Problem 1
Consider gold atoms arranged on an fcc lattice. The lattice constant of metallic gold is 4 Å. If the lattice constant were to be increased, the crystal would go through a metal-insulator transition.

(a) Describe how you could calculate the lattice constant at which this metal-insulator takes place. (Hint: the factor $1.19 a_0 \approx 1$ Å is for metallic hydrogen.)

(b) How could you measure plasmons on the metal side of the transition and excitons on the insulating side of the transition?

(c) Plot the plasma frequency as a function of the lattice constant and the exciton energy as a function of the lattice constant.

Problem 2
(a) Describe an experiment you could perform to determine the real and imaginary parts of the dielectric function of an insulator at $\omega = 0$.

(b) Sketch the real and imaginary parts of the dielectric function of an insulator.

(c) Sketch the real and imaginary parts of the dielectric function of a metal.

Problem 3
The quantum Hall effect can be observed in a MOSFET.

(a) Explain what inversion in a MOSFET is and how it can be used to obtain a 2-D metal.

(b) Sketch the density of states for a 2-D metal in a magnetic field.

(c) Sketch the resistivity and the Hall resistivity as a function of the magnetic field.

(d) Why do the Shubnikov-de Hass oscillations disappear for low magnetic fields?

Problem 4
Ferromagnetism can be described by Landau's theory of second order phase transitions.

(a) What is the order parameter for ferromagnetism? Sketch the temperature dependence of the order parameter.

(b) How could you observe this phase transition in an experiment?

(c) How could you calculate the magnetic susceptibility from Landau theory? Sketch the temperature dependence of the magnetic susceptibility.