Subperiodic groups, line groups and their applications

Understanding the symmetries described by subperiodic groups - frieze, rod and layer groups - has been instrumental in predicting various properties (band structures, optical absorption, Raman spectra, diffraction patterns, topological properties etc.) of 'low-dimensional' crystals. This knowledge is crucial in the tailored design of materials for specific applications across electronics, photonics and materials engineering. However, there are materials that have the property of being periodic only in one direction and whose symmetry cannot be described by the subperiodic rod groups. Describing the symmetry of these materials necessitates the application of line group theory. This paper gives an overview of subperiodic groups while briefly introducing line groups in order to acquaint the crystallographic community with these symmetries and direct them to pertinent literature. Since line groups are generally not sub-periodic, they have thus far remained outside the realm of symmetries traditionally considered in crystallography, although there are numerous 'one-dimensional' crystals (i.e. monoperiodic structures) possessing line group symmetry.