Shelling of homogeneous media

Grimm, Uwe and Baake, Michael (2004). Shelling of homogeneous media. Ferroelectrics, 305(1) pp. 173–178.

DOI: https://doi.org/10.1080/00150190490462685

Abstract

A homogeneous medium is characterised by a point set in Euclidean space (for the atomic positions, say), together with some self-averaging property. Crystals and quasicrystals are homogeneous, but also many structures with disorder still are. The corresponding shelling is concerned with the number of points on shells around an arbitrary, but fixed centre. For non-periodic point sets, where the shelling depends on the chosen centre, a more adequate quantity is the averaged shelling, obtained by averaging over points of the set as centres. For homogeneous media, such an average is still well defined, at least almost surely (in the probabilistic sense). Here, we present a two-step approach for planar model sets.

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