 Solution

The reciprocal-space volume inside a spherical Fermi surface is ${\mathcal{V}}_{\mathrm{F}}=4\pi k_{\mathrm{F}}^3/3$. The first Brillouin zone has volume $(2\pi {)}^3$ divided by the primitive real-space cell volume:Since $k_{\mathrm{F}}^3=3{\pi }^2(4/a^3)$,Numerically,The half-zone volume reflects one electron per primitive cell and two spin states per Bloch wavevector: a completely filled single band would require two electrons per cell. Volume alone does not establish containment inside the polyhedral zone. Here the nearest reciprocal-lattice points are of type $(111)$, so their zone faces are a distance $\pi \sqrt{3}/a$ from the origin. The free-electron radius obeysso these ideal monovalent spheres fit inside the first zone. Real band structure can distort a Fermi surface into zone-face contacts or necks; that is additional physics beyond the free sphere.