Mathematical diffraction of aperiodic structures

TL;DR

This paper reviews mathematical diffraction theory for aperiodic structures, focusing on pure point and continuous diffraction, especially in non-periodic systems like quasicrystals, highlighting recent advances and key examples.

Contribution

It provides a comprehensive overview of mathematical diffraction for aperiodic structures, emphasizing the distinction between pure point and continuous diffraction and illustrating with characteristic examples.

Findings

Characterization of pure point diffraction in aperiodic structures

Analysis of continuous diffraction and diffuse scattering

Summary of key mathematical results and examples

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.