Radius ratios, and coordination

Table 5.2. Spatial arrangement of rigid spheres in relation to radius ratio and coordination number...

Coordination numbers in different crystals depend on the sizes and shapes of the ions or atoms, their electronic structures, and, in some cases, on the temperature and pressure under which they were formed. An oversimplified and approximate approach to predicting coordination numbers uses the radius ratio, r+/r. Simple calculation from tables of ionic radii allows prediction of possible structnres by modeling the ions as hard spheres. For hard spheres, the ideal size for a smaller cation in an octahedral hole of an anion lattice is a radius of 0.414r. Calculations for other geometries result in the radius ratios and coordination number predictions shown in Table 7.1. 

Table 5 Ion radius ratios and predicted and found coordination numbers for the alkali metal halides...

Rule 1 Ionic radii and radius ratio. A coordinated polyhedron of anions is formed about each cation, the cation-anion distance being determined by the radius sum and the coordination number of the cation by the radius ratio. (Pauling, 1929)... 

Three factors are critical in determining the structure of ceramic compounds crystal stoichiometry, the radius ratio, and the propensity for covalency and tetrahedral coordination. 

Table 2 Limiting Radius Ratios, Cation Coordination Numbers, and Ionic Arrangements in Ionic Solids...

EXERCISE 7.3 Fluorite (CaF2) has fluoride ions in a simple cubic array and calcium ions in alternate body centers, with r+/r = 0.97. What coordination numbers of the two ions are predicted by the radius ratio What coordination numbers are observed Predict the coordination number of Ca + in CaCl2 and CaBr2. 

In the Table 2.4 the coordination numbers dependency of cation to anion radius ratio and in Table 2.5 the ionic radii of the elements the most often encountered in cement chemistry, are given. It should be remembered that the ionic radii, given in Table 2.5, depend on oxidation degree, and are decreasing with its rise. 

Correlation among Radius Ratios, Maximum Coordination Numbers, and Types of Holes Occupied... 

The asymmetry of distribution of phases about the so-called ideal radius ratio of 1.225 is apparent from Figure 9. Far fewer phases occur at lower radius ratios and furthermore, the atoms are not compressed sufficiently for contacts in the 16-12 coordination to be formed... 

For halides the cation should have a charge of 2+ rather than 4+ for tetrahedral coordination. The only fluoride compound capable of containing two-coordinate F and four-coordinate cations is Bep2. For ZrF, the radius ratio rule predicts that Zr" " is eight-coordinate if all fluorine atoms are two - c o o rdinate. 

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... 

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