Aperiodic crystals and beyond

Abstract

Crystals are paradigms of ordered structures. While order was once seen as synonymous with lattice periodic arrangements, the discoveries of incommensurate crystals and quasicrystals led to a more general perception of crystalline order, encompassing both periodic and aperiodic crystals. The current definition of crystals rests on their essentially point-like diffraction. Considering a number of recently investigated toy systems, with particular emphasis on non-crystalline ordered structures, the limits of the current definition are explored.

33 pages, many pictures

References in corpus (27)

• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties
• Quasicrystals: A matter of definition
• Order in glassy systems
• Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces
• What is a crystal?
• Symmetry Breaking and Order in the Age of Quasicrystals
• Pure Point spectrum for measure dynamical systems on locally compact Abelian groups
• Diffraction of random tilings: some rigorous results
• Homometric model sets and window covariograms
• Mathematical diffraction of aperiodic structures
• Can Kinematic Diffraction Distinguish Order from Disorder?
• Kinematic Diffraction from a Mathematical Viewpoint
• Spectral and topological properties of a family of generalised Thue-Morse sequences
• Dynamical versus diffraction spectrum for structures with finite local complexity
• 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
• Diffraction of limit periodic point sets
• Squirals and beyond: Substitution tilings with singular continuous spectrum
• Stationary processes with pure point diffraction
• Examples of substitution systems and their factors
• Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra
• Pinwheel patterns and powder diffraction
• Scaling of the Thue-Morse diffraction measure
• Close-packed dimers on the line: diffraction versus dynamical spectrum
• Pure Glass in Finite Dimensions
• Statistical mechanics of glass transition in lattice molecule models
• Ordered amorphous spin system

Cited by in corpus (5)

• Anomalous Light Transport Induced by Deeply Subwavelength Quasiperiodicity in Multilayered Dielectric Metamaterials
• Fraudulent White Noise: Flat power spectra belie arbitrarily complex processes
• Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Optical computing of quantum revivals