PHY.K02 Molecular and Solid State Physics
05.03.2021

Name   Matrikelnr.

(This was an online exam. The students had access to the internet.)

Problem 1: Lennard-Jones potential

Van der Waals bonds are often described by a Lennard-Jones potential with the form, $U(R)= 4\epsilon\left[\left(\frac{\sigma}{R}\right)^{12}-\left(\frac{\sigma}{R}\right)^{6}\right]$, where $\epsilon=$  meV and $\sigma=$  Å are characteristic constants for the bond and $R$ is the bond length (the distance between the atoms).

(a) Find the interaction force between the two atoms at a distance of $R=$  Å. A positive force pushes the atoms apart and a negative force pulls them together.

$F=$  N.

(b) Find the equilibrium spacing $R_0$ between the two atoms.

$R_0=$  Å

Problem 2: Diffraction

X-rays with an wavelength of 0. Å are used to analyze a crystal with a sodium chloride crystal structure with a lattice constant of  Å.

The diffraction peaks are labeled using the conventional (cubic) unit cell. If the primary beam is directed in the positve $x$-direction, what are $\vec{k}$, $\vec{k}'$ and $\vec{G}$ for the reflection?

$\vec{k}=$ $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

$\vec{G}$ = $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

$\vec{k}'=$ $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

Problem 3: NaCl

A conventional unit cell of NaCl is shown below. The Bravais lattice is fcc. The lattice contants are given in the figure.

(a) A crystal of NaCl is  mm ×  mm ×  mm. How many $k$ states are there in the first Brillouin zone?

Number of $k$ states =

(b) How many optical branches are there of the phonon dispersion?

Number of optical branches =

(c) How many phonon modes does this crystal have?

phonon modes =

(d) The speed of sound at low frequencies is $c=$  m/s. What is the the wavelength of a phonon mode with a frequency of $\omega = $ 0000 rad/s travelling in the [111] direction?

$\lambda =$ m

(e) At  K, what is the mean number of phonons in a phonon state with a frequency of $\omega = $  $\times 10^{12}$ rad/s?

phonons =

Problem 4: Electrons

Near the top of the valence band of an intrinsic semiconductor, the electron dispersion relation is,

$E = $ -$k^2$ [J].

Near the bottom of the conduction the electron dispersion relation is,

$E = $ + $k^2$ [J].

(a) What is the chemical potential at $T=0$ K?

$\mu =$ eV

(b) What is the effective mass of the holes in terms of the free electron mass $m_e$?

$m_h^* =$ $m_e$.

(c) An electron is in $k$-state $\vec{k}= $  $\hat{k}_x$ m-1. The wavefunction at point $\vec{r}$ is $\psi (\vec{r}) = \psi_0$. What is the wavefunction at $\vec{r}+a\,\hat{x}$? Here $a=3$Å is the lattice constant.

$\psi (\vec{r}+a\,\hat{x}) =$ ( + $i$ ) $\psi_0$