From: hackbard Date: Wed, 14 Nov 2007 15:48:42 +0000 (+0100) Subject: added 1_02s, first excercise only! X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=2d7b9c20db0fa96841f8708528d3afca5c6ae63d;p=lectures%2Flatex.git added 1_02s, first excercise only! --- diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex new file mode 100644 index 0000000..8eb5881 --- /dev/null +++ b/solid_state_physics/tutorial/1_02s.tex @@ -0,0 +1,116 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics I}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + WS 2007/08\\ + \vspace{8pt} + {\Large\bf Tutorial 2} +\end{center} + +\section{Phonons 1} +\begin{enumerate} + \item \begin{itemize} + \item $r_i=r_{i0}+u_i$\\ + $\rho=r_2-r_1=r_{20}+u_2-r_{10}-u_1=(r_{20}-r_{10})+(u_2-u_1) + =\rho_0+\sigma$ + \item $\Phi-\Phi_0=\frac{D}{2}(\rho-\rho_0)^2 + =\frac{D}{2}(\rho^2+\rho_0^2-2\rho_0\rho)$\\ + $\rho^2=\rho_0^2+\sigma^2+2\rho_0\sigma$ + $\Rightarrow$ $\rho=\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}$\\ + $\Rightarrow$ $\Phi-\Phi_0=\frac{D}{2} + [2\rho_0^2+\sigma^2+2\rho_0\sigma- + 2\rho_0\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}]$ + \end{itemize} + \item $\sigma \parallel \rho_0$: + \begin{enumerate} + \item \begin{flushleft} + \includegraphics[height=6cm]{elongation_p01.eps} + \includegraphics[height=6cm]{elongation_p02.eps} + \includegraphics[height=6cm]{elongation_p03.eps} + \end{flushleft} + \item $\sigma = \sigma_{\parallel}$:\\ + $\rho_0 \sigma_{\parallel} = |\rho_0| |\sigma_{\parallel}|$\\ + $\Phi-\Phi_0=\frac{D}{2}\left(2\rho_0^2+\sigma_{\parallel}^2+ + 2\rho_0\sigma_{\parallel}- + 2\rho_0\sqrt{(\rho_0+\sigma_{\parallel})^2}\right) + =\frac{D}{2}\sigma_{\parallel}^2$ + \end{enumerate} + \item $\sigma \perp \sigma_0$: + \begin{enumerate} + \item \begin{flushleft} + \includegraphics[height=5.3cm]{elongation_n01.eps} + \includegraphics[height=5.3cm]{elongation_n02.eps} + \includegraphics[height=5.3cm]{elongation_n03.eps} + \end{flushleft} + \item $\sigma=\sigma_{\perp}$:\\ + $\sigma_{\perp} \rho_0 = 0$\\ + $\Phi-\Phi_0=\frac{D}{2}\left[2\rho_0^2+\sigma_{\perp}^2- + 2\rho_0\sqrt{\rho_0^2+\sigma_{\perp}^2}\right]$ + + \item $\sigma_{\perp} = \alpha \rho_0$, $\alpha \ll 1$\\ + $\sqrt{\rho_0^2+\sigma_{\perp}^2}= + \sqrt{\rho_0^2+\alpha^2\rho_0^2}= + \rho_0\sqrt{1+\alpha^2}= + \rho_0(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)$\\ + $\Rightarrow \Phi-\Phi_0= + \frac{D}{2}\left[\rho_0^2\left(2+\alpha^2- + 2(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)\right)\right]= + \frac{D}{2}\left[\rho_0^2(\frac{\alpha^4}{4}+\ldots)\right]$\\ + $\Rightarrow \Phi-\Phi_0\stackrel{\alpha\ll 1}{=} + \frac{D}{2}\rho_0^2\frac{\alpha^4}{4}= + \frac{D}{2}\sigma_{\perp}^2\frac{\alpha^2}{4}$ + \item $\sigma_{\parallel}$, $\sigma_{\perp} \ll \rho_0$\\ + $\Rightarrow$ potential contribution of $\sigma_{\perp}$ + compared to contribution of $\sigma_{\parallel}$ + negligible small. + \end{enumerate} + \item \begin{itemize} + \item As long as the displacements and thus the elongation is small + compared to the equilibrium state the change in the potential + due to the perpendicular elongation is negligible small. + \item Regarding a possible existence of perpendicular elongation + the model of the linear chain is unproblematic. + \item In a real crystal couplings in other directions exist. + These can only be neglected if they are small compared to the + coupling of the considered direction. + \end{itemize} +\end{enumerate} + +\section{Phonons 2} +\begin{enumerate} +\item Derive the dispersion relation for a linear chain with two different + alternating types of atoms. +\item Discuss the two solutions for $\omega^2$. +\end{enumerate} + +\end{document}