### D.63 Fermi-Dirac integrals at low temperature

This note finds the basic Fermi-Dirac integrals for the free-electron gas at low temperature. To summarize the main text, the number of particles and total energy per unit volume are to be found from

where the Fermi-Dirac distribution and the density of states are:

and the number of spin states 2 for systems of electrons. This may be rewritten in terms of the scaled energies

to give

To find the number of particles per unit volume for small but nonzero temperature, in the final integral change integration variable to , then take the integral apart as

and clean it up, by dividing top and bottom of the center integral by the exponential and then inverting the sign of in the integral, to give

In the second integral, the range that is not killed off by the exponential in the bottom is very small for large and you can therefore approximate as , or using a Taylor series if still higher precision is required. (Note that the Taylor series only includes odd terms. That makes the final expansions proceed in powers of 1/.) The range of integration can be extended to infinity, since the exponential in the bottom is exponentially large beyond 1. For the same reason, the third integral can be ignored completely. Note that ​12, see [40, 18.81-82, p. 132] for this and additional integrals.

Finding the number of particles per unit volume this way and then solving the expression for the Fermi level gives

 (D.39)

This used the approximations that and is small, so

The integral in the expression for the total energy per unit volume goes exactly the same way. That gives the average energy per particle as

 (D.40)

To get the specific heat at constant volume, divide by and differentiate with respect to temperature: