Free-electron state counting in different dimensions ($g(E_F)=dn/(2E_F)$)

Two spin states per wavevector and parabolic dispersion give $n=2v_d{k}_F^d/(2\pi {)}^d$, where $v_d$ is the unit-ball volume. Thus $E_F={\hslash }^2[n{a}_d{]}^{2/d}/(2m)$ 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.