Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 5 / Unlettered 1 / Solution

For the mutually orthogonal direct primitive lattice translation vectors ${\mathbf{a}}_i$, define the reciprocal lattice basis byThus $\mathbf{G}={\sum }_i{h}_i{\mathbf{b}}_i$, with integer $h_i$, satisfies $\exp (i\mathbf{G}\cdot \mathbf{R})=1$ for every lattice vector $\mathbf{R}$.