Abstract
Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.
Highlights
The discovery of quasicrystals in the early 1980s (Shechtman et al, 1984) led to a reconsideration of the fundamental concept of a crystal [see Grimm (2015) and references therein], and highlighted the need for a mathematically robust treatment of the diffraction of systems that exhibit aperiodic order
It was established that regular model sets (Moody, 2000), meaning systems obtained by projection from higher-dimensional lattices via cut-and-project mechanisms with ‘nice’ windows, have pure point diffraction (Schlottmann, 2000; Richard & Strungaru, 2017a)
The result on the pure point nature of diffraction holds for rather general setups, including cut-and-project schemes with non-Euclidean internal spaces
Summary
The discovery of quasicrystals in the early 1980s (Shechtman et al, 1984) led to a reconsideration of the fundamental concept of a crystal [see Grimm (2015) and references therein], and highlighted the need for a mathematically robust treatment of the diffraction of systems that exhibit aperiodic order. This is followed by a brief review of the standard approach to diffraction, where we exploit the description of the Fibonacci point set as a cut-and-project set and the general results for the diffraction of regular model sets. This makes it possible to efficiently compute the diffraction of certain cut-and-project systems with complicated windows, such as windows with fractal boundaries, as are commonly found in inflation structures It turns out that this can dinstinguish between generic and inflation-invariant choices for the window in the cut-and-project scheme
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