Symmetry Analysis of Some Basic Structural Properties of Incommensurately Modulated Crystals by Projective Representations of Unimodular Groups

Projective representations of the unimodular groups are applied in the general symmetry theory of incommensurately modulated crystals. It is demonstrated that the algebraic condition relevant for the existence of line groups can be directly connected to those related to linear projective groups. As a concrete application example, the formalisms of the thermodynamic modelling of the structural phase transitions and the Fourier‐analysis formulae of the scattering processes in kinematic approximation are discussed by use of this group‐theoretical technique, introduced newly in this study for symmetry analyses of the general structural features of incommensurate systems. The symmetries of the infinitesimally long discrete chain‐type systems are applied in a novel manner by use of the ray‐representations of groups for structural studies of the incommensurate crystals. After thermodynamic analysis of structural phase transitions resulting in such types of condensed matter systems, the kinematic diffraction formalism is extended to be applicable in the diffuse X‐ray scattering experiments.