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{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-30T08:23:45","documenter_version":"1.8.0"}}
{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-31T03:04:49","documenter_version":"1.8.0"}}
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · CGcoefficient.jl</title><meta name="title" content="Home · CGcoefficient.jl"/><meta property="og:title" content="Home · CGcoefficient.jl"/><meta property="twitter:title" content="Home · CGcoefficient.jl"/><meta name="description" content="Documentation for CGcoefficient.jl."/><meta property="og:description" content="Documentation for CGcoefficient.jl."/><meta property="twitter:description" content="Documentation for CGcoefficient.jl."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" 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type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href><img src="assets/logo.svg" alt="CGcoefficient.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href>CGcoefficient.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Install"><span>Install</span></a></li><li><a class="tocitem" href="#Usage"><span>Usage</span></a></li><li><a class="tocitem" href="#About"><span>About</span></a></li></ul></li><li><a class="tocitem" href="api/">API</a></li><li><span class="tocitem">Formula</span><ul><li><a class="tocitem" href="wigner/">Wigner Symbols</a></li><li><a class="tocitem" href="lsjj/">LS-jj recoupling</a></li><li><a class="tocitem" href="moshinsky/">Moshinsky</a></li><li><a class="tocitem" href="pfrational/">Prime Factorization</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/0382/CGcoefficient.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/0382/CGcoefficient.jl/blob/master/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Home"><a class="docs-heading-anchor" href="#Home">Home</a><a id="Home-1"></a><a class="docs-heading-anchor-permalink" href="#Home" title="Permalink"></a></h1><p>A package to calculate CG-coefficient, Racha coefficient, and Wigner 3j, 6j, 9j symbols. We offer three version API for all these coeddicients.</p><ul><li><strong>normal version</strong>: <code>CG, threeJ, SixJ, Racah, nineJ</code>. Their parameters can be <code>Integer</code> or <code>Rational</code> (half integer), and their result is simplified by default.</li><li><strong>double parameters version</strong>: <code>dCG, d3j, d6j, dRacah d9j</code>. Their parameters means double of the real angular momentum, so can only be <code>Integer</code>. The result is not simplified. We will explain what is <code>simplify</code> later.</li><li><strong>float version</strong>: <code>fCG, f3j, f6j, fRacah, f9j</code>. Their parameters is same as double parameter version. They use <code>Float64</code> for calculation, so they are not exact but are fast for numeric calculation. Because these function use stored <code>binomial</code> result for speed up calculation, you should reserve space before calculate large angular momentum coefficients. See <code>wigner_init_float</code> function for details.</li></ul><h2 id="Install"><a class="docs-heading-anchor" href="#Install">Install</a><a id="Install-1"></a><a class="docs-heading-anchor-permalink" href="#Install" title="Permalink"></a></h2><p>Just install with Julia REPL and enjoy it.</p><pre><code class="language-julia-repl hljs">pkg&gt; add CGcoefficient</code></pre><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><pre><code class="language-julia hljs">using CGcoefficient
sixJ(1,2,3,4,5,6)</code></pre><p class="math-container">\[\frac{1}{3}\sqrt{\frac{2}{715}}\]</p><p>In a markdown enviroment, such as jupyter notebook, it will give you a latex output.</p><pre><code class="language-julia hljs">d6j(2,4,6,8,10,12)</code></pre><p class="math-container">\[\frac{572}{3}\sqrt{\frac{1}{116968280}}\]</p><p>The <code>d</code> version functions do not simplify the result for the seek of speed, because <code>simplify</code> needs prime factorization which is slow. You can simplify the result explicitly,</p><pre><code class="language-julia hljs">simplify(d6j(2,4,6,8,10,12))</code></pre><p class="math-container">\[\frac{1}{3}\sqrt{\frac{2}{715}}\]</p><p>You can also do some arithmetics with the result, thus do arithmetics using the <code>SqrtRational</code> type. The result is also not simplified</p><pre><code class="language-julia hljs">x = sixJ(1,2,3,4,5,6) * exact_sqrt(1//7) * exact_sqrt(1//13) * iphase(2+3+5+6)
simplify(x)</code></pre><p class="math-container">\[\frac{1}{39}\sqrt{\frac{2}{385}}\]</p><p>In a console enviroment it will give out a text output.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; nineJ(1,2,3,5,4,3,6,6,0)</code><code class="nohighlight hljs ansi" style="display:block;">1//39√(2//385)</code></pre><p>You can also use <code>print</code> function to force print a text output.</p><pre><code class="language-julia hljs">print(Racah(1,2,3,2,1,2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1//5√(1//21)</code></pre><h2 id="About"><a class="docs-heading-anchor" href="#About">About</a><a id="About-1"></a><a class="docs-heading-anchor-permalink" href="#About" title="Permalink"></a></h2><p>This package is inspired by Ref <sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. See <a href="https://github.com/ManyBodyPhysics/CENS/blob/master/MBPT/VEffective/bhf-modules.f90">CENS-MBPT</a> for details.</p><p>The idea is to simplify 3nj Symbols to sum combinations of binomial coefficients. We can calculate binomial coefficients by Pascal&#39;s Triangle, and store them first. Then we calculate 3nj Symbols using the stored binomial coefficients.</p><p>In this package, we just use the builtin <code>binomial</code> function for exact calculation. Only the float version uses stored <code>binomial</code>s.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. <a href="https://github.com/ManyBodyPhysics/CENS">https://github.com/ManyBodyPhysics/CENS</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="api/">API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Monday 30 December 2024 08:23">Monday 30 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
simplify(x)</code></pre><p class="math-container">\[\frac{1}{39}\sqrt{\frac{2}{385}}\]</p><p>In a console enviroment it will give out a text output.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; nineJ(1,2,3,5,4,3,6,6,0)</code><code class="nohighlight hljs ansi" style="display:block;">1//39√(2//385)</code></pre><p>You can also use <code>print</code> function to force print a text output.</p><pre><code class="language-julia hljs">print(Racah(1,2,3,2,1,2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1//5√(1//21)</code></pre><h2 id="About"><a class="docs-heading-anchor" href="#About">About</a><a id="About-1"></a><a class="docs-heading-anchor-permalink" href="#About" title="Permalink"></a></h2><p>This package is inspired by Ref <sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. See <a href="https://github.com/ManyBodyPhysics/CENS/blob/master/MBPT/VEffective/bhf-modules.f90">CENS-MBPT</a> for details.</p><p>The idea is to simplify 3nj Symbols to sum combinations of binomial coefficients. We can calculate binomial coefficients by Pascal&#39;s Triangle, and store them first. Then we calculate 3nj Symbols using the stored binomial coefficients.</p><p>In this package, we just use the builtin <code>binomial</code> function for exact calculation. Only the float version uses stored <code>binomial</code>s.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. <a href="https://github.com/ManyBodyPhysics/CENS">https://github.com/ManyBodyPhysics/CENS</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="api/">API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Tuesday 31 December 2024 03:04">Tuesday 31 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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