final version

diff --git a/solid_state_physics/tutorial/2_03.tex b/solid_state_physics/tutorial/2_03.tex
index b4a3571d3eed75259ae7ff0e171ef09f843cc76c..c33fe435ab15dd6771439425d4dd13816bc223d9 100644 (file)
--- a/solid_state_physics/tutorial/2_03.tex
+++ b/solid_state_physics/tutorial/2_03.tex
@@ -48,13 +48,15 @@ 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.
+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 quitly at
+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
@@ -73,7 +75,7 @@ of the ions whose equlibrium sites are ${\bf R}$.
 \begin{enumerate}
  \item Show that the energy density can be rewritten to read:
        \[
-   u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
+   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
        \]
  \item Show that the potential contribution to the energy
        in the harmonic approximation is given by
@@ -111,13 +113,56 @@ $\Phi_{\mu v}({\bf r})=
        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
- \item
+ \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
+       $\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
+       Write down the resulting expression for the specific heat.
 \end{enumerate}
 
 \end{document}