Diffraction from quasicrystals: Geometric structure factor

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Abstract
A quasicrystal may be described by its quasilattice and its atomic decoration. We discuss two different commonly used methods for decorating quasilattices which lead to inequivalent real-space structures, the tile-decoration method and the hyperlattice-decoration-and-projection method. It is shown that diffraction patterns of such quasicrystals cannot be generally split into the intrinsic structure factor, due to the quasilattice, and the geometric structure factor, due to the decoration. For the hyperlattice decoration the zero-wave-vector limit cannot separate the quasilattice and the decoration contributions. However, such separation does occur for certain simple sequences of wave vectors and the tile-decorated quasilattices. We point to the ambiguities in choosing the ‘‘unit tiles’’ of a quasicrystal and we emphasize that the number of ‘‘atoms’’ per tile can be fractional. We have focused on two-dimensional pentagonal quasicrystals but some of our conclusions and results survive generalizations to other cases.

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