 Crystal basis structure factor

The crystal basis structure factor is the coherent sum of scattering contributions from the atomic basis of a crystal at a chosen reciprocal lattice vector $\mathbf{G}$:The positions ${\mathbf{r}}_s$ lie within a cell and $f_s$ are the corresponding scattering amplitudes. Fractional coordinates give $F_{hkl}={\sum }_s{f}_s{e}^{2\pi i(h{x}_s+k{y}_s+l{z}_s)}$.

The Bravais lattice sum fixes possible Bragg-peak positions, while this basis sum controls their amplitudes and can cancel some exactly. Two identical sites separated by half a cubic body diagonal give a factor $1+(-1{)}^{h+k+l}$, extinguishing odd sums. Replacing one site by a different species generally prevents that cancellation. The intensity involves $|F{|}^2$, so amplitudes must be added before squaring.