Consider a gas of N free electrons, in a region of volume V , in the ultrarelativistic regime. The energy spectrum is given by

Using  the  energy  spectrum ε→_k =  hks,  the  number  of particles and the internal energy of this fermion gas are given by

and

where ƒ (ε) is the Fermi-Dirac distribution.

At T = 0, we have

The asymptotic form of the specific heat at low tempera- tures comes from a straightforward application of Sommerfeld´s method. We should note that

and

where

as can be checked in a good integral table (in Gradshteyn and Ryzhik, for example).

We then write

From the first equation, we write an expansion for the chemical potential potential µ in terms of 1/ [βε_T ] = (T/TF),

Inserting this expansion of µ into the expression for the internal energy, we finally have

Note that there is no linear power of (T/T_F ).