Deformed model sets and distorted Penrose tilings

Abstract

In this article, we show how the mathematical concept of a deformed model set can be used to gain insight into the diffraction pattern of quasicrystalline structures. We explain what a deformed model set is, what its characteristic features are and how it relates to certain disorder phenomena in solids. We then apply this concept to distorted Penrose tilings, i.e., Penrose tilings where we apply size-effect-like distortions. While the size effect in crystals only operates on the diffuse scattering, there is also an intensity transfer on the Bragg peaks in distorted Penrose tilings. The persistence of pure point diffraction in distorted Penrose tilings can be explained by interpreting such tilings as deformed model sets.

Deformed Penrose tilings

“…The Bragg intensities have been affected enormously, but in addition some diffuse scattering has appeared. Subsequently much of this diffuse scattering has been shown to originate from the residual randomness inherent in the MC simulation process [14]; see later also. Figure 3.…”

“…It is therefore of interest to ask the question: is the distorted Penrose tiling that was produced by the size-effect relaxation, an example of a deformed model set? This question was addressed by Sing and Welberry [14] as described in the following section. When a Penrose tiling is subject to the size-effect relaxation described above the end product is a set of cartesian coordinates for each of the vertices of the tiling.…”

Deformed Penrose tilings

“…A variety of structures arises due to the ability of these chains to form double and triple chain blocks by sharing octahedral edges. We will present the structure determination of three such tunnel compounds with approximately the same A/Mn ratio, SrMn 3 O 6 [1], CaMn 3 O 6 and Sr 0.9 Mn 3 (O,F) 6 , where the shape of the channels and the ordering pattern of the A-cations vary. We have solved their structures with transmission electron microscopy (TEM) and then refined them from XRD and/or NPD data.…”

“…Replacing Sr by Ca or adding F to the compound thus results in different structures, which will be compared and discussed. In a recent study [1] using Monte Carlo (MC) simulation of a model quasicrystal it was shown that the kinds of local distortions that result from 'size-effect'-like relaxations are in fact very similar to some mathematical constructions called 'deformed model sets' [2][3][4]. Of particular interest is the fact that it has been proven that these deformed model sets are pure 'point diffractive', i.e.…”

“…they give diffraction patterns that have sharp Bragg peaks and no diffuse scattering. It should be noted that the MC simulations in [1] themselves gave diffraction patterns displaying some diffuse scattering but this can be attributed to the fact that the simulations by their very nature did include a certain amount of randomness. Examples of 'deformed model sets' on the other hand may be constructed by simple prescription and hence contain no randomness.…”