Three-dimensional parabolic interband joint density of states ($g_j(E)=\frac{(2m_r/{\hslash }^2{)}^{3/2}}{2{\pi }^2}\sqrt{E-E_g}$)

For isotropic direct-gap parabolic bands, $m_r^{-1}=m_e^{\ast -1}+m_h^{\ast -1}$ and transition energy is $E_g+{\hslash }^2{k}^2/(2m_r)$. Differentiating the two-spin state count $k^3/(3{\pi }^2)$ per volume gives the displayed density for $E>E_g$, and zero below. Density per angular frequency is $\hslash g_j(\hslash \omega )$, not the same dimensioned quantity.