Ratio, radius

Coordination numbers in different crystals depend on the sizes and shapes of the ions or atoms, their electronic structures, and, in some cases, the temperature and pressure under which they were formed. A simple, but at best approximate, approach to predicting coordination munbers uses the radius ratio, r+/r. Simple calculation from tables of 

FIGURE 7-12 Structures of Calcium Carbonate, CaC03. (a) Calcite. (b) Two views of aragonite. 

For hard spheres, the ideal size for a smaller cation in an octahedral hole of an anion lattice is a radius of 0.414r. Similar calculations for other geometries result in the radius ratios (r+/r ) shown in Table 7-1. 

Limiting Values Number Geometry Ionic Compounds 

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

TABLE 7.1 Radius Ratios (r /r ) and Predicted Coordination Numbers 

The Na cation fits easily into the octahedral holes of the Cr lattice, which is ccp. 

The zinc ion radius varies more with coordination number. The radius ratios are r+/r = 74/170 = 0.435 for the CN = 4andr+/r = 88/170 = 0.518 for the CN = 6 radius. Both predict CN = 6, but the smaller one is close to the tetrahedral limit of 0.414. Experimentally, the Zn cation fits into the tetrahedral holes of the lattice, which is either ccp (zinc blende) or hep (wurtzite). 

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The co-ordination number in ionic compounds is determined by the radius ratio - a measure of the necessity to minimize cationic contacts. More subtle effects are the Jahn-Teller effect (distortions due to incomplete occupancy of degenerate orbitals) and metal-metal bonding. 

This then is the limiting radius ratio for six nearest neighbours— when the anion is said to have a co-ordination number of 6. Similar calculations give the following limiting values ... 

For eight nearest neighbours (a co-ordination number of 8) the radius ratio r /r must not be less than 0.73. 

The small lithium Li" and beryllium Be ions have high charge-radius ratios and consequently exert particularly strong attractions on other ions and on polar molecules. These attractions result in both high lattice and hydration energies and it is these high energies which account for many of the abnormal properties of the ionic compounds of lithium and beryllium. 

Coordination Number Orbitals Hybridized Geometrical Arrangement Minimum Radius Ratio... 

The residual shear stress distribution in the assembled cylinders, prior to the appHcation of internal pressure, may be calculated, from pressure P, generated across the interface. The resulting shear stress distribution in the compound cylinder, when subjected to an internal pressure may be calculated from the sum of the residual stress distribution and that which would have been generated elastically in a simple cylinder of the same overall radius ratio as that of the compound cylinder. 

In practice compound shrinkage is often used to prestress a high strength or corrosion-resistant liner. The optimum radius ratios of components of different yield strengths have been shown (37,38) to be... 

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 addition to the Zachariasen and radius ratio rules, for oxides the electronegativity of the predominant cation should be between 1.7 and 2.1 (7). If the cation electronegativity is too high, the compound tends to form molecules or discrete polyatomic ions rather than a connected network. For example, CrO satisfies the radius ratio rule, but the highly electronegative Cr ions promote the formation of discrete dichromate(VI) ions, Cr202 , in the presence of other oxides. 

Modifiers in glass are compounds that tend to donate anions to the network, whereas the cations occupy "holes" in the disordered stmcture. These conditions cause the formation of nonbridging anions, or anions that are connected to only one network-forming cation, as shown in Figure 2. Modifier compounds usually contain cations with low charge-to-radius ratios (Z/r), such as alkali or alkaline-earth ions. 

Vaned diffuser ioss. Vaned diffuser losses are based on conical diffuser test results. They are a function of the impeller blade loading and the vaneless space radius ratio. They also take into account the blade incidence angle and skin friction from the vanes. 

Type of Velocity Diagram Hub-tip Radius Ratio Number Of Stall Zones Rate, Stall Speed, abs/ Rotor Speed Fluctuation during stall, i pVIpV Radial Extent of Stall Zone Type of Stall ... 

Lest I leave the erroneous impression here that colloid science, in spite of the impossibility of defining it, is not a vigorous branch of research, I shall conclude by explaining that in the last few years, an entire subspeciality has sprung up around the topic of colloidal (pseudo-) crystals. These are regular arrays that are formed when a suspension (sol) of polymeric (e.g., latex) spheres around half a micrometre in diameter is allowed to settle out under gravity. The suspension can include spheres of one size only, or there may be two populations of different sizes, and the radius ratio as well as the quantity proportions of the two sizes are both controllable variables. Crystals such as AB2, AB4 and AB13 can form (Bartlett et al. 1992, Bartlett and van... 

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

The above qualitative conclusions made on the basis of the results of [116, 124-127] correlate with the results of [129,130] in which the calculation is based on composite models with nucleus-shell inclusions. The authors illustrate this with the calculation of a system consisting of a hard nucleus and elastomeric shell in a matrix of intermediate properties, and a system where the nucleus and matrix properties are identical whereas the shell is much more rigid. The method may, however, be also applied to systems with inclusions where the nucleus is enclosed in a multi layer shell. Another, rather unexpected, result follows from [129,130] for a fixed inclusions concentration, the relative modulus of the system decreases with increasing nucleus radius/inclusion radius ratio, that is with decreasing shell thickness. 

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... 

When the radius ratio of an ionic compound is less than about 0.4, corresponding to cations that are significantly smaller than the anion, the small tetrahedral holes may be occupied. An example is the zinc-blende structure (which is also called the sphalerite structure), named after a form of the mineral ZnS (Fig. 5.43). This structure is based on an expanded cubic close-packed lattice of the big S2 anions, with the small Zn2+ cations occupying half the tetrahedral holes. Each Zn2+ ion is surrounded by four S2 ions, and each S2" ion is surrounded by four Zn2+ ions so the zinc-blende structure has (4,4)-coordination. 

EXAMPLE 5.4 Sample exercise predicting a structure from the radius ratio rule... 

Ions stack together in the regular crystalline structure corresponding to lowest energy. The structure adopted depends on the radius ratio of cation and anion. Covalent character in an ionic bond itnposes a directional character on the bonding. 

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