Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 1 / c

Write $F_x(\mathbf{h})=f_x{S}_x(h,k,l)$, where the four-site orbit has structure factorA cyclic permutation of $h,k,l$ fixes the first term and permutes the remaining three. Thus $S_x(k,l,h)=S_x(h,k,l)$. Substituting $-h,-k,l$ exchanges the first and second exponentials and the third and fourth, with the common coefficient $(-1{)}^{h+l}$:The two forms of this phase, $e^{+i\pi (h+l)}$ and $e^{-i\pi (h+l)}$, agree for integer indices. Multiplication by a unit-modulus phase preserves intensity, irrespective of whether $f_x$ is real or complex. Since the phase is independent of $x$ and atomic species, it also multiplies the total structure factor of the crystal.