Radius ratio, minimum

Coordination Number Orbitals Hybridized Geometrical Arrangement Minimum Radius Ratio... 

Polyhedron Codrdination number Minimum radius ratio... 

Coordination Number Minimum Radius Ratio Coordination Polyhedron... 

Table 4. Coordination polyhedra, coordination numbers, radius ratio q and corresponding minimum cavity radii for oxygen and nitrogen ligand sites3)...

Coordination Polyhedron Coordination Number Minimum radius ratio e Minimum cavity radius (A) Oxygen sites (fu, = 1.40 A) Nitrogen sites (rw = 1.50 A)... 

Cavity size is certainly an important factor in determining the striking preference of ligands 28,29,30 respectively for cations Li+, Na+ and K+, whose size most closely fits the intramolecular spherical cavity (Table 6). The cavities of these ligands are larger than the minimum cavity determined by the minimum radius ratio (Table 4). In the systems 31—33... 

Tablb 13-18.—Values of the Minimum Radius Ratio fob Stability of Va ious Coordination Polyhrdra... 

Coordination number Minimum radius ratio Coordination polyhedron... 

Q Calculate the minimum radius ratio expected for a coordination number of eight. 

Figure 12.4. Space-filling optimal helix, with a pitch radius ratio of 2.512. This value is determined by requiring that the local radius of curvature of the axis of the helix is equal to half the minimum distance of closest approach between different turns of the helix. The corresponding tube (which can be thought of as being inflated uniformly around the axis) is space-filling. Strikingly, the same geometry is found, within 3%, for oc-helices in the native-state structures of proteins [30].

An ionic solid should achieve maximum electrostatic stability when (i) each ion is surrounded by as many as possible ions of opposite charge, and (ii) the anioncation distance is as short as possible. There is, however, a play-off between these two factors. Consider an octahedral hole in a close-packed array of anions (see Topic D3) The minimum radius of the hole, obtained when the anions are in contact, is 0.414 times the anion radius. A cation smaller than this will not be able achieve the minimum possible anion-cation distance in octahedral coordination, and a structure with lower coordination (e.g. tetrahedral) may be preferred. These considerations lead to the radius ratio rules, which predict the likely CN for the smaller ion (usually the cation) in terms of the ratio r where... 

We have already considered in detail the structures of the alkali halides, all of which crystallize with either the sodium chloride or the caesium chloride arrangement ( 3.04 and 3.05). All of these compounds are essentially ionic, and the degree of ionic character depends on the difference in electronegativity of the atoms concerned it is thus a maximum in caesium fluoride and a minimum in lithium iodide. As we have seen, the radius ratio r+jr is the primary factor in determining whether a given halide possesses the sodium chloride or the caesium... 

We should mention that in the few cases in which the variation in electron density in a crystal has been accurately determined (e.g. NaCl), the minimum electron density does not in fact occur at distances from the nuclei indicated by the ionic radii in general use e.g. in LiF and NaCl, the minima are found at 92 and 118 pm from the nucleus of the cation, whereas tabulated values of / l + and rj4a+ are 76 and 102 pm, respectively. Such data make it clear that discussing lattice structures in terms of the ratio of the ionic radii is, at best, only a rough guide. For this reason, we restrict our discussion of radius ratio rules to that in Box 5.4. 

Rule I. You can draw a polyhedron of anions around every cation in a crystalline lattice, such that (i) the interionic separation can be determined as the sum of the ionic radii, according to Equation (12.10), and (ii) the coordination number of the cation can be determined using the radius ratio rule (vide infra). The radius ratio rule sets the minimum r+/r ratio that can exist for a cation with a given ccxjrdina-tion number. This ratio can be determined from a geometrical consideration of the minimum cationic radius necessary to keep the anions in a particular coordination geometry from just touching each other. 

Solving for the radius ratio, r/R, one obtains the minimum radius ratio necessary to keep the anions from touching one another ... 

The minimum radius ratios for other coordination geometries are listed in Table 12.6. 

Example 12-6. Prove that the minimum radius ratio for a cation having cubic coordination is 0.732. 

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