diff --git a/src/external/libGbGLF/doc/GbG-LF.pdf b/src/external/libGbGLF/doc/GbG-LF.pdf new file mode 100644 index 00000000..f7493f5c Binary files /dev/null and b/src/external/libGbGLF/doc/GbG-LF.pdf differ diff --git a/src/external/libGbGLF/doc/GbG-LF.tex b/src/external/libGbGLF/doc/GbG-LF.tex new file mode 100644 index 00000000..96cf3370 --- /dev/null +++ b/src/external/libGbGLF/doc/GbG-LF.tex @@ -0,0 +1,184 @@ +\documentclass[twoside]{article} + +\usepackage[english]{babel} +%\usepackage{a4} +\usepackage{amssymb,amsmath,bm} +\usepackage{graphicx,tabularx} +\usepackage{fancyhdr} +\usepackage{array} +\usepackage{float} +\usepackage{hyperref} +\usepackage{xspace} +\usepackage{rotating} +\usepackage{dcolumn} +\usepackage{geometry} +\usepackage{color} + +\geometry{a4paper,left=20mm,right=20mm,top=20mm,bottom=20mm} + +% \setlength{\topmargin}{10mm} +% \setlength{\topmargin}{-13mm} +% % \setlength{\oddsidemargin}{0.5cm} +% % \setlength{\evensidemargin}{0cm} +% \setlength{\oddsidemargin}{1cm} +% \setlength{\evensidemargin}{1cm} +% \setlength{\textwidth}{15cm} +\setlength{\textheight}{23.8cm} + +\pagestyle{fancyplain} +\addtolength{\headwidth}{0.6cm} +\fancyhead{}% +\fancyhead[RE,LO]{\bf \textsc{GapIntegrals}}% +\fancyhead[LE,RO]{\thepage} +\cfoot{--- A.~Suter -- \today~ ---} +\rfoot{\includegraphics[width=2cm]{PSI-Logo_narrow.jpg}} + +\DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it} + +\newcommand{\mean}[1]{\langle #1 \rangle} +\newcommand{\ie}{\emph{i.e.}\xspace} +\newcommand{\musrfithead}{MUSRFIT\xspace} +\newcommand{\musrfit}{\textsc{musrfit}\xspace} + +\newcolumntype{d}[1]{D{.}{.}{#1}} +\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}} + +\begin{document} +% Header info -------------------------------------------------- +\thispagestyle{empty} +\noindent +\begin{tabular}{@{\hspace{-0.2cm}}l@{\hspace{6cm}}r} +\noindent\includegraphics[width=3.4cm]{PSI-Logo_narrow.jpg} & + {\Huge\sf Memorandum} +\end{tabular} +% +\vskip 1cm +% +\begin{tabular}{@{\hspace{-0.5cm}}ll@{\hspace{4cm}}ll} +Date: & \today & & \\[3ex] +From: & A. Suter & & \\ +E-Mail: & \verb?andreas.suter@psi.ch? && +\end{tabular} +% +\vskip 0.3cm +\noindent\hrulefill +\vskip 1cm +% +\section*{Homogenous Disorder Model: GbG in Longitudinal Fields}% + +Noakes and Kalvius \cite{noakes1997} derived a phenomenological model for +homogenous disorder: Gaussian-broadened Gaussian disorder (see also +Ref.\,\cite{yaouanc2011}). In both mentioned references only the zero field +case and the weak transverse field case are discussed. Here I briefly summarize +the longitudinal field (LF) case under the assumption that the applied field doesn't +polarize the impurties, \ie the applied field is ``innocent''. + +The Gauss-Kubo-Toyabe LF polarization function is + +\begin{eqnarray}\label{eq:GKT_LF} + P_{Z,{\rm GKT}}^{\rm LF} &=& 1 - 2 \frac{\sigma^2}{\omega_{\rm ext}^2}\left[ 1 - \cos(\omega_{\rm ext} t)\,\exp\left(-1/2 (\sigma t)^2\right) \right] + \label{eq:GKT_LF_1}\\ + & & + 2 \frac{\sigma^2}{\omega_{\rm ext}^3} \int_0^t \sin(\omega_{\rm ext} \tau)\,\exp\left(-1/2 (\omega_{\rm ext} \tau)^2\right) d\tau. \label{eq:GKT_LF_2} +\end{eqnarray} + +\noindent The Gaussian disorder is assumed to have the funtional form + +\begin{equation}\label{eq:GaussianDisorder} + \varrho = \frac{1}{\sqrt{2\pi}}\,\frac{1}{\sigma_1} \exp\left( -\frac{1}{2} \, \left[ \frac{\sigma - \sigma_0}{\sigma_1} \right]^2 \right). +\end{equation} + +\noindent In Ref.\cite{yaouanc2011} a slightly different notation is used: $\sigma \to \Delta_{\rm G}$, $\sigma_0 \to \Delta_{0}$, and +$\sigma_1 \to \Delta_{\rm GbG}$. + +\noindent The GbG LF polarizatio function is given by + +\begin{equation} + P_{Z,{\rm GbG}}^{\rm LF} = \int_0^\infty d\sigma \left\{ \varrho \cdot P_{Z,{\rm GKT}}^{\rm LF} \right\}. +\end{equation} + +\noindent Assuming that $\sigma_0 \gg \sigma_1$ this can be approximated by + +\begin{equation} + P_{Z,{\rm GbG}}^{\rm LF} \simeq \int_{-\infty}^\infty d\sigma \left\{ \varrho \cdot P_{Z,{\rm GKT}}^{\rm LF} \right\}. +\end{equation} + +\noindent Integrating + +\begin{equation*} + P_{Z,{\rm GbG}}^{\rm LF, (1)} = \int_{-\infty}^\infty d\sigma \left\{ \varrho \cdot P_{Z,{\rm GKT}}^{\rm LF, (1)} \right\}, +\end{equation*} + +\noindent where $P_{Z,{\rm GKT}}^{\rm LF, (1)}$ is given by Eq.(\ref{eq:GKT_LF_1}), leads to + +\begin{equation}\label{eq:GbG_LF_1} + P_{Z,{\rm GbG}}^{\rm LF, (1)} = 1 - 2 \frac{\sigma_0^2+\sigma_1^2}{\omega_{\rm ext}^2} + + 2 \frac{\sigma_0^2 + \sigma_1^2 (1 + \sigma_1^2 t^2)}{\omega_{\rm ext}^2 (1 + \sigma_1^2 t^2)^{5/2}}\, \cos(\omega_{\rm ext} t)\, + \exp\left[-\frac{1}{2} \frac{\sigma_0^2 t^2}{1+\sigma_1^2 t^2}\right], +\end{equation} + +\noindent and Eq.(\ref{eq:GKT_LF_2}) leads to the non-analytic integral + +\begin{eqnarray} + P_{Z,{\rm GbG}}^{\rm LF, (2)} &=& \int_{-\infty}^\infty d\sigma \left\{ \varrho \cdot P_{Z,{\rm GKT}}^{\rm LF, (2)} \right\} \nonumber \\ + &=& \int_0^t d\tau \left\{ \frac{\sigma_0^4 + 3 \sigma_1^4 (1 + \sigma_1^2 \tau^2)^2 + 6 \sigma_0^2 \sigma_1^2 (1+\sigma_1^2 \tau^2)}{\omega_{\rm ext}^3 (1+\sigma_1^2 \tau^2)^{9/2}} + \sin(\omega_{\rm ext} \tau)\, \exp\left[-\frac{1}{2} \frac{\sigma_0^2 t^2}{1+\sigma_1^2 t^2}\right] \right\}. \label{eq:GbG_LF_2} +\end{eqnarray} + +\noindent The full GbG LF polarization function is hence + +\begin{equation} + P_{Z,{\rm GbG}}^{\rm LF} = P_{Z,{\rm GbG}}^{\rm LF, (1)} + P_{Z,{\rm GbG}}^{\rm LF, (2)} +\end{equation} + + +\subsection*{The GbG LF Polarization Function as a User Function in \musrfithead} + +Eqs.(\ref{eq:GbG_LF_1})\&(\ref{eq:GbG_LF_2}) are implemented in \musrfit as user function. The current implementation is far from being efficient but stable. +The typical call from within the msr-file would be + +\begin{verbatim} +############################################################### +FITPARAMETER +# Nr. Name Value Step Pos_Error Boundaries + 1 PlusOne 1 0 none + 2 MinusOne -1 0 none + 3 Alpha 0.78699 -0.00036 0.00036 0 none + 4 Asy 0.06682 0.00027 none 0 0.33 + 5 Sig0 0.3046 -0.0087 0.0093 0 100 + 6 Rb 1.0000 0.0027 none 0 1 + 7 Field0 0 0 none + 8 Field1 20.03 0 none + 9 Field2 99.32 0 none + +############################################################### +THEORY +asymmetry fun1 +userFcn libGbGLF PGbGLF map2 5 fun2 (field sigma0 Rb) + +############################################################### +FUNCTIONS +fun1 = map1 * par4 +fun2 = par5 * par6 +\end{verbatim} + +\noindent where \texttt{PGbGLF} takes 3 arguments: + +\begin{enumerate} + \item field in Gauss + \item $\sigma_0$ in ($1/\mu s$) + \item $R_b = \sigma_1 / \sigma_0$ +\end{enumerate} + +\noindent \textbf{Be aware that we explicitly assumed $\sigma_1 \ll \sigma_0$, \ie $R_b < 1$.} + +\bibliographystyle{plain} +\begin{thebibliography}{1} + +\bibitem{noakes1997} D.~R.~Noakes, G.~M.~Kalvius, Phys.~Rev.~B, \textbf{56}, 2352 +(1997). +\bibitem{yaouanc2011} A.~Yaouanc, P.~Dalmas~de~R\'{e}otier, ``Muon Spin +Rotation, Relaxation, and Resonance'', Oxford University Press (2011). + +\end{thebibliography} + + +\end{document} diff --git a/src/external/libGbGLF/doc/PSI-Logo_narrow.jpg b/src/external/libGbGLF/doc/PSI-Logo_narrow.jpg new file mode 100644 index 00000000..9bda1610 Binary files /dev/null and b/src/external/libGbGLF/doc/PSI-Logo_narrow.jpg differ