Holes in closest packing

Figure 3.35. Position of the holes in closest packing. Unit cell projections are shown for the cubic and hexagonal sphere closest packing. Coordinates of the spheres and of the tetrahedral and octahedral holes are given. The values indicated inside the drawing correspond to the third coordinate (along the vertical axis) when two values are given, these correspond to two positions along the same vertical line.

There are three types of holes in closest packed structures ... 

A FIGURE 23.8 Octahedral Holes in Closest-Packed Crystals Octahedral holes are found in a closest-packed stmcture. The octahedral hole is surrounded by six of the atoms in the closest-packed structure. 

We noted earlier that certain pairs of ions of similar size can together form c.p. layers AX3 and that of the two possible AX3 layers with non-adjacent A atoms only one is found in complex halides and oxides A B3,X3 ,c- This layer (Fig. 4.15(d)) can be stacked in closest packing to form octahedral Xg holes between the layers without bringing A ions into contact. Figure 4.30 shows two such layers and the positions of octahedral coordination for the B atoms between the layers. The number of these Xg holes is equal to the number of A atoms in the structure. All of these positions or a proportion of them may be occupied in a complex halide or oxide by cations B carrying a suitable charge to give an electrically neutral crystal A jB X3 e, where A and X are, for example, and F , Cs" and Cl , Ba and 0 ", etc. The formula depends on the proportion of B positions occupied ... 

In any crystal structure, the close-packed or closest-packed planes are the lowest energy planes. On all other planes, the density of atoms is lower, and the interatomic distance and the energy of the plane are greater. Contrary to intuitive expectations, the diameter of the largest holes or interstices between atoms in the close-packed f.c.c. structure is considerably greater than the diameter of the largest interstices between atoms in the non-close-packed b.c.c. structure. 

In the following, we start by assuming purely ionic structures. In spinel the oxide ions form a cubic closest-packing. Two-thirds of the metal ions occupy octahedral interstices, the rest tetrahedral ones. In a normal spinel the A ions are found in the tetrahedral interstices and the M ions in the octahedral interstices we express this by the subscripts T and O, for example Mgr[Al2](904. Since tetrahedral holes are smaller than octahedral holes, the A ions should be smaller than the M ions. Remarkably, this condition is not fulfilled in many spinels, and just as remarkable is the occurrence of inverse spinels which have half of the M ions occupying tetrahedral sites and the other half occupying octahedral sites while the A ions occupy the remaining octahedral sites. Table 17.3 summarizes these facts and also includes a classification according to the oxidation states of the metal ions. 

I shall take the simple view that most metal oxide structures are derivatives of a closest packed 02 lattice with the metal ions occupying tetrahedral or octahedral holes in a manner which is principally determined by size, charge (and hence stoichiometry) and d configuration (Jj). The presence of d electrons can lead to pronounced crystal field effects or metal-metal bonding. The latter can lead to clustering of metal atoms within the lattice with large distortions from idealized (ionic) geometries. 

Figure 3.34. Holes in the closest packing of equal spheres. Two superimposed layers of spheres are shown (continuous and dotted lines). Tetrahedral (T) and octahedral (O) holes are indicated.

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. 

On page 528 the statement is made that "crystalline [Naclosest packed, large, complex cations with sodide anions in the octahedral holes (Fig. 2-50a). Review the discussion of closest packing in Chapter A and label the atoms in Fig. 12.50 as belonging to layers A and B and indicate the octahedral holes. 

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). 

Consider a closest packed lattice of sulfide ions. Zinc ions tend to occupy tetrahedral holes in such a framework since they are quite small (74 pm) compared with the larger sulfide ions (170 pm) ff the sulfide ions form a ccp array, the resulting structure is zinc blende if they form an hep array, the resulting structure is wurtzite. See Fig. 4.15. 

The use of radius ratios to rationalize structures and to predict coordination numbers may be illustrated as follows.27 Consider beryllium sulfide, in which rn,a /r5i- = 59 pm/170 pm = 0.35. We should thus expect a coordination number of 4 as the Be2+ ion fits most readily into the tetrahedral holes of the closest packed lattice, and indeed this is found experimentally BcS adopts a wurtzite structure. 

In the same way we can predict that sodium ions will prefer octahedral holes m a closest packed lattice of chloride ions (rNB Acr 116 pm/167 pm = 0.69), forming the well-known sodium chloride lattice with a coordination number of 6 (Fig. 4.1a). 

Consider the molecule CIFjOj (with chlorine the central atom). How many isomers are possible Which is the most stable Assign point group designations to each of the isomers. 6.id The Structure for AliBr (Fig. 6. Ih) is assumed by both Al2Br6and ALCUin the gas phase. In the solid, however, the structures can best be described as closest packed arrays of halogen atoms (or ions) with aluminum atoms (or ions) in tetrahedral or octahedral holes. In solid aluminum bromide the aluminum atoms arc found in pairs in adjacent tetrahedral holes. In solid aluminum chloride, atoms are found in one-lhird of the octahedral holes... 

The conductivity may be compared with that of a 35% aqueous solution of sulfuric acid. 0.8 fl-1 cm-1. The structure consists of a complex (not a simple closest-packed) arrangement of iodide ions with Rb ions in octahedral holes and Ag+ ions in tetrahedral holes. Of the 56 tetrahedral sites available to the Ag+ ions, only 16 are occupied, leaving many vacancies. The relatively small size of the silver ion (114 pm) compared with the rubidium (166 pm) and iodide (206 pm) ions give the silver ion more mobility in the relatively rigid latice of the latter ions. Furthermore, the vacant sites are arranged in channels, down which the Ag+ can readily move (Fig. 7.16). 

The isopoly anions may be considered to be portions of a closest packed array of oxide ions with the metal ions occupying the octahedral holes. The edge-sharing array found in [V n02S]6 consists of ten octahedra stacked as shown in Fig. 16.10a. This seems to be the largest stacked-octahedral isopoly anion cluster compatible with metal-metal repulsions, and the remaining edge-shared structures represent portions of this unit.46... 

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