Quasicrystal defined

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. 

Figure 25 A QSD projection on the AX — AR plan for nine quasicrystals (numbering given in Table 1 of Villars ). The dashed lines show segments of the boimdary of the AX — AR zone defining the quasicrystal conditions based on cI162 compounds. (Reprinted with permission from K.M. Rabe, J.C. Phillips, P. Villars, and I.D. Brown, Phys. Rev., 1992, B45, 7650. American Physical Society. http //pra.aps.org)...

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. 

Molecular quantum potential and non-local interaction depend on molecular size and the nature of intramolecular cohesion. Macromolecular assemblies such as polymers, biopolymers, liquids, glasses, crystals and quasicrystals are different forms of condensed matter with characteristic quanmm potentials. The one property they have in common is non-local long-range interaction, albeit of different intensity. Without enquiring into the mechanism of their formation, various forms of condensed matter are considered to have well-defined electronic potential energies that depend on the nuclear framework. A regular array of nuclei in a structure such as diamond maximizes cohesive interaction between nuclei and electrons, precisely balanced by the quantum potential, almost as in an atom. 

A quasicrystal is a quasiperiodic structure that is ordered but aperiodic, i.e., it lacks translational order. Its unit cell has defined angles and distances with respect to other unit cells but long-range translational order is missing. In addition to intriguing structural characteristics, the transport properties of quasicrystals are expected to be unique due to the missing translational order. For example, Man et al. showed that photonic quasicrystals are excellent candidates for photonic band gap materials, where the existence of photonic waves is forbidden in photonic band gap ranges [35]. 

The concept of quaternion groups i.s useful for the understanding of icosahedral quasicrystals and shows how the existence of icosahedral quasicrystals is a natural consequence of the use of quaternions to represent symmetry groups. In this connection a real quaternion is defined as an ordered quadruple of four real numbers w,x, y, z). subject to the following rules of addition and multiplication where q = (w, x, y, z) and q = (w, r. /, c ) ... 

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