Ionic radius criterion

Although the ionic radius criterion of Goldschmidt continues to serve as a useful principle of crystal chemistry, attention has been drawn to limitations of it (Bums and Fyfe, 1967b Bums, 1973). As noted earlier, the magnitude of the ionic radius and the concept of radius ratio (i.e. cation radius/anion radius) has proven to be a valuable guide for determining whether an ion may occupy a specific coordination site in a crystal structure. However, subtle differences between ionic radii are often appealed to in interpretations of trace element distributions during mineral formation. 

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

Ionic radii estimated independently by Goldschmidt and Pauling were obtained from interatomic distances in simple ionic crystals such as halides and oxides of metals at laboratory temperatures. These tables of ionic radii (Goldschmidt, 1954 p. 88 Pauling, 1960, p. 514), as well as the more recent compilations (Ahrens, 1952 Shannon and Prewitt, 1969 Whittaker and Muntus, 1970 Shannon, 1976), are based on average interatomic distances within a particular coordination site. The values are strictly valid only for ions in regular octahedral, cubic, or tetrahedral coordinations in these simple crystal 

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In ferromagnesian silicates, therefore, Ni2+ ions are expected to be enriched over Mg2+ in smallest octahedral sites, with the other divalent transition metal ions favouring larger sites in the crystal structures. Thus, based on the ionic radius criterion alone, the olivine Ml and pyroxene Ml sites would be expected be enriched in Ni2+, with the other divalent cations showing preferences for the larger olivine M2 and pyroxene M2 sites. Similarly, in aluminosilicates, all trivalent transition metal ions are predicted to show preferences for the largest [A106] octahedron. 

The ionic radius criterion for interpreting geochemical distributions of trace elements was given a boost in the early 1970 s when correlations were shown to exist between ionic radii and partition coefficients of some trace elements (Onuma et al., 1968 Higuchi and Nagasawa, 1969 Jensen, 1973). The influence of cation radius and charge on trace element distribution patterns was demonstrated by measurements of the distribution coefficient, >, defined by... 

In the case of a neutral non-ionic chelating agent we have neutral carrier-selective electrodes transport is achieved by selective complexa-tion of certain ions. The best-known electrode of this kind is the potassium-selective electrode, whose membrane consists of a valinomycin macrocycle immobilized in phenylether. The important criterion appears to be the size of the cavity in the centre of the macrocycle and interferences are from cations with similar hydrated ionic radius, such as Rb+ and Cs+. 

In (a) and (c) there would be no great difference between the characters of the A-S and B—S bonds in a particular compound, while in (b) the B and S atoms form a covalent complex which may be finite or infinite in one, two, or three dimensions. By analogy with oxides we should describe (a) and (c) as complex sulphides and (b) as thio-salts. Compounds of type (c) are not found in oxy-compounds, and moreover the criterion for isomorphous replacement is different from that applicable to complex oxides because of the more ionic character of the bonding in the latter. In ionic compounds the possibility of isomorphous replacement depends largely on ionic radius, and the chemical properties of a particular ion are of minor importance. So we find the following ions replacing one another in oxide structures Fe, Mg , Mn , Zn, in positions of octahedral coordination, while Na" " more often replaces Ca (which has approximately the same size) than K , to which it is more closely related chemically. In sulphides, on the other hand, the criterion is the formation of the same number of directed bonds, and we find atoms such as Cu, Fe, Mo, Sn, Ag, and Hg replacing Zn in zinc-blende and closely related structures. 

The main criterion for the elements to crystallize in this network is their ionic radius. The ionic radii of various divalent and trivalent metal ions arc given in Table 2. It can be clearly inferred from this table that ions such as Be are too small and ions like Cd arc too large, to be incorporated into HT-like network [31 ]. 

Several chromogenic reagents used for the determination of the trivalent rare earths include 8-hydroxyquinoline (e s 5230), Alizarin S (e s lO ), Arsenazo I (e s 10 ), Arsenazo III (e s 5 x lO ). Arsenazo III has been reported by Savvin (1964, 1964a) to form 1 1 complexes with the rare earths and to be more selective than Arsenazo I and II. Cations which have a radius of less than 0.7-0.8 A show no color reaction with Arsenazo III. These include Be, Zn, Al, Ga, In, Ge, Ti, and Sn. Sc, however, with an ionic radius of 0.81 (compared to 0.95 for In) does form a colored complex at pH of 1 to 2 as do the other rare earths, which suggests the ionic radius is not the only criterion to be considered. Arsenazo III also is an extremely sensitive reagent for tetravalent cations and the reaction with these cations can be carried out in strongly acidic media. Molar absorptivities as high as 1.5 x 10 /mole-cm have been reported. 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity... 

It is seen that the agreement is not very good in general, the lithium salts showing especially large deviations, and that no set of ionic radii could reproduce the experimental values satisfactorily, since these values do not satisfy the criterion of additivity. The difference between the observed values for Li+—1 and Li+—F is 1.01 A, and that between Rb-1—1 and Rb+—F is only 0.84 A these quantities, representing the difference in radius of I and F, should be equal. 

As has been shown in the preceding section, it is this distance criterion that has been used in defining self-consistent sets of ionic radii of which the tabulation of Shannon is the most complete one. With respect to the radius ratio, the quantitative significance is low, at best (see Section 1.1), and a materials scientist is better off in interpreting this part of the first rule as being mostly qualitative in nature. 

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