Which distributions of matter diffract? - Some answers

TL;DR

This review explores which matter distributions produce pure point diffraction spectra, discussing mathematical conditions, model sets, lattice systems, and stochastic point sets, with examples like the paperfolding sequence.

Contribution

It provides a comprehensive overview of conditions for pure point diffraction and connects mathematical frameworks with specific models and stochastic systems.

Findings

Cut and project schemes explain pure point diffraction

Model sets and lattice substitution systems often yield pure point spectra

Stochastic point sets have diverse diffraction properties

Abstract

This review revolves around the question which general distribution of scatterers (in a Euclidean space) results in a pure point diffraction spectrum. Firstly, we treat mathematical diffration theory and state conditions under which such a distribution has pure point diffraction. We explain how a cut and project scheme naturally appears in this context and then turn our attention to the special situation of model sets and lattice substitution systems. As an example, we analyse the paperfolding sequence. In the last part, we summarize some aspects of stochastic point sets, with focus both on structure and diffraction.