= Anisotropic ideal-gas density of states
{title2=$N(\varepsilon,\gamma)=\sqrt\gamma\,N(\varepsilon,1)$}
For the continuum dispersion $\varepsilon_{\mathbf k}=\hbar^2(k_x^2+k_y^2+k_z^2/\gamma)/(2m)$ 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, $N=\sqrt\gamma(2m/\hbar^2)^{3/2}\sqrt\varepsilon/(4\pi^2)$. 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.
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