= Solution
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 $-\pi/a\leq k_j\leq\pi/a$, hence
$$
\boxed{V_{\mathrm{BZ}}=\left(\frac{2\pi}{a}\right)^3=\frac{(2\pi)^3}{V_c}.}
$$
It 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.
Back to article page
