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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 2}
40 \section{Band structure: indirect band gap of silicon}
41 Some facts about silicon:
43 \item Lattice constant: $a=5.43 \times 10^{-10} \, m$.
44 \item Silicon has an indirect band gap.
46 \item The minimum of the conduction band is located at
47 $k=0.85 \frac{2 \pi}{a}$.
48 \item The maximum of the valance band is located at $k=0$.
49 \item The energy gap is $E_g=1.12 \, eV$.
53 \item Calculate the wavelength of the light necessary to lift an electron from
54 the valence to the conduction band.
55 What is the momentum of such a photon?
56 \item Calculate the phonon momentum necessary for the transition.
57 Compare the momentum values of phonon and photon.
61 Consider two masses $M_1$ and $M_2$ with their idle positions
62 $r_{10}$ and $r_{20}$ connected by a spring with spring constant $D$.
63 The equilibrium distance vector is $\rho_{0}=r_{20}-r_{10}$.
64 Denote the deflection by $u_1$ and $u_2$, the deflected positions by
65 $r_1$ and $r_2$ and their distance vector by $\rho=r_2-r_1$.
66 The vector of elongation is thus given by $\sigma = u_2 -u_1$.
68 \item Write down a potential $\Phi - \Phi_0$ as a function of
69 $\rho_0$ and $\sigma$. Therefor prove and use the relation
71 \item Discuss the case $\sigma \parallel \rho_0$.
73 \item Sketch examples for elongations $u_1$ and $u_2$.
74 \item Express the potential $\Phi-\Phi_0$ as a function of
75 $\sigma = \sigma_{\parallel}$.
77 \item Discuss the case $\sigma \perp \sigma_0$.
79 \item Sketch examples for elongations $u_1$ and $u_2$.
80 \item Express the potential $\Phi-\Phi_0$ as a function of
81 $\rho_0$ and $\sigma = \sigma_{\perp}$.
82 \item Examine the case $\sigma_{\perp} \ll \rho_0$.
83 {\bf Hint:} Use $\sigma_{\perp} = \alpha \rho_0$ and
85 \item Compare the potential contribution of $\sigma_{\parallel}$ and
88 \item Discuss the model of two masses deflected along the same direction
89 as a possible model for the dynamic behaviour of atoms in a crystal
90 keeping earlier results in mind.