X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_04s.tex;fp=solid_state_physics%2Ftutorial%2F2_04s.tex;h=287c275bde4c50d7960e253074f52cd985143810;hp=0000000000000000000000000000000000000000;hb=9c6ed4d9ce5cdc917ceab10ae57a50ba2891f9fd;hpb=98cbe939b498d77688a0f40cf01180e8ad76a9a6 diff --git a/solid_state_physics/tutorial/2_04s.tex b/solid_state_physics/tutorial/2_04s.tex new file mode 100644 index 0000000..287c275 --- /dev/null +++ b/solid_state_physics/tutorial/2_04s.tex @@ -0,0 +1,160 @@ +\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 4 - proposed solutions} +\end{center} + +\vspace{4pt} + +\section{Legendre transformation and Maxwell relations} + +\begin{enumerate} + \item Legendre transformation: + \begin{eqnarray} + dg &=& df - \sum_{i=r+1}^{n} d(u_ix_i)\nonumber\\ + &=& df - \sum_{i=r+1}^{n} (x_idu_i + u_idx_i)\nonumber\\ + &=& \sum_{i=1}^r u_idx_i - \sum_{i=r+1}^n x_idu_i\nonumber + \end{eqnarray} + \[ + \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n) + \] + \item Use $T=\left.\frac{\partial E}{\partial S}\right|_V$ and + $-p=\left.\frac{\partial E}{\partial V}\right|_S$.\\ + Start with internal energy $E=E(S,V)$: + \[ + \Rightarrow dE=\frac{\partial E}{\partial S}dS + + \frac{\partial E}{\partial V}dV = + TdS - pdV + \] + Enthalpy $H=E+pV$: + \[ + \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp + \] + \[ + \Rightarrow + \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and } + \left.\frac{\partial H}{\partial p}\right|_S=V + \] + Helmholtz free energy $F=E-TS$: + \[ + \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT + \] + \[ + \Rightarrow + \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and } + \left.\frac{\partial F}{\partial T}\right|_V=-S + \] + Gibbs free energy $G=H-TS=E+pV-TS$: + \[ + \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT + \] + \[ + \Rightarrow + \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and } + \left.\frac{\partial G}{\partial T}\right|_p=-S + \] + \item Maxwell relations:\\ + Enthalpy: $dH=TdS+Vdp$ + \[ + \frac{\partial}{\partial S} + \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p= + \frac{\partial}{\partial p} + \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S + \Rightarrow + \left.\frac{\partial V}{\partial S}\right|_p= + \left.\frac{\partial T}{\partial p}\right|_S + \] + Helmholtz free energy: $dF=-pdV-SdT$ + \[ + \frac{\partial}{\partial V} + \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T= + \frac{\partial}{\partial T} + \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V + \Rightarrow + \left.-\frac{\partial S}{\partial V}\right|_T= + \left.-\frac{\partial p}{\partial T}\right|_V + \] + \item For a thermodynamic potential $\Phi(X,Y)$ the following identity + expressing the permutability of derivatives holds: + \[ + \frac{\partial^2 \Phi}{\partial X \partial Y} = + \frac{\partial^2 \Phi}{\partial Y \partial X} + \] + Derive the Maxwell relations by taking the mixed derivatives of the + potentials in (b) with respect to the variables they depend on. + Exchange the sequence of derivation and use the identities gained in (b). +\end{enumerate} + +\section{Thermal expansion of solids} + +It is well known that solids change their length $L$ and volume $V$ respectively +if there is a change in temperature $T$ or in pressure $p$ of the system. +The following exercise shows that +thermal expansion cannot be described by rigorously harmonic crystals. + +\begin{enumerate} + \item The coefficient of thermal expansion of a solid is given by + $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$. + Show that the coefficient of thermal expansion of the volume + $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$ + equals $3\alpha_L$ for isotropic materials. + \item Find an expression for the pressure as a function of the free energy + $F=E-TS$. + Rewrite this equation to express the pressure entirely in terms of + the internal energy $E$. + Evaluate the pressure by using the harmonic form of the internal energy. + {\bf Hint:} + Step 2 introduced an integral over the temperature $T'$. + Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$. + Use integration by parts with respect to $x$. + \item The normal mode frequencies of a rigorously harmonic crystal + are unaffected by a change in volume. + What does this imply for the pressure + (Which variables does the pressure depend on)? + Draw conclusions for the coefficient of thermal expansion. + \item Find an expression for $C_p-C_V$ in terms of temperature $T$, + volume $V$, the coefficient of thermal expansion $\alpha_V$ and + the inverse bulk modulus (isothermal compressibility) + $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\ + $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity + for constant pressure and + $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity + for constant volume. +\end{enumerate} + +\end{document}