Problem 1
To plot the frequency response of the magnetic susceptibility, consider how a paramagnet would respond if an magnetic field were pulsed on. Plot the real and imaginary parts of the magnetic susceptibility of a paramagnet. Explain why the susceptibility has the form that you have drawn.

Plot the real and imaginary parts of the magnetic susceptibility of a diamagnet.

Problem 2
GaAs is a direct band gap semiconductor with a bandgap of $E_g=1.424$ eV. The electrons at the minimum of the conduction band have an effective mass of $m^*= 0.067m_e$ the light holes at the top of the valence band have an effective mass of $m_{lh}^*= 0.082m_e$, and the heavy holes at the top of the valence band have an effective mass of $m_{hh}^*= 0.45m_e$. Here $m_e$ is the mass of a free electron.

(a) Sketch the electron dispersion relation and the electron density of states near the bottom of the conduction band and the top of the valence band.

(b) Describe how you could use this information to calculate temperature dependence of the chemical potential of undoped GaAs.

(c) Describe how you can use this information to plot the optical absorption of GaAs.

Problem 3

(a) In the Nerst effect, a sample is placed in a magnetic field and a temperature gradient that are perpendicular to each other. The electric field in the direction perpendicular to both the magnetic field and the temperature gradient is measured. How could the Boltzmann equation be used to calculate this electric field?

(b) How does the Nerst effect depend on the electron and phonon dispersion relations?

(c) What is the relaxation time approximation? What changes in a relaxation time?

Problem 4
Barium titanate is a ferroelectric that undergoes a first order structural phase transition from a rhombohedral phase to a monoclinic phase at about -80 C.

(a) What could be used as the order parameter? What symmetry is broken?

(b) Explain briefly what the Landau theory of first order phase transitions is.

(c) Sketch the specific heat and the electric susceptibility of barium titanate as a function of temperature. Where does the latent heat appear in these plots?

(d) Sketch the pyroelectric coefficient as a function of temperature for barium titanate.