Thermodynamics of the Ideal Fermi gas

Introduction

To find the chemical potential μ of an ideal Fermi gas for T > 0 , we need to find the value of μ that yields the desired number of particles. We have

where g(ε) is the density of states for a system of electrons:

It is convenient to let ε = xε_F, μ = μ*ε, and T* = kT/ε_F, where ε_F is the usual Fermi energy.

Then we can rewrite the expression for N as

If we substitute the expression for ε_F, we can rewrite the condition for μ* as

    Similarly, the mean energy E can be expressed as

If we make the same substitutions as before, we find

The applet/application does the integrals for μ* and e* numerically.

Problems

1. Start with T* = 0.2 and find μ* such that the first integral is satisfied. Click on the Accept Parameters button when you are satisfied that the value of μ* satisfies the integral condition. Does μ* initially increase or decrease as T* is increased from zero? What is the sign of μ* for T* >> 1?
2. At what value of T* is μ* ≅ 0?
3. Each time you compute a satisfactory value of μ* for a given value of T*, the program plots the corresponding value of e* by evaluating the necessary integral. How does e* vary with T* for T* << 1 (T << T_F)? Use your results for the mean energy to determine the temperature dependence of the specific heat.

References

The properties of the ideal Fermi gas are discussed in almost all texts on statistical mechanics.

Updated 2 May 2007.