Count energy states across the whole Brillouin zone

For a large periodic sample, each independent wavevector occupies a reciprocal-space cell of volume (2π)ᵈ divided by the real-space sample measure. Divide the total density of states by that measure to obtain an intensive quantity. Sum over every band and any genuinely independent spin or valley channels. If spin-split bands are already listed separately, multiplying by an additional spin factor would count them twice. A filled orbital band integrates to one state per primitive cell per spin.

At a nonzero temperature the density of states still describes available one-electron states; occupation is a separate factor supplied by the Fermi function. A sharp peak does not say every state there is occupied, and a zero density at an isolated energy does not necessarily imply a finite insulating gap. Disorder broadening, interactions, and experimental spectral matrix elements can further separate a measured signal from the clean independent-band density defined here.

In one dimension, change variables from wavevector to energy separately on each monotonic branch. The delta-function identity contributes the reciprocal absolute energy slope at every root. Both positive and negative roots must be included for a symmetric parabolic band. The absolute value prevents a negative group velocity from producing a negative density of states. Since velocity is energy slope divided by hbar, small velocity naturally enhances the state count per energy.

For a quadratic minimum, the resulting density diverges as inverse square root of energy above the edge. This divergence is integrable: the accumulated count goes to zero as the square root of that energy. In two dimensions a quadratic isotropic band instead gives a constant density above its edge, while in three dimensions it rises as a square root. The dimension changes the number of wavevectors on an energy shell and cannot be repaired by simply multiplying the one-dimensional expression by a constant.

In a three-dimensional crystal a conventional high-symmetry path is a set of lines with zero volume in the Brillouin zone. It is useful for visualizing symmetry and crossings but cannot by itself determine an integral density of states. Use a full-zone integration method and verify its wavevector convergence. Smearing a sparse mesh can hide sampling errors by producing a smooth curve; smoothness is not evidence that the integrated count or a narrow feature has converged.

The function lab isolates the slope of a quadratic. Reset and set coefficient to 1 and position to 1; predict value 1 and slope 2. Reset and set coefficient to 2 and position to 1; predict value 2 and slope 4. For a common physical energy one would also need to change the root position before evaluating a density of states. This adjustable parabola has no state counting, spin degeneracy, or Brillouin-zone integration.