Pauling's Rules

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

Pauling's Rules

The anions around a cation define a coordination polyhedron. The best-known example is the silica tetrahedron. The distance between cations and anions is determined by the sum of their ionic radii. The ratio of their radii determines the coordination number, or number of anions surrounding the cation. In a silica tetrahedron the coordination number of the Si atom is 4.

Since electrostatic attraction depends only on distance, cations tend to pull anions equally from all directions. Thus the polyhedra tend to be highly symmetrical. The Platonic solids (left), especially tetrahedra, cubes and octahedra, dominate, along with a few other very symmetrical solids.

It is relatively uncommon to find irregular polyhedra. It happens most often among the sulfides and semi-metals where covalent bonding and ellipsoidal ions frequently are found. Also, after major ions are accounted for, minor ions sometimes tend to get stuffed into whatever spaces are left over. Finally, if ions in surrounding polyhedra aren't arranged perfectly symmetrically, polyhedra may be distorted somewhat.

Electrostatic valency principle: The total strength of the valency bonds that reach an anion from all its neighboring cations is equal to the charge of the anion. For example, in olivine, (Mg,Fe)₂SiO₄, the SiO₄ tetrahedra are isolated units. The charge on the Si ion is +4, on Mg or Fe +2 and on O -2. Since 4 oxygens surround a silicon, there must be a charge of -1 on each oxygen to be balanced by the Mg and Fe atoms.
Cation coordination polyhedra tend to be linked (share anions) at corners first, then edges, and faces last of all because of the electrostatic repulsion between cations. The repulsion is especially great for cations with large charge and small coordination number, like Si. Silica tetrahedra almost never link any other way than at their corners.

Why the emphasis on cations? Because cations are small, and electrostatic attraction is an inverse-square force. Any inverse-square force exerted by a sphere can be regarded as coming from a point at the center of the sphere. If we consider an oxygen ion with -2 charge and ionic radius 1.4 Angstroms, and a magnesium ion of +2 charge and radius 0.66 Angstroms, the electrostatic attraction exerted by the magnesium ion on its "surface" is more than four times that of the oxygen ion. Cations attract anions more strongly than anions attract cations.
In a crystal containing different cations, cations with large charge and small coordination number tend not to share polyhedral elements. Think of Bowen's Series: among the ferromagnesian minerals, olivine (isolated tetrahedra) tends to form first, followed by pyroxene (single chains), amphibole (double chains) and biotite (sheets). The silica tetrahedra don't link until they have no other choice. (In the feldspars, the other ingredient in Bowen's Series, the Ca, Na and K cations are so big, and two of the three have charges of only +1, that they repel Si ions less, so Rule 3 doesn't operate quite as powerfully.) Basically this is a consequence of electrostatic repulsion between cations - they try to stay as far apart as possible.
In crystals the number of structurally distinct sites tends to be small.

Example: Olivine

The Packed-Sphere Representation

This is the atomic structure of olivine. Oxygen atoms (blue) are very nearly close-packed. Iron or magnesium atoms are yellow and silicon is purple. Some silicons rest in a triangular void with the fourth atom of the tetrahedron on top (the tetrahedron points up). Others rest on top of an oxygen with a triangle of three oxygen atoms on top (the tetrahedron points down).

Advantage: the near close-packing (it's not quite exact) of the oxygens is very evident.

Disadvantages: only one layer is visible and the coordination of the cations is not very clear.

The polyhedral Representation

The structure of olivine in a polyhedral format. Every vertex represents an oxygen atom. Iron or magnesium atoms are at the centers of the octahedra, silicon at the centers of the tetrahedra. Blue tetrahedra sit over openings between the octahedra and correspond to the purple tetrahedra in the next layer down.

Advantages: three-dimensional structure and atomic coordinations are much clearer. Two layers are shown and part of a third so that the layering sequence becomes apparent.

Disadvantage: the close-packing of the oxygen atoms is harder to see.

Suggested Exercise: superimpose the two diagrams and verify that they do in fact represent the same thing.

Pauling's Rules Applied to Olivine

All the cations occur at the centers of octahedra or tetrahedra. Although the oxygens are close-packed, olivine is orthorhombic because of the way the cations fill the voids between the oxygen sheets. There are actually two kinds of octahedra. The octahedra that form the central spine of the zigzag chains are slightly distorted because adjacent octahedra share edges with a silica tetrahedron, and the Si +4 cation repels the Mg cations in the octahedra. The octahedra at the ends of the zigzags are regular..
Every oxygen links to one silica tetrahedron and three ferromagnesian octahedra. The silicon atom has a charge of +4 but is shared by four oxygens, so each oxygen feels an attraction of +4 x 1/4 = +1. Each ferromagnesian ion has a charge of +2, shared among six neighbors, so each oxygen feels an attraction of +2 x 1/6 = +1/3 from each cation. Since there are three octahedra meeting at each oxygen atom, the total effect of all three neighboring cations is +1/3 x 3 = +1. Thus the total cation charge reaching each oxygen is +2, balancing the -2 of the oxygen ion.
Does it seem like we're double counting? We're not. Each ferromagnesian ion is shared by six neighbors, attracting each with a net charge of +1/3, but each oxygen is also adjacent to three cations, so their combined effect is 3 x +1/3 = +1. Sometimes it's easier to do this sort of accounting if we pair specific anions and cations together rather than distribute the charges among neighbors.
Note that neither diagram shows a tetrahedron meeting three octahedra. We'd have to show additional layers or build a three-dimensional model. But we can reason out that it must be so from Pauling's Second Rule. The vacant points of tetrahedra in the polyhedral view would connect to the joins between three octahedra in the next layer.
There are no face-to face polyhedra. Silica tetrahedra do sit on the faces of octahedral openings, but the openings are vacant. There are no cations there.
Silica tetrahedra (small radius, +4 charge, coordination number 4) are not linked at all. The ferromagnesian octahedra (larger radius, +2 charge, coordination number 6) link edge to edge.
All oxygen atoms are shared by a tetrahedron and three octahedra. All silicon atoms are in tetrahedra. All ferromagnesian atoms are in octahedra of two slightly different types. There are only four different settings for atoms: the environments of all oxygen and silicon atoms are identical, and those of ferromagnesian ions nearly identical.

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Created 18 September 1999, Last Update