We have constructed a generalized Penrose tiling by the cut-and-project method and compared its structure with that of the regular Penrose tiling. We derived the structure factor of the generalized Penrose tiling and applied it to calculate the diffraction pattern of a non-decorated structure.
The generalized Penrose tiling is an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter sın łangle 0,1). The formula for structure factor, calculated within the average unit cell approach, works in physical space only and is directly dependent on the s parameter. It allows to straightforwardly change the long-range order of the refined structure just by changing the s parameter and keeping the tile decoration unchanged. The possibility and viability of using the shift as one of the refinement parameters during structure refinement was tested for a numerically generated simple binary decagonal quasicrystal.
Quasicrystals are aperiodic structures with no periodicity both in direct and reciprocal space. The diffraction pattern of quasicrystals consists however of the periodic series of peaks in the scattering vector space. The intensities of the peaks of all series reduced in a proper way build up the so-called envelope function common for the whole pattern. The Fourier transformed envelope gives the average unit cell which is the statistical distribution of atomic positions in physical space. The distributions lifted to high dimensions correspond to atomic surfaces - the basic concept of structural quasicrystals modeling within higher-dimensional approach.
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