Free-electron state counting in different dimensions
= Free-electron state counting in different dimensions
{title2=$g(E_F)=dn/(2E_F)$}
Two spin states per <wavevector> and <parabolic dispersion> give $n=2v_dk_F^d/(2\pi)^d$, where $v_d$ is the unit-ball volume. Thus $E_F=\hbar^2[na_d]^{2/d}/(2m)$ with $a_1=\pi/2$, $a_2=2\pi$, $a_3=3\pi^2$. Differentiating the cumulative count yields $g(E_F)=dn/(2E_F)$, normalized per length, area or volume as appropriate.