Group representation theory and crystal-structure determination

Group-representation theory (GRT) is a bedrock layer of modern chemistry and physics. Its reach includes any physical or mathematical system that exhibits any kind of symmetry, even the universe itself, and provides a unique parameter set for describing order parameters that break the parent symmetries of a system. In the field of chemical and materials physics, GRT has long been an essential tool for understanding band structures and spectroscopic data, and for classifying continuous phase transitions. Here, we'll present GRT as the ideal basis for the experimental determination of atomic structures arising from symmetry loss in a crystal. The scope includes all types of commensurate and incommensurate order parameters (e.g. atomic displacements, magnetic moments, molecular/polyhedral rotations, compositional site orderings, lattice strains, thermal ellipsoids) at diffraction-relevant length scales.