Quasicrystals fivefold symmetry

The other important symmetry-related discovery was the quasicrystals. Both the truncated icosahedral structure of buckminsterfullerene and the regular but nonperiodic network of the quasicrystals are related to fivefold symmetry. In spite of this intimate connection between them at an intellectual level, their stories did not cross. The conceptual linkage between them is provided by Fuller s physical geometry and this is also what relates them to the icosahedral structure of viruses (see, Section 9.5.2 on Icosahedral Packing). 

F. Denoyer, X-Ray Diffraction Study of Slowly Solidified Icosahedral Alloys. Ini. Hargittai, ed., Quasicrystals, Networks, and Molecules of Fivefold Symmetry, VCH, New York, 1990, pp. 69-82. 

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. 

Quasicrystals, Networks, and Molecules of Fivefold Symmetry, VCH, New York, 1990. 

Figure Bl.8.6. An electron diffraction pattern looking down the fivefold symmetry axis of a quasicrystal. Because Friedel s law introduces a centre of symmetry, the symmetry of the pattern is tenfold. (Courtesy of L Bendersky.)...

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. 

Quasicrystals represent the third type of aperiodic materials. Quasiperiodicity may occur in one, two, or three dimensions of physical space and is associated with special irrational numbers such as the golden mean r = (1 -h /5)/2, and = 2 -F V3. The most remarkable feature of quasicrystals is the appearance of noncrystal-lographic point group symmetries in their diffraction patterns, such as 8/mmm, lO/mmm, l2/mmm, and 2lm35. The golden mean is related to fivefold symmetry via the relation r = 2 cos( r/5) r can be considered as the most irrational number, since it is the irrational number that has the worst approximation by a truncated continued fraction,... 

Notice that rotation symmetry only exists for n = l, 2, 3, 4, and 6 five- and sevenfold rotations are not allowed because bodies with these symmetries cannot fill all space for the same reason that you cannot tile a floor with pentagons or with septagons. Icosahedral quasicrystals with fivefold symmetry can form, but cannot grow into crystalline solids in the strict sense of the word. Even so, such quasicrystals are extremely interesting both theoretically as well as from applications that utilize their unusual properties, as discussed later. 

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

A pecuhar sohd phase, which has been discovered not too long ago [172], is the quasi-crystalline phase. Quasi-crystals are characterized by a fivefold or icosahedral symmetry which is not of crystallographic type and therefore was assumed to be forbidden. In addition to dislocations which also exist in normal crystals, quasi-crystals show new types of defects called phasons. Computer simulations of the growth of quasicrystals [173] are still somewhat scarce, but an increasing number of quasi-crystalline details are studied by simulations, including dislocations and phasons, anomalous self-diffusion, and crack propagation [174,175]. 

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