Final Exam

Due: Dec 7th 2015 11:59 PM

Open books and open notes.

Use of internet/solution books is NOT allowed.

You have to work alone.

Anyone caught cheating will be given F grade.

1. We have already calculated expression for magnetization for paramagnets.

a) Derive expression for the internal energy and magnetic heat capacity in a constant magnetic

field. (4 points)

b) Derive limiting forms of specific heat at high and low temperatures. (4 points)

2. a) Write down primitive vectors for FCC lattice (2 points)

b) Calculate the ratio of volume of primitive cell of FCC to conventional cell. (2 points)

c) Write basis vectors assuming FCC to be simple cubic with basis. (2 points)

c) Write down primitive vectors for BCC lattice. (2 points)

d) Write basis vectors assuming BCC to be simple cubic with basis. (2 points)

3. Consider a pure dielectric with cubic symmetry and dielectric constant ϵ r=10.

a) Dope this dielectric with a single impurity atom that provides a 1 electron to the system.

Derive the expression for the radius of this electron around this ionized defect. Calculate the

value of this radius assuming that mass to be 1/10th the free electron mass. (4 points)

b) Applying electric field to this material displaces this electron “cloud” by some distance x.

Derive the expression for polarizability of this atom. Calculate the numerical value of

polarizability. (hint: Electric field due to dipole at a distance r >> x is E= p4 π ϵ 0r

3 ) (4points)

c) Derive an expression for density of doping that would result in metallic behavior of the

material. Calculate the numerical value for this doping. (6 points)

4. Calculate the heat capacity in 2D materials. Assume that the motion is restricted in the plane

and that the temperature is low i.e. T≪ΘD. Also assume there is no out of plane interaction.

(4 points)