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Ionic Solids

Efficient Packing

The crystal structures of ionic solids composed solely of monatomic ions is governed by two relatively simple factors.

  1. The most efficient packing of the ions is desired.
  2. Because like charges repel, it is necessary to prevent two anions or two cations from coming in contact with each other.

Just as covalent bonding and non-spherical atoms created exceptions to the closest-packing rule for metals, these same issues can be important in the packing of ionic solids. Consequently there are many exceptions to the simple rules discussed here.

To pack monatomic ions together to form an ionic solid, first select the largest ion, which is frequently the anion. Pack these large ions together in a closest-packed fashion (either hcp or ccp). Next insert the smaller ion (usually the cation) in the holes in the closest-packed structure for the larger ion.

It is important that ions of like charge do not come in contact; therefore the smaller ion must be larger than the size of the hole in which it is being place. If the smaller ion is smaller than the hole, it will fit into the hole and the larger ions will remain in physical contact. If the smaller ion is larger than the hole, the smaller ion will push the larger ions apart and thus prevent the high-energy situation of having two ions of like charge in physical contact.



Holes in Closest-Packed Structures

What sort of holes are available in a closest-packed structure?

Both ccp and hcp have many trigonal holes. However, the rhole/r ratio for a tetrahedral hole (0.2247) is very small and relatively few ionic solids have cation-anion radius ratios smaller than this. Consequently it is rarely necessary to utilize trigonal holes. The vast majority of ionic solids have ions in tetrahedral, octahedral, or cubic environments.

The links below display cubic closest-packed and hexagonal closest-packed structures and illustrate the types of holes present in these structures.

Cubic Closest-Packed Structure                                         Hexagonal Closest-Packed Structure




Sizes of Holes

In viewing the exercises involving trigonal, tetrahedral, octahedral, and cubic holes, it should have been apparent that the holes differ in size. Specifically, the ratio rhole/r was smallest for the trigonal hole and largest for the cubic hole.

The virtual reality display below shows the relative sizes of the largest spheres that can fit into a hole of each geometry. The light blue sphere on the far right shows the size of the lattice atoms whose packing creates the hole.

Trigonal Hole
coordination number = 3
Tetrahedral Hole
coordination number = 4
Octahedral Hole
coordination number = 6
Cubic Hole
coordination number = 8
Atom

This virtual reality display requires Java3D. If the display is not visible, consult the Java3D FAQ.




Structure of Ionic Solids

Consider an ionic solid composed of monatomic cations and anions and assume the anion is larger than the cation.

In the interests of efficient packing, the anions will pack together in a ccp or hcp structure. The cations will then be inserted into holes in the closest-packed structure. It is important that the cation be larger than the hole into which it is inserted so that the anions in the structure are pushed apart and thus are not in physical contact. In order to achieve the most efficient packing, the cation will go in the smallest hole possible so long as the cation is larger than the hole.

For a tetrahedral hole, rhole/r = 0.225. For an octahedral hole, this ratio is 0.414. For a cubic hole, the ratio is 0.732.

Suppose rcation/ranion = 0.38. If the cation is placed in a cubic or octahedral hole, the anions will not be pushed apart, because the cation is smaller than the hole. In this case the cation will go into a tetrahedral hole.

Suppose rcation/ranion = 0.68. If the cation is placed in a cubic hole, the anions will not be pushed apart, because the cation is smaller than the hole. The cation must go into a tetrahedral or octahedral hole. Placing the cation in a tetrahedral hole will result in less efficient packing (fewer atoms per unit volume) than placing the cation in an octahedral hole, because the tetrahedral hole is smaller and thus the anions will be pushed further apart. As a result the cation will end up in octahedral holes.

If rcation/ranion > 0.732, the cation will prefer to go in cubic holes. Observant readers, however, will recall that there are no cubic holes in either the hexagonal closest-packed structure or the cubic closest-packed structure. Two possibilities exist for coping with this situation. First, the anions might form a cubic structure rather than a ccp or hcp structure. Second, although there are no cubic holes in the ccp structure, you may recall that if all tetrahedral holes are filled with cations, the cations themselves create a cubic hole for the anion. The ccp structure might, therefore, be used if rcation/ranion is near unity.

The table below outlines the various possibilities. Links are provided to pages that illustrate the various examples.

Size Ratio Preferred
Hole Geometry
Coordination
Number
Lattice Structure
for Larger Ion
Examples
0.225 to 0.414 Tetrahedral 4 hexagonal closest-packed
cubic closest-packed
Wurtzite (ZnS)
Zinc Blende (ZnS)
0.414 to 0.732 Octahedral 6 hexagonal closest-packed
cubic closest-packed
Rutile (TiO2)
Sodium Chloride
0.732 to 1.000 Cubic 8 cubic
cubic closest-packed
Cesium Chloride
Fluorite (CaF2)



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