M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 68
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Courtesy ofDorothy Hodgkin [37].9.5.1. Sphere PackingThe most efficient packing results in the greatest possible density.The density is the fraction of the total space occupied by the packingunits. Only those modes of packing are considered here in which eachsphere is in contact with at least six neighbors. The densities of somepackings are given in Table 9-5. There are stable arrangements withTable 9-5. Densities of Sphere PackingaCoordination numberName of packing6Simple cubic8Simple hexagonal8Body-centered cubic10Body-centered tetragonal12Closest packingaAfter A. F. Wells, Structural Inorganic Chemistry, 5thClarendon Press, Oxford, 1984.Density0.52360.60460.68020.69810.7405Edition,9.5.
Dense Packing443smaller numbers of neighbors, meaning lower coordination numbers,when directed bonds are present. In our discussion, however, the existence of chemical bonds is not a prerequisite at all.For three-dimensional six-coordination, the most symmetricalpacking is when the spheres are at the points of a simple cubic lattice(Figure 9-26a). Each sphere is in contact with six others situatedat the vertices of an octahedron. In order to increase clarity, theatoms are shown separated in the figure.
The packing is more realistically represented when the spheres touch each other. Already Kepler(Figure 9-9), and later Dalton (Figures 9-10, 9-24), employed suchrepresentations.(a)(b)(c)(d)Figure 9-26. Examples of sphere packing after Wells [39]. Reproduced withpermission. (a) Simple cubic; (b) The somewhat distorted cubic packing of arsenic;(c) Simple hexagonal; (d) Body-centered cubic.The structure of crystalline arsenic provides an example of somewhat distorted simple cubic packing.
It is illustrated in Figure 9-26b.The atoms are in the positions of the cubic structure. Each has threenearest and three more distant neighbors. The layers formed by thenearest bonded atoms may also be derived from a plane of hexagons.These layers buckle as the bond angle decreases from 120◦ .4449 CrystalsThe simple hexagonal sphere packing is shown in Figure 9-26c.The coordination number is eight. It is not very important forcrystal structures. Figure 9-26d shows the body-centered packing with8-coordination. For the central atom the six next nearest neighborsare at the centers of neighboring unit cells.
In terms of polyhedraldomains, a truncated octahedron is adopted here. The central atom, infact, has a coordination number of 14.It may often be convenient to describe the crystal structure interms of the domains of the atoms [40]. The domain is the polyhedron enclosed by planes drawn midway between the atom and eachneighbor, these planes being perpendicular to the lines connecting theatoms.
The number of faces of the polyhedral domain is the coordination number of the atom and the whole structure is a space-fillingarrangement of such polyhedra.The closest packing of equal circles on a plane surface has alreadybeen considered. The closest packing of spheres on a plane surfaceposes a similar problem. Again, the densest arrangement is whena sphere is in contact with six others. Layers of spheres may thenbe superimposed in various ways. The closest packing is when eachsphere touches three others in each adjacent layer, the total numberof contacts then being 12. Closest packing is thus based on closestpacked layers. Figure 9-27 illustrates this. The spheres in one layer areFigure 9-27.
Closest packing of ABC layers after Wells [41]. Reproduced withpermission.9.5. Dense Packing445labeled A, and a similar layer can be placed above the first so that thecenters of the spheres in the upper layer are vertically above the positions B (or C). The third layer can be placed in two ways. The centersof the spheres may lie above either the C or the A positions. The twosimplest sequences of layers are then ABABAB ... and ABCABC ...They will have the same density (0.7405).The packing based on the sequence ABABAB... is called hexagonalclosest packing and is illustrated by Figure 9-28a.
Each sphere has12 neighbors situated at the vertices of a coordination polyhedron.The packing based on the sequence ABCABC... is called cubic closestpacking. It is illustrated in Figure 9-28b, and is characterized by cubicsymmetry.(a)(b)Figure 9-28. Close packing of spheres after Shubnikov and Koptsik [42].
(a)Hexagonal closest packing; (b) Cubic closest packing. Used with permission.The closest packing of equal spheres is achieved in an arrangement in which each sphere touches three others in each adjacent layer.The total number of neighbors is then 12. Although the packing inany layer is evidently the densest possible packing, this is not necessarily true of the space-filling arrangements resulting from stackingsuch layers. Thus, consider the addition of a fourth sphere to the mostclosely packed triangular arrangement [43]. The maximum number ofcontacts is three in the emerging tetrahedral group. The space-fillingarrangement would require each tetrahedron to have faces commonwith four other tetrahedra. However, regular tetrahedra are not suitable to fill space without gaps or overlaps because the angle of thetetrahedron, 70◦ 32 , is not an exact submultiple of 360◦ .Alternatively, continue placing spheres around a central one, allspheres having the same radius.
The maximum number that can be4469 Crystalsplaced in contact with the first sphere is 12. However, there is a littlemore room around the central sphere than just for 12, but not enoughfor a 13th sphere. Because of the extra room there is an infinite numberof ways of arranging the 12 spheres [44].The question of densest packing of spheres has been an intriguingproblem in mathematics for centuries; it has been labeled “one of theoldest math problems in the world” [45].9.5.2.
Icosahedral PackingThe most symmetrical arrangement is to place the 12 spheres at thevertices of a regular icosahedron, which is the only regular polyhedron with 12 vertices. Thus, the icosahedral packing is the mostsymmetrical. However, it is not the densest packing. Also, it is not acrystallographic packing in terms of classical crystallography. Whenicosahedra are packed together they will not form a plane, but willgradually curve up and will eventually form a closed system as isillustrated in Figure 9-29 [46].Figure 9-29. Icosahedral polyoma virus drawn after Adolph et al.
[47].Buckminster Fuller recognized early the importance of icosahedral construction and its great stability in geodesic shapes as well asin viruses. He may have not had the rigorous scientific bases aboutnucleic acids and about the viruses, but had a fertile imagination andconnected seemingly distant pieces of information about structures.This is what he wrote [48]:This simple formula governing the rate at whichballs are agglomerated around other balls or shells9.5.
Dense Packing447in closest packing is an elegant manifest of thereliably incisive transactions, formings, and transformings of Universe. I made that discovery in thelate 1930s and published it in 1944. The molecular biologists have confirmed and developed myformula by virtue of which we can predict thenumber of nodes in the external protein shells ofall the viruses, within which shells are housed theDNA-RNA-programmed design controls of all thebiological species and of all the individuals withinthose species. Although the polio virus is quitedifferent from the common cold virus, and bothare different from other viruses, all of them employfrequency to the second power times ten plus twoin producing those most powerful structural enclosures of all the biological regeneration of life.
It isthe structural power of these geodesic-sphere shellsthat makes so lethal those viruses unfriendly toman. They are almost indestructible.Indeed, the discoverers of virus structures, Donald Caspar andAaron Klug stated thatthe solution we have found ... was, in fact,inspired by the geometrical principles appliedby Buckminster Fuller in the construction ofgeodesic domes ... The resemblance of the designof geodesic domes ... to icosahedral viruses hadattracted our attention at the time of the polioviruswork ...
Fuller has pioneered in the developmentof a physically orientated geometry based on theprinciples of efficient design [49].The length of an edge of a regular icosahedron is some 5% greaterthan the distance from the center to vertex. Thus, the sphere ofthe outer shell of 12 makes contact only with the central sphere.Conversely, if each sphere of an icosahedral group of 12, all touchingthe central sphere, is in contact with its 5 neighbors, then the centralsphere must have a radius of some 10% smaller than the radius ofthe outer spheres. The relative size considerations are important in the4489 Crystalsstructures of free molecules as well if the central atom or group ofatoms is surrounded by 12 ligands [50].An interesting case, and a step forward from the isolated moleculetowards more extended systems is when an icosahedron of 12 spheresabout a central sphere is surrounded by a second icosahedral shellexactly twice the size of the first [51]. This shell will contain 42spheres and will lie over the first so that spheres will be in contactalong the fivefold axes.
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