Disvections: mismatches, dislocations, and non-Abelian properties of quasicrystals

Abstract

Complete dislocations in quasicrystals are intersections of dislocations in a high-dimensional lattice with an irrational cut that represents the physical space. We study the properties which proceed from that definition, either by methods relevant to the Volterra process, or by topological methods. The Volterra process applies in the high-dimensional lattice in a rather trivial way, but its restriction to the quasicrystal introduces very unusual geometrical properties, described in terms of phason deformations and of their singularities (mismatches). For example, the motion of a defect is generically non-commutative: the `landscape' of mismatches carried by complete dislocations or disclinations in motion depends on the path which is followed by the defect between two positions, when two such paths surround a defect. The same type of argument applies, mutatis mutandis, to the question of the intersection of two defects. Other properties, of a more metallurgical nature, ensue from that use of the Volterra process, like the existence of stacking faults in QCs, bound by incomplete dislocations, and the relationship between mismatches and the reshuffling of atoms. Now, if one wishes to describe these remarkable properties with the sole use of physical observables (i.e. without mentioning the high-dimensional lattice), it appears that the natural language is the language of the topological theory of defects in the quasilattice. In particular it is shown that the group which classifies the dislocations is non-Abelian, a property in a direct relationship with the above-mentioned non-commutativity. We give the name of disvections to complete dislocations, because of their relationship with Cartan's transvections, which are translations with non-Abelian characters in a hyperbolic space.