Quasicrystals modeling

The activated character of the dependence tj(T) shown in (1.123) is often considered as a feature suggesting a quasicrystal model of the liquid. Data taken from liquid-vapour co-existence curves are frequently analysed in coordinates In tj from 1/T in order to determine t/o- The point that tj(n, T) is a function not only of the temperature T, but also of the density n is ignored. The density along the co-existence curve is... 

There are three known thermodynamically stable blue phases BPi and BPjj, and BPin (or the foggy phase). The structure of the first two is already established BPj is a body-centered cubic phase (symmetry group O or /4i32) and BPu is simple cubic (symmetry group or P4232) [42]. The foggy phase, BPm, can probably be described using one of the quasicrystal models [43]. 

In this paper we will briefly review the findings of an earlier study of the properties of quasicrystals using SHG and present further experimental data to probe their structure and nonlinear optical properties. Finally, a model describing the behavior of quasicrystals in an external electric field is developed. 

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. 

Here the principles of constructing a 3D structure model from several HREM images of projections of inorganic crystals will be presented. Some of the principles may also be applied to non-periodic objects. A complex quasicrystal approximant v-AlCrFe is used as an example (Zou et al., 2003). Procedures for ab initio structure determination by 3D reconstruction are described in detail. The software CRISP, ELD. Triple and 3D-Map are used for 3D reconstruction. The 3D reconstruction method was demonstrated on the silicate mineral (Wenk et al. 1992). It was also applied to solve the 3D structures of a series mesoporous materials (Keneda etal. 2002). 

Recently "quasicrystals having the shape of a triacontahedron have been discovered in specially prepared alloys of aluminum and other metals. A triacontahedron e a regular polyhedron with 30 identical, diamond-shaped feces (Fig. 3.35). Quasicrystals seemingly defy the rules of symmetry that do not allow a periodic structure having unit cells with five-fold symmetry. What is the symmetry of a triacontahedron Can you make a model of it similar to the polyhedra given in Appendix H ... 

The dependence of anion exclusion volume on particle thickness in the infinite-plane model arises solely from the geometric decrease in exclusion-specific surface area with an increase in particle thickness via quasicrystal formation [Eq. (13)]. Thus, on the infinite-plane model, the anion exclusion volume simply has an inverse relation to particle thickness. Reduction of the anion exclusion volume from an increase in particle thickness is, however, more complicated and more significant on the disk model (Fig. 6). An increase in particle thickness (or the number of unit layers... 

FIG. 6 Comparison between MGC theory (open symbols) and the disk model (filled symbols) for the dependence of Vcx on the electrical double layer thickness [Eq. (14)]. The number of unit layers in a quasicrystal is a fixed parameter for each set of curves [34],... 

Therefore, in addition to reducing the anion exclusion-specific surface area on the basal plane (as in the infinite-plane model), an increase in particle thickness can increase the area for positive anion adsorption on the edge surface. In the vicinity of a positively charged edge surface of a thick particle, a large quantity of anions can be so adsorbed. The net anion exclusion thus may decrease dramatically, and even become negative (i.e., net positive anion adsorption), as the average number of unit layers in a quasicrystal, nav, exceeds 4 (i.e., ho > 2.5 nm) for dilute to moderate electrolyte concentrations (Fig. 6). Thus, a monotonic decrease in anion exclusion... 

Alan Mackay made the connection with crystallography [139], He designed a pattern of circles based on a quasi-lattice to model a possible atomic structure. An optical transformation then created a simulated diffraction pattern exhibiting local tenfold symmetry (see, in the Introduction). In this way, Mackay virtually predicted the existence of what was later to be known as quasicrystals, and issued a warning that such structures may be encountered but may stay unrecognized if unexpected ... 

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A related model for quasicrystal structures has also been presented by Aslanov. ... 

The above discussion considers a model for the stmcture of the crystalline cubic R-AlsCuTis. Now consider possible perturbations in this model to give the quasicrystalline T2-Al5CuTi3. In this coimection, the 54-vertex Mackay icosahedron (Figure 2) appears as a structural unit in certain quasicrystals. The Mackay icosahedron has a shell stmcture consisting of the following layers ... 

This model is a member of a series of structures, all built on a simple principle of interpenetration of polyhedra of five-fold point symmetry. The simplest member of the family is shown in Fig. (2.18(a)). Higher members, formed by successive interpenetrations of icosahedra fill space (Fig. 2.18(b)) and a perfectly ordered quasicrystal structure is obtained, there is no need to use oblate or prolate polyhedra they are generated by a single structure building principle [24]. 

We think that the structural principles that imderlie quasicrystals (discussed in Chapter 2) are relevant in order to describe the superstructure of tropocollagen molecules, which must accommodate the wide variety of tissue structures that can be formed. There are no possibilities of packing triple helices in a space-filling arrangement. A structure with five-fold symmetry in the plane and with perfect periodicity in the perpendicular direction, however, which is consistent with the quarter stagger pentafibril model, is in fact a quasicrystal of so-called T-type (also known as the decagonal phase). 

The structure of amorphous metals, quasicrystals, and crystalline inter-metallic compounds can be modelled by atom clusters with icosahedral arrangement [3.113-117]. The differences between the various phases result from a different arrangement of the individual atom clusters. Therefore, it is evident that there exists a close relation between the different states of matter, and that the different phases corresponding to minima of the free enthalpy can be quite easily transformed into each other. For example, rapid cooling from the melt results in an amorphous alloy for high quenching rates, and a quasicrystalline... 

Figure 8.26 Icosahedra, arranged with the same orientation, but not on a lattice, can be taken as one model of a quasicrystal alloy...

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,... 

Both quasicrystals icosahedral Al-Pd-Mn (i-AlPdMn) and Al-Cu-Fe (i-AlCuFe) that belong to the F-phase are best modelled through a single model of the atomic positions that we introduce in this section. 

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. 

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