Commit 7542891e by Tobias WEBER

### continued with lecture (crystal coords)

parent 33652a66
 ... ... @@ -64,3 +64,4 @@ stack.hh *.aux *.pdf *.synctex.* *.toc
 % % collection of useful formulas % preparation of the TAS lecture % @author Tobias Weber % @date 13-jul-2018 % @license see 'LICENSE' file % \documentclass{article} \documentclass[english]{book} \usepackage{amsmath} \usepackage{tensor} \usepackage{bm} \usepackage{graphicx} \usepackage{siunitx} \usepackage{babel} \usepackage[a4paper]{geometry} \geometry{tmargin=2.5cm, bmargin=2.5cm, lmargin=2cm, rmargin=2cm} \begin{document} Useful formulas and derivations, T. Weber, tweber@ill.fr, July 13, 2018. \title{Notes on Triple Axis Spectroscopy} \author{T. Weber, tweber@ill.fr} \maketitle \tableofcontents % ------------------------------------------------------------------------------------------------------------------------------------ % ==================================================================================================================================== \chapter{Crystal Coordinates and TAS Angles} % ------------------------------------------------------------------------------------------------------------------------------------ \section{Fractional Coordinates} \begin{center} ... ... @@ -57,9 +66,9 @@ Inserting $\left| a \right>$ and $\left| b \right>$ into Eq. \ref{ab} gives: Using the cross product between $\left| a \right>$ and $\left| b \right>$, we get: \left\Vert \left| a \right> \times \left| b \right> \right\Vert = \left\Vert \left( \begin{array}{c} 0 \\ 0 \\ a_1 b_2 \end{array} \right) \right\Vert = ab \sin \gamma, \label{crossab} \left\Vert \left| a \right> \times \left| b \right> \right\Vert = \left\Vert \left( \begin{array}{c} 0 \\ 0 \\ a_1 b_2 \end{array} \right) \right\Vert = ab \sin \gamma, \label{crossab} b_2 = b \sin \gamma, \boxed{ \left| b \right> = \left( \begin{array}{c} b \cos \gamma \\ b \sin \gamma \\ 0 \end{array} \right). } \label{bvec} ... ... @@ -84,10 +93,10 @@ The last component, $c_3$, can be obtained from the vector length normalisation, \left< c | c \right > = c_1^2 + c_2^2 + c_3^2 = c^2, c_3^2 = c^2 - c_1^2 - c_2^2, c_3^2 = c^2 \left[1 - \cos^2 \beta - \left(\frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \right)^2 \right], \boxed{ \left| c \right> = \left( \begin{array}{c} c \cdot \cos \beta \\ c \cdot \frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \\ c \cdot \sqrt{ 1 - \cos^2 \beta - \left(\frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \right)^2 } \boxed{ \left| c \right> = \left( \begin{array}{c} c \cdot \cos \beta \\ c \cdot \frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \\ c \cdot \sqrt{ 1 - \cos^2 \beta - \left(\frac{\cos \alpha - \cos \gamma \cos \beta}{\sin \gamma} \right)^2 } \end{array} \right). } \label{avec} ... ... @@ -100,26 +109,54 @@ The crystallographic $A$ matrix, which transforms real-space fractional to lab c \end{array} \right). The $B$ matrix, which transforms reciprocal-space relative lattice units (rlu) to lab coordinates (1/A), is: The $B$ matrix, which transforms reciprocal-space relative lattice units (rlu) to lab coordinates (1/\AA), is: B = 2 \pi A^{-t}, where $-t$ denotes the transposed inverse. The metric tensor corresponding to the coordinate system defined by the $B$ matrix is: \left(g_{ij}\right) = \left<\bm{b_i} | \bm{b_j} \right> = B^T B, where the reciprocal basis vectors $\left| \bm{b_i} \right>$ form the columns of $B$. \subsection*{Example: Lengths and Angles in the Reciprocal Lattice} Having a metric makes it straightforward to calculate lengths and angles. The length of a reciprocal lattice vector $\left| G \right>$ seen from the lab system is (in 1/\AA{} units): \left\Vert \left< G | G \right> \right\Vert = \sqrt{\left< G | G \right>} = \sqrt{G_i G^j} = \sqrt{g_{ij} G^i G^j}. The angle $\theta$ between two Bragg peaks $\left| G \right>$ and $\left| H \right>$ is given by their dot product: \frac{\left< G | H \right>}{\left\Vert \left< G | G \right> \right\Vert \cdot \left\Vert \left< H | H \right> \right\Vert} = \cos \theta, % % \frac{G_i H^j }{\left\Vert \left< G | G \right> \right\Vert \cdot \left\Vert \left< H | H \right> \right\Vert} = \cos \theta, % \frac{g_{ij} G^i H^j }{\sqrt{g_{ij} G^i G^j} \sqrt{g_{ij} H^i H^j}} = \cos \theta. % ------------------------------------------------------------------------------------------------------------------------------------ % ------------------------------------------------------------------------------------------------------------------------------------ \section{TAS Angles and Scattering Triangle} \begin{figure} \begin{center} \includegraphics[width = 0.2 \textwidth]{triangle} \includegraphics[width = 0.5 \textwidth]{tas} \hspace{1.5cm} \includegraphics[trim=0 -2cm 0 0, width=0.25\textwidth]{triangle} \end{center} \caption{Triple-axis layout and scattering triangle.} \end{figure} \subsection*{Monochromator Angles $a_1$, $a_2$ and Analyser Angles $a_5$, $a_6$} The monochromator (and analyser) angles follow directly from Bragg's equation: 2 k_{i,f} \sin a_{1,5} = 2 \pi / d_{m,a}, \boxed{ a_{1,5} = \arcsin \left( \frac{\pi}{d_{m,a} \cdot k_{i,f}} \right). } 2 d_{m,a}\sin a_{1,5} = n \lambda_{i,f}, 2 k_{i,f} \sin a_{1,5} = 2 \pi n / d_{m,a}, \boxed{ a_{1,5} = \arcsin \left( \frac{\pi n}{d_{m,a} \cdot k_{i,f}} \right). } Fulfilling the Bragg condition, the angles $a_2$ and $a_6$ are simply: $a_{2,6} = 2 \cdot a_{1,5}.$ ... ... @@ -161,5 +198,27 @@ The sign, $\sigma_{\mathrm{side}}$, of $\xi$ depends on which side of the orient \paragraph*{Special case} Special case for cubic crystals, $g_{ij} = \delta_{ij} \cdot \left( 2\pi / a \right)^2$: \xi = \sigma_{\mathrm{side}} \cdot \arccos \left( \frac{ Q_i a^i }{ \sqrt{Q_i Q^i} \sqrt{a_i a^i} } \right) % ------------------------------------------------------------------------------------------------------------------------------------ % ==================================================================================================================================== % ==================================================================================================================================== \chapter{Neutron Scattering Cross-Sections} % ==================================================================================================================================== % ==================================================================================================================================== \chapter{Triple-Axis Resolution Ellipsoid} % ==================================================================================================================================== \end{document}
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