Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 2 / A9 / a / Solution

In a face-centered cubic lattice, nearest neighbours touch along a face diagonal. Four radii span that diagonal, so $\sqrt{2}a=4r$ andThusThe primitive vectors enclose volume $a^3/4$, containing one atom; the conventional cubic cell of volume $a^3$ contains four. Keeping these cell conventions distinct avoids a factor-of-four error in the electron density.