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% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 2}
\end{center}

\section{Band structure: indirect band gap of silicon}
Some facts about silicon:
\begin{itemize}
 \item Lattice constant: $a=5.43 \times 10^{-10} \, m$.
 \item Silicon has an indirect band gap.
       \begin{itemize}
        \item The minimum of the conduction band is located at
	      $k=0.85 \frac{2 \pi}{a}$.
	\item The maximum of the valance band is located at $k=0$.
	\item The energy gap is $E_g=1.12 \, eV$.
       \end{itemize}
\end{itemize}
\begin{enumerate}
 \item Calculate the wavelength of the light necessary to lift an electron from
       the valence to the conduction band.
       What is the momentum of such a photon?
 \item Calculate the phonon momentum necessary for the transition.
       Compare the momentum values of phonon and photon.
\end{enumerate}

\section{Phonons}
Consider two masses $M_1$ and $M_2$ with their idle positions
$r_{10}$ and $r_{20}$ connected by a spring with spring constant $D$.
The equilibrium distance vector is $\rho_{0}=r_{20}-r_{10}$.
Denote the deflection by $u_1$ and $u_2$, the deflected positions by
$r_1$ and $r_2$ and their distance vector by $\rho=r_2-r_1$.
The vector of elongation is thus given by $\sigma = u_2 -u_1$.
\begin{enumerate}
 \item Write down a potential $\Phi - \Phi_0$ as a function of
       $\rho_0$ and $\sigma$. Therefor prove and use the relation
       $\rho=\rho_0+\sigma$.
 \item Discuss the case $\sigma \parallel \rho_0$.
       \begin{enumerate}
        \item Sketch examples for elongations $u_1$ and $u_2$.
        \item Express the potential $\Phi-\Phi_0$ as a function of
	      $\sigma = \sigma_{\parallel}$.
       \end{enumerate}
 \item Discuss the case $\sigma \perp \sigma_0$.
       \begin{enumerate}
        \item Sketch examples for elongations $u_1$ and $u_2$.
        \item Express the potential $\Phi-\Phi_0$ as a function of
	      $\rho_0$ and $\sigma = \sigma_{\perp}$.
	\item Examine the case $\sigma_{\perp} \ll \rho_0$.
	      {\bf Hint:} Use $\sigma_{\perp} = \alpha \rho_0$ and
	                  $\alpha \ll 1.$
	\item Compare the potential contribution of $\sigma_{\parallel}$ and
	      $\sigma_{\perp}$.
       \end{enumerate}
 \item Discuss the model of two masses deflected along the same direction
       as a possible model for the dynamic behaviour of atoms in a crystal
       keeping earlier results in mind.
\end{enumerate}

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