Lattices octahedral holes

KEY TERMS face-centered cubic crystal lattice octahedral hole... 

Figure 2.9 Illustration of a porous TiCl particle in which the porous particles are made up of TijCl units containing lattice octahedral holes providing porosity on the molecular level. Reprinted from [44] with permission from Elsevier Publishing.

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. 

Therefore the relationship between these interconvertible structures originates from a cubic anion lattice of 32 0 ions in the cell. With 32 Fe ions in the octahedral holes stoichiometric FeO is formed. Replacement of a number of Fe ions with two-thirds of their number of Fe ions maintains electrical neutrality but provides non-stoichiometric Fei 0. Continual replacement in this way to leave 24 Fe atoms in the cubic cell produces Fej04, and... 

Such simple considerations led Scholten and Konvalinka to confirm the form of the dependence of the reaction velocity on the pressure, as had been observed in their experiments. Taking into account a more realistic situation, on the polycrystalline hydride surface with which a hydrogen molecule is dealing when colliding and subsequently being dissociatively adsorbed, we should assume rather a different probability of an encounter with a hydride center of a /3-phase lattice, an empty octahedral hole, or a free palladium atom—for every kind of crystallite orientation on the surface, even when it is represented, for the sake of simplicity, by only the three low index planes. 

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. 

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 ... 

Figure 5.18.1 The NaCl crystal structure consisting of two interpenetrating face-centered cubic lattices. The face-centered cubic arrangement of sodium cations (the smaller spheres) is readily apparent with the larger spheres (representing chloride anions) filling what are known as the octahedral holes of the lattice. Calcium oxide also crystallizes in the sodium chloride structure.

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... 

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. 

M-M multiple bonding has long been known in metal oxide structures. The first Mo=Mo bond was seen in one crystalline form of Mo02 which has a distorted rutile structure wherein the Mo(4+) ions occupy adjacent octahedral holes throughout the lattice ( 4). The octahedra are distorted because of the short Mo-Mo distances 2.51 X. La. RejO has a fluorite type structure in which 02 is substituted for F and four of the five Ca2 sites are occupied by La3 ions. The remaining Ca2 site is occupied by an (Re=Re)8 unit with an Re-Re distance 2.259(1) A... 

Chapter 1 is an introduction to crystal stractures and the iorric model. It introduces many of the crystal structures that appear in later chapters and discusses the concepts of iorric radii and lattice energies. Ideas such as close-packed structures and tetrahedral and octahedral holes are covered here these are used later to explain a number of solid state properties. 

Point defects. Point defects (Fig. 5.1) are limited to a single point in the lattice, although the lattice will buckle locally so that the influence of point defects may spread quite far. A Frenkel defect consists of a misplaced interstitial atom and a lattice vacancy (the site the atom should have occupied). For example, silver bromide, which has the NaCl structure, has substantial numbers of Ag+ ions in tetrahedral holes in the ccp Br array, instead of in the expected octahedral holes. Frenkel defects are especially common in salts containing large, polarizable anions like bromide or iodide. 

When discussing metal alloys (Section 4.3), we saw that atoms of non-metallic elements such as H, B, C, and N can be inserted into the interstices (tetrahedral and octahedral holes) of a lattice of metal atoms to form metal-like compounds that are usually nonstoichiometric and have considerable technological importance. These interstitial compounds are commonly referred to as metal hydrides, borides, carbides, or nitrides, but the implication that they contain the anions H, B3, C4, or N3- is misleading. To clarify this point, we consider first the properties of truly ionic hydrides, carbides, and nitrides. 

Hagg found that metals can accommodate interstitial nonmetal atoms of radius up to 59% of that of the metal atoms. Show that, in this limiting case, accommodation of the nonmetal atoms in the octahedral holes of a face-centered cubic metal lattice should result in an expansion of the unit cell dimension by 12.4%. [Hint Review the radius ratio rules in Section 4.5.]... 

The structures of the hydrides, oxides and nitrides in this group are rather peculiar, for they can always be described as lattices, as found in pure metals, with the negative ions inserted in the octahedral holes of these structures. In the case of TiN, TiO and, in general, all compounds AB, all octahedral holes are occupied, and the structure is that of the sodium chloride type. There are nitrides of other types, too, e.g. A2N, A3N, etc., in which cases only a part of the octahedral holes are occupied. 

Now the ions formed by hydrogen and the elements of the second period are so small that, on placing them in the octahedral holes of the metal, the distances between the metal atoms are not increased. There is practically no dilatation of the metal lattices and the metal atoms remain at the distances they were in the lattice of the metal, itself. Since, in the lower compounds, not all valency electrons of the metal are used in bond formation, there are electrons available to form the same kinds of bonds they form in the pure metal, or, in other words, in the formula for the heat of formation of the ionic compound... 

The examples in Table III, show that the hydrogen atoms occupy tetrahedral holes at the beginning of the transition series. As we move along the transition series, we observe the interstitial hydride shift toward octahedral holes and the hydrides of the heavier elements become progressively unstable. Palladium is exceptional since it is the only heavy element of group VIII that gives a simple hydride. Hydride formation is accompanied in most cases by a change in metallic lattice type and in all cases by a considerable increase in metal-metal distances. 

Radius Ratio It is not difficult to calculate the size of the octahedral hole in a lattice of closest... 

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). 

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). 

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