Density of states ($g(E)$)

The density of states $g(E)$ is defined so that $g(E)dE$ counts one-particle states with energies in an interval of width $dE$. A sum over closely spaced states then becomes an integral weighted by $g(E)$.

For a spin-degenerate band, the total density of states includes both spin species and is twice the single-spin density of states. Per lattice site, $\nu (ϵ)=2N^{-1}{\sum }_k\delta (ϵ-{ϵ}_k)$. Using a separate explicit spin sum then requires a compensating factor one half when converting momentum sums. The normalization matters in the Stoner criterion.

The single-spin density of states counts momentum states for one spin species. For degenerate up and down species, the spin-summed density of states is twice this value. Specify separately whether the density is per unit volume or extensive. In the BCS gap equation, the momentum sum labels opposite-spin pairs once and therefore uses the single-spin density per volume when the interaction has normalization $V/\mathcal{V}$.