Lattice

0.1 Density of States of Gas with Energy Function 

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The density of states $g(ϵ)$ for a spin-$S$ gas with spin degeneracy $g_S$ in a $D$-dimensional box with energy relation $ϵ(\mathbf{p})$ is
$$g(ϵ)=g_s{\left(\frac{L}{2\pi \hslash }\right)}^D\int d^{D}p\delta (ϵ-ϵ(p))=g_s{\left(\frac{L}{2\pi \hslash }\right)}^D\int d{\mathrm{\Omega }}_D\frac{p(ϵ{)}^{D-1}}{|\frac{\partial ϵ}{\partial p}(p(ϵ))|}$$