Trigonal holes

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. 

Trigonal holes are formed by three spheres in the same layer [Fig. 16.36(a)],... 

In fact, trigonal holes are so small that they are never occupied in binary ionic compounds. Whether the tetrahedral or octahedral holes in a given binary ionic solid are occupied depends mainly on the relative sizes of the anion and cation. Next, we will determine the sizes of the octahedral and tetrahedral holes and consider guidelines for their occupation by ions. 

The holes that exist among closest packed uniform spheres, (a) The trigonal hole formed by three spheres in a given plane, (b) The tetrahedral hole formed when a sphere occupies a dimple in an adjacent layer, (c) The octahedral hole formed by six spheres in two adjacent layers. 

Dent and Kokes consider that wurtzite derives from isotropically expanded, hexagonal close-packed layers of oxide ions, with correspondingly expanded zinc layers in which zinc ions occupy one half of the tetrahedral holes between oxide layers. This expansion increases the radius of the trigonal holes in the oxide layers such that, at 0.058 nm, they can almost accommodate a zinc ion. The structure is quite open and consists of straight channels of octahedral sites, each 0.20 nm in diameter, separated by trigonal squeeze points , 0.12 nm in diameter. 

In fact, trigonal holes are so small that they are never occupied in binary ionic compounds. Whether the tetrahedral or octahedral holes in a given binary ionic solid are occupied depends mainly on the relative sizes of the anion and cation. For example, in zinc sulfide the ions (ionic radius = 180 pm) are arranged in a cubic closest packed structure with the smaller ions (ionic radius = 70 pm) in the tetrahedral holes. The locations of the tetrahedral holes in the face-centered cubic unit cell of the ccp structure are shown in Fig. 10.36(a). Note from this figure that there are eight tetrahedral holes in the unit cell. Also recall from the discussion in Section 10.4 that there are four net spheres in the face-centered cubic unit cell. Thus there are twice as many tetrahedral holes as packed anions in the closest packed structure. Zinc sulfide must have the same number of S ions and Zn ions to achieve electrical neutrality. Thus in the zinc sulfide structure only half the tetrahedral holes contain Zn ions, as shown in Fig. 10.36(c). 

Consider a cation in a trigonal hole. What size ion will just fit in the hole if the packed spheres have radius R7... 

A trigonal hole, one in the center of a triangle of spheres, is so very small that it is seldomly occupied.)... 

Similar calculations can be used for tetrahedral and trigonal holes, for which r = 0.225 R and r = 0.155 R, respectively. These calculations show that in the cubic closest packed structure, the octahedral hole is bigger than the tetrahedral hole. 

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