Commit 5235c196 by Tobias WEBER

### continued with formulas

parent f35db047
 ... ... @@ -7,6 +7,9 @@ \documentclass{article} \usepackage{amsmath} \usepackage{graphicx} \usepackage[a4paper]{geometry} \geometry{tmargin=2.5cm, bmargin=2.5cm, lmargin=2cm, rmargin=2cm} ... ... @@ -16,29 +19,36 @@ Collection of useful formulas, T. Weber, July 13, 2018. \section{Scattering Triangle} \begin{center} \includegraphics[width = 0.2 \textwidth]{triangle} \end{center} \subsection*{Scattering Angle $a_4$} \left| Q \right> = \left| k_i \right> - \left| k_f \right> \\ \left< Q | Q \right> = \left( \left< k_i \right| - \left< k_f \right| \right) \cdot \left( \left| k_i \right> - \left| k_f \right> \right) \left< Q | Q \right> = \left< k_i | k_i \right> + \left< k_f | k_f \right> - 2 \left< k_i | k_f \right> Q^2 = k_i^2 + k_f^2 - 2 k_i k_f \cos a_4 a_4 = \arccos \left( \frac{k_i^2 + k_f^2 - Q^2}{2 k_i k_f} \right) \left| Q \right> = \left| k_i \right> - \left| k_f \right> \\ \left< Q | Q \right> = \left( \left< k_i \right| - \left< k_f \right| \right) \cdot \left( \left| k_i \right> - \left| k_f \right> \right) \left< Q | Q \right> = \left< k_i | k_i \right> + \left< k_f | k_f \right> - 2 \left< k_i | k_f \right> Q^2 = k_i^2 + k_f^2 - 2 k_i k_f \cos a_4 \boxed{ a_4 = \arccos \left( \frac{k_i^2 + k_f^2 - Q^2}{2 k_i k_f} \right) } \subsection*{Rocking Angle $a_3$} \boxed{ a_3 = 180^{\circ} - \left( \psi + \xi \right) } Angle $\psi$ between $\left| k_i \right>$ and $\left| Q \right>$, in units of \AA{}$^{-1}$, as before: \left| k_f \right> = \left| k_i \right> - \left| Q \right> \left< k_f | k_f \right> = \left( \left< k_i \right| - \left< Q \right| \right) \cdot \left( \left| k_i \right> - \left| Q \right> \right) \left< k_f | k_f \right> = \left< k_i | k_i \right> + \left< Q | Q \right> - 2 \left< k_i | Q \right> k_f^2 = k_i^2 + Q^2 - 2 k_i Q \cos \psi \boxed{ \psi = \arccos \left( \frac{k_i^2 + Q^2 - k_f^2}{2 k_i Q} \right) } Angle $\xi$ between $\left| Q \right>$ and orientation vector $\left| a \right>$ (i.e. $ax$, $ay$, $az$), in units of rlu; $g_{ij} = \left| b_i \left> \right< b_j \right|$ is the covariant metric of the reciprocal lattice with basis $\left| b_i \right>$: \xi = \arccos \left( \frac{ \left< Q | a \right> }{ \sqrt{\left< Q | Q \right>} \sqrt{\left< a | a \right>} } \right) \boxed{ \xi = \arccos \left( \frac{ Q^i g_{ij} a^j }{ \sqrt{Q^i g_{ij} Q^j} \sqrt{a^i g_{ij} a^j} } \right) } Special case for cubic crystals, $g_{ij} = \delta_{ij} 2\pi / a$: \xi = \arccos \left( \frac{ Q_i a^i }{ \sqrt{Q_i Q^i} \sqrt{a_i a^i} } \right) \end{document}

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