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 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
$\exp(-E/k_{\text{B}}T)$, 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 $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
${\bf u}({\bf R})$ 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:
\[
- u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
+ w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
\]
\item Show that the potential contribution to the energy
in the harmonic approximation is given by
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}
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 -\\
- Models of Debye and Einstein}
+ 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
+ $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
+ Write down the resulting expression for the specific heat.
\end{enumerate}
\end{document}
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