Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A9 / b / ii / Solution

Construct Bragg planes by taking the midpoints between the reciprocal origin and the other reciprocal lattice points. In one dimension these are points at $k=m\pi /a$, for nonzero integer $m$.

The first Brillouin zone is the interval bounded by the nearest two such points:The second Brillouin zone consists of the next intervals on both sides:Boundary points may be assigned to one adjacent zone to avoid double counting. The first zone is reached from the origin without crossing a Bragg point; the second is reached after crossing one. Each zone has total length $2\pi /a$. In particular, the two separated pieces of the second zone are not the entire translated first-zone interval centered at $2\pi /a$.