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% 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
\[
c_V = \frac{\partial w}{\partial T}
\]
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}

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 $w$ is given by
\[
w=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
\qquad \beta=\frac{1}{k_{\text{B}}T}
\]
in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
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 ions whose equlibrium sites are ${\bf R}$.
\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}{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}
       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:}
       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 energy contribution) 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}


\section{Specific heat in the quantum theory of the harmonic crystal -\\
         The Debye model}

\begin{enumerate}
 \item
 \item
\end{enumerate}

\end{document}
