191 lines
7.1 KiB
TeX
191 lines
7.1 KiB
TeX
\documentclass[twoside]{article}
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\usepackage[english]{babel}
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%\usepackage{a4}
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\usepackage{amssymb,amsmath,bm}
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\usepackage{graphicx,tabularx}
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\usepackage{fancyhdr}
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\usepackage{array}
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\usepackage{float}
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\usepackage{hyperref}
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\usepackage{xspace}
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\usepackage{rotating}
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\usepackage{dcolumn}
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\usepackage{geometry}
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\usepackage{color}
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\geometry{a4paper,left=20mm,right=20mm,top=20mm,bottom=20mm}
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% \setlength{\topmargin}{10mm}
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% \setlength{\topmargin}{-13mm}
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% % \setlength{\oddsidemargin}{0.5cm}
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% % \setlength{\evensidemargin}{0cm}
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% \setlength{\oddsidemargin}{1cm}
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% \setlength{\evensidemargin}{1cm}
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% \setlength{\textwidth}{15cm}
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\setlength{\textheight}{23.8cm}
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\pagestyle{fancyplain}
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\addtolength{\headwidth}{0.6cm}
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\fancyhead{}%
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\fancyhead[RE,LO]{\bf \textsc{LineProfile}}%
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\fancyhead[LE,RO]{\thepage}
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\cfoot{--- J.~A.~Krieger -- \today~ ---}
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\rfoot{\includegraphics[width=2cm]{PSI-Logo_narrow.jpg}}
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\DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it}
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\newcommand{\mean}[1]{\langle #1 \rangle}
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\newcommand{\lem}{LE-$\mu$SR\xspace}
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\newcommand{\lemhead}{LE-$\bm{\mu}$SR\xspace}
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\newcommand{\musr}{$\mu$SR\xspace}
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\newcommand{\musrhead}{$\bm{\mu}$SR\xspace}
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\newcommand{\trimsp}{\textsc{trim.sp}\xspace}
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\newcommand{\musrfithead}{MUSRFIT\xspace}
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\newcommand{\musrfit}{\textsc{musrfit}\xspace}
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\newcommand{\gapint}{\textsc{GapIntegrals}\xspace}
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\newcommand{\YBCO}{YBa$_{2}$Cu$_{3}$O$_{7-\delta}$\xspace}
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\newcommand{\YBCOhead}{YBa$_{\bm{2}}$Cu$_{\bm{3}}$O$_{\bm{7-\delta}}$\xspace}
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\newcolumntype{d}[1]{D{.}{.}{#1}}
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\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}
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\begin{document}
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% Header info --------------------------------------------------
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\thispagestyle{empty}
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\noindent
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\begin{tabular}{@{\hspace{-0.2cm}}l@{\hspace{6cm}}r}
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\noindent\includegraphics[width=3.4cm]{PSI-Logo_narrow.jpg} &
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{\Huge\sf Memorandum}
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\end{tabular}
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%
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\vskip 1cm
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%
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\begin{tabular}{@{\hspace{-0.5cm}}ll@{\hspace{4cm}}ll}
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Date: & \today & & \\[3ex]
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From: & J.~A.~Krieger & \\
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E-Mail: & \verb?jonas.krieger@psi.ch? &&
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\end{tabular}
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%
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\vskip 0.3cm
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\noindent\hrulefill
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\vskip 1cm
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%
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\section*{\musrfithead plug-in for simple $\beta$-NMR resonance line shapes}%
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This library contains useful functions to fit NMR and $\beta$-NMR line shapes.
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The functional form of the powder averages was taken from
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\href{http://dx.doi.org/10.1007/978-3-642-68756-3_2}{M. Mehring, Principles
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of High Resolution NMR in Solids (Springer 1983)}.
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%
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The \texttt{libLineProfile} library currently contains the following functions:
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\begin{description}
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\item[LineGauss]
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\begin{equation}
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A(f)=e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}}
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\end{equation}
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Gaussian line shape around $f_0$ with width $\sigma$ and height~$1$.\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile LineGauss 1 2?\\[1.5ex]
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Parameters: $f_0$, $\sigma$.
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\item[LineLaplace]
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\begin{equation}
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A(f)=e^{-2\ln 2 \left|\frac{f-f_0}{\sigma}\right|}
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\end{equation}
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Laplaceian line shape around $f_0$ with width $\sigma$ and
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height~$1$.\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile LineLaplace 1 2?
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\\[1.5ex]
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Parameters: $f_0$, $\sigma$.
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\item[LineLorentzian]
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\begin{equation}
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A(f)=
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\frac{\sigma^2}{4(f-f_0)^2+\sigma^2}
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\end{equation}
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Lorentzian line shape around $f_0$ with width $\sigma$ and
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height~$1$.\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile LineLorentzian 1 2?
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\\[1.5ex]
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Parameters: $f_0$, $\sigma$.
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\item[LineSkewLorentzian]
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\begin{equation}
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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)}}
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\end{equation}
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Skewed Lorentzian line shape around $f_0$ with width $\sigma$,
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height~$1$ and skewness parameter $a$.\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile LineSkewLorentzian 1 2 3?
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\\[1.5ex]
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Parameters: $f_0$, $\sigma$, $a$.
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\item[LineSkewLorentzian2]
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\begin{equation}
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A(f)= \left\{\begin{matrix}\frac{{\sigma_1}^2}{4{(f-f_0)}^2+{\sigma_1}^2},&f<f_0\\[9pt] \frac{{\sigma_2}^2}{4{(f-f_0)}^2+{\sigma_2}^2},&f>f_0\end{matrix}\right.
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\end{equation}
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Skewed Lorentzian line shape around $f_0$ with height~$1$ and widths $\sigma_1$,
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and $\sigma_2$.\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile LineSkewLorentzian2 1 2 3?
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\\[1.5ex]
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Parameters: $f_0$, $\sigma_1$, $\sigma_2$.
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\item[PowderLineAxialLor]
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\begin{equation}
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A(f)= I_{\mathrm ax}(f)\circledast\left( \frac{\sigma^2}{4f^2+\sigma^2} \right)
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\end{equation}
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Powder average of a axially symmetric interaction, convoluted with a Lorentzian.
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\begin{equation}\label{eq:Iax}
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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.
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\end{equation}
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The maximal height of the curve is normalized to $\sim$1.
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\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile PowderLineAxialLor 1 2 3?
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\\[1.5ex]
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Parameters: $f_\parallel$, $f_\perp$, $\sigma$.
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\item[PowderLineAxialGss]
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\begin{equation}
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A(f)= I_{\mathrm ax}(f)\circledast\left(e^{-\frac{4\ln 2 (f-f_0)^2}{
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\sigma^2}} \right)
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\end{equation}
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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.
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\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile PowderLineAxialGss 1 2 3?
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\\[1.5ex]
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Parameters: $f_\parallel$, $f_\perp$, $\sigma$.
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\item[PowderLineAsymLor]
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\begin{equation}
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A(f)= I(f)\circledast\left( \frac{\sigma^2}{4f^2+\sigma^2} \right)
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\end{equation}
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Powder average of a asymmetric interaction, convoluted with a Lorentzian.
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Assume without loss of generality that $f_1<f_2<f_3$, then
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\begin{align}\label{eq:Iasym}
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I(f)&=\left\{\begin{matrix}
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\frac{K(m)}{\pi\sqrt{(f-f_1)(f_3-f_2)}},& f_3\geq f>f_2 \\[9pt]
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\frac{K(m)}{\pi\sqrt{(f_3-f)(f_2-f_1)}},& f_2>f\geq f_1\\[9pt]
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0 & \text{otherwise}
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\end{matrix} \right. \\
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m&=\left\{\begin{matrix}
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\frac{(f_2-f_1)(f_3-f)}{(f_3-f_2)(f-f_1)},& f_3\geq f>f_2 \\[6pt]
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\frac{(f-f_1)(f_3-f_2)}{(f_3-f)(f_2-f_1)},& f_2>f\geq f_1\\[6pt]
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\end{matrix} \right. \\\label{eq:Kofm}
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K(m)&=\int_0^{\pi/2}\frac{\mathrm d\varphi}{\sqrt{1-m^2\sin^2{\varphi}}},
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\end{align}
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where $K(m)$ is the complete elliptic integral of the first kind.
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Note that $f_1<f_2<f_3$ is not required by the code.
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The maximal height of the curve is normalized to $\sim$1.
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\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile PowderLineAsymLor 1 2 3 4?
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\\[1.5ex]
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Parameters: $f_1$, $f_2$,$f_3$, $\sigma$.
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\item[PowderLineAsymGss]
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\begin{equation}
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A(f)= I(f)\circledast\left(e^{-\frac{4\ln 2 (f-f_0)^2}{ \sigma^2}} \right)
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\end{equation}
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Powder average of a asymmetric interaction {(Eq.~\ref{eq:Iasym}\,-\,\ref{eq:Kofm})}, convoluted with a Gaussian.
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The maximal height of the curve is normalized to $\sim$1.
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\\[1.5ex]
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\musrfit theory line: \verb?userFcn libLineProfile PowderLineAsymGss 1 2 3 4?
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\\[1.5ex]
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Parameters: $f_1$, $f_2$,$f_3$, $\sigma$.
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\end{description}
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\end{document}
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