Quasicrystals penrose tiling pattern

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. 

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 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. 

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]. 

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