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

For direct-lattice points $x_n=na$, a lattice-periodic plane wave must have the same phase at every translated lattice point:Thus $Ga=2\pi m$, with integer $m$, and the reciprocal lattice isIts spacing is $2\pi /a$. These plane waves execute an integer number of complete cycles within a direct-lattice period, so their wavelength magnitudes are $a/|m|$ for $m\ne 0$.

A general electronic Bloch wave need not itself be lattice-periodic: wavevectors differing by $G_m$ have the same translation phase. The reciprocal lattice labels that equivalence, while a large finite specimen with periodic boundary conditions has a much finer grid of allowed electronic wavevectors.