Crystal radii of cations

AR (=Ra-Rc) for selected binary compounds. Da and are the D parameters of anions and cations, respectively, while Rg and Rc are Shannon s crystal radii of anions and... 

Ionic radii are quoted in Tables 2.3 and 2.5 for a large number of cations including those of the elements in groups 13, 14, 15, and 16, which do not form predominately ionic bonds. These values were obtained by subtracting the fluoride or oxide ion radius obtained from predominantly ionic solids from the length of a bond that is not predominantly ionic. The very small values for the radii of cations obtained in this way do not bear much relation to the real size of the atom in the crystal or molecule. 

Now use Coulomb s law to compare the strengths of the ionic bonds in crystals of magnesium oxide and lithium fluoride. The sizes of the four ions are taken from the tabulation of radii of cations and anions in Table 5-4. 

Since the electron distribution function for an ion extends indefi-finitely, it is evident that no single characteristic size can be assigned to it. Instead, the apparent ionic radius will depend upon the physical property under discussion and will differ for different properties. We are interested in ionic radii such that the sum of two radii (with certain corrections when necessary) is equal to the equilibrium distance between the corresponding ions in contact in a crystal. It will be shown later that the equilibrium interionic distance for two ions is determined not only by the nature of the electron distributions for the ions, as shown in Figure 13-1, but also by the structure of the crystal and the ratio of radii of cation and anion. We take as our standard crystals those with the sodium chloride arrangement, with the ratio of radii of cation and anion about 0.75 and with the amount of ionic character of the bonds about the same as in the alkali halogenides, and calculate crystal radii of ions such that the sum of two radii gives the equilibrium interionic distance in a standard crystal. 

Anion Contact and Double Repulsion.2 —The explanations of the deviations from additivity are indicated by Figure 13-6, in which the circles have radii corresponding to the crystal radii of the ions and are drawn with the observed interionic distances. It is seen that for LiCl, LiBr, and Lil the anions are in mutual contact, as suggested in 1920 by Land6.14 A simple calculation shows that if the ratio p = r+/r of the radii of cation and anion falls below /2 — 1 = 0.414 anion-anion contact will occur rather than cation-anion contact (the ions being considered as rigid spheres). A comparison of apparent anion radii in these crystals and crystal radii from Table 13-8 is given in Table 13-7. 

It is convenient to give Eg a value (n g = 0.262) such as to make F(p) equal to unity for p = 0.75 this causes Rq to be equal to the sum of the standard radii of cation and anion for crystals with this radius ratio, which was selected as standard in Section 13-2 because it is ap-... 

Crystals with the Rutile and the.Fluorite Structures Interionic Distances for Substances of Unsymmetrical Valence Type.—In a crystal of a substance of unsymmetrical valence type, such as fluorite, CaFs (Fig. 13-10), the equilibrium cation-anion interionic distance cannot be expected necessarily to be given by the sum of the crystal radii of the bivalent calcium ion and the univalent fluoride ion. The sum of the univalent radii of calcium and fluoride, 2.54 A, would give the equilibrium interionic distance in a hypothetical crystal with attractive and repulsive forces corresponding to the sodium chloride arrangement. 

Similar substantially constant differences are obtained with other pairs of alkali halides of B 1 structure, having either a cation or an anion in common. As a result, the conclusion was reached that each ion makes a specific contribution toward an experimentally observed r0, well-nigh irrespective of the nature of the other ion with which it is associated in the lattice. In other words, characteristic radii should be attributable to the ions (1,2). However, a knowledge of the internuclear distances in the crystals is not sufficient by itself to determine absolute values for crystal radii of ions, and various criteria have been used to assign the size of a particular ion or the relative sizes of a pair of alkali and halide ions. 

The concept of atomic or ionic size is one that has been debated for many years. The structure map of Figure 1 used the crystal radii of Shannon and Prewitt and these are generally used today in place of Pauling s radii. Shannon and Prewitt s values come from examination of a large database of interatomic distances, assuming that intemuclear separations are given simply by the sum of anion and cation radii. Whereas this is reasonably frue for oxides and fluorides, it is much more difficult to generate a self-consistent set of radii for sulfides, for example. A set of radii independent of experimental input would be better. The pseudopotential radius is one such estimate of atomic or orbital size. 

Figure 1. Valence of core cations in their aquo complexes plotted against crystal radii of the cations. The radii are mostly from Shannon and Prewitt (22). (0) Cations (O) hydroxy cations and hydroxy anions (X) oxy cations oxy anions.

If the central cations are regularly surrounded by the oxygen atoms so that the outer spherical surfaces of the atoms just touch each other, the ratios of the radii of cation and oxygen atoms are 0.73 for 8 coordination and 1.00 for 12 coordination. The relevant ratios in the biphenyl crystals are the following rNa+/rQ = 0.83, rK+/rQ = 1.06, rRb+/r0 = 117, where r is the radius of the indicated species. Hence 8 coordination for Na+ is feasible,... 

The accumulation of lattice constants gave rise to a growing Hbrary of interatomic (and interionic) distances, providing atomic and ionic radii. In 1929 Pauling published five principles (rules) that formed the first rational basis for understanding aystal structures. For example, the ratio of the ionic radii of cations to anions determines coordination number in crystals coordination number 6 for each chlorine and sodium ion in NaCl coordination number 8 for each ion in CsCl. 

The corresponding data for the enthalpies of solvation (A goiv(ion)) for aqueous solutions are given in Appendix 2.11.3. More recent applications of the Born equation to ion solvation in a series of solvents have been reported by Izmailov,Khomutov, and Criss and Luksha. The results for the non-aqueous systems are summarised in Appendices 2.11.4 (free energies) and 2.11.5 (enthalpies). It should be noted that Izmailov assumed 5 = 0 and used Pauling s radii Khomutov used the crystal radii of Gourary and Adrian and assumed d = 0.74 A for cations and 0.42 A for anions. 

Indeed, the number of modifications of the Bom equation is hardly countable. Rashin and Honig, as example, used the covalent radii for cations and the crystal radii for anions as the cavity radii, on the basis of electron density distributions in ionic crystals. On the other hand, Stokes put forward that the ion s radius in the gas-phase might be appreciably larger than that in solution (or in a crystal lattice of the salt of the ion). Therefore, the loss in self-energy of the ion in the gas-phase should be the dominant contributor. He could show indeed that the Bom equation works well if the vdW radius of the ion is used, as calculated by a quantum mechanical scaling principle applied to an isoelectronic series centering around the crystal radii of the noble gases. More recent accounts of the subject are avail-able. ... 

With the expression r(M) = 1.39/7 - 1.22 A, the crystal radii of the M-cations used to construct the average bond lengths, , in Figure 1 were estimated. These... 

When publishing his tables of effective ionic radii. Shannon referred to the results of the charge density studies and included a second table of crystal radii where the radius of each cation had been increased by addition of 14 pm and the radius of each anion reduced by the same amount. It should be clear that addition of the crystal radii of the cation and anion will produce the same interatomic distance as addition of the traditional effective ionic radii, but Shannon felt that crystal radii correspond more closely to the physical size of the ions in a solid. ... 

Fig. 6. Comparison of polyvalent radii and univalent radii of cations and anions according to Pauling s simple model. The univalent radii are the ones used in crystals containing polyvalent ions to make radius ratio arguments.

One of the most important parameters that defines the structure and stability of inorganic crystals is their stoichiometry - the quantitative relationship between the anions and the cations [134]. Oxygen and fluorine ions, O2 and F, have very similar ionic radii of 1.36 and 1.33 A, respectively. The steric similarity enables isomorphic substitution of oxygen and fluorine ions in the anionic sub-lattice as well as the combination of complex fluoride, oxyfluoride and some oxide compounds in the same system. On the other hand, tantalum or niobium, which are the central atoms in the fluoride and oxyfluoride complexes, have identical ionic radii equal to 0.66 A. Several other cations of transition metals are also sterically similar or even identical to tantalum and niobium, which allows for certain isomorphic substitutions in the cation sublattice. 

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