properties Paper Read this if: You want the best results for GW170817’s source, our best measurement of the Hubble constant, or limits on the post-merger signal Favourite part: Look how tiny the uncertainties are! As time progresses, we often refine our analyses of gravitational-wave data. This can be because we’ve had time to recalibrate data from our detectors, because better analysis techniques have been developed, or just because we’ve had time to allow more computationally intensive analyses to finish. This paper is our first attempt at improving our inferences about GW170817. The results use an improved calibration of Virgo data, and analyses more of the signal (down to a low frequency of 23 Hz, instead of 30 Hz, which gives use about an extra 1500 cycles), uses improved models of the waveforms, and includes a new analysis looking at the post-merger signal. The results update those given in the GW170817 Discovery Paper, the GW170817 Hubble Constant Paper and the GW170817 Post-merger Paper. Inspiral Our initial analysis was based upon quick to calculate post-Newtonian waveform known as TaylorF2. We thought this should be a conservative choice: any results with more complicated waveforms should give tighter results. This worked out. We try several different waveform models, each based upon the point particle waveforms we use for analysing binary black hole signals with extra bits to model the tidal deformation of neutron stars. The results are broadly consistent, so I’ll concentrate on discussing our preferred results calculated using IMRPhenomPNRT waveform (which uses IMRPhenomPv2 as a base and adds on numerical-relativity calibrated tides). As in the GW170817 Discovery Paper, we perform the analysis with two priors on the binary spins, one with spins up to 0.89 (which should safely encompass all possibilities for neutron stars), and one with spins of up to 0.05 (which matches observations of binary neutron stars in our Galaxy). The first analysis we did was to check the location of the source. Reassuringly, we are still perfectly consistent with the location of AT 2017gfo (phew!). The localization is much improved, the 90% sky area is down to just $16~\mathrm{deg^2}$! Go Virgo! Having established that it still makes sense that AT 2017gfo pin-points the source location, we use this as the position in subsequent analyses. We always use the sky position of the counterpart and the redshift of the host galaxy (Levan et al. 2017), but we don’t typically use the distance. This is because we want to be able to measure the Hubble constant, which relies on using the distance inferred from gravitational waves. We use the distance from Cantiello et al. (2018) [citation note] for one calculation: an estimation of the inclination angle. The inclination is degenerate with the distance (both affect the amplitude of the signal), so having constraints on one lets us measure the other with improved precision. Without the distance information, we find that the angle between the binary’s total angular momentum and the line of sight is $152^{+21}_{-27}~\mathrm{deg}$ for the high-spin prior and $146^{+25}_{-27}~\mathrm{deg}$ with the low-spin prior. The difference between the two results is because of the spin angular momentum slightly shifts the direction of the total angular momentum. Incorporating the distance information, for the high-spin prior the angle is $153^{+15}_{-11}~\mathrm{deg}$ (so the misalignment angle is $27^{+11}_{-15}~\mathrm{deg}$), and for the low-spin prior it is $151^{+15}_{-11}~\mathrm{deg}$ (misalignment $29^{+11}_{-15}~\mathrm{deg}$) [citation note]. Estimated orientation and magnitude of the two component spins. The left pair is for the high-spin prior and so magnitudes extend to 0.89, and the right pair are for the low-spin prior and extend to 0.05. In each, the distribution for the more massive component is on the left, and for the smaller component on the right. The probability is binned into areas which have uniform prior probabilities. The low-spin prior truncates the posterior distribution, but this is less of an issue for the high-spin prior. Results are shown at a point in the inspiral corresponding to a gravitational-wave frequency of $100~\mathrm{Hz}$. Parts of Figure 8 and 9 of the GW170817 Properties Paper. Main results include: • The luminosity distance is $38.7_{-14.3}^{+7.4}~\mathrm{Mpc}$ with the the low-spin prior and $40.8_{-12.3}^{+5.6}~\mathrm{Mpc}$ with the high-spin prior. The difference is for the same reason as the difference in inclination measurements. The results are consistent with the distance to NGC 4993 [citation note]. • The chirp mass redshifted to the detector-frame is measured to be $1.1975^{+0.0001}_{-0.0001} M_\odot$ with the low-spin prior and $1.1976^{+0.0001}_{-0.0001} M_\odot$ with the high-spin. This corresponds to a physical chirp mass of $1.186_{-0.001}^{+0.001} M_\odot$. • The spins are not well constrained. We get the best measurement along the direction of the orbital angular momentum. For the low-spin prior, this is enough to disfavour the spins being antialigned, but that’s about it. For the high-spin prior, we rule out large spins aligned or antialigned, and very large spins in the plane. The aligned components of the spin are best described by the effective inspiral spin parameter $\chi_\mathrm{eff}$, for the low-spin prior it is $0.00^{+0.02}_{-0.01}$ and for the high-spin prior it is $0.02^{+0.08}_{-0.02}$. • Using the low-spin prior, the component masses are $m_1 = 1.36$$1.60 M_\odot$ and $m_2 = 1.16$$1.36 M_\odot$, and for the high-spin prior they are $m_1 = 1.36$$1.89 M_\odot$ and $m_2 = 1.00$$1.36 M_\odot$. These are largely consistent with our previous results. There are small shifts, but the biggest change is that the errors are a little smaller. Estimated masses for the two neutron stars in the binary using the high-spin (left) and low-spin (right) priors. The two-dimensional plot follows a line of constant chirp mass which is too narrow to resolve on this scale. Results are shown for four different waveform models. TaylorF2 (used in the initial analysis), IMRPhenomDNRT and SEOBNRT have aligned spins, while IMRPhenomPNRT includes spin precession. IMRPhenomPNRT is used for the main results.Figure 5 of the GW170817 Properties Paper. For the Hubble constant, we find $H_0 = 70^{+19}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ with the low-spin prior and $H_0 = 70^{+13}_{-7}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ with the high-spin prior. Here, we quote maximum a posterior value and narrowest 68% intervals as opposed to the the usual median and symmetric 90% credible interval. You might think its odd that the uncertainty is smaller when using the wider high-spin prior, but this is just another consequence of the difference in the inclination measurements. The values are largely in agreement with our initial values. The best measured tidal parameter is the combined dimensionless tidal deformability $\tilde{\Lambda}$. With the high-spin prior, we can only set an upper bound of $\tilde{\Lambda} < 630$ . With the low-spin prior, we find that we are still consistent with zero deformation, but the distribution peaks away from zero. We have $\tilde{\Lambda} = 300^{+500}_{-190}$ using the usual median and symmetric 90% credible interval, and $\tilde{\Lambda} = 300^{+420}_{-230}$ if we take the narrowest 90% interval. This looks like we have detected matter effects, but since we’ve had to use the low-spin prior, which is only appropriate for neutron stars, this would be a circular argument. More details on what we can learn about tidal deformations and what neutron stars are made of, under the assumption that we do have neutron stars, are given in the GW170817 Equation-of-state Paper. Post-merger Previously, in the GW170817 Post-merger Paper, we searched for a post-merger signal. We didn’t find anything. Now, we try to infer the shape of the signal, assuming it is there (with a peak within $250~\mathrm{ms}$ of the coalescence time). We still don’t find anything, but now we set much tighter upper limits on what signal there could be there. For this analysis, we use data from the two LIGO detectors, and from GEO 600! We don’t use Virgo data, as it is not well behaved at these high frequencies. We use BayesWave to try to constrain the signal. Noise amplitude spectral density for the detectors used, prior and posterior strain upper limits, and selected numerical simulations as a function of frequency. The signal upper limits are Bayesian 90% credible bounds for the signal in Hanford, but is derived from a coherent analysis of all three indicated detectors. Figure 13 of the GW170817 Properties Paper. While the upper limits are much better, they are still about 12–215 times larger than expectations from simulations. Therefore, we’d need to improve our detector sensitivity by about a factor of 3.5–15 to detect a similar signal. Fingers crossed! The GW170817 Equation-of-state Paper Synopsis: GW170817 Equation-of-state Paper Read this if: You want to know what neutron stars are made of Favourite part: The beautiful butterfly plots Usually in our work, we like to remain open minded and not make too many assumptions. In our analysis of GW170817, as presented in the GW170817 Properties Paper, we have remained agnostic about the components of the binary, seeing what the data tell us. However, from the electromagnetic observations, there is solid evidence that the source is a binary neutron star system. In this paper, we take it as granted that the source is made of two neutron stars, and that these neutron stars are made of similar stuff™ [citation note], to see what we can learn about the properties of neutron stars. When a two neutron stars get close together, they become distorted by each other’s gravity. Tides are raised, kind of like how the Moon creates tides on Earth. Creating tides takes energy out of the orbit, causing the inspiral to proceed faster. This is something we can measure from the gravitational wave signal. Tides are larger when the neutron stars are bigger. The size of neutron stars and how easy they are the stretch and squash depends upon their equation of state. We can use the measurements of the neutron star masses and amount of tidal deformation to infer their size and their equation of state. The signal is analysed as in the GW170817 Properties Paper (IMRPhenomPNRT waveform, low-spin prior, position set to match AT 2017gfo). However, we also add in some information about the composition of neutron stars. Calculating the behaviour of this incredibly dense material is difficult, but there are some relations (called universal relations) between the tidal deformability of neutron stars and their radii which are insensitive to the details of the equation of state. One relates symmetric and antisymmetric combinations of the tidal deformations of the two neutron stars as a function of the mass ratio, allows us to calculate consistent tidal deformations. Another relates the tidal deformation to the compactness (mass divided by radius) allows us to convert tidal deformations to radii. The analysis includes the uncertainty in these relations. In addition to this, we also use a parametric model of the equation of state to model the tidal deformations. By sampling directly in terms of the equation of state, it is easy to impose constraints on the allowed values. For example, we impose that the speed of sound inside the neutron star is less than the speed of light, that the equation of state can support neutron stars of that mass, that it is possible to explain the most massive confirmed neutron star (we use a lower limit for this mass of $1.97 M_\odot$), as well as it being thermodynamically stable. Accommodating the most massive neutron star turns out to be an important piece of information. The plot below shows the inferred tidal deformation parameters for the two neutron stars. The two techniques, using the equation-of-state insensitive relations and using the parametrised equation-of-state model without included the constraint of matching the $1.97 M_\odot$ neutron star, give similar results. For a $1.4 M_\odot$ neutron star, these results indicate that the tidal deformation parameter would be $\Lambda_{1.4} = 190^{+390}_{-120}$. We favour softer equations of state over stiffer ones [citation note]. I think this means that neutron stars are more huggable. Probability distributions for the tidal parameters of the two neutron stars. The tidal deformation of the more massive neutron star $\Lambda_1$ must be greater than that for the smaller neutron star $\Lambda_2$. The green shading and (50% and 90%) contours are calculated using the equation-of-state insensitive relations. The blue contours are for the parametrised equation-of-state model. The orange contours are from the GW170817 Properties Paper, where we don’t assume a common equation of state. The black lines are predictions from a selection of different equations of state Figure 1 of the GW170817 Equation-of-state Paper. We can translate our results into estimates on the size of the neutron stars. The plots below show the inferred radii. The results for the parametrised equation-of-state model now includes the constraint of accommodating a $1.97 M_\odot$ neutron star, which is the main reason for the difference in the plots. Using the equation-of-state insensitive relations we find that the radius of the heavier ($m_1 = 1.36$$1.62M_\odot$) neutron star is $R_1 = 10.8^{+2.0}_{-1.7}~\mathrm{km}$ and the radius of the lighter ($m_2 = 1.15$$1.36M_\odot$) neutron star is $R_2 = 10.7^{+2.1}_{-1.5}~\mathrm{km}$. With the parametrised equation-of-state model, the radii are $R_1 = 11.9^{+1.4}_{-1.4}~\mathrm{km}$ ($m_1 = 1.36$$1.58M_\odot$) and $R_2 = 11.9^{+1.4}_{-1.4}~\mathrm{km}$ ($m_2 = 1.18$$1.36M_\odot$). Posterior probability distributions for neutron star masses and radii (blue for the more massive neutron star, orange for the lighter). The left plot uses the equation-of-state insensitive relations, and the right uses the parametrised equation-of-state model. In the one-dimensional plots, the dashed lines indicate the priors. The lines in the top left indicate the size of a Schwarzschild Black hole and the Buchadahl limit for the collapse of a neutron star. Figure 3 of the GW170817 Equation-of-state Paper. When I was an undergraduate, I remember learning that neutron stars were about $15~\mathrm{km}$ in radius. We now know that’s not the case. If you want to investigate further, you can download the posterior samples from these analyses. Bonus notes Standard sirens In astronomy, we often use standard candles, objects like type IA supernovae of known luminosity, to infer distances. If you know how bright something should be, and how bright you measure it to be, you know how far away it is. By analogy, we can infer how far away a gravitational-wave source is by how loud it is. It is thus not a candle, but a siren. Sean Carrol explains more about this term on his blog. Nature I know… Nature published the original Schutz paper on measuring the Hubble constant using gravitational waves; therefore, there’s a nice symmetry in publishing the first real result doing this in Nature too. Globular clusters Instead of a binary neutron star system forming from a binary of two stars born together, it is possible for two neutron stars to come close together in a dense stellar environment like a globular cluster. A significant fraction of binary black holes could be formed this way. Binary neutron stars, being less massive, are not as commonly formed this way. We wouldn’t expect GW170817 to have formed this way. In the GW170817 Progenitor Paper, we argue that the probability of GW170817’s source coming from a globular cluster is small—for predicted rates, see Bae, Kim & Lee (2014). Levan et al. (2017) check for a stellar cluster at the site of AT 2017gfo, and find nothing. The smallest 30% of the Milky Way’s globular clusters would evade this limit, but these account for just 5% of the stellar mass in globular clusters, and a tiny fraction of dynamical interactions. Fong et al. (2019) perform some detailed observations looking for a globular cluster, and also find nothing. This excludes a cluster down to $1.3\ times 10^4 M_\odot$, which is basically all (99.996%) of them. Therefore, it’s unlikely that a cluster is the source of this binary. Citation notes Merger rates From our gravitational-wave data, we estimate the current binary neutron star merger rate density is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Several electromagnetic observers performed their own rate estimates from the frequency of detection (or lack thereof) of electromagnetic transients. Kasliwal et al. (2017) consider transients seen by the Palomar Transient Factory, and estimate a rate density of approximately $320~\mathrm{Gpc^{-3}\,yr^{-1}}$ (3-sigma upper limit of $800~\mathrm{Gpc^{-3}\,yr^{-1}}$), towards the bottom end of our range, but their rate increases if not all mergers are as bright as AT 2017gfo. Siebert et al. (2017) works out the rate of AT 2017gfo-like transients in the Swope Supernova Survey. They obtain an upper limit of $16000~\mathrm{Gpc^{-3}\,yr^{-1}}$. They use to estimate the probability that AT 2017gfo and GW170817 are just a chance coincidence and are actually unrelated. The probability is $9 \times 10^{-6}$ at 90% confidence. Smartt et al. (2017) estimate the kilonova rate from the ATLAS survey, they calculate a 95% upper limit of $30000~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely above our range. Yang et al. (2017) calculates upper limits from the DLT40 Supernova survey. Depending upon the reddening assumed, this is between $93000^{+16000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $109000^{+28000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Their figure 3 shows that this is well above expected rates. Zhang et al. (2017) is interested in the rate of gamma-ray bursts. If you know the rate of short gamma-ray bursts and of binary neutron star mergers, you can learn something about the beaming angle of the jet. The smaller the jet, the less likely we are to observe a gamma-ray burst. In order to do this, they do their own back-of-the-envelope for the gravitational-wave rate. They get $1100_{-910}^{+2500}~\mathrm{Gpc^{-3}\,yr^{-1}}$. That’s not too bad, but do stick with our result. If you’re interested in the future prospects for kilonova detection, I’d recommend Scolnic et al. (2017). Check out their Table 2 for detection rates (assuming a rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$): LSST and WFIRST will see lots, about 7 and 8 per year respectively. Using later observational constraints on the jet structure, Gupta & Bartos (2018) use the short gamma-ray burst rate to estimate a binary neutron star merger rate of $500~\mathrm{Gpc^{-3}\,yr^{-1}}$. They project that around 30% of gravitational-wave detections will be accompanied by gamma-ray bursts, once LIGO and Virgo reach design sensitivity. Della Valle et al. (2018) calculate an observable kilonova rate of $352_{-281}^{+810}~\mathrm{Gpc^{-3}\,yr^{-1}}$. To match up to our binary neutron star merger rate, we either need only a fraction of binary neutron star mergers to produce kilonova or for them to only be observable for viewing angles of less than $40^\circ$. Their table 2 contains a nice compilation of rates for short gamma-ray bursts. The electromagnetic story Some notes on an incomplete overview of papers describing the electromagnetic discovery. A list of the first wave of papers was compiled by Maria Drout, Stefano Valenti, and Iair Arcavi as a starting point for further reading. Independently of our gravitational-wave detection, a short gamma-ray burst GRB 170817A was observed by Fermi-GBM (Goldstein et al. 2017). Fermi-LAT did not see anything, as it was offline for crossing through the South Atlantic Anomaly. At the time of the merger, INTEGRAL was following up the location of GW170814, fortunately this meant it could still observe the location of GW170817, and following the alert they found GRB 170817A in their data (Savchenko et al. 2017). Following up on our gravitational-wave localization, an optical transient AT 2017gfo was discovered. The discovery was made by the One-Meter Two-Hemisphere (1M2H) collaboration using the Swope telescope at the Las Campanas Observatory in Chile; they designated the transient as SSS17a (Coulter et al. 2017). That same evening, several other teams also found the transient within an hour of each other: • The Distance Less Than 40 Mpc (DLT40) search found the transient using the PROMPT 0.4-m telescope at the Cerro Tololo Inter-American Observatory in Chile; they designated the transient DLT17ck (Valenti et al. 2017). • The VINROUGE collaboration (I think, they don’t actually identify themselves in their own papers) found the transient using VISTA at the European Southern Observatory in Chile (Tanvir et al. 2017). Their paper also describes follow-up observations with the Very Large Telescope, the Hubble Space Telescope, the Nordic Optical Telescope and the Danish 1.54-m Telescope, and has one of my favourite introduction sections of the discovery papers. • The MASTER collaboration followed up with their network of global telescopes, and it was their telescope at the San Juan National University Observatory in Argentina which found the transient (Lipunov et al. 2017); they, rather catchily denote the transient as OTJ130948.10-232253.3. • The Dark Energy Survey and the Dark Energy Camera GW–EM (DES and DECam) Collaboration found the transient with the DECam on the Blanco 4-m telescope, which is also at the Cerro Tololo Inter-American Observatory in Chile (Soares-Santos et al. 2017). • The Las Cumbres Observatory Collaboration used their global network of telescopes, with, unsurprisingly, their 1-m telescope at the Cerro Tololo Inter-American Observatory in Chile first imaging the transient (Arcavi et al. 2017). Their observing strategy is described in a companion paper (Arcavi et al. 2017), which also describes follow-up of GW170814. From these, you can see that South America was the place to be for this event: it was night at just the right time. There was a huge amount of follow-up across the infrared–optical–ultraviolet range of AT 2017gfo. Villar et al. (2017) attempts to bring these together in a consistent way. Their Figure 1 is beautiful. Assembled lightcurves from ultraviolet, optical and infrared observations of AT 2017gfo. The data points are the homogenized data, and the lines are fitted kilonova models. The blue light initially dominates but rapidly fades, while the red light undergoes a slower decay. Figure 1 of Villar et al. (2017). Hinderer et al. (2018) use numerical relativity simulations to compare theory and observations for gravitational-wave constraints on the tidal deformation and the kilonova lightcurve. They find that observations could be consistent with a neutron star–black hole binary and well as a binary neutron star. Coughline & Dietrich (2019) come to a similar conclusion. I think it’s unlikely that there would be a black hole this low mass, but it’s interesting that there are some simulations which can fit the observations. AT 2017gfo was also the target of observations across the electromagnetic spectrum. An X-ray afterglow was observed 9 days post merger, and 16 days post merger, just as we thought the excitement was over, a radio afterglow was found: The afterglow will continue to brighten for a while, so we can expect a series of updates: • Pooley, Kumar & Wheeler (2017) observed with Chandra 108 and 111 days post merger. Ruan et al. (2017) observed with Chandra 109 days post merger. The large gap in the the X-ray observations from the initial observations is because the Sun got in the way. • Mooley et al. (2017) update the GROWTH radio results up to 107 days post merger (the largest span whilst still pre-empting new X-ray observations), observing with the Very Large Array, Australia Telescope Compact Array and Giant Meterewave Radio Telescope. Excitingly, the afterglow has also now been spotted in the optical: • Lyman et al. (2018) observed with Hubble 110 (rest-frame) days post-merger (which is when the Sun was out of the way for Hubble). At this point the kilonova should have faded away, but they found something, and this is quite blue. The conclusion is that it’s the afterglow, and it will peak in about a year. • Margutti et al. (2018) brings together Chandra X-ray observations, Very Large Array radio observations and Hubble optical observations. The Hubble observations are 137 days post merger, and the Chandra observations are 153 days and 163 days post-merger. They find that they all agree (including the tentative radio signal at 10 days post-merger). They argue that the emission disfavours on-axis jets and spherical fireballs. Evolution of radio, optical and X-ray spectral energy density of the counterpart to GW170817. The radio and X-ray are always dominated by the afterglow, as indicated by them following the same power law. At early times, the optical is dominated by the kilonova, but as this fades, the afterglow starts to dominate. Figure. 1 of Margutti et al. (2018). • D’Avanzo et al. (2018) observed in X-ray 135 days post-merger with XMM-Newton. They find that the flux is faded compared to the previous trend. They suggest that we’re just at the turn-over, so this is consistent with the most recent Hubble observations. • Resmi et al. (2018) observed at low radio frequencies with the Giant Meterwave Radio Telescope. They saw the signal at $1390~\mathrm{MHz}$ after 67 days post-merger, but this evolves little over the duration of their observations (to day 152 post-merger), also suggesting a turn-over. • Dobie et al. (2018) observed in radio 125–200 days post-merger with the Very Large Array and Australia Telescope Compact Array, and they find that the afterglow is starting to fade, with a peak at 149 ± 2 days post-merger. • Nynka et al. (2018) made X-ray observations at 260 days post-merger. They conclude the afterglow is definitely fading, and that this is not because of passing of the synchrotron cooling frequency. • Mooley et al. (2018) observed in radio to 298 days. They find the turn-over around 170 days. They argue that results support a narrow, successful jet. • Troja et al. (2018) observed in radio and X-ray to 359 days. The fading is now obvious, and starting to reveal something about the jet structure. Their best fits seems to favour a structured relativistic jet or a wide-angled cocoon. • Lamb et al. (2018) observed in optical to 358 days. They infer a peak around 140–160 days. Their observations are well fit either by a Gaussian structured jet or a two-component jet (with the second component being the cocoon), although the two-component model doesn’t fit early X-ray observations well. They conclude there must have been a successful jet of some form. Radio, optical and X-ray observations to 358 days after merger. The coloured lines show fitted Gaussian jet models. Figure 3 of Lamb et al. (2018). • Fong et al. (2019) observe in optical to 584 days post-merger, combined with observation in radio to 585 days post-merger and in X-ray 583 days post-merger. These observations favour a structured jet over a quasi-spherical outflow. Hajela et al. (2019) extend the radio and X-ray observations even further, out to 743 days post-merger. Left: Optical afterglow observed until 584 days post-merger together with predictions for a structured jet and a quasi-spherical outflow (Wu & MacFadyen 2018). Right: Radio, optical and X-ray observations to 535 days, 534 days and 533 days post-merger-respectively. Triangles denote upper limits. Figures 2 and 3 of Fong et al. (2019). The story of the most ambitious cross-over of astronomical observations might now becoming to an end. Shapiro delay Using the time delay between GW170817 and GRB 170817A, a few other teams also did their own estimation of the Shapiro delay before they knew what was in our GW170817 Gamma-ray Burst Paper. • Wang et al. (2017) consider the Milky Way potential and large scale structure to estimate $-4 \times 10^{-9} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM}$. • Boran et al. (2017) consider all the galaxies in the GLADE catalogue which are within a radius of $400~\mathrm{kpc}$ of the line of sight, and derive $\|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}\| \leq 3.9 \times 10 ^{-9}$. • Wei et al. (2017) estimate $\|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}\| \leq 5.9 \times 10 ^{-8}$ using the Milky Way’s potential and $\|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}\| \leq 9.2 \times 10 ^{-11}$ using the Virgo cluster’s potential. Our estimate of $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ is the most conservative. Comparison to other gamma-ray bursts Are the electromagnetic counterparts to GW170817 similar to what has been observed before? Yue et al. (2017) compare GRB 170817A with other gamma-ray bursts. It is low luminosity, but it may not be alone. There could be other bursts like it (perhaps GRB 070923, GRB 080121 and GRB 090417A), if indeed they are from nearby sources. They suggest that GRB 130603B may be the on-axis equivalent of GRB 170817A [citation note]; however, the non-detection of kilonovae for several bursts indicates that there needs to be some variation in their properties too. This agree with the results of Gompertz et al. (2017), who compares the GW170817 observations with other kilonovae: it is fainter than the other candidate kilonovae (GRB 050709, GRB 060614, GRB 130603B and tentatively GRB 160821B), but equally brighter than upper limits from other bursts. There must be a diversity in kilonovae observations. Fong et al. (2017) look at the diversity of afterglows (across X-ray to radio), and again find GW170817’s counterpart to be faint. This is probably because we are off-axis. The most comprehensive study is von Kienlin et al. (2019) who search ten years of Fermi archives and find 13 GRB 170817A-like short gamma-ray bursts: GRB 081209A, GRB 100328A, GRB 101224A, GRB 110717A; GRB 111024C, GRB 120302B, GRB 120915A, GRB 130502A, GRB 140511A, GRB 150101B, GRB 170111B, GRB 170817A and GRB 180511A. There is a range behaviours in these, with the shorter GRBs showing fast variability. Future observations will help unravel how much variation there is from viewing different angles, and how much intrinsic variation there is from the source—perhaps some short gamma-ray bursts come from neutron star–black hole binaries? Inclination, jets and ejecta Pretty much every observational paper has a go at estimating the properties of the ejecta, the viewing angle or something about the structure of the jet. I may try to pull these together later, but I’ve not had time yet as it is a very long list! Most of the inclination measurements assumed a uniform top-hat jet, which we now know is not a good model. In my non-expert opinion, the later results seem more interesting. With very-long baseline interferometry radio observations to 230 days post-merger, Mooley et al. (2018) claim that while the early radio emission was powered by the wide cocoon of a structured jet, the later emission is dominated by a narrow, energetic jet. There was a successful jet, so we would have seen something like a regular short gamma-ray burst on axis. They estimate that the jet opening angle is $< 5~\mathrm{deg}$, and that we are viewing it at an angle of $20 \pm 5~\mathrm{deg}$. With X-ray and radio observations to 359 days, Troja et al. (2018) estimate (folding in gravitational-wave constraints too) that the viewing angle is $22 \pm 6~\mathrm{deg}$, and the width of a Gaussian structured jet would be $3.4 \pm 1.1~\mathrm{deg}$. Hubble constant and misalignment Guidorzi et al. (2017) try to tighten the measurement of the Hubble constant by using radio and X-ray observations. Their modelling assumes a uniform jet, which doesn’t look like a currently favoured option [citation note], so there is some model-based uncertainty to be included here. Additionally, the jet is unlikely to be perfectly aligned with the orbital angular momentum, which may add a couple of degrees more uncertainty. Mandel (2018) works the other way and uses the recent Dark Energy Survey Hubble constant estimate to bound the misalignment angle to less than $28~\mathrm{deg}$, which (unsurprisingly) agrees pretty well with the result we obtained using the Planck value. Finstad et al. (2018) uses the luminosity distance from Cantiello et al. (2018) [citation note] as a (Gaussian) prior for an analysis of the gravitational-wave signal, and get a misalignment $32^{+10}_{-13}\pm 2~\mathrm{deg}$ (where the errors are statistical uncertainty and an estimate of systematic error from calibration of the strain). Hotokezaka et al. (2018) use the inclination results from Mooley et al. (2018) [citation note] (together with the updated posterior samples from the GW170817 Properties Paper) to infer a value of $h = 0.689^{+0.047}_{-0.046}$ (quoting median and 68% symmetric credible interval). Using different jet models changes their value for the Hubble constant a little; the choice of spin prior does not (since we get basically all of the inclination information from their radio observations). The results is still consistent with Planck and SH0ES, but is closer to the Planck value. Posterior probability distribution for the Hubble constant inferred from GW170817 using only gravitational waves (GWs), and folding in models for the power-law jet (PLJ) model and very-long baseline interferometry (VLBI) radio observations. The lines symmetric mark 68% intervals. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SH0ES). Figure 2 of Hotokezaka et al. (2018) Dhawan et al. (2019) use broadband photometry of the kilonova to estimate the observation angle as $32.5^{+11.7}_{-9.7}~\mathrm{deg}$. Combining this with results from the Hubble Constant Paper they find $h = 0.724^{+0.079}_{-0.073}$. NGC 4993 properties In the GW170817 Progenitor Paper we used component properties for NGC 4993 from Lim et al. (2017): a stellar mass of $(10^{10.454}/h^2) M_\odot$ and a dark matter halo mass of $(10^{12.2}/h) M_\odot$, where we use the Planck value of $h = 0.679$ (but conclusions are similar using the SH0ES value for this). Blanchard et al. (2017) estimate a stellar mass of about $\log(M_\ast/M_\odot) = 10.65^{+0.03}_{-0.03}$. They also look at the star formation history, 90% were formed by $6.8^{+2.2}_{-0.8}~\mathrm{Gyr}$ ago, and the median mass-weighted stellar age is $13.2^{+0.5}_{-0.9}~\mathrm{Gyr}$. From this they infer a merger delay time of $6.8$$13.6~\mathrm{Gyr}$. From this, and assuming that the system was born close to its current location, they estimate that the supernova kick $V_\mathrm{kick} \leq 200~\mathrm{km\,s^{-1}}$, towards the lower end of our estimate. They use $h = 0.677$. Im et al. (2017) find a mean stellar mass of $0.3$$1.2 \times 10^{11} M_\odot$ and the mean stellar age is greater than about $3~\mathrm{Gyr}$. They also give a luminosity distance estimate of $38.4 \pm 8.9~\mathrm{Mpc}$, which overlaps with our gravitational-wave estimate. I’m not sure what value of $h$ they are using. Levan et al. (2017) suggest a stellar mass of around $1.4 \times 10^{11} M_\odot$. They find that 60% of stars by mass are older than $5~\mathrm{Gyr}$ and that less than 1% are less than $0.5~\mathrm{Gyr}$ old. Their Figure 5 has some information on likely supernova kicks, they conclude it was probably small, but don’t quantify this. They use $h = 0.696$. Pan et al. (2017) find $\log(M_\ast/M_\odot) = 10.49^{+0.08}_{-0.20}$. They calculate a mass-weighted mean stellar age of $10.97~\mathrm{Gyr}$ and a likely minimum age for GW170817’s source system of $2.8~\mathrm{Gyr}$. They use $h = 0.7$. Troja et al. (2017) find a stellar mass of $\log(M_\ast/M_\odot) \sim 10.88$, and suggest an old stellar population of age $> 2~\mathrm{Gyr}$. Ebrová & Bílek (2018) assume a distance of $41.0~\mathrm{kpc}$ and find a halo mass of $1.939 \times 10^{12} M_\odot$. They suggest that NGC 4993 swallowed a smaller late-type galaxy somewhere between $0.2~\mathrm{Gyr}$ and $1~\mathrm{Gyr}$ ago, most probably around $0.4~\mathrm{Gyr}$ ago. The consensus seems to be that the stellar population is old (and not much else). Fortunately, the conclusions of the GW170817 Progenitor Paper are pretty robust for delay times longer than $1~\mathrm{Gyr}$ as seems likely. A couple of other papers look at the distance of the galaxy: • Hjoth et al. (2017) combine a redshift measurement from MUSE, and a fundamental plane estimate based upon Hubble observations, to obtain an distance of $41.0 \pm 3.1~\mathrm{Mpc}$. • Cantiello et al. (2018) use Hubble observations to estimate the distance using surface brightness fluctuations. They obtain a distance of $40.7 \pm 1.4 \pm 1.9~\mathrm{Mpc}$. This implies a value for the Hubble constant of $h = 0.719 \pm 0.071$. The values are consistent with our gravitational-wave estimates. The remnant’s fate We cannot be certain what happened to the merger remnant from gravitational-wave observations alone. However, electromagnetic observations do give some hints here. Evans et al. (2017) argue that their non-detection of X-rays when observing with Swift and NuSTAR indicates that there is no neutron star remnant at this point, meaning we must have collapsed to form a black hole by 0.6 days post-merger. This isn’t too restricting in terms of the different ways the remnant could collapse, but does exclude a stable neutron star remnant. MAXI also didn’t detect any X-rays 4.6 hours after the merger (Sugita et al. 2018). Pooley, Kumar & Wheeler (2017) consider X-ray observations of the afterglow. They calculate that if the remnant was a hypermassive neutron star with a large magnetic field, the early (10 day post-merger) luminosity would be much higher (and we could expect to see magnetar outbursts). Therefore, they think it is more likely that the remnant is a black hole. However, Piro et al. (2018) suggest that if the the spin-down of the neutron star remnant is dominated by losses due to gravitational wave emission, rather than electromagnetic emission, then the scenario is still viable. They argue that a tentatively identified X-ray flare seen 155 days post-merger, could be evidence of dissipation of the the neutron star’s toroidal magnetic field. Kasen et al. (2017) use the observed red component of the kilonova to argue that the remnant must have collapsed to a black hole in $< 10~\mathrm{ms}$. A neutron star would irradiate the ejecta with neutrinos, lower the neutron fraction and making the ejecta bluer. Since it is red, the neutrino flux must have been shut off, and the neutron star must have collapsed. We are in case b in their figure below. Cartoon of the different components of matter ejected from neutron star mergers. Red colours show heavy r-process elements and blue colours light r-process elements. There is a tidal tail of material forming a torus in the orbital plane, roughly spherical winds from the accretion disk, and material squeezed into the polar reasons during the collision. In case a, we have a long-lived neutron star, and its neutrino irradiation leads to blue ejecta. In case b the neutron star collapses, cutting off the neutrino flux. In case c, there is a neutron star–black hole merger, and we don’t have the polar material from the collision. Figure 1 of Kasen et al. (2017); also see Figure 1 of Margalit & Metzger (2017). Ai et al. (2018) find that there are some corners of parameter space for certain equations of state where a long-lived neutron star is possible, even given the observations. Therefore, we should remain open minded. Margalit & Metzger (2017) and Bauswein et al. (2017) note that the relatively large amount of ejecta inferred from observations [citation note] is easier to explain when there is delayed (on timescales of $> 10~\mathrm{ms}$). This is difficult to resolve unless neutron star radii are small ($\lesssim 11~\mathrm{km}$). Metzger, Thompson & Quataert (2018) derive how this tension could be resolved if the remnant was a rapidly spinning magnetar with a life time of $0.1$$1~\mathrm{s}$Matsumoto et al. (2018), suggest that the optical emission is powered by the the jet and material accreting onto the central object, rather than r-process decay, and this permits much smaller amounts of ejecta, which could also solve the issue. Yu & Dai (2017) suggest that accretion onto a long-lived neutron star could power the emission, and would only require a single opacity for the ejecta. Li et al. (2018) put forward a similar theory, arguing that both the high ejecta mass and low opacity are problems for the standard r-process explanation, but fallback onto a neutron star could work. However, Margutti et al. (2018) say that X-ray emission powered by a central engine is disfavoured at all times. In conclusion, it seems probable that we ended up with a black hole, and we had an a unstable neutron star for a short time after merger, but I don’t think it’s yet settled how long this was around. Gill, Nathanail & Rezzolla (2019) considered how long it would take to produce the observed amount of ejecta, and the relative amounts of red and blue eject, as well as the delay time between the gravitational-wave measurement of the merger and the observation of the gamma-ray burst, to estimate how long it took the remnant to collapse to a black hole. They find a lifetime of $= 0.98^{+0.31}_{-0.26}~\mathrm{s}$. Twin stars We might not have two neutron stars with the same equation of state if they can undergo a phase transition. This would be kind of of like if one one made up of fluffer marshmallow, and the other was made up of gooey toasted marshmallow: they have the same ingredient, but in one the type of stuff has changed, giving it different physical properties. Standard neutron stars could be made of hadronic matter, kind of like a giant nucleus, but we could have another type where the hadrons break down into their component quarks. We could therefore have two neutron stars with similar masses but with very different equations of state. This is referred to as the twin star scenario. Hybrid stars which have quark cores surrounded by hadronic outer layers are often discussed in this context. Neutron star equation of state Several papers have explored what we can deduce about the nature of neutron star stuff™ from gravitational wave or electromagnetic observations the neutron star coalescence. It is quite a tricky problem. Below are some investigations into the radii of neutron stars and their tidal deformations; these seem compatible with the radii inferred in the GW170817 Equation-of-state Paper. Bauswein et al. (2017) argue that the amount of ejecta inferred from the kilonova is too large for there to have been a prompt collapse to a black hole [citation note]. Using this, they estimate that the radius of a non-rotating neutron star of mass $1.6~\mathrm{M_\odot}$ has a radius of at least $10.68_{-0.04}^{+0.15}~\mathrm{km}$. They also estimate that the radius for the maximum mass nonrotating neutron star must be greater than $9.60_{-0.03}^{+0.14}~\mathrm{km}$. Köppel, Bovard & Rezzolla (2019) calculate a similar, updated analysis, using a new approach to fit for the maximum mass of a neutron star, and they find a radius for $1.6~\mathrm{M_\odot}$ is greater than $10.90~\mathrm{km}$, and for $1.4~\mathrm{M_\odot}$ is greater than $10.92~\mathrm{km}$. Annala et al. (2018) combine our initial measurement of the tidal deformation, with the requirement hat the equation of state supports a $2 M_\odot$ neutron star (which they argue requires that the tidal deformation of a $1.4 M_\odot$ neutron star is at least $120$). They argue that the latter condition implies that the radius of a $1.4 M_\odot$ neutron star is at least $9.9~\mathrm{km}$ and the former that it is less than $13.6~\mathrm{km}$. Radice et al. (2018) combine together observations of the kilonova (the amount of ejecta inferred) with gravitational-wave measurements of the masses to place constraints on the tidal deformation. From their simulations, they argue that to explain the ejecta, the combined dimensionless tidal deformability must be $\tilde{\Lambda} > 400$. This is consistent with results in the GW170817 Properties Paper, but would eliminate the main peak of the distribution we inferred from gravitational waves alone. However, Kuichi et al. (2019) show that it is possible to get the required ejecta for smaller tidal deformations, depending upon assumptions about the maximum neutron star mass (higher masses allow smaller tidal deformations)mand asymmetry of the binary components. Lim & Holt (2018) perform some equation-of-state calculations. They find that their particular method (chiral effective theory) is already in good agreement with estimates of the maximum neutron star mass and tidal deformations. Which is nice. Using their models, they predict that for GW170817’s chirp mass $\tilde{\Lambda} = 532^{+106}_{-119}$. Raithel, Özel & Psaltis (2018) argue that for a given chirp mass, $\tilde{\Lambda}$ is only a weak function of component masses, and depends mostly on the radii. Therefore, from our initial inferred value, they put a 90% upper limit on the radii of $13~\mathrm{km}$. Most et al. (2018) consider a wide range of parametrised equations of state. They consider both hadronic (made up of particles like neutrons and protons) equation of states, and ones where they undergo phase transitions (with hadrons breaking into quarks), which could potentially mean that the two neutron stars have quite different properties [citation note]. A number of different constraints are imposed, to give a selection of potential radius ranges. Combining the requirement that neutron stars can be up to $2.01 M_\odot$ (Antoniadis et al. 2013), the maximum neutron star mass of $2.17 M_\odot$ inferred by Margalit & Metzger (2017), our initial gravitational-wave upper limit on the tidal deformation and the lower limit from Radice et al. (2018), they estimate that the radius of a $1.4 M_\odot$ neutron star is $12.00$$13.45~\mathrm{km}$ for the hadronic equation of state. For the equation of state with the phase transition, they do the same, but without the tidal deformation from Radice et al. (2018), and find the radius of a $1.4 M_\odot$ neutron star is $8.53$$13.74~\mathrm{km}$. Paschalidis et al. (2018) consider in more detail the idea equations of state with hadron–quark phase transitions, and the possibility that one of the components of GW170817’s source was a hadron–quark hybrid star. They find that the initial tidal measurements are consistent with this. Burgio et al. (2018) further explore the possibility that the two binary components have different properties. They consider both there being a hadron–quark phase transition, and also that one star is hadronic and the other is a quark star (made up of deconfined quarks, rather than ones packaged up inside hadrons). X-ray observations indicate that neutron stars have radii in the range $9.9$$11.2~\mathrm{km}$, whereas most of the radii inferred for GW170817’s components are larger. This paper argues that this can be resolved if one of the components of GW170817’s source was a hadron–quark hybrid star or a quark star. De et al. (2018) perform their own analysis of the gravitational signal, with a variety of different priors on the component masses. They assume that the two neutron stars have the same radii. In the GW170817 Equation-of-state Paper we find that the difference can be up to about $2~\mathrm{km}$, which I think makes this an OK approximation; Zhao & Lattimer (2018) look at this in more detail. Within their approximation, they estimate the neutron stars to have a common radius of $8.9$$13.2~\mathrm{km}$. Malik et al. (2018) use the initial gravitational-wave upper bound on tidal deformation and the lower bound from Radice et al. (2018) in combination with several equations of state (calculated using relativistic mean field and of Skyrme Hartree–Fock recipes, which sound delicious). For a $1.4 M_\odot$ neutron star, they obtain a tidal deformation in the range $344$$859$ and the radius in the range $11.82$$13.72~\mathrm{km}$. Radice & Dai (2018) do their own analysis of our gravitational-wave data (using relative binning) and combine this with an analysis of the electromagnetic observations using models for the accretion disc. They find that the areal radius of a $1.4 M_\odot$ is $12.2^{+1.0}_{-0.8} \pm 0.2~\mathrm{km}$. These results are in good agreement with ours, their inclusion of electromagnetic data pushes their combined results towards larger values for the tidal deformation. Montaña et al. (2018) consider twin star scenarios [citation note] where we have a regular hadronic neutron star and a hybrid hadron–quark star. They find the data are consistent with neutron star–neutron star, neutron star–hybrid star or hybrid star–hybrid star binaries. Their Table II is a useful collection of results for the radius of a $1.4 M_\odot$ neutron star, including the possibility of phase transitions. Coughlin et al. (2018) use our LIGO–Virgo results and combine them with constraints from the observation of the kilonova (combined with fits to numerical simulations) and the gamma-ray burst. The electromagnetic observations give some extra information of the tidal deformability, mass ratio and inclination. They use the approximation that the neutron stars have equal radii. They find that the tidal deformability $\tilde{\Lambda}$ has a 90% interval $279$$822$ and the neutron star radius is $11.1$$13.4~\mathrm{km}$. Zhou, Chen & Zhang (2019) use data from heavy ion collider experiments, which constrains the properties of nuclear density stuff™ at one end of the spectrum, the existence of $2 M_\odot$ neutron stars, and our GW170817 Equation-of-state Paper constraints on the tidal deformation to determine that the radius of a $1.4 M_\odot$ neutron star is $11.1$$13.3~\mathrm{km}$. Kumar & Landry (2019) use the GW170817 Equation-of-state Paper constraints, and combine these of electromagnetic constraints to get an overall tidal deformability measurement. They use of observations of X-ray bursters from Özel et al. (2016) which give mass and radius measurements, and translate these using universal relations. Their overall result is the tidal deformability of a $1.4 M_\odot$ neutron star is $112^{+46}_{-33}$. Gamba, Read & Wade (2019) estimate the systematic error in the GW170817 Equation-of-state Paper results for the neutron star radius which may have been introduced from assumptions about the crust’s equation of state. They find that the error could be $0.3~\mathrm{km}$ (about 3%). GW170817—The pot of gold at the end of the rainbow Advanced LIGO and Advanced Virgo have detected their first binary neutron star inspiral. Remarkably, this event was observed not just with gravitational waves, but also across the electromagnetic spectrum, from gamma-rays to radio. This discovery confirms the theory that binary neutron star mergers are the progenitors of short gamma-ray bursts and kilonovae, and may be the primary source of heavy elements like gold. In this post, I’ll go through some of the story of GW170817. As for GW150914, I’ll write another post on the more technical details of our papers, once I’ve had time to catch up on sleep. Discovery The second observing run (O2) of the advanced gravitational-wave detectors started on 30 November 2016. The first detection came in January—GW170104. I was heavily involved in the analysis and paper writing for this. We finally finished up in June, at which point I was thoroughly exhausted. I took some time off in July [bonus note], and was back at work for August. With just one month left in the observing run, it would all be downhill from here, right? August turned out to be the lava-filled, super-difficult final level of O2. As we have now announced, on August 14, we detected a binary black hole coalescence—GW170814. This was the first clear detection including Virgo, giving us superb sky localization. This is fantastic for astronomers searching for electromagnetic counterparts to our gravitational-wave signals. There was a flurry of excitement, and we thought that this was a fantastic conclusion to O2. We were wrong, this was just the save point before the final opponent. On August 17, we met the final, fire-ball throwing boss. Text messages from our gravitational-wave candidate event database GraceDB. The final message is for GW170817, or as it was known at the time, G298048. It certainly caught my attention. The messages above are for GW170814, that was picked up multiple times by our search algorithms. It was a busy week. At 1:58 pm BST my phone buzzed with a text message, an automated alert of a gravitational-wave trigger. I was obviously excited—I recall that my exact thoughts were “What fresh hell is this?” I checked our online event database and saw that it was a single-detector trigger, it was only seen by our Hanford instrument. I started to relax, this was probably going to turn out to be a glitch. The template masses, were low, in the neutron star range, not like the black holes we’ve been finding. Then I saw the false alarm rate was better than one in 9000 years. Perhaps it wasn’t just some noise after all—even though it’s difficult to estimate false alarm rates accurately online, as especially for single-detector triggers, this was significant! I kept reading. Scrolling down the page there was an external coincident trigger, a gamma-ray burst (GRB 170817A) within a couple of seconds… We’re gonna need a bigger author list. Credit: Zanuck/Brown Productions Short gamma-ray bursts are some of the most powerful explosions in the Universe. I’ve always found it mildly disturbing that we didn’t know what causes them. The leading theory has been that they are the result of two neutron stars smashing together. Here seemed to be the proof. The rapid response call was under way by the time I joined. There was a clear chirp in Hanford, you could be see it by eye! We also had data from Livingston and Virgo too. It was bad luck that they weren’t folded into the online alert. There had been a drop out in the data transfer from Italy to the US, breaking the flow for Virgo. In Livingston, there was a glitch at the time of the signal which meant the data wasn’t automatically included in the search. My heart sank. Glitches are common—check out Gravity Spy for some examples—so it was only a matter of time until one overlapped with a signal [bonus note], and with GW170817 being such a long signal, it wasn’t that surprising. However, this would complicate the analysis. Fortunately, the glitch is short and the signal is long (if this had been a high-mass binary black hole, things might not have been so smooth). We were able to exorcise the glitch. A preliminary sky map using all three detectors was sent out at 12:54 am BST. Not only did we defeat the final boss, we did a speed run on the hard difficulty setting first time [bonus note]. Spectrogram of Livingston data showing part of GW170817’s chirp (which sweeps upward in frequncy) as well as the glitch (the big blip at about $-0.6~\mathrm{s}$). The lower panel shows how we removed the glitch: the grey line shows gating window that was applied for preliminary results, to zero the affected times, the blue shows a fitted model of the glitch that was subtracted for final results. You can clearly see the chirp well before the glitch, so there’s no danger of it being an artefect of the glitch. Figure 2 of the GW170817 Discovery Paper The three-detector sky map provided a great localization for the source—this preliminary map had a 90% area of ~30 square degrees. It was just in time for that night’s observations. The plot below shows our gravitational-wave localizations in green—the long band is without Virgo, and the smaller is with all three detectors—as with GW170814, Virgo makes a big difference. The blue areas are the localizations from Fermi and INTEGRAL, the gamma-ray observatories which measured the gamma-ray burst. The inset is something new… Localization of the gravitational-wave, gamma-ray, and optical signals. The main panel shows initial gravitational-wave 90% areas in green (with and without Virgo) and gamma-rays in blue (the IPN triangulation from the time delay between Fermi and INTEGRAL, and the Fermi GBM localization). The inset shows the location of the optical counterpart (the top panel was taken 10.9 hours after merger, the lower panel is a pre-merger reference without the transient). Figure 1 of the Multimessenger Astronomy Paper. That night, the discoveries continued. Following up on our sky location, an optical counterpart (AT 2017gfo) was found. The source is just on the outskirts of galaxy NGC 4993, which is right in the middle of the distance range we inferred from the gravitational wave signal. At around 40 Mpc, this is the closest gravitational wave source. After this source was reported, I think about every single telescope possible was pointed at this source. I think it may well be the most studied transient in the history of astronomy. I think there are ~250 circulars about follow-up. Not only did we find an optical counterpart, but there was emission in X-ray and radio. There was a delay in these appearing, I remember there being excitement at our Collaboration meeting as the X-ray emission was reported (there was a lack of cake though). The figure below tries to summarise all the observations. As you can see, it’s a mess because there is too much going on! The timeline of observations of GW170817’s source. Shaded dashes indicate times when information was reported in a Circular. Solid lines show when the source was observable in a band: the circles show a comparison of brightnesses for representative observations. Figure 2 of the Multimessenger Astronomy Paper. The observations paint a compelling story. Two neutron stars insprialled together and merged. Colliding two balls of nuclear density material at around a third of the speed of light causes a big explosion. We get a jet blasted outwards and a gamma-ray burst. The ejected, neutron-rich material decays to heavy elements, and we see this hot material as a kilonova [bonus material]. The X-ray and radio may then be the afterglow formed by the bubble of ejected material pushing into the surrounding interstellar material. Science What have we learnt from our results? Here are some gravitational wave highlights. We measure several thousand cycles from the inspiral. It is the most beautiful chirp! This is the loudest gravitational wave signal yet found, beating even GW150914. GW170817 has a signal-to-noise ratio of 32, while for GW150914 it is just 24. Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. The signal is clearly visible in the two LIGO detectors as the upward sweeping chirp. It is not visible in Virgo because of its lower sensitivity and the source’s position in the sky. The Livingston data have the glitch removed. Figure 1 of the GW170817 Discovery Paper. The signal-to-noise ratios in the Hanford, Livingston and Virgo were 19, 26 and 2 respectively. The signal is quiet in Virgo, which is why you can’t spot it by eye in the plots above. The lack of a clear signal is really useful information, as it restricts where on the sky the source could be, as beautifully illustrated in the video below. While we measure the inspiral nicely, we don’t detect the merger: we can’t tell if a hypermassive neutron star is formed or if there is immediate collapse to a black hole. This isn’t too surprising at current sensitivity, the system would basically need to convert all of its energy into gravitational waves for us to see it. From measuring all those gravitational wave cycles, we can measure the chirp mass stupidly well. Unfortunately, converting the chirp mass into the component masses is not easy. The ratio of the two masses is degenerate with the spins of the neutron stars, and we don’t measure these well. In the plot below, you can see the probability distributions for the two masses trace out bananas of roughly constant chirp mass. How far along the banana you go depends on what spins you allow. We show results for two ranges: one with spins (aligned with the orbital angular momentum) up to 0.89, the other with spins up to 0.05. There’s nothing physical about 0.89 (it was just convenient for our analysis), but it is designed to be agnostic, and above the limit you’d plausibly expect for neutron stars (they should rip themselves apart at spins of ~0.7); the lower limit of 0.05 should safely encompass the spins of the binary neutron stars (which are close enough to merge in the age of the Universe) we have estimated from pulsar observations. The masses roughly match what we have measured for the neutron stars in our Galaxy. (The combinations at the tip of the banana for the high spins would be a bit odd). Estimated masses for the two neutron stars in the binary. We show results for two different spin limits, $\chi_z$ is the component of the spin aligned with the orbital angular momentum. The two-dimensional shows the 90% probability contour, which follows a line of constant chirp mass. The one-dimensional plot shows individual masses; the dotted lines mark 90% bounds away from equal mass. Figure 4 of the GW170817 Discovery Paper. If we were dealing with black holes, we’d be done: they are only described by mass and spin. Neutron stars are more complicated. Black holes are just made of warped spacetime, neutron stars are made of delicious nuclear material. This can get distorted during the inspiral—tides are raised on one by the gravity of the other. These extract energy from the orbit and accelerate the inspiral. The tidal deformability depends on the properties of the neutron star matter (described by its equation of state). The fluffier a neutron star is, the bigger the impact of tides; the more compact, the smaller the impact. We don’t know enough about neutron star material to predict this with certainty—by measuring the tidal deformation we can learn about the allowed range. Unfortunately, we also didn’t yet have good model waveforms including tides, so for to start we’ve just done a preliminary analysis (an improved analysis was done for the GW170817 Properties Paper). We find that some of the stiffer equations of state (the ones which predict larger neutron stars and bigger tides) are disfavoured; however, we cannot rule out zero tides. This means we can’t rule out the possibility that we have found two low-mass black holes from the gravitational waves alone. This would be an interesting discovery; however, the electromagnetic observations mean that the more obvious explanation of neutron stars is more likely. From the gravitational wave signal, we can infer the source distance. Combining this with the electromagnetic observations we can do some cool things. First, the gamma ray burst arrived at Earth 1.7 seconds after the merger. 1.7 seconds is not a lot of difference after travelling something like 85–160 million years (that’s roughly the time since the Cretaceous or Late Jurassic periods). Of course, we don’t expect the gamma-rays to be emitted at exactly the moment of merger, but allowing for a sensible range of emission times, we can bound the difference between the speed of gravity and the speed of light. In general relativity they should be the same, and we find that the difference should be no more than three parts in $10^{15}$. Second, we can combine the gravitational wave distance with the redshift of the galaxy to measure the Hubble constant, the rate of expansion of the Universe. Our best estimates for the Hubble constant, from the cosmic microwave background and from supernova observations, are inconsistent with each other (the most recent supernova analysis only increase the tension). Which is awkward. Gravitational wave observations should have different sources of error and help to resolve the difference. Unfortunately, with only one event our uncertainties are rather large, which leads to a diplomatic outcome. Posterior probability distribution for the Hubble constant $H_0$ inferred from GW170817. The lines mark 68% and 95% intervals. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SHoES). Figure 1 of the Hubble Constant Paper. Finally, we can now change from estimating upper limits on binary neutron star merger rates to estimating the rates! We estimate the merger rate density is in the range $1540^{+3200}_{-1220}~\mathrm{Gpc^{-3}\,yr^{-1}}$ (assuming a uniform of neutron star masses between one and two solar masses). This is surprisingly close to what the Collaboration expected back in 2010: a rate of between $10~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $10000~\mathrm{Gpc^{-3}\,yr^{-1}}$, with a realistic rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$. This means that we are on track to see many more binary neutron stars—perhaps one a week at design sensitivity! Summary Advanced LIGO and Advanced Virgo observed a binary neutron star insprial. The rest of the astronomical community has observed what happened next (sadly there are no neutrinos). This is the first time we have such complementary observations—hopefully there will be many more to come. There’ll be a huge number of results coming out over the following days and weeks. From these, we’ll start to piece together more information on what neutron stars are made of, and what happens when you smash them together (take that particle physicists). Also: I’m exhausted, my inbox is overflowing, and I will have far too many papers to read tomorrow. GW170817 Discovery Paper: GW170817: Observation of gravitational waves from a binary neutron star inspiral Multimessenger Astronomy Paper: Multi-messenger observations of a binary neutron star merger Data release: LIGO Open Science Center If you’re looking for the most up-to-date results regarding GW170817, check out the O2 Catalogue Paper. Bonus notes Inbox zero Over my vacation I cleaned up my email. I had a backlog starting around September 2015. I think there were over 6000 which I sorted or deleted. I had about 20 left to deal with when I got back to work. GW170817 undid that. Despite doing my best to keep up, there are over a 1000 emails in my inbox… Worst case scenario Around the start of O2, I was asked when I expected our results to be public. I said it would depend upon what we found. If it was only high-mass black holes, those are quick to analyse and we know what to do with them, so results shouldn’t take long, now we have the first few out of the way. In this case, perhaps a couple months as we would have been generating results as we went along. However, the worst case scenario would be a binary neutron star overlapping with non-Gaussian noise. Binary neutron stars are more difficult to analyse (they are longer signals, and there are matter effects to worry about), and it would be complicated to get everyone to be happy with our results because we were doing lots of things for the first time. Obviously, if one of these happened at the end of the run, there’d be quite a delay… I think I got that half-right. We’re done amazingly well analysing GW170817 to get results out in just two months, but I think it will be a while before we get the full O2 set of results out, as we’ve been neglecting otherthings (you’ll notice we’ve not updated our binary black hole merger rate estimate since GW170104, nor given detailed results for testing general relativity with the more recent detections). At the time of the GW170817 alert, I was working on writing a research proposal. As part of this, I was explaining why it was important to continue working on gravitational-wave parameter estimation, in particular how to deal with non-Gaussian or non-stationary noise. I think I may be a bit of a jinx. For GW170817, the glitch wasn’t a big problem, these type of blips can be removed. I’m more concerned about the longer duration ones, which are less easy to separate out from background noise. Don’t say I didn’t warn you in O3. Parameter estimation rota The duty of analysing signals to infer their source properties was divided up into shifts for O2. On January 4, the time of GW170104, I was on shift with my partner Aaron Zimmerman. It was his first day. Having survived that madness, Aaron signed back up for the rota. Can you guess who was on shift for the week which contained GW170814 and GW170817? Yep, Aaron (this time partnered with the excellent Carl-Johan Haster). Obviously, we’ll need to have Aaron on rota for the entirety of O3. In preparation, he has already started on paper drafting Methods Section: Chained ROTA member to a terminal, ignored his cries for help. Detections followed swiftly. The lightest elements (hydrogen, helium and lithium) we made during the Big Bang. Stars burn these to make heavier elements. Energy can be released up to around iron. Therefore, heavier elements need to be made elsewhere, for example in the material ejected from supernova or (as we have now seen) neutron star mergers, where there are lots of neutrons flying around to be absorbed. Elements (like gold and platinum) formed by this rapid neutron capture are known as r-process elements, I think because they are beloved by pirates. A couple of weeks ago, the Nobel Prize in Physics was announced for the observation of gravitational waves. In December, the laureates will be presented with a gold (not chocolate) medal. I love the idea that this gold may have come from merging neutron stars. Here’s one we made earlier. Credit: Associated Press/F. Vergara Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignments Gravitational waves allow us to infer the properties of binary black holes (two black holes in orbit about each other), but can we use this information to figure out how the black holes and the binary form? In this paper, we show that measurements of the black holes’ spins can help us this out, but probably not until we have at least 100 detections. Black hole spins Black holes are described by their masses (how much they bend spacetime) and their spins (how much they drag spacetime to rotate about them). The orientation of the spins relative to the orbit of the binary could tell us something about the history of the binary [bonus note]. We considered four different populations of spin–orbit alignments to see if we could tell them apart with gravitational-wave observations: 1. Aligned—matching the idealised example of isolated binary evolution. This stands in for the case where misalignments are small, which might be the case if material blown off during a supernova ends up falling back and being swallowed by the black hole. 2. Isotropic—matching the expectations for dynamically formed binaries. 3. Equal misalignments at birth—this would be the case if the spins and orbit were aligned before the second supernova, which then tilted the plane of the orbit. (As the binary inspirals, the spins wobble around, so the two misalignment angles won’t always be the same). 4. Both spins misaligned by supernova kicks, assuming that the stars were aligned with the orbit before exploding. This gives a more general scatter of unequal misalignments, but typically the primary (bigger and first forming) black hole is more misaligned. These give a selection of possible spin alignments. For each, we assumed that the spin magnitude was the same and had a value of 0.7. This seemed like a sensible idea when we started this study [bonus note], but is now towards the upper end of what we expect for binary black holes. Hierarchical analysis To measurement the properties of the population we need to perform a hierarchical analysis: there are two layers of inference, one for the individual binaries, and one of the population. From a gravitational wave signal, we infer the properties of the source using Bayes’ theorem. Given the data $d_\alpha$, we want to know the probability that the parameters $\mathbf{\Theta}_\alpha$ have different values, which is written as $p(\mathbf{\Theta}_\alpha\|d_\alpha)$. This is calculated using $\displaystyle p(\mathbf{\Theta}_\alpha\|d_\alpha) = \frac{p(d_\alpha \| \mathbf{\Theta}_\alpha) p(\mathbf{\Theta}_\alpha)}{p(d_\alpha)}$, where $p(d_\alpha \| \mathbf{\Theta}_\alpha)$ is the likelihood, which we can calculate from our knowledge of the noise in our gravitational wave detectors, $p(\mathbf{\Theta}_\alpha)$ is the prior on the parameters (what we would have guessed before we had the data), and the normalisation constant $p(d_\alpha)$ is called the evidence. We’ll use the evidence again in the next layer of inference. Our prior on the parameters should actually depend upon what we believe about the astrophysical population. It is different if we believed that Model 1 were true (when we’d only consider aligned spins) than for Model 2. Therefore, we should really write $\displaystyle p(\mathbf{\Theta}_\alpha\|d_\alpha, \lambda) = \frac{p(d_\alpha \| \mathbf{\Theta}_\alpha,\lambda) p(\mathbf{\Theta}_\alpha,\lambda)}{p(d_\alpha\|\lambda)}$, where $\lambda$ denotes which model we are considering. This is an important point to remember: if you our using our LIGO results to test your theory of binary formation, you need to remember to correct for our choice of prior. We try to pick non-informative priors—priors that don’t make strong assumptions about the physics of the source—but this doesn’t mean that they match what would be expected from your model. We are interested in the probability distribution for the different models: how many binaries come from each. Given a set of different observations $\{d_\alpha\}$, we can work this out using another application of Bayes’ theorem (yay) $\displaystyle p(\mathbf{\lambda}\|\{d_\alpha\}) = \frac{p(\{d_\alpha\} \| \mathbf{\lambda}) p(\mathbf{\lambda})}{p(\{d_\alpha\})}$, where $p(\{d_\alpha\} \| \mathbf{\lambda})$ is just all the evidences for the individual events (given that model) multiplied together, $p(\mathbf{\lambda})$ is our prior for the different models, and $p(\{d_\alpha\})$ is another normalisation constant. Now knowing how to go from a set of observations to the probability distribution on the different channels, let’s give it a go! Results To test our approach made a set of mock gravitational wave measurements. We generated signals from binaries for each of our four models, and analysed these as we would for real signals (using LALInference). This is rather computationally expensive, and we wanted a large set of events to analyse, so using these results as a guide, we created a larger catalogue of approximate distributions for the inferred source parameters $p(\mathbf{\Theta}_\alpha\|d_\alpha)$. We then fed these through our hierarchical analysis. The GIF below shows how measurements of the fraction of binaries from each population tightens up as we get more detections: the true fraction is marked in blue. Probability distribution for the fraction of binaries from each of our four spin misalignment populations for different numbers of observations. The blue dot marks the true fraction: and equal fraction from all four channels. The plot shows that we do zoom in towards the true fraction of events from each model as the number of events increases, but there are significant degeneracies between the different models. Notably, it is difficult to tell apart Models 1 and 3, as both have strong support for both spins being nearly aligned. Similarly, there is a degeneracy between Models 2 and 4 as both allow for the two spins to have very different misalignments (and for the primary spin, which is the better measured one, to be quite significantly misaligned). This means that we should be able to distinguish aligned from misaligned populations (we estimated that as few as 5 events would be needed to distinguish the case that all events came from either Model 1 or Model 2 if those were the only two allowed possibilities). However, it will be more difficult to distinguish different scenarios which only lead to small misalignments from each other, or disentangle whether there is significant misalignment due to big supernova kicks or because binaries are formed dynamically. The uncertainty of the fraction of events from each model scales roughly with the square root of the number of observations, so it may be slow progress making these measurements. I’m not sure whether we’ll know the answer to how binary black hole form, or who will sit on the Iron Throne first. arXiv: 1703.06873 [astro-ph.HE] Journal: Monthly Notices of the Royal Astronomical Society471(3):2801–2811; 2017 Birmingham science summary: Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignment (by Simon) If you like this you might like: Farr et al. (2017)Talbot & Thrane (2017), Vitale et al. (2017), Trifirò et al. (2016), Minogue (2000) Bonus notes Spin misalignments and formation histories If you have two stars forming in a binary together, you’d expect them to be spinning in roughly the same direction, rotating the same way as they go round in their orbit (like our Solar System). This is because they all formed from the same cloud of swirling gas and dust. Furthermore, if two stars are to form a black hole binary that we can detect gravitational waves from, they need to be close together. This means that there can be tidal forces which gently tug the stars to align their rotation with the orbit. As they get older, stars puff up, meaning that if you have a close-by neighbour, you can share outer layers. This transfer of material will tend to align rotate too. Adding this all together, if you have an isolated binary of stars, you might expect that when they collapse down to become black holes, their spins are aligned with each other and the orbit. Unfortunately, real astrophysics is rarely so clean. Even if the stars were initially rotating the same way as each other, they doesn’t mean that their black hole remnants will do the same. This depends upon how the star collapses. Massive stars explode as supernova, blasting off their outer layers while their cores collapse down to form black holes. Escaping material could carry away angular momentum, meaning that the black hole is spinning in a different direction to its parent star, or material could be blasted off asymmetrically, giving the new black hole a kick. This would change the plane of the binary’s orbit, misaligning the spins. Alternatively, the binary could be formed dynamically. Instead of two stars living their lives together, we could have two stars (or black holes) come close enough together to form a binary. This is likely to happen in regions where there’s a high density of stars, such as a globular cluster. In this case, since the binary has been randomly assembled, there’s no reason for the spins to be aligned with each other or the orbit. For dynamically assembled binaries, all spin–orbit misalignments are equally probable. This project was led by Simon Stevenson. It was one of the first things we started working on at the beginning of his PhD. He has now graduated, and is off to start a new exciting life as a postdoc in Australia. We got a little distracted by other projects, most notably analysing the first detections of gravitational waves. Simon spent a lot of time developing the COMPAS population code, a code to simulate the evolution of binaries. Looking back, it’s impressive how far he’s come. This paper used a simple approximation to to estimate the masses of our black holes: we called it the Post-it note model, as we wrote it down on a single Post-it. Now Simon’s writing papers including the complexities of common-envelope evolution in order to explain LIGO’s actual observations. GW170104 and me On 4 January 2017, Advanced LIGO made a new detection of gravitational waves. The signal, which we call GW170104 [bonus note], came from the coalescence of two black holes, which inspiralled together (making that characteristic chirp) and then merged to form a single black hole. On 4 January 2017, I was just getting up off the sofa when my phone buzzed. My new year’s resolution was to go for a walk every day, and I wanted to make use of the little available sunlight. However, my phone informed me that PyCBC (one or our search algorithms for signals from coalescing binaries) had identified an interesting event. I sat back down. I was on the rota to analyse interesting signals to infer their properties, and I was pretty sure that people would be eager to see results. They were. I didn’t leave the sofa for the rest of the day, bringing my new year’s resolution to a premature end. Since 4 January, my time has been dominated by working on GW170104 (you might have noticed a lack of blog posts). Below I’ll share some of my war stories from life on the front line of gravitational-wave astronomy, and then go through some of the science we’ve learnt. (Feel free to skip straight to the science, recounting the story was more therapy for me). Time–frequency plots for GW170104 as measured by Hanford (top) and Livingston (bottom). The signal is clearly visible as the upward sweeping chirp. The loudest frequency is something between E3 and G♯3 on a piano, and it tails off somewhere between D♯4/E♭4 and F♯4/G♭4. Part of Fig. 1 of the GW170104 Discovery Paper. The story In the second observing run, the Parameter Estimation group have divided up responsibility for analysing signals into two week shifts. For each rota shift, there is an expert and a rookie. I had assumed that the first slot of 2017 would be a quiet time. The detectors were offline over the holidays, due back online on 4 January, but the instrumentalists would probably find some extra tinkering they’d want to do, so it’d probably slip a day, and then the weather would be bad, so we’d probably not collect much data anyway… I was wrong. Very wrong. The detectors came back online on time, and there was a beautifully clean detection on day one. My partner for the rota was Aaron Zimmerman. 4 January was his first day running parameter estimation on live signals. I think I would’ve run and hidden underneath my duvet in his case (I almost did anyway, and I lived through the madness of our first detection GW150914), but he rose to the occasion. We had first results after just a few hours, and managed to send out a preliminary sky localization to our astronomer partners on 6 January. I think this was especially impressive as there were some difficulties with the initial calibration of the data. This isn’t a problem for the detection pipelines, but does impact the parameters which we infer, particularly the sky location. The Calibration group worked quickly, and produced two updates to the calibration. We therefore had three different sets of results (one per calibration) by 6 January [bonus note]! Producing the final results for the paper was slightly more relaxed. Aaron and I conscripted volunteers to help run all the various permutations of the analysis we wanted to double-check our results [bonus note]. Recovered gravitational waveforms from analysis of GW170104. The broader orange band shows our estimate for the waveform without assuming a particular source (wavelet). The narrow blue bands show results if we assume it is a binary black hole (BBH) as predicted by general relativity. The two match nicely, showing no evidence for any extra features not included in the binary black hole models. Figure 4 of the GW170104 Discovery Paper. I started working on GW170104 through my parameter estimation duties, and continued with paper writing. Ahead of the second observing run, we decided to assemble a team to rapidly write up any interesting binary detections, and I was recruited for this (I think partially because I’m not too bad at writing and partially because I was in the office next to John Veitch, one of the chairs of the Compact Binary Coalescence group,so he can come and check that I wasn’t just goofing off eating doughnuts). We soon decided that we should write a paper about GW170104, and you can decide whether or not we succeeded in doing this rapidly… Being on the paper writing team has given me huge respect for the teams who led the GW150914 and GW151226 papers. It is undoubtedly one of the most difficult things I’ve ever done. It is extremely hard to absorb negative remarks about your work continuously for months [bonus note]—of course people don’t normally send comments about things that they like, but that doesn’t cheer you up when you’re staring at an inbox full of problems that need fixing. Getting a collaboration of 1000 people to agree on a paper is like herding cats while being a small duckling. On of the first challenges for the paper writing team was deciding what was interesting about GW170104. It was another binary black hole coalescence—aren’t people getting bored of them by now? The signal was quieter than GW150914, so it wasn’t as remarkable. However, its properties were broadly similar. It was suggested that perhaps we should title the paper “GW170104: The most boring gravitational-wave detection”. One potentially interesting aspect was that GW170104 probably comes from greater distance than GW150914 or GW151226 (but perhaps not LVT151012) [bonus note]. This might make it a good candidate for testing for dispersion of gravitational waves. Dispersion occurs when different frequencies of gravitational waves travel at different speeds. A similar thing happens for light when travelling through some materials, which leads to prisms splitting light into a spectrum (and hence the creation of Pink Floyd album covers). Gravitational waves don’t suffered dispersion in general relativity, but do in some modified theories of gravity. It should be easier to spot dispersion in signals which have travelled a greater distance, as the different frequencies have had more time to separate out. Hence, GW170104 looks pretty exciting. However, being further away also makes the signal quieter, and so there is more uncertainty in measurements and it is more difficult to tell if there is any dispersion. Dispersion is also easier to spot if you have a larger spread of frequencies, as then there can be more spreading between the highest and lowest frequencies. When you throw distance, loudness and frequency range into the mix, GW170104 doesn’t always come out on top, depending upon the particular model for dispersion: sometimes GW150914’s loudness wins, other times GW151226’s broader frequency range wins. GW170104 isn’t too special here either. Even though GW170104 didn’t look too exciting, we started work on a paper, thinking that we would just have a short letter describing our observations. The Compact Binary Coalescence group decided that we only wanted a single paper, and we wouldn’t bother with companion papers as we did for GW150914. As we started work, and dug further into our results, we realised that actually there was rather a lot that we could say. I guess the moral of the story is that even though you might be overshadowed by the achievements of your siblings, it doesn’t mean that you’re not awesome. There might not be one outstanding feature of GW170104, but there are lots of little things that make it interesting. We are still at the beginning of understanding the properties of binary black holes, and each new detection adds a little more to our picture. I think GW170104 is rather neat, and I hope you do too. As we delved into the details of our results, we realised there was actually a lot of things that we could say about GW170104, especially when considered with our previous observations. We ended up having to move some of the technical details and results to Supplemental Material. With hindsight, perhaps it would have been better to have a companion paper or two. However, I rather like how packed with science this paper is. The paper, which Physical Review Letters have kindly accommodated, despite its length, might not be as polished a classic as the GW150914 Discovery Paper, but I think they are trying to do different things. I rarely ever refer to the GW150914 Discovery Paper for results (more commonly I use it for references), whereas I think I’ll open up the GW170104 Discovery Paper frequently to look up numbers. Although perhaps not right away, I’d quite like some time off first. The weather’s much better now, perfect for walking… Success! The view across Lac d’Annecy. Taken on a stroll after the Gravitational Wave Physics and Astronomy Workshop, the weekend following the publication of the paper. The science Advanced LIGO’s first observing run was hugely successful. Running from 12 September 2015 until 19 January 2016, there were two clear gravitational-wave detections, GW1501914 and GW151226, as well as a less certain candidate signal LVT151012. All three (assuming that they are astrophysical signals) correspond to the coalescence of binary black holes. The second observing run started 30 November 2016. Following the first observing run’s detections, we expected more binary black hole detections. On 4 January, after we had collected almost 6 days’ worth of coincident data from the two LIGO instruments [bonus note], there was a detection. The searches The signal was first spotted by an online analysis. Our offline analysis of the data (using refined calibration and extra information about data quality) showed that the signal, GW170104, is highly significant. For both GstLAL and PyCBC, search algorithms which use templates to search for binary signals, the false alarm rate is estimated to be about 1 per 70,000 years. The signal is also found in unmodelled (burst) searches, which look for generic, short gravitational wave signals. Since these are looking for more general signals than just binary coalescences, the significance associated with GW170104 isn’t as great, and coherent WaveBurst estimates a false alarm rate of 1 per 20,000 years. This is still pretty good! Reconstructions of the waveform from unmodelled analyses also match the form expected for binary black hole signals. The search false alarm rates are the rate at which you’d expect something this signal-like (or more signal-like) due to random chance, if you data only contained noise and no signals. Using our knowledge of the search pipelines, and folding in some assumptions about the properties of binary black holes, we can calculate a probability that GW170104 is a real astrophysical signal. This comes out to be greater than $1 - (3\times10^5) = 0.99997$. The source As for the previous gravitational wave detections, GW170104 comes from a binary black hole coalescence. The initial black holes were $31.2^{+8.4}_{-6.0} M_\odot$ and $19.4^{+5.3}_{-5.9} M_\odot$ (where $1 M_\odot$ is the mass of our Sun), and the final black hole was $48.7^{+5.7}_{-4.6} M_\odot$. The quoted values are the median values and the error bars denote the central 90% probable range. The plot below shows the probability distribution for the masses; GW170104 neatly nestles in amongst the other events. Estimated masses for the two black holes in the binary $m_1 \geq m_2$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours for all events. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 2 of the GW170104 Discovery Paper. GW150914 was the first time that we had observed stellar-mass black holes with masses greater than around $25 M_\odot$. GW170104 has similar masses, showing that our first detection was not a fluke, but there really is a population of black holes with masses stretching up into this range. Black holes have two important properties: mass and spin. We have good measurements on the masses of the two initial black holes, but not the spins. The sensitivity of the form of the gravitational wave to spins can be described by two effective spin parameters, which are mass-weighted combinations of the individual spins. • The effective inspiral spin parameter $\chi_\mathrm{eff}$ qualifies the impact of the spins on the rate of inspiral, and where the binary plunges together to merge. It ranges from +1, meaning both black holes are spinning as fast as possible and rotate in the same direction as the orbital motion, to −1, both black holes spinning as fast as possible but in the opposite direction to the way that the binary is orbiting. A value of 0 for $\chi_\mathrm{eff}$ could mean that the black holes are not spinning, that their rotation axes are in the orbital plane (instead of aligned with the orbital angular momentum), or that one black hole is aligned with the orbital motion and the other is antialigned, so that their effects cancel out. • The effective precession spin parameter $\chi_\mathrm{p}$ qualifies the amount of precession, the way that the orbital plane and black hole spins wobble when they are not aligned. It is 0 for no precession, and 1 for maximal precession. We can place some constraints on $\chi_\mathrm{eff}$, but can say nothing about $\chi_\mathrm{p}$. The inferred value of the effective inspiral spin parameter is $-0.12^{+0.21}_{-0.30}$. Therefore, we disfavour large spins aligned with the orbital angular momentum, but are consistent with small aligned spins, misaligned spins, or spins antialigned with the angular momentum. The value is similar to that for GW150914, which also had a near-zero, but slightly negative $\chi_\mathrm{eff}$ of $-0.06^{+0.14}_{-0.14}$. Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ and effective precession spin $\chi_\mathrm{p}$ parameter. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models, as well as the prior probability distribution. The dotted lines mark the edge of our 90% probability intervals. We learn basically nothing about precession. Part of Figure 3 of the GW170104 Discovery Paper. Converting the information about $\chi_\mathrm{eff}$, the lack of information about $\chi_\mathrm{p}$, and our measurement of the ratio of the two black hole masses, into probability distributions for the component spins gives the plots below [bonus note]. We disfavour (but don’t exclude) spins aligned with the orbital angular momentum, but can’t say much else. Estimated orientation and magnitude of the two component spins. The distribution for the more massive black hole is on the left, and for the smaller black hole on the right. The probability is binned into areas which have uniform prior probabilities, so if we had learnt nothing, the plot would be uniform. Part of Figure 3 of the GW170104 Discovery Paper. One of the comments we had on a draft of the paper was that we weren’t making any definite statements about the spins—we would have if we could, but we can’t for GW170104, at least for the spins of the two inspiralling black holes. We can be more definite about the spin of the final black hole. If two similar mass black holes spiral together, the angular momentum from the orbit is enough to give a spin of around $0.7$. The spins of the component black holes are less significant, and can make it a bit higher of lower. We infer a final spin of $0.64^{+0.09}_{-0.20}$; there is a tail of lower spin values on account of the possibility that the two component black holes could be roughly antialigned with the orbital angular momentum. Estimated mass $M_\mathrm{f}$ and spin$a_\mathrm{f}$ for the final black hole. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 6 of the GW170104 Supplemental Material (Figure 11 of the arXiv version). If you’re interested in parameter describing GW170104, make sure to check out the big table in the Supplemental Material. I am a fan of tables [bonus note]. Merger rates Adding the first 11 days of coincident data from the second observing run (including the detection of GW170104) to the results from the first observing run, we find merger rates consistent with those from the first observing run. To calculate the merger rates, we need to assume a distribution of black hole masses, and we use two simple models. One uses a power law distribution for the primary (larger) black hole and a uniform distribution for the mass ratio; the other uses a distribution uniform in the logarithm of the masses (both primary and secondary). The true distribution should lie somewhere between the two. The power law rate density has been updated from $31^{+42}_{-21}~\mathrm{Gpc^{-3}\,yr^{-1}}$ to $32^{+33}_{-20}~\mathrm{Gpc^{-3}\,yr^{-1}}$, and the uniform in log rate density goes from $97^{+135}_{-67}~\mathrm{Gpc^{-3}\,yr^{-1}}$ to $103^{+110}_{-63}~\mathrm{Gpc^{-3}\,yr^{-1}}$. The median values stay about the same, but the additional data have shrunk the uncertainties a little. Astrophysics The discoveries from the first observing run showed that binary black holes exist and merge. The question is now how exactly they form? There are several suggested channels, and it could be there is actually a mixture of different formation mechanisms in action. It will probably require a large number of detections before we can make confident statements about the the probable formation mechanisms; GW170104 is another step towards that goal. There are two main predicted channels of binary formation: • Isolated binary evolution, where a binary star system lives its life together with both stars collapsing to black holes at the end. To get the black holes close enough to merge, it is usually assumed that the stars go through a common envelope phase, where one star puffs up so that the gravity of its companion can steal enough material that they lie in a shared envelope. The drag from orbiting inside this then shrinks the orbit. • Dynamical evolution where black holes form in dense clusters and a binary is created by dynamical interactions between black holes (or stars) which get close enough to each other. It’s a little artificial to separate the two, as there’s not really such a thing as an isolated binary: most stars form in clusters, even if they’re not particularly large. There are a variety of different modifications to the two main channels, such as having a third companion which drives the inner binary to merge, embedding the binary is a dense disc (as found in galactic centres), or dynamically assembling primordial black holes (formed by density perturbations in the early universe) instead of black holes formed through stellar collapse. All the channels can predict black holes around the masses of GW170104 (which is not surprising given that they are similar to the masses of GW150914). The updated rates are broadly consistent with most channels too. The tightening of the uncertainty of the rates means that the lower bound is now a little higher. This means some of the channels are now in tension with the inferred rates. Some of the more exotic channels—requiring a third companion (Silsbee & Tremain 2017; Antonini, Toonen & Hamers 2017) or embedded in a dense disc (Bartos et al. 2016; Stone, Metzger & Haiman 2016; Antonini & Rasio 2016)—can’t explain the full rate, but I don’t think it was ever expected that they could, they are bonus formation mechanisms. However, some of the dynamical models are also now looking like they could predict a rate that is a bit low (Rodriguez et al. 2016; Mapelli 2016; Askar et al. 2017; Park et al. 2017). Assuming that this result holds, I think this may mean that some of the model parameters need tweaking (there are more optimistic predictions for the merger rates from clusters which are still perfectly consistent), that this channel doesn’t contribute all the merging binaries, or both. The spins might help us understand formation mechanisms. Traditionally, it has been assumed that isolated binary evolution gives spins aligned with the orbital angular momentum. The progenitor stars were probably more or less aligned with the orbital angular momentum, and tides, mass transfer and drag from the common envelope would serve to realign spins if they became misaligned. Rodriguez et al. (2016) gives a great discussion of this. Dynamically formed binaries have no correlation between spin directions, and so we would expect an isotropic distribution of spins. Hence it sounds quite simple: misaligned spins indicates dynamical formation (although we can’t tell if the black holes are primordial or stellar), and aligned spins indicates isolated binary evolution. The difficulty is the traditional assumption for isolated binary evolution potentially ignores a number of effects which could be important. When a star collapses down to a black hole, there may be a supernova explosion. There is an explosion of matter and neutrinos and these can give the black hole a kick. The kick could change the orbital plane, and so misalign the spin. Even if the kick is not that big, if it is off-centre, it could torque the black hole, causing it to rotate and so misalign the spin that way. There is some evidence that this can happen with neutron stars, as one of the pulsars in the double pulsar system shows signs of this. There could also be some instability that changes the angular momentum during the collapse of the star, possibly with different layers rotating in different ways [bonus note]. The spin of the black hole would then depend on how many layers get swallowed. This is an area of research that needs to be investigated further, and I hope the prospect of gravitational wave measurements spurs this on. For GW170104, we know the spins are not large and aligned with the orbital angular momentum. This might argue against one variation of isolated binary evolution, chemically homogeneous evolution, where the progenitor stars are tidally locked (and so rotate aligned with the orbital angular momentum and each other). Since the stars are rapidly spinning and aligned, you would expect the final black holes to be too, if the stars completely collapse down as is usually assumed. If the stars don’t completely collapse down though, it might still be possible that GW170104 fits with this model. Aside from this, GW170104 is consistent with all the other channels. Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ for all events. To indicate how much (or little) we’ve learnt, the prior probability distribution for GW170104 is shown (the other priors are similar).All of the events have $\|\chi_\mathrm{eff}\| < 0.35$ at 90% probability. Figure 5 of the GW170104 Supplemental Material (Figure 10 of the arXiv version). This is one of my favourite plots [bonus note]. If we start looking at the population of events, we do start to notice something about the spins. All of the inferred values of $\chi_\mathrm{eff}$ are close to zero. Only GW151226 is inconsistent with zero. These values could be explained if spins are typically misaligned (with the orbital angular momentum or each other) or if the spins are typically small (or both). We know that black holes spins can be large from observations of X-ray binaries, so it would be odd if they are small for binary black holes. Therefore, we have a tentative hint that spins are misaligned. We can’t say why the spins are misaligned, but it is intriguing. With more observations, we’ll be able to confirm if it is the case that spins are typically misaligned, and be able to start pinning down the distribution of spin magnitudes and orientations (as well as the mass distribution). It will probably take a while to be able to say anything definite though, as we’ll probably need about 100 detections. Tests of general relativity As well as giving us an insight into the properties of black holes, gravitational waves are the perfect tools for testing general relativity. If there are any corrections to general relativity, you’d expect them to be most noticeable under the most extreme conditions, where gravity is strong and spacetime is rapidly changing, exactly as in a binary black hole coalescence. For GW170104 we repeated tests previously performed. Again, we found no evidence of deviations. We added extra terms to to the waveform and constrained their potential magnitudes. The results are pretty much identical to at the end of the first observing run (consistent with zero and hence general relativity). GW170104 doesn’t add much extra information, as GW150914 typically gives the best constraints on terms that modify the post-inspiral part of the waveform (as it is louder), while GW151226 gives the best constraint on the terms which modify the inspiral (as it has the longest inspiral). We also chopped the waveform at a frequency around that of the innermost stable orbit of the remnant black hole, which is about where the transition from inspiral to merger and ringdown occurs, to check if the low frequency and high frequency portions of the waveform give consistent estimates for the final mass and spin. They do. We have also done something slightly new, and tested for dispersion of gravitational waves. We did something similar for GW150914 by putting a limit on the mass of the graviton. Giving the graviton mass is one way of adding dispersion, but we consider other possible forms too. In all cases, results are consistent with there being no dispersion. While we haven’t discovered anything new, we can update our gravitational wave constraint on the graviton mass of less than $7.7 \times 10^{-23}~\mathrm{eV}/c^2$. The search for counterparts We don’t discuss observations made by our astronomer partners in the paper (they are not our results). A number (28 at the time of submission) of observations were made, and I expect that there will be a series of papers detailing these coming soon. So far papers have appeared from: • AGILE—hard X-ray and gamma-ray follow-up. They didn’t find any gamma-ray signals, but did identify a weak potential X-ray signal occurring about 0.46 s before GW170104. It’s a little odd to have a signal this long before the merger. The team calculate a probability for such a coincident to happen by chance, and find quite a small probability, so it might be interesting to follow this up more (see the INTEGRAL results below), but it’s probably just a coincidence (especially considering how many people did follow-up the event). • ANTARES—a search for high-energy muon neutrinos. No counterparts are identified in a ±500 s window around GW170104, or over a ±3 month period. • AstroSat-CZTI and GROWTH—a collaboration of observations across a range of wavelengths. They don’t find any hard X-ray counterparts. They do follow up on a bright optical transient ATLASaeu, suggested as a counterpart to GW170104, and conclude that this is a likely counterpart of long, soft gamma-ray burst GRB 170105A. • ATLAS and Pan-STARRS—optical follow-up. They identified a bright optical transient 23 hours after GW170104, ATLAS17aeu. This could be a counterpart to GRB 170105A. It seems unlikely that there is any mechanism that could allow for a day’s delay between the gravitational wave emission and an electromagnetic signal. However, the team calculate a small probability (few percent) of finding such a coincidence in sky position and time, so perhaps it is worth pondering. I wouldn’t put any money on it without a distance estimate for the source: assuming it’s a normal afterglow to a gamma-ray burst, you’d expect it to be further away than GW170104’s source. • Borexino—a search for low-energy neutrinos. This paper also discusses GW150914 and GW151226. In all cases, the observed rate of neutrinos is consistent with the expected background. • CALET—a gamma-ray search. This paper includes upper limits for GW151226, GW170104, GW170608, GW170814 and GW170817. • DLT40—an optical search designed for supernovae. This paper covers the whole of O2 including GW170608, GW170814, GW170817 plus GW170809 and GW170823. • Fermi (GBM and LAT)—gamma-ray follow-up. They covered an impressive fraction of the sky localization, but didn’t find anything. • INTEGRAL—gamma-ray and hard X-ray observations. No significant emission is found, which makes the event reported by AGILE unlikely to be a counterpart to GW170104, although they cannot completely rule it out. • The intermediate Palomar Transient Factory—an optical survey. While searching, they discovered iPTF17cw, a broad-line type Ic supernova which is unrelated to GW170104 but interesting as it an unusual find. • Mini-GWAC—a optical survey (the precursor to GWAC). This paper covers the whole of their O2 follow-up including GW170608. • The Owens Valley Radio Observatory Long Wavelength Array—a search for prompt radio emission. • TOROS—optical follow-up. They identified no counterparts to GW170104 (although they did for GW170817). If you are interested in what has been reported so far (no compelling counterpart candidates yet to my knowledge), there is an archive of GCN Circulars sent about GW170104. Summary Advanced LIGO has made its first detection of the second observing run. This is a further binary black hole coalescence. GW170104 has taught us that: • The discoveries of the first observing run were not a fluke. There really is a population of stellar mass black holes with masses above $25 M_\odot$ out there, and we can study them with gravitational waves. • Binary black hole spins may be typically misaligned or small. This is not certain yet, but it is certainly worth investigating potential mechanisms that could cause misalignment. • General relativity still works, even after considering our new tests. • If someone asks you to write a discovery paper, run. Run and do not look back. Title: GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2 Journal: Physical Review Letters; 118(22):221101(17); 2017 (Supplemental Material) arXiv: 1706.01812 [gr-qc] Data release: LIGO Open Science Center Science summary: GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2 If you’re looking for the most up-to-date results regarding GW170104, check out the O2 Catalogue Paper. Bonus notes Naming Gravitational wave signals (at least the short ones, which are all that we have so far), are named by their detection date. GW170104 was discovered 2017 January 4. This isn’t too catchy, but is at least better than the ID number in our database of triggers (G268556) which is used in corresponding with our astronomer partners before we work out if the “GW” title is justified. Previous detections have attracted nicknames, but none has stuck for GW170104. Archisman Ghosh suggested the Perihelion Event, as it was detected a few hours before the Earth reached its annual point closest to the Sun. I like this name, its rather poetic. More recently, Alex Nitz realised that we should have called GW170104 the Enterprise-D Event, as the USS Enterprise’s registry number was NCC-1701. For those who like Star Trek: the Next Generation, I hope you have fun discussing whether GW170104 is the third or fourth (counting LVT151012) detection: “There are four detections! The 6 January sky map I would like to thank the wi-fi of Chiltern Railways for their role in producing the preliminary sky map. I had arranged to visit London for the weekend (because my rota slot was likely to be quiet… ), and was frantically working on the way down to check results so they could be sent out. I’d also like to thank John Veitch for putting together the final map while I was stuck on the Underground. Binary black hole waveforms The parameter estimation analysis works by matching a template waveform to the data to see how well it matches. The results are therefore sensitive to your waveform model, and whether they include all the relevant bits of physics. In the first observing run, we always used two different families of waveforms, to see what impact potential errors in the waveforms could have. The results we presented in discovery papers used two quick-to-calculate waveforms. These include the effects of the black holes’ spins in different ways • SEOBNRv2 has spins either aligned or antialigned with the orbital angular momentum. Therefore, there is no precession (wobbling of orientation, like that of a spinning top) of the system. • IMRPhenomPv2 includes an approximate description of precession, packaging up the most important information about precession into a single parameter $\chi_\mathrm{p}$. For GW150914, we also performed a follow-up analysis using a much more expensive waveform SEOBNRv3 which more fully includes the effect of both spins on precession. These results weren’t ready at the time of the announcement, because the waveform is laborious to run. For GW170104, there were discussions that using a spin-aligned waveform was old hat, and that we should really only use the two precessing models. Hence, we started on the endeavour of producing SEOBNRv3 results. Fortunately, the code has been sped up a little, although it is still not quick to run. I am extremely grateful to Scott Coughlin (one of the folks behind Gravity Spy), Andrea Taracchini and Stas Babak for taking charge of producing results in time for the paper, in what was a Herculean effort. I spent a few sleepless nights, trying to calculate if the analysis was converging quickly enough to make our target submission deadline, but it did work out in the end. Still, don’t necessarily expect we’ll do this for a all future detections. Since the waveforms have rather scary technical names, in the paper we refer to IMRPhenomPv2 as the effective precession model and SEOBNRv3 as the full precession model. On distance Distance measurements for gravitational wave sources have significant uncertainties. The distance is difficult to measure as it determined from the signal amplitude, but this is also influences by the binary’s inclination. A signal could either be close and edge on or far and face on-face off. Estimated luminosity distance $D_\mathrm{L}$ and binary inclination angle $\theta_{JN}$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 4 of the GW170104 Supplemental Material (Figure 9 of the arXiv version). The uncertainty on the distance rather awkwardly means that we can’t definitely say that GW170104 came from a further source than GW150914 or GW151226, but it’s a reasonable bet. The 90% credible intervals on the distances are 250–570 Mpc for GW150194, 250–660 Mpc for GW151226, 490–1330 Mpc for GW170104 and 500–1500 Mpc for LVT151012. Translating from a luminosity distance to a travel time (gravitational waves do travel at the speed of light, our tests of dispersion are consistent wit that!), the GW170104 black holes merged somewhere between 1.3 and 3.0 billion years ago. This is around the time that multicellular life first evolved on Earth, and means that black holes have been colliding longer than life on Earth has been reproducing sexually. Time line A first draft of the paper (version 2; version 1 was a copy-and-paste of the Boxing Day Discovery Paper) was circulated to the Compact Binary Coalescence and Burst groups for comments on 4 March. This was still a rough version, and we wanted to check that we had a good outline of the paper. The main feedback was that we should include more about the astrophysical side of things. I think the final paper has a better balance, possibly erring on the side of going into too much detail on some of the more subtle points (but I think that’s better than glossing over them). A first proper draft (version 3) was released to the entire Collaboration on 12 March in the middle of our Collaboration meeting in Pasadena. We gave an oral presentation the next day (I doubt many people had read the paper by then). Collaboration papers are usually allowed two weeks for people to comment, and we followed the same procedure here. That was not a fun time, as there was a constant trickle of comments. I remember waking up each morning and trying to guess how many emails would be in my inbox–I normally low-balled this. I wasn’t too happy with version 3, it was still rather rough. The members of the Paper Writing Team had been furiously working on our individual tasks, but hadn’t had time to look at the whole. I was much happier with the next draft (version 4). It took some work to get this together, following up on all the comments and trying to address concerns was a challenge. It was especially difficult as we got a series of private comments, and trying to find a consensus probably made us look like the bad guys on all sides. We released version 4 on 14 April for a week of comments. The next step was approval by the LIGO and Virgo executive bodies on 24 April. We prepared version 5 for this. By this point, I had lost track of which sentences I had written, which I had merely typed, and which were from other people completely. There were a few minor changes, mostly adding technical caveats to keep everyone happy (although they do rather complicate the flow of the text). The paper was circulated to the Collaboration for a final week of comments on 26 April. Most comments now were about typos and presentation. However, some people will continue to make the same comment every time, regardless of how many times you explain why you are doing something different. The end was in sight! The paper was submitted to Physical Review Letters on 9 May. I was hoping that the referees would take a while, but the reports were waiting in my inbox on Monday morning. The referee reports weren’t too bad. Referee A had some general comments, Referee B had some good and detailed comments on the astrophysics, and Referee C gave the paper a thorough reading and had some good suggestions for clarifying the text. By this point, I have been staring at the paper so long that some outside perspective was welcome. I was hoping that we’d have a more thorough review of the testing general relativity results, but we had Bob Wald as one of our Collaboration Paper reviewers (the analysis, results and paper are all reviewed internally), so I think we had already been held to a high standard, and there wasn’t much left to say. We put together responses to the reports. There were surprisingly few comments from the Collaboration at this point. I guess that everyone was getting tired. The paper was resubmitted and accepted on 20 May. One of the suggestions of Referee A was to include some plots showing the results of the searches. People weren’t too keen on showing these initially, but after much badgering they were convinced, and it was decided to put these plots in the Supplemental Material which wouldn’t delay the paper as long as we got the material submitted by 26 May. This seemed like plenty of time, but it turned out to be rather frantic at the end (although not due to the new plots). The video below is an accurate representation of us trying to submit the final version. I have an email which contains the line “Many Bothans died to bring us this information” from 1 hour and 18 minutes before the final deadline. After this, things were looking pretty good. We had returned the proofs of the main paper (I had a fun evening double checking the author list. Yes, all of them). We were now on version 11 of the paper. Of course, there’s always one last thing. On 31 May, the evening before publication, Salvo Vitale spotted a typo. Nothing serious, but annoying. The team at Physical Review Letters were fantastic, and took care of it immediately! There’ll still be one more typo, there always is… Looking back, it is clear that the principal bottle-neck in publishing the results is getting the Collaboration to converge on the paper. I’m not sure how we can overcome this… Actually, I have some ideas, but none that wouldn’t involve some form of doomsday device. Detector status The sensitivities of the LIGO Hanford and Livinston detectors are around the same as they were in the first observing run. After the success of the first observing run, the second observing run is the difficult follow up album. Livingston has got a little better, while Hanford is a little worse. This is because the Livingston team concentrate on improving low frequency sensitivity whereas the Hanford team focused on improving high frequency sensitivity. The Hanford team increased the laser power, but this introduces some new complications. The instruments are extremely complicated machines, and improving sensitivity is hard work. The current plan is to have a long commissioning break after the end of this run. The low frequency tweaks from Livingston will be transferred to Hanford, and both sites will work on bringing down other sources of noise. While the sensitivity hasn’t improved as much as we might have hoped, the calibration of the detectors has! In the first observing run, the calibration uncertainty for the first set of published results was about 10% in amplitude and 10 degrees in phase. Now, uncertainty is better than 5% in amplitude and 3 degrees in phase, and people are discussing getting this down further. Spin evolution As the binary inspirals, the orientation of the spins will evolve as they precess about. We always quote measurements of the spins at a point in the inspiral corresponding to a gravitational wave frequency of 20 Hz. This is most convenient for our analysis, but you can calculate the spins at other points. However, the resulting probability distributions are pretty similar at other frequencies. This is because the probability distributions are primarily determined by the combination of three things: (i) our prior assumption of a uniform distribution of spin orientations, (ii) our measurement of the effective inspiral spin, and (iii) our measurement of the mass ratio. A uniform distribution stays uniform as spins evolve, so this is unaffected, the effective inspiral spin is approximately conserved during inspiral, so this doesn’t change much, and the mass ratio is constant. The overall picture is therefore qualitatively similar at different moments during the inspiral. Footnotes I love footnotes. It was challenging for me to resist having any in the paper. Gravity waves It is possible that internal gravity waves (that is oscillations of the material making up the star, where the restoring force is gravity, not gravitational waves, which are ripples in spacetime), can transport angular momentum from the core of a star to its outer envelope, meaning that the two could rotate in different directions (Rogers, Lin & Lau 2012). I don’t think anyone has studied this yet for the progenitors of binary black holes, but it would be really cool if gravity waves set the properties of gravitational wave sources. I really don’t want to proof read the paper which explains this though. Colour scheme For our plots, we use a consistent colour coding for our events. GW150914 is blue; LVT151012 is green; GW151226 is red–orange, and GW170104 is purple. The colour scheme is designed to be colour blind friendly (although adopting different line styles would perhaps be more distinguishable), and is implemented in Python in the Seaborn package as colorblind. Katerina Chatziioannou, who made most of the plots showing parameter estimation results is not a fan of the colour combinations, but put a lot of patient effort into polishing up the plots anyway. Search for transient gravitational waves in coincidence with short-duration radio transients during 2007–2013 Gravitational waves give us a new way of observing the Universe. This raises the possibility of multimessenger astronomy, where we study the same system using different methods: gravitational waves, light or neutrinos. Each messenger carries different information, so by using them together we can build up a more complete picture of what’s going on. This paper looks for gravitational waves that coincide with radio bursts. None are found, but we now have a template for how to search in the future. On a dark night, there are two things which almost everyone will have done: wondered at the beauty of the starry sky and wondered exactly what was it that just went bump… Astronomers do both. Transient astronomy is about figuring out what are the things which go bang in the night—not the things which make suspicious noises, but objects which appear (and usually disappear) suddenly in the sky. Most processes in astrophysics take a looooong time (our Sun is four-and-a-half billion years old and is just approaching middle age). Therefore, when something happens suddenly, flaring perhaps over just a few seconds, you know that something drastic must be happening! We think that most transients must be tied up with a violent event such as an explosion. However, because transients are so short, it can difficult to figure out exactly where they come from (both because they might have faded by the time you look, and because there’s little information to learn from a blip in the first place). Radio transients are bursts of radio emission of uncertain origin. We’ve managed to figure out that some come from microwave ovens, but the rest do seem to come from space. This paper looks at two types: rotating radio transients (RRATs) and fast radio bursts (FRBs). RRATs look like the signals from pulsars, except that they don’t have the characteristic period pattern of pulsars. It may be that RRATs come from dying pulsars, flickering before they finally switch off, or it may be that they come from neutron stars which are not normally pulsars, but have been excited by a fracturing of their crust (a starquake). FRBs last a few milliseconds, they could be generated when two neutron stars merge and collapse to form a black hole, or perhaps from a highly-magnetised neutron star. Normally, when astronomers start talking about magnetic fields, it means that we really don’t know what’s going on [bonus note]. That is the case here. We don’t know what causes radio transients, but we are excited to try figuring it out. This paper searches old LIGO, Virgo and GEO data for any gravitational-wave signals that coincide with observed radio transients. We use a catalogue of RRATs and FRBs from the Green Bank Telescope and the Parkes Observatory, and search around these times. We use a burst search, which doesn’t restrict itself to any particular form of gravitational-wave; however, the search was tuned for damped sinusoids and sine–Gaussians (generic wibbles), cosmic strings (which may give an indication of how uncertain we are of where radio transients could come from), and coalescences of binary neutron stars or neutron star–black hole binaries. Hopefully the search covers all plausible options. Discovering a gravitational wave coincident with a radio transient would give us much welcomed information about the source, and perhaps pin down their origin. Search results for gravitational waves coincident with radio transients. The probabilities for each time containing just noise (blue) match the expected background distribution (dashed). This is consistent with a non-detection. The search discovered nothing. Results match what we would expect from just noise in the detectors. This is not too surprising since we are using data from the first-generation detectors. We’ll be repeating the analysis with the upgraded detectors, which can find signals from larger distances. If we are lucky, multimessenger astronomy will allow us to figure out exactly what needs to go bump to create a radio transient. arXiv: 1605.01707 [astro-ph.HE] Journal: Physical Review D; 93(12):122008(14); 2016 Science summary: Searching for gravitational wave bursts in coincidence with short duration radio bursts Favourite thing that goes bump in the night: Heffalumps and Woozles [probably not the cause of radio transients] Bonus note Magnetism and astrophysics Magnetic fields complicate calculations. They make things more difficult to model and are therefore often left out. However, we know that magnetic fields are everywhere and that they do play important roles in many situations. Therefore, they are often invoked as an explanation of why models can’t explain what’s going on. I learnt early in my PhD that you could ask “What about magnetic fields?” at the end of almost any astrophysics seminar (it might not work for some observational talks, but then you could usually ask “What about dust?” instead). Handy if ever you fall asleep… A black hole Pokémon The world is currently going mad for Pokémon Go, so it seems like the perfect time to answer the most burning of scientific questions: what would a black hole Pokémon be like? Type: Dark/Ghost Black holes are, well, black. Their gravity is so strong that if you get close enough, nothing, not even light, can escape. I think that’s about as dark as you can get! After picking Dark as a primary type, I thought Ghost was a good secondary type, since black holes could be thought of as the remains of dead stars. This also fit well with black holes not really being made of anything—they are just warped spacetime—and so are ethereal in nature. Of course, black holes’ properties are grounded in general relativity and not the supernatural. In the games, having a secondary type has another advantage: Dark types are weak against Fighting types. In reality, punching or kicking a black hole is a Bad Idea™: it will not damage the black hole, but will certainly cause you some difficulties. However, Ghost types are unaffected by Fighting-type moves, so our black hole Pokémon doesn’t have to worry about them. Height: 0’04″/0.1 m Real astrophysical black holes are probably a bit too big for Pokémon games. The smallest Pokémon are currently the electric bug Joltik and fairy Flabébé, so I’ve made our black hole Pokémon the same size as these. It should comfortably fit inside a Pokéball. Measuring the size of a black hole is actually rather tricky, since they curve spacetime. When talking about the size of a black hole, we normally think in terms of the Schwarzschild radius. Named after Karl Schwarzschild, who first calculated the spacetime of a black hole (although he didn’t realise that at the time), the Schwarzschild radius correspond to the event horizon (the point of no return) of a non-spinning black hole. It’s rather tricky to measure the distance to the centre of a black hole, so really the Schwarzschild radius gives an idea of the circumference (the distance around the edge) of the event horizon: this is 2π times the Schwarschild radius. We’ll take the height to really mean twice the Schwarzschild radius (which would be the Schwarzschild diameter, if that were actually a thing). Weight: 7.5 × 1025 lbs/3.4 × 1025 kg Although we made our black hole pocket-sized, it is monstrously heavy. The mass is for a black hole of the size we picked, and it is about 6 times that of the Earth. That’s still quite small for a black hole (it’s 3.6 million times less massive than the black hole that formed from GW150914’s coalescence). With this mass, our Pokémon would have a significant effect on the tides as it would quickly suck in the Earth’s oceans. Still, Pokémon doesn’t need to be too realistic. Our black hole Pokémon would be by far the heaviest Pokémon, despite being one of the smallest. The heaviest Pokémon currently is the continent Pokémon Primal Groudon. This is 2,204.4 lbs/999.7 kg, so about 34,000,000,000,000,000,000,000 times lighter. Within the games, having such a large weight would make our black hole Pokémon vulnerable to Grass Knot, a move which trips a Pokémon. The heavier the Pokémon, the more it is hurt by the falling over, so the more damage Grass Knot does. In the case of our Pokémon, when it trips it’s not so much that it hits the ground, but that the Earth hits it, so I think it’s fair that this hurts. Gender: Unknown Black holes are beautifully simple, they are described just by their mass, spin and electric charge. There’s no other information you can learn about them, so I don’t think there’s any way to give them a gender. I think this is rather fitting as the sun-like Solrock is also genderless, and it seems right that stars and black holes share this. Ability: Sticky Hold Hidden ability: Soundproof Sticky Hold prevents a Pokémon’s item from being taken. (I’d expect wild black hole Pokémon to be sometimes found holding Stardust, from stars they have consumed). Due to their strong gravity, it is difficult to remove an object that is orbiting a black hole—a common misconception is that it is impossible to escape the pull of a black hole, this is only true if you cross the event horizon (if you replaced the Sun with a black hole of the same mass, the Earth would happily continue on its orbit as if nothing had happened). Soundproof is an ability that protects Pokémon from sound-based moves. I picked it as a reference to sonic (or acoustic) black holes. These are black hole analogues—systems which mimic some of the properties of black holes. A sonic black hole can be made in a fluid which flows faster than its speed of sound. When this happens, sound can no longer escape this rapidly flowing region (it just gets swept away), just like light can’t escape from the event horizon or a regular black hole. Sonic black holes are fun, because you can make them in the lab. You can them use them to study the properties of black holes—there is much excitement about possibly observing the equivalent of Hawking radiation. Predicted by Stephen Hawking (as you might guess), Hawking radiation is emitted by black holes, and could cause them to evaporate away (if they didn’t absorb more than they emit). Hawking radiation has never been observed from proper black holes, as it is very weak. However, finding the equivalent for sonic black holes might be enough to get Hawking his Nobel Prize… Moves: Start — Gravity Start — Crunch The starting two moves are straightforward. Gravity is the force which governs black holes; it is gravity which pulls material in and causes the collapse of stars. I think Crunch neatly captures the idea of material being squeezed down by intense gravity. Level 16 — Vacuum Wave Vacuum Wave sounds like a good description of a gravitational wave: it is a ripple in spacetime. Black holes (at least when in a binary) are great sources of gravitational waves (as GW150914 and GW151226 have shown), so this seems like a sensible move for our Pokémon to learn—although I may be biased. Why at level 16? Because Einstein first predicted gravitational waves from his theory of general relativity in 1916. Level 18 — Discharge Black holes can have an electric charge, so our Pokémon should learn an Electric-type move. Charged black holes can have some weird properties. We don’t normally worry about charged black holes for two reasons. First, charged black holes are difficult to make: stuff is usually neutral overall, you don’t get a lot of similarly charged material in one place that can collapse down, and even if you did, it would quickly attract the opposite charge to neutralise itself. Second, if you did manage to make a charged black hole, it would quickly lose its charge: the strong electric and magnetic fields about the black hole would lead to the creation of charged particles that would neutralise the black hole. Discharge seems like a good move to describe this process. Why level 18? The mathematical description of charged black holes was worked out by Hans Reissner and Gunnar Nordström, the second paper was published in 1918. Level 19 —Light Screen In general relativity, gravity bends spacetime. It is this warping that causes objects to move along curved paths (like the Earth orbiting the Sun). Light is affected in the same way and gets deflected by gravity, which is called gravitational lensing. This was the first experimental test of general relativity. In 1919, Arthur Eddington led an expedition to measure the deflection of light around the Sun during a solar eclipse. Black holes, having strong gravity, can strongly lens light. The graphics from the movie Interstellar illustrate this beautifully. Below you can see how the image of the disc orbiting the black hole is distorted. The back of the disc is visible above and below the black hole! If you look closely, you can also see a bright circle inside the disc, close to the black hole’s event horizon. This is known as the light ring. It is where the path of light gets so bent, that it can orbit around and around the black hole many times. This sounds like a Light Screen to me. Light-bending around the black hole Gargantua in Interstellar. The graphics use proper simulations of black holes, but they did fudge a couple of details to make it look extra pretty. Credit: Warner Bros./Double Negative. Level 29 — Dark Void Level 36 — Hyperspace Hole These are three moves which with the most black hole-like names. Dark Void might be “black hole” after a couple of goes through Google Translate. Hyperspace Hole might be a good name for one of the higher dimensional black holes theoreticians like to play around with. (I mean, they like to play with the equations, not actually the black holes, as you’d need more than a pair of safety mittens for that). Shadow Ball captures the idea that a black hole is a three-dimensional volume of space, not just a plug-hole for the Universe. Non-rotating black holes are spherical (rotating ones bulge out at the middle, as I guess many of us do), so “ball” fits well, but they aren’t actually the shadow of anything, so it falls apart there. I’ve picked the levels to be the masses of the two black holes which inspiralled together to produce GW150914, measured in units of the Sun’s mass, and the mass of the black hole that resulted from their merger. There’s some uncertainty on these measurements, so it would be OK if the moves were learnt a few levels either way. Level 63 — Whirlpool Level 63 — Rapid Spin When gas falls into a black hole, it often spirals around and forms into an accretion disc. You can see an artistic representation of one in the image from Instellar above. The gas swirls around like water going down the drain, making Whirlpool and apt move. As it orbits, the gas closer to the black hole is moving quicker than that further away. Different layers rub against each other, and, just like when you rub your hands together on a cold morning, they heat up. One of the ways we look for black holes is by spotting the X-rays emitted by these hot discs. As the material spirals into a black hole, it spins it up. If a black hole swallows enough things that were all orbiting the same way, it can end up rotating extremely quickly. Therefore, I thought our black hole Pokémon should learn Rapid Spin as the same time as Whirlpool. I picked level 63, as the solution for a rotating black hole was worked out by Roy Kerr in 1963. While Schwarzschild found the solution for a non-spinning black hole soon after Einstein worked out the details of general relativity in 1915, and the solution for a charged black hole came just after these, there’s a long gap before Kerr’s breakthrough. It was some quite cunning maths! (The solution for a rotating charged black hole was quickly worked out after this, in 1965). Level 77 — Hyper Beam Another cool thing about discs is that they could power jets. As gas sloshes around towards a black hole, magnetic fields can get tangled up. This leads to some of the material to be blasted outwards along the axis of the field. We’ve some immensely powerful jets of material, like the one below, and it’s difficult to imagine anything other than a black hole that could create such high energies! Important work on this was done by Roger Blandford and Roman Znajek in 1977, which is why I picked the level. Hyper Beam is no exaggeration in describing these jets. Jets from Centaurus A are bigger than the galaxy itself! This image is a composite of X-ray (blue), microwave (orange) and visible light. You can see the jets pushing out huge bubbles above and below the galaxy. We think the jets are powered by the galaxy’s central supermassive black hole. Credit: ESO/WFI/MPIfR/APEX/NASA/CXC/CfA/A.Weiss et al./R.Kraft et al. After using Hyper Beam, a Pokémon must recharge for a turn. It’s an exhausting move. A similar thing may happen with black holes. If they accrete a lot of stuff, the radiation produced by the infalling material blasts away other gas and dust, cutting off the black hole’s supply of food. Black holes in the centres of galaxies may go through cycles of feeding, with discs forming, blowing away the surrounding material, and then a new disc forming once everything has settled down. This link between the black hole and its environment may explain why we see a trend between the size of supermassive black holes and the properties of their host galaxies. Level 100 — Spacial Rend Level 100 — Roar of Time To finish off, since black holes are warped spacetime, a space move and a time move. Relativity say that space and time are two aspects of the same thing, so these need to be learnt together. It’s rather tricky to imagine space and time being linked. Wibbly-wobbly, timey-wimey, spacey-wacey stuff gets quickly gets befuddling. If you imagine just two space dimension (forwards/backwards and left/right), then you can see how to change one to the other by just rotating. If you turn to face a different way, you can mix what was left to become forwards, or to become a bit of right and a bit of forwards. Black holes sort of do the same thing with space and time. Normally, we’re used to the fact that we a definitely travelling forwards in time, but if you stray beyond the event horizon of a black hole, you’re definitely travelling towards the centre of the black hole in the same inescapable way. Black holes are the masters when it comes to manipulating space and time. There we have it, we can now sleep easy knowing what a black hole Pokémon would be like. Well almost, we still need to come up with a name. Something resembling a pun would be traditional. Suggestions are welcome. The next games in the series are Pokémon Sun and Pokémon Moon. Perhaps with this space theme Nintendo might consider a black hole Pokémon too?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 471, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.824741542339325, "perplexity": 1033.8977237771585}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540548544.83/warc/CC-MAIN-20191213043650-20191213071650-00507.warc.gz"}
https://www.gradesaver.com/textbooks/science/physics/CLONE-afaf42be-9820-4186-8d76-e738423175bc/chapter-2-exercises-and-problems-page-32/73	## Essential University Physics: Volume 1 (4th Edition) Clone Using the known relationships and known equations, we can write the following equation: $x(t-1)-x(t)=-\frac{1}{2}g(t-1)^2+ \frac{1}{2}gt^2$ Using substitution, it follows: $t^2-(t-1)^2=\frac{t^2}{4}$ We now simplify this equation: $t^2-8t+4=0$ Using the quadratic formula, we find: $t=7.464 \ seconds$ Now, we find: $\Delta x = \frac{1}{2}gt^2=\frac{1}{2}(9.81)(7.464)^2=\fbox{273.27 meters}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982337355613708, "perplexity": 525.6597032642314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986675409.61/warc/CC-MAIN-20191017145741-20191017173241-00033.warc.gz"}
https://brilliant.org/problems/will-it-ever-stop/	# Will it ever stop? A vertical conducting rod of mass $$m$$ and length $$L$$ is free to slide on two smooth horizontal conducting rails as shown, where there is a magnetic field of intensity $$B$$ directed inside the screen and the resistance of the resistor is $$R$$. The rod is given an initial velocity of $$v_0$$. Find the total distance travelled by the rod. Assume that no other forces other than the magnetic force are acting on the rod. ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.958707869052887, "perplexity": 150.34453185197913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00302.warc.gz"}
http://www.researchgate.net/researcher/7219756_G_F_Bertsch	# G. F. Bertsch University of Washington Seattle, Seattle, Washington, United States Are you G. F. Bertsch? ## Publications (313)1111.28 Total impact • Source ##### Article: Numerical pump-probe experiments of laser-excited silicon in nonequilibrium phase [Hide abstract] ABSTRACT: We calculate the dielectric response of crystalline silicon following irradiation by a high-intensity laser pulse, modeling the dynamics by the time-dependent Kohn-Sham equations in the presence of the laser field. Pump-probe measurements of the response are numerically simulated by including both pump and probe externals fields in the simulation. As expected, the excited silicon shows features of an electron-hole plasma of nonequilibrium phase in its response, characterized by a negative divergence in the real part of the dielectric function at small frequencies. The response to the probe pulse depends on the polarization of the pump pulse. We also find that the imaginary part of the dielectric function can be negative, particularly for the parallel polarization of pump and probe fields. We compare the calculated response with a simple Drude model. The real part of the dielectric function is well fitted by the model, treating the effective mass as a fitting parameter while taking electron density from the calculation. The fitted effective masses are consistent with carrier-band dispersions. 01/2014; 89(6). • Source ##### Article: Combinatorial level densities by the real-time method G. F. Bertsch, L. M. Robledo [Hide abstract] ABSTRACT: Levels densities of independent-particle Hamiltonians can be calculated easily by using the real-time representation of the evolution operator together with the fast Fourier transform. We describe the method and implement it with a set of Python programs. Examples are provided for the total and partial levels densities of a heavy deformed nucleus (Dy-164). The partial level densities that may be calculated are the projected ones on neutron number, proton number, azimuthal angular momentum, and parity. Computer Physics Communications. 01/2014; • Source ##### Article: Spin constraints on nuclear energy density functionals L. M. Robledo, R. N. Bernard, G. F. Bertsch [Hide abstract] ABSTRACT: The Gallagher-Moszkowski rule in the spectroscopy of odd-odd nuclei imposes a new spin constraint on the energy functionals for self-consistent mean field theory. The commonly used parameterization of the effective three-body interaction in the Gogny and Skyrme families of energy functionals is ill-suited to satisfy the spin constraint. In particular, the Gogny parameterization of the three-body interaction has the opposite spin dependence to that required by the observed spectra. The two-body part has a correct sign, but in combination the rule is violated as often as not. We conclude that a new functional form is needed for the effective three-body interaction that can take into better account the different spin-isospin channels of the interaction. 10/2013; 89(2). • ##### Dataset: JChemPhys.134.144106 K-M Lee, K Yabana, G F Bertsch • ##### Dataset: JChemPhys.134.144106 K-M Lee, K Yabana, G F Bertsch • ##### Article: Real-Time TDDFT simulation for coherent phonon generation in crystalline solids [Hide abstract] ABSTRACT: We have been developing a theoretical framework to describe electron dynamics in a crystalline solid under an ultrashort laser pulse. We rely upon the time-dependent density functional theory, solving the time-dependent Kohn-Sham equation in real-time and real-space. Using our method, it is possible to describe both linear and nonlinear light-matter interactions in a unified way. In my presentation, I will focus on the application to coherent phonon generation, a coherent atomic oscillation over a macroscopic volume. I will show applications to two material, semiconductor Si and semimetal Sb. For Si, we have found that the TDDFT is capable of describe two distinct mechanisms of the coherent phonon generation. When the laser frequency is below the direct bandgap, virtual electronic excitation induces impulsive force to atoms. When the laser frequency is above the gap, real electronic excitation causes the atomic motion. For Sb, we study the frequency dependence of the coherent phonon generation and compare our results with phenomenological theories. 03/2013; • ##### Article: Erratum: "Nonadiabatic generation of coherent phonons" [J. Chem. Phys. 137, 22A527 (2012)]. The Journal of Chemical Physics 01/2013; 138(2):029903. · 3.12 Impact Factor • Source • Source ##### Article: Pairing gaps in Hartree-Fock Bogoliubov theory with the Gogny D1S interaction L. M. Robledo, R. Bernard, G. F. Bertsch [Hide abstract] ABSTRACT: As part of a program to study odd-A nuclei in the Hartree-Fock-Bogoliubov (HFB) theory, we have developed a new calculational tool to find the HFB minima of odd-A nuclei based on the gradient method and using interactions of Gogny's form. The HFB minimization includes both time-even and time-odd fields in the energy functional, avoiding the commonly used "filling approximation". Here we apply the method to calculate neutron pairing gaps in some representative isotope chains of spherical and deformed nuclei, namely the Z=8,50 and 82 spherical chains and the Z=62 and 92 deformed chains. We find that the gradient method is quite robust, permitting us to carry out systematic surveys involving many nuclei. We find that the time-odd field does not have large effect on the pairing gaps calculated with the Gogny D1S interaction. Typically, adding the T-odd field as a perturbation increases the pairing gap by ~100 keV, but the re-minimization brings the gap back down. This outcome is very similar to results reported for the Skyrme family of nuclear energy density functionals. Comparing the calculated gaps with the experimental ones, we find that the theoretical errors have both signs implying that the D1S interaction has a reasonable overall strength. However, we find some systematic deficiencies comparing spherical and deformed chains and comparing the lighter chains with the heavier ones. The gaps for heavy spherical nuclei are too high, while those for deformed nuclei tend to be too low. The calculated gaps of spherical nuclei show hardly any A-dependence, contrary to the data. Inclusion of the T-odd component of the interaction does not change these qualitative findings. Physical Review C 10/2012; 86(6). · 3.72 Impact Factor • Source ##### Article: Nonadiabatic generation of coherent phonons [Hide abstract] ABSTRACT: The time-dependent density functional theory (TDDFT) is the leading computationally feasible theory to treat excitations by strong electromagnetic fields. Here the theory is applied to coherent optical phonon generation produced by intense laser pulses. We examine the process in the crystalline semimetal antimony (Sb), where nonadiabatic coupling is very important. This material is of particular interest because it exhibits strong phonon coupling and optical phonons of different symmetries can be observed. The TDDFT is able to account for a number of qualitative features of the observed coherent phonons, despite its unsatisfactory performance on reproducing the observed dielectric functions of Sb. A simple dielectric model for nonadiabatic coherent phonon generation is also examined and compared with the TDDFT calculations. The Journal of Chemical Physics 08/2012; 137(22):22A527. · 3.12 Impact Factor • Source ##### Article: Electromagnetic transition strengths in soft deformed nuclei L. M. Robledo, G. F. Bertsch [Hide abstract] ABSTRACT: Spectroscopic observables such as electromagnetic transitions strengths can be related to the properties of the intrinsic mean-field wave function when the latter are strongly deformed, but the standard rotational formulas break down when the deformation decreases. Nevertheless there is a well-defined, non-zero, spherical limit that can be evaluated in terms of overlaps of mean-field intrinsic deformed wave functions. We examine the transition between the spherical limit and strongly deformed one for a range of nuclei comparing the two limiting formulas with exact projection results. We find a simple criterion for the validity of the rotational formula depending on $<\Delta \vec{J}^2>$, the mean square fluctuation in the angular momentum of the intrinsic state. We also propose an interpolation formula which describes the transition strengths over the entire range of deformations, reducing to the two simple expressions in the appropriate limits. Physical Review C 06/2012; 86(5). · 3.72 Impact Factor • Source ##### Article: Pairing in finite systems: beyond the HFB theory L. M. Robledo, G. F. Bertsch [Hide abstract] ABSTRACT: The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number. 05/2012; • Source ##### Article: Nuclear correlations and the r process. A Arcones, G F Bertsch [Hide abstract] ABSTRACT: We show that long-range correlations for nuclear masses have a significant effect on the synthesis of heavy elements by the r process. As calculated by Delaroche et al. [Phys. Rev. C 81, 014303 (2010)], these correlations suppress magic number effects associated with minor shells. This impacts the calculated abundances before the third r-process peak (at mass number A≈195), where the abundances are low and form a trough. This trough and the position of the third abundance peak are strongly affected by the masses of nuclei in the transition region between deformed and spherical. Based on different astrophysical environments, our results demonstrate that a microscopic theory of nuclear masses including correlations naturally smoothens the separation energies, thus reducing the trough and improving the agreement with observed solar system abundances. Physical Review Letters 04/2012; 108(15):151101. · 7.73 Impact Factor • Source ##### Article: Nuclear pairing: basic phenomena revisited G. F. Bertsch [Hide abstract] ABSTRACT: I review the phenomena associated with pairing in nuclear physics, most prominently the ubiquitous presence of odd-even mass differences and the properties of the excitation spectra, very different for even-even and odd-A nuclei. There are also significant dynamical effects of pairing, visible in the inertias associated with nuclear rotation and large-amplitude shape deformation. 03/2012; • Source ##### Article: Pairing dynamics in particle transport G. Scamps, D. Lacroix, G. F. Bertsch, K. Washiyama [Hide abstract] ABSTRACT: We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the Hamiltonian. In contrast, the theories based on the BCS approximation are more problematic. In the usual formulation of TDHF+BCS, the equation of continuity is violated and one sees unphysical oscillations in particle densities. This can be ameliorated by freezing the occupation numbers during the evolution in TDHF+BCS, but there are other problems with the BCS that make it doubtful for reaction dynamics. We also compare different numerical implementations of the time-dependent HFB equations. The equations of motion for the $U$ and $V$ Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient. Finally, we compare the time-dependent HFB solutions with numerically exact solutions of the two-particle Schrodinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schrodinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory. Physical Review C 02/2012; 85(3). · 3.72 Impact Factor • Source ##### Article: Time-dependent density functional theory for strong electromagnetic fields in crystalline solids [Hide abstract] ABSTRACT: We apply the coupled dynamics of time-dependent density functional theory and Maxwell equations to the interaction of intense laser pulses with crystalline silicon. As a function of electromagnetic field intensity, we see several regions in the response. At the lowest intensities, the pulse is reflected and transmitted in accord with the dielectric response, and the characteristics of the energy deposition are consistent with two-photon absorption. The absorption process begins to deviate from that at laser intensities of ∼1013 W/cm2, where the energy deposited is of the order of 1 eV per atom. Changes in the reflectivity are seen as a function of intensity. When it passes a threshold of about 3×1012 W/cm2, there is a small decrease. At higher intensities, above 2×1013 W/cm2, the reflectivity increases strongly. This behavior can be understood qualitatively in a model treating the excited electron-hole pairs as a plasma. Physical Review B 01/2012; 85(4):045134. · 3.66 Impact Factor • Source ##### Article: Symmetry restoration in Hartree-Fock-Bogoliubov based theories. G F Bertsch, L M Robledo [Hide abstract] ABSTRACT: We present a Pfaffian formula for projection and symmetry restoration for wave functions of the general Bogoliubov form, including quasiparticle excited states and linear combinations of them. This solves a long-standing problem in calculating states of good symmetry, arising from the sign ambiguity of the commonly used determinant formula. A simple example is given of projecting a good particle number and angular momentum from a Bogoliubov wave function in the Fock space of a single j-shell. Physical Review Letters 01/2012; 108(4):042505. · 7.73 Impact Factor • ##### Conference Paper: Theoretical investigation for coherent phonon generation studied with first-principles calculation [Hide abstract] ABSTRACT: We investigate mechanisms of coherent phonon generation in time-dependent density functional theory. It is shown that stimulated Raman and displacive excitation mechanisms are understood in a unified way. Lasers and Electro-Optics (CLEO), 2012 Conference on; 01/2012 • Source ##### Article: Energy spectrum and effective mass using a nonlocal 3-body interaction Alexandros Gezerlis, G. F. Bertsch [Hide abstract] ABSTRACT: We recently proposed a nonlocal form for the 3-body induced interaction that is consistent with the Fock space representation of interaction operators but leads to a fractional power dependence on the density. Here we examine the implications of the nonlocality for the excitation spectrum. In the two-component weakly interacting Fermi gas, we find that it gives an effective mass that is comparable to the one in many-body perturbation theory. Applying the interaction to nuclear matter, it predicts a huge enhancement to the effective mass. Since the saturation of nuclear matter is partly due to the induced 3-body interaction, fitted functionals should treat the effective mass as a free parameter, unless the 2- and 3-body contributions are determined from basic theory. Physical Review C 11/2011; 85(3). · 3.72 Impact Factor • Source ##### Article: Global systematics of octupole excitations in even-even nuclei L. M. Robledo, G. F. Bertsch [Hide abstract] Physical Review C 07/2011; 84(5). · 3.72 Impact Factor #### Publication Stats 11k Citations 1,111.28 Total Impact Points #### Institutions • ###### University of Washington Seattle • • Institute for Nuclear Theory • • Department of Physics Seattle, Washington, United States • ###### University of Tsukuba • Centre for Computational Sciences Tsukuba, Ibaraki-ken, Japan • ###### Universität Basel • Department of Physics Basel, BS, Switzerland • ###### University of Alcalá • Department of Physics and Mathematics • ###### Trinity Washington University Washington, Washington, D.C., United States Japan • ###### Michigan State University • Department of Physics and Astronomy East Lansing, MI, United States • ###### University of Everett Washington Seattle, Washington, United States • ###### McGill University • Department of Physics • ###### The University of Tennessee Medical Center at Knoxville Knoxville, Tennessee, United States • ###### Oak Ridge National Laboratory • Physics Division Oak Ridge, FL, United States • ###### University of Tennessee Knoxville, Tennessee, United States • ###### Argonne National Laboratory • Division of Physics Lemont, Illinois, United States • ###### Joint Institute for Heavy Ion Research Oak Ridge, Tennessee, United States • ###### University of Copenhagen • Niels Bohr Institute Copenhagen, Capital Region, Denmark	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.917999804019928, "perplexity": 2339.741885364923}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657131545.81/warc/CC-MAIN-20140914011211-00026-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/1305.1740/	# Radio pulsars: the search for truth V S Beskin, Ya N Istomin, A A Philippov P N Lebedev Physical Institute, Leninskii prosp. 53, Moscow, 119991, Moscow Institute of Physics and Technology, Institutsky per, 9, Dolgoprudny, Moscow region, 141700, Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Peyton Hall, Princeton, NJ 08544 Usp. Fiz. Nauk 183, 179–194 (2013) [in Russian] English translation: Physics – Uspekhi, 56, 164–179 (2013) Translated by G Pontecorvo; edited by A Semikhatov Abstract. It was as early as the 1980s that A V Gurevich and his group proposed a theory to explain the magnetosphere of radio pulsars and the mechanism by which they produce coherent radio emission. The theory has been sharply criticized and is currently rarely mentioned when discussing the observational properties of radio pulsars, even though all the criticisms were in their time disproven in a most thorough and detailed manner. Recent results show even more conclusively that the theory has no internal inconsistencies. New observational data also demonstrate the validity of the basic conclusions of the theory. Based on the latest results on the effects of wave propagation in the magnetosphere of a neuron star, we show that the developed theory does indeed allow quantitative predictions of the evolution of neutron stars and the properties of the observed radio emission. ## 1 Introduction Thirty years have passed since the group led by A V Gurevich at the Theoretical Physics Department of the Lebedev Physical Institute, Russian Academy of Sciences, published its first article [1] on the theory of the magnetosphere of radio pulsars, 25 years since the publication of their article [2] dealing with the development of the theory of radio emission of pulsars, and, finally, 20 years since the publication of mongraph [3], in which a consistent theory was formulated of the principal processes responsible for the observed activity of radio pulsars. In hindsight, we would like to make some comments, which, we hope, can be useful at the present stage of development of the theory. Total success was not achieved, however, in spite of a number of tactical gains (for instance, the key role of the secondary electron-positron plasma filling the magnetosphere of a neutron star was fully appreciated [14,17], as also was the importance of the current mechanism of energy losses, i.e., of energy losses due to longitudinal currents flowing in the magnetosphere of a pulsar [10]). In particular, it is only in recent years that processes occurring in the magnetosphere of an oblique rotator have started to be actively discussed [18-21]; previously, most work was devoted to the axisymmetric magnetosphere [22-29]. No common standpoint regarding the mechanism of coherent radio emission exists yet [30-32]. By the late 1980s, research activities on the theory of the pulsar magnetosphere and the theory of radio emission were drastically reduced. Two or three serious publications per year (!) certainly made no real difference. Actually, a period of deep stagnation set in. On the one hand, most theorists were not able to propose a model that could provide readily testable predictions, while, on the other hand, the existence of a simple hollow-cone geometric model [33] (in which the directivity pattern of the radio emission repeats the density profile of the secondary electron-positron plasma outflowing along the open magnetic field lines) permitted interpreting observational data without turning to theory. As a result, the connection between theory and observations was practically lost. Perhaps, precisely the atmosphere of general failure (although there most likely existed purely opportunistic reasons, also) resulted in a series of studies, performed in the 1980s by the group led by Gurevich [1, 2, 34, 35], in which the authors succeeded in formulating a consistent theory of the magnetosphere and of the radio emission of pulsars, which was met, mildly speaking, without friendliness. Here, we only present a few quotations from articles and reviews on book [3] that was the conclusion of a decade of work111Incidentally, the fact that critical reviews continued to appear until quite recently [36, 37] rather serves as an argument in favor of our conclusions.. ”We conclude that their computation of the dielectric tensor of a plasma in a strong magnetic field is wrong” [38]. ”It has been claimed that this instability is spurious” [39]. ”This theory is known to be incorrect. It contains several fatal flows” [40]. Regretfully, such peremptory criticism made any serious discussion of the results of our work simply impossible, although practically all the main critical pronouncements were given detailed explanations [30, 41, 42], demonstrating their judgements to be unjustified. Therefore, as before, we believe in the validity of our conclusions, which can be formulated as follows. I. Theory of the magnetosphere of radio pulsars. 1. The plasma filling the pulsar magnetosphere totally screens the magnetodipole radiation of a neutron star. As a result, all the energy losses must be due to longitudinal currents circulating within the pulsar magnetosphere (and closing up on the surface of the neutron star). 2. For a local Goldreich current (see Section 2.2), current losses should be significantly smaller for an orthogonal rotator than for the axisymmetric one, which follows from the expression we found for current losses for an arbitrary inclination angle of the magnetic dipole axis to the axis of rotation. 3. The inclination angle of the magnetic dipole axis to the axis of rotation should increase with time, unlike the magnetodipole losses. 4. Quite a small longitudinal current is realized in the pulsar magnetosphere, which results in the inevitable appearance outside the light cylinder of a region where the electric field is greater than the magnetic field. Inside this region, effective acceleration of the plasma flowing out of the pulsar magnetosphere becomes possible. 1. The dielectric permittivity tensor of plasma in an inhomogeneous medium has been found, in particular, for the relativistic plasma moving along curved magnetic field lines. This tensor permits correctly describing the interaction of particles with waves propagating in an inhomogeneous plasma. 2. The analysis of such a tensor reveals the instability of ’curvature-plasma’ waves, which can serve as the base instability for the maser mechanism providing coherence of the observed emission. 3. By taking the nonlinear interaction of waves into account, the excitation level has been determined for transverse waves capable to escape from the magnetosphere of a neutron star, and, thus, the intensity of the radio emission of pulsars has been established. 4. Based on a consistent analysis of wave propagation in the magnetosphere of a pulsar, which, for instance, takes the refraction of an ordinary wave into account, quantitative predictions have been made concerning the main observational characteristics of pulsars (the frequency dependence of the width of mean profiles, the statistics of pulsars with single or double profiles, etc.). The purpose of this article is to show that at present, sufficient material has been accumulated to assert with confidence that the principal theoretical points of our theory not only have not become obsolete (which could well have happened owing to the impetuous development of numerical methods) but also can be a basis for understanding the phenomenon of radio pulsars. Moreover, we show that the recently obtained observational data confirm the validity of the main theoretical conclusions formulated over 20 years ago. ## 2 Pulsar magnetosphere ### 2.1 Screening of the magnetodipole radiation A uniformly magnetized sphere rotating in the vacuum is known to lose rotational energy owing to magnetodipole losses [43]: Wtot=−JrΩ˙Ω=16B20Ω4R6c3sin2χ. (1) Here, is the radius of the sphere, is the magnetic field at the magnetic pole, is the moment of inertia, is the inclination angle of the magnetic axis to the rotation axis, and is the angular velocity of rotation. This mechanism is quite universal, and hence it would be natural to assume expression (1) for magnetodipole losses to also hold in the case of a magnetosphere filled with secondary electron-positron plasma. Therefore, estimate (1) still expresses the common view of the rotation deceleration mechanism of radio pulsars. However, this conclusion, seemingly evident at first glance, turned out to have no foundation. To be more precise, we showed that in the framework of a force-free approximation (a secondary plasma, whose energy density is negligible compared to the energy density of the magnetic field, fully screens the longitudinal electric field) and in the case of a zero longitudinal (parallel to the magnetic field) electric current, the flux of the Poynting vector through the surface of a light cylinder, , vanishes [1]. From a mathematical standpoint, this is because the toroidal component of the magnetic field on the surface of the light cylinder must vanish (actually, this conclusion was obtained in 1975 in Ref. [44]): Bφ(RL)=0. (2) From a physical standpoint, this means that the plasma filling the pulsar magnetosphere completely screens the magnetodipole radiation of the neutron star. In other words, in the case of a zero longitudinal current, the magnetospheric plasma emission is in a phase precisely opposite to the phase of the pulsar magnetodipole radiation. Consequently, all the energy losses should be due to longitudinal currents circulating inside the magnetosphere of the neutron star and closing up on its surface. These energy losses can be determined by the formula , where K=1c∫[r×[Js×B]]dS (3) is the decelerating moment of the Ampere force due to surface currents . We recall that it is possible to obtain an analytic solution for an oblique rotator because in the case of a zero longitudinal current, the equation describing the magnetosphere of a neutron star is linear; it is also extremely important here that the boundary condition at infinity (along the rotation axis) was used. It follows that the energy losses can be written as [1] Wtot=f2∗4B20Ω4R6c3i0cosχ, (4) where is the dimensionless longitudinal current normalized to the so-called Goldreich current (or the Golreich-Julian current), jGJ=ΩB2π, (5) where is a coefficient that depends weakly on the inclination angle . We note that the conclusion concerning the complete screening of the magnetodipole radiation was, naturally, not obvious. Therefore, not surprisingly, it remains far from being adopted by everyone. It is interesting that now, after 30 years have passed, we have been surprised to learn from many participants in those discussions that the main criticism seems to have concerned our alleged claim that pulsars lose no rotation energy at all. But we never made any such statement and could not have done so. The main conclusion in Ref. [1] is formula (4) for current energy losses, which clearly points to the deceleration mechanism. At present, screening of the magnetodipole radiation of a neutron star can be confidently said to indeed take place. First of all, in 1999, the group of L Mestel [45] performed studies equivalent to those presented in Ref. [1] and fully confirmed our conclusions: in the case of a zero longitudinal current, the energy losses of an oblique rotator are equal to zero. Fig. 1 shows the structure of the magnetic field of an orthogonal rotator in the equatorial plane, obtained in Ref. [45] (the corresponding cross section remained in the draft copies of Ref. [1]). It is clearly seen that the magnetic field lines indeed approach the light cylinder at a right angle. However, the most important recent result is that magnetodipole losses are also absent in the solution for the magnetosphere of an oblique rotator constructed by Spitkovsky on the basis of numerical simulation [20]. First of all, this follows from the assertion concerning the split-monopole asymptotic form of the solution obtained, which is close to the model of radial magnetized wind [18]. Such flows only involve stationary magnetized wind (independent of time outside a thin current sheet), in which, however, the energy flux is related to the flux of the Poynting vector. But the main point is that the absence of magnetodipole losses is also confirmed by a straightforward analysis of the structure of electromagnetic fields in Ref. [20]. Indeed, in the case of a vacuum rotator, for any inclination angle , a variable-in-time component of the magnetic field must be present on the rotation axis, with the amplitude B⊥=\|¨m\|c2r, (6) where is the magnetic moment of the star (with ), is its second time derivative, and . However, as can be seen from Fig. 2, the variable component of the magnetic field in the Spitkovsky solution decreases much more rapidly, like . In our opinion, the absence of variable fields decreasing as in the numerical solution for an oblique rotator is the most convincing proof of the total screening of magnetodipole radiation in the case of a magnetosphere completely filled with plasma. ### 2.2 Current losses One more important consequence of the theory of current losses is that for a local longitudinal Goldreich current, the rotational energy losses should decrease as the inclination angle increases [1, 3]. The point is that, besides the factor related to the scalar product , a significant dependence of the current losses in Eqn (4) on the inclination angle is also involved in the quantity . Indeed, in the definition of the dimensionless current , the denominator contains the Goldreich current for the axisymmetric case, while in the case of nonzero angles , the Goldreich-Julian charge density in the vicinity of the magnetic poles ρGJ=−ΩB2πc≈−ΩB2πccosχ (7) itself depends on the angle . On the other hand, it is natural to expect the longitudinal current to be bounded by the value . At any rate, both in the Ruderman-Sutherland model [14] with the particle escape from the surface of a neutron star being hindered and in the Arons model [46], in which the escape of particles is free, the longitudinal current is indeed determined by the relation . But then, in the case of an oblique rotator, the dimensionless current has the upper bound . As a result, the current losses must decrease as the angle increases, at least like . In particular, if (when is to be substituted by its characteristic value in the range of the polar cap, ), we obtain Wtot=c⊥B20Ω4R6c3(ΩRc)iA. (8) In the case of a local Goldreich current, , while the coefficient already depends not only on the profile of the asymmetric longitudinal current but also on the shape of the polar cap. In discussions of this issue, the following reasoning is standardly used as an argument against the decrease in losses occurring as increases. In expression (3) for the decelerating moment, an increase in the angle is indeed accompanied by the surface current decreasing as . But the characteristic distance between the axis and the polar cap increases as , and hence, even in the case of a local Goldreich current, the losses depend weakly on the inclination angle. However, as has been demonstrated by a more precise analysis in Ref. [1], the above reasoning, which seems obvious at first glance, does not take the real structure of surface currents in the polar cap region into account. As shown in Fig. 3, the currents that close up should actually be arranged such that the current averaged over the polar cap surface vanish. Consequently, in determining the deceleration rate of a radio pulsar, it is necessary to consider higher-order effects, such as the effect of the curved surface of a neutron star. But if the averaged surface current within the polar cap indeed vanishes, then, as shown in Fig. 3, a surface current comparable in value to the volume current flowing in the magnetosphere should flow along the separatrix separating the regions of closed and open field lines. In the case of an orthogonal rotator (and for a circular polar cap, when the result can be obtained analytically), the inverse current should amount to of the volume current flowing in the region of open field lines. A remarkable event was that numerical simulation [47] of the magnetosphere of an oblique rotator actually revealed inverse currents flowing along the separatrix. True, the inverse current only amounted to 20% of the volume current. But such a discrepancy between the results of simulation and theoretical predictions can most likely be explained by the radius of the star being set in simulations only to half the radius of the light cylinder, when the magnetic field near the magnetic poles already differs strongly from the dipole field. Finally, we note that no contradiction actually exists, either, between relation (4) and the expression Wtot≈14B20Ω4R6c3(1+sin2χ), (9) obtained by Spitkovsky for an oblique rotator; approximate formula (9) was obtained in Ref. [20] for a magnetosphere in which the longitudinal current was actually significantly larger than the local Goldreich current (see Ref. [48] for the details), which is consistent with the condition (correspondingly, ). A longitudinal current exceeding the local Goldreich current was necessary for constructing a smooth solution containing the magnetohydrodynamic (MHD) wind outflowing to infinity (see Section 2.4). It is interesting that one more possibility was recently revealed for directly testing the validity of expression (4) for current losses (and at the same time the validity of the assertion that the pulsar magnetosphere completely screens the magnetodipole radiation of a neutron star). The possibility of implementing such a test is related to the unusual properties of the pulsar B193124 [49]. Unlike the radiation of other radio pulsars, the radiation of B193124 is strongly variable. This pulsar is in an active state for days, then its radio emission is switched off in less than 10 s, and it is no longer observable for the next days. It is important that the absolute value of the rotation deceleration of B193124 is different in the ’on’ and ’off’ states: ˙Ωon = −1.02×10−14[s−2], (10) ˙Ωoff = −0.68×10−14[s−2], (11) with ˙Ωon˙Ωoff≈1.5. (12) Later, the pulsar J18320031 was also found to exhibit similar behavior ( days, days, and in this case, also, the ratio ), as did the pulsar J18410500 (in that case, [50]). It is natural to assume that the difference between and for these pulsars occurs simply because deceleration in the switched-on state is related to current losses, while in the switched-off state (when the magnetosphere is not filled with plasma), it is due to the emission of a magnetodipole wave [51, 52]. Then, using relations (1) and (4), we obtain ˙Ωon˙Ωoff=3f2∗2cot2χ, (13) which yields a reasonable value for the inclination angle . On the other hand, if relation (9) is applied for the switched-on state, we arrive at ˙Ωon˙Ωoff=32(1+sin2χ)sin2χ. (14) Clearly, this quantity cannot be equal to 1.5 or 2.5 for any value of the inclination angle222For this reason, relation (9) was somewhat corrected in Ref. [53].. If such an interpretation of the observations corresponds to reality, it follows that the longitudinal current flowing in the magnetosphere does not actually exceed the local Goldreich current. ### 2.3 Evolution of the inclination angle Determining the evolution of the inclination angle could serve as one more test. For current losses (more exactly, for local Goldreich current, and for inclination angle ), the decelerating moment is directed opposite to the magnetic moment of the neutron star , and therefore the Euler equation leads to the conservation of the projection of the rotation angular velocity onto the axis perpendicular to . Hence, the following quantity must be conserved during the evolution [3]: Ωsinχ=const. (15) Consequently, in the case of current losses, the angle between the axis of rotation and the magnetic axis should increase (but not decrease, as in the case of magnetodipole radiation), and the characteristic time of its evolution should coincide with the characteristic dynamical age of the pulsar, [34]. Regretfully, no method has been found to determine the direction of evolution of the inclination angle for individual pulsars (see, however, Ref. [54]). On the other hand, the prediction indicating an increase in the angle is known not to contradict observations statistically [34, 55]. The last assertion requires clarification. Observations reveal average statistical inclination angles indisputably decrease as the period of pulsars increases and its derivative decreases [56]. Therefore, the average statistical inclination angle decreases as the dynamic age increases. Correspondingly, pulsars with larger periods can be observed to exhibit relatively larger widths of the mean pulses [57] (where is the width of thedirectivity pattern). But this by no means implies that the inclination angle for each concrete pulsar decreases with time. Such a behavior of the average inclination angle can also be realized when the angles of each pulsar increase in accordance with (15). Indeed, as can be seen from Fig. 4, for the given values of the pulsar period and the magnetic field , the production of particles is suppressed precisely at angles close to . This is because Goldreich-Julian charge density (7) decreases significantly at such angles, which in turn leads to a decrease in the electric potential drop near the surface of the neutron star. As a result, stable production of secondary particles becomes impossible there. Therefore, owing to such a dependence of the pulsar extinction line on , the average inclination angle can also decrease as the dynamic age increases, for example, in the case of pulsars uniformly distributed over the plane. A detailed analysis, already carried out in Ref. [34] on the basis of a kinetic equation describing the distribution of pulsars (see also later studies [58, 59]) confirmed that the observed distribution of pulsars over the inclination angle does not contradict hypothesis (15) of the increase in the angle for any individual pulsar. In any case, it is quite clear, that the inclination angle is a key hidden parameter: without taking it into account, it is impossible to construct a consistent theory of the evolution of radio pulsars. Regretfully, with a few exceptions (see, e.g., Ref. [60]), modern theorists (so-called scenario producers) describing the evolution of neutron stars [61-63] do not take the influence of the inclination angle evolution on the observed pulsar distribution into account. ### 2.4 Light surface Starting from the 1970s, the magnetosphere of a pulsar has been discussed almost exclusively in the force-free approximation [23, 64-66]. The reason is that the plasma filling the magnetosphere of a neutron star is less important (secondary) than the magnetic field; therefore (at least within the light cylinder), the particle energy density can be neglected. In the force-free approximation, the structure of the magnetosphere is described by the so-called pulsar equation, i.e., an elliptic equation for the magnetic flux function. In Section 2.1, in discussing the solution of a similar equation for the zero longitudinal current, we noted that in the case of numerical simulation of an axisymmetric magnetosphere, it is necessary to introduce an additional condition for the external boundary of the integration region [24-29]. Such a condition is usually chosen in the form that the magnetic field lines be radial. In this approach, precisely such an additional condition fixes the longitudinal current flowing within the magnetosphere. Therefore, it is not surprising that the current is close to the critical current (5) obtained analytically by F C Michel [64] for the quasispherical wind. A very important property of this solution is that the energy in the wind is carried by the crossed fields and , which form a radial flux of the electromagnetic energy (the Poynting vector flux), and the electric field is smaller than the magnetic field as far as infinity. Otherwise, the freezing-in condition , which always serves as the cornerstone in the approach considered, would be violated. On the other hand, as is readily verified, implementation of the condition is possible only if the longitudinal electric current flowing in the magnetosphere is suffiently large. Indeed, in the case of a quasispherical wind outside the light cylinder, the electric field Eθ=ΩrsinθcBp (16) and the toroidal magnetic field Bφ=2Icrsinθ (17) decrease with the distance as (while the poloidal field decreases as ). Therefore, for the light surface to recede to infinity, it is necessary that the toroidal magnetic field on the light cylinder be of the same order of magnitude as the poloidal field. Implementation of this condition is possible precisely when the total current outflowing beyond the light cylinder is not less than the Goldreich current , where is the radius of the polar cap. We stress that in all numerical simulations, no restrictions were imposed on the longitudinal current outflowing from the neutron star surface. Therefore, it is not surprising that the longitudinal current obtained as a solution of the problem turned out to be of the order of . We recall that in the complete MHD version, where taking the finiteness of particle masses into account results in the appearance of an additional critical (fast magnetosonic) surface, the longitudinal current is no longer a free parameter [67]. The value of the longitudinal current is close to . Therefore, most reseachers currently consider the existence of a strongly magnetized wind for which the condition is satisfied to be practically proven [20, 21]. We stress that the issue here concerns scales comparable to the radius of the light cylinder (); at larger scales, as follows, for instance, from an analysis of the interaction of the pulsar wind with supernova remnants [68], the main energy must already be carried by particles. As is known, within the theory of strongly magnetized wind, such an acceleration required for a quasispherical outflow cannot be obtained [67, 69-71]. Generally speaking, the rigorous analytic conclusion that the longitudinal current is close to the critical one only concerned stationary axisymmetric flows. But numerical calculations recently performed for nonstationary force-free configurations [53] have shown unambiguously that the system undergoes quite rapid evolution precisely toward a state with a current close to the critical current. And such a state corresponds to the minimum-energy configuration (for example, minimum energy is exhibited by configurations in which the singular point separating the regions of closed and open field lines is on the light cylinder, but not inside the magnetosphere [29, 72]333Solutions in which the singular point is inside the light cylinder are, most likely, affected by the limited time available for numerical calculations (A Tchekhovskoy, private communication).). Thus, the existence of quite a strong longitudinal current has been confirmed once again; in any case, no restrictions were imposed on the longitudinal current either. The following problem arises here, however. As noted, all the theories of stationary particle production in the magnetosphere of a neutron star [3, 14, 46, 73, 74] unambiguously testify in favor of the longitudinal current density not possibly being larger than the local Goldreich current, which, as can be seen from its definition (7), depends on the inclination angle jGJ=ΩB2πcosχ. (18) For example, in the case of an orthogonal rotator, the local Goldreich-Julian charge density in the vicinity of the magnetic poles must be times smaller than in the case of an axisymmetric magnetosphere. Hence, the longitudinal current flowing along open field lines should also be smaller in the same proportion (for ordinary pulsars with a period s, this current is nearly 100 times smaller for an orthogonal rotator). Therefore, in constructing the solution for an oblique dipole [20], it was necessary to assume the longitudinal current in the region of the magnetic poles to be significantly higher than the local Goldreich current (A Spitkovsky, private communication). Therefore, the value of the longitudinal current flowing in a neutron star’s magnetosphere turns out to be the key issue, without resolving which it is impossible to move toward the understanding of the structure of the magnetosphere of radio pulsars. The problem lies in whether the region of plasma generation at the surface of a neutron star can provide the longitudinal current sufficiently large for the existence of an MHD wind from an oblique rotator. If the necessary current can be created (see, e.g., Ref. [75], where one-dimensional, nonstationary regime was considered), nothing can prevent the production of the MHD wind in which the main part of the energy is carried by the electromagnetic field: such is the opinion of most reseachers. But if the generation of a sufficiently high longitudinal current turns out to be impossible for some reason, then, in the vicinity of the light cylinder, a ’light surface’ inevitably arises in which the electric field becomes equal to the magnetic field. Precisely such a structure was predicted by us in Ref. [1]. The appearance of a light surface in the magnetosphere of a radio pulsar radically alters the properties of the pulsar wind because, close to the light surface, closure of the current and effective particle acceleration inevitably occur. Solving the equations of two-fluid hydrodynamics (precisely describing the difference in motion between electrons and positrons) in the case of the simplest cylindrical geometry reveals [1] that a significant part of the energy carried within the light surface by the electromagnetic field in the thin transition layer close to the light surface, Δr∼λ−1RL (19) is transferred to plasma particles ( is the production multiplicity of particles close to the surface of a neutron star). In the same layer, as shown in Fig. 5, practically total closure of the longitudinal current circulating in the magnetosphere occurs. As a result, a natural explanation is also found for the high efficiency of particle acceleration. Subsequently, a similar result was also obtained for a more realistic geometry, when the poloidal magnetic field near the surface of the light cylinder is close to a monopole field [76]. In particular, it has been confirmed that the particle energy immediately beyond the light surface is by the order of magnitude given by Ee ∼ eB0R1λ(ΩRc)2 ∼ 104MeV(λ103)−1(B01012G)(P1s)−2, but does not exceed the value , at which the effects of radiation friction become essential. Quantity (20) practically corresponds to the total energy transfer from the Poynting vector flux to the flux of accelerated particles. In (20), is times smaller than the energy corresponding to the maximum potentials difference of different magnetic field lines in the magnetized wind. Here, is the characteristic size of the central engine; in radio pulsars, is equal to the size of the polar cap . As a result, we can express the Lorentz factor as γmax=σ, (21) where is the so-called magnetization parameter444Recently, the notation for the magnetization parameter has become popular, with standing for the ratio of the electromagnetic energy flux to the energy flux carried by particles. introduced by Michel [77] in 1969. As was shown in Ref. [48], the magnetization parameter can be represented in the very simple form σ∼1λ(WtotWA)1/2, (22) where is the total energy losses and erg s is a universal constant; it corresponds to the minimum energy losses of the central engine which can accelerate particles to relativistic energy. Because the particle production multiplicity for radio pulsars is known [73, 74, 78], the value of can also be found.For most pulsars, the value of lies in the range , and it can reach only in the youngest sources (the Crab and Vela pulsars). We note that the parameter is very convenient for determining the key parameters of strongly magnetized winds. For example, the radius of the fast magnetsonic surface is expressed simply as rf∼σ1/3RL. (23) Therefore, our theory predicts effective particle acceleration in the region of the light cylinder up to energies corresponding to the Lorentz factor . Clearly, such an acceleration is only possible within the fast magnetosonic surface, ; as was noted, if the magnetized wind is free to reach the fast magnetosonic surface, then the longitudinal current is comparable in value to the Goldreich current. Clearly, the effective acceleration of particles should result in the generation of hard radiation, which, in principle, could be detected. In [3], the synchrotron losses of accelerated particles are estimated, and both the total energy release and the energy range of radiation are shown to depend very strongly on the period of the pulsar. The energies of emitted photons can reach several tens of MeV only in the case of the youngest pulsars (Crab, Vela), while the radiation of most radio pulsars due to the synchrotron mechanism must lie in the optical range. The energy released in all ranges has turned out to be quite low, which has allowed observing these sources with the aid of existing detectors. On the other hand, as is well known, another important channel in which energy is released and which is capable of resulting in the production of -quanta of even higher energies is the inverse Compton scattering of thermal X-ray photons emitted from the surface of a neutron star. This process has recently been regarded as the main process for the generation of photons of energies in the TeV range for the widest class of objects, such as active galactic nuclei [79,80], galactic sources in the TeV range [81], and, naturally, radio pulsars [82]. In those cases where the ’central engine’ is indeed a rapidly rotating neutron star, the Lorentz factor of electrons (or positrons) necessary for shifting the observed photons and soft -quanta toward the TeV energies corresponds precisely to values . Thus, for the pulsar B125963 (which is in a double system containing a Be-star), observations agree best with the value [83]. As can be readily estimated from relations (21) and (22), precisely this value is the characteristic value of the magnetization parameter for B125963 . We especially draw attention to the work of the group of F Aharonian published in Nature [82], the title of which is precisely the following: ”Abrupt acceleration of a ’cold’ ultrarelativistic wind from the Crab pulsar.” It is shown in Ref. [82], for example, that the observed intensity of TeV photons can be explained, if a rapid acceleration of particles occurs at distances of the order of as a result of which the particles acquire energies corresponding to Lorentz factors . As was noted above, the value corresponds precisely to estimate (22) for the magnetization parameter of the Crab pulsar. Moreover, the scale of is certainly smaller than the radius of the fast magnetosonic surface (see Eqn 23). A detailed comparison of theoretical predictions and observational data is beyond the scope of this article, however555In our opinion, the distance from the acceleration region to the light cylinder amounting to for the Crab pulsar may be overestimated.. Nevertheless, it must be noted that after it became clear that the existence of a large potential difference between the magnetic field lines in the magnetized wind does not lead directly to any effective particle acceleration up to ultra-high energies (see, e.g., Ref. [84]), the possibility of direct electrostatic acceleration of particles is not being taken into consideration (see, e.g., Ref. [85]). However, as was shown above, this process could well be realized in the case where, for some reason or other, the longitudinal current flowing in the magnetosphere of a compact object is quite small. ## 3 Theory of radio emission ### 3.1 Formulation of the problem As is well known, one of the methods for determining the instability increment of waves in a plasma consists in analyzing the dispersion equation, for which it is necessary to determine the dielectric permittivity tensor of the medium. The procedure for calculating the dielectric permittivity tensor of an inhomogeneous anisotropic plasma in the approximation of geometric optics on the basis of the standard approach, expounded, for example, in Ref. [86], is described in detail in [2]. In the same work, a study is presented of the collective interaction in which electromagnetic waves associated with curvature radiation are simultaneously amplified by the Cherenkov mechanism. We emphasize that this effect is absent in the vacuum. Most likely because the calculation procedure is quite complicated, erroneous assertions have been made in a number of publications [87-89]. The objections put forward in Ref. [89] were later withdrawn [90] by the author. As regards Refs [87, 88], which are still cited in papers on the relevant topic, they contain numerous arithmetic errors, which have all been revealed and described in detail in Ref. [42]. In addition, another method for dealing with the problem of collective curvature radiation was proposed in [36-38, 91]. In these studies, a model problem, which could be solved ’exactly’, was considered in a cylindrical geometry. The magnetic field in this problem is assumed to be precisely circular, while the relativistic plasma moves along the circular magnetic field lines owing to the centrifugal drift, , directed parallel to the cylindrical axis. Here, is the cyclotron radius and is the curvature radius. But as we now show, this approach cannot be used in analyzing the collective curvature radiation either (in more detail see [96]). As in [36-38], we consider the electromagnetic fields in the wave to be of the form [E,B]=[E(ρ),B(ρ)]×exp{−iωt+isφ+ikzz}, (24) where, is the wave frequency, is an integer defining the azimuthal component of the wave vector, and is the component of the wave vector parallel to the cylinder axis. In the approach considered, the amplitudes are assumed to depend only on the coordinate . Moreover, not the vectors and , but their cylindrical components and depend only on the cylindrical radius . This means that the polarization in the wave follows the magnetic field, turning from one point to another, which is possible only in the case of a well-defined boundary condition, for instance, for a system inside a metal cylinder. Under these conditions, we arrive at a one-dimensional problem, which can indeed be readily solved. However, as is not difficult to show, the wave considered within such an approach has nothing to do with curvature radiation. To show this, we consider a particle moving along a circular trajectory of radius with a constant velocity . Such motion corresponds to an infinite magnetic field. Then the emitted energy is equal to the work performed by the field of the wave on the electric current of the particle. The electric current is given by j=evδ(φ−Ωt)δ(z)δ(ρ−ρ0)ρeφ, (25) where . For the polarization chosen, ∫jEdr=evEφ(ρ0)exp{−iωt+isΩt}. (26) It hence follows that radiation is possible only when . But this is the condition for Cherenkov, but not curvature, radiation. A wave with such polarization cannot be generated by the curvature mechanism. The difference between curvature and Cherenkov waves is that the interaction time of bremsstrahlung and the irradiating particle is finite. A freely propagating wave with a nearly constant polarization deviates from the direction of motion of the particle. As a result, a nonzero projection of the electric field of the wave arises in the direction of the electric current of the particle, i.e., an exchange of energy between the wave and the particle becomes possible. The process lasts a finite time , which can be found from the condition . For relativistic particles () we have . ### 3.2 Polarization of the curvature wave It is well known that the radiation field of the electric current and of the charge density of a moving particle of charge is described by retarded potentials [43] A = 1c∫j(t′)Rdr, (27) Φ = ∫ρe(t′)Rdr, (28) where is the so-called retarded time and is the distance from the charge to the observer who is at the point with coordinates at the moment , R = [ρ2+z2+ρ20−2ρρ0cos(φ′−φ)]1/2, (29) φ′ = Ωt′. (30) We now introduce the Fourier transform of potentials (27) and (28) with respect to time: Aω = 12π∫A(t)exp{iωt}dt, (31) Φω = 12π∫Φ(t)exp{iωt}dt. (32) It is convenient in what follows to pass from integration over to integration over the retarded time , and then over the angle . As a result, in Cartesian coordiates () we obtain the vector potential and the scalar potential in the form [Aω;Φω]=eρ02πcexp{iωφ/Ω}[−Ks;Kc;0;cvK0], (33) where , , and are functions of only the coordinates and : K0 = ∫exp{iω(R/c+Ω−1)α}R+vρsinα/cdα, Ks = ∫exp{iω(R/c+Ω−1)α}sinαR+vρsinα/cdα, (34) Kc = ∫exp{iω(R/c+Ω−1)α}cosαR+vρsinα/cdα, R = (ρ2+z2+ρ20−2ρρ0cosα)1/2. We stress that expression (33) is valid not only in the wave zone but also at any point . The dependence on the angle is given by the factor . Hence, owing to the periodicity in the angle we simply obtain . An important fact follows from relation (33): the field of the irradiated wave is a superposition of three harmonics: , and . For example, we present the following expression for the azimuthal component of the electric field : Eφω=iωv(−ρ0ρΦω+vcAφω)= (35) −ieρ0ω2πv2exp{isφ}[ρ0ρK0−v2c2(Kssinφ+Kccosφ)]. The first term in the right-hand part of (35), which is proportional to the scalar potential , is not significant in the wave zone , but is significant in the vicinity of the particle trajectory, . Owing to the presence of this term, a particle which is in resonance with one of the three harmonics, for instance, with th (), is knocked out of resonance by the adjacent harmonics . Here, the component changes sign in a time . The synchronicity condition determines the time as τ≃1/Ωγ=ρ0/cγ, (36) which coincides with the formation time of curvature radiation. Hence, the emitted curvature wave comprises three harmonics, and , with a fixed relation between the amplitudes. Precisely this circumstance provides the curvature mechanism of radiation. The neighboring harmonics, , arise owing to modulation of the field of the emitted wave by the electric current of the particle whith the harmonic . It can now be understood why simply dealing with collective curvature radiation in a cylindrical geometry with a single cylindrical harmonic does not reveal any significant amplification of waves [36-38, 91]. The chosen polarization excludes the curvature radiation. ### 3.3 Propagation of the triplet of harmonics It was shown in Section 3.2 that the curvature radiation of a single charged particle in the vacuum cannot be described by a single cylindrical harmonic . In the problem of the collective curvature radiation of waves, modulation of the electric current of particles occurs at the same time as their excitation; therefore, the resonance azimuthal harmonic mixes with the harmonics of the modulation of the particle electric current, giving rise to harmonics with all possible values of . Below, we show that all azimuthal harmonics contribute to the response of the medium to the electromagnetic field. But here, we show that propagation of the triplet of cylindrical harmonics differs significantly from the propagation of a single harmonic, which is usually discussed in the literature. We consider the simple cylindrical one-dimensional problem of the emission from a flux of cold relativistic plasma particles with charge and mass , which move in the plane along an infinite azimuthal magnetic field . In this case, the particles can only move in the -direction with a velocity at an arbitrary cylindrical radius . We assume the nonperturbed particle number density and velocity to be constant, i.e., independent of the radius . The electric current then has only a component along , while the magnetic field of the wave has only the component (). The electric field in the wave has two nonzero components, and (). The dependence of the wave field on the coordinates is given by [Eρ;Eφ;Bz]=[Eρ(ρ);Eφ(ρ);Bz(ρ)]exp{−iωt+isφ}. (37) From the Maxwell equations, we obtain dE(σ)φdρ=iσρE(σ)ρ−iρσω2c2E(σ)ρ−E(σ)φρ, (38) dE(σ)ρdρ=−iσρE(σ)φ+4πωσρj(σ)φ−E(σ)ρρ, (39) where the index corresponds to one of the three harmonics, or . For simplicity, we introduce the dimensionless variable , and the quantity Λ=ω2pω2γ3, (40) where is the plasma frequency, is the Lorentz factor of plasma particles, and Gσ=4πj(σ)φΛω (41) is the dimensionless current. In the new variables, Eqns (38) and (39) become dE(σ)φdr = iσrE(σ)ρ−irσE(σ)ρ−E(σ)φr, (42) dE(σ)ρdr = −iσrE(σ)φ+ΛσrGσ−E(σ)ρr. (43) As was noted, we here consider the interaction of three waves, and . It is very important that the propagation of these waves is not independent: coupling between the waves is realized by means of the electrostatic field with the lowest azimuthal harmonic . The electrostatic field turns out to be the result of nonlinear coupling of the high-frequency harmonics and . The propagation equations of the mode in the same notation have the form dEφdr = irEρ−Eφr, (44) dEρdr = −i1rEφ+ΛZ−Eρr, (45) where . We stress that the second term in the right-hand part of Eqn (42) is absent in Eqn (44) because the field proportional to is static (). To determine the response of the medium to the electromagnetic fields, we can use the continuity equation and the Euler equation: ∂ne∂t+∇(nev)=0, (46) (∂∂t+v∇)p=e(E+[vc,B]). (47) It is easy to verify that only the -component of the Euler equation is significant, while the radial component provides the equilibrium configuration across the infinite magnetic field. We represent the velocity of plasma particles and the flux number density in terms of the expansion in powers of the wave field amplitudes: vφ=v(0)φ+δv(1)φ+δv(2)φ+..., (48) ne=n(0)e+δn(1)e+δn(2)e+.... (49) The linear response can be readily found as n(1)e = n(0)ekv(1)φω−kv(0)φ, (50) v(1)φ = ieEφmeγ3(ω−kv(0)φ), (51) where . To find the nonlinear response of the medium, it is necessary to take the nonlinear relation between and into account: δpφ=meγ3δvφ−32mev(0)φγ5(δvφ)2c2. (52) Straightforward calculation gives Gs = 11−sv(0)φ/r⎡⎣iEsφ1−sv(0)φ/r+αrv(0)φ (53) × ⎛⎝As,s−1Es−1φE1φ1−(s−1)v(0)φ/r − As,s+1Es+1φE1∗φ1−(s+1)v(0)φ/r⎞⎠⎤⎦, Gs−1 = 11−(s−1)v(0)φ/r⎡⎣iEs−1φ1−(s−1)v(0)φ/r (54) − αrv(0)φAs,s−1EsφE1∗φ1−sv(0)φ/r⎤⎦, Gs+1 = 11−(s+1)v(0)φ/r[iEsφ1−(s+1)v(0)φ/r (55) + αrv(0)φAs,s+1EsφE1φ1−sv(0)φ/r⎤⎦, Z = 1(v(0)φ)2[iE1ϕ1/r+ (56) + α⎛⎝Es+1φEs∗φ(1−(s+1)v(0)φ/r)(1−sv(0)φ/r) + EsφE(s−1)∗φ(1−sv(0)φ/r)(1−(s−1)v(0)φ/r)⎞⎠⎤⎦, Aa,b = 11−av(0)φ/r+11−bv(0)φ/r−3γ2, where and the velocity of plasma particles is expressed in terms of the speed of light . Similar quantities can be found in Ref. [3] for plane waves. The equations were analyzed numerically with the initial condition , corresponding to the normal mode in the vacuum for a cylindrical geometry; here, is the Bessel function of order . The singularity in Eqns (53)–(56) was smoothed out, as usual, by adding a small term to the resonance denominators in (50) and (51). Equations (42)–(45) were numerically solved for and at two different values of and . In the first case, we neglected the nonlinear terms in (53)–(56), while the second case corresponded to the fully nonlinear problem. The results of calculations for both cases are presented in Fig. 6. To illustrate the influence of the nonlinear current better, we have chosen the amplitude of the modes and to be 20 times larger than the amplitude of the s mode. Actually, the mode couples to the entire continuum of modes, and hence the above model assumption is reasonable. Fig. 6 reveals to be 2.5 times larger in the fully nonlinear problem than when the nonlinear current is neglected. Thus, we have shown that the triplet of cylindrical harmonics, better correponding to the curvature mechanism, is amplified more effectively than a single cylindrical harmonic. This means, inter alia, that the true polarization of collective curvature modes can only be obtained by calculating the dielectric permittivity tensor of the plasma flowing in a strong curved magnetic field. Here, the solution of the wave equations not only yields the dispersion relation for normal modes but also determines their polarization. We note that initially, it is totally unclear what polarization corresponds to nonstable modes. At first glance, the essentially nonlinear problem discussed above is not directly relevant to the amplification problem in the linear approximation. We included the nonlinearity only in order to examine the self-consistent coupling of modes and . Even in the case of a weak nonlinearity, the presence of adjacent modes alters the amplification of the mode significantly. It is also absolutely clear that the coupling of harmonics with the low-frequency harmonic results in the appearance of all possible azimuthal harmonics. ### 3.4 Calculation of the dielectric permittivity tensor In this section, we show that the asymptotic form of the dielectric permittivity tensor obtained in Ref. [2] for large values of the magnetic field curvature radius can be found by straightforward summation of responses (50) and (51) to individual cylindrical modes. We first note that in the case of an infinite toroidal field, there is only a response to the toroidal component of the wave electric field [92]. Here, we only consider the case of a stationary medium; therefore, the time dependence can be chosen to be of the form . Summation over all cylindrical modes yields Dφ(ρ,φ)=Eφ(ρ,φ)−∞∑s=−∞Eφ(ρ,s)K(ρ,s)exp{isφ}, (57) where (see, e.g., Ref. [36]) K(ρ,s)=4πe2ω∫vφω−svφ/ρ∂f(0)∂pφdpφ (58) and is the nonperturbed particle distribution function. Applying the Fourier transformation Eφ(ρ,s)=12π2π∫0Eφ(ρ,φ′)exp{−isφ′}dφ′ (59) and passing to a Cartesian coordinate system, we find Dx=Ex+12π∫ρ′dρ′dφ′ρ′∞∑s=−∞Eφ(ρ′,φ′)δ(ρ−ρ′) ×K(ρ,s)exp{is(φ−φ′)}sinφ, (60) Dy=Ey−12π∫ρ′dρ′dφ′ρ′∞∑s=−∞Eφ(ρ′,φ′)δ(ρ−ρ′) ×K(ρ,s)exp{is(φ−φ′)}cosφ. (61) We choose a local coordinate sytem in the particle orbit plane with the axis directed along the magnetic field. From the expressions presented above, we obtain the components of the kernel of the dielectric permittivity operator: εyy(r,r′)=1−12π1ρ′∞∑s=−∞δ(ρ−ρ′)K(ρ,s) ×exp{is(φ−φ′)}cosφcosφ′; (62) εyx(r,r′)=12π1ρ′∞∑s=−∞δ(ρ−ρ′)K(ρ,s) ×exp{is(φ−φ′)}cosφsinφ′; (63) εxy(r,r′)=12π1ρ′∞∑s=−∞δ(ρ−ρ′)K(ρ,s) ×exp{is(φ−φ′)}sinφcosφ′, (64) εxx(r,r′)=1−12π1ρ′∞∑s=−∞δ(ρ−ρ′)K(ρ,s) ×exp{is(φ−φ′)}sinφsinφ′, (65) which determine the material relation Di(r)=∫εij(r,r′)Ej(r′)dr′. (66) We note that the kernel found satisfies the necessary symmetry property εij(r,r′,ω)=εji(r′,r,−ω), (67) resulting from the condition . Further, it is well known that for calculating the dielectric permittivity tensor, only the symmetrized form of must be used [93, 94]: εij(ω,k,η→r)=∫εij(ω,ξ,η)exp{−ikξ}dξ. (68) Here, by definition, and . It is important to note that only this tensor correctly describes the interaction between waves and particles in a medium with slowly varying parameters (see, e.g., Refs [35, 86]; in problems dealing with cosmological plasma, this approach was applied in Ref. [95]). Substituting the components of the kernel, we now obtain εxx(ω,k,η)=1−12π∫dξexp{−ikξ}1\|η−ξ/2\|× ∞∑s=−∞δ(\|η+ξ/2\|−\|η−ξ/2\|)K(\|η+ξ/2\|,s)× exp{is(φ−φ′)}sinφsinφ′, (69) εxy(ω,k,η)=12π∫dξexp{−ikξ}1\|η−ξ/2\|× ∞∑s=−∞δ(\|η+ξ/2\|−\|η−ξ/2\|)K(\|η+ξ/2\|,s)× exp{is(φ−φ′)}sinφcosφ′, (70) εyx(ω,k,η)=12π∫dξexp{−ikξ}1\|η−ξ/2\|× ∞∑s=−∞δ(\|η+ξ/2\|−\|η−ξ/2\|)K(\|η+ξ/2\|,s)× exp{is(φ−φ′)}cosφsinφ′, (71) εyy(ω,k,η)=1−12π∫dξexp{−ikξ}1\|η−ξ/2\|× ∞∑s=−∞δ(\|η+ξ/2\|−\|η−ξ/2\|)K(\|η+ξ/2\|,s)× exp{is(φ−φ′)}cosφcosφ′. (72) The angles and in expressions (69)–(72) are functions of the polar angles and of the vectors and , sinφ=\|η\|sinαη+(\|ξ\|/2)sinαξ\|η+ξ/2\|, (73) cosφ′=\|η\|cosαη−(\|ξ\|/2)cosαξ\|η−ξ/2\|. (74) As a result, the integration in (69)–(72) reduces to integration over the components of the vector that are perpendicular to . On the other hand, the the delta function in relations (69)–(72) are given by δ(...)=δ(θ−π/2)(\|η+ξ/2\|−\|η−ξ/2\|)′θ +δ(θ+π/2)(\|η+ξ/2\|−\|η−ξ/2\|)′θ, (75) where is the angle between vectors and . Therefore, the integration over the angles is carried out in a trivial manner. Finally, performing the transition we obtain . In accordance with (75), we can therefore write , where is the component of the wave vector directed along the magnetic field. The property of the final result being independent of is very important. Precisely it provides the same symmetry property as in a homogeneous medium [42]: εij(−ω,−k,−B,r)=εji(ω,k,B,r). (76) If transformation (68) is neglected, the necessary symmetry cannot be obtained [36] (the authors of Ref. [36] explained the dependence of the dielectric tensor on by not being a Killing vector). Finally, we use the Taylor expansion in and the reduction of summation to a delta function: ∑(...)1ω	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9626667499542236, "perplexity": 557.3706957366853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362605.52/warc/CC-MAIN-20211203060849-20211203090849-00445.warc.gz"}
https://bettergradesfast.com/academy/fractions-adding-subtracting-primary-math-ebook/	Adding and Subtracting Fractions \| Primary Math eBook Adding and Subtracting Fractions with Like Denominators When adding or subtracting fractions with the same denominators, we look at the numerators and add/subtract them. Example: 2/9 + 4/9 = 6/9 Subtract 2/8 from ⅞. 7 - 2 = 5, therefore: ⅞ - 2/8 = ⅝ Adding and Subtracting Fractions with Unlike Denominators When adding/subtracting fractions with different denominators, you must first change all fractions so that they have the same denominator. Then, you add their changed numerators as was shown above. Example: Multiples of 8: 8, 16, 24, 32, etc. Multiples of 4: 4, 8, 12, etc. LCM = 8 8 x 1 = 8 4 x 2 = 8 Now, we multiply the corresponding numerators: If necessary, we change the answer into a mixed number (if the answer is an improper fraction). 8/8 = 1 Subtract 2/9 from ½. Multiples of 9: 9, 18, 27, 36, etc. Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 LCM = 18 9 x 2 = 18 2 x 9 = 18 Note: When adding or subtracting mixed numbers, you must first turn all mixed numbers into improper fractions. Then, follow the steps as described above to calculate the answer.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9248198866844177, "perplexity": 677.0228976791934}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305420.54/warc/CC-MAIN-20220128043801-20220128073801-00227.warc.gz"}
http://mathoverflow.net/questions/71074/non-deterministic-turing-machines	# non-deterministic turing machines I have one simple question: There is a set, which can be decided in polynomial time by a (one-band) non-deterministic Turing Machine. Why should there exist one (one-band) non-deterministic Turing Machine, which decides the same set, but with the additional property: There exists one natural number k, such that all the possible calculations last exactly n^k steps, if the input has the length n? This is one "without-loss-of-generality-assumption" in the proof of a theorem. (namely the proof of the Theorem of Fagin which says: "ESO captures NP") I cannot understand it. How can we manipulate the first machine to get the second machine? - Since the language lies in P, the time is bounded above by n^k for some k. If we need exactly n^k, we can assume that our machine makes additional moves. – AlB Jul 23 '11 at 17:18 It is not always possible to do. Take $n=1$ and suppose that the accepting computations on a 1-letter input have length at least 2. – Mark Sapir Jul 23 '11 at 17:23 ok, I forgot to say that n>1. – tibet Jul 23 '11 at 17:37 to the comment of AIB: Yes this is the problem. But how does the machine know the number of steps. It has to count the number of steps during the calculation and if it stops earlier then n^k (by input length n) then it should make theese additional (dummy) steps and then stop. But how? Given one Language in NP, how can I show that there is one NTM which stops exactly after n^k steps for every input of lentgh n? – tibet Jul 23 '11 at 17:44 @tibet, the overhead of step counting is tolerable, it could be a logarithmic counter, etc. But the exact details must be tedious, especially if you want to do that on a one-tape machine. – AlB Jul 23 '11 at 17:57 This is possible, but it is somewhat tricky to do. Here is an outline of one way to do it... Start with your original one-tape Turing machine $M_0$ which runs in time $\leq k + n^k$ (say) on input of length $n$. First create a two-tape Turing machine $M_1$ which simulates $M_0$ on one tape and keeps track of a step-counter on the other tape. The counter is initially set to value $k + n^k$ and is decremented at each simulation step. When the simulation of $M_0$ terminates, $M_1$ keeps doing dummy moves until the counter is exhausted. Thus $M_1$ runs in exactly the same time on every input of length $n$. Finally, we simulate $M_1$ on a one-tape Turing machine $M_2$ as follows. Think of even cells as belonging to the first tape of $M_1$ and odd cells as belonging to the second tape of $M_1$. To keep track of where the two $M_1$ heads, each symbol will now have a plain and a red variant; there will be only two red variants at any given time and they will mark the two head positions. It is straightforward to simulate $M_1$ on such a tape, but the simpler ways do not simulate each step of $M_1$ in a constant number of steps since switching from one head to the other requires a variable number of moves. To remedy this, first note that $M_1$ uses less than $\ell + n^\ell$ cells of the tape for some $\ell$ that can be effectively estimated from $k$ and the above transformations. When it starts, $M_2$ reads the input length $n$ and places a freshly minted marker on the $(\ell + n^\ell)$-th cell, beyond any cell required to simulate $M_1$. Whenever $M_2$ simulates a step of $M_1$, it proceeds as follows: • Starting at the base of the tape, $M_2$ finds the appropriate tape head (red symbol in even/odd position). • $M_2$ then performs the appropriate action to simulate $M_1$, these each take a fixed finite amount of steps which may vary from operation to operation. Once this is completed, $M_2$ dances around a little so that it returns to the original tape position exactly 1001 steps after it arrived there. • Then $M_2$ moves right until it finds the marker at position $\ell + n^\ell$ at which point it turns around and returns to the base of the tape. Although this is very inefficient, $M_2$ does correctly simulate $M_1$ and it takes exactly the same amount of time to simulate each step of $M_1$. Furthermore, $M_2$ still runs in polynomial time, though the polynomial is much worse than the original $k + n^k$. Edit: This answer was simplified from its original version. - A k-band NTM which runs in time t(n) can be simuleted by a 1-band NTM which uses time t(n)^2. I think the proof uses the same argument as yours. – tibet Jul 24 '11 at 21:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8172101974487305, "perplexity": 335.65934953325717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275645.9/warc/CC-MAIN-20160524002115-00036-ip-10-185-217-139.ec2.internal.warc.gz"}
https://worldwidescience.org/topicpages/g/guided+ion+beam.html	#### Sample records for guided ion beam 1. Guiding center simulations of strong ion beams with applications to the Counterstreaming Ion Torus International Nuclear Information System (INIS) Tull, C. 1978-03-01 In the proposed Counterstreaming Ion Torus (CIT) steady state rather than pulsed operation may be possible if all of the plasma power density is provided by neutral beam injection. After the neutral beams have penetrated the magnetic field, strong ion beam currents are produced. A major concern with the relatively strong counterstreaming ion currents is the effect of the beam self-magnetic fields on the macroscopic equilibrium of the system. Pinching and self focusing of the individual beams may occur, or the repulsive interaction of the two oppositely directed beam currents may destroy the equilibrium entirely. We investigate this macroscopic behavior of the ion beams with a guiding center plasma particle simulation model and we describe a model we have developed to simulate steady state behavior in an ideal CIT configuration 2. Understanding focused ion beam guided anodic alumina nanopore development International Nuclear Information System (INIS) Chen Bo; Lu, Kathy; Tian Zhipeng 2011-01-01 Graphical abstract: Display Omitted Highlights: → We study the effect of FIB patterning on pore evolution during anodization. → FIB patterned concaves with 1.5 nm depth can effectively guide nanopore growth. → The edge effect of FIB guided patterns causes nanopores to bend. → Anodization window is enlarged to 50-80 V for 150 nm interpore distance hexagonal arrays. - Abstract: Focused ion beam (FIB) patterning in combination with anodization has shown great promise in creating unique pore patterns. This work is aimed to understand the effect of the FIB patterned sites in guiding anodized pore development. Highly ordered porous anodic alumina has been created with the guidance of FIB created patterns on electropolished aluminum followed by oxalic acid anodization. Shallow concaves created by the FIB with only 1.5 nm depth can effectively guide the growth of ordered nanopore patterns. With the guidance of the FIB pattern, the anodization rate is much faster and the nanopore growth direction bends at the boundary of the FIB patterned and un-patterned regions. FIB patterning also enlarges the anodization window; ordered nanopore arrays with 150 nm interpore distances can be produced under an applied potential from 50 V to 80 V. The fundamental understanding of these unique processes is discussed. 3. A sextupole ion beam guide to improve the efficiency and beam quality at IGISOL International Nuclear Information System (INIS) Karvonen, P.; Moore, I.D.; Sonoda, T.; Kessler, T.; Penttilae, H.; Peraejaervi, K.; Ronkanen, P.; Aystoe, J. 2008-01-01 The laser ion source project at the IGISOL facility, Jyvaeskylae, has motivated the development and construction of an rf sextupole ion beam guide (SPIG) to replace the original skimmer electrode. The SPIG has been tested both off-line and on-line in proton-induced fission, light-ion and heavy-ion induced fusion-evaporation reactions and, in each case, has been directly compared to the skimmer system. For both fission and light-ion induced fusion, the SPIG has improved the mass-separated ion yields by a factor of typically 4-8. Correspondingly, the transmission efficiency of both systems has been studied in simulations with and without space charge effects. The transport capacity of the SPIG has been experimentally determined to be ∼10 12 ions s -1 before space charge effects start to take effect. A direct comparison with the simulation has been made using data obtained via light-ion fusion evaporation. Both experiment and simulation show an encouraging agreement as a function of current extracted from the ion guide. 4. Beam cooling using a gas-filled RFQ ion guide CERN Document Server Henry, S; De Saint-Simon, M; Jacotin, M; Képinski, J F; Lunney, M D 1999-01-01 A radiofrequency quadrupole mass filter is being developed for use as a high-transmission beam cooler by operating it in buffer gas at high pressure. Such a device will increase the sensitivity of on-line experiments that make use of weakly produced radioactive ion beams. We present simulations and some preliminary measurements for a device designed to cool the beam for the MISTRAL RF mass spectrometer on- line at ISOLDE. The work is carried out partly within the frame of the European Community research network: EXOTRAPS. (9 refs). 5. Photonic guiding structures in lithium niobate crystals produced by energetic ion beams Science.gov (United States) Chen, Feng 2009-10-01 A range of ion beam techniques have been used to fabricate a variety of photonic guiding structures in the well-known lithium niobate (LiNbO3 or LN) crystals that are of great importance in integrated photonics/optics. This paper reviews the up-to-date research progress of ion-beam-processed LiNbO3 photonic structures and reports on their fabrication, characterization, and applications. Ion beams are being used with this material in a wide range of techniques, as exemplified by the following examples. Ion beam milling/etching can remove the selected surface regions of LiNbO3 crystals via the sputtering effects. Ion implantation and swift ion irradiation can form optical waveguide structures by modifying the surface refractive indices of the LiNbO3 wafers. Crystal ion slicing has been used to obtain bulk-quality LiNbO3 single-crystalline thin films or membranes by exfoliating the implanted layer from the original substrate. Focused ion beams can either generate small structures of micron or submicron dimensions, to realize photonic bandgap crystals in LiNbO3, or directly write surface waveguides or other guiding devices in the crystal. Ion beam-enhanced etching has been extensively applied for micro- or nanostructuring of LiNbO3 surfaces. Methods developed to fabricate a range of photonic guiding structures in LiNbO3 are introduced. Modifications of LiNbO3 through the use of various energetic ion beams, including changes in refractive index and properties related to the photonic guiding structures as well as to the materials (i.e., electro-optic, nonlinear optic, luminescent, and photorefractive features), are overviewed in detail. The application of these LiNbO3 photonic guiding structures in both micro- and nanophotonics are briefly summarized. 6. Gradient and alternating diameter nanopore templates by focused ion beam guided anodization International Nuclear Information System (INIS) Chen Bo; Lu, Kathy; Tian Zhipeng 2010-01-01 Ordered arrays of anodic alumina nanopores with uniform pore diameters have been fabricated by self-organized anodization of aluminum. However, gradient or alternating diameter nanopore arrays with designed interpore distances have not been possible. In this study, focused ion beam lithography is used to fabricate hexagonally arranged concaves with different diameters in designed arrangements on aluminum surfaces. The patterns are then used to guide the further growth of alumina nanopores in the subsequent oxalic acid anodization. Gradient and alternating nanopore arrangements have been attained by FIB patterning guided oxalic acid anodization. The fundamental understanding of the process is discussed. 7. Guided ion beam and theoretical studies of the bond energy of SmS+ Science.gov (United States) Armentrout, P. B.; Demireva, Maria; Peterson, Kirk A. 2017-12-01 Previous work has shown that atomic samarium cations react with carbonyl sulfide to form SmS+ + CO in an exothermic and barrierless process. To characterize this reaction further, the bond energy of SmS+ is determined in the present study using guided ion beam tandem mass spectrometry. Reactions of SmS+ with Xe, CO, and O2 are examined. Results for collision-induced dissociation processes with all three molecules along with the endothermicity of the SmS+ + CO → Sm+ + COS exchange reaction are combined to yield D0(Sm+-S) = 3.37 ± 0.20 eV. The CO and O2 reactions also yield a SmSO+ product, with measured endothermicities that indicate D0(SSm+-O) = 3.73 ± 0.16 eV and D0(OSm+-S) = 1.38 ± 0.27 eV. The SmS+ bond energy is compared with theoretical values characterized at several levels of theory, including CCSD(T) complete basis set extrapolations using all-electron basis sets. Multireference configuration interaction calculations with explicit spin-orbit calculations along with composite thermochemistry using the Feller-Peterson-Dixon method and all-electron basis sets were also explored for SmS+, and for comparison, SmO, SmO+, and EuO. 8. Ion beam monitoring International Nuclear Information System (INIS) McKinney, C.R. 1980-01-01 An ion beam analyzer is specified, having an ion source for generating ions of a sample to be analyzed, means for extracting the sample ions, means for focusing the sample ions into a beam, separation means positioned along the ion beam for selectively deflecting species of ions, and means for detecting the selected species of ions. According to the specification, the analyzer further comprises (a) means for disabling at least a portion of the separation means, such that the ion beam from the source remains undeflected; (b) means located along the path of the undeflected ion beam for sensing the sample ions; and (c) enabling means responsive to the sensing means for automatically re-enabling the separation means when the sample ions reach a predetermined intensity level. (author) 9. Ion Beam Propulsion Study Science.gov (United States) 2008-01-01 The Ion Beam Propulsion Study was a joint high-level study between the Applied Physics Laboratory operated by NASA and ASRC Aerospace at Kennedy Space Center, Florida, and Berkeley Scientific, Berkeley, California. The results were promising and suggested that work should continue if future funding becomes available. The application of ion thrusters for spacecraft propulsion is limited to quite modest ion sources with similarly modest ion beam parameters because of the mass penalty associated with the ion source and its power supply system. Also, the ion source technology has not been able to provide very high-power ion beams. Small ion beam propulsion systems were used with considerable success. Ion propulsion systems brought into practice use an onboard ion source to form an energetic ion beam, typically Xe+ ions, as the propellant. Such systems were used for steering and correction of telecommunication satellites and as the main thruster for the Deep Space 1 demonstration mission. In recent years, "giant" ion sources were developed for the controlled-fusion research effort worldwide, with beam parameters many orders of magnitude greater than the tiny ones of conventional space thruster application. The advent of such huge ion beam sources and the need for advanced propulsion systems for exploration of the solar system suggest a fresh look at ion beam propulsion, now with the giant fusion sources in mind. 10. Ion beam diagnosis International Nuclear Information System (INIS) Strehl, P. 1994-04-01 This report is an introduction to ion beam diagnosis. After a short description of the most important ion beam parameters measurements of the beam current by means of Faraday cups, calorimetry, and beam current transformers and measurements of the beam profile by means of viewing screens, profile grids and scanning devices, and residual gas ionization monitors are described. Finally measurements in the transverse and longitudinal phase space are considered. (HSI) 11. Focusing peculiarities of ion-channel guiding on a relativistic electron beam in a free-electron laser with a three-dimensional wiggler International Nuclear Information System (INIS) Ouyang, Zhengbiao; Zhang, Shi-Chang 2014-01-01 In a free-electron laser the ‘natural focusing’ effect of a three-dimensional wiggler is too weak to confine the transport of a relativistic electron beam when the beam has a high current and consequently an external focusing system is often needed. In this paper we study the focusing peculiarities of an ion-channel guide field on an electron beam. Nonlinear simulations of an electron beam transport show that, compared to an axial guide magnetic field, the ion-channel guide field results in smaller velocity–space and configuration–space spreads. The intrinsic mechanism of this physical phenomenon is that the ion-channel guide field confines the trajectory of the electron motion resulting in a smaller instantaneous curvature radius and a slighter curvature-center excursion than an axial guide magnetic field does. It is also found that, unlike with an axial guide magnetic field, over-focusing may occur if the ion-channel guide field is too strong. (paper) 12. Electron Beam Ion Sources CERN Document Server Zschornacka, G.; Thorn, A. 2013-12-16 Electron beam ion sources (EBISs) are ion sources that work based on the principle of electron impact ionization, allowing the production of very highly charged ions. The ions produced can be extracted as a DC ion beam as well as ion pulses of different time structures. In comparison to most of the other known ion sources, EBISs feature ion beams with very good beam emittances and a low energy spread. Furthermore, EBISs are excellent sources of photons (X-rays, ultraviolet, extreme ultraviolet, visible light) from highly charged ions. This chapter gives an overview of EBIS physics, the principle of operation, and the known technical solutions. Using examples, the performance of EBISs as well as their applications in various fields of basic research, technology and medicine are discussed. 13. Nanostructures by ion beams Science.gov (United States) Schmidt, B. Ion beam techniques, including conventional broad beam ion implantation, ion beam synthesis and ion irradiation of thin layers, as well as local ion implantation with fine-focused ion beams have been applied in different fields of micro- and nanotechnology. The ion beam synthesis of nanoparticles in high-dose ion-implanted solids is explained as phase separation of nanostructures from a super-saturated solid state through precipitation and Ostwald ripening during subsequent thermal treatment of the ion-implanted samples. A special topic will be addressed to self-organization processes of nanoparticles during ion irradiation of flat and curved solid-state interfaces. As an example of silicon nanocrystal application, the fabrication of silicon nanocrystal non-volatile memories will be described. Finally, the fabrication possibilities of nanostructures, such as nanowires and chains of nanoparticles (e.g. CoSi2), by ion beam synthesis using a focused Co+ ion beam will be demonstrated and possible applications will be mentioned. 14. Intense ion beam generator International Nuclear Information System (INIS) Humphries, S. Jr.; Sudan, R.N. 1977-01-01 Methods and apparatus for producing intense megavolt ion beams are disclosed. In one embodiment, a reflex triode-type pulsed ion accelerator is described which produces ion pulses of more than 5 kiloamperes current with a peak energy of 3 MeV. In other embodiments, the device is constructed so as to focus the beam of ions for high concentration and ease of extraction, and magnetic insulation is provided to increase the efficiency of operation 15. IFR channel-guiding of spinning beams International Nuclear Information System (INIS) O'Brien, K.J. 1986-06-01 A simple model is adopted to study the Ion Focussed Regime (IFR) laser channel-guiding of a spinning relativistic electron beam. It is discovered that spinning beams precess about the IFR axis as they damp; whereas, nonspinning beams remain planarly polarized 16. Cluster ion beam facilities International Nuclear Information System (INIS) Popok, V.N.; Prasalovich, S.V.; Odzhaev, V.B.; Campbell, E.E.B. 2001-01-01 A brief state-of-the-art review in the field of cluster-surface interactions is presented. Ionised cluster beams could become a powerful and versatile tool for the modification and processing of surfaces as an alternative to ion implantation and ion assisted deposition. The main effects of cluster-surface collisions and possible applications of cluster ion beams are discussed. The outlooks of the Cluster Implantation and Deposition Apparatus (CIDA) being developed in Guteborg University are shown 17. Ion beam neutralization with ferroelectrically generated electron beams Energy Technology Data Exchange (ETDEWEB) Herleb, U; Riege, H [European Organization for Nuclear Research, Geneva (Switzerland). LHC Division 1997-12-31 A technique for ion beam space-charge neutralization with pulsed electron beams is described. The intensity of multiply-charged ions produced with a laser ion source can be enhanced or decreased separately with electron beam trains of MHz repetition rate. These are generated with ferroelectric cathodes, which are pulsed in synchronization with the laser ion source. The pulsed electron beams guide the ion beam in a similar way to the alternating gradient focusing of charged particle beams in circular accelerators such as synchrotrons. This new neutralization technology overcomes the Langmuir-Child space-charge limit and may in future allow ion beam currents to be transported with intensities by orders of magnitude higher than those which can be accelerated today in a single vacuum tube. (author). 6 figs., 10 refs. 18. Reactions of O(+) With C(n)H(2n+2), n=2-4: A Guided-Ion Beam Study National Research Council Canada - National Science Library Levandier, D 2004-01-01 We have measured absolute reaction cross sections for the interaction of 0+ with ethane, propane, and n-butane at collision energies in the range from near thermal to approximately 20 eV, using the guided-ion beam (GIB) technique... 19. Ion-beam technologies Energy Technology Data Exchange (ETDEWEB) Fenske, G.R. [Argonne National Lab., IL (United States) 1993-01-01 This compilation of figures and diagrams reviews processes for depositing diamond/diamond-like carbon films. Processes addressed are chemical vapor deposition (HFCVD, PACVD, etc.), plasma vapor deposition (plasma sputtering, ion beam sputtering, evaporation, etc.), low-energy ion implantation, and hybrid processes (biased sputtering, IBAD, biased HFCVD, etc.). The tribological performance of coatings produced by different means is discussed. 20. Ion beam studies International Nuclear Information System (INIS) Freeman, J.H.; Chivers, D.J.; Gard, G.A.; Temple, W. 1977-04-01 A description of techniques for the production of intense beams of heavy ions is given. A table of recommended operational procedures for most elements is included. The ionisation of boron is considered in some detail because of its particular importance as a dopant for ion implantation. (author) 1. Biomaterials modification by ion beam International Nuclear Information System (INIS) Zhang Tonghe; Yi Zhongzhen; Zhang Xu; Wu Yuguang 2001-01-01 Ion beam technology is one of best ways for the modification of biomaterials. The results of ion beam modification of biomaterials are given. The method and results of improved biocompatibility are indicated by ion beam technology. The future development of ion beam modification of biomaterials is discussed 2. Negative ion beam processes International Nuclear Information System (INIS) Hayward, T.D.; Lawrence, G.P.; Bentley, R.F.; Malanify, J.J.; Jackson, J.A. 1975-06-01 Los Alamos Scientific Laboratory fiscal year 1975 work on production of intense, very bright, negative hydrogen (H - ), ion beams and conversion of a high-energy (a few hundred MeV) negative beam into a neutral beam are described. The ion source work has used a cesium charge exchange source that has produced H - ion beams greater than or equal to 10 mA (about a factor of 10 greater than those available 1 yr ago) with a brightness of 1.4 x 10 9 A/m 2 -rad 2 (about 18 times brighter than before). The high-energy, neutral beam production investigations have included measurements of the 800-MeV H - -stripping cross section in hydrogen gas (sigma/sub -10/, tentatively 4 x 10 -19 cm 2 ), 3- to 6-MeV H - -stripping cross sections in a hydrogen plasma (sigma/sub -10/, tentatively 2 to 4 x 10 -16 cm 2 ), and the small-angle scattering that results from stripping an 800-MeV H - ion beam to a neutral (H 0 ) beam in hydrogen gas. These last measurements were interrupted by the Los Alamos Meson Physics Facility shutdown in December 1974, but should be completed early in fiscal year 1976 when the accelerator resumes operation. Small-angle scattering calculations have included hydrogen gas-stripping, plasma-stripping, and photodetachment. Calculations indicate that the root mean square angular spread of a 390-MeV negative triton (T - ) beam stripped in a plasma stripper may be as low as 0.7 μrad 3. Ion beam texturing Science.gov (United States) Hudson, W. R. 1977-01-01 A microscopic surface texture was created by sputter-etching a surface while simultaneously sputter-depositing a lower sputter yield material onto the surface. A xenon ion-beam source was used to perform the texturing process on samples as large as 3-cm diameter. Textured surfaces have been characterized with SEM photomicrographs for a large number of materials including Cu, Al, Si, Ti, Ni, Fe, stainless steel, Au, and Ag. A number of texturing parameters are studied including the variation of texture with ion-beam powder, surface temperature, and the rate of texture growth with sputter etching time. 4. Activation of CH4 by Th(+) as studied by guided ion beam mass spectrometry and quantum chemistry. Science.gov (United States) Cox, Richard M; Armentrout, P B; de Jong, Wibe A 2015-04-06 The reaction of atomic thorium cations with CH4 (CD4) and the collision-induced dissociation (CID) of ThCH4(+) with Xe are studied using guided ion beam tandem mass spectrometry. In the methane reactions at low energies, ThCH2(+) (ThCD2(+)) is the only product; however, the energy dependence of the cross-section is inconsistent with a barrierless exothermic reaction as previously assumed on the basis of ion cyclotron resonance mass spectrometry results. The dominant product at higher energies is ThH(+) (ThD(+)), with ThCH3(+) (ThCD3(+)) having a similar threshold energy. The latter product subsequently decomposes at still higher energies to ThCH(+) (ThCD(+)). CID of ThCH4(+) yields atomic Th(+) as the exclusive product. The cross-sections of all product ions are modeled to provide 0 K bond dissociation energies (in eV) of D0(Th(+)-H) ≥ 2.25 ± 0.18, D0(Th(+)-CH) = 6.19 ± 0.16, D0(Th(+)-CH2) ≥ 4.54 ± 0.09, D0(Th(+)-CH3) = 2.60 ± 0.30, and D0(Th(+)-CH4) = 0.47 ± 0.05. Quantum chemical calculations at several levels of theory are used to explore the potential energy surfaces for activation of methane by Th(+), and the effects of spin-orbit coupling are carefully considered. When spin-orbit coupling is explicitly considered, a barrier for C-H bond activation that is consistent with the threshold measured for ThCH2(+) formation (0.17 ± 0.02 eV) is found at all levels of theory, whereas this barrier is observed only at the BHLYP and CCSD(T) levels otherwise. The observation that the CID of the ThCH4(+) complex produces Th(+) as the only product with a threshold of 0.47 eV indicates that this species has a Th(+)(CH4) structure, which is also consistent with a barrier for C-H bond activation. This barrier is thought to exist as a result of the mixed ((4)F,(2)D) electronic character of the Th(+) J = (3)/2 ground level combined with extensive spin-orbit effects. 5. Ion beam inertial fusion International Nuclear Information System (INIS) Bangerter, R.O. 1995-01-01 About twenty years ago, A. W. Maschke of Brookhaven National Laboratory and R. L. Martin of Argonne National Laboratory recognized that the accelerators that have been developed for high energy and nuclear physics are, in many ways, ideally suited to the requirements of inertial fusion power production. These accelerators are reliable, they have a long operating life, and they can be efficient. Maschke and Martin noted that they can focus ion beams to small focal spots over distances of many meters and that they can readily operate at the high pulse repetition rates needed for commercial power production. Fusion, however, does impose some important new constraints that are not important for high energy or nuclear physics applications. The most challenging new constraint from a scientific standpoint is the requirement that the accelerator deliver more than 10 14 W of beam power to a small quantity (less than 100 mg) of matter. The most challenging constraint from an engineering standpoint is accelerator cost. Maschke showed theoretically that accelerators could produce adequate work. Heavy-ion fusion is widely recognized to be a promising approach to inertial fusion power production. It provides an excellent opportunity to apply methods and technology developed for basic science to an important societal need. The pulsed-power community has developed a complementary, parallel approach to ion beam fusion known as light-ion fusion. The talk will discuss both heavy-ion and light-ion fusion. It will explain target physics requirements and show how they lead to constraints on the usual accelerator parameters such as kinetic energy, current, and emittance. The talk will discuss experiments that are presently underway, specifically experiments on high-current ion sources and injectors, pulsed-power machines recirculating induction accelerators, and transverse beam combining. The talk will give a brief description of a proposed new accelerator called Elise 6. Low energy ion beam dynamics of NANOGAN ECR ion source Energy Technology Data Exchange (ETDEWEB) Kumar, Sarvesh, E-mail: [email protected]; Mandal, A. 2016-04-01 A new low energy ion beam facility (LEIBF) has been developed for providing the mass analyzed highly charged intense ion beams of energy ranging from a few tens of keV to a few MeV for atomic, molecular and materials sciences research. The new facility consists of an all permanent magnet 10 GHz electron cyclotron resonance (ECR) ion source (NANOGAN) installed on a high voltage platform (400 kV) which provides large currents of multiply charged ion beams. Higher emittance at low energy of intense ion beam puts a tremendous challenge to the beam optical design of this facility. The beam line consists of mainly the electrostatic quadrupoles, an accelerating section, analyzing cum switching magnet and suitable beam diagnostics including vacuum components. The accelerated ion beam is analyzed for a particular mass to charge (m/q) ratio as well as guided to three different lines along 75°, 90° and 105° using a large acceptance analyzing cum switching magnet. The details of transverse beam optics to all the beam lines with TRANSPORT and GICOSY beam optics codes are being described. Field computation code, OPERA 3D has been utilized to design the magnets and electrostatic quadrupoles. A theoretical estimation of emittance for optimized geometry of ion source is given so as to form the basis of beam optics calculations. The method of quadrupole scan of the beam is used to characterize the emittance of the final beam on the target. The measured beam emittance increases with m/q ratios of various ion beams similar to the trend observed theoretically. 7. Ion beam analysis International Nuclear Information System (INIS) Bethge, K. 1995-01-01 Full text: Ion beam analysis is an accelerator application area for the study of materials and the structure of matter; electrostatic accelerators of the Van de Graaff or Dynamitron type are often used for energies up to a few MeV. Two types of machines are available - the single-ended accelerator type with higher beam currents and greater flexibility of beam management, or the tandem accelerator, limited to atomic species with negative ions. The accelerators are not generally installed at specialist accelerator laboratories and have to be easy to maintain and simple to operate. The most common technique for industrial research is Rutherford Back Scattering Spectrometry (RBS). Helium ions are the preferred projectiles, since at elevated energies (above 3 MeV) nuclear resonance scattering can be used to detect photons associated with target molecules containing elements such as carbon, nitrogen or oxygen. Due to the large amount of available data on nuclear reactions in this energy range, activation analysis (detecting trace elements by irradiating the sample) can be performed with charged particles from accelerators over a wider range of atoms than with the conventional use of neutrons, which is more suited to light elements. Resonance reactions have been used to detect trace metals such as aluminium, titanium and vanadium. Hydrogen atoms are vital to the material performance of several classes of materials, such as semiconductors, insulators and ceramics. Prudent selection of the projectile ion aids the analysis of hydrogen composition; the technique is then a simple measurement of the emitted gamma radiation. Solar cell material and glass can be analysed in this way. On a world-wide basis, numerous laboratories perform ion beam analysis for research purposes; considerable work is carried out in cooperation between scientific laboratories and industry, but only a few laboratories provide a completely commercial service 8. Ion Beam Extraction by Discrete Ion Focusing DEFF Research Database (Denmark) 2010-01-01 An apparatus (900) and methods are disclosed for ion beam extraction. In an implementation, the apparatus includes a plasma source (or plasma) (802) and an ion extractor (804). The plasma source is adapted to generate ions and the ion extractor is immersed in the plasma source to extract a fracti... 9. Focused ion beam technology International Nuclear Information System (INIS) Gamo, K. 1993-01-01 Focussed ion beam (FIB) technology has the advantage of being a maskless process compatible with UHV processing. This makes it attractive for use in in situ processing and has been applied to the fabrication of various mesoscopic structures. The present paper reviews these results whilst putting emphasis on in situ processing by a combined FIB and molecular beam epitaxy system. The typical performance of present FIB systems is also presented. In order to utilize the potential advantages of FIB processing, reduction of damage and improvement of throughput are important, and much effort has been devoted to developing processing techniques which require a reduced dose. The importance of low-energy FIB is discussed. (author) 10. Heavy ion beam probing International Nuclear Information System (INIS) Hickok, R.L. 1980-07-01 This report consists of the notes distributed to the participants at the IEEE Mini-Course on Modern Plasma Diagnostics that was held in Madison, Wisconsin in May 1980. It presents an overview of Heavy Ion Beam Probing that briefly describes the principles and discuss the types of measurements that can be made. The problems associated with implementing beam probes are noted, possible variations are described, estimated costs of present day systems, and the scaling requirements for large plasma devices are presented. The final chapter illustrates typical results that have been obtained on a variety of plasma devices. No detailed calculations are included in the report, but a list of references that will provide more detailed information is included 11. Cornell electron beam ion source International Nuclear Information System (INIS) Kostroun, V.O.; Ghanbari, E.; Beebe, E.N.; Janson, S.W. 1981-01-01 An electron beam ion source (EBIS) for the production of low energy, multiply charged ion beams to be used in atomic physics experiments has been designed and constructed. An external high perveance electron gun is used to launch the electron beam into a conventional solenoid. Novel features of the design include a distributed sputter ion pump to create the ultrahigh vacuum environment in the ionization region of the source and microprocessor control of the axial trap voltage supplies 12. Ion beam assisted film growth CERN Document Server Itoh, T 2012-01-01 This volume provides up to date information on the experimental, theoretical and technological aspects of film growth assisted by ion beams.Ion beam assisted film growth is one of the most effective techniques in aiding the growth of high-quality thin solid films in a controlled way. Moreover, ion beams play a dominant role in the reduction of the growth temperature of thin films of high melting point materials. In this way, ion beams make a considerable and complex contribution to film growth. The volume will be essential reading for scientists, engineers and students working in thi 13. Computer experiments on ion beam cooling and guiding in fair-wind gas cell and extraction RF-funnel system International Nuclear Information System (INIS) 2004-01-01 Here we present results of the further development of two novel ideas in the field of slow RI-beams production. They are a fair-wind gas cell concept for big-size high-pressure buffer gas cells and a new approach to the extraction system. For this purpose, detailed gas dynamic simulations based on the solution of a full system of time-dependent Navier-Stokes equations have been performed for both the fair-wind gas cell of 500 mm length at 1 bar helium buffer gas pressure and the RF-funnel extraction system at low buffer gas pressure. The results of gas dynamic calculations were used for detailed microscopic Monte Carlo ion-beam trajectory simulations under the combined effect of the buffer gas flow and electric fields of the RF-funnels. The obtained results made it apparent that the use of the fair-wind gas cell concept and extraction RF-funnels look very promising for production of high-quality low-energy RI-beams 14. Ion beam generation and focusing International Nuclear Information System (INIS) Miller, P.A.; Mendel, C.W.; Swain, D.W.; Goldstein, S.A. 1975-01-01 Calculations have shown that efficiently generated and focused ion beams could have significant advantages over electron beams in achieving ignition of inertially-confined thermonuclear fuel. Efficient ion beam generation implies use of a good ion source and suppression of net electron current. Net electron flow can be reduced by allowing electrons to reflex through a highly transparent anode or by use of transverse magnetic fields (either beam self-fields or externally applied fields). Geometric focusing can be achieved if the beam is generated by appropriately shaped electrodes. Experimental results are presented which demonstrate ion beam generation in both reflexing and pinched-flow diodes. Spherically shaped electrodes are used to concentrate a proton beam, and target response to proton deposition is studied 15. Cooling of molecular ion beams International Nuclear Information System (INIS) Wolf, A.; Krohn, S.; Kreckel, H.; Lammich, L.; Lange, M.; Strasser, D.; Grieser, M.; Schwalm, D.; Zajfman, D. 2004-01-01 An overview of the use of stored ion beams and phase space cooling (electron cooling) is given for the field of molecular physics. Emphasis is given to interactions between molecular ions and electrons studied in the electron cooler: dissociative recombination and, for internally excited molecular ions, electron-induced ro-vibrational cooling. Diagnostic methods for the transverse ion beam properties and for the internal excitation of the molecular ions are discussed, and results for phase space cooling and internal (vibrational) cooling are presented for hydrogen molecular ions 16. Electrohydrodynamic emitters of ion beams International Nuclear Information System (INIS) Dudnikov, V.G.; Shabalin, A.L. 1990-01-01 Physical processes determining generation of ion beams with high emission current density in electrohydrodynamic emitters are considered. Electrohydrodynamic effects developing in ion emission features and kinetics of ion interaction in beams with high density are discussed. Factors determining the size of the emission zone, emission stability at high and low currents, cluster generation, increase of energy spread and decrease of brightness are analyzed. Problems on practical provision of stable EHD emitter functioning are considered. 94 refs.; 8 figs.; 1 tab 17. Intense beams of light ions International Nuclear Information System (INIS) Camarcat, Noel 1985-01-01 Results of experiments performed in order to accelerate intense beams of light and heavier ions are presented. The accelerating diodes are driven by existing pulsed power generators. Optimization of the generator structure is described in chapter I. Nuclear diagnostics of the accelerated light ion beams are presented in chapter II. Chapter III deals with the physics of intense charged particle beams. The models developed are applied to the calculation of the performances of the ion diodes described in the previous chapters. Chapter IV reports preliminary results on a multiply ionized carbon source driven by a 0.1 TW pulsed power generator. (author) [fr 18. Intense electron and ion beams CERN Document Server Molokovsky, Sergey Ivanovich 2005-01-01 Intense Ion and Electron Beams treats intense charged-particle beams used in vacuum tubes, particle beam technology and experimental installations such as free electron lasers and accelerators. It addresses, among other things, the physics and basic theory of intense charged-particle beams; computation and design of charged-particle guns and focusing systems; multiple-beam charged-particle systems; and experimental methods for investigating intense particle beams. The coverage is carefully balanced between the physics of intense charged-particle beams and the design of optical systems for their formation and focusing. It can be recommended to all scientists studying or applying vacuum electronics and charged-particle beam technology, including students, engineers and researchers. 19. Ion beam stabilization in ion implantation equipment International Nuclear Information System (INIS) Pina, L. 1973-01-01 The results are presented of experimental efforts aimed at ion beam current stabilization in an equipment for ion implantation in solids. The related problems of power supplies are discussed. Measured characteristics of laboratory equipment served the determination of the parameters to be required of the supplies as well as the design and the construction of the supplies. The respective wiring diagram is presented. (J.K.) 20. Control of colliding ion beams International Nuclear Information System (INIS) Salisbury, W.W. 1985-01-01 This invention relates to a method and system for enhancing the power-producing capability of a nuclear fusion reactor, and more specifically to methods and structure for enhancing the ion density in a directed particle fusion reactor. In accordance with the invention, oppositely directed ion beams constrained to helical paths pass through an annular reaction zone. The object is to produce fusion reactions due to collisions between the ion beams. The reaction zone is an annulus as between an inner-cylindrical electrode and an outer-cylindrical coaxial electrode. The beams are enhanced in ion density at spaced points along the paths by providing spline structures protruding from the walls of the electrodes into the reaction zone. This structure causes variations in the electric field along the paths followed by the ion beams. Such fields cause the beams to be successively more and less concentrated as the beams traverse the reaction zone. Points of high concentration are the points at which fusion-producing collisions are most likely to take place 1. Ion beams in materials processing and analysis CERN Document Server Schmidt, Bernd 2012-01-01 This book covers ion beam application in modern materials research, offering the basics of ion beam physics and technology and a detailed account of the physics of ion-solid interactions for ion implantation, ion beam synthesis, sputtering and nano-patterning. 2. Ion guiding and losses in insulator capillaries International Nuclear Information System (INIS) Juhasz, Z.; Sulik, B.; Vikor, Gy.; Biri, S.; Fekete, E.; Ivan, I.; Gall, F.; Toekesi, K.; Matefi-Tempfli, S.; Matefi-Tempfli, M. 2007-01-01 Complete text of publication follows. Not long ago it was discovered that insulating capillaries can guide slow ions, so that the ions avoid close contact with the capillary walls and preserve their initial charge state. This phenomenon did not only give a new puzzle for theoreticians but opened the way for new possible applications where ions are manipulated (deflected, focused and directed to different patterns on the irradiated media) with small capillary devices. The most important question for such applications is how large fraction of the ions can be guided to the desired direction. It is already known that the ion guiding is due to the charging up of the inner capillary walls by earlier ion impact events. In tilted capillaries one side of the capillary walls charges up. This deflects the later arriving ions, so that some of them pass through the capillaries nearly parallel with respect to their axes. The angle where the transmission drops to 1/e of the direct transmission at 0 deg is the guiding angle, which characterize the guiding ability. At 0 deg the ideal 100 percent transmission for the ions, which enter the capillaries, is reduced due to the mirror charge attraction and geometrical imperfections. These losses appear in the transmission for tilted capillaries with similar magnitude, since after the deflection region, which usually restricted to the close surroundings of the capillary openings, the guided ions pass through the rest of the capillaries as in non-tilted samples. In our experimental studies with Al 2 O 3 capillaries we found that around 90 percent of the incoming ions are lost. To understand these significant losses, the effects of the mirror charge attraction and geometrical imperfections have been calculated classically. The mirror charge potential was taken from.The model of the capillaries used in the calculations can be seen in Figure 1. The calculations have shown that the effects of mirror charge attraction and the angular 3. Maskless, resistless ion beam lithography Energy Technology Data Exchange (ETDEWEB) Ji, Qing [Univ. of California, Berkeley, CA (United States) 2003-01-01 As the dimensions of semiconductor devices are scaled down, in order to achieve higher levels of integration, optical lithography will no longer be sufficient for the needs of the semiconductor industry. Alternative next-generation lithography (NGL) approaches, such as extreme ultra-violet (EUV), X-ray, electron-beam, and ion projection lithography face some challenging issues with complicated mask technology and low throughput. Among the four major alternative NGL approaches, ion beam lithography is the only one that can provide both maskless and resistless patterning. As such, it can potentially make nano-fabrication much simpler. This thesis investigates a focused ion beam system for maskless, resistless patterning that can be made practical for high-volume production. In order to achieve maskless, resistless patterning, the ion source must be able to produce a variety of ion species. The compact FIB system being developed uses a multicusp plasma ion source, which can generate ion beams of various elements, such as O2+, BF2+, P+ etc., for surface modification and doping applications. With optimized source condition, around 85% of BF2+, over 90% of O2+ and P+ have been achieved. The brightness of the multicusp-plasma ion source is a key issue for its application to maskless ion beam lithography. It can be substantially improved by optimizing the source configuration and extractor geometry. Measured brightness of 2 keV He+ beam is as high as 440 A/cm2 • Sr, which represents a 30x improvement over prior work. Direct patterning of Si thin film using a focused O2+ ion beam has been investigated. A thin surface oxide film can be selectively formed using 3 keV O2+ ions with the dose of 1015 cm-2. The oxide can then serve as a hard mask for patterning of the Si film. The 4. Maskless, resistless ion beam lithography International Nuclear Information System (INIS) Ji, Qing 2003-01-01 As the dimensions of semiconductor devices are scaled down, in order to achieve higher levels of integration, optical lithography will no longer be sufficient for the needs of the semiconductor industry. Alternative next-generation lithography (NGL) approaches, such as extreme ultra-violet (EUV), X-ray, electron-beam, and ion projection lithography face some challenging issues with complicated mask technology and low throughput. Among the four major alternative NGL approaches, ion beam lithography is the only one that can provide both maskless and resistless patterning. As such, it can potentially make nano-fabrication much simpler. This thesis investigates a focused ion beam system for maskless, resistless patterning that can be made practical for high-volume production. In order to achieve maskless, resistless patterning, the ion source must be able to produce a variety of ion species. The compact FIB system being developed uses a multicusp plasma ion source, which can generate ion beams of various elements, such as O 2 + , BF 2 + , P + etc., for surface modification and doping applications. With optimized source condition, around 85% of BF 2 + , over 90% of O 2 + and P + have been achieved. The brightness of the multicusp-plasma ion source is a key issue for its application to maskless ion beam lithography. It can be substantially improved by optimizing the source configuration and extractor geometry. Measured brightness of 2 keV He + beam is as high as 440 A/cm 2 · Sr, which represents a 30x improvement over prior work. Direct patterning of Si thin film using a focused O 2 + ion beam has been investigated. A thin surface oxide film can be selectively formed using 3 keV O 2 + ions with the dose of 10 15 cm -2 . The oxide can then serve as a hard mask for patterning of the Si film. The process flow and the experimental results for directly patterned poly-Si features are presented. The formation of shallow pn-junctions in bulk silicon wafers by scanning focused P 5. Ion beam analysis fundamentals and applications CERN Document Server Nastasi, Michael; Wang, Yongqiang 2015-01-01 Ion Beam Analysis: Fundamentals and Applications explains the basic characteristics of ion beams as applied to the analysis of materials, as well as ion beam analysis (IBA) of art/archaeological objects. It focuses on the fundamentals and applications of ion beam methods of materials characterization.The book explains how ions interact with solids and describes what information can be gained. It starts by covering the fundamentals of ion beam analysis, including kinematics, ion stopping, Rutherford backscattering, channeling, elastic recoil detection, particle induced x-ray emission, and nucle 6. Intense-proton-beam transport through an insulator beam guide International Nuclear Information System (INIS) Hanamori, Susumu; Kawata, Shigeo; Kikuchi, Takashi; Fujita, Akira; Chiba, Yasunobu; Hikita, Taisuke; Kato, Shigeru 1998-01-01 In this paper we study intense-proton-beam transport through an insulator guide. In our previous papers (Jpn. J. Appl. Phys. 34 (1995) L520, Jpn. J. Appl. Phys. 35 (1996) L1127) we proposed a new system for intense-electron-beam transport using an insulator guide. In contrast to the electron beam, an intense-proton beam tends to generate a virtual anode, because of the large proton mass. The virtual anode formation at the initial stage is prevented by prefilled plasma in this system. During and after this, electrons are extracted from the plasma generated at the insulator surface by the proton beam space charge and expand over the transport area. The proton beam charge is effectively neutralized by the electrons. Consequently, the proton beam propagates efficiently through the insulator beam guide. The electron extraction is self-regulated by the net space charge of the proton beam. (author) 7. Nanostructuring by ion beam International Nuclear Information System (INIS) Valbusa, U.; Boragno, C.; Buatier de Mongeot, F. 2003-01-01 In metals, the surface curvature dependence of the sputtering yield and the presence of an extra energy barrier whenever diffusing adatoms try to descend step edges, produce a similar surface instability, which builds up regular patterns. By tuning the competition between these two mechanisms, it is possible to create self-organized structures of the size of few nanometers. Height, lateral distance and order of the structures change with the deposition parameters like ion energy, dose, incident angle and substrate temperature. The paper offers an overview of the experiments carried out and foresees possible applications of these results in the area of material science 8. Materials Science with Ion Beams CERN Document Server Bernas, Harry 2010-01-01 This book introduces materials scientists and designers, physicists and chemists to the properties of materials that can be modified by ion irradiation or implantation. These techniques can help design new materials or to test modified properties; novel applications already show that ion-beam techniques are complementary to others, yielding previously unattainable properties. Also, ion-beam interactions modify materials at the nanoscale, avoiding the often detrimental results of lithographic or chemical techniques. Here, the effects are related to better-known quasi-equilibrium thermodynamics, and the consequences to materials are discussed with concepts that are familiar to materials science. Examples addressed concern semiconductor physics, crystal and nanocluster growth, optics, magnetism, and applications to geology and biology. 9. Focused ion beam technology and ultimate applications International Nuclear Information System (INIS) Gierak, Jacques 2009-01-01 In this topical review, the potential of the focused ion beam (FIB) technology and ultimate applications are reviewed. After an introduction to the technology and to the operating principles of liquid metal ion sources (LMIS), of ion optics and instrument architectures, several applications are described and discussed. First, the application of FIB for microcircuit inspection, metrology and failure analysis is presented. Then, we introduce and illustrate some advanced patterning schemes we propose as next generation FIB processing examples. These patterning schemes are (i) local defect injection or smoothing in magnetic thin film direct patterning, (ii) functionalization of graphite substrates to guide organization of clusters, (iii) local and selective epitaxy of III–V semiconductor quantum dots and (iv) FIB patterned solid-state nanopores for biological molecules manipulation and analysis. We conclude this work by giving our vision of the future developments for FIB technology. (topical review) 10. Ion-beam Plasma Neutralization Interaction Images International Nuclear Information System (INIS) Igor D. Kaganovich; Edward Startsev; S. Klasky; Ronald C. Davidson 2002-04-01 Neutralization of the ion beam charge and current is an important scientific issue for many practical applications. The process of ion beam charge and current neutralization is complex because the excitation of nonlinear plasma waves may occur. Computer simulation images of plasma neutralization of the ion beam pulse are presented 11. Ion-beam Plasma Neutralization Interaction Images Energy Technology Data Exchange (ETDEWEB) Igor D. Kaganovich; Edward Startsev; S. Klasky; Ronald C. Davidson 2002-04-09 Neutralization of the ion beam charge and current is an important scientific issue for many practical applications. The process of ion beam charge and current neutralization is complex because the excitation of nonlinear plasma waves may occur. Computer simulation images of plasma neutralization of the ion beam pulse are presented. 12. Ion beam modification of polymers International Nuclear Information System (INIS) Sofield, C.J.; Sugden, S.; Ing, J.; Bridwell, L.B.; Wang, Y.Q. 1993-01-01 The implantation of polymers has received considerable attention in recent years, primarily to examine doping of conducting polymers and to increase the surface conductivity (by many orders of magnitude) of highly insulating polymers. The interest in these studies was partly motivated by possible applications to microelectronic device fabrication. More recently it has been observed that ion implantation can under some conditions lead to the formation of a hard (e.g. as hard as steel, ca. 3 MPa) and conducting surface layer. This paper will review the ion beam modification of polymers resulting from ion implantation with reference to fundamental ion-solid interactions. This leads us to examine whether or not implantation of polymers is a contradiction in terms. (Author) 13. A specialized bioengineering ion beam line International Nuclear Information System (INIS) Yu, L.D.; Sangyuenyongpipat, S.; Sriprom, C.; Thongleurm, C.; Suwanksum, R.; Tondee, N.; Prakrajang, K.; Vilaithong, T.; Brown, I.G.; Wiedemann, H. 2007-01-01 A specialized bioengineering ion beam line has recently been completed at Chiang Mai University to meet rapidly growing needs of research and application development in low-energy ion beam biotechnology. This beam line possesses special features: vertical main beam line, low-energy (30 keV) ion beams, double swerve of the beam, a fast pumped target chamber, and an in-situ atomic force microscope (AFM) system chamber. The whole beam line is situated in a bioclean environment, occupying two stories. The quality of the ion beam has been studied. It has proved that this beam line has significantly contributed to our research work on low-energy ion beam biotechnology 14. Transport of intense ion beams International Nuclear Information System (INIS) Lambertson, G.; Laslett, L.J.; Smith, L. 1977-01-01 The possibility of using intense bursts of heavy ions to initiate an inertially confined fusion reaction has stimulated interest in the transport of intense unneutralized heavy ion beams by quadrupole or solenoid systems. This problem was examined in some detail, using numerical integration of the coupled envelope equations for the quadrupole case. The general relations which emerge are used to develop examples of high energy transport systems and as a basis for discussing the limitations imposed by a transport system on achievable intensities for initial acceleration 15. Materials research with ion beams International Nuclear Information System (INIS) Meyer, J.D. 1988-01-01 This report gives a series of helpful programs which are used in materials research with ion beams. In this context algorithms which can substitute table books are dealt with. This is true for the programs DEDX and PRAL; they are used in order to determine the energy loss of ions in solid bodies, their working range and straggling. Furthermore, simulator routines and analyzers are described. The program TRIM simulates the physical phenomena which occur with the penetration of high-energy ions into solid bodies. In this context electronic excitations, phonons and lattice distortions which are caused by the ions are dealt with. For the experimental ion implantation it is interesting to know the final distribution of the simulated ions in the solid body. The program RBS simulates the Rutherford spectrum of ions which are scattered from a solid body which may consist of up to nine elements and up to one hundred layers. The unknown composition of a solid body can be determined in direct comparison with the experimental spectrum. The program NRA determines concentration and penetrative distribution of an impurity by means of the experimental nuclear reaction spectrum of this impurity. All programs are written in FORTRAN 77. (orig./MM) [de 16. Ion beam sputter implantation method International Nuclear Information System (INIS) King, W.J. 1978-01-01 By means of ion beam atomizing or sputtering an integrally composed coating, the composition of which continuously changes from 100% of the substrate to 100% of the coating, can be surfaced on a substrate (e.g. molten quartz on plastic lenses). In order to do this in the facility there is directed a primary beam of accelerated noble gas ions on a target from the group of the following materials: SiO 2 , Al 2 O 3 , Corning Glass 7070, Corning Glass 7740 or borosilicate glass. The particles leaving the target are directed on the substrate by means of an acceleration potential of up to 10 KV. There may, however, be coated also metal layers (Ni, Co) on a mylar film resulting in a semireflecting metal film. (RW) [de 17. Development of focused ion beam systems with various ion species International Nuclear Information System (INIS) Ji Qing; Leung, K.-N.; King, Tsu-Jae; Jiang Ximan; Appleton, Bill R. 2005-01-01 Conventional focused ion beam systems employ a liquid-metal ion source (LMIS) to generate high-brightness beams, such as Ga + beams. Recently there has been an increased need for focused ion beams in areas like biological studies, advanced magnetic-film manufacturing and secondary-ion mass spectroscopy (SIMS). In this article, status of development on focused ion beam systems with ion species such as O 2 + , P + , and B + will be reviewed. Compact columns for forming focused ion beams from low energy (∼3keV), to intermediate energy (∼35keV) are discussed. By using focused ion beams, a SOI MOSFET is fabricated entirely without any masks or resist 18. A fast beam-ion instability Energy Technology Data Exchange (ETDEWEB) Stupakov, G V [Stanford Linear Accelerator Center, Menlo Park, CA (United States) 1996-08-01 The ionization of residual gas by an electron beam in an accelerator generates ions that can resonantly couple to the beam through a wave propagating in the beam-ion system. Results of the study of a beam-ion instability are presented for a multi-bunch train taking into account the decoherence of ion oscillations due to the ion frequency spread and spatial variation of the ion frequency. It is shown that the combination of both effects can substantially reduce the growth rate of the instability. (author) 19. Large area ion and plasma beam sources Energy Technology Data Exchange (ETDEWEB) Waldorf, J. [IPT Ionen- und Plasmatech. GmbH, Kaiserslautern (Germany) 1996-06-01 In the past a number of ion beam sources utilizing different methods for plasma excitation have been developed. Nevertheless, a widespread use in industrial applications has not happened, since the sources were often not able to fulfill specific demands like: broad homogeneous ion beams, compatibility with reactive gases, low ion energies at high ion current densities or electrical neutrality of the beam. Our contribution wants to demonstrate technical capabilities of rf ion and plasma beam sources, which can overcome the above mentioned disadvantages. The physical principles and features of respective sources are presented. We report on effective low pressure plasma excitation by electron cyclotron wave resonance (ECWR) for the generation of dense homogeneous plasmas and the rf plasma beam extraction method for the generation of broad low energy plasma beams. Some applications like direct plasma beam deposition of a-C:H and ion beam assisted deposition of Al and Cu with tailored thin film properties are discussed. (orig.). 20. Large area ion and plasma beam sources International Nuclear Information System (INIS) Waldorf, J. 1996-01-01 In the past a number of ion beam sources utilizing different methods for plasma excitation have been developed. Nevertheless, a widespread use in industrial applications has not happened, since the sources were often not able to fulfill specific demands like: broad homogeneous ion beams, compatibility with reactive gases, low ion energies at high ion current densities or electrical neutrality of the beam. Our contribution wants to demonstrate technical capabilities of rf ion and plasma beam sources, which can overcome the above mentioned disadvantages. The physical principles and features of respective sources are presented. We report on effective low pressure plasma excitation by electron cyclotron wave resonance (ECWR) for the generation of dense homogeneous plasmas and the rf plasma beam extraction method for the generation of broad low energy plasma beams. Some applications like direct plasma beam deposition of a-C:H and ion beam assisted deposition of Al and Cu with tailored thin film properties are discussed. (orig.) 1. The most reactive third-row transition metal: Guided ion beam and theoretical studies of the activation of methane by Ir+ Science.gov (United States) Li, Feng-Xia; Zhang, Xiao-Guang; Armentrout, P. B. 2006-09-01 The potential energy surface for activation of methane by the third-row transition metal cation, Ir+, is studied experimentally by examining the kinetic energy dependence of reactions of Ir+ with methane, IrCH2+ with H2 and D2, and collision-induced dissociation of IrCH2+ with Xe using guided ion beam tandem mass spectrometry. A flow tube ion source produces Ir+ in its electronic ground state term and primarily in the ground spin-orbit level. We find that dehydrogenation to form IrCH2+ + H2 is exothermic, efficient, and the only process observed at low energies for reaction of Ir+ with methane, whereas IrH+ dominates the product spectrum at higher energies. We also observe the IrH2+ product, which provides evidence that methane activation proceeds via a dihydride (H)2IrCH2+ intermediate. The kinetic energy dependences of the cross sections for several endothermic reactions are analyzed to give 0 K bond dissociation energies (in eV) of D0(Ir+-2H) > 5.09 +/- 0.07, D0(Ir+-C) = 6.59 +/- 0.05, D0(Ir+-CH) = 6.91 +/- 0.23, and D0(Ir+-CH3) = 3.25 +/- 0.18. D0(Ir+-CH2) = 4.92 +/- 0.03 eV is determined by measuring the forward and reverse reaction rates for Ir++CH4[right harpoon over left]IrCH2++H2 at thermal energy. Ab initio calculations at the B3LYP/HW+/6-311++G(3df,3p) level performed here show reasonable agreement with the experimental bond energies and with the few previous experimental and theoretical values available. Theory also provides the electronic structures of the product species as well as intermediates and transition states along the reactive potential energy surfaces. We also compare this third-row transition metal system with the first-row and second-row congeners, Co+ and Rh+. Differences in reactivity and mechanisms can be explained by the lanthanide contraction and relativistic effects that alter the relative size of the valence s and d orbitals. 2. Neurosurgical applications of ion beams Science.gov (United States) Fabrikant, Jacob I.; Levy, Richard P.; Phillips, Mark H.; Frankel, Kenneth A.; Lyman, John T. 1989-04-01 The program at Donner Pavilion has applied nuclear medicine research to the diagnosis and radiosurgical treatment of life-threatening intracranial vascular disorders that affect more than half a million Americans. Stereotactic heavy-charged-particle Bragg peak radiosurgery, using narrow beams of heavy ions, demonstrates superior biological and physical characteristics in brain over X-and γ-rays, viz., improved dose distribution in the Bragg peak and sharp lateral and distal borders and less scattering of the beam. Examination of CNS tissue response and alteration of cerebral blood-flow dynamics related to heavy-ion Bragg peak radiosurgery is carried out using three-dimensional treatment planning and quantitative imaging utilizing cerebral angiography, computerized tomography (CT), magnetic resonance imaging (MRI), cine-CT, xenon X-ray CT and positron emission tomography (PET). Also under examination are the physical properties of narrow heavy-ion beams for improving methods of dose delivery and dose distribution and for establishing clinical RBE/LET and dose-response relationships for human CNS tissues. Based on the evaluation and treatment with stereotactically directed narrow beams of heavy charged particles of over 300 patients, with cerebral angiography, CT scanning and MRI and PET scanning of selected patients, plus extensive clinical and neuroradiological followup, it appears that Stereotactic charged-particle Bragg peak radiosurgery obliterates intracranial arteriovenous malformations or protects against rebleeding with reduced morbidity and no mortality. Discussion will include the method of evaluation, the clinical research protocol, the Stereotactic neuroradiological preparation, treatment planning, the radiosurgery procedure and the protocol for followup. Emphasis will be placed on the neurological results, including the neuroradiological and clinical response and early and late delayed injury in brain leading to complications (including vasogenic edema 3. Revised data taking schedule with ion beams CERN Document Server Gazdzicki, Marek; Aduszkiewicz, A; Andrieu, B; Anticic, T; Antoniou, N; Argyriades, J; Asryan, A G; Baatar, B; Blondel, A; Blumer, J; Boldizsar, L; Bravar, A; Brzychczyk, J; Bubak, A; Bunyatov, S A; Choi, K U; Christakoglou, P; Chung, P; Cleymans, J; Derkach, D A; Diakonos, F; Dominik, W; Dumarchez, J; Engel, R; Ereditato, A; Feofilov, G A; Fodor, Z; Ferrero, A; Gazdzicki, M; Golubeva, M; Grebieszkow, K; Grzeszczuk, A; Guber, F; Hasegawa, T; Haungs, A; Igolkin, S; Ivanov, A S; Ivashkin, A; Kadija, K; Katrynska, N; Kielczewska, D; Kikola, D; Kisiel, J; Kobayashi, T; Kolesnikov, V I; Kolev, D; Kolevatov, R S; Kondratiev, V P; Kowalski, S; Kurepin, A; Lacey, R; Laszlo, A; Lyubushkin, V V; Majka, Z; I Malakhov, A; Marchionni, A; Marcinek, A; Maris, I; Matveev, V; Melkumov, G L; Meregaglia, A; Messina, M; Mijakowski, P; Mitrovski, M; Montaruli, T; Mrówczynski, St; Murphy, S; Nakadaira, T; Naumenko, P A; Nikolic, V; Nishikawa, K; Palczewski, T; Pálla, G; Panagiotou, A D; Peryt, W; Planeta, R; Pluta, J; Popov, B A; Posiadala, M; Przewlocki, P; Rauch, W; Ravonel, M; Renfordt, R; Röhrich, D; Rondio, E; Rossi, B; Roth, M; Rubbia, A; Rybczynski, M; Sadovskii, A; Sakashita, K; Schuster, T; Sekiguchi, T; Seyboth, P; Shibata, M; Sissakian, A N; Skrzypczak, E; Slodkowski, M; Sorin, A S; Staszel, P; Stefanek, G; Stepaniak, J; Strabel, C; Ströbele, H; Susa, T; Szentpétery, I; Szuba, M; Tada, M; Taranenko, A; Tsenov, R; Ulrich, R; Unger, M; Vassiliou, M; Vechernin, V V; Vesztergombi, G; Wlodarczyk, Z; Wojtaszek, A; Zipper, W; CERN. Geneva. SPS and PS Experiments Committee; SPSC 2009-01-01 This document presents the revised data taking schedule of NA61 with ion beams. The revision takes into account limitations due to the new LHC schedule as well as final results concerning the physics performance with secondary ion beams. It is proposed to take data with primary Ar and Xe beams in 2012 and 2014, respectively, and to test and use for physics a secondary B beam from primary Pb beam fragmentation in 2010, 2011 and 2013. 4. Ion beam processes in Si International Nuclear Information System (INIS) Holland, O.W.; Narayan, J.; Fathy, D. 1984-07-01 Observation of the effects of implants of energetic ions at high dose rates into Si have produced some exciting and interesting results. The mechanism whereby displacement damage produced by ions self-anneals during high dose rate implantation is discussed. It is shown that ion beam annealing (IBA) offers in certain situations unique possibilities for damage annealing. Annealing results of the near surface in Si with a buried oxide layer, formed by high dose implantation, are presented in order to illustrate the advantages offered by IBA. It is also shown that ion irradiation can stimulate the epitaxial recrystallization of amorphous overlayers in Si. The nonequilibrium alloying which results from such epitaxial processes is discussed as well as mechanisms which limit the solid solubility during irradiation. Finally, a dose rate dependency for the production of stable damage by ion irradiation at a constant fluence has been observed. For low fluence implants, the amount of damage is substantially greater in the case of high flux rather than low flux implantation 5. Ion density in ionizing beams International Nuclear Information System (INIS) Knuyt, G.K.; Callebaut, D.K. 1978-01-01 The equations defining the ion density in a non-quasineutral plasma (chasma) are derived for a number of particular cases from the general results obtained in paper 1. Explicit calculations are made for a fairly general class of boundaries: all tri-axial ellipsoids, including cylinders with elliptic cross-section and the plane parallel case. The results are very simple. When the ion production and the beam intensity are constant then the steady state ion space charge is also constant in space, it varies over less than 10% for the various geometries, it may exceed the beam density largely for comparatively high pressures (usually still less than about 10 -3 Torr), it is tabulated for a number of interesting cases and moreover it can be calculated precisely and easily by some simple formulae for which also approximations are elaborated. The total potential is U =-ax 2 -by 2 -cz 2 , a, b and c constants which can be calculated immediately from the space charge density and the geometry; the largest coefficient varies at most over a factor four for various geometries; it is tabulated for a number of interesting cases. (author) 6. ECR ion source based low energy ion beam facility Mass analyzed highly charged ion beams of energy ranging from a few keV to a few MeV plays an important role in various aspects of research in modern physics. In this paper a unique low energy ion beam facility (LEIBF) set up at Nuclear Science Centre (NSC) for providing low and medium energy multiply charged ion ... 7. Pseudo ribbon metal ion beam source International Nuclear Information System (INIS) Stepanov, Igor B.; Ryabchikov, Alexander I.; Sivin, Denis O.; Verigin, Dan A. 2014-01-01 The paper describes high broad metal ion source based on dc macroparticle filtered vacuum arc plasma generation with the dc ion-beam extraction. The possibility of formation of pseudo ribbon beam of metal ions with the parameters: ion beam length 0.6 m, ion current up to 0.2 A, accelerating voltage 40 kV, and ion energy up to 160 kV has been demonstrated. The pseudo ribbon ion beam is formed from dc vacuum arc plasma. The results of investigation of the vacuum arc evaporator ion-emission properties are presented. The influence of magnetic field strength near the cathode surface on the arc spot movement and ion-emission properties of vacuum-arc discharge for different cathode materials are determined. It was shown that vacuum-arc discharge stability can be reached when the magnetic field strength ranges from 40 to 70 G on the cathode surface 8. Pseudo ribbon metal ion beam source. Science.gov (United States) Stepanov, Igor B; Ryabchikov, Alexander I; Sivin, Denis O; Verigin, Dan A 2014-02-01 The paper describes high broad metal ion source based on dc macroparticle filtered vacuum arc plasma generation with the dc ion-beam extraction. The possibility of formation of pseudo ribbon beam of metal ions with the parameters: ion beam length 0.6 m, ion current up to 0.2 A, accelerating voltage 40 kV, and ion energy up to 160 kV has been demonstrated. The pseudo ribbon ion beam is formed from dc vacuum arc plasma. The results of investigation of the vacuum arc evaporator ion-emission properties are presented. The influence of magnetic field strength near the cathode surface on the arc spot movement and ion-emission properties of vacuum-arc discharge for different cathode materials are determined. It was shown that vacuum-arc discharge stability can be reached when the magnetic field strength ranges from 40 to 70 G on the cathode surface. 9. Applications of capillary optics for focused ion beams International Nuclear Information System (INIS) Umezawa, Kenji 2014-01-01 This article introduces applications of focused ion beams (∼1 μm) with glass capillaries systems. A first report on the interaction between ion beams and glass capillaries was published in 1996. The guiding capabilities of glass capillaries were discovered due to ion reflection from inner wall of glass surfaces. Meanwhile, the similar optics have been already realized in focusing X-rays using glass capillaries. The basic technology of X-rays optics using glass capillaries had been developed in the 1980's and 1900's. Also, low energy atom scattering spectroscopy for insulator material analysis will be mentioned. (author) 10. Fusion at counterstreaming ion beams - ion optic fusion (IOF) International Nuclear Information System (INIS) Gryzinski, M. 1981-01-01 The results of investigation are briefly reviewed in the field of ion optic fusion performed at the Institute of Nuclear Research in Swierk. The ion optic fusion concept is based on the possibility of obtaining fusion energy at highly ordered motion of ions in counterstreaming ion beams. For this purpose TW ion beams must be produced and focused. To produce dense and charge-neutralized ion beams the selective conductivity and ballistic focusing ideas were formulated and used in a series of RPI devices with low-pressure cylindrical discharge between grid-type electrodes. 100 kA, 30 keV deuteron beams were successfully produced and focused into the volume of 1 cm 3 , yielding 10 9 neutrons per 200 ns shot on a heavy ice target. Cylindrically convergent ion beams with magnetic anti-defocusing were proposed in order to reach a positive energy gain at reasonable energy level. (J.U.) 11. Ion-beam plasma and propagation of intense compensated ion beams International Nuclear Information System (INIS) Gabovich, M.D. 1977-01-01 Discussed are the results of investigation of plasma properties recieved by neutralization of intensive ion beam space charge. Considered is the process of ion beam compensation by charges, formed as a result of gas ionization by this beam or by externally introduced ones. Emphasis is placed on collective phenomena in ion-beam plasma, in particular on non-linear effects limiting amplitude of oscillations. It is shown, that not only dinamic decompensation but the Coulomb collisions of ions with electrons as well as other collective oscillations significantly affects the propagation of compensated ion beams. All the processes are to be taken into account at solving the problem of obtaining ''superdense'' compensated beams 12. Ion-beam plasma and propagation of intense compensated ion beams Energy Technology Data Exchange (ETDEWEB) Gabovich, M D [AN Ukrainskoj SSR, Kiev. Inst. Fiziki 1977-02-01 Discussed are the results of investigation of plasma properties received by neutralization of intense ion beam space charge. Considered is the process of ion beam compensation by charges, formed as a result of gas ionization by this beam or by externally introduced ones. Emphasis is placed on collective phenomena in ion-beam plasma, in particular on non-linear effects limiting amplitude of oscillations. It is shown that not only dynamic decompensation but the Coulomb collisions of ions with electrons as well as other collective oscillations significantly affects the propagation of compensated ion beams. All the processes are to be taken into account in solving the problem of obtaining ''superdense'' compensated beams. 13. Conical pinched electron beam diode for intense ion beam source International Nuclear Information System (INIS) Matsukawa, Yoshinobu; Nakagawa, Yoshiro 1982-01-01 For the purpose of improvement of the pinched electron beam diode, the production of an ion beam by a diode with electrodes in a conical shape was studied at low voltage operation (--200 kV). The ion beam is emitted from a small region of the diode apex. The mean ion beam current density near the axis at 12 cm from the diode apex is two or three times that from an usual flat parallel diode with the same dimension and impedance. The brightness and the power brightness at the otigin are 450 MA/cm 2 sr and 0.12 TW/cm 2 sr respectively. (author) 14. Tool steel ion beam assisted nitrocarburization International Nuclear Information System (INIS) Zagonel, L.F.; Alvarez, F. 2007-01-01 The nitrocarburization of the AISI-H13 tool steel by ion beam assisted deposition is reported. In this technique, a carbon film is continuously deposited over the sample by the ion beam sputtering of a carbon target while a second ion source is used to bombard the sample with low energy nitrogen ions. The results show that the presence of carbon has an important impact on the crystalline and microstructural properties of the material without modification of the case depth 15. Beam-plasma instability in ion beam systems used in neutral beam generation International Nuclear Information System (INIS) Hooper, E.B. Jr. 1977-02-01 The beam-plasma instability is analyzed for the ion beams used for neutral beam generation. Both positive and negative ion beams are considered. Stability is predicted when the beam velocity is less than the electron thermal velocity; the only exception occurs when the electron density accompanying a negative ion beam is less than the ion density by nearly the ratio of electron to ion masses. For cases in which the beam velocity is greater than the electron thermal velocity, instability is predicted near the electron plasma frequency 16. ORNL positive ion neutral beam program International Nuclear Information System (INIS) Whealton, J.H.; Haselton, H.H.; Barber, G.C. 1978-01-01 The neutral beam group at Oak Ridge National Laboratory has constructed neutral beam generators for the ORMAK and PLT devices, is presently constructing neutral beam devices for the ISX and PDX devices, and is contemplating the construction of neutral beam systems for the advanced TNS device. These neutral beam devices stem from the pioneering work on ion sources of G. G. Kelley and O. B. Morgan. We describe the ion sources under development at this Laboratory, the beam optics exhibited by these sources, as well as some theoretical considerations, and finally the remainder of the beamline design 17. Cooled heavy ion beams at the ESR International Nuclear Information System (INIS) Steck, M.; Beckert, K.; Bosch, F.; Eickhoff, H.; Franzke, B.; Klepper, O.; Nolden, F.; Reich, H.; Schlitt, B.; Spaedtke, P.; Winkler, T. 1996-01-01 The storage ring ESR has been used in various operational modes for experiments with electron cooled heavy ion beams. Besides the standard storage mode including injection and beam accumulation the deceleration of highly charged ions has been demonstrated. Beams of highly charged ions have been injected and accumulated and finally decelerated to a minimum energy of 50 MeV/u. An ultraslow extraction method using charge changing processes is now also available for cooled beams of highly charged ions. For in ring experiments the internal gas jet and the cold electron beam of the cooling system are applied as targets. High precision mass spectrometry by Schottky noise detection has been demonstrated. Operation at transition energy has been achieved with cooled beams opening the field for experiments which require an isochronous revolution of the ions. (orig.) 18. Microwave bessel beams generation using guided modes KAUST Repository Salem, Mohamed; Kamel, Aladin Hassan; Niver, Edip 2011-01-01 A novel method is devised for Bessel beams generation in the microwave regime. The beam is decomposed in terms of a number of guided transverse electric modes of a metallic waveguide. Modal expansion coefficients are computed from the modal power orthogonality relation. Excitation is achieved by means of a number of inserted coaxial loop antennas, whose currents are calculated from the excitation coefficients of the guided modes. The efficiency of the method is evaluated and its feasibility is discussed. Obtained results can be utilized to practically realize microwave Bessel beam launchers. © 2006 IEEE. 19. Microwave bessel beams generation using guided modes KAUST Repository Salem, Mohamed 2011-06-01 A novel method is devised for Bessel beams generation in the microwave regime. The beam is decomposed in terms of a number of guided transverse electric modes of a metallic waveguide. Modal expansion coefficients are computed from the modal power orthogonality relation. Excitation is achieved by means of a number of inserted coaxial loop antennas, whose currents are calculated from the excitation coefficients of the guided modes. The efficiency of the method is evaluated and its feasibility is discussed. Obtained results can be utilized to practically realize microwave Bessel beam launchers. © 2006 IEEE. 20. Ion-Beam-Excited Electrostatic Ion Cyclotron Waves DEFF Research Database (Denmark) Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens 1976-01-01 Self-excited electrostatic ion cyclotron waves were observed in an ion-beam-plasma system produced in a DP-operated Q-machine. The frequency of the waves showed the theoretically predicted variation with the magnetic field.......Self-excited electrostatic ion cyclotron waves were observed in an ion-beam-plasma system produced in a DP-operated Q-machine. The frequency of the waves showed the theoretically predicted variation with the magnetic field.... CERN Document Server Kellerbauer, A G; Dilling, J; Henry, S; Herfurth, F; Kluge, H J; Lamour, E; Moore, R B; Scheidenberger, C; Schwarz, S; Sikler, G; Szerypo, J 2002-01-01 A linear radiofrequency quadrupole ion guide and beam buncher has been installed at the ISOLTRAP mass spectrometry experiment at the ISOLDE facility at CERN. The apparatus is being used as a beam cooling, accumulation, and bunching system. It operates with a buffer gas that cools the injected ions and converts the quasicontinuous 60- keV beam from the ISOLDE facility to 2.5-keV beam pulses with improved normalized transverse emittance. Recent measurements suggest a capture efficiency of the ion guide of up to 40% and a cooling and bunching efficiency of at least 12% which is expected to still be increased. The improved ISOLTRAP setup has so far been used very successfully in three on-line experiments. (12 refs). 2. Radiation effects of ion beams on polymers International Nuclear Information System (INIS) Tagawa, Seiichi 1993-01-01 Recent progress in the radiation effects of ion beams on polymers are reviewed briefly. Our recent work on the radiation effects of ion beams on polystyrene thin films on silicon wafers and time resolved emission studies on polymers are described. (orig.) 3. Ion Beams in Nanoscience and Technology CERN Document Server Hellborg, Ragnar 2010-01-01 Energetic ion beam irradiation is the basis of a wide plethora of powerful research- and fabrication-techniques for materials characterisation and processing on a nanometre scale. This book is suitable for practitioners, researchers and graduate students working in the field of ion beams and application 4. Ion-Ion Plasmas Produced by Electron Beams Science.gov (United States) Fernsler, R. F.; Leonhardt, D.; Walton, S. G.; Meger, R. A. 2001-10-01 The ability of plasmas to etch deep, small-scale features in materials is limited by localized charging of the features. The features charge because of the difference in electron and ion anisotropy, and thus one solution now being explored is to use ion-ion plasmas in place of electron-ion plasmas. Ion-ion plasmas are effectively electron-free and consist mainly of positive and negative ions. Since the two ion species behave similarly, localized charging is largely eliminated. However, the only way to produce ion-ion plasmas at low gas pressure is to convert electrons into negative ions through two-body attachment to neutrals. While the electron attachment rate is large at low electron temperatures (Te < 1 eV) in many of the halogen gases used for processing, these temperatures occur in most reactors only during the afterglow when the heating fields are turned off and the plasma is decaying. By contrast, Te is low nearly all the time in plasmas produced by electron beams, and therefore electron beams can potentially produce ion-ion plasmas continuously. The theory of ion-ion plasmas formed by pulsed electron beams is examined in this talk and compared with experimental results presented elsewhere [1]. Some general limitations of ion-ion plasmas, including relatively low flux levels, are discussed as well. [1] See the presentation by D. Leonhardt et al. at this conference. 5. Beam emittance measurements on multicusp ion sources Energy Technology Data Exchange (ETDEWEB) Sarstedt, M.; Lee, Y.; Leung, K.N. [and others 1995-08-01 Multicusp ion sources are used for various applications. Presently, the implementation of this type of ion source planned for the development of an ion beam lithography machine, which will be used for the projection of sub-0.2 {mu}m patterns onto a wafer substrate. Since, for this application, a very good beam quality and a small ion energy spread are required, emittance measurements have been performed on a multicusp ion source for various source conditions. It is shown that the installation of proper capacitors between the extraction electrodes is necessary to avoid rf-pickup, which otherwise leads to a distortion of the beam emittance. The influence of the magnetic filter field on the beam emittance has been investigated, and the beam emittance of a dc filament-discharge plasma has also been compared to that of an rf-generated plasma. 6. Beam emittance measurements on multicusp ion sources International Nuclear Information System (INIS) Sarstedt, M.; Lee, Y.; Leung, K.N. 1995-08-01 Multicusp ion sources are used for various applications. Presently, the implementation of this type of ion source planned for the development of an ion beam lithography machine, which will be used for the projection of sub-0.2 μm patterns onto a wafer substrate. Since, for this application, a very good beam quality and a small ion energy spread are required, emittance measurements have been performed on a multicusp ion source for various source conditions. It is shown that the installation of proper capacitors between the extraction electrodes is necessary to avoid rf-pickup, which otherwise leads to a distortion of the beam emittance. The influence of the magnetic filter field on the beam emittance has been investigated, and the beam emittance of a dc filament-discharge plasma has also been compared to that of an rf-generated plasma 7. Modeling of ion beam surface treatment Energy Technology Data Exchange (ETDEWEB) Stinnett, R W [Quantum Manufacturing Technologies, Inc., Albuquerque, NM (United States); Maenchen, J E; Renk, T J [Sandia National Laboratories, Albuquerque, NM (United States); Struve, K W [Mission Research Corporation, Albuquerque, NM (United States); Campbell, M M [PASTDCO, Albuquerque, NM (United States) 1997-12-31 The use of intense pulsed ion beams is providing a new capability for surface engineering based on rapid thermal processing of the top few microns of metal, ceramic, and glass surfaces. The Ion Beam Surface Treatment (IBEST) process has been shown to produce enhancements in the hardness, corrosion, wear, and fatigue properties of surfaces by rapid melt and re-solidification. A new code called IBMOD was created, enabling the modeling of intense ion beam deposition and the resulting rapid thermal cycling of surfaces. This code was used to model the effect of treatment of aluminum, iron, and titanium using different ion species and pulse durations. (author). 3 figs., 4 refs. 8. Intense ion beams for inertial confinement fusion International Nuclear Information System (INIS) Mehlhorn, T.A. 1997-01-01 Intense beams of light of heavy ions are being studied as inertial confinement fusion (ICF) drivers for high yield and energy. Heavy and light ions have common interests in beam transport, targets, and alternative accelerators. Self-pinched transport is being jointly studied. This article reviews the development of intense ion beams for ICF. Light-ion drivers are highlighted because they are compact, modular, efficient and low cost. Issues facing light ions are: (1) decreasing beam divergence; (2) increasing beam brightness; and (3) demonstrating self-pinched transport. Applied-B ion diodes are favored because of efficiency, beam brightness, perceived scalability, achievable focal intensity, and multistage capability. A light-ion concept addressing these issues uses: (1) an injector divergence of ≤ 24 mrad at 9 MeV; (2) two-stage acceleration to reduce divergence to ≤ 12 mrad at 35 MeV; and (3) self-pinched transport accepting divergences up to 12 mrad. Substantial progress in ion-driven target physics and repetitive ion diode technology is also presented. Z-pinch drivers are being pursued as the shortest pulsed power path to target physics experiments and high-yield fusion. However, light ions remain the pulsed power ICF driver of choice for high-yield fusion energy applications that require driver standoff and repetitive operation. 100 refs 9. Beam brilliance investigation of high current ion beams at GSI heavy ion accelerator facility. Science.gov (United States) 2014-02-01 In this work the emittance measurements of high current Ta-beam provided by VARIS (Vacuum Arc Ion Source) ion source are presented. Beam brilliance as a function of beam aperture at various extraction conditions is investigated. Influence of electrostatic ion beam compression in post acceleration gap on the beam quality is discussed. Use of different extraction systems (single aperture, 7 holes, and 13 holes) in order to achieve more peaked beam core is considered. The possible ways to increase the beam brilliance are discussed. 10. Mutation induction by ion beams in plants Energy Technology Data Exchange (ETDEWEB) Tanaka, Atsushi [Japan Atomic Energy Research Inst., Takasaki, Gunma (Japan). Takasaki Radiation Chemistry Research Establishment 2001-03-01 The effect of ion beams such as C, He, and Ne ions was investigated on the mutation induction in plants with the expectation that ion beams of high linear energy transfer (LET) can frequently produce large DNA alternation such as inversion, translocation and large deletion rather than point mutation. Mutation frequency was investigated using Arabidopsis visible phenotype loci and was 8 to 33 fold higher for 220 MeV carbon ions than for electrons. Mutation spectrum was investigated on the flower color of chrysanthemum cv to find that flower mutants induced by ion beams show complex and stripe types rather than single color. Polymerase chain reaction analysis was performed to investigate DNA alteration of mutations. In conclusion, the characteristics of ion beams for the mutation induction are 1) high frequency, 2) broad mutation spectrum, and 3) novel mutants. (S. Ohno) 11. Mutation induction by ion beams in plants International Nuclear Information System (INIS) Tanaka, Atsushi 2001-01-01 The effect of ion beams such as C, He, and Ne ions was investigated on the mutation induction in plants with the expectation that ion beams of high linear energy transfer (LET) can frequently produce large DNA alternation such as inversion, translocation and large deletion rather than point mutation. Mutation frequency was investigated using Arabidopsis visible phenotype loci and was 8 to 33 fold higher for 220 MeV carbon ions than for electrons. Mutation spectrum was investigated on the flower color of chrysanthemum cv to find that flower mutants induced by ion beams show complex and stripe types rather than single color. Polymerase chain reaction analysis was performed to investigate DNA alteration of mutations. In conclusion, the characteristics of ion beams for the mutation induction are 1) high frequency, 2) broad mutation spectrum, and 3) novel mutants. (S. Ohno) 12. Constraints on ion beam handling for intersecting beam experiments International Nuclear Information System (INIS) Kruse, T. 1981-01-01 The intense synchrotron radiation beams from the NSLS uv or x-ray storage rings still do not compare in monochromatized photon flux with a laser beam, a fact which becomes apparent in considering reaction rates for interaction of photon and ion beams. There are two prototypical interaction geometries, parallel and perpendicular. Calculations should properly be done in the rest frame of the ion beam; however, expected beta values are small, so the lab frame will be employed and aberration and Doppler shift effects neglected 13. Beam-beam observations in the Relativistic Heavy Ion Collider Energy Technology Data Exchange (ETDEWEB) Luo, Y. [Brookhaven National Laboratory (BNL), Upton, NY (United States); Fischer, W. [Brookhaven National Laboratory (BNL), Upton, NY (United States); White, S. [Brookhaven National Laboratory (BNL), Upton, NY (United States) 2015-06-24 The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory has been operating since 2000. Over the past decade, thanks to the continuously increased bunch intensity and reduced βs at the interaction points, the maximum peak luminosity in the polarized proton operation has been increased by more than two orders of magnitude. In this article, we first present the beam-beam observations in the previous RHIC polarized proton runs. Then we analyze the mechanisms for the beam loss and emittance growth in the presence of beam-beam interaction. The operational challenges and limitations imposed by beam-beam interaction and their remedies are also presented. In the end, we briefly introduce head-on beam-beam compensation with electron lenses in RHIC. 14. Beam losses in heavy ion drivers CERN Document Server Mustafin, E R; Hofmann, I; Spiller, P J 2002-01-01 While beam loss issues have hardly been considered in detail for heavy ion fusion scenarios, recent heavy ion machine developments in different labs (European Organization for Nuclear Research (CERN), Gesellschaft fur Schwerionenforschung (GSI), Institute for Theoretical and Experimental Physics (ITEP), Relativistic Heavy-Ion Collider (RHIC)) have shown the great importance of beam current limitations due to ion losses. Two aspects of beam losses in heavy ion accelerators are theoretically considered: (1) secondary neutron production due to lost ions, and (2) vacuum pressure instability due to charge exchange losses. Calculations are compared and found to be in good agreement with measured data. The application to a Heavy-Ion Driven Inertial Fusion (HIDIF) scenario is discussed. 12 Refs. 15. Pulsed high current ion beam processing equipment International Nuclear Information System (INIS) Korenev, S.A.; Perry, A. 1995-01-01 A pulsed high voltage ion source is considered for use in ion beam processing for the surface modification of materials, and deposition of conducting films on different substrates. The source consists of an Arkad'ev-Marx high voltage generator, a vacuum ion diode based on explosive ion emission, and a vacuum chamber as substrate holder. The ion diode allows conducting films to be deposited from metal or allow sources, with ion beam mixing, onto substrates held at a pre-selected temperature. The main variables can be set in the ranges: voltage 100-700 kV, pulse length 0.3 μs, beam current 1-200 A depending on the ion chosen. The applications of this technology are discussed in semiconductor, superconductor and metallizing applications as well as the direction of future development and cost of these devices for commercial application. 14 refs., 6 figs 16. Cobalt alloy ion sources for focused ion beam implantation Energy Technology Data Exchange (ETDEWEB) Muehle, R.; Doebeli, M. [Paul Scherrer Inst. (PSI), Villigen (Switzerland); Zimmermann, P. [Eidgenoessische Technische Hochschule, Zurich (Switzerland) 1997-09-01 Cobalt alloy ion sources have been developed for silicide formation by focused ion beam implantation. Four eutectic alloys AuCo, CoGe, CoY and AuCoGe were produced by electron beam welding. The AuCo liquid alloy ion source was investigated in detail. We have measured the emission current stability, the current-voltage characteristics, and the mass spectrum as a function of the mission current. (author) 1 fig., 2 refs. 17. High current density ion beam measurement techniques International Nuclear Information System (INIS) Ko, W.C.; Sawatzky, E. 1976-01-01 High ion beam current measurements are difficult due to the presence of the secondary particles and beam neutralization. For long Faraday cages, true current can be obtained only by negative bias on the target and by summing the cage wall and target currents; otherwise, the beam will be greatly distorted. For short Faraday cages, a combination of small magnetic field and the negative target bias results in correct beam current. Either component alone does not give true current 18. Production of ion beam by conical pinched electron beam diode International Nuclear Information System (INIS) Matsukawa, Y.; Nakagawa, Y. 1982-01-01 Some properties of the ion beam produced by pinched electron beam diode having conical shape electrodes and organic insulator anode was studied. Ion energy is about 200keV and the peak diode current is about 30 kA. At 11cm from the diode apex, not the geometrical focus point, concentrated ion beam was obtained. Its density is more than 500A/cm 2 . The mean ion current density within the radius of 1.6cm around the axis from conical diode is two or three times that from an usual pinched electron beam diode with flat parallel electrodes of same dimension and impedance under the same conditions. (author) 19. Intense non-relativistic cesium ion beam International Nuclear Information System (INIS) Lampel, M.C. 1984-02-01 The Heavy Ion Fusion group at Lawrence Berkeley Laboratory has constructed the One Ampere Cesium Injector as a proof of principle source to supply an induction linac with a high charge density and high brightness ion beam. This is studied here. An electron beam probe was developed as the major diagnostic tool for characterizing ion beam space charge. Electron beam probe data inversion is accomplished with the EBEAM code and a parametrically adjusted model radial charge distribution. The longitudinal charge distribution was not derived, although it is possible to do so. The radial charge distribution that is derived reveals an unexpected halo of trapped electrons surrounding the ion beam. A charge fluid theory of the effect of finite electron temperature on the focusing of neutralized ion beams (Nucl. Fus. 21, 529 (1981)) is applied to the problem of the Cesium beam final focus at the end of the injector. It is shown that the theory's predictions and assumptions are consistent with the experimental data, and that it accounts for the observed ion beam radius of approx. 5 cm, and the electron halo, including the determination of an electron Debye length of approx. 10 cm 20. Modified betatron for ion beam fusion International Nuclear Information System (INIS) Rostoker, N.; Fisher, A. 1986-01-01 An intense neutralized ion beam can be injected and trapped in magnetic mirror or tokamak geometry. The details of the process involve beam polarization so that the beam crosses the fringing fields without deflection and draining the polarization when the beam reaches the plasma. Equilibrium requires that a large betatron field be added in tokamak geometry. In mirror geometry a toroidal field must be added by means of a current along the mirror axis. In either case, the geometry becomes that of the modified betatron which has been studied experimentally and theoretically in recent years. We consider beams of d and t ions with a mean energy of 500 kev and a temperature of about 50 kev. The plasma may be a proton plasma with cold ions. It is only necessary for beam trapping or to carry currents. The ion energy for slowing down is initially 500 kev and thermonuclear reactions depend only on the beam temperature of 50 kev which changes very slowly. This new configuration for magnetic confinement fusion leads to an energy gain of 10--20 for d-t reactions whereas previous studies of beam target interaction predicted a maximum energy gain of 3--4. The high beam energy available with pulsed ion diode technology is also essential for advanced fuels. 16 refs., 3 figs 1. Multicharged and intense heavy ion beam sources International Nuclear Information System (INIS) Kutner, V.B. 1981-01-01 The cyclotron plasma-are source (PIG), duoplasmatron (DP), laser source (LS), electron beam ion source (EBIS) and electron cyclotron resonance source (ECRS) from the viewpoint of generating intense and high charge state beams are considered. It is pointed out that for the last years three types of multicharged ion sources-EBIS, ECR and LS have been essentially developed. In the EBIS source the Xe 48+ ions are produced. The present day level of the development of the electron-beam ionization technique shows that by means of this technique intensive uranium nuclei beams production becomes a reality. On the ECR source Xe 26+ approximately 4x10 10 h/s, Asub(r)sup(12+) approximately 10 12 h/s intensive ion beams are produced. In the laser source a full number of C 6+ ions during one laser pulse constitutes not less than 10 10 from the 5x10mm 2 emission slit. At the present time important results are obtained pointing to the possibility to separate the ion component of laser plasma in the cyclotron central region. On the PIG source the Xe 15+ ion current up to 10μA per pulse is produced. In the duoplasmatron the 11-charge state of xenon ion beams is reached [ru 2. Intense pulsed heavy ion beam technology International Nuclear Information System (INIS) Masugata, Katsumi; Ito, Hiroaki 2010-01-01 Development of intense pulsed heavy ion beam accelerator technology is described for the application of materials processing. Gas puff plasma gun and vacuum arc discharge plasma gun were developed as an active ion source for magnetically insulated pulsed ion diode. Source plasma of nitrogen and aluminum were successfully produced with the gas puff plasma gun and the vacuum arc plasma gun, respectively. The ion diode was successfully operated with gas puff plasma gun at diode voltage 190 kV, diode current 2.2 kA and nitrogen ion beam of ion current density 27 A/cm 2 was obtained. The ion composition was evaluated by a Thomson parabola spectrometer and the purity of the nitrogen ion beam was estimated to be 86%. The diode also operated with aluminum ion source of vacuum arc plasma gun. The ion diode was operated at 200 kV, 12 kA, and aluminum ion beam of current density 230 A/cm 2 was obtained. The beam consists of aluminum ions (Al (1-3)+ ) of energy 60-400 keV, and protons (90-130 keV), and the purity was estimated to be 89%. The development of the bipolar pulse accelerator (BPA) was reported. A double coaxial type bipolar pulse generator was developed as the power supply of the BPA. The generator was tested with dummy load of 7.5 ohm, bipolar pulses of -138 kV, 72 ns (1st pulse) and +130 kV, 70 ns (2nd pulse) were successively generated. By applying the bipolar pulse to the drift tube of the BPA, nitrogen ion beam of 2 A/cm 2 was observed in the cathode, which suggests the bipolar pulse acceleration. (author) 3. Electron beam based transversal profile measurements of intense ion beams International Nuclear Information System (INIS) El Moussati, Said 2014-01-01 A non-invasive diagnostic method for the experimental determination of the transverse profile of an intense ion beam has been developed and investigated theoretically as well as experimentally within the framework of the present work. The method is based on the deflection of electrons when passing the electromagnetic field of an ion beam. To achieve this an electron beam is employed with a specifically prepared transversal profile. This distinguish this method from similar ones which use thin electron beams for scanning the electromagnetic field [Roy et al. 2005; Blockland10]. The diagnostic method presented in this work will be subsequently called ''Electron-Beam-Imaging'' (EBI). First of all the influence of the electromagnetic field of the ion beam on the electrons has been theoretically analyzed. It was found that the magnetic field causes only a shift of the electrons along the ion beam axis, while the electric field only causes a shift in a plane transverse to the ion beam. Moreover, in the non-relativistic case the magnetic force is significantly smaller than the Coulomb one and the electrons suffer due to the magnetic field just a shift and continue to move parallel to their initial trajectory. Under the influence of the electric field, the electrons move away from the ion beam axis, their resulting trajectory shows a specific angle compared to the original direction. This deflection angle practically depends just on the electric field of the ion beam. Thus the magnetic field has been neglected when analysing the experimental data. The theoretical model provides a relationship between the deflection angle of the electrons and the charge distribution in the cross section of the ion beam. The model however only can be applied for small deflection angles. This implies a relationship between the line-charge density of the ion beam and the initial kinetic energy of the electrons. Numerical investigations have been carried out to clarify the 4. Ion beam processing of bio-ceramics Science.gov (United States) Ektessabi, A. M. 1995-05-01 Thin films of bio-inert (TiO 2+α, Al 2O 3+α) and bio-active (compounds of calcium and phosphorus oxides, hydroxyapatite) were deposited on the most commonly used implant materials such as titanium and stainless steel, using a dual-ion-beam deposition system. Rutherford backscattering spectroscopy was carried out for quantitative measurement of the interfacial atomic mixing and the composition of the elements. The experimental results show that by controlling the ion beam energy and current, thin films with very good mechanical properties are obtained as a result of the ion beam mixing within the film and at the interface of the film and substrate. 5. Guide for External Beam Radiotherapy. Procedures 2007 International Nuclear Information System (INIS) Ardiet, Jean-Michel; Bourhis, Jean; Eschwege, Francois; Gerard, Jean-Pierre; Martin, Philippe; Mazeron, Jean-Jacques; Barillot, Isabelle; Bey, Pierre; Cosset, Jean-Marc; Thomas, Olivier; Bolla, Michel; Bourguignon, Michel; Godet, Jean-Luc; Krembel, David; Valero, Marc; Bara, Christine; Beauvais-March, Helene; Derreumaux, Sylvie; Vidal, Jean-Pierre; Drouard, Jean; Sarrazin, Thierry; Lindecker-Cournil, Valerie; Robin, Sun Hee Lee; Thevenet, Nicolas; Depenweiller, Christian; Le Tallec, Philippe; Ortholan, Cecile; Aimone, Nicole; Baldeschi, Carine; Cantelli, Andree; Estivalet, Stephane; Le Prince, Cyrille; QUERO, Laurent; Costa, Andre; Gerard, Jean-Pierre; Ardiet, Jean-Michel; Bensadoun, Rene-Jean; Bourhis, Jean; Calais, Gilles; Lartigau, Eric; Ginot, Aurelie; Girard, Nicolas; Mornex, Francoise; Bolla, Michel; Chauvet, Bruno; Maingon, Philippe; Martin, Etienne; Azria, David; Gerard, Jean-Pierre; Grehange, Gilles; Hennequin, Christophe; Peiffert, Didier; Toledano, Alain; Belkacemi, Yazid; Courdi, Adel; Belliere, Aurelie; Peignaux, Karine; Mahe, Marc; Bondiau, Pierre-Yves; Kantor, Guy; Lepechoux, Cecile; Carrie, Christian; Claude, Line 2007-01-01 In order to optimize quality and security in the delivery of radiation treatment, the French SFRO (Societe francaise de radiotherapie oncologique) is publishing a Guide for Radiotherapy. This guide is realized according to the HAS (Haute Autorite de sante) methodology of 'structured experts consensus'. This document is made of two parts: a general description of external beam radiation therapy and chapters describing the technical procedures of the main tumors to be irradiated (24). For each procedure, a special attention is given to dose constraints in the organs at risk. This guide will be regularly updated 6. Plasma ion sources and ion beam technology in microfabrications International Nuclear Information System (INIS) Ji, Lili 2007-01-01 For over decades, focused ion beam (FIB) has been playing a very important role in microscale technology and research, among which, semiconductor microfabrication is one of its biggest application area. As the dimensions of IC devices are scaled down, it has shown the need for new ion beam tools and new approaches to the fabrication of small-scale devices. In the meanwhile, nanotechnology has also deeply involved in material science research and bioresearch in recent years. The conventional FIB systems which utilize liquid gallium ion sources to achieve nanometer scale resolution can no longer meet the various requirements raised from such a wide application area such as low contamination, high throughput and so on. The drive towards controlling materials properties at nanometer length scales relies on the availability of efficient tools. In this thesis, three novel ion beam tools have been developed and investigated as the alternatives for the conventional FIB systems in some particular applications. An integrated focused ion beam (FIB) and scanning electron microscope (SEM) system has been developed for direct doping or surface modification. This new instrument employs a mini-RF driven plasma source to generate focused ion beam with various ion species, a FEI two-lens electron (2LE) column for SEM imaging, and a five-axis manipulator system for sample positioning. An all-electrostatic two-lens column has been designed to focus the ion beam extracted from the source. Based on the Munro ion optics simulation, beam spot sizes as small as 100 nm can be achieved at beam energies between 5 to 35 keV if a 5 (micro)m-diameter extraction aperture is used. Smaller beam spot sizes can be obtained with smaller apertures at sacrifice of some beam current. The FEI 2LE column, which utilizes Schottky emission, electrostatic focusing optics, and stacked-disk column construction, can provide high-resolution (as small as 20 nm) imaging capability, with fairly long working distance 7. Heavy ion beams from the new Hungarian ECR ion source International Nuclear Information System (INIS) Biri, S.; Valek, A.; Ditroi, F.; Koivisto, H.; Arje, J.; Stiebing, K.; Schmidt, L. 1998-01-01 The first beams of highly charged ions in Hungary were obtained in fall of 1996. The new 14.5 GHz ECR ion source of ATOMKI produced beams of multiply charged ions with remarkable intensities at first experiments. Since then, numerous further developments were carried out. An external electrondonor electrode drastically increased the plasma density and, consequently, the intensity of highly charged ions. These upgrades concentrated mainly on beams from gaseous elements and were carried out by the ECRIS team of ATOMKI. Another series of experiments - ionising from solids - however, was done in the framework of an international collaboration. The first metal ion beam has been extracted from the ECRIS in November 1997 using the known method of Metal Ions from Volatile Compounds (MIVOC). The possibility to put the MIVOC chamber inside the ion source was also tested and the dosing regulation problem of metal vapours inside the ion source was solved. As a result, beams of more than 10 μA of highly charged Fe and Ni ions were produced. (author) 8. Diffuse ions produced by electromagnetic ion beam instabilities International Nuclear Information System (INIS) Winske, D.; Leroy, M.M. 1984-01-01 The evolution of the electromagnetic ions beam instability driven by the reflected ion component backstreaming away from the earth's how shock into the foreshock region is studied by means computer simulation. The linear the quasi-linear states of the instability are found to be in good agreement with known results for the resonant model propagating parallel to the beam along the magnetic field and with theory developed in this paper for the nonresonant mode, which propagates antiparallel to the beam direction. The quasi-linear stage, which produces large amplitude 8Bapprox.B, sinusoidal transverse waves and ''intermediate'' ion distribution, is terminated by a nonlinear phase in which strongly nonlinear, compressive waves and ''diffuse'' ion distributions are produced. Additional processes by which the diffuse ions are accelerated to observed high energies are not addressed. The results are discussed in terms of the ion distributions and hydromagnetic waves observed in the foreshock of the earth's bow shock and of interplanetary shocks 9. Consideration of beam plasma ion-source International Nuclear Information System (INIS) Sano, Fumimichi; Kusano, Norimasa; Ishida, Yoshihiro; Ishikawa, Junzo; Takagi, Toshinori 1976-01-01 Theoretical and experimental analyses and their comparison were made on the plasma generation and on the beam extraction for the beam plasma ion-source. The operational principle and the structure of the ion-source are explained in the first part. Considerations are given on the electron beam-plasma interaction and the resulting generation of high frequency or microwaves which in turn increases the plasma density. The flow of energy in this system is also explained in the second part. The relation between plasma density and the imaginary part of frequency is given by taking the magnetic flux density, the electron beam energy, and the electron beam current as parameters. The relations between the potential difference between collector and drift tube and the plasma density or the ion-current are also given. Considerations are also given to the change of the plasma density due to the change of the magnetic flux density at drift tube, the change of the electron beam energy, and the change of the electron beam current. The third part deals with the extraction characteristics of the ion beam. The structure of the multiple-aperture electrode and the relation between plasma density and the extracted ion current are explained. (Aoki, K.) 10. Colliding-beams polarized ion source International Nuclear Information System (INIS) Trainor, T.A.; Douglas, J.G.; Badt, D.; Christiensen, C.; Herron, A.; Leach, D.; Olsen, J.; Osborne, J.L.; Zeps, V. 1985-01-01 This ion source was to be purchased from ANAC, Inc., a New Zealand-based supplier of beam optics hardware and atomic beam polarized ion sources in December 1982. Shortly before scheduled delivery ANAC went into receivership. During 1983 little work was done on the project as various steps were taken by us, first to get the ion source completed at ANAC, and then, failing that, to obtain the existing parts. In early 1984 we began work to finish the ion source in Seattle. The project is nearly complete, and this article presents progress to date. 2 refs 11. Beam Angular Divergence Effects in Ion Implantation International Nuclear Information System (INIS) Horsky, T. N.; Hahto, S. K.; Bilbrough, D. G.; Jacobson, D. C.; Krull, W. A.; Goldberg, R. D.; Current, M. I.; Hamamoto, N.; Umisedo, S. 2008-01-01 An important difference between monomer ion beams and heavy molecular beams is a significant reduction in beam angular divergence and increased on-wafer angular accuracy for molecular beams. This advantage in beam quality stems from a reduction in space-charge effects within the beam. Such improved angular accuracy has been shown to have a significant impact on the quality and yield of transistor devices [1,12]. In this study, B 18 H x + beam current and angular divergence data collected on a hybrid scanned beam line that magnetically scans the beam across the wafer is presented. Angular divergence is kept below 0.5 deg from an effective boron energy of 200 eV to 3000 eV. Under these conditions, the beam current is shown analytically to be limited by space charge below about 1 keV, but by the matching of the beam emittance to the acceptance of the beam line above 1 keV. In addition, results of a beam transport model which includes variable space charge compensation are presented, in which a drift mode B 18 H x + beam is compared to an otherwise identical boron beam after deceleration. Deceleration is shown to introduce significant space-charge blow up resulting in a large on-wafer angular divergence. The divergence effects introduced by wafer charging are also discussed. 12. Accelerated ion beam research at ATOMKI International Nuclear Information System (INIS) Kiss, A.Z. 2009-01-01 The paper summarizes the studies on accelerated ion beams at ATOMKI and their technical background, their use from chemical analysis to biological, medical, geological, archaeological applications, their advance from material science to micromachining. (TRA) 13. Intense pulsed ion beams for fusion applications International Nuclear Information System (INIS) Humphries, S. Jr. 1980-04-01 The subject of this review paper is the field of intense pulsed ion beam generation and the potential application of the beams to fusion research. Considerable progress has been made over the past six years. The ion injectors discussed utilize the introduction of electrons into vacuum acceleration gaps in conjunction with high voltage pulsed power technology to achieve high output current. Power levels from injectors exceeding 1000 MW/cm 2 have been obtained for pulse lengths on the order of 10 -7 sec. The first part of the paper treats the physics and technology of intense ion beams. The second part is devoted to applications of intense ion beams in fusion research. A number of potential uses in magnetic confinement systems have been proposed 14. Ion beam techniques in arts and archaeology International Nuclear Information System (INIS) Qin Guangyong; Pan Xianjia; Sun Zhongtian; Gao Zhengyao 1991-01-01 The ion beam techniques used in studies of arts and archaeology are compared with other analytical techniques. Some examples are specially selected to illustrate the achievements and trends of the techniques in this field 15. Radioactive heavy ion secondary beams International Nuclear Information System (INIS) Bimbot, R. 1987-01-01 The production of secondary radioactive beams at GANIL using the LISE spectrometer is reviewed. The experimental devices, and secondary beam characteristics are summarized. Production of neutron rich secondary beams was studied for the systems Ar40 + Be at 44 MeV/u, and 018 + Be at 45 and 65 MeV/u. Partial results were also obtained for the system Ne22 + Ta at 45 MeV/u. Experiments using secondary beams are classified into two categories: those which correspond to fast transfer of nuclei from the production target to a well shielded observation point; and those in which the radioactive beam interacts with a secondary target 16. Applications of focused ion beams in microelectronics International Nuclear Information System (INIS) Broughton, C.; Beale, M.I.J.; Deshmukh, V.G.I. 1986-04-01 We present the conclusions of the RSRE programme on the application of focused ion beams in microelectronics and review the literature published in this field. We discuss the design and performance of focused beam implanters and the viability of their application to semiconductor device fabrication. Applications in the areas of lithography, direct implantation and micromachining are discussed in detail. Comparisons are made between the use of focused ion beams and existing techniques for these fabrication processes with a strong emphasis placed on the relative throughputs. We present results on a novel spot size measurement technique and the effect of beam heating on resist. We also present the results of studies into implantation passivation of resist to oxygen plasma attack as basis for a dry development lithography scheme. A novel lithography system employing flood electron exposure from a photocathode which is patterned by a focused ion beam which can also be used to repair mask defects is considered. (author) 17. Radioactive ion beam facilities at INFN LNS International Nuclear Information System (INIS) Rifuggiato, D; Calabretta, L; Celona, L; Chines, F; Cosentino, L; Cuttone, G; Finocchiaro, P; Pappalardo, A; Re, M; Rovelli, A 2011-01-01 Radioactive ion beams are produced at INFN- Laboratori Nazionali del Sud (LNS) by means of the two operating accelerators, the Tandem and the Superconducting Cyclotron (CS), originally designed to accelerate stable beams. Both the ISOL (Isotope Separation On Line) and the IFF (In-Flight Fragmentation) methods are exploited to produce RIBs in two different ways at different energies: in the first case, the Cyclotron is the primary accelerator and the Tandem accelerates the secondary beams, while in the second case radioactive fragments are produced by the Cyclotron beam in a thin target with energies comparable to the primary beam energy. The ISOL facility is named EXCYT (Exotics at the Cyclotron and Tandem) and was commissioned in 2006, when the first radioactive beam ( 8 Li) has been produced. The IFF installation is named FRIBs (in Flight Radioactive Ion Beams), and it has started to produce radioactive beams in 2001, placing a thin target in the extraction beam line of the Cyclotron. The development of both facilities to produce and accelerate radioactive ion beams at LNS, is briefly described, with some details on the future prospects that are presently under consideration or realization. 18. Uses of laser optical pumping to produce polarized ion beams International Nuclear Information System (INIS) Anderson, L.W. 1983-01-01 Laser optical pumping can be used to produce polarized alkali atom beams or polarized alkali vapor targets. Polarized alkali atom beams can be converted into polarized alkali ion beams, and polarized alkali vapor targets can be used to produce polarized H - or 3 He - ion beams. In this paper the authors discuss how the polarized alkali atom beams and polarized alkali vapor targets are used to produce polarized ion beams with emphasis on the production of polarized negative ion beams 19. Beam modulation for heavy ion radiotherapy International Nuclear Information System (INIS) Kanai, T.; Minohara, S.; Sudou, M. 1993-01-01 The first clinical trial of heavy ion radiation therapy is scheduled in 1994 by using the heavy ion medical accelerator in Chiba (HIMAC). In order to start the clinical trial, first, it is necessary to know the physical characteristics of high energy heavy ions in human bodies, for example, dose and linear energy transfer (LET) distribution. Also the knowledge on the biological effectiveness of heavy ions is required. Based on these biophysical properties of heavy ions, monoenergetic heavy ion beam should be modulated so as to make the spread Bragg peak suitable to heavy ion radiation therapy. In order to establish a methodology to obtain the most effective spread Bragg peak for heavy ion radiation therapy, a heavy ion irradiation port at the RIKEN ring cyclotron facility was constructed. By using a 135 MeV/u carbon beam, the biophysical properties of the heavy ions were investigated, and a range modulator was designed to have uniform biological response in the spread Bragg peak. The physical and biological rationality of the spread Bragg peak were investigated. The dose, LET and biological effect of a monoenergetic heavy ion beam, the design of the range modulator, and the distributions of LET and biological dose for the spread Bragg peak are reported. (K.I.) 20. Beam dynamics in heavy ion induction LINACS International Nuclear Information System (INIS) Smith, L. 1981-10-01 Interest in the use of an induction linac to accelerate heavy ions for the purpose of providing the energy required to initiate an inertially confined fusion reaction has stimulated a theoretical effort to investigate various beam dynamical effects associated with high intensity heavy ion beams. This paper presents a summary of the work that has been done so far; transverse, longitudinal and coupled longitudinal transverse effects are discussed 1. Calculation of ballistic focusing of ion beams International Nuclear Information System (INIS) Astrelin, V.T.; Syresin, E.M. 1984-01-01 The motion of ions passing from the homogeneous magnetic field into a conical one is treated analytically in paraxial approximation. Further ions transform into neutral particles at the recharging target which is placed in the conical area of field. The optimal conditions for maximum compression of the beams of neutral particles are investigated. An influence of the initial angular spread on the beam compression is analysed. The computation results together with the those of analytical treatment are presented 2. Ion beam processing of advanced electronic materials International Nuclear Information System (INIS) Cheung, N.W.; Marwick, A.D.; Roberto, J.B. 1989-01-01 This report contains research programs discussed at the materials research society symposia on ion beam processing of advanced electronic materials. Major topics include: shallow implantation and solid-phase epitaxy; damage effects; focused ion beams; MeV implantation; high-dose implantation; implantation in III-V materials and multilayers; and implantation in electronic materials. Individual projects are processed separately for the data bases 3. Construction of ion beam pulse radiolysis system Energy Technology Data Exchange (ETDEWEB) Chitose, Norihisa; Katsumura, Yosuke; Domae, Masafumi; Ishigure, Kenkichi; Murakami, Takeshi [Tokyo Univ. (Japan) 1996-10-01 An ion beam pulse radiolysis system has been constructed at HIMAC facility. Ion beam of 24 MeV He{sup 2+} with the duration longer than 1 {mu}s is available for irradiation. Three kinds of aqueous solutions, (C{sub 6}H{sub 5}){sub 2}CO, NaHCO{sub 3} and KSCN, were irradiated and the absorption signals were observed. (author) 4. Intense ion beam research at Los Alamos International Nuclear Information System (INIS) Rej, D.J.; Bartsch, R.R.; Davis, H.A.; Faehl, R.J.; Gautier, D.C.; Greenly, J.B.; Henins, I.; Linton, T.W.; Muenchausen, R.E.; Waganaar, W.J. 1992-01-01 Two new interdisciplinary programs are underway at Los Alamos involving the physics and technology of intense light ion beams. In contrast to high-power ICF applications, the LANL effort concentrates on the development of relatively low-voltage (50 to 800 kV) and long-pulsewidth (0.1 to 1 μs) beams. The first program involves the 1.2 MV, 300-kJ Anaconda generator which has been fitted with an extraction ion diode. Long pulsewidth ion beams have been accelerated, propagated, and extracted for a variety of magnetic field conditions. The primary application of this beam is the synthesis of novel materials. Initial experiments on the congruent evaporative deposition of metallic and ceramic thin films are reported. The second program involves the development of a 120-keV, 50-kA, 1-μs proton beam for the magnetic fusion program as an ion source for an intense diagnostic neutral beam. Ultra-bright, pulsed neutral beams will be required to successfully measure ion temperatures and thermalized alpha particle energy distributions in large, dense, ignited tokamaks such as ITER 5. Intense ion beam research at Los Alamos International Nuclear Information System (INIS) Rej, D.J.; Bartsch, R.R.; Davis, H.A.; Faehl, R.J.; Gautier, D.C.; Greenly, J.B.; Henins, I.; Linton, T.W.; Muenchausen, R.E.; Waganaar, W.J. 1993-01-01 Two new interdisciplinary programs are underway at Los Alamos involving the physics and technology of intense light ion beams. In contrast to high-power ICF applications, the LANL effort concentrates on the development of relatively low-voltage (50 to 800 kV) and long pulsewidth (0.1 to 1 μs) beams. The first program involves the 1.2 MV, 300-kJ Anaconda generator which has been fitted with an extraction ion diode. Long pulsewidth ion beams have been accelerated, propagated, and extracted for a variety of magnetic field conditions. The primary application of this beam is the synthesis of novel materials. Initial experiments on the congruent evaporative deposition of metallic and ceramic thin films are reported. The second program involves the development of a 120-keV, 50-kA, 1-μs proton beam for the magnetic fusion program as an ion source for an intense diagnostic neutral beam. Ultra-bright, pulsed neutral beams will be required to successfully measure ion temperatures and thermalized alpha particle distributions in large, dense, ignited tokamaks such as ITER 6. National Centre for Radioactive Ion Beams (NCRIB) International Nuclear Information System (INIS) Chintalapudi, S.N. 1999-01-01 A dedicated National Centre for RIB (NCRIB) proposed discussed at several forums is presented. The production of (RIB) radioactive ion beams and applications of beams leading to competitive studies in nuclear structure, nuclear reactions, condensed matter, bio-science and radioactive isotope production etc. are mentioned 7. High energy density in matter produced by heavy ion beams International Nuclear Information System (INIS) 1989-07-01 This Annual Report summarizes research activities carried out in 1988 in the framework of the government-funded program 'High Energy Density in Matter produced by Heavy Ion Beams'. It addresses fundamental problems of the generation of heavy ion beams and the investigation of hot dense plasmas produced by these beams. Its initial motivation and its long-term goal is the feasibility of inertial confinement fusion by intense heavy ion beams. Two outstanding events deserve to be mentioned explicity, the Heavy Ion Inertial Fusion Conference held in Darmstadt and organized by GSI end of June and the first heavy ion beam injected into the new SIS facility in November. The former event attracted more than hundred scientists for three days to the 4th Conference in this field. This symposium showed the impressive progress since the last conference in Washington two years ago. In particular the first beams in MBE-4 at LBL and results of beam plasma interaction experiments at GSI open new directions for future investigations. The ideas for non-Lionvillean injection into storage rings presented by Carlo Rubbia will bring the discussion of driver scenarios into a new stage. The latter event is a milestone for both machine and target experiments. It characterizes the beginning of the commissioning phase for the new SIS/ESR facility which will be ready for experiments at the end of this year. The commissioning of SIS is on schedule and first experiments can start at the beginning of 1990. A status report of the accelerator project is included. Theoretical activities were continued as in previous years, many of them providing guide lines for future experiments, in particular for the radiation transport aspects and for beam-plasma interaction. (orig.) 8. A synchronous beam sweeper for heavy ions International Nuclear Information System (INIS) Bogaty, J.M. 1989-01-01 The Argonne Tandem Linac Accelerator System (ATLAS) facility at Argonne National Laboratory provides a wide range of accelerated heavy ions from the periodic table. Frequently, the beam delivery rate of 12 MHz is too fast for the type of experiment on line. Reaction by-products from a target bombardment may have a decay interval much longer than the dead time between beam bunches. To prevent data from being corrupted by incoming ions a beam sweeper was developed which synchronously eliminates selected beam bunches to suit experimental needs. As the SWEEPER is broad band (DC to 6 MHz) beam delivery rates can be instantaneously changed. Ion beam bunches are selectively kicked out by an electrostatic dipole electrode pulsed to 2 kVDC. The system has been used for almost three years with several hundred hours of operating time logged to date. Beam bunch delivery rates of 6 MHz down to 25 kHz have been provided. Since this is a non-resonant system any beam delivery rate from 6 MHz down to zero can be set. In addition, burst modes have been used where beam is supplied in 12 MHz bursts and then shut down for a period of time set by the user. 3 figs 9. A pencil beam algorithm for helium ion beam therapy Energy Technology Data Exchange (ETDEWEB) Fuchs, Hermann; Stroebele, Julia; Schreiner, Thomas; Hirtl, Albert; Georg, Dietmar [Christian Doppler Laboratory for Medical Radiation Research for Radiation Oncology, Medical University of Vienna, 1090 Vienna (Austria); Department of Radiation Oncology, Medical University of Vienna/AKH Vienna, 1090 Vienna (Austria) and Comprehensive Cancer Center, Medical University of Vienna/AKH Vienna, 1090 Vienna (Austria); Department of Radiation Oncology, Medical University of Vienna/AKH Vienna (Austria) and Comprehensive Cancer Center, Medical University of Vienna/AKH Vienna, 1090 Vienna (Austria); PEG MedAustron, 2700 Wiener Neustadt (Austria); Department of Nuclear Medicine, Medical University of Vienna, 1090 Vienna (Austria); Christian Doppler Laboratory for Medical Radiation Research for Radiation Oncology, Medical University of Vienna, 1090 Vienna (Austria); Department of Radiation Oncology, Medical University of Vienna/AKH Vienna, 1090 Vienna (Austria) and Comprehensive Cancer Center, Medical University of Vienna/AKH Vienna, 1090 Vienna (Austria) 2012-11-15 Purpose: To develop a flexible pencil beam algorithm for helium ion beam therapy. Dose distributions were calculated using the newly developed pencil beam algorithm and validated using Monte Carlo (MC) methods. Methods: The algorithm was based on the established theory of fluence weighted elemental pencil beam (PB) kernels. Using a new real-time splitting approach, a minimization routine selects the optimal shape for each sub-beam. Dose depositions along the beam path were determined using a look-up table (LUT). Data for LUT generation were derived from MC simulations in water using GATE 6.1. For materials other than water, dose depositions were calculated by the algorithm using water-equivalent depth scaling. Lateral beam spreading caused by multiple scattering has been accounted for by implementing a non-local scattering formula developed by Gottschalk. A new nuclear correction was modelled using a Voigt function and implemented by a LUT approach. Validation simulations have been performed using a phantom filled with homogeneous materials or heterogeneous slabs of up to 3 cm. The beams were incident perpendicular to the phantoms surface with initial particle energies ranging from 50 to 250 MeV/A with a total number of 10{sup 7} ions per beam. For comparison a special evaluation software was developed calculating the gamma indices for dose distributions. Results: In homogeneous phantoms, maximum range deviations between PB and MC of less than 1.1% and differences in the width of the distal energy falloff of the Bragg-Peak from 80% to 20% of less than 0.1 mm were found. Heterogeneous phantoms using layered slabs satisfied a {gamma}-index criterion of 2%/2mm of the local value except for some single voxels. For more complex phantoms using laterally arranged bone-air slabs, the {gamma}-index criterion was exceeded in some areas giving a maximum {gamma}-index of 1.75 and 4.9% of the voxels showed {gamma}-index values larger than one. The calculation precision of the 10. High Precision Beam Diagnostics for Ion Thrusters NARCIS (Netherlands) Van Reijen, B.; Koch, N.; Lazurenko, A.; Weis, S.; Schirra, M.; Genovese, A.; Haderspeck, J.; Gill, E.K.A. 2011-01-01 The Thales diagnostic equipment for ion beam characterization consists of a gridded and single orifice retarding potential analyzer (RPA) and an energy selective mass spectrometer (ESMS). During the development phase of these sensors considerable effort was put into the removal of ion optical 11. Ion beam source construction and applications International Nuclear Information System (INIS) Torab, S.I.R. 2011-01-01 The aim of this thesis is to improve the performance of a new shape cold cathode Penning ion source to be suitable for some applications. In this work, many trials have been made to reach the optimum dimensions of the new shape of cold Molybdenum cathode Penning ion source with radial extraction. The high output ion beam can be extracted in a direction transverse to the discharge region. The new shape cold cathode Penning ion source consists of Copper cylindrical hollow anode of 40 mm length, 12 mm diameter and has two similar cone ends of 15 mm length, 22 mm upper cone diameter and 12 mm bottom cone diameter. The two movable Molybdenum cathodes are fixed in Perspex insulator and placed symmetrically at two ends of the anode. The Copper emission disc of 2 mm thickness and has central aperture of different diameters is placed at the middle of the anode for ion beam exit. The inner surface of the emission disc is isolated from the anode by Perspex insulator except an area of diameter 5 mm to confine the electrical discharge in this area. A movable Faraday cup is placed at different distances from the emission electrode aperture and used to collect the output ion beam from the ion source. The working gases are admitted to the ion source through a hole in the anode via a needle valve which placed between the gas cylinder and the ion source. The optimum anode- cathode distance, the uncovered area diameter of the emission disc, the central aperture diameter of the emission electrode, the distance between emission electrode and Faraday cup have been determined using Argon gas. The optimum distances of the ion source were found to be equal to 6 mm, 5 mm, 2.5 mm, and 3 cm respectively where stable discharge current and maximum output ion beam current at low discharge current can be obtained. The discharge characteristics, ion beam characteristics, and the efficiency of the ion source have been measured at different operating conditions and different gas pressures using 12. Experimental studies with radioactive ion beams International Nuclear Information System (INIS) Sastry, D.L.; Sree Krishna Murty, G.; Chandrasekhar Rao, M.V.S. 1991-01-01 The sources of information presented are essentially taken from the papers reported at several international seminars and those appeared in the Journal of Nuclear Instruments and Methods in Physics Research. Production and usage of radioactive ion beams (RIB) in research have received the attention of scientists all over the world during the past six years. The first radioactive ion beams ( 19 Ne) were produced at Bevalac for the purpose of medical research using a primary beam of energy 800 MeV/a.m.u. (author). 19 refs., 2 figs., 3 tabs 13. Barium ion beam. Annual progress report International Nuclear Information System (INIS) Lazar, N.; Dandl, R.; Rynn, N.; Wickham, M. 1985-01-01 The barium ion beam Zeeman diagnostic is an in situ nonperturbing diagnostic designed to measure both the plasma electric and magnetic fields in devices such as STM and EBT. The diagnostic satisfies the requirements of high precision, spatial resolution and nonperturbation of the plasma. The technique uses resonance absorption of light from a single moded laser in a beam of energetic barium ions to measure the Zeeman effect in the absorption spectrum (to measure changes in the magnetic field) and to observe the changes in beam velocity by the Doppler shift of the absorption lines 14. The role of space charge compensation for ion beam extraction and ion beam transport (invited) International Nuclear Information System (INIS) Spädtke, Peter 2014-01-01 Depending on the specific type of ion source, the ion beam is extracted either from an electrode surface or from a plasma. There is always an interface between the (almost) space charge compensated ion source plasma, and the extraction region in which the full space charge is influencing the ion beam itself. After extraction, the ion beam is to be transported towards an accelerating structure in most cases. For lower intensities, this transport can be done without space charge compensation. However, if space charge is not negligible, the positive charge of the ion beam will attract electrons, which will compensate the space charge, at least partially. The final degree of Space Charge Compensation (SCC) will depend on different properties, like the ratio of generation rate of secondary particles and their loss rate, or the fact whether the ion beam is pulsed or continuous. In sections of the beam line, where the ion beam is drifting, a pure electrostatic plasma will develop, whereas in magnetic elements, these space charge compensating electrons become magnetized. The transport section will provide a series of different plasma conditions with different properties. Different measurement tools to investigate the degree of space charge compensation will be described, as well as computational methods for the simulation of ion beams with partial space charge compensation 15. Cellular radiobiology of heavy-ion beams International Nuclear Information System (INIS) Tobias, C.A.; Blakely, E.A.; Ngo, F.Q.H.; Roots, R.J.; Yang, T.C. 1981-01-01 Progress is reported in the following areas of this research program: relative biological effectiveness and oxygen enhancement ratio of silicon ion beams; heavy ion effects on the cell cycle; the potentiation effect (2 doses of high LET heavy-ion radiations separated by 2 to 3 hours); potentially lethal damage in actively growing cells and plateau growth cells; radiation induced macromolecular lesions and cellular radiation chemistry; lethal effects of dual radiation; and the development of a biophysical repair/misrepair model 16. Fast ion beam-laser interactions International Nuclear Information System (INIS) Berry, H.G.; Young, L.; Engstroem, L.; Hardis, J.E.; Somerville, L.P.; Ray, W.J.; Kurtz, C. 1985-01-01 The authors are using collinear laser excitation of fast ion beams to study a number of atomic structure problems. The problems include the determination of fine and hyperfine structure in light positive and negative ions, plus measurements of absolute wavelengths of light from two-electron ions. In addition the authors intend to use a similar experimental arrangement to study excitation and decay of high Rydberg states first in the absence of fields and then in crossed electric and magnetic fields 17. Ion beam heating of thin silicon membranes International Nuclear Information System (INIS) Tissot, P.E.; Hart, R.R. 1993-01-01 For silicon membranes irradiated by an ion beam in a vacuum environment, such as the masks used for ion beam lithography and the membranes used for thin film self-annealing, the heat transfer modes are radiation and limited conduction through the thin membrane. The radiation component depends on the total hemispherical emissivity which varies with the thickness and temperature of the membrane. A semiempirical correlation for the absorption coefficient of high resistivity silicon was derived and the variation of the total emissivity with temperature was computed for membranes with thicknesses between 0.1 and 10 μm. Based on this result, the temperatures reached during exposure to ion beams of varying intensities were computed. A proper modeling of the emissivity is shown to be important for beam heating of thin silicon membranes. (orig.) 18. Light ion beam transport research at NRL International Nuclear Information System (INIS) Hinshelwood, D.D.; Boller, J.R.; Cooperstein, G. 1996-01-01 Transport of light ion beams through low-pressure background gas is under investigation at NRL in support of the light-ion ICF program at Sandia National Laboratories. Scaling experiments and the field solver/orbit code ATHETA have been used to design and construct a focusing, extraction applied-B diode for transport experiments. An active anode source has been developed to provide a high proton fraction in the ion beam and a fast ion turn-on time. A very sensitive Zeeman diagnostic is being developed to determine the net current distribution in the beam/transport system. Both analytical and numerical techniques using several codes are being applied to transport modeling, leading to the capability of full system studies. (author). 1 tab., 5 figs., 10 refs 19. Light ion beam transport research at NRL Energy Technology Data Exchange (ETDEWEB) Hinshelwood, D D; Boller, J R; Cooperstein, G [Naval Research Lab., Washington, DC (United States). Plasma Physics Div.; and others 1997-12-31 Transport of light ion beams through low-pressure background gas is under investigation at NRL in support of the light-ion ICF program at Sandia National Laboratories. Scaling experiments and the field solver/orbit code ATHETA have been used to design and construct a focusing, extraction applied-B diode for transport experiments. An active anode source has been developed to provide a high proton fraction in the ion beam and a fast ion turn-on time. A very sensitive Zeeman diagnostic is being developed to determine the net current distribution in the beam/transport system. Both analytical and numerical techniques using several codes are being applied to transport modeling, leading to the capability of full system studies. (author). 1 tab., 5 figs., 10 refs. 20. Negative ion beam extraction in ROBIN International Nuclear Information System (INIS) Bansal, Gourab; Gahlaut, Agrajit; Soni, Jignesh; Pandya, Kaushal; Parmar, Kanu G.; Pandey, Ravi; Vuppugalla, Mahesh; Prajapati, Bhavesh; Patel, Amee; Mistery, Hiren; Chakraborty, Arun; Bandyopadhyay, Mainak; Singh, Mahendrajit J.; Phukan, Arindam; Yadav, Ratnakar K.; Parmar, Deepak 2013-01-01 Highlights: ► A RF based negative hydrogen ion beam test bed has been set up at IPR, India. ► Ion source has been successfully commissioned and three campaigns of plasma production have been carried out. ► Extraction system (35 kV) has been installed and commissioning has been initiated. Negative ion beam extraction is immediate milestone. -- Abstract: The RF based single driver −ve ion source experiment test bed ROBIN (Replica Of BATMAN like source in INDIA) has been set up at Institute for Plasma Research (IPR), India in a technical collaboration with IPP, Garching, Germany. A hydrogen plasma of density 5 × 10 12 cm −3 is expected in driver region of ROBIN by launching 100 kW RF power into the driver by 1 MHz RF generator. The cesiated source is expected to deliver a hydrogen negative ion beam of 10 A at 35 kV with a current density of 35 mA/cm 2 as observed in BATMAN. In first phase operation of the ROBIN ion source, a hydrogen plasma has been successfully generated (without extraction system) by coupling 80 kW RF input power through a matching network with high power factor (cos θ > 0.8) and different plasma parameters have been measured using Langmuir probes and emission spectroscopy. The plasma density of 2.5 × 10 11 cm −3 has been measured in the extraction region of ROBIN. For negative hydrogen ion beam extraction in second phase operation, extraction system has been assembled and installed with ion source on the vacuum vessel. The source shall be first operated in volume mode for negative ion beam extraction. The commissioning of the source with high voltage power supply has been initiated 1. Electron and ion beam transport to fusion targets International Nuclear Information System (INIS) Freeman, J.R.; Baker, L.; Miller, P.A.; Mix, L.P.; Olsen, J.N.; Poukey, J.W.; Wright, T.P. 1979-01-01 ICF reactors have been proposed which incorporate a gas-filled chamber to reduce x-ray and debris loading of the first wall. Focused beams of either electrons or ions must be transported efficiently for 2-4 m to a centrally located fusion target. Laser-initiated current-carrying plasma discharge channels provide the guiding magnetic field and the charge- and current-neutralizing medium required for beam propagation. Computational studies of plasma channel formation in air using a 1-D MHD model with multigroup radiation diffusion have provided a good comparison with the expansions velocity and time dependent refractivity profile determined by holographic interferometry. Trajectory calculations have identified a beam expansion mechanism which combines with the usual ohmic dissipation to reduce somewhat the transported beam fluence for electrons. Additional trajectory calculations have been performed for both electrons and light ions to predict the limits on the particle current density which can be delivered to a central target by overlapping the many independently-generated beams. Critical features of the use of plasma channels for transport and overlap of charged particle beams are being tested experimentally with up to twelve electron beams from the Proto II accelerator 2. Detection systems for radioactive ion beams International Nuclear Information System (INIS) Savajols, H. 2002-01-01 Two main methods are used to produce radioactive ion beams: -) the ISOL method (isotope separation on-line) in which the stable beam interacts with a thick target, the reaction products diffuse outside the target and are transferred to a source where they are ionized, a mass separator and a post-accelerator drive the selected radioactive ions to the right energy; -) the in-flight fragmentation method in which the stable beam interacts with a thin target, the reaction products are emitted from the target with a restricted angular distribution and a velocity close to that of the incident beam, the experimenter has to take advantage from the reaction kinetics to get the right particle beam. Characteristic time is far longer with the ISOL method but the beam intensity is much better because of the use of a post-accelerator. In both cases, the beam intensity is lower by several orders of magnitude than in the case of a stable beam. This article presents all the constraints imposed by radioactive beams to the detection systems of the reaction products and gives new technical solutions according to the type of nuclear reaction studied. (A.C.) 3. Proposed LLNL electron beam ion trap International Nuclear Information System (INIS) Marrs, R.E.; Egan, P.O.; Proctor, I.; Levine, M.A.; Hansen, L.; Kajiyama, Y.; Wolgast, R. 1985-01-01 The interaction of energetic electrons with highly charged ions is of great importance to several research fields such as astrophysics, laser fusion and magnetic fusion. In spite of this importance there are almost no measurements of electron interaction cross sections for ions more than a few times ionized. To address this problem an electron beam ion trap (EBIT) is being developed at LLNL. The device is essentially an EBIS except that it is not intended as a source of extracted ions. Instead the (variable energy) electron beam interacting with the confined ions will be used to obtain measurements of ionization cross sections, dielectronic recombination cross sections, radiative recombination cross sections, energy levels and oscillator strengths. Charge-exchange recombinaion cross sections with neutral gasses could also be measured. The goal is to produce and study elements in many different charge states up to He-like xenon and Ne-like uranium. 5 refs., 2 figs 4. Ion acceleration in modulated electron beams International Nuclear Information System (INIS) Bonch-Osmolovskij, A.G.; Dolya, S.N. 1977-01-01 A method of ion acceleration in modulated electron beams is considered. Electron density and energy of their rotational motion are relatively low. However the effective ion-accelerating field is not less than 10 MeV/m. The electron and ion numbers in an individual bunch are also relatively small, although the number of produced bunches per time unit is great. Some aspects of realization of the method are considered. Possible parameters of the accelerator are given. At 50 keV electron energy and 1 kA beam current a modulation is realized at a wave length of 30 cm. The ion-accelerating field is 12 MeV/m. The bunch number is 2x10 3 in one pulse at a gun pulse duration of 2 μs. With a pulse repetition frequency of 10 2 Hz the number of accelerated ions can reach 10 13 -10 14 per second International Nuclear Information System (INIS) Kiuchi, Masato; Chayahara, Akiyoshi; Kinomura, Atsushi; Ensinger, Wolfgang 1994-01-01 The nitridation of vanadium by ion beam irradiation is studied by the ion implantation method and the dynamic mixing method. The nitrogen ion implantation was carried out into deposited V(110) films. Using both methods, three phases are formed, i.e. α-V, β-V 2 N, and δ-VN. Which phases are formed is related to the implantation dose or the arrival ratio. The orientation of the VN films produced by the dynamic ion beam mixing method is (100) and that of the VN films produced by the ion implantation method is (111). The nitridation of vanadium is also discussed in comparison with that of titanium and chromium. ((orig.)) 6. Magnetic Field Measurements in Beam Guiding Magnets CERN Document Server Henrichsen, K N 1998-01-01 Electromagnets used as beam guiding elements in particle accelerators and colliders require very tight tole-rances on their magnetic fields and on their alignment along the particle path. This article describes the methods and equipment used for magnetic measurements in beam transport magnets. Descriptions are given of magnetic resonance techniques, various induction coil methods, Hall generator measurements, the fluxgate magnetometer as well as the recently developed method of beam based alignment. References of historical nature as well as citations of recent work are given. The present commercial availability of the different sensors and asso-ciated equipment is indicated. Finally we shall try to analyze possible future needs for developments in those fields. 7. Characterization of ion beam induced nanostructures International Nuclear Information System (INIS) Ghatak, J.; Satpati, B.; Umananda, M.; Kabiraj, D.; Som, T.; Dev, B.N.; Akimoto, K.; Ito, K.; Emoto, T.; Satyam, P.V. 2006-01-01 Tailoring of nanostructures with energetic ion beams has become an active area of research leading to the fundamental understanding of ion-solid interactions at nanoscale regime and with possible applications in the near future. Rutherford backscattering spectrometry (RBS), high resolution transmission electron microscopy (HRTEM) and asymmetric X-ray Bragg-rocking curve experimental methods have been used to characterize ion-induced effects in nanostructures. The possibility of surface and sub-surface/interface alloying at nano-scale regime, ion-beam induced embedding, crater formation, sputtering yield variations for systems with isolated nanoislands, semi-continuous and continuous films of noble metals (Au, Ag) deposited on single crystalline silicon will be reviewed. MeV-ion induced changes in specified Au-nanoislands on silicon substrate are tracked as a function of ion fluence using ex situ TEM. Strain induced in the bulk silicon substrate surface due to 1.5 MeV Au 2+ and C 2+ ion beam irradiation is determined by using HRTEM and asymmetric Bragg X-ray rocking curve methods. Preliminary results on 1.5 MeV Au 2+ ion-induced effects in nanoislands of Co deposited on silicon substrate will be discussed 8. Reaching for highest ion beam intensities through laser ion acceleration and beam compression Energy Technology Data Exchange (ETDEWEB) Schumacher, Dennis; Brabetz, Christian; Blazevic, Abel; Bagnoud, Vincent; Weih, Simon [GSI Helmholtzzentrum fuer Schwerionenforschung (Germany); Jahn, Diana; Ding, Johannes; Roth, Markus [TU Darmstadt (Germany); Kroll, Florian; Schramm, Ulrich; Cowan, Tom [Helmholtzzentrum Dresden Rossendorf (Germany); Collaboration: LIGHT-Collaboration 2016-07-01 Laser ion acceleration provides access to ion sources with unique properties. To use these capabilities the LIGHT collaboration (Laser Ion Generation Handling and Transport) was founded. The aim of this collaboration is the beam transport and manipulation of laser accelerated ions with conventional accelerator structures. Therefor a dedicated beam line has been build up at GSI Helmholtzzentrum fuer Schwerionenforschung. With this beam line the manipulation of the transversal and also the longitudinal beam parameters has been achieved. It has been shown that laser generated ion beams can be transported over more than 6 meters and pulses shorter than 300 ps can be generated at this distance. This Talk will give an overview over the recent developments and plans of the LIGHT collaboration. 9. Multiple Electron Stripping of Heavy Ion Beams International Nuclear Information System (INIS) Mueller, D.; Grisham, L.; Kaganovich, I.; Watson, R. L.; Horvat, V.; Zaharakis, K. E.; Peng, Y. 2002-01-01 One approach being explored as a route to practical fusion energy uses heavy ion beams focused on an indirect drive target. Such beams will lose electrons while passing through background gas in the target chamber, and therefore it is necessary to assess the rate at which the charge state of the incident beam evolves on the way to the target. Accelerators designed primarily for nuclear physics or high energy physics experiments utilize ion sources that generate highly stripped ions in order to achieve high energies economically. As a result, accelerators capable of producing heavy ion beams of 10 to 40 Mev/amu with charge state 1 currently do not exist. Hence, the stripping cross-sections used to model the performance of heavy ion fusion driver beams have, up to now, been based upon theoretical calculations. We have investigated experimentally the stripping of 3.4 Mev/amu Kr 7+ and Xe +11 in N2; 10.2 MeV/amu Ar +6 in He, N2, Ar and Xe; 19 MeV/amu Ar +8 in He, N2, Ar and Xe; 30 MeV He 1 + in He, N2, Ar and Xe; and 38 MeV/amu N +6 in He, N2, Ar and Xe. The results of these measurements are compared with the theoretical calculations to assess their applicability over a wide range of parameters 10. Mutation induction by ion beams in arabidopsis International Nuclear Information System (INIS) Tanaka, Atsushi 1999-01-01 An investigation was made on characteristics of ion beams for the biological effects and the induction of mutation using Arabidopsis plant as a model plant for the molecular genetics. Here, the characteristics of mutation at the molecular level as well as new mutants induced by ion beams were described. The ast and sep1 were obtained from the offspring of 1488 carbon ion-irradiated seeds respectively. The uvi1-uvi4 mutants were also induced from 1280 M 1 lines. Thus, ion beams can induce not only known mutants such as tt, gl and hy but also novel mutants with high frequency. Even in the tt phenotype, two new mutant loci other than known loci were found. In chrysanthemum, several kinds of single, complex or stripped flower-color mutants that have been never induced by γirradiation, indicating that ion beams could produce a variety of mutants with the same phenotype. In conclusion, ion beams for the mutation induction are characterized by 1) to induce mutants with high frequency, 2) to show broad mutation spectrum and 3) to produce novel mutants. For these reasons, chemical mutagens such as EMS and low LET ionizing radiation such as X-rays and γ-rays will predominantly induce many but small modifications or DNA damages on the DNA strands. As the result, several point mutations will be produced on the genome. On the contrary, ion beams as a high LET ionizing radiation will not cause so many but large and irreparable DNA damage locally, resulting in production of limited number of null mutation. (M.N.) 11. Mutation induction by ion beams in arabidopsis Energy Technology Data Exchange (ETDEWEB) Tanaka, Atsushi [Japan Atomic Energy Research Inst., Takasaki, Gunma (Japan). Takasaki Radiation Chemistry Research Establishment 1999-07-01 An investigation was made on characteristics of ion beams for the biological effects and the induction of mutation using Arabidopsis plant as a model plant for the molecular genetics. Here, the characteristics of mutation at the molecular level as well as new mutants induced by ion beams were described. The ast and sep1 were obtained from the offspring of 1488 carbon ion-irradiated seeds respectively. The uvi1-uvi4 mutants were also induced from 1280 M{sub 1} lines. Thus, ion beams can induce not only known mutants such as tt, gl and hy but also novel mutants with high frequency. Even in the tt phenotype, two new mutant loci other than known loci were found. In chrysanthemum, several kinds of single, complex or stripped flower-color mutants that have been never induced by {gamma}irradiation, indicating that ion beams could produce a variety of mutants with the same phenotype. In conclusion, ion beams for the mutation induction are characterized by 1) to induce mutants with high frequency, 2) to show broad mutation spectrum and 3) to produce novel mutants. For these reasons, chemical mutagens such as EMS and low LET ionizing radiation such as X-rays and {gamma}-rays will predominantly induce many but small modifications or DNA damages on the DNA strands. As the result, several point mutations will be produced on the genome. On the contrary, ion beams as a high LET ionizing radiation will not cause so many but large and irreparable DNA damage locally, resulting in production of limited number of null mutation. (M.N.) 12. Development and application of ion beam diagnostics International Nuclear Information System (INIS) Pfister, Jochen 2010-01-01 At GSI - Helmholtz Centre for Heavy Ion Research in Darmstadt/Germany the HITRAP project is in the commissioning phase. This world-wide unique facility consists of a linear decelerator for heavy, highly charged ions including atomic physics precision experiments. During commissioning of the cavities, transverse emittances were measured using the single-shot pepperpot method as well as the multi-gradient method. The extraction emittance of the experimental storage ring (ESR) was determined. Furthermore, the phase space distribution of an decelerated beam at an intermediate energy of 500keV/u was measured behind the IH-structure. New algorithms have been integrated into the analysis of digital images. The longitudinal bunch structure measurements of the ion beam at the entry point into the decelerator and the operation of the Double-drift Buncher is shown. The design, development and the first commissioning of a new single-shot pepperpot emittance meter for very low beam currents and beam energies in the order of some hundred nA is described, making it possible to measure the beam behind the deceleration cavities. In addition, transverse beam dynamics calculations were performed, which supported the hands-on commissioning of the accelerator. It is described how the entire beam line from the ESR to the radio-frequency quadrupole can be optimized using the new routine for transverse effects of the bunching and deceleration, which was successfully integrated into the software COSY Infinity. (orig.) 13. Ion beams in silicon processing and characterization International Nuclear Information System (INIS) Chason, E.; Picraux, S.T.; Poate, J.M.; Borland, J.O.; Current, M.I.; Diaz de la Rubia, T.; Eaglesham, D.J.; Holland, O.W.; Law, M.E.; Magee, C.W.; Mayer, J.W.; Melngailis, J.; Tasch, A.F. 1997-01-01 General trends in integrated circuit technology toward smaller device dimensions, lower thermal budgets, and simplified processing steps present severe physical and engineering challenges to ion implantation. These challenges, together with the need for physically based models at exceedingly small dimensions, are leading to a new level of understanding of fundamental defect science in Si. In this article, we review the current status and future trends in ion implantation of Si at low and high energies with particular emphasis on areas where recent advances have been made and where further understanding is needed. Particularly interesting are the emerging approaches to defect and dopant distribution modeling, transient enhanced diffusion, high energy implantation and defect accumulation, and metal impurity gettering. Developments in the use of ion beams for analysis indicate much progress has been made in one-dimensional analysis, but that severe challenges for two-dimensional characterization remain. The breadth of ion beams in the semiconductor industry is illustrated by the successful use of focused beams for machining and repair, and the development of ion-based lithographic systems. This suite of ion beam processing, modeling, and analysis techniques will be explored both from the perspective of the emerging science issues and from the technological challenges. copyright 1997 American Institute of Physics 14. The quest for crystalline ion beams CERN Document Server Schramm, U; Bussmann, M; Habs, D 2002-01-01 The phase transition of an ion beam into its crystalline state has long been expected to dramatically influence beam dynamics beyond the limitations of standard accelerator physics. Yet, although considerable improvement in beam cooling techniques has been made, strong heating mechanisms inherent to existing high-energy storage rings have prohibited the formation of the crystalline state in these machines up to now. Only recently, laser cooling of low-energy beams in the table-top rf quadrupole storage ring PAaul Laser cooLing Acceleration System (PALLAS) has lead to the experimental realization of crystalline beams. In this article, the quest for crystalline beams as well as their unique properties as experienced in PALLAS will be reviewed. 15. Filamentation of a converging heavy ion beam International Nuclear Information System (INIS) Lee, E.P.; Buchanan, H.L.; Rosenbluth, M.N. 1980-01-01 A major concern in the use of heavy ion beams as igniters in pellet fusion systems is the vulnerability of the beam to the transverse flamentation instability. The undesirable consequence of this mode is the transverse heating of the beam to the extent that convergence on the pellet becomes impossible. This work considers the case of a beam injected into a gas filled reactor vessel, where finite pulse length and propagation distance play an important role in limiting growth. Two geometries are analyzed: a nonconverging case where the radius at injection is nearly equal to the desired radius at the pellet, and a converging case in which the injection radius is large and the beam is pre-focused to converge at the target. It is found that a cold beam will be severely disrupted if the product of the magnetic plasma frequency and the propagation distance is much larger than unity 16. Beam-guiding system for Rutherford-scattering diagnostic at TEXTOR International Nuclear Information System (INIS) Cosler, A; Bertschinger, G.; Kemmereit, E.; Ven, H.W. van der; Barbian, E.P.; Blokland, A.A.E. van 1988-01-01 A beam-guiding system for a neutral beam probe diagnostic has been developed for implementation at TEXTOR. Energetic helium atoms scattered on the plasma ions provide information about the local ion temperature. Time resolution is attained by sampling scattered particles measured individually by a time-of-flight analyser. The mechanical supports have been designed for lateral and angular movement of the beam-guiding system to be used for radial scanning of the torus and for optimization of the scattering angle. The parameters of the probing beam itself can be controlled jby a small beam profile diagnsotic. Provisions are made to observe separately the radial or axial component of the ion velocity distribution. (author). 10 refs.; 7 figs 17. Ion beam modification of buckminsterfullerene International Nuclear Information System (INIS) Prawer, S.; Nugent, K.W.; McCulloch, D.G.; Leong, W.H.; Hoffman, A.; Kalish, R. 1995-01-01 The response of thin films of buckminsterfullerene (C 60 ) to energetic xenon ion impact is investigated. The diagnostics employed include Fourier Transform Infrared and Raman Spectroscopies, Cross-Sectional Transmission Electron Microscopy, and Atomic Force Microscopy. By combining the information obtained from these diagnostics with that from the dependence of the conductivity on ion dose, it is concluded that each C 60 molecule completely disintegrates when hit by an energetic ion. The cross-section for the destruction of about 7 x 10 -13 cm 2 for irradiation with 620 keV Xe ions. The disintegration occurs when C atoms are knocked-out of the molecule either directly by the impinging ion or by an energetic knock-on C atom with the damage cascade. This process is quite different from the Coulomb Explosion mechanism previously proposed in the literature. For very low ions doses ( 11 Xe/cm 2 ) most of the C 60 molecules remain intact; however this dose is sufficient to completely disrupt the ordering of the C 60 molecules in the van der Waals bonded C 60 solid. Disruption of the lattice ordering at such low doses is considered to be attributable to the weakness of the van der Waals forces which bind the C 60 clusters together into the molecular solid. 13 refs., 7 figs 18. Cold guided beams of polar molecules International Nuclear Information System (INIS) Motsch, Michael 2010-01-01 This thesis reports on experiments characterizing cold guided beams of polar molecules which are produced by electrostatic velocity filtering. This filtering method exploits the interaction between the polar molecules and the electric field provided by an electrostatic quadrupole guide to extract efficiently the slow molecules from a thermal reservoir. For molecules with large and linear Stark shifts such as deuterated ammonia (ND 3 ) or formaldehyde (H 2 CO), fluxes of guided molecules of 10 10 -10 11 molecules/s are produced. The velocities of the molecules in these beams are in the range of 10-200 m/s and correspond to typical translational temperatures of a few Kelvin. The maximum velocity of the guided molecules depends on the Stark shift, the molecular mass, the geometry of the guide, and the applied electrode voltage. Although the source is operated in the near-effusive regime, the number density of the slowest molecules is sensitive to collisions. A theoretical model, taking into account this velocity-dependent collisional loss of molecules in the vicinity of the nozzle, reproduces the density of the guided molecules over a wide pressure range. A careful adjustment of pressure allows an increase in the total number of molecules, whilst yet minimizing losses due to collisions of the sought-for slow molecules. This is an important issue for future applications. Electrostatic velocity filtering is suited for different molecular species. This is demonstrated by producing cold guided beams of the water isotopologs H 2 O, D 2 O, and HDO. Although these are chemically similar, they show linear and quadratic Stark shifts, respectively, when exposed to external electric fields. As a result, the flux of HDO is larger by one order of magnitude, and the flux of the individual isotopologs shows a characteristic dependence on the guiding electric field. The internal-state distribution of guided molecules is studied with a newly developed diagnostic method: depletion 19. Ion beam processing of bio-ceramics International Nuclear Information System (INIS) Ektessabi, A.M. 1995-01-01 Thin films of bio-inert (TiO 2+α , Al 2 O 3+α ) and bio-active (compounds of calcium and phosphorus oxides, hydroxy-apatite) were deposited on the most commonly used implant materials such as titanium and stainless steel, using a dual-ion-beam deposition system. Rutherford backscattering spectroscopy was carried out for quantitative measurement of the interfacial atomic mixing and the composition of the elements. The experimental results show that by controlling the ion beam energy and current, thin films with very good mechanical properties are obtained as a result of the ion beam mixing within the film and at the interface of the film and substrate. (orig.) 20. Variable-spot ion beam figuring International Nuclear Information System (INIS) Wu, Lixiang; Qiu, Keqiang; Fu, Shaojun 2016-01-01 This paper introduces a new scheme of ion beam figuring (IBF), or rather variable-spot IBF, which is conducted at a constant scanning velocity with variable-spot ion beam collimated by a variable diaphragm. It aims at improving the reachability and adaptation of the figuring process within the limits of machine dynamics by varying the ion beam spot size instead of the scanning velocity. In contrast to the dwell time algorithm in the conventional IBF, the variable-spot IBF adopts a new algorithm, which consists of the scan path programming and the trajectory optimization using pattern search. In this algorithm, instead of the dwell time, a new concept, integral etching time, is proposed to interpret the process of variable-spot IBF. We conducted simulations to verify its feasibility and practicality. The simulation results indicate the variable-spot IBF is a promising alternative to the conventional approach. 1. Charge neutralization of small ion beam clumps Energy Technology Data Exchange (ETDEWEB) Welch, D R [Mission Research Corp., Albuquerque, NM (United States); Olson, C L; Hanson, D L [Sandia National Labs., Albuquerque, NM (United States) 1997-12-31 The mega-ampere currents associated with light ion fusion (LIF) require excellent charge neutralization to prevent divergence growth. As the size and space-charge potential of a beam clump or beamlet become small (submillimeter size and kilovolt potentials), the neutralization becomes increasingly difficult. Linear theory predicts that plasma electrons cannot neutralize potentials < {phi}{sub crit} = (1/2)m{sub e}v{sub i}{sup 2}/e, where m{sub e} is the electron mass and v{sub i} is the ion beam velocity. A non-uniform beam would, therefore, have regions with potentials sufficient to add divergence to beam clumps. The neutralization of small beamlets produced on the SABLE accelerator and in numerical simulation has supported the theory, showing a plateau in divergence growths as the potential in the beamlet exceeds {phi}{sub crit}. (author). 1 tab., 2 figs., 4 refs. 2. A Study on the Ion Beam Extraction using Duo-PiGatron Ion source for Vertical Type Ion Beam Facility Energy Technology Data Exchange (ETDEWEB) Kim, Bom Sok; Lee, Chan young; Lee, Jae Sang [KAERI, Daejeon (Korea, Republic of) 2015-05-15 In Korea Multipurpose Accelerator Complex (KOMAC), we have started ion beam service in the new beam utilization building since March this year. For various ion beam irradiation services, we are developed implanters such as metal (150keV/1mA), gaseous (200keV/5mA) and high current ion beam facility (20keV/150mA). One of the new one is a vertical type ion beam facility without acceleration tube (60keV/20mA) which is easy to install the sample. After the installation is complete, it is where you are studying the optimal ion beam extraction process. Detailed experimental results will be presented. Vertical Type Ion Beam Facility without acceleration tube of 60keV 20mA class was installed. We successfully extracted 60keV 20mA using Duo- PiGatron Ion source for Vertical Type Ion Beam Facility. Use the BPM and Faraday-cup, is being studied the optimum conditions of ion beam extraction. 3. Spatially-Resolved Ion Trajectory Measurements During Cl2 Reactive Ion Beam Etching and Ar Ion Beam Etching International Nuclear Information System (INIS) Vawter, G. Allen; Woodworth, Joseph R.; Zubrzycki, Walter J. 1999-01-01 The angle of ion incidence at the etched wafer location during RIBE and IBE using Cl 2 , Ar and O 2 ion beams has been characterized using an ion energy and angle analyzer. Effects of beam current and accelerator grid bias on beam divergence and the spatial uniformity of the spread of incident angles are measured. It is observed that increased total beam current can lead to reduced current density at the sample stage due to enhanced beam divergence at high currents. Results are related to preferred etch system design for uniform high-aspect-ratio etching across semiconductor wafers 4. Calculation of ion storage in electron beams with account of ion-ion interactions International Nuclear Information System (INIS) Perel'shtejn, Eh.A.; Shirkov, G.D. 1979-01-01 Ion storage in relativistic electron beams was calculated taking account of ion-ion charge exchange and ionization. The calculations were made for nitrogen ion storage from residual gas during the compression of electron rings in the adhezator of the JINR heavy ion accelerator. The calculations were made for rings of various parameters and for various pressures of the residual gas. The results are compared with analogous calculations made without account of ion-ion processes. It is shown that at heavy loading of a ring by ions ion-ion collisions play a significant part, and they should be taken into account while calculating ion storage 5. Ion beam dump for JT-60 NBI International Nuclear Information System (INIS) Kuriyama, Masaaki; Horiike, Hiroshi; Matsuda, Shinzaburo; Morita, Hiroaki; Shibanuma, Kiyoshi 1981-10-01 The design of the active cooling type ion beam dump for JT-60 NBI which receives the total beam power of 5.6 MW for 10 sec continuously is described. It is composed of array of many finned tubes which is made of oxygen free copper with 0.2% silver content. The safety margin against thermal and mechanical troubles is estimated by the heat transfer and the thermal stress calculation. (author) 6. Ion beam pulse radiolysis system at HIMAC Energy Technology Data Exchange (ETDEWEB) Chitose, N; Katsumura, Y; Domae, M; Ishigure, K [Tokyo Univ. (Japan); Murakami, T 1997-03-01 An ion beam pulse radiolysis system has been constructed at HIMAC facility. Ion beam of 24MeV He{sup 2+} with the duration longer than 1 {mu}s is available for irradiation. Three kinds of aqueous solutions, (C{sub 6}H{sub 5}){sub 2}CO, NaHCO{sub 3}, and KSCN, were irradiated and the absorption signals corresponding to (C{sub 6}H{sub 5}){sub 2}CO{sup -}, CO{sub 3}{sup -}, and (SCN){sub 2}{sup -} respectively were observed. Ghost signals which interfere with the measurement are also discussed. (author) 7. Ion beam techniques for analyzing polymers irradiated by ions International Nuclear Information System (INIS) Rickards, J.; Zironi, E.P.; Andrade, E.; Dominguez, B. 1992-01-01 In the study of the effects of ion beam irradiation of polymers very large doses can be administered in short times. Thousands of MGy can be produced in a small volume of a sample in a few minutes by bombarding with typical ion beam currents. For instance, in an experiment done to observe the effects of 750 keV proton irradiation PVC, using a collimator of 1 mm diameter, 1 μC of charge integration deposits a dose of 50 MGy. The use of ion beams also opens up the possibility of using the same beam for irradiation and for analysis of the effects, using the well known ion beam analysis techniques. PIXE allows the measurement of chlorine in PVC. Polymers containing fluorine can be measured with the resonant nuclear reaction (RNR) technique, which is specific only to certain elements. The amount of hydrogen in the sample and its profile can be obtained using energy recoil detection analysis (ERDA); carbon, oxygen, and nitrogen can be measured and profiled using Rutherford backscattering (RBS) and also using the (d,p) and (d, α) nuclear reactions (NR). Loss of mass is one effect that can be studied using these techniques. It was studied in two different polymers, PVC and CR-39, in order to determine carbon buildup during ion irradiation. It was concluded that carbon builds up following different mechanisms in these two materials, due to the different possibilities of forming volatile compounds. It is also suggested that CR-39 should be a good material for ion beam lithography. (author) 8. High-powered pulsed-ion-beam acceleration and transport Energy Technology Data Exchange (ETDEWEB) Humphries, S. Jr.; Lockner, T.R. 1981-11-01 The state of research on intense ion beam acceleration and transport is reviewed. The limitations imposed on ion beam transport by space charge effects and methods available for neutralization are summarized. The general problem of ion beam neutralization in regions free of applied electric fields is treated. The physics of acceleration gaps is described. Finally, experiments on multi-stage ion acceleration are summarized. 9. High-powered pulsed-ion-beam acceleration and transport International Nuclear Information System (INIS) Humphries, S. Jr.; Lockner, T.R. 1981-11-01 The state of research on intense ion beam acceleration and transport is reviewed. The limitations imposed on ion beam transport by space charge effects and methods available for neutralization are summarized. The general problem of ion beam neutralization in regions free of applied electric fields is treated. The physics of acceleration gaps is described. Finally, experiments on multi-stage ion acceleration are summarized 10. Treatment planning with ion beams International Nuclear Information System (INIS) Foss, M.H. 1985-01-01 Ions have higher linear energy transfer (LET) near the end of their range and lower LET away from the end of their range. Mixing radiations of different LET complicates treatment planning because radiation kills cells in two statistically independent ways. In some cases, cells are killed by a single-particle, which causes a linear decrease in log survival at low dosage. When the linear decrease is subtracted from the log survival curve, the remaining curve has zero slope at zero dosage. This curve is the log survival curve for cells that are killed only by two or more particles. These two mechanisms are statistically independent. To calculate survival, these two kinds of doses must be accumulated separately. The effect of each accumulated dosage must be read from its survival curve, and the logarithms of the two effects added to get the log survival. Treatment plans for doses of protons, He 3 ions, and He 4 ions suggest that these ions will be useful therapeutic modalities 11. Fullerene genesis by ion beams International Nuclear Information System (INIS) Gamaly, E.G.; Chadderton, L.T.; Commonwealth Scientific and Industrial Research Organization, Lindfield, NSW 1995-01-01 Clearly detectable quantities of molecular fullerene (C 60 ), the most recently discovered allotrope of carbon, have been observed in graphite following irradiation with heavy projectile ions at energies of about 1 GeV using high pressure chromatography. Similar experiments using lower ion energies gave no corresponding signal, indicating an absence of fullerene. This clear difference suggests that there exists an energy threshold for fullerene genesis. Beginning with a microscopic description of deposition and transfer of energy from the ion to the target, a theoretical model is developed for interpretation of these and similar experiments. An important consequence is a description of the formation of large carbon clusters in the hot dense 'primeval soup' of single carbon atoms by means of random 'sticky' collisions. The ion energy threshold is seen as arising, physically, from a balance in the competition between the rate of primary energy deposition and the rate of system cooling. Rate equations for the basic clustering process allow calculations of the time-dependent number densities for the different carbon clusters produced. An important consequence of the theory is that it is established that the region for the specific phase transition from graphite to fullerene lies in the same pressure regime on the phase diagram as does the corresponding transition for graphite to diamond. (author) 12. Towards a magnetic field separation in Ion Beam Sputtering processes Energy Technology Data Exchange (ETDEWEB) Malobabic, Sina, E-mail: [email protected] [Laser Zentrum Hannover e.V., Hollerithallee 8, 30419 Hannover (Germany); Quest: Centre of Quantum Engineering and Space-Time Research, Leibniz Universität Hannover (Germany); Jupé, Marco [Laser Zentrum Hannover e.V., Hollerithallee 8, 30419 Hannover (Germany); Quest: Centre of Quantum Engineering and Space-Time Research, Leibniz Universität Hannover (Germany); Kadhkoda, Puja [Laser Zentrum Hannover e.V., Hollerithallee 8, 30419 Hannover (Germany); Ristau, Detlev [Laser Zentrum Hannover e.V., Hollerithallee 8, 30419 Hannover (Germany); Quest: Centre of Quantum Engineering and Space-Time Research, Leibniz Universität Hannover (Germany) 2015-10-01 Defects embedded in coatings due to particle contamination are considered as a primary factor limiting the quality of optical coatings in Ion Beam Sputtering. An approach combining the conventional Ion Beam Sputtering process with a magnetic separator in order to remove these particles from film growth is presented. The separator provides a bent axial magnetic field that guides the material flux towards the substrate positioned at the exit of the separator. Since there is no line of sight between target and substrate, the separator prevents that the particles generated in the target area can reach the substrate. In this context, optical components were manufactured that reveal a particle density three times lower than optical components which were deposited using a conventional Ion Beam Sputtering process. - Highlights: • We use bent magnetic fields to guide and separate the sputtered deposition material. • No line of sight between substrate and target prevents thin films from particles. • The transport efficiency of binary and ternary oxides is investigated. • The defect statistics of manufactured dielectric ternary multilayers are evaluated. • The phase separation leads to a drastically reduction of particle contamination. 13. Ion-beam texturing of uniaxially textured Ni films International Nuclear Information System (INIS) Park, S.J.; Norton, D.P.; Selvamanickam, Venkat 2005-01-01 The formation of biaxial texture in uniaxially textured Ni thin films via Ar-ion irradiation is reported. The ion-beam irradiation was not simultaneous with deposition. Instead, the ion beam irradiates the uniaxially textured film surface with no impinging deposition flux, which differs from conventional ion-beam-assisted deposition. The uniaxial texture is established via a nonion beam process, with the in-plane texture imposed on the uniaxial film via ion beam bombardment. Within this sequential ion beam texturing method, grain alignment is driven by selective etching and grain overgrowth 14. Production of highly charged ion beams from ECR ion sources International Nuclear Information System (INIS) Xie, Z.Q. 1997-09-01 Electron Cyclotron Resonance (ECR) ion source development has progressed with multiple-frequency plasma heating, higher mirror magnetic fields and better technique to provide extra cold electrons. Such techniques greatly enhance the production of highly charged ions from ECR ion sources. So far at cw mode operation, up to 300 eμA of O 7+ and 1.15 emA of O 6+ , more than 100 eμA of intermediate heavy ions for charge states up to Ar 13+ , Ca 13+ , Fe 13+ , Co 14+ and Kr 18+ , and tens of eμA of heavy ions with charge states to Kr 26+ , Xe 28+ , Au 35+ , Bi 34+ and U 34+ have been produced from ECR ion sources. At an intensity of at least 1 eμA, the maximum charge state available for the heavy ions are Xe 36+ , Au 46+ , Bi 47+ and U 48+ . An order of magnitude enhancement for fully stripped argon ions (I ≥ 60 enA) also has been achieved. This article will review the ECR ion source progress and discuss key requirement for ECR ion sources to produce the highly charged ion beams 15. Realization of a scanning ion beam monitor International Nuclear Information System (INIS) Pautard, C. 2008-07-01 During this thesis, a scanning ion beam monitor has been developed in order to measure on-line fluence spatial distributions. This monitor is composed of an ionization chamber, Hall Effect sensors and a scintillator. The ionization chamber set between the beam exit and the experiment measures the ion rate. The beam spot is localized thanks to the Hall Effect sensors set near the beam sweeping magnets. The scintillator is used with a photomultiplier tube to calibrate the ionization chamber and with an imaging device to calibrate the Hall Effect sensors. This monitor was developed to control the beam lines of a radiobiology dedicated experimentation room at GANIL. These experiments are held in the context of the research in hadron-therapy. As a matter of fact, this new cancer treatment technique is based on ion irradiations and therefore demands accurate knowledge about the relation between the dose deposit in biological samples and the induced effects. To be effective, these studies require an on-line control of the fluence. The monitor has been tested with different beams at GANIL. Fluence can be measured with a relative precision of ±4% for a dose rate ranging between 1 mGy/s and 2 Gy/s. Once permanently set on the beam lines dedicated to radiobiology at GANIL, this monitor will enable users to control the fluence spatial distribution for each irradiation. The scintillator and the imaging device are also used to control the position, the spot shape and the energy of different beams such as those used for hadron-therapy. (author) 16. Improvement of herbage by heavy ion beams International Nuclear Information System (INIS) Xie Hongmei; Hao Jifang; Wei Zengquan; Xie Zhongkui; Li Fengqin; Wang Yajun 2004-01-01 Herbage seeds of legume and grass were irradiated in penetration by 80 MeV/u 20 Ne 10+ ions. The results of field tests and observations of the root-tip cells showed that growth of the seedling was obviously weakened with increasing doses. Frequencies of chromosomal aberration and micronucleus increased significantly with increasing doses. According to the field growth tests, radiation sensitivity of grass herbage to the heavy ion beams was much higher than leguminous herbage, and suitable dose of the heavy ion irradiation for the grass and leguminous herbage is 20-30 Gy and 150 Gy, respectively 17. Physics with fast molecular-ion beams International Nuclear Information System (INIS) Kanter, E.P. 1980-01-01 Fast (MeV) molecular-ion beams provide a unique source of energetic projectile nuclei which are correlated in space and time. The recognition of this property has prompted several recent investigations of various aspects of the interactions of these ions with matter. High-resolution measurements on the fragments resulting from these interactions have already yielded a wealth of new information on such diverse topics as plasma oscillations in solids and stereochemical structures of molecular ions as well as a variety of atomic collision phenomena. The general features of several such experiments will be discussed and recent results will be presented 18. Profiling hydrogen in materials using ion beams International Nuclear Information System (INIS) Ziegler, J.F.; Wu, C.P.; Williams, P. 1977-01-01 Over the last few years many ion beam techniques have been reported for the profiling of hydrogen in materials. Nine of these were evaluated using similar samples of hydrogen ion-implanted into silicon. When possible the samples were analyzed using two or more techniques to confirm the ion-implanted accuracy. The results of this analysis which has produced a consensus profile of H in silicon which is useful as a calibration standard are reported. The analytical techniques used have capabilities ranging from very high depth resolution (approximately 50 A) and high sensitivity (less than 1 ppM) to deep probes for hydrogen which can sample throughout thin sheets 19. Temperature-dependent ion beam mixing International Nuclear Information System (INIS) Rehn, L.E.; Alexander, D.E. 1993-08-01 Recent work on enhanced interdiffusion rates during ion-beam mixing at elevated temperatures is reviewed. As discussed previously, expected increase in ion-beam mixing rates due to 'radiation-enhanced diffusion' (RED), i.e. the free migration of isolated vacancy and interstitial defects, is well documented in single-crystal specimens in the range of 0.4 to 0.6 of absolute melting temperature. In contrast, the increase often observed at somewhat lower temperatures during ion-beam mixing of polycrystalline specimens is not well understood. However, sufficient evidence is available to show that this increase reflects intracascade enhancement of a thermally-activated process that also occurs without irradiation. Recent evidence is presented which suggests that this process is Diffusion-induced Grain-Boundary Migration (DIGM). An important complementary conclusion is that because ion-beam mixing in single-crystal specimens exhibits no significant temperature dependence below that of RED, models that invoke only irradiation-specific phenomena, e.g., cascade-overlap, thermal-spikes, or liquid-diffusion, and hence which predict no difference in mixing behavior between single- or poly-crystalline specimens, cannot account for the existing results 20. National Centre for Radioactive Ion Beams (NCRIB) International Nuclear Information System (INIS) Chintalapudi, S.N. 1999-01-01 Radioactive Ion (nuclear) Beams have become prolific recently. Nuclear physics and associated subjects have staged a comeback to almost the beginning with the advent of RIB. A dedicated National Centre for RIB (NCRIB) proposed, discussed at several forums and under serious consideration is described 1. BEARS: Radioactive ion beams at LBNL International Nuclear Information System (INIS) Powell, J.; Guo, F.Q.; Haustein, P.E. 1998-01-01 BEARS (Berkeley Experiments with Accelerated Radioactive Species) is an initiative to develop a radioactive ion-beam capability at Lawrence Berkeley National Laboratory. The aim is to produce isotopes at an existing medical cyclotron and to accelerate them at the 88 inch Cyclotron. To overcome the 300-meter physical separation of these two accelerators, a carrier-gas transport system will be used. At the terminus of the capillary, the carrier gas will be separated and the isotopes will be injected into the 88 inch Cyclotron's Electron Cyclotron Resonance (ECR) ion source. The first radioactive beams to be developed will include 20-min 11 C and 70-sec 14 O, produced by (p,n) and (p,α) reactions on low-Z targets. A test program is currently being conducted at the 88 inch Cyclotron to develop the parts of the BEARS system. Preliminary results of these tests lead to projections of initial 11 C beams of up to 2.5 x 10 7 ions/sec and 14 O beams of 3 x 10 5 ions/sec 2. Treatment Planning for Ion Beam Therapy Science.gov (United States) Jäkel, Oliver The special aspects of treatment planning for ion beams are outlined in this chapter, starting with positioning and immobilization of the patient, describing imaging and segmentation, definition of treatment parameters, dose calculation and optimization, and, finally, plan assessment, verification, and quality assurance. CERN Document Server Bollen, G; Dezfuli, A M G; Henry, S; Herfurth, F; Kellerbauer, A G; Kim, T; Kluge, H J; Kohl, A; Lamour, E; Lunney, M D; Moore, R B; Quint, W; Schwarz, S; Varfalvy, P; Vermeeren, L 1998-01-01 ISOLTRAP is a Penning trap spectrometer at the on-line mass separator ISOLDE at CERN for the mass determination of radioisotopes. It consists of three electromagnetic traps in tandem; a Paul trap for ISOLDE beam collection, a Penning trap for cooling and purification and a high-precision Penning trap for the measurement of masses by cyclotron resonance. The Paul trap, which collects radionuclide ions using only electric fields and a noble buffer gas, has been essential for the masses of radionuclides that cannot be surface ionized. The success with this system has led to the present program to increase the collection efficiency by replacing the Paul trap by a radiofrequency quadrupole ion guide operating as a buncher. This system would also provide a DC ISOLDE beam of emittance approaching 1$\\pi$ -mm-mrad. (3 refs). 4. Surrey Ion Beam Centre: the EPSRC MRF for ion beam applications - 01002 International Nuclear Information System (INIS) Webb, R.P. 2016-01-01 The SIBC (Surrey Ion Beam Centre) is an element of the Virtual Ion Beam Centre that coordinates 3 U.K. experimental facilities: SIBC (University of Surrey) for implantation and ion beam applications, Miami and MEIS facility (University of Huddersfield) and gamma ray and neutron irradiation emulation facility (University of Manchester). The SIBC works actively with industry, developing bespoke processes and services, particularly for the photonics industry and provides ion beam facilities to about 20 companies across the world. It operates a stringent quality control program and is one of the few ion beam laboratories in the world to operate under ISO 9001 certification. The equipment of SIBC is presented and some applications of ion beam analysis concerning the identification of gunshot residues, the determination of the origin of a painting, the analysis of proteins are described. Different techniques such as PIXE (Particle Induced X-ray Emission), RBS (Rutherford Backscattering Spectroscopy), NRA (Nuclear Reaction Analysis), SIMS (Secondary Ion Mass Spectrometry) are also explained in the slides of the presentation that have been added at the end of the paper 5. Ion beam analysis of metal ion implanted surfaces International Nuclear Information System (INIS) Evans, P.J.; Chu, J.W.; Johnson, E.P.; Noorman, J.T.; Sood, D.K. 1993-01-01 Ion implantation is an established method for altering the surface properties of many materials. While a variety of analytical techniques are available for the characterisation of implanted surfaces, those based on particle accelerators such as Rutherford backscattering (RBS) and nuclear reaction analysis (NRA) provide some of the most useful and powerful for this purpose. Application of the latter techniques to metal ion implantation research at ANSTO will be described with particular reference to specific examples from recent studies. Where possible, the information obtained from ion beam analysis will be compared with that derived from other techniques such as Energy Dispersive X-ray (EDX) and Auger spectroscopies. 4 refs., 5 figs 6. Ion beam analysis of metal ion implanted surfaces Energy Technology Data Exchange (ETDEWEB) Evans, P J; Chu, J W; Johnson, E P; Noorman, J T [Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW (Australia); Sood, D K [Royal Melbourne Inst. of Tech., VIC (Australia) 1994-12-31 Ion implantation is an established method for altering the surface properties of many materials. While a variety of analytical techniques are available for the characterisation of implanted surfaces, those based on particle accelerators such as Rutherford backscattering (RBS) and nuclear reaction analysis (NRA) provide some of the most useful and powerful for this purpose. Application of the latter techniques to metal ion implantation research at ANSTO will be described with particular reference to specific examples from recent studies. Where possible, the information obtained from ion beam analysis will be compared with that derived from other techniques such as Energy Dispersive X-ray (EDX) and Auger spectroscopies. 4 refs., 5 figs. 7. Ion beam analysis of metal ion implanted surfaces Energy Technology Data Exchange (ETDEWEB) Evans, P.J.; Chu, J.W.; Johnson, E.P.; Noorman, J.T. [Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW (Australia); Sood, D.K. [Royal Melbourne Inst. of Tech., VIC (Australia) 1993-12-31 Ion implantation is an established method for altering the surface properties of many materials. While a variety of analytical techniques are available for the characterisation of implanted surfaces, those based on particle accelerators such as Rutherford backscattering (RBS) and nuclear reaction analysis (NRA) provide some of the most useful and powerful for this purpose. Application of the latter techniques to metal ion implantation research at ANSTO will be described with particular reference to specific examples from recent studies. Where possible, the information obtained from ion beam analysis will be compared with that derived from other techniques such as Energy Dispersive X-ray (EDX) and Auger spectroscopies. 4 refs., 5 figs. 8. The TMX heavy ion beam probe International Nuclear Information System (INIS) Hallock, G.A. 1994-01-01 A heavy ion beam probe has been used to measure the radial space potential distribution in the central cell of TMX. This was the first beam probe system to utilize computer control, CAMAC instrumentation, and fast time response for broadband fluctuation capabilities. The fast time response was obtained using off-line processing of the energy analyzer detector signals and wideband transimpedance amplifiers. The on-axis space potential was found to be 300--400 V, with φ e /T ec ∼8. The radial potential profile is parabolic when gas box fueling is used. The frequency of observed fluctuations was found to agree with the E x B plasma rotation frequency during the discharge. The measured Tl ++ secondary ion current level is consistent with calculations, given reasonable assumptions for beam attenuation 9. High repetition rate intense ion beam source International Nuclear Information System (INIS) Hammer, D.A.; Glidden, S.C.; Noonan, B. 1992-01-01 This final report describes a ≤ 150kV, 40kA, 100ns high repetition rate pulsed power system and intense ion beam source which is now in operation at Cornell University. Operation of the Magnetically-controlled Anode Plasma (MAP) ion diode at > 100Hz (burst mode for up to 10 pulse bursts) provides an initial look at repetition rate limitations of both the ion diode and beam diagnostics. The pulsed power systems are capable of ≥ 1kHz operation (up to 10 pulse bursts), but ion diode operation was limited to ∼100Hz because of diagnostic limitations. By varying MAP diode operating parameters, ion beams can be extracted at a few 10s of keV or at up to 150keV, the corresponding accelerating gap impedance ranging from about 1Ω to about 10Ω. The ability to make hundreds of test pulses per day at an average repetition rate of about 2 pulses per minute permits statistical analysis of diode operation as a function of various parameters. Most diode components have now survived more than 10 4 pulses, and the design and construction of the various pulsed power components of the MAP diode which have enabled us to reach this point are discussed. A high speed data acquisition system and companion analysis software capable of acquiring pulse data at 1ms intervals (in bursts of up to 10 pulses) and processing it in ≤ min is described 10. Ion-Beam-Excited Electrostatic Ion Cyclotron Instability DEFF Research Database (Denmark) Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens 1977-01-01 The stability limits of the ion‐beam‐excited, electrostatic, ion cyclotron instability were investigated in a Q‐machine plasma where the electrons could be heated by microwaves. In agreement with theory, the beam energy necessary for excitation decreased with increasing electron temperature.... 11. Ion-Beam-Excited, Electrostatic, Ion Cyclotron Instability DEFF Research Database (Denmark) Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens 1977-01-01 The stability limits of the ion‐beam‐excited, electrostatic, ion cyclotron instability were investigated in a Q‐machine plasma where the electrons could be heated by microwaves. In agreement with theory, the beam energy necessary for excitation decreased with increasing electron temperature.... 12. Ions kinematics in an electrostatic ion beam trap Energy Technology Data Exchange (ETDEWEB) Attia, D 2004-06-01 In this study, I have tried to provide a better understanding of the dynamics of ions inside an electrostatic ion beam trap. The electrostatic ion trap allows to store ions moving between two electrostatic mirrors. Although the trap has been developed already seven years ago, no direct measurement of the transversal velocity distribution of the ions has been performed. Such quantity is central for understanding the conditions under which a beam should be produced (mainly emittance) in order to be trapped by such a device. The data I have obtained during the course of this work are based on an experimental technique which relies on the direct imaging of the particles exiting the trap, as well as on numerical simulations of the ion trajectories inside the trap. I have personally been involved in the hardware development of the imaging system, the data acquisition and analysis of the data as well as il all numerical calculations presented here. These results allow us to obtain, for the first time, experimental information on the transverse phase space of the trap, and contribute to the overall understanding of the ion motion in this system. (author) 13. Study on broad beam heavy ion CT International Nuclear Information System (INIS) Ohno, Yumiko; Kohno, Toshiyuki; Sasaki, Hitomi; Nanbu, S.; Kanai, Tatsuaki 2003-01-01 To achieve the heavy ion radiotherapy more precisely, it is important to know the distribution of the electron density in a human body, which is highly related to the range of charged particles. From a heavy ion CT image, we can directly obtain the 2-D distribution of the electron density in a sample. For this purpose, we have developed a broad beam heavy ion CT system. The electron density was obtained using some kinds of solutions targets. Also the dependence of the spatial resolution on the target size and the kinds of beams was estimated in this work using cylinders targets of 40, 60 and 80 mm in diameter, each of them has a hole of 10 mm in diameter at the center of it. (author) 14. Target design for heavy ion beam fusion International Nuclear Information System (INIS) Meyer-ter-Vehn, J.; Metzler, N. 1981-07-01 Target design for Heavy Ion Beam Fusion and related physics are discussed. First, a modified version of the Kidder-Bodner model for pellet gain is presented and is used to define the working point (Esub(beam) = 4.8 MJ, Gain 83) for a reactor size target. Secondly, stopping of heavy ions in hot dense plasma is investigated and numerical results for stopping powers and ranges of 10 GeV Bi-ions in Pb, Li, and PbLi-alloy are given. Finally, results of an explicit implosion calculation, using the 1-D code MINIHY, are discussed in detail. The hydrodynamic efficiency is found to be about 5%. Special attention is given to the shock sequence leading to the ignition configuration. Also the growth of Rayleigh-Taylor instability at the absorber-pusher interface is estimated. (orig.) 15. Negative ion based neutral beams for plasma heating International Nuclear Information System (INIS) Prelec, K. 1978-01-01 Neutral beam systems based on negative ions have been considered because of a high expected power efficiency. Methods for the production, acceleration and neutralization of negative ions will be reviewed and possibilities for an application in neutral beam lines explored 16. Research and development of advanced materials using ion beam Energy Technology Data Exchange (ETDEWEB) Namba, Susumu [Nagasaki Inst. of Applied Science, Nagasaki (Japan) 1997-03-01 A wide range of research and development activities of advanced material synthesis using ion beams will be discussed, including ion beam applications to the state-of-the-art electronics from giant to nano electronics. (author) 17. Ion beam heating for fast ignition International Nuclear Information System (INIS) Gus'kov, S.Yu.; Limpouch, J.; Klimo, O. 2010-01-01 Complete text of publication follows. The characteristics features of the formation of the spatial distribution of the energy transferred to the plasma from a beam of ions with different initial energies, masses and charges under fast ignition conditions are determined. The motion of the Bragg peak is extended with respect to the spatial distribution of the temperature of the ion-beam-heated medium. The parameters of the ion beams are determined to initiate different regimes of fast ignition of thermonuclear fuel precompressed to a density of 300-500 g/cm 3 - the edge regime, in which the ignition region is formed at the outer boundary of the fuel, and the internal regime, in which the ignition region is formed in central parts of the fuel. The conclusion on the requirements for fast ignition by light and heavy ion beams is presented. It is shown that the edge heating with negative temperature gradient is described by a self-similar solution. Such a temperature distribution is the reason of the fact that the ignited beam energy at the edge heating is larger than the minimal ignition energy by factor 1.65. The temperature Bragg peak may be produced by ion beam heating in the reactor scale targets with pR-parameter larger than 3-4 g/cm 2 . In particular, for central ignition of the targets with pR-parameters in the range of 4-8 g/cm 2 the ion beam energy should be, respectively, from 5 to 7 times larger than the minimal ignition energy. The work by S.Ye. Gus'kov, D.V. Il'in, and V.E. Sherman was supported by the Ministry of Education and Science of the Russian Federation under the program 'Development of the Scientific Potential of High Education for 2009-2010' (project no. 2.1.1/1505) and the Russian Foundation for Basic Research (project no. 08-02-01394 a ). The work by J. Limpouch and O. Klimo was supported by the Czech Ministry of Education (project no. LC528, MSM6840770022). 18. Current neutralization of converging ion beams International Nuclear Information System (INIS) Mosher, D. 1978-01-01 It is desired to consider the problem of current neutralization of heavy ion beams traversing gas backgrounds in which the conductivity changes due to beam heating and beam convergence. The procedure is to determine Green's-function solutions to the magnetic-diffusion equation derived from Maxwell's equations and an assumed scaler-plasma conductivity sigma for the background-electron current density j/sub e/. The present calculation is more general than some previously carried out in that arbitrary time variations for the beam current j/sub b/ and conductivity are allowed and the calculation is valid for both weak and strong neutralization. Results presented here must be combined with an appropriate energy-balance equation for the heated background in order to obtain the neutralization self-consistently 19. Applications of ion beam analysis workshop. Workshop handbook International Nuclear Information System (INIS) 1995-01-01 A workshop on applications of ion beam analysis was held at ANSTO, immediate prior to the IBMM-95 Conference in Canberra. It aims was to review developments and current status on use of ion beams for analysis, emphasizing the following aspects: fundamental ion beam research and secondary effects of ion beams; material sciences, geological, life sciences, environmental and industrial applications; computing codes for use in accelerator research; high energy heavy ion scattering and recoil; recent technological development using ion beams. The handbook contains the workshop's program, 29 abstracts and a list of participants 20. Image-projection ion-beam lithography International Nuclear Information System (INIS) Miller, P.A. 1989-01-01 Image-projection ion-beam lithography is an attractive alternative for submicron patterning because it may provide high throughput; it uses demagnification to gain advantages in reticle fabrication, inspection, and lifetime; and it enjoys the precise deposition characteristics of ions which cause essentially no collateral damage. This lithographic option involves extracting low-mass ions (e.g., He + ) from a plasma source, transmitting the ions at low voltage through a stencil reticle, and then accelerating and focusing the ions electrostatically onto a resist-coated wafer. While the advantages of this technology have been demonstrated experimentally by the work of IMS (Austria), many difficulties still impede extension of the technology to the high-volume production of microelectronic devices. We report a computational study of a lithography system designed to address problem areas in field size, telecentricity, and chromatic and geometric aberration. We present a novel ion-column-design approach and conceptual ion-source and column designs which address these issues. We find that image-projection ion-beam technology should in principle meet high-volume-production requirements. The technical success of our present relatively compact-column design requires that a glow-discharge-based ion source (or equivalent cold source) be developed and that moderate further improvement in geometric aberration levels be obtained. Our system requires that image predistortion be employed during reticle fabrication to overcome distortion due to residual image nonlinearity and space-charge forces. This constitutes a software data preparation step, as do correcting for distortions in electron lithography columns and performing proximity-effect corrections. Areas needing further fundamental work are identified 1. The application of ion beams to corrosion science International Nuclear Information System (INIS) Ashworth, V.; Grant, W.A.; Proctor, R.P.M. 1976-01-01 Briefly, the paper provides some basic information on the use of ion beams for surface alloying and surface analysis. After a brief historical review of those fields in which the techniques are already widely applied the important features of typical ion beam machines are described. The basic processes that occur when an ion beam strikes a solid are then considered. Selected ion beam analysis techniques are then discussed. Attention is drawn, wherever possible, to applications in corrosion science and engineering. (author) 2. Development of a beam ion velocity detector for the heavy ion beam probe Energy Technology Data Exchange (ETDEWEB) Fimognari, P. J., E-mail: [email protected]; Crowley, T. P.; Demers, D. R. [Xantho Technologies, LLC, Madison, Wisconsin 53705 (United States) 2016-11-15 In an axisymmetric plasma, the conservation of canonical angular momentum constrains heavy ion beam probe (HIBP) trajectories such that measurement of the toroidal velocity component of secondary ions provides a localized determination of the poloidal flux at the volume where they originated. We have developed a prototype detector which is designed to determine the beam angle in one dimension through the detection of ion current landing on two parallel planes of detecting elements. A set of apertures creates a pattern of ion current on wires in the first plane and solid metal plates behind them; the relative amounts detected by the wires and plates determine the angle which beam ions enter the detector, which is used to infer the toroidal velocity component. The design evolved from a series of simulations within which we modeled ion beam velocity changes due to equilibrium and fluctuating magnetic fields, along with the ion beam profile and velocity dispersion, and studied how these and characteristics such as the size, cross section, and spacing of the detector elements affect performance. 3. Development of a beam ion velocity detector for the heavy ion beam probe International Nuclear Information System (INIS) Fimognari, P. J.; Crowley, T. P.; Demers, D. R. 2016-01-01 In an axisymmetric plasma, the conservation of canonical angular momentum constrains heavy ion beam probe (HIBP) trajectories such that measurement of the toroidal velocity component of secondary ions provides a localized determination of the poloidal flux at the volume where they originated. We have developed a prototype detector which is designed to determine the beam angle in one dimension through the detection of ion current landing on two parallel planes of detecting elements. A set of apertures creates a pattern of ion current on wires in the first plane and solid metal plates behind them; the relative amounts detected by the wires and plates determine the angle which beam ions enter the detector, which is used to infer the toroidal velocity component. The design evolved from a series of simulations within which we modeled ion beam velocity changes due to equilibrium and fluctuating magnetic fields, along with the ion beam profile and velocity dispersion, and studied how these and characteristics such as the size, cross section, and spacing of the detector elements affect performance. 4. Inertial confinement fusion with light ion beams International Nuclear Information System (INIS) VanDevender, J.P.; Cook, D.L. 1986-01-01 The Particle Beam Fusion Accelerator II (PBFA II) is presently under construction and is the only existing facility with the potential of igniting thermonuclear fuel in the laboratory. The accelerator will generate up to 5 megamperes of lithium ions at 30 million electron volts and will focus them onto an inertial confinement fusion (ICF) target after beam production and focusing have been optimized. Since its inception, the light ion approach to ICF has been considered the one that combines low cost, high risk, and high payoff. The beams are of such high density that their self-generated electric and magnetic fields were thought to prohibit high focal intensities. Recent advances in beam production and focusing demonstrate that these self-forces can be controlled to the degree required for ignition, break-even, and high gain experiments. ICF has been pursued primarily for its potential military applications. However, the high efficiency and cost-effectiveness of the light ion approach enhance its potential for commercial energy application as well 5. Electron-ion recombination in merged beams International Nuclear Information System (INIS) Wolf, A.; Habs, D.; Lampert, A.; Neumann, R.; Schramm, U.; Schuessler, T.; Schwalm, D. 1993-01-01 Detailed studies of recombination processes between electrons and highly charged ions have become possible by recent improvements of merged-beams experiments. We discuss in particular measurements with stored cooled ion beams at the Test Storage Ring (TSR) in Heidelberg. The cross section of dielectronic recombination was measured with high energy resolution for few-electron systems up to the nuclear charge of Cu at a relative energy up to 2.6 keV. At low energy (∼0.1 eV) total recombination rates of several ions were measured and compared with calculated radiative recombination rates. Laser-stimulated recombination of protons and of C 6+ ions was investigated as a function of the photon energy using visible radiation. Both the total recombination rates and the stimulated recombination spectra indicate that in spite of the short interaction time in merged beams, also collisional capture of electrons into weakly bound levels (related to three-body recombination) could be important 6. Inertial fusion with heavy ion beams International Nuclear Information System (INIS) Bock, R.; Hofmann, I.; Arnold, R. 1984-01-01 The underlying principle of inertial confinement is the irradiation of a small pellet filled with DT-fuel by laser or particle beams in order to compress the fuel and ignite it. As 'drivers' for this process large laser installations and light-ion devices have been built since then and the results obtained during the past few years have increased our confidence, that the ignition conditions might be reached. Further conditions, however, have to be fulfilled for operating a power plant. In particular, the driver needs to have enough efficiency to be economical, and for a continuous energy production a high repetition rate and availability is required. It is less than ten years since it was realized that heavy ion beams might be a promising candidate for achieving inertial confinement fusion (ICF). Due to the evolution of high-energy and heavy-ion physics during the past 25 years, accelerators have attained a high technical and technological standard and an excellent operational reliability. Nevertheless, the heavy ion driver for a fusion power plant requires beam specifications exceeding those of existing accelerators considerably. (Auth.) 7. Ion beam biotechnology and its application to maize breeding International Nuclear Information System (INIS) Yu Lixia; Li Wenjian; Dong Xicun; Zhou Libin; Ma Shuang 2008-01-01 Since the mid of 1980's, ion beam had been widely used in mutagenic breeding of various crops. Ion beam biotechnology had provided a new way for improving corn variety and creating new germplasm resources, and had promoted the development of maize breeding. The ion beam characteristics, the mutagenic mechanism and its application in maize breeding were described. (authors) 8. MEV Energy Electrostatic Accelerator Ion Beam Emittance Measurement OpenAIRE I.G. Ignat’ev; M.I. Zakharets; S.V. Kolinko; D.P. Shulha 2014-01-01 The testing equipment was designed, manufactured and tried out permitting measurements of total current, current profile and emittance of an ion beam extracted from the ion beam. MeV energy electrostatic accelerator ion H + beam emittance measurement results are presented. 9. Ion beam figuring of silicon aspheres Science.gov (United States) Demmler, Marcel; Zeuner, Michael; Luca, Alfonz; Dunger, Thoralf; Rost, Dirk; Kiontke, Sven; Krüger, Marcus 2011-03-01 Silicon lenses are widely used for infrared applications. Especially for portable devices the size and weight of the optical system are very important factors. The use of aspherical silicon lenses instead of spherical silicon lenses results in a significant reduction of weight and size. The manufacture of silicon lenses is more challenging than the manufacture of standard glass lenses. Typically conventional methods like diamond turning, grinding and polishing are used. However, due to the high hardness of silicon, diamond turning is very difficult and requires a lot of experience. To achieve surfaces of a high quality a polishing step is mandatory within the manufacturing process. Nevertheless, the required surface form accuracy cannot be achieved through the use of conventional polishing methods because of the unpredictable behavior of the polishing tools, which leads to an unstable removal rate. To overcome these disadvantages a method called Ion Beam Figuring can be used to manufacture silicon lenses with high surface form accuracies. The general advantage of the Ion Beam Figuring technology is a contactless polishing process without any aging effects of the tool. Due to this an excellent stability of the removal rate without any mechanical surface damage is achieved. The related physical process - called sputtering - can be applied to any material and is therefore also applicable to materials of high hardness like Silicon (SiC, WC). The process is realized through the commercially available ion beam figuring system IonScan 3D. During the process, the substrate is moved in front of a focused broad ion beam. The local milling rate is controlled via a modulated velocity profile, which is calculated specifically for each surface topology in order to mill the material at the associated positions to the target geometry. The authors will present aspherical silicon lenses with very high surface form accuracies compared to conventionally manufactured lenses. 10. NSUF Ion Beam Investment Options Workshop Report Energy Technology Data Exchange (ETDEWEB) Heidrich, Brenden John [Idaho National Lab. (INL), Idaho Falls, ID (United States) 2016-03-01 The workshop that generated this data was convened to develop a set of recommendations (a priority list) for possible funding in the area of US domestic ion beam irradiation capabilities for nuclear energy-focused RD&D. The results of this workshop were intended for use by the Department of Energy - Office of Nuclear Energy (DOE-NE) for consideration of support for these facilities. The workshop considered, as part of the initial potential future support discussions, input submitted through the Office of Nuclear Energy Request for Information (RFI) (DE-SOL-0008318, April 13, 2015), but welcomed discussion (and presentation) of other options, whether specific or general in scope. Input from users, including DOE-NE program interests and needs for ion irradiation RD&D were also included. Participants were selected from various sources: RFI respondents, NEUP/NEET infrastructure applicants, universities with known expertise in nuclear engineering and materials science and other developed sources. During the three days from March 22-24, 2016, the workshop was held at the Idaho National Laboratory Meeting Center in the Energy Innovation Laboratory at 775 University Drive, Idaho Falls, ID 83401. Thirty-one members of the ion beam community attended the workshop, including 15 ion beam facilities, six representatives of Office of Nuclear Energy R&D programs, an industry representative from EPRI and the chairs of the NSUF User’s Organization and the NSUF Scientific Review Board. Another four ion beam users were in attendance acting as advisors to the process, but did not participate in the options assessment. Three members of the sponsoring agency, the Office of Science and Technology Innovation (NE-4) also attended the workshop. 11. Ion beam irradiation effects on aromatic polymers International Nuclear Information System (INIS) Shukushima, Satoshi; Ueno, Keiji 1995-01-01 We studied the optical and thermal properties of aromatic polymer films which had been irradiated with 1 MeV H + , H 2 + and He + ions. The examined aromatic polymers were polyetherether ketone(PEEK), polyetherimide(PEI), polyether sulfon(PES), polysulfon(PSF), and polyphenylene sulfide(PPS). The optical densities at 300nm of PES and PSF greatly increased after the irradiation. The optical densities at 400nm of all the examined polymer lineally increased with the irradiation dose. The PEEK film which had been irradiated with 1 MeV H + was not deformed above melting point. This demonstrates that cross-linking occurs in PEEK films by ion beam irradiation. As for the effects, depending on the mass of the irradiated ions, it was found that the ions with a high mass induced larger effects on the aromatic polymers for the same absorption energy. (author) 12. Surface generation of negative hydrogen ion beams International Nuclear Information System (INIS) Bommel, P.J.M. van. 1984-01-01 This thesis describes investigations on negative hydrogen ion sources at the ampere level. Formation of H - ions occurs when positive hydrogen ions capture two electrons at metal surfaces. The negative ionization probability of hydrogen at metal surfaces increases strongly with decreasing work function of the surface. The converters used in this study are covered with cesium. Usually there are 'surface plasma sources' in which the hydrogen source plasma interacts with a converter. In this thesis the author concentrates upon investigating a new concept that has converters outside the plasma. In this approach a positive hydrogen ion beam is extracted from the plasma and is subsequently reflected from a low work function converter surface. (Auth.) 13. Deflagration wave formed by ion beam, 2 International Nuclear Information System (INIS) Abe, T.; Kasuya, K.; Niu, K.; Tamba, M. 1979-06-01 Analyses are given for structures of deflagration waves formed by ion beams in spherical targets. The singularity at the sonic point disappears in the spherical target if the beam pressure is in balance with the plasma pressure. The expanding supersonic flow of the background plasma can be connected with the subsonic flow in the core of the target through the deflagration wave. The length and the strength of the deflagration wave in the spherical target is comparable with the corresponding ones in the slab target. (author) 14. Light-ion beam for microelectronic applications International Nuclear Information System (INIS) Hirsch, L.; Tardy, P.; Wantz, G.; Huby, N.; Moretto, P.; Serani, L.; Natali, F.; Damilano, B.; Duboz, J.Y.; Reverchon, J.L. 2005-01-01 In this paper we describe the structure and the composition of (Al,Ga)N/GaN Bragg reflectors obtained from Rutherford backscattering spectroscopy. Bragg reflectors constitute a part of blue (λ = 450 nm) resonant cavity light emitting diodes. To improve the measurement accuracy, three tilt angles have been used (10 deg. , 25 deg. and 50 deg. ). In a second part of the paper, ion beam induced charges study has been carried out, with a 2 MeV 4 He + micro-beam, on metal-semiconductor-metal UV photodetectors. Results have been taken into account for the design of the photodetector electrodes 15. Design study of primary ion provider for relativistic heavy ion collider electron beam ion source. Science.gov (United States) Kondo, K; Kanesue, T; Tamura, J; Okamura, M 2010-02-01 Brookhaven National Laboratory has developed the new preinjector system, electron beam ion source (EBIS) for relativistic heavy ion collider (RHIC) and National Aeronautics and Space Administration Space Radiation Laboratory. Design of primary ion provider is an essential problem since it is required to supply beams with different ion species to multiple users simultaneously. The laser ion source with a defocused laser can provide a low charge state and low emittance ion beam, and is a candidate for the primary ion source for RHIC-EBIS. We show a suitable design with appropriate drift length and solenoid, which helps to keep sufficient total charge number with longer pulse length. The whole design of primary ion source, as well as optics arrangement, solid targets configuration and heating about target, is presented. 16. Neutralization principles for the Extraction and Transport of Ion Beams CERN Document Server Riege, H 2000-01-01 The strict application of conventional extraction techniques of ion beams from a plasma source is characterized by a natural intensity limit determined by space charge.The extracted current may be enhanced far beyond this limit by neutralizing the space charge of the extracted ions in the first extraction gap of the source with electrons injected from the opposite side. The transverse and longitudinal emittances of a neutralized ion beam, hence its brightness, are preserved. Results of beam compensation experiments, which have been carried out with a laser ion source, are resumed for proposing a general scheme of neutralizing ion sources and their adjacent low-energy beam transport channels with electron beams. Many technical applications of high-mass ion beam neutralization technology may be identified: the enhancement of ion source output for injection into high-intensity, low-and high-energy accelerators, or ion thrusters in space technology, for the neutral beams needed for plasma heating of magnetic conf... 17. Time resolved ion beam induced charge collection International Nuclear Information System (INIS) Sexton W, Frederick; Walsh S, David; Doyle L, Barney; Dodd E, Paul 2000-01-01 Under this effort, a new method for studying the single event upset (SEU) in microelectronics has been developed and demonstrated. Called TRIBICC, for Time Resolved Ion Beam Induced Charge Collection, this technique measures the transient charge-collection waveform from a single heavy-ion strike with a -.03db bandwidth of 5 GHz. Bandwidth can be expanded up to 15 GHz (with 5 ps sampling windows) by using an FFT-based off-line waveform renormalization technique developed at Sandia. The theoretical time resolution of the digitized waveform is 24 ps with data re-normalization and 70 ps without re-normalization. To preserve the high bandwidth from IC to the digitizing oscilloscope, individual test structures are assembled in custom high-frequency fixtures. A leading-edge digitized waveform is stored with the corresponding ion beam position at each point in a two-dimensional raster scan. The resulting data cube contains a spatial charge distribution map of up to 4,096 traces of charge (Q) collected as a function of time. These two dimensional traces of Q(t) can cover a period as short as 5 ns with up to 1,024 points per trace. This tool overcomes limitations observed in previous multi-shot techniques due to the displacement damage effects of multiple ion strikes that changed the signal of interest during its measurement. This system is the first demonstration of a single-ion transient measurement capability coupled with spatial mapping of fast transients 18. Time resolved ion beam induced charge collection Energy Technology Data Exchange (ETDEWEB) SEXTON,FREDERICK W.; WALSH,DAVID S.; DOYLE,BARNEY L.; DODD,PAUL E. 2000-04-01 Under this effort, a new method for studying the single event upset (SEU) in microelectronics has been developed and demonstrated. Called TRIBICC, for Time Resolved Ion Beam Induced Charge Collection, this technique measures the transient charge-collection waveform from a single heavy-ion strike with a {minus}.03db bandwidth of 5 GHz. Bandwidth can be expanded up to 15 GHz (with 5 ps sampling windows) by using an FFT-based off-line waveform renormalization technique developed at Sandia. The theoretical time resolution of the digitized waveform is 24 ps with data re-normalization and 70 ps without re-normalization. To preserve the high bandwidth from IC to the digitizing oscilloscope, individual test structures are assembled in custom high-frequency fixtures. A leading-edge digitized waveform is stored with the corresponding ion beam position at each point in a two-dimensional raster scan. The resulting data cube contains a spatial charge distribution map of up to 4,096 traces of charge (Q) collected as a function of time. These two dimensional traces of Q(t) can cover a period as short as 5 ns with up to 1,024 points per trace. This tool overcomes limitations observed in previous multi-shot techniques due to the displacement damage effects of multiple ion strikes that changed the signal of interest during its measurement. This system is the first demonstration of a single-ion transient measurement capability coupled with spatial mapping of fast transients. 19. Neutralized ion beam modification of cellulose membranes for study of ion charge effect on ion-beam-induced DNA transfer Science.gov (United States) Prakrajang, K.; Sangwijit, K.; Anuntalabhochai, S.; Wanichapichart, P.; Yu, L. D. 2012-02-01 Low-energy ion beam biotechnology (IBBT) has recently been rapidly developed worldwide. Ion-beam-induced DNA transfer is one of the important applications of IBBT. However, mechanisms involved in this application are not yet well understood. In this study plasma-neutralized ion beam was applied to investigate ion charge effect on induction of DNA transfer. Argon ion beam at 7.5 keV was neutralized by RF-driven plasma in the beam path and then bombarded cellulose membranes which were used as the mimetic plant cell envelope. Electrical properties such as impedance and capacitance of the membranes were measured after the bombardment. An in vitro experiment on plasmid DNA transfer through the cellulose membrane was followed up. The results showed that the ion charge input played an important role in the impedance and capacitance changes which would affect DNA transfer. Generally speaking, neutral particle beam bombardment of biologic cells was more effective in inducing DNA transfer than charged ion beam bombardment. 20. Beam analysis spectrometer for relativistic heavy ions International Nuclear Information System (INIS) Schimmerling, W.; Subramanian, T.S.; McDonald, W.J.; Kaplan, S.N.; Sadoff, A.; Gabor, G. 1983-01-01 A versatile spectrometer useful for measuring the mass, charge, energy, fluence and angular distribution of primaries and fragments associated with relativistic heavy ion beams is described. The apparatus is designed to provide accurate physical data for biology experiments and medical therapy planning as a function of depth in tissue. The spectrometer can also be used to measure W, the average energy to produce an ion pair, range-energy, dE/dx, and removal cross section data of interest in nuclear physics. (orig.) 1. Measurement of ultra-low ion energy of decelerated ion beam using a deflecting electric field Energy Technology Data Exchange (ETDEWEB) Thopan, P.; Suwannakachorn, D.; Tippawan, U. [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Yu, L.D., E-mail: [email protected] [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400 (Thailand) 2015-12-15 In investigation on ultra-low-energy ion bombardment effect on DNA, an ion beam deceleration lens was developed for high-quality ultra-low-energy ion beam. Measurement of the ion energy after deceleration was necessary to confirm the ion beam really decelerated as theoretically predicted. In contrast to conventional methods, this work used a simple deflecting electrostatic field after the deceleration lens to bend the ion beam. The beam bending distance depended on the ion energy and was described and simulated. A system for the measurement of the ion beam energy was constructed. It consisted of a pair of parallel electrode plates to generate the deflecting electrical field, a copper rod measurement piece to detect ion beam current, a vernier caliper to mark the beam position, a stepping motor to translate the measurement rod, and a webcam-camera to read the beam bending distance. The entire system was installed after the ion-beam deceleration lens inside the large chamber of the bioengineering vertical ion beam line. Moving the measurement rod across the decelerated ion beam enabled to obtain beam profiles, from which the beam bending distance could be known and the ion beam energy could be calculated. The measurement results were in good agreement with theoretical and simulated results. 2. Production of microbunched beams of very highly charged ions with an electron beam ion source International Nuclear Information System (INIS) Stoeckli, M.P. 1998-01-01 Electron beam ion sources produce very highly charged ions most efficiently in a batch mode as the confinement time can be directly optimized for the production of the desired charge state. If, after confinement, the voltage of the ion-confining downstream dam is lowered rapidly, all ions escape and form an ion beam pulse with a length of a few tens of μs. Raising the main trap voltage while maintaining a constant dam voltage in a open-quotes spill-over expulsionclose quotes reduces the energy spread of the expelled ions. The longer time periods of open-quotes slow-,close quotes open-quotes leaky batch mode-,close quotes and open-quotes direct current (dc) batch mode-close quotes expulsions allow for increasing the ion beam duty cycle. Combining the rapid expulsion with one of the latter methods allows for the expulsion of the ions of a single batch in many small microbunches with variable intervals, maintaining the low energy spread and the increased duty cycle of slow expulsions. Combining the open-quotes microbunchingclose quotes with open-quotes dc batch mode productionclose quotes and a multitrap operation will eventually allow for the production of equally intense ion bunches over a wide range of frequencies without any deadtime, and with minimal compromise on the most efficient production parameters. copyright 1998 American Institute of Physics 3. Ion-acoustic solitons in a plasma with electron beam International Nuclear Information System (INIS) Esfandyari, A. R.; Khorram, S. 2001-01-01 Ion-acoustic solitons in a collisionless plasma consisting of warm ions, hot isothermal electrons and a electron beam are studied by using the reductive perturbation method. The basic set of fluid equations is reduced to Korteweg-de Vries and modified Korteweg-de Vries temperature and electron beam on ion acoustic equations. The effect of ion solitons are investigated 4. Condensed matter physics with radioactive ion beams International Nuclear Information System (INIS) Haas, H. 1996-01-01 An overview of the present uses of radioactive ion beams from ISOLDE for condensed matter research is presented. As simple examples of such work, tracer studies of diffusion processes with radioisotopes and blocking/channeling measurements of emitted particles for lattice location are discussed. Especially the application of nuclear hyperfine interaction techniques such as PAC or Moessbauer spectroscopy has become a powerful tool to study local electronic and structural properties at impurities. Recently, interesting information on impurity properties in semiconductors has been obtained using all these methods. The extreme sensitivity of nuclear techniques makes them also well suited for investigations of surfaces, interfaces, and biomolecules. Some ideas for future uses of high energy radioactive ion beams beyond the scope of the present projects are outlined: the study of diffusion in highly immiscible systems by deep implantation, nuclear polarization with the tilted-foil technique, and transmutation doping of wide-bandgap semiconductors. (orig.) 5. Bayesian analysis of ion beam diagnostics International Nuclear Information System (INIS) Toussaint, U. von; Fischer, R.; Dose, V. 2001-01-01 Ion beam diagnostics are routinely used for quantitative analysis of the surface composition of mixture materials up to a depth of a few μm. Unfortunately, advantageous properties of the diagnostics, like high depth resolution in combination with a large penetration depth, no destruction of the surface, high sensitivity for large as well as for small atomic numbers, and high sensitivity are mutually exclusive. Among other things, this is due to the ill-conditioned inverse problem of reconstructing depth distributions of the composition elements. Robust results for depth distributions are obtained with adaptive methods in the framework of Bayesian probability theory. The method of adaptive kernels allows for distributions which contain only the significant information of the data while noise fitting is avoided. This is achieved by adaptively reducing the degrees of freedom supporting the distribution. As applications for ion beam diagnostics Rutherford backscattering spectroscopy and particle induced X-ray emission are shown 6. Ion beam therapy fundamentals, technology, clinical applications CERN Document Server 2012-01-01 The book provides a detailed, up-to-date account of the basics, the technology, and the clinical use of ion beams for radiation therapy. Theoretical background, technical components, and patient treatment schemes are delineated by the leading experts that helped to develop this field from a research niche to its current highly sophisticated and powerful clinical treatment level used to the benefit of cancer patients worldwide. Rather than being a side-by-side collection of articles, this book consists of related chapters. It is a common achievement by 76 experts from around the world. Their expertise reflects the diversity of the field with radiation therapy, medical and accelerator physics, radiobiology, computer science, engineering, and health economics. The book addresses a similarly broad audience ranging from professionals that need to know more about this novel treatment modality or consider to enter the field of ion beam therapy as a researcher. However, it is also written for the interested public an... 7. Ion sources development at GANIL for radioactive beams and high charge state ions International Nuclear Information System (INIS) Leroy, R.; Barue, C.; Canet, C.; Dupuis, M.; Flambard, J.L.; Gaubert, G.; Gibouin, S.; Huguet, Y.; Jardin, P.; Lecesne, N.; Leherissier, P.; Lemagnen, F.; Pacquet, J.Y.; Pellemoine-Landre, F.; Rataud, J.P.; Saint-Laurent, M.G.; Villari, A.C.C.; Maunoury, L. 2001-01-01 The GANIL laboratory has in charge the production of ion beams for nuclear and non nuclear physics. This article reviews the last developments that are underway in the fields of radioactive ion beam production, increase of the metallic ion intensities and production of highly charges ion beams. (authors) 8. Simulations of multistage intense ion beam acceleration International Nuclear Information System (INIS) Slutz, S.A.; Poukey, J.W. 1992-01-01 An analytic theory for magnetically insulated, multistage acceleration of high intensity ion beams, where the diamagnetic effect due to electron flow is important, has been presented by Slutz and Desjarlais. The theory predicts the existence of two limiting voltages called V 1 (W) and V 2 (W), which are both functions of the injection energy qW of ions entering the accelerating gap. As the voltage approaches V 1 (W), unlimited beam-current density can penetrate the gap without the formation of a virtual anode because the dynamic gap goes to zero. Unlimited beam current density can penetrate an accelerating gap above V 2 (W), although a virtual anode is formed. It was found that the behavior of these limiting voltages is strongly dependent on the electron density profile. The authors have investigated the behavior of these limiting voltages numerically using the 2-D particle-in-cell (PIC) code MAGIC. Results of these simulations are consistent with the superinsulated analytic results. This is not surprising, since the ignored coordinate eliminates instabilities known to be important from studies of single stage magnetically insulated ion diodes. To investigate the effect of these instabilities the authors have simulated the problem with the 3-D PIC code QUICKSILVER, which indicates behavior that is consistent with the saturated model 9. Ion accumulation and space charge neutralization in intensive electron beams for ion sources and electron cooling International Nuclear Information System (INIS) Shirkov, G.D. 1996-01-01 The Electron Beam Ion Sources (EBIS), Electron Beam Ion Traps (EBIT) and electron beams for electron cooling application have the beam parameters in the same ranges of magnitudes. EBIS and EBIT produce and accumulate ions in the beam due to electron impact ionization. The cooling electron beam accumulates positive ions from the residual gas in the accelerator chamber during the cooling cycle. The space charge neutralization of cooling beam is also used to reduce the electron energy spread and enhance the cooling ability. The advanced results of experimental investigations and theoretical models of the EBIS electron beams are applied to analyze the problem of beam neutralization in the electron cooling techniques. The report presents the analysis of the most important processes connected with ion production, accumulation and losses in the intensive electron beams of ion sources and electron cooling systems for proton and ion colliders. The inelastic and elastic collision processes of charged particles in the electron beams are considered. The inelastic processes such as ionization, charge exchange and recombination change the charge states of ions and neutral atoms in the beam. The elastic Coulomb collisions change the energy of particles and cause the energy redistribution among components in the electron-ion beams. The characteristic times and specific features of ionization, beam neutralization, ion heating and loss in the ion sources and electron cooling beams are determined. The dependence of negative potential in the beam cross section on neutralization factor is studied. 17 refs., 5 figs., 1 tab 10. Broad beam ion sources and some surface processes International Nuclear Information System (INIS) Neumann, H.; Scholze, F.; Tarz, M.; Schindler, A.; Wiese, R.; Nestler, M.; Blum, T. 2005-01-01 Modern broad-beam multi-aperture ion sources are widely used in material and surface technology applications. Customizing the generated ion beam properties (i. e. the ion current density profile) for specific demands of the application is a main challenge in the improvement of the ion beam technologies. First we introduce ion sources based on different plasma excitation principles shortly. An overview of source plasma and ion beam measurement methods deliver input data for modelling methods. This beam profile modelling using numerical trajectory codes and the validation of the results by Faraday cup measurements as a basis for ion beam profile design are described. Furthermore possibilities for ex situ and in situ beam profile control are demonstrated, like a special method for in situ control of a linear ion source beam profile, a grid modification for circular beam profile design and a cluster principle for broad beam sources. By means of these methods, the beam shape may be adapted to specific technological demands. Examples of broad beam source application in ion beam figuring of optical surfaces, modification of stainless steel, photo voltaic processes and deposition of EUVL-multilayer stacks are finally presented. (Author) 11. Development of the Holifield Radioactive Ion Beam Facility International Nuclear Information System (INIS) Tatum, B.A. 1997-01-01 The Holifield Radioactive Ion Beam Facility (HRIBF) construction project has been completed and the first radioactive ion beam has been successfully accelerated. The project, which began in 1992, has involved numerous facility modifications. The Oak Ridge Isochronous Cyclotron has been converted from an energy booster for heavy ion beams to a light ion accelerator with internal ion source. A target-ion source and mass analysis system have been commissioned as key components of the facility's radioactive ion beam injector to the 25MV tandem electrostatic accelerator. Beam transport lines have been completed, and new diagnostics for very low intensity beams have been developed. Work continues on a unified control system. Development of research quality radioactive beams for the nuclear structure and nuclear astrophysics communities continues. This paper details facility development to date 12. Ion beam deposited epitaxial thin silicon films International Nuclear Information System (INIS) Orrman-Rossiter, K.G.; Al-Bayati, A.H.; Armour, D.G.; Donnelly, S.E.; Berg, J.A. van den 1991-01-01 Deposition of thin films using low energy, mass-separated ion beams is a potentially important low temperature method of producing epitaxial layers. In these experiments silicon films were grown on Si (001) substrates using 10-200 eV 28 Si + and 30 Si + ions at substrate temperatures in the range 273-1073 K, under ultrahigh-vacuum conditions (deposition pressure -7 Pa). The film crystallinity was assessed in situ using medium energy ion scattering (MEIS). Films of crystallinity comparable to bulk samples were grown using 10-40 eV 28 Si + and 30 Si + ions at deposition temperatures in the range 623-823 K. These experiments confirmed the role of key experimental parameters such as ion energy, substrate temperature during deposition, and the surface treatment prior to deposition. It was found that a high temperature in situ anneal (1350-1450 K) gave the best results for epitaxial nucleation, whereas low energy (20-40 eV) Cl + ion bombardment resulted in amorphous film growth. The deposition energy for good epitaxial growth indicates that it is necessary to provide enough energy to induce local mobility but not to cause atomic displacements leading to the buildup of stable defects, e.g. divacancies, below the surface layer of the growing film. (orig.) 13. Mutation induction of orchids by ion beams International Nuclear Information System (INIS) Affrida Abu Hassan; Zaiton Ahmad; Sakinah Ariffin; Oono, Yutaka; Hase, Yoshihiro; Shikazono; Naoya; Narumi, Issay; Tanaka, Atsushi 2010-01-01 Mutation induction using ionizing radiation provides an effective alternative means for improvement of orchids. In this study, ion beams were used because they have much higher linear energy transfer (LET) than X-rays or gamma rays, and subsequently lead to higher mutation frequency and broad mutation spectrum. The proto corm-like bodies (PLBs) of three orchid species (Dendrobium crumenatum, Dendrobium mirbellianum) were irradiated at various doses with 320 MeV 12 C 6+ ions accelerated by Azimuthally Varying Field (AVF) cyclotron at JAEAs Takasaki Ion Accelerators for Advanced Radiation Application (TIARA). The optimum irradiation condition and the effect of irradiation on each species were studied, particularly on flower colour and morphology, flowering habit and insect resistance. Dose effects on plantlet regeneration for each species were also obtained. Some morphological changes were observed in flowers of Dendrobium crumenatum, whilst one insect resistant mutant was obtained in Dendrobium mirbellianum. (author) 14. Laboratory of ion beam applications at ATOMKI International Nuclear Information System (INIS) Borbely-Kiss, I.; Huszank, R.; Kertesz, Zs.; Kiss, A.Z.; Koltay, E.; Rajta, I.; Simon, A.; Szabo, Gy.; Szikszai, Z.; Szilasi, S.Z.; Szoboszlai, Z.; Uzonyi, I. 2008-01-01 Introduction. The Laboratory of Ion Beam Applications of ATOMKI is devoted to applications of atomic and nuclear physics in the fields of environmental research, biomedicine, geology, materials and surface science (including ion beam induced damage investigations and proton beam lithography) and cultural heritage research. We perform our work in the frame of various projects and collaborations: EU, IAEA, R and D, OTKA, etc. Our laboratory provides service for external (national and international) and internal users and contributes to higher education, as well. The Laboratory is based on the home-made 5 MV Van de Graaff (VdG) electrostatic accelerator of the institute. The accelerator was put into operation in 1971 and in the beginning it supplied ion beams exclusively for nuclear physics. A few years later with the measurements of K-shell ionization cross sections the door became open also for basic atomic physics. In parallel with this basic study, the application of proton induced X-ray emission (PIXE) for the elemental analysis of biological (hair, erythrocyte and blood plasma) samples and atmospheric aerosols also started. The first paper on PIXE, a methodological one, was published in 1978. The experience gained on these applications and later on archaeology led to the construction of complex PIXE chambers, which were sold, together with the corresponding know-how, to institutions in China, Portugal, Bangladesh, Jordan, North Korea, Singapore, Cuba and Mexico through the International Atomic Energy Agency (IAEA). For the evaluation of PIXE spectra the laboratory has been continuously developing its own computer programme package. The first version of this continuous development was published in 1988. In the meantime a second IBA analysis method, the proton induced gamma ray emission (PIGE), was introduced in the laboratory and was applied simultaneously with PIXE. Application of deuteron induced gamma ray emission (DIGE) started more than a decade later. A 15. Active and passive beam application design guide for global application CERN Document Server Rimmer, Julian 2015-01-01 The Active and Passive Beam Application Design Guide is the result of collaboration by worldwide experts to give system designers a current, authoritative guide on successfully applying active and passive beam technology. Active and Passive Beam Application Design Guide provide energy-efficient methods of cooling, heating, and ventilating indoor areas, especially spaces that require individual zone control and where internal moisture loads are moderate. The systems are simple to operate, with low maintenance requirements. This book is an essential resource for consulting engineers, architects, owners, and contractors who are involved in the design, operation, and installation of these systems. Building on REHVA’s Chilled Beam Application Guidebook, this new guide provides up-to-date tools and advice for designing, commissioning, and operating chilled-beam systems to achieve a determined indoor climate, and includes examples of active and passive beam calculations and selections. Dual units (SI and I-P) are... 16. Coherent electromagnetic radiation of a combined electron-ion beam Energy Technology Data Exchange (ETDEWEB) Pankratov, S G; Samoshenkov, Yu K [Vsesoyuznyj Nauchno-Issledovatel' skij Inst. Optiko-Fizicheskikh Izmerenij, Moscow (USSR) 1977-07-01 The intensity of coherent electromagnetic radiation due to interaction of a modulated electron beam with a modulated ion beam is calculated. It is shown that the radiation intensity has a sharp maximum at the frequency equal to the difference of the modulation frequency of the electron and ion beams. The results obtained are compared with those corresponding to the scattering of a modulated electron beam on randomly distributed gas ions. 17. Development of ion/proton beam equipment for industrial uses Energy Technology Data Exchange (ETDEWEB) Choi, Byung Ho; Lee, J. H.; Cho, Y. S.; Joo, P. K.; Kang, S. S.; Song, W. S.; Kim, H. J.; Chang, G. H.; Bang, S. W 1999-12-01 KAERI has possessed design and fabrication technologies of various ion sources including Duoplasmatron and DuoPiGatron developed by R and D projects of the long-term nuclear technology development program. In order to industrialize ion beam equipments utilizing these ion sources, a technology transfer project for a technology transfer project for a domestic firm has been performed. Under this project, engineers of the firm have been trained through classroom lectures of ion beam principles and OJT, an ion/proton beam equipment (DEMO equipment) has been designed, assembled and commissioned jointly with the engineers. Quality of the ion sources has been quantified, and technologies for ion beam equipment construction, functional test and application research have been developed. The DEMO equipment, which consists of an ion source, power supplies, vacuum, cooling and target systems, has been fabricated and tested to secure stability and reliability for industrial uses. Various characteristic tests including high voltage insulation, beam extraction, beam current measuring, etc. have been performed. This DEMO can be utilized for ion sources development as well as ion beam process development for various industrial products. Engineers of the firm have been trained for the industrialization of ion beam equipment and joined in beam application technology development to create industrial needs of beam equipment. (author) 18. Bond energies of ThO+ and ThC+: A guided ion beam and quantum chemical investigation of the reactions of thorium cation with O2 and CO Science.gov (United States) Cox, Richard M.; Citir, Murat; Armentrout, P. B.; Battey, Samuel R.; Peterson, Kirk A. 2016-05-01 Kinetic energy dependent reactions of Th+ with O2 and CO are studied using a guided ion beam tandem mass spectrometer. The formation of ThO+ in the reaction of Th+ with O2 is observed to be exothermic and barrierless with a reaction efficiency at low energies of k/kLGS = 1.21 ± 0.24 similar to the efficiency observed in ion cyclotron resonance experiments. Formation of ThO+ and ThC+ in the reaction of Th+ with CO is endothermic in both cases. The kinetic energy dependent cross sections for formation of these product ions were evaluated to determine 0 K bond dissociation energies (BDEs) of D0(Th+-O) = 8.57 ± 0.14 eV and D0(Th+-C) = 4.82 ± 0.29 eV. The present value of D0 (Th+-O) is within experimental uncertainty of previously reported experimental values, whereas this is the first report of D0 (Th+-C). Both BDEs are observed to be larger than those of their transition metal congeners, TiL+, ZrL+, and HfL+ (L = O and C), believed to be a result of lanthanide contraction. Additionally, the reactions were explored by quantum chemical calculations, including a full Feller-Peterson-Dixon composite approach with correlation contributions up to coupled-cluster singles and doubles with iterative triples and quadruples (CCSDTQ) for ThC, ThC+, ThO, and ThO+, as well as more approximate CCSD with perturbative (triples) [CCSD(T)] calculations where a semi-empirical model was used to estimate spin-orbit energy contributions. Finally, the ThO+ BDE is compared to other actinide (An) oxide cation BDEs and a simple model utilizing An+ promotion energies to the reactive state is used to estimate AnO+ and AnC+ BDEs. For AnO+, this model yields predictions that are typically within experimental uncertainty and performs better than density functional theory calculations presented previously. 19. Bond energies of ThO{sup +} and ThC{sup +}: A guided ion beam and quantum chemical investigation of the reactions of thorium cation with O{sub 2} and CO Energy Technology Data Exchange (ETDEWEB) Cox, Richard M; Citir, Murat; Armentrout, P. B., E-mail: [email protected] [Department of Chemistry, University of Utah, Salt Lake City, Utah 84112-0850 (United States); Battey, Samuel R.; Peterson, Kirk A. [Department of Chemistry, Washington State University, Pullman, Washington 99164-4630 (United States) 2016-05-14 Kinetic energy dependent reactions of Th{sup +} with O{sub 2} and CO are studied using a guided ion beam tandem mass spectrometer. The formation of ThO{sup +} in the reaction of Th{sup +} with O{sub 2} is observed to be exothermic and barrierless with a reaction efficiency at low energies of k/k{sub LGS} = 1.21 ± 0.24 similar to the efficiency observed in ion cyclotron resonance experiments. Formation of ThO{sup +} and ThC{sup +} in the reaction of Th{sup +} with CO is endothermic in both cases. The kinetic energy dependent cross sections for formation of these product ions were evaluated to determine 0 K bond dissociation energies (BDEs) of D{sub 0}(Th{sup +}–O) = 8.57 ± 0.14 eV and D{sub 0}(Th{sup +}–C) = 4.82 ± 0.29 eV. The present value of D{sub 0} (Th{sup +}–O) is within experimental uncertainty of previously reported experimental values, whereas this is the first report of D{sub 0} (Th{sup +}–C). Both BDEs are observed to be larger than those of their transition metal congeners, TiL{sup +}, ZrL{sup +}, and HfL{sup +} (L = O and C), believed to be a result of lanthanide contraction. Additionally, the reactions were explored by quantum chemical calculations, including a full Feller-Peterson-Dixon composite approach with correlation contributions up to coupled-cluster singles and doubles with iterative triples and quadruples (CCSDTQ) for ThC, ThC{sup +}, ThO, and ThO{sup +}, as well as more approximate CCSD with perturbative (triples) [CCSD(T)] calculations where a semi-empirical model was used to estimate spin-orbit energy contributions. Finally, the ThO{sup +} BDE is compared to other actinide (An) oxide cation BDEs and a simple model utilizing An{sup +} promotion energies to the reactive state is used to estimate AnO{sup +} and AnC{sup +} BDEs. For AnO{sup +}, this model yields predictions that are typically within experimental uncertainty and performs better than density functional theory calculations presented previously. 20. Guiding effect of bent macroscopic quartz tube for high current electron beam International Nuclear Information System (INIS) Zhang Mingwu; Chen Jing; Wu Yehong; Yang Bian; Wang Wei; Xue Yingli; Yu Deyang; Cai Xiaohong 2012-01-01 By using an incident electron beam with the high current and high energy, the guiding effect of the bent macroscopic quartz tube for the electron beam has been investigated. The angular distributions of outgoing electrons depending on the current and energy of incident electrons were measured. The dependences of electron transmitted fraction on energy and current of incident electrons are also shown. As the incident electron energy increasing, the electron transmitted fraction increases, but it decreases while the incident electron current increasing. The results have been compared with the present data. This work presents, the process of guiding electrons is essentially different from that of guiding highly charged ions, the guiding electron beam was caused by both elastic and inelastic collisions between electrons and inner walls of quartz tube, rather than self-organized charging effect on the surface of inner wall of quartz tube. (authors) 1. Broad ion beam serial section tomography Energy Technology Data Exchange (ETDEWEB) Winiarski, B., E-mail: [email protected] [School of Materials, University of Manchester, Manchester M13 9PL (United Kingdom); Materials Division, National Physical Laboratory, Teddington TW11 0LW (United Kingdom); Gholinia, A. [School of Materials, University of Manchester, Manchester M13 9PL (United Kingdom); Mingard, K.; Gee, M. [Materials Division, National Physical Laboratory, Teddington TW11 0LW (United Kingdom); Thompson, G.E.; Withers, P.J. [School of Materials, University of Manchester, Manchester M13 9PL (United Kingdom) 2017-01-15 Here we examine the potential of serial Broad Ion Beam (BIB) Ar{sup +} ion polishing as an advanced serial section tomography (SST) technique for destructive 3D material characterisation for collecting data from volumes with lateral dimensions significantly greater than 100 µm and potentially over millimetre sized areas. Further, the associated low level of damage introduced makes BIB milling very well suited to 3D EBSD acquisition with very high indexing rates. Block face serial sectioning data registration schemes usually assume that the data comprises a series of parallel, planar slices. We quantify the variations in slice thickness and parallelity which can arise when using BIB systems comparing Gatan PECS and Ilion BIB systems for large volume serial sectioning and 3D-EBSD data acquisition. As a test case we obtain 3D morphologies and grain orientations for both phases of a WC-11%wt. Co hardmetal. In our case we have carried out the data acquisition through the manual transfer of the sample between SEM and BIB which is a very slow process (1–2 slice per day), however forthcoming automated procedures will markedly speed up the process. We show that irrespective of the sectioning method raw large area 2D-EBSD maps are affected by distortions and artefacts which affect 3D-EBSD such that quantitative analyses and visualisation can give misleading and erroneous results. Addressing and correcting these issues will offer real benefits when large area (millimetre sized) automated serial section BIBS is developed. - Highlights: • In this work we examine how microstructures can be reconstructed in three-dimensions (3D) by serial argon broad ion beam (BIB) milling, enabling much larger volumes (>250×250×100µm{sup 3}) to be acquired than by serial section focused ion beam-scanning electron microscopy (FIB-SEM). • The associated low level of damage introduced makes BIB milling very well suited to 3D-EBSD acquisition with very high indexing rates. • We explore 2. Focused ion beam milling of carbon fibres International Nuclear Information System (INIS) Huson, Mickey G.; Church, Jeffrey S.; Hillbrick, Linda K.; Woodhead, Andrea L.; Sridhar, Manoj; Van De Meene, Allison M.L. 2015-01-01 A focused ion beam has been used to mill both individual carbon fibres as well as fibres in an epoxy composite, with a view to preparing flat surfaces for nano-indentation. The milled surfaces have been assessed for damage using scanning probe microscopy nano-indentation and Raman micro-probe analysis, revealing that FIB milling damages the carbon fibre surface and covers surrounding areas with debris of disordered carbon. The debris is detected as far as 100 μm from the milling site. The energy of milling as well as the orientation of the beam was varied and shown to have an effect when assessed by Raman spectroscopy. - Highlights: • Focused ion beam (FIB) milling was used to mill flat surfaces on carbon fibres. • Raman spectroscopy showed amorphous carbon was generated during FIB milling. • The amorphous debris is detected as far as 100 μm from the milling site. • This surface degradation was confirmed by nano-indentation experiments. 3. Ion beam induces nitridation of silicon International Nuclear Information System (INIS) Petravic, M.; Williams, J.S.; Conway, M. 1998-01-01 High dose ion bombardment of silicon with reactive species, such as oxygen and nitrogen, has attracted considerable interest due to possible applications of beam-induced chemical compounds with silicon. For example, high energy oxygen bombardment of Si is now routinely used to form buried oxide layers for device purposes, the so called SIMOX structures. On the other hand, Si nitrides, formed by low energy ( 100 keV) nitrogen beam bombardment of Si, are attractive as oxidation barriers or gate insulators, primarily due to the low diffusivity of many species in Si nitrides. However, little data exists on silicon nitride formation during bombardment and its angle dependence, in particular for N 2 + bombardment in the 10 keV range, which is of interest for analytical techniques such as SIMS. In SIMS, low energy oxygen ions are more commonly used as bombarding species, as oxygen provides stable ion yields and enhances the positive secondary ion yield. Therefore, a large body of data can be found in the literature on oxide formation during low energy oxygen bombardment. Nitrogen bombardment of Si may cause similar effects to oxygen bombardment, as nitrogen and oxygen have similar masses and ranges in Si, show similar sputtering effects and both have the ability to form chemical compounds with Si. In this work we explore this possibility in some detail. We compare oxide and nitride formation during oxygen and nitrogen ion bombardment of Si under similar conditions. Despite the expected similar behaviour, some large differences in compound formation were found. These differences are explained in terms of different atomic diffusivities in oxides and nitrides, film structural differences and thermodynamic properties. (author) 4. Nanodevices produced with focussed ion beams International Nuclear Information System (INIS) Doetsch, U.; Wieck, A.D. 1998-01-01 In directly writing the 30 nm focus of a focussed Ga-ion beam (FIB) with an energy of 100 keV we define insulating lines in two-dimensional electronic layers in semiconductors. Ga ions act in GaAs and silicon as deep impurities or p-type doping, respectively. In this way the insulation by such written lines is due to lateral depletion within npn-like interfaces. In writing two FIB lines with a close spacing we define conducting channels between them. In applying a voltage of several Volts to the adjacent areas of the channel relative to it we can tune the effective width of the channel in the range of a few 100 nm to zero and obtain thus a one-dimensional field-effect-transistor-type structure. This transistor exhibits a pure lateral field effect and is thus topologically very different to current transistor concepts. Due to its particular geometry it is called in-plane-gate (IPG) transistor, since the gate and the channel are in the same plane. The fabrication of this type of transistor is thus completely maskless and does not require any alignment procedures since gate, source and drain are all written in the same writing process. Due to the computer-control of the beam deflection even more complex structures are just a question of software and do not need a set of specific masks or photoresist like in the classical lithography. The required line ion dose is of the order of 10 6 cm -1 which means that there are about 100 ions per μm implanted. For devices with maximum micron dimensions only a few hundred ions need thus to be implanted. (orig.) 5. Performance with lead ions of the LHC beam dump system CERN Document Server Bruce, R; Jensen, L; Lefèvre, T; Weterings, W 2007-01-01 The LHC beam dump system must function safely with 208Pb82+ions. The differences with respect to the LHC proton beams are briefly recalled, and the possible areas for performance concerns discussed, in particular the various beam intercepting devices and the beam instrumentation. Energy deposition simulation results for the most critical elements are presented, and the conclusions drawn for the lead ion operation. The expected performance of the beam instrumentation systems are reviewed in the context of the damage potential of the ion beam and the required functionality of the various safety and post-operational analysis requirements. 6. A radioactive ion beam facility using photofission CERN Document Server Diamond, W T 1999-01-01 Use of a high-power electron linac as the driver accelerator for a Radioactive Ion Beam (RIB) facility is proposed. An electron beam of 30 MeV and 100 kW can produce nearly 5x10 sup 1 sup 3 fissions/s from an optimized sup 2 sup 3 sup 5 U target and about 60% of this from a natural uranium target. An electron beam can be readily transmitted through a thin window at the exit of the accelerator vacuum system and transported a short distance through air to a water-cooled Bremsstrahlung-production target. The Bremsstrahlung radiation can, in turn, be transported through air to the isotope-production target. This separates the accelerator vacuum system, the Bremsstrahlung target and the isotope-production target, reducing remote handling problems. The electron beam can be scanned over a large target area to reduce the power density on both the Bremsstrahlung and isotope-production targets. These features address one of the most pressing technological challenges of a high-power RIB facility, namely the production o... 7. Teeth characterization using ion beam analysis International Nuclear Information System (INIS) Rizzutto, M.A.; Added, N.; Tabacniks, M.H.; Falla-Sotelo, F.; Curado, J.F.; Francci, C.; Markarian, R.A.; Quinelato, A.; Youssef, F.; Mori, M.; Youssef, M. 2006-01-01 A collaboration project between the School of Dentistry and the Institute of Physics of the University of Sao Paulo has been established to measure elemental concentrations in teeth by proton induced X-ray emission (PIXE) and heavy ion elastic recoil detection analysis (HI-ERDA) techniques. Data on trace elements in human, bovine and swine teeth, analyzed by PIXE with a 2.4 MeV proton beam, were compared and concentrations for several elements were obtained with tens of μg/g sensitivity. HI-ERDA measurements employing a 52 MeV Cl beam were done to evaluate changes in elementary concentration in dental enamel after bleaching treatment with different products in 25 bovine incisors teeth. This nondestructive technique allowed the measurements of Ca, P, O and C concentrations above the limit of 100 μg/g. (author) 8. Deflagration wave formed by ion beam, 3 International Nuclear Information System (INIS) Niu, Keishiro; Abe, Takashi; Tamba, Moritake. 1980-01-01 An analysis is given for the structure of the deflagration wave which is formed in a target bombarded by an ion beam. Stationary deflagration and/or detonation waves are formed at the surface of the target in a case in which the reaction energy of direct fusion and/or the beam energy deposited in the target are less than a critical value. On the other hand, no solution for stationary wave exists, if the energy deposited in the wave exceeds a critical value. In the latter case, the time-dependent fundamental equations reduce approximately to a self-similar type of equations. Numerical integrations are carried out for this type of differential equations, and an example of self-similar deflagration wave numerically obtained is plotted in the figures. (author) 9. High-resolution electron collision spectroscopy with multicharged ions in merged beams Energy Technology Data Exchange (ETDEWEB) Lestinsky, M. 2007-04-18 The Heidelberg ion storage ring Tsr is currently the only ring equipped with two independent devices for the collinear merging of a cold electron beam with stored ions. This greatly improves the potential of electron-ion collision experiments, as the ion beam can be cooled with one electron beam, while the other one is used as a dedicated target for energy-resolved electron collision processes, such as recombination. The work describes the implementation of this system for rst electron collision spectroscopy experiments. A detection system has been realized including an ion detector and specroscopic beam-control software and instrumentation. Moreover, in order to improve the spectroscopic resolution systematical studies of intrinsic relaxation processes in the electron beam have been carried out. These include the dependence on the electron beam density, the magnetic guiding eld strength, and the acceleration geometry. The recombination measurements on low-lying resonances in lithiumlike Sc{sup 18+} yield a high-precision measurement of the 2s-2p{sub 3/2} transition energy in this system. Operation of the two-electron-beam setup at high collision energy ({approx}1000 eV) is established using resonances of hydrogenlike Mg{sup 11+}, while the unique possibility of modifying the beam-merging geometry con rms its importance for the electron-ion recombination rate at lowest relative energy, as demonstrated on F{sup 6+}. (orig.) 10. High-resolution electron collision spectroscopy with multicharged ions in merged beams International Nuclear Information System (INIS) Lestinsky, M. 2007-01-01 The Heidelberg ion storage ring Tsr is currently the only ring equipped with two independent devices for the collinear merging of a cold electron beam with stored ions. This greatly improves the potential of electron-ion collision experiments, as the ion beam can be cooled with one electron beam, while the other one is used as a dedicated target for energy-resolved electron collision processes, such as recombination. The work describes the implementation of this system for rst electron collision spectroscopy experiments. A detection system has been realized including an ion detector and specroscopic beam-control software and instrumentation. Moreover, in order to improve the spectroscopic resolution systematical studies of intrinsic relaxation processes in the electron beam have been carried out. These include the dependence on the electron beam density, the magnetic guiding eld strength, and the acceleration geometry. The recombination measurements on low-lying resonances in lithiumlike Sc 18+ yield a high-precision measurement of the 2s-2p 3/2 transition energy in this system. Operation of the two-electron-beam setup at high collision energy (∼1000 eV) is established using resonances of hydrogenlike Mg 11+ , while the unique possibility of modifying the beam-merging geometry con rms its importance for the electron-ion recombination rate at lowest relative energy, as demonstrated on F 6+ . (orig.) 11. Development of a focused ion beam micromachining system Energy Technology Data Exchange (ETDEWEB) Pellerin, J.G.; Griffis, D.; Russell, P.E. 1988-12-01 Focused ion beams are currently being investigated for many submicron fabrication and analytical purposes. An FIB micromachining system consisting of a UHV vacuum system, a liquid metal ion gun, and a control and data acquisition computer has been constructed. This system is being used to develop nanofabrication and nanomachining techniques involving focused ion beams and scanning tunneling microscopes. 12. Plasma and ion beam processing at Los Alamos International Nuclear Information System (INIS) Rej, D.J.; Davis, H.A.; Henins, I. 1994-01-01 Efforts are underway at Los Alamos National Laboratory to utilize plasma and intense ion beam science and technology of the processing of advanced materials. A major theme involves surface modification of materials, e.g., etching, deposition, alloying, and implantation. In this paper, we concentrate on two programs, plasma source ion implantation and high-intensity pulsed ion beam deposition 13. A beam profile monitor for heavy ion beams at high impact energies International Nuclear Information System (INIS) Hausmann, A.; Stiebing, K.E.; Bethge, K.; Froehlich, O.; Koehler, E.; Mueller, A.; Rueschmann, G. 1994-01-01 A beam profile monitor for heavy ion beams has been developed for the use in experiments at the Heavy Ion Synchrotron SIS at Gesellschaft fuer Schwerionenforschung Darmstadt (GSI). Four thin scintillation fibres are mounted on one wheel and scan the ion beam sequentially in two linearly independent directions. They are read out via one single photomultiplier common to all four fibres into one time spectrum, which provides all information about beam position, beam extension, time structure and lateral homogeneity of the beam. The system operates in a wide dynamic range of beam intensities. ((orig.)) 14. High spin studies with radioactive ion beams International Nuclear Information System (INIS) Garrett, J.D. 1992-01-01 The variety of new research possibilities afforded by the culmination of the two frontier areas of nuclear structure: high spin and studies far from nuclear stability (utilizing intense radioactive ion beams) are discussed. Topics presented include: new regions of exotic nuclear shape (e.g. superdeformation, hyperdeformation, and reflection-asymmetric shapes); the population of and consequences of populating exotic nuclear configurations; and complete spectroscopy (i.e. the overlap of state of the art low-and high-spin studies in the same nucleus) 15. High spin studies with radioactive ion beams Energy Technology Data Exchange (ETDEWEB) Garrett, J D [Oak Ridge National Lab., TN (United States) 1992-08-01 The variety of new research possibilities afforded by the culmination of the two frontier areas of nuclear structure: high spin and studies far from nuclear stability (utilizing intense radioactive ion beams) are discussed. Topics presented include: new regions of exotic nuclear shape (e.g. superdeformation, hyperdeformation, and reflection-asymmetric shapes); the population of and consequences of populating exotic nuclear configurations; and, complete spectroscopy (i.e. the overlap of state of the art low- and high-spin studies in the same nucleus). (author). 47 refs., 8 figs. 16. Probing surface magnetism with ion beams International Nuclear Information System (INIS) Winter, H. 2007-01-01 Ion beams can be used to probe magnetic properties of surfaces by a variety of different methods. Important features of these methods are related to trajectories of atomic projectiles scattered from the surface of a solid target and to the electronic interaction mechanisms in the surface region. Both items provide under specific conditions a high sensitivity for the detection of magnetic properties in the region at the topmost layer of surface atoms. This holds in particular for scattering under planar surface channeling conditions, where under grazing impact atoms or ions are reflected specularly from the surface without penetration into the subsurface region. Two different types of methods are employed based on the detection of the spin polarization of emitted or captured electrons and on spin blocking effects for capture into atomic terms. These techniques allow one to probe the long range and short range magnetic order in the surface region 17. Electron temperature effects for an ion beam source International Nuclear Information System (INIS) Uramoto, Joshin. 1979-05-01 A hydrogen high temperature plasma up to 200 eV is produced by acceleration of electrons in a hot hollow cathode discharge and is used as an ion beam source. Then, two characteristics are observed: A rate of the atomic ion (H + ) number increases above 70%. A perveance of the ion beam increases above 30 times compared with that of a cold plasma, while a floating potential of an ion acceleration electrode approaches an ion acceleration potential (- 500 V) according as an increment of the electron temperature. Moreover, a neutralized ion beam can be produced by only the negative floating electrode without an external power supply. (author) 18. Preliminary results of spatially resolved ECR ion beam profile investigations International Nuclear Information System (INIS) Panitzsch, L.; Stalder, M.; Wimmer-Schweingruber, R.F. 2012-01-01 The profile of an ion beam produced in an Electron Cyclotron Resonance Ion Source (ECRIS) can vary greatly depending on the source settings and the ion-optical tuning. Strongly focussed ion beams form circular structures (hollow beams) as predicted by simulations and observed in experiments. Each of the rings is predicted to be dominated by ions with same or at least similar m/q-ratios due to ion-optical effects. To check this we performed a series of preliminary investigations to test the required tuning capabilities of our ion source. This includes beam focussing (A) and beam steering (B) using a 3D-movable extraction. Having tuned the source to deliver a beam of strongly focussed ions of different ion species and having steered this beam to match the transmittance area of the sector magnet we also recorded the ion charge state distribution of the strongly focussed beam profile at different, spatially limited positions (C). The preliminary results will be introduced within this paper: it appears that our 3D-movable extraction is very efficient to steer and to focus the beam strongly. The paper is followed by the slides of the presentation. (authors) 19. Getting Ready for Ion-Beam Therapy Research in Austria - Building-up Research in Parallel with a Facility International Nuclear Information System (INIS) Georg, Dietmar; Knaeusl; Kuess, Peter; Fuchs, Hermann; Poetter, Richard; Schreiner, Thomas 2015-01-01 With participation in ion-beam projects funded nationally or by the European Commission (EC), ion-beam research activities were started at the Medical University of Vienna in parallel with the design and construction of the ion-beam center MedAustron in Wiener Neustadt, 50 km from the Austrian capital. The current medical radiation physics research activities that will be presented comprise: (1) Dose calculation and optimization: ion-beam centers focus mostly on proton and carbon-ion therapy. However, there are other ion species with great potential for clinical applications. Helium ions are currently under investigation from a theoretical physics and biology perspective. (2) Image guided and adaptive ion-beam therapy: organ motion and anatomic changes have a severe influence in ion-beam therapy since variations in heterogeneity along the beam path have a significant impact on the particle range. Ongoing research focuses on possibilities to account for temporal variations of the anatomy during radiotherapy. Both during and between fractions also considering temporal variations in tumor biology. Furthermore, research focuses on particle therapy positron emission tomography (PT-PET) verification and the detection of prompt gammas for on-line verification of ion-beam delivery. (3) Basic and applied dosimetry: an end-to-end procedure was designed and successfully tested in both scanned proton and carbon-ion beams, which may also serve as a dosimetric credentialing procedure for clinical trials in the future. (Author) 20. Recent radioactive ion beam program at RIKEN and related topics Recent experimental programs at RIKEN concerning RI beams are reviewed. RIKEN has the ring cyclotron (RRC) with high intense heavy-ion beams and large acceptance fragment separator, RIPS. The complex can provide high intense RI-beams. By using the high intense RI-beams, a variety of experiments have been ... 1. Important atomic physics issues for ion beam fusion International Nuclear Information System (INIS) Bangerter, R.O. 1985-01-01 This paper suggests several current atomic physics questions important to ion beam fusion. Among the topics discussed are beam transport, beam-target interaction, and reactor design. The major part of the report is discussion concerning areas of research necessary to better understand beam-target interactions 2. Exploration of the Singlet O2 Oxidation of 8-Oxoguanine by Guided-Ion Beam Scattering and Density Functional Theory: Changes of Reaction Intermediates, Energetics, and Kinetics upon Protonation/Deprotonation and Hydration. Science.gov (United States) Sun, Yan; Lu, Wenchao; Liu, Jianbo 2017-02-09 8-Oxo-7,8-dihydro-2'-deoxyguanosine (8-oxodGuo) is one of the most common DNA lesions resulting from reactive oxygen species and ionizing radiation, and is involved in mutagenesis, carcinogenesis, and cell death. Notably, 8-oxodGuo is more reactive toward singlet (a 1 Δ g ) O 2 than the undamaged guanosine, and the lesions arising from the secondary oxidation of 8-oxodGuo are more mutagenic. Herein the 1 O 2 oxidation of free base 8-oxoguanine (8-oxoG) was investigated at different initial conditions including protonated [8-oxoG + H] + , deprotonated [8-oxoG - H] - , and their monohydrates. Experiment was carried out on a guided-ion beam scattering tandem mass spectrometer. Measurements include the effects of collision energy (E col ) on reaction cross sections over a center-of-mass E col range from 0.1 to 0.5 eV. The aim of this study is to quantitatively probe the sensitivity of the early stage of 8-oxoG oxidation to ionization and hydration. Density functional theory and Rice-Ramsperger-Kassel-Marcus calculations were performed to identify the intermediates and the products along reaction pathways and locate accessible reaction potential energy surfaces, and to rationalize reaction outcomes from energetic and kinetic points of view. No product was observed for the reaction of [8-oxoG + H] + ·W 0,1 (W = H 2 O) because insurmountable barriers block the addition of 1 O 2 to reactant ions. Neither was [8-oxoG - H] - reactive with 1 O 2 , in this case due to the rapid decay of transient intermediates to starting reactants. However, the nonreactivity of [8-oxoG - H] - was inverted by hydration; as a result, 4,5-dioxetane of [8-oxoG - H] - was captured as the main oxidation product. Reaction cross section for [8-oxoG - H] - ·W + 1 O 2 decreases with increasing E col and becomes negligible above 0.3 eV, indicating that the reaction is exothermic and has no barriers above reactants. The contrasting oxidation behaviors of [8-oxoG + H] + ·W 0,1 and [8-oxoG - H] - ·W 0 3. Large area negative ion source for high voltage neutral beams International Nuclear Information System (INIS) Poulsen, P.; Hooper, E.B. Jr. 1979-11-01 A source of negative deuterium ions in the multi-ampere range is described that is readily extrapolated to reactor size, 10 amp or more of neutral beam, that is of interest in future experiments and reactors. The negative ion source is based upon the double charge exchange process. A beam of positive ions is created and accelerated to an energy at which the attachment process D + M → D - + M + proceeds efficiently. The positive ions are atomically neutralized either in D 2 or in the charge exchange medium M. Atomic species make a second charge exchange collision in the charge target to form D - . For a sufficiently thick target, the beam reaches an equilibrium fraction of negative ions. For reasons of efficiency, the target is typically alkali metal vapor; this experiment uses sodium. The beam of negative ions can be accelerated to high (>200 keV) energy, the electrons stripped from the ions, and a high energy neutral beam formed 4. Proceedings of national seminar on physics with radioactive ion beams International Nuclear Information System (INIS) Chintalapudi, S.N.; Shyam, R. 1991-01-01 This volume containing the proceedings of the national seminar on physics with radioactive ion beams gives a broad overview of the developments taking place in the area of nuclear physics and accelerator physics with special emphasis on the utilization of radioactive ion beams for various studies. Topics covered include studies on nuclear structure and nuclear astrophysics and the wide ranging applications of radioactive ion beams in these and other areas of nuclear sciences. Papers relevant to INIS are indexed separately 5. Commissioning of the ion beam buncher and cooler for LEBIT Energy Technology Data Exchange (ETDEWEB) Sun, T.; Bollen, G.; Ringle, R.; Schury, P. [Michigan State University, NSCL, East Lansing, MI (United States); Michigan State University, Department of Physics and Astronomy, East Lansing, MI (United States); Schwarz, S.; Lawton, D. [Michigan State University, NSCL, East Lansing, MI (United States) 2005-09-01 A radiofrequency-quadrupole ion accumulator and buncher has been set-up for the low-energy-beam and ion-trap (LEBIT) facility, which is in its final commissioning phase at the NSCL/MSU. The buncher is a cryogenic system with separated cooling and accumulation stages, optimized for excellent beam quality and high performance. The completed set-up of the LEBIT ion buncher is presented as well as first experimental results on pulse forming and beam properties. (orig.) 6. Commissioning of the ion beam buncher and cooler for LEBIT International Nuclear Information System (INIS) Sun, T.; Bollen, G.; Ringle, R.; Schury, P.; Schwarz, S.; Lawton, D. 2005-01-01 A radiofrequency-quadrupole ion accumulator and buncher has been set-up for the low-energy-beam and ion-trap (LEBIT) facility, which is in its final commissioning phase at the NSCL/MSU. The buncher is a cryogenic system with separated cooling and accumulation stages, optimized for excellent beam quality and high performance. The completed set-up of the LEBIT ion buncher is presented as well as first experimental results on pulse forming and beam properties. (orig.) 7. An ion beam deceleration lens for ultra-low-energy ion bombardment of naked DNA Energy Technology Data Exchange (ETDEWEB) Thopan, P.; Prakrajang, K. [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Thongkumkoon, P. [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400 (Thailand); Suwannakachorn, D. [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Yu, L.D., E-mail: [email protected] [Plasma and Beam Physics Research Facility, Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200 (Thailand); Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400 (Thailand) 2013-07-15 Highlights: ► An ion beam deceleration lens was designed and constructed. ► The deceleration lens was installed and tested. ► The decelerated ion beam energy was measured using an electrical field. ► Decelerated ultra-low-energy ion beam bombarded naked DNA. ► Ion beam with energy of a few tens of eV could break DNA strands. -- Abstract: Study of low-energy ion bombardment effect on biological living materials is of significance. High-energy ion beam irradiation of biological materials such as organs and cells has no doubt biological effects. However, ion energy deposition in the ion-bombarded materials dominantly occurs in the low-energy range. To investigate effects from very-low-energy ion bombardment on biological materials, an ion beam deceleration lens is necessary for uniform ion energy lower than keV. A deceleration lens was designed and constructed based on study of the beam optics using the SIMION program. The lens consisted of six electrodes, able to focus and decelerate primary ion beam, with the last one being a long tube to obtain a parallel uniform exiting beam. The deceleration lens was installed to our 30-kV bioengineering-specialized ion beam line. The final decelerated-ion energy was measured using a simple electrostatic field to bend the beam to range from 10 eV to 1 keV controlled by the lens parameters and the primary beam condition. In a preliminary test, nitrogen ion beam at 60 eV decelerated from a primary 20-keV beam bombarded naked plasmid DNA. The original DNA supercoiled form was found to change to relaxed and linear forms, indicating single or double strand breaks. The study demonstrated that the ion bombardment with energy as low as several-tens eV was possible to break DNA strands and thus potential to cause genetic modification of biological cells. 8. An ion beam deceleration lens for ultra-low-energy ion bombardment of naked DNA International Nuclear Information System (INIS) Thopan, P.; Prakrajang, K.; Thongkumkoon, P.; Suwannakachorn, D.; Yu, L.D. 2013-01-01 Highlights: ► An ion beam deceleration lens was designed and constructed. ► The deceleration lens was installed and tested. ► The decelerated ion beam energy was measured using an electrical field. ► Decelerated ultra-low-energy ion beam bombarded naked DNA. ► Ion beam with energy of a few tens of eV could break DNA strands. -- Abstract: Study of low-energy ion bombardment effect on biological living materials is of significance. High-energy ion beam irradiation of biological materials such as organs and cells has no doubt biological effects. However, ion energy deposition in the ion-bombarded materials dominantly occurs in the low-energy range. To investigate effects from very-low-energy ion bombardment on biological materials, an ion beam deceleration lens is necessary for uniform ion energy lower than keV. A deceleration lens was designed and constructed based on study of the beam optics using the SIMION program. The lens consisted of six electrodes, able to focus and decelerate primary ion beam, with the last one being a long tube to obtain a parallel uniform exiting beam. The deceleration lens was installed to our 30-kV bioengineering-specialized ion beam line. The final decelerated-ion energy was measured using a simple electrostatic field to bend the beam to range from 10 eV to 1 keV controlled by the lens parameters and the primary beam condition. In a preliminary test, nitrogen ion beam at 60 eV decelerated from a primary 20-keV beam bombarded naked plasmid DNA. The original DNA supercoiled form was found to change to relaxed and linear forms, indicating single or double strand breaks. The study demonstrated that the ion bombardment with energy as low as several-tens eV was possible to break DNA strands and thus potential to cause genetic modification of biological cells 9. Ion spectroscopy for improvement of the physical beam model for therapy planning in ion beam therapy Energy Technology Data Exchange (ETDEWEB) Arico, Giulia 2016-11-23 Helium and carbon ions enable a more conformal dose distribution, narrower penumbra and higher relative biological effectiveness than photon and proton radiotherapy. However, they may undergo nuclear fragmentation in the patient tissues and the arising secondary fragments affect the delivered biological dose distributions. Currently there is a lack of data regarding ion nuclear fragmentation. One reason is the large size (up to some meters) of the experimental setups required for the investigations. In this thesis a new method is presented, which makes use of versatile pixelated semiconductor detectors (Timepix). This method is based on tracking of single particles and pattern recognition of their signals in the detectors. Measurements were performed at the HIT facility. The mixed radiation field arising from 430 MeV/u carbon ion beams and 221 MeV/u helium ion beams in water and in PMMA targets was investigated. The amounts of primary (carbon or helium) ions detected behind targets with the same water equivalent thickness (WET) were found to be in agreement within the statistical uncertainties. However, more fragments (differences up to 20% in case of H) and narrower lateral particle distributions were measured behind the PMMA than the water targets. The spectra of ions behind tissue surrogates and corresponding water targets with the same WET were analysed. The results obtained with adipose and inner bone surrogates and with the equivalent water phantoms were found to be consistent within the uncertainties. Significant differences in the results were observed in the case of lung and cortical bone surrogates when compared to the water phantoms. The experimental results were compared to FLUKA Monte Carlo simulations. This comparison could contribute to enhance the ion interaction models currently implemented for {sup 12}C and {sup 4}He ion beams. 10. Materials processing with intense pulsed ion beams International Nuclear Information System (INIS) Rej, D.J.; Davis, H.A.; Olson, J.C. 1996-01-01 We review research investigating the application of intense pulsed ion beams (IPIBs) for the surface treatment and coating of materials. The short range (0.1-10 μm) and high-energy density (1-50 J/cm 2 ) of these short-pulsed (≤ 1 μs) beams (with ion currents I = 5 - 50 kA, and energies E = 100 - 1000 keV) make them ideal to flash-heat a target surface, similar to the more familiar pulsed laser processes. IPIB surface treatment induces rapid melt and solidification at up to 10 10 K/s to cause amorphous layer formation and the production of non-equilibrium microstructures. At higher energy density the target surface is vaporized, and the ablated vapor is condensed as coatings onto adjacent substrates or as nanophase powders. Progress towards the development of robust, high-repetition rate IPIB accelerators is presented along with economic estimates for the cost of ownership of this technology 11. Guiding of slow neon and molecular hydrogen ions through nanocapillaries in PET International Nuclear Information System (INIS) Stolterfoht, N.; Hellhammer, R.; Sobocinski, P.; Pesic, Z.D.; Bundesmann, J.; Sulik, B.; Shah, M.B.; Dunn, K.; Pedregosa, J.; McCullough, R.W. 2005-01-01 The transmission profiles of atomic 3keV Ne 7+ ions and molecular 1keV H 2 + and H 3 + ions passing through nanocapillaries were studied. Capillaries with a diameter of 100nm and a length of 10μm in insulating PET polymers were used. The high aspect ratio of 100 is achieved by the method of etching ion tracks produced by high-energy xenon impact. The angular distributions of the transmitted projectiles show that the majority of ions are transported in their initial charge state along the capillary axis even when the capillaries are tilted with respect to the incident beam direction. This result indicates ion-guiding, which is produced by charge-up effects influencing the ion trajectories in a self-supporting manner. The guiding effects are found to be different for highly charged neon and singly charged molecular hydrogen. Negligible fragmentation of the molecular ions was observed 12. Influence of ion beam and geometrical parameters on properties of Si thin films grown by Ar ion beam sputtering Energy Technology Data Exchange (ETDEWEB) Bundesmann, Carsten; Feder, Rene; Neumann, Horst [Leibniz-Institut fuer Oberflaechenmodifizierung e.V., Leipzig (Germany) 2012-07-01 Ion beam sputtering (IBS) offers, in contrast to other physical vapour deposition techniques, such as magnetron sputtering or electron beam evaporation, the opportunity to change the properties of the layer forming particles (sputtered and scattered particles) by varying ion beam parameters (ion species, ion energy) and geometrical parameters (ion incidence angle, emission angle). Consequently, these effects can be utilized to tailor thin film properties [1]. The goal is to study systematically the correlations between the primary and secondary parameters and, at last, the effects on the properties of Si thin films, such as optical properties, stress, surface topography and composition. First experimental results are presented for Ar-ion sputtering of Si. 13. Progress toward a microsecond duration, repetitively pulsed, intense- ion beam International Nuclear Information System (INIS) Davis, H.A.; Olson, J.C.; Reass, W.A.; Coates, D.M.; Hunt, J.W.; Schleinitz, H.M.; Greenly, J.B. 1996-01-01 A number of intense ion beams applications are emerging requiring repetitive high-average-power beams. These applications include ablative deposition of thin films, rapid melt and resolidification for surface property enhancement, advanced diagnostic neutral beams for the next generation of Tokamaks, and intense pulsed-neutron sources. We are developing a 200-250 keV, 15 kA, 1 μs duration, 1-30 Hz intense ion beam accelerator to address these applications 14. From field evaporation to focused ion beams International Nuclear Information System (INIS) Forbes, R.G. 2004-01-01 Full text: This paper report various items of recent progress in the theory of field evaporation and the theory of the liquid-metal ion source. The research has, in part, been driven by a desire to find out how to reduce the beam-spot size in a focused ion beam machine, which is developing as a significant tool of nanotechnology. A major factor in determining beam spot size seems to be the behavior of the liquid-metal ion source (LMIS), and one route might be to reduce the minimum emission current of a LMIS, if this is possible. Theories of LMIS minimum emission current have been re-examined. Some progress has been made, but development of more accurate theory has been constrained by several factors, include the long-known limitations of the present theory of field evaporation (FEV). This, in turn, has stimulated a wider re-examination of FEV theory. As part of some general theoretical remarks, the following items of recent progress will be covered. Various results concerning the prediction of the field F e at which the activation energy Q for field evaporation is zero, including calculations in which vacuum electrostatic energy changes are taken into account, and another look at the views of Kingham and Tsong concerning escape charge-state. Some years ago, the following approximate formula was derived for the dependence of FEV activation energy on field F: Q=B(F e /F - 1) 2 . It has recently been possible to show that the parameter B can be estimated as B= βYΩ/8, where Y is Young's modulus, Ω is the atomic volume, and β is a correction factor of order. In the framework of the charge-draining mechanism, another look at how the activation-energy hump can be modelled, in order to predict/explain the conditions under which FEV becomes dominated by ion tunnelling rather than field evaporation. A review of the changes in LMIS theory that result from applying the equation of continuity to the metal/vacuum interface, including modifications to the theory of minimum 15. Recent US advances in ion-beam-driven high energy density physics and heavy ion fusion International Nuclear Information System (INIS) Logan, B.G.; Bieniosek, F.M.; Celata, C.M.; Coleman, J.; Greenway, W.; Henestroza, E.; Kwan, J.W.; Lee, E.P.; Leitner, M.; Roy, P.K.; Seidl, P.A.; Vay, J.-L.; Waldron, W.L.; Yu, S.S.; Barnard, J.J.; Cohen, R.H.; Friedman, A.; Grote, D.P.; Kireeff Covo, M.; Molvik, A.W.; Lund, S.M.; Meier, W.R.; Sharp, W.; Davidson, R.C.; Efthimion, P.C.; Gilson, E.P.; Grisham, L.; Kaganovich, I.D.; Qin, H.; Sefkow, A.B.; Startsev, E.A.; Welch, D.; Olson, C. 2007-01-01 During the past two years, significant experimental and theoretical progress has been made in the US heavy ion fusion science program in longitudinal beam compression, ion-beam-driven warm dense matter, beam acceleration, high brightness beam transport, and advanced theory and numerical simulations. Innovations in longitudinal compression of intense ion beams by >50X propagating through background plasma enable initial beam target experiments in warm dense matter to begin within the next two years. We are assessing how these new techniques might apply to heavy ion fusion drivers for inertial fusion energy 16. Performance of positive ion based high power ion source of EAST neutral beam injector International Nuclear Information System (INIS) Hu, Chundong; Xie, Yahong; Xie, Yuanlai; Liu, Sheng; Xu, Yongjian; Liang, Lizhen; Jiang, Caichao; Li, Jun; Liu, Zhimin 2016-01-01 The positive ion based source with a hot cathode based arc chamber and a tetrode accelerator was employed for a neutral beam injector on the experimental advanced superconducting tokamak (EAST). Four ion sources were developed and each ion source has produced 4 MW @ 80 keV hydrogen beam on the test bed. 100 s long pulse operation with modulated beam has also been tested on the test bed. The accelerator was upgraded from circular shaped to diamond shaped in the latest two ion sources. In the latest campaign of EAST experiment, four ion sources injected more than 4 MW deuterium beam with beam energy of 60 keV into EAST 17. Biological effect of penetration controlled irradiation with ion beams Energy Technology Data Exchange (ETDEWEB) Tanaka, Atsushi; Shimizu, Takashi; Kikuchi, Masahiro; Kobayashi, Yasuhiko; Watanabe, Hiroshi [Japan Atomic Energy Research Inst., Takasaki, Gunma (Japan). Takasaki Radiation Chemistry Research Establishment; Yamashita, Takao 1997-03-01 To investigate the effect of local irradiation with ion beams on biological systems, technique for penetration controlled irradiation has been established. The range in a target was controlled by changing the distance from beam window in the atmosphere, and could be controlled linearly up to about 31 {mu}m in biological material. In addition, the effects of the penetration controlled irradiations with 1.5 MeV/u C and He ions were examined using tobacco pollen. The increased frequency of leaky pollen produced by ion beams suggests that the efficient pollen envelope damages would be induced at the range-end of ion beams. (author) 18. Design of a negative ion neutral beam system for TNS International Nuclear Information System (INIS) Easoz, J.R. 1978-05-01 A conceptual design of a neutral beam line based on the neutralization of negative deuterium ions is presented. This work is a detailed design of a complete neutral beam line based on using negative ions from a direct extraction source. Anticipating major technological advancements, beam line components have been scaled including the negative ion sources and components for the direct energy recovery of charged beams and high speed cryogenic pumping. With application to the next step in experimental fusion reactors (TNS), the neutral beam injector system that has been designed provides 10 MW of 200 keV neutral deuterium atoms. Several arms are required for plasma ignition 19. Development of Emittance Analysis Software for Ion Beam Characterization International Nuclear Information System (INIS) 2007-01-01 Transverse beam emittance is a crucial property of charged particle beams that describes their angular and spatial spread. It is a figure of merit frequently used to determine the quality of ion beams, the compatibility of an ion beam with a given beam transport system, and the ability to suppress neighboring isotopes at on-line mass separator facilities. Generally, a high-quality beam is characterized by a small emittance. In order to determine and improve the quality of ion beams used at the Holifield Radioactive Ion Beam Facility (HRIBF) for nuclear physics and nuclear astrophysics research, the emittances of the ion beams are measured at the off-line Ion Source Test Facilities. In this project, emittance analysis software was developed to perform various data processing tasks for noise reduction, to evaluate root-mean-square emittance, Twiss parameters, and area emittance of different beam fractions. The software also provides 2D and 3D graphical views of the emittance data, beam profiles, emittance contours, and RMS. Noise exclusion is essential for accurate determination of beam emittance values. A Self-Consistent, Unbiased Elliptical Exclusion (SCUBEEx) method is employed. Numerical data analysis techniques such as interpolation and nonlinear fitting are also incorporated into the software. The software will provide a simplified, fast tool for comprehensive emittance analysis. The main functions of the software package have been completed. In preliminary tests with experimental emittance data, the analysis results using the software were shown to be accurate 20. DEVELOPMENT OF EMITTANCE ANALYSIS SOFTWARE FOR ION BEAM CHARACTERIZATION Energy Technology Data Exchange (ETDEWEB) 2007-01-01 Transverse beam emittance is a crucial property of charged particle beams that describes their angular and spatial spread. It is a fi gure of merit frequently used to determine the quality of ion beams, the compatibility of an ion beam with a given beam transport system, and the ability to suppress neighboring isotopes at on-line mass separator facilities. Generally a high quality beam is characterized by a small emittance. In order to determine and improve the quality of ion beams used at the Holifi eld Radioactive Ion beam Facility (HRIBF) for nuclear physics and nuclear astrophysics research, the emittances of the ion beams are measured at the off-line Ion Source Test Facilities. In this project, emittance analysis software was developed to perform various data processing tasks for noise reduction, to evaluate root-mean-square emittance, Twiss parameters, and area emittance of different beam fractions. The software also provides 2D and 3D graphical views of the emittance data, beam profi les, emittance contours, and RMS. Noise exclusion is essential for accurate determination of beam emittance values. A Self-Consistent, Unbiased Elliptical Exclusion (SCUBEEx) method is employed. Numerical data analysis techniques such as interpolation and nonlinear fi tting are also incorporated into the software. The software will provide a simplifi ed, fast tool for comprehensive emittance analysis. The main functions of the software package have been completed. In preliminary tests with experimental emittance data, the analysis results using the software were shown to be accurate. 1. Prototype ion source for JT-60 neutral beam injectors International Nuclear Information System (INIS) Akiba, M. 1981-01-01 A prototype ion source for JT-60 neutral beam injectors has been fabricated and tested. Here, we review the construction of the prototype ion source and report the experimental results about the source characteristics that has been obtained at this time. The prototype ion source is now installed at the prototype unit of JT-60 neutral beam injection units and the demonstration of the performances of the ion source and the prototype unit has just started 2. Ballistic-neutralized chamber transport of intense heavy ion beams International Nuclear Information System (INIS) Rose, D.V.; Welch, D.R.; Oliver, B.V.; Clark, R.E.; Sharp, W.M.; Friedman, A. 2001-01-01 Two-dimensional particle-in-cell simulations of intense heavy ion beams propagating in an inertial confinement fusion (ICF) reactor chamber are presented. The ballistic-neutralized transport scheme studied uses 4 GeV Pb +1 ion beams injected into a low-density, gas-filled reactor chamber and the beam is ballistically focused onto an ICF target before entering the chamber. Charge and current neutralization of the beam is provided by the low-density background gas. The ballistic-neutralized simulations include stripping of the beam ions as the beam traverses the chamber as well as ionization of the background plasma. In addition, a series of simulations are presented that explore the charge and current neutralization of the ion beam in an evacuated chamber. For this vacuum transport mode, neutralizing electrons are only drawn from sources near the chamber entrance 3. Collective ion acceleration by relativistic electron beams in plasmas International Nuclear Information System (INIS) Galvez, M.; Gisler, G. 1991-01-01 A two-dimensional fully electromagnetic particle-in-cell code is used to simulate the interaction of a relativistic electron beam injected into a finite-size background neutral plasma. The simulations show that the background electrons are pushed away from the beam path, forming a neutralizing ion channel. Soon after the beam head leaves the plasma, a virtual cathode forms which travels away with the beam. However, at later times a second, quasi-stationary, virtual cathode forms. Its position and strength depends critically on the parameters of the system which critically determines the efficiency of the ion acceleration process. The background ions trapped in the electrostatic well of the virtual cathode are accelerated and at later times, the ions as well as the virtual cathode drift away from the plasma region. The surfing of the ions in the electrostatic well produces an ion population with energies several times the initial electron beam energy. It is found that optimum ion acceleration occurs when the beam-to-plasma density ratio is near unity. When the plasma is dense, the beam is a weak perturbation and accelerates few ions, while when the plasma is tenuous, the beam is not effectively neutralized, and a virtual cathode occurs right at the injection plane. The simulations also show that, at the virtual cathode position, the electron beam is pinched producing a self-focusing phenomena 4. An electron cyclotron resonance ion source based low energy ion beam platform International Nuclear Information System (INIS) Sun, L. T.; Shang, Y.; Ma, B. H.; Zhang, X. Z.; Feng, Y. C.; Li, X. X.; Wang, H.; Guo, X. H.; Song, M. T.; Zhao, H. Y.; Zhang, Z. M.; Zhao, H. W.; Xie, D. Z. 2008-01-01 To satisfy the requirements of surface and atomic physics study in the field of low energy multiple charge state ion incident experiments, a low energy (10 eV/q-20 keV/q) ion beam platform is under design at IMP. A simple test bench has been set up to test the ion beam deceleration systems. Considering virtues such as structure simplicity, easy handling, compactness, cost saving, etc., an all-permanent magnet ECRIS LAPECR1 [Lanzhou all-permanent magnet electron cyclotron resonance (ECR) ion source No. 1] working at 14.5 GHz has been adopted to produce intense medium and low charge state ion beams. LAPECR1 source has already been ignited. Some intense low charge state ion beams have been produced on it, but the first test also reveals that many problems are existing on the ion beam transmission line. The ion beam transmission mismatches result in the depressed performance of LAPECR1, which will be discussed in this paper. To obtain ultralow energy ion beam, after being analyzed by a double-focusing analyzer magnet, the selected ion beam will be further decelerated by two afocal deceleration lens systems, which is still under design. This design has taken into consideration both ions slowing down and also ion beam focusing. In this paper, the conceptual design of deceleration system will be discussed 5. An electron cyclotron resonance ion source based low energy ion beam platform. Science.gov (United States) Sun, L T; Shang, Y; Ma, B H; Zhang, X Z; Feng, Y C; Li, X X; Wang, H; Guo, X H; Song, M T; Zhao, H Y; Zhang, Z M; Zhao, H W; Xie, D Z 2008-02-01 To satisfy the requirements of surface and atomic physics study in the field of low energy multiple charge state ion incident experiments, a low energy (10 eV/q-20 keV/q) ion beam platform is under design at IMP. A simple test bench has been set up to test the ion beam deceleration systems. Considering virtues such as structure simplicity, easy handling, compactness, cost saving, etc., an all-permanent magnet ECRIS LAPECR1 [Lanzhou all-permanent magnet electron cyclotron resonance (ECR) ion source No. 1] working at 14.5 GHz has been adopted to produce intense medium and low charge state ion beams. LAPECR1 source has already been ignited. Some intense low charge state ion beams have been produced on it, but the first test also reveals that many problems are existing on the ion beam transmission line. The ion beam transmission mismatches result in the depressed performance of LAPECR1, which will be discussed in this paper. To obtain ultralow energy ion beam, after being analyzed by a double-focusing analyzer magnet, the selected ion beam will be further decelerated by two afocal deceleration lens systems, which is still under design. This design has taken into consideration both ions slowing down and also ion beam focusing. In this paper, the conceptual design of deceleration system will be discussed. 6. Optics of ion beams for the neutral beam injection system on HL-2A Tokamak Energy Technology Data Exchange (ETDEWEB) Zou, G. Q.; Lei, G. J.; Cao, J. Y.; Duan, X. R. [Southwestern Institute of Physics, Chengdu, 610041 (China) 2012-07-15 The ion beam optics for the neutral beam injection system on HL-2A Tokomak is studied by two- dimensional numerical simulation program firstly, where the emitting surface is taken at 100 Debye lengths from the plasma electrode. The mathematical formulation, computation techniques are described. Typical ion orbits, equipotential contours, and emittance diagram are shown. For a fixed geometry electrode, the effect of plasma density, plasma potential and plasma electron temperature on ion beam optics is examined, and the calculation reliability is confirmed by experimental results. In order to improve ion beam optics, the application of a small pre-acceleration voltage ({approx}100 V) between the plasma electrode and the arc discharge anode is reasonable, and a lower plasma electron temperature is desired. The results allow optimization of the ion beam optics in the neutral beam injection system on HL-2A Tokomak and provide guidelines for designing future neutral beam injection system on HL-2M Tokomak. 7. Optics of ion beams for the neutral beam injection system on HL-2A Tokamak. Science.gov (United States) Zou, G Q; Lei, G J; Cao, J Y; Duan, X R 2012-07-01 The ion beam optics for the neutral beam injection system on HL-2A Tokomak is studied by two- dimensional numerical simulation program firstly, where the emitting surface is taken at 100 Debye lengths from the plasma electrode. The mathematical formulation, computation techniques are described. Typical ion orbits, equipotential contours, and emittance diagram are shown. For a fixed geometry electrode, the effect of plasma density, plasma potential and plasma electron temperature on ion beam optics is examined, and the calculation reliability is confirmed by experimental results. In order to improve ion beam optics, the application of a small pre-acceleration voltage (∼100 V) between the plasma electrode and the arc discharge anode is reasonable, and a lower plasma electron temperature is desired. The results allow optimization of the ion beam optics in the neutral beam injection system on HL-2A Tokomak and provide guidelines for designing future neutral beam injection system on HL-2M Tokomak. 8. PIXE and ion beam analysis in forensics International Nuclear Information System (INIS) Bailey, Melanie; Warmenhoven, John; Chrislopher, Matt; Kirkby, Karen; Palitsin, Vladimir; Grime, Geoff; Jeynes, Chris; Jones, Brian; Wenn, Roger 2013-01-01 Full text: University of Surrey has, for the past four years, collaborated with police institutions from across Europe and the rest of the world lo scope potential applications of ion beam analysis (IBA) in forensic science. In doing this we have consulted practitioners across a range of forensic disciplines, and critically compared IBA with conventional characterisation techniques to investigate the areas in which IBA can add evidential value. In this talk, the results of this feasibility study will be presented, showing the types of sample for which IBA shows considerable promise. We will show how a combination of PIXE with other IBA techniques (EBS, PIGE, MeV-SIMS) can be used to give unprecedented characterisation of forensic samples and comment on the significance of these results for forensic casework. We will also show cases where IBA not appear to add any significant improvement over conventional techniques. (author) 9. Prototyping of beam position monitor for medium energy beam transport section of RAON heavy ion accelerator Energy Technology Data Exchange (ETDEWEB) Jang, Hyojae, E-mail: [email protected]; Jin, Hyunchang; Jang, Ji-Ho; Hong, In-Seok [Rare Isotope Science Project, Institute for Basic Science, Daejeon (Korea, Republic of) 2016-02-15 A heavy ion accelerator, RAON is going to be built by Rare Isotope Science Project in Korea. Its target is to accelerate various stable ions such as uranium, proton, and xenon from electron cyclotron resonance ion source and some rare isotopes from isotope separation on-line. The beam shaping, charge selection, and modulation should be applied to the ions from these ion sources because RAON adopts a superconducting linear accelerator structure for beam acceleration. For such treatment, low energy beam transport, radio frequency quadrupole, and medium energy beam transport (MEBT) will be installed in injector part of RAON accelerator. Recently, development of a prototype of stripline beam position monitor (BPM) to measure the position of ion beams in MEBT section is under way. In this presentation, design of stripline, electromagnetic (EM) simulation results, and RF measurement test results obtained from the prototyped BPM will be described. 10. Beam-plasma discharge in a Kyoto beam-plasma-ion source International Nuclear Information System (INIS) Ishikawa, J.; Takagi, T. 1983-01-01 A beam-plasma type ion source employing an original operating principle has been developed by the present authors. The ion source consists of an ion extraction region with an electron gun, a thin long drift tube as the plasma production chamber, and a primary electron beam collector. An electron beam is effectively utilized for the dual purpose of high density plasma production as a result of beam-plasma discharge, and high current ion beam extraction with ion space-charge compensation. A high density plasma of the order of 10 11 --10 13 cm -3 was produced by virtue of the beam-plasma discharge which was caused by the interaction between a space-charge wave on the electron beam and a high frequency plasma wave. The plasma density then produced was 10 2 --10 3 times the density produced only by collisional ionization by the electron beam. In order to obtain a stable beam-plasma discharge, a secondary electron beam emitted from the electron collector should be utilized. The mechanism of the beam-plasma discharge was analyzed by use of a linear theory in the case of the small thermal energy of the electron beam, and by use of a quasilinear theory in the case of the large thermal energy. High current ion beams of more than 0.1 A were extracted even at a low extraction voltage of 1--5 kV 11. The emittance of high current heavy ion beams International Nuclear Information System (INIS) White, N.R.; Devaney, A.S. 1989-01-01 Ion implantation is the main application for high current heavy ion beams. Transfer ratio is defined as the ratio of the total ion current leaving the ion source to the current delivered to the endstation. This ratio is monitored and logged and its importance is explained. It is also affected by other factors, such as the isotopic and molecular composition of the total ion beam. The transfer ratio reveals the fraction of ions which are intercepted by parts of the beamline system. The effects of these ions are discussed in two categories: processing purity and reliability. In discussing the emittance of ribbon beams, the two orthogonal planes are usually considered separately. Longitudinal emittance is determined by slot length and by plasma ion temperature. It has already been revealed that the longitudinal divergence of the beams from BF3 is perhaps double that of the beam from arsenic vapour or argon, at the same total perveance from the ion source. This poses the question: why is the ion temperature higher for BF3 than for As or Ar? The transverse emittance is in practical terms dominated by the divergence. It is the most fruitful area for improvement in most real-world systems. There is an intrinsic divergence arising from initial ion energies within the plasma, and there is emittance growth that can occur as a result of aberration in the beam extraction optics. (N.K.) 12. Effects of beam, target and substrate potentials in ion beam processing International Nuclear Information System (INIS) Harper, J.M.E. 1982-01-01 Ion beam etching and deposition are normally carried out with beam, target and substrate potentials near ground potential. In this paper, the effects of intentional or unintentional changes in these potentials are described. Examples include beam neutralization, a single extraction grid, substrate bias, and target bias. Each example is described in terms of beam plasma parameters. (Auth.) 13. Focused ion beam (FIB) milling of electrically insulating specimens using simultaneous primary electron and ion beam irradiation International Nuclear Information System (INIS) Stokes, D J; Vystavel, T; Morrissey, F 2007-01-01 There is currently great interest in combining focused ion beam (FIB) and scanning electron microscopy technologies for advanced studies of polymeric materials and biological microstructures, as well as for sophisticated nanoscale fabrication and prototyping. Irradiation of electrically insulating materials with a positive ion beam in high vacuum can lead to the accumulation of charge, causing deflection of the ion beam. The resultant image drift has significant consequences upon the accuracy and quality of FIB milling, imaging and chemical vapour deposition. A method is described for suppressing ion beam drift using a defocused, low-energy primary electron beam, leading to the derivation of a mathematical expression to correlate the ion and electron beam energies and currents with other parameters required for electrically stabilizing these challenging materials 14. Application of ECR ion source beams in atomic physics Energy Technology Data Exchange (ETDEWEB) Meyer, F.W. 1987-01-01 The availability of intense, high charge state ion beams from ECR ion sources has had significant impact not only on the upgrading of cyclotron and synchrotron facilities, but also on multicharged ion collision research, as evidenced by the increasing number of ECR source facilities used at least on a part time basis for atomic physics research. In this paper one such facility, located at the ORNL ECR source, and dedicated full time to the study of multicharged ion collisions, is described. Examples of applications of ECR ion source beams are given, based on multicharged ion collision physics studies performed at Oak Ridge over the last few years. 21 refs., 18 figs., 2 tabs. 15. Ion beam analysis and modern materials science International Nuclear Information System (INIS) Feldman, Leonard C. 2012-01-01 Full text: Modern research has provided the means of creating materials structures controlled at the atomic scale. Familiar examples include the formation of hetero-structures grown with atomic precision, nanostructures with designed electronic properties and new organic structures employing the richness of organic chemistry. The current forefront of such materials research includes the creation of new materials for energy and electronics applications. The electron transport properties of these diverse materials, and hence their performance, is invariably linked by the basic interactions at the interface. Interfaces are the critical component, and least understood aspect, of almost all such materials-based structures. Ion beam analysis, and its role in interfacial definition, will be described in the context of a number of such forefront projects underway at the Rutgers Institute for Advanced Materials, Devices and Nanotechnology (IAMDN). These include: 1) quantitative analysis of self-assembled monolayers on organic single crystals resulting in enhanced surface mobility and more effective organic field effect transistors, 2) monolayer scale interfacial analysis of complex oxide hetero-structures to elucidate the properties of the enhanced two-dimensional electron mobility and 3) characterization of the semiconductor- dielectric interface in the SiC/SiO2 system, with application for energy efficient power transmission. Despite extraordinary advances in synthesis, interface properties continue as an uncontrolled region of hetero-materials formation. Their understanding requires the detailed analysis of a complement of tools including ion beam analysis. Fellow Researchers: R. A. Bartynski, L.C.Feldman, E. Garfunkel, T. Gustafsson, H.D. Lee, D. Mastrogiovanni, V. Podzorov, L. S. Wielunski, J. R. Williams(Auburn), G. Liu, J. Williams, S. Dhar. (author) 16. Molecular-beam epitaxial growth and ion-beam analysis systems for functional materials research International Nuclear Information System (INIS) Takeshita, H.; Aoki, Y.; Yamamoto, S.; Naramoto, H. 1992-01-01 Experimental systems for molecular beam epitaxial growth and ion beam analysis have been designed and constructed for the research of inorganic functional materials such as thin films and superlattices. (author) 17. Beam diagnostics and data acquisition system for ion beam transport line used in applied research International Nuclear Information System (INIS) Skuratov, V.A.; Didyk, A.Yu.; Arkhipov, A.V.; Illes, A.; Bodnar, K.; Illes, Z.; Havancsak, K. 1999-01-01 Ion beam transport line for applied research on U-400 cyclotron, beam diagnostics and data acquisition system for condensed matter studies are described. The main features of Windows-based real time program are considered 18. Energy spread in ion beam analysis International Nuclear Information System (INIS) Szilagyi, E. 2000-01-01 In ion beam analysis (IBA) the depth profiles are extracted from the experimentally determined energy profiles. The spectra, however, are subject to finite energy resolution of both extrinsic and intrinsic origin. Calculation of those effects such as instrumental beam, geometry and detection-related energy and angular spreads as well as energy straggling, multiple scattering and Doppler effects in the sample itself is not trivial, especially since it involves treatment of non-independent random processes. A proper account for energy spread is vital in IBA not only for correct extraction of elemental and isotopic depth profiles from the measured spectra, but already prior to data acquisition, in optimising experimental conditions to reach the required depth resolution at a certain depth. After a short review of the literature on the different energy spread contributions experimental examples are given from resonance, RBS, elastic BS and ERDA practice in which an account for energy spread contributions is essential. Some further examples illustrate extraction of structural information (roughness, pore size, etc.) from elaborated depth resolution calculation for such layer structures 19. Energy spread in ion beam analysis Energy Technology Data Exchange (ETDEWEB) Szilagyi, E. E-mail: [email protected] 2000-03-01 In ion beam analysis (IBA) the depth profiles are extracted from the experimentally determined energy profiles. The spectra, however, are subject to finite energy resolution of both extrinsic and intrinsic origin. Calculation of those effects such as instrumental beam, geometry and detection-related energy and angular spreads as well as energy straggling, multiple scattering and Doppler effects in the sample itself is not trivial, especially since it involves treatment of non-independent random processes. A proper account for energy spread is vital in IBA not only for correct extraction of elemental and isotopic depth profiles from the measured spectra, but already prior to data acquisition, in optimising experimental conditions to reach the required depth resolution at a certain depth. After a short review of the literature on the different energy spread contributions experimental examples are given from resonance, RBS, elastic BS and ERDA practice in which an account for energy spread contributions is essential. Some further examples illustrate extraction of structural information (roughness, pore size, etc.) from elaborated depth resolution calculation for such layer structures. 20. Generation of an intense ion beam by a pinched relativistic electron beam International Nuclear Information System (INIS) 1976-01-01 The pinched electron beam of a pulsed electron accelerator is used to generate an intense beam of ions. A foil anode and vacuum drift tube are used. The space charge field of the pinched beam in the tube accelerates ions from the foil anode. Ion currents of 10 kA at a density of 5kA/cm 2 with pulse length of 50 ns are obtained using a 5 kJ, 450 kV, 3 Ω diode. (author) 1. Wave Propagation in an Ion Beam-Plasma System DEFF Research Database (Denmark) Jensen, T. D.; Michelsen, Poul; Juul Rasmussen, Jens 1979-01-01 The spatial evolution of a velocity- or density-modulated ion beam is calculated for stable and unstable ion beam plasma systems, using the linearized Vlasov-Poisson equations. The propagation properties are found to be strongly dependent on the form of modulation. In the case of velocity... 2. High energy density in matter produced by heavy ion beams International Nuclear Information System (INIS) 1987-08-01 This annual report summarizes the results of research carried out in 1986 within the framework of the program 'High Energy Density in Matter Produced by Heavy Ion Beams' which is funded by the Federal Ministry for Research and Technology. Its initial motivation and its ultimate goal is the question whether inertial confinement can be achieved by intense beams of heavy ions. (orig./HSI) 3. Generation and focusing of intense ion beams with an inverse pinch ion diode International Nuclear Information System (INIS) Hashimoto, Yoshiyuki; Sato, Morihiko; Yatsuzuka, Mitsuyasu; Nobuhara, Sadao 1992-01-01 Generation and focusing of ion beams using an inverse pinch ion diode with a flat anode has been studied. The ion beams generated with the inverse pinch ion diode were found to be focused at 120 mm from the anode by the electrostatic field in the diode. The energy and maximum current density of the ion beams were 180 keV and 420 A/cm 2 , respectively. The focusing angle of the ion beams was 4.3deg. The beam brightness was estimated to be 1.3 GW/cm 2 ·rad 2 . The focusing distance of the ion beams was found to be controllable by changing the diameters of the anode and cathode. (author) 4. Guided transmission of slow Ne ions through the nanochannels of highly ordered anodic alumina DEFF Research Database (Denmark) Mátéfi-Tempfli, Stefan; Mátéfi-Tempfli, M.; Piraux, L. 2006-01-01 as a suspended membrane of about 15νm thickness on the aluminium frame to which it belongs. The AlO capillaries were bombarded with 3keV Ne ions. The first results unambiguously show the existence of ion guiding observed at 5° and 7.5° tilt angles of the capillaries compared to the beam direction. To the best... 5. Ion beam induced stress formation and relaxation in germanium Energy Technology Data Exchange (ETDEWEB) Steinbach, T., E-mail: [email protected] [Institut für Festkörperphysik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena (Germany); Reupert, A.; Schmidt, E.; Wesch, W. [Institut für Festkörperphysik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena (Germany) 2013-07-15 Ion irradiation of crystalline solids leads not only to defect formation and amorphization but also to mechanical stress. In the past, many investigations in various materials were performed focusing on the ion beam induced damage formation but only several experiments were done to investigate the ion beam induced stress evolution. Especially in microelectronic devices, mechanical stress leads to several unwanted effects like cracking and peeling of surface layers as well as changing physical properties and anomalous diffusion of dopants. To study the stress formation and relaxation process in semiconductors, crystalline and amorphous germanium samples were irradiated with 3 MeV iodine ions at different ion fluence rates. The irradiation induced stress evolution was measured in situ with a laser reflection technique as a function of ion fluence, whereas the damage formation was investigated by means of Rutherford backscattering spectrometry. The investigations show that mechanical stress builds up at low ion fluences as a direct consequence of ion beam induced point defect formation. However, further ion irradiation causes a stress relaxation which is attributed to the accumulation of point defects and therefore the creation of amorphous regions. A constant stress state is reached at high ion fluences if a homogeneous amorphous surface layer was formed and no further ion beam induced phase transition took place. Based on the results, we can conclude that the ion beam induced stress evolution seems to be mainly dominated by the creation and accumulation of irradiation induced structural modification. 6. Monte Carlo program for the cold neutron beam guide International Nuclear Information System (INIS) Yoshiki, H. 1985-02-01 A Monte Carlo program for the transport of cold neutrons through beam guides has been developed assuming that the neutrons follow the specular reflections. Cold neutron beam guides are normally used to transport cold neutrons (4 ∼ 10 Angstrom) to experimental equipments such as small angle scattering apparatus, TOF measuring devices, polarized neutron spectrometers, and ultra cold neutron generators, etc. The beam guide is about tens of meters in length and is composed from a meter long guide elements made up from four pieces of Ni coated rectangular optical glass. This report describes mathematics and algorithm employed in the Monte Carlo program together with the display of the results. The source program and input data listings are also attached. (Aoki, K.) 7. Ion Beams: A Powerful Tool for Making New Functional Materials International Nuclear Information System (INIS) Dev, B. N. 2010-01-01 It is well known that ion beams play an important role in semiconductor industry, which utilizes ion implantation and irradiation for materials modification. Ion sputtering technique is used to fabricate multifunctional coatings and multilayers. Using ion implantation, there is a continued effort for fabrication of quantum bit structures for future quantum computers. Availability of focused ion beams (FIBs) has widened the applications of ion beams and nanostructured functional materials are being fabricated using FIBs. Various quantum structures can be fabricated using FIB. Ferromagnetism can either be induced or destroyed in special layered structures using ion irradiation. The magnetic exchange bias phenomenon is of tremendous utility in magnetic recording. Issues of lateral diffusion in nanoscale doping of semiconductors by FIB and an example of exchange bias enhancement by ion irradiation are discussed. 8. Lifetime obtained by ion beam assisted deposition Energy Technology Data Exchange (ETDEWEB) Chakaroun, M. [XLIM-MINACOM-UMR 6172, Faculte des Sciences et Techniques, 123 av. Albert Thomas, 87060 Limoges cedex (France); Antony, R. [XLIM-MINACOM-UMR 6172, Faculte des Sciences et Techniques, 123 av. Albert Thomas, 87060 Limoges cedex (France)], E-mail: [email protected]; Taillepierre, P.; Moliton, A. [XLIM-MINACOM-UMR 6172, Faculte des Sciences et Techniques, 123 av. Albert Thomas, 87060 Limoges cedex (France) 2007-09-15 We have fabricated green organic light-emitting diodes based on tris-(8-hydroxyquinoline)aluminium (Alq3) thin films. In order to favor the charge carriers transport from the anode, we have deposited a N,N'-diphenyl-N,N'-bis (3-methylphenyl)-1,1'-diphenyl-4,4'-diamine (TPD) layer (hole transport layer) on a ITO anode. Cathode is obtained with a calcium layer covered with a silver layer. This silver layer is used to protect the other layers against oxygen during the OLED use. All the depositions are performed under vacuum and the devices are not exposed to air during their realisation. In order to improve the silver layer characteristics, we have realized this layer with the ion beam assisted deposition process. The aim of this process is to densify the layer and then reduce the permeation of H{sub 2}O and O{sub 2}. We have used argon ions to assist the silver deposition. All the OLEDs optoelectronic characterizations (I = f(V), L = f(V)) are performed in the ambient air. We compare the results obtained with the assisted layer with those obtained with a classical cathode realized by thermal unassisted evaporation. We have realized lifetime measurements in the ambient air and we discuss about the assisted layer influence on the OLEDs performances. 9. Can one crystallize a heavy ion beam? International Nuclear Information System (INIS) Hasse, R.W. 1990-05-01 We study the possibility of obtaining liquid or crystalline ordered structures in a cooled heavy ion beam in a storage ring. First the structure of very cold ions confined in a cylindrically symmetric static potential is explored by means of molecular dynamics calculations. Liquid like structures are obtained for the ratio of average Coulomb to thermal energies and Γ ≅ 10 and crystalline structures like strings, zigzags, helices, tetrehedra, intertwined helices, polygons, etc. emerge for Γ > 25. For larger densities, the particles arrange in cylindrical shells and form equilateral triangles on their surfaces arranged in hexagons which are characteristic of two-dimensional Coulomb solids. The molecular dynamics results are compared to results of energy minimization of these structures or of geometrical models. Realistic molecular dynamics calculations in the lattice of the Experimental Storage Ring at GSI Darmstadt including the effects of the bending, focussing and defocussing magnets, of the free sections and of the electron cooler revealed that such structures at higher densities are easily destroyed by heating through shearing forces. Therefore the dynamics of the simple Coulomb string is explored in more detail. The potential energy for large amplitude longitudinal and transverse vibrations is calculated and the dispersion relations and response functions in the harmonic limit are given and possible excitation mechanisms are discussed. (orig.) 10. Ion beam sputter coatings for laser technology Science.gov (United States) Ristau, Detlev; Gross, Tobias 2005-09-01 The initial motivation for the development of Ion Beam Sputtering (IBS) processes was the need for optical coatings with extremely low optical scatter losses for laser gyros. Especially, backscattering of the gyro-mirrors couples the directional modes in the ring resonator leading to the lock in effect which limits the sensitivity of the gyro. Accordingly, the first patent on IBS was approved for an aircraft company (Litton) in 1978. In the course of the rapid development of the IBS-concept during the last two decades, an extremely high optical quality could be achieved for laser coatings in the VIS- and NIR-spectral region. For example, high reflecting coatings with total optical losses below 1 ppm were demonstrated for specific precision measurement applications with the Nd:YAG-laser operating at 1.064 μm. Even though the high quality level of IBS-coatings had been confirmed in many applications, the process has not found its way into the production environment of most optical companies. Major restrictions are the relatively low rate of the deposition process and the poor lateral homogeneity of the coatings, which are related to the output characteristics of the currently available ion sources. In the present contribution, the basic principles of IBS will be discussed in the context of the demands of modern laser technology. Besides selected examples for special applications of IBS, aspects will be presented for approaches towards rapid manufacturing of coatings and the production of rugate filters on the basis of IBS-techniques. 11. Reconstruction of negative hydrogen ion beam properties from beamline diagnostics Energy Technology Data Exchange (ETDEWEB) Ruf, Benjamin 2014-09-25 For the experimental fusion reactor ITER, which should show the feasibility of sustaining a fusion plasma with a positive power balance, some technology still has to be developed, amongst others also the plasma heating system. One heating technique is the neutral beam injection (NBI). A beam of fast deuterium atoms is injected into the fusion plasma. By heavy particle collisions the beam particles give their energy to the plasma. A NBI system consists of three major components. First, deuterium ions are generated in a low temperature, low pressure plasma of an ion source. At ITER, the requirements on the beam energy of 1 MeV cause the necessity of negative charged deuterium ions. Secondly, the ions are accelerated within an acceleration system with several grids, where the plasma grid is the first grid. The grids are on different descending high voltage potentials. The source itself is on the highest negative potential. Thirdly, the fast deuterium ions have to be neutralised. This thesis deals with the second step in the mentioned beam system, the ion acceleration and beam formation. The underlying experiments and measurements were carried out at the testbeds BATMAN (BAvarianTest MAchine for Negative ions) and ELISE (Extraction from a Large Ion Source Experiment) at the Max-Planck-Institut fuer Plasmaphysik Garching (IPP Garching). The main goal of this thesis is to provide a tool which allows the determination of the beam properties. These are beam divergence, stripping losses and beam inhomogeneity. For this purpose a particle trajectory code has been developed from scratch, namely BBC-NI (Bavarian Beam Code for Negative Ions). The code is able to simulate the whole beam and the outcome of several beam diagnostic tools. The data obtained from the code together with the measurements of the beam diagnostic tools should allow the reconstruction of the beam properties. The major beam diagnostic tool, which is used in this thesis, is the beam emission spectroscopy 12. Study of tapered glass capillary focusing MeV ion beam International Nuclear Information System (INIS) Gong Zhiyu; Yan Sha; Ma Hongji; Nie Rui; Xue Jianming; Wang Yugang 2012-01-01 In recent years, tapered glass capillary ion beam focusing is developing rapidly. It is attractive for simple, compact, low cost and easy use. However, the focusing mechanism for MeV ion beams is still indistinct. We present several experimental results of focusing 2 MeV He + beam. Ion beams were focused by tapered glass capillaries with various outlet inner diameters from several micron to hundred micron. The current densities, angle divergences and energy spectra of the transmitted ion beams are measured. The results proved that 2 MeV He + ions can focused and guided by our capillaries. The energy spectra show that a great part of transmitted ions experienced obvious energy loss, which is different from results of others research groups. We discussed the reason and charged it to the larger incident angle. Considered the incident ions with larger incident angle, the charge will distribute in a layer of micro meter depth in the capillary’s inner wall, but not the surface. The energy loss and many other spectra characters can be explained in this way. 13. Status of radioactive ion beams at the HRIBF CERN Document Server Stracener, D W 2003-01-01 Radioactive Ion Beams (RIBs) at the Holifield Radioactive Ion Beam Facility (HRIBF) are produced using the isotope separation on-line technique and are subsequently accelerated up to a few MeV per nucleon for use in nuclear physics experiments. The first RIB experiments at the HRIBF were completed at the end of 1998 using sup 1 sup 7 F beams. Since then other proton-rich ion beams have been developed and a large number of neutron-rich ion beams are now available. The neutron-rich radioactive nuclei are produced via proton-induced fission of uranium in a low-density matrix of uranium carbide. Recently developed RIBs include sup 2 sup 5 Al from a silicon carbide target and isobarically pure beams of neutron-rich Ge, Sn, Br and I isotopes from a uranium carbide target. 14. Lithium ion beam driven hohlraums for PBFA II International Nuclear Information System (INIS) Dukart, R.J. 1994-01-01 In our light ion inertial confinement fusion (ICF) program, fusion capsules are driven with an intense x-ray radiation field produced when an intense beam of ions penetrates a radiation case and deposits energy in a foam x-ray conversion region. A first step in the program is to generate and measure these intense fields on the Particle Beam Fusion Accelerator II (PBFA II). Our goal is to generate a 100-eV radiation temperature in lithium ion beam driven hohlraums, the radiation environment which will provide the initial drive temperature for ion beam driven implosion systems designed to achieve high gain. In this paper, we describe the design of such hohlraum targets and their predicted performance on PBFA II as we provide increasing ion beam intensities 15. Laser cooling of a magnetically guided ultra cold atom beam Energy Technology Data Exchange (ETDEWEB) Aghajani-Talesh, Anoush 2014-07-01 This thesis examines two complimentary methods for the laser cooling of a magnetically guided ultra-cold atom beam. If combined, these methods could serve as a starting point for high-through put and possibly even continuous production of Bose-Einstein condensates. First, a mechanism is outlined to harvest ultra cold atoms from a magnetically guided atom beam into an optical dipole trap. A continuous loading scheme is described that dissipates the directed kinetic energy of a captured atom via deceleration by a magnetic potential barrier followed by optical pumping to the energetically lowest Zeeman sublevel. The application of this scheme to the transfer of ultra cold chromium atoms from a magnetically guided atom beam into a deep optical dipole trap is investigated via numerical simulations of the loading process. Based on the results of the theoretical studies the feasibility and the efficiency of our loading scheme, including the realisation of a suitable magnetic field configuration, are analysed. Second, experiments were conducted on the transverse laser cooling of a magnetically guided beam of ultra cold chromium atoms. Radial compression by a tapering of the guide is employed to adiabatically heat the beam. Inside the tapered section heat is extracted from the atom beam by a two-dimensional optical molasses perpendicular to it, resulting in a significant increase of atomic phase space density. A magnetic offset field is applied to prevent optical pumping to untrapped states. Our results demonstrate that by a suitable choice of the magnetic offset field, the cooling beam intensity and detuning, atom losses and longitudinal heating can be avoided. Final temperatures below 65 μK have been achieved, corresponding to an increase of phase space density in the guided beam by more than a factor of 30. 16. Laser cooling of a magnetically guided ultra cold atom beam International Nuclear Information System (INIS) Aghajani-Talesh, Anoush 2014-01-01 This thesis examines two complimentary methods for the laser cooling of a magnetically guided ultra-cold atom beam. If combined, these methods could serve as a starting point for high-through put and possibly even continuous production of Bose-Einstein condensates. First, a mechanism is outlined to harvest ultra cold atoms from a magnetically guided atom beam into an optical dipole trap. A continuous loading scheme is described that dissipates the directed kinetic energy of a captured atom via deceleration by a magnetic potential barrier followed by optical pumping to the energetically lowest Zeeman sublevel. The application of this scheme to the transfer of ultra cold chromium atoms from a magnetically guided atom beam into a deep optical dipole trap is investigated via numerical simulations of the loading process. Based on the results of the theoretical studies the feasibility and the efficiency of our loading scheme, including the realisation of a suitable magnetic field configuration, are analysed. Second, experiments were conducted on the transverse laser cooling of a magnetically guided beam of ultra cold chromium atoms. Radial compression by a tapering of the guide is employed to adiabatically heat the beam. Inside the tapered section heat is extracted from the atom beam by a two-dimensional optical molasses perpendicular to it, resulting in a significant increase of atomic phase space density. A magnetic offset field is applied to prevent optical pumping to untrapped states. Our results demonstrate that by a suitable choice of the magnetic offset field, the cooling beam intensity and detuning, atom losses and longitudinal heating can be avoided. Final temperatures below 65 μK have been achieved, corresponding to an increase of phase space density in the guided beam by more than a factor of 30. 17. Application of electron beam, ion beam and positron beam to polymer sciences International Nuclear Information System (INIS) Tagawa, Seiichi 1999-01-01 Full text: Particle beams are finding increasing application in material sciences and the interest covers both applied as well as fundamental investigations. In the present talk application of electron and ion beams in several polymers such as polysilanes, polystyrene, polyolefins, polymethylmethacrylates and related polymers will be presented. It includes among other investigations (such as product analysis) pulse radiolysis studies and effect of LET on polymers. Importance of positron studies in material sciences especially bulk polymers is well documented. A relatively new technique, namely, positron beam application especially in thin film polymers is a new and emerging areas. The interest ranges from applied aspects as well as fundamental understanding of surfaces and interfaces. The present talk will detail the development of a pulsed positron beam using LINAC at Institute of Scientific and Industrial Research (ISIR) as well as its applications to polymer thin films 18. Underling modification in ion beam induced Si wafers International Nuclear Information System (INIS) Hazra, S.; Chini, T.K.; Sanyal, M.K.; Grenzer, J.; Pietsch, U. 2005-01-01 Subsurface (amorphous-crystalline interface) structure of keV ion beam modified Si(001) wafers was studied for the first time using non-destructive technique and compared with that of the top one. Ion-beam modifications of the Si samples were done using state-of-art high-current ion implanter facility at Saha Institute of Nuclear Physics by changing energy, dose and angle of incidence of the Ar + ion beam. To bring out the underlying modification depth-resolved x-ray grazing incidence diffraction has been carried out using synchrotron radiation facility, while the structure of the top surface was studied through atomic force microscopy 19. Electromagnetic ion beam instability upstream of the earth's bow shock International Nuclear Information System (INIS) Gary, S.P.; Gosling, J.T.; Forslund, D.W. 1981-01-01 The linear theory of the electromagnetic ion beam instability for arbitrary angles of propagation has been studied. The parameters considered in the theory are typical of the solar wind upstream of the earth's bow shock when a 'reflected' proton beam is present. Maximum growth occurs for propagation parallel to the ambient field B, but this instability also displays significant growth at wave-vectors oblique to B, Oblique, unstable modes seem to be the likely source of the compressive magnetic fluctuations recently observed in conjunction with 'diffuse' ion population. An energetic ion beam does not directly give rise to linear growth of either ion acoustic or whistler mode instabilities 20. Heavy ion particle beam interaction with a hot ionized target International Nuclear Information System (INIS) Dei-Cas, R.; Bardy, J.; Beuve, M.A.; Laget, J.P.; Menier, A.; Renaud, M. 1983-03-01 The present status of the experimental facility consisting of a heavy ion beam travelling through a laser created plasma target is described. Some aspects such as laser-tandem coupling, beam performances, constraints on the plasma parameter ranges, plasma and beam diagnostics are analyzed 1. Beam Tools for Geant4 (User's Guide) International Nuclear Information System (INIS) V.Daniel Elvira, Paul Lebrun and Panagiotis Spentzouris email [email protected] 2002-01-01 Geant4 is a tool kit developed by a collaboration of physicists and computer professionals in the high energy physics field for simulation of the passage of particles through matter. The motivation for the development of the Beam Tools is to extend the Geant4 applications to accelerator physics. The Beam Tools are a set of C++ classes designed to facilitate the simulation of accelerator elements: r.f. cavities, magnets, absorbers, etc. These elements are constructed from Geant4 solid volumes like boxes, tubes, trapezoids, or spheers. There are many computer programs for beam physics simulations, but Geant4 is ideal to model a beam through a material or to integrate a beam line with a complex detector. There are many such examples in the current international High Energy Physics programs. For instance, an essential part of the RandD associated with the Neutrino Source/Muon Collider accelerator is the ionization cooling channel, which is a section of the system aimed to reduce the size of the muon beam in phase space. The ionization cooling technique uses a combination of linacs and light absorbers to reduce the transverse momentum and size of the beam, while keeping the longitudinal momentum constant. The MuCool/MICE (muon cooling) experiments need accurate simulations of the beam transport through the cooling channel in addition to a detailed simulation of the detectors designed to measure the size of the beam. The accuracy of the models for physics processes associated with muon ionization and multiple scattering is critical in this type of applications. Another example is the simulation of the interaction region in future accelerators. The high luminosity and background environments expected in the Next Linear Collider (NLC) and the Very Large Hadron Collider (VLHC) pose great demand on the detectors, which may be optimized by means of a simulation of the detector-accelerator interface 2. A high charge state heavy ion beam source for HIF International Nuclear Information System (INIS) Eylon, S.; Henestroza, E. 1995-04-01 A high current low emittance high charge state heavy ion beam source is being developed. This is designed to deliver HIF (heavy ion fusion) driver accelerator scale beam. Using high-charge-state beam in a driver accelerator for HIF may increase the acceleration efficiency, leading to a reduction in the driver accelerator size and cost. The proposed source system which consists of the gas beam electron stripper followed by a high charge state beam separator, can be added to existing single charge state, low emittance, high brightness ion sources and injectors. We shall report on the source physics design using 2D beam envelope simulations and experimental feasibility studies' results using a neutral gas stripper and a beam separator at the exit of the LBL 2 MV injector 3. Measurements of Beam Ion Loss from the Compact Helical System International Nuclear Information System (INIS) Darrow, D.S.; Isobe, M.; Kondo, Takashi; Sasao, M. 2010-01-01 Beam ion loss from the Compact Helical System (CHS) has been measured with a scintillator-type probe. The total loss to the probe, and the pitch angle and gyroradius distributions of that loss, have been measured as various plasma parameters were scanned. Three classes of beam ion loss were observed at the probe position: passing ions with pitch angles within 10o of those of transition orbits, ions on transition orbits, and ions on trapped orbits, typically 15o or more from transition orbits. Some orbit calculations in this geometry have been performed in order to understand the characteristics of the loss. Simulation of the detector signal based upon the following of orbits from realistic beam deposition profiles is not able to reproduce the pitch angle distribution of the losses measured. Consequently it is inferred that internal plasma processes, whether magnetohydrodynamic modes, radial electric fields, or plasma turbulence, move previously confined beam ions to transition orbits, resulting in their loss. 4. Negative ions as a source of low energy neutral beams Energy Technology Data Exchange (ETDEWEB) Fink, J.H. 1980-01-01 Little consideration has been given to the impact of recent developments in negative ion source technology on the design of low energy neutral beam injectors. However, negative ion sources of improved operating efficiency, higher gas efficiency, and smaller beam divergence will lead to neutral deuterium injectors, operating at less than 100 keV, with better operating efficiencies and more compact layouts than can be obtained from positive ion systems. 5. Negative ions as a source of low energy neutral beams International Nuclear Information System (INIS) Fink, J.H. 1980-01-01 Little consideration has been given to the impact of recent developments in negative ion source technology on the design of low energy neutral beam injectors. However, negative ion sources of improved operating efficiency, higher gas efficiency, and smaller beam divergence will lead to neutral deuterium injectors, operating at less than 100 keV, with better operating efficiencies and more compact layouts than can be obtained from positive ion systems 6. Design of a negative ion neutral beam system for TNS International Nuclear Information System (INIS) Easoz, J.R.; Sink, D.A. 1979-01-01 A design is presented that suggests that a negative ion neutral beam based on direct extraction is applicable to TNS, assuming technological advancements in several areas. Improvements in negative ion sources, direct energy conversion of charged beams, and high speed cryogenic pumping are needed. The increase in efficiency over a positive ion system and the encouraging results of the first attempt at a total design justify increased effort in the development of the above mentioned areas 7. Space-charge compensation of highly charged ion beam from laser ion source International Nuclear Information System (INIS) Kondrashev, S.A.; Collier, J.; Sherwood, T.R. 1996-01-01 The problem of matching an ion beam delivered by a high-intensity ion source with an accelerator is considered. The experimental results of highly charged ion beam transport with space-charge compensation by electrons are presented. A tungsten thermionic cathode is used as a source of electrons for beam compensation. An increase of ion beam current density by a factor of 25 is obtained as a result of space-charge compensation at a distance of 3 m from the extraction system. The process of ion beam space-charge compensation, requirements for a source of electrons, and the influence of recombination losses in a space-charge-compensated ion beam are discussed. (author) 8. New beam for the CERN fixed target heavy ion programme CERN Document Server Hill, C E; O'Neill, M 2002-01-01 The physicists of the CERN heavy ion community (SPS fixed target physics) have requested lighter ions than the traditional lead ions, to scale their results and to check their theories. Studies have been carried out to investigate the behaviour of the ECR4 for the production of an indium beam. Stability problems and the low melting point of indium required some modifications to the oven power control system which will also benefit normal lead ion production. Present results of the source behaviour and the ion beam characteristics will be presented. 9. Dielectronic recombination measurements using the Electron Beam Ion Trap International Nuclear Information System (INIS) Knapp, D.A. 1991-01-01 We have used the Electron Beam Ion Trap at LLNL to study dielectronic recombination in highly charged ions. Our technique is unique because we observe the x-rays from dielectronic recombination at the same time we see x-rays from all other electron-ion interactions. We have recently taken high-resolution, state-selective data that resolves individual resonances 10. Recent radioactive ion beam program at RIKEN and related topics Keywords. RIKEN; radioactive ion beams; magic numbers. PACS No. 21.10.-k. 1. Introduction. In RIKEN, there are several heavy ion accelerators. Main accelerator is the RIKEN ring cyclotron (RRC) with K = 540, that has been operated from 1986. The RRC has two injectors; one is heavy ion linear accelerator that has been ... 11. Modeling space charge in beams for heavy-ion fusion International Nuclear Information System (INIS) Sharp, W.M. 1995-01-01 A new analytic model is presented which accurately estimates the radially averaged axial component of the space-charge field of an axisymmetric heavy-ion beam in a cylindrical beam pipe. The model recovers details of the field near the beam ends that are overlooked by simpler models, and the results compare well to exact solutions of Poisson's equation. Field values are shown for several simple beam profiles and are compared with values obtained from simpler models 12. Dynamics of heavy ion beams during longitudinal compression International Nuclear Information System (INIS) Ho, D.D.M.; Bangerter, R.O.; Lee, E.P.; Brandon, S.; Mark, J.W.K. 1987-01-01 Heavy ion beams with initially uniform line charge density can be compressed longitudinally by an order of magnitude in such a way that the compressed beam has uniform line charge density and velocity-tilt profiles. There are no envelope mismatch oscillations during compression. Although the transverse temperature varies along the beam and also varies with time, no substantial longitudinal and transverse emittance growth has been observed. Scaling laws for beam radius and transport system parameters are given 13. Ion beam modification of biological materials in nanoscale Science.gov (United States) Yu, L. D.; Anuntalabhochai, S. 2012-07-01 Ion interaction with biological objects in nanoscale is a novel research area stemming from applications of low-energy ion beams in biotechnology and biomedicine. Although the ion beam applications in biotechnology and biomedicine have achieved great successes, many mechanisms remain unclear and many new applications are to be explored. We have carried out some research on exploring the mechanisms and new applications besides attaining ion beam induction of mutation breeding and gene transformation. In the studies on the mechanisms, we focused our investigations on the direct interaction in nanoscale between ions and biological living materials. Our research topics have included the low-energy ion range in DNA, low-energy ion or neutral beam bombardment effect on DNA topological form change and mutation, low-energy ion or neutral beam bombardment effect on the cell envelope and gene transformation, and molecular dynamics simulation of ultra-low-energy ion irradiation of DNA. In the exploration of new applications, we have started experiments on ion irradiation or bombardment, in the nanoscaled depth or area, of human cells for biomedical research. This paper introduces our experiments and reports interesting results. 14. Ion beam modification of biological materials in nanoscale International Nuclear Information System (INIS) Yu, L.D.; Anuntalabhochai, S. 2012-01-01 Ion interaction with biological objects in nanoscale is a novel research area stemming from applications of low-energy ion beams in biotechnology and biomedicine. Although the ion beam applications in biotechnology and biomedicine have achieved great successes, many mechanisms remain unclear and many new applications are to be explored. We have carried out some research on exploring the mechanisms and new applications besides attaining ion beam induction of mutation breeding and gene transformation. In the studies on the mechanisms, we focused our investigations on the direct interaction in nanoscale between ions and biological living materials. Our research topics have included the low-energy ion range in DNA, low-energy ion or neutral beam bombardment effect on DNA topological form change and mutation, low-energy ion or neutral beam bombardment effect on the cell envelope and gene transformation, and molecular dynamics simulation of ultra-low-energy ion irradiation of DNA. In the exploration of new applications, we have started experiments on ion irradiation or bombardment, in the nanoscaled depth or area, of human cells for biomedical research. This paper introduces our experiments and reports interesting results. 15. Time evolution of ion guiding through nanocapillaries in a PET polymer International Nuclear Information System (INIS) Stolterfoht, N.; Hellhammer, R.; Pesic, Z.D.; Hoffmann, V.; Bundesmann, J.; Petrov, A.; Fink, D.; Sulik, B.; Shah, M.; Dunn, K.; Pedregosa, J.; McCullough, R.W. 2004-01-01 The time evolution of transmitting 1.6 keV H + and 3 keV Ne 7+ ions through nanocapillaries (100 nm diameter and 10 μm length) in PET insulators was studied. By measuring the angular distribution of the transmitted projectiles it is shown that the majority of ions are transported in their initial charge state along the capillary axis even when the capillaries are tilted with respect to the incident beam direction. The results indicate ion guiding effects, which are produced by charge-up effects influencing the ion trajectories in a self-organizing manner. The data analysis reveals that a certain fraction of capillaries is inclined with respect to the foil normal. Emphasis is given to unravel the influence of the capillary inclination on the guided transmission of the different ions species 16. Development of intense pulsed heavy ion beam diode using gas puff plasma gun as ion source International Nuclear Information System (INIS) Ito, H.; Higashiyama, M.; Takata, S.; Kitamura, I.; Masugata, K. 2006-01-01 A magnetically insulated ion diode with an active ion source of a gas puff plasma gun has been developed in order to generate a high-intensity pulsed heavy ion beam for the implantation process of semiconductors and the surface modification of materials. The nitrogen plasma produced by the plasma gun is injected into the acceleration gap of the diode with the external magnetic field system. The ion diode is operated at diode voltage approx. =200 kV, diode current approx. =2 kA and pulse duration approx. =150 ns. A new acceleration gap configuration for focusing ion beam has been designed in order to enhance the ion current density. The experimental results show that the ion current density is enhanced by a factor of 2 and the ion beam has the ion current density of 27 A/cm 2 . In addition, the coaxial type Marx generator with voltage 200 kV and current 15 kA has been developed and installed in the focus type ion diode. The ion beam of ion current density approx. =54 A/cm 2 is obtained. To produce metallic ion beams, an ion source by aluminum wire discharge has been developed and the aluminum plasma of ion current density ∼70 A/cm 2 is measured. (author) 17. Important atomic physics issues for ion beam fusion International Nuclear Information System (INIS) Bangerter, Roger. 1986-01-01 The nearly endless variety of interesting and challenging problems makes physics research enjoyable. Most of us would choose to be physicists even if physics had no practical applications. However, physics does have practical applications. This workshop deals with one of those applications, namely ion beam fusion. Not all interesting and challenging atomic physics questions are important for ion beam fusion. This paper suggests some questions that may be important for ion beam fusion. It also suggests some criteria for determining if a question is only interesting, or both interesting and important. Importance is time dependent and, because of some restrictions on the flow of information, also country dependent. In the early days of ion beam fusion, it was important to determine if ion beam fusion made sense. Approximate answers and bounds on various parameters were required. Accurate, detailed answers were not needed. Because of the efforts of many people attending this workshop, we now know that ion beam fusion does make some sense. We must still determine if ion beam fusion truly makes good sense. If it does make good sense, we must determine how to make it work. Accurate detailed answers are becoming increasingly important. (author) 18. A contemporary guide to beam dynamics International Nuclear Information System (INIS) Forest, E.; Hirata, Kohji 1992-09-01 A methodological discussion is given for single particle beam dynamics in circular machines. The discussions are introductory, but (or, even therefore) we avoid to rely on too much simplified concepts. We treat things from a very general and fundamental point of view, because this is the easiest and rightest way to teach how to simulate particle motion and how to analyze its results. We give some principles of particle tracking free from theoretical prejudices. We also introduce some transparent methods to deduce the necessary information from the tracking: many of the traditional beam-dynamics concepts can be abstracted from them as approximate quantities which are valid in certain limiting cases 19. A contemporary guide to beam dynamics International Nuclear Information System (INIS) Forest, E.; Hirata, Kohji. 1992-08-01 A methodological discussion is given for single particle beam dynamics in circular machines. The discussions are introductory, but (or, even therefore) we avoid to rely on too much simplified concepts. We treat things from a very general and fundamental point of view, because this is the easiest and rightest way to teach how to simulate particle motion and how to analyze its results. We give some principles of particle tracking free from theoretical prejudices. We also introduce some transparent methods to deduce the necessary information from the tracking: many of the traditional beam-dynamics concepts can be abstracted from them as approximate quantities which are valid in certain limiting cases. (author) 20. Gabor lens focusing of a negative ion beam International Nuclear Information System (INIS) Palkovic, J.A.; Mills, F.E.; Schmidt, C.; Young, D.E. 1989-05-01 Gabor or plasma lenses have previously been used to focus intense beams of positive ions at energies from 10 keV to 5 MeV. It is the large electrostatic field of the non-neutral plasma in the Gabor lens which is responsible for the focusing. Focusing an ion beam with a given sign of charge in a Gabor lens requires a non-neutral plasma with the opposite sign of charge as the beam. A Gabor lens constructed at Fermilab has been used to focus a 30 keV proton beam with good optical quality. We discuss studies of the action of a Gabor lens on a beam of negative ions. A Gabor lens has been considered for matching an H/sup /minus// beam into an RFQ in the redesign of the low energy section of the Fermilab linac. 9 refs., 3 figs., 1 tab 1. DIAGNOSTICS FOR ION BEAM DRIVEN HIGH ENERGY DENSITY PHYSICS EXPERIMENTS International Nuclear Information System (INIS) Bieniosek, F.M.; Henestroza, E.; Lidia, S.; Ni, P.A. 2010-01-01 Intense beams of heavy ions are capable of heating volumetric samples of matter to high energy density. Experiments are performed on the resulting warm dense matter (WDM) at the NDCX-I ion beam accelerator. The 0.3 MeV, 30-mA K + beam from NDCX-I heats foil targets by combined longitudinal and transverse neutralized drift compression of the ion beam. Both the compressed and uncompressed parts of the NDCX-I beam heat targets. The exotic state of matter (WDM) in these experiments requires specialized diagnostic techniques. We have developed a target chamber and fielded target diagnostics including a fast multi-channel optical pyrometer, optical streak camera, laser Doppler-shift interferometer (VISAR), beam transmission diagnostics, and high-speed gated cameras. We also present plans and opportunities for diagnostic development and a new target chamber for NDCX-II. 2. Ion beam collimating grid to reduce added defects Science.gov (United States) Lindquist, Walter B.; Kearney, Patrick A. 2003-01-01 A collimating grid for an ion source located after the exit grid. The collimating grid collimates the ion beamlets and disallows beam spread and limits the beam divergence during transients and steady state operation. The additional exit or collimating grid prevents beam divergence during turn-on and turn-off and prevents ions from hitting the periphery of the target where there is re-deposited material or from missing the target and hitting the wall of the vessel where there is deposited material, thereby preventing defects from being deposited on a substrate to be coated. Thus, the addition of a collimating grid to an ion source ensures that the ion beam will hit and be confined to a specific target area. 3. A quadrupole ion trap as low-energy cluster ion beam source CERN Document Server Uchida, N; Kanayama, T 2003-01-01 Kinetic energy distribution of ion beams was measured by a retarding field energy analyzer for a mass-selective cluster ion beam deposition system that uses a quadrupole ion trap as a cluster ion beam source. The results indicated that the system delivers a cluster-ion beam with energy distribution of approx 2 eV, which corresponded well to the calculation results of the trapping potentials in the ion trap. Using this deposition system, mass-selected hydrogenated Si cluster ions Si sub n H sub x sup + were actually deposited on Si(111)-(7x7) surfaces at impact kinetic energy E sub d of 3-30 eV. Observation by using a scanning tunneling microscope (STM) demonstrated that Si sub 6 H sub x sup + cluster ions landed on the surface without decomposition at E sub d =3 eV, while the deposition was destructive at E sub d>=18 eV. (author) 4. Ion source for ion beam deposition employing a novel electrode assembly Science.gov (United States) Hayes, A. V.; Kanarov, V.; Yevtukhov, R.; Hegde, H.; Druz, B.; Yakovlevitch, D.; Cheesman, W.; Mirkov, V. 2000-02-01 A rf inductively coupled ion source employing a novel electrode assembly for focusing a broad ion beam on a relatively small target area was developed. The primary application of this ion source is the deposition of thin films used in the fabrication of magnetic sensors and optical devices. The ion optics consists of a three-electrode set of multiaperture concave dished grids with a beam extraction diameter of 150 mm. Also described is a variation in the design providing a beam extraction diameter of 120 mm. Grid hole diameters and grid spacing were optimized for low beamlet divergence and low grid impingement currents. The radius of curvature of the grids was optimized to obtain an optimally focused ion beam at the target location. A novel grid fabrication and mounting design was employed which overcomes typical limitations of such grid assemblies, particularly in terms of maintaining optimum beam focusing conditions after multiple cycles of operation. Ion beam generation with argon and xenon gases in energy ranges from 0.3 to 2.0 keV was characterized. For operation with argon gas, beam currents greater than 0.5 A were obtained with a beam energy of 800 eV. At optimal beam formation conditions, beam profiles at distances about equal to the radius of curvature were found to be close to Gaussian, with 99.9% of the beam current located within a 150 mm target diameter. Repeatability of the beam profile over long periods of operation is also reported. 5. Beam optics study of a negative ion source for neutral beam injection application at ASIPP Energy Technology Data Exchange (ETDEWEB) Wei, Jiang-Long; Liang, Li-Zhen [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031 (China); Jiang, Cai-Chao [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031 (China); Graduate school, University of Science and Technology of China, Hefei 230026 (China); Xie, Ya-Hong, E-mail: [email protected] [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031 (China); Hu, Chun-Dong; Li, Jun; Gu, Yu-Ming; Chen, Yu-Qian [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031 (China); Li, Jing-Yong; Wu, Ming-Shan [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031 (China); Graduate school, University of Science and Technology of China, Hefei 230026 (China) 2017-04-15 In order to study the generation and extraction of negative ions for neutral beam injection application, a negative ion source is being designed and constructed at Institute of Plasma Physics, Chinese Academy of Sciences (ASIPP). Through a four electrode grids system inside the accelerator, a negative ion beam will be extracted and accelerated up to −60 kV on a reduced scale extraction area of 12 × 50 cm{sup 2} (the area of PG apertures is 185 cm{sup 2}). The beam optics is a key issue for the accelerator design, and greatly determine the source experimental performance in term of beam current, heat load on the grid, beam divergence, and so on. In this paper, the trajectories of electrons and negative ions were simulated in the electrode grids of the negative ion source. The filter capability of electron deflection magnet on the co-extracted electrons is evaluated and confirmed. The negative ion beam optics was designed according to the calculated results of beam divergence and beam radius along the beamlet in different acceleration voltages. The deflection effect of the electron deflection magnet on the negative ion beam was investigated in the single beamlet case and multi-beamlets case. 6. Beam-optics study of the gantry beam delivery system for light-ion cancer therapy International Nuclear Information System (INIS) Pavlovic, M. 1995-12-01 Ion optics considerations on the granty-like beam delivery system for light-ion cancer therapy are presented. A low-angle active beam scanning in two directions is included in the preliminary gantry design. The optical properties of several gantry modifications are discussed. (orig.) 7. Ion beam characterisation of nanometre structures Energy Technology Data Exchange (ETDEWEB) 1995-08-01 Ion beam analysis methods have been applied to the study of technologically important issues in III-V nanometre structure science. In the first application, the incorporation of hydrogen in GaAs during electron cyclotron resonance etching was studied using the {sup 1}H({sup 15}N,{alpha}{gamma}){sup 12}C reaction analysis method. The major part of the work was carried out using mass and energy dispersive Recoil Spectrometry (RS). RS was used to study reactions of thin metal films InP reactions. The metals investigated include Cr, Ti, Ni, Pd and Pt and the reactions as a function of temperature were studied to elucidate suitable compounds for contacts and metallization. Using {sup 127}I in the 0.5A to 0.7A MeV region as the projectile, the depth profiles for the different elements were obtained. Complementary measurements with X-ray diffraction to obtain chemical phase information as well as scanning electron microscopy to study the surface morphology were also carried out. 59 refs, 15 figs. 8. Ion beam characterisation of nanometre structures International Nuclear Information System (INIS) 1995-08-01 Ion beam analysis methods have been applied to the study of technologically important issues in III-V nanometre structure science. In the first application, the incorporation of hydrogen in GaAs during electron cyclotron resonance etching was studied using the 1 H( 15 N,αγ) 12 C reaction analysis method. The major part of the work was carried out using mass and energy dispersive Recoil Spectrometry (RS). RS was used to study reactions of thin metal films InP reactions. The metals investigated include Cr, Ti, Ni, Pd and Pt and the reactions as a function of temperature were studied to elucidate suitable compounds for contacts and metallization. Using 127 I in the 0.5A to 0.7A MeV region as the projectile, the depth profiles for the different elements were obtained. Complementary measurements with X-ray diffraction to obtain chemical phase information as well as scanning electron microscopy to study the surface morphology were also carried out. 59 refs, 15 figs 9. The production of accelerated radioactive ion beams International Nuclear Information System (INIS) Olsen, D.K. 1993-01-01 During the last few years, substantial work has been done and interest developed in the scientific opportunities available with accelerated radioactive ion beams (RIBs) for nuclear physics, astrophysics, and applied research. This interest has led to the construction, development, and proposed development of both first- and second-generation RIB facilities in Asia, North America, and Europe; international conferences on RIBs at Berkeley and Louvain-la-Neuve; and many workshops on specific aspects of RIB production and science. This paper provides a discussion of both the projectile fragmentation, PF, and isotope separator on-line, ISOL, approach to RIB production with particular emphasis on the latter approach, which employs a postaccelerator and is most suitable for nuclear structure physics. The existing, under construction, and proposed facilities worldwide are discussed. The paper draws heavily from the CERN ISOLDE work, the North American IsoSpin Laboratory (ISL) study, and the operating first-generation RIB facility at Louvain-la-Neuve, and the first-generation RIB project currently being constructed at ORNL 10. Basic aspects of ion beam mixing International Nuclear Information System (INIS) Averback, R.S. 1985-07-01 Irradiation of solids with energetic particles results in the reorganization of constituent target atoms, i.e., ion beam mixing (IM). At low temperatures, IM is characterized by prompt (10 -10 s) diffusion processes which are localized in the vicinity of the displacement cascade. Mixing at low temperatures can cause the system to depart far from the equilibrium state. At elevated temperatures, the diffusion of radiation-induced defects extends the mixing to longer times and greater distances. These delayed IM processes tend to return the system toward equilibrium. Recent experimental progress has led to a qualitative understanding of the fundamental aspects of IM in both temperature regimes. This has been achieved through systematic measurements of the influences of temperature, dose, dose-rate, cascade energy density, and chemical interactions on IM. The results of these experiments will be reviewed and compared to IM models based on collisional, thermal spike, and radiation-enhanced diffusion processes. The relation of IM to other fundamental radiation damage effects will also be discussed. 75 refs., 8 figs., 2 tabs 11. Edge effect correction using ion beam figuring. Science.gov (United States) Yang, Bing; Xie, Xuhui; Li, Furen; Zhou, Lin 2017-11-10 The edge effect is regarded as one of the most difficult technical issues for fabricating large primary mirrors, as it can greatly reduce the key performance of the optical system. Ion beam figuring (IBF) has the advantage of no edge effect, so we can use it to remove high points on the edge and improve surface accuracy. The edge local correction method (ELCM) of IBF processes only the surface edge zone, and is very different from the current full caliber figuring method (FCFM). Therefore, it is necessary to study the ELCM of IBF. In this paper, the key factors of ELCM are analyzed, such as dwell time algorithm, edge data extension methods, and the outward dimension of the starting figuring point. At the same time, the distinctions between ELCM and FCFM are compared. Finally, a 142 mm diameter fused silica mirror is fabricated to verify the validity of the theoretical of ELCM. The experimental results indicate that the figuring precision and efficiency can be obviously improved by ELCM. 12. Calculations of Neutral Beam Ion Confinement for the National Spherical Torus Experiment International Nuclear Information System (INIS) Redi, M.H.; Darrow, D.S.; Egedal, J.; Kaye, S.M.; White, R.B. 2002-01-01 The spherical torus (ST) concept underlies several contemporary plasma physics experiments, in which relatively low magnetic fields, high plasma edge q, and low aspect ratio combine for potentially compact, high beta and high performance fusion reactors. An important issue for the ST is the calculation of energetic ion confinement, as large Larmor radius makes conventional guiding center codes of limited usefulness and efficient plasma heating by RF and neutral beam ion technology requires minimal fast ion losses. The National Spherical Torus Experiment (NSTX) is a medium-sized, low aspect ratio ST, with R=0.85 m, a=0.67 m, R/a=1.26, Ip1.4 MA, Bt*0.6 T, 5 MW of neutral beam heating and 6 MW of RF heating. 80 keV neutral beam ions at tangency radii of 0.5, 0.6 and 0.7 m are routinely used to achieve plasma betas above 30%. Transport analyses for experiments on NSTX often exhibit a puzzling ion power balance. It will be necessary to have reliable beam ion calculations to distinguish among the source and loss channels, and to explore the possibilities for new physics phenomena, such as the recently proposed compressional Alfven eigenmode ion heating 13. Device for guiding a subthermal neutron beam and focussing device made of micro-neutron guides International Nuclear Information System (INIS) Marx, D. 1977-01-01 The invention concerns a device for guiding, in particular for diverting, a subthermal neutron beam with curved boundary surfaces at least in one level, whose sides towards the neutron beam are covered with at least one coating which reflects the subthermal neutrons completely. (orig./RW) [de 14. Verification of high efficient broad beam cold cathode ion source Energy Technology Data Exchange (ETDEWEB) Abdel Reheem, A. M., E-mail: [email protected] [Accelerators and Ion Sources Department, Nuclear Research Center, Atomic Energy Authority, P.N.13759, Cairo (Egypt); Radiation Physics Department, National Center for Radiation Research and Technology (NCRRT), Atomic Energy Authority (AEA), Cairo (Egypt); Ahmed, M. M. [Physics Department, Faculty of Science, Helwan University, Cairo (Egypt); Abdelhamid, M. M.; Ashour, A. H. [Radiation Physics Department, National Center for Radiation Research and Technology (NCRRT), Atomic Energy Authority (AEA), Cairo (Egypt) 2016-08-15 An improved form of cold cathode ion source has been designed and constructed. It consists of stainless steel hollow cylinder anode and stainless steel cathode disc, which are separated by a Teflon flange. The electrical discharge and output characteristics have been measured at different pressures using argon, nitrogen, and oxygen gases. The ion exit aperture shape and optimum distance between ion collector plate and cathode disc are studied. The stable discharge current and maximum output ion beam current have been obtained using grid exit aperture. It was found that the optimum distance between ion collector plate and ion exit aperture is equal to 6.25 cm. The cold cathode ion source is used to deposit aluminum coating layer on AZ31 magnesium alloy using argon ion beam current which equals 600 μA. Scanning electron microscope and X-ray diffraction techniques used for characterizing samples before and after aluminum deposition. 15. Hydrodynamic motion of a heavy-ion-beam-heated plasma International Nuclear Information System (INIS) Jacoby, J.; Hoffmann, D.H.H.; Mueller, R.W.; Mahrt-Olt, K.; Arnold, R.C.; Schneider, V.; Maruhn, J. 1990-01-01 The first experimental study is reported of a plasma produced by a heavy-ion beam. Relevant parameters for heating with heavy ions are described, temperature and density of the plasma are determined, and the hydrodynamic motion in the target induced by the beam is studied. The measured temperature and the free-electron density are compared with a two-dimensional hydrodynamic-model calculation. In accordance with the model, a radial rarefaction wave reaching the center of the target was observed and the penetration velocity of the ion beam into the xenon-gas target was measured 16. Modeling of neutral beam ion loss from CHS plasmas International Nuclear Information System (INIS) Darrow, D.S.; Isobe, Mitsutaka; Sasao, Mamiko; Kondo, T. 2000-01-01 Beam ion loss measurements from Compact Helical System (CHS) plasmas under a variety of conditions show a strong loss of ions in the range of pitch angles corresponding to transition orbits at the probe location. A numerical model has been developed which includes the beam ion orbits, and details of the detector, plasma, vessel, and neutral beam geometry. From this, the expected classical (i.e. collisionless single particle orbit) signal at the detector can be computed. Preliminary comparisons between the experimental data and model predictions indicate that the classical behavior of the orbits and the machine geometry are insufficient to explain the observations. (author) 17. A second-generation ion beam buncher and cooler International Nuclear Information System (INIS) Schwarz, S.; Bollen, G.; Lawton, D.; Neudert, A.; Ringle, R.; Schury, P.; Sun, T. 2003-01-01 A radiofrequency quadrupole (RFQ) ion accumulator and buncher has been designed for the low-energy beam and ion-trap (LEBIT) facility which is being set up at the NSCL/MSU. The LEBIT buncher will be a cryogenic system. Compared to room-temperature systems an improved beam quality and overall efficiency are expected. It will feature a novel electrode structure with a drastically reduced number of electrodes for simplified operation. Its design is presented and Monte-Carlo type ion-trajectory calculations are discussed which predict excellent beam quality and high performance 18. A second-generation ion beam buncher and cooler Energy Technology Data Exchange (ETDEWEB) Schwarz, S. E-mail: [email protected]; Bollen, G.; Lawton, D.; Neudert, A.; Ringle, R.; Schury, P.; Sun, T 2003-05-01 A radiofrequency quadrupole (RFQ) ion accumulator and buncher has been designed for the low-energy beam and ion-trap (LEBIT) facility which is being set up at the NSCL/MSU. The LEBIT buncher will be a cryogenic system. Compared to room-temperature systems an improved beam quality and overall efficiency are expected. It will feature a novel electrode structure with a drastically reduced number of electrodes for simplified operation. Its design is presented and Monte-Carlo type ion-trajectory calculations are discussed which predict excellent beam quality and high performance. 19. A second-generation ion beam buncher and cooler CERN Document Server Schwarz, S; Lawton, D; Neudert, A; Ringle, R; Schury, P; Sun, T 2003-01-01 A radiofrequency quadrupole (RFQ) ion accumulator and buncher has been designed for the low-energy beam and ion-trap (LEBIT) facility which is being set up at the NSCL/MSU. The LEBIT buncher will be a cryogenic system. Compared to room-temperature systems an improved beam quality and overall efficiency are expected. It will feature a novel electrode structure with a drastically reduced number of electrodes for simplified operation. Its design is presented and Monte-Carlo type ion-trajectory calculations are discussed which predict excellent beam quality and high performance. 20. Polarization Studies in Fast-Ion Beam Spectroscopy International Nuclear Information System (INIS) Trabert, E 2001-01-01 In a historical review, the observations and the insight gained from polarization studies of fast ions interacting with solid targets are presented. These began with J. Macek's recognition of zero-field quantum beats in beam-foil spectroscopy as indicating alignment, and D.G. Ellis' density operator analysis that suggested the observability of orientation when using tilted foils. Lastly H. Winter's studies of the ion-beam surface interaction at grazing incidence yielded the means to produce a high degree of nuclear orientation in ion beams 1. Characterization of a cryogenic ion guide at IGISOL NARCIS (Netherlands) Saastamoinen, A.; Moore, I. D.; Ranjan, M.; Dendooven, P.; Penttila, H.; Perajarvi, K.; Popov, A.; Aysto, J. 2012-01-01 A small volume cryogenic ion guide has been characterized at the IGISOL facility, Jyvaskyla, as a prototype to verify whether there are any major obstacles to the use of high-density cryogenic helium gas for the stopping and extraction of high-energy ions from a large volume cryogenic ion catcher. 2. Optical guiding and beam bending in free-electron lasers International Nuclear Information System (INIS) Scharlemann, E.T. 1987-01-01 The electron beam in a free-electron laser (FEL) can act as an optical fiber, guiding or bending the optical beam. The refractive and gain effects of the bunched electron beam can compensate for diffraction, making possible wigglers that are many Rayleigh ranges (i.e., characteristic diffraction lengths) long. The origin of optical guiding can be understood by examining gain and refractive guiding in a fiber with a complex index of refraction, providing a mathematical description applicable also to the FEL, with some extensions. In the exponential gain regime of the FEL, the electron equations of motion must be included, but a self-consistent description of exponential gain with diffraction fully included becomes possible. The origin of the effective index of refraction of an FEL is illustrated with a simple example of bunched, radiating dipoles. Some of the properties of the index of refraction are described. The limited experimental evidence for optical beam bending is summarized. The evidence does not yet provide conclusive proof of the existence of optical guiding, but supports the idea. Finally, the importance of refractive guiding for the performance of a high-gain tapered-wiggler FEL amplifier is illustrated with numerical simulations 3. Facilities for radiotherapy with ion beams status and worldwide developments CERN Document Server Wolf, B H 1999-01-01 Forty-five years after the first ion beam therapy in Berkeley around 25,000 cancer patients worldwide have been treated successfully. Ion accelerators, designed for nuclear research, delivered most of this treatment. The first hospital-based facility started operation in 1998 at Loma Linda California, the first for heavier ions at Chiba, Japan in 1994 and the first commercially delivered facilities started operation in 1998 at Kashiwa, Japan. In 2000, the Harvard Medical Centre, Boston, US, will commence operation and several new facilities are planned or under construction worldwide, although none in Australia. This paper will discuss the physical and biological advantages of ion beams over x-rays and electrons. In the treatment of cancer patients ion beam therapy is especially suited for localised tumours in radiation sensitive areas like skull or spine. Heavier ions are also effective in anoxic tumour cells (found around the normally oxygenated cell population). An additional advantage of the heavier carbo... 4. Design of the radioactive ion beam facility at the LNS International Nuclear Information System (INIS) Migneco, E.; Alba, R.; Calabretta, L.; Ciavola, G.; Cuttone, G.; Di Giacomo, M.; Gammino, S.; Gmaj, P.; Moscatello, M.H.; Raia, G. 1992-01-01 At the Laboratorio Nazionale del Sud the existing 15 MV Tandem will be coupled to the Superconducting Cyclotron booster, which will provide light and heavy ion beams in the energy range 100-20 MeV/n. Using these beams, secondary radioactive beams can be produced by projectile fragmentation. A fragment separator will collect the secondary beam produced at energies near that of the projectile and deliver it into the experimental areas. The possibility of using an ECRIS source for the axial injection into the Cyclotron and producing radioactive ions on a thick source placed inside the Tandem preinjector is also discussed. (author) 7 refs.; 2 figs.; 1 tab 5. High harmonic ion cyclotron heating in DIII-D: Beam ion absorption and sawtooth stabilization International Nuclear Information System (INIS) Heidbrink, W.W.; Fredrickson, E.D.; Mau, T.K.; Petty, C.C.; Pinsker, R.I.; Porkolab, M.; Rice, B.W. 1999-01-01 Combined neutral beam injection and fast wave heating at the fourth cyclotron harmonic produce an energetic deuterium beam ion tail in the DIII-D tokamak. When the concentration of thermal hydrogen exceeds ∼ 5%, the beam ion absorption is suppressed in favour of second harmonic hydrogen absorption. As theoretically expected, the beam absorption increases with beam ion gyro-radius; also, central absorption at the fifth harmonic is weaker than central absorption at the fourth harmonic. For central heating at the fourth harmonic, an energetic, perpendicular, beam population forms inside the q = 1 surface. The beam ion tail transiently stabilizes the sawtooth instability but destabilizes toroidicity induced Alfven eigenmodes (TAEs). Saturation of the central heating correlates with the onset of the TAEs. Continued expansion of the q = 1 radius eventually precipitates a sawtooth crash; complete magnetic reconnection is observed. (author) 6. Determination of the meniscus shape of a negative ion beam from an experimentally obtained beam profile Science.gov (United States) Ichikawa, M.; Kojima, A.; Chitarin, G.; Agostinetti, P.; Aprile, D.; Baltador, C.; Barbisan, M.; Delogu, R.; Hiratsuka, J.; Marconato, N.; Nishikiori, R.; Pimazzoni, A.; Sartori, E.; Serianni, G.; Tobari, H.; Umeda, N.; Veltri, P.; Watanabe, K.; Yoshida, M.; Antoni, V.; Kashiwagi, M. 2017-08-01 In order to understand the physics mechanism of a negative ion extraction in negative ion sources, an emission surface of the negative ions around an aperture at a plasma grid, so-called a meniscus, has been analyzed by an inverse calculation of the negative ion trajectory in a two dimensional beam analysis code. In this method, the meniscus is defined as the final position of the negative ion trajectories which are inversely calculated from the measured beam profile to the plasma grid. In a case of the volume-produced negative ions, the calculated meniscus by the inverse calculation was similar to that obtained in conventional beam simulation codes for positive ion extractions such as BEAMORBT and SLACCAD. The negative ion current density was uniform along the meniscus. This indicates that the negative ions produced in the plasma are transported to the plasma grid uniformly as considered in the transportation of the positive ions. However, in a surface production case of negative ions, where the negative ions are generated near the plasma grid with lower work function by seeding cesium, the current density in the peripheral region of the meniscus close to the plasma grid surface was estimated to be 2 times larger than the center region, which suggested that the extraction process of the surface-produced negative ions was much different with that for the positive ions. Because this non-uniform profile of the current density made the meniscus shape strongly concave, the beam extracted from the peripheral region could have a large divergence angle, which might be one of origins of so-called beam halo. This is the first results of the determination of the meniscus based on the experiment, which is useful to improve the prediction of the meniscus shape and heat loads based on the beam trajectories including beam halo. 7. Ion beam modification of solids ion-solid interaction and radiation damage CERN Document Server Wesch, Werner 2016-01-01 This book presents the method of ion beam modification of solids in realization, theory and applications in a comprehensive way. It provides a review of the physical basics of ion-solid interaction and on ion-beam induced structural modifications of solids. Ion beams are widely used to modify the physical properties of materials. A complete theory of ion stopping in matter and the calculation of the energy loss due to nuclear and electronic interactions are presented including the effect of ion channeling. To explain structural modifications due to high electronic excitations, different concepts are presented with special emphasis on the thermal spike model. Furthermore, general concepts of damage evolution as a function of ion mass, ion fluence, ion flux and temperature are described in detail and their limits and applicability are discussed. The effect of nuclear and electronic energy loss on structural modifications of solids such as damage formation, phase transitions and amorphization is reviewed for ins... 8. Surface characterization after subaperture reactive ion beam etching Energy Technology Data Exchange (ETDEWEB) Miessler, Andre; Arnold, Thomas; Rauschenbach, Bernd [Leibniz-Institut fuer Oberflaechenmodifizierung (IOM), Leipzig (Germany) 2010-07-01 In usual ion beam etching processes using inert gas (Ar, Xe, Kr..) the material removal is determined by physical sputtering effects on the surface. The admixture of suitable gases (CF{sub 4}+O{sub 2}) into the glow discharge of the ion beam source leads to the generation of reactive particles, which are accelerated towards the substrate where they enhance the sputtering process by formation of volatile chemical reaction products. During the last two decades research in Reactive Ion Beam Etching (RIBE) has been done using a broad beam ion source which allows the treatment of smaller samples (diameter sample < diameter beam). Our goal was to apply a sub-aperture Kaufman-type ion source in combination with an applicative movement of the sample with respect to the source, which enables us to etch areas larger than the typical lateral dimensions of the ion beam. Concerning this matter, the etching behavior in the beam periphery plays a decisive role and has to be investigated. We use interferometry to characterize the final surface topography and XPS measurements to analyze the chemical composition of the samples after RIBE. 9. Materials science education: ion beam modification and analysis of materials Science.gov (United States) Zimmerman, Robert; Muntele, Claudiu; Ila, Daryush 2012-08-01 The Center for Irradiation of Materials (CIM) at Alabama A&M University (http://cim.aamu.edu) was established in 1990 to serve the University in its research, education and services to the need of the local community and industry. CIM irradiation capabilities are oriented around two tandem-type ion accelerators with seven beam lines providing high-resolution Rutherford backscattering spectrometry, MeV focus ion beam, high-energy ion implantation and irradiation damage studies, particle-induced X-ray emission, particle-induced gamma emission and ion-induced nuclear reaction analysis in addition to fully automated ion channeling. One of the two tandem ion accelerators is designed to produce high-flux ion beam for MeV ion implantation and ion irradiation damage studies. The facility is well equipped with a variety of surface analysis systems, such as SEM, ESCA, as well as scanning micro-Raman analysis, UV-VIS Spectrometry, luminescence spectroscopy, thermal conductivity, electrical conductivity, IV/CV systems, mechanical test systems, AFM, FTIR, voltammetry analysis as well as low-energy implanters, ion beam-assisted deposition and MBE systems. In this presentation, we will demonstrate how the facility is used in material science education, as well as providing services to university, government and industry researches. 10. Modification of graphene by ion beam Science.gov (United States) Gawlik, G.; Ciepielewski, P.; Jagielski, J.; Baranowski, J. 2017-09-01 Ion induced defect generation in graphene was analyzed using Raman spectroscopy. A single layer graphene membrane produced by chemical vapor deposition (CVD) on copper foil and then transferred on glass substrate was subjected to helium, carbon, nitrogen, argon and krypton ions bombardment at energies from the range 25 keV to 100 keV. A density of ion induced defects and theirs mean size were estimated by using Raman measurements. Increasing number of defects generated by ion with increase of ion mass and decrease of ion energy was observed. Dependence of ion defect efficiency (defects/ion) on ion mass end energy was proportional to nuclear stopping power simulated by SRIM. No correlation between ion defect efficiency and electronic stopping power was observed. 11. Overview of Light-Ion Beam Therapy International Nuclear Information System (INIS) Chu, William T. 2006-01-01 compared to those in conventional (photon) treatments. Wilson wrote his personal account of this pioneering work in 1997. In 1954 Cornelius Tobias and John Lawrence at the Radiation Laboratory (former E.O. Lawrence Berkeley National Laboratory) of the University of California, Berkeley performed the first therapeutic exposure of human patients to hadron (deuteron and helium ion) beams at the 184-Inch Synchrocyclotron. By 1984, or 30 years after the first proton treatment at Berkeley, programs of proton radiation treatments had opened at: University of Uppsala, Sweden, 1957; the Massachusetts General Hospital-Harvard Cyclotron Laboratory (MGH/HCL), USA, 1961; Dubna (1967), Moscow (1969) and St Petersburg (1975) in Russia; Chiba (1979) and Tsukuba (1983) in Japan; and Villigen, Switzerland, 1984. These centers used the accelerators originally constructed for nuclear physics research. The experience at these centers has confirmed the efficacy of protons and light ions in increasing the tumor dose relative to normal tissue dose, with significant improvements in local control and patient survival for several tumor sites. M.R. Raju reviewed the early clinical studies. In 1990, the Loma Linda University Medical Center in California heralded in the age of dedicated medical accelerators when it commissioned its proton therapy facility with a 250-MeV synchrotron. Since then there has been a relatively rapid increase in the number of hospital-based proton treatment centers around the world, and by 2006 there are more than a dozen commercially-built facilities in use, five new facilities under construction, and more in planning stages. In the 1950s larger synchrotrons were built in the GeV region at Brookhaven (3-GeV Cosmotron) and at Berkeley (6-GeV Bevatron), and today most of the world's largest accelerators are synchrotrons. With advances in accelerator design in the early 1970s, synchrotrons at Berkeley and Princeton accelerated ions with atomic numbers between 6 and 18, at 12. Overview of Light-Ion Beam Therapy Energy Technology Data Exchange (ETDEWEB) Chu, William T. 2006-03-16 treatment volume compared to those in conventional (photon) treatments. Wilson wrote his personal account of this pioneering work in 1997. In 1954 Cornelius Tobias and John Lawrence at the Radiation Laboratory (former E.O. Lawrence Berkeley National Laboratory) of the University of California, Berkeley performed the first therapeutic exposure of human patients to hadron (deuteron and helium ion) beams at the 184-Inch Synchrocyclotron. By 1984, or 30 years after the first proton treatment at Berkeley, programs of proton radiation treatments had opened at: University of Uppsala, Sweden, 1957; the Massachusetts General Hospital-Harvard Cyclotron Laboratory (MGH/HCL), USA, 1961; Dubna (1967), Moscow (1969) and St Petersburg (1975) in Russia; Chiba (1979) and Tsukuba (1983) in Japan; and Villigen, Switzerland, 1984. These centers used the accelerators originally constructed for nuclear physics research. The experience at these centers has confirmed the efficacy of protons and light ions in increasing the tumor dose relative to normal tissue dose, with significant improvements in local control and patient survival for several tumor sites. M.R. Raju reviewed the early clinical studies. In 1990, the Loma Linda University Medical Center in California heralded in the age of dedicated medical accelerators when it commissioned its proton therapy facility with a 250-MeV synchrotron. Since then there has been a relatively rapid increase in the number of hospital-based proton treatment centers around the world, and by 2006 there are more than a dozen commercially-built facilities in use, five new facilities under construction, and more in planning stages. In the 1950s larger synchrotrons were built in the GeV region at Brookhaven (3-GeV Cosmotron) and at Berkeley (6-GeV Bevatron), and today most of the world's largest accelerators are synchrotrons. With advances in accelerator design in the early 1970s, synchrotrons at Berkeley and Princeton accelerated ions with atomic numbers 13. Ion sources for initial use at the Holifield radioactive ion beam facility International Nuclear Information System (INIS) Alton, G.D. 1994-01-01 The Holifield Radioactive Ion Beam Facility (HRIBF) now under construction at the Oak Ridge National Laboratory will use the 25-MV tandem accelerator for the acceleration of radioactive ion beams to energies appropriate for research in nuclear physics; negative ion beams are, therefore, required for injection into the tandem accelerator. Because charge exchange is an efficient means for converting initially positive ion beams to negative ion beams, both positive and negative ion sources are viable options for use at the facility; the choice of the type of ion source will depend on the overall efficiency for generating the radioactive species of interest. A high-temperature version of the CERN-ISOLDE positive ion source has been selected and a modified version of the source designed and fabricated for initial use at the HRIBF because of its low emittance, relatively high ionization efficiencies and species versatility, and because it has been engineered for remote installation, removal and servicing as required for safe handling in a high-radiation-level ISOL facility. Prototype plasma-sputter negative ion sources and negative surfaceionization sources are also under design consideration for generating negative radioactive ion beams from high electron-affinity elements. A brief review of the HRIBF will be presented, followed by a detailed description of the design features, operational characteristics, ionization efficiencies, and beam qualities (emittances) of these sources 14. Ion current reduction in pinched electron beam diodes International Nuclear Information System (INIS) Quintenz, J.P.; Poukey, J.W. 1977-01-01 A new version of a particle-in-cell diode code has been written which permits the accurate treatment of higher-current diodes with greater physical dimensions. Using this code, we have studied ways to reduce the ion current in large-aspect-ratio pinched electron beam diodes. In particular, we find that allowing the ions to reflex in such diodes lowers the ion to electron current ratio considerably. In a 3-MV R/d=24 case this ratio was lowered by a factor of 6--8 compared with the corresponding nonreflexing-ion diode, while still producing a superpinched electron beam 15. High energy density in matter produced by heavy ion beams International Nuclear Information System (INIS) 1986-05-01 In this report the activities of the GSI Darmstadt (FRG) during 1985 concerning inertial confinement fusion by heavy ion beams. Short communications and abstracts are presented concerning a Z-pinch experiment, heavy ion pumped lasers and X-ray spectroscopy, the study of ion-ion collisions, a RFQ development and beam transport studies, accelerator theory, targets for SIS/ESR experiments, the rayleigh-Taylor instability, studies on the equation of state for matter under high pressure, as well as the development of computer codes. (HSI) 16. Ion beam microanalysis of human hair follicles International Nuclear Information System (INIS) Kertesz, Zs.; Szikszai, Z.; Telek, A.; Biro, T.; Debrecen Univ. 2006-01-01 Complete text of publication follows. Hair follicle (HF) is an appendage organ of the skin which is of importance to the survival of mammals and still maintains significance for the human race - not just biologically, but also through cosmetic and commercial considerations. However data on the composition of hair follicles are scarce and mostly limited to the hair shaft. In addition, to the best of our knowledge, no data are available concerning the distribution of elements in human hair follicle with various growth and cycling phases. In this study [1] we provided detailed quantitative elemental distribution of organ-cultured hair follicle in anagen and catagen growth phases using ion microscopy in order to reach a better understanding of the function, development, and cyclic activity of the hair follicle. The microprobe analysis was carried out at the scanning ion microprobe facilities at the ATOMKI Debrecen, and at the Jozef Stefan Institute, Ljubljana, Slovenia, using combined STIM and PIXE ion beam analytical techniques. Human anagen hair follicles were isolated from skin obtained from females undergoing face-lift surgery. Cultured anagen HFs were treated by either vehicle or by 10 μM capsaicin for 5 days. Elemental distributions and absolute concentrations were determined along 5 capsaicin treated (catagen), and 4 control (anagen) hair follicles. The investigated length varied between 1.5 and 2 mm. Average elemental concentration values of the whole sample and the different morphological parts were also determined. Concentrations for most of the elements were found to be the same in the corresponding parts of the anagen and the catagen hair follicles. However, significant differences were observed in the Ca concentration between the anagen and catagen HFs. With respect to the distribution of Ca, in anagen (control) HFs, the following concentrations were measured (given in μg/g dry weight): dermal papilla, ∼500; matrix of the bulb, 1000-1500; outer/ inner 17. Efficient Injection of Electron Beams into Magnetic Guide Fields International Nuclear Information System (INIS) Chorny, V.; Cooperstein, G.; Dubyna, V.; Frolov, O.; Harper-Slaboszewicz, V.; Hinshelwood, D.; Schneider, R.; Solovyov, V.; Tsepilov, H.; Vitkovitsky, I.; Ware, K. 1999-01-01 Preliminary experimental and modeling study of injection and transport of high current electron beams in current-neutralized background gas has been performed. Initial analysis of the results indicates that high current triaxial ring diode operates very reproducibly in the pinch mode. High current density beam can be injected efficiently into the drift region, using azimuthal guide field with reduced intensity near the injection region. This was shown to improve the effectiveness of capturing the beam for the transport. The transport length was insufficient to measure losses, such as would arise from scattering with the background gas 18. Auroral ion beams and ion acoustic wave generation by fan instability Energy Technology Data Exchange (ETDEWEB) 1996-04-01 Satellite observations indicate that efficient energy transport among various plasma particles and between plasma waves and plasma particles is taking place in auroral ion beam regions. These observations show that two characteristic wave types are associated with the auroral ion beam regions: electrostatic hydrogen cyclotron waves with frequencies above hydrogen gyrofrequency, and low frequency waves with frequencies below hydrogen gyrofrequency. We speculate that the low frequency waves can be ion acoustic waves generated through the fan instability. The presence of a cold background ion component is necessary for the onset of this instability. A cold ion component has been directly observed and has been indirectly suggested from observations of solitary wave structures. The wave-particle interaction during the development of the fan instability results in an efficient ion beam heating in the direction perpendicular to the ambient magnetic field. The fan instability development and the ion beam heating is demonstrated in a numerical particle simulation. 23 refs, 16 figs. 19. Physics with energetic radioactive ion beams International Nuclear Information System (INIS) Henning, W.F. 1996-01-01 Beams of short-lived, unstable nuclei have opened new dimensions in studies of nuclear structure and reactions. Such beams also provide key information on reactions that take place in our sun and other stars. Status and prospects of the physics with energetic radioactive beams are summarized 20. Graphene defects induced by ion beam Science.gov (United States) Gawlik, Grzegorz; Ciepielewski, Paweł; Baranowski, Jacek; Jagielski, Jacek 2017-10-01 The CVD graphene deposited on the glass substrate was bombarded by molecular carbon ions C3+ C6+ hydrocarbon ions C3H4+ and atomic ions He+, C+, N+, Ar+, Kr+ Yb+. Size and density of ion induced defects were estimated from evolution of relative intensities of Raman lines D (∼1350 1/cm), G (∼1600 1/cm), and D‧ (∼1620 1/cm) with ion fluence. The efficiency of defect generation by atomic ions depend on ion mass and energy similarly as vacancy generation directly by ion predicted by SRIM simulations. However, efficiency of defect generation in graphene by molecular carbon ions is essentially higher than summarized efficiency of similar group of separate atomic carbon ions of the same energy that each carbon ion in a cluster. The evolution of the D/D‧ ratio of Raman lines intensities with ion fluence was observed. This effect may indicate evolution of defect nature from sp3-like at low fluence to a vacancy-like at high fluence. Observed ion graphene interactions suggest that the molecular ion interacts with graphene as single integrated object and should not be considered as a group of atomic ions with partial energy. 1. The beam diagnostic instruments in Beijing radioactive ion-beam facilities isotope separator on-line International Nuclear Information System (INIS) Ma, Y.; Cui, B.; Ma, R.; Tang, B.; Chen, L.; Huang, Q.; Jiang, W. 2014-01-01 The beam diagnostic instruments for Beijing Radioactive Ion-beam Facilities Isotope Separator On-Line are introduced [B. Q. Cui, Z. H. Peng, Y. J. Ma, R. G. Ma, B. Tang, T. Zhang, and W. S. Jiang, Nucl. Instrum. Methods 266, 4113 (2008); T. J. Zhang, X. L. Guan, and B. Q. Cui, in Proceedings of APAC 2004, Gyeongju, Korea, 2004, http://www.jacow.org , p. 267]. For low intensity ion beam [30–300 keV/1 pA–10 μA], the beam profile monitor, the emittance measurement unit, and the analyzing slit will be installed. For the primary proton beam [100 MeV/200 μA], the beam profile scanner will be installed. For identification of the nuclide, a beam identification unit will be installed. The details of prototype of the beam diagnostic units and some experiment results will be described in this article 2. Selection and design of ion sources for use at the Holifield radioactive ion beam facility International Nuclear Information System (INIS) Alton, G.D.; Haynes, D.L.; Mills, G.D.; Olsen, D.K. 1994-01-01 The Holifield Radioactive Ion Beam Facility now under construction at the Oak Ridge National Laboratory will use the 25 MV tandem accelerator for the acceleration of radioactive ion beams to energies appropriate for research in nuclear physics; negative ion beams are, therefore, required for injection into the tandem accelerator. Because charge exchange is an efficient means for converting initially positive ion beams to negative ion beams, both positive and negative ion sources are viable options for use at the facility. The choice of the type of ion source will depend on the overall efficiency for generating the radioactive species of interest. Although direct-extraction negative ion sources are clearly desirable, the ion formation efficiencies are often too low for practical consideration; for this situation, positive ion sources, in combination with charge exchange, are the logical choice. The high-temperature version of the CERN-ISOLDE positive ion source has been selected and a modified version of the source designed and fabricated for initial use at the facility because of its low emittance, relatively high ionization efficiencies, and species versatility, and because it has been engineered for remote installation, removal, and servicing as required for safe handling in a high-radiation-level ISOL facility. The source will be primarily used to generate ion beams from elements with intermediate to low electron affinities. Prototype plasma-sputter negative ion sources and negative surface-ionization sources are under design consideration for generating radioactive ion beams from high-electron-affinity elements. The design features of these sources and expected efficiencies and beam qualities (emittances) will be described in this report 3. Effect of ion beam irradiation on metal particle doped polymer ... and converts polymeric structure into hydrogen depleted carbon network. ... Composite materials; ion beam irradiation; dielectric properties; X-ray diffraction. ..... Coat. Technol. 201 8225. Raja V, Sharma A K and Narasimha V V R 2004 Mater. 4. Modification of bamboo surface by irradiation of ion beams International Nuclear Information System (INIS) Wada, M.; Nishigaito, S.; Flauta, R.; Kasuya, T. 2003-01-01 When beams of hydrogen ions, He + and Ar + were irradiated onto bamboo surface, gas release of hydrogen, water, carbon monoxide and carbon dioxide were enhanced. Time evolution of the gas emission showed two peaks corresponding to release of adsorbed gas from the surface by sputtering, and thermal desorption caused by the beam heating. The difference in etched depths between parenchyma lignin and vascular bundles was measured by bombarding bamboo surface with the ion beams in the direction parallel to the vascular bundles. For He + and Ar + , parenchyma lignin was etched more rapidly than vascular bundles, but the difference in etched depth decreased at a larger dose. In the case of hydrogen ion bombardment, vascular bundles were etched faster than parenchyma lignin and the difference in etched depth increased almost in proportion to the dose. The wettability of outer surface of bamboo was improved most effectively by irradiation of a hydrogen ion beam 5. Modeling and computer simulation of ion beam synthesis of nanostructures Energy Technology Data Exchange (ETDEWEB) Strobel, M. 1999-11-01 The following topics were dealt with: ion beam synthesis of nanoclusters, kinetic three dimensional lattice Monte Carlo method, Ostwald ripening, redistribution of implanted impurities, buried layer formation, comparisation to experimental results. 6. Nanoscale insights into ion-beam cancer therapy CERN Document Server 2017-01-01 This book provides a unique and comprehensive overview of state-of-the-art understanding of the molecular and nano-scale processes that play significant roles in ion-beam cancer therapy. It covers experimental design and methodology, and reviews the theoretical understanding of the processes involved. It offers the reader an opportunity to learn from a coherent approach about the physics, chemistry and biology relevant to ion-beam cancer therapy, a growing field of important medical application worldwide. The book describes phenomena occurring on different time and energy scales relevant to the radiation damage of biological targets and ion-beam cancer therapy from the molecular (nano) scale up to the macroscopic level. It illustrates how ion-beam therapy offers the possibility of excellent dose localization for treatment of malignant tumours, minimizing radiation damage in normal tissue whilst maximizing cell-killing within the tumour, offering a significant development in cancer therapy. The full potential ... 7. Ion-beam modification of properties of metals and alloys International Nuclear Information System (INIS) Khodasevich, V.V.; Uglov, V.V.; Ponaryadov, V.V.; Zhukova, S.I. 2002-01-01 Physical fundaments for ion-beam modification and plasma-vacuum synthesis of new types of coatings and compounds in technically important metals and alloys were development as well as corresponding installation and technologies were created. (authors) 8. Use of energetic ion beams in materials synthesis and processing International Nuclear Information System (INIS) Appleton, B.R. 1992-01-01 A brief review of the use energetic ion beams and related techniques for the synthesis, processing, and characterization of materials is presented. Selected opportunity areas are emphasized with examples, and references are provided for more extensive coverage. (author) 9. Dust particle diffusion in ion beam transport region Energy Technology Data Exchange (ETDEWEB) Miyamoto, N.; Okajima, Y.; Romero, C. F.; Kuwata, Y.; Kasuya, T.; Wada, M., E-mail: [email protected] [Graduate school of Science and Engineering, Doshisha University, Kyotanabe, Kyoto 610-0321 (Japan) 2016-02-15 Dust particles of μm size produced by a monoplasmatron ion source are observed by a laser light scattering. The scattered light signal from an incident laser at 532 nm wavelength indicates when and where a particle passes through the ion beam transport region. As the result, dusts with the size more than 10 μm are found to be distributed in the center of the ion beam, while dusts with the size less than 10 μm size are distributed along the edge of the ion beam. Floating potential and electron temperature at beam transport region are measured by an electrostatic probe. This observation can be explained by a charge up model of the dust in the plasma boundary region. 10. Bootstrap current of fast ions in neutral beam injection heating International Nuclear Information System (INIS) Huang Qianhong; Gong Xueyu; Yang Lei; Li Xinxia; Lu Xingqiang; Yu Jun 2012-01-01 The bootstrap current of fast ions produced by the neutral beam injection is investigated in a large aspect ratio tokamak with circular cross-section under specific parameters. The bootstrap current density distribution and the total bootstrap current are figured out. In addition, the beam bootstrap current always accompanies the electron return current due to the parallel momentum transfer from fast ions. With the electron return current considered, the net current density obviously decreases due to electron return current, at the same time the peak of current moves towards the centre plasma. Numerical results show that the value of the net current depends sensitively not only on the angle of the neutral beam injection but also on the ratio of the velocity of fast ions to the critical velocity: the value of net current is small for the neutral beam parallel injection but increases multipliedly for perpendicular injection, and increases with beam energy increasing. (authors) 11. Current neutralization in ballistic transport of light ion beams International Nuclear Information System (INIS) Hubbard, R.F.; Slinker, S.P.; Lampe, M.; Joyce, G.; Ottinger, P. 1992-01-01 Intense light ion beams are being considered as drivers to ignite fusion targets in the Laboratory Microfusion Facility (LMF). Ballistic transport of these beams from the diode to the target is possible only if the beam current is almost completely neutralized by plasma currents. This paper summarizes related work on relativistic electron beam and heavy ion beam propagation and describes a simple simulation model (DYNAPROP) which has been modified to treat light ion beam propagation. DYNAPROP uses an envelope equation to treat beam dynamics and uses rate equations to describe plasma and conductivity generation. The model has been applied both to the high current, 30 MeV Li +3 beams for LMF as well as low current, 1.2 MeV proton beams which are currently being studied on GAMBLE B at the Naval Research Laboratory. The predicted ratio of net currents to beam current is ∼0.1--0.2 for the GAMBLE experiment and ∼0.01 for LMF. The implications of these results for LMF and the GAMBLE experiments art discussed in some detail. The simple resistive model in DYNAPROP has well-known limitations in the 1 torr regime which arise primarily from the neglect of plasma electron transport. Alternative methods for treating the plasma response are discussed 12. A practical guide to handling laser diode beams CERN Document Server Sun, Haiyin 2015-01-01 This book offers the reader a practical guide to the control and characterization of laser diode beams. Laser diodes are the most widely used lasers, accounting for 50% of the global laser market. Correct handling of laser diode beams is the key to the successful use of laser diodes, and this requires an in-depth understanding of their unique properties. Following a short introduction to the working principles of laser diodes, the book describes the basics of laser diode beams and beam propagation, including Zemax modeling of a Gaussian beam propagating through a lens. The core of the book is concerned with laser diode beam manipulations: collimating and focusing, circularization and astigmatism correction, coupling into a single mode optical fiber, diffractive optics and beam shaping, and manipulation of multi transverse mode beams. The final chapter of the book covers beam characterization methods, describing the measurement of spatial and spectral properties, including wavelength and linewidth meas... 13. Using Target Ablation for Ion Beam Quality Improvement International Nuclear Information System (INIS) Zhao Shuan; Chen Jia-Er; Lin Chen; Ma Wen-Jun; Yan Xue-Qing; Wang Jun-Jie 2016-01-01 During the laser foil interaction, the output ion beam quality including the energy spread and beam divergence can be improved by the target ablation, due to the direct laser acceleration (DLA) electrons generated in the ablation plasma. The acceleration field established at the target rear by these electrons, which is highly directional and triangle-envelope, is helpful for the beam quality. With the help of the target ablation, both the beam divergence and energy spread will be reduced. If the ablation is more sufficient, the impact of DLA-electron-caused field will be strengthened, and the beam quality will be better, confirmed by the particle-in-cell simulation. (paper) 14. Advanced characterization of materials using swift ion beams Energy Technology Data Exchange (ETDEWEB) Tabacniks, Manfredo H. [Universidade de Sao Paulo (USP), SP (Brazil) 2011-07-01 Swift ion beams are powerful non destructive tools for material analysis especially thin films. In spite of their high energy, usually several MeV/u, little energy is deposited by the ion on the sample. Energetic ions also use to stop far away (or outside) the inspected volume, hence producing negligible damage to the sample. Ion beam methods provide quantitative trace element analysis of any atomic element (and some isotopes) in a sample and are able to yield elemental depth profiles with spatial resolution of the order of 10mm. Relying on nuclear properties of the atoms, these methods are insensitive to the chemical environment of the element, consequently not limited by matrix effects. Ion beam methods are multielemental, can handle insulating materials, are quick (an analysis usually takes less than 15 minutes), and need little (if any) sample preparation. Ion beams are also sensitive to surface roughness and sample porosity and can be used to quickly inspect these properties in a sample. The Laboratory for Ion Beam Analysis of the University of Sao Paulo, LAMFI, is a multi-user facility dedicated to provide Ion Beam Methods like PIXE, RBS, FRS and NRA techniques for the analysis of materials and thin films. Operating since 1994, LAMFI is being used mostly by many researchers from within and outside USP, most of them non specialists in ion beam methods, but in need of ion beam analysis to carry out their research. At LAMFI, during the last 9 years, more than 50% of the accelerator time was dedicated to analysis, usually PIXE or RBS. 21% was down time and about 14% of the time was used for the development of ion beam methods which includes the use of RBS for roughness characterization exploring the shading of the beam by structures on the surface and by modeling the RBS spectrum as the product of a normalized RBS spectrum and a height density distribution function of the surface. Single element thick target PIXE analysis is being developed to obtain the thin 15. Advanced characterization of materials using swift ion beams International Nuclear Information System (INIS) Tabacniks, Manfredo H. 2011-01-01 Swift ion beams are powerful non destructive tools for material analysis especially thin films. In spite of their high energy, usually several MeV/u, little energy is deposited by the ion on the sample. Energetic ions also use to stop far away (or outside) the inspected volume, hence producing negligible damage to the sample. Ion beam methods provide quantitative trace element analysis of any atomic element (and some isotopes) in a sample and are able to yield elemental depth profiles with spatial resolution of the order of 10mm. Relying on nuclear properties of the atoms, these methods are insensitive to the chemical environment of the element, consequently not limited by matrix effects. Ion beam methods are multielemental, can handle insulating materials, are quick (an analysis usually takes less than 15 minutes), and need little (if any) sample preparation. Ion beams are also sensitive to surface roughness and sample porosity and can be used to quickly inspect these properties in a sample. The Laboratory for Ion Beam Analysis of the University of Sao Paulo, LAMFI, is a multi-user facility dedicated to provide Ion Beam Methods like PIXE, RBS, FRS and NRA techniques for the analysis of materials and thin films. Operating since 1994, LAMFI is being used mostly by many researchers from within and outside USP, most of them non specialists in ion beam methods, but in need of ion beam analysis to carry out their research. At LAMFI, during the last 9 years, more than 50% of the accelerator time was dedicated to analysis, usually PIXE or RBS. 21% was down time and about 14% of the time was used for the development of ion beam methods which includes the use of RBS for roughness characterization exploring the shading of the beam by structures on the surface and by modeling the RBS spectrum as the product of a normalized RBS spectrum and a height density distribution function of the surface. Single element thick target PIXE analysis is being developed to obtain the thin 16. Ion beams from high-current PF facilities Energy Technology Data Exchange (ETDEWEB) Sadowski, M [Soltan Inst. for Nuclear Studies, Otwock-Swierk (Poland) 1997-12-31 Pulsed beams of fast deuterons and impurity or admixture ions emitted from high-current PF-type facilities operated in different laboratories are dealt with. A short comparative analysis of time-integrated and time-resolved studies is presented. Particular attention is paid to the microstructure of such ion beams, and to the verification of some theoretical models. (author). 5 figs., 19 refs. 17. Mutation induced with ion beam irradiation in rose Energy Technology Data Exchange (ETDEWEB) Yamaguchi, H. E-mail: [email protected]; Nagatomi, S.; Morishita, T.; Degi, K.; Tanaka, A.; Shikazono, N.; Hase, Y 2003-05-01 The effects of mutation induction by ion beam irradiation on axillary buds in rose were investigated. Axillary buds were irradiated with carbon and helium ion beams, and the solid mutants emerged after irradiation by repeated cutting back. In helium ion irradiation, mutations were observed in plants derived from 9 buds among 56 irradiated buds in 'Orange Rosamini' and in plants derived from 10 buds among 61 irradiated buds in 'Red Minimo'. In carbon ion, mutations were observed in plants derived from 12 buds among 88 irradiated buds in 'Orange Rosamini'. Mutations were induced not only in higher doses but also in lower doses, with which physiological effect by irradiation was hardly observed. Irradiation with both ion beams induced mutants in the number of petals, in flower size, in flower shape and in flower color in each cultivar. 18. Focused ion beam machining and deposition for nanofabrication Energy Technology Data Exchange (ETDEWEB) Davies, S T; Khamsehpour, B [Warwick Univ., Coventry (United Kingdom). Dept. of Engineering 1996-05-01 Focused ion beam micromatching (FIBM) and focused ion beam deposition (FIBD) enable spatially selective, maskless, patterning and processing of materials at extremely high levels of resolution. State-of-the-art focused ion beam (FIB) columns based on high brightness liquid metal ion source (LMIS) technology are capable of forming probes with dimensions of order 10 nm with a lower limit on spot size set by the inherent energy spread of the LMIS and the chromatic aberration of ion optical systems. The combination of high lateral and depth resolution make FIBM and FIBD powerful tools for nanotechnology applications. In this paper we present some methods of controlling FIBM and FIBD processes for nanofabrication purposes and discuss their limitations. (author). 19. Beam-front dynamics and ion acceleration in drifting intense relativistic electron beams International Nuclear Information System (INIS) Alexander, K.F.; Hintze, W. 1976-01-01 Collective ion acceleration at the injection of a relativistic electron beam into a low-pressure gas or a plasma is discussed and its strong dependence on the beam-front dynamics is shown. A simple one-dimensional model taking explicitly into account the motion and ionizing action of the ions in the beam-front region is developed for the calculation of the beam drift velocity. The obtained pressure dependence is in good agreement with experimental data. The energy distribution is shown of the ions accelerated in the moving potential well of the space charge region. Scaling laws for the beam-front dynamics and ion acceleration are derived. (J.U.) 20. The synchrotron and its related technology for ion beam therapy International Nuclear Information System (INIS) Hiramoto, Kazuo; Umezawa, Masumi; Saito, Kazuyoshi; Tootake, Satoshi; Nishiuchi, Hideaki; Hara, Shigemistu; Tanaka, Masanobu; Matsuda, Koji; Sakurabata, Hiroaki; Moriyama, Kunio 2007-01-01 Hitachi has developed several new technologies for the synchrotron and its related system to realize reliable and flexible operation of a proton therapy system. Especially important among them are a non-resonant RF acceleration cavity using FINEMET core with multiple power feeding and radio frequency driven beam extraction technique (RF-DE) for a synchrotron. Various treatment operations such as variable acceleration energy or respiration gating became possible and simple due to the above technique. For beam transport, a beam steering method for the beam, using transfer matrix realizes quick and precise correction of the beam orbit. A compact microwave ion source has also been developed for the injector to obtain further higher reliability and availability. Most of these technologies are also effective to enhance the reliability and flexibility of other ion beam therapy systems 1. Examination of fracture surfaces using focused ion beam milling International Nuclear Information System (INIS) Cairney, J.M.; Munroe, P.R.; Schneibel, J.H. 2000-01-01 Composite materials consisting of an iron aluminide matrix with composition approximately Fe-40at%Al, reinforced with a volume fraction of 40--70% ceramic particles (TiC, WC, TiB 2 or ZrB 2 ), are currently being developed. Focused ion beam milling is a relatively new tool to materials science. It uses a high resolution (<5nm), energetic beam of gallium ions to selectively sputter regions of a material, whilst also functioning as a scanning ion microscope. The milling accuracy is of the order of the beam size allowing very precise sectioning to be carried out. The focused ion beam can be used to prepare highly localized cross sections which reveal the internal sub-structure of materials, avoiding detrimental processes such as deformation, or closing of existing cracks by mechanical abrasion. An area is milled from the sample such that, upon tilting, the internal structure can be imaged. The focused ion beam therefore offers a unique opportunity to examine cross-sections of the fracture surfaces in FeAl-based composites. In the present study, the focused ion beam was used to obtain cross-sections of fracture surfaces in two composite materials, in order to examine the extent of interfacial debonding and matrix deformation, thus providing more information about the mode of fracture. These cross-sections were prepared at regions where significant debonding was observed 2. Frequency threshold for ion beam formation in expanding RF plasma Science.gov (United States) Chakraborty Thakur, Saikat; Harvey, Zane; Biloiu, Ioana; Hansen, Alex; Hardin, Robert; Przybysz, William; Scime, Earl 2008-11-01 We observe a threshold frequency for ion beam formation in expanding, low pressure, argon helicon plasma. Mutually consistent measurements of ion beam energy and density relative to the background ion density obtained with a retarding field energy analyzer and laser induced fluorescence indicate that a stable ion beam of 15 eV appears for source frequencies above 11.5 MHz. Reducing the frequency increases the upstream beam amplitude. Downstream of the expansion region, a clear ion beam is seen only for the higher frequencies. At lower frequencies, large electrostatic instabilities appear and an ion beam is not observed. The upstream plasma density increases sharply at the same threshold frequency that leads to the appearance of a stable double layer. The observations are consistent with the theoretical prediction that downstream electrons accelerated into the source by the double layer lead to increased ionization, thus balancing the higher loss rates upstream [1]. 1. M. A. Lieberman, C. Charles and R. W. Boswell, J. Phys. D: Appl. Phys. 39 (2006) 3294-3304 3. Beam dynamics of mixed high intensity highly charged ion Beams in the Q/A selector Energy Technology Data Exchange (ETDEWEB) Zhang, X.H., E-mail: [email protected] [Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China); Yuan, Y.J.; Yin, X.J.; Qian, C.; Sun, L.T. [Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China); Du, H.; Li, Z.S.; Qiao, J.; Wang, K.D. [Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China); University of Chinese Academy of Sciences, Beijing 100049 (China); Zhao, H.W.; Xia, J.W. [Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China) 2017-06-11 Electron cyclotron resonance (ECR) ion sources are widely used in heavy ion accelerators for their advantages in producing high quality intense beams of highly charged ions. However, it exists challenges in the design of the Q/A selection systems for mixed high intensity ion beams to reach sufficient Q/A resolution while controlling the beam emittance growth. Moreover, as the emittance of beam from ECR ion sources is coupled, the matching of phase space to post accelerator, for a wide range of ion beam species with different intensities, should be carefully studied. In this paper, the simulation and experimental results of the Q/A selection system at the LECR4 platform are shown. The formation of hollow cross section heavy ion beam at the end of the Q/A selector is revealed. A reasonable interpretation has been proposed, a modified design of the Q/A selection system has been committed for HIRFL-SSC linac injector. The features of the new design including beam simulations and experiment results are also presented. 4. Structuring of silicon with low energy focused ion beams Energy Technology Data Exchange (ETDEWEB) Nebiker, P.W.; Doebeli, M. [Paul Scherrer Inst. (PSI), Villigen (Switzerland); Muehle, R. [Eidgenoessische Technische Hochschule, Zurich (Switzerland) 1997-09-01 The defect production in silicon induced by focused ion beam irradiation as a function of energy and projectile mass has been investigated and compared to the measured sputter yield. The aim was to find optimal beam parameters for the structuring of semiconductors with a minimum amount of defects produced per removed atom. (author) 2 figs., 2 refs. 5. Therapy tumor with the heavy ions beam International Nuclear Information System (INIS) Dang Bingrong; Wei Zengquan; Li Wenjian 2002-01-01 As physical characteristic of heavy ions Bragg peak, therapy tumor with heavy ions is becoming advanced technology. So, many countries have developed the technology and used to treat tumor, the societal and economic effects are beneficial to people. The authors show the development, present situation and information of research in world of advanced radiotherapy with heavy ions 6. Evaporative cooling of highly charged ions in EBIT [Electron Beam Ion Trap]: An experimental realization International Nuclear Information System (INIS) Schneider, M.B.; Levine, M.A.; Bennett, C.L.; Henderson, J.R.; Knapp, D.A.; Marrs, R.E. 1988-01-01 Both the total number and trapping lifetime of near-neon-like gold ions held in an electron beam ion trap have been greatly increased by a process of 'evaporative cooling'. A continuous flow of low-charge-state ions into the trap cools the high-charge-state ions in the trap. Preliminary experimental results using titanium ions as a coolant are presented. 8 refs., 6 figs., 2 tabs 7. A high charge state heavy ion beam source for heavy ion fusion International Nuclear Information System (INIS) Eylon, S.; Henestroza, E. 1996-01-01 A high current, low emittance, high charge state heavy ion beam source is being developed. This is designed to deliver a heavy ion fusion (HIF) driver accelerator scale beam. Using a high charge state beam in a driver accelerator for HIF may increase the acceleration efficiency, leading to a reduction in the driver accelerator size and cost. The proposed source system, which consists of a gas beam electron stripper followed by a high charge state beam separator, can be added to existing single charge state, low emittance, high brightness ion sources and injectors. We shall report on the source physics design using 3D beam simulations and experimental feasibility study results using a neutral gas stripper and a beam separator at the exit of the LBL 2 MV injector. (orig.) 8. Metal negative ion beam extraction from a radio frequency ion source Energy Technology Data Exchange (ETDEWEB) Kanda, S.; Yamada, N.; Kasuya, T.; Romero, C. F. P.; Wada, M. 2015-04-08 A metal ion source of magnetron magnetic field geometry has been designed and operated with a Cu hollow target. Radio frequency power at 13.56 MHz is directly supplied to the hollow target to maintain plasma discharge and induce self-bias to the target for sputtering. The extraction of positive and negative Cu ion beams have been tested. The ion beam current ratio of Cu{sup +} to Ar{sup +} has reached up to 140% when Ar was used as the discharge support gas. Cu{sup −} ion beam was observed at 50 W RF discharge power and at a higher Ar gas pressure in the ion source. Improvement of poor RF power matching and suppression of electron current is indispensable for a stable Cu{sup −} ion beam production from the source. 9. Biomaterial imaging with MeV-energy heavy ion beams International Nuclear Information System (INIS) Seki, Toshio; Wakamatsu, Yoshinobu; Nakagawa, Shunichiro; Aoki, Takaaki; Ishihara, Akihiko; Matsuo, Jiro 2014-01-01 The spatial distribution of several chemical compounds in biological tissues and cells can be obtained with mass spectrometry imaging (MSI). In conventional secondary ion mass spectrometry (SIMS) with keV-energy ion beams, elastic collisions occur between projectiles and atoms of constituent molecules. The collisions produce fragments, making the acquisition of molecular information difficult. In contrast, ion beams with MeV-energy excite near-surface electrons and enhance the ionization of high-mass molecules; hence, SIMS spectra of fragment-suppressed ionized molecules can be obtained with MeV-SIMS. To compare between MeV and conventional SIMS, we used the two methods based on MeV and Bi 3 -keV ions, respectively, to obtain molecular images of rat cerebellum. Conventional SIMS images of m/z 184 were clearly observed, but with the Bi 3 ion, the distribution of the molecule with m/z 772.5 could be observed with much difficulty. This effect was attributed to the low secondary ion yields and we could not get many signal counts with keV-energy beam. On the other hand, intact molecular ion distributions of lipids were clearly observed with MeV-SIMS, although the mass of all lipid molecules was higher than 500 Da. The peaks of intact molecular ions in MeV-SIMS spectra allowed us to assign the mass. The high secondary ion sensitivity with MeV-energy heavy ions is very useful in biomaterial analysis 10. Acceleration of beam ions during major radius compression in TFTR International Nuclear Information System (INIS) Wong, K.L.; Bitter, M.; Hammett, G.W. 1985-09-01 Tangentially co-injected deuterium beam ions were accelerated from 82 keV up to 150 keV during a major radius compression experiment in TFTR. The ion energy spectra and the variation in fusion yield were in good agreement with Fokker-Planck code simulations. In addition, the plasma rotation velocity was observed to rise during compression 11. Induction linac drivers for commercial heavy-ion beam fusion International Nuclear Information System (INIS) Keefe, D. 1987-11-01 This paper discusses induction linac drivers necessary to accelerate heavy ions at inertial fusion targets. Topics discussed are: driver configurations, the current-amplifying induction linac, high current beam behavior and emittance growth, new considerations for driver design, the heavy ion fusion systems study, and future studies. 13 refs., 6 figs., 1 tab 12. Production of intense negative ion beams in magnetically insulated diodes International Nuclear Information System (INIS) Lindenbaum, H. 1988-01-01 Production of intense negative ion beams in magnetically insulated diodes was studied in order to develop an understanding of this process by measuring the ion-beam parameters as a function of diode and cathode plasma conditions in different magnetically insulated diodes. A coral diode, a racetrack diode, and an annular diode were used. The UCI APEX pulse line, with a nominal output of 1MV, 140kA, was used under matched conditions with a pulse length of 50 nsec. Negative-ion intensity and divergence were measured with Faraday cups and CR-39 track detectors. Cathode plasma was produced by passive dielectric cathodes and later, by an independent plasma gun. Negative-ion currents had an intensity of a few A/cm 2 with a divergence ranging between a few tenths milliradians for an active TiH 2 plasma gun and 300 milliradians for a passive polyethelene cathode. Negative ions were usually emitted from a few hot spots on the cathode surface. These hot spots are believed to cause transverse electrical fields in the diode gap responsible for the beam divergence. Mass spectrometry measurements showed that the ion beam consists of mainly H - ions when using a polyethelene or a TiH 2 cathodes, and mainly of negative carbon ions when using a carbon cathode 13. Effects of ion beam irradiation on semiconductor devices Energy Technology Data Exchange (ETDEWEB) Nashiyama, Isamu; Hirao, Toshio; Itoh, Hisayoshi; Ohshima, Takeshi [Japan Atomic Energy Research Inst., Takasaki, Gunma (Japan). Takasaki Radiation Chemistry Research Establishment 1997-03-01 Energetic heavy-ion irradiation apparatus has been developed for single-event effects (SEE) testing. We have applied three irradiation methods such as a scattered-ion irradiation method, a recoiled-atom irradiation method, and a direct-beam irradiation method to perform SEE testing efficiently. (author) 14. Study of heliumlike neon using an electron beam ion trap International Nuclear Information System (INIS) Wargelin, B.J.; Kahn, S.M.; Beiersdorfer, P. 1992-01-01 The 2-to-1 spectra of several astrophysically abundant He-like ions are being studied using the Electron Beam Ion Trap (EBIT) at Lawrence Livermore National Laboratory. Spectra are recorded for a broad range of plasma parameters, including electron density, energy, and ionization balance. We describe the experimental equipment and procedure and present some typical data 15. Generation of monoenergetic ion beams via ionization dynamics (Conference Presentation) Science.gov (United States) Lin, Chen; Kim, I. Jong; Yu, Jinqing; Choi, Il Woo; Ma, Wenjun; Yan, Xueqing; Nam, Chang Hee 2017-05-01 The research on ion acceleration driven by high intensity laser pulse has attracted significant interests in recent decades due to the developments of laser technology. The intensive study of energetic ion bunches is particularly stimulated by wide applications in nuclear fusion, medical treatment, warm dense matter production and high energy density physics. However, to implement such compact accelerators, challenges are still existing in terms of beam quality and stability, especially in applications that require higher energy and narrow bandwidth spectra ion beams. We report on the acceleration of quasi-mono-energetic ion beams via ionization dynamics in the interaction of an intense laser pulse with a solid target. Using ionization dynamics model in 2D particle-in-cell (PIC) simulations, we found that high charge state contamination ions can only be ionized in the central spot area where the intensity of sheath field surpasses their ionization threshold. These ions automatically form a microstructure target with a width of few micron scale, which is conducive to generate mono-energetic beams. In the experiment of ultraintense (< 10^21 W/cm^2) laser pulses irradiating ultrathin targets each attracted with a contamination layer of nm-thickness, high quality < 100 MeV mono-energetic ion bunches are generated. The peak energy of the self-generated micro-structured target ions with respect to different contamination layer thickness is also examined This is relatively newfound respect, which is confirmed by the consistence between experiment data and the simulation results. 16. Atomic physics measurements in an electron Beam Ion Trap International Nuclear Information System (INIS) Marrs, R.E.; Beiersdorfer, P.; Bennett, C. 1989-01-01 An electron Beam Ion Trap at Lawrence Livermore National Laboratory is being used to produce and trap very-highly-charged ions (q ≤ 70/+/) for x-ray spectroscopy measurements. Recent measurements of transition energies and electron excitation cross sections for x-ray line emission are summarized. 13 refs., 10 figs 17. Initial stages of the ion beam mixing process International Nuclear Information System (INIS) Traverse, A.; Le Boite, M.G.; Nevot, L.; Pardo, B.; Corno, J. 1987-01-01 The grazing x-ray reflectometry technique, performed on irradiated periodic multilayers, was used to study the early stages of the ion beam mixing process. We present our first results, obtained on NiAu samples irradiated with He ions. The experimental fluence dependence of the effective diffusion coefficient is in good agreement with a calculation based on a purely ballistic process 18. ION-BEAM CHANNELING IN A QUASI-CRYSTAL NARCIS (Netherlands) VANVOORTHUYSEN, EHD; SMULDERS, PJM; WERKMAN, RD; DEBOER, JL; VANSMAALEN, S 1992-01-01 We have observed ion-beam channeling in a quasicrystal. For 1-MeV He-4+ ions in icosahedral Al-Cu-Fe the maximum effect found is 36%. The full width at half maximum of the observed dips is 1.3-degrees. The effect persists up to great depths (> 200 nm), thus showing a high degree of ordering in this 19. Two dimensional simulation of ion beam-plasm interaction \| Echi ... African Journals Online (AJOL) Hybrid plasma simulation is a model in which different components of the plasma are treated differently. In this work the ions are treated as particles while the electrons are treated as a neutralizing background fluid through which electric signals may propagate. Deuterium ion beams incident on the tritium plasma interact ... 20. An ion beam analysis software based on ImageJ International Nuclear Information System (INIS) Udalagama, C.; Chen, X.; Bettiol, A.A.; Watt, F. 2013-01-01 The suit of techniques (RBS, STIM, ERDS, PIXE, IL, IF,…) available in ion beam analysis yields a variety of rich information. Typically, after the initial challenge of acquiring data we are then faced with the task of having to extract relevant information or to present the data in a format with the greatest impact. This process sometimes requires developing new software tools. When faced with such situations the usual practice at the Centre for Ion Beam Applications (CIBA) in Singapore has been to use our computational expertise to develop ad hoc software tools as and when we need them. It then became apparent that the whole ion beam community can benefit from such tools; specifically from a common software toolset that can be developed and maintained by everyone with freedom to use and allowance to modify. In addition to the benefits of readymade tools and sharing the onus of development, this also opens up the possibility for collaborators to access and analyse ion beam data without having to depend on an ion beam lab. This has the virtue of making the ion beam techniques more accessible to a broader scientific community. We have identified ImageJ as an appropriate software base to develop such a common toolset. In addition to being in the public domain and been setup for collaborative tool development, ImageJ is accompanied by hundreds of modules (plugins) that allow great breadth in analysis. The present work is the first step towards integrating ion beam analysis into ImageJ. Some of the features of the current version of the ImageJ ‘ion beam’ plugin are: (1) reading list mode or event-by-event files, (2) energy gates/sorts, (3) sort stacks, (4) colour function, (5) real time map updating, (6) real time colour updating and (7) median and average map creation 1. An ion beam analysis software based on ImageJ Energy Technology Data Exchange (ETDEWEB) Udalagama, C., E-mail: [email protected] [Centre for Ion Beam Applications (CIBA), Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117 542 (Singapore); Chen, X.; Bettiol, A.A.; Watt, F. [Centre for Ion Beam Applications (CIBA), Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117 542 (Singapore) 2013-07-01 The suit of techniques (RBS, STIM, ERDS, PIXE, IL, IF,…) available in ion beam analysis yields a variety of rich information. Typically, after the initial challenge of acquiring data we are then faced with the task of having to extract relevant information or to present the data in a format with the greatest impact. This process sometimes requires developing new software tools. When faced with such situations the usual practice at the Centre for Ion Beam Applications (CIBA) in Singapore has been to use our computational expertise to develop ad hoc software tools as and when we need them. It then became apparent that the whole ion beam community can benefit from such tools; specifically from a common software toolset that can be developed and maintained by everyone with freedom to use and allowance to modify. In addition to the benefits of readymade tools and sharing the onus of development, this also opens up the possibility for collaborators to access and analyse ion beam data without having to depend on an ion beam lab. This has the virtue of making the ion beam techniques more accessible to a broader scientific community. We have identified ImageJ as an appropriate software base to develop such a common toolset. In addition to being in the public domain and been setup for collaborative tool development, ImageJ is accompanied by hundreds of modules (plugins) that allow great breadth in analysis. The present work is the first step towards integrating ion beam analysis into ImageJ. Some of the features of the current version of the ImageJ ‘ion beam’ plugin are: (1) reading list mode or event-by-event files, (2) energy gates/sorts, (3) sort stacks, (4) colour function, (5) real time map updating, (6) real time colour updating and (7) median and average map creation. 2. Emittance growth due to space charge compensation and beam intensity instabilities in negative ion beams Directory of Open Access Journals (Sweden) C. A. Valerio-Lizarraga 2018-03-01 Full Text Available The need to extract the maximum beam intensity with low transversal emittance often comes with the drawback of operating the ion source to limits where beam current instabilities arise, such fluctuations can change the beam properties producing a mismatch in the following sections of the machine. The space charge compensation (SCC generated by the beam particles colliding with the residual gas reaches a steady state after a build-up time. This paper shows how once in the steady state, the beam ends with a transversal emittance value bigger than the case without compensation. In addition, we study how the beam intensity variation can disturb the SCC dynamics and its impact on the beam properties. The results presented in this work come from 3-D simulations using tracking codes taking into account the secondary ions to estimate the degree of the emittance growth due to space charge and SCC. 3. Magnetic fusion with high energy self-colliding ion beams International Nuclear Information System (INIS) Rostoker, N.; Wessel, F.; Maglich, B.; Fisher, A. 1992-06-01 Field-reversed configurations of energetic large orbit ions with neutralizing electrons have been proposed as the basis of a fusion reactor. Vlasov equilibria consisting of a ring or an annulus have been investigated. A stability analysis has been carried out for a long thin layer of energetic ions in a low density background plasma. There is a growing body of experimental evidence from tokamaks that energetic ions slow down and diffuse in accordance with classical theory in the presence of large non-thermal fluctuations and anomalous transport of low energy (10 keV) ions. Provided that major instabilities are under control, it seems likely that the design of a reactor featuring energetic self-colliding ion beams can be based on classical theory. In this case a confinement system that is much better than a tokamak is possible. Several methods are described for creating field reversed configurations with intense neutralized ion beams 4. METI/NEDO Projects on Cluster Ion Beam Process Technology International Nuclear Information System (INIS) Yamada, Isao; Matsuo, Jiro; Toyoda, Noriaki 2003-01-01 Since the initial study of gas cluster ion beams (GCIB) was started in the Ion Beam Engineering Experimental Laboratory of Kyoto University, more than 15 years have passed. Some of the results of that study have already been applied for industrial use. Unique characteristics of gas cluster ion bombardment have been found to offer potential for various other industrial applications. The impact of an accelerated cluster ion upon a target surface imparts very high energy densities into the impact area and produces non-linear effects that are not associated with the impacts of atomic ions. Among prospective applications for these effects are included shallow ion implantation, high rate sputtering, surface cleaning and smoothing, and low temperature thin film formation 5. Magnetic fusion with high energy self-colliding ion beams International Nuclear Information System (INIS) Restoker, N.; Wessel, F.; Maglich, B.; Fisher, A. 1993-01-01 Field-reversed configurations of energetic large orbit ions with neutralizing electrons have been proposed as the basis of a fusion reactor. Vlasov equilibria consisting of a ring or an annulus have been investigated. A stability analysis has been carried out for a long thin layer of energetic ions in a low density background plasma. There is a growing body of experimental evidence from tokamaks that energetic ions slow down and diffuse in accordance with classical theory in the presence of large non-thermal fluctuations and anomalous transport of low energy (10 keV) ions. Provided that major instabilities are under control, it seems likely that the design of a reactor featuring energetic self-colliding ion beams can be based on classical theory. In this case a confinement system that is much better than a tokamak is possible. Several methods are described for creating field reversed configurations with intense neutralized ion beams 6. Ion collection efficiency of ionization chambers in electron beams International Nuclear Information System (INIS) Garcia, S.; Cecatti, E.R. 1984-01-01 When ionization chambers are used in pulsed radiation beams the high-density of ions produced per pulse permits ion recombination, demanding the use of a correction factor. An experimental technique using the charge collected at two different voltages permits the calculation of the ion collection efficiency. The ion collection efficiency of some common ionization chambers in pulsed electron beams were studied as a function of electron energy, dose rate and depth. Accelerators with magnetic scanning system, in which the instantaneous dose rate is much greater than the average dose rate, present a smaller collection efficiency than accelerators with scattering foil. The results lead to the introduction of a correction factor for ion recombination that is the reciprocal of the ion collection efficiency. It is also suggested a simple technique to connect an external variable DC power supply in a Baldwin Farmer dosemeter. (Author) [pt 7. Investigation of fullerene ions in crossed-beams experiments International Nuclear Information System (INIS) Hathiramani, D.; Scheier, P.; Braeuning, H.; Trassl, R.; Salzborn, E.; Presnyakov, L.P.; Narits, A.A.; Uskov, D.B. 2003-01-01 Employing the crossed-beams technique, we have studied the interaction of fullerene ions both with electrons and He 2+ -ions. Electron-impact ionization cross sections for C 60 q+ (q=1,2,3) have been measured at electron energies up to 1000 eV. Unusual features in shape and charge state dependence have been found, which are not observed for atomic ions. The evaporative loss of neutral C 2 fragments in collisions with electrons indicates the presence of two different mechanisms. In a first-ever ion-ion crossed-beams experiment involving fullerene ions a cross section of (1.05 ± 0.06) x 10 -15 cm 2 for charge transfer in the collision C 60 + + He 2+ at 117.2 keV center-of-mass energy has been obtained 8. The cooling of confined ions driven by laser beams International Nuclear Information System (INIS) Reyna, L.G. 1993-01-01 We finalize the dynamics of confined ions driven by a quantized radiation field. The ions can absorb photons from an incident laser beam and relax back to the ground state by either induced emissions or spontaneous emissions. Here we assume that the absorption of photons is immediately followed by spontaneous emissions, resulting in single-level ions perturbed by the exchange of momentum with the radiation field. The probability distribution of the ions is calculated using singular expansions in the low noise asymptotic limit. The present calculations reproduce the quantum results in the limit of heavy particles in static traps, and the classical results of ions in radio-frequency confining wells 9. ISOL science at the Holifield Radioactive Ion Beam Facility Energy Technology Data Exchange (ETDEWEB) Beene, James R [ORNL; Bardayan, Daniel W [ORNL; Galindo-Uribarri, Alfredo {nmn} [ORNL; Gross, Carl J [ORNL; Jones, K. L. [University of Tennessee, Knoxville (UTK); Liang, J Felix [ORNL; Nazarewicz, Witold [ORNL; Stracener, Daniel W [ORNL; Tatum, B Alan [ORNL; Varner Jr, Robert L [ORNL 2011-01-01 The Holi eld Radioactive Ion Beam Facility, located in Oak Ridge, Tennessee, is operated as a National User Facility for the U.S. Department of Energy, producing high quality ISOL beams of short-lived, radioactive nuclei for studies of exotic nuclei, astrophysics research, and various societal applications. The primary driver, the Oak Ridge Isochronous Cyclotron, produces rare isotopes by bombarding highly refractory targets with light ions. The radioactive isotopes are ionized, formed into a beam, mass selected, injected into the 25-MV Tandem, accelerated, and used in experiments. This article reviews HRIBF and its science. 10. Technical Aspects of Delivering Simultaneous Dual and Triple Ion Beams to a Target at the Michigan Ion Beam Laboratory Science.gov (United States) Toader, O.; Naab, F.; Uberseder, E.; Kubley, T.; Taller, S.; Was, G. The Michigan Ion Beam Laboratory (MIBL) at the University of Michigan in Ann Arbor, Michigan, USA, plays a significant role in supporting the mission of the U.S. DOE Office of Nuclear Energy. MIBL is a charter laboratory of the NSUF (National Scientific User Facility - US DoE) and hosts users worldwide. The laboratory has evolved from a single accelerator laboratory to a highly versatile facility with three accelerators (3 MV Tandem, a 400 kV Ion Implanter and a 1.7 MV Tandem), seven beam lines and five target chambers that together, provide unique capabilities to capture the extreme environment experienced by materials in reactor systems. This capability now includes simultaneous multiple (dual, triple) ion irradiations, an irradiation accelerated corrosion cell, and soon, in-situ dual beam irradiation in a transmission electron microscope (TEM) for the study of radiation damage coupled with injection of transmutation elements. The two beam lines that will connect to the 300 kV FEI Tecnai G2 F30 microscope are expected to be operational by the end of 2017. Multiple simultaneous ion beam experiments involving light and heavy ions are already in progress. This paper will outline the current equipment and will focus on the new capability of running dual and triple ion beam experiments. 11. Arc-based smoothing of ion beam intensity on targets International Nuclear Information System (INIS) Friedman, Alex 2012-01-01 By manipulating a set of ion beams upstream of a target, it is possible to arrange for a smoother deposition pattern, so as to achieve more uniform illumination of the target. A uniform energy deposition pattern is important for applications including ion-beam-driven high energy density physics and heavy-ion beam-driven inertial fusion energy (“heavy-ion fusion”). Here, we consider an approach to such smoothing that is based on rapidly “wobbling” each of the beams back and forth along a short arc-shaped path, via oscillating fields applied upstream of the final pulse compression. In this technique, uniformity is achieved in the time-averaged sense; this is sufficient provided the beam oscillation timescale is short relative to the hydrodynamic timescale of the target implosion. This work builds on two earlier concepts: elliptical beams applied to a distributed-radiator target [D. A. Callahan and M. Tabak, Phys. Plasmas 7, 2083 (2000)] and beams that are wobbled so as to trace a number of full rotations around a circular or elliptical path [R. C. Arnold et al., Nucl. Instrum. Methods 199, 557 (1982)]. Here, we describe the arc-based smoothing approach and compare it to results obtainable using an elliptical-beam prescription. In particular, we assess the potential of these approaches for minimization of azimuthal asymmetry, for the case of a ring of beams arranged on a cone. It is found that, for small numbers of beams on the ring, the arc-based smoothing approach offers superior uniformity. In contrast with the full-rotation approach, arc-based smoothing remains usable when the geometry precludes wobbling the beams around a full circle, e.g., for the X-target [E. Henestroza, B. G. Logan, and L. J. Perkins, Phys. Plasmas 18, 032702 (2011)] and some classes of distributed-radiator targets. 12. Characterisation Of The Beam Plasma In High Current, Low Energy Ion Beams For Implanters International Nuclear Information System (INIS) Fiala, J.; Armour, D. G.; Berg, J. A. van der; Holmes, A. J. T.; Goldberg, R. D.; Collart, E. H. J. 2006-01-01 The effective transport of high current, positive ion beams at low energies in ion implanters requires the a high level of space charge compensation. The self-induced or forced introduction of electrons is known to result in the creation of a so-called beam plasma through which the beam propagates. Despite the ability of beams at energies above about 3-5 keV to create their own neutralising plasmas and the development of highly effective, plasma based neutralising systems for low energy beams, very little is known about the nature of beam plasmas and how their characteristics and capabilities depend on beam current, beam energy and beamline pressure. These issues have been addressed in a detailed scanning Langmuir probe study of the plasmas created in beams passing through the post-analysis section of a commercial, high current ion implanter. Combined with Faraday cup measurements of the rate of loss of beam current in the same region due to charge exchange and scattering collisions, the probe data have provided a valuable insight into the nature of the slow ion and electron production and loss processes. Two distinct electron energy distribution functions are observed with electron temperatures ≥ 25 V and around 1 eV. The fast electrons observed must be produced in their energetic state. By studying the properties of the beam plasma as a function of the beam and beamline parameters, information on the ways in which the plasma and the beam interact to reduce beam blow-up and retain a stable plasma has been obtained 13. Direct deposition of gold on silicon with focused ion beams Energy Technology Data Exchange (ETDEWEB) Nebiker, P.W.; Doebeli, M. [Paul Scherrer Inst. (PSI), Villigen (Switzerland); Muehle, R. [Eidgenoessische Technische Hochschule, Zurich (Switzerland) 1997-09-01 Irradiation with ions at very low energies (below 500 eV) no longer induces a removal of substrate material, but the ions are directly deposited on the surface. In this way, gold has been deposited on silicon with focused ion beam exposure and the properties of the film have been investigated with atomic force microscopy and Auger electron spectroscopy. (author) 3 figs., 1 ref. 14. Ion Dynamics at Shocks: Ion Reflection and Beam Formation at Quasi-perpendicular Shocks International Nuclear Information System (INIS) Kucharek, Harald; Moebius, Eberhard 2005-01-01 The physics of collisionless shocks is controlled by the ion dynamics. The generation of gyrating ions by reflection as well as the formation of field-aligned ion beams are essential parts of this dynamic. On the one hand reflection is most likely the first interaction of ions with the shock before they undergo the downstream thermalization process. On the other hand field-aligned ion beams, predominately found at the quasi-perpendicular bow shock, propagate into the distant foreshock region and may create wave activity. We revisit ion reflection, the source and basic production mechanism of field-aligned ion beams, by using multi-spacecraft measurements and contrast these observations with existing theories. Finally, we propose an alternative production mechanism 15. Laser cooling and ion beam diagnosis of relativistic ions in a storage ring International Nuclear Information System (INIS) Schroeder, S. 1990-08-01 Particle accelerator and storage ring technology has reached an advanced state, so that different heavy ion storage rings are coming into operation by now, capable of storing even fully stripped ions up to U 92+ . The main purpose of these machines are the accumulation of ions and the ability of improving the beam quality, that is the phase space density of the stored beams. This beam cooling is done successfully by the well established stochastic and electron cooling techniques. A new cooling method, the laser cooling, is taken over from atomic beam and ion trap experiments, where it has yielded extremely low temperatures of atomic samples. As a canditate at storage rings 7 Li + ions are stored in the Heidelberg TSR at 13.3 MeV. The ion beam properties of the metastable fraction like momentum spread, storage time and the influence of residual gas scattering are investigated by colinear laser spectroscopy in the experimental section of the TSR. An optical pumping experiment using two dye laser systems yields information about ion kinematics and velocity mixing processes in the ring. Lifetimes in the order of 100 ms for velocity classes marked in this way show that laser cooling can be applied to the stored 7 Li + beam. In an experimental situation of two strong counterpropagating laser beams, both tuned near resonance, a dramatic reduction of the ion beam momentum spread is observed. With a special geometrical control of laser and ion beam the longitudinal beam temperature is reduced from 260 K to at least 3 K with very high collection efficiency. (orig./HSI) [de 16. Ion-beam nanopatterning: experimental results with chemically-assisted beam Science.gov (United States) Pochon, Sebastien C. R. 2018-03-01 The need for forming gratings (for example used in VR headsets) in materials such as SiO2 has seen a recent surge in the use of Ion beam etching techniques. However, when using an argon-only beam, the selectivity is limited as it is a physical process. Typically, gases such as CHF3, SF6, O2 and Cl2 can be added to argon in order to increase selectivity; depending on where the gas is injected, the process is known as Reactive Ion Beam Etching (RIBE) or Chemically Assisted Ion Beam Etching (CAIBE). The substrate holder can rotate in order to provide an axisymmetric etch rate profile. It can also be tilted over a range of angles to the beam direction. This enables control over the sidewall profile as well as radial uniformity optimisation. Ion beam directionality in conjunction with variable incident beam angle via platen angle setting enables profile control and feature shaping during nanopatterning. These hardware features unique to the Ion Beam etching methods can be used to create angled etch features. The CAIBE technique is also well suited to laser diode facet etch (for optoelectronic devices); these typically use III-V materials like InP. Here, we report on materials such as SiO2 etched without rotation and at a fixed platen angle allowing the formation of gratings and InP etched at a fixed angle with rotation allowing the formation of nanopillars and laser facets. 17. Shunting arc plasma source for pure carbon ion beam Energy Technology Data Exchange (ETDEWEB) Koguchi, H.; Sakakita, H.; Kiyama, S.; Shimada, T.; Sato, Y.; Hirano, Y. [Energy Technology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568 (Japan) 2012-02-15 A plasma source is developed using a coaxial shunting arc plasma gun to extract a pure carbon ion beam. The pure carbon ion beam is a new type of deposition system for diamond and other carbon materials. Our plasma device generates pure carbon plasma from solid-state carbon material without using a hydrocarbon gas such as methane gas, and the plasma does not contain any hydrogen. The ion saturation current of the discharge measured by a double probe is about 0.2 mA/mm{sup 2} at the peak of the pulse. 18. Shunting arc plasma source for pure carbon ion beam. Science.gov (United States) Koguchi, H; Sakakita, H; Kiyama, S; Shimada, T; Sato, Y; Hirano, Y 2012-02-01 A plasma source is developed using a coaxial shunting arc plasma gun to extract a pure carbon ion beam. The pure carbon ion beam is a new type of deposition system for diamond and other carbon materials. Our plasma device generates pure carbon plasma from solid-state carbon material without using a hydrocarbon gas such as methane gas, and the plasma does not contain any hydrogen. The ion saturation current of the discharge measured by a double probe is about 0.2 mA∕mm(2) at the peak of the pulse. 19. Electron-ion recombination rates for merged-beams experiments International Nuclear Information System (INIS) Pajek, M. 1994-01-01 Energy dependence of the electron-ion recombination rates are studied for different recombination processes (radiative recombination, three-body recombination, dissociative recombination) for Maxwellian relative velocity distribution of arbitrary asymmetry. The results are discussed in context of the electron-ion merged beams experiments in cooling ion storage rings. The question of indication of a possible contribution of the three-body recombination to the measured recombination rates versus relative energy is particularly addressed. Its influence on the electron beam temperature derived from the energy dependence of recombination rate is discussed 20. Generation of high brightness ion beam from insulated anode PED International Nuclear Information System (INIS) Matsukawa, Yoshinobu 1988-01-01 Generation and focusing of a high density ion beam with high brightness from a organic center part of anode of a PED was reported previously. Mass, charge and energy distribution of this beam were analyzed. Three kind of anode were tried. Many highly ionized medium mass ions (up to C 4+ , O 6+ ) accelarated to several times of voltage difference between anode and cathode were observed. In the case of all insulator anode the current carried by the medium mass ions is about half of that carried by protons. (author) 1. Accelerator complex for a radioactive ion beam facility at ATLAS International Nuclear Information System (INIS) Nolen, J.A. 1995-01-01 Since the superconducting heavy ion linac ATLAS is an ideal post-accelerator for radioactive beams, plans are being developed for expansion of the facility with the addition of a driver accelerator, a production target/ion source combination, and a low q/m pre-accelerator for radioactive ions. A working group including staff from the ANL Physics Division and current ATLAS users are preparing a radioactive beam facility proposal. The present paper reviews the specifications of the accelerators required for the facility 2. Source of the backstreaming ion beams in the foreshock region International Nuclear Information System (INIS) Tanaka, M.; Goodrich, C.C.; Winske, D.; Papadopoulos, K. 1983-01-01 A new source mechanism is proposed for the 'reflected' ion beams observed in the foreshock region of the earth's bow shock. In our model the beams originate in the magnetosheath downstream of the qausi-perpendicular portion of the shock. The quasi-perpendicular shock transition is characterized by two downstream ion populations including high-energy gyrating ions in addition to the directly transmitted anisotropic ions. We show by particle simulations that this highly anisotropic downstream ion distribution (T/sub perpendicular//T/sub parallel/ >>1) can excite electromagnetic ion cyclotron waves which, in turn, pitch angle scatter the gyrating ions in a few ion gyroperiods. As a result, some ions acquire large parallel velocities and move fast enough along the convecting downstream magnetic field to escape back across the bow shock into the upstream region. The distribution of escaping ions calculated by using the pitch-angle-scattered ions, as a source, becomes a beam with a large temperature anisotropy T/sub perpendicular/ approx.3--5 T/sub parallel/ and a mean velocity along the magnetic field of about twice that of the solar wind velocity. A significant result is the presence of the maximum angle theta/sub n/B = theta/sub c/ above which no ions can escape, where theta/sub n/B is the angle between the shock normal and the interplanetary magnetic field. A wide peak of constant escaping ion flux is formed below theta/sub c/ whose number density is 1--2% of that of the solar wind. These results are in general agreement with the ISEE observations of the 'reflected' ions 3. LET effects of high energy ion beam irradiation on polysilanes Energy Technology Data Exchange (ETDEWEB) Seki, Shu; Kanzaki, Kenichi; Tagawa, Seiichi; Yoshida, Yoichi [Osaka Univ., Ibaraki (Japan). Inst. of Scientific and Industrial Research; Kudoh, Hisaaki; Sugimoto, Masaki; Sasuga, Tsuneo; Seguchi, Tadao; Shibata, Hiromi 1997-03-01 Thin films of poly(di-n-hexylsilane) were irradiated with 2-20 MeV H{sup +} and He{sup +} ion beams. The beams caused heterogeneous reactions of crosslinking and main chain scission in the films. The relative efficiency of the crosslinking was drastically changed in comparison with that of main chain scission. The anomalous change in the molecular weight distribution was analyzed with increasing irradiation fluence, and the ion beam induced reaction radius; track radius was determined for the radiation sources by the function of molecular weight dispersion. Obtained values were 59{+-}15 A and 14{+-}6 A for 2 MeV He{sup +} and 20 MeV H{sup +} ion beams respectively. (author) 4. Holifield Radioactive Ion Beam Facility Development and Status CERN Document Server Tatum, Alan 2005-01-01 The Holifield Radioactive Ion Beam Facility (HRIBF) is a national user facility dedicated to nuclear structure, reactions, and nuclear astrophysics research with radioactive ion beams (RIBs) using the isotope separator on-line (ISOL) technique. An integrated strategic plan for physics, experimental systems, and RIB production facilities have been developed and implementation of the plan is under way. Specific research objectives are defined for studying the nature of nucleonic matter, the origin of elements, solar physics, and synthesis of heavy elements. Experimental systems upgrade plans include new detector arrays and beam lines, and expansion and upgrade of existing devices. A multifaceted facility expansion plan includes a \$4.75M High Power Target Laboratory (HPTL), presently under construction, to provide a facility for testing new target materials, target geometries, ion sources, and beam preparation techniques. Additional planned upgrades include a second RIB production system (IRIS2), an external axi... 5. Generation and transport of laser accelerated ion beams Energy Technology Data Exchange (ETDEWEB) Schmidt, Peter; Boine-Frankenheim, Oliver [Technische Univ. Darmstadt (Germany); GSI Helmholtzzentrum fuer Schwerionenforschung GmbH, Darmstadt (Germany); Kornilov, Vladimir; Spaedtke, Peter [GSI Helmholtzzentrum fuer Schwerionenforschung GmbH, Darmstadt (Germany); Collaboration: LIGHT-Collaboration 2013-07-01 Currently the LIGHT- Project (Laser Ion Generation, Handling and Transport) is performed at the GSI Helmholtzzentrum fuer Schwerionenforschung GmbH Darmstadt. Within this project, intense proton beams are generated by laser acceleration, using the TNSA mechanism. After the laser acceleration the protons are transported through the beam pipe by a pulsed power solenoid. To study the transport a VORPAL 3D simulation is compared with CST simulation. A criterion as a function of beam parameters was worked out, to rate the importance of space charge. Furthermore, an exemplary comparison of the solenoid with a magnetic quadrupole-triplet was carried out. In the further course of the LIGHT-Project, it is planned to generate ion beams with higher kinetic energies, using ultra-thin targets. The acceleration processes that can appear are: RPA (Radiation Pressure Acceleration) and BOA (Break-Out Afterburner). Therefore the transport of an ion distribution will be studied, as it emerges from a RPA acceleration. 6. Bootstrap current of fast ions in neutral beam injection heating International Nuclear Information System (INIS) Huang Qianhong; Gong Xueyu; Li Xinxia; Yu Jun 2012-01-01 The bootstrap current of fast ions produced by neutral beam injection (NBI) is investigated in a large-aspect-ratio tokamak with circular cross-section under specific parameters. The bootstrap current density distribution and the total bootstrap current are reported. In addition, the beam bootstrap current always accompanies the electron return current due to the parallel momentum transfer from fast ions. With the electron return current taken into consideration, the net current density obviously decreases; at the same time, the peak of the current moves towards the central plasma. Numerical results show that the value of the net current depends sensitively not only on the angle of the NBI but also on the ratio of the velocity of fast ions to the critical velocity: the value of the net current is small for neutral beam parallel injection, but increases severalfold for perpendicular injection, and increases with increasing beam energy. (paper) 7. Final project report for NEET pulsed ion beam project Energy Technology Data Exchange (ETDEWEB) Kucheyev, S. O. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States) 2018-01-11 The major goal of this project was to develop and demonstrate a novel experimental approach to access the dynamic regime of radiation damage formation in nuclear materials. In particular, the project exploited a pulsed-ion-beam method in order to gain insight into defect interaction dynamics by measuring effective defect interaction time constants and defect diffusion lengths. This project had the following four major objectives: (i) the demonstration of the pulsed ion beam method for a prototypical nuclear ceramic material, SiC; (ii) the evaluation of the robustness of the pulsed beam method from studies of defect generation rate effects; (iii) the measurement of the temperature dependence of defect dynamics and thermally activated defect-interaction processes by pulsed ion beam techniques; and (iv) the demonstration of alternative characterization techniques to study defect dynamics. As we describe below, all these objectives have been met. 8. Linac4 low energy beam measurements with negative hydrogen ions Energy Technology Data Exchange (ETDEWEB) Scrivens, R., E-mail: [email protected]; Bellodi, G.; Crettiez, O.; Dimov, V.; Gerard, D.; Granemann Souza, E.; Guida, R.; Hansen, J.; Lallement, J.-B.; Lettry, J.; Lombardi, A.; Midttun, Ø.; Pasquino, C.; Raich, U.; Riffaud, B.; Roncarolo, F.; Valerio-Lizarraga, C. A.; Wallner, J.; Yarmohammadi Satri, M.; Zickler, T. [CERN, 1211 Geneva 23 (Switzerland) 2014-02-15 Linac4, a 160 MeV normal-conducting H{sup −} linear accelerator, is the first step in the upgrade of the beam intensity available from the LHC proton injectors at CERN. The Linac4 Low Energy Beam Transport (LEBT) line from the pulsed 2 MHz RF driven ion source, to the 352 MHz RFQ (Radiofrequency Quadrupole) has been built and installed at a test stand, and has been used to transport and match to the RFQ a pulsed 14 mA H{sup −} beam at 45 keV. A temporary slit-and-grid emittance measurement system has been put in place to characterize the beam delivered to the RFQ. In this paper a description of the LEBT and its beam diagnostics is given, and the results of beam emittance measurements and beam transmission measurements through the RFQ are compared with the expectation from simulations. 9. Drag of ballistic electrons by an ion beam Energy Technology Data Exchange (ETDEWEB) Gurevich, V. L.; Muradov, M. I., E-mail: [email protected] [Russian Academy of Sciences, Ioffe Physicotechnical Institute (Russian Federation) 2015-12-15 Drag of electrons of a one-dimensional ballistic nanowire by a nearby one-dimensional beam of ions is considered. We assume that the ion beam is represented by an ensemble of heavy ions of the same velocity V. The ratio of the drag current to the primary current carried by the ion beam is calculated. The drag current turns out to be a nonmonotonic function of velocity V. It has a sharp maximum for V near v{sub nF}/2, where n is the number of the uppermost electron miniband (channel) taking part in conduction and v{sub nF} is the corresponding Fermi velocity. This means that the phenomenon of ion beam drag can be used for investigation of the electron spectra of ballistic nanostructures. We note that whereas observation of the Coulomb drag between two parallel quantum wires may in general be complicated by phenomena such as tunneling and phonon drag, the Coulomb drag of electrons of a one-dimensional ballistic nanowire by an ion beam is free of such spurious effects. 10. Microfabricated Ion Beam Drivers for Magnetized Target Fusion Science.gov (United States) Persaud, Arun; Seidl, Peter; Ji, Qing; Ardanuc, Serhan; Miller, Joseph; Lal, Amit; Schenkel, Thomas 2015-11-01 Efficient, low-cost drivers are important for Magnetized Target Fusion (MTF). Ion beams offer a high degree of control to deliver the required mega joules of driver energy for MTF and they can be matched to several types of magnetized fuel targets, including compact toroids and solid targets. We describe an ion beam driver approach based on the MEQALAC concept (Multiple Electrostatic Quadrupole Array Linear Accelerator) with many beamlets in an array of micro-fabricated channels. The channels consist of a lattice of electrostatic quadrupoles (ESQ) for focusing and of radio-frequency (RF) electrodes for ion acceleration. Simulations with particle-in-cell and beam envelope codes predict >10x higher current densities compared to state-of-the-art ion accelerators. This increase results from dividing the total ion beam current up into many beamlets to control space charge forces. Focusing elements can be biased taking advantage of high breakdown electric fields in sub-mm structures formed using MEMS techniques (Micro-Electro-Mechanical Systems). We will present results on ion beam transport and acceleration in MEMS based beamlets. Acknowledgments: This work is supported by the U.S. DOE under Contract No. DE-AC02-05CH11231. 11. Application of ion beams for polymeric carbon based biomaterials International Nuclear Information System (INIS) Evelyn, A.L. 2001-01-01 Ion beams have been shown to be quite suitable for the modification and analysis of carbon based biomaterials. Glassy polymeric carbon (GPC), made from cured phenolic resins, has a high chemical inertness that makes it useful as a biomaterial in medicine for drug delivery systems and for the manufacture of heart valves and other prosthetic devices. Low and high-energy ion beams have been used, with both partially and fully cured phenolic resins, to enhance biological cell/tissue growth on, and to increase tissue adhesion to GPC surfaces. Samples bombarded with energetic ion beams in the keV to MeV range exhibited increased surface roughness, measured using optical microscopy and atomic force microscopy. Ion beams were also used to perform nuclear reaction analyses of GPC encapsulated drugs for use in internal drug delivery systems. The results from the high energy bombardment were more dramatic and are shown in this paper. The interaction of energetic ions has demonstrated the useful application of ion beams to enhance the properties of carbon-based biomaterials 12. Development of a dc, broad beam, Mevva ion source International Nuclear Information System (INIS) Brown, I.G.; Dickinson, M.R.; Galvin, J.E.; MacGill, R.A. 1991-09-01 We are developing an embodiment of metal vapor vacuum arc (Mevva) ion source which will operate dc and have very large area beam. In preliminary testing, a dc titanium ion beam was formed with a current of approximately 0.6 amperes at an extraction voltage of 9kV (about 18 keV ion energy, by virtue of the ion charge state distribution) and using an 18 cm diameter set of multi-aperture. Separately, we have tested and formed beam from a 50 cm diameter (2000 cm 2 ) set of grids using a pulsed plasma gun. This configuration appears to be very efficient in terms of plasma utilization, and we have formed beams with diameter 33 cm (FWHM) and ion current up to 7 amperes at an extraction voltage of 50 kV (about 100 keV mean ion energy) and up to 20 amperes peak at the current overshoot part of the beam pulse. Here we describe this Part Of our Mevva development program and summarize the results obtained to-date 13. Study of beam optics and beam halo by integrated modeling of negative ion beams from plasma meniscus formation to beam acceleration International Nuclear Information System (INIS) Miyamoto, K.; Okuda, S.; Hatayama, A.; Hanada, M.; Kojima, A. 2013-01-01 To understand the physical mechanism of the beam halo formation in negative ion beams, a two-dimensional particle-in-cell code for simulating the trajectories of negative ions created via surface production has been developed. The simulation code reproduces a beam halo observed in an actual negative ion beam. The negative ions extracted from the periphery of the plasma meniscus (an electro-static lens in a source plasma) are over-focused in the extractor due to large curvature of the meniscus. 14. Study of beam optics and beam halo by integrated modeling of negative ion beams from plasma meniscus formation to beam acceleration Energy Technology Data Exchange (ETDEWEB) Miyamoto, K. [Naruto University of Education, 748 Nakashima, Takashima, Naruto-cho, Naruto-shi, Tokushima 772-8502 (Japan); Okuda, S.; Hatayama, A. [Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522 (Japan); Hanada, M.; Kojima, A. [Japan Atomic Energy Agency, 801-1 Mukouyama, Naka 319-0913 (Japan) 2013-01-14 To understand the physical mechanism of the beam halo formation in negative ion beams, a two-dimensional particle-in-cell code for simulating the trajectories of negative ions created via surface production has been developed. The simulation code reproduces a beam halo observed in an actual negative ion beam. The negative ions extracted from the periphery of the plasma meniscus (an electro-static lens in a source plasma) are over-focused in the extractor due to large curvature of the meniscus. 15. Ion beam and dual ion beam sputter deposition of tantalum oxide films Science.gov (United States) Cevro, Mirza; Carter, George 1994-11-01 Ion beam sputter deposition (IBS) and dual ion beam sputter deposition (DIBS) of tantalum oxide films was investigated at room temperature and compared with similar films prepared by e-gun deposition. Optical properties ie refractive index and extinction coefficient of IBS films were determined in the 250 - 1100 nm range by transmission spectrophotometry and at (lambda) equals 632.8 nm by ellipsometry. They were found to be mainly sensitive to the partial pressure of oxygen used as a reactive gas in the deposition process. The maximum value of the refractive index of IBS deposited tantalum oxide films was n equals 2.15 at (lambda) equals 550 nm and the extinction coefficient of order k equals 2 X 10-4. Films deposited by e-gun deposition had refractive index n equals 2.06 at (lambda) equals 550 nm. Films deposited using DIBS ie deposition assisted by low energy Ar and O2 ions (Ea equals 0 - 300 eV) and low current density (Ji equals 0 - 40 (mu) A/cm2) showed no improvement in the optical properties of the films. Preferential sputtering occurred at Ea(Ar) equals 300 eV and Ji equals 20 (mu) A/cm2 and slightly oxygen deficient films were formed. Different bonding states in the tantalum-oxide films were determined by x-ray spectroscopy while composition of the film and contaminants were determined by Rutherford scattering spectroscopy. Tantalum oxide films formed by IBS contained relatively high Ar content (approximately equals 2.5%) originating from the reflected argon neutrals from the sputtering target while assisted deposition slightly increased the Ar content. Stress in the IBS deposited films was measured by the bending technique. IBS deposited films showed compressive stress with a typical value of s equals 3.2 X 109 dyn/cm2. Films deposited by concurrent ion bombardment showed an increase in the stress as a function of applied current density. The maximum was s approximately equals 5.6 X 109 dyn/cm2 for Ea equals 300 eV and Ji equals 35 (mu) A/cm2. All 16. Ion-beam and dual-ion-beam sputter deposition of tantalum oxide films Science.gov (United States) Cevro, Mirza; Carter, George 1995-02-01 Ion-beam sputter deposition (IBS) and dual-ion-beam sputter deposition (DIBS) of tantalum oxide films was investigated at room temperature and compared with similar films prepared by e-gun deposition. The optical properties, i.e., refractive index and extinction coefficient, of IBS films were determined in the 250- to 1100-nm range by transmission spectrophotometry and at (lambda) equals 632.8 nm by ellipsometry. They were found to be mainly sensitive to the partial pressure of oxygen used as a reactive gas in the deposition process. The maximum value of the refractive index of IBS deposited tantalum oxide films was n equals 2.15 at (lambda) equals 550 nm and the extinction coefficient of order k equals 2 X 10-4. Films deposited by e-gun deposition had refractive index n 2.06 at (lambda) equals 550 nm. Films deposited using DIBS, i.e., deposition assisted by low energy Ar and O2 ions (Ea equals 0 to 300 eV) and low current density (Ji equals 0 to 40 (mu) A/cm2), showed no improvement in the optical properties of the films. Preferential sputtering occurred at Ea(Ar) equals 300 eV and Ji equals 20 (mu) A/cm2 and slightly oxygen deficient films were formed. Different bonding states in the tantalum-oxide films were determined by x-ray spectroscopy, whereas composition of the film and contaminants were determined by Rutherford backscattering spectroscopy (RBS). Tantalum oxide films formed by IBS contained relatively high Ar content (approximately equals 2.5%) originating from the reflected argon neutrals from the sputtering target whereas assisted deposition slightly increased the Ar content. Stress in the IBS-deposited films was measured by the bending technique. IBS-deposited films showed compressive stress with a typical value of s equals 3.2 X 109 dyn/cm2. Films deposited by concurrent ion bombardment showed an increase in the stress as a function of applied current density. The maximum was s approximately equals 5.6 X 109 dyn/cm2 for Ea equals 300 eV and Ji equals 17. New cultivar produced by heavy-ion beam irradiation Energy Technology Data Exchange (ETDEWEB) Kanaya, Takeshi; Miyazaki, Kiyoshi; Suzuki, Kenichi; Iwaki, Kazunari [Suntory Flowers Ltd., Higashiomi, Shiga (Japan); Ichida, Hiroyuki; Hayashi, Yoriko; Saito, Hiroyuki; Ryuto, Hiromichi; Fukunishi, Nobuhisa; Abe, Tomoko [RIKEN, Nishina Center, Wako, Saitama (Japan) 2007-03-15 The RIKEN accelerator research facility (RARF) is the one of the biggest facilities to accelerate heavy ions in all over the world since 1986. We started our trials in plant breeding since 1993. Soon we found that the ion beam is highly effective in the cause of mutagenesis of tobacco embryos during the fertilization without damage to other plant tissue. RIKEN and Suntory Flowers Ltd. have jointly developed some new ornamental varieties of Verbena and Petunia using ion-beam irradiation. We already put 5 new flower cultivars on the market in Japan, USA, Canada and EU since 2002. We report here a new variety of Torenia obtained by ion-beam irradiation. (author) 18. Longitudinal dynamics of laser-cooled fast ion beams DEFF Research Database (Denmark) Weidemüller, M.; Eike, B.; Eisenbarth, U. 1999-01-01 We present recent results of our experiments on laser cooling of fast stored ion beams at the Heidelberg Test Storage Ring. The longitudinal motion of the ions is directly cooled by the light pressure force, whereas efficient transverse cooling is obtained indirectly by longitudinal-transverse co......We present recent results of our experiments on laser cooling of fast stored ion beams at the Heidelberg Test Storage Ring. The longitudinal motion of the ions is directly cooled by the light pressure force, whereas efficient transverse cooling is obtained indirectly by longitudinal....... When applying laser cooling in square-well buckets over long time intervals, hard Coulomb collisions suddenly disappear and the longitudinal temperature drops by about a factor of three. The observed longitudinal behaviour of the beam shows strong resemblance with the transition to an Coulomb... 19. New cultivar produced by heavy-ion beam irradiation International Nuclear Information System (INIS) Kanaya, Takeshi; Miyazaki, Kiyoshi; Suzuki, Kenichi; Iwaki, Kazunari; Ichida, Hiroyuki; Hayashi, Yoriko; Saito, Hiroyuki; Ryuto, Hiromichi; Fukunishi, Nobuhisa; Abe, Tomoko 2007-01-01 The RIKEN accelerator research facility (RARF) is the one of the biggest facilities to accelerate heavy ions in all over the world since 1986. We started our trials in plant breeding since 1993. Soon we found that the ion beam is highly effective in the cause of mutagenesis of tobacco embryos during the fertilization without damage to other plant tissue. RIKEN and Suntory Flowers Ltd. have jointly developed some new ornamental varieties of Verbena and Petunia using ion-beam irradiation. We already put 5 new flower cultivars on the market in Japan, USA, Canada and EU since 2002. We report here a new variety of Torenia obtained by ion-beam irradiation. (author) 20. Advances in ion beam intensity at Sandia National Laboratories International Nuclear Information System (INIS) Mehlhorn, T.A.; Bailey, J.E.; Coats, R.S. 1995-01-01 In 1993 lithium beam intensities ≥1 TW/cm 2 were achieved and lithium-driven target experiments at the ∼1,400 TW/g level were performed on the Particle Beam Fusion Accelerator II (PBFA II) at Sandia National Laboratories. Hohlraum radiation temperatures of up to 60 eV were achieved using this lithium beam. The 1995 Light-Ion ICF Program milestone of achieving a 100 eV radiation temperature in an ion-driven hohlraum will require a lithium beam intensity of 5 ± 1 TW/cm 2 on a 4 mm diameter cylindrical target; this will require both an increase in coupled lithium power and a decrease in total lithium beam divergence. The lithium beam power has been limited to ∼5--6 TW by a so-called ''parasitic load.'' This parasitic current loss in the ion diodes has recently been identified as being carried by ions that are accelerated from plasmas that are formed when high voltage electrons are lost to anodes with many monolayers of hydrocarbon surface contamination. Control of anode and cathode plasmas on the SABRE accelerator using RF-discharge cleaning, anode heating, and cryogenic cooling of the cathode have increased the efficiency of the production of lithium current by a factor of 2--3. A new ion diode incorporating glow discharge cleaning and titanium gettering pumps has been installed in PBFA II and will be tested in December, 1994. Anode heaters should be available in January, 1995. Circuit model calculations indicate that one can more than double the coupled lithium ion power on PBFA II by eliminating the parasitic current. LiF source divergence presently dominates the total beam divergence. Progress in lithium beam focal intensity using diode cleaning techniques coupled with an active lithium source is reported 1. New experimental initiatives using very highly charged ions from an 'electron beam ion trap' International Nuclear Information System (INIS) Schneider, D. 1996-01-01 A short review of the experimental program in highly-charged heavy ion physics conducted at the Lawrence Livermore National Laboratory Electron Beam Ion Trap (EBIT) facility is presented. The heavy-ion research, involving ions up to fully stripped U 92+ , includes precision x-ray spectroscopy and lifetime studies, electron impact ionization and excitation cross section measurements. The investigations of ion-surface interactions following the impact of high-Z highly charged ions on surfaces are aimed to study the neutralization dynamics effecting the ion and the response of the surface as well. (author) 2. HIGH ENERGY DENSITY PHYSICS EXPERIMENTS WITH INTENSE HEAVY ION BEAMS International Nuclear Information System (INIS) Bieniosek, F.M.; Henestroza, E.; Leitner, M.; Logan, B.G.; More, R.M.; Roy, P.K.; Ni, P.; Seidl, P.A.; Waldron, W.L.; Barnard, J.J. 2008-01-01 The US heavy ion fusion science program has developed techniques for heating ion-beam-driven warm dense matter (WDM) targets. The WDM conditions are to be achieved by combined longitudinal and transverse space-charge neutralized drift compression of the ion beam to provide a hot spot on the target with a beam spot size of about 1 mm, and pulse length about 1-2 ns. As a technique for heating volumetric samples of matter to high energy density, intense beams of heavy ions are capable of delivering precise and uniform beam energy deposition dE/dx, in a relatively large sample size, and the ability to heat any solid-phase target material. Initial experiments use a 0.3 MeV K+ beam (below the Bragg peak) from the NDCX-I accelerator. Future plans include target experiments using the NDCX-II accelerator, which is designed to heat targets at the Bragg peak using a 3-6 MeV lithium ion beam. The range of the beams in solid matter targets is about 1 micron, which can be lengthened by using porous targets at reduced density. We have completed the fabrication of a new experimental target chamber facility for WDM experiments, and implemented initial target diagnostics to be used for the first target experiments in NDCX-1. The target chamber has been installed on the NDCX-I beamline. The target diagnostics include a fast multi-channel optical pyrometer, optical streak camera, VISAR, and high-speed gated cameras. Initial WDM experiments will heat targets by compressed NDCX-I beams and will explore measurement of temperature and other target parameters. Experiments are planned in areas such as dense electronegative targets, porous target homogenization and two-phase equation of state 3. Special design issues. Ion beam driver-reaction chamber interfaces International Nuclear Information System (INIS) Moir, R.W.; Peterson, R.R.; Kessler, G. 1995-01-01 Design issues of the interface between ion beam drivers and the reaction chamber for heavy ion beam and light ion beam inertial fusion drivers are discussed. The interface must provide for radiation protection of final focusing magnets, pumping of evaporated material and non-condensable gas that enter the beam ports, thermal insulation, heat removal, a.o.. Beam ports and focal magnets must be protected by neutronically thick shielding between the beam path and the magnet conductor. The required thickness of the shielding determines the minimum spacing between individual beams in a cluster of beams. The cone angle of this cluster can affect target performance. The beamlines are subjected to evaporated material, debris, and rapidly moving droplets. The reaction chambers used here are HYLIFE-II for indirect, HIBALL-II for direct drive. The light ion beam interface is based on the LIBRA and LIBRA-LiTE studies. In the case of HYLIFE-II, liquid jets must be demonstrated with a thickness of 0.5 m and with an edge that comes to within 10 mm of the beam edges to protect the ports. Design of compact focal arrays with enough shielding to give magnets an adequate lifetime must be achieved. As shielding is added the size of the beam array will grow and the target will drop. For HIBALL neutron shielding of the focal magnets provides an adequate lifetime. Replaceable special INPORT units will have to be developed in the region of the beam ports. For light ions transport issues have led to structures being placed close enough to the target that they experience a higher neutron damage rate and must be replaced once or twice a year, which would require remote maintenance. Light ion concepts could greatly benefit from a self-pinched transport scheme, though the details are unclear and the effect on availability is uncertain. Light and heavy ions have similar problems in keeping the gas in the drivers at a low density. Both will require active means to preserve this low density, while 4. Multiple-ion-beam time-of-flight mass spectrometer International Nuclear Information System (INIS) Rohrbacher, Andreas; Continetti, Robert E. 2001-01-01 An innovative approach to increase the throughput of mass spectrometric analyses using a multiple-ion-beam mass spectrometer is described. Two sample spots were applied onto a laser desorption/ionization target and each spot was simultaneously irradiated by a beam of quadrupled Nd:YLF laser radiation (261.75 nm) to produce ions by laser-desorption ionization. Acceleration of the ions in an electric field created parallel ion beams that were focused by two parallel einzel lens systems. After a flight path of 2.34 m, the ions were detected with a microchannel plate-phosphor screen assembly coupled with a charge coupled device camera that showed two resolved ion beams. Time-of-flight mass spectra were also obtained with this detector. Experiments were performed using both metal atom cations (Ti + and Cr + ) produced by laser desorption/ionization and the molecular ions of two different proteins (myoglobin and lysozyme), created by matrix assisted laser desorption/ionization using an excess of nicotinic acid as matrix 5. High-energy acceleration of an intense negative ion beam International Nuclear Information System (INIS) Takeiri, Y.; Ando, A.; Kaneko, O. 1995-02-01 A high-current H - ion beam has been accelerated with the two-stage acceleration. A large negative hydrogen ion source with an external magnetic filter produces more than 10 A of the H - ions from the grid area of 25cm x 50cm with the arc efficiency of 0.1 A/kW by seeding a small amount of cesium. The H - ion current increases according to the 3/2-power of the total beam energy. A 13.6 A of H - ion beam has been accelerated to 125 keV at the operational gas pressure of 3.4 mTorr. The optimum beam acceleration is achieved with nearly the same electric fields in the first and the second acceleration gaps on condition that the ratio of the first acceleration to the extraction electric fields is adjusted for an aspect ratio of the extraction gap. The ratio of the acceleration drain current to the H - ion current is more than 1.7. That is mainly due to the secondary electron generated by the incident H - ions on the extraction grid and the electron suppression grid. The neutralization efficiency was measured and agrees with the theoretical calculation result. (author) 6. Improving depth resolutions in positron beam spectroscopy by concurrent ion-beam sputtering Science.gov (United States) John, Marco; Dalla, Ayham; Ibrahim, Alaa M.; Anwand, Wolfgang; Wagner, Andreas; Böttger, Roman; Krause-Rehberg, Reinhard 2018-05-01 The depth resolution of mono-energetic positron annihilation spectroscopy using a positron beam is shown to improve by concurrently removing the sample surface layer during positron beam spectroscopy. During ion-beam sputtering with argon ions, Doppler-broadening spectroscopy is performed with energies ranging from 3 keV to 5 keV allowing for high-resolution defect studies just below the sputtered surface. With this technique, significantly improved depth resolutions could be obtained even at larger depths when compared to standard positron beam experiments which suffer from extended positron implantation profiles at higher positron energies. Our results show that it is possible to investigate layered structures with a thickness of about 4 microns with significantly improved depth resolution. We demonstrated that a purposely generated ion-beam induced defect profile in a silicon sample could be resolved employing the new technique. A depth resolution of less than 100 nm could be reached. 7. Advanced ion beam calorimetry for the test facility ELISE International Nuclear Information System (INIS) Nocentini, R.; Fantz, U.; Franzen, P.; Fröschle, M.; Heinemann, B.; Riedl, R.; Ruf, B.; Wünderlich, D.; Bonomo, F.; Pimazzoni, A.; Pasqualotto, R. 2015-01-01 The negative ion source test facility ELISE (Extraction from a Large Ion Source Experiment) is in operation since beginning of 2013 at the Max-Planck-Institut für Plasmaphysik (IPP) in Garching bei München. The large radio frequency driven ion source of ELISE is about 1×1 m 2 in size (1/2 the ITER source) and can produce a plasma for up to 1 h. Negative ions can be extracted and accelerated by an ITER-like extraction system made of 3 grids with an area of 0.1 m 2 , for 10 s every 3 minutes. A total accelerating voltage of up to 60 kV is available, i.e. a maximum ion beam power of about 1.2 MW can be produced. ELISE is equipped with several beam diagnostic tools for the evaluation of the beam characteristics. In order to evaluate the beam properties with a high level of detail, a sophisticated diagnostic calorimeter has been installed in the test facility at the end of 2013, starting operation in January 2014. The diagnostic calorimeter is split into 4 copper plates with separate water calorimetry for each of the plates. Each calorimeter plate is made of 15×15 copper blocks, which act as many separate inertial calorimeters and are attached to a copper plate with an embedded cooling circuit. The block geometry and the connection with the cooling plate are optimized to accurately measure the time-averaged power of the 10 s ion beam. The surface of the blocks is covered with a black coating that allows infrared (IR) thermography which provides a 2D profile of the beam power density. In order to calibrate the IR thermography, 48 thermocouples are installed in as many blocks, arranged in two vertical and two horizontal rows. The paper describes the beam calorimetry in ELISE, including the methods used for the IR thermography, the water calorimetry and the analytical methods for beam profile evaluation. It is shown how the maximum beam inhomogeneity amounts to 13% in average. The beam divergence derived by IR thermography ranges between 1° and 4° and 8. Performance test results of ion beam transport for SST-1 neutral beam injector Energy Technology Data Exchange (ETDEWEB) Jana, M R; Mattoo, S K [Institute for Plasma Research Bhat, Gandhinagar-382428, Gujarat (India); Uhlemann, R, E-mail: [email protected] [Forschungszentrum Juelich, Institute fur Energieforschung IEF-4, Plasmaphysik D-52425 Juelich (Germany) 2010-02-01 A neutral beam injector is built at IPR to heat the plasma of SST-1 and its upgrade. It delivers a maximum beam power of 1.7 MW for 55 kV Hydrogen beam or 80 kV Deuterium beam. At lower beam voltage, the delivered power falls to 500 kW at 30 kV Hydrogen beam which is adequate to heat SST-1 plasma ions to {approx} 1 keV. Process of acceleration of ions to the required beam voltage, conversion of ions to neutrals and removal of un-neutralized ions and the beam diagnostic systems occupy a large space. The consequence is that linear extent of the neutral beam injector is at least a few meters. Also, port access provides a very narrow duct. Even a very good injector design and fabrication practices keep beam divergence at a very low but finite value. The result is beam transport becomes an important issue. Since a wide area beam is constructed by hundreds of beam lets, it becomes essential they be focused in such a way that beam transport loss is minimized. Horizontal and vertical focal lengths are two parameters, in addition to beam divergence, which give a description of the beam transport. We have obtained these two parameters for our injector by using beam transport code; making several hundred simulation runs by varying optical parameters of the beam. The selected parameters set has been translated into the engineering features of the extractor grid set of the ion source. Aperture displacement technique is used to secure the horizontal beam focusing at 5.4 m. Combination of both aperture displacement and inclining of two grid halves to {approx} 17 mrad are secured for vertical beam focusing at 7 m from earth grid of the ion source. The gaps between the design, engineered and performance tested values usually arise due to lack of exercising control over fabrication processes or due to inaccuracies in the assumption made in the model calculations of beam optics and beam transport. This has been the case with several injectors, notably with JET injector. To overcome 9. Near spherical illumination of ion-beam and laser targets International Nuclear Information System (INIS) Mark, J.W.K. 1985-01-01 A procedure is developed for reducing energy-deposition asymmetry in spherical targets driven directly by ion or laser beams. This work is part of a strategy for achieving illumination symmetry in such targets, which is proposed as an alternative to those in the literature. This strategy allows an axially symmetric placement of beamlets, which would be convenient for some driven or reactor scenarios. It also allows the use of beam currents or energy fluxes and beam transverse profiles to help reduce deposition asymmetry with fewer beamlets. In the ideal limit of thin deposition layers and controlled beam profiles, at most six beamlets are needed for target symmetry 10. Ion beam induced luminescence from diamond using an MeV ion microprobe Energy Technology Data Exchange (ETDEWEB) Bettiol, A A; Jamieson, D N; Prawer, S; Allen, M G [Melbourne Univ., Parkville, VIC (Australia). School of Physics 1994-12-31 Analysis of the luminescence induced by a MeV ion beam offers the potential to provide useful information about the chemical properties of atoms in crystals to complement the information provided by more traditional Ion Beam Analysis (IBA) such as Rutherford Backscattering Spectrometry (RBS), ion channeling and Particle Induced X-ray Emission (PIXE). Furthermore, the large penetration depth of the MeV ion beam offers several advantages over the relatively shallow penetration of keV electrons typically employed in cathodoluminescence. An Ion Beam Induced Luminescence (IBIL) detection system was developed for the Melbourne microprobe that allows the spatial mapping of the luminescence signal along with the signals from RBS and PIXE. Homoepitaxial diamond growth has been studied and remarkable shifts in the characteristic blue luminescence of diamond towards the green were observed in the overgrowth. This has been tentatively identified as being due to transition metal inclusions in the epitaxial layers. 8 refs., 2 refs. 11. A linear radiofrequency ion trap for accumulation, bunching, and emittance improvement of radioactive ion beams International Nuclear Information System (INIS) Herfurth, F.; Dilling, J.; Kellerbauer, A. 2000-05-01 An ion beam cooler and buncher has been developed for the manipulation of radioactive ion beams. The gas-filled linear radiofrequency ion trap system is installed at the Penning trap mass spectrometer ISOLTRAP at ISOLDE/CERN. Its purpose is to accumulate the 60-keV continuous ISOLDE ion beam with high efficiency and to convert it into low-energy low-emittance ion pulses. The efficiency was found to exceed 10% in agreement with simulations. A more than 10-fold reduction of the ISOLDE beam emittance can be achieved. The system has been used successfully for first on-line experiments. Its principle, setup and performance will be discussed. (orig.) 12. Ion beam induced luminescence from diamond using an MeV ion microprobe Energy Technology Data Exchange (ETDEWEB) Bettiol, A.A.; Jamieson, D. N.; Prawer, S.; Allen, M.G. [Melbourne Univ., Parkville, VIC (Australia). School of Physics 1993-12-31 Analysis of the luminescence induced by a MeV ion beam offers the potential to provide useful information about the chemical properties of atoms in crystals to complement the information provided by more traditional Ion Beam Analysis (IBA) such as Rutherford Backscattering Spectrometry (RBS), ion channeling and Particle Induced X-ray Emission (PIXE). Furthermore, the large penetration depth of the MeV ion beam offers several advantages over the relatively shallow penetration of keV electrons typically employed in cathodoluminescence. An Ion Beam Induced Luminescence (IBIL) detection system was developed for the Melbourne microprobe that allows the spatial mapping of the luminescence signal along with the signals from RBS and PIXE. Homoepitaxial diamond growth has been studied and remarkable shifts in the characteristic blue luminescence of diamond towards the green were observed in the overgrowth. This has been tentatively identified as being due to transition metal inclusions in the epitaxial layers. 8 refs., 2 refs. 13. Electrical shielding box measurement of the negative hydrogen beam from Penning ion gauge ion source. Science.gov (United States) Wang, T; Yang, Z; Dong, P; long, J D; He, X Z; Wang, X; Zhang, K Z; Zhang, L W 2012-06-01 The cold-cathode Penning ion gauge (PIG) type ion source has been used for generation of negative hydrogen (H(-)) ions as the internal ion source of a compact cyclotron. A novel method called electrical shielding box dc beam measurement is described in this paper, and the beam intensity was measured under dc extraction inside an electrical shielding box. The results of the trajectory simulation and dc H(-) beam extraction measurement were presented. The effect of gas flow rate, magnetic field strength, arc current, and extraction voltage were also discussed. In conclusion, the dc H(-) beam current of about 4 mA from the PIG ion source with the puller voltage of 40 kV and arc current of 1.31 A was extrapolated from the measurement at low extraction dc voltages. 14. Guided ultrasonic wave beam skew in silicon wafers Science.gov (United States) Pizzolato, Marco; Masserey, Bernard; Robyr, Jean-Luc; Fromme, Paul 2018-04-01 In the photovoltaic industry, monocrystalline silicon wafers are employed for solar cells with high conversion efficiency. Micro-cracks induced by the cutting process in the thin wafers can lead to brittle wafer fracture. Guided ultrasonic waves would offer an efficient methodology for the in-process non-destructive testing of wafers to assess micro-crack density. The material anisotropy of the monocrystalline silicon leads to variations of the guided wave characteristics, depending on the propagation direction relative to the crystal orientation. Selective guided ultrasonic wave excitation was achieved using a contact piezoelectric transducer with custom-made wedges for the A0 and S0 Lamb wave modes and a transducer holder to achieve controlled contact pressure and orientation. The out-of-plane component of the guided wave propagation was measured using a non-contact laser interferometer. The phase slowness (velocity) of the two fundamental Lamb wave modes was measured experimentally for varying propagation directions relative to the crystal orientation and found to match theoretical predictions. Significant wave beam skew was observed experimentally, especially for the S0 mode, and investigated from 3D finite element simulations. Good agreement was found with the theoretical predictions based on nominal material properties of the silicon wafer. The important contribution of guided wave beam skewing effects for the non-destructive testing of silicon wafers was demonstrated. 15. Consideration of fluctuation in secondary beam intensity of heavy ion beam probe measurements Energy Technology Data Exchange (ETDEWEB) Fujisawa, A.; Iguchi, H.; Lee, S.; Hamada, Y. 1997-01-01 Heavy ion beam probes have capability to detect local electron density fluctuation in the interior of plasmas through the detected beam intensity fluctuation. However, the intensity fluctuation should suffer a certain degree of distortion from electron density and temperature fluctuations on the beam orbits, and as a result the signal can be quite different from the local density fluctuation. This paper will present a condition that the intensity fluctuation can be regarded as being purely local electron density fluctuation, together with discussion about the contamination of the fluctuation along the beam orbits to the beam intensity fluctuation. (author) 16. Evaluation of Negative-Ion-Beam Driver Concepts for Heavy Ion Fusion International Nuclear Information System (INIS) Grisham, Larry R. 2002-01-01 We evaluate the feasibility of producing and using atomically neutral heavy ion beams produced from negative ions as drivers for an inertial confinement fusion reactor. Bromine and iodine appear to be the most attractive elements for the driver beams. Fluorine and chlorine appear to be the most appropriate feedstocks for initial tests of extractable negative ion current densities. With regards to ion sources, photodetachment neutralizers, and vacuum requirements for accelerators and beam transport, this approach appears feasible within existing technology, and the vacuum requirements are essentially identical to those for positive ion drivers except in the target chamber. The principal constraint is that this approach requires harder vacuums in the target chamber than do space-charge-neutralized positive ion drivers. With realistic (but perhaps pessimistic) estimates of the total ionization cross section, limiting the ionization of a neutral beam to less than 5% while traversing a four -meter path would require a chamber pressure of no more than 5 x 10 -5 torr. Alternatively, even at chamber pressures that are too high to allow propagation of atomically neutral beams, the negative ion approach may still have appeal, since it precludes the possibly serious problem of electron contamination of a positive ion beam during acceleration, drift compression, and focusing 17. Beam structure and transverse emittance studies of high-energy ion beams International Nuclear Information System (INIS) Saadatmand, K.; Johnson, K.F.; Schneider, J.D. 1991-01-01 A visual diagnostic technique has been developed to monitor and study ion beam structure shape and size along a transport line. In this technique, a commercially available fluorescent screen is utilized in conjunction with a video camera. This visual representation of the beam structure is digitized and enhanced through use of false-color coding and displayed on a TV monitor for on-line viewing. Digitized information is stored for further off-line processing (e.g., extraction of beam profiles). An optional wire grid placed upstream of the fluor screen adds the capability of transverse emittance (or angular spread) measurement to this technique. This diagnostic allows real-time observation of the beam response to parameter changes (e.g., evolution of the beam structure, shifts in the beam intensity at various spatial locations within the beam perimeter, and shifts in the beam center and position). 3 refs., 5 figs 18. The Heidelberg CSR: Stored Ion Beams in a Cryogenic Environment International Nuclear Information System (INIS) Wolf, A.; Hahn, R. von; Grieser, M.; Orlov, D. A.; Fadil, H.; Welsch, C. P.; Andrianarijaona, V.; Diehl, A.; Schroeter, C. D.; Crespo Lopez-Urrutia, J. R.; Weber, T.; Mallinger, V.; Schwalm, D.; Ullrich, J.; Rappaport, M.; Urbain, X.; Haberstroh, Ch.; Quack, H.; Zajfman, D. 2006-01-01 A cryogenic electrostatic ion storage ring CSR is under development at the Max-Planck Institute for Nuclear Physics in Heidelberg, Germany. Cooling of the ultrahigh vacuum chamber is envisaged to lead to extremely low pressures as demonstrated by cryogenic ion traps. The ring will apply electron cooling with electron beams of a few eV up to 200 eV. Through long storage times of 1000 s as well as through the low wall temperature, internal cooling of infrared-active molecular ions to their rotational ground state will be possible and their collisions with merged collinear beams of electrons and neutral atoms can be detected with high energy resolution. In addition storage of slow highly charged ions is foreseen. Using a fixed in-ring gas target and a reaction microscope, collisions of the stored ions at a speed of the order of the atomic unit can be kinematically reconstructed. The layout and the cryogenic concept are introduced 19. Ions for LHC Beam Physics and Engineering Challenges CERN Document Server Maury, Stephan; Baggiolini, Vito; Beuret, Andre; Blas, Alfred; Borburgh, Jan; Braun, Hans Heinrich; Carli, Christian; Chanel, Michel; Fowler, Tony; Gilardoni, S S; Gourber-Pace, Marine; Hancock, Steven; Hill, Charles E; Hourican, Michael; Jowett, John M; Kahle, Karsten; Kuchler, Detlef; Mahner, Edgar; Manglunki, Django; Martini, Michel; Paoluzzi, Mauro M; Pasternak, Jaroslaw; Pedersen, Flemming; Raich, Uli; Rossi, Carlo; Royer, Jean Pierre; Schindl, Karlheinz; Scrivens, Richard; Sermeus, Luc; Shaposhnikova, Elena; Tranquille, Gerard; Vretenar, Maurizio; Zickler, Thomas 2005-01-01 The first phase of the heavy ion physics program at the LHC aims to provide lead-lead collisions at energies of 5.5 TeV per colliding nucleon pair and ion-ion luminosity of 1027 cm-2s-1. The transformation of CERN’s ion injector complex (Linac3-LEIR-PS-SPS) presents a number of beam physics and engineering challenges, which are described in this paper. In the LHC itself, there are fundamental performance limitations due to various beam loss mechanisms. To study these without risk of damage there will be an initial period of operation with a reduced number of nominal intensity bunches. While reducing the work required to commission the LHC with ions in 2008, this will still enable early physics discoveries. 20. Numerical study of neutron beam divergence in a beam-fusion scenario employing laser driven ions Science.gov (United States) Alejo, A.; Green, A.; Ahmed, H.; Robinson, A. P. L.; Cerchez, M.; Clarke, R.; Doria, D.; Dorkings, S.; Fernandez, J.; McKenna, P.; Mirfayzi, S. R.; Naughton, K.; Neely, D.; Norreys, P.; Peth, C.; Powell, H.; Ruiz, J. A.; Swain, J.; Willi, O.; Borghesi, M.; Kar, S. 2016-09-01 The most established route to create a laser-based neutron source is by employing laser accelerated, low atomic-number ions in fusion reactions. In addition to the high reaction cross-sections at moderate energies of the projectile ions, the anisotropy in neutron emission is another important feature of beam-fusion reactions. Using a simple numerical model based on neutron generation in a pitcher-catcher scenario, anisotropy in neutron emission was studied for the deuterium-deuterium fusion reaction. Simulation results are consistent with the narrow-divergence (∼ 70 ° full width at half maximum) neutron beam recently served in an experiment employing multi-MeV deuteron beams of narrow divergence (up to 30° FWHM, depending on the ion energy) accelerated by a sub-petawatt laser pulse from thin deuterated plastic foils via the Target Normal Sheath Acceleration mechanism. By varying the input ion beam parameters, simulations show that a further improvement in the neutron beam directionality (i.e. reduction in the beam divergence) can be obtained by increasing the projectile ion beam temperature and cut-off energy, as expected from interactions employing higher power lasers at upcoming facilities. 1. The influence of beam divergence on ion-beam induced surface patterns International Nuclear Information System (INIS) Kree, R.; Yasseri, T.; Hartmann, A.K. 2009-01-01 We present a continuum theory and a Monte Carlo model of self-organized surface pattern formation by ion-beam sputtering including effects of beam profiles. Recently, it has turned out that such secondary ion-beam parameters may have a strong influence on the types of emerging patterns. We first discuss several cases, for which beam profiles lead to random parameters in the theory of pattern formation. Subsequently we study the evolution of the averaged height profile in continuum theory and find that the typical Bradley-Harper scenario of dependence of ripple patterns on the angle of incidence can be changed qualitatively. Beam profiles are implemented in Monte Carlo simulations, where we find generic effects on pattern formation. Finally, we demonstrate that realistic beam profiles, taken from experiments, may lead to qualitative changes of surface patterns. 2. Intense Ion Beams for Warm Dense Matter Physics International Nuclear Information System (INIS) Heimbucher, Lynn; Coleman, Joshua Eugene 2008-01-01 The Neutralized Drift Compression Experiment (NDCX) at Lawrence Berkeley National Laboratory is exploring the physical limits of compression and focusing of ion beams for heating material to warm dense matter (WDM) and fusion ignition conditions. The NDCX is a beam transport experiment with several components at a scale comparable to an inertial fusion energy driver. The NDCX is an accelerator which consists of a low-emittance ion source, high-current injector, solenoid matching section, induction bunching module, beam neutralization section, and final focusing system. The principal objectives of the experiment are to control the beam envelope, demonstrate effective neutralization of the beam space-charge, control the velocity tilt on the beam, and understand defocusing effects, field imperfections, and limitations on peak intensity such as emittance and aberrations. Target heating experiments with space-charge dominated ion beams require simultaneous longitudinal bunching and transverse focusing. A four-solenoid lattice is used to tune the beam envelope to the necessary focusing conditions before entering the induction bunching module. The induction bunching module provides a head-to-tail velocity ramp necessary to achieve peak axial compression at the desired focal plane. Downstream of the induction gap a plasma column neutralizes the beam space charge so only emittance limits the focused beam intensity. We present results of beam transport through a solenoid matching section and simultaneous focusing of a singly charged K + ion bunch at an ion energy of 0.3 MeV. The results include a qualitative comparison of experimental and calculated results after the solenoid matching section, which include time resolved current density, transverse distributions, and phase-space of the beam at different diagnostic planes. Electron cloud and gas measurements in the solenoid lattice and in the vicinity of intercepting diagnostics are also presented. Finally, comparisons of 3. Intense Ion Beam for Warm Dense Matter Physics Energy Technology Data Exchange (ETDEWEB) Coleman, Joshua Eugene [Univ. of California, Berkeley, CA (United States) 2008-01-01 The Neutralized Drift Compression Experiment (NDCX) at Lawrence Berkeley National Laboratory is exploring the physical limits of compression and focusing of ion beams for heating material to warm dense matter (WDM) and fusion ignition conditions. The NDCX is a beam transport experiment with several components at a scale comparable to an inertial fusion energy driver. The NDCX is an accelerator which consists of a low-emittance ion source, high-current injector, solenoid matching section, induction bunching module, beam neutralization section, and final focusing system. The principal objectives of the experiment are to control the beam envelope, demonstrate effective neutralization of the beam space-charge, control the velocity tilt on the beam, and understand defocusing effects, field imperfections, and limitations on peak intensity such as emittance and aberrations. Target heating experiments with space-charge dominated ion beams require simultaneous longitudinal bunching and transverse focusing. A four-solenoid lattice is used to tune the beam envelope to the necessary focusing conditions before entering the induction bunching module. The induction bunching module provides a head-to-tail velocity ramp necessary to achieve peak axial compression at the desired focal plane. Downstream of the induction gap a plasma column neutralizes the beam space charge so only emittance limits the focused beam intensity. We present results of beam transport through a solenoid matching section and simultaneous focusing of a singly charged K+ ion bunch at an ion energy of 0.3 MeV. The results include a qualitative comparison of experimental and calculated results after the solenoid matching section, which include time resolved current density, transverse distributions, and phase-space of the beam at different diagnostic planes. Electron cloud and gas measurements in the solenoid lattice and in the vicinity of intercepting diagnostics are also presented. Finally 4. Negative-ion-based neutral beams for fusion International Nuclear Information System (INIS) Cooper, W.S.; Anderson, O.A.; Chan, C.F. 1987-10-01 To maximize the usefulness of an engineering test reactor (e.g., ITER, TIBER), it is highly desirable that it operate under steady-state conditions. The most attractive option for maintaining the circulating current needed in the center of the plasma is the injection of powerful beams of neutral deuterium atoms. The beam simultaneously heats the plasma. At the energies required, in excess of 500 keV, such beams can be made by accelerating D - ions and then removing the electron. Sources are being developed that generate the D - ions in the volume of a specially constructed plasma discharge, without the addition of cesium. These sources must operate with minimum gas flow, to avoid stripping the D - beam, and with minimum electron output. We are designing at LBL highly efficient electrostatic accelerators that combine electric strong-focusing with dc acceleration and offer the possibility of varying the beam energy at constant current while minimizing breakdown. Some form of rf acceleration may also be required. To minimize irradiation of the ion sources and accelerators, the D - beam can be transported through a maze in the neutron shielding. The D - ions can be converted to neutrals in a gas or plasma target, but advances in laser and mirror technology may make possible very efficient photodetachment systems by the time an ETR becomes operational. 9 refs., 4 figs 5. Performance test of electron cyclotron resonance ion sources for the Hyogo Ion Beam Medical Center Science.gov (United States) Sawada, K.; Sawada, J.; Sakata, T.; Uno, K.; Okanishi, K.; Harada, H.; Itano, A.; Higashi, A.; Akagi, T.; Yamada, S.; Noda, K.; Torikoshi, M.; Kitagawa, A. 2000-02-01 Two electron cyclotron resonance (ECR) ion sources were manufactured for the accelerator facility at the Hyogo Ion Beam Medical Center. H2+, He2+, and C4+ were chosen as the accelerating ions because they have the highest charge to mass ratio among ion states which satisfy the required intensity and quality. The sources have the same structure as the 10 GHz ECR source at the Heavy Ion Medical Accelerator in Chiba except for a few improvements in the magnetic structure. Their performance was investigated at the Sumitomo Heavy Industries factory before shipment. The maximum intensity was 1500 μA for H2+, 1320 μA for He2+, and 580 μA for C4+ at the end of the ion source beam transport line. These are several times higher than required. Sufficient performance was also observed in the flatness and long-term stability of the pulsed beams. These test results satisfy the requirements for medical use. 6. Sawtooth activity of the ion cloud in an electron-beam ion trap International Nuclear Information System (INIS) 2003-01-01 The dynamics of an ensemble of highly charged Ar and Ba ions in an electron-beam ion trap (EBIT) was studied by recording time-resolved x-ray spectra emitted from trapped ions. Sawtoothlike signatures manifest in the spectra for a variety of EBIT operating conditions indicating a sudden collapse of the ion inventory in the trap. The collapse occurs on a time scale of approximately 100 ms and the evolution of the sawteeth is very sensitive to parameters such as electron-beam current and axial trap depth. Analysis of the measurements is based on a time-dependent calculation of the trapping process showing that sawtooth activity is caused by the feedback between the low-Z argon and high-Z barium ions. This unexpected behavior demonstrates the importance of nonlinear effects in electron-beam traps containing more than a single ion species 7. On- and off-line monitoring of ion beam treatment Energy Technology Data Exchange (ETDEWEB) Parodi, Katia, E-mail: [email protected] 2016-02-11 Ion beam therapy is an emerging modality for high precision radiation treatment of cancer. In comparison to conventional radiation sources (photons, electrons), ion beams feature major dosimetric advantages due to their finite range with a localized dose deposition maximum, the Bragg peak, which can be selectively adjusted in depth. However, due to several sources of treatment uncertainties, full exploitation of these dosimetric advantages in clinical practice would require the possibility to visualize the stopping position of the ions in vivo, ideally in real-time. To this aim, different imaging methods have been proposed and investigated, either pre-clinically or even clinically, based on the detection of prompt or delayed radiation following nuclear interaction of the beam with the irradiated tissue. However, the chosen or ad-hoc developed instrumentation has often relied on technologies originally conceived for different applications, thus compromising on the achievable performances for the sake of cost-effectiveness. This contribution will review major examples of used instrumentation and related performances, identifying the most promising detector developments for next generation devices especially dedicated to on-line monitoring of ion beam treatment. Moreover, it will propose an original combination of different techniques in a hybrid detection scheme, aiming to make the most of complementary imaging methods and open new perspectives of image guidance for improved precision of ion beam therapy. 8. Simulation of ion beam scattering in a gas stripper Energy Technology Data Exchange (ETDEWEB) Maxeiner, Sascha, E-mail: [email protected]; Suter, Martin; Christl, Marcus; Synal, Hans-Arno 2015-10-15 Ion beam scattering in the gas stripper of an accelerator mass spectrometer (AMS) enlarges the beam phase space and broadens its energy distribution. As the size of the injected beam depends on the acceleration voltage through phase space compression, the stripper becomes a limiting factor of the overall system transmission especially for low energy AMS system in the sub MV region. The spatial beam broadening and collisions with the accelerator tube walls are a possible source for machine background and energy loss fluctuations influence the mass resolution and thus isotope separation. To investigate the physical processes responsible for these effects, a computer simulation approach was chosen. Monte Carlo simulation methods are applied to simulate elastic two body scattering processes in screened Coulomb potentials in a (gas) stripper and formulas are derived to correctly determine random collision parameters and free path lengths for arbitrary (and non-homogeneous) gas densities. A simple parametric form for the underlying scattering cross sections is discussed which features important scaling behaviors. An implementation of the simulation was able to correctly model the data gained with the TANDY AMS system at ETH Zurich. The experiment covered transmission measurements of uranium ions in helium and beam profile measurements after the ion beam passed through the He-stripper. Beam profiles measured up to very high stripper densities could be understood in full system simulations including the relevant ion optics. The presented model therefore simulates the fundamental physics of the interaction between an ion beam and a gas stripper reliably. It provides a powerful and flexible tool for optimizing existing AMS stripper geometries and for designing new, state of the art low energy AMS systems. 9. Production of highly charged ion beams with SECRAL International Nuclear Information System (INIS) Sun, L. T.; Zhao, H. W.; Zhang, X. Z.; Feng, Y. C.; Li, J. Y.; Guo, X. H.; Ma, H. Y.; Zhao, H. Y.; Ma, B. H.; Wang, H.; Li, X. X.; Jin, T.; Xie, D. Z.; Lu, W.; Cao, Y.; Shang, Y. 2010-01-01 Superconducting electron cyclotron resonance ion source with advanced design in Lanzhou (SECRAL) is an all-superconducting-magnet electron cyclotron resonance ion source (ECRIS) for the production of intense highly charged ion beams to meet the requirements of the Heavy Ion Research Facility in Lanzhou (HIRFL). To further enhance the performance of SECRAL, an aluminum chamber has been installed inside a 1.5 mm thick Ta liner used for the reduction of x-ray irradiation at the high voltage insulator. With double-frequency (18+14.5 GHz) heating and at maximum total microwave power of 2.0 kW, SECRAL has successfully produced quite a few very highly charged Xe ion beams, such as 10 e μA of Xe 37+ , 1 e μA of Xe 43+ , and 0.16 e μA of Ne-like Xe 44+ . To further explore the capability of the SECRAL in the production of highly charged heavy metal ion beams, a first test run on bismuth has been carried out recently. The main goal is to produce an intense Bi 31+ beam for HIRFL accelerator and to have a feel how well the SECRAL can do in the production of very highly charged Bi beams. During the test, though at microwave power less than 3 kW, more than 150 e μA of Bi 31+ , 22 e μA of Bi 41+ , and 1.5 e μA of Bi 50+ have been produced. All of these results have again demonstrated the great capability of the SECRAL source. This article will present the detailed results and brief discussions to the production of highly charged ion beams with SECRAL. 10. Intense ion beam neutralization using underdense background plasma Energy Technology Data Exchange (ETDEWEB) Berdanier, William [Department of Physics, The University of Texas at Austin, Austin, Texas 78712 (United States); Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Roy, Prabir K. [Lawrence Berkeley National Laboratory, Berkeley, California 94720 (United States); Kaganovich, Igor [Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States) 2015-01-15 Producing an overdense background plasma for neutralization purposes with a density that is high compared to the beam density is not always experimentally possible. We show that even an underdense background plasma with a small relative density can achieve high neutralization of intense ion beam pulses. Using particle-in-cell simulations, we show that if the total plasma electron charge is not sufficient to neutralize the beam charge, electron emitters are necessary for effective neutralization but are not needed if the plasma volume is so large that the total available charge in the electrons exceeds that of the ion beam. Several regimes of possible underdense/tenuous neutralization plasma densities are investigated with and without electron emitters or dense plasma at periphery regions, including the case of electron emitters without plasma, which does not effectively neutralize the beam. Over 95% neutralization is achieved for even very underdense background plasma with plasma density 1/15th the beam density. We compare results of particle-in-cell simulations with an analytic model of neutralization and find close agreement with the particle-in-cell simulations. Further, we show experimental data from the National Drift Compression experiment-II group that verifies the result that underdense plasma can neutralize intense heavy ion beams effectively. 11. Beam instrumentation for the BNL Heavy Ion Transfer Line International Nuclear Information System (INIS) Witkover, R.L.; Buxton, W.; Castillo, V.; Feigenbaum, I.; Lazos, A.; Li, Z.G.; Smith, G.; Stoehr, R. 1987-01-01 The Heavy Ion Transfer Line (HITL) was constructed to transport beams from the BNL Tandem Van de Graaff (TVDG) to be injected into the AGS. Because the beam line is approximately 2000 feet long and the particle rigidity is so low, 20 beam monitor boxes were placed along the line. The intensity ranges from 1 to 100 nanoAmps for the dc trace beam used for line set-up, to over 100 μA for the pulsed beam to be injected into the AGS. Profiles are measured using multiwire arrays (HARPS) while Faraday cups and beam transformers monitor the intensity. The electronics stations are operated through 3 Instrumentation Controllers networked to Apollo workstations in the TVDG and AGS control rooms. Details of the detectors and electronics designs and performance will be given 12. Imaging instrument for positron emitting heavy ion beam injection International Nuclear Information System (INIS) Llacer, J.; Chatterjee, A.; Jackson, H.C.; Lin, J.C.; Zunzunegui, M.V. 1978-10-01 The design and performance of an instrument for the imaging of coincidence annihilation gamma rays emitted from the end point of the trajectories of radioactive high-energy heavy ions is described. The positron-emitting heavy ions are the result of nuclear fragmentation of accelerated heavy ions used in cancer therapy or diagnostic medicine. The instrument constructed is capable of locating the ion beam trajectory end point within 1 mm for an injected activity of 200 nanoCi in a measurement time of 1 sec in some favorable conditions. Limited imaging in three dimensions is also demonstrated 13. First experiments with the Greifswald electron-beam ion trap Science.gov (United States) Schabinger, B.; Biedermann, C.; Gierke, S.; Marx, G.; Radtke, R.; Schweikhard, L. 2013-09-01 The former Berlin electron-beam ion trap (EBIT) was moved to Greifswald. In addition to x-ray studies the setup will be used for the investigation of interaction processes between highly charged ions and atomic clusters such as charge exchange and fragmentation. The EBIT setup has now been reassembled and highly charged ions have been produced from Xe-Ar gas mixtures to study the ‘sawtooth effect’. In addition, the layout of the extraction beamline, the interaction region and product analysis for interaction studies with highly charged ions are presented. 14. Mixed ion beams near transition energy International Nuclear Information System (INIS) Hancock, S. 1991-01-01 The standard derivations of the energy and phase of the synchronous particle in a proton accelerator assume, as if by definition, that said synchronous particle lies on the central orbit of the machine. This is manifestly unjustified in the particular case of the acceleration near transition of a mixture of ions, when a small difference in charge-to-mass ratio can produce a large radial separation of the different ion species. The development of a simple derivation of the parameters of the synchronous particle that involves no such a priori constraint has yielded some surprises; not, least, a belated explanation for an apparent anomaly encountered in 1987, when a mixture of oxygen and sulphur ions was accelerated in the CERN Proton Synchrotron for the first time. These ideas are supported by measurements performed in 1990 during a second ion run 15. Self-pinched transport of intense ion beams International Nuclear Information System (INIS) Ottinger, P.F.; Neri, J.M.; Stephanakis, S.J. 1999-01-01 Electron beams with substantial net currents have been routinely propagated in the self-pinched mode for the past two decades. However, as the physics of gas breakdown and beam neutralization is different for ion beams, previous predictions indicated insufficient net current for pinching so that ion beam self-pinched transport (SPT) was assumed impossible. Nevertheless, recent numerical simulations using the IPROP code have suggested that ion SPT is possible. These results have prompted initial experiments to investigate SPT of ion beams. A 100-kA, 1.2-MeV, 3-cm-radius proton beam, generated on the Gamble II pulsed-power accelerator at NRL, has been injected into helium in the 30- to 250-mTorr regime to study this phenomenon. Evidence of self-pinched ion beam transport was observed in the 35- to 80-mTorr SPT pressure window predicted by IPROP. Measured signals from a time- and space-resolved scattered proton diagnostic and a time-integrated Li(Cu) nuclear activation diagnostic, both of which measure protons striking a 10-cm diameter target 50 cm into the transport region, are significantly larger in this pressure window than expected for ballistic transport. These results are consistent with significant self-magnetic fields and self-pinching of the ion beam. On the other hand, time-integrated signals from these same two diagnostics are consistent with ballistic transport at pressures above and below the SPT window. Interferometric electron line-density measurements, acquired during beam injection into the helium gas, show insignificant ionization below 35 mTorr, a rapidly rising ionization fraction with pressure in the SPT window, and a plateau in ionization fraction at about 2% for pressures above 80 mTorr. These and other results are consistent with the physical picture for SPT. IPROP simulations, which closely model the Gamble II experimental conditions, produce results that are in qualitative agreement with the experimental results. The advantages of SPT for 16. Radioactive ion beam facilities in Europe International Nuclear Information System (INIS) Blumenfeld, Y. 2008-01-01 The past two decades have seen extraordinarily rapid development of radioactive beam physics throughout the world and in particular in Europe. The important scientific advances have stemmed from a large number of facilities. Previously existing stable beam machines have been adapted to produce rare isotope beams and dedicated facilities have come on-line. This talk gives an overview of the present European installations highlighting their complementary nature. The European roadmap calls for the construction of two next generation facilities: FAIR making use of projectile fragmentation and EURISOL based on the ISOL technique. The future FAIR facility will be described and the path towards EURISOL presented in the light of the construction of 'intermediate' generation facilities SPIRAL2, HIE ISOLDE and SPES and results from the ongoing EURISOL Design Study. 17. X-ray spectroscopy of hydrogen-like ions in an electron beam ion trap Energy Technology Data Exchange (ETDEWEB) Tarbutt, M.R.; Crosby, D.; Silver, J.D. [Univ. of Oxford, Clarendon Lab. (United Kingdom); Myers, E.G. [Dept. of Physics, Florida State Univ., Tallahassee, FL (United States); Nakamura, N.; Ohtani, S. [ICORP, JST, Chofu, Tokyo (Japan) 2001-07-01 The X-ray emission from highly charged hydrogen-like ions in an electron beam ion trap is free from the problems of satellite contamination and Doppler shifts inherent in fast-beam sources. This is a favourable situation for the measurement of ground-state Lamb shifts in these ions. We present recent progress toward this goal, and discuss a method whereby wavelength comparison between transitions in hydrogenlike ions of different nuclear charge Z, enable the measurement of QED effects without requiring an absolute calibration. 18. Investigations on transport and storage of high ion beam intensities International Nuclear Information System (INIS) 2009-01-01 In the framework of this thesis the intense low energy ion beam transport was investigated. Especially, the beam transport in toroidal magnetic field configurations was discussed, as it may allow the accumulation of high intensive beams in the future. One of the specific tasks is to design an injection system that can be used for the proposed low energy accumulator ring. A simulation code (TBT) was written to describe the particle motion in curved segments. Particle in Cell techniques were utilized to simulate a multi particle dynamics. A possibility of reading an external data file was made available so that a measured distribution can be used to compare simulation results with measured ones. A second order cloud in cell method was used to calculate charge density and in turn to solve Poisson's equation. Further simulations were performed to study the self field effects on beam transport. Experiments were performed to compare the simulation results and gain practical experience. The preparatory experiments consisted of building and characterization of the ion source in a first step. Along with the momentum spectrometer and emittance scanner the beam properties were studied. Low mass ion beams He + and mixed p, H 2+ , H 3+ beams were analyzed. In the second stage, beams were transported through a solenoid and the phase space distribution was measured as a function of the magnetic field for different beam energies. The phase-space as distributions measured in a first stage were simulated backward and then again forward transported through the solenoid. The simulated results were then compared with the measured distribution. The LINTRA transport program was used. The phase-space distribution was further simulated for transport experiments in a toroidal magnetic field. The transport program that was used to simulate the beam in the toroid was also used to design the injection system. The injection system with its special field configurations was designed to perform 19. Investigations on transport and storage of high ion beam intensities Energy Technology Data Exchange (ETDEWEB) 2009-08-25 In the framework of this thesis the intense low energy ion beam transport was investigated. Especially, the beam transport in toroidal magnetic field configurations was discussed, as it may allow the accumulation of high intensive beams in the future. One of the specific tasks is to design an injection system that can be used for the proposed low energy accumulator ring. A simulation code (TBT) was written to describe the particle motion in curved segments. Particle in Cell techniques were utilized to simulate a multi particle dynamics. A possibility of reading an external data file was made available so that a measured distribution can be used to compare simulation results with measured ones. A second order cloud in cell method was used to calculate charge density and in turn to solve Poisson's equation. Further simulations were performed to study the self field effects on beam transport. Experiments were performed to compare the simulation results and gain practical experience. The preparatory experiments consisted of building and characterization of the ion source in a first step. Along with the momentum spectrometer and emittance scanner the beam properties were studied. Low mass ion beams He{sup +} and mixed p, H{sup 2+}, H{sup 3+} beams were analyzed. In the second stage, beams were transported through a solenoid and the phase space distribution was measured as a function of the magnetic field for different beam energies. The phase-space as distributions measured in a first stage were simulated backward and then again forward transported through the solenoid. The simulated results were then compared with the measured distribution. The LINTRA transport program was used. The phase-space distribution was further simulated for transport experiments in a toroidal magnetic field. The transport program that was used to simulate the beam in the toroid was also used to design the injection system. The injection system with its special field configurations was 20. Electron Accelerators for Radioactive Ion Beams Energy Technology Data Exchange (ETDEWEB) Lia Merminga 2007-10-10 The summary of this paper is that to optimize the design of an electron drive, one must: (a) specify carefully the user requirements--beam energy, beam power, duty factor, and longitudinal and transverse emittance; (b) evaluate different machine options including capital cost, 10-year operating cost and delivery time. The author is convinced elegant solutions are available with existing technology. There are several design options and technology choices. Decisions will depend on system optimization, in-house infrastructure and expertise (e.g. cryogenics, SRF, lasers), synergy with other programs.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8536374568939209, "perplexity": 3888.241332226325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201329.40/warc/CC-MAIN-20190318132220-20190318153641-00076.warc.gz"}
http://mathoverflow.net/questions/120314?sort=votes	MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4). ## Smoothing of piecewise Euclidean Riemannian metrics Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on $M$, which is singular on (part of) the codimension $2$ skeleton of $T$. Is it possible to approximate $g_0$ by a smooth Riemannian metric? The approximation should in particular change length of curves and the volume by arbitrarily small amounts. I guess the answer is positive and well-known, but I did manage to find a reference (in particular, several works ask the smoothing to satisfy certain curvature assumptions, which I do not). Is there a reference or are there obstruction to smoothing? - You need to assume that $M$ is smooth and $T$ is a smooth triangulation: not every PL manifold has a smooth structure. As for your question, Ontaneda had to address similar issues in sections 7-8 of front.math.ucdavis.edu/1110.6374 – Igor Belegradek Jan 30 at 14:13 @Igor Belegradek: thanks, I did thought about smooth manifold only, but forgot to write it. Now edited. As for the triangulation, any assumption that does not prevent it to exist is fine. – Benoît Kloeckner Jan 30 at 16:02 Thanks also for the reference, I'll definitely have a look at it. – Benoît Kloeckner Jan 30 at 16:10 @Benoît, what I meant is even if $M$ is smooth it may be that your triangulation is non-smoothable or has a smoothing of $M$ that is different from the original one. Thurston in [Three-dimensional geometry and topology, Princeton Mathematical Series, 35] discusses the issue (around page 197, if memory serves) and sketches that any low-dimensional PL manifold is smoothable by a direct geometric argument. This would be a starting point, and then one would have smooth the metric, but without matching the smoothing with the original smooth structure on $M$ there is no way to proceed. – Igor Belegradek Jan 30 at 17:10 You cannot change lengths by arbitrarily small amounts, even in the 2-dimensional case. Consider a cone singularity and a loop near the apex wrapping many times around it. – Sergei Ivanov Jan 31 at 10:01 show 2 more comments	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8826053738594055, "perplexity": 437.797480205889}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709101476/warc/CC-MAIN-20130516125821-00035-ip-10-60-113-184.ec2.internal.warc.gz"}
https://jpmccarthymaths.com/2010/12/13/ma-1008-taylor-series-of-latex-ln-x/	Section 8.8, Q. 4 Find the Taylor series expansion of the function $\ln (1+x)$ about the point $x=1$. (This question was asked at Friday’ tutorial but, with one eye on the answer given, I was unable to do it. Having looked at the problem again I’m sure that the question should have been:) Find the Taylor series expansion of the function $\ln x$ about the point $x=1$. (I have indicated this issue to Prof. Stynes) Solution The Taylor series of any infinitely differentiable function about a point $x=a$ is given by the power series: $f(x)\approx \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ Computing the first few derivatives of $f(x)=\ln x$: $\left.f^{(0)}(x)=\ln x\right\|_{x=1}=0$ $\left.f'(x)=x^{-1}\right\|_{x=1}=1$ $\left.f''(x)=(-1)x^{-2}\right\|_{x=1}=-1$ $\left.f'''(x)=(-1)(-2)x^{-3}\right\|_{x=1}=2$ $\left.f^{(iv)}(x)=(-1)(-2)(-3)x^{-4}\right\|_{x=1}=-6$ $\vdots$ $\left.f^{(n)}(x)=(-1)^{n+1}(n-1)!x^{-n}\right\|_{x=1}=(-1)^{n+1}(n-1)!$ This is valid for $n\geq 1$. At $n=0$, $f^{(0)}(1)=f(1)=0$. Hence we have; $f(x)\approx \sum_{n=1}^\infty \frac{(-1)^{n+1}(n-1)!}{n!}(x-1)^n$ $f(x)\approx \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(x-1)^n$ $\Box$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 20, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9058692455291748, "perplexity": 241.98550856022896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257731.70/warc/CC-MAIN-20190524184553-20190524210553-00269.warc.gz"}
https://www.sarthaks.com/2829282/particle-traveling-through-earths-atmosphere-speed-earth-bound-observer-distance-travels	# A particle is traveling through the Earth’s atmosphere at a speed of 0.866c. To an earth-bound observer, the distance. It travels 2.50 km. How far doe 104 views in Physics closed A particle is traveling through the Earth’s atmosphere at a speed of 0.866c. To an earth-bound observer, the distance. It travels 2.50 km. How far does the particle travel in the particle’s frame of reference? 1. 1.5 km 2. 1.25 km 3. 2.00 km 4. 0.25 km by (24.2k points) selected Correct Answer - Option 2 : 1.25 km CONCEPT: • According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant. • However special theory of relativity leads us to the concept of variation of mass with velocity. • It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest. • Some Interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations. Length Contraction: The distance from the earth to a star measured by an observer in a moving spaceship would seem smaller than the distance measured by an observer on earth. i.e: (i-e S’ < S). $L = \frac{L'}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}⇒ L'=L\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}}$ L' < L since v <c Calculation: Given: L0 = 2.50 km, ν = 0.866c $L = {L_o}\sqrt {l - \frac{{{v^2}}}{{{C^2}}}}$ $L= \left( {2.50 \times {{10}^3}} \right)\sqrt {1 - \frac{{{{\left( {0.866c} \right)}^2}}}{{{c^2}}}}$ $L = \left( {2.50 \times {{10}^3}} \right)\sqrt {1 - \left( {0.75} \right)}$ L = 2.50 × 103 × 0.5 L = 1.25 km Important points: Time Dilation: According to classical physics, time is an absolute quantity. But according to the special theory of relativity, Time is not an absolute quantity. It depends upon the motion of the frame of reference. If the interval of time (say ticking of a clock) between two signals in an inertial frame S be t, then the time interval between these very two signals in another inertial frame S’ moving with respect to the first will be given by $t' = \frac{t}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$ This means that t’ has increased or dilated. In other words, the clock will go slow. Variation of mass: The mass is also not invariant. If a body at rest has a mass m0 it's mass when it moves with a velocity v, increases to m given by: $m = \frac{{{m_o}}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.896600604057312, "perplexity": 544.7483710900385}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00729.warc.gz"}
http://en.wikipedia.org/wiki/Rudin-Shapiro_sequence	# Rudin–Shapiro sequence (Redirected from Rudin-Shapiro sequence) In mathematics the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence is an infinite automatic sequence named after Marcel Golay, Walter Rudin and Harold S. Shapiro, who independently investigated its properties.[1] ## Definition Each term of the Rudin–Shapiro sequence is either +1 or −1. The nth term of the sequence, bn, is defined by the rules: $a_n=\textstyle\sum \varepsilon_i \varepsilon_{i+1}$ $b_n=(-1)^{a_n}$ where the εi are the digits in the binary expansion of n. Thus an counts the number of (possibly overlapping) occurrences of the sub-string 11 in the binary expansion of n, and bn is +1 if an is even and −1 if an is odd.[2][3][4] For example, a6 = 1 and b6 = −1 because the binary representation of 6 is 110, which contains one occurrence of 11; whereas a7 = 2 and b7 = +1 because the binary representation of 7 is 111, which contains two (overlapping) occurrences of 11. Starting at n = 0, the first few terms of the an sequence are: 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... (sequence A014081 in OEIS) and the corresponding terms bn of the Rudin–Shapiro sequence are: +1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ... (sequence A020985 in OEIS) ## Properties The Rudin–Shapiro sequence can be generated by a four state automaton.[5] There is a recursive definition[3] $\begin{cases} b_{2n} & = b_n \\ b_{2n+1} & = (-1)^n b_n \end{cases}$ The values of the terms an and bn in the Rudin–Shapiro sequence can be found recursively as follows. If n = m.2k where m is odd then $a_n = \begin{cases} a_{(m-1)/4} & \text{if } m = 1 \mod 4 \\ a_{(m-1)/2} + 1 & \text{if } m = 3 \mod 4 \end{cases}$ $b_n = \begin{cases} b_{(m-1)/4} & \text{if } m = 1 \mod 4 \\ -b_{(m-1)/2} & \text{if } m = 3 \mod 4 \end{cases}$ Thus a108 = a13 + 1 = a3 + 1 = a1 + 2 = a0 + 2 = 2, which can be verified by observing that the binary representation of 108, which is 1101100, contains two sub-strings 11. And so b108 = (−1)2 = +1. The Rudin-Shapiro word +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 −1 −1 +1 −1 ..., which is created by concatenating the terms of the Rudin–Shapiro sequence, is a fixed point of the morphism or string substitution rules +1 +1 +1 +1 +1 −1 +1 −1 +1 +1 −1 +1 −1 +1 −1 −1 +1 −1 −1 −1 −1 −1 −1 +1 as follows: +1 +1 +1 +1 +1 −1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 −1 −1 +1 −1 ... It can be seen from the morphism rules that the Rudin–Shapiro string contains at most four consecutive +1s and at most four consecutive −1s. The sequence of partial sums of the Rudin–Shapiro sequence, defined by $s_n = \sum_{k=0}^n b_k \, ,$ with values 1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 4, ... (sequence A020986 in OEIS) can be shown to satisfy the inequality $\sqrt{\frac{3n}{5}} < s_n < \sqrt{6n} \text{ for } n \ge 1 \, .$[1]	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8558204770088196, "perplexity": 894.7862135272417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776431152.42/warc/CC-MAIN-20140707234031-00006-ip-10-180-212-248.ec2.internal.warc.gz"}
https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/1805-variation-of-parameters-method-of-solving-non-homogeneous-differential-equations.html?tmpl=component&print=1&page=	## Variation of Parameters Method of Solving Non Homogeneous Differential Equations We start from the nonhomogeneous differential equationThe associated homogeneous equationhas fundamental – linearly independent – solutionsandand then the general solution of the associated homogeneous equation iswhereandare constants. The general solution of the original nonhomogeneous equation iswhereis a particular solution of the original nonhomogeneous equation. The method of variation of parameters looks for a particular solutionof the formwhich means finding the functionsandBy substitutinginto the original nonhomogeneous equation we obtain the simultaneous equations Solving these equations simultaneously gives and whereis the determinant of the matrix – this determinant is called the Wronskian. Then and Summary Find two fundamental solutions of the homogeneous equation Write down the form of the particular solution Findand	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9724714756011963, "perplexity": 648.0785345951284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823630.63/warc/CC-MAIN-20171020025810-20171020045810-00573.warc.gz"}
https://www.susannebradley.com/publication/ssbfp15/	# Biomechanical Simulation and Control of Hands and Tendinous Systems ### Abstract The tendons of the hand and other biomechanical systems form a complex network of sheaths, pulleys, and branches. By modeling these anatomical structures, we obtain realistic simulations of coordination and dynamics that were previously not possible. First, we introduce Eulerian-on-Lagrangian discretization of tendon strands, with a new selective quasistatic formulation that eliminates unnecessary degrees of freedom in the longitudinal direction, while maintaining the dynamic behavior in transverse directions. This formulation also allows us to take larger time steps. Second, we introduce two control methods for biomechanical systems: first, a general-purpose learning-based approach requiring no previous system knowledge, and a second approach using data extracted from the simulator. We use various examples to compare the performance of these controllers. Type Publication ACM Transactions on Graphics Date	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8309023976325989, "perplexity": 1549.8400657089512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00095.warc.gz"}
https://crypto.stackexchange.com/questions/71258/what-can-be-said-about-the-self-power-map-on-groups-based-on-dlp	Introduction I've been playing with group representation theory some time, concretely representing groups as permutation groups (Cayley's theorem), where the group $$G$$ has an embedding into the symmetric group. The set of symbols is composed by the elements of $$G$$. In my case, $$X=U$$, where $$U$$ are the units of $$Z_p^$$, so $$\vert G \vert = \vert U \vert = p-1$$. First, if we analyze the group action induced by multiplication of elements $$a,b\in G$$ we observe that every $$a$$ has a $$b$$ that reaches $$c$$, so it's transitive. Second, there does not exist an $$a$$ that fixes $$b$$. It's only acceptable when $$a$$ is the identity element: $$ab= b \iff a= e_G$$. Satisfies the Free or Fixed Point action, thus it's also faithful. Self-power map explained Then define $$\phi : G \to Sym(G)$$ as the map that sends elements from $$G$$ (units) to elements (permutations) in the permutation group $$P$$ which is isomorphic to $$G$$. If we finally select $$G=Z_p^$$, let $$g$$ be a generator in $$G$$ that we want to represent as permutation $$\sigma_g \in P$$. Then we must construct the permutation by multiplying $$g$$ to all the units in $$G$$. $$\forall 1\leq x \leq p-1 \quad \sigma_g = \phi_g(x) = g\cdot x \pmod p$$ As $$g$$ is a generator $$\sigma_g$$ will have an unique cycle comprised of $$p-1$$ elements. To represent $$\sigma_g$$ in cycle notation, the following expression can be useful: $$\sigma_g = (1,\phi_g(1),\phi_g(\phi_g(1)),\cdots,\phi_g(1)^{p-1}) = (g^0,g^1,g^2, \cdots, g^{p-1})$$ Clearly, this cycle defines the exponential permutation, where in position $$i$$ you can find $$g^i \mod p$$. The self power map $$\sigma_g^$$ is given finding the cycle of the representation of $$\sigma_g$$ as a one-line permutation where $$g^0=g^{p-1}=1$$ is the last element in the one line permutation. $$\sigma_g^ = g^1,g^2,\cdots, g^{p-1} = g,g^2,\cdots,1 = (1,g,g^g,g^{g^{\cdots{g}}},\cdots)$$ We can construct $$\sigma_g^$$ directly if we define $$\alpha_g : G \to Sym(G)$$ where $$\alpha_g(x) = g^x \pmod p$$, so the self-power map of $$g$$ that's $$\sigma_g^$$ is defined as follows: $$\sigma_g^* = (1, \alpha_g(1), \alpha_g(\alpha_g(1)), \cdots) = (1,g^1, g^g \pmod p, g^{g^g \pmod p}\pmod p, \cdots)$$ The cycle notation of the self-power map $$\sigma_g^$$ of a generator $$ = G$$ normally has more than one cycle, being fixed points and transpositions interesting topics that I've found, specially when dealing with safe primes. It is important to remark that every exponent on i.e $$g^{g^{g}}$$ is reduced modulo $$p$$ as shown in the cycle construction of $$\sigma_g^$$. (NOTE: There's a strong symmetry between units and exponents when you view the inverse of $$\sigma_g^$$ as a 2-line matrix representation). Conclusion We have found the self-power map of $$g$$, however, the last definition is very vague, as $$\sigma_g^$$ can have more than one cycle.. In my study, I made multiple C++ programs to measure which cycle types are encountered. Also my record was solving Dlog with a 60 bit random prime. To implement such a solver, I used the fact that every generator is generated by other generator if the exponent is coprime to the group's order, so if we can't find $$x$$ in the self-power map of $$g$$ then we can try to find $$x$$ in the self-power map of other generator $$h$$. There is also another method, relying on the multiplicative inverse of the exponent $$x$$. Let $$g^x\equiv h\pmod p$$, then if we cannot find $$g$$ in the self power map of $$h$$, if $$x$$ is originally coprime, $$h$$ is a generator, so select a new generator $$s$$, which is $$h^y \equiv g^{xy} \pmod p$$. Try to cycle until recover $$g$$ from the sequence, this yields an $$w$$ s.t $$yxw \equiv 1 \pmod{p-1}$$ so $$s^w \equiv g^{yxw} \equiv g \pmod p$$. These methods are easily translated to multi-threading environments, select a limit so the program stop the cycling and selects a new generator. But since the self power map is a cyclic structure, that tells me that the average cycle length of the cycle where $$x$$ or $$w$$ lies is big enough when selecting the self power map of a new generator. And the average cycle length of a cycle escalates rapidly when the group's order is bigger. Question: Has the self power map been studied for estimating the average complexity for solving a DLOG instance? • Your paragraph "Clearly, this cycle defines the exponential permutation, where in position i you can find gimodp. But this is a cycle yet. Represent is a one-line permutation and fint it's cycle:" and the equations following it are unclear and have missing symbols Jun 25, 2019 at 21:22 • Because your post is unclear I can hazard just a guess, but you may be looking for a generic algorithm. There are some known results about them, including a lower bound on their complexity. My understanding of going from $G$ to Sym(G) as you are doing is that this usually results in trying to find a function from a cyclif group into the group of units. But I'm not entirely confident of that. Jul 20, 2019 at 3:02 • @kodlu: I've corrected the fact that $1$ should be the last entry on the one-line permutation of $\sigma_g$, which was the first element, and that was my mistake. Now it should be more clear, but still, note that the self-power map can have more than one cycle (moreover, I do a remark in my post about this). Excuse me, but I find overwhelming to put a complete description of the topic in this Q&A site, since the bigger the post the less reads I'd receive and that tends to make things unclear. Jul 22, 2019 at 12:29 • @theREALyumdub: The use of the morphism $G \to Sym(G)$ is motivated by the fact that a generator of $G$ defines a cycle generator on $Sym(G)$, so DLOG can be reduced to cycle finding. I'm interested on the complexity of cycling back to the congruence/residue we are interested on. As I say on my post there are multiple ways to do this, but all depend on cycling and I'd like to know what's the expectancy when dealing with crypto's standard numbers for a prime $p$ which are quite big. Jul 22, 2019 at 12:33 • @kub0x Is this part of a graduate study or did you come up with it on your own? Maybe if you provided some context I could better answer the question. Jul 22, 2019 at 20:29 The binary operation $$a*b = \underbrace{a^{a^{a^{\dots}}}}$$ where the chain is repeated b times, has been studied before. If this is what you mean by "Self-Power Map," then Tetration (and the associated hyperoperations) have been studied for a while, at least around the dawn of computers as machines.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 69, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9198012948036194, "perplexity": 335.62181666070444}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00001.warc.gz"}
https://www.computer.org/csdl/trans/td/2011/02/ttd2011020337-abs.html	Issue No. 02 - February (2011 vol. 22) ISSN: 1045-9219 pp: 337-351 Driss Guerchi , UAE University, Al-Ain Leila Ismail , UAE University, Al-Ain ABSTRACT Convolution represents a major computational load for many scientific and engineering applications, including seismic surface simulations and seismic imaging. Since convolution presents a heavy computational load, increasing its efficiency can significantly enhance the performance of associated applications. In this work, we present an in-depth analysis of the convolution algorithm and its complexity in order to develop adequate parallel algorithms. The implementation of these algorithms and their evaluation on the IBM Cell Broadband Engine (BE) processor reveals the gains and losses achieved by parallelizing the direct convolution. The performance results show that despite the complexity of the convolution processing, a speedup gain of at least 71.4 is obtained. The parallel vectorized algorithm requires the development effort of considering three independent vectorization strategies. Given the wide availability of Cell processors, the proposed parallelization approach can be widely adopted by any convolution-based application. INDEX TERMS Parallel computing, IBM Cell BE, convolution, performance. CITATION Driss Guerchi, Leila Ismail, "Performance Evaluation of Convolution on the Cell Broadband Engine Processor", IEEE Transactions on Parallel & Distributed Systems, vol. 22, no. , pp. 337-351, February 2011, doi:10.1109/TPDS.2010.70	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8129381537437439, "perplexity": 2548.4840865804513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280280.6/warc/CC-MAIN-20170116095120-00095-ip-10-171-10-70.ec2.internal.warc.gz"}
https://hal.univ-lorraine.fr/hal-04026064	An improved algorithm for the solution of kriging equations in a global neighbourhood - Archive ouverte HAL Access content directly Conference Papers Year : ## An improved algorithm for the solution of kriging equations in a global neighbourhood N. Kuka • Function : Author #### Abstract This paper deals with the solution of linear algebraic equations encountered in geostatistics. The kriging estimator in a global neighbourhood requires the inversion of a large symmetric positive definite matrix K or the search of the solution by any other direct or iterative method. Classical algorithms such as Gauss elimination, Jacobi methods or UtDU decomposition have been extensively used in the past (Journel and Huijbregts, 1981; Davis and Grivet, 1984). However, some of these methods require the knowledge of the second members and do not take into account the accuracy of the solution. An alternative method combining a direct and an iterative procedure is presented here. The basic idea is a UtUDU decomposition of the kriging matrix, then an iterative improvement of the original solution until a given accuracy is reached. The effective computer time depends on n^3 (more precisely n^3 /6 + n^2 (i+1) where n being the number of equations, i the number of iterations usually 1 to 2) which is similar to direct method (n^3 /6 + n^2), however the accuracy of the solution is guaranteed. This technique could be applied to solve direct or dual kriging systems. A similar technique based on the LU decomposition could be extended to the non stationary case. #### Domains Sciences of the Universe [physics] ### Dates and versions hal-04026064 , version 1 (13-03-2023) ### Identifiers • HAL Id : hal-04026064 , version 1 ### Cite N. Kuka, Jean-Jacques Royer. An improved algorithm for the solution of kriging equations in a global neighbourhood. 2nd Codata Conference on Geomathematics and Geostatistics, 1992, Leeds, United Kingdom. pp.77-82. ⟨hal-04026064⟩ ### Export BibTeX TEI Dublin Core DC Terms EndNote Datacite 0 View	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8394625186920166, "perplexity": 1572.8705164067935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00331.warc.gz"}
http://fds.duke.edu/db/aas/math/faculty/bray/publications/287075	Department of Mathematics Search \| Help \| Login \| \| Math @ Duke ....................... ....................... Webpage ## Publications [#287075] of Hubert Bray Papers Published 1. Bray, HL; Jauregui, JL, A geometric theory of zero area singularities in general relativity, Asian Journal of Mathematics, vol. 17 no. 3 (September, 2013), pp. 525-560, ISSN 1093-6106 [arXiv:0909.0522v1], [doi] (last updated on 2018/07/22) Abstract: The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such \zero area singularities" in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also dene the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically at manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose inequality [9]. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the positive mass theorem that allows for certain types of incomplete metrics. © 2013 International Press. [email protected] ph: 919.660.2800 fax: 919.660.2821 Mathematics Department Duke University, Box 90320 Durham, NC 27708-0320	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.888812243938446, "perplexity": 965.8489237744681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593223.90/warc/CC-MAIN-20180722120017-20180722140017-00373.warc.gz"}
https://www.jbstudies.com/2022/09/ncert-solution-class12-maths-chapter5-continuity-and-differentiability-exercise-5.8.html	# NCERT Solutions Class 12 maths Chapter-5 (Continuity And Differentiability)Exercise 5.8 NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-5 (Continuity And Differentiability)Exercise 5.8 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter. We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination. ### Exercise 5.8 Question 1. Verify Rolle’s theorem for the function f(x) = x2+ 2x – 8, x [– 4, 2]. Solution: Now f(x) = x² + 2x – 8 is a polynomial So, f(x) is continuous in the interval [-4,2] and differentiable in the interval (- 4,2) f(-4) = (-4)² + 2(-4) – 8 = 16 – 8 – 8 = 0 f(2) = 2² + 4 – 8 = 8 – 8 = 0 f(-4) = f(2) As Conditions of Rolle’s theorem are satisfied. Then there exists some c in (-4, 2) such that f′(c) = 0 f'(x) = 2x + 2 f’ (c) = 2c + 2 = 0 c = – 1, and -1 [-4,2] Hence, f’ (c) = 0 at c = – 1. Question 2. Examine if Rolle’s theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s theorem from these example? (i) f(x) = [x] for x [5, 9] Solution: In the interval [5, 9], Now, f (x) = [x] which is neither continuous nor derivable at Integers. f (x) is neither continuous nor derivable at x = 6,7,8 Hence, Rolle’s theorem is NOT applicable (ii) f(x) = [x] for x [– 2, 2] Solution: In the interval [– 2, 2], Now, f (x) = [x] which is neither continuous nor derivable at Integers. f (x) is neither continuous nor derivable at x = -1,0,1 Hence, Rolle’s theorem is NOT applicable (iii) f(x) = x2– 1 for x [1, 2] Solution: Now f(x) = x² – 1 is a polynomial So, f(x) is continuous in the interval [1, 2] and differentiable in the interval (1,2) f(1) = (1)² – 1 = 0 f(2) = 2² – 1 = 3 f(-4) ≠ f(2) As Conditions of Rolle’s theorem are NOT satisfied. Hence, Rolle’s theorem is NOT applicable Question 3. If f : [– 5, 5] → R is a differentiable function and if f′(x) does not vanish anywhere, then prove that f(– 5) ≠ f(5). Solution: For Rolle’s theorem f is continuous in [a, b] ………(1) f is derivable in [a, b] ………(2) f (a) = f (b) ………(3) then f’ (c)=0, c (a, b) So as, f is continuous and derivable but f ‘(c) ≠ 0 It concludes, f(a) ≠ f(b) f(-5) ≠ f(5) Question 4. Verify Mean Value Theorem, if f(x) = x2– 4x – 3 in the interval [a, b], where a = 1 and b = 4. Solution: Now f(x) = x² – 4x -3 is a polynomial So, f(x) is continuous in the interval [1,4] and differentiable in the interval (1,4) f(1) = (1)² – 4(1) – 3 = -6 f(4) = 4² – 4(4) – 3 = -3 f′(c) = 2c – 4 As Conditions of Mean Value Theorem are satisfied. Then there exists some c in (1,4) such that f′(c) = = 1 2c – 4 = 1 c = 5/2 and c = 5/2 ∈ (1,4) Question 5. Verify Mean Value Theorem, if f(x) = x3– 5x2– 3x in the interval [a, b], where a = 1 and b = 3. Find all c (1, 3) for which f′(c) = 0. Solution: Now f(x) = x3– 5x2– 3x is a polynomial So, f(x) is continuous in the interval [1,3] and differentiable in the interval (1,3) f(1) = (1)3– 5(1)2– 3(1) = -7 f(3) = 33– 5(3)2– 3(3) = -27 f′(c) = 3c2 – 5(2c) – 3 f′(c) = 3c2 – 10c – 3 As Conditions of Mean Value Theorem are satisfied. Then there exists some c in (1,3) such that f′(c) = 3c2 – 10c – 3 = -10 3c2 – 10c + 7 = 0 3c2 – 3c – 7c + 7 = 0 3c (c-1) – 7(c -1) = 0 (3c -7) (c-1) = 0 c = 7/3 or c = 1 As, 1 ∉ (1,3) So, c = 7/3 ∈ (1,3) According to the Rolle’s Theorem As, f(3) ≠ f(1), Then there does not exist some c ∈ (1,3) such that f′(c) = 0 Question 6. Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2. (i) f(x) = [x] for x [5, 9] Solution: In the interval [5, 9], Now, f (x) = [x] which is neither continuous nor derivable at Integers. f (x) is neither continuous nor derivable at x = 6,7,8 Hence, Mean value theorem is NOT applicable (ii) f(x) = [x] for x [– 2, 2] Solution: In the interval [– 2, 2], Now, f (x) = [x] which is neither continuous nor derivable at Integers. f (x) is neither continuous nor derivable at x = -1,0,1 Hence, Mean value theorem is NOT applicable (iii) f(x) = x2– 1 for x [1, 2] Solution: Now f(x) = x² – 1 is a polynomial So, f(x) is continuous in the interval [1,2] and differentiable in the interval (1,2) f(1) = (1)² – 1 = 0 f(2) = 2² -1 = 3 f′(c) = 2c As Conditions of Mean Value Theorem are satisfied. Then there exists some c in (1,2) such that f′(c) = = 3 2c = 3 c = 3/2 and c = 3/2 ∈ (1,4)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9663980603218079, "perplexity": 1143.4851718958291}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335257.60/warc/CC-MAIN-20220928145118-20220928175118-00040.warc.gz"}
http://blitiri.blogspot.com/2017/08/surprise-and-belief-update.html	## 18 August 2017 ### Surprise and belief update In a previous post, I started discussing a paper [1] on the (un)surprising nature of a long streak of heads in a coin toss. My conclusion was that the surprise is not intrinsic to the particular sequence of throws, but rather residing in its relation with our prior information. I will detail this reasoning here, before returning to the paper itself. Let us accept as prior information the null hypothesis $$H_0$$ "the coin is unbiased". The conditional probabilities of throwing heads or tails are then equal: $$P(H\|H_0) = P(T\|H_0)=1/2$$. With the same prior, the probability of any sequence $$S_k$$ of 92 throws is the same: $$P(S_k\|H_0) = 2^{-92}$$, where $$k$$ ranges from $$1$$ to $$2^{92}$$. Assume now that the sequence we actually get consists of all heads: $$S_1 = \lbrace HH \ldots H\rbrace$$ What is the (posterior) probability of getting heads on the 93rd throw? Let us consider two options: 1. We can hold steadfast to our initial estimate of lack of bias $$P(H\|H_0) = 1/2$$. 2. We can update our "belief value" and say something like: "although my initial assessment was that the coin is unbiased [and the process of throwing is really random and I'm not hallucinating etc.], having thrown 92 heads in a row is good evidence to the contrary and on next throw I'll probably also get heads". Thus, $$P(H\|H_0 S_1) > 1/2$$ and in fact much closer to 1. How close exactly depends on the strength of our initial confidence in $$H_0$$, but I will not do the calculation here (I sketched it in the previous post). I would say that most rational persons would choose option 2 and abandon $$H_0$$; holding on to it (choice 1) would require an extremely strong confidence in our initial assessment. Note that for a sequence $$S_2$$ consisting of 46 heads and 46 tails (in any order) the distinction above is moot, since $$P(H\|H_0 S_2) =P(H\|H_0) = 1/2$$. The distinction between $$S_1$$ and $$S_2$$ is not their prior probability [2] but the way they challenge (and update) our belief. Back to Martin Smith's paper now: what makes him adopt the first choice? I think the most revealing phrase is the following: When faced with this result, of course it is sensible to check [...] whether the coins are double-headed or weighted or anything of that kind. Having observed a run of 92 heads in a row, one should regard it as very likely that the coins are double-headed or weighted. But, once these realistic possibilities have been ruled out, and we know they don’t obtain, any remaining urge to find some explanation (no matter how farfetched) becomes self-defeating.[italics in the text] As I understand it, he implicitly distinguishes between two kinds of propositions: observations (such as $$S_1$$) and checks (which are "of the nature of" $$H_0$$, although they can occur after the fact) and bestows upon the second category a protected status: these types of conclusions, e.g. "the coin is unbiased" survive even in the face of overwhelming evidence to the contrary (at least when it results from observation.) There is however no basis for this distinction: checks are also empirical findings: by visual inspection, I conclude that the coin does indeed exhibit two different faces; by more elaborate experiments I deduce that the center of mass is indeed in the geometrical center of the coin, within experimental precision; by some unspecified method I conclude that the "throwing process" is indeed random; by pinching myself I decide that I am not dreaming etc. At this point, however, the common sense remark is: "if you want to check the coin against bias, the easiest way would be to throw it about 92 times and count the heads". If we estimate the probability of the observations (given our prior belief) we should also update our belief in light of the observations. Recognizing this symmetry gives quantitative meaning to the "surprise" element, which is higher for some sequences than for others. 1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017. 2. We only considered here the probabilities before and after the 92 throws. One might also update one's belief after each individual throw, so that $$P(H)$$ would increase gradually.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9070417881011963, "perplexity": 693.5959957877499}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887981.42/warc/CC-MAIN-20180119125144-20180119145144-00670.warc.gz"}
http://mathoverflow.net/questions/90523/on-the-least-prime-in-arithmetic-progressions	# On the least prime in arithmetic progressions My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that $$p(a, q) \ll q^L$$ for some absolute constant $L$. Wiki page for this theorem lists a number of papers that estimate $L$ with the most recent result by Xylouris who proved that $L \leq 5.2$. It is also known that the Generalized Riemann Hypothesis implies $$p(a, q) \ll (q\log q)^2 \text{,}$$ while in 1978, Heath-Brown conjectured even tighter bound: $$p(a, q) \ll q(\log q)^2 \text{.}$$ I'm wondering whether this last bound, if true (it is still an open problem), implies something non-trivial about $L$-functions? - I'd guess that you could try to plug in the relevant values into the explicit formula (for example see equation (2) here: math.ubc.ca/~gerg/teaching/613-Winter2011/LinnikTheorem.pdf) and do the computations, but I'll leave a more authoritative statement on this to the experts. – Timothy Foo Mar 8 '12 at 7:02 I think I agree. If you knew there were lots of small primes in every arithmetic progression - essentially the desired asymptotic number with a small error term - then that would probably improve the known zero-free region for Dirichlet $L$-functions, up to a proof of the generalized Riemann hypothesis if the error term were good enough. But just one small prime in each residue class, I'm not sure that would give us any leverage. – Greg Martin Mar 13 '12 at 18:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9590694308280945, "perplexity": 215.1611904057371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207932182.13/warc/CC-MAIN-20150521113212-00008-ip-10-180-206-219.ec2.internal.warc.gz"}
https://people.maths.bris.ac.uk/~matyd/GroupNames/288/C4xD36.html	Copied to clipboard ## G = C4×D36order 288 = 25·32 ### Direct product of C4 and D36 Series: Derived Chief Lower central Upper central Derived series C1 — C18 — C4×D36 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×D36 — C4×D36 Lower central C9 — C18 — C4×D36 Upper central C1 — C2×C4 — C42 Generators and relations for C4×D36 G = < a,b,c \| a4=b36=c2=1, ab=ba, ac=ca, cbc=b-1 > Subgroups: 656 in 141 conjugacy classes, 58 normal (30 characteristic) C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×D4, Dic9, C36, C36, D18, D18, C2×C18, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C22×D9, C4×D12, C4⋊Dic9, D18⋊C4, C4×C36, C2×C4×D9, C2×D36, C4×D36 Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, D12, C22×S3, C4×D4, D18, S3×C2×C4, C2×D12, C4○D12, C4×D9, D36, C22×D9, C4×D12, C2×C4×D9, C2×D36, D365C2, C4×D36 Smallest permutation representation of C4×D36 On 144 points Generators in S144 (1 85 127 70)(2 86 128 71)(3 87 129 72)(4 88 130 37)(5 89 131 38)(6 90 132 39)(7 91 133 40)(8 92 134 41)(9 93 135 42)(10 94 136 43)(11 95 137 44)(12 96 138 45)(13 97 139 46)(14 98 140 47)(15 99 141 48)(16 100 142 49)(17 101 143 50)(18 102 144 51)(19 103 109 52)(20 104 110 53)(21 105 111 54)(22 106 112 55)(23 107 113 56)(24 108 114 57)(25 73 115 58)(26 74 116 59)(27 75 117 60)(28 76 118 61)(29 77 119 62)(30 78 120 63)(31 79 121 64)(32 80 122 65)(33 81 123 66)(34 82 124 67)(35 83 125 68)(36 84 126 69) (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144) (1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 110)(9 109)(10 144)(11 143)(12 142)(13 141)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 120)(35 119)(36 118)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73) G:=sub<Sym(144)\| (1,85,127,70)(2,86,128,71)(3,87,129,72)(4,88,130,37)(5,89,131,38)(6,90,132,39)(7,91,133,40)(8,92,134,41)(9,93,135,42)(10,94,136,43)(11,95,137,44)(12,96,138,45)(13,97,139,46)(14,98,140,47)(15,99,141,48)(16,100,142,49)(17,101,143,50)(18,102,144,51)(19,103,109,52)(20,104,110,53)(21,105,111,54)(22,106,112,55)(23,107,113,56)(24,108,114,57)(25,73,115,58)(26,74,116,59)(27,75,117,60)(28,76,118,61)(29,77,119,62)(30,78,120,63)(31,79,121,64)(32,80,122,65)(33,81,123,66)(34,82,124,67)(35,83,125,68)(36,84,126,69), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>; G:=Group( (1,85,127,70)(2,86,128,71)(3,87,129,72)(4,88,130,37)(5,89,131,38)(6,90,132,39)(7,91,133,40)(8,92,134,41)(9,93,135,42)(10,94,136,43)(11,95,137,44)(12,96,138,45)(13,97,139,46)(14,98,140,47)(15,99,141,48)(16,100,142,49)(17,101,143,50)(18,102,144,51)(19,103,109,52)(20,104,110,53)(21,105,111,54)(22,106,112,55)(23,107,113,56)(24,108,114,57)(25,73,115,58)(26,74,116,59)(27,75,117,60)(28,76,118,61)(29,77,119,62)(30,78,120,63)(31,79,121,64)(32,80,122,65)(33,81,123,66)(34,82,124,67)(35,83,125,68)(36,84,126,69), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) ); G=PermutationGroup([[(1,85,127,70),(2,86,128,71),(3,87,129,72),(4,88,130,37),(5,89,131,38),(6,90,132,39),(7,91,133,40),(8,92,134,41),(9,93,135,42),(10,94,136,43),(11,95,137,44),(12,96,138,45),(13,97,139,46),(14,98,140,47),(15,99,141,48),(16,100,142,49),(17,101,143,50),(18,102,144,51),(19,103,109,52),(20,104,110,53),(21,105,111,54),(22,106,112,55),(23,107,113,56),(24,108,114,57),(25,73,115,58),(26,74,116,59),(27,75,117,60),(28,76,118,61),(29,77,119,62),(30,78,120,63),(31,79,121,64),(32,80,122,65),(33,81,123,66),(34,82,124,67),(35,83,125,68),(36,84,126,69)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,117),(2,116),(3,115),(4,114),(5,113),(6,112),(7,111),(8,110),(9,109),(10,144),(11,143),(12,142),(13,141),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(21,133),(22,132),(23,131),(24,130),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,120),(35,119),(36,118),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]]) 84 conjugacy classes class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 18 18 2 1 1 1 1 2 2 2 2 18 18 18 18 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 84 irreducible representations dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 D9 C4×S3 D12 D18 C4○D12 C4×D9 D36 D36⋊5C2 kernel C4×D36 C4⋊Dic9 D18⋊C4 C4×C36 C2×C4×D9 C2×D36 D36 C4×C12 C36 C2×C12 C18 C42 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 2 3 2 3 4 4 9 4 12 12 12 Matrix representation of C4×D36 in GL3(𝔽37) generated by 6 0 0 0 6 0 0 0 6 , 36 0 0 0 4 8 0 29 12 , 1 0 0 0 6 20 0 26 31 G:=sub<GL(3,GF(37))\| [6,0,0,0,6,0,0,0,6],[36,0,0,0,4,29,0,8,12],[1,0,0,0,6,26,0,20,31] >; C4×D36 in GAP, Magma, Sage, TeX C_4\times D_{36} % in TeX G:=Group("C4xD36"); // GroupNames label G:=SmallGroup(288,83); // by ID G=gap.SmallGroup(288,83); # by ID G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,6725,292,9414]); // Polycyclic G:=Group<a,b,c\|a^4=b^36=c^2=1,ab=ba,ac=ca,cbc=b^-1>; // generators/relations ׿ × 𝔽	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9959438443183899, "perplexity": 3558.2928950596743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038067870.12/warc/CC-MAIN-20210412144351-20210412174351-00233.warc.gz"}
https://chem.libretexts.org/Core/Organic_Chemistry/Alkynes/Reactivity_of_Alkynes/Hydration_of_Alkynes_and_Tautomerism	# Hydration of Alkynes and Tautomerism As with alkenes, the addition of water to alkynes requires a strong acid, usually sulfuric acid, and is facilitated by mercuric sulfate. However, unlike the additions to double bonds which give alcohol products, addition of water to alkynes gives ketone products ( except for acetylene which yields acetaldehyde ). ### keto-enol Tautomerization The explanation for this deviation lies in enol-keto tautomerization, illustrated by the following equation. The initial product from the addition of water to an alkyne is an enol (a compound having a hydroxyl substituent attached to a double-bond), and this immediately rearranges to the more stable keto tautomer. Tautomers are defined as rapidly interconverted constitutional isomers, usually distinguished by a different bonding location for a labile hydrogen atom (colored red here) and a differently located double bond. The equilibrium between tautomers is not only rapid under normal conditions, but it often strongly favors one of the isomers ( acetone, for example, is 99.999% keto tautomer ). Even in such one-sided equilibria, evidence for the presence of the minor tautomer comes from the chemical behavior of the compound. Tautomeric equilibria are catalyzed by traces of acids or bases that are generally present in most chemical samples. The three examples shown below illustrate these reactions for different substitutions of the triple-bond. The tautomerization step is indicated by a red arrow. For terminal alkynes the addition of water follows the Markovnikov rule, as in the second example below, and the final product ia a methyl ketone ( except for acetylene, shown in the first example ). For internal alkynes ( the triple-bond is within a longer chain ) the addition of water is not regioselective. If the triple-bond is not symmetrically located ( i.e. if R & R' in the third equation are not the same ) two isomeric ketones will be formed. HC≡CH + H2O + HgSO4 & H2SO4 → [ H2C=CHOH ] → H3C-CH=O RC≡CH + H2O + HgSO4 & H2SO4 → [ RC(OH)=CH2 ] → RC(=O)CH3 RC≡CR' + H2O + HgSO4 & H2SO4 → [ RHC=C(OH)R' + RC(OH)=CHR' ] → RCH2-C(=O)R' + RC(=O)-CH2R' Two factors have an important influence on the enol-keto tautomerizations described here. The first is the potential energy difference between the tautomeric isomers. This factor determines the position of the equilibrium state. The second factor is the activation energy for the interconversion of one tautomer to the other. This factor determines the rate of rearrangement. Since the potential energy or stability of a compound is in large part a function of its covalent bond energies, we can estimate the relative energy of keto and enol tautomers by considering the bonds that are changed in the rearrangement. From the following diagram, we see that only three significant changes occur, and the standard bond energies for those changes are given to the right of the equation. The keto tautomer has a 17.5 kcal/mole advantage in bond energy, so its predominance at equilibrium is expected. The rapidity with which enol-keto tautomerization occurs suggests that the activation energy for this process is low. We have noted that the rearrangement is acid & base catalyzed, and very careful experiments have shown that interconversion of tautomers is much slower if such catalysts are absent. A striking example of the influence of activation energy on such transformations may be seen in the following hypothetical rearrangement. Here we have substituted a methyl group (colored maroon) for the proton of a conventional tautomerism, and the methyl shifts from oxygen to carbon just as the proton does in going from an enol to a ketone. H2C=CH-O-CH3 X> CH3-CH2-CH=O The potential energy change for this rearrangement is even more advantageous than for enol-keto tautomerism, being estimated at over 25 kcal/mole from bond energy changes. Despite this thermodynamic driving force, the enol ether described above is completely stable to base treatment, and undergoes rapid acid-catalyzed hydrolysis with loss of methanol, rather than rearrangement. The controlling difference in this case must be a prohibitively high activation energy for the described rearrangement, combined with lower energy alternative reaction paths.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8432647585868835, "perplexity": 4342.957566590393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549425737.60/warc/CC-MAIN-20170726002333-20170726022333-00497.warc.gz"}
https://giasutamtaiduc.com/math-formulas-for-class-12.html	# ✅ Math Formulas For Class 12 ⭐️⭐️⭐️⭐️⭐ 5/5 - (1 bình chọn) Mục Lục ## Double Angle and Half Angle Formulas ### Relations And Functions Definition/Theorems 1. Empty relation holds a specific relation R in X as: R = φ ⊂ X × X. 2. A Symmetric relation R in X satisfies a certain relation as: (a, b) ∈ R implies (b, a) ∈ R. 3. A Reflexive relation R in X can be given as: (a, a) ∈ R; for all ∀ a ∈ X. 4. A Transitive relation R in X can be given as: (a, b) ∈ R and (b, c) ∈ R, thereby, implying (a, c) ∈ R. 5. A Universal relation is the relation R in X can be given by R = X × X. 6. Equivalence relation R in X is a relation that shows all the reflexive, symmetric and transitive relations. Properties ### Inverse Trigonometric Functions Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here. Properties/Theorems The domain and range of inverse trigonometric functions are given below: Formulas ### Matrices Definition/Theorems Elementary Operations ### Determinants Definition/Theorems 1. The determinant of a matrix A = [a11]1 × 1 can be given as: \|a11\| = a11. 2. For any square matrix A, the \|A\| will satisfy the following properties: • (i) \|A′\| = \|A\|, where A′ = transpose of A. • (ii) If we interchange any two rows (or columns), then sign of determinant changes. • (iii) If any two rows or any two columns are identical or proportional, then the value of the determinant is zero. • (iv) If we multiply each element of a row or a column of a determinant by constant k, then the value of the determinant is multiplied by k. Formulas ### Continuity And Differentiability Definition/Properties 1. A function is said to be continuous at a given point if the limit of that function at the point is equal to the value of the function at the same point. Formulas Given below are the standard derivatives: ### Integrals Definition/Properties Formulas – Standard Integrals ### Formulas – Integrals (Special Functions) Formulas – Integration by Parts 1. The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function} ### Vector Algebra Definition/Properties 1. Vector is a certain quantity that has both the magnitude and the direction. The position vector of a point P (x, y, z) is given by: Formulas ### Three Dimensional Geometry Definition/Properties 1. Direction cosines of a line are the cosines of the angle made by a particular line with the positive directions on coordinate axes. 2. Skew lines are lines in space which are neither parallel nor intersecting. These lines lie in separate planes. 3. If l, m and n are the direction cosines of a line, then l2 + m2 + n2 = 1. Formulas ### Probability Definition/Properties 1. The conditional probability of an event E holds the value of the occurrence of the event F as: ## Frequently Asked Questions – FAQs ### What are the basic maths formulas for class 12th? The basic formulas that are introduced for class 12th students are for the topics: Algebra Geometry Matrices Calculus Linear Programming Probability ### How many formulas are there in Maths? There are many formulas in Maths for which we cannot keep a record. Because for each and every concept there are formulas to find the solutions for mathematical problems. Also, for each grade the level of formulas are different. ### What is the importance of Maths formulas? The importance of learning Maths formulas is that it helps us to solve problems easily. We should have to put the values of entities in the given formula and simplify them. For example, to find the average of a given set of values, we have to need to know first the sum of all those values and number of values. Hence, the average will be equal to the ratio of the sum of values and the number of values. ### What is the formula for integrating trigonometry ratio? ∫cos(a) da = Sin a + C ∫sin (a) da = -Cos a + C ∫sec^2a da = tan a + C ### What is the integration for exponential function? The value of exponential function e^x remains the same with constant even after integration. ∫e^x dx = e^x + C Ques: How many formulas are present in the class 12 CBSE Maths? Ans: It is almost next to impossible to keep a record of all the formulas given in the Maths book of class 12 CBSE. As for each and every theory and concept, given in the book, there exist one or more formulas to help find the solutions for the given mathematical problems. As well the level of formulas increases with each grade making class 12 Mathematics the most difficult on the school level. Ques: Give a list of basic maths formulas used in CBSE class 12th? Ans: A list containing the basic maths formulas that have been introduced for the students of class 12th have been listed in this above article. The basic list of topics include: 1. Algebra 2. Matrices 3. Geometry 4. Linear Programming 5. Calculus 6. Probability Ques: What is the formula used for the trigonometric ratio integration? Ans: ∫sin (x) dx = -Cos x + C ∫cos(x) dx = Sin x + C ∫sec^2x dx = tan x + C, etc. Ques: Why are the mathematical formulas important? Ans: The mathematical formulas are important because it helps in solving the mathematical problems with utmost ease. Hence it is important to learn these mathematical formulas for solving the problems in a given time span and in an efficient manner. Mathematical formulas are in the generalized form and at the time of solving the mathematical problems, all we need to do is put the value of entities in the formula given and make the whole process easier and swifter. Ques: Where can I find the complete class 12 Mathematics formula for the NCERT book? Ans: Students can find the compiled list of formulas in this article on the embibe platform for free. Students can also find direct links to Class 12 Mathematics Notes, Solutions, practice papers, mock tests, important questions, and much more. Students can go through this article to find the complete NCERT Solutions for Class 12 Mathematics along with the links of complete NCERT Book Class 12 Mathematics & important study material. Ques: Which is the best solution for NCERT Class 12 Mathematics? Ans: Students can find 100 percent accurate Solutions for the NCERT Class 12 Mathematics on the Embibe platform. This article contains the complete solution which has been solved by expert mathematics teachers associated with embibe. We, at Embibe, provide solutions for all the questions given in the Class 12 Mathematics textbook after taking the CBSE Board guidelines under strict consideration from the latest NCERT book for Class 12 Mathematics. Ques: What is the formula used for the exponential function integration? Ans: If an exponential function is integrated, the function will remain unchanged with a constant being added to it. Hence, ∫e^x dx = e^x + Constant © Math Formulas ⭐️⭐️⭐️⭐️⭐	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9818671345710754, "perplexity": 688.7378698861647}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358953.29/warc/CC-MAIN-20211130050047-20211130080047-00493.warc.gz"}
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/10%3A_Trees/10.01%3A_What_is_a_Tree	$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ # 10.1: What is a Tree? $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ ## Definition What distinguishes trees from other types of graphs is the absence of certain paths called cycles. Recall that a path is a sequence of consecutive edges in a graph, and a circuit is a path that begins and ends at the same vertex. Definition $$\PageIndex{1}$$: Cycle A cycle is a circuit whose edge list contains no duplicates. It is customary to use $$C_n$$ to denote a cycle with $$n$$ edges. The simplest example of a cycle in an undirected graph is a pair of vertices with two edges connecting them. Since trees are cycle-free, we can rule out all multigraphs having at least one pair of vertices connected with two or more edges from consideration as trees. Trees can either be undirected or directed graphs. We will concentrate on the undirected variety in this chapter. Definition $$\PageIndex{2}$$: Tree An undirected graph is a tree if it is connected and contains no cycles or self-loops. Example $$\PageIndex{1}$$: Some Trees and Non-Trees 1. Graphs i, ii and iii in Figure $$\PageIndex{1}$$ are all trees, while graphs iv, v, and vi are not trees. 2. A $$K_2$$ is a tree. However, if $$n\geq 3\text{,}$$ a $$K_n$$ is not a tree. 3. In a loose sense, a botanical tree is a mathematical tree. There are usually no cycles in the branch structure of a botanical tree. 4. The structures of some chemical compounds are modeled by a tree. For example, butane Figure $$\PageIndex{2}$$ consists of four carbon atoms and ten hydrogen atoms, where an edge between two atoms represents a bond between them. A bond is a force that keeps two atoms together. The same set of atoms can be linked together in a different tree structure to give us the compound isobutane Figure $$\PageIndex{3}$$. There are some compounds whose graphs are not trees. One example is benzene Figure $$\PageIndex{4}$$. One type of graph that is not a tree, but is closely related, is a forest. Definition $$\PageIndex{3}$$: Forest A forest is an undirected graph whose components are all trees. Example $$\PageIndex{2}$$: A Forest The top half of Figure $$\PageIndex{1}$$ can be viewed as a forest of three trees. Graph (vi) in this figure is also a forest. ## Conditions for a graph to be a tree We will now examine several conditions that are equivalent to the one that defines a tree. The following theorem will be used as a tool in proving that the conditions are equivalent. Lemma $$\PageIndex{1}$$ Let $$G = (V, E)$$ be an undirected graph with no self-loops, and let $$v_a, v_b\in V\text{.}$$ If two different simple paths exist between $$v_a$$ and $$v_b\text{,}$$ then there exists a cycle in $$G\text{.}$$ Proof Let $$p_1= \left(e_1, e_2, \ldots , e_m \right)$$ and $$p_2=\left(f_1,f_2,\ldots , f_n\right)$$ be two different simple paths from $$v_a$$ to $$v_b\text{.}$$ The first step we will take is to delete from $$p_1$$ and $$p_2$$ the initial edges that are identical. That is, if $$e_1= f_1\text{,}$$ $$e_2= f_2, \dots \text{,}$$ $$e_{j}= f_j\text{,}$$ and $$e_{j+1}\neq f_{j+1}$$ delete the first $$j$$ edges of both paths. Once this is done, both paths start at the same vertex, call it $$v_c\text{,}$$ and both still end at $$v_b\text{.}$$ Now we construct a cycle by starting at $$v_c$$ and following what is left of $$p_1$$ until we first meet what is left of $$p_2\text{.}$$ If this first meeting occurs at vertex $$v_d\text{,}$$ then the remainder of the cycle is completed by following the portion of the reverse of $$p_2$$ that starts at $$v_d$$ and ends at $$v_c\text{.}$$ Theorem $$\PageIndex{1}$$: Equivalent Conditions for a Graph to be a Tree Let $$G = (V, E)$$ be an undirected graph with no self-loops and $$\lvert V \rvert =n\text{.}$$ The following are all equivalent: 1. $$G$$ is a tree. 2. For each pair of distinct vertices in $$V\text{,}$$ there exists a unique simple path between them. 3. $$G$$ is connected, and if $$e\in E\text{,}$$ then $$(V, E-\{e\})$$ is disconnected. 4. $$G$$ contains no cycles, but by adding one edge, you create a cycle. 5. $$G$$ is connected and $$\lvert E \rvert =n-1 \text{.}$$ Proof Proof Strategy. Most of this theorem can be proven by proving the following chain of implications: $$(1) \Rightarrow (2)\text{,}$$ $$(2) \Rightarrow (3)\text{,}$$ $$(3)\Rightarrow (4)\text{,}$$ and $$(4) \Rightarrow (1)\text{.}$$ Once these implications have been demonstrated, the transitive closure of $$\Rightarrow$$ on $${1, 2, 3, 4}$$ establishes the equivalence of the first four conditions. The proof that Statement 5 is equivalent to the first four can be done by induction, which we will leave to the reader. $$(1) \Rightarrow (2)$$ (Indirect). Assume that $$G$$ is a tree and that there exists a pair of vertices between which there is either no path or there are at least two distinct paths. Both of these possibilities contradict the premise that $$G$$ is a tree. If no path exists, $$G$$ is disconnected, and if two paths exist, a cycle can be obtained by Theorem $$\PageIndex{1}$$. $$(2) \Rightarrow (3)\text{.}$$ We now use Statement 2 as a premise. Since each pair of vertices in $$V$$ are connected by exactly one path, $$G$$ is connected. Now if we select any edge $$e$$ in $$E\text{,}$$ it connects two vertices, $$v_1$$ and $$v_2\text{.}$$ By (2), there is no simple path connecting $$v_1$$ to $$v_2$$ other than $$e\text{.}$$ Therefore, no path at all can exist between $$v_1$$ and $$v_2$$ in $$(V, E - \{e\})\text{.}$$ Hence $$(V, E - \{e\})$$ is disconnected. $$(3)\Rightarrow (4)\text{.}$$ Now we will assume that Statement 3 is true. We must show that $$G$$ has no cycles and that adding an edge to $$G$$ creates a cycle. We will use an indirect proof for this part. Since (4) is a conjunction, by DeMorgan's Law its negation is a disjunction and we must consider two cases. First, suppose that $$G$$ has a cycle. Then the deletion of any edge in the cycle keeps the graph connected, which contradicts (3). The second case is that the addition of an edge to $$G$$ does not create a cycle. Then there are two distinct paths between the vertices that the new edge connects. By Lemma $$\PageIndex{1}$$, a cycle can then be created, which is a contradiction. $$(4) \Rightarrow (1)$$ Assume that $$G$$ contains no cycles and that the addition of an edge creates a cycle. All that we need to prove to verify that $$G$$ is a tree is that $$G$$ is connected. If it is not connected, then select any two vertices that are not connected. If we add an edge to connect them, the fact that a cycle is created implies that a second path between the two vertices can be found which is in the original graph, which is a contradiction. The usual definition of a directed tree is based on whether the associated undirected graph, which is created by “erasing” its directional arrows, is a tree. In Section 10.3 we will introduce the rooted tree, which is a special type of directed tree. ## Exercises Exercise $$\PageIndex{1}$$ Given the following vertex sets, draw all possible undirected trees that connect them. 1. $$\displaystyle V_a= \{\text{right},\text{left}\}$$ 2. $$\displaystyle V_b = \{+,-,0\}$$ 3. $$V_c = \{\text{north}, \text{south}, \text{east}, \text{west}\}\text{.}$$ The number of trees are: (a) 1, (b) 3, and (c) 16. The trees that connect $$V_c$$ are: Exercise $$\PageIndex{2}$$ Are all trees planar? If they are, can you explain why? If they are not, you should be able to find a nonplanar tree. Exercise $$\PageIndex{3}$$ Prove that if $$G$$ is a simple undirected graph with no self-loops, then $$G$$ is a tree if and only if $$G$$ is connected and $$\lvert E \rvert = \lvert V \rvert - 1\text{.}$$ Hint Use induction on $$\lvert E\rvert \text{.}$$ Exercise $$\PageIndex{4}$$ 1. Prove that if $$G = (V, E)$$ is a tree and $$e \in E\text{,}$$ then $$(V, E - \{e\})$$ is a forest of two trees. 2. Prove that if $$\left(V_1,E_1\right.$$) and $$\left(V_2,E_2\right)$$ are disjoint trees and $$e$$ is an edge that connects a vertex in $$V_1$$ to a vertex in $$V_2\text{,}$$ then $$\left(V_1\cup V_2, E_1\cup E_2\cup \{e\}\right)$$ is a tree. Exercise $$\PageIndex{5}$$ 1. Prove that any tree with at least two vertices has at least two vertices of degree 1. 2. Prove that if a tree has $$n$$ vertices, $$n \geq 4\text{,}$$ and is not a path graph, $$P_n\text{,}$$ then it has at least three vertices of degree 1. 1. Assume that $$(V,E)$$ is a tree with $$\|V\|≥2$$, and all but possibly one vertex in $$V$$ has degree two or more. \begin{aligned}2\|E\|=\sum\limits_{v\in V}^{}\text{deg}(v)\geq 2\|V\|-1 &\Rightarrow \|E\|\geq \|V\|-\frac{1}{2} \\ &\Rightarrow \|E\|\geq \|V\| \\ &\Rightarrow (V,E)\text{ is not a tree.}\end{aligned} 2. The proof of this part is similar to part a in that we can infer $$2\|E\|≥2\|V\|−1$$, using the fact that a non-chain tree has at least one vertex of degree three or more.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8675229549407959, "perplexity": 148.89331349315086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517018.29/warc/CC-MAIN-20220517063528-20220517093528-00499.warc.gz"}
https://chemistry.stackexchange.com/questions/41189/estimation-of-the-bond-angle-of-water	# Estimation of the bond angle of water We know from experimental data that $\ce{H-O-H}$ bond angle in water is approximately 104.5 degrees. If its two lone pairs were bonds (which is unfortunately impossible) also $\ce{O-H}$ bonds and a perfect tetrahedron resulted, then VSEPR theory would predict that the bond angle would be 109.5 degrees - this number can be easily derived using the geometry of a tetrahedron. However, how would people give estimates of the actual bonding angle of water, which is caused by a slightly greater repulsion by the lone pairs than there would be if they were bonds? What physics would be involved in the calculation? Thanks! how would people give estimates of the actual bonding angle of water What physics would be involved in the calculation Background That's a very good question. In many cases Coulson's Theorem can be used to relate bond angles to the hybridization indices of the bonds involved. $$\ce{1+\lambda_{i} \lambda_{j} cos(\theta_{ij})=0}$$ where $\ce{\lambda_{i}}$ represents the hybridization index of the $\ce{C-i}$ bond (the hybridization index is the square root of the bond hybridization) and $\ce{\theta_{ij}}$ represents the $\ce{i-C-j}$ bond angle. For example, in the case of methane, each $\ce{C-H}$ bond is $\ce{sp^3}$ hybridized and the hybridization index of each $\ce{C-H}$ bond is $\sqrt3$. Using Coulson's theorem we find that the $\ce{H-C-H}$ bond angle is 109.5 degrees. Unlike methane, water is not a perfectly tetrahedral molecule, so the oxygen bonding orbitals and the oxygen lone pair orbitals will not be exactly $\ce{sp^3}$ hybridized. Since addition of s-character to an orbital stabilizes electrons in that orbital (because the s orbital is lower in energy than the p orbital) and since the electron density is higher in the lone pair orbital than the $\ce{O-H}$ orbital (because the electrons in the lone pair orbital are not being shared with another atom), we might expect that oxygen lone pair orbital will have more s-character and the oxygen $\ce{O-H}$ orbital will have less s-character. If we examine the case where the $\ce{O-H}$ bond is $\ce{sp^4}$ hybridized, we find from Coulson's Theorem that the $\ce{H-O-H}$ angle is predicted to be around 104.5°. So indeed, in agreement with our prediction, removing s-character from the oxygen $\ce{O-H}$ orbital gives rise to the observed bond angle. Note on reality For over 50 years students have been told that water is roughly $\ce{sp^3}$ hybridized. The general description is that there are two equivalent O-H sigma bonds and two equivalent lone pairs orbitals. The lone pair - lone pair repulsion is greater than the sigma bond - sigma bond repulsion, so the hybridization changes as described above and the lone pair-O-lone pair angle opens up slightly and the $\ce{H-O-H}$ angle closes down to the observed 104.5 degrees. With the advent of photoelectron spectroscopy it was found that the two lone pairs in water were not equivalent (2 signals were observed for the lone pairs). Now, the hybridization of water is described as follows: • 2 $\ce{sp^4}$ O-H sigma bonds • one lone pair in a p orbital • and the second lone pair in an $\ce{sp}$ orbital As you said, lone pairs result in slightly more repulsion than bonds, because the lone pairs themselves expand a little around the atom. Common practice that I've learned is just to subtract about 2.5 degrees from the bond angle for every lone pair. For instance, a molecule such as NCl3 has 4 negative charge centers, giving it a tetrahedral group geometry, but it has one lone pair, giving it bond angles of about 107 degrees. Following this pattern, water would have bond angles of about 104.5 degrees. Here's a picture that sort of shows how the lone pairs expand, reducing the bond angles slightly.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8834151029586792, "perplexity": 1059.2033716736216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027320734.85/warc/CC-MAIN-20190824105853-20190824131853-00462.warc.gz"}
http://mathhelpforum.com/calculus/147733-find-dy-dx-following.html	# Thread: find dy/dx of the following 1. ## find dy/dx of the following Hi all, Should be an easy one. Find dy/dx of: x^2 + y^2 = 49 2. Originally Posted by Joel Hi all, Should be an easy one. Find dy/dx of: x^2 + y^2 = 49 I could do this but that'd defy the point of helping. If you use implicit differentiation and the chain rule you'll be on the road to success. If you want more help please post what you've done 3. Originally Posted by Joel Hi all, Should be an easy one. Find dy/dx of: x^2 + y^2 = 49 Dear Joel, Derivate the left and right sides of the equation, and use the chain rule. $x^2+y^2=49$ $\frac{d}{dx}\left(x^2+y^2\right)=\frac{d(49)}{dx}$ Hope you can continue from here. x^2 + y^2 = 49 y^2 = 49 - x^2 y = +- root(49 - x^2) y = +- ( 49 - x^2 ) ^1/2 y' = 1/2 (49 - x^2)^-1/2 . (-2x) am I anywhere near on the right track ??? 5. Originally Posted by Joel x^2 + y^2 = 49 y^2 = 49 - x^2 y = +- root(49 - x^2) y = +- ( 49 - x^2 ) ^1/2 y' = 1/2 (49 - x^2)^-1/2 . (-2x) am I anywhere near on the right track ??? I would stick with implicit differentiation rather than trying to solve the quadratic equation. $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(49)$ The first and last terms should be easy enough, for the middle term you can differentiate y as though it was x and then multiply by $\frac{dy}{dx}$ (which is the chain rule) For example $\frac{d}{dx} \left(\frac{1}{3}y^3\right) = y^2 \frac{dy}{dx}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8571813702583313, "perplexity": 1673.200146837969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189245.97/warc/CC-MAIN-20170322212949-00466-ip-10-233-31-227.ec2.internal.warc.gz"}
https://en.wikipedia.org/wiki/Diffusion	# Diffusion A diffusion is a process in physics. Some particles are dissolved in a glass of water. At first, the particles are all near one corner of the glass. If the particles all randomly move around ("diffuse") in the water, then the particles will eventually become distributed randomly and uniformly, and organized (but diffusion will still continue to occur, just that there will be no net flux). Time lapse video of diffusion of a dye dissolved in water into a gel. Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a statistical trend that the solute fills the container more and more uniformly. Bottom: With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic force pushing molecules rightward, but there appears to be one in the bottom panel. This apparent force is called an entropic force. 3D rendering of diffusion of purple dye in water. Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential). This is also referred to as the movement of a substance down a concentration gradient. A gradient is the change in the value of a quantity (e.g., concentration, pressure, temperature) with the change in another variable (usually distance). For example, a change in concentration over a distance is called a concentration gradient, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is a called a temperature gradient. The word diffusion is derived from the Latin word, "diffundere", which means "to spread out" (if a substance is “spreading out”, it is moving from an area of high concentration to an area of low concentration). A distinguishing feature of diffusion is that it results in mixing or mass transport, without requiring bulk motion (bulk flow). Thus, diffusion should not be confused with convection, or advection, which are other transport phenomena that utilize bulk motion to move particles from one place to another. ## Diffusion vs. bulk flow An example of a situation in which bulk flow and diffusion can be differentiated is the mechanism by which oxygen enters the body during external respiration (breathing). The lungs are located in the thoracic cavity, which is expanded as the first step in external respiration. This expansion leads to an increase in volume of the alveoli in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air outside the body (relatively high pressure) and the alveoli (relatively low pressure). The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal (i.e., the movement of air by bulk flow stops once there is no longer a pressure gradient). The air arriving in the alveoli has a higher concentration of oxygen than the “stale” air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the capillaries that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of carbon dioxide in the alveoli decreases (air has a very low concentration of carbon dioxide compared to the blood in the body). This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli. The blood is then transported around the body by the pumping action of the heart. As the left ventricle of the heart contracts, the volume decreases, which causes the pressure in the ventricle to increase. This creates a pressure gradient between the heart and the capillaries, and blood moves through blood vessels by bulk flow (down the pressure gradient). As the thoracic cavity contracts during expiration, the volume of the alveoli decreases and creates a pressure gradient between the alveoli and the air outside the body, and air moves by bulk flow down the pressure gradient. ## Diffusion in the context of different disciplines Diffusion furnaces used for thermal oxidation The concept of diffusion is widely used in: physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas and of price values). However, in each case, the object (e.g., atom, idea, etc.) that is undergoing diffusion is “spreading out” from a point or location at which there is a higher concentration of that object. There are two ways to introduce the notion of diffusion: either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or a physical and atomistic one, by considering the random walk of the diffusing particles.[1] In the phenomenological approach, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion. According to Fick's laws, the diffusion flux is proportional to the negative gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Some time later, various generalizations of Fick's laws were developed in the frame of thermodynamics and non-equilibrium thermodynamics.[2] From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein.[3] The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals. In biology, the terms "net movement" or "net diffusion" are often used when considering the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes and if there is a higher concentration of oxygen outside the cell than inside, oxygen molecules will diffuse into the cell. However, because the movement of molecules is random, occasionally oxygen molecules will move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the probability that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) will be into the cell. In other words, there will be a net movement of oxygen molecules down the concentration gradient. ## History of diffusion in physics In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, Pliny the Elder had previously described the cementation process which produces steel from the element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colours of stained glass or earthenware and Chinese ceramics. In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:[4] "...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time.” The measurements of Graham contributed to James Clerk Maxwell deriving, in 1867, the coefficient of diffusion for CO2 in air. The error rate is less than 5%. In 1855, Adolf Fick, the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion. He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating a formalism that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827). Robert Boyle demonstrated diffusion in solids in the 17th century[5] by penetration of Zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. William Chandler Roberts-Austen, the well-known British metallurgist, and former assistant of Thomas Graham, studied systematically solid state diffusion on the example of gold in lead in 1896. :[6] "... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals." In 1858, Rudolf Clausius introduced the concept of the mean free path. In the same year, James Clerk Maxwell developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion was developed by Albert Einstein, Marian Smoluchowski and Jean-Baptiste Perrin. Ludwig Boltzmann, in the development of the atomistic backgrounds of the macroscopic transport processes, introduced the Boltzmann equation, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.[7] Yakov Frenkel (sometimes, Jakov/Jacov Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data. Some time later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.[6] Henry Eyring, with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion.[8] The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.[9] ## Basic models of diffusion ### Diffusion flux Each model of diffusion expresses the diffusion flux through concentrations, densities and their derivatives. Flux is a vector ${\displaystyle \mathbf {J} }$. The transfer of a physical quantity ${\displaystyle N}$ through a small area ${\displaystyle \Delta S}$ with normal ${\displaystyle \nu }$ per time ${\displaystyle \Delta t}$ is ${\displaystyle \Delta N=(\mathbf {J} ,\nu )\Delta S\Delta t+o(\Delta S\Delta t)\,,}$ where ${\displaystyle (\mathbf {J} ,\nu )}$ is the inner product and ${\displaystyle o(...)}$ is the little-o notation. If we use the notation of vector area ${\displaystyle \Delta \mathbf {S} =\nu \Delta S}$ then ${\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\Delta t+o(\Delta \mathbf {S} \Delta t)\,.}$ The dimension of the diffusion flux is [flux]=[quantity]/([time]·[area]). The diffusing physical quantity ${\displaystyle N}$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, ${\displaystyle n}$, the diffusion equation has the form ${\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}$ where ${\displaystyle W}$ is intensity of any local source of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the no-flux boundary conditions can be formulated as ${\displaystyle (\mathbf {J} (x),\nu (x))=0}$ on the boundary, where ${\displaystyle \nu }$ is the normal to the boundary at point ${\displaystyle x}$. ### Fick's law and equations Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient: ${\displaystyle \mathbf {J} =-D\nabla n\ ,\;\;J_{i}=-D{\frac {\partial n}{\partial x_{i}}}\ .}$ The corresponding diffusion equation (Fick's second law) is ${\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D\nabla n(x,t))=D\Delta n(x,t)\ ,}$ where ${\displaystyle \Delta }$ is the Laplace operator, ${\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}$ ### Onsager's equations for multicomponent diffusion and thermodiffusion Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, ${\displaystyle -\nabla n}$. In 1931, Lars Onsager[10] included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport, ${\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}$ where ${\displaystyle \mathbf {J} _{i}}$ is the flux of the ith physical quantity (component) and ${\displaystyle X_{j}}$ is the jth thermodynamic force. The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s (he used the term "force" in quotation marks or "driving force"): ${\displaystyle X_{i}={\rm {grad}}{\frac {\partial s(n)}{\partial n_{i}}}\ ,}$ where ${\displaystyle n_{i}}$ are the "thermodynamic coordinates". For the heat and mass transfer one can take ${\displaystyle n_{0}=u}$ (the density of internal energy) and ${\displaystyle n_{i}}$ is the concentration of the ith component. The corresponding driving forces are the space vectors ${\displaystyle X_{0}={\rm {grad}}{\frac {1}{T}}\ ,\;\;\;X_{i}=-{\rm {grad}}{\frac {\mu _{i}}{T}}\;(i>0),}$ because ${\displaystyle {\rm {d}}s={\frac {1}{T}}{\rm {d}}u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}{\rm {d}}n_{i}}$ where T is the absolute temperature and ${\displaystyle \mu _{i}}$ is the chemical potential of the ith component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients. For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium: ${\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\partial n_{k}}}\right\|_{n=n^{}}{\rm {grad}}n_{k}\ ,}$ where the derivatives of s are calculated at equilibrium n. The matrix of the kinetic coefficients ${\displaystyle L_{ij}}$ should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth). The transport equations are ${\displaystyle {\frac {\partial n_{i}}{\partial t}}=-{\rm {div}}\mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}{\rm {div}}X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\partial n_{k}}}\right\|_{n=n^{}}\right]\Delta n_{k}\ .}$ Here, all the indexes i, j, k=0,1,2,... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix ${\displaystyle D_{ik}}$of the diffusion (i,k>0), thermodiffusion (i>0, k=0 or k>0, i=0) and thermal conductivity (i=k=0) coefficients. Under isothermal conditions T=const. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, ${\displaystyle -(1/T)\nabla \mu _{j}}$, and the matrix of diffusion coefficients is ${\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right\|_{n=n^{}}}$ (i,k>0). There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations ${\displaystyle \sum _{j}L_{ij}X_{j}}$ can be measured. For example, in the original work of Onsager[10] the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics[11] this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not effect the measurable quantities. ### Nondiagonal diffusion must be nonlinear The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form ${\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\Delta c_{j}\,.}$ If the matrix of diffusion coefficients is diagonal then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, ${\displaystyle D_{12}\neq 0}$, and consider the state with ${\displaystyle c_{2}=\ldots =c_{n}=0}$. At this state, ${\displaystyle \partial c_{2}/\partial t=D_{12}\Delta c_{1}}$. If ${\displaystyle D_{12}\Delta c_{1}(x)<0}$ at some points then ${\displaystyle c_{2}(x)}$ becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.[9] ### Einstein's mobility and Teorell formula The Einstein relation (kinetic theory) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)[12] ${\displaystyle D=\mu \,k_{B}T}$ where D is the diffusion constant; μ is the "mobility"; kB is Boltzmann's constant; T is the absolute temperature. Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation ${\displaystyle {\mathfrak {m}}}$. The mobility—based approach was further applied by T. Teorell.[13] In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula: the flux is equal to mobility×concentration×force per gram ion. This is the so-called Teorell formula. The force under isothermal conditions consists of two parts: 1. Diffusion force caused by concentration gradient: ${\displaystyle -RT{\frac {1}{n}}\nabla n=-RT\nabla (\ln(n/n^{\rm {eq}})).}$ 2. Electrostatic force caused by electric potential gradient: ${\displaystyle q\nabla \varphi .}$ Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential. The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein – Teorell approach, If for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule. The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is[9] ${\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\mbox{external force per gram particle}}))\,,}$ where μ is the chemical potential, μ0 is the standard value of the chemical potential. The expression ${\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}$ is the so-called activity. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form[9] ${\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\mbox{external force per gram particle}}))\,.}$ The standard derivation of the activity includes a normalization factor and for small concentrations ${\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}$, where ${\displaystyle n^{\ominus }}$ is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity, ${\displaystyle n/n^{\ominus }}$, ${\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\mbox{external force per gram particle}}))]}$ #### Teorell formula for multicomponent diffusion The Teorell formula with combination of Onsager's definition of the diffusion force gives ${\displaystyle \mathbf {J} _{i}={\mathfrak {m_{i}}}a_{i}\sum _{j}L_{ij}X_{j}\,,}$ where ${\displaystyle {\mathfrak {m_{i}}}}$ is the mobility of the ith component, ${\displaystyle a_{i}}$ is its activity, ${\displaystyle L_{ij}}$ is the matrix of the coefficients, ${\displaystyle X_{j}}$ is the thermodynamic diffusion force, ${\displaystyle X_{j}=-{\rm {\nabla }}{\frac {\mu _{j}}{T}}}$. For the isothermal perfect systems, ${\displaystyle X_{j}=-R{\frac {{\rm {\nabla }}n_{j}}{n_{j}}}}$. Therefore, the Einstein-Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion: ${\displaystyle {\frac {\partial n_{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {n_{i}}{n_{j}}}\nabla n_{j}\right)\,.}$ where ${\displaystyle D_{ij}}$ is the matrix of coefficients. The Chapman-Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation. ### Jumps on the surface and in solids Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places. Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure. The system includes several reagents ${\displaystyle A_{1},A_{2},\ldots A_{m}}$ on the surface. Their surface concentrations are ${\displaystyle c_{1},c_{2},\ldots c_{m}}$. The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free paces is ${\displaystyle z=c_{0}}$. The sum of all ${\displaystyle c_{i}}$ (including free places) is constant, the density of adsorption places b. The jump model gives for the diffusion flux of ${\displaystyle A_{i}}$ (i=1,...,n): ${\displaystyle \mathbf {J} _{i}=-D_{i}[z\nabla c_{i}-c_{i}\nabla z]\,.}$ The corresponding diffusion equation is:[9] ${\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\mathrm {div} \mathbf {J} _{i}=D_{i}[z\Delta c_{i}-c_{i}\Delta z]\,.}$ Due to the conservation law, ${\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}$ and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because ${\displaystyle (b-c)\nabla c-c\nabla (b-c)=b\nabla c}$. For two and more components the equations are nonlinear. If all particles can exchange their positions with their closest neighbours then a simple generalization gives ${\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\nabla c_{i}-c_{i}\nabla c_{j}]}$ ${\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\Delta c_{i}-c_{i}\Delta c_{j}]}$ where ${\displaystyle D_{ij}=D_{ji}\geq 0}$ is a symmetric matrix of coefficients which characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration ${\displaystyle c_{0}}$. Various versions of these jump models are also suitable for simple diffusion mechanisms in solids. ### Diffusion in porous media For diffusion in porous media the basic equations are:[14] ${\displaystyle \mathbf {J} =-D\nabla n^{m}}$ ${\displaystyle {\frac {\partial n}{\partial t}}=D\Delta n^{m}\,,}$ where D is the diffusion coefficient, n is the concentration, m>0 (usually m>1, the case m=1 corresponds to Fick's law). For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is ${\displaystyle v=-{\frac {k}{\mu }}\nabla p}$ where k is the permeability of the medium, μ is the viscosity and p is the pressure. The flux J=nv and for ${\displaystyle p\sim n^{\gamma }}$ Darcy's law gives the equation of diffusion in porous media with m=γ+1. For underground water infiltration the Boussinesq approximation gives the same equation with m=2. For plasma with the high level of radiation the Zeldovich-Raizer equation gives m>4 for the heat transfer. ## Diffusion in physics ### Elementary theory of diffusion coefficient in gases Random collisions of particles in a gas. The diffusion coefficient ${\displaystyle D}$ is the coefficient in the Fick's first law ${\displaystyle J=-D{\partial n}/{\partial x}}$, where J is the diffusion flux (amount of substance) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length]. Let us consider two gases with molecules of the same diameter d and mass m (self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient ${\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}$ where kB is the Boltzmann constant, T is the temperature, P is the pressure, ${\displaystyle \ell }$ is the mean free path, and vT is the mean thermal speed: ${\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}$ We can see that the diffusion coefficient in the mean free path approximation grows with T as T3/2 and decreases with P as 1/P. If we use for P the ideal gas law P=RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T1/2 and for given temperature it decreases with the total concentration as 1/n. For two different gases, A and B, with molecular masses mA, mB and molecular diameters dA, dB, the mean free path estimate of the diffusion coefficient of A in B and B in A is: ${\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}$ ### The theory of diffusion in gases based on Boltzmann's equation In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, ${\displaystyle f_{i}(x,c,t)}$, where t is the time moment, x is position and c is velocity of molecule of the ith component of the mixture. Each component has its mean velocity ${\displaystyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}$. If the velocities ${\displaystyle C_{i}(x,t)}$ do not coincide then there exists diffusion. In the Chapman-Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:[7] • individual concentrations of particles, ${\displaystyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}$ (particles per volume), • density of moment ${\displaystyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}$ (mi is the ith particle mass), • density of kinetic energy ${\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right)}$. The kinetic temperature T and pressure P are defined in 3D space as ${\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc}$; ${\displaystyle P=k_{\rm {B}}nT}$, where ${\displaystyle n=\sum _{i}n_{i}}$ is the total density. For two gases, the difference between velocities, ${\displaystyle C_{1}-C_{2}}$ is given by the expression:[7] ${\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\}}$, where ${\displaystyle F_{i}}$ is the force applied to the molecules of the ith component and ${\displaystyle k_{T}}$ is the thermodiffusion ratio. The coefficient D12 is positive. This is the diffusion coefficient. Four terms in the formula for C1-C2 describe four main effects in the diffusion of gases: 1. ${\displaystyle \nabla \left({\frac {n_{1}}{n}}\right)}$ describes the flux of the first component from the areas with the high ratio n1/n to the areas with lower values of this ratio (and, analogously the flux of the second component from high n2/n to low n2/n because n2/n=1-n1/n); 2. ${\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}$ describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion; 3. ${\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}$ describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient. 4. ${\displaystyle k_{T}{\frac {1}{T}}\nabla T}$ describes thermodiffusion, the diffusion flux caused by the temperature gradient. All these effects are called diffusion because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a bulk transport and differ from advection or convection. In the first approximation,[7] • ${\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}$ for rigid spheres; • ${\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}$ for repulsing force ${\displaystyle \kappa _{12}r^{-\nu }}$. The number ${\displaystyle A_{1}({\nu })}$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book[7]) We can see that the dependence on T for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration n for a given temperature has always the same character, 1/n. In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity V is the mass average velocity. It is defined through the momentum density and the mass concentrations: ${\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}$ where ${\displaystyle \rho _{i}=m_{i}n_{i}}$ is the mass concentration of the ith species, ${\displaystyle \rho =\sum _{i}\rho _{i}}$ is the mass density. By definition, the diffusion velocity of the ith component is ${\displaystyle v_{i}=C_{i}-V}$, ${\displaystyle \sum _{i}\rho _{i}v_{i}=0}$. The mass transfer of the ith component is described by the continuity equation ${\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}$ where ${\displaystyle W_{i}}$ is the net mass production rate in chemical reactions, ${\displaystyle \sum _{i}W_{i}=0}$. In these equations, the term ${\displaystyle \nabla (\rho _{i}V)}$ describes advection of the ith component and the term ${\displaystyle \nabla (\rho _{i}v_{i})}$ represents diffusion of this component. In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.[15] For the diffusion velocities in multicomponent gases (N components) they used ${\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\nabla (\ln T)\right)\,;}$ ${\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\nabla (\ln P)+\mathbf {g} _{j}\,;}$ ${\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}$ Here, ${\displaystyle D_{ij}}$ is the diffusion coefficient matrix, ${\displaystyle D_{i}^{(T)}}$ is the thermal diffusion coefficient, ${\displaystyle f_{i}}$ is the body force per unite mass acting on the ith species, ${\displaystyle X_{i}=P_{i}/P}$ is the partial pressure fraction of the ith species (and ${\displaystyle P_{i}}$ is the partial pressure), ${\displaystyle Y_{i}=\rho _{i}/\rho }$ is the mass fraction of the ith species, and ${\displaystyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1}$. As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes. ### Diffusion of electrons in solids Main article: Diffusion_current When the density of electrons in solids is not in equilibrium, diffusion of electrons will occur. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light is shining on one end (see right figure), electron will diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as diffusion current. Diffusion current can also be described by Fick's first law ${\displaystyle J=-D{\partial n}/{\partial x}\,,}$ where J is the diffusion current density (amount of substance) per unit area per unit time, n (for ideal mixtures) is the electron density, x is the position [length]. ## Random walk (random motion) The apparent random motion of atoms, ions or molecules explained. Substances appear to move randomly due to collisions with other substances. From the iBook "Cell Membrane Transport", free license granted by IS3D, LLC, 2014. One common misconception is that individual atoms, ions or molecules move “randomly”, which they do not. In the animation on the right, the ion on in the left panel has a “random” motion, but this motion is not random as it is the result of “collisions” with other ions. As such, the movement of a single atom, ion, or molecule within a mixture just appears to be random when viewed in isolation. The movement of a substance within a mixture by “random walk” is governed by the kinetic energy within the system that can be affected by changes in concentration, pressure or temperature. ### Separation of diffusion from convection in gases While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task. Under normal conditions, molecular diffusion dominates only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection, and to study diffusion on the larger scale, special efforts are needed. Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it will soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because the temperature inhomogeneity. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).[citation needed] In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass. ## References 1. ^ J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2, 1.1–1.10. 2. ^ S.R. De Groot, P. Mazur (1962). Non-equilibrium Thermodynamics. North-Holland, Amsterdam. 3. ^ A. Einstein (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (PDF). Ann. Phys. 17 (8): 549–560. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806. 4. ^ Diffusion Processes, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971. 5. ^ L.W. Barr (1997), In: Diffusion in Materials, DIMAT 96, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9. 6. ^ a b H. Mehrer; N.A. Stolwijk (2009). "Heroes and Highlights in the History of Diffusion" (PDF). Diffusion Fundamentals 11 (1): 1–32. 7. S. Chapman, T. G. Cowling (1970) The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press (3rd edition), ISBN 052140844X. 8. ^ J.F. Kincaid; H. Eyring; A.E. Stearn (1941). "The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State". Chem. Rev. 28 (2): 301–365. doi:10.1021/cr60090a005. 9. A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". Mathematical Modelling of Natural Phenomena 6 (5): 184–262. arXiv:1012.2908. doi:10.1051/mmnp/20116509. 10. ^ a b Onsager, L. (1931). "Reciprocal Relations in Irreversible Processes. I". Physical Review 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405. 11. ^ L.D. Landau, E.M. Lifshitz (1980). Statistical Physics. Vol. 5 (3rd ed.). Butterworth-Heinemann. ISBN 978-0-7506-3372-7. 12. ^ S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, ISBN 0815320515. 13. ^ T. Teorell (1935). "Studies on the "Diffusion Effect" upon Ionic Distribution. Some Theoretical Considerations". Proceedings of the National Academy of Sciences of the United States of America 21 (3): 152–61. Bibcode:1935PNAS...21..152T. doi:10.1073/pnas.21.3.152. PMC 1076553. PMID 16587950. 14. ^ J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033. 15. ^ S. H. Lam (2006). "Multicomponent diffusion revisited" (PDF). Physics of Fluids 18 (7): 073101. Bibcode:2006PhFl...18g3101L. doi:10.1063/1.2221312. 16. ^ D. Ben-Avraham and S. Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems (PDF). Cambridge University Press. ISBN 0521622786. 17. ^ Weiss, G. (1994). Aspects and Applications of the Random Walk. North-Holland. ISBN 0444816062.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 130, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102715253829956, "perplexity": 1025.4537996989313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824757.62/warc/CC-MAIN-20160723071024-00212-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.deepmind.com/publications/exact-sampling-of-determinantal-point-processes-with-sublinear-time-preprocessing	Publication Exact sampling of determinantal point processes with sublinear time preprocessing Abstract We study the complexity of sampling from a distribution over all index subsets of the set {1,...,n} with the probability of a subset S proportional to the determinant of the submatrix LS of some n×n p.s.d. matrix L, where LS corresponds to the entries of L indexed by S. Known as a determinantal point process, this distribution is used in machine learning to induce diversity in subset selection. In practice, we often wish to sample multiple subsets S with small expected size k=E[\|S\|]n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose, we propose a new algorithm which, given access to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is npoly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We also give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with npoly(k) time preprocessing. Authors' notes	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9517124891281128, "perplexity": 533.7789655541318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572304.13/warc/CC-MAIN-20220816120802-20220816150802-00604.warc.gz"}
http://mathoverflow.net/questions/46479/multiple-factors-of-the-character-of-a-representation/46484	# Multiple factors of the character of a representation In algebra, various objects admit a unique decomposition into irreducible elements. For instance integers $n\ge1$, univariate polynomials $p\in k[X]$ (even multivariate ones), or characters in representation theory of finite groups. In each situation, an irreducible occurs with a multiplicity. It is interesting, from a theoretical point of view, to have a reduction to the situation where every multiplicity is $1$ (or $0$ if you insist to write the product/sum with all irreducibles of the structure). This can be done explicitly in the case of polynomials, by dividing $p$ by the g.c.d. of $p$ and $p'$, the latter being calculated with the help of the Euclid algorithm. Is there something similar for characters in representation theory of finite groups ? Suppose we know only the cardinals of conjugacy classes of $G$, together with the table of multiplication of these classes. But we don't know the table of characters. Given a character $\chi$, is it possible for instance to split it as a sum $\chi_1+\cdots+\chi_r$, where $\chi_\ell$ gathers the irreducible characters entering in $\chi$ with multiplicity $\ell$ ? Perhaps the question should be restricted to complex characters; who knows ? Even a weaker property could be interesting, provided it is associated with a finite-time algorithm. Of course, I have in mind to apply such a property to the regular representation. Then $\chi_\ell$ would be $\ell$ times the sum of irreducible characters of degree $\ell$. - At least we can do this with REPRESENTATIONS. In fact, if $V$ is a representation of $G$, then $\mathrm{End}_G\left(V\right)$ is a direct product of matrix rings $M_{n_i}\left(\mathbb C\right)$ (with $n_i$ being the multiplicities of the irreps in $V$), and $V$ can be seen as a direct sum of their standard representations $\mathbb C^{n_i}$. Now, while it is hard (if possible at all) to actually (algorithmically) decompose the ring $\mathrm{End}_G\left(V\right)$ into the direct product $M_{n_i}\left(\mathbb C\right)$, we can still find our which part of $V$ is the direct sum of all ... – darij grinberg Nov 18 '10 at 13:40 ... irreps of multiplicity $1$ - namely, this is the subset $V_1$ of $V$ consisting of all $v\in V$ such that $\left(AB-BA\right)v=0$ for all $A,B\in\mathrm{End}_G\left(V\right)$. Why? Because it is the subset of $V$ on which all $M_{n_i}\left(\mathbb C\right)$ with $n_i\geq 2$ act as zero. Similarly we can obtain the direct sum of all irreps of multiplicity $\leq 2$ in $V$ - it is the subset $V_2$ of $V$ consisting of all $v\in V$ such that ... – darij grinberg Nov 18 '10 at 13:43 ... $\left(ABCD–BACD−ABDC+BADC−ACBD+CABD+ACDB\right.$ $\left. –CADB+ADBC–DABC−ADCB+DACB+CDAB−CDBA−DCAB+DCBA \right.$ $\left.–BDAC+BDCA+DBAC–DBCA+BCAD−BCDA−CBAD+CBDA\right)v=0$ for all $A,B,C,D\in \mathrm{End}_G\left(V\right)$. What I used here is the Amitsur-Levitzki identity, or, more precisely, the fact that it is an identity for $M_2$ but not for $M_3$. Similarly we can find, for each $k\in \mathbb N$, the direct sum of all irreps which occur with multiplicity $\leq k$ in $V$. This is not exactly the direct sum of all irreps with multiplicity $= k$ in $V$, but now you can – darij grinberg Nov 18 '10 at 13:45 ... split $V_{k-1}$ away from $V_k$ by using Maschke's theorem (totally constructive) and get it. Now, of course, this is not what you are asking for because you want to do it with characters rather than representations. Is there a way to construct a representation from its character constructively, without decomposing it into irreducibles? – darij grinberg Nov 18 '10 at 13:47 Oh, and I'm working over an algebraically closed field of characteristic $0$ all the time. Over a non-algebraically closed one, I don't think we can do it algorithmically. In fact, consider a cyclic group. Whether some irreps of degree $1$ over $\overline K$ are actually defined over $K$ or not depends on whether $K$ has some roots of unity, which we cannot know per se. But if we allow a roots-of-unity oracle, then I suspect that we can do pretty much all of representation theory constructively, including computing all the irreps. – darij grinberg Nov 18 '10 at 13:53 If you wish to speak in terms of algorithms, then as Jim says, the multiplication on $A = Z(\mathbb{C}G)$ determines the character table. Use the basis of class sums, and for each class sum $C$, write down the matrix (in terms of the standard basis) for the linear transformation of $A$ given by multiplication by $C$, say $M(C)$. Simultaneously diagonalizing the $M(C)$ (which theory tells us can be done) leads to a basis of mutually orthogonal idempotents of $A$. This easily yields the character table, which allows decomposition of all characters. – Geoff Robinson Dec 1 '14 at 23:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9381550550460815, "perplexity": 121.6985527486548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257821671.5/warc/CC-MAIN-20160723071021-00266-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/volume-of-tetrahedron.397463/	# Volume of tetrahedron 1. Apr 22, 2010 1. The problem statement, all variables and given/known data Find the volume of the tetrahedron with vertices at $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$ 3. The attempt at a solution I worked out the triple integral and found out that the volume is $\frac{1}{6}$? Is this correct? I know there is probably a much quicker way working the volume by just using the volume formula for tetrahedron. However, I am not sure which value to substitute to the formula, so could you just tell me whether this answer is right or not? Thanks! 2. Apr 22, 2010 ### Mr.Miyagi The vertices you give do not make a tetrahedron. Try to draw the points in three dimensions. You'll see the volume is just half that of a cube with length 1. So the volume you're seeking should be 1/2. 3. Apr 22, 2010 ### Dick For a tetrahedron, like a cone, the area is (1/3)(area of the base)height. So (1/3)(1/2)1=1/6, yes. 4. Apr 22, 2010 ### HallsofIvy Staff Emeritus Mr. Miyagi is wrong. That is in fact a tetrahedron, it is 1/6 of a cube, not 1/2, and its volume is, indeed, 1/6. More generally, the volume of the tetrahedron is with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c) is (abc)/6. 5. Apr 22, 2010 ### Mr.Miyagi Ugh, sorry about that... Is it too late to claim temporary insanity? :uhh: Thanks for correcting it so quickly. Similar Discussions: Volume of tetrahedron	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8477517366409302, "perplexity": 724.200827603239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825057.91/warc/CC-MAIN-20171022022540-20171022042540-00766.warc.gz"}
https://math.stackexchange.com/questions/1416324/applying-the-definition-of-lebesgue-integral-to-specific-functions	# Applying the definition of Lebesgue Integral to specific functions I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I have in mind. 1. I know that the integral of $f(x) = x^{-\frac{1}{2}}$ over $[0,1]$ cannot be obtained through Riemann Integration. Fair enough, but how do we actually compute it through Lebesgue Integration? [Here I mean, from the very basics of the theory, without using calculus shortcuts, with the addition that we are allowed to do this from some theorem] 2. I (sort of know) that we cannot compute the integral of a gaussian function $e^{-x^{2}}$ by using standard Riemann techniques. Can the Lebesgue integral accomplish it? If so, how? PS: I do hope what I wrote make sense. If it is not the case, please feel free to point me out any conceptual mistake. • The improper Riemann Integral exists. But I take it that you mean strictly Riemann integrable and not in the improper sense that all of us know, love, then take for granted. – Mark Viola Aug 31 '15 at 21:52 • Indeed, you are right. :) – Kolmin Aug 31 '15 at 21:58 let $f(x) = x^{-\frac{1}{2}}$ on $(0,1]$ and $f(0)=0$. The $f$ is defined on $[0,1]$. Now define $f_n(x)=f(x)\cdot\chi _{(\frac{1}{n},1]}(x)$, Then, $f_n\nearrow f$ a.e. and so the Monotone Convergence Theorem applies to say that $\int_{[0,1]}fdt=\lim_{n\to \infty}\int _{[0,1]}f_ndt$ and now just observe that the RHS of this is the usual (convergent) Improper Riemann integral.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9881908893585205, "perplexity": 253.06589339527952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314641.41/warc/CC-MAIN-20190819032136-20190819054136-00184.warc.gz"}
http://proceedings.mlr.press/v37/sharma15.html	On Greedy Maximization of Entropy Dravyansh Sharma, Ashish Kapoor, Amit Deshpande Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:1330-1338, 2015. Abstract Submodular function maximization is one of the key problems that arise in many machine learning tasks. Greedy selection algorithms are the proven choice to solve such problems, where prior theoretical work guarantees (1 - 1/e) approximation ratio. However, it has been empirically observed that greedy selection provides almost optimal solutions in practice. The main goal of this paper is to explore and answer why the greedy selection does significantly better than the theoretical guarantee of (1 - 1/e). Applications include, but are not limited to, sensor selection tasks which use both entropy and mutual information as a maximization criteria. We give a theoretical justification for the nearly optimal approximation ratio via detailed analysis of the curvature of these objective functions for Gaussian RBF kernels.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9302904009819031, "perplexity": 391.0124957326011}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056900.32/warc/CC-MAIN-20210919190128-20210919220128-00319.warc.gz"}
https://www.codecogs.com/library/maths/calculus/differential/the-d-operator.php	I have forgotten • https://me.yahoo.com # The D operator Solving Differential Equations using the D operator View other versions (5) ## Theory Of Differential Operator (differential Module) ### Definition A differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science). The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include: and if generalize Note is an operator and must therefore always be followed by some expression on which it operates. ### Simple Equivalents • means but • Similarly and ## The D Operator And The Fundamental Laws Of Algebra The following differential equation: may be expressed as: or This can be factorised to give: Examples But is it justifiable to treat D in this way? Algebraic procedures depend upon three laws. • The Distributive Law: • The Commutative Law: • The Index Law: If D satisfies these Laws, then it can be used as an Algebraic operator(or a linear operator). However: • only when u is a constant. Thus we can see that D does satisfy the Laws of Algebra very nearly except that it is not interchangeable with variables. In the following analysis we will write are constants and is a positive integer. As has been seen, we can factorise this or perform any operation depending upon the fundamental laws of Algebra. We can now apply this principle to a number of applications. ## The Use Of The D Operator To Find The Complementary Function For Linear Equations It is required to solve the following equations: Example: ##### Example - Simple example Problem Solve the following equation:- Workings Using the D operator this can be written as:- Solution Integrating using as the factor ## Three Useful Formulae Based On The Operator D ### Equation A Let represent a polynomial function Since and From which it can be seen that: Example: ##### Example - Equation A example Problem Workings This can be re-written as: Solution We can put D = 4 ### Equation B Where is any function of x Applying Leibniz's theorem for the differential coefficient of a product. Similarly and so on therefore Example: ##### Example - Equation B example Problem Find the Particular Integral of: Workings We have used D as if it were an algebraic constant but it is in fact an operator where Solution ### Equation C - Trigonometrical Functions And so on Therefore similarly Example: ##### Example - Trigonometric example Problem Find the Particular Integral of:- Workings This can be re-written as:- Using equation 1 we can put If we multiply the top and bottom of this equation by But Solution But since ## Linear First Order D Equations With Constant Coefficients These equations have on the right hand side This equation is Using an Integrating Factor of the equation becomes:- Which is the General Solution. ## Linear Second Order D Equations With Constant Coefficients Where are the roots of the quadratic equation. i.e. the auxiliary equation. Where is an arbitrary Constant This equation can be re-written as:- Integrating • Thus when we can write the General Solution as:- Where A and B are arbitrary Constants. Example: ##### Example - Linear second order example Problem Workings The roots of this equation are:- Therefore the General Solution is • The Special Case where From Equation (41) or • The roots of the Auxiliary Equation are complex. If the roots of the are complex then the General Solution will be of the form , and the solution will be given by:- Solution The roots of this equation are :- ## Physical Examples Example: ##### Example - Small oscilations Problem Show that if satisfies the differential equation with k < n and if when The complete period of small oscillations of a simple pendulum is 2 secs. and the angular retardation due to air resistance is 0.04 X the angular velocity of the pendulum. The bob is held at rest so the the string makes a small angle with the downwards vertical and then let go. Show that after 10 complete oscillations the string will make an angle of about 40' with the vertical.(LU) Workings Using the "D" operator we can write When t = 0 = 0 and = 0 and Solution At t = 0 We have been given that k = 0.02 and the time for ten oscillations is 20 secs.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 71, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.984228789806366, "perplexity": 1026.2478342358236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363791.16/warc/CC-MAIN-20211209091917-20211209121917-00394.warc.gz"}
http://www.mathematicalfoodforthought.com/	## Saturday, July 7, 2007 ### What's Your Function? Topic: Algebra. Level: AMC/AIME. Problem: Given two positive reals $\alpha$ and $\beta$, show that there is a continuous function $f$ that satisfies $f(x) = f(x+\alpha)+f(x+\beta)$. Solution: There are several special cases that are interesting to look at before we make a guess as to what type of function $f$ will be. First we consider the case $\alpha = \beta = 1$. This immediately gives $f(x) = 2f(x+1)$. It should not be too difficult to guess that $f(x) = 2^{-x}$ is a solution to this, as well as any constant multiple of it. Now try $\alpha = -1$ and $\beta = -2$, resulting in $f(x) = f(x-1)+f(x-2)$. Looks a lot like Fibonacci, right? In fact, one possible function is just $f(x) = \phi^x$. That's pretty convenient. Notice how both of these illuminating examples are exponential functions, which leads us to guess that our function will be exponential as well. So, following this track, we set $f(x) = a^x$ so we simply need to solve $a^x = a^{x+\alpha}+a^{x+\beta}$ $a^x = a^x\left(a^{\alpha}+a^{\beta}\right)$. Unfortunately, the equation $a^{\alpha}+a^{\beta} = 1$ does not always have a solution (take $\alpha = 1$ and $\beta = -1$ for example). But that's ok and I'll worry about it some other time. In any case we have found a function for whenever the equation $a^{\alpha}+a^{\beta} = 1$ has a solution in the reals. ## Friday, June 29, 2007 ### Addition At Its Finest. Topic: Calculus/S&S. Problem: Evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{n(n+1)}$ where $x$ is a real number with $\|x\| < 1$. Solution: Looking at that all too common denominator, we do a partial fraction decomposition in hopes of telescoping series. The summation becomes $\displaystyle \sum_{n=1}^{\infty} \left(\frac{x^n}{n}-\frac{x^n}{n+1}\right)$. Common Taylor series knowledge tells us that $\displaystyle \ln{(1-x)} = -\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots\right) = -\sum_{n=1}^{\infty} \frac{x^n}{n}$, which convenient fits the first part of the summation. As for the second part, we get $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{n+1} = \frac{1}{x} \sum_{n=1}^{\infty} \frac{x^{n+1}}{n+1} = \frac{-\ln{(1-x)}-x}{x}$ from the same Taylor series. Combining the results, our answer is then $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{n(n+1)} = 1-\ln{(1-x)}+\frac{\ln{(1-x)}}{x}$. QED. -------------------- Comment: Even though the trick at the beginning didn't actually get much to telescope, the idea certainly made it easier to recognize the Taylor series. Algebraic manipulations are nifty to carry around and can be applied in problems wherever you go. -------------------- Practice Problem: Show that $\displaystyle \int_0^{\frac{\pi}{2}} \ln{(\tan{x})} = 0$. ## Monday, June 18, 2007 ### The Smaller The Better. Topic: Calculus. Problem: Given a complicated function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, find an approximate local minimum. Solution: The adjective complicated is only placed so that we assume there is no easy way to solve $\bigtriangledown f = 0$ to immediately give the solution. We seek an algorithm that will lead us to a local minimum (hopefully a global minimum as well). We start at an arbitrary point $X_0 = (x_1, x_2, \ldots, x_n)$. Consider the following process (for $k = 0, 1, 2, \ldots$), known as gradient descent: 1. Calculate (approximately) $\bigtriangledown f(X_k)$. 2. Set $X_{k+1} = X_k - \gamma_k \bigtriangledown f(X_k)$, where $\gamma_k$ is a constant that can be determined by a linesearch. It is well-known that the direction of the gradient is the direction of maximum increase and the direction opposite the gradient is the direction of maximum decrease. Hence this algorithm is based on the idea that we always move in the direction that will decrease $f$ the most. Sounds pretty good, right? Well, unfortunately gradient descent converges very slowly so it is only really useful for smaller optimization problems. Fortunately, there exist other algorithms but obviously they are more complex, such as the nonlinear conjugate gradient method or Newton's method, the latter of which involves the computation of the inverse of the Hessian matrix, which is a pain. ## Wednesday, June 6, 2007 ### Colorful! Topic: Calculus. Theorem: (Green's Theorem) Let $R$ be a simply connected plane region whose boundary is a simple, closed, piecewise smooth curve $C$ oriented counterclockwise. If $f(x, y)$ and $g(x, y)$ are continuous and have continuous first partial derivatives on some open set containing $R$, then $\displaystyle \oint_C f(x, y) dx + g(x, y) dy = \int_R \int \left(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right) dA$. -------------------- Problem: Evaluate $\displaystyle \oint_C x^2y dx + (y+xy^2) dy$, where $C$ is the boundary of the region enclosed by $y = x^2$ and $x = y^2$. Solution: First, verify that this region satisfies all of the requirements for Green's Theorem - indeed, it does. So we may apply the theorem with $f(x, y) = x^2y$ and $g(x, y) = y+xy^2$. From these, we have $\frac{\partial g}{\partial x} = y^2$ and $\frac{\partial f}{\partial y} = x^2$. Then we obtain $\displaystyle \oint_C x^2y dx + (y+xy^2) dy = \int_R \int (y^2-x^2) dA$. But clearly this integral over the region $R$ can be represented as $\displaystyle \int_0^1 \int_{x^2}^{\sqrt{x}} (y^2-x^2) dy dx$, so it remains a matter of calculation to get the answer. First, we evaluate the inner integral to get $\displaystyle \int_0^1 \int_{x^2}^{\sqrt{x}} (y^2-x^2) dy dx = \int_0^1 \left[\frac{y^3}{3}-x^2y\right]_{x^2}^{\sqrt{x}} dx = \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx$. Then finally we have $\displaystyle \int_0^1 \left(\frac{x^{3/2}}{3}-x^{5/2}-\frac{x^6}{3}+x^4\right)dx = \left[\frac{2x^{5/2}}{15}-\frac{2x^{7/2}}{7}-\frac{x^7}{21}+\frac{x^5}{5}\right]_0^1 = \frac{2}{15}-\frac{2}{7}-\frac{1}{21}+\frac{1}{5} = 0$. QED. -------------------- Comment: To me, Green's Theorem is a very interesting result. It's not at all obvious that a line integral along the boundary of a region is equivalent to an integral of some partial derivatives in the region itself. A simplified proof of the result can be obtained by proving that $\displaystyle \oint_C f(x, y) dx = -\int_R \int \frac{\partial f}{\partial y} dA$ and $\displaystyle \oint_C g(x, y) dy = \int_R \int \frac{\partial g}{\partial x} dA$. -------------------- Practice Problem: Let $R$ be a plane region with area $A$ whose boundary is a piecewise smooth simple closed curve $C$. Show that the centroid $(\overline{x}, \overline{y})$ of $R$ is given by $\displaystyle \overline{x} = \frac{1}{2A} \oint_C x^2 dy$ and $\displaystyle \overline{y} = -\frac{1}{2A} \oint_C y^2 dx$. ## Monday, June 4, 2007 ### More Integrals... whine. Topic: Calculus. Definition: (Jacobian) If $T$ is the transformation from the $uv$-plane to the $xy$-plane defined by the equations $x = x(u, v)$ and $y = y(u, v)$, then the Jacobian of $T$ is denoted by $J(u, v)$ or by $\partial(x, y)/\partial(u, v)$ and is defined by $J(u, v) = \frac{\partial(x, y)}{\partial(u, v)} = \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} - \frac{\partial y}{\partial u} \cdot \frac{\partial x}{\partial v}$, i.e. the determinant of the matrix of the partial derivatives (also known as the Jacobian matrix). Naturally, this can be generalized to more variables. -------------------- Theorem: If the transformation $x = x(u, v)$, $y = y(u, v)$ maps the region $S$ in the $uv$-plane into the region $R$ in the $xy$-plane, and if the Jacobian $\partial(x, y)/\partial(u, v)$ is nonzero and does not change sign on $S$, then (with appropriate restrictions on the transformation and the regions) it follows that $\displaystyle \int_R \int f(x, y) dA_{xy} = \int_S \int f(x(u, v), y(u, v)) \left\|\frac{\partial(x, y)}{\partial(u, v)} \right\| dA_{uv}$. -------------------- Problem: Evaluate $\displaystyle \int_R \int e^{(y-x)/(y+x)} dA$, where $R$ is the region in the first quadrant enclosed by the trapezoid with vertices $(0, 1); (1, 0); (0, 4); (4, 0)$. Solution: The bounding lines can be written as $x = 0$, $y = 0$, $y = -x+1$, and $y = -x+4$. Now consider the transformation $u = y+x$ and $v = y-x$. In the $uv$-plane, the bounding lines of the new region $S$ can now be written as $u = 1$, $u = 4$, $v = u$, and $v = -u$. We can write $x$ and $y$ as functions of $u$ and $v$: simply $x = \frac{u-v}{2}$ and $y = \frac{u+v}{2}$. So the Jacobian $\displaystyle \frac{\partial(x, y)}{\partial(u, v)} = \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} - \frac{\partial y}{\partial u} \cdot \frac{\partial x}{\partial v} = \frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \left(-\frac{1}{2} \right) = \frac{1}{2}$. Then our original integral becomes $\displaystyle \int_R \int e^{(y-x)/(y+x)} dA = \frac{1}{2} \int_S \int e^{v/u} dA$. And this is equivalent to $\displaystyle \frac{1}{2} \int_S \int e^{v/u} dA = \frac{1}{2} \int_1^4 \int_{-u}^u e^{v/u} dv du = \frac{1}{2} \int_1^4 \big[ u e^{v/u} \big]_{v=-u}^u du = \frac{1}{2} \int_1^4 u\left(e-\frac{1}{e}\right) du = \frac{15}{4}\left(e-\frac{1}{e}\right)$. QED. -------------------- Comment: Note that the above theorem is probably very important in multivariable calculus, as it is the equivalent to $u$-substitution in one variable, which we all know is the ultimate integration technique. It functions in the same way, giving you a lot more flexibility on the function you are integrating and the region you are integrating on. -------------------- Practice Problem: Evaluate $\displaystyle \int_R \int (x^2-y^2) dA$, where $R$ is the rectangular region enclosed by the lines $y = -x$, $y = 1-x$, $y = x$, $y = x+2$. ## Monday, May 28, 2007 ### One By One, We're Making It Fun. Topic: Calculus/S&S. Theorem: (Stolz-Cesaro) Let $\{a_n\}$ and $\{b_n\}$ be sequences of real numbers such that $\{b_n\}$ is positive, strictly increasing, and unbounded. If the limit $\displaystyle \lim_{n \rightarrow \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n} = L$ exists, then the following limit also exists and we have $\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L$. -------------------- Theorem: (Summation by Parts) If $\{f_k\}$ and $\{g_k\}$ are two sequences, then $\displaystyle \sum_{k=m}^n f_k(g_{k+1}-g_k) = [f_{n+1}g_{n+1}-f_mg_m]-\sum_{k=m}^n g_{k+1}(f_{k+1}-f_k)$. -------------------- Problem: Let $\{a_n\}$ be a sequence of real numbers such that $\displaystyle \sum_{k=0}^{\infty} a_k$ converges. Show that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n+1} \sum_{k=0}^n k \cdot a_k = 0$. Solution: Define the sequence $\{b_n\}$ by $\displaystyle b_n = \sum_{k=0}^n a_k$ and let $\displaystyle \lim_{n \rightarrow \infty} b_n = L$. Then, by summation by parts with $\{f_n\} = \{n\}$ and $\{g_n\} = \{b_n\}$, we have $\displaystyle \sum_{k=0}^n k \cdot a_k = \sum_{k=0}^n k \cdot (b_{k+1}-b_k) = (n+1)b_{n+1}-\sum_{k=0}^n b_{k+1}$. The summation we wish to take the limit of is then $\displaystyle \frac{1}{n+1} \sum_{k=0}^n k \cdot a_k = b_{n+1}-\frac{1}{n+1} \sum_{k=0}^n b_{k+1}$. But since, by Stolz-Cesaro, $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n+1} \sum_{k=0}^n b_{k+1} = \lim_{n \rightarrow \infty} b_{n+1} = L$, we obtain $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n+1} \sum_{k=0}^n k \cdot a_k = \lim_{n \rightarrow \infty} \left(b_n-\frac{1}{n+1} \sum_{k=0}^n b_{k+1}\right) = L-L = 0$. QED. -------------------- Comment: Summation by parts is a very useful technique to change sums around so that they are easier to evaluate. If you hadn't noticed, it is the discrete analogue of integration by parts and is in fact very similar. Stolz-Cesaro is powerful as well and seems like a discrete analogue to L'Hopital (but I'm not sure about this one). Applying well-known calculus ideas to discrete things can often turn into neat results. -------------------- Practice Problem: If $\{a_n\}$ is a decreasing sequence such that $\displaystyle \lim_{n \rightarrow \infty} a_n = 0$, show that $\displaystyle \sum_{k=1}^{\infty} a_k \cdot \sin{(kx)}$ converges for all $x$. ## Saturday, May 12, 2007 ### Bigger Means Better. Topic: Algebra/Inequalities/Sets. Level: Olympiad. Definition: A set $S$ is said to be convex if, for any $X, Y \in S$ and $\lambda \in [0,1]$, we have $\lambda X+(1-\lambda)Y \in S$. -------------------- Definition: Let $f: D \rightarrow \mathbb{R}$ be a real-valued function defined on a convex set $D$. We say that $f$ is convex if, for all $X, Y \in D$ and $\lambda \in [0, 1]$, we have $\lambda f(X) + (1-\lambda) f(Y) \ge f(\lambda X+(1-\lambda)Y)$. -------------------- Theorem: (Jensen's Inequality) Let $f$ be a real-valued, convex function defined on a convex set $D$. Given $x_1, x_2, \ldots, x_n \in D$ and nonnegative reals $w_1, w_2, \ldots, w_n$ such that $w_1+w_2+\cdots+w_n = 1$, we have $w_1f(x_1)+w_2f(x_2)+\cdots+w_nf(x_n) \ge f(w_1x_1+w_2x_2+\cdots+w_nx_n)$ or, more concisely, $\displaystyle \sum_{i=1}^n w_if(x_i) \ge f\left(\sum_{i=1}^n w_ix_i\right)$. Proof: We proceed by induction on $n$. Base Case - $n = 1$, we have $f(x_1) \ge f(x_1)$ which is trivially true. $n = 2$, we have $\lambda_1 f(x_1)+\lambda_2 f(x_2) \ge f(\lambda_1 x_1+\lambda_2 x_2)$ with $\lambda_1+\lambda_2 = 1$ which is true by the definition of convexity. Induction Step - Suppose $\displaystyle \sum_{i=1}^k w_if(x_i) \ge f\left(\sum_{i=1}^k w_ix_i\right)$. We wish to show $\displaystyle \sum_{i=1}^{k+1} u_if(x_i) \ge f\left(\sum_{i=1}^{k+1} u_ix_i\right)$ for nonnegative reals $u_1, u_2, \ldots, u_{k+1}$ that sum to $1$. Well, $\displaystyle f\left(\sum_{i=1}^{k+1} u_ix_i\right) \le u_{k+1}f(x_{k+1})+(1-u_{k+1})f\left(\frac{1}{1-u_{k+1}} \sum_{i=1}^k u_ix_i\right)$ by the definition of convexity. But since $\displaystyle \frac{1}{1-u_{k+1}} \sum_{i=1}^k u_i = \sum_{i=1}^k \frac{u_i}{1-u_{k+1}} = 1$, by our inductive hypothesis (the $k$ term case) we must have $\displaystyle f\left(\frac{1}{1-u_{k+1}} \sum_{i=1}^k u_ix_i\right) \le \sum_{i=1}^k \frac{u_i}{1-u_{k+1}} f(x_i)$. Combining this into the above inequality, we obtain $\displaystyle f\left(\sum_{i=1}^{k+1} u_ix_i\right) \le u_{k+1}f(x_{k+1})+(1-u_{k+1})\sum_{i=1}^k \frac{u_i}{1-u_{k+1}} f(x_i) = \sum_{i=1}^{k+1} u_if(x_i)$ as desired, completing the induction. QED. -------------------- Definition: The convex hull of a set $S$ is defined to be the smallest convex set containing $S$. It is the set of all points that can be written as $\displaystyle \sum_{i=1}^n a_ix_i$, where $n$ is a natural number, $a_1, a_2, \ldots, a_n$ are nonnegative reals that sum to $1$, and $x_1, x_2, \ldots, x_n \in S$. -------------------- Definition: A vertex of a set $D$ is a point $v \in D$ such that for all $x \in D$ not equal to $v$ and $\lambda > 1$ we have $\lambda v+(1-\lambda)x \not\in D$. -------------------- Theorem: Let $V_D = \{v_1, v_2, \ldots, v_k\}$ be the set of vertices of a compact convex set $D$. The convex hull of $V_D$ is $D$. Example: The set of vertices of a convex polygon is, in fact, its vertices as traditionally defined in geometry. Any point inside the polygon can be written as a convex combination of its vertices. -------------------- Theorem: If $f$ is a real-valued, convex function defined on a compact convex set $D$, then $\displaystyle \max_{x \in D} f(x) = \max_{x \in V_D} f(x)$, where $V_D$ is the set of vertices of $D$. Proof: We will show that $\displaystyle f(x) \le \max_{x \in V_D} f(x)$ for all $x \in D$. Let $\displaystyle x = \sum_{i=1}^k \lambda_i v_i$ where $\lambda_1, \lambda_2, \ldots, \lambda_k$ are nonnegative reals that sum to $1$ and $V_D = \{v_1, v_2, \ldots, v_k\}$. This is possible by the preceding theorem. Then by Jensen's Inequality we get $\displaystyle \sum_{i=1}^k \lambda_i f(v_i) \ge f\left(\sum_{i=1}^k \lambda_i v_i\right) = f(x)$. And since $\displaystyle \max_{x \in V_D} f(x) \ge \sum_{i=1}^k \lambda_i f(v_i)$, we have $\displaystyle \max_{x \in V_D} f(x) \ge f(x)$ for all $x \in D$. Furthermore, since $V_D$ is a subset of $D$, we know that this maximum is attained, thus $\displaystyle \max_{x \in D} f(x) = \max_{x \in V_D} f(x)$ as desired. QED. -------------------- Comment: This is a very useful result, as it allows us to limit our search for the maximum of a function to its set of vertices, which is usually a considerably smaller set, though it may still be infinite (think sphere). In any case, this works well for constrained optimization problems in which the domain is limited to a polygon, the easiest application of this theorem. -------------------- Practice Problem: Let $x$ and $y$ be real numbers satisfying $\|2x-y\| \le 3$ and $\|y-3x\| \le 1$. Find the maximum value of $f(x, y) = (x-y)^2$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9856210947036743, "perplexity": 126.14602904389488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251705142.94/warc/CC-MAIN-20200127174507-20200127204507-00174.warc.gz"}
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-1	Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki. Zapraszamy na https://bibliotekanauki.pl PL EN Preferencje Język Widoczny [Schowaj] Abstrakt Liczba wyników • # Artykuł - szczegóły ## Fundamenta Mathematicae 2006 \| 189 \| 2 \| 99-109 ## z⁰-Ideals and some special commutative rings EN ### Abstrakty EN In a commutative ring R, an ideal I consisting entirely of zero divisors is called a torsion ideal, and an ideal is called a z⁰-ideal if I is torsion and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We prove that in large classes of rings, say R, the following results hold: every z-ideal is a z⁰-ideal if and only if every element of R is either a zero divisor or a unit, if and only if every maximal ideal in R (in general, every prime z-ideal) is a z⁰-ideal, if and only if every torsion z-ideal is a z⁰-ideal and if and only if the sum of any two torsion ideals is either a torsion ideal or R. We give a necessary and sufficient condition for every prime z⁰-ideal to be either minimal or maximal. We show that in a large class of rings, the sum of two z⁰-ideals is either a z⁰-ideal or R and we also give equivalent conditions for R to be a PP-ring or a Baer ring. 99-109 wydano 2006 ### Twórcy autor • Department of Mathematics, Bu Ali Sina University, Hamedan, Iran • Institute for Studies, in Theoretical Physics and Mathematics (IPM), Tehran, Iran	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8841745853424072, "perplexity": 1272.100174403514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103347800.25/warc/CC-MAIN-20220628020322-20220628050322-00664.warc.gz"}
http://math.stackexchange.com/questions/796358/question-about-paschs-postulate-line-going-through-all-three-sides-of-a-triang	# Question about Pasch's Postulate, line going through all three sides of a triangle I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which aggravates my confusion. To start, this is his exact wording for Pasch's Postulate: "The Posutulate of Pasch: Given a triangle $\triangle ABC$, and a line $L$ in the same plane. If $L$ contains a point $E$, between $A$ and $C$, then $L$ intersects either $\overline{AB}$ or $\overline{BC}$." This could mean one of two things: Either that the line $L$ must intersect exactly one of either $\overline{AB}$ or $\overline{BC}$; or that the line $L$ must intersect at least one of either $\overline{AB}$ or $\overline{BC}$ (it can't turn around and leave). When I look at the other resources (mostly wikipedia), I lean towards the former. However, in the same section, the reader is asked to prove that: "If $L$ contains no vertex of the triangle, then $L$ cannot intersect all of the three sides" from Pasch's Postulate. If the former were true, the proof of this statement would be "Pasch's Postulate QED," which seems a little bit too easy. However, I've spent hours trying to figure out how to do it with the latter defintion (it could intersect both), but I just get so confused because my lines aren't straight. I'm completely at a loss. So my questions is: Which of my two interpretations is correct? Can this problem be proved without using the former definition? EDIT: I've realized (with the help of one Andre Nicolas in the comments), that clearly the former definition doesn't work because the line could pass through $B$. However, that leaves me with the same problem: How does one go about proving that the line does not go through all three sides of the triangle without passing through a vertex? - The line could intersect both by going through $B$. So it is "at least one." But usually only one. – André Nicolas May 15 '14 at 20:58 @Andre That's a good point. Which suggests the latter interpretation is correct, but then I'm back to this problem I cannot for the life of me solve. How do I prove it doesn't go through both of the others, and not pass through a vertex? – weirdesky May 15 '14 at 21:04 In the instructor's manual for the fourth edition of Greenberg's Euclidean and Non-Euclidean Geometry, there is a list of about 73 statements that are equivalent to the parallel postulate, in the presence of the rest of the axioms presented. There may be several things called Pasch's postulate, not sure. Meanwhile, I do not have Moise's book, so I cannot immediately help. – Will Jagy May 15 '14 at 21:22 I do not know what axioms of betweeness Moise uses. The proof will lie there. – André Nicolas May 15 '14 at 23:10 David Hilbert uses Pasch's axiom in his book Foundations of Geometry (1902 – English translation by E. J. Townsend, 1950) . It is numbered II.5 and is stated as: Let $A, B, C$ be three points that do not lie on a line and let $\mathit a$ be a line in the plane $ABC$ which does not meet any of the points $A, B,C$. If the line a passes through a point of the segment $\overline {AB}$, it also passes through a point of the segment $\overline {AC}$, or through a point of segment $\overline {BC}$. The fact that both segments $\overline {AC}$ and $\overline {BC}$ are not intersected by the line $\mathit a$ is proved in Supplement I,1, written by P. Bernays. We stay with a “standard” reading of Pasch’s axiom, considering the “or” as inclusive, and we have to prove that : If $A, B, C$ are noncollinear points and $\mathit l \cap \{ A,B,C \} = \emptyset$, then $\mathit l$ cannot intersect all three sides of $\Delta ABC$. Note. I will use the relation $between(x,z,y)$, $x, y, z$ points, to say that $z$ is between $x$ and $y$. Proof Suppose $\{ D \} = \mathit l \cap \overline {AB}$ (thus : $between(A, D, B)$), $\{ E \} = \mathit l \cap \overline {AC}$ (thus : $between(A, E, C)$) and $\{ F \} = \mathit l \cap \overline {BC}$ (thus : $between(B, F, C)$), and suppose $between(D, E, F)$. Now $\overline {BD} = \overline {AB}$ and $\overline {BF} = \overline {BC}$, so $B, D$, and $F$ are not collinear. Now $\overline {AC} \cap \overline {DF} = \{ E \}$, because $E \in \overline {AC} \cap \mathit l$ and if there was an $X \in \overline {AC} \cap \overline {DF}$ with $X \ne E$, then $\overline {AC}$ and $\mathit l$ would have two points in common and they would coincide (Axiom I.1). If we apply Pasch’s axiom to $\Delta DBF$ and line $\overline {AC}$, we must have either : $\overline {AC} \cap \overline {BD} \ne \emptyset$ or $\overline {AC} \cap \overline {BF} \ne \emptyset$. But $\overline {AC} \cap \overline {BD} \subset \overline {AC} \cap \overline {BA} = \{ A \}$, and $A \notin \overline {BD}$ (since $between(A, D, B)$); so $\overline {AC} \cap \overline {BD} = \emptyset$. Also, $\overline {AC} \cap \overline {BF} \subset \overline {AC} \cap \overline {BC} = \{ C \}$, and $C \notin \overline {BF}$ (since $between(B, F, C)$); so $\overline {AC} \cap \overline {BF} = \emptyset$. Hence our assumptions contradict Pasch’s axiom. - Yep, thank you. I found exactly this proof in "Hilbert's Axioms of Plane Order" by Wylie when the site was down. It took me hours (literally) to find it. Very clear proof, though, thank you. I'd vote you up, but I can't. – weirdesky May 16 '14 at 13:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8352274894714355, "perplexity": 269.11013955004245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398468971.92/warc/CC-MAIN-20151124205428-00019-ip-10-71-132-137.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/81915/history-of-the-notation-mathbb-z-n	# History of the notation $\mathbb Z_n$ This question was motivated by Martin's comment in Free $\mathbb{Z}_2$-actions match at some point When was the notation $\mathbb Z_n$ introduced for $n$-adic integers and by whom? When was it introduced for integers modulo $n$ and by whom? I tried searching on Google without much luck. Added. Martin brings up the related point when was the subscript notation $A_f$ introduced for localizing a ring at the monoid generated by $f$ (which is a third possible interpretation of $\mathbb Z_n$)? - It is pretty common to denote the cyclic group of order $n$ by $C_n\dots$ – Igor Rivin Nov 25 '11 at 22:11 I looked in Hensel's original book Theorie der Algebraischen Zahlen and he writes ${\mathbf Q}_p$ as $K(p)$ and, as far as I could tell, he has no specific notation for $p$-adic integers. He just refers to them when needed with the words "$p$-adic integers" (in German). In Hasse's book Number Theory the $p$-adic numbers are $P_p$ and the $p$-adic integers are $\Gamma_p$. – KConrad Nov 25 '11 at 23:52 Borevich-Shafarevich's book "Number Theory" (1960s) denotes the $p$-adic integers as $O_p$, $p$-adic numbers as $R_p$ (since they write $R$ for the rational numbers). Dwork's paper on rationality of the zeta-function (Amer. J. Math 1960) denotes $p$-adic numbers as $Q'$ and $p$-adic integers as ${\mathfrak O}'$. Lubin's paper on formal groups (Ann. Math. 80, 1964) uses ${\mathbf Z}_p$ for the $p$-adic integers and ${\mathbf Q}_p$ for the $p$-adic numbers. Someone should look at the first edition of Serre's Corps Locaux (1962). – KConrad Nov 26 '11 at 0:34 @Benjamin Steinberg: I think it'd be more accurate to say $C_n$ is a notation for the cyclic group abstractly (using multiplication for the group law notation, as usual in abstract groups) rather than the cyclic group multiplicatively, because many people might want to write that as $\mu_n$. – KConrad Nov 26 '11 at 1:38 I have the 1968 edition of Corps Locaux, which is a photographic reproduction of the first edition. Serre Uses ${\mathbf{Q}}_p$ for the $p$-adic numbers, and ${\mathbf{Z}}_p$ for their integers. I'm pretty sure that I learned this notation from Lang, in course I took with him at Columbia, in 1956-57. – Lubin Nov 26 '11 at 5:56 show 23 more comments I fished around in Google scholar and found so many examples that I don't feel like listing any of the links. Nonetheless, a clear picture emerges of an answer that I found a bit surprising: The notation $\mathbb{Z}_p$ for the $p$-adic integers evolved in three separate parts. I should also explain that the real science of etymology is about the evolution of words or notation, not just "when did it first happen". The subscript notation not only for the $p$-adic integers, but more generally for $p$-adic completions, already appears in several papers in the 1930s and 1940s. For instance Carl Ludwig Siegel says in 1941, "$R$ is the field of rational numbers, $R_p$ the field of $p$-adic numbers, where $p$ denotes any prime number, $R_\infty$ the field of real numbers; moreover $J$ is the ring of integral numbers and $J_p$ the ring of $p$-adic integers". Of course, no one would use this notation today! The use of $Z$ for the integers has a semi-separate history. I even found an old paper, but more recent than this one by Siegel, that used $Z$ for the integers but $R$ for the $p$-adic integers, with no subscript. Generally the notation for $p$-adic integers and $p$-adic numbers standardized at $Z_p$ and $Q_p$ in the 1950s. Quite possibly Bourbaki, Algebra, deserves credit for standardizing $Z$ and $Q$ for integers and rationals. Blackboard bold notation ($\mathbb{Z}$ and $\mathbb{Q}$) came last, at least in print. Despite its name, it's no longer obvious to me that blackboard bold actually first came from blackboards or from typewriters. It's sometimes also credited to Bourbaki, but this seems to be wrong. There is a historical account by Lee Rudolph (in comp.text.tex) that credits certain typewriter models in the 1960s for producing blackboard bold typography for the integers, etc. If that is where it started, then the notation seemed to catch on fairly quickly, although there were holdouts that used ordinary bold for decades after that. (But, before blackboard bold was fashionable, it wasn't even standard to make the set of integers bold $\mathbf{Z}$ instead of just $Z$.) As an aside, the collision of notation between the $p$-adic integers and the integers mod $p$ is unfortunate. I really prefer to write $\mathbb{Z}/n$ for the integers mod $n$, because it is then written exactly as it reads. Also, partly since it is such a commonly used object, I see no need for extra parentheses, or an extra $\mathbb{Z}$, and certainly just using an $n$ subscript is bad. I'm optimistic that this notation is the way of the future and it would be an interesting separate question in history of notation. (Sorry, I didn't see the entire string of comments before I wrote all of this. The comments make most of these remarks, but it seems useful to combine them into one historical summary.) - Did you see anything indicating when some for of Z subscript n appeared (blackboard or not) for integers mod n? The comments above seemed to have already landed the subscript notation for completions at the door of Hasse in the thirties. – Benjamin Steinberg Nov 27 '11 at 2:30 One should check what notation, if any, van der Waerden used in Moderne Algebra (1930) for the integers mod $n$ since whatever convention he chose influenced the next generations in algebra but not necessarily other parts of mathematics. (Incidentally, in his book Number Theory (1950), Hasse wrote $Z_\infty$ for the complex numbers.) – KConrad Nov 27 '11 at 2:46 I found a paper by one John Moore (1957) that uses $Z_n$ for integers mod $n$ and $Z^p$ for $p$-adic integers. So that's one solution, apparently obsolete, for the collision-of-notation problem. Now I see a couple of other papers in the 1950s. It seems that the standard of $Z$ for the integers, which appeared in that period, swept in the collision of notation immediately and unintentionally. Nakayama (1957) writes $Z(p)$ for $\mathbb{Z}/p$, integers mod $p$, while Floyd (1951) writes $I_p$ for the same. – Greg Kuperberg Nov 27 '11 at 3:01 Greg: It's okay not to give links, but could you indicate the journal and not just the year? It makes it easier to check things. – KConrad Nov 27 '11 at 3:31 – Greg Kuperberg Nov 27 '11 at 3:39 show 1 more comment	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8706421852111816, "perplexity": 579.5388811086018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011198370/warc/CC-MAIN-20140305091958-00007-ip-10-183-142-35.ec2.internal.warc.gz"}
http://www.lmfdb.org/GaloisGroup/25T21	# Properties Label 25T21 Order $$200$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $D_5\wr C_2$ ## Group action invariants Degree $n$ : $25$ Transitive number $t$ : $21$ Group : $D_5\wr C_2$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,22,3,24,5,21,2,23,4,25)(6,17,8,19,10,16,7,18,9,20)(11,12,13,14,15), (1,16,17,7,8,23,24,14,15,5)(2,6,18,22,9,13,25,4,11,20)(3,21,19,12,10) $\|\Aut(F/K)\|$: $1$ ## Low degree resolvents \|G/N\|Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 8: $D_{4}$ Resolvents shown for degrees $\leq 47$ Degree 5: None ## Low degree siblings 10T19, 10T21 x 2, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 40T167 x 2, 40T170 Siblings are shown with degree $\leq 47$ A number field with this Galois group has no arithmetically equivalent fields. ## Conjugacy Classes Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $10$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $4, 4, 4, 4, 4, 4, 1$ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $10$ $2$ $( 2,11)( 3,21)( 4, 6)( 5,16)( 7,14)( 8,24)(10,19)(13,22)(15,17)(18,25)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $20$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ $10, 10, 5$ $20$ $10$ $( 1, 2,12,13,23,24, 9,10,20,16)( 3,22,14, 8,25,19, 6, 5,17,11)( 4, 7,15,18,21)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $20$ $10$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$ $10, 10, 5$ $20$ $10$ $( 1, 3,23,25,20,17,12,14, 9, 6)( 2,13,24,10,16)( 4, 8,21, 5,18,22,15,19, 7,11)$ $5, 5, 5, 5, 5$ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$ ## Group invariants Order: $200=2^{3} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [200, 43] Character table: 2 3 2 3 2 2 1 1 1 1 1 1 . 1 1 5 2 1 . . 1 2 1 1 2 1 1 2 2 2 1a 2a 2b 4a 2c 5a 10a 10b 5b 10c 10d 5c 5d 5e 2P 1a 1a 1a 2b 1a 5b 5b 5e 5a 5a 5d 5c 5e 5d 3P 1a 2a 2b 4a 2c 5b 10c 10d 5a 10a 10b 5c 5e 5d 5P 1a 2a 2b 4a 2c 1a 2a 2c 1a 2a 2c 1a 1a 1a 7P 1a 2a 2b 4a 2c 5b 10c 10d 5a 10a 10b 5c 5e 5d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 1 X.3 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 X.4 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 X.5 2 . -2 . . 2 . . 2 . . 2 2 2 X.6 4 -2 . . . A C . A C . -1 B B X.7 4 -2 . . . A C . A C . -1 B B X.8 4 . . . -2 B . C B . C -1 A A X.9 4 . . . -2 B . C B . C -1 A A X.10 4 . . . 2 B . -C B . -C -1 A A X.11 4 . . . 2 B . -C B . -C -1 A A X.12 4 2 . . . A -C . A -C . -1 B B X.13 4 2 . . . A -C . A -C . -1 B B X.14 8 . . . . -2 . . -2 . . 3 -2 -2 A = -2E(5)-E(5)^2-E(5)^3-2E(5)^4 = (3-Sqrt(5))/2 = 1-b5 B = 2E(5)^2+2E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9237069487571716, "perplexity": 906.3805374590962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986660829.5/warc/CC-MAIN-20191015231925-20191016015425-00459.warc.gz"}
http://www.math.iisc.ac.in/seminars/2022/2022-09-07-shiva-chidambaram.html	Number Theory Seminar Venue: LH-3 The modularity lifting theorem of Boxer-Calegari-Gee-Pilloni established for the first time the existence of infinitely many modular abelian surfaces $A / \mathbb{Q}$ upto twist with $\text{End}_{\mathbb{C}}(A) = \mathbb{Z}$. We render this explicit by first finding some abelian surfaces whose associated mod-$p$ representation is residually modular and for which the modularity lifting theorem is applicable, and then transferring modularity in a family of abelian surfaces with fixed $3$-torsion representation. Let $\rho: G_{\mathbb{Q}} \rightarrow GSp(4,\mathbb{F}_3)$ be a Galois representation with cyclotomic similitude character. Then, the transfer of modularity happens in the moduli space of genus $2$ curves $C$ such that $C$ has a rational Weierstrass point and $\mathrm{Jac}(C)[3] \simeq \rho$. Using invariant theory, we find explicit parametrization of the universal curve over this space. The talk will feature demos of relevant code in Magma. Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ; E-mail: chair.math[at]iisc[dot]ac[dot]in Last updated: 20 Mar 2023	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8684699535369873, "perplexity": 634.488548671777}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943555.25/warc/CC-MAIN-20230320175948-20230320205948-00483.warc.gz"}
https://stats.stackexchange.com/questions/51244/better-precision-worse-recall	# better precision worse recall is it possible for an algorithm A to have a better precision but worse recall (or better recall but worse precision) than another algorithm B? Although I know that precision and recall are different things, it seems to me that if algorithm A has a better precision (or recall) than algorithm B then algorithm A will also have a better recall (or precision) than B. Thanks Ahmet Typically, there is a tradeoffs between precision and recall, so yes. Here's a simple example. Imagine you have predicted probabilities from a logistic regression, and you are choosing a classification threshold. A higher threshold will typically have better precision and worse recall than a low threshold. • I know that there is a trade-off between precision and recall. My question is not about the relation between precision and recall for a single algorithm but for two algorithms. But I think the current formulation of my problem is not clear, I will try to correct it Mar 3 '13 at 16:35 • @Ahmet Yılmaz: The point I am making is: algorithm A may "prioritize" precision, while algorithm B may prioritize recall. It is not at all surprising to get a result like yours. Try comparing the 2 algorithm on a non-threshold dependent metric, like area under the ROC curve. – Zach Mar 3 '13 at 19:50 To make an extreme example... Algorithm A: always say yes (ie, label all examples as positive). Algorithm B: only say yes in the one instance you are absolutely sure of. Algorithm A has perfect recall (but usually pretty bad precision) and algorithm B has perfect precision--assuming that one instance was right--(but awful recall). • I agree with you, thank you for your answers. However, the point that I was try to make is somewhat different. But I think it will take some thinking to reformulate my question. Thanks any way. Mar 3 '13 at 20:02 • @AhmetYılmaz Nothing says "Thanks" like an upvote. Mar 3 '13 at 20:06 Yes In information retrieval, precision and recall are used to evaluate search algorithms. For example, if I were to have a fixed test database of 1000 books. Of those 1000 only 10 are from the the author Steven King. A high precision search algorithm would only return books written by Steven King but it would probably not return all of them, maybe seven of them. A high recall algorithm would return all 10 of the Steven King books but it would also return books by other authors that have name King or Steven. Balancing both precision and recall is a key concept in Information retrieval. Typically the weighted harmonic mean of precision over recall is used. It is called the F-Measure. • In a way, isn't that similar to false-positives and false-negatives? Or specificity and sensitivity? Is the difference only semantics? Mar 5 '13 at 7:10 Refer to this sample Precision-Recall curve plotted for evaluating the classification performance of 3 different algorithms on the same test set. Precision is plotted on the Y-axis and Recall on the X-axis, percentages have been expressed in decimals. The curve is plotted by steadily increasing the cutoff value from 0 on the right-most side (recall=100%) to the highest value on the left-most side of the graph (recall=0%). For simplicity, we'll refer to the algorithms as "red", "green" and "blue". Now, when we compare points A and B, the recall reduced by 20% when the cutoff was increased from point A to point B, while the precision increased by 4% since false-positive rate was reduced. But the corresponding precision for algorithm "green" was much worse indicating a poorer classification performance. When we examine the performance at point C, the "red" algorithm performs much better (higher precision) than the other two for the same level of recall. Based on the nature of the classification task and the "cost" of false positives vs. false negatives, an appropriate algorithm may be chosen. For example, if a recall of 30% is the minimum acceptable for the task at hand, then algorithm "red" should be chosen since it has the highest precision, i.e. it will pick up more true positives for the same number of predicted positives. Lets ignore algorithm "red" for a moment and consider only the "blue" and "green" classifiers. If the minimum acceptable recall is 70%, then algorithm "blue" would be chosen since it is more precise at that rate of recall. But if the minimum acceptable recall is 20%, then algorithm "green" should be chosen since it has higher precision at that recall rate. It is possible for an algorithm to have better precision and recall than another algorithm and this can be indicated by plotting their curves together. For the same algorithm, usually if cutoff is raised precision increases while recall decreases. However, a lot depends on the algorithm being used and how well is it able to classify the dataset based on the attributes provided.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8392131924629211, "perplexity": 719.7066443085433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00211.warc.gz"}
https://testbook.com/question-answer/find-the-equation-of-a-line-perpendicular-to-the-l--6076f88d4b0e5ccbbff0b252	# Find the equation of a line perpendicular to the line x + 2y - 4 = 0 and passing through the point (-1, 3). 1. 2x - y + 5 = 0 2. 2x + 2y - 5 = 0 3. x - 2y - 5 = 0 4. None of these. 5. 2x + y + 5 = 0 Option 1 : 2x - y + 5 = 0 ## Detailed Solution Concept: If two lines are perpendicular, then the product of their slopes is equal to -1. The equation of a line is typically written as y = mx + c where m is the slope and c is the y-intercept. Calculation: The equation of the given line x + 2y - 4 = 0 can also be written as: y = -x/2 + 2 The slope of this line is equal to -1/2. Let the slope of the line perpendicular to it be m: We must have (-1/2)(m) = -1. ⇒ m = 2. So the equation of the required line is y = 2x + c. Since this line passes through (-1, 3), we must have: 3 = 2(-1) + c ⇒ c = 5 The equation is y = 2x + 5 or 2x - y + 5 = 0	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.846066415309906, "perplexity": 236.8797420089848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304686.15/warc/CC-MAIN-20220124220008-20220125010008-00070.warc.gz"}
https://encyclopedia.pub/10	Probability Associated with Anderson-Darling Statistic Created by: Lorentz Jäntschi The subject of this entry is on the regard of one of the order statistics: the Anderson-Darling statistic. We provided a method of calculation for the probability associated with the Cumulative Density Function (CDF) of the Anderson-Darling statistic. Our study shown that the value of the probability is affected by the sample size. As consequence, we constructed a function to provide an estimate for the associated probability depending on both the value of the statistic and the sample size. Table of Content [Hide] A commonly known fact among all order statistics is that is very difficult to extract the probability associated with the statistic for a simple fact: their CDF (cumulative density function) - depicted in the figure for a range of probabilities from 0.500 to 0.995 (with a step of 0.5%) and for a range of sample sizes from n = 2 to n = 42 - have no analytical expression (e.g. we do not possess a mathematical function which to express it analytically). Unfortunately, the same case applies for their PDF (probability density function) making unavailable also the numeric integration methods. The only way of associating the probability (e.g., α = 1 - p) with the statistic is from Monte-Carlo experiments, and this is the way in which all were reported (see [1], [2][3] and [4] for instance) and are on the use today. Unfortunately, it is an inconvenience to use the raw data from Monte-Carlo simulations, namely we may have access to certain thresholds (for instance the value of the statistic corresponding to α = 5%), but it is not possible to extract the probability associated with a particular value of the statistic (for a simple reason: it is not tabulated). For some instances, when the statistic it is in range of tabulated data, the interpolation may provide satisfactory results - but also here precautions should be taken because the probability does not vary linearly with the statistic. We run a computationally Monte-Carlo simulation with the Anderson-Darling statistic (named AD in the next) big and precise enough to obtain a good estimate of the probability associated with the statistic. The simulation strategy, the estimation strategy and the obtained function expressing the probability associated with the AD statistic are given in [5]. $\hat{p} = \Bigg( \sum_{i=0}^{4}{\sum_{j=0}^{4}{b_{i,j}x^{i/4}}n^{-j}} \Bigg)^{-1}, x = e^{AD}$ where AD is the calculated Anderson-Darling statistic on the sample of size n: $AD = - n - \sum_{i=1}^{n}{ \frac{(2 \cdot i-1)\cdot ln(p_{i} \cdot (1-p_{n+1-i}))}{n} }$ and is the ordered (ascending) probability measuring that i-th (ordered) observation to belong to a certain distribution. The values for the coefficients are given in the following table. j = 0 j = 1 j = 2 j =3 j = 4 i = 0 5.6737 -38.9087 88.7461 -179.5470 199.3247 i = 1 -13.5729 83.6500 -181.6768 347.6606 -367.4883 i = 2 12.0750 -70.3770 139.8035 -245.6051 243.5784 i = 3 -7.3190 30.4792 -49.9105 76.7476 -70.1764 i = 4 3.7309 -6.1885 7.3420 -9.3021 7.7018 The proposed model is intended to be used (the applicability domain is) when p > 0.5 (e.g., α < 0.5). The AD statistic (as any other order statistic) can be used to test (to measure the agreement) any distribution, not only the well-known normal (Gauss) one. The use of AD statistic requires an estimate of the parameters of the distribution being tested. Certain precautions should be made, namely if the estimation of (some of) the parameters are not made with Maximum Likelihood Estimation method (see [6]; the usually alternative being the Central Moments method, see [7]), then a corresponding number must be subtracted from sample size value before to extract the probability associated with the statistic. References 1. Kolmogorov, A Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari 1933, 4, 83-91, N.A.. 2. Smirnov, N. Table for estimating the goodness of fit of empirical distributions. Annals of Mathematical Statistics 1948, 19, 279-281, 10.1214/aoms/1177730256. 3. Anderson, T.W.; Darling, D.A. Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes. Annals of Mathematical Statistics 1952, 23, 193-212, 10.1214/aoms/1177729437. 4. Anderson, T.W.; Darling, D.A. A Test of Goodness-of-Fit. Journal of the American Statistical Association 1954, 49, 765-769, 10.2307/2281537. 5. Jäntschi, L.; Bolboacă S.D. Computation of probability associated with Anderson-Darling statistic. Mathematics 2018, 6(6), 88, 10.3390/math6060088. 6. Fisher, R.A. On an Absolute Criterion for Fitting Frequency Curves. Messenger of Mathematics 1912, 41, 155-160, N.A.. 7. Fisher, R.A. On the Mathematical Foundations of Theoretical Statistics. Philosophical Transactions of the Royal Society A 1922, 222, 309-368, 10.1098/rsta.1922.0009.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9088836908340454, "perplexity": 882.008914235476}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672431.45/warc/CC-MAIN-20191016235542-20191017023042-00415.warc.gz"}
https://economics.stackexchange.com/questions/20513/is-there-a-discrepancy-between-the-ratio-and-total-expenditure-methods-of-measur	# Is there a discrepancy between the ratio and total expenditure methods of measuring elasticity? According to the Expenditure method of measuring price elasticity of demand, if the two expenditures in comparison are constant (in spite of changes in price), the demand is considered to be unitary elastic. However, there are several examples where this does not seem to hold true. For instance, Price: 10 ; Demand: 12 ; Expenditure: 120 Price: 08 ; Demand: 15 ; Expenditure: 120 Here, though the expenditure is equal, if we calculate the percentage changes in price and demand, they come to be 20% and 25% respectively. So as per the ratio (or percentage method), the elasticity would be equal to 25% / 20%, which is not equal to 1. Then, how can we conclude that the expenditure method gives the right results? If something rises by $25\%$ then you then need a $20\%$ reduction in the new larger number to get back where you started: $1.25 \times 0.8=1$. This asymmetry means comparing large percentage changes can produced oddities Another approach, closer to the idea of approaching the derivative as a limit, is to use logarithms and calculate the elasticity as a ratio of logarithms, something like $$\frac{\log\left(\frac{Q_2}{Q_1}\right)}{\log\left(\frac{P_2}{P_1}\right)}= \frac{\log(Q_2)-\log(Q_1)}{\log(P_2)-\log(P_1)}$$ and with your example this would give $\frac{\log(15)-\log(12)}{\log(8)-\log(10)} = \frac{\log(1.25)}{\log(0.8)} =-1$ as expected	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9841428995132446, "perplexity": 563.1433834347959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607118.51/warc/CC-MAIN-20200122131612-20200122160612-00017.warc.gz"}
http://rcd.ics.org.ru/authors/detail/327-ivan_bizyaev	0 2013 Impact Factor # Ivan Bizyaev ## Publications: Bizyaev I. A., Borisov A. V., Mamaev I. S. The Hess–Appelrot Case and Quantization of the Rotation Number 2017, vol. 22, no. 2, pp. 180-196 Abstract This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder. Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196 DOI:10.1134/S156035471702006X Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S. Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups 2016, vol. 21, no. 6, pp. 759-774 Abstract This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals. Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals Citation: Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774 DOI:10.1134/S1560354716060125 Borisov A. V., Mamaev I. S., Bizyaev I. A. The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity 2016, vol. 21, no. 5, pp. 556-580 Abstract In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems. Keywords: celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A., The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580 DOI:10.1134/S1560354716050075 Borisov A. V., Mamaev I. S., Bizyaev I. A. Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period 2016, vol. 21, no. 4, pp. 455-476 Abstract In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations. Keywords: nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476 DOI:10.1134/S1560354716040055 Bizyaev I. A., Borisov A. V., Mamaev I. S. The Dynamics of Vortex Sources in a Deformation Flow 2016, vol. 21, no. 3, pp. 367-376 Abstract This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse. Keywords: integrability, vortex sources, reduction, deformation flow Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376 DOI:10.1134/S1560354716030084 Bizyaev I. A., Borisov A. V., Mamaev I. S. Dynamics of the Chaplygin Sleigh on a Cylinder 2016, vol. 21, no. 1, pp. 136-146 Abstract This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical. Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146 DOI:10.1134/S1560354716010081 Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A. Qualitative Analysis of the Dynamics of a Wheeled Vehicle 2015, vol. 20, no. 6, pp. 739-751 Abstract This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined. Keywords: nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system Citation: Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751 DOI:10.1134/S156035471506009X Bizyaev I. A., Borisov A. V., Kazakov A. O. Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors 2015, vol. 20, no. 5, pp. 605-626 Abstract In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks. Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor Citation: Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626 DOI:10.1134/S1560354715050056 Borisov A. V., Mamaev I. S., Bizyaev I. A. The Jacobi Integral in Nonholonomic Mechanics 2015, vol. 20, no. 3, pp. 383-400 Abstract In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface. Keywords: nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400 DOI:10.1134/S1560354715030107 Bizyaev I. A., Borisov A. V., Mamaev I. S. The Dynamics of Three Vortex Sources 2014, vol. 19, no. 6, pp. 694-701 Abstract In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented. Keywords: integrability, vortex sources, shape sphere, reduction, homothetic configurations Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Three Vortex Sources, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701 DOI:10.1134/S1560354714060070 Bizyaev I. A., Borisov A. V., Mamaev I. S. Superintegrable Generalizations of the Kepler and Hook Problems 2014, vol. 19, no. 3, pp. 415-434 Abstract In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form. Keywords: superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434 DOI:10.1134/S1560354714030095 Bizyaev I. A., Borisov A. V., Mamaev I. S. The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside 2014, vol. 19, no. 2, pp. 198-213 Abstract In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found. Keywords: nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213 DOI:10.1134/S156035471402004X Borisov A. V., Mamaev I. S., Bizyaev I. A. The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere 2013, vol. 18, no. 3, pp. 277-328 Abstract In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior. Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328 DOI:10.1134/S1560354713030064	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8474792242050171, "perplexity": 574.4806727832798}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122041.70/warc/CC-MAIN-20170423031202-00203-ip-10-145-167-34.ec2.internal.warc.gz"}
http://clay6.com/qa/69659/the-phase-difference-between-two-waves-reaching-a-point-is-frac-what-is-the	# The phase difference between two waves reaching a point is $\frac{\pi}{2}$ . What is the resultant amplitude if the individual amplitude are 3 mm and 4 mm ? $\begin{array}{1 1} 5\;mm \\ 50\;mm \\ 5\;cm \\ 500\;cm \end{array}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9615942239761353, "perplexity": 272.05097681295945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151638.93/warc/CC-MAIN-20210725045638-20210725075638-00032.warc.gz"}
https://www.physicsoverflow.org/18956/what-lagrangian-from-which-klein-gordon-equation-derived-qft	# What is the Lagrangian from which the Klein-Gordon equation is derived in QFT? + 1 like - 0 dislike 169 views 1. Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it? 2. What is the canonical conjugate momentum? I derive the same result as in two sources separately, but with opposite sign, and I am starting to suspect that the error could be in the Lagrangian I am departing from. 3. Is there any difference in the answers to that two questions if you choose (+---) or (-+++)? If so, which one? This post imported from StackExchange Physics at 2014-06-14 12:57 (UCT), posted by SE-user Eduardo Guerras Valera retagged Jun 14, 2014 + 6 like - 0 dislike 1. Yes. The standard scalar field which all QFT books (e.g. Peskin & Schroeder, Zee) start with yields the KG equation. For that reason it is also called the Klein-Gordon field. The Lagrangian (density) is \begin{align} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. \end{align} Here the metric is (+ - - -). 2. By definition it is $\pi = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}$. This gives $\pi = \partial_0 \phi$. 3. It is purely convention, there is no right choice. The only difference in using a different metric will be in how we write things down - any quantities that involve contraction with the metric $\eta_{\mu \nu}$ will change by a minus sign. For example in the Lagrangian, using the metric (- + + +), the first term is changed to $-\frac{1}{2} \partial_\mu \phi \partial^\mu \phi$. But this is still equal to $\frac{1}{2}(\partial_t^2 \phi - \nabla^2 \phi)$ regardless of which metric we use. This post imported from StackExchange Physics at 2014-06-14 12:57 (UCT), posted by SE-user nervxxx answered Feb 3, 2013 by (210 points) Thanks a lot for your answer, it has put me on the right track. Eqs (1.14) and (1.15) in the preprint of Srednicki have the key to the changed sign of question 2. I found it thank to your indications. This post imported from StackExchange Physics at 2014-06-14 12:57 (UCT), posted by SE-user Eduardo Guerras Valera Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8358871340751648, "perplexity": 1048.1901528693513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250595282.35/warc/CC-MAIN-20200119205448-20200119233448-00054.warc.gz"}
http://link.springer.com/article/10.1007%2FBF01414710	, Volume 3, Issue 3, pp 159-165 Shock wave interaction with cellular materials Purchase on Springer.com \$39.95 / €34.95 / £29.95* Rent the article at a discount Rent now * Final gross prices may vary according to local VAT. Abstract The equations governing the head-on collision of a planar shock wave with a cellular material and a numerical scheme for solving the set of the governing equations were outlined. In addition, the condition for the transmitted compression waves to transform into a shock wave, inside the cellular material was introduced. It was proved analytically that a cellular material cannot be used as a means of reducing the pressure load acting on the end-wall of the shock tube. In subsequent papers, the interaction of planar shock waves with specific cellular materials, e.g., foams and honeycombs will be described in detail. This article was processed using Springer-Verlag TEX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9067676663398743, "perplexity": 1139.5968791275482}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00336-ip-10-147-4-33.ec2.internal.warc.gz"}
https://de.maplesoft.com/support/help/maple/view.aspx?path=License%2FDejaVuFonts	DejaVu Fonts License - Maple Programming Help Fonts are (c) Bitstream (see below). DejaVu changes are in public domain. Glyphs imported from Arev fonts are (c) Tavmjong Bah (see below). Permission is hereby granted, free of charge, to any person obtaining a copy of the fonts accompanying this license ("Fonts") and associated documentation files (the "Font Software"), to reproduce and distribute the Font Software, including without limitation the rights to use, copy, merge, publish, distribute, and/or sell copies of the Font Software, and to permit persons to whom the Font Software is furnished to do so, subject to the following conditions: The above copyright and trademark notices and this permission notice shall be included in all copies of one or more of the Font Software typefaces. The Font Software may be modified, altered, or added to, and in particular the designs of glyphs or characters in the Fonts may be modified and additional glyphs or characters may be added to the Fonts, only if the fonts are renamed to names not containing either the words "Bitstream" or the word "Vera". This License becomes null and void to the extent applicable to Fonts or Font Software that has been modified and is distributed under the "Bitstream Vera" names. The Font Software may be sold as part of a larger software package but no copy of one or more of the Font Software typefaces may be sold by itself. THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL BITSTREAM OR THE GNOME FOUNDATION BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM OTHER DEALINGS IN THE FONT SOFTWARE. Except as contained in this notice, the names of Gnome, the Gnome Foundation, and Bitstream Inc., shall not be used in advertising or otherwise to promote the sale, use or other dealings in this Font Software without prior written authorization from the Gnome Foundation or Bitstream Inc., respectively. For further information, contact: fonts at gnome dot org.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8273943066596985, "perplexity": 4127.754596608648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998513.14/warc/CC-MAIN-20190617163111-20190617185111-00055.warc.gz"}
http://mathhelpforum.com/calculus/129874-integration-parts-practice-print.html	Integration by Parts practice Printable View • Feb 20th 2010, 10:11 PM Keithfert488 Integration by Parts practice Hi I was wondering if I got this right so could someone check for me? $\int x\sin x\cos xdx$ I used integration by parts with $u=\sin x\cos x$ and $dv=xdx$ and got... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}\int x^2\cos 2xdx$ Then, I used parts again with $u=x^2$ and $dv=\cos 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}-\int x\sin 2xdx)$ Parts yet again with $u=x$ and $dv=\sin 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}+(\frac{x\cos 2x}{2}-\frac{1}{2}\int \cos 2xdx))$ $=\frac{x^2\sin 2x}{4}-\frac{1}{2}(\frac{x^2\sin 2x}{2}+\frac{x\cos 2x}{2}-\frac{\sin 2x}{4})+C$ $=\frac{x^2\sin 2x}{4}-\frac{x^2\sin 2x}{4}-\frac{x\cos 2x}{4}+\frac{\sin 2x}{8}+C$ $=\frac{\sin 2x-2x\cos 2x}{8}+C$ • Feb 20th 2010, 10:36 PM Prove It Quote: Originally Posted by Keithfert488 Hi I was wondering if I got this right so could someone check for me? $\int x\sin x\cos xdx$ I used integration by parts with $u=\sin x\cos x$ and $dv=xdx$ and got... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}\int x^2\cos 2xdx$ Then, I used parts again with $u=x^2$ and $dv=\cos 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}-\int x\sin 2xdx)$ Parts yet again with $u=x$ and $dv=\sin 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}+(\frac{x\cos 2x}{2}-\frac{1}{2}\int \cos 2xdx))$ $=\frac{x^2\sin 2x}{4}-\frac{1}{2}(\frac{x^2\sin 2x}{2}+\frac{x\cos 2x}{2}-\frac{\sin 2x}{4})+C$ $=\frac{x^2\sin 2x}{4}-\frac{x^2\sin 2x}{4}-\frac{x\cos 2x}{4}+\frac{\sin 2x}{8}+C$ $=\frac{\sin 2x-2x\cos 2x}{8}+C$ Rewrite the integral as $\frac{1}{2}\int{x\sin{2x}\,dx}$. Let $u = x$ so that $du = 1$ Let $dv = \sin{2x}$ so that $v = -\frac{1}{2}\cos{2x}$. So $\frac{1}{2}\int{x\sin{2x}\,dx} = -\frac{1}{2}x\cos{2x} - \int{-\frac{1}{2}\cos{2x}\cdot 1\,dx}$ $= -\frac{1}{2}x\cos{2x} + \frac{1}{2}\int{\cos{2x}\,dx}$ $= -\frac{1}{2}x\cos{2x} + \frac{1}{2}\sin{2x} + C$. • Feb 20th 2010, 10:36 PM mr fantastic Quote: Originally Posted by Keithfert488 Hi I was wondering if I got this right so could someone check for me? $\int x\sin x\cos xdx$ I used integration by parts with $u=\sin x\cos x$ and $dv=xdx$ and got... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}\int x^2\cos 2xdx$ Then, I used parts again with $u=x^2$ and $dv=\cos 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}-\int x\sin 2xdx)$ Parts yet again with $u=x$ and $dv=\sin 2xdx$ ... $\frac{x^2\sin x\cos x}{2}-\frac{1}{2}(\frac{x^2\sin x}{2}+(\frac{x\cos 2x}{2}-\frac{1}{2}\int \cos 2xdx))$ $=\frac{x^2\sin 2x}{4}-\frac{1}{2}(\frac{x^2\sin 2x}{2}+\frac{x\cos 2x}{2}-\frac{\sin 2x}{4})+C$ $=\frac{x^2\sin 2x}{4}-\frac{x^2\sin 2x}{4}-\frac{x\cos 2x}{4}+\frac{\sin 2x}{8}+C$ $=\frac{\sin 2x-2x\cos 2x}{8}+C$ There are several ways you can check this, including: 1. Using WolframAlpha. 2. Differentiate your answer and see if it works. • Feb 20th 2010, 10:40 PM Keithfert488 whoa I've never seen WolframAlpha. That's awesome!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 47, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846928715705872, "perplexity": 1333.0677767499747}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280504.74/warc/CC-MAIN-20170116095120-00324-ip-10-171-10-70.ec2.internal.warc.gz"}
http://physicstasks.eu/1797/heating-the-log-cabin	## Heating the Log Cabin a) How much heat transfers through side walls of a log cabin during one winter day? Length of the cabin is 10 m, width is 7 m, height of the walls is 3.5 m and their thickness is 50 cm. Average outdoor temperature is -10 °C and the indoor temperature is kept at 18 °C. b) How much wood has to be burnt up during one day in a stove, whose thermal efficiency is 30%, so that the indoor temperature is being kept constant? c) How much would an electric heating of the cabin cost? Efficiency of electric heating is practically 100% and an average price of electric energy is for example 4.30 CZK/kWh. d) How high must the volumetric flow rate of water in the radiator be if water temperature at the entry into the radiator is 80 °C and temperature of the water leaving the radiator is 70 °C? Assume that the roof is so well thermally isolated that we can neglect heat loss through the roof. • #### Hint Transferred heat can be calculated from Fourier’s law of heat conduction. • #### Notation a = 10 m length of the cabin b = 7 m width of the cabin h = 3.5 m height of the cabin walls d = 50 cm = 0.50 m thickness of the cabin walls t1 = -10 oC average outdoor temperature t2 = 18 oC indoor temperature t3 = 80 oC water temperature at the entry into the radiator t4 = 70 oC temperature of the water leaving the radiator τ = 1 d = 86 400 s time η = 30% = 0.3 wood stove efficiency Q = ? heat transferred through the walls m = ? mass of wood to be burnt qV = ? volumetric flow rate of water in the radiator From the Handbook of Chemistry and Physics: λ = 0.15 W m-1K-1 thermal conductivity of wood H = 15 MJ kg-1 heat of combustion for wood cwater = 4 180 J kg-1 K-1 specific heat capacity of water • #### Analysis The heat transferred through a homogeneous board (in our case, the wall; inhomogeneity of the wall such as windows and the door are not considered) is proportional to the area of the wall (board), time, during which the heat is being transferred, and the temperature difference between the ends of the board (under the condition that the temperature difference is constant). On the contrary, the transferred heat is inversely proportional to the thickness of the board (wall). For some materials their ability to transfer the heat is specified by so-called thermal conductivity. The greater it is the more heat is being transferred. To calculate the heat transferred through the walls during one day, we have all the required values. To keep the indoor temperature constant, we have to supply the same heat that is being transferred to the environment. To calculate the required amount of wood, we will use its heat of combustion, i.e. the heat we receive by burning 1 kg of wood. Using the fact that the water has to supply so much heat so that the indoor temperature is kept constant, we calculate the required volumetric flow rate of water in the radiator. Supplied heat is proportional to the difference between the temperature of flowing in water and the water leaving the radiator. • #### Solution a) For transferring heat it holds that: $Q=\lambda \frac{S \tau}{d} \Delta t\,,$ where λ is the thermal conductivity of wood, S is the total area of the walls, for which it is true that $S=(2a+2b)h\,,$ where d is the thickness of the walls, τ is the period of time during which the heat has been transferring and Δt = t2t1 is the difference between the temperature inside and outside the cabin. By substitution we obtain the relation, which we can substitute given values into: $Q=\lambda\frac{\left(2a+2b\right)h\tau}{d}\left(t_2-t_1\right)\,,$ $Q=0.15\cdot\frac{\left(2\cdot{10}+2\cdot{7}\right)\cdot3.5\cdot{86400}}{0.5}\cdot\left[18-(-10)t_1\right]\,\mathrm{J}\,,$ $Q\dot{=}8.6\cdot{10}^{7}\,\mathrm{J}=86\,\mathrm{MJ}\,.$ b) Burning the wood whose mass is m, we obtain the heat Hm. However, only η = 30% of this heat we use for heating the cabin (wood stove efficiency). The heat used for heating the cabin has to equal the heat transferred through the walls, therefore $Q=\eta Hm\,,$ Hence we will express the required mass of wood m and calculate it by substitution: $m=\frac{Q}{\eta H}=\frac{86\,\mathrm{MJ}}{0.3\cdot{15}\,\mathrm{MJ\cdot kg^{-1}}}\dot{=}19\,\mathrm{kg}\,.$ c) First we derive the conversion relationship between the units of energy: $1\,\mathrm{kWh}=1\,\mathrm{kW}\cdot1\,\mathrm{h}=1000\,\mathrm{W}\cdot\,3600\,\mathrm{s}=3.6\cdot{10^6}\,\mathrm{Ws}=3.6\,\mathrm{MJ}\,.$ From the conversion relationship we see that 3.6 MJ of electric energy cost 4.30 CZK. It means that the amount of money we would have paid for the electric heating for one day is: $\frac{86}{3.6}\cdot4.30\,\mathrm{CZK}=103\,\mathrm{CZK}\,.$ d) If we denote the volumetric flow rate of water in the radiator as qV, then the volume of the water passed through the radiator during the time τ is qVτ. This water cools down and supplies the heat $Q_V=cq_V\tau \left(t_3-t_4\right)\,.$ Then we compare the heat supplied by water and the heat transferred through the walls $Q_V=Q\,,$ $c_{\mathrm{water}}q_V\tau\left(t_3-t_4\right)=\lambda\frac{\left(2a+2b\right)h\tau}{d}\left(t_2-t_1\right)\,.$ and express the unknown volumetric flow rate and substitute given values: $q_V=\lambda\frac{\left(2a+2b\right)h\left(t_2-t_1\right)}{c_{\mathrm{water}}d\left(t_3-t_4\right)}\,,$ $q_V=0.15\cdot\frac{\left(2\cdot{10}+2\cdot{7}\right)\cdot{3.5}\cdot\left[18-(-10)\right]}{4180\cdot{0.5}\cdot\left(80-70\right)}\,\mathrm{kg\cdot s^{-1}}\,,$ $q_V\dot{=}0.024\,\mathrm{kg\cdot s^{-1}}=86\,\mathrm{kg\cdot h^{-1}}\,.$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8334745764732361, "perplexity": 544.8350423532514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217901.91/warc/CC-MAIN-20180820234831-20180821014831-00679.warc.gz"}
https://www.physicsforums.com/threads/help-rates-of-change.147016/	Homework Help: Help! rates of change 1. Dec 6, 2006 lufbrajames 1. The problem statement, all variables and given/known data We need to know the temperature of the copper bit of a soldering iron varies with time after the power has been switched on. This is a first step to determine how long it takes for the temperature of the bit to reach the operating temperature at wich it can melt the solder. We assume all heat produced goes directly to the bit and none is lost to the air. i.e. the temperature of the bit, Theta = Theta (t), depends only on time. 3 laws of physics: the rate of energy storage in the bit is the product of the mass m of copper, the specific heat c of copper and the rate of change in the bit. the rate of loss of heat from the bit to the air has the form kA(theta - Thate a) where theta a is the temprature of the air, A is the (constant) cross section of the bit, and k is a constant The heat traveling from the barrel to the bit is the sum of the heat loss from the bit and the heat stored in the bit (consesrvation of energy) 1) To which value do you expect the temprature of the bit to settle? 2) Sketch a graph of Theta with t (time) 3) Write down the differential equation which describes the cooling process. 4)Given that the solution tof the equation dtheta/dt + a theta = b where a and b are cpnstant, is theta = b/a + Ce^-at where c is a constant, write down the solution of your eqaution in part 3 which satisfies the initial condition theta = theta 0 at t = 0 2. Relevant equations 3. The attempt at a solution 1) the temperaturewill settle at theta a the temperature of the air. 2)i drew a graph that showed the temperature drop rapidly at first and then steadyout to nearly level at theta a 3) (this is where i get really stuck!) i got the following dtheta/dt + k = A(theta - theta 0) 4) theta = A(theta - theta 0) / k 2. Dec 7, 2006 Kurdt Staff Emeritus Question 1 and 2 seem ok to me. For question 3 you want something like this: $$\frac{d\theta}{dt}=kA(\theta-\theta_a)$$ For question 4 put the above equation in the same form as the one they give you and you can imply the solution. Or you could always try and solve it yourself but its a tricky one. 3. Dec 9, 2006 lufbrajames Thanks, im still a bit confused about 4 from this equation: $$\frac{d\theta}{dt}=kA(\theta-\theta_a)$$ i get $$\theta=\frac{b}{k}+Ce^kt$$ when trying to put it into the form given in quation 4... is this at all right? 4. Dec 9, 2006 marlon Both integration variables need to be on one side each !!! k and A are constants $$\frac{d\theta}{dt}=kA(\theta-\theta_a)$$ $$\frac{d\theta}{\theta-\theta_a}=kAdt$$ Integrate left and right hand side to come to (make sure you take into account given initial conditions for theta and t !!!) : $$ln ( \theta-\theta_a ) =kAt$$ and if ln(a) = b <-> a = exp(b) you should be able to take it from here marlon 5. Dec 9, 2006 lufbrajames Thanks for the help so far, i think i mite be getting in now i have: $$\frac{d\theta}{dt}=kA(\theta-\theta_a)$$ $$\frac{d\theta}{\theta-\theta_a}=kAdt$$ $$ln ( \theta-\theta_a ) =kAt$$ $$( \theta-\theta_a ) = e^{kAt}$$ and becuase it must satisfy $$\theta = \theta_0$$ and t = 0 then: $$e^{kA} = \theta-\theta_a$$ Is this correct? Last edited: Dec 9, 2006 6. Dec 9, 2006 estel You're missing a constant of integration, and theta has disappered. Also, jumping back to question 1 - if the powe has been turned ON, the bit won't settle to the ambient temperature, it will stop at the temperature such that the electrical energy being converted to thermal energy = energy lost to the air. Might want to check that. 7. Dec 9, 2006 lufbrajames $$e^{kA} = \theta-\theta_a + C$$ ?? also the first question says when the electricity supply is switched off, sory that i missed that bit out. 8. Dec 9, 2006 estel put the constant on the side integrated with respect to t, so when you exponentiate, you get a constant multiplier on that side Qexp(kAt) 9. Dec 10, 2006 lufbrajames ok i think i understand $$Ce^{kAt} = \theta-\theta_a$$ i also have to sketch a graph of the solution and compare it to my original sketch in part 2... would'nt they be exactly the same??? 10. Dec 10, 2006 lufbrajames after sketching the graph, the next part says Now we switch on the electricity supply. 1. Let W be the (constan) heat per second supplied to the solderin iron, produce a differential equation for $$\theta$$ 2. what is the steady-state temperature of the bit? 3 sketch the graph of $$\theta$$ against time 4 if $$\theta =\theta_0$$ at t = 0 find the solution for theta as a function of time, you may assume that theta_0 = theta_a in order to sketch the graph of temperature with time --------- i think i have the answer to part 1 $$\frac{d\theta}{dt} = kAW-(\theta - \theta_a)$$ does steady-state temperature mean average? or the temperature at which it stops rising? for the 3rd part i beleive the graph should climb slowly at first then shoot up rapidly??? this is what i have for part 4: $$\frac{d\theta}{dt} = kAW-(\theta - \theta_a)$$ $$\frac{d\theta}{\theta - \theta_a} = kAW-dt$$ $$ln(\theta-\theta_a) = kAW-t$$ $$(\theta-\theta_a) = Ce^{kAW-t}$$ Last edited: Dec 10, 2006 11. Dec 11, 2006 lufbrajames anyone? im rly stugling with this topic 12. Dec 11, 2006 Kurdt Staff Emeritus For this bit you should get: $$( \theta-\theta_a ) = Ce^{kAt}$$ then when $$\theta = \theta_0$$ at t=0 one would get, $$C= \theta_0 - \theta_a$$ For question 1 of the second set you're on the right lines so the rate of change of temperature wrt time will be the heat input minus the heat lost. $$\frac{d\theta}{dt} = W -kA(\theta - \theta_a)$$ For part two steady state means when an equilibrium is reached so in other words when the temperature does not change any more. Therefore you want to make $$\frac{d\theta}{dt}=0$$. Part 3 will show the temperature rising then reaching a plateau. For part 4 you will have to solve the equation and apply the conditions given. Last edited: Dec 11, 2006 13. Dec 12, 2006 lufbrajames What do you mean solve the equation? there are no figures givend except for: $$\theta = \theta_0$$ at $$t = 0$$ Assume $$\theta_0 = \theta_a$$ 14. Dec 12, 2006 Kurdt Staff Emeritus The second set of questions asks you to come up with a differential equation in question 1. For part four it asks you to solve the equation (constructed in part 1) to find $$\theta(t)$$ with the conditions given. 15. Dec 12, 2006 lufbrajames oh rite, like i did before, How do i rearange $$\theta - \theta_a = Ce^{W-kAt}$$ to get A, i have figures for W K and $$\theta_a$$ and $$\theta$$ 16. Dec 12, 2006 Kurdt Staff Emeritus I'm not sure you've solved the equation properly. You should be coming out with: $$\theta(t) = \frac{W+kA\theta_a}{kA} +C e^{-kAt}$$ Last edited: Dec 12, 2006 17. Dec 12, 2006 lufbrajames $$\theta(t) = \frac{W+kA\theta_a}{kA} +C e^{kAt}$$ how did you get this? does this mean that earlier in my thread, i have also solved an equation wrong? $$\frac{d\theta}{dt}=kA(\theta-\theta_a)$$ and solving this i got: $$Ce^{kAt} = \theta-\theta_a$$ Last edited: Dec 12, 2006 18. Dec 12, 2006 Kurdt Staff Emeritus The equation you got previously was fine. What I did is rather than attempt to solve it normally which is tricky, is to compare it to the equation in question 4 of the first set and infer the solution. We can write the differential equation as: $$\frac{d\theta}{dt}+kA\theta=W+kA\theta_a$$ For the equation in your first post we see that $$a= kA$$ and $$b=W+kA\theta_a$$ 19. Dec 12, 2006 Kurdt Staff Emeritus Oh I made a mistake in the post above. The equation should be $$\theta(t) = \frac{W+kA\theta_a}{kA} +C e^{-kAt}$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8979139924049377, "perplexity": 854.3243349507773}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221213247.0/warc/CC-MAIN-20180818001437-20180818021437-00176.warc.gz"}
http://math.stackexchange.com/questions/185637/what-are-the-subobjects-of-a-manifold	# What are the subobjects of a manifold? Categorically a subobject of an object $a$ of some category $A$ is an object $a'$ with a monic morphism to $a$, ie $a'\to a$, upto isomorphism. When $A$ is either a Topological or Differential manifold, what are the subobjects of a manifold, and are they the same as submanifolds? - That depends. What morphisms are you choosing? – Qiaochu Yuan Aug 23 '12 at 2:21 @Yuan: The real intent of the question is how to categorically select the 'usual' submanifolds, ie embeddings, immersed submanifolds etc which from peoples comments seem to require Diff being concrete; and to see whether the usual categorical subobjects result in anything useful here. – Mozibur Ullah Aug 23 '12 at 21:10 Monic morphisms are injections in concrete categories like the category of manifolds, so a subobject of a manifold $M$ is another manifold with an injection into $M$. This isn't quite the same thing as a submanifold, as usually submanifolds are required to be embedded, meaning that they inherit their topology from the larger manifold. For an example of a manifold that is injected into a larger manifold but isn't embedded, let $M$ be a torus (considered as a quotient space of $\mathbb{R}^2$ under integer translations), and let $L$ be the real line, mapped to a line of irrational slope in $\mathbb{R}^2$ and then projected to the torus. This causes the line to "wrap around" infinitely without touching itself. This map is an injection, thus $L$ with this map is a subobject, but the image isn't a manifold, since any neighborhood of a point contains infinitely many "nearby" lines. Edit: As pointed out below, I was erroneous when I said monic morphisms are injective in all concrete categories. It is true for manifolds, however, as proved in Makoto Kato's answer. - Ok, so to get an embedded manifold, we must stipulate that it has the initial topology wrt to the injection. That works for topological manifolds, but not for differential manifolds. – Mozibur Ullah Aug 23 '12 at 0:05 "Monic morphisms are injections in concrete categories" This is not true in general. – Makoto Kato Aug 23 '12 at 0:46 A sufficient condition for a concrete category $C$ to have the property that the monics are precisely the injective morphisms is that the forgetful functor $C \to \text{Set}$ is representable. This is true for the category of manifolds and for many other common concrete categories, but is false in general. In general we can only conclude that injections are monic. – Qiaochu Yuan Aug 23 '12 at 2:18 @MakotoKato Thanks for pointing out my mistake, I've noted it in the bottom of the answer. To MoziburUllah: Yes, in the differentiable category you also require that the map is an immersion. I can't actually come up with with a smooth injective map of manifolds that isn't an immersion off the top of my head, though. – MartianInvader Aug 23 '12 at 18:16 @MartianInvader As for an example of a smooth injective map of manifolds that is not an immersion, please see my answer. – Makoto Kato Aug 23 '12 at 18:33 Let $\mathcal{C}$ be the category of toplogical(resp. differential) manifolds. The objects of $\mathcal{C}$ are topological(resp. differential) manifolds and the morphisms of $\mathcal{C}$ are continuous(resp. smooth) maps. Let $f\colon X \rightarrow Y$ be a morphism in $\mathcal{C}$. We claim $f$ is a monomorphism if and only if $f$ is injective. Suppose $f$ is a monomorphism. Let $x, y$ be distinct points of $X$. Let $p$ be a $0$-dimensional object in $\mathcal{C}$. There exists the unique morphism $g\colon p \rightarrow X$ such that $g(p) = x$. Similarly there exists the unique morphism $h\colon p \rightarrow X$ such that $h(p) = y$. Since $g \neq h$, $fg \neq fh$. Hence $f(x) \neq f(y)$. Hence $f$ is an injective map. Conversly suppose $f$ is injective. Clearly $f$ is a monomorphism. "Are they the same as submanifolds?" Generally no. Counter-example: Let $f\colon \mathbb{R} \rightarrow \mathbb{R}^2$ be the map defined by $f(x) = (x^3, 0)$. $f$ is smooth and injective, but is not an immersion($f'(0) = 0$). Hence $\mathbb{R}$ cannot be identified with a submanifold of $\mathbb{R}^2$ by $f$. - +1 Well explained! Perhaps it might be helpful for some less experienced readers if you would specify the objects and morphisms in the categories of topological/differential manifolds. – magma Aug 23 '12 at 12:55 I always learned that a map isn't considered smooth if the derivative is ever zero. Otherwise, you could make a path with corners and call it "smooth" by having all its derivatives be zero at each corner. – MartianInvader Aug 23 '12 at 18:37 @MartianInvader If you don't allow a smooth function to have the zero derivative, you can't call a constant function smooth. In your case, a path with corners is not differentiable at the corners. Regards, – Makoto Kato Aug 23 '12 at 19:09 @Kato: From what I understand, an immersion implies locally the image is a submanifold. From this it doesn't follow that if it isn't an immersion then the image cannot be locally a submanifold. It seems to me it could go either way? – Mozibur Ullah Aug 23 '12 at 21:29 @MoziburUllah An injective immersion is identified with the canonical injection of a submanifold. Conversely the canonical injection of a submanifold is an immersion. Regards, – Makoto Kato Aug 23 '12 at 21:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9238726496696472, "perplexity": 288.05490317523515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119645676.15/warc/CC-MAIN-20141024030045-00079-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.jobilize.com/trigonometry/test/solving-application-problems-with-geometric-sequences-by-openstax	# 13.3 Geometric sequences (Page 3/6) Page 3 / 6 Given a geometric sequence with ${a}_{2}=4$ and ${a}_{3}=32$ , find ${a}_{6}.$ ${a}_{6}=16,384$ ## Writing an explicit formula for the n Th term of a geometric sequence Write an explicit formula for the $n\text{th}$ term of the following geometric sequence. The first term is 2. The common ratio can be found by dividing the second term by the first term. $\frac{10}{2}=5$ The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula. $\begin{array}{l}{a}_{n}={a}_{1}{r}^{\left(n-1\right)}\hfill \\ {a}_{n}=2\cdot {5}^{n-1}\hfill \end{array}$ The graph of this sequence in [link] shows an exponential pattern. Write an explicit formula for the following geometric sequence. ${a}_{n}=-{\left(-3\right)}^{n-1}$ ## Solving application problems with geometric sequences In real-world scenarios involving arithmetic sequences, we may need to use an initial term of ${a}_{0}$ instead of ${a}_{1}.\text{\hspace{0.17em}}$ In these problems, we can alter the explicit formula slightly by using the following formula: ${a}_{n}={a}_{0}{r}^{n}$ ## Solving application problems with geometric sequences In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year. 1. Write a formula for the student population. 2. Estimate the student population in 2020. 1. The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04. Let $P$ be the student population and $n$ be the number of years after 2013. Using the explicit formula for a geometric sequence we get 2. We can find the number of years since 2013 by subtracting. $2020-2013=7$ We are looking for the population after 7 years. We can substitute 7 for $n$ to estimate the population in 2020. ${P}_{7}=284\cdot {1.04}^{7}\approx 374$ The student population will be about 374 in 2020. A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week. 1. Write a formula for the number of hits. 2. Estimate the number of hits in 5 weeks. 1. The number of hits will be about 333. Access these online resources for additional instruction and practice with geometric sequences. ## Key equations recursive formula for $nth$ term of a geometric sequence ${a}_{n}=r{a}_{n-1},n\ge 2$ explicit formula for $\text{\hspace{0.17em}}nth\text{\hspace{0.17em}}$ term of a geometric sequence ${a}_{n}={a}_{1}{r}^{n-1}$ ## Key concepts • A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant. • The constant ratio between two consecutive terms is called the common ratio. • The common ratio can be found by dividing any term in the sequence by the previous term. See [link] . • The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See [link] and [link] . • A recursive formula for a geometric sequence with common ratio $r$ is given by $\text{\hspace{0.17em}}{a}_{n}=r{a}_{n–1}\text{\hspace{0.17em}}$ for $n\ge 2$ . • As with any recursive formula, the initial term of the sequence must be given. See [link] . • An explicit formula for a geometric sequence with common ratio $r$ is given by $\text{\hspace{0.17em}}{a}_{n}={a}_{1}{r}^{n–1}.$ See [link] . • In application problems, we sometimes alter the explicit formula slightly to $\text{\hspace{0.17em}}{a}_{n}={a}_{0}{r}^{n}.\text{\hspace{0.17em}}$ See [link] . Find the possible value of 8.5 using moivre's theorem which of these functions is not uniformly cintinuous on (0, 1)? sinx which of these functions is not uniformly continuous on 0,1 solve this equation by completing the square 3x-4x-7=0 X=7 Muustapha =7 mantu x=7 mantu 3x-4x-7=0 -x=7 x=-7 Kr x=-7 mantu 9x-16x-49=0 -7x=49 -x=7 x=7 mantu what's the formula Modress -x=7 Modress new member siame what is trigonometry deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent Thomas solve for me this equational y=2-x what are you solving for Alex solve x Rubben you would move everything to the other side leaving x by itself. subtract 2 and divide -1. Nikki then I got x=-2 Rubben it will b -y+2=x Alex goodness. I'm sorry. I will let Alex take the wheel. Nikki ouky thanks braa Rubben I think he drive me safe Rubben how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m More example of algebra and trigo What is Indices If one side only of a triangle is given is it possible to solve for the unkown two sides? cool Rubben kya Khushnama please I need help in maths Okey tell me, what's your problem is? Navin the least possible degree ? (1+cosA)(1-cosA)=sin^2A good Neha why I'm sending you solved question Mirza Teach me abt the echelon method Khamis exact value of cos(π/3-π/4) What is differentiation?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 26, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702627062797546, "perplexity": 657.6576806733442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145260.40/warc/CC-MAIN-20200220162309-20200220192309-00215.warc.gz"}
https://math.stackexchange.com/questions/2341199/explaining-why-the-solutions-of-a-determinant-form-a-line	Explaining why the solutions of a determinant form a line Consider two distinct points \begin{bmatrix}a_1\\b_1\end{bmatrix} and \begin{bmatrix}a_2\\b_2\end{bmatrix} in the plane. Explain why the solutions \begin{bmatrix}x_1\\x_1\end{bmatrix} of the equation det(A) = 0 form a line and why this line goes through the two points \begin{bmatrix}a_1\\b_1\end{bmatrix} and \begin{bmatrix}a_2\\b_2\end{bmatrix}. A = \begin{bmatrix}1&1&1\\x_1&a_1&b_1\\x_2&a_2&b_2\end{bmatrix}. So far I just took the determinant of the matrix and got: $a_1b_2 + b_1x_2 + x_1a_2 - a_1x_2 - a_2b_1 - b_2x_1$ and set it equal to zero as per the question. So I'm left with: $a_1b_2 + b_1x_2 + x_1a_2 - a_1x_2 - a_2b_1 - b_2x_1 = 0$ I'm not sure where to go from here. • This is best explained considering the projective plane: the point $(x_1,x_2)$, when you embed the affine plane in the projective plane via $(x_1,x_2)\longmapsto (x_1:x_2:1)$ is on the line defined by the points $(a_1,a_2)$ and $(b_1,b_2)$ if and only if the corresponding projective points are collinear. – Bernard Jun 29 '17 at 22:16 Use another characterisation of $\det A$ being zero: \begin{align} \det A = 0 &⇔ \text{the columns of $A$ are linearly dependent} \\ &⇔ ∃λ,μ,ν ∈ F \colon\quad λ\begin{bmatrix}x_1 \\ x_2\end{bmatrix} + μ\begin{bmatrix}a_1 \\ a_2\end{bmatrix} + ν\begin{bmatrix}b_1 \\ b_2\end{bmatrix} = 0 \quad\text{and}\quad λ + μ + ν = 0\\ &\hphantom{⇔}\text{and not all of $λ, μ, ν$ are zero}. \end{align} Here, $F$ is the underlying field (probably $F = ℝ$) and the last equivalence comes from splitting up the nontrivial linear combinations of the columns into their first and the last two components respectively. If $λ = 0$, then $μ = -ν$ and both are nonzero, so $\begin{bmatrix}a_1 \\ a_2\end{bmatrix} = \begin{bmatrix}b_1 \\ b_2\end{bmatrix}$, which is absurd since both points are assumed to be distinct. What happens if $λ ≠ 0$? Furthermore, what happens if $\begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ lies on the line through $\begin{bmatrix}a_1 \\ a_2\end{bmatrix}$ and $\begin{bmatrix}b_1 \\ b_2\end{bmatrix}$?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.998629093170166, "perplexity": 170.55779864484688}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525312.3/warc/CC-MAIN-20190717141631-20190717163631-00464.warc.gz"}
https://www.arxiv-vanity.com/papers/1801.01199/	# Strong CO+ and N+2 Emission in Comet C/2016 R2 (Pan-STARRS) Anita L. Cochran McDonald Observatory, University of Texas at Austin Adam J. McKay NASA Goddard Space Flight Center/Universities Space Research Association ###### Abstract We report on imaging and spectroscopic observations of comet C/2016 R2 (Pan-STARRS) obtained with the 0.8 m and 2.7 m telescopes of McDonald Observatory in November and December 2017 respectively. The comet was at a heliocentric distance greater than 3 au during both sets of observations. The images showed a well-developed tail with properties that suggested it was an ion tail. The spectra confirmed that we were observing well-developed bands of CO and N. The N detection was unequivocally cometary and was one of the strongest bands of N detected in a comet spectrum. We derived the ratio of these two ions and from that we were able to derive that N/CO = 0.15. This is the highest such ratio reported for a comet. 111This paper includes data taken at The McDonald Observatory of The University of Texas at Austin ## 1 Introduction Comets represent leftovers from the origins of the Solar System and are an amalgam of various ices and dust. When perturbed into the inner Solar System, they get heated and the ices sublime forming the coma and any tails. The most volatile species are sublimed first and are generally exhausted from near the surface leaving less volatile ices to sublime in subsequent solar passages. Spectra of comets consist of emissions from the sublimed ices, including neutrals and ions, along with a continuum of solar light reflected off the dust. Comet C/2016 R2 (Pan-STARRS) (hereafter R2) was discovered by the Pan-STARRS telescope on 7 September 2016. With an orbital period of years and semi-major axis of AU, this comet came from the Oort cloud, but was not a dynamically new comet. Its perihelion distance will be at 2.6 AU in May 2018. In this letter we report on optical imaging and spectroscopic observations of this comet obtained at The University of Texas McDonald Observatory in November–December 2017. ## 2 Observations We imaged R2 using the narrow-band Hale-Bopp comet filters (Farnham et al., 2000) on 15 November 2017 UT using the Prime Focus Corrector Camera on the 0.76 m telescope at McDonald Observatory. This camera has a 4646 arcmin field-of-view with 1.35 arcsec pixels. A log of observations is given in Table 1. Images were obtained with filters intended to isolate emissions of OH, CN, C and two continuum regions in the blue and NUV. Figure 1 shows the images obtained with the CN and C filters. Of note in this figure is the extremely well developed tail seen in both filters but substantially stronger and more developed in the CN image. It is rare to see a well developed tail at such a large heliocentric distance. In addition, it is extremely unusual to see the tail more developed in the CN filter. Our first idea was that this was an ion tail rather than a dust tail, but even that is uncommon at such a large heliocentric distance. Consultation with others in the field also suggested that the tail must be ionic (Schleicher, Farnham, Knight, personal communications, 2017). In order to confirm our hypothesis of ionic emission contaminating our narrow-band imaging, we followed up the images with spectroscopy. Spectra were obtained of R2 using the Tull 2DCoude spectrograph (Tull et al., 1995) on the Harlan J. Smith 2.7 m telescope of McDonald Observatory on 8–10 December 2017 UT. Details of the observations are in Table 1. The spectrograph was used with a 1.2 arcsec wide by 8 arcsec tall slit centered on the optocenter for all observations, yielding a resolving power () of 60,000. The brightness of R2 was around V total. It was immediately evident that this comet’s spectrum was different than most comets when we read out the first spectrum on 8 December. Missing was the strong CN band at 3880 Å that is normally one of the strongest emission features observed in optical spectra of comets, and something we have observed with even slightly fainter comets when they are closer to the Sun. Missing also was any hint of the other usual molecules, C, C, CH, or NH. Instead there was a well-developed series of bands scattered from approximately 3700 Å to 5100 Å. Most of the bands either degraded to the red or were peaked near the center of the band. There was one band that degraded blueward near 3900 Å. The majority of the detected strong bands can be attributed to CO. We detected the CO (4,0), (3,0), (2,0), (1,0), (4,2), (3,2), (2,1) and (1,1) A – X bands. Figure 2 shows the ladder of the CO (2,0) band, though we actually observed both ladders. As can be seen from the figure, the CO bands are quite complex, with many perturbations and satellite lines. The wavelengths for these bands come from Kuo et al. (1986), Haridass et al. (1992) and Haridass et al. (2000). The blue degrading band can be attributed to the B –X (0,0) band of N with a bandhead at 3914Å. Figure 3 shows this band with the P and R branches marked (since the band is a transition it does not have a Q branch). The wavelengths and structure of the band comes from Dick et al. (1978). In this figure note that we see R-branch lines up through J=16 (and quite possibly to J=18). Note also, that the odd J-level R-branch lines are weaker than the even ones, as would be expected for this homonuclear molecule. There is also evidence for the lowest J-levels of the much weaker (0,1) band with bandhead at 4278 Å. When N is observed in a cometary spectrum it is often erroneously attributed to being from the comet; N is excited in the Earth’s atmosphere, especially near dusk and dawn, and it is the telluric lines that are most often what are detected. Cochran et al. (2000) derived very tight upper limits for N for comets 122P/deVico and C/1995 O1 (Hale-Bopp). Other ionic species, including CO, were observed in these cometary spectra. This raises the question of why we believe that the N seen in R2’s spectrum is cometary in nature. The evidence is actually quite strong. First, the comet was observed in the middle of the night, when we would not expect much telluric emission. Second, other comets observed on the same nights did not show this band at all. Third, the band was observed in all spectra on all three nights. Finally, the lines are precisely at the correct wavelength for the cometary Doppler shift and are not coincident with the telluric restframe. This last statement relies on the high resolving power of the coudé spectra for certainty. In addition to these clearly defined bands, we see evidence of the forbidden oxygen transitions of O (D) and possibly O (S). We also see some additional emission lines that we have yet to identify. However, we can eliminate emissions due to CO, CH and HO. ## 3 Analysis and Implications Since R2 cannot be on its first passage into the inner Solar System, it is surprising to detect two such volatile species and nothing else. But these two species are interesting to observe as we expect preferentially for C to be bound into CO or CO and N into N when they formed in the outer Solar System (Lewis & Prinn, 1980; Mousis et al., 2012; Charnley & Rodgers, 2002). With our identifications of the ions of these species, we can determine the N/CO ratio of the ices in this comet. From the band intensity and some physical constants, one can compute the column density as N=Iν′ν′′/gν′ν′′ where N is the column density, I is the integrated band intensity and is the excitation factor. From this it follows that yields the ratio of the column density of N/CO. We measured the band intensity simply by marking a continuum and summing all of the flux above that continuum. For CO we measured both ladders of the (2,0) band separately and summed them. For N, we measured the whole P-branch and R(1) – R(5) together and then added in the additional flux contributions of R(6) – R(16). The N and CO (2,0) bands are close in wavelength but they are still 7 orders apart on the CCD and well off of the grating blaze. Thus, it is likely that the throughput is slightly different for the two orders. We did observe flux standards on each night. However, these two bands are at the wavelengths of the Balmer decrement in the A stars typically used for standards. Thus, there are no calibrations of this region. Instead, we used the solar spectrum from the daytime sky, obtained through the same spectrograph via a ground-glass port, in order to figure the relative flux of these orders when compared with the atlas of Kurucz et al. (1984). We determined we needed to increase the N flux by a factor of 2.0 to have comparable throughput to the CO order. The excitation factor used for CO was photons sec mol (Magnani & A’Hearn, 1986). The excitation factor for N was photons sec mol (Lutz et al., 1993). Putting together these various factors, we found that N/CO = 0.15. Converting from the quantity of the ions to the quantity of the neutrals is dependent on our understanding of the photodestruction branching ratios that are not well understood. One possible source of CO is that some of it comes from dissociation of CO. However, in that case we would expect to see CO in our spectra and we definitely do not detect any. Thus, we assume all the CO comes from the ionization of CO and convert our measured ratio of N/CO to N/CO. Wyckoff & Theobald (1989) argued that one must multiply the ion ratio by two to derive the neutral ratio, while Lutz et al. (1993) argued that no such factor is necessary. The argument of Lutz et al. is consistent with the solar photoionization rates given in Table 3 of Huebner & Mukherjee (2015) so we adopted this argument. Thus, assuming that CO and N are ionized in proportion to the amount of neutrals, this means that N/CO = 0.15. Our measured ratio is much higher than the upper limits on this ratio found for deVico and Hale-Bopp using the same instrument and techniques (Cochran et al., 2000). Their limits ranged from for deVico to for Hale-Bopp. Indeed, it is much higher than other observations of N/CO, as listed in Table III of Cochran et al. Korsun et al. (2014) measured the N/CO in comet C/2002 VQ94 (LINEAR), a comet active at AU, as 0.06. Feldman (2015) placed a 3- upper limit on N/CO of 0.027 for comet C/2001 Q4 (NEAT) using FUSE observations. Ivanova et al. (2016) measured N/CO as 0.013 for comet 29P/Schwassmann-Wachmann 1 at 5.25 au, though the N feature is not well defined in these low-resolution spectra. Using a mass spectrometer on the Rosetta spacecraft in orbit with comet 67P/Churyumov-Gerasimenko, Rubin et al. (2015) measured an N/CO ratio of . This is certainly the most robust measure of this ratio since it was measured in situ, though the closeness of these species in mass makes the measurement subject to model interpretation. Thus, comet R2 shows an N/CO ratio at least a factor of 2 greater than any comet measured so far. Wierzchos & Womack (2017) reported on submillimeter observations of R2, including a detection of the CO J=2–1 rotational line and a non-detection of the HCN J=3–2 rotational transition. They conclude that this comet appears to be very CO-rich. We could potentially use the O (D) 6300Å lines as a proxy to determine the abundance of water. However, as CO can also contribute photons to this line, the derived water value would be suspect. Therefore, we leave detailed analysis of the O (D) 6300Å line in terms of the production of water for a future publication. The ratio of N/CO trapped in the cometary ices is not necessarily identical with the amount in the solar nebula. Owen & Bar-Nun (1995) used studies of deposition of gases into amorphous water ice in the laboratory to show that ices incorporated into comets at around 50 K would have N/CO if N/CO is in the solar nebula. Our measurement of N/CO is within a factor of 2 of their derived value, though ours is higher. Indeed, only some of the older photographic data related by Arpigny (personal communication) and shown in Cochran et al. (2000) come close to the Owen and Bar-Nun prediction. It must be remembered that the ratio of species seen in the gas phase is not necessarily representative of the ratio of the ices in the nucleus. However, CO and N are not terribly reactive with other species so chemical reactions probably do not alter this ratio much. It also means we do not expect to see the ratio change with heliocentric distance. Additionally, Bar-Nun et al. (1988) showed that CO and N should be released in the same proportion as they exist in the ices. Owen and Bar-Nun predicted that there would be higher values of N/CO for dynamically new comets than for older comets. However, as pointed out earlier, R2 has a period of around 20,000 years and a semi-major axis of around 1500 AU suggesting that it has been near the Sun prior to this apparition. Note that the comets with measured N/CO ratios represent a variety of dynamical types, from Jupiter Family comets such as 67P and 29P, to dynamically new comets such as C/1940 R2 (Cunningham). There is no clear trend of the magnitude of this ratio with dynamical type. Comet C/2016 R2 (Pan-STARRS) showed an unusual optical spectrum with strong CO and N emissions and none of the usual neutrals seen in most cometary spectra. This intriguing object showed the strongest and clearest N emissions ever detected with modern digital spectra. Faced with this unusual spectrum, we have alerted many members of the cometary community and they (and we) are requesting follow-up observations with a variety of instruments at all wavelengths in order to try to understand this unusual comet. ALC was supported by NASA Grant NNX17A186G. AJM is funded through the NASA Postdoctoral Program, administered by the Universities Space Research Association. \facilitiesMcD:0.8m, Smith (2DCoude) ## References • Bar-Nun et al. (1988) Bar-Nun, A., Kleinfeld, I., & Kochavi, E. 1988, Phys. Rev. B, 38, 7749 • Charnley & Rodgers (2002) Charnley, S. B. & Rodgers, S. D. 2002, Ap. J., 569, L133 • Cochran et al. (2000) Cochran, A. L., Cochran, W. D., & Barker, E. S. 2000, Icarus, 146, 583 • Dick et al. (1978) Dick, K. A., Benesch, W., Crosswihite, H. M., Tilford, S. G., Gottscho, R. A., & Field, R. W. 1978, J. Mol. Spec, 69, 95 • Farnham et al. (2000) Farnham, T. L., Schleicher, D. G., & A’Hearn, M. F. 2000, Icarus, 147, 180 • Feldman (2015) Feldman, P. D. 2015, Ap. J., 812, 15 • Haridass et al. (1992) Haridass, C., Prasad, C. V. V., & Paddi Reddy, S. 1992, Ap. J., 388, 669 • Haridass et al. (2000) —. 2000, J. Mol. Spec., 199, 180 • Huebner & Mukherjee (2015) Huebner, W. F. & Mukherjee, J. 2015, Plan. and Space Sci., 106, 11 • Ivanova et al. (2016) Ivanova, O. V., Luk‘yanyk, I. V., Kiselev, N. N., Afanasiev, V. L., Picazzio, E., Cavichia, O., de Almeida, A. A., & Andrievsky, S. M. 2016, Plan. and Space Sci., 121, 10 • Korsun et al. (2014) Korsun, P. P., Rousselot, P., Kulyk, I. V., Afanasiev, V. L., & Ivanova, O. V. 2014, Icarus, 232, 88 • Kuo et al. (1986) Kuo, C.-H., Milkman, I. W., Steimie, T. C., & Moseley, J. T. 1986, J. Chem. Phys., 85, 4269 • Kurucz et al. (1984) Kurucz, R. L., Furenlid, I., & Brault, J. 1984, Solar flux atlas from 296 to 1300 nm (Sunspot, New Mexico: National Solar Observatory Atlas) • Lewis & Prinn (1980) Lewis, J. S. & Prinn, R. G. 1980, Ap. J., 238, 357 • Lutz et al. (1993) Lutz, B. L., Womack, M., & Wagner, R. M. 1993, Ap. J., 407, 402 • Magnani & A’Hearn (1986) Magnani, L. & A’Hearn, M. F. 1986, Ap. J., 302, 477 • Mousis et al. (2012) Mousis, O., Guilbert-Lepoutre, A., Lunine, J. I., Cochran, A. L., Waite, J. H., Petit, J.-M., & Rousselot, P. 2012, Ap. J., 757, 146 • Owen & Bar-Nun (1995) Owen, T. & Bar-Nun, A. 1995, Icarus, 116, 215 • Tull et al. (1995) Tull, R. G., MacQueen, P. J., Sneden, C., & Lambert, D. L. 1995, Pub. A.S.P., 107, 251 • Wierzchos & Womack (2017) Wierzchos, K. & Womack, M. 2017, Central Bureau Electronic Telegrams, 4464 • Wyckoff & Theobald (1989) Wyckoff, S. & Theobald, J. 1989, Adv. Space Res., 9, 157	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9371154308319092, "perplexity": 3317.6532184544863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155268.80/warc/CC-MAIN-20210805000836-20210805030836-00638.warc.gz"}
https://www.giancolianswers.com/giancoli-physics-7th-edition-solutions/chapter-2/problem-42	## You are here For the 2nd part of the question, why can't you use the formula Vf=Vo+2at? Thanks. Hi sheumangutman, Thanks for the question. I think you mean to suggest $v_f = v_o + at$, correct? That approach would be equally fine, provided you establish that $v_f = -v_o$, which is true since it returns back to the original launch height. This means your suggested formula, with a substitution for $v_f$, would become $-v_o = v_o + at$ which rearranges to $t = \dfrac{-2v_o}{a}$ which is the same formula shown at 1:20 in the video. You have to be a little cautious with your suggested formula, however, since it works only when you can tell how the final velocity compares with the initial velocity. $v_f = -v_o$ only when something returns to it's original height. If something fell into a hole after being launched upwards, meaning it returns to a different height, you would be better off using the $d=v_ot + \dfrac{1}{2}at^2$ formula. For this problem, however, the formula you suggest would be just fine. All the best, Mr. Dychko For part c) What specific factors make this an estimate? Hi Icbishop, did you post this question on the wrong video? I don't see a part c) for this problem. Cheers, Mr. Dychko Well I double checked and I do indeed have a part c on #42 chapter 2. It is the 7th Edition as well. The question also matches up with the solution provided. Ah, right you are Icbishop. Sorry about that, I just checked the video without looking at the textbook. I've added some notes to the quick answer above the video. Cheers, Mr. Dychko By dividing by "t" instead of factoring aren't we losing a physically meaningful answer t=0 (the time the ball was hit)? Hi idan, you are making a valid point. You have a sharp eye! Mathematically, it would be more correct to factor and then find the roots of the resulting equation, thereby discovering the answer $t=0$. However, since this question is asking for the "time in the air", $t=0$, as you know, is an extraneous solution, so the effort of finding the solution $t=0$ and then discarding it as extraneous isn't worth the effort. The technique in the video is perfectly fine for a physics class, since math is just a tool for finding solutions to the physical problem, which in this case is "how much time is the ball in the air", not "at what times is the ball at height zero". For the latter, one must follow the technique you suggest by factoring since $t=0$ would be a non-trivial solution, but as it is, $t=0$ is not a solution for determining the "time in the air" problem. In a math class there's no question that to answer "solve this equation", one must factor and include $t=0$ as a solution. All the best, Mr. Dychko Question #39 a ball is thrown straight up with a speed of 36 m/s. how long does it take to return to its starting point. Question is my teacher teaches out of 7th edition I have 6th edition but can not find this question in my book or on this website.It says on top of my page ch# 2 and page ref 2-7 but cant find it also how come you don't post question you just go to answers only . need to see question sometimes so I can find which edition it is in. Hi EddieG, thanks for the question. I would love to post the questions, but I've avoided that since it would be a copyright issue with the publisher since I didn't create the questions. Perhaps you can ask your teacher to photocopy just the problems from their 7th Edition text for your reference? All the best, Mr. Dychko	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.832145094871521, "perplexity": 377.4258364934951}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583657557.2/warc/CC-MAIN-20190116175238-20190116201238-00484.warc.gz"}

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