Radius-ratio rule

Predicted Dependence of Structure Type on Cation/Anion Radius Ratio for Binary Ionic Compounds 

For example, the ionic radius of is usually quoted as 72 pm, and for magnesium oxide, sulfide, and telluride we have 

This example was chosen to show that the radius ratio structure predictions work quite well except near the limiting radius ratio values. Here, r+Jr- is borderline for MgS, and we predicted the wrong structure. This is not surprising, since the concept of fixed ionic radii is not well founded. 

First of all, we cannot measure r+ or r directly. As explained in Section 4.1, X-ray diffraction measurements give us internuclear spacings corresponding to (r+ + r ) and some noncontact separations, rather than r+ or r as such. If, however, we can guess any one or r on the basis of some theory or other, then we can estimate all the others. 

Further, ions are not hard, billiard ball like spheres. Since the wave functions that describe the electronic distribution in an atom or ion do not suddenly drop to zero amplitude at some particular radius, we must consider the surfaces of our supposedly spherical ions to be somewhat fuzzy. A more subtle complication is that the apparent radius of an ion increases (typically by some 6 pm for each increment) whenever the coordination number increases. Shannon has compiled a comprehensive set of ionic radii that take this into account. Selected Shannon-type ionic radii are given in Appendix F these are based on a radius for of 140 pm for six coordination, which is close to the traditionally accepted value, whereas Shannon takes the reference value as 126 pm on the grounds that it gives more realistic ionic sizes. For most purposes, this distinction does not mat- 

---

For halides the cation should have a charge of 2+ rather than 4+ for tetrahedral coordination. The only fluoride compound capable of containing two-coordinate F and four-coordinate cations is Bep2. For ZrF, the radius ratio rule predicts that Zr" " is eight-coordinate if all fluorine atoms are two - c o o rdinate. 

In addition to the Zachariasen and radius ratio rules, for oxides the electronegativity of the predominant cation should be between 1.7 and 2.1 (7). If the cation electronegativity is too high, the compound tends to form molecules or discrete polyatomic ions rather than a connected network. For example, CrO satisfies the radius ratio rule, but the highly electronegative Cr ions promote the formation of discrete dichromate(VI) ions, Cr202 , in the presence of other oxides. 

EXAMPLE 5.4 Sample exercise predicting a structure from the radius ratio rule... 

The dominant features which control the stoichiometry of transition-metal complexes relate to the relative sizes of the metal ions and the ligands, rather than the niceties of electronic configuration. You will recall that the structures of simple ionic solids may be predicted with reasonable accuracy on the basis of radius-ratio rules in which the relative ionic sizes of the cations and anions in the lattice determine the structure adopted. Similar effects are important in determining coordination numbers in transition-metal compounds. In short, it is possible to pack more small ligands than large ligands about a metal ion of a given size. 

A lattice is a three-dimensional array, and there are eight systems known. Inorganic substances are usually defined by one crystal system by the so-called radius-ratio rule [22], but organic compounds often have the capability of existing in more than one crystal form, a phenomenon referred to as polymorphism. 

An important reason for the exceptions to the radius ratio predictions is that ions are not hard spheres but somewhat compressible, hence do not have a truly constant radius. Another reason for the inadequacy of the radius ratio rules, particularly when the anions are much larger than the cations, is that some structures are determined by the close packing of the anions, leaving the cations in holes between the anions. In such a case more anions may be packed around a cation of a given fixed radius than are predicted by the radius ratio, so that although the anions are touching each other, they are not touching the cation. However, if... 

The radius ratio is considered important because the central ion must be prevented from rattling around in a cavity (see Orgel, 1966). However, the radius ratio is not a rigorous prognosticator, since the concept applies to hard spheres. It has already been noted that ions may be polarizable and deformable, sometimes with a tendency to directional covalent bond formation. These properties affect models based on hard spheres and the extent to which the radius ratio determines the coordination number of a particular ligand. Thus, .. . we can accept the radius ratio rule as a useful, if imperfect, tool in our arsenal for predicting and understanding the behavior of ionic compounds. (Huheey, 1983). 

One reason for the failure of the radius ratio rules is that ions do not behave like hard spheres. Even those that are hard in the Pearson (1973) sense can still be compressed. This is clearly seen in the way the bond length varies with the bond valence. If cation anion bonds can be compressed, so can the distance between the 0 ions in the first coordination sphere. The stronger the cation anion bonds, therefore, the closer the anions in the first coordination sphere can be pulled together (Shannon el al. 1975). 

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.]... 

It should be kept dearly in mind that the radius ratio rules apply strictly only to the packing of hard spheres of known size. As this is seldom the case, it is surprising that the rules work as well as they do. Anions are not hard like billiard balls, but polarizable under the influence of cations. To whatever extent such polarization or covalency occurs, errors are apt to result from application or the radius ratio rules. Covalent honds are directed in space unlike electrostatic attractions, and so certain orientations are preferred. 

Rg. 4.18 Actual crystal structures of the alkali halides (as shown by the symbols) contrasted with the predictions cl the radius ratio rule. Tie figure is divided into three regions by the lines rjr. 0.414 and r+/h- a 0.732, predicting coordination number 4 (wurizite or zinc blende, upper left), coordination number 6 (rock salt, NaCl, middle), and coordination number 8 (CsCI, lower right). The crystal radius of lithium, and to a lesser extent that of sodium, changes with coordination number, so both ihe radii with C.N. 4 (left) and C.N = 6 Iright) have been plotted. 

Perform radius ratio calculations to show winch alkali halides violate the radius ratio rule. 

Even if there are exceptions to the radius ratio rule, or if exact data are hard to come by, it is still a valid guiding principle. Cite three independent examples of pairs of compounds illustrating structural differences resulting from differences in ionic radii. 

An analysis of 227 compounds indicated that the radius ratio rule worked about iwo-thirds of (he lime. Particularly troublesome wen Croup IB (t ) and llB(l2 chakogBmdcsfifce HgS. Naihan, 1 C J. Chen EJuc. 1985.62,215-218. 

Trivalent yttrium and lanthanide metals, except for promethium, have been complexed to octaethylporphyrin by heating at 210 °C in an imidazole melt.17 The complexes obtained as hydroxides, Mm(OEP)(OH), are unstable in acidic media. As the charge radius ratio rule predicts, the early lanthanide metalloporphyrins, MIU(OEP)(OH) (M = La, Ce, PR, Nd), are demetallated during purification, and the middle series (M = Sm, Eu, Gd, Tb, Dy) in 1 % acetic acid in methanol, while the last five (M = Ho, Er, Tm, Yb, Lu) survive in 2% acetic acid in methanol but are dissociated in dilute hydrochloric acid. The Mnl(OEP)(OH) appears to coordinate more than one equivalent of pyridine and piperidine, and dimerizes in noncoordinating solvents such as benzene and dichloromethane at 10 4 M concentration. The dimer is considered to be a di-p-hydroxo-bridged species, different from the p-oxo dimer, Scin(OEP) 20 (Scheme 6). 

The Octet Rule. To the isomorphism exhibited in Table 1 may be added the radius-ratio rules. The statement in crystal chemistry that rattling is bad , that the number of anions about a cation should not be so great as to create a cavity larger than the cation 72>, corresponds to the statement in covalent chemistry that the number of electron-pairs about an atomic core should not be so great as to exceed the number of low-lying, available, valence-shell orbitals. 

The alkali metals react with many other elements directly to make binary solids. The alkali halides are often regarded as the most typical ionic solids. Their lattice energies agree closely with calculations although their structures do not all conform to the simple radius ratio rules, as all have the rock salt (NaCl) structure at normal temperature and pressure, except CsCl, CsBr and Csl, which have the eight-coordinate CsCl structure. The alkali halides are all moderately soluble in water, LiF being the least so. 

Pauling subsequently introduced three mles governing ionic sfructures (Pauling, 1928, 1929). The first is known as the radius ratio rule. The idea is that the relative sizes of the ions determine the sfructure adopted by an ionic compound. Pauling proposed specific values for the ratios of the cation radius to the anion radius as lower limits for different coordination types. These values are given in Table 3.5. 

Unfortunately, the radius ratio rules are incorrect in their prediction of coordination numbers about as often as they are correct. Usually, it overestimates the coordination number of the cation. This model essentially regards ions as hard, incompressible spheres, in which covalent bonding is not considered. The directionahty, or overlap requirements, of the covalent bonding contribution probably plays as significant a role as ion size fitting. 

If, on the other hand, the radius ratio rules are violated on the downside, that is to say the hard sphere cation and anion no longer touch, the structure is longer stable and another structure should be adopted. For instance, if r+/r < 0.224, then tetrahedral coordination is no longer possible but a threefold planar triangular coordination can be found by locating the cation in the trigonal hole centered in the plane of the close-packed layer. Such coordination should be stable for 0.15 < r+/r < 0.224. In fact, however, trigonal coordination is rare in chemical systems other than those of boron. 

While radius ratio rules generally predict correct structure types for the alkali halides in only about 50% of the cases, the MEG calculations (not even shell stabilized) correctly predict the NaCl structure to be more stable for 15 of the 16 compounds studied. Calculated structural parameters were also in good agreement with experiment, and calculated pressures for the NaCl CsCl transition were in fair agreement with the limited experimental data available. These results indicate that preferred coordination numbers and structural types in alkali halides can be accu-... 

---