 Solution

Use fractional coordinates and the structure factor conventionFor real atomic scattering factors, $F(-\mathbf{h})=F(\mathbf{h}{)}^{\ast }$, immediately proving Friedel's law. If an inversion centre is chosen as origin, each site at ${\mathbf{r}}_j$ has an identical partner at $-{\mathbf{r}}_j$, with the same possibly complex atomic scattering factor. Each pair contributes $2f_j\cos (2\pi \mathbf{h}\cdot {\mathbf{r}}_j)$; sites fixed by inversion modulo a lattice translation also have even phase factors at reciprocal-lattice vectors. Henceeven for scalar anomalous X-ray scattering in a centrosymmetric structure. This last equality is an equality of amplitudes, not generally complex conjugacy.