Past exam of the physics A course of the University of Cambridge / 2015 / Paper 2 / Section A / 5

Write $a=V_c^{1/3}$ for the primitive cubic lattice spacing. With wavevector convention $e^{i\mathbf{k}\cdot \mathbf{r}}$, reciprocal-lattice primitive vectors have length $2\pi /a$. The first Brillouin zone is the cube $-\pi /a\le k_j\le \pi /a$, henceIt has the same volume as any primitive reciprocal unit cell. A reciprocal coordinate defined in cycles per metre would instead give $1/V_c$; that is a different convention from this $k$-space convention.