Closest-Packed Structures

One of the driving forces for crystallization is the maximum occupancy of space. This is particularly true for metallic solids, whose crystalline structures can often be considered as the packing of identically sized spheres. Metallic bonding is considered to be nondirectional. Unlike the nonmetals, which can fill their valence shells by sharing only a few pairs of electrons, the metals require much larger coordination numbers in order to satisfy their outermost valence shells. As a result, the metals 

The bottoms of the closest-packed spheres in the second (red) layer will occupy indentations in the first (black) layer. In this case, all of the upward-pointing indentations are occupied. 

Because each of the corner A atoms of a hep unit cell are shared between eight different unit cells, while the intervening 6 atom is contained within a single unit cell, the hep contains a total of two atoms (one A and one 6) per unit cell. 

Example 11 -2. Prove that the atoms of both the cubic and hep lattices occupy the same amount of space and that they both pack with 74% efficiency. 

Solution. Let us first consider the case of the cep or face-centered cubic unit cell. The closest contact between atoms in this crystalline lattice occurs along the face-diagonals of a cube, as shown in the given diagram. 

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The Distribution of Spherons in Layers.—Several theoretical and empirical arguments indicate that the nature of spheron-spheron interactions is not such as to limit the ligancy of a spheron to a fixed value, but that, instead, maximum stability is achieved when each spheron ligates about itself the maximum number of neighbors aggregates of spherons, like aggregates of argonon (noble-gas) atoms or metal atoms, assume a closest-packed structure. 

Figure 5.15 Anion-centered polyhedron (rhombic dodecahedron) found in the cubic closest-packed structure (a) oriented with respect to cubic axes, the c axis is vertical (b) oriented with [111] vertical (c) cation positions occupied in the sodium chloride, NaCl, structure and (id) cation positions occupied in the zinc blende (sphalerite) cubic ZnS structure.

Figure 5.16 Anion-centered polyhedron found in the hexagonal closest-packed structure (a) oriented with the hexagonal c axis vertical (b) cation positions occupied in the ideal corundum, A1203, structure (c) cation positions occupied in the idealized rutile, Ti02, structure and (d) cation positions occupied in the wurtzite hexagonal ZnS structure the central cation is omitted for clarity.

AH and AS correspond to half of the values derived from the slope of the line in Fig. 28. A comparison of the dissociation data of PdO to those reported for Ir02 and Ru02 is given in Table 7. Ru02l which lias the closest packed structure, also is the most stable oxide. 

Some metals crystallize in more than one structural type, which means that there are two alio tropic modifications. The metals marked do not conform precisely to the closest-packed structure, but deviate slightly from it. Uranium, manganese, gallium and indium have very abnormal structures, and the last two are transitional between metallic and non-metallic elements of the carbon group. The picture presented by the metallic structures is utterly different from that of elements of the four last groups of the periodic system. The homopolar bonds of these latter strive to produce a state in which the number of neighbours of each atom is determined by its valency. In the other elements, however, forces appear to be acting that tend to surround each atom with as many other atoms as possible. 

Closest-packed Structures.—If the stability of a metal crystal were determined by the number of bonds formed at a minimum interatomic distance, with no contribution of longer bonds, the structures with closest packing would be the most stable for the metallic elements. These structures, which have been described in the preceding section, involve contact between each atom and the 12 nearest neighbor. (The next interatomic distances are 41 percent larger and presumably have little significance.)... 

The close approximation of metal atoms in these crystals to mutually attracting spheres is further shown by the values observed for the axial ratio c/aot the hexagonal closest-packed structures, as tabulated below. 

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. 

The synthesis of macroscopic amounts of buckybail led to the study of many interesting properties of this molecule which continue unabated as this hook goes to press.44 The C6t) molecule is nearly spherical, and while the molecules themselves pack nicely in a cuhic closest packed structure, each molecule has essentially the free rotation of a ball bearing, and because of this disorder the structure could not be determined at room temperature.43 Being nearly spherical and lacking bond polarities... 

This brings us to a class of compounds too often overlooked in the discussion of simple ionic compounds the transition metal halides. In general, these compounds (except fluorides) crystallize in structures that are hard to reconcile with the structures of simple ionic compounds seen previously (Figs. 4.1-4.3). For example, consider the cadmium iodide structure (Fig. 7.8). It is true that the cadmium atoms occupy octahedral holes in a hexagonal closest packed structure of iodine atoms, but in a definite layered structure that can be described accurately only in terms of covalent bonding and infinite layer molecules. 

If all the packing atoms are no longer neutral (e.gn half are cations and half are anions), the closest packed structures are no longer the most stable, as can be seen from the similar two-dimensional case (see above). However, these structures may still be useful when considered as limiting cases for certain ionic crystals. Consider lithium iodide, in which the iodide anions are so much larger than the lithium cations that they may be assumed to touch or nearly touch. They can be considered to provide the framework for the crystal. The much smaller lithium ions can then fit irto the small interstices between the anions. If they expand the lattice slightly to remove the anion-anion contact, the anionic repulsion will be reduced and the crystal stabilized, but the simple model based on a closest packed system of anions may still be taken as the limiting case and a useful approximation. 

Where the lithium ions fit best will be determined by their size relative to the iodide ions. Note from above that there are two types of interstices in a closest packed structure. These represent tetrahedral (f) and octahedral (o) holes because the coordination of a small ion fitted into them is either tetrahedral or octahedral (see Fig. 4.12). The octahedral holes are considerably larger than the tetrahedral holes and can accommodate larger cations without severe distortion of the structure. In lithium iodide the lithium ions fit into the octahedral holes in a cubic closest packed lattice of iodide ions. The resulting structure is the same as found in sodium chloride and is face-centered (note that face-centered cubic and cubic closest packed describe the same lattice). 

Wells, A. F. Structural Inorganic Chemistry, 5th ed4 Clarendon Oxford, 1984. For a comprehensive and detailed discussion of (he broad usefulness of classifying structures in terms of closest-packed structures, see Douglas, B. E, McDaniel. D. H. Alexander. J. J. Concepts and Models of Inorganic Chenustry, 2od cd Wiley New York. 1983 pp 198-208. 

With larger cations, such as cesium, the radius ratio (rCl /rcl-= 181 pm/ 167 pm = 1.08) increases beyond the acceptable limit for a coordination number of 6 the coordination number of the cations (and anions) increases to 8, and the cesium chloride lattice (Fig. 4.1b) results. As we have seen, although this is an efficient structure for cations and anions of about the same size, it cannot be directly related to a closest packed structure of anions... 

Mg crystallizes in the hexagonal closest packed structure and has a density 1.74 g/cm3. (a) What is the volume of the unit cell (b) What is the distance between nearest neighbors (c) How many nearest neighbors does each atom have ... 

The 17 rare-earth metals are known to adopt five crystalline forms. At room temperature, nine exist in the hexagonal closest packed structure, four in the double c-axis hep (dhep) structure, two in the cubic closest packed structure and one in each of the body-centered cubic packed and rhombic (Sm-type) structures, as listed in Table 18.1.1. This distribution changes with temperature and pressure as many of the elements go through a number of structural phase transitions. All of the crystal structures, with the exception of bep, are closest packed, which can be defined by the stacking sequence of the layers of close-packed atoms, and are labeled in Fig. 18.1.1. 

Besides the mobility on the cluster surface in solution the ball-like character of the huge molecules obviously prevents crystallization. If we assume a closest packed structure for the M55 cluster sphere in a crystal, then there would exist octahedral lattice vacancies with a diameter of 8-9 Al There is some hope to find ligands causing inter-molecular interactions and thus facilitating crystallization. 

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