Defects in quasicrystals

Abstract

There are several incentives to identify the analogues of dislocations and disclinations in quasicrystals. Penrose patterns are two-dimensional systems, and according to phase space counting fluctuations of the atomic displacements are so strong that exact long-range order is reduced to quasi-longrange order just as in periodic crystals. The necessity of higher dimensions is motivated in Section 2 both in a continuum model and a microscopic model. In Section 3, dislocations and disclinations are constructed via the Volterra process in the hypercubic lattices. Suitable projection proves, that the singularities are not only accompanied by deformations in real space ("phonon strain"), but also by deformations in phase space ("phason strain"). In the microscopic model the phason strain manifests itself in the form of exceptional vertices, called mistakes. Thus disclinations and dislocations are accompanied by clouds of mistakes and become "dressed" like polarons. Also in Section 3 methods of algebraic topology are applied to the classification of singularities in quasiperiodic systems. It is ascertained that disclination dipoles are equivalent to single dislocations. Finally, dislocation motion is demonstrated by the example of a dislocation dipole, whose constituents slowly move apart.