\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{LineProfile}}% \fancyhead[LE,RO]{\thepage} \cfoot{--- J.~A.~Krieger -- \today~ ---} \rfoot{\includegraphics[width=2cm]{PSI-Logo_narrow.jpg}} \DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it} \newcommand{\mean}[1]{\langle #1 \rangle} \newcommand{\lem}{LE-$\mu$SR\xspace} \newcommand{\lemhead}{LE-$\bm{\mu}$SR\xspace} \newcommand{\musr}{$\mu$SR\xspace} \newcommand{\musrhead}{$\bm{\mu}$SR\xspace} \newcommand{\trimsp}{\textsc{trim.sp}\xspace} \newcommand{\musrfithead}{MUSRFIT\xspace} \newcommand{\musrfit}{\textsc{musrfit}\xspace} \newcommand{\gapint}{\textsc{GapIntegrals}\xspace} \newcommand{\YBCO}{YBa$_{2}$Cu$_{3}$O$_{7-\delta}$\xspace} \newcommand{\YBCOhead}{YBa$_{\bm{2}}$Cu$_{\bm{3}}$O$_{\bm{7-\delta}}$\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: & J.~A.~Krieger & \\ E-Mail: & \verb?jonas.krieger@psi.ch? && \end{tabular} % \vskip 0.3cm \noindent\hrulefill \vskip 1cm % \section*{\musrfithead plug-in for simple $\beta$-NMR resonance line shapes}% This library contains useful functions to fit NMR and $\beta$-NMR line shapes. The functional form of the powder averages was taken from \href{http://dx.doi.org/10.1007/978-3-642-68756-3_2}{M. Mehring, Principles of High Resolution NMR in Solids (Springer 1983)}. % The \texttt{libLineProfile} library currently contains the following functions: \begin{description} \item[LineGauss] \begin{equation} A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \end{equation} Gaussian line shape around $f_0$ with width $\sigma$ and height~$1$.\\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile LineGauss 1 2?\\[1.5ex] Parameters: $f_0$, $\sigma$. \item[LineLaplace] \begin{equation} A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{\sigma}\right|} \end{equation} Laplaceian line shape around $f_0$ with width $\sigma$ and height~$1$.\\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile LineLaplace 1 2? \\[1.5ex] Parameters: $f_0$, $\sigma$. \item[LineLorentzian] \begin{equation} A(f)= \frac{\sigma^2}{4(f-f_0)^2+\sigma^2} \end{equation} Lorentzian line shape around $f_0$ with width $\sigma$ and height~$1$.\\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile LineLorentzian 1 2? \\[1.5ex] Parameters: $f_0$, $\sigma$. \item[LineSkewLorentzian] \begin{equation} A(f)= \frac{\sigma*\sigma_a}{4(f-f_0)^2+\sigma_a^2}, \quad \sigma_a=\frac{2\sigma}{1+e^{a(f-f_0)}} \end{equation} Skewed Lorentzian line shape around $f_0$ with width $\sigma$, height~$1$ and skewness parameter $a$.\\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile LineSkewLorentzian 1 2 3? \\[1.5ex] Parameters: $f_0$, $\sigma$, $a$. \item[LineSkewLorentzian2] \begin{equation} A(f)= \left\{\begin{matrix}\frac{{\sigma_1}^2}{4{(f-f_0)}^2+{\sigma_1}^2},&ff_0\end{matrix}\right. \end{equation} Skewed Lorentzian line shape around $f_0$ with height~$1$ and widths $\sigma_1$, and $\sigma_2$.\\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile LineSkewLorentzian2 1 2 3? \\[1.5ex] Parameters: $f_0$, $\sigma_1$, $\sigma_2$. \item[PowderLineAxialLor] \begin{equation} A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{\sigma^2}{4f^2+\sigma^2} \right) \end{equation} Powder average of a axially symmetric interaction, convoluted with a Lorentzian. \begin{equation}\label{eq:Iax} 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. \end{equation} The maximal height of the curve is normalized to $\sim$1. \\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile PowderLineAxialLor 1 2 3? \\[1.5ex] Parameters: $f_\parallel$, $f_\perp$, $\sigma$. \item[PowderLineAxialGss] \begin{equation} A(f)= I_{\mathrm ax}(f)\circledast\left(e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right) \end{equation} Powder average of a axially symmetric interaction (Eq.~\ref{eq:Iax}), convoluted with a Gaussian. The maximal height of the curve is normalized to $\sim$1. \\[1.5ex] \musrfit theory line: \verb?userFcn libLineProfile PowderLineAxialGss 1 2 3? \\[1.5ex] Parameters: $f_\parallel$, $f_\perp$, $\sigma$. \item[PowderLineAsymLor] \begin{equation} A(f)= I(f)\circledast\left( \frac{\sigma^2}{4f^2+\sigma^2} \right) \end{equation} Powder average of a asymmetric interaction, convoluted with a Lorentzian. Assume without loss of generality that $f_1f_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. \\\label{eq:Kofm} K(m)&=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}}, \end{align} where $K(m)$ is the complete elliptic integral of the first kind. Note that $f_1