Deformed Penrose tilings

Welberry, Thomas Richard and Sing, Bernd (2007). Deformed Penrose tilings. Philosophical Magazine, 87(18-21) pp. 2877–2886.




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.

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