Perfect icosahedra

Since we will consider polymer chains with lengths in the icosahedral regime, it is instme-tive to review properties of perfect icosahedra. The surface of the icosahedron consists of 

Interpreting the process of structure formation of a small system of LJ particles as a conformational transition, it is particularly interesting to consider the structural behavior at the surface as it is for these system sizes more relevant than the bulk effects. In the thermodynamic limit, of course, the phase transition will be driven mainly by the crystal formation in the bulk (fee structure in the case of LJ particles). However, the systems we are going to study in this chapter are so small that the general aspects of nucleation transitions for very large systems are not valid anymore. The crystallization of small systems depends extremely on the precise system size - a lesson that has already been taught in Section 5.2 -and this is due to surface effects. 

For this reason it is instructive to study the surface-to-volume ratio of these icosahedral structures. As can be read from the numbers listed in Table 6.1, the surface-to-volume ratio rsv = not surprisingly decreases with the number of particles A , but 

The number of particles in the bulk is easier to calculate by the recursive relation 

In the limit of large systems ( oo), N and A A. Thus, in this limit, the surface-to-volume ratio vanishes like 

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Note that the structures depicted in Fig. 5 are not self-similar because the angle of rotation of the faces differs for each layer. The layers should, therefore, not be called shells as they are called in the case of pure alkaline earth-metal clusters. With increasing size, the shape of the cluster will converge asymptotically to that of a perfect icosahedron. 

These are shown in Figure 14.22. The B12H.12 anion, known officially as the dodecahydrododecaborate anion, is a perfect icosahedron and is one of the most stable molecular ions known. It is also highly water-soluble and incredibly heat-resistant. Table 14.6 shows the relationship between the closo, nido, and arachno structural classes starting with four of the closo dianions, (n = 6, 7,... 

Assuming perfect stoichiometric structures, the stabilization of the boron frameworks of MB2, MB4, MBg, MBj2 and elemental B requires the addition of two electrons from each metal atom. Whatever the Bj2 unit, icosahedron or cubooctahe-dron, 26 electrons are required for internal bonding and 12 for external bonding. Since the 12 B possesses only 36 electrons, the metal must supply two electrons to each Bi2 group. The results for YB,2 are consistent with this model measurements indicate that one electron per Y is delocalized in the conduction band. ... 

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

The structure necessitated the throwing of all caution to the wind (the Greek icosahedron) and it was proposed immediately by Kroto, Heath. O Brien. Curl, and Smalley Nature. 318. 162 (1987). After all, it was surely too perfect a solution to be wrong. We named Cm after Buckminster Fuller, which has turned out to he a highly appropriate name."... 

In Chapter 2 (Section 2.9) we see how the cluster bonding requirements for the icosahedron, plus two-center and three-center inter-cluster bonds perfectly uses the three available valence electrons and four available valence orbitals in a covalently bonded cluster network. Once one has these advanced bonding models in hand, then the explanation of the B network structure is no more difficult than that of the C diamond structure. One purpose of this text is to provide these advanced models, but for now the solution to the problem remains hidden. Hey, a little suspense always helps the story line. At this empirical stage of the presentation you have learned that the nature of bonding (distribution of electrons) is expressed in geometry. The tricky bit is to interpret the empirical nuclear position in terms of a useful (simplest one that answers the question asked) model for the distribution of valence electrons. 

The tetrahedron is one of the perfect Platonic bodies, which are shown in Figure 4.3 and are known by the names tetrahedron, octahedron, cube, dodecahedron, and icosahedron. These bodies were thought by the Greeks to symbolize the fom elements and a fifth one called Quintessence. The tetrahedron symbolized Fire, the octahedron Air, the cube Earth, and the icosahedron Water. According to this philosophy, the... 

As creator, God is unable to make a perfect world with an imperfect material. Thus, Platon believed that all the substances are composed by air (octahedral units), earth (eubie units), fire (tetrahedral units) and water (icosahedron units). 

The Cgo structure, the most stable of this family, is a perfect truncated icosahedron, a geometric solid having 20 6-membered rings and the considerable symmetry of the point group I), 6 C5 axes, 10 C3 axes, and 15 Q axes. The symmetry was first established by measuring the C NMR spectrum of in which only one peak was observed, verifying that all 60 carbon atoms were chemically equivalent. 

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