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:
Our goal is to find the value of μ that gives the desired density ρ = N/V. Because the integral in Eq. (1) cannot be done analytically except at low temperatures, we will use numerical methods to evaluate it.
It is convenient to let ε = xεF, μ = μ*ε, and T* = kT/εF, where εF is the usual Fermi energy.
Then we can rewrite the expression for N as
Similarly, the mean energy E can be expressed as
If we make the same substitutions as before, we find
The program evaluates the integrals for μ* and e* numerically.
The properties of the ideal Fermi gas are discussed in almost all texts on statistical mechanics.
Updated 28 December 2009.