 For the three-dimensional case the density of states formula_11 is;

In three-dimensional space the Brillouin zone boundaries are planes.The dispersion relations show conics of the free-electron energy dispersion parabolas for all possible reciprocal lattice vectors.This results in a very complicated set intersecting of curves when the dispersion relations are calculated because there is a large number of possible angles between evaluation trajectories, first and higher order Brillouin zone boundaries and dispersion parabola intersection cones.