Using a volume integral on  dimensions, the density of states is:
The Fermi energy is obtained by looking for the number density of particles:
To get:
where  is the corresponding d-dimensional volume,  is the dimension for the internal Hilbert space. For the case of spin-½, every energy is twice-degenerate, so in this case .
A particular result is obtained for , where the density of states becomes a constant (does not depend on the energy):