Icosahedral packing

For icosahedral packing, the transition from an inner core of one spheron to one of two spherons would be expected to take place between V = 90 and N = 92. The effect of the shell structure (completed mantle at 31 rather than 32 spherons) may explain why the transition occurs over the range 88 to 92 rather than more sharply at 90 to 92. 

A core of 13 spherons with an inner core of a single spheron (icosahedral packing) may occur for N = 90 (45 spherons), but not for N s 92. The energy values reported by Barber et al.4 indicate that the change to prolate deformation is about half completed at N = 90. Possibly these nuclei show resonance between the prolate structure and the icosahedral spherical structure.5... 

The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

I have assumed that this equation applies to structures with two or more spheres in the central layer (as well as with one. as in icosahedral packing), and have applied it in the calculation of the ranges of values of the neutron number N in which successive subsubshells are occupied (12). (In this calculation the difference in radius of the different kinds of spherons is taken into consideration.) The assignment of quantum numbers is made with use of the following assumptions (14) ... 

The arrangement of 22 spherons around an inner tetrahedron of four spherons shown in Fig. 7 involves icosahedral packing each of the four inner spherons is surrounded by an icosahedron of 12, three of which are the three other inner spherons. This structure (26 spherons, 52 neutrons) with one spheron missing may be assigned to magic number 50. The complete structure, with 26 spherons, corresponds to the stable nucleus as discussed in the following section. 

The most stable core configuration for an inner core of one spheron is KM, with an outer core of nine spherons. For icosahedral packing the outer core contains 12 spherons. These structures, which we may take as defining the limits of stability for a one-spheron inner core, have been discussed in the preceding section. 

The other important symmetry-related discovery was the quasicrystals. Both the truncated icosahedral structure of buckminsterfullerene and the regular but nonperiodic network of the quasicrystals are related to fivefold symmetry. In spite of this intimate connection between them at an intellectual level, their stories did not cross. The conceptual linkage between them is provided by Fuller s physical geometry and this is also what relates them to the icosahedral structure of viruses (see, Section 9.5.2 on Icosahedral Packing). 

Figure 9-30. Icosahedral packing of spheres showing the third shell [53], This is popularly called the Mackay icosahedron. ...

FIG. 4.4. (a) Relation between cuboctahedron and icosahedron, (b) Icosahedral packing of equal spheres, showing the third layer (n = 3). (AC 1962 15 916). 

If all the B atoms of each icosahedron were to form a sixth bond to a B atom of another icosahedron there would be formed a radiating icosahedral packing which is not periodic (Chapter 4). The structure of rhombohedral Bios n y be visualized as built of units (Bs4)(Bi oBBi 0). The Bg4 unit consists of an icosahedron bonded to... 

We also choose to study two of the clusters that do not have an icosahedral global minimum LJ38 and LJ75. For these clusters the second-lowest energy minimum is based upon icosahedral packing and we expect these surfaces to have two funnels one that leads to the low-energy icosahedral minima and one that leads to the global minimum. 

Because icosahedral packing seems to be somewhat less favorable energetically than cuboctahedral, such particles could be transformed to the more stable f.c.c. packing, whereas the opposite transformation is hardly possible under HREM, ED experimental conditions. This seems to be a realistic explanation of the disagreement found, and the occurence of at least some part of the icosahedral (or multiply-twinned) metal cores is beyond any doubt. 

The most symmetrical arrangement is to place the 12 spheres at the vertices of a regular icosahedron, which is the only regular polyhedron with 12 vertices. Thus, the icosahedral packing is the most symmetrical. However, it is... 

An interesting case, and a step forward from the isolated molecule toward more extended systems, is when an icosahedron of 12 spheres about a central sphere is surrounded by a second icosahedral shell exactly twice the size of the first [9-31]. This shell will contain 42 spheres and will lie over the first so that spheres will be in contact along the fivefold axes. Further layers can be added in the same fashion. The third layer is shown in Figure 9-30 as an example of icosahedral packing of equal spheres. The layers of spheres succeed each other in cubic close-packing sequence on each triangular face. Each sphere which is... 

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