Past exam of the physics A course of the University of Cambridge / 2017 / Paper 2 / Section A / 4

In a conventional face-centered cubic lattice cell of side $a_F$, touching nearest-neighbour hard spheres lie along a face diagonal. Four radii span that diagonal, so $\sqrt{2}{a}_F=4r$ and $a_F=2\sqrt{2}r$. In a conventional body-centered cubic lattice cell, the contact lies on the body diagonal: $\sqrt{3}{a}_B=4r$, giving $a_B=4r/\sqrt{3}$. ThereforeThese are the nonprimitive cubic unit cells requested. They contain four and two atoms respectively, so their volume ratio should not be confused with the volume-per-atom ratio, which is half as large, $3\sqrt{6}/8\simeq 0.9186$. The common sphere radius cancels.