Quasicrystals tiling

In the same paper (Yamamoto 1996) an authoritative description is given of several interrelated topics such as super-space group determination, structure determination, indexing of diffraction patterns of quasicrystals, polygonal tiling, icosahedral tiling, structure factor calculation, description of quasicrystal structures, cluster models of quasicrystals. 

In structure, the quasicrystal relates to the Penrose tile structures (polygon), originally proposed by Roger Penrose, a mathematician at Oxford University. See Crystal. 

Oblate and prolate rhombohedrons that can be combined to form three-dimensional tiling necessary for a quasicrystal. 

Electron diffraction pattern of an AIMn quasicrystal along the fivefold axis (left) and a computed Fourier pattern of a three-dimensional Penrose tiling (right). From C. Janot, Quasicrystals, A Primer, 2nd ed. (London Oxford Univ. Press, 1994), p. 3, figure 1.24. 

Figure 1.11 Portion of a Penrose tiling based on two rhombuses. Penrose tilings are nonperiodic tilings of the plane and are two-dimensional analogs of quasicrystals. (Diagram created by the free Windows application Bob s Rhombus Walker, v. 3.0.19, JKS Software, Stamford, CT.)...

Nevertheless, it must be pointed out that Pauling s explanation has not been generally accepted. Indeed, alternative and widely accepted rationalizations invoking such concepts as Penrose tiling and icosahedral crystals in hyperspaces have been advanced to explain the anomalous existence of the quasicrystals [95-103]. 

In this chapter, the concepts associated with classical crystallography are gradually weakened. Initially the effects of introducing small defects into a crystal are examined. These require almost no modification of the ideas already presented. However, structures with enormous unit cells pose more severe problems, and incommensurate structures are known in which a diffraction pattern is best quantified by recourse to higher dimensional space. Finally, classical crystallographic ideas break down when quasicrystals are examined. These structures, related to the Penrose tilings described in Chapter 2, can no longer be described in terms of the Bravais lattices described earlier. 

Quasicrystals can also be considered as three-dimensional analogues of Penrose tilings. 

Quasicrystals or quasiperiodic crystals are metallic alloys which yield sharp diffraction patterns that display 5-, 8-, 10- or 12-fold symmetry rotational axes - forbidden by the rules of classical crystallography. The first quasicrystals discovered, and most of those that have been investigated, have icosahedral symmetry. Two main models of quasicrystals have been suggested. In the first, a quasicrystal can be regarded as made up of icosahedral clusters of metal atoms, all oriented in the same way, and separated by variable amounts of disordered material. Alternatively, quasicrystals can be considered to be three-dimensional analogues of Penrose tilings. In either case, the material does not possess a crystallographic unit cell in the conventional sense. 

We studied the structure of the bulk terminations according to a model M ) of icosahedral quasicrystals of an F-phase (see Refs. [3-5]). The model in a physical space is based on an icosahedral tiling [6] projected from the Dg,... 

The model is a superposition of three icosahedral quasilattices, q, a and b, of atomic positions in the physical space E, as explained in A model of icosahedral quasicrystals based on the tiling 7 (2r) , These are defined in the Table 12-1. 

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

Fig. 9.1. Translational symmetiy radsting in crystals and its lack in quasi-crystals despite a perfect long-range order, (a) The translational symmetiy in the NaCl crystal build of Na" " and Cl ions (b) the Penrose tiling as an example of a 2-D quasicrystaL without translational symmetry (e) a medieval Arabian mosaic as an example of a long-range, non-translatimial order.

A very effective demonstration of enhanced plasmonic effects from laterally illuminated MWCNTs has been demonstrated by the diffraction patterns observed from a 2D Penrose tiled quasicrystal structure [29]. This stmcture has been seen... 

M.A. Kaliteevski, S. Brand, R.A. Abram, T.E. Krauss, R. De La Rue, P. Millar, Two-dimensional Penrose-tiled photonic quasicrystals from diffraction pattern to band structure. Nanotechnology 11, 274 (2000)... 

Nanoparticles can be ordered at the detector surface or elsewhere using self-assembly techniques or pattering by top-down approach. They can be distributed in regular patterns, thus accurately controlling the interparticle distance or they can be randomly scattered. A possible way to implement nanoparticles for plasmonic enhancement is to arrange them in a quasicrystal pattern (for instance Penrose tiling), which ensures an isotropic photonic response of the strucmre [325]. 

Fig. 3 Quasicrystalline structures observed in BCP SA. (a) TEM image of a 2D 12-fold quasicrystal derived from a star block copolymer (the inset shows its EFT pattern), (b) Transcribed tiling pattern (reprinted with permission from [34] Copyright 2010 American Physics Society), (c) TEM image and (d, e) unit cell of the Frank-Kasper sigma phase obtained from tetra-BCP SA (c-e reprinted with permission from [33] Copyright 2010 AAAS)...

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