A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns

Details A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns

2006-10-12

arxiv.org/abs/math/0610408v1

Journal of Geometry and Physics 57 (2007) 1331-1343

Mathematics

Spectral Theory

16 pages, 8 figures

Scientific paper

10.1016/j.geomphys.2006.10.009

Diffraction images with continuous rotation symmetry arise from amorphous systems, but also from regular crystals when investigated by powder diffraction. On the theoretical side, pinwheel patterns and their higher dimensional generalisations display such symmetries as well, in spite of being perfectly ordered. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. An alternative substitution rule for the pinwheel tiling, with two different prototiles, permits the derivation of several combinatorial and spectral properties of this still somewhat enigmatic example. These results are compared with properties of the square lattice and its powder diffraction.

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