Circular-cap carrier count in a square Brillouin zone
= Circular-cap carrier count in a square Brillouin zone
For $s=Q/(2R)$, each face cap has area $C=R^2[\arccos s-s\sqrt{1-s^2}]$. The total second-band electron area is $4C$; the first-band <hole> area is $Q^2-\pi R^2+4C$. Multiplying either area by the <spin degeneracy> two and dividing by $(2\pi)^2$ gives the carrier densities. This is an infinitesimal-potential count, not an exact finite-gap formula.