Anion-cation distance

In lithium chloride, bromide and iodide, magnesium sulfide and selenide and strontium chloride the inter-atomic distances depend on the anion radius alone, for the anions are in mutual contact the observed anion-anion distances agree satisfactorily with the calculated radii. In lithium fluoride, sodium chloride, bromide and iodide and magnesium oxide the observed anion-cation distances are larger than those calculated because of double repulsion the anions are approaching mutual contact, and the repulsive forces between them as well as those between anion and cation are operative. 

Figure 6.12 The structures of some typical color centers in alkali halide crystals (such as NaCl). The defects are represented on a plane of the alkali halide crystal. The circles represent the lattice ions and a is the anion-cation distance.

Because of their importance as basic primary centers, we will now discuss the optical bands associated with the F centers in alkali halide crystals. The simplest approximation is to consider the F center - that is, an electron trapped in a vacancy (see Figure 6.12) - as an electron confined inside a rigid cubic box of dimension 2a, where a is the anion-cation distance (the Cr -Na+ distance in NaCl). Solving for the energy levels of such an electron is a common problem in quantum mechanics. The energy levels are given by... 

Inserting the values of the constants in Equation (6.4), this can be written in terms of the anion-cation distance as... 

The anion-cation distance in potassium chloride (KCl) is 0.315 nm. (a) Using the simple model of an electron in a rigid box, estimate the wavelength peaks of the two lowest energy transitions for the F centers in KCl. (b) Now determine the wavelength peak of the lowest energy transition from the experimental tit to expression (6.5) and comment on the differences relative to the result obtained in (a). 

Figure 2.7 Energies and surface transition versus the reciprocal of the anion-cation distance. Reprinted from ref [46] with permission from Francis, Taylor Ltd.

The experimentally measured anion-cation distances in highly ionic solids can be interpreted on the assumption that each ion has a nearly fixed radius. For example, the difference in anion-cation distance between the halides NaX and KX is close to 36 pm irrespective of the anion X, and it is natural to attribute this to the difference in radii between Na+ and K+. To separate the observed distances into the sum of two ionic radii is, however, difficult to do in an entirely satisfactory way. One procedure is to look for the minimum value in the electron density distribution between neighboring ions, but apart from the experimental difficulties involved such measurements do not really support the assumption of constant radius. Sets of ionic radii are therefore all based ultimately on somewhat arbitrary assumptions. Several different sets have been derived, the most widely used being those of Shannon and Prewitt, based on the assumed radius of 140 pm for O in six-coordination. Values for a selection of ions are shown in Table 1. 

An ionic solid should achieve maximum electrostatic stability when (i) each ion is surrounded by as many as possible ions of opposite charge, and (ii) the anioncation distance is as short as possible. There is, however, a play-off between these two factors. Consider an octahedral hole in a close-packed array of anions (see Topic D3) The minimum radius of the hole, obtained when the anions are in contact, is 0.414 times the anion radius. A cation smaller than this will not be able achieve the minimum possible anion-cation distance in octahedral coordination, and a structure with lower coordination (e.g. tetrahedral) may be preferred. These considerations lead to the radius ratio rules, which predict the likely CN for the smaller ion (usually the cation) in terms of the ratio r where... 

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