Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 58 / b / iv / Solution

Moving the origin by $\mathbf{t}$ changes a structure-factor phase by $-2\pi \mathbf{h}\cdot \mathbf{t}$ in the corresponding coordinate convention. The triplet phaseis unchanged because the three origin shifts cancel. In a centrosymmetric origin convention, the associated sign product $S(\mathbf{h})S(\mathbf{k})S(\mathbf{h}+\mathbf{k})$ is therefore invariant.

For example shifting $x$ by half a period flips both $S(10)$ and $S(11)$, leaving $S(01)$ unchanged; shifting $y$ flips the other corresponding pair. A general origin may make the factors complex, but the phase-invariant statement remains valid.