Radius-ratio effect

Cations in octahedral holes surrounded by anions. In (a), the anions all contact the cation but in (b), the anions contact only each other. 

Singly Charged Doubly Charged Triply Charged  

The type of analysis shown for the tetrahedral case leads to the conclusion that the cation cannot touch all the anions unless rc 0.225 ra. As a result, when rc/ra is in the range 0.225-0.414 a tetrahedral arrangement of anions around the cation is expected. Table 3.5 summarizes the results of similar calculations for other arrangements of ions. 

Of course, the examples shown in Table 3.6 are those where a large covalent contribution is expected. However, these data show conclusively that a hard sphere ionic model does not account for all the properties of some materials even though they are predominantly ionic. 

The value for rc/ra for CsCl is 0.934 and as expected the structure has eight Cl- ions surrounding each Cs+ ion. However, it is interesting to note that even CsCl has the sodium chloride structure at temperatures above 445 °C. Some of the other alkali halides that normally have the sodium chloride structure exhibit the CsCl structure when subjected to very high pressure. 

---

With exception of LiF which is between NaF and KF with respect to p, but between RbF and CsF with respect to SjU. This might be due to the "radius-ratio-effect emphasized by Pauling (Ref. 7, p. 529 in the 1960 edition). However, the claim that this effect justifies the neglect of ionic polarization cannot be brought in agreement with the large deviation shown by the "corrected boiling point of CsF in l.c. Fig. 13—9. 

Fig. 13.8.—The observed melting points of the alkali halogenides (left) and values corrected for the radius-ratio effect (right.)...

For compounds in which the radius ratio is small, another term may be added to (5) to include also the appreciable effect of anion-anion repulsion. Pauling has indeed proposed such a treatment, analogous to Born s, but refined to include radius-ratio effects in doing so, he has been able to predict just how much the interionic distances in each of the alkali halides should depart from strict additivity. In using his modified treatment further for calculation of lattice energies, he has been able to show that the anomalies in melting points and boiling points, mentioned earlier in this chapter, may be correlated, at least semiquantitatively, with the radius ratios. 

Note that nitrates have three oxygen atoms surrounding each nitrogen whereas phosphates have four oxygens about each phosphorus. This is probably not a radius ratio effect but is instead connected with the inability of phosphorus to form double bonds of the usual type. When nitrogen, a first-row element, forms four bonds, one or two may be double, but when phosphorus forms four bonds, all must be single. A number of workers feel that there is some overlap between the 3d orbitals of phosphorus and the 2p orbitals of oxygen in phosphates, and that this leads to double-bond character" of a sort, but this question is still open. 

The structures and stabilities of the ionic salts are determined in part by the lattice energies and by radius ratio effects. Thus the Li+ ion is usually tetrahedrally surrounded by water molecules or negative ions, although Li(H20) f has also been found. On the other hand, the large Cs+ ion can accommodate eight near-neighbor Cl ions, and its structure is different from that of NaCl, where the smaller cation Na+ can accommodate only six near-neighbors. The Na+ ion appears to be 6-coordinate in some nonaqueous solvents. 

B. Ionic Substances — Lattice geometries, lattice energies, ionic radii and radius/ ratio effects... 

The second factor is the radius-ratio effect. Only nearest-neighbor interactions were included in the repulsion part of Equation (5.3). But there is one situation where next-nearest neighbors must be considered this is the case where one ion, usually the anion, is so much bigger than the other that they are in contact, or strongly overlapping. The values of Rq in Table 5.1 and values of Rq may be used to assess this possibility. 

A AE mix of 5-30 kcal/mol is calculated. It is small for compounds like NaCl and CaO, and large for cases such as CuCl, BeO or MnS. With these corrections, the ionic model gives very good agreement with A exp for all CN6 compounds, except CdO. BeO is predicted to be 95 percent ionic. It has CN4 because of the radius-ratio effect. This is also the case for MgTe. 

Only BeO has CN4, yet it has a value of 7 that shows ionic bonding. This agrees with the calculation of the cohesive energy presented ealier. CN4 is forced upon BeO by the radius ratio effect. The value of 7 forBeO may be compared with the 7 for ZnO, which is just what is expected for covalent bonding. 

The biological function of Group lA and IIA cations of the periodic table is reviewed against the background of their chemistry. Utilization of these cations arises from an ability to form different types of complex compounds, which is dependent upon the radius-ratio effect. If the details of their biochemistry are to be understood, new probe methods for following the cations in biological systems must be devised. Some possibilities based upon the principle of isomorphous replacement are described and tested. 

---