From: hackbard Date: Thu, 8 Nov 2007 17:58:17 +0000 (+0100) Subject: tutorial 2 X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=24dbdf53cbe784ede66bfe0dbdc6655e28c2d2f6;p=lectures%2Flatex.git tutorial 2 --- diff --git a/solid_state_physics/tutorial/1_02.tex b/solid_state_physics/tutorial/1_02.tex new file mode 100644 index 0000000..55733e6 --- /dev/null +++ b/solid_state_physics/tutorial/1_02.tex @@ -0,0 +1,93 @@ +\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{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}