Energy-Derived Ionic Radii

There are several methods for determining ionic radii from physical characteristics of atoms and crystals. Thus, Fumi and Tosi [209] derived ionic radii (similar to the bonded ones) for alkali halides, using the Born model of crystal lattice energy with experimental interatomic distances, compressibilities and polarizabilities. Rossein-sky [210] calculated ionic radii from ionization potentials and electron affinities of atoms, his results were close to Pauling s. Important conclusions can also be drawn from the behaviour of solids under pressure. Considering metal as an assembly of cations immersed into electron gas, its compressibility at extremely high pressures 

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Ionic radius. The wide variation of metal-oxygen distances within individual coordination sites and between different sites in crystal structures of silicate minerals warns against too literal use of the radius of a cation, derived from interatomic distances in simple structures. Relationships between cation radius and phenocryst/glass distribution coefficients for trace elements are often anomalous for transition metal ions (Cr3+, V3+, Ni2+), which may be attributed to the influence of crystal field stabilization energies. 

One of the diflSculties in applying the Born equation is that the effective radius of the ion is not known further, the calculations assume the dielectric constant of the solvent to be constant in the neighborhood of the ion. The treatment has been modified by Webb who allowed for the variation of dielectric constant and also for the work required to compress the solvent in the vicinity of the ion further, by expressing the effective ionic radius as a function of the partial molal volume of the ion, it was possible to derive values of the free energy of solvation without making any other assumptions concerning the effective ionic radius. 

Because this treatment makes no explicit provision for atomic displacements and electrostatic forces exerted by nearest neighbor atoms, it cannot be properly quantified. It is of interest, however, to see if the energies derived from it bear any relationship to those observed. Dielectric constants of silicates at room temperature are generally in the range 5-12 (Shannon, 1993), while the radius of the defect should be larger than the ionic radius of Th " (Figure 13), because the... 

In deriving theoretical values for inter-ionic distances in ionic crystals the sum of the univalent crystal radii for the two ions should be taken, and corrected by means of Equation 13, with z given a value dependent on the ratio of the Coulomb energy of the crystal to that of a univalent sodium chloride type crystal. Thus, for fluorite the sum of the univalent crystal radii of calcium ion and fluoride ion would be used, corrected by Equation 13 with z placed equal to y/2, for the Coulomb energy of the fluorite crystal (per ion) is just twice that of the univalent sodium chloride structure. This procedure leads to the result 1.34 A. (the experimental distance is 1.36 A.). However, usually it is permissible to use the sodium chloride crystal radius for each ion, that is, to put z = 2 for the calcium... 

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