Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section D / 43 / b / iii / 3

The twelve face-centred cubic neighbours are $(a/\sqrt{2})(\pm 1,\pm 1,0)$ and permutations of the zero coordinate. Eight vectors have a nonzero given component, each contributing $a^2/2$, while off-diagonal terms cancel between signs. ThusFor a cubic neighbour shell this can also be checked from ${\sum }_j{r}_{j,\alpha }{r}_{j,\beta }=(z{a}^2/3){\delta }_{\alpha \beta }$ with coordination number $z=12$.