Deformed Penrose tilings

Abstract

Monte Carlo (MC) simulation of a model quasicrystal (2D Penrose rhomb tiling) shows that the kinds of local distortions that result from size-effect-like relaxations are in fact very similar to mathematical constructions called deformed model sets. Of particular interest is the fact that these deformed model sets are pure point-diffractive, i.e. they give diffraction patterns that have sharp Bragg peaks and no diffuse scattering. Although the aforementioned MC simulations give diffraction patterns displaying some diffuse scattering, this can be attributed to the fact that the simulations include a certain amount of unavoidable randomness. Examples of simple deformed model sets have been constructed by simple prescription and hence contain no randomness. In this case the diffraction patterns show no diffuse scattering. It is demonstrated that simple deformed model sets can be constructed, based on the 2D Penrose rhomb tiling, by using deformations which are defined in terms of the local environment of each vertex. The resulting model sets are all topologically equivalent to the Penrose tiling (same connectedness), are perfectly quasicrystalline but show an enormous variation in the Bragg peak intensities. For the examples described, which are based on nearest-neighbour environments, 12 deformation parameters may be chosen independently. If more distant neighbours are taken into account further sets of parameters may be defined. One example of these simple deformed tilings, which shows great similarity to the size-effect-distorted tiling, is discussed in detail.

Structural transformations in quasicrystals induced by higher dimensional lattice transitions

“…It is useful to discuss the relation of our method with other higher dimensional approaches used in the literature to study experimentally observed transitions between icosahedral quasicrystals and cubic phases in three dimensions (Kramer 1987;Litvin et al 1987;Li et al 1989a,b;Torres et al 1989;Li & Cheng 1990;Aragón & Torres 1991;Mukhopadhya et al 1991;Cheng et al 1992;Sun 1993), and for deformations of Penrose tilings of the plane (cf., for instance, Ishii 1993; Sing & Welberry 2006;Welberry & Sing 2007). As mentioned in §6, icosahedral quasicrystals can be obtained by projection of an SC lattice in six dimensions on a three-dimensional subspace that carries an irreducible representation of the icosahedral group.…”

Software package for structure analysis of quasicrystals∗

“…Recently it was demonstrated that the size effect in a fictitious quasicrystal also leads to such a model, where the ODs are subdivided into small fragments like ones shown in figure 5 [21].…”