Mathematical diffraction of aperiodic structures

Abstract

Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details.

25 pages, lots of figures

References in corpus (9)

• Pure point diffraction and cut and project schemes for measures: The smooth case
• Homometric model sets and window covariograms
• Can Kinematic Diffraction Distinguish Order from Disorder?
• A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns
• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
• Cluster Model of Decagonal Tilings
• Homometric Point Sets and Inverse Problems
• The singular continuous diffraction measure of the Thue-Morse chain
• A comment on the relation between diffraction and entropy

Cited by in corpus (12)

• Imaging Quasi-Periodic Electronic States in a Synthetic Penrose Tiling
• Aperiodic crystals and beyond
• A short guide to pure point diffraction in cut-and-project sets
• Josephson effect in a Fibonacci quasicrystal
• Spectrum of a Rudin-Shapiro-like sequence
• Topological superconductivity in Fibonacci quasicrystals
• Patterns and quasipatterns from the superposition of two hexagonal lattices
• On the notions of symmetry and aperiodicity for Delone sets
• Atomistic mechanisms of dynamics in a two-dimensional dodecagonal quasicrystal
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Hamiltonian Cycles on Ammann-Beenker Tilings
• Squiral diffraction