tutorial 2

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+\documentclass[a4paper,11pt]{article}
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+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{ae}
+\usepackage{aecompl}
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+\usepackage{pstricks}
+\usepackage{pst-node}
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+
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+
+\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{Band structure: indirect band gap of silicon}
+Some facts about silicon:
+\begin{itemize}
+ \item Lattice constant: $a=5.43 \times 10^{-10} \, m$.
+ \item Silicon has an indirect band gap.
+       \begin{itemize}
+        \item The minimum of the conduction band is located at
+             $k=0.85 \frac{2 \pi}{a}$.
+       \item The maximum of the valance band is located at $k=0$.
+       \item The energy gap is $E_g=1.12 \, eV$.
+       \end{itemize}
+\end{itemize}
+\begin{enumerate}
+ \item Calculate the wavelength of the light necessary to lift an electron from
+       the valence to the conduction band.
+       What is the momentum of such a photon?
+ \item Calculate the phonon momentum necessary for the transition.
+       Compare the momentum values of phonon and photon.
+\end{enumerate}
+
+\section{Phonons}
+Consider two masses $M_1$ and $M_2$ with their idle positions
+$r_{10}$ and $r_{20}$ connected by a spring with spring constant $D$.
+The equilibrium distance vector is $\rho_{0}=r_{20}-r_{10}$.
+Denote the deflection by $u_1$ and $u_2$, the deflected positions by
+$r_1$ and $r_2$ and their distance vector by $\rho=r_2-r_1$.
+The vector of elongation is thus given by $\sigma = u_2 -u_1$.
+\begin{enumerate}
+ \item Write down a potential $\Phi - \Phi_0$ as a function of
+       $\rho_0$ and $\sigma$. Therefor prove and use the relation
+       $\rho=\rho_0+\sigma$.
+ \item Discuss the case $\sigma \parallel \rho_0$.
+       \begin{enumerate}
+        \item Sketch examples for elongations $u_1$ and $u_2$.
+        \item Express the potential $\Phi-\Phi_0$ as a function of
+             $\sigma = \sigma_{\parallel}$.
+       \end{enumerate}
+ \item Discuss the case $\sigma \perp \sigma_0$.
+       \begin{enumerate}
+        \item Sketch examples for elongations $u_1$ and $u_2$.
+        \item Express the potential $\Phi-\Phi_0$ as a function of
+             $\rho_0$ and $\sigma = \sigma_{\perp}$.
+       \item Examine the case $\sigma_{\perp} \ll \rho_0$.
+             {\bf Hint:} Use $\sigma_{\perp} = \alpha \rho_0$ and
+                         $\alpha \ll 1.$
+       \item Compare the potential contribution of $\sigma_{\parallel}$ and
+             $\sigma_{\perp}$.
+       \end{enumerate}
+ \item Discuss the model of two masses deflected along the same direction
+       as a possible model for the dynamic behaviour of atoms in a crystal
+       keeping earlier results in mind.
+\end{enumerate}
+
+\end{document}