Ionic radii coordination number-radius ratio

Before the discussion of particular dependencies which are encountered in clinker solid solutions it will be useful to remind general principles of solid solutions formation. In crystal chemistry significant importance have the interionic distances in the lattice, which is the basis for ionic radius determination . The radii ratio of neighbouring ions in the lattice is deciding of the numbers of the closest neighbours, then of so called coordination number of ion. For example silicon ion is as a mle surrounded by four oxygen ions (CN=4), located in the tetrahedron comers . 

The formulated principals correlating crystal structure features with the X Nb(Ta) ratio do not take into account the impact of the second cation. Nevertheless, substitution of a second cation in compounds of similar types can change the character of the bonds within complex ions. Specifically, the decrease in the ionic radius of the second (outer-sphere) cation leads not only to a decrease in its coordination number but also to a decrease in the ionic bond component of the complex [277]. 

In an ionic solid, the coordination number means the number of ions of opposite charge immediately surrounding a specific ion. In the rock-salt structure, the coordination numbers of the cations and the anions are both 6, and the structure overall is described as having (6,6)-coordination. In this notation, the first number is the cation coordination number and the second is that of the anion. The rock-salt structure is found for a number of other minerals having ions of the same charge number, including KBr, Rbl, MgO, CaO, and AgCl. It is common whenever the cations and anions have very different radii, in which case the smaller cations can fit into the octahedral holes in a face-centered cubic array of anions. The radius ratio, p (rho), which is defined as... 

The dominant features which control the stoichiometry of transition-metal complexes relate to the relative sizes of the metal ions and the ligands, rather than the niceties of electronic configuration. You will recall that the structures of simple ionic solids may be predicted with reasonable accuracy on the basis of radius-ratio rules in which the relative ionic sizes of the cations and anions in the lattice determine the structure adopted. Similar effects are important in determining coordination numbers in transition-metal compounds. In short, it is possible to pack more small ligands than large ligands about a metal ion of a given size. 

Several additional, more complicated structure types are known for ionic compounds. For example, according to the radius ratio, one could expect the rutile type for strontium iodide (rSr2+ /i = 0.54). In fact, the structure consists of Sr2+ ions with a coordination number of 7 and anions having two different coordination numbers, 3 and 4. 

A coordination polyhedron of anions is formed around every cation. The cation-anion distances are determined by the sum of the ionic radii, and the coordination number of the cation by the radius ratio. 

The so-called radius ratio principle establishes that, for a cation/anion radius ratio lower than 0.414, the coordination of the complex is 4. The coordination numbers rise to 6 for ratios between 0.414 and 0.732 and to 8 for ratios higher than 0.732. Actually the various compounds conform to this principle only qualitatively. Tossell (1980) has shown that, if Ahrens s ionic radii are adopted, only 60% of compounds conform to the radius ratio principle. ... 

The radius ratio is considered important because the central ion must be prevented from rattling around in a cavity (see Orgel, 1966). However, the radius ratio is not a rigorous prognosticator, since the concept applies to hard spheres. It has already been noted that ions may be polarizable and deformable, sometimes with a tendency to directional covalent bond formation. These properties affect models based on hard spheres and the extent to which the radius ratio determines the coordination number of a particular ligand. Thus, .. . we can accept the radius ratio rule as a useful, if imperfect, tool in our arsenal for predicting and understanding the behavior of ionic compounds. (Huheey, 1983). 

Twelve anions can be arranged around a cation when the radius ratio is 0.95 to 1.00. However, unlike the three structure types considered so far, geometrically the coordination number 12 does not allow for any arrangement which has cations surrounded only by anions and anions only by cations simultaneously. This kind of coordination therefore does not occur among ionic compounds. When becomes larger than 1, as for RbF... 

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... 

The correct structure is predicted for BeO, MgO, and CaO, but for SrO and BaO the predicted coordination number is eight, although the structures found are six coordinate. Crystals adopt the structure that has the most favorable lattice energy, and the failure of the radius ratio concept in this case leads us to examine the assumptions on which it is based. Ionic radii are not known with great accuracy and they change with different coordination numbers (Table 7). Also, ions are not necessarily spherical, or inelastic (see Structure Property Maps for Inorganic Solids) ... 

Cation Ionic radius (A) Charge-to-radius ratio Metal-oxygen distance (A) Average coordination number... 

---