Phasons and the plastic deformation of quasicrystals

The plastic deformation of quasicrystals (QC) is ruled by two types of singularities of the QC order, singularities of the `phonon' strain field, and singularities of the `phason' strain field. In the framework of the general topological theory of defects, in which the QC is defined as an irrational subset of a crystal of higher dimension, both types of defects appear as distinct components of the same entity, called a disvection [2]. Each of them can also be given a description in terms of more classical concepts, within a detailed analysis of the Volterra process: it can be shown that (a) the phonon singularity breaks some symmetry of translation, represented by its Burgers vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} projected from a high dimensional crystalline lattice onto the physical space; it is therefore akin to a perfect dislocation; (b) the phason singularities (there are many attached to each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-dislocation), that we call matching faults, are dipoles of dislocations whose Burgers vectors are of a special type; they do break not only a particular symmetry of translation but also the class of local isomorphism (in the jargon of QCs) of the QC. In fact, such dipoles, if they open up into loops, bound stacking faults - thus a phason singularity is an imperfect dislocation. A mismatch is nothing else than an elementary matching fault. It is suggested that it is the simultaneous presence of perfect dislocations and of phason singularities, and their interplay, that are at the origin of the peculiar characters of the plastic deformation of quasicrystals, namely the brittle-ductile transition followed by a stage of work softening; in particular the brittle-ductile transition could be related to a cooperative transition of the Kosterlitz-Thouless type which affects the dipoles and turn them into (imperfect) dislocation loops.