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Crystallography studies crystal structures and their symmetries. A structure determination connects measurements such as crystal scattering patterns to a model of the unit cell and its atomic basis of a crystal.

X-ray crystallographic structure inference makes that connection explicit: compare proposed crystal structures through the positions and wave intensities of their predicted reflections. The main difficulty is that a diffraction pattern encodes wave interference, so it must be interpreted through a model rather than read as a direct picture of the chemical atoms.

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]