Commit 817abb9b by Tobias WEBER

### formulas

parent baae6623
 % % preparation of the TAS lecture % useful formulas % @author Tobias Weber % @date 13-jul-2018 % @license see 'LICENSE' file ... ... @@ -24,10 +24,10 @@ \begin{document} \title{Notes on Triple Axis Spectroscopy} \title{Useful formulas} \author{T. Weber, tweber@ill.fr} \maketitle \tableofcontents %\tableofcontents ... ... @@ -208,98 +208,7 @@ Special case for cubic crystals, $g_{ij} = \delta_{ij} \cdot \left( 2\pi / a \ri % ==================================================================================================================================== \chapter{Neutron Scattering} \begin{center} \includegraphics[width = 0.75 \textwidth]{recip} \end{center} \section{Nuclear Scattering} The cross-sections for neutron scattering are derived in \cite[Ch. 2]{Squires2012} of which we summarise the most important results here. An excellent overview can also be found in \cite[Ch. 2]{Shirane2002}. The double-differential cross-section is the number of neutrons that are scattered into a solid angle$d\Omega$and energy range$\left[ E_f, E_f + dE \right]$, normalised to time$t$, Flux$\Phi$,$d\Omega$, and$dE$\cite[p. 10]{Squires2012}, \left(\frac{d^2 \sigma}{d\Omega dE_f}\right)_{i \rightarrow f} = \frac{1}{\Phi \cdot d\Omega \cdot dE_f} \cdot R_{i \rightarrow f}, where the transition rate$R_{i \rightarrow f}$is given by Fermi's Golden Rule \cite[p. 509]{Merzbacher1998}: R_{i \rightarrow f} = \frac{2\pi}{\hbar} \left| \left< i | V | f \right> \right|^2 \cdot N_f, with the number of final states,$N_f$, in the range$\left[ E_f, E_f + dE \right]$. Neutron-nuclear scattering is described using the Fermi pseudo-potential$V$\cite[p. 15]{Squires2012}: V \left( \bm{r} \right) = \frac{2\pi \hbar^2}{m_n} \cdot b \cdot \delta \left( \bm{r} \right), V \left( \bm{Q} \right) = \frac{2\pi \hbar^2}{m_n} \cdot b. Summing over all available final states$\left| f \right>$and thermally averaging over the initial states$\left| i \right>$, results in \cite[p. 20]{Squires2012}: \boxed{ \frac{d^2 \sigma}{d\Omega dE_f} = \frac{1}{2\pi \hbar} \cdot \frac{k_f}{k_i} \cdot \sum_{kl}{b_k b_l \int{dt \cdot \exp{ \left( -\imath \omega t \right)} \cdot \left< \exp{\left( -\imath \bm{Q} \cdot \bm{\hat{R}_k} \left(t=0 \right) \right)} \exp{\left(\imath \bm{Q} \cdot \bm{\hat{R}_l} \left(t \right) \right)} \right>_{\mathrm{therm.}} }}, } with the Heisenberg operators$\bm{\hat{R}_l} \left(t \right)$. Averaging the scattering lengths$b_k b_l$into an effective$b\$ leads to two distinct contributions to the scattering cross section \cite[p. 22]{Squires2012}: \begin{split} \frac{d^2 \sigma}{d\Omega dE_f} & = \frac{1}{2\pi \hbar} \cdot \frac{k_f}{k_i} \cdot \left< b \right> ^2 \cdot \sum_{kl}{ \int{dt \cdot \exp{ \left( -\imath \omega t \right)} \cdot \left< \exp{\left( -\imath \bm{Q} \cdot \bm{\hat{R}_k} \left(t=0 \right) \right)} \exp{\left(\imath \bm{Q} \cdot \bm{\hat{R}_l} \left(t \right) \right)} \right>_{\mathrm{therm.}} }} \\ & + \frac{1}{2\pi \hbar} \cdot \frac{k_f}{k_i} \cdot \underbrace{\left( \left< b^2 \right> - \left< b \right>^2 \right)}_{\equiv b_{inc}^2} \cdot \sum_{k}{ \int{dt \cdot \exp{ \left( -\imath \omega t \right)} \cdot \left< \exp{\left( -\imath \bm{Q} \cdot \bm{\hat{R}_k} \left(t=0 \right) \right)} \exp{\left(\imath \bm{Q} \cdot \bm{\hat{R}_k} \left(t \right) \right)} \right>_{\mathrm{therm.}} }}. \end{split} The first term of the sum is the coherent contribution (i.e. collective phenomena: phonons, Bragg peaks), the second one the incoherent contribution (spin- and nuclear-incoherent peaks, diffuse scattering). % ------------------------------------------------------------------------------------------------------------------------------------ \section{Magnetic Scattering} % ==================================================================================================================================== % ==================================================================================================================================== \chapter{Triple-Axis Resolution Ellipsoid} % ==================================================================================================================================== \bibliographystyle{alpha} \bibliography{\jobname.bib} %\bibliographystyle{alpha} %\bibliography{\jobname.bib} \end{document}
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