Icosahedral model

The fivefold symmetry discovered by Shechtman is modelled in terms of the stacking of icosahedra and the term icosahedral symmetry is sometimes used. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

Hybrid density functional calculations have been carried out for AU-O2, Au-CO, Aui3, AU13-O2, Au -CO, AU13-H2, and AU55 clusters to discuss the catalytic behavior of Au clusters with different sizes and structures for CO oxidation [179]. From these calculations, it was found that O2 and CO could adsorb onto several Au model systems. Especially, icosahedral Aun cluster has a relatively weak interaction with O2 while both icosahedral and cubooctahedral Aui3 clusters have interactions with CO. These findings suggest that the surfaces of the Au clusters are the active sites for the catalytic reactions on the supported and unsupported Au catalysts. 

Ludwig s (2001) review discusses water clusters and water cluster models. One of the water clusters discussed by Ludwig is the icosahedral cluster developed by Chaplin (1999). A fluctuating network of water molecules, with local icosahedral symmetry, was proposed by Chaplin (1999) it contains, when complete, 280 fully hydrogen-bonded water molecules. This structure allows explanation of a number of the anomalous properties of water, including its temperature-density and pressure-viscosity behaviors, the radial distribution pattern, the change in water properties on supercooling, and the solvation properties of ions, hydrophobic molecules, carbohydrates, and macromolecules (Chaplin, 1999, 2001, 2004). 

A special group of particles that are often produced are the icosahedral (I5) and decahedral (D5) structures shown in Fig. 9. These particles have a fivefold symetry axis which is forbidden for infinite crystals. Yang (1 0) has described these particles using a non-Fcc model. The particles are composed by five (D5) and twenty (I5) tetrahedral units in twin relationship. However the units have a non-Fcc structure. The decahedral is composed by body-centered orthorhombic units and the icosahedral by rhombohedral... 

Particles of face-centered cubic metals of diameter 5 nm of more have been studied extensively by high resolution electron microscopy, diffraction and other methods. It has been shown that such particles are usually multiply twinned, often conforming approximately the idealized models of decahedral and icosahedral particles consisting of clusters of five or twenty tetrahedrally... 

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. 

Rayment, I. (1983). Molecular replacement method at low resolution optimum strategy and intrinsic limitations as determined by calculations on icosahedral virus models. Acta Crystallogr. A 39,102-116. 

The Hu and Bentley model is the only one that tries to describe VLP production and assembling in baculovirus infected insect cells [105]. Nevertheless, regarding VLP assembly, the formalism presented is completely theoretical and based on the assembly pathway of icosahedral viruses. From a process development point of view, this model does not generate enough output to make it applicable to bioreaction operational parameters definition. However, it can be used as the basis for a more structured approach to the VLP assembling process in baculovirus infected insect cells. 

The canyons are depressions approximately 15 to 20 A deep that encircle each icosahedral five-fold axis (Figure 1). When first seen in HRV 14, these canyons were postulated to be the site at which a cellular receptor would bind. Subsequent electron-microscopic data revealed that ICAM-1 does indeed bind in the canyon as predicted, although in a somewhat different orientation than early models [22,23]. These canyons allow the receptor binding sites to escape immunological surveillance because the canyons are too narrow to allow an immunoglobulin to contact the canyon floor. Directly underneath the floor of the canyon lies a second important structure, the VP1 hydrophobic pocket. 

The boron hydrides, including the polyhedral boranes, heteroboranes, and their metaUa derivatives, encompass an amazingly diverse area of chemistry. This class contains the most extensive array of structurally characterized cluster compounds known. Included here are many novel clusters possessing idealized molecular geometries ranging over every point group symmetry from identity (C[) to icosahedral (I[). Because boron hydride clusters may be considered in some respects to be progenitorial models of metal clusters, their development has provided a framework for the development of cluster chemistry in... 

Simple tetrahedral or octahedral models are useful in connection with basic structural questions (as, for example, the first time you try to convince yourself that the two enantiomers of CHFCIBr or of [Co(en)3]+3 are really nonsuperimposable). If stick models are rot available, such simple models can be constructed in a few minutes from paper. In addition, models having bond angles not normally found in ball-and-stick kits—for example, the icosahedral boranes and carboranes—can also be reudily constructed from paper. Paper models are especially useful when large numbers of models are necessary as, for example, in constructing models of the iso-end heteropolyanions. 

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