Free-electron Fermi circle in a square lattice ($k_{F}a=\sqrt{2\pi z}$)

For a two-dimensional free electron gas with two spin states and $z$ electrons per square cell of side $a$, counting occupied wavevectors gives $z/a^2=2\pi k_F^2/(2\pi {)}^2$. The square first Brillouin zone has half-width $\pi /a$. A monovalent circle lies inside its faces; a divalent circle crosses its faces but not its corners. A real periodic potential can deform and reconnect the crossings, as in Fermi-surface reconstruction in a weak square potential; valence alone is insufficient to prove insulation.