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\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 6 - proposed solutions}
\end{center}

\section{Indirect band gap of silicon}

\begin{enumerate}
 \item \begin{itemize}
        \item Photon wavelength:\\
	      $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
	          \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
	      $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
	                          =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
	\item Photon momentum:\\
	      $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
	        =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
       \end{itemize}
 \item Phonon momentum necessary for transition:\\
       $p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
         =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
       $\rightarrow$ Phonon momentum 3 orders of magnitude below
                     the momentum necessary for transition!
 \item 
\end{enumerate}

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