Cubic closest packed holes

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. 

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). 

The stmcture of fluorite, CaF2, may be regarded as being composed of cubic closest packed Ca2+ cations with all the tetrahedral interstices occupied by F-anions, as shown in Fig. 10.1.6. The stoichiometric formula is consistent with the fact that the number of tetrahedral holes is twice that of the number of closest packed atoms. 

The structure of sodium chloride, which is the prototype for most of the alkali halides, is best described as a cubic closest packed array of Cl- ions with the Na+ ions in all of the octahedral holes [see Fig. 16.42(b)]. The relative sizes of these ions are such that rua 0.66i ci-> so this solid obeys the guidelines given previously. Note that the CP ions are forced apart by the Na+ ions, which are too large for the octahedral holes in the closest packed array of CP ions. Since the number of octahedral holes is the same as the number of packed spheres, all the octahedral holes must be filled with Na+ ions to achieve the required 1 1 stoichiometry. Most other alkali halides also have the sodium chloride structure. In fact, all the halides of lithium, sodium, potassium, and rubidium have this structure. Cesium fluoride has the sodium chloride structure but because of the large size of Cs+ ions, in this case the Cs ions form a cubic closest packed arrangement with the F ions in all the octahedral holes. On the other hand, cesium chloride, in which the Cs+ and CP ions are almost the same size, has a simple cubic structure of CP ions, with each Cs+ ion in the cubic hole in the center of each cube. The compounds cesium bromide and cesium iodide also have this latter structure. 

The compounds Na20, CdS, and Zrl4 all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case ... 

The first is that in the cubic CaF2 (and U2O), and ZnS structures the atomic positions are exactly those of cubic closest packing and certain of the tetrahedral interstices. In PfS and PbO this is not so. In the ideal PtS (PdO) structure, with c.c.p. metal atoms, S would be in regular tetrahedral holes but the four S atoms... 

In this structure (Fig. 6.1(a)) the A and X atoms alternate in a simple cubic sphere packing, each atom being surrounded by 6 others at the vertices of a regular octahedron. An alternative description, that the A (X) atoms occupy octahedral holes in a cubic closest packing of X (A) atoms is realistic only for compounds such as LiCl in which each Cl is actually in contact with 12 Cl atoms. This structure was derived by Barlow (1898) as a possible structure for crystals composed of... 

The positions of the Ca ions in the fluorite structure are those of cubic closest packing and the positions of the F ions correspond to all the tetrahedral holes. However, the ions which are actually in contact are the F ions (F—F, 2-70 A, twice the radius of F ) whereas the shortest distance between Ca " ions is 3 8 A, which may be compared with the radius of Ca (1-0 A). The structure is therefore... 

From the fact that the smallest kind of interstice between spheres in contact is a tetrahedral hole it follows that we should expect to find coordination polyhedra with only triangular faces, in contrast to those in, for example, cubic closest packing which have square in addition to triangular faces. Moreover, it seems likely that the preferred coordination polyhedra will be those in which five or six triangular faces (and hence five or six edges) meet at each vertex, since the faces are then most nearly equilateral. It follows from Euler s relation (p. 61) that for such a polyhedron, 1)5 -h Oug = 12, where 1)5 and are the numbers of vertices at which five or six edges meet, so that starting from the icosahedron (vg = 12) we may add 6-fold vertices to form polyhedra with more than twelve vertices. 

In fact, trigonal holes are so small that they are never occupied in binary ionic compounds. Whether the tetrahedral or octahedral holes in a given binary ionic solid are occupied depends mainly on the relative sizes of the anion and cation. For example, in zinc sulfide the ions (ionic radius = 180 pm) are arranged in a cubic closest packed structure with the smaller ions (ionic radius = 70 pm) in the tetrahedral holes. The locations of the tetrahedral holes in the face-centered cubic unit cell of the ccp structure are shown in Fig. 10.36(a). Note from this figure that there are eight tetrahedral holes in the unit cell. Also recall from the discussion in Section 10.4 that there are four net spheres in the face-centered cubic unit cell. Thus there are twice as many tetrahedral holes as packed anions in the closest packed structure. Zinc sulfide must have the same number of S ions and Zn ions to achieve electrical neutrality. Thus in the zinc sulfide structure only half the tetrahedral holes contain Zn ions, as shown in Fig. 10.36(c). 

The structure of sodium chloride can be described in terms of a cubic closest packed array of Cr ions with Na ions in all the octahedral holes. The locations of the octahedral holes in the face-centered cubic unit cell are shown in Fig. 10.37(a). The easiest octahedral hole to find in this structure is the one at the center of the cube. Note that this hole is surrounded by six spheres, as is required to form an octahedron. The remaining octahedral holes are shared with other unit cells and are more difficult to visualize. However, it can be shown that the number of octahedral holes in the ccp structure is the same as the number of packed anions. Figure 10.37(b) shows the structure for sodium chloride that results from Na ions filling all the octahedral holes in a ccp array of Cl ions. 

What is the formula for the compound that crystallizes with a cubic closest packed array of sulfur ions, and that contains zinc ions in of the tetrahedral holes and aluminum ions in of the octahedral holes ... 

A mineral crystallizes in a cubic closest packed array of oxygen ions with aluminum ions in some of the octahedral holes and magnesium ions in some of the tetrahedral holes. Deduce the formula of this mineral and predict the fraction of octahedral holes and tetrahedral holes that are filled by the various cations. 

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