Joint density of states ($J(\Omega )={\sum }_{v,c}\int \frac{d^{d}k}{(2\pi {)}^d}\delta (E_c(\mathbf{k})-E_v(\mathbf{k})-\Omega )$)

For filled valence and empty conduction energy bands, the joint density of states counts pairs of states at a prescribed interband energy difference. In a direct optical transition the photon momentum is usually negligible compared with a Brillouin zone dimension, so initial and final states have nearly the same crystal momentum. The absorption spectrum weights this count by transition matrix elements; its onset and structure can constrain the band gap and electronic band structure. Indirect transitions additionally involve phonons to supply crystal momentum.