69
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

Deformed Penrose tilings

&
Pages 2877-2886 | Received 30 Aug 2006, Accepted 25 Oct 2006, Published online: 02 Dec 2010
 

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.

Acknowledgements

TRW would like to acknowledge the support of the Australian Research Council and the Australian Partnership for Advanced Computing. BS acknowledges financial support by the Cusanuswerk and the German Research Council, Collaborative Research Centre 701. BS thanks the Research School of Chemistry and the Cusanuswerk for financing a profitable stay and trip to Canberra. BS and TRW also thank Michael Baake for valuable discussions.

Log in via your institution

Log in to Taylor & Francis Online

PDF download + Online access

  • 48 hours access to article PDF & online version
  • Article PDF can be downloaded
  • Article PDF can be printed
USD 61.00 Add to cart

Issue Purchase

  • 30 days online access to complete issue
  • Article PDFs can be downloaded
  • Article PDFs can be printed
USD 786.00 Add to cart

* Local tax will be added as applicable

Related Research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.