MAS.020 Introduction to Solid State Physics
21.11.2024

Name Matrikelnr.

Problem 1

The conventional unit cell of MgO is shown to the right. The oxygen atoms are red and the magnesium atoms are green.

(a) What is the Bravais lattice of MgO?

(b) How many oxygen atoms and how many magnesium atoms are there in the conventional unit cell?

(d) Which planes are equivalent to the (111) plane?

Solution

Problem 2

In a diffraction experiment on a single crystal, the incoming x-rays have a wave vector $\vec{k} = 10^{10} \hat{x}$ 1/m and diffraction peaks are measured at

$\vec{k}' = 1.1\times 10^{10} \hat{x}$
$\vec{k}' = 0.9\times 10^{10} \hat{x}$
$\vec{k}' = 10^{10} \hat{x} + 10^9 \hat{y}$
$\vec{k}' = 10^{10} \hat{x} - 10^9 \hat{y}$
$\vec{k}' = 10^{10} \hat{x} + 10^9 \hat{z}$
$\vec{k}' = 10^{10} \hat{x} - 10^9 \hat{z}$

(a) What are the primitive lattice vectors in reciprocal space for this crystal?

(b) What are the primitive lattice vectors in real space for this crystal?

(d) What additional information would you need to determine how the atoms are arranged in the primitive unit cell?

Solution

Problem 3

This question is about calculating the phonon dispersion relation.

(a) The atomic vibrations are modeled as a coupled mass-spring system. How do we know which masses to use and how do we know what spring constants to use?

(b) The mass-spring system is described by coupled differential equations. As long as the springs are linear, the form of the solution to these equations is known. What do you know about these solutions? Give the form of the solutions if possible.

(d) How does the number of branches in the phonon dispersion relation depend on the number of atoms in the primitive unit cell?

(e) Describe how the phonon density of states is calculated.

Solution

(a) The masses are the masses of the atoms in the crystal. We can look up the atomic masses. The spring constants are extracted from a band structure calculation. The energy will be a minimum for a certain bond length $d_0$. For small changes in the bond length near the minimum, the energy will increase like $E= \frac{1}{2}k_{eff}(d-d_0)^2$. Using band structure calculations to calculate the energy for different bond lengths $d$, it is possible to extract the effective spring constant $k_{eff}$.

(b) The normal mode solutions must be eigenfunctions of the translation operator. All of the atoms in a normal mode oscillate with the same frequency. The displacement from equilibrium of atom $lmn$ has the form,

$$\vec{u}_{lmn} = \vec{u}_{\vec{k}}\exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right).$$

(c) There are as many $\vec{k}$ values in the first Brillouin zone as there are primitive unit cells in the crystal.

(d) There are $3p$ phonon modes per $\vec{k}$ value where $p$ is the number of atoms in the primitive unit cell.

(e) The phonon density of states is calculated by selecting a uniform distribution of $\vec{k}$ states in the first Brillouin zone and calculating the frequencies $\omega$ of all of the modes at that $\vec{k}$ state. The density of states is a histogram showing how often each frequency appears in this calculation. It is necessary to include about one million $\vec{k}$ states in this calculation to get a reasonable density of states.

Problem 4

(a) What are the allowed electron wavefunctions $\psi$ in the free electron model?

(b) What is the energy associated with the wavefunction $\psi$ from part (a)?

(d) How many electrons fit into a band in an $E\text{ vs. }k$ dispersion relationship?

Solution

MAS.020 Introduction to Solid State Physics 21.11.2024

MAS.020 Introduction to Solid State Physics
21.11.2024