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@ -8,7 +8,7 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Documentation of user libs (user functions) &mdash; musrfit 1.8.2 documentation</title>
<title>Documentation of user libs (user functions) &mdash; musrfit 1.9.0 documentation</title>
@ -58,6 +58,10 @@
<div class="version">
1.9
</div>
@ -95,6 +99,7 @@
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="#nonlocal-superconductivity-related-meissner-screening-functions-as-libs">Nonlocal superconductivity related Meissner screening functions (AS libs)</a></li>
<li class="toctree-l2"><a class="reference internal" href="#depth-resolved-information-as-libs">Depth resolved information (AS libs)</a></li>
<li class="toctree-l2"><a class="reference internal" href="#functions-to-analyze-bgr-nmr-data-bnmr-libs">Functions to analyze β-NMR data (BNMR libs)</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#libbnmr">libBNMR</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#functions">Functions</a></li>
@ -498,11 +503,120 @@ The expected name of the <code class="docutils literal notranslate"><span class=
</div>
</div>
<div class="section" id="nonlocal-superconductivity-related-meissner-screening-functions-as-libs">
<h2>Nonlocal superconductivity related Meissner screening functions (AS libs)<a class="headerlink" href="#nonlocal-superconductivity-related-meissner-screening-functions-as-libs" title="Permalink to this headline"></a></h2>
<p>To be written yet …</p>
<span id="nonlocal-libs"></span><span id="index-14"></span><h2>Nonlocal superconductivity related Meissner screening functions (AS libs)<a class="headerlink" href="#nonlocal-superconductivity-related-meissner-screening-functions-as-libs" title="Permalink to this headline"></a></h2>
<p>This library allows to calculate the magnetic field profile <span class="math notranslate nohighlight">\(B(z)\)</span> for nonlocal superconductors.
For details see <a class="reference external" href="http://dx.doi.org/10.1103/PhysRevLett.95.197201">A. Suter, et al., PRB 72, 024506 (2005)</a>, and references therein.</p>
<p>The provided function calculates the muon spin polarization</p>
<div class="math notranslate nohighlight">
\[P(t, E) = \int n(z, E)\, \cos(\gamma_\mu B(z) t + \phi) \, dz,\]</div>
<p>where <span class="math notranslate nohighlight">\(B(z)\)</span> is calculated in the limit of specular reflection.
The corresponding user function is called as</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libPNL_PippardFitter</span> <span class="n">PNL_PippardFitter</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span> <span class="mi">5</span> <span class="mi">6</span> <span class="mi">7</span> <span class="mi">8</span> <span class="mi">9</span>
</pre></div>
</div>
<p>with the parameters</p>
<ol class="arabic simple">
<li>implantation energy in (keV).</li>
<li>reduced temperature <span class="math notranslate nohighlight">\(t=T/T_c\)</span>.</li>
<li>thickness in (nm).</li>
<li>electron mean path, <span class="math notranslate nohighlight">\(\ell\)</span> in (nm).</li>
<li>superconducting coherence length, <span class="math notranslate nohighlight">\(\xi\)</span> in (nm).</li>
<li>London penetration length, <span class="math notranslate nohighlight">\(\lambda_{\rm L}\)</span> in (nm).</li>
<li>external magnetic field strength in (G).</li>
<li>the effective detector phase, <span class="math notranslate nohighlight">\(\varphi\)</span> in <span class="math notranslate nohighlight">\((^\circ)\)</span>.</li>
<li>a “dead layer” thickness in (nm).</li>
</ol>
<p>Typically this function needs to be multiplied by a Gaussian in order to take into account: nuclear dipole broadening, partial trapped flux, etc.</p>
<p>In order to find the muon stopping profile, <span class="math notranslate nohighlight">\(n(z,E)\)</span>, needed for the calculation, the library needs to find the corresponding trimsp
rge-files (muon stoppping profiles). For this the library reads at start-up the following xml-file (example):</p>
<div class="highlight-xml notranslate"><div class="highlight"><pre><span></span><span class="cp">&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;</span>
<span class="nt">&lt;nonlocal</span> <span class="na">xmlns=</span><span class="s">&quot;http://nemu.web.psi.ch/musrfit/nonlocal&quot;</span><span class="nt">&gt;</span>
<span class="nt">&lt;comment&gt;</span>
nonlocal_startup.xml
<span class="nt">&lt;/comment&gt;</span>
<span class="nt">&lt;nonlocal_par&gt;</span>
<span class="nt">&lt;fourier_points&gt;</span>262144<span class="nt">&lt;/fourier_points&gt;</span>
<span class="nt">&lt;/nonlocal_par&gt;</span>
<span class="nt">&lt;trim_sp&gt;</span>
<span class="nt">&lt;data_path&gt;</span>./profiles/<span class="nt">&lt;/data_path&gt;</span>
<span class="nt">&lt;rge_fln_pre&gt;</span>Sn_E<span class="nt">&lt;/rge_fln_pre&gt;</span>
<span class="nt">&lt;energy_list&gt;</span>
<span class="nt">&lt;energy&gt;</span>1000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>2000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>4000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>6000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>8000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>10000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>12000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>14100<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>18000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>22000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>25000<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;energy&gt;</span>27300<span class="nt">&lt;/energy&gt;</span>
<span class="nt">&lt;/energy_list&gt;</span>
<span class="nt">&lt;/trim_sp&gt;</span>
<span class="nt">&lt;/nonlocal&gt;</span>
</pre></div>
</div>
<p>Here the number of Fourier points needed in the calculation can be defined (<code class="docutils literal notranslate"><span class="pre">fourier_points</span></code>). The <code class="docutils literal notranslate"><span class="pre">trim_sp</span></code> section
contains all the information needed to load the proper muon stopping profiles. <code class="docutils literal notranslate"><span class="pre">data_path</span></code> is the path to the needed rge-files.
<code class="docutils literal notranslate"><span class="pre">rge_fln_pre</span></code> is the rge-file prefix, and <code class="docutils literal notranslate"><span class="pre">energy</span></code> are all the energy tags. E.g. <code class="docutils literal notranslate"><span class="pre">./profile/Sn_E1000.rge</span></code> would be the first
muon stopping profile for an energy of <span class="math notranslate nohighlight">\(E=1000\)</span> (eV).</p>
<p>The name of the xml-file has to be <code class="docutils literal notranslate"><span class="pre">nonlocal_startup.xml</span></code> and needs to be placed in the directory where the analysis takes place, i.e.
in the directory of all the msr-files.</p>
</div>
<div class="section" id="depth-resolved-information-as-libs">
<span id="depthprof-lib"></span><span id="index-15"></span><h2>Depth resolved information (AS libs)<a class="headerlink" href="#depth-resolved-information-as-libs" title="Permalink to this headline"></a></h2>
<p>A method to extract depth-resolved information from the implantation energy dependence of the experimental parameters in a low-energy
muon spin spectroscopy experiment. For details see <a class="reference external" href="https://doi.org/10.1063/1.5126529">A. F. A. Simões, et al. Review of Scientific Instruments. 2020; 91(2): 023906 (7 pp.)</a>.</p>
<p>If you have a layered material (e.g. <span class="math notranslate nohighlight">\(N\)</span> layers), properties like the asymmetry might depend on the layer in which the muons are stopped.
For instance there might be different probabilities for muonium formation depending on the material, charge transfer layers, etc.</p>
<p>Since the muon stopping distribution, <span class="math notranslate nohighlight">\(n(z)\)</span>, has some finite range, these properties will be smeared out. For the following we define
the stopping probability, <span class="math notranslate nohighlight">\(p_{i}(E)\)</span> for a finite slice <span class="math notranslate nohighlight">\(i\)</span>, ranging from <span class="math notranslate nohighlight">\(z \in [a, b]\)</span> as</p>
<div class="math notranslate nohighlight">
\[p_{i}(E) = \int_a^b n(z,E) \, dz\]</div>
<p>Furthermore it is assumes that there is a sharp transition between the layers of the property of interest, e.g. the diamagnetic fraction, <span class="math notranslate nohighlight">\(f_i\)</span>.
Hence the measured property as function of energy is</p>
<div class="math notranslate nohighlight">
\[f(E) = \sum_{i=1}^N p_{i}(E) \cdot f_i.\]</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Currently it is recommended to read in the data in ASCII or DAT format as a non-μSR fit <a class="reference internal" href="user-manual.html#non-musr-fit"><span class="std std-ref">(fit type 8)</span></a>.</p>
</div>
<p>The user library for the depth profile analysis looks for a <strong>3 layer material</strong>, assuming one is looking for the diamagnetic fraction, like</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1">###############################################################</span>
<span class="n">FITPARAMETER</span>
<span class="c1"># Nr. Name Value Step Pos_Error Boundaries</span>
<span class="mi">1</span> <span class="n">f1</span> <span class="mf">0.54540</span> <span class="mf">0.00072</span> <span class="n">none</span> <span class="mi">0</span> <span class="mi">1</span>
<span class="mi">2</span> <span class="n">f2</span> <span class="mf">0.23957</span> <span class="mf">0.00048</span> <span class="n">none</span>
<span class="mi">3</span> <span class="n">f3</span> <span class="mf">0.05615</span> <span class="mf">0.00047</span> <span class="n">none</span> <span class="mi">0</span> <span class="mi">1</span>
<span class="mi">4</span> <span class="n">x1</span> <span class="mf">63.5000</span> <span class="mf">0.0014</span> <span class="n">none</span>
<span class="mi">5</span> <span class="n">x2</span> <span class="mf">101.5001</span> <span class="mf">0.0044</span> <span class="n">none</span>
<span class="c1">###############################################################</span>
<span class="n">THEORY</span>
<span class="n">userFcn</span> <span class="n">libPDepthProfile</span> <span class="n">PDepthProfile</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span> <span class="mi">5</span>
</pre></div>
</div>
<p>Here, <span class="math notranslate nohighlight">\(f1\)</span> is the diamagnetic fraction of the first layer, etc. <span class="math notranslate nohighlight">\(x1\)</span> is the thickness of the first layer, etc.</p>
<p>In order to find the muon stopping profile, <span class="math notranslate nohighlight">\(n(z,E)\)</span>, needed for the calculation, the library needs to find the corresponding trimsp
rge-files (muon stoppping profiles). For this the library reads at start-up the following xml-file (example):</p>
<div class="highlight-xml notranslate"><div class="highlight"><pre><span></span><span class="cp">&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;</span>
<span class="nt">&lt;depthProf</span> <span class="na">xmlns=</span><span class="s">&quot;http://nemu.web.psi.ch/musrfit/depthProf&quot;</span><span class="nt">&gt;</span>
<span class="nt">&lt;comment&gt;</span>
TrimSp information
<span class="nt">&lt;/comment&gt;</span>
<span class="nt">&lt;trim_sp&gt;</span>
<span class="nt">&lt;data_path&gt;</span>./TRIMSP/<span class="nt">&lt;/data_path&gt;</span>
<span class="nt">&lt;rge_fln_pre&gt;</span>SiO2_70nm2.0_30nm2.2_SiC_E<span class="nt">&lt;/rge_fln_pre&gt;</span>
<span class="nt">&lt;energy_vect</span> <span class="na">start=</span><span class="s">&quot;1000&quot;</span> <span class="na">stop=</span><span class="s">&quot;22000&quot;</span> <span class="na">step=</span><span class="s">&quot;1000&quot;</span><span class="nt">/&gt;</span>
<span class="nt">&lt;/trim_sp&gt;</span>
<span class="nt">&lt;/depthProf&gt;</span>
</pre></div>
</div>
</div>
<div class="section" id="functions-to-analyze-bgr-nmr-data-bnmr-libs">
<span id="bnmr-libs"></span><span id="index-14"></span><h2>Functions to analyze β-NMR data (BNMR libs)<a class="headerlink" href="#functions-to-analyze-bgr-nmr-data-bnmr-libs" title="Permalink to this headline"></a></h2>
<span id="bnmr-libs"></span><span id="index-16"></span><h2>Functions to analyze β-NMR data (BNMR libs)<a class="headerlink" href="#functions-to-analyze-bgr-nmr-data-bnmr-libs" title="Permalink to this headline"></a></h2>
<p>This is a collection of <code class="docutils literal notranslate"><span class="pre">C++</span></code> classes using the <code class="docutils literal notranslate"><span class="pre">musrfit</span></code> <a class="reference internal" href="user-manual.html#id38"><span class="std std-ref">user-functions</span></a>
interface in order to facilitate the usage in conjunction with <code class="docutils literal notranslate"><span class="pre">musrfit</span></code>. It consists of two libraries:</p>
<ul class="simple">
@ -514,11 +628,11 @@ interface in order to facilitate the usage in conjunction with <code class="docu
<p class="last">Currently it is recommended to read in the data in ASCII format as a non-μSR fit <a class="reference internal" href="user-manual.html#non-musr-fit"><span class="std std-ref">(fit type 8)</span></a>.</p>
</div>
<div class="section" id="libbnmr">
<span id="index-15"></span><h3>libBNMR<a class="headerlink" href="#libbnmr" title="Permalink to this headline"></a></h3>
<span id="index-17"></span><h3>libBNMR<a class="headerlink" href="#libbnmr" title="Permalink to this headline"></a></h3>
<p>In β-NMR the SLR is usually measured by implanting a pulse of <span class="math notranslate nohighlight">\(^8\)</span>Li with a length <span class="math notranslate nohighlight">\(t_0\)</span> into the sample.
The asymmetry is measured both during the pulse and afterwards. For a a general spin relaxation function <span class="math notranslate nohighlight">\(f(t)\)</span> the time evolution of the asymmetry is then given by [<a class="reference external" href="http://dx.doi.org/10.1103/PhysRevLett.96.147601">Z. Salman, et al., PRL 96, 147601 (2006)</a>]:</p>
<div class="math notranslate nohighlight" id="slr">
<span id="index-16"></span>\[\begin{split}P(t) = \left\{\begin{matrix}
<span id="index-18"></span>\[\begin{split}P(t) = \left\{\begin{matrix}
\frac{\int_0^t e^{-(t-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^t e^{-t'/\tau_{\mathrm{Li}}}dt' } &amp; t\leq t_0\\[6pt]
\frac{\int_0^{t_0}e^{-(t_0-t')/\tau_{\mathrm{Li}}}f(t-t')dt'}{\int_0^{t_0}e^{-t'/\tau_{\mathrm{Li}}}dt'} &amp; t&gt; t_0,
\end{matrix}\right.\end{split}\]</div>
@ -526,7 +640,7 @@ The asymmetry is measured both during the pulse and afterwards. For a a general
<div class="section" id="functions">
<h4>Functions<a class="headerlink" href="#functions" title="Permalink to this headline"></a></h4>
<p>The <code class="docutils literal notranslate"><span class="pre">libBNMR</span></code> library currently contains the following functions:</p>
<p id="index-17"><strong>Exponential relaxation</strong></p>
<p id="index-19"><strong>Exponential relaxation</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libBNMR</span> <span class="n">ExpRlx</span> <span class="mi">1</span> <span class="mi">2</span>
</pre></div>
</div>
@ -536,7 +650,7 @@ The asymmetry is measured both during the pulse and afterwards. For a a general
<li>relaxation rate <span class="math notranslate nohighlight">\(\lambda\)</span> (s<span class="math notranslate nohighlight">\(^{-1}\)</span>)</li>
</ol>
<p>This function implements <span class="math notranslate nohighlight">\(f(t)=e^{-\lambda t}\)</span>.</p>
<p id="index-18"><strong>Stretched exponential relaxation</strong></p>
<p id="index-20"><strong>Stretched exponential relaxation</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libBNMR</span> <span class="n">SExpRlx</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
</pre></div>
</div>
@ -550,19 +664,19 @@ The asymmetry is measured both during the pulse and afterwards. For a a general
</div>
</div>
<div class="section" id="liblineprofile">
<span id="index-19"></span><h3>libLineProfile<a class="headerlink" href="#liblineprofile" title="Permalink to this headline"></a></h3>
<span id="index-21"></span><h3>libLineProfile<a class="headerlink" href="#liblineprofile" title="Permalink to this headline"></a></h3>
<p>In addition to some simple line shapes <code class="docutils literal notranslate"><span class="pre">libLineProfile</span></code> contains functions to fit chemical shift anisotropies in the powder average.
Their functional form can be found in <a class="reference external" href="http://dx.doi.org/10.1007/978-3-642-68756-3_2">M. Mehring, Principles of High Resolution NMR in Solids (Springer 1983)</a>.</p>
<p>For an axially symmetric interaction it is given by:</p>
<div class="math notranslate nohighlight" id="iax">
<span id="index-20"></span>\[\begin{split}I_{\mathrm ax}(f)=\left\{\begin{matrix} \frac{1}{2\sqrt{(f_\parallel-f_\perp)(f-f_\perp)}}&amp; f\in(f_\perp,f_\parallel)\cup(f_\parallel,f_\perp)\\[6pt] 0 &amp; \text{otherwise}\end{matrix} \right.\end{split}\]</div>
<span id="index-22"></span>\[\begin{split}I_{\mathrm ax}(f)=\left\{\begin{matrix} \frac{1}{2\sqrt{(f_\parallel-f_\perp)(f-f_\perp)}}&amp; f\in(f_\perp,f_\parallel)\cup(f_\parallel,f_\perp)\\[6pt] 0 &amp; \text{otherwise}\end{matrix} \right.\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(f_\parallel\)</span> and <span class="math notranslate nohighlight">\(f_\perp\)</span> are the frequencies that would be observed if the field is oriented paralell or perpendicular to the symmetry axis, respectively.</p>
<div class="line-block">
<div class="line">In case of a completely anisotropic interaction, the powder average can be described by the frequencies along the three principle axis <span class="math notranslate nohighlight">\(f_1,f_2,f_3\)</span>.</div>
<div class="line">Assume without loss of generality that <span class="math notranslate nohighlight">\(f_1&lt;f_2&lt;f_3\)</span>, then</div>
</div>
<div class="math notranslate nohighlight" id="ianiso">
<span id="index-21"></span>\[\begin{split}I(f)&amp;=\left\{\begin{matrix}
<span id="index-23"></span>\[\begin{split}I(f)&amp;=\left\{\begin{matrix}
\frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},&amp; f_3\geq f&gt;f_2 \\[9pt]
\frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},&amp; f_2&gt;f\geq f_1\\[9pt]
0 &amp; \text{otherwise}
@ -578,7 +692,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
<div class="section" id="id1">
<h4>Functions<a class="headerlink" href="#id1" title="Permalink to this headline"></a></h4>
<p>The <code class="docutils literal notranslate"><span class="pre">libLineProfile</span></code> library currently contains the following functions:</p>
<p id="index-22"><strong>Gaussian</strong></p>
<p id="index-24"><strong>Gaussian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">LineGauss</span> <span class="mi">1</span> <span class="mi">2</span>
</pre></div>
</div>
@ -593,7 +707,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
</div>
<div class="math notranslate nohighlight">
\[A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}\]</div>
<p id="index-23"><strong>Lorentzian</strong></p>
<p id="index-25"><strong>Lorentzian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">LineLorentzian</span> <span class="mi">1</span> <span class="mi">2</span>
</pre></div>
</div>
@ -608,7 +722,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
</div>
<div class="math notranslate nohighlight">
\[A(f)= \frac{w^2}{4(f-f_0)^2+w^2}\]</div>
<p id="index-24"><strong>Laplacian</strong></p>
<p id="index-26"><strong>Laplacian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">LineLaplace</span> <span class="mi">1</span> <span class="mi">2</span>
</pre></div>
</div>
@ -623,7 +737,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
</div>
<div class="math notranslate nohighlight">
\[A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{w}\right|}\]</div>
<p id="index-25"><strong>Skewed Lorentzian</strong></p>
<p id="index-27"><strong>Skewed Lorentzian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">LineSkewLorentzian</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
</pre></div>
</div>
@ -639,7 +753,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
</div>
<div class="math notranslate nohighlight">
\[A(f)= \frac{w w_a}{4(f-f_0)^2+w_a^2}, \quad w_a=\frac{2w}{1+e^{a(f-f_0)}}\]</div>
<p id="index-26"><strong>Skewed Lorentzian 2</strong></p>
<p id="index-28"><strong>Skewed Lorentzian 2</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">LineSkewLorentzian2</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
</pre></div>
</div>
@ -655,7 +769,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
</div>
<div class="math notranslate nohighlight">
\[\begin{split}A(f)= \left\{\begin{matrix}\frac{{w_1}^2}{4{(f-f_0)}^2+{w_1}^2},&amp;f\leq f_0\\[9pt] \frac{{w_2}^2}{4{(f-f_0)}^2+{w_2}^2},&amp;f&gt;f_0\end{matrix}\right.\end{split}\]</div>
<p id="index-27"><strong>Powder average of an axially symmetric interaction convoluted with a Lorentzian</strong></p>
<p id="index-29"><strong>Powder average of an axially symmetric interaction convoluted with a Lorentzian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">PowderLineAxialLor</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
</pre></div>
</div>
@ -672,7 +786,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
<div class="math notranslate nohighlight">
\[A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]</div>
<p>with <span class="math notranslate nohighlight">\(I_{\mathrm ax}(f)\)</span> defined <a class="reference internal" href="#iax"><span class="std std-ref">above</span></a>.</p>
<p id="index-28"><strong>Powder average of an axially symmetric interaction convoluted with a Gaussian</strong></p>
<p id="index-30"><strong>Powder average of an axially symmetric interaction convoluted with a Gaussian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">PowderLineAxialGss</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span>
</pre></div>
</div>
@ -689,7 +803,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
<div class="math notranslate nohighlight">
\[A(f)= I_{\mathrm ax}(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)\]</div>
<p>with <span class="math notranslate nohighlight">\(I_{\mathrm ax}(f)\)</span> defined <a class="reference internal" href="#iax"><span class="std std-ref">above</span></a>.</p>
<p id="index-29"><strong>Powder average of an anisotropic interaction convoluted with a Lorentzian</strong></p>
<p id="index-31"><strong>Powder average of an anisotropic interaction convoluted with a Lorentzian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">PowderLineAsymLor</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span>
</pre></div>
</div>
@ -707,7 +821,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
<div class="math notranslate nohighlight">
\[A(f)= I(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]</div>
<p>with <span class="math notranslate nohighlight">\(I(f)\)</span> defined <a class="reference internal" href="#ianiso"><span class="std std-ref">above</span></a>. Note that <span class="math notranslate nohighlight">\(f_1&lt;f_2&lt;f_3\)</span> is not required by the code.</p>
<p id="index-30"><strong>Powder average of an anisotropic interaction convoluted with a Gaussian</strong></p>
<p id="index-32"><strong>Powder average of an anisotropic interaction convoluted with a Gaussian</strong></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">userFcn</span> <span class="n">libLineProfile</span> <span class="n">PowderLineAsymGss</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span>
</pre></div>
</div>
@ -750,8 +864,8 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
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<p>
&copy; Copyright 2022, Andreas Suter.
Last updated on Dec 12, 2022.
&copy; Copyright 2023, Andreas Suter.
Last updated on Feb 17, 2023.
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@ -773,7 +887,7 @@ K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\en
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