Incommensurability in crystals

Abstract The aim of this review is to present aperiodic crystals from a unifying point of view, showing why it is justified to call them crystals, despite the lack of a three-dimensional lattice periodicity, and to discuss in what sense they differ from periodic crystals in structure, symmetry and other physical properties. The extension of the concept of crystal has been based, during the last two decades, on investigation of incommensurate crystal phases. Among these, the most important ones are the modulated crystals. Their crystallographic nature, already apparent in the diffraction pattern, could be made explicit on the symmetry level by embedding in a higher-dimensional Euclidean space. The recent discovery of quasicrystal phases (representing a fairly different class of aperiodic crystals than the modulated ones) can also be approached in a similar way. Furthermore, it now appears that another class of incommensurate crystals, the so-called composite structures, represents a kind of intermediate case between the other two. In § 1 basic concepts are presented, together with a number of compounds given as illustration for typical incommensurate crystal phases. In § 2 we deal with the general formalism allowing a crystallographic symmetry characterization. It is intended as a first approach to crystal-structure determination, which justifies the emphasis on the diffraction pattern and on the modulated-crystal case. The appropriate generalization to the quasicrystal case is considered in § 3. The crystallographic nature of the incommensurate phases is apparent in their growth forms. It has been known for centuries that crystal morphology is essentially based on lattice periodicity. It is therefore fascinating to discover how nature solves the problem in the incommensurate crystal case; we discuss this in § 4. The origin of incommensurability is the subject of § 5 (on a phenomenological level) and of § 6 (on a microscopic level). In § 7 the basic concepts needed in crystal-structure determination are discussed in more detail than was appropriate in § 2. In § 8 we discuss those physical properties which are more closely related to incommensurability, fitting nevertheless into the general frame of crystal physics. In § 9 we deal with defects, again in particular with the additional ones due to the more complex structure of aperiodic crystals, making clear at the same time why defects play an even more important role than in periodic crystals.