 Solution

For elastic scattering, $E_f=E_i$ and ${\mathbf{k}}_f-{\mathbf{k}}_i=\mathbf{G}$, where $\mathbf{G}$ is a reciprocal lattice vector. In the ideal static crystal, coherent scattering obeys this lattice selection rule; the crystal as a whole can take the mechanical recoil with negligible energy. crystal momentum is defined modulo $\hslash \mathbf{G}$, not as unconstrained free-particle momentum.