Symmetry in Diffraction Patterns

Abstract

Symmetry in the atomic positions within the unit cell leads to symmetry in the intensities of the diffracted beams. The intensity-weighted reciprocal lattice is introduced as a tool to visualise a single-crystal X-ray diffraction pattern. The symmetry of the diffraction pattern is the point symmetry of this object. It comprises the point symmetry parts of all symmetry operations in the space group. Space symmetry operations (screws, glides and centring translations) are distinguished from point symmetry operations by additional systematic intensity absences amongst specific groups of diffracted beams. The combination of the point group of a diffraction pattern and any systematic absences reveals the crystal's space group. Under normal scattering conditions, the diffraction pattern always contains an inversion centre, so I(hkl)=I(h̄k̄l̄). This inversion centre combines with the true point group of the diffraction pattern to give its Laue group, which is usually what is measured. Under conditions where significant anomalous scattering occurs, the true point group can be measured and it is possible to determine the absolute structure (handedness) of a non-centrosymmetric crystal structure. This allows absolute configuration to be determined for chiral API molecules that crystallise in non-centrosymmetric space groups.