the density of states

The density of states answers a bookkeeping question that turns out to control the thermodynamics of a solid: how many quantum states are crammed into each thin slice of energy? Think of the energy axis as a staircase; the density of states is how tightly packed the steps are at each height. Where the steps bunch up, electrons have many seats to occupy; where they thin out, few.

Precisely, the density of states g(E) is defined so that g(E) dE is the number of single-particle states with energy between E and E + dE. Written as a sum over states, g(E) = sum over n of delta(E - E_n), where delta is the Dirac delta function. For a free-electron gas in three dimensions it grows as the square root of energy, g(E) proportional to sqrt(E); crucially, its shape depends on dimensionality (a step in 2D, a 1/sqrt(E) spike in 1D). The total number of electrons is recovered by weighting with the occupation: N = integral of g(E) f(E) dE, using the Fermi-Dirac distribution f. Where a band is flat, so that the gradient grad_k E vanishes, g(E) develops sharp kinks or divergences called van Hove singularities.

The density of states is the bridge from the microscopic band structure to macroscopic, measurable quantities: the electronic heat capacity is proportional to g(E_F), the value right at the Fermi energy; optical absorption, tunnelling spectra, and magnetic susceptibility all read it out. The one caveat to keep straight is that it is a per-energy count of states, not of electrons, and how many of those states are actually filled is a separate question answered by the occupation function.