Diffraction from quasi-crystals and disordered twinned aggregates

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Following the discovery of systems with diffraction patterns exhibiting icosahedral symmetry by Shechtman, Blecht, Gratias & Cahn [Phys. Rev. Lett. (1984), 53, 1951–1953], which is forbidden in classical crystallography, interest has centred on Penrose tiling and its generalizations as a model for the explanation of these patterns, although Pauling [Nature (London) (1985), 317, 512–514; Phys. Rev. Lett. (1987), 58, 365–368] has proposed an alternative explanation in terms of crystal twinning. In an attempt to reconcile these two points of view a computer algorithm has been developed, following the methods outlined by Wolny, Pytlik & Lebech [J. Phys. C (1988), 21, 2267–2277], to grow a number of different structures which all tile the two-dimensional (2D) plane without defects. With the same two types of basic unit, it is shown that it is possible, on the one hand, to produce structures similar to the Penrose tiling pattern and, on the other, different types of twinned and disordered structures, with a continuous sequence of intermediates. Optical diffraction patterns of these structures have been obtained for comparison with real quasi-crystals.