In found the density of states and the chemical potential

Chapter 7, Problem 31P

(choose chapter or problem)

Problem 31P

In Problem you found the density of states and the chemical potential for a two-dimensional Fermi gas. Calculate the heat capacity of this gas in the limit kT ≪ ϵF. Also show that the heat capacity has the expected behavior when kT ≫ ϵF. Sketch the heat capacity as a function of temperature.

Problem:

Consider a free Fermi gas in two dimensions, confined to a square area A = L2.

(a) Find the Fermi energy (in terms of N and A), and show that the average energy of the particles is ϵF/2

(b) Derive a formula for the density of states. You should find that it is a constant, independent of ϵ.

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when. kT ≪ ϵF and when T is much higher.

(d) Because g(ϵ) is a constant for this system, it is possible to carry out the integral for the number of particles analytically. Do so, and solve for µ as a function of N. Show that the resulting formula has the expected qualitative behavior.

(e) Show that in the high-temperature limit, kT ≫ ϵF, the chemical potential of this system is the same as that of an ordinary ideal gas.