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\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 internal energy density of the system.
In the following the contribution to the specific heat due to the
degrees of freedom of the lattice ions is calculated.

\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 quietly 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:
       \[
   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
       \]
 \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}

As found in exercise 1, the specific heat of a classical harmonic crystal
is not depending on temeprature.
However, as temperature drops below room temperature
the specific heat of all solids is decreasing as $T^3$ in insulators
and $AT+BT^3$ in metals.
This can be explained in a quantum theory of the specific heat of
a harmonic crystal, in which the energy density $w$ is given by
\[
w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
\]
\begin{enumerate}
 \item Show that the energy density can be rewritten to read:
       \[
   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
       \]
 \item Evaluate the expression of the energy density.
       {\bf Hint:}
       The energy levels of a harmonic crystal of N ions
       can be regarded as 3N independent oscillators,
       whose frequencies are those of the 3N classical normal modes.
       The contribution to the total energy of a particular normal mode
       with angular frequency $\omega_s({\bf k})$ 
       ($s$: branch, ${\bf k}$: wave vector) is given by
       $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
       excitation number $n_{{\bf k}s}$ being restricted to integers greater
       or equal zero.
       The total energy is given by the sum over the energies of the individual
       normal modes.
       Use the totals formula of the geometric series to expcitly calculate
       the sum of the exponential functions.
 \item Separate the above result into a term vanishing as $T$ goes to zero and
       a second term giving the energy of the zero-point vibrations of the
       normal modes.
 \item Write down an expression for the specific heat.
       Consider a large crystal and thus replace the sum over the discrete
       wave vectors with an integral.
 \item Debye replaced all branches of the vibrational spectrum with three
       branches, each of them obeying the dispersion relation
       $w=ck$.
       Additionally the integral is cut-off at a radius $k_{\text{D}}$
       to have a total amount of N allowed wave vectors.
       Determine $k_{\text{D}}$.
       Evaluate the simplified integral and introduce the
       Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
       and the Debye temperature $\Theta_{\text{D}}$ which is given by
       $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
       Write down the resulting expression for the specific heat.
\end{enumerate}

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