PHY.K02 Molecular and Solid State Physics
27.11.2020

Name Matrikelnr.

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

Problem 1: Molecule

What is the molecular orbital Hamitonian of ? Here $\vec{r}_a$ is the position of one of the atomic nuclei and $\vec{r}_b$ is the position of the other one. Slater's rules were used to determine the nuclear charge.

                                                           

How many molecular orbitals are occupied in the ground state of ?

orbitals

How many vibrational modes does have?

vibrational modes

Problem 2: CaTiO3

CaTiO3 has a cubic perovskite crystal structure. The lattice contants are given in the figure.

(a) A crystal of CaTiO3 is  mm ×  mm ×  mm. How many $k$ states are there in the first Brillouin zone?

Number of $k$ states =

(b) How many optical branches of there of the phonon dispersion of CaTiO3 ?

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 =

(f) What is the length of the reciprocal lattice vector $\vec{G}$?

$|\vec{G}|=$ m-1

(g) A 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 $a$ is the lattice constant.

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

Problem 3: Bond potential

Suppose you have a single crystal of a solid and an x-ray diffractometer with a pressure cell. The pressure cell uses hydraulic fluid to press on the crystal from all sides.

(a) Describe an experiment you could perform to measure the form of the bond potential between the atoms.

(b) How could you calculate the form of the bond potential theoretically?

(c) What other experiment could be performed to determine the form of the bond potential?

(d) How can you determine which elements are present in a crystal from an x-ray diffraction experiment?

(e) Why does the photon emitted when a diatomic molecule makes a transition between vibrational states depend on the rotational state?