MAS.020 Introduction to Solid State Physics
22.11.2022

Name Matrikelnr.

Problem 1

The unit cell of a crystal is shown below.

(a) What is the Bravais lattice?

(b) What are the primitive reciprocal lattice vectors of this crystal?

(d) What is the space group and the point group of this crystal?

Solution

(a) $a \ne b \ne c$ and all of the angles are 90° so this is an orthorhombic crystal. The space group is given as Pmmm #47. We can also look up space group 47 at <a href="http://lampz.tugraz.at/~hadley/ss1/crystalstructure/sg2bravais.html">http://lampz.tugraz.at/~hadley/ss1/crystalstructure/sg2bravais.html</a> where it gives the space group as simple orthorhombic.

(b) The reciprocal primitive lattice vectors are $\vec{b}_1 = \frac{2\pi}{a} \,\hat{x}$, $\vec{b}_2 = \frac{2\pi}{b} \,\hat{y}$, $\vec{b}_3 = \frac{2\pi}{c} \,\hat{z}$.

(c) The number of optical branches is $3p-3$ where $p$ is the number of atoms in the unit cell. $p = $1 blue + 2 green + 8 × 1/8 corners + 20 × 1/4 edges + 8 × 1/2 faces = 13.

There are 36 optical branches.

(d) The space group is 47 and the point group can be looked up at <a href="http://lampz.tugraz.at/~hadley/ss2/crystalphysics/crystalclasses/crystalclasses.html">http://lampz.tugraz.at/~hadley/ss2/crystalphysics/crystalclasses/crystalclasses.html</a>. The point group is orthorhombic-dipyramidal = mmm = Vh = D2h.

Problem 2

(a) In a diffraction experiment, x-rays with wavevector of $\vec{k}=0.5\times 10^{10}\,\hat{x}$ m^-1 are directed at a crystal and a diffracted peak is observed at $\vec{k}'= 0.3\times 10^{10}\,\hat{x}+0.4\times 10^{10}\,\hat{y}$ What is the reciprocal lattice vector associated with this diffraction peak?

(b) What is the largest reciprocal lattice vector that can be measured using x-rays with a wave vector of $\vec{k}=0.5\times 10^{10}\,\hat{x}$ m^-1?

(c) Suppose many reciprocal lattice vectors have been measured. How could you use this data to determine the Bravais lattice?

(d) If x-rays with a shorter wavevector of $\vec{k}=0.2\times 10^{10}\,\hat{x}$ m^-1 instead of $\vec{k}=0.5\times 10^{10}\,\hat{x}$ m^-1 are used in the diffraction experiment, how would the reciprocal lattice vectors change?

Solution

Problem 3

(a) What is the integral of the phonon density of states over all frequencies?

(b) How do you calculate the phonon contribution to the specific heat?

(c) How can you calculate the values of the spring constants that are needed to calculate the phonon dispersion relation?

(d) How can you measure the values of the spring constants in an experiment?

Solution

(a) The integral over all modes is 3 times the density of atoms, $3n=\int\limits_{-\infty}^{\infty}D(\omega)d\omega$.

(b) To calculate the phonon contribution to the specific heat, calculate the internal energy density,

$$ u = \int\limits_{-\infty}^{\infty}\hbar\omega D(\omega) f_{BE}(\omega)d\omega,$$

where $\hbar\omega$ is the energy of state $\omega$ and $f_{BE}(\omega)$ is the Bose-Einstein function.

The specific heat is the derivative of the internal energy density with respect to temperature,

$$c_v = \frac{du}{dT}.$$

(c) The spring constants are calculated by performing electronic band structure calculations for different arrangements of the atoms. There will be a minimum in the energy for some atomic arrangement and the energy will rise if an atom is moved away from the minimum. If this minimum is fit by a parabola, the curvature of the parabola will determine the linear spring forces that are needed for the phonon dispersion calculation.

(d) Experimentally you can compress the crystal and see how much force is necessary to make it smaller. This could be done in a pressure cell inside an x-ray diffractometer. The diffractometer can measure the lattice constant to about one part in $10^5$.

Problem 4
An electronic band structure is shown below. There are four electrons per primitive unit cell.

(a) How many electrons per primitive unit cell are there in the lower band between -2 eV and -8 eV?

(b) How would you calculate the chemical potential? Draw approximately the chemical potential into the band structure.

(d) What do the letters on the horizontal axis represent? What are the units of the horizontal axis?

Solution

MAS.020 Introduction to Solid State Physics 22.11.2022

MAS.020 Introduction to Solid State Physics
22.11.2022