Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell model of quasicrystals

Abstract

The quasi-unit cell picture describes the atomic structure of quasicrystals in terms of a single, repeating cluster which overlaps neighbors according to specific overlap rules. In this paper, we discuss the precise relationship between a general atomic decoration in the quasi-unit cell picture atomic decorations in the Penrose tiling and in related tiling pictures. Using these relations, we obtain a simple, practical method for determining the density, stoichiometry and symmetry of a quasicrystal based on the atomic decoration of the quasi-unit cell taking proper account of the sharing of atoms between clusters.

Looking for alternatives to the superspace description of icosahedral quasicrystals

“…It is known that the structure of quasicrystals may be described in terms of repeating quasi-unit cells in the same way as periodic crystals are described by unit cell translations [20]. A quasi-unit cell approach implies: choosing an appropriate set of basic tiles, decorating the basic tiles in a specific way, and then covering the space with identically decorated quasi-unit cells both with or without overlappings.…”

“…We believe that the tiling approach [20][21][22][23] could be a reasonable alternative to the superspace description, although it is also not free of controversy. Let us give a few quotations: 'Quasicrystals cannot be defined as packing of identical unit cells' [10]; 'The atomic decoration of the tiles may not be uniform all over the tiling' [24]; 'The problem of determining the quasi-lattice cannot be generally separated from determining the quasicrystal structure' [24].…”

Quasicrystals—The impact of N.G. de Bruijn

“…Further aspects of the decagon approach were discussed in [49,50]. This is followed by a number of studies that discuss clusters of different types of atoms and the related chemistry filling the decagon in with atoms, thus explaining how it comes about [51,52,53,54]. Even comparisons with experiments are made in some of these papers [51,55].…”

Growth of a One Dimensional Quasiperiodic Covering with Locally Determined Decorations

“…This new type of local growth algorithm may help us to answer the old puzzle of how quasicrystals grow with quasiperiodic order. Atomic structures of many quasicrystals, especially those that show high-quality quasiperiodic ordering, are well described by using quasi-unit-cell models [20,21,22] based on the covering with overlapping tiles. The overlap corresponds physically to the sharing of atoms by neighboring clusters.…”