To finish the calculation for DOS find the number of states per unit sample volume at an energy E inside an interval [ E , E + d E ] . The general form of DOS of a system is given as D n ( E ) = d Ω Ω n ( E ) d E The scheme sketched so far only applies to monotonically rising and spherically symmetric dispersion relations. In general the dispersion relation E ( k ) is not spherically symmetric and in many cases it isn't continuously rising either. To express D as a function of E the inverse of the dispersion relation E ( k ) has to be substituted into the expression of Ω Ω n ( k ) as a function of k to get the expression of Ω Ω n ( E ) as a function of the energy. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. More detailed derivations are available.