Threefold coordinated cations

The structure of ionic crystals usually corresponds to the maximum possible coordination number. If the ions are assumed to be hard spheres, there must be contact between ions of opposite signs and no contact between ions of like sign. The coordination depends on the ratio of anion-to-cation diameters. Figure 13.5 shows that the critical condition for threefold coordination is R/(R + r) = cos 30° = /3/2. Therefore, to prevent contact between like ions,... 

In the bulk, cations and anions are sixfold coordinated. The total energy of the crystal is nearly completely given by the Madelung (electrostatic) energy. On the surfaces of the microcrystals, different local coordinations are encountered, with coordination numbers varying from five on the (100), (010), and (001) faces to four on edges and steps and three on corners. Threefold coordinated ions are also present on reconstructed (111) faces. 

Water molecules are four-coordinated in the ices but may be three-coordinated in small molecule hydrates. In the ices and in the ice-like clathrate hydrate structures, the water molecules are always four-coordinated. In small molecule hydrates, they display four- and threefold coordination. They always donate two hydrogen bonds, but may accept one or two bonds, or be connected to one or two cations. 

If, on the other hand, the radius ratio rules are violated on the downside, that is to say the hard sphere cation and anion no longer touch, the structure is longer stable and another structure should be adopted. For instance, if r+/r < 0.224, then tetrahedral coordination is no longer possible but a threefold planar triangular coordination can be found by locating the cation in the trigonal hole centered in the plane of the close-packed layer. Such coordination should be stable for 0.15 < r+/r < 0.224. In fact, however, trigonal coordination is rare in chemical systems other than those of boron. 

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