Physics 6311: Statistical Mechanics - Homework 12

Problem 1: Liquid ${}^3$Helium (8 points)

Liquid ${}^3$He is approximately a Fermi gas (spin 1/2). The density is 0.081 g/cm${}^3$.

a)

Calculate the Fermi energy (at zero temperature). Also calculate the Fermi velocity (the velocity corresponding to the Fermi energy).

b)

At roughly what temperatures do you expect the fermionic character of ${}^3$He to be important?

Problem 2: Fermi gas with gap in the density of states (8 points)

A system of noninteracting fermions has a density of states that behaves as $g(𝜖)\sim c|𝜖-{𝜖}_F|$ close to the Fermi energy (this can happen, e.g., in certain exotic superconductors). Qualitatively discuss the low-temperature specific heat of such a system (what is the leading power in $T$?). Use the Sommerfeld expansion for guidance; you do not have to carry out the actual calculation.

Problem 3: Velocity distribution of the Fermi gas (10 points)

For an ideal Fermi gas at zero temperature, derive the probability density of the particle velocities and compare it to the Maxwell distribution of a classical ideal gas of the same total energy (per particle).

[Hint: You will need to find the correct temperature for the classical gas.]

Problem 4: Relativistic Fermi gas (14 points)

Consider an ideal Fermi gas of $N$ spin-1/2 particles in a cubic box of size $L$. Their dispersion relation is $𝜖=c|p|$, i.e., they behave ultra-relativistically.

a)

Determine the zero-temperature behavior of this Fermi gas. Find the the Fermi energy, the total energy, and the pressure as functions of the particle density.

b)

Use the Sommerfeld expansion to find the leading low-temperature specific heat.