Consider a gas of free electrons, in a d-dimensional space, within a hypercubic container of side L

Given the spectrum of energy

we write the thermodynamic number of fermions in d dimen- sions,

Due to the spherical symmetry of the integrand, we have

where it can be shown (see Appendix 4) that

Check that C₂ = 2π, C₃ = 4π, and so on. Introducing the change of variables

and the definition of the ”Fermi-Dirac distribution”,

we have

with the ”density of states”

In two dimensions, note that D (ε) is just a constant.