From: hackbard Date: Tue, 20 May 2008 18:59:46 +0000 (+0200) Subject: initial checkin X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=c912342ad8d60890a85fa387abd7a663bed31d32;p=lectures%2Flatex.git initial checkin --- diff --git a/solid_state_physics/tutorial/2_03.tex b/solid_state_physics/tutorial/2_03.tex new file mode 100644 index 0000000..e751a49 --- /dev/null +++ b/solid_state_physics/tutorial/2_03.tex @@ -0,0 +1,106 @@ +\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})} +\renewcommand{\labelenumii}{\arabic{enumii})} +\renewcommand{\labelenumiii}{\roman{enumiii})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics II}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + SS 2008\\ + \vspace{8pt} + {\Large\bf Tutorial 3} +\end{center} + +\vspace{8pt} + +The specific heat (capacity) is the measure of the energy +required to increase the temperature of a unit quantity of a substance +by a certain temperature interval. +Thus, the specific heat at constant volume $V$ is given by +\[ +c_V = \frac{\partial u}{\partial T} +\] +in which $u$ is the energy density of the system. + +\section{Specific heat in the classical theory of the harmonic crystal -\\ + The law of Dulong and Petit} + +In the classical theory of the harmonic crystal equilibrium properties +can no longer be evaluated by simply assuming that each ion sits quitly at +its Bravais lattice site {\bf R}. +From now on expectation values have to be claculated by +integrating over all possible ionic configurations weighted by +$\exp(-E/k_{\text{B}}T)$, where $E$ is the energy of the configuration. +Thus, the energy density $u$ is given by +\[ +u=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)}, +\qquad \beta=\frac{1}{k_{\text{B}}T} +\] +in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$ +is the volume elemnt in crystal phase space. +${\bf u}({\bf R})$ and ${\bf P}({\bf R})$ are the 3N canonical coordinates +(here: deviations from equlibrium sites) +and 3N canonical momenta +of the ion whose equlibrium site is ${\bf R}$. +\begin{enumerate} + \item Show that the energy density can be rewritten to read: + \[ + u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H). + \] + \item Show that the potential contribution to the energy + in the harmonic approximation is given by + \begin{eqnarray} + U&=&U_{\text{eq}}+U_{\text{harm}} \nonumber \\ + U_{\text{eq}}&=&\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'}) + \nonumber \\ + U_{\text{harm}}&=&\frac{1}{2}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}} + [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'}) + [u_v({\bf R})-u_v({\bf R'})] \nonumber + \end{eqnarray} + in which +$\Phi_{\mu v}({\bf r})= + \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}$ + and $\Phi({\bf r})$ is the potential contribution of two atoms + separated by ${\bf r}$. + {\bf Hint:} + +\end{enumerate} + + +\section{Specific heat in the quantum theory of the harmonic crystal -\\ + Models of Debye and Einstein} + +\begin{enumerate} + \item + \item +\end{enumerate} + +\end{document}