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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 - proposed solutions}
\end{center}

\vspace{8pt}

\section{Specific heat in the classical theory of the harmonic crystal -\\
         The law of Dulong and Petit}

\begin{enumerate}
 \item Energy:
       \begin{eqnarray}
       w&=&-\frac{1}{V}\frac{\partial}{\partial \beta}
       ln \int d\Gamma \exp(-\beta H)
       =-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
       \frac{\partial}{\partial \beta} \int d\Gamma \exp(-\beta H)\nonumber\\
       &=&-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
       \int d\Gamma \frac{\partial}{\partial \beta} \exp(-\beta H)\nonumber\\
       &=&-\frac{1}{V}\frac{1}{\int d\Gamma \exp(-\beta H)}
       \int d\Gamma \exp(-\beta H) (-H) \qquad \textrm{ q.e.d.} \nonumber
       \end{eqnarray}
 \item Potential energy:
       \[
       U=\frac{1}{2}\sum_{{\bf RR'}}\Phi({\bf r}({\bf R})-{\bf r}({\bf R'}))
        =\frac{1}{2}\sum_{{\bf RR'}}
         \Phi({\bf R}-{\bf R'}+{\bf u}({\bf R})-{\bf u}({\bf R'}))
       \]
       Using Taylor and
       $U_{\text{eq}}=\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})$:
       \[
       U=U_{\text{eq}}+
         \frac{1}{2}\sum_{{\bf RR'}}({\bf u}({\bf R})-{\bf u}({\bf R'}))
	 \nabla\Phi({\bf R}-{\bf R'})+
	 \frac{1}{4}\sum_{{\bf RR'}}
	 [({\bf u}({\bf R})-{\bf u}({\bf R'})) \nabla]^2
	 \Phi({\bf R}-{\bf R'}) + \mathcal{O}(u^3)
       \]
       Linear term:\\
       The coefficient of ${\bf u}({\bf R})$ is
       $\sum_{\bf R'}\nabla\Phi({\bf R}-{\bf R'})$
       which is minus the force excerted on atom ${\bf R}$
       by all other atoms in equlibrium positions.
       There is no net force on any atom in equlibrium.
       The linear term is zero.\\\\
       Harmonic term:\\
       $(a\nabla)^2 \Phi=
        a\nabla a\nabla \Phi=
	a\nabla \sum_u a_u \frac{\partial\Phi}{\partial r_u}=
	\sum_v \frac{\partial \sum_u a_u
	\frac{\partial\Phi}{\partial r_u}}{\partial r_v} a_v=
	\sum_{uv}\frac{\partial}{\partial r_v} a_u
	\frac{\partial \Phi}{\partial r_u} a_v=
	\sum_{uv}a_u \frac{\partial^2\Phi}{\partial r_u \partial r_v} a_v$\\
       \[\Rightarrow
       U_{\text{harm}}=\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
       [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
       [u_v({\bf R})-u_v({\bf R'})],
       \quad \Phi_{\mu v}({\bf r})=
        \frac{\partial^2 \Phi({\bf r})}{\partial r_{\mu}\partial r_v}.
       \]
 \item Change of variables:
       \[
       {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
       {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
       \]
       \[
       \Rightarrow
       d{\bf u}({\bf R})=\beta^{-3/2}d\bar{{\bf u}}({\bf R}), \qquad
       d{\bf P}({\bf R})=\beta^{-3/2}d\bar{{\bf P}}({\bf R}), \qquad
       \]
       Kinetic energy contribution:
       \[
       H_{\text{kin}}=\frac{{\bf P}({\bf R})^2}{2M}
       \]
       Integral (using change of variables):
       \begin{eqnarray}
       \int d\Gamma \exp(-\beta H)&=&
       \int d\Gamma \exp\left[-\beta\left(\sum \frac{{\bf P}({\bf R})^2}{2M}+
       U_{\text{eq}} + U_{\text{harm}}\right)\right]\nonumber\\
       &=&
       \exp(-\beta U_{\text{eq}})\beta^{-3N}
       \LARGE(\int\prod_{{\bf R}}d\bar{{\bf u}}({\bf R})d\bar{{\bf P}}({\bf R})
       \nonumber\\
       &&\times \exp\left[
       -\sum\frac{1}{2M}{\bf P}({\bf R})^2
       -\frac{1}{4}\sum
       [\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})]
       \Phi_{\mu v}({\bf R}-{\bf R'})
       [\bar{u}_v({\bf R})-\bar{u}_v({\bf R'})]
       \right]\LARGE)\nonumber
       \end{eqnarray}
       \[
       \Rightarrow w=-\frac{1}{V}\frac{\partial}{\partial \beta}
       ln\left((\exp(-\beta U_{\text{eq}})\beta^{-3N} \times \text{const}
       \right)
       =\frac{U_{\text{eq}}}{V}+3\frac{N}{V}k_{\text{B}}T
       =u_{\text{eq}}+3nk_{\text{B}}T
       \]
       \[
       \Rightarrow
       c_{\text{V}}=\frac{\partial w}{\partial T}=3nk_{\text{B}}
       \]
\end{enumerate}

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

\[
w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
\]
\begin{enumerate}
 \item Energy: $\rightarrow$ 1(a)
       \[
   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
       \]
 \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}
