p17p35ujimn9ejc9cap81i1j7e4.pdf

D GENERALIZATIONS ABOUT CRYSTAL STRUCTURES BASED ON PERIODICITY
Figure 1.8. Geometric configurations with different radius ratios. (a) shows a stable configuration.
The ions in (b) have the minimum radius ratio for stability in this arrangement. The
radius ratio of the ions in (c) makes this an unstable configuration.
Stable and unstable configurations are illustrated in Fig. 1.8. Basically, a configuration
is stable until anion–anion repulsions force longer and less stable
anion–cation bond distances (as in 1.8c). The critical or minimum stable radius
ratio is defined by the point when the cation contacts all of the neighboring
anions, and the anions just contact one another, as shown in Fig. 1.8(b). For any
given coordination number, there is a minimum stable radius ratio that can be
derived through simple geometric arguments (see Example 1.2). Atoms with
radius ratios as shown in 1.8(c) would be more stable in a configuration with a
lower coordination number. The minimum radius ratios for selected geometries
are summarized in Fig. 1.10.
Example 1.2 Calculating minimum stable radius ratios
Determine the minimum stable radius ratio for octahedral (six-fold)
coordination.
1. First, assume that the cation in the center contacts the surrounding anions and that
the anions just contact one another (this is the minimum stable configuration). Using
Fig. 1.9, we find a plane that contains both cation–anion and anion–anion contacts.
One such plane is the equatorial plane (see Fig. 1.9b).
2. Next, we note that the sides of the isosceles triangles in Fig. 1.9 have the lengths:
abr and cr r . Based on these geometric relationships, the radius ratio can
be easily determined:
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