The radius ratio

Key point: The radius ratio indicates the likely coordination numbers of the ions in a binary compound. A parameter that figures widely in the literature of inorganic chemistry, particularly in introductory texts, is the ‘radius ratio’, (gamma), of the ions. The radius ratio is the ratio of the radius of the smaller ion (r_small) to that of the larger (r_large):

In most cases, r_small is the cation radius and r_large is the anion radius. The minimum radius ratio that can support a given coordination number is then calculated by considering the geometrical problem of packing together spheres of different sizes (Table 3.6). It is argued that, if the radius ratio falls below the minimum given, then ions of opposite charge will not be in contact and ions of like charge will touch. According to a simple electrostatic argument, a lower coordination number, in which the contact of oppositely charged ions is restored, then becomes favourable. Another way of looking at this argument is that as the radius of the M ion increases, more anions can pack around it, so giving a larger number of favourable Coulombic interactions. In this respect, compare CsCl, and its (8,8) coordination, with NaCl, and its (6,6)-coordination. We can use our previous calculations of hole size (Example 3.4), to put these ideas on a firmer footing. A cation of radius between 0.225r and 0.414r can occupy a tetrahedral hole in a close-packed or slightly expanded close-packed array of anions of radius r.

However, once the radius of a cation reaches 0.414r, the anions are forced so far apart that octahedral coordination becomes possible and most favourable. Note that 0.225r represents the size of the smallest ion that will fit in a tetrahedral hole and that cations between 0.225r and 0.414r will push the anions apart. However, the coordination number cannot increase to 6 with good contacts between cation and anions until the radius goes above 0.414r. Similar arguments apply for the tetrahedral holes that can be filled by ions with sizes up to 0.225r. These concepts of ion packing based on radius ratios can often be used to predict which structure is most likely for any particular choice of cation and anion (Table 3.6). In practice, the radius ratio is most reliable when the cation coordination number is 8, and less reliable with six- and four-coordinate cations because directional covalent bonding becomes more important for these lower coordination numbers.

■ A brief illustration. To predict the crystal structure of TlCl we note that the ionic radii are r(Tl⁺) =159pm and r(cl^-)= 181 pm, giving γ= 0.88. We can therefore predict that TlCl is likely to adopt a caesium-chloride structure with (8,8)-coordination. That is the structure found in practice.

 ■The ionic radii used in these calculations are those obtained by consideration of structures under normal conditions. At high pressures, different structures may be preferred, especially those with higher coordination numbers and greater density. Thus, many simple compounds transform between the simple (4,4)-, (6,6)-, and (8,8)-coordination structures under pressure. Examples of this behaviour include most of the lighter alkali metal halides, which change from a (6,6)-coordinate rock-salt structure to an (8,8) coordinate caesium-chloride structure at 5 kbar (the rubidium halides) or 10–20 kbar (the sodium and potassium halides). The ability to predict the structures of compounds under pressure is important for understanding the behaviour of ionic compounds under such conditions. Calcium oxide, for instance, is predicted to transform from the rock salt to the caesium-chloride structure at around 600 kbar, the pressure in the Earth’s lower mantle.

Similar arguments involving the relative ionic radii of cations and anions and their preferred coordination numbers (that is, preferences for octahedral, tetrahedral, or cubic geometries) can be applied throughout structural solid-state chemistry and aid the pre diction of which ions might be incorporated into a particular structure type. For more complex stoichiometries, such as the ternary compounds with the perovskite and spinel structure types, the ability to predict which combinations of cations and anions will yield a specific structure type has proved very useful. One example is that for the high temperature superconducting cuprates (Section 24.8), the design of a particular structure feature, such as Cu ⁺² in octahedral coordination to oxygen, can be achieved using ionic radii considerations.

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مواضيع ذات صلة

The Born–Mayer equation

Structure maps

Ionic radii

Characteristic structures of ionic solids

Atomic radii of metals

Nonclose-packed structures

Polytypism

The close packing of spheres

Lattices and unit cells

The orbitals

Bonding and antibonding orbitals

Hybridization