 X-ray crystallographic structure inference

X-ray crystallographic structure inference is an inverse problem: infer a crystal structure from measured X-ray diffraction. A proposed unit cell and atomic basis of a crystal predict a pattern; structure determination asks which proposals survive comparison with that pattern. The useful separation is between directions and strengths. Peak positions constrain the reciprocal lattice and hence interplanar spacings; relative wave intensities constrain the motif through the crystal basis structure factor. Knowing a repeat distance does not locate all the chemical atoms within that repeat.

The lattice constant from two indexed reflections example isolates the geometric inference. Basis intensities and phase ambiguity then holds the translations fixed while changing the motif. Even complete, noiseless wave intensity data need not determine all wave phases: this is the phase problem. Chemical composition, symmetry, and additional measurements can supply constraints beyond the original pattern.^[ref]