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-{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-17T00:49:26","documenter_version":"1.1.1"}}
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previews/PR178/anisotropy.html

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 \hat{\mathcal{O}}_{6,\pm2} & =\phi_{\pm}(\hat{S}_{+}^{2}\pm \hat{S}_{-}^{2})(33\hat{S}_{z}^{4}-(18X+123)\hat{S}_{z}^{2}+X^{2}+10X+102)+\mathrm{h.c.}\\
 \hat{\mathcal{O}}_{6,\pm1} & =\phi_{\pm}(\hat{S}_{+}\pm \hat{S}_{-})(33\hat{S}_{z}^{5}-(30X-15)\hat{S}_{z}^{3}+(5X^{2}-10X+12)\hat{S}_{z})+\mathrm{h.c.}\\
 \hat{\mathcal{O}}_{6,0} & =231\hat{S}_{z}^{6}-(315X-735)\hat{S}_{z}^{4}+(105X^{2}-525X+294)\hat{S}_{z}^{2}-5X^{3}+40X^{2}-60X
-\end{align*}\]</p><p>Stevens operators <span>$\hat{\mathcal{O}}_{k,q}$</span> for odd <span>$k$</span> are disallowed from the single-ion anisotropy under the assumption of time-reversal symmetry. Computer-generated tables of Stevens operators with larger k are available from C. Rudowicz and C. Y. Chung, J. Phys.: Condens. Matter 16, 5825 (2004).</p><p>For each <span>$k$</span> value, the collection of operators <span>$\{\hat{\mathcal{O}}_{k,q&#39;}\}$</span> for <span>$q&#39; = -k, \dots, k$</span> is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform <span>$\hat{\mathcal{O}}_{k,q}$</span> into a linear combination of <span>$\hat{\mathcal{O}}_{k,q&#39;}$</span> where <span>$q&#39;$</span> varies but <span>$k$</span> remains fixed. </p><p>In taking the large-<span>$S$</span> limit, each dipole operator is replaced by its expectation value <span>$\mathbf{s} = \langle \hat{\mathbf{S}} \rangle$</span>, and only leading-order terms are retained. The operator <span>$\hat{\mathcal{O}}_{k,q}$</span> becomes a homogeneous polynomial <span>$O_{k,q}(\mathbf{s})$</span> of order <span>$k$</span> in the spin components. One can see these polynomials by constructing <a href="library.html#Sunny.stevens_matrices-Tuple{Any}"><code>stevens_matrices</code></a> with the argument <code>S = Inf</code>. Due to the normalization constraint, each dipole can be expressed in polar angles, <span>$(\theta, \phi)$</span>. Then the Stevens functions <span>$O_{k,q}(\mathbf{s})$</span> correspond to the spherical harmonic functions <span>$Y_l^m(\theta, \phi)$</span> where <span>$l=k$</span> and <span>$m=q$</span>, and modulo <span>$k$</span> and <span>$q$</span>-dependent rescaling factors.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="structure-factor.html">« Structure Factor Calculations</a><a class="docs-footer-nextpage" href="library.html">Library 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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.1 on <span class="colophon-date" title="Tuesday 17 October 2023 00:49">Tuesday 17 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{align*}\]</p><p>Stevens operators <span>$\hat{\mathcal{O}}_{k,q}$</span> for odd <span>$k$</span> are disallowed from the single-ion anisotropy under the assumption of time-reversal symmetry. Computer-generated tables of Stevens operators with larger k are available from C. Rudowicz and C. Y. Chung, J. Phys.: Condens. Matter 16, 5825 (2004).</p><p>For each <span>$k$</span> value, the collection of operators <span>$\{\hat{\mathcal{O}}_{k,q&#39;}\}$</span> for <span>$q&#39; = -k, \dots, k$</span> is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform <span>$\hat{\mathcal{O}}_{k,q}$</span> into a linear combination of <span>$\hat{\mathcal{O}}_{k,q&#39;}$</span> where <span>$q&#39;$</span> varies but <span>$k$</span> remains fixed. </p><p>In taking the large-<span>$S$</span> limit, each dipole operator is replaced by its expectation value <span>$\mathbf{s} = \langle \hat{\mathbf{S}} \rangle$</span>, and only leading-order terms are retained. The operator <span>$\hat{\mathcal{O}}_{k,q}$</span> becomes a homogeneous polynomial <span>$O_{k,q}(\mathbf{s})$</span> of order <span>$k$</span> in the spin components. One can see these polynomials by constructing <a href="library.html#Sunny.stevens_matrices-Tuple{Any}"><code>stevens_matrices</code></a> with the argument <code>S = Inf</code>. Due to the normalization constraint, each dipole can be expressed in polar angles, <span>$(\theta, \phi)$</span>. Then the Stevens functions <span>$O_{k,q}(\mathbf{s})$</span> correspond to the spherical harmonic functions <span>$Y_l^m(\theta, \phi)$</span> where <span>$l=k$</span> and <span>$m=q$</span>, and modulo <span>$k$</span> and <span>$q$</span>-dependent rescaling factors.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="structure-factor.html">« Structure Factor Calculations</a><a class="docs-footer-nextpage" href="library.html">Library 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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.1 on <span class="colophon-date" title="Tuesday 17 October 2023 02:25">Tuesday 17 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>