Structure icosahedral

The maximum number of rigid spheres that can be brought into contact with another sphere with the same radius is twelve. The corresponding coordination polyhedra, seen in the cubic and hexagonal closest-packed structures, have eight triangular faces and six square faces. 

There are many known structures of intermetallic compounds that involve icosahedral coordination about the smaller atoms. Usually these structures are complex, with 20, 52, 58, 162, 184, or more atoms in a cubic unit of structure. Many of the crystals are cubic. The icosahedron has 12 fivefold axes of symmetry, 20 threefold axes, and 30 twofold axes the fivefold axes cannot be retained in the crystal, but some of the others can be (a maximum of four threefold axes in a cubic crystal). 

A simple icosahedral structure 1 is that of MoAlx, WAlia, and (Mnr Cr)Alu. In this structure, based on a body-centered cubic lattice, 

Two other allotropic forms have been reported by Basinski and 

Christian,32 one with structure Al, stable from 1100° to 1130°C, and one with structure A2, stable from 1130° to 1240°C. The lattice constants, corrected to room temperature, correspond to R (L 12) about 1.30 A. Another form, obtained by quenching, has been assigned the A6 structure, with 8 bonds at 2.582 A and 4 at —2.669 A, corresponding to R(L 12) — 1.306 A. The calculated valence for the manganese atoms in these three forms is 4.5 hence these atoms may be considered to be similar to the larger atoms in a-manganese and 0-manganese. 

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Miehle W, Kandler O, Leisner T and Echt O 1989 Mass spectrometric evidence for icosahedral structure in large rare gas clusters Ar, Kr, Xe J. Chem. Phys. 91 5940... 

Figure 16.6 A T = 3 icosahedral virus structure contains 180 subunits in its protein shell. Each asymmetric unit (one such unit is shown in thick lines) contains three protein subunits A, B, and C. The icosahedral structure is viewed along a threefold axis, the same view as in Figure 16.5. One asymmetric unit is shown in dark colors.

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

Our optimized cluster geometries agree well with other reported results for most part. One noticeable exception, however, is the case of Niia for which we find a capped distorted trigonal anti-prism structure with a central atom to be the most stable geometry, while the previously proposed distorted icosahedral structure was found to be unstable. We note that no full geometry unrestricted Optimization was done in obtaining the structure of Ref. 13. 

Fig. 5. Limiting stable arrangements of spherons around a central spheron. (Left) Nine outer spherons (KM structure) (right) 12 outer spherons (icosahedral structure).

Figure 4.16 From the left to the right Planar structure for Auio, tetrahedral structure for AU20 and icosahedral structure for W Au -]2.

In Chapter 4, the icosahedral structure of the B12 molecule was shown. Although all of the symmetry elements of a molecule having this structure will not be enumerated, the symmetry type is known as Ih. 

A black, crystalline form of boron having a density of 2.34 g/cm3 results. Boron exists in unit cells that have an icosahedral structure that has 20 faces that are equilateral triangles meeting at 12 vertices with a boron atom at each vertex, as shown in Figure 13.1. 

In order to describe derivatives of B12 or B12H122 having icosahedral structures, it is necessary to have a way to designate positions of atoms or substituent groups. In order to do that, the positions are identified by a numbering system that is illustrated as shown in Figure 13.5. 

Figure 3 Clusters with perfect icosahedral structure one shell (N = 13), two shells (N = 55), and three shells (N = 147).

Amongst the cioso-hydroborate anions the closo-dodecaborate B 2H 22 is the best known because many derivatives can be made by maintaining its icosahedral structure in substitution reactions. Thus the replacement of its hydrogen atoms... 

An additional piece of information provided by this calculation is that the mean square displacement of the gold atoms on all the sites except for the PPhj-coordinated sites are primarily, though not exclusively, radial. The mean square displacement of the gold atoms on the PPh 3-coordinated sites, on the other hand, have an in-plane (i.e. tangential) mean-square displacement which is not only larger than in the radial direction [94, 100], but also exhibits significant inplane anisotropy. This may be related to an incompletely developed soft mode in the cluster [101], which might be considered as a precursor to a transition from the observed close-packed structure to the icosahedral structure, predicted theoretically by Sawada et al. [102]. 

Energy calculations for small clusters of atoms indicate that a cluster of 55 atoms should be reasonably stable (Mackay 1962, Allpress and Sanders 1970, Hore and Pal 1972). In addition, calculations suggest that the 55-atom cluster will take up an icosahedral structure in preference to the cubic cuboctahedral structure (Hore and Pal 1972). 

The larger atoms, magnesium, have ligancy 16 (12 Cu and 4 Mg), as seen in Figure 11-13. This increase in ligancy for magnesium, relative to that (12) for the element, and the assumption of icosahedral coordination by copper cause a decrease in volume of the compound relative to the elements. The volume decrease is 6.7 percent. A part of it may be the result of electron transfer (Sec. 11-12), but similar decreases are found in general for icosahedral structures. 

Fig 4. Frontal view uf truncated icosahedral structure of Cm cluster... 

The structure of water is related to the structures of hexagonal boat-form and chair-form ice that exist at atmospheric pressure. The structure may be folded, in three dimensions, to form an icosahedral network, based on the regular arrangement of 14-molecule units. Twenty of the 14-molecule units, comprising in total 280 molecules of water, may form a 3nm diameter icosahedral structures which may be strained with increased size (Doye and Wales, 2001). Icosahedron means a solid figure having twenty faces. The stability of the network is finely balanced, being able to fluctuate between an expanded low density structure and a more dense... 

Figure 4 Calculated binding energies of Ag and Pd clusters as a function of cluster size for (a) fc.c. and (b) icosahedral structures (after Baetzold and Hamilton31)...

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