From 6755c668d425e5d27c0c4b8f504d1eac24d0c9ba Mon Sep 17 00:00:00 2001 From: hackbard Date: Wed, 21 May 2008 00:23:36 +0200 Subject: [PATCH] finished excercise 1 --- solid_state_physics/tutorial/2_03.tex | 31 +++++++++++++++++++++------ 1 file changed, 24 insertions(+), 7 deletions(-) diff --git a/solid_state_physics/tutorial/2_03.tex b/solid_state_physics/tutorial/2_03.tex index e751a49..bddd192 100644 --- a/solid_state_physics/tutorial/2_03.tex +++ b/solid_state_physics/tutorial/2_03.tex @@ -46,9 +46,9 @@ 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 \[ -c_V = \frac{\partial u}{\partial T} +c_V = \frac{\partial w}{\partial T} \] -in which $u$ is the energy density of the system. +in which $w$ is the energy density of the system. \section{Specific heat in the classical theory of the harmonic crystal -\\ The law of Dulong and Petit} @@ -59,9 +59,9 @@ 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 +Thus, the energy density $w$ is given by \[ -u=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)}, +w=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)}, \qquad \beta=\frac{1}{k_{\text{B}}T} \] in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$ @@ -69,7 +69,7 @@ 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 ${\bf R}$. +of the ions whose equlibrium sites are ${\bf R}$. \begin{enumerate} \item Show that the energy density can be rewritten to read: \[ @@ -81,7 +81,7 @@ of the ion whose equlibrium site is ${\bf R}$. 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_{\text{harm}}&=&\frac{1}{4}\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} @@ -91,7 +91,24 @@ $\Phi_{\mu v}({\bf r})= and $\Phi({\bf r})$ is the potential contribution of two atoms separated by ${\bf r}$. {\bf Hint:} - + Write down the potential energy for the instantaneous positions + ${\bf r}({\bf R})$, with ${\bf u}({\bf R})={\bf r}({\bf R})-{\bf R}$. + Apply Taylor approximation to $\Phi({\bf r}+{\bf a})$ with + ${\bf r}={\bf R}-{\bf R'}$ and + ${\bf a}={\bf u}({\bf R})-{\bf u}({\bf R'})$ + and only retain terms quadratic in $u$. + \item Use the evaluated potential to calculate the energy density + (do not forget the kinetic contribution to energy) and + the specific heat $c_{\text{V}}$. + {\bf Hint:} + Use the following change of variables + \[ + {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad + {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R}) + \] + to extract the temperature dependence of the integral. + Does this also work for anharmonic terms? + Which parts of the integral do not contribute to $w$ and why? \end{enumerate} -- 2.39.2