Quasiperiodic crystals

A. Yamamoto, Crystallography of quasiperiodic crystals. Acta Crystallogr. A 52 (1996) 509. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

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 QC (a quasiperiodic crystal) is a form of soHd matter that exhibits order without periodicity. QCs are often associated with classically forbidden rotational symmetries, although strictly speaking, this is not a necessary feature [1]. In this chapter, we are concerned with QCs composed of metal atoms, although quasiperiodicity has also been discovered in block copolymers [4], Hquid crystals [5], and nanoparticle superlattices [6]. Many excellent reviews and books are available that provide an introduction to aU aspects of QCs [7-12]. 

Quasicrystals are solid materials exhibiting diffraction patterns with apparently sharp spots containing symmetry axes such as fivefold or eightfold axes, which are incompatible with the three-dimensional periodicity associated with crystal lattices. Many such materials are aluminum alloys, which exhibit diffraction patterns with fivefold symmetry axes such materials are called icosahedral quasicrystals. " Such quasicrystals " may be defined to have delta functions in their Fourier transforms, but their local point symmetries are incompatible with the periodic order of traditional crystallography. Structures with fivefold symmetry exhibit quasiperiodicity in two dimensions and periodicity in the third. Quasicrystals are thus seen to exhibit a lower order than in true crystals but a higher order than truly amorphous materials. 

Quasiperiodic tilings are most widely used for the description of quasi-crystals. With appropriate atomic decorations of the vertices, they serve as structure models which explain physical properties of quasi crystals [39]. FVom a theoretical point of view, they are idealisations of real substances on which the usual models of statistical physics like the Ising model may be studied [40-42]. Quasiperiodic tilings arose before the discovery of quasi-crystals, however, more as an object of aesthetic interest in geometry [43,44]. 

An approach that can be generally applied to various multilayer structures is the well-known transfer matrix technique [87, 180]. It can be used regardless of the geometry of the layers or the intended application of the multilayer. Besides being usable for the calculation of quarterwave Bragg mirrors, it can be used without modification to accurately determine the properties of antireflection coatings, step-down structures, various quasiperiodic, aperiodic and random stractures, but also 2D and 3D photonic crystals as well [241]... 

A quasicrystal (QC) is a type of solid that is well-ordered but not periodic. QCs are often associated with classically forbidden rotational symmetries, although strictly speaking, this is not necessary [1]. The discovery of QCs led to a refinement of the definition of a crystal as any soHd with an essentially discrete diffraction pattern. [2]. This transferred the definition of the concept from real to reciprocal space and in doing so broadened the scope of the term to encompass both periodic and quasiperiodic materials. Thus a QC is a nonperiodic crystalline material. 

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