Problem 1
A free-electron gas is a system on non-interacting fermions. How could you calculate the magnetic susceptibility of a free-electron gas? Make a sketch of the magnetic susceptibility vs. magnetic field $B$. At high magnetic fields, the magnetization approaches a constant. Why does this happen?

Problem 2
NaCl is an ionic crystal and an electrical insulator.

(a) Sketch the dielectric function for this material. How could the Kramers-Kronig relation be used to check the form of the dielectric function?

(b) Polaritons are observed in NaCl. Sketch the polariton dispersion relation for the ionic crystal. Use the same scale for the frequency axis as in part (a).

(c) Sketch the reflectance for the ionic crystal for normally incident light. Use the same scale for the frequency axis as in part (a).

(d) What are polarons? How could you measure the the properties of polarons in NaCl?

Problem 3

The Seebeck Effect relates the gradient of the electrochemical potential to the gradient of the temperature.

(a) What is the electrochemical potential?

(b) What rank tensor is the Seebeck coefficient?

(c) How can you calculate the Seebeck coefficient?

(d) If you wanted to calculate the Seebeck coefficient for a free electron gas, what dispersion relation would you use?

Problem 4
(a) Explain how the exchange energy leads to ferromagnetism.

(b) What is mean field theory? How is mean field theory used to calculate the Curie-Weiss law?

(c) How was Landau's theory of phase transitions used to calculate the Curie-Weiss law?