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

The dominant contributions are $2F(10)\cos (2\pi x)+2F(01)\cos (2\pi y)$. Their extrema occur at $x=0$ or $\frac{1}{2}$ and independently $y=0$ or $\frac{1}{2}$, modulo unit translations. The four candidate positions areWhich is a maximum rather than a minimum or saddle depends on the two signs. Higher Fourier terms sharpen these broad maxima into atomic peaks.

The sign choices that make each dominant cosine positive at the selected maximum are:

Equivalently, at a selected maximum $(x_0,y_0)$, the signs are $(-1{)}^{2x_0}$ and $(-1{)}^{2y_0}$.

The third term is $2F(11)\cos [2\pi (x+y)]$. To reinforce a maximum at $(x_0,y_0)$ from the preceding table, its sign should beHence the direct-methods triplet relationship is $S(10)S(01)S(11)\simeq +1$ for the strong reflections under consideration. This maximizes constructive density at shared atomic positions. For weak or competing reflections the relation is probabilistic rather than an unconditional algebraic theorem.

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.

Set $\mathbf{k}=\mathbf{h}$ in the direct-methods triplet relationship. It becomes $S(\mathbf{h}{)}^{2}S(2\mathbf{h})\simeq +1$, and since the square of a nonzero sign is one, $S(2\mathbf{h})\simeq +1$. This recovers the direct-methods pair relationship, with the same strong-amplitude qualifications.