Problem 1

A linear monoatomic chain is composed of $N$ masses which are separated by the lattice constant $a = 0.5$ nm. The chain has a length $L$ $( L = N a)$ of 1 cm. The masses have a mass of $6 \times 10^{-26}$ kg and the sound velocity $v =$ is 1200 m/s.

a) Calculate the density of states $D(\lambda)$ – where $\lambda$ is the wavelength of the phonons – in the long wavelength limit. Start from the density of states $D(k) = \frac{1}{\pi}$ – with $k$ as the wavevector of the phonons. Give the units of $D(\lambda)$.

Solution

b) How many states are possible in the interval $\lambda_1 = 2 \times 10^{-4}$ m and $\lambda_2 = 4 \times 10^{-4}$ m?

Solution

c) Find all allowed (quantized) values of the wavevectors $k$ by using periodic boundary conditions. The displacement of atom $l$ for the normal mode $k$ is,

$u_l = u_ke^{i(kla - \omega t)}$.

Solution

d) Calculate the effective spring constant $C$. Take the long wavelength limit of the dispersion relation $\omega (k)$ and compare it with the sound velocity. The phonon dispersion relation is,

$\omega = \sqrt{\frac{4C}{m}}\left|\sin \frac{ka}{2} \right|$.

Problem 2

Consider a monovalent metal in a simple cubic lattice ($a =0.2$ nm) with only one atom per primitive unit cell. The volume of the crystal is 1 cm³. The system has a Fermi energy $E_F$ of $1.45 \times 10^{-18}$ J (= 9.107 eV).

a) Calculate the electron density of the free electrons. Give units of your result.

b) How many states $N$ are occupied at a temperature $T = 300$ K in the energy interval $E_F -2 k_BT$ to $E_F +2k_BT$? Give only the formula based on the density of states $D(E)$ for calculating the number of states, and explain the different quantities which have to be considered. There is no need to calculate a final value.

$D(E) = \frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2}\sqrt{E}$.

c) Draw the dispersion relation $E(k)$ in one of the principle directions $x$, $y$, or $z$, using the empty lattice approximation. Give the limiting values of $k$ at the Brillouin zone boundary. What is the maximum energy and the maximum wavenumber $k$ of the highest occupied electron state at a temperature $T = 0$ K?

d) What is the maximum possible $k$ value and the maximum possible energy of an electron for the whole Brillouin zone (three-dimensional lattice) within the empty lattice approximation?

Useful fundamental parameters could be: $m_e = 9.11 \times 10^{-31}$ kg and $\hbar= 1.054 \times 10^{-34}$ Js.

Problem 3

a) Construct the generating matrix for a twofold rotation axis (a rotation of 180° ) about the $y$-axis.

Solution

b) Give the two different group elements of this point group.

Solution

c) Pyroelectricity describes how the electric polarization changes as the temperature changes. The pyroelectric coefficients form a rank 1 tensor $\pi_i = \frac{\partial P_i}{\partial T}$ where $P_i$ is the polarization in different crystallographic directions $i$ and $T$ is the temperature. Give the independent tensor elements.

Solution

d) Does such a crystal show piezoelectric properties? Why or why not?

Solution