 Solution

There are $L$ primitive cells and therefore $L$ lattice points. The specified reciprocal-space density $L{A}_c/(2\pi {)}^2$ counts distinct allowed wavevectors; each supports two electron spin states. Filling the disk at zero temperature givesUsing $A_c=2a^2$ and $k_F^2={\pi }^2/(2a^2)$,The circle fills a fraction $\pi /4$ of the first-zone area, while one spin-degenerate band has two states per primitive cell; these give the same result. A single-spin count would instead be $\pi /4$, omitting the physical twofold spin degeneracy of this model. The circle has only just reached the sides, so the zone corners remain unfilled: first contact does not mean filling the entire first band.