From 2d7b9c20db0fa96841f8708528d3afca5c6ae63d Mon Sep 17 00:00:00 2001
From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Wed, 14 Nov 2007 16:48:42 +0100
Subject: [PATCH] added 1_02s, first excercise only!

---
 solid_state_physics/tutorial/1_02s.tex | 116 +++++++++++++++++++++++++
 1 file changed, 116 insertions(+)
 create mode 100644 solid_state_physics/tutorial/1_02s.tex

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}
-- 
2.20.1