Appendix Ionic Radii

Of the several systems of ionic radii which have been proposed to account for the approximately additive relationships which exist between the observed interatomic distances in ionic crystals, the one which has come into most general use is that of Pauling (111,114)- This is not, as is variously supposed, either a set of empirical radii derived purely from the experimental data which it is, in turn, supposed to reproduce, or one which has been derived purely from theoretical considerations. It is a semiempirical system in the sense that from a very limited set of experimental data, certain relationships are derived using approximate theories of atomic and crystal structure, which adequately account for a much wider set of data. 

Since no precise physical significance can be attached to the concept of atomic or ionic radius (the electronic wave functions approach zero asymptotically), the radii to be assigned are those which reproduce the equilibrium interatomic distances in ionic crystals. These distances depend on the balance between the attractive and repulsive forces, and thus not only on the electron distributions of the ions but also on the crystal structure and the radius ratios. Pauling assumes that the relative sizes of a pair of isoelec-tronic ions are inversely proportional to the effective nuclear charges operating on the outmost electron shell that is 

At equilibrium dV/dR = 0 and R2 = (nB/Az2y,(-n 1). If the ions were to enter into Coulomb attraction as if they were monovalent, with the repulsion coefficient unchanged, the equilibrium interatomic distance would be 

Approximate values of n are known for various types of ions and are used to calculate the crystal radii of Table III.7 

Since the ionic radii of Table III have been obtained with reference to the NaCl type of structure as standard, it is not to be expected that they should apply to other types without corrections to take into account possible variation of A and B. For two different structures with Madelung constants A and /U, and repulsion coefficients Bx and B we have 

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The ionic radius sum has been multiplied by 0.95 to allow for the change from 6- to 4-coordination (see Appendix). 

Table 1 Appendix A2 Electronegativity (ij), metallic (ionic) valency and metallic (ionic) radius of the elements... 

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. Shannon10 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 O2- 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-... 

Changes of coordination number A guiding principle of crystal chemistry is that the coordination number of a cation depends on the radius ratio, RJR, where Rc and / a are the ionic radii of the cation and anion, respectively. Octahedrally coordinated cations are predicted when 0.414 < 7 c// a < 0.732, while four-fold (tetrahedral) and eight- to twelvefold (cubic to dodecahedral) coordinations are favoured for radius ratios below 0.414 and above 0.732, respectively. The ionic radii summarized in Appendix 3... 

Factors that influence ionic size include the coordination number of the ion, the covalent character of the bonding, distortions of regular crystal geometries, and delocalization of electrons (metallic or semiconducting character, described in Chapter 7). The radius of the anion is also influenced by the size and charge of the cation (the anion exerts a smaller influence on the radius of the cation). The table in Appendix B-1 shows the effect of coordination number. 

The sizes of atoms and ions influence how they interact in chemical compounds. Although atomic radius is not a precisely defined concept, these sizes can be estimated in several ways. If the electron density is known from theory or experiment, a contour surface of fixed electron density can be drawn, as demonstrated in Section 5.1 for one-electron atoms. Alternatively, if the atoms or ions in a crystal are assumed to be in contact with one another, a size can be defined from the measured distances between their centers (this approach is explored in greater detail in Chapter 21). These and other measures of size are reasonably consistent with each other and allow for the tabulation of sets of atomic and ionic radii, many of which are listed in Appendix F. 

Values of ionic radii for selected ions are listed in Appendix 6. Ionic radii are sometimes quoted for species such as Si" and Cl, but such data are highly artificial. The sums of the appropriate ionization energies of Si and Cl (9950 and 39 500kJmoU respectively) make it inconceivable that such ions exist in stable species. Nonetheless, a value for the radius of Cl can be calculated by subtracting... 

All alkali metal hydrides (see Sections 9.7 and 10.4) crystallize with the NaCl lattice. From diffraction data and the ionic radii of the metal ions Appendix 6) the radius of can be estimated using equation 9.3 it varies from 130 pm (in LiH) to 154 pm (in CsH) and can be considered similar to that of F (133 pm). 

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