+ \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.