Past exam of the condensed matter physics course of the University of Oxford / 2019 / 2 / d / Solution

The leading coefficient fixes the atom count:With molar mass $40\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, the sample mass isOne atom belongs to each primitive unit cell of monatomic argon on a face-centered cubic lattice, but the conventional cube contains four atoms. If its side is $a$, $N=4V/a^3$, and nearest neighbors lie across half a face diagonal. Thereforeusing $V=21\times 10^{-6}{\mathrm{m}}^3$. The corresponding conventional lattice constant is $a\simeq 5.18\text{A˚}$. Treating the source's one-atom cell as the conventional cube would omit the factor four and give a wrong nearest-neighbor distance.