Past exam of the materials science course of the University of Cambridge / 2019 / Paper 1 / 3 / c

Summing phase-weighted atomic scattering factors over the four positions givesIf $h,k$ have different parity, the reflection vanishes. If they have equal parity, this becomes $2[f_{\mathrm{Au}}+(-1{)}^{h+l}{f}_{\mathrm{Cu}}]$. The difference reflections reveal chemical order; they disappear when all sites instead have the same average scattering factor.

Ordering reflections by $d_{hkl}^{-2}=(h^2+k^2)/a^2+l^2/c^2$ with $c\simeq 0.9a$, the first three nonzero families are $001$, $110$ and $111$. The $100$ family is extinct. Their structure factors areThe atomic scattering factors in each expression are evaluated at that reflection's scattering vector, so equal symbolic forms need not mean equal numerical amplitudes at different angles. In particular, the low-angle superlattice peaks need not be the strongest peaks.

The first peak is $001$, with $d_{001}=c$; the second is $110$, with $d_{110}=a/\sqrt{2}$. Applying Bragg's law with $\lambda =0.193604\mathrm{nm}$ givesThus $c/a=0.927$, consistent with the approximate ratio used to order the reflections. These are the lengths of the supplied larger basal cell; the primitive basal lattice parameter is $a/\sqrt{2}=0.2798\mathrm{nm}$.