update of the docu. Added the beta-NMR docu written by Jonas Krieger.

doc/html/_sources/index.txt

@@ -3,8 +3,8 @@
    You can adapt this file completely to your liking, but it should at least
    contain the root `toctree` directive.
 
-Welcome to musrfit documentation!
+Welcome to the musrfit documentation!
-=================================
+=====================================
 
 .. toctree::
    :maxdepth: 2

doc/html/_sources/user-libs.txt

@@ -407,3 +407,341 @@ Nonlocal superconductivity related Meissner screening functions (AS libs)
 -------------------------------------------------------------------------
 
 To be written yet ...
+  
+.. index:: BNMR-libs
+.. _BNMR-libs:
+
+Functions to analyze |bgr|-NMR data (BNMR libs) 
+-------------------------------------------------------------------------
+
+This is a collection of ``C++`` classes using the ``musrfit`` :ref:`user-functions <user-functions>` 
+interface in order to facilitate the usage in conjunction with ``musrfit``. It consists of two libraries: 
+
+* ``libBNMR`` contains functions to fit spin lattice relaxation (SLR) data.
+* ``libLineProfile`` contains functions to fit resonance lineshapes.
+
+
+.. note:: 
+
+  Currently it is recommended to read in the data in ASCII format as a non-|mgr|\SR fit  :ref:`(fit type 8) <non-musr-fit>`.
+
+.. index:: libBNMR
+
+libBNMR
+++++++++++
+
+In |bgr|-NMR the SLR is usually measured by implanting a pulse of :math:`^8`\ Li with a length :math:`t_0` into the sample. 
+The asymmetry is measured both during the pulse and afterwards. For a a general spin relaxation function :math:`f(t)` the time evolution of the asymmetry is then given by [`Z. Salman, et al., PRL 96, 147601 (2006) <http://dx.doi.org/10.1103/PhysRevLett.96.147601>`_]: 
+
+
+
+.. index:: SLR
+.. _SLR:
+
+.. math::  
+   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' }       &  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'}    &  t> t_0,
+  \end{matrix}\right.
+
+where :math:`\tau_{\mathrm{Li}}=1.21`\ s is the :math:`^8`\ Li lifetime.
+
+
+Functions
+^^^^^^^^^^^^
+The ``libLineProfile`` library currently contains the following functions:
+
+
+
+
+.. index:: ExpRlx
+
+**Exponential relaxation**
+
+::
+
+  userFcn libBNMR ExpRlx 1 2
+
+The parameters are:
+
+#. pulse length :math:`t_0` (ms)
+#. relaxation rate :math:`\sigma` (ms\ :math:`^{-1}`\ )
+
+This function implements :math:`f(t)=e^{-\sigma t}`.
+
+.. index:: SExpRlx
+
+**Stretched exponential relaxation**
+
+::
+
+  userFcn libBNMR SExpRlx 1 2 3
+
+The parameters are:
+
+#. pulse length :math:`t_0` (ms)
+#. relaxation rate :math:`\sigma` (ms\ :math:`^{-1}`\ )
+#. stretching exponent :math:`\beta`
+
+This function implements :math:`f(t)=e^{-(\sigma t)^{\beta}}`.
+
+
+  
+.. index:: libLineProfile
+
+libLineProfile
++++++++++++++++++
+In addition to some simple line shapes ``libLineProfile`` contains functions to fit chemical shift anisotropies in the powder average.
+Their functional form can be found in  `M. Mehring, Principles of High Resolution NMR in Solids (Springer 1983) <http://dx.doi.org/10.1007/978-3-642-68756-3_2>`_.
+
+For an axially symmetric interaction it is given by:
+
+.. index:: Iax
+.. _Iax:
+
+.. math::
+
+      I_{\mathrm ax}(f)=\left\{\begin{matrix} \frac{1}{2\sqrt{(f_\parallel-f_\perp)(f-f_\perp)}}& f\in(f_\perp,f_\parallel)\cup(f_\parallel,f_\perp)\\[6pt] 0 & \text{otherwise}\end{matrix} \right.
+
+where :math:`f_\parallel` and :math:`f_\perp` are the frequencies that would be observed if the field is oriented paralell or perpendicular to the symmetry axis, respectively.
+
+
+| In case of a completely anisotropic interaction, the powder average can be described by the frequencies along the three principle axis :math:`f_1,f_2,f_3`.
+| Assume without loss of generality that :math:`f_1<f_2<f_3`, then
+
+.. index:: Ianiso
+.. _Ianiso:
+
+.. math::  
+      I(f)&=\left\{\begin{matrix}
+        \frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},& f_3\geq f>f_2 \\[9pt]
+        \frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},& f_2>f\geq f_1\\[9pt]
+        0 & \text{otherwise}
+      \end{matrix} \right.  \\
+ 	\\
+      m&=\left\{\begin{matrix}
+        \frac{(f_2-f_1)(f_3-f)}{(f_3-f_2)(f-f_1)},& f_3\geq f>f_2 \\[6pt]
+        \frac{(f-f_1)(f_3-f_2)}{(f_3-f)(f_2-f_1)},& f_2>f\geq f_1\\[6pt]
+      \end{matrix} \right.  \\
+ 	\\
+      K(m)&=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},
+
+
+:math:`K(m)` is the complete elliptic integral of the first kind.
+
+
+
+Functions
+^^^^^^^^^^^^
+The ``libLineProfile`` library currently contains the following functions:
+
+
+
+
+.. index:: LineGauss
+
+**Gaussian**
+
+::
+
+  userFcn  libLineProfile LineGauss 1 2
+
+The parameters are:
+
+#. center of the line :math:`f_0`
+#. FWHM of the line :math:`\sigma`
+
+| The height of the peak is 1.
+| The functional form is given by
+
+.. math::
+   A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}
+
+
+.. index:: LineLorentzian
+
+**Lorentzian**
+
+::
+
+  userFcn  libLineProfile LineLorentzian 1 2
+
+The parameters are:
+
+#. center of the line :math:`f_0`
+#. FWHM of the line :math:`w`
+
+| The height of the peak is 1. 
+| The functional form is given by
+
+.. math::
+    A(f)= \frac{w^2}{4(f-f_0)^2+w^2}
+
+
+.. index:: LineLaplace
+
+**Laplacian**
+
+::
+
+  userFcn  libLineProfile LineLaplace 1 2
+
+The parameters are:
+
+#. center of the line :math:`f_0`
+#. FWHM of the line :math:`w`
+
+| The height of the peak is 1. 
+| The functional form is given by
+
+.. math::
+ A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{w}\right|}
+
+
+
+.. index:: LineSkewLorentzian
+
+**Skewed Lorentzian**
+
+::
+
+  userFcn  libLineProfile LineSkewLorentzian 1 2 3
+
+The parameters are:
+
+#. center of the line :math:`f_0`
+#. width of the line :math:`w`
+#. skewness parameter :math:`a`
+
+| The height of the peak is 1. 
+| The functional form is given by
+
+.. math::
+    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)}}
+
+
+
+.. index:: LineSkewLorentzian2
+
+**Skewed Lorentzian 2**
+
+::
+
+  userFcn  libLineProfile LineSkewLorentzian2 1 2 3
+
+The parameters are:
+
+#. center of the line :math:`f_0`
+#. width left of the center :math:`w_1`
+#. width right of the center :math:`w_2`
+
+| The height of the peak is 1. 
+| The functional form is given by
+
+.. math::
+    A(f)=  \left\{\begin{matrix}\frac{{w_1}^2}{4{(f-f_0)}^2+{w_1}^2},&f\leq f_0\\[9pt] \frac{{w_2}^2}{4{(f-f_0)}^2+{w_2}^2},&f>f_0\end{matrix}\right.
+
+
+
+.. index:: PowderLineAxialLor
+
+
+**Powder average of an axially symmetric interaction convoluted with a Lorentzian**
+
+::
+
+  userFcn  libLineProfile PowderLineAxialLor 1 2 3
+
+The parameters are:
+
+#.  frequency for the field oriented paralell to the symmetry axis :math:`f_\parallel`
+#.  frequency for the field oriented perpendicular to the symmetry axis :math:`f_\parallel`
+#.  FWHM of the Lorentzian :math:`w`
+
+| The height of the peak is :math:`\sim`\ 1. 
+| The functional form is given by
+
+.. math::
+    A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)
+
+with :math:`I_{\mathrm ax}(f)` defined :ref:`above <Iax>`.
+
+    
+
+.. index:: PowderLineAxialGss
+
+
+**Powder average of an axially symmetric interaction convoluted with a Gaussian**
+
+::
+
+  userFcn  libLineProfile PowderLineAxialGss 1 2 3
+
+The parameters are:
+
+#.  frequency for the field oriented paralell to the symmetry axis :math:`f_\parallel`
+#.  frequency for the field oriented perpendicular to the symmetry axis :math:`f_\parallel`
+#. FWHM of the Gaussian :math:`\sigma`
+
+| The height of the peak is :math:`\sim`\ 1. 
+| The functional form is given by
+
+.. math::
+    A(f)= I_{\mathrm ax}(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)
+
+with :math:`I_{\mathrm ax}(f)` defined :ref:`above <Iax>`.
+
+  
+
+.. index:: PowderLineAsymLor
+
+
+**Powder average of an anisotropic interaction convoluted with a Lorentzian**
+
+::
+
+  userFcn  libLineProfile PowderLineAsymLor 1 2 3 4
+
+The parameters are:
+
+#.   :math:`f_1`
+#.   :math:`f_1`
+#.   :math:`f_3` frequencies along the principal axes
+#. FWHM of the Lorentzian :math:`w`
+
+| The height of the peak is :math:`\sim`\ 1. 
+| The functional form is given by
+
+.. math::
+    A(f)= I(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)
+
+with :math:`I(f)` defined :ref:`above <Ianiso>`. Note that :math:`f_1<f_2<f_3` is not required by the code.
+  
+    
+    
+.. index:: PowderLineAsymGss
+
+
+**Powder average of an anisotropic interaction convoluted with a Gaussian**
+
+::
+
+  userFcn  libLineProfile PowderLineAsymGss 1 2 3 4
+
+The parameters are:
+
+#.   :math:`f_1`
+#.   :math:`f_1`
+#.   :math:`f_3` frequencies along the principal axes
+#. FWHM of the Gaussian :math:`\sigma`
+
+| The height of the peak is :math:`\sim`\ 1. 
+| The functional form is given by
+
+.. math::
+    A(f)= I(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)
+
+with :math:`I(f)` defined :ref:`above <Ianiso>`. Note that :math:`f_1<f_2<f_3` is not required by the code.
+
+
+

doc/html/acknowledgement.html

@@ -113,7 +113,7 @@ extremely competent way to deal with his projects as well as to deal with the ch
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
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doc/html/any2many.html

@@ -107,7 +107,7 @@ For a detailed description see <a class="reference internal" href="user-manual.h
     </div>
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-      Last updated on Jul 03, 2018.
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doc/html/bugtracking.html

@@ -98,7 +98,7 @@ or send an e-mail to A. Suter at PSI.</p>
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
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doc/html/cite.html

@@ -26,7 +26,7 @@
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+    <link rel="prev" title="Welcome to the musrfit documentation!" href="index.html" /> 
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@@ -39,7 +39,7 @@
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              accesskey="N">next</a> |</li>
         <li class="right" >
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+          <a href="index.html" title="Welcome to the musrfit documentation!"
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@@ -76,7 +76,7 @@
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   <h4>Previous topic</h4>
   <p class="topless"><a href="index.html"
-                        title="previous chapter">Welcome to musrfit documentation!</a></p>
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@@ -112,14 +112,14 @@
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              >next</a> |</li>
         <li class="right" >
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+          <a href="index.html" title="Welcome to the musrfit documentation!"
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doc/html/genindex.html

@@ -52,9 +52,11 @@
  | <a href="#B"><strong>B</strong></a>
  | <a href="#C"><strong>C</strong></a>
  | <a href="#D"><strong>D</strong></a>
+ | <a href="#E"><strong>E</strong></a>
  | <a href="#F"><strong>F</strong></a>
  | <a href="#G"><strong>G</strong></a>
  | <a href="#H"><strong>H</strong></a>
+ | <a href="#I"><strong>I</strong></a>
  | <a href="#L"><strong>L</strong></a>
  | <a href="#M"><strong>M</strong></a>
  | <a href="#N"><strong>N</strong></a>
@@ -138,17 +140,21 @@
   <dt><a href="user-manual.html#index-28">background-single-histo</a>
   </dt>
 
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
       
   <dt><a href="user-libs.html#index-1">BMW-libs</a>
   </dt>
 
+  </dl></td>
+  <td style="width: 33%" valign="top"><dl>
       
   <dt><a href="user-libs.html#index-13">BMWlibs-XML</a>
   </dt>
 
       
+  <dt><a href="user-libs.html#index-14">BNMR-libs</a>
+  </dt>
+
+      
   <dt><a href="setup-standard.html#index-2">boost-c++</a>
   </dt>
 
@@ -219,6 +225,16 @@
   </dl></td>
 </tr></table>
 
+<h2 id="E">E</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%" valign="top"><dl>
+      
+  <dt><a href="user-libs.html#index-17">ExpRlx</a>
+  </dt>
+
+  </dl></td>
+</tr></table>
+
 <h2 id="F">F</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%" valign="top"><dl>
@@ -323,21 +339,65 @@
   </dl></td>
 </tr></table>
 
+<h2 id="I">I</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%" valign="top"><dl>
+      
+  <dt><a href="user-libs.html#index-21">Ianiso</a>
+  </dt>
+
+  </dl></td>
+  <td style="width: 33%" valign="top"><dl>
+      
+  <dt><a href="user-libs.html#index-20">Iax</a>
+  </dt>
+
+  </dl></td>
+</tr></table>
+
 <h2 id="L">L</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%" valign="top"><dl>
       
+  <dt><a href="user-libs.html#index-15">libBNMR</a>
+  </dt>
+
+      
   <dt><a href="user-libs.html#index-2">libFitPofB</a>
   </dt>
 
       
+  <dt><a href="user-libs.html#index-19">libLineProfile</a>
+  </dt>
+
+      
   <dt><a href="setup-standard.html#index-6">libxml2</a>
   </dt>
 
+      
+  <dt><a href="user-manual.html#index-23">lifetime</a>
+  </dt>
+
   </dl></td>
   <td style="width: 33%" valign="top"><dl>
       
-  <dt><a href="user-manual.html#index-23">lifetime</a>
+  <dt><a href="user-libs.html#index-22">LineGauss</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-24">LineLaplace</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-23">LineLorentzian</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-25">LineSkewLorentzian</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-26">LineSkewLorentzian2</a>
   </dt>
 
   </dl></td>
@@ -692,6 +752,24 @@
   <dt><a href="user-manual.html#index-38">packing</a>
   </dt>
 
+      
+  <dt><a href="user-libs.html#index-30">PowderLineAsymGss</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-29">PowderLineAsymLor</a>
+  </dt>
+
+  </dl></td>
+  <td style="width: 33%" valign="top"><dl>
+      
+  <dt><a href="user-libs.html#index-28">PowderLineAxialGss</a>
+  </dt>
+
+      
+  <dt><a href="user-libs.html#index-27">PowderLineAxialLor</a>
+  </dt>
+
   </dl></td>
 </tr></table>
 
@@ -741,6 +819,10 @@
   </dt>
 
       
+  <dt><a href="user-libs.html#index-18">SExpRlx</a>
+  </dt>
+
+      
   <dt><a href="user-manual.html#index-64">single-histogram-fit</a>
   </dt>
 
@@ -751,6 +833,10 @@
   </dt>
 
       
+  <dt><a href="user-libs.html#index-16">SLR</a>
+  </dt>
+
+      
   <dt><a href="setup-standard.html#index-1">supported-operating-systems</a>
   </dt>
 
@@ -901,7 +987,7 @@
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     <div class="footer">
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+      Last updated on Aug 23, 2018.
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doc/html/index.html

@@ -6,7 +6,7 @@
   <head>
     <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
     
-    <title>Welcome to musrfit documentation! &mdash; musrfit 1.4.0 documentation</title>
+    <title>Welcome to the musrfit documentation! &mdash; musrfit 1.4.0 documentation</title>
     
     <link rel="stylesheet" href="_static/nature.css" type="text/css" />
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@@ -46,8 +46,8 @@
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-  <div class="section" id="welcome-to-musrfit-documentation">
+  <div class="section" id="welcome-to-the-musrfit-documentation">
-<h1>Welcome to musrfit documentation!<a class="headerlink" href="#welcome-to-musrfit-documentation" title="Permalink to this headline">¶</a></h1>
+<h1>Welcome to the musrfit documentation!<a class="headerlink" href="#welcome-to-the-musrfit-documentation" title="Permalink to this headline">¶</a></h1>
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 <ul>
 <li class="toctree-l1"><a class="reference internal" href="cite.html">How to Cite <tt class="docutils literal"><span class="pre">musrfit</span></tt>?</a></li>
@@ -68,6 +68,7 @@
 <li class="toctree-l1"><a class="reference internal" href="user-libs.html">Documentation of user libs (user functions)</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="user-libs.html#meissner-profiles-vortex-lattice-related-functions-bmw-libs">Meissner-Profiles / Vortex-Lattice related functions (BMW libs)</a></li>
 <li class="toctree-l2"><a class="reference internal" href="user-libs.html#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="user-libs.html#functions-to-analyze-bgr-nmr-data-bnmr-libs">Functions to analyze β-NMR data (BNMR libs)</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="setup-standard.html">Setting up <tt class="docutils literal"><span class="pre">musrfit</span></tt> on Different Platforms</a><ul>
@@ -141,7 +142,7 @@
         <div class="sphinxsidebarwrapper">
   <h3><a href="#">Table Of Contents</a></h3>
   <ul>
-<li><a class="reference internal" href="#">Welcome to musrfit documentation!</a></li>
+<li><a class="reference internal" href="#">Welcome to the musrfit documentation!</a></li>
 <li><a class="reference internal" href="#indices-and-tables">Indices and tables</a></li>
 </ul>
 
@@ -184,7 +185,7 @@
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doc/html/msr2data.html

@@ -439,7 +439,7 @@ fit serves as template for the second and so on. The template field stays empty
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@@ -257,7 +257,7 @@ SCRIPT COMMANDS:
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       Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3.
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doc/html/musr-root.html

@@ -928,7 +928,7 @@ the entry has been added. The last token, <tt class="docutils literal"><span cla
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
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doc/html/musredit.html

@@ -555,7 +555,7 @@ the corresponding fit parameter value, except the phases where the step will be
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
       Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3.
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doc/html/search.html

@@ -94,7 +94,7 @@
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
       Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3.
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doc/html/setup-dks.html

@@ -359,7 +359,7 @@ The only thing you need <tt class="docutils literal"><span class="pre">DKS</span
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
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doc/html/setup-standard.html

@@ -1486,7 +1486,7 @@ $ musrview test-histo-ROOT-NPP.msr
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
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doc/html/tutorial.html

@@ -448,7 +448,7 @@ For a complete description please refer to the manuals of <a class="reference in
     </div>
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doc/html/user-libs.html

@@ -370,6 +370,233 @@ The expected name of the <tt class="docutils literal"><span class="pre">RGE</spa
 <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>
 </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>
+<p>This is a collection of <tt class="docutils literal"><span class="pre">C++</span></tt> classes using the <tt class="docutils literal"><span class="pre">musrfit</span></tt> <a class="reference internal" href="user-manual.html#id20"><em>user-functions</em></a>
+interface in order to facilitate the usage in conjunction with <tt class="docutils literal"><span class="pre">musrfit</span></tt>. It consists of two libraries:</p>
+<ul class="simple">
+<li><tt class="docutils literal"><span class="pre">libBNMR</span></tt> contains functions to fit spin lattice relaxation (SLR) data.</li>
+<li><tt class="docutils literal"><span class="pre">libLineProfile</span></tt> contains functions to fit resonance lineshapes.</li>
+</ul>
+<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 format as a non-μSR fit  <a class="reference internal" href="user-manual.html#non-musr-fit"><em>(fit type 8)</em></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>
+<p>In β-NMR the SLR is usually measured by implanting a pulse of <span class="math">\(^8\)</span>Li with a length <span class="math">\(t_0\)</span> into the sample.
+The asymmetry is measured both during the pulse and afterwards. For a a general spin relaxation function <span class="math">\(f(t)\)</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" id="slr">
+<span id="index-16"></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>
+<p>where <span class="math">\(\tau_{\mathrm{Li}}=1.21\)</span>s is the <span class="math">\(^8\)</span>Li lifetime.</p>
+<div class="section" id="functions">
+<h4>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h4>
+<p>The <tt class="docutils literal"><span class="pre">libLineProfile</span></tt> library currently contains the following functions:</p>
+<p id="index-17"><strong>Exponential relaxation</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn libBNMR ExpRlx 1 2
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>pulse length <span class="math">\(t_0\)</span> (ms)</li>
+<li>relaxation rate <span class="math">\(\sigma\)</span> (ms<span class="math">\(^{-1}\)</span>)</li>
+</ol>
+<p>This function implements <span class="math">\(f(t)=e^{-\sigma t}\)</span>.</p>
+<p id="index-18"><strong>Stretched exponential relaxation</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn libBNMR SExpRlx 1 2 3
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>pulse length <span class="math">\(t_0\)</span> (ms)</li>
+<li>relaxation rate <span class="math">\(\sigma\)</span> (ms<span class="math">\(^{-1}\)</span>)</li>
+<li>stretching exponent <span class="math">\(\beta\)</span></li>
+</ol>
+<p>This function implements <span class="math">\(f(t)=e^{-(\sigma t)^{\beta}}\)</span>.</p>
+</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>
+<p>In addition to some simple line shapes <tt class="docutils literal"><span class="pre">libLineProfile</span></tt> 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" 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>
+<p>where <span class="math">\(f_\parallel\)</span> and <span class="math">\(f_\perp\)</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">\(f_1,f_2,f_3\)</span>.</div>
+<div class="line">Assume without loss of generality that <span class="math">\(f_1&lt;f_2&lt;f_3\)</span>, then</div>
+</div>
+<div class="math" id="ianiso">
+<span id="index-21"></span>\[\begin{split}I(f)&amp;=\left\{\begin{matrix}
+  \frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},&amp; f_3\geq f&gt;f_2 \\[9pt]
+  \frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},&amp; f_2&gt;f\geq f_1\\[9pt]
+  0 &amp; \text{otherwise}
+\end{matrix} \right.  \\
+  \\
+m&amp;=\left\{\begin{matrix}
+  \frac{(f_2-f_1)(f_3-f)}{(f_3-f_2)(f-f_1)},&amp; f_3\geq f&gt;f_2 \\[6pt]
+  \frac{(f-f_1)(f_3-f_2)}{(f_3-f)(f_2-f_1)},&amp; f_2&gt;f\geq f_1\\[6pt]
+\end{matrix} \right.  \\
+  \\
+K(m)&amp;=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},\end{split}\]</div>
+<p><span class="math">\(K(m)\)</span> is the complete elliptic integral of the first kind.</p>
+<div class="section" id="id1">
+<h4>Functions<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h4>
+<p>The <tt class="docutils literal"><span class="pre">libLineProfile</span></tt> library currently contains the following functions:</p>
+<p id="index-22"><strong>Gaussian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile LineGauss 1 2
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>center of the line <span class="math">\(f_0\)</span></li>
+<li>FWHM of the line <span class="math">\(\sigma\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is 1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}\]</div>
+<p id="index-23"><strong>Lorentzian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile LineLorentzian 1 2
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>center of the line <span class="math">\(f_0\)</span></li>
+<li>FWHM of the line <span class="math">\(w\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is 1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)= \frac{w^2}{4(f-f_0)^2+w^2}\]</div>
+<p id="index-24"><strong>Laplacian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile LineLaplace 1 2
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>center of the line <span class="math">\(f_0\)</span></li>
+<li>FWHM of the line <span class="math">\(w\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is 1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{w}\right|}\]</div>
+<p id="index-25"><strong>Skewed Lorentzian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile LineSkewLorentzian 1 2 3
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>center of the line <span class="math">\(f_0\)</span></li>
+<li>width of the line <span class="math">\(w\)</span></li>
+<li>skewness parameter <span class="math">\(a\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is 1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)=  \frac{w w_a}{4(f-f_0)^2+w_a^2}, \quad w_a=\frac{2w}{1+e^{a(f-f_0)}}\]</div>
+<p id="index-26"><strong>Skewed Lorentzian 2</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile LineSkewLorentzian2 1 2 3
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>center of the line <span class="math">\(f_0\)</span></li>
+<li>width left of the center <span class="math">\(w_1\)</span></li>
+<li>width right of the center <span class="math">\(w_2\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is 1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[\begin{split}A(f)=  \left\{\begin{matrix}\frac{{w_1}^2}{4{(f-f_0)}^2+{w_1}^2},&amp;f\leq f_0\\[9pt] \frac{{w_2}^2}{4{(f-f_0)}^2+{w_2}^2},&amp;f&gt;f_0\end{matrix}\right.\end{split}\]</div>
+<p id="index-27"><strong>Powder average of an axially symmetric interaction convoluted with a Lorentzian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile PowderLineAxialLor 1 2 3
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>frequency for the field oriented paralell to the symmetry axis <span class="math">\(f_\parallel\)</span></li>
+<li>frequency for the field oriented perpendicular to the symmetry axis <span class="math">\(f_\parallel\)</span></li>
+<li>FWHM of the Lorentzian <span class="math">\(w\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is <span class="math">\(\sim\)</span>1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]</div>
+<p>with <span class="math">\(I_{\mathrm ax}(f)\)</span> defined <a class="reference internal" href="#iax"><em>above</em></a>.</p>
+<p id="index-28"><strong>Powder average of an axially symmetric interaction convoluted with a Gaussian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile PowderLineAxialGss 1 2 3
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li>frequency for the field oriented paralell to the symmetry axis <span class="math">\(f_\parallel\)</span></li>
+<li>frequency for the field oriented perpendicular to the symmetry axis <span class="math">\(f_\parallel\)</span></li>
+<li>FWHM of the Gaussian <span class="math">\(\sigma\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is <span class="math">\(\sim\)</span>1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)= I_{\mathrm ax}(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)\]</div>
+<p>with <span class="math">\(I_{\mathrm ax}(f)\)</span> defined <a class="reference internal" href="#iax"><em>above</em></a>.</p>
+<p id="index-29"><strong>Powder average of an anisotropic interaction convoluted with a Lorentzian</strong></p>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile PowderLineAsymLor 1 2 3 4
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li><span class="math">\(f_1\)</span></li>
+<li><span class="math">\(f_1\)</span></li>
+<li><span class="math">\(f_3\)</span> frequencies along the principal axes</li>
+<li>FWHM of the Lorentzian <span class="math">\(w\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is <span class="math">\(\sim\)</span>1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)= I(f)\circledast\left( \frac{w^2}{4f^2+w^2} \right)\]</div>
+<p>with <span class="math">\(I(f)\)</span> defined <a class="reference internal" href="#ianiso"><em>above</em></a>. Note that <span class="math">\(f_1&lt;f_2&lt;f_3\)</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>
+<div class="highlight-python"><div class="highlight"><pre><span></span>userFcn  libLineProfile PowderLineAsymGss 1 2 3 4
+</pre></div>
+</div>
+<p>The parameters are:</p>
+<ol class="arabic simple">
+<li><span class="math">\(f_1\)</span></li>
+<li><span class="math">\(f_1\)</span></li>
+<li><span class="math">\(f_3\)</span> frequencies along the principal axes</li>
+<li>FWHM of the Gaussian <span class="math">\(\sigma\)</span></li>
+</ol>
+<div class="line-block">
+<div class="line">The height of the peak is <span class="math">\(\sim\)</span>1.</div>
+<div class="line">The functional form is given by</div>
+</div>
+<div class="math">
+\[A(f)= I(f)\circledast\left( e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)\]</div>
+<p>with <span class="math">\(I(f)\)</span> defined <a class="reference internal" href="#ianiso"><em>above</em></a>. Note that <span class="math">\(f_1&lt;f_2&lt;f_3\)</span> is not required by the code.</p>
+</div>
+</div>
+</div>
 </div>
 
 
@@ -386,6 +613,11 @@ The expected name of the <tt class="docutils literal"><span class="pre">RGE</spa
 </ul>
 </li>
 <li><a class="reference internal" href="#nonlocal-superconductivity-related-meissner-screening-functions-as-libs">Nonlocal superconductivity related Meissner screening functions (AS libs)</a></li>
+<li><a class="reference internal" href="#functions-to-analyze-bgr-nmr-data-bnmr-libs">Functions to analyze β-NMR data (BNMR libs)</a><ul>
+<li><a class="reference internal" href="#libbnmr">libBNMR</a></li>
+<li><a class="reference internal" href="#liblineprofile">libLineProfile</a></li>
+</ul>
+</li>
 </ul>
 </li>
 </ul>
@@ -435,7 +667,7 @@ The expected name of the <tt class="docutils literal"><span class="pre">RGE</spa
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
       Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3.
     </div>
   </body>

doc/html/user-manual.html

@@ -2162,7 +2162,7 @@ In case this cannot be ensured, the parallelization can be disabled by <em>&#821
     </div>
     <div class="footer">
         &copy; Copyright 2018, Andreas Suter.
-      Last updated on Jul 03, 2018.
+      Last updated on Aug 23, 2018.
       Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2.3.
     </div>
   </body>