= 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 $a\hat{\mathbf x},a\hat{\mathbf y},a\hat{\mathbf z}$ yields
$$
\boxed{\mathbf G=\frac{2\pi}{a}(h,k,l),\quad h,k,l\in\mathbb Z}.
$$
Thus $\mathbf G$ 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.
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