Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction. Issue 5 (9th July 2020)

• Record Type: Journal Article
• Authors: Baake, Michael; Grimm, Uwe
• Is Part Of: Acta crystallographica. Volume 76:Issue 5(2020:Sep.)
• Page Start: 559
• Page End: 570
• Publication Date: 2020-07-09
• Subjects: quasicrystals -- projection method -- inflation rules -- diffraction -- hyperuniformity; Crystallography -- Periodicals; Condensed matter -- Periodicals; 548
• DOI: 10.1107/S2053273320007421
• Languages: English
• ISSNs: 2053-2733

Abstract

Abstract : Averaged quantities such as mean shelling numbers, scaling behaviour or diffraction for cut‐and‐project sets can conveniently be computed in internal space, also for systems with fractally bounded windows. 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.