Tetrahedral holes

Compounds that have the empirical formulas MCr02 and DCr204 where M is a monovalent and D a divalent cation, are known as chromites. These are actually mixed oxides and probably are better written as M20-Cr203 and D0-Cr203, respectively. The oxides of D are largely spinels, ie, the oxygen atoms define a close-packed cubic array having the octahedral holes occupied by the Cr(III) cation and the tetrahedral holes occupied by D (54). Chromite ore is an important member of this class of oxides. 

The higher solubility of carbon in y-iron than in a-iroii is because the face-ceiiued lattice can accommodate carbon atoms in slightly expanded octahedral holes, but the body-centred lattice can only accommodate a much smaller carbon concentration in specially located, distorted tetrahedral holes. It follows that the formation of fenite together with cementite by eutectoid composition of austenite, leads to an increase in volume of the metal with accompanying compressive stresses at die interface between these two phases. 

Fig. 16.1. Ionic ceramics, (a) The rocksalt, or NoCl, structure, (b) Magnesia, MgO, has the rocksalt structure. It can be thought of as an f.c.c. packing with Mg ions in the octahedral holes. ( ) Cubic zirconia ZrOj an f.c.c. packing of Zr with O in the tetrahedral holes, (d) Alumina, AljOj a c.p.h. packing of oxygen with Al in two-thirds of the octahedral holes.

Fig. 16.2. Both the f.c.c. and the c.p.h. structures are close-packed. Both contain one octahedral hole per atom, and two tetrahedral holes per atom. The holes in the f.c.c. structures ore shown here.

This "packing" argument may seem an unnecessary complication. But its advantage comes now. Consider cubic zirconia, ZrOj, an engineering ceramic of growing importance. The structure (Fig. 16.1c) looks hard to describe, but it isn t. It is simply an f.c.c. packing of zirconium with the ions in the tetrahedral holes. Since there are two tetrahedral holes for each atom of the f.c.c. structure, the formula works out at ZrOj. 

There are many other ionic oxides with structures which are more complicated than these. We will not go into them here. But it is worth knowing that most can be thought of as a dense (f.c.c. or c.p.h.) packing of oxygen, with various metal ions arranged, in an orderly fashion, in the octahedral or the tetrahedral holes. 

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

Many metals have close-packed structures, with the atoms stacked in either a hexagonal or a cubic arrangement close-packed atoms have a coordination number of 12. Close-packed structures have one octahedral and tivo tetrahedral holes per atom. 

When the radius ratio of an ionic compound is less than about 0.4, corresponding to cations that are significantly smaller than the anion, the small tetrahedral holes may be occupied. An example is the zinc-blende structure (which is also called the sphalerite structure), named after a form of the mineral ZnS (Fig. 5.43). This structure is based on an expanded cubic close-packed lattice of the big S2 anions, with the small Zn2+ cations occupying half the tetrahedral holes. Each Zn2+ ion is surrounded by four S2 ions, and each S2" ion is surrounded by four Zn2+ ions so the zinc-blende structure has (4,4)-coordination. 

FIGURE 5.43 Hie zinc-blende (sphalerite) structure, rhe tour zinc ions (pink) form a tetrahedron within a face-centered cubic unit cell composed of sulfide ions (vellow).The zinc ions occupy half the tetrahedral holes between the sulfide ions, and the parts or the unit cell occupied by zinc ions are shaded blue. The detail shows how each zinc ion is surrounded by four sulfide ions each sulfide ion is similarly surrounded by four zinc ions. 

A commonly occurring mineral has a cubic unit cell in which the metal cations, M, occupy the corners and face centers. Inside the unit cell, anions, A. occupy all the tetrahedral holes created by the cations. What is the empirical formula of the M Aj compound ... 

Tetrahedral and octahedral interstitial holes are formed by the vacancies left when anions pack in a ccp array, (a) Which hole can accommodate the larger ions (b) What is the size ratio of the largest metal cation that can occupy an octahedral hole to the largest that can occupy a tetrahedral hole while maintaining the close-packed nature of the anion lattice (c) If half the tetrahedral holes are occupied, what will be the empirical formula of the compound MVAV, where M represents the cations and A the anions ... 

Ziegler-Natta catalyst A stereospecific catalyst for polymerization reactions, consisting of titanium tetrachloride and triethylaluminum. zinc-blende structure A crystal structure in which the cations occupy half the tetrahedral holes in a nearly close packed cubic lattice of anions also known as sphalerite structure. 

According to the calculation of Fig. 7.2 (p. 53), a sphere with radius 0.225 fits into the tetrahedral hole enclosed by four spheres of radius 1. 

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. 

Spinels have a crystal structure in which there is a face-centered cubic arrangement of O2 ions. There are two types of structures in which cations have octahedral or tetrahedral arrangements of anions surrounding them. In the spinel structure, it is found that the +3 ions are located in octahedral holes and the tetrahedral holes are occupied by the +2 ions. A different structure is possible for these ions. That structure has half of the +3 metal ions located in the tetrahedral holes while the other half of these ions and the +2 ions are located in the octahedral holes. In order to indicate the population of the two types of lattice sites, the formula for the compound is grouped with the tetrahedral hole population indicated first (the position normally occupied by the +2 ion, A) followed by the groups populating the octahedral holes. Thus, the formula AB204 becomes B(AB)04 in order to correctly... 

Only two oxides of cobalt have been characterized, CoO and Co304 (which is actually ConConl04). The latter has a structure in which Co2+ ions are located in tetrahedral holes and Co3+ ions are located in octahedral holes of a spinel structure. Decomposition of either Co(OH)2 or CoC03 produces CoO, and decomposition of Co(N03)2 can be used to produce Co304. 

In many of the transition metals, such as titanium, vanadium and molybdenum, carbon, nitrogen and oxygen atoms can fit into octahedral holes, and hydrogen into the tetrahedral holes. The fit here is estimated by assuming the atoms all have incompressible radii, and the contact must be such that the interstitial atoms do not rattle around in the holes. 

Similar considerations may be made with reference to the other simple close-packed structure, that is to the hexagonal Mg-type structure. In this case two basic derived structures can be considered the NiAs type with occupied octahedral holes and the wurtzite (ZnS) type with one set of occupied tetrahedral holes (compare with the data given with an origin shift in 7.4.2.3.2). For a few more comments about interstices and interstitial structures see 3.8.4. See Fig. 3.35. 

The importance of the geometrical factor in determining the stability of these phases has been pointed out (Pearson 1972). In a simplified description, Laves phases AM2 of the MgCu2 type may be presented as cubic face-centred packing of large spheres A which form tetrahedral holes that are occupied by tetrahedra of smaller spheres M. The ideal value of the radius ratio rA/rM is 1.225. The values experimentally observed for the various Laves types range from 1.05 to 1.7. 

In the schemes of Fig. 7.10, typical sections of a few adjacent cells of this structure are shown these are also compared with those of a number of related hexagonal structures, some of which are described in the following paragraphs. Notice that important filled-up derivatives can be considered among the ordered structures derived from Mg. Typical examples are the hP4-NiAs type with occupied octahedral holes and the wurtzite (hP4-ZnS) type with one set of occupied tetrahedral holes. 

Sphalerite and wurtzite structures general remarks. Compounds isostructural with the cubic cF8-ZnS sphalerite include AgSe, A1P, AlAs, AlSb, BAs, GaAs, InAs, BeS, BeSe, BeTe, BePo, CdS, CdSe, CdTe, CdPo, HgS, HgSe, HgTe, etc. The sphalerite structure can be described as a derivative structure of the diamond-type structure. Alternatively, we may describe the same structure as a derivative of the cubic close-packed structure (cF4-Cu type) in which a set of tetrahedral holes has been filled-in. This alternative description would be especially convenient when the atomic diameter ratio of the two species is close to 0.225 see the comments reported in 3.7.3.1. In a similar way the closely related hP4-ZnO... 

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