Octahedral and tetrahedral interstices

Octahedral and Tetrahedral Interstices in the Cubic Closest-packing... 

The required local charge balance between cations and anions which is expressed in Pauling s rule causes the distribution of cations and anions among the octahedral and tetrahedral interstices of the sphere packing. Other distributions of the cations are not compatible with Pauling s rule. 

Table III. Comparison of Octahedral and Tetrahedral Interstices in the Monohydrido Trianion of Compound 4 with Those in Nickel Metal and in NiH0 6...

The shortest cation- -cation distance R is between type I and type II cations, which occupy octahedral and tetrahedral interstices that share a common face. Thus for Rx < Re, there are bonding and antibonding hg bands associated with cation--cation bonds of a three-layer II—I—II block. The next-shortest distance R is within a type I plane, where the tetrahedral interstices share a common edge. If R2 < Rn there are bonding and antibonding ea bands associated with the type I planes. The third shortest distance Rz = II —II is between type II cations. Their octahedra share a common edge. Since Ri < Rz, the t20 electrons tend to concentrate in the Ri bonds, especially if Ri < Rc < Rz. There are three important cation-anion-cation interactions to be considered 180° IIt-anion-IIt, 125° IL-anion-I, and 90° II -anion-Il6. 

Various schemes have been proposed for the classification of the different alumina structures (Lippens and Steggerda, 1970). One approach was to focus attention on the temperatures at which they are formed, but it is perhaps more logical to look for differences in the oxide lattice. On this basis, one can distinguish broadly between the a-series with hexagonal close-packed lattices (i.e. ABAB...) and the y-series with cubic close-packed lattices (i.e. ABCABC...). Furthermore, there is little doubt that both y- and j/-A1203 have a spinel (MgAl204) type of lattice. The unit cell of spinel is made up of 32 cubic close-packed O2" ions and therefore 21.33 Al3+ ions have to be distributed between a total of 24 possible cationic sites. Differences between the individual members of the y-series are likely to be due to disorder of the lattice and in the distribution of the cations between octahedral and tetrahedral interstices. 

The structures that have been determined so far include those of the pure, electrically insulating Qo (ref. 5) and the heavily doped KfcCfto (ref. 6). At room temperature, solid forms a face-centred cubic (f.c.c.) lattice with 10.0 A intercluster separ-ation KeCftQ has a body-centred cubic structure, with K. atoms in distorted tetrahedral sites. No superconductivity was observed in that material. Cs6Cso has the same structure. We have found that the superconductor K3Q0 has a f.c.c. structure derived from that of C o by incorporating K ions into all of the octahedral and tetrahedral interstices of the host lattice. 

The picture of the closest sphere packing for the alum crystal (see Fig. 4.4) correlates to a rough educational reduction. The expert knows that the substance potassium aluminum sulfate dodecahydrate is composed of hydrated potassium ions, whose octahedral and tetrahedral interstices are filled with aluminum ions and with sulfate ions. This model concept may be introduced later and can be used in future lessons. Nevertheless, it is considered reasonable, at first, to introduce the alum particles and to choose reduced concrete models. 

The fee and hep arrangements offer both octahedral and tetrahedral interstices, making them good hosts for cations, since two size ranges can be incorporated. Both fee and hep arrangements can be stabilized by filling just the tet-... 

Note the Y-AI2O3 crystal structure is best described as a defect spinel structure comprised of a 2 X 2 X 2 fee array of oxide ions with 21/3 aluminum ions divided over the octahedral and tetrahedral interstices. By contrast, a-Al203 is an HCP array of 0 , with AP in 2/3 of the octahedral sites. For thermal transformatiems of alumina, see Stumpf, H. C. Russell, A. S. Newstane, J. W. Tucker, C. M. Ind. Eng. Chem. 1950,42, 1398. 

Other octahedral and tetrahedral interstices are located at positions within the unit cell that are equivalent to these representative ones. 

In three-dimensional closest packing, the spherical atoms are located in position 4(a). There are two types of interstices octahedral and tetrahedral holes which occupy positions 4(b) and 8(c), respectively. The number of tetrahedral holes is twice that of the spheres, while the number of octahedral holes is equal to that of the spheres. The positions of the holes are shown in Fig. 10.1.1. 

There are two types of interstices in the hep structure octahedral and tetrahedral holes, as in the ccp structure. However, the hep and ccp structures differ in the linkage of interstices. In ccp, neighboring octahedral interstices share edges, and the tetrahedral interstices behave likewise. In hep, neighboring octahedral interstices share faces, and a pair of tetrahedral interstices share a common face to form a trigonal bipyramid, so these two tetrahedral interstices cannot be filled by small atoms simultaneously. Figure 10.2.1 shows the positions of the octahedral interstices (a) and tetrahedral interstices (b). Table 10.2.1 lists the pure metals with hep structure and the parameters of the unit cell. 

In order to arrive at the important sodium chloride structure, one could look at and discuss the interstices in the cubic closest sphere packing model [2] interstices that are octahedral and tetrahedral can be found (see E5.8). If one fills the octahedral interstices with smaller spheres, one ends up with the sodium chloride structure (see Fig. 5.13). The larger spheres symbolize the chloride ions, the smaller spheres the sodium ions. The structure can be described as the cubic closest packing of chloride ions, whose octahedral interstices are completely filled by sodium ions. For other salt structures, only part of the octahedral sites are filled, as in aluminum oxide where the ratio of ions 2 3 applies [2],... 

A bcc lattice has two octahedral interstices and three pseudotetrahedral interstices per atom. The ccp and hep lattices have one octahedral and two tetrahedral interstitial sites per atom. In metals the interstitial octahedral and tetrahedral sites between the positive metal ions are not all fully occupied some of them are only... 

In close-packed arrays of anions X, both kinds of interstices, octahedral and tetrahedral ones, may simultaneously be occupied by cations M. In order to obtain a layer structure occupation of the tetrahedra and octahedra has to be such that periodically an empty layer alternates with a three-dimensional block. An example with only one kind of mixed layer is shown in Figure 25. 

There are four spheres, four octahedral interstices and eight tetrahedral interstices per unit cell. Therefore, their numerical relations are the same as for hexagonal closest-packing, as well as for any other stacking variant of closest-packings one octahedral and two tetrahedral interstices per sphere. Moreover, the sizes of these interstices are the same in all closest-packings of spheres. 

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. 

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