 Solution

Translating a scattering centre by any lattice translation $\mathbf{R}$ changes the scattered amplitude by $e^{i\mathbf{G}\cdot \mathbf{R}}$, up to the immaterial overall sign convention. constructive interference requires $\mathbf{G}\cdot \mathbf{R}=2\pi$ times an integer. Applying this to $a\hat{\mathbf{x}},a\hat{\mathbf{y}},a\hat{\mathbf{z}}$ yieldsThus $\mathbf{G}$ lies in the reciprocal lattice. elastic scattering additionally requires $|{\mathbf{k}}_i+\mathbf{G}|=|{\mathbf{k}}_i|=2\pi /\lambda$; not every reciprocal vector is accessible for a fixed incident beam.