This paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals. It analyses diffraction in the limit of the infinite system, through an appropriately defined autocorrelation. The main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by Bombieri and Taylor.

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On diffraction by aperiodic structures

Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative: 


discreteness;


extensiveness;


finiteness of local complexity;


repetitivity;


diffractivity;


aperiodicity;


existence of exotic symmetry (optional)

Model Sets: A Survey

Quantum Mechanics was born in 1900, when Planck [PL00] investigated the laws of black body radiation. He found the correct formula for the power spectrum in terms of the light frequency. Einstein’s interpretation in 1905 by means of energy quanta [EI05] was confirmed by his interpretation of the photoelectric effect. However it took years before Quantum Mechanics became such a solid body of knowledge that it could not be avoided by any reasonable physicist. There is no doubt today that it is a fundamental theory of matter, and that it has changed daily life through new technology, in a way which has never been known before in human history.

Gap Labelling Theorems for Schrödinger Operators

An overview is given of the crystallography of aperiodic crystals, in particular of incommensurately modulated structures and incommensurate intergrowth compounds. The symmetry, the description of the structure and diffraction are described. An extensive discussion is given of crystallographic methods for determination of incommensurate crystal structures. A series of specific compounds is discussed in detail, covering the various ways of incommensurateness. Special attention is given to the commensurate approximation and to the superspace description of superstructures. An historical introduction and a section on the relation between physical properties and incommensurate structure are included.

Incommensurate crystal structures

This contribution deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular, we focus on their interplay with various physically motivated equivalence concepts such as local indistinguishability and local equivalence. Various discrete patterns with non-crystallographic symmetries are described in detail, and some of their magic properties are introduced. This perfectly ordered world is augmented by a brief introduction to the stochastic world of random tilings.

A Guide to Mathematical Quasicrystals

This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodic patterns. We present the ideas introduced in [9] and developed in [10].

A. Katz

Papers

Quasiperiodic Patterns with Icosahedral Symmetry