Curved crystals, defects and disorder

Abstract

The structure of media whose short-range order cannot be extended crystallographically in flat space (disordered metallic and covalent systems, Frank-Kasper phases, quasicrystals, blue phases and various molecular phases) is describable in terms of distributions of defects in a crystal situated in a space of constant curvature. This paper reviews the geometrical and topological properties of curved crystals and of their defects, the relations between defects and the mapping on flat space, and how certain universal properties generally attributed to disordered systems can be inferred from this model. A number of topics which have not yet received full recognition as belonging to the subject of frustration and curvature are emphasized, such as the role of other defects than disclinations (in particular disvections), the curved space description of incommensurate structures such as quasicrystals, and in these latter the concept of phase.