add necessary docu for the Gaussian Lorentzian approach.

src/external/LF_GL/docu/dynamicGL_LF.bib

@@ -0,0 +1,62 @@
+@Article{ Hayano79,
+	title = "Zero- and low-field spin relaxation studied by positive muons",
+	author = "R. S. Hayano and Y. J. Uemura and J. Imazato and N. Nishida and T. Yamazaki and R. Kubo",
+	journal = "Phys. Rev. B",
+	volume = "20",
+	year = "1979",
+	pages = "850"
+}
+
+@Article{ Uemura85,
+	title = "Muon-spin relaxation in \emph{AuFe} and \emph{CuMn} spin glasses",
+	author = "Y. J. Uemura and T. Yamazaki and D. R. Harshman and M. Senba and E. J. Ansaldo",
+	journal = "Phys. Rev. B",
+	volume = "31",
+	year = "1985",
+	pages = "546"
+}
+
+@Article{ DalmasDeReotier92,
+	title = "Quantum calculation of the muon depolarization function: effect of spin dynamics in nuclear dipole systems",
+	author = "P. {Dalams de R\'{e}otier} and A. Yaouanc",
+	journal = "J. Phys.: Condens. Matter",
+	volume = "4",
+	year = "1992",
+	pages = "4533"
+}
+
+@Article{ Keren94,
+	title = "Generalization of the Abragam relaxation function to a longitudinal field",
+	author = "A. Keren",
+	journal = "Phys. Rev. B",
+	volume = "50",
+	year = "1994",
+	pages = "10039"
+}
+
+@Article{ Larkin00,
+	title = "Exponential field distribution in $\mathrm{Sr(Cu_{1-x}Zn_x)_2O_3}$",
+	author = "M. I. Larkin and Y. Fudamoto and I. M. Gat and A. Kinkhabwala and K. M. Kojima and G. M. Luke and J. Merrin and B. Nachumi and Y. J. Uemura and M. Azuma and T. Saito and M. Takano",
+	journal = "Physica B",
+	volume = "289-290",
+	year = "00",
+	pages = "153"
+}
+
+@Article{ McMullen78,
+	title = "Positive-muon spin depolarization in solids",
+	author = "T. McMullen and E. Zaremba",
+	journal = "Phys. Rev. B",
+	volume = "18",
+	year = "1978",
+	pages = "3026"
+}
+
+@Book { Yaouanc11,
+	title = "Muon Spin Rotation, Relaxation, and Resonance",
+	author = "A. Yaouanc and P. Dalmas de R\'{e}otier",
+	year = 2011,
+	publisher = "Oxford University Press",
+	address = "Oxford"
+}
+

src/external/LF_GL/docu/dynamicGL_LF.tex

@@ -0,0 +1,135 @@
+\documentclass[twoside]{article}
+
+\usepackage[english]{babel}
+\usepackage{a4}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{array}
+\usepackage{float}
+\usepackage{hyperref}
+\usepackage{xspace}
+\usepackage{rotating}
+\usepackage{dcolumn}
+
+\setlength{\topmargin}{10mm}
+\setlength{\topmargin}{-13mm}
+% \setlength{\oddsidemargin}{0.5cm}
+% \setlength{\evensidemargin}{0cm}
+\setlength{\oddsidemargin}{1cm}
+\setlength{\evensidemargin}{1cm}
+\setlength{\textwidth}{14.5cm}
+\setlength{\textheight}{23.8cm}
+
+\pagestyle{fancyplain}
+\addtolength{\headwidth}{0.6cm}
+\fancyhead{}%
+\fancyhead[RE,LO]{\bf Static and Dynamic LF Functions}%
+\fancyhead[LE,RO]{\thepage}
+\cfoot{--- Andreas Suter -- \today ---}
+\rfoot{\includegraphics[width=2cm]{psi_01_sp.pdf}}
+
+\DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it}
+
+\newcommand{\mean}[1]{\langle #1 \rangle}
+\newcommand{\lem}{LE-$\mu$SR\xspace}
+\newcommand{\musr}{$\mu$SR\xspace}
+\newcommand{\etal}{\emph{et al.\xspace}}
+
+\newcolumntype{d}[1]{D{.}{.}{#1}}
+
+\DeclareMathOperator\erf{erf}
+
+\begin{document}
+% Header info --------------------------------------------------
+\thispagestyle{empty}
+\noindent
+\begin{tabular}{@{\hspace{-0.7cm}}l@{\hspace{6cm}}r}
+\noindent\includegraphics[width=3.4cm]{psi_01_sp.pdf} &
+  {\Huge\sf Memorandum}
+\end{tabular}
+%
+\vskip 1cm
+%
+\begin{tabular}{@{\hspace{-0.5cm}}ll@{\hspace{4cm}}ll}
+Datum:   & \today        &     & \\[3ex]
+Von:     & Andreas Suter & An: & \\
+Telefon: & +41\, (0)56\, 310\, 4238        &     & \\
+Raum:    & WLGA / 119    & cc: & \\
+e-mail:  & \verb?andreas.suter@psi.ch? & & \\
+\end{tabular}
+%
+\vskip 0.3cm
+\noindent\hrulefill
+\vskip 1cm
+%
+
+\section*{Dynamic Gaussian-Lorentzian LF}
+
+For details about the Gaussian-, and Lorentzian dynamics in LF see \cite{Hayano79,Uemura85,DalmasDeReotier92,Keren94,Yaouanc11}.
+
+Here, the focus is only on the strong collision model, where the local dynamics is given by a Gaussian distribution. The system, however, is dilute so that the overall distribution is Lorentzian. This has been discussed the first time in the context of spin-glasses \cite{Uemura85}, where a Laplace approach has been followed. This approach is unfavorable numerically.
+Here I follow the approach given in \cite{Yaouanc11}, Chapter 6.4.
+
+The measured macroscopic longitudinal polarization function is given as
+
+\begin{equation}\label{eq:P_ZL}
+ P_Z^{\rm L}(t, \Delta_{\rm L, ZF}, B, \nu) = \int_0^\infty P_Z^{\rm G}(t, \underbrace{r \cdot \Delta_{\rm L, ZF} }_{\Delta_{\rm G, ZF}}, B, \nu) \cdot \underbrace{\sqrt{\frac{2}{\pi}} \, \frac{1}{r^2}\, \exp\left(-\frac{1}{2 r^2}\right)}_{=: f(r)}\, dr,
+\end{equation}
+
+\noindent where $P_Z^{\rm G}(t, \Delta_{\rm G, ZF}, B, \nu)$ is the ``local'' Gaussian dynamic LF function, $B$ the applied LF-field, $\nu$ the hopping frequency, and
+$$
+  r = \frac{\Delta_{\rm G, ZF}}{\Delta_{\rm L, ZF}}.
+$$
+
+\noindent The function $P_Z^{\rm L}(t, \Delta_{\rm L, ZF}, B=0, \nu=0)$ can exactly be calculated and the resulting ZF function is
+
+\begin{equation}
+  P_Z^{\rm L}(t, \Delta_{\rm L, ZF}, B=0, \nu=0) = \frac{1}{3} + \frac{2}{3} \left( 1 - \gamma_\mu \Delta_{\rm L, ZF}\cdot t \right) \cdot \exp(-\gamma_\mu \Delta_{\rm L, ZF}\cdot t ),
+\end{equation}
+whereas $P_Z^{\rm G}(t, \Delta_{\rm L, ZF}, B=0, \nu=0)$ is
+\begin{equation}
+  P_Z^{\rm G}(t, \Delta_{\rm G, ZF}, B=0, \nu=0) = \frac{1}{3} + \frac{2}{3} \left( 1 - \left[\gamma_\mu \Delta_{\rm G, ZF}\cdot t\right]^2 \right) \cdot \exp(-1/2\, (\gamma_\mu \Delta_{\rm G, ZF}\cdot t)^2 ).
+\end{equation}
+
+\noindent In its general form $P_Z^{\rm G}(t, \Delta_{\rm G, ZF}, B, \nu)$ cannot be calculated analytically. The numerical evaluation used in \texttt{musrfit} is following the time domain approach as given in \cite{DalmasDeReotier92}
+
+\noindent Eq.(\ref{eq:P_ZL}) can be written as a generalized Riemann sum as
+\begin{equation}\label{eq:P_ZL_Riemann}
+ P_Z^{\rm L}(t, \Delta_{\rm L, ZF}, B, \nu) \simeq \sum_k P_Z^{\rm G}(t, r_k \cdot \Delta_{\rm L, ZF}, B, \nu) \cdot
+ \sqrt{\frac{2}{\pi}} \, \frac{1}{r_k^2}\, \exp\left(-\frac{1}{2 r_k^2}\right)\, \Delta r_k,
+\end{equation}
+where $\Delta r_k$ are \emph{non}-equidistant sampling intervals. $r_k$ are values somewhere within a sampling interval $\Delta r_k$.
+
+\noindent To understand the following it is useful to have a look at $f(r)$ (see Eq.(\ref{eq:P_ZL}) and Fig.(\ref{fig:f_r})).
+\newpage
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=0.7\textwidth]{f_r.pdf}
+ \caption{}\label{fig:f_r}
+\end{figure}
+
+\noindent This shows, that it will be important to have narrow enough $r$-intervals in the region from $0.2$ and $5.0$. The chosen sampling is given by the $r$-points
+\begin{eqnarray*}
+ \vec{r} &=& (0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.25, 1.5, 1.75, 2.0, 2.5, 3.0, 4.0, 5.0, 7.5, 10.0, \\
+  & & 12.8125, 15.625, 18.4375, 21.25, 26.875, 32.5, 43.75, 55.0, 77.5, 100.0).
+\end{eqnarray*}
+The intervals are given by
+$$
+  \Delta r_k = \vec{r}(k+1) - \vec{r}(k).
+$$
+
+\noindent The calculation of $P_Z^{\rm G}(t, r \cdot \Delta_{\rm L, ZF}, B, \nu)$ is quite costly, and hence the following approach has been chosen. Within an $\Delta r_k$-Interval we approximate Eq.(\ref{eq:P_ZL_Riemann}) by
+
+\begin{eqnarray}\label{eq:P_ZL_num}
+  P_Z^{\rm L}(t, \Delta_{\rm L, ZF}, B, \nu) &\simeq& \sum_k P_Z^{\rm G}(t, r_k \cdot \Delta_{\rm L, ZF}, B, \nu) \cdot
+ \sqrt{\frac{2}{\pi}} \, \int_{\vec{r}(k)}^{\vec{r}(k+1)} \frac{1}{r^2}\, \exp\left(-\frac{1}{2 r^2}\right)\, dr \nonumber \\
+ &=& \sum_k P_Z^{\rm G}(t, r_k \cdot \Delta_{\rm L, ZF}, B, \nu) \cdot \left[ \erf\left(\frac{1}{\sqrt{2}\, \vec{r}(k)}\right) - \erf\left(\frac{1}{\sqrt{2}\, \vec{r}(k+1)}\right) \right],
+\end{eqnarray}
+where $\erf(\cdot)$ is the Gaussian error function.
+
+\bibliographystyle{unsrt}
+\bibliography{dynamicGL_LF.bib}
+
+\end{document}

src/external/LF_GL/docu/f_r.C

@@ -0,0 +1,19 @@
+void f_r()
+{
+  TCanvas *c1 = new TCanvas("c1", "c1", 800, 600);
+  TF1 *fr = new TF1("fr", "TMath::Sqrt(2.0/TMath::Pi())*1.0/TMath::Power(x, 2.0)*TMath::Exp(-0.5/TMath::Power(x, 2.0))", 0.0, 10);
+  fr->SetNpx(1500);
+  fr->GetXaxis()->SetTitle("#it{r}");
+  fr->GetXaxis()->SetTitleSize(0.05);
+  fr->GetXaxis()->SetTitleOffset(0.8);
+  fr->GetXaxis()->CenterTitle(kTRUE);
+  fr->GetYaxis()->SetTitle("#it{f(r)}");
+  fr->GetYaxis()->SetTitleSize(0.05);
+  fr->GetYaxis()->SetTitleOffset(0.8);
+  fr->GetYaxis()->CenterTitle(kTRUE);
+  fr->SetTitle("");
+  fr->SetLineColor(4);
+  fr->Draw();
+
+  c1->SetGrid();
+}