 Solution

ID: 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

t

changes a structure-factor phase by

- 2 π h \cdot t

in the corresponding coordinate convention. The triplet phase

ϕ_{h} + ϕ_{k} - ϕ_{h + k}

is unchanged because the three origin shifts cancel. In a centrosymmetric origin convention, the associated sign product

S (h) S (k) S (h + 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.

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