Problem 1
Many observable properties of a metal can be calculated from the $E$ vs. $k$ dispersion relation.

(a) How could the electron dispersion relation of a metal be measured?

(b) How could the electronic contribution to the specific heat be calculated from the dispersion relation?

(c) How could the electrical conductivity be calculated from the dispersion relation?

(d) How could the dielectric function be calculated from the dispersion relation?

Problem 2
(a) How can you measure the Fermi surface of a metal?

(b) What do you expect the fermi surface of a bcc metal with valence 1 would look like?

(c) Why does the longitudinal resistivity $\rho_{xx}$ go to zero in the Quantum Hall effect?

Problem 3
The dielectric constant of materials used in a supercapacitor can be high ($\epsilon_r$~1000). Explain why the dielectric contant is temperature dependent. The index of refraction $n=\mathcal{Re}\left[\sqrt{\epsilon_r} \right]$ is not so large for these materials ($n$~2). Why is this so?

Problem 4
The pyroelectric coefficient describes how the polarization of a crystal changes as the temperature changes.

(a) Assume Landau's theory of a second order phase transition can be used to describe polarization of the crystal. Sketch the pyroelectric coefficient as a function of temperature.

(b) If you measured the pryoelectric coefficient as a function of temperature and the electric susceptibilty as a function of temperature you would be able to predict the specific heat as a function of temperature. Explain how this is possible.