Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 34E / b

The listed points are precisely integer combinations of $a_1,a_2,a_3$ (the face-centred cubic lattice), as follows by sorting the parity classes of doubled Cartesian coordinates.

$|a_1\cdot (a_2\times a_3)|=a^3/2$.

Using $b_i=2\pi (a_j\times a_k)/V$ gives $b_1=(2\pi /a)(0,1,1)$, $b_2=(2\pi /a)(1,0,1)$, $b_3=(2\pi /a)(1,1,0)$. The reciprocal cell volume is $(2\pi {)}^3/V=16{\pi }^3/a^3$.