Closest packing arrangement

In Activity 4.1, students described and modeled crystalline solids according to the way their atoms packed into a regular structure. We found that their atoms could pack into face-centered, body-centered, or hexagonal-closest arrangements. Another way to understand the structure of a crystalline solid is to consider the bonding forces between its structural units. 

Several comments are warranted about the apatite structure before its description is proffered. As noted in Table 1, the atoms lie on or near four (00/) planes in the atomic arrangement. Ca2, P, 01, 02, and the X anion (where X= F, OH, Cl) lie on (or are disordered about) special positions on the mirror planes at z = V4 and /4. Intercalated approximately halfway between these planes are Cal (in a special position at z = 0, -1/2) and 03, in the general position with z values of -0.07 and -0.57. On the basis of the layer structure of the atomic arrangement, O Keeffe and Hyde (1985) noted the similarity between the cation positions in the apatite structure and the MnsSis intermetallic phase, and offered a description of apatite as a ca//on-closest-packed atomic arrangement. Dai et al. (1991) and Dai and Harlow (1991) elucidated that intriguing cation-closest packing relationship in an examination of arsenate apatites, and their work is worthy of detailed review. 

The adherence to close-packed structural arrangements lends support to the idea that these compounds can be used as models for metal surface chemistry—with respect to chemisorbed species and their mobility and reactions of substrates on these surfaces. It also indicates a marked deviation from the behavior of boranes and their derivatives. Structures based upon some polyhedra favored by boron, such as the pentagonal bipyramid, triangulated dodecahedron, and especially the icosahedron, are absent so far in metal-carbonyl cluster chemistry. In this connection, it has been mentioned that [M(CO)3],g compounds should be the closest analogs to On skeletal electron counting... 

Further, attention may be drawn to the fact that all curves converge to a point at which Voc C is about 0.80. Theoretically this should be 0.74, corresponding to the closest arrangement which can exist on packing rigid spheres (one surrounded by 12). This may be due to the fact that the coils are not rigid spheres. 

In the hexagonal closest packed (hep) arrangement, shown in Figure 5.3c, the coordination consists of six atoms in the same plane as the atom under consideration plus three atoms from both of the adjacent planes, making a total of 12. 

I have found that the assumption that in atomic nuclei the nucleons are in large part aggregated into clusters arranged in closest packing leads to simple explanations of many properties of nuclei. Some aspects of the closest-packing theory of nuclear structure are presented in the following paragraphs.1... 

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.

The nature of spheron-spheron interactions is such that maximum stability is achieved when each spheron ligates about itself the maximum number of neighbors, to produce a nucleus with a closest-packed structure. A simple argument (12) leads to the conclusion that the spherons in a nucleus are arranged in concentric layers. The packing radius of a spheron varies from 1.28 f for the dineutron to 1.62 f for the helion. The radius (to nucleon density half that of the inner region) of the largest nucleus is 6.8 f... 

In crystalline C60 the molecules have a face-centered cubic arrangement, i.e. they are packed as in a cubic closest-packing of spheres as they are nearly spherical, the molecules spin in the crystal. The crystals are as soft as graphite. Similar to the intercalation com-... 

In a-B12 the icosahedra are arranged as in a cubic closest-packing of spheres (Fig. 11.16). In one layer of icosahedra every icosahedron is surrounded by six other icosahedra that are linked by three-center two-electron bonds. Every boron atom involved contributes an average of electrons to these bonds, which amounts to -6 = 4 electrons per icosahedron. Every icosahedron is surrounded additionally by six icosahedra of the two adjacent layers, to which it is bonded by normal B-B bonds this requires 6 electrons per icosahedron. In total, this adds up exactly to the above-mentioned 10 electrons for the inter-icosahedron bonds. 

The cluster condensation can be carried on the chains of octahedra sharing edges can be joined to double-strands and finally to layers of octahedra (Fig. 13.18). Every layer consists of metal atoms in two planes arranged in the same way as two adjacent layers of atoms in a closest-packing of spheres. This is simply a section from a metal structure. The X atoms occupy positions between the metal layers and act as insulating layers. Substances like ZrCl that have this structure have metallic properties in two dimensions. 

As in ionic compounds, the atoms in a binary intermetallic compound show a tendency, albeit less pronounced, to be surrounded by atoms of the other kind as far as possible. However, it is not possible to fulfill this condition simultaneously for both kinds of atoms if they form a closest-packed arrangement. For compositions MXn with n < 3 it cannot be fulfilled for either the M or the X atoms in every case every atom has to have some adjacent atoms of the same kind. Only with a higher content of X atoms, beginning with MX3 (n > 3), are atomic arrangements possible in which every M atom is surrounded solely by X atoms the X atoms, however, must continue to have other X atoms as neighbors. 

The CsCl type offers the simplest way to combine atoms of two different elements in the same arrangement as in body-centered cubic packing the atom in the center of the unit cell is surrounded by eight atoms of the other element in the vertices of the unit cell. In this way each atom only has adjacent atoms of the other element. This is a condition that cannot be fulfilled in a closest-packing of spheres (cf. preceding section). 

In cubic closest-packing, consideration of the face-centered unit cell is a convenient way to get an impression of the arrangement of the interstices. The octahedral interstices are situated in the center of the unit cell and in the middle of each of its edges [Fig. 17.3(a)], The octahedra share vertices in the three directions parallel to the unit cell edges. They share edges in the directions diagonal to the unit cell faces. There are no face-sharing octahedra. 

Relative arrangement of the octahedra in hexagonal and in cubic closest-packing in the direction of stacking of the hexagonal layers... 

We focus attention here on the binary compounds MX, the X atoms being arranged in a closest-packed manner and the M atoms occupying the octahedral interstices. Since the... 

---