Anisotropic ideal-gas density of states ($N(\epsilon ,\gamma )=\sqrt{\gamma }N(\epsilon ,1)$)

For the continuum dispersion ${\epsilon }_{\mathbf{k}}={\hslash }^2(k_x^2+k_y^2+k_z^2/\gamma )/(2m)$ with $\gamma >0$, substituting $q_z=k_z/\sqrt{\gamma }$ makes the energy isotropic and gives Jacobian $\sqrt{\gamma }$. Every momentum-state counting integral therefore acquires that factor. Per internal state and unit volume, $N=\sqrt{\gamma }(2m/{\hslash }^2{)}^{3/2}\sqrt{\epsilon }/(4{\pi }^2)$. The result assumes an unrestricted three-dimensional continuum; increasing this mass anisotropy is not the same operation as imposing a two-dimensional Brillouin zone and a finite interlayer band width.