Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 34B (Applications of Quantum Mechanics)

A Bravais lattice is the integer span of three linearly independent primitive vectors. Its reciprocal lattice consists of vectors $G$ satisfying $G\cdot r\in 2\pi \mathbb{Z}$ for every lattice vector $r$.

Translations by lattice vectors commute with the Hamiltonian. Their simultaneous eigenvalues have the form $e^{ik\cdot r}$, so an energy eigenfunction may be chosen withThus ${\psi }_k=e^{ik\cdot x}{u}_k$ with lattice-periodic $u_k$, which is Bloch theorem. Wavevectors differing by a reciprocal-lattice vector describe the same translation character, and the Brillouin zone is a fundamental Wigner-Seitz cell of the reciprocal lattice.

TakeTheir integer span consists exactly of the cubic points together with the body-centre points, so the BCC set is a Bravais lattice. Using the inverse matrix supplied in the question, reciprocal primitive vectors areThese generate a face-centred cubic lattice; thus the reciprocal lattice of BCC is FCC, with the displayed scale.