Pinwheel patterns and powder diffraction

Baake, Michael; Frettlöh, Dirk and Grimm, Uwe (2007). Pinwheel patterns and powder diffraction. Philosophical Magazine, 87(18-21) pp. 2831–2838.

DOI: https://doi.org/10.1080/14786430601057953

Abstract

Pinwheel patterns and their higher dimensional generalizations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. We present first steps and results towards a general framework to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with a toy model for the powder diffraction of the square lattice.

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