Problem 1 a) What is the largest $k$-vector within the first Brillouin zone of this crystal structure? b) How many $k$-points are inbetween the $k$-states of $k_1 = 5 \times 10^9$ m-1 and $k_2 = 6 \times 10^9$ m-1? c) Give the wavevector $k_F$ of the electron at the Fermi energy $E_F$ at $T = 0$ Kelvin. d) Are all occupied electronic states within the first Brillouin zone at a temperature $T = 0$ K? Why or why not? Problem 2 a) Calculate the effective spring constant $C$, if the speed of sound is $1.08 \times 10^{4}$ m/s. b) What is the maximum normal mode frequency $\omega_{\text{max}}$? c) Give a formula for calculating the specific heat of the phonons. d) Is the specific heat dependent on the mass of the atoms? Why or why not? Problem 3 b) The pyroelectric vector is $\pi_i= \frac{\partial P_i}{\partial T}$ where $P_i$ is the electric polarisation and $T$ as the temperature. Show that the pyroelectric effect is not compatible with a crystal which has inversion symmetry. c) At 25°C, diamond has a high thermal conductivity of 2350 W m-1 K-1. Can this property be explained by the Wiedemann - Franz law? Explain why or why not?
Consider a simple cubic lattice with a lattice constant of 0.3 nm and one atom per unit cell. Each atom contributes with a single electron to the electron density. The density of states is $D(k)=\frac{k^2}{\pi^2}$ and the dispersion relation is given by $E=\frac{\hbar^2k^2}{2m}$. The size of the crystal is 1 cm³.
Consider a linear monoatomic chain with an atomic separation of 0.485 nm and a mass of each atom of $m = 6.81 \times 10^{-26}$ kg. The dispersion relation is given by $\omega=\sqrt{\frac{4C}{m}}|\sin( ka/2)|$.
a) Give the transformation matrix for the symmetry operation of a 2-fold rotation axis (rotation angle is 180°) around the $y$-axis. How many elements does the point group have which includes only this rotation matrix as a generating matrix?