the structure factor
A single atom's scattering (its form factor f) is only half the story. A unit cell usually holds several atoms, each at its own position, and the waves they scatter arrive at the detector with different phases depending on where the atoms sit. The structure factor, written F_hkl, is the grand total: it adds up the waves from every atom in the cell for a given reflection (hkl), keeping track of both how strongly each scatters (its f) and how far out of step it is (its phase). It is the single most important quantity in structure determination.
The formula makes the bookkeeping explicit: F_hkl = sum over all atoms j of f_j times exp(2 pi i (h x_j + k y_j + l z_j)), where (x_j, y_j, z_j) are atom j's fractional coordinates in the cell. The exponential is just a compact way of writing the phase — how many wavelengths of path difference atom j contributes for the (hkl) reflection. F is therefore a complex number with a size (amplitude) and a direction (phase). When atoms scatter in step their contributions add and F is large; when they scatter out of step they subtract and F can shrink, even all the way to zero.
Two consequences flow from this. First, the measured brightness of a reflection is the squared size of F: the diffracted intensity is proportional to |F_hkl|^2. Second, when the atoms are arranged so that F comes out to exactly zero for a whole class of (hkl), those reflections are missing — the systematic absences that fingerprint centring and symmetry. The catch that makes crystallography hard is the phase problem: an experiment measures |F|^2 and so recovers the amplitude of F but throws away its phase, and you need both to rebuild the atomic positions.
In body-centred iron, atoms sit at (0,0,0) and (1/2,1/2,1/2). Then F_hkl = f times [1 + exp(pi i (h+k+l))]. When h+k+l is even, the bracket is 1 + 1 = 2 and the reflection is strong; when h+k+l is odd, it is 1 - 1 = 0 and the reflection vanishes. So (110) appears but (100) does not.
Summing two atoms' waves gives F; for BCC it forces all h+k+l-odd reflections to zero — a systematic absence read straight off the structure factor.
The structure factor holds both amplitude and phase, but a diffraction experiment records only the intensity |F|^2 and loses the phase. Recovering that lost phase is the phase problem, the central difficulty of solving a structure from diffraction data.