Pauling's rules
[2]: 524 [3] For the coordination numbers and corresponding polyhedra in the table below, Pauling mathematically derived the minimum radius ratio for which the cation is in contact with the given number of anions (considering the ions as rigid spheres).
If the cation is smaller, it will not be in contact with the anions which results in instability leading to a lower coordination number.
The central diagram shows the minimal radius ratio.
Similar geometrical proofs yield the minimum radius ratios for the highly symmetrical cases C.N.
[5] If the radius ratio is less than the minimum, two anions will tend to depart and the remaining four will rearrange into a tetrahedral geometry where they are all in contact with the cation.
The radius ratio rules are a first approximation which have some success in predicting coordination numbers, but many exceptions do exist.
[3] In a set of over 5000 oxides, only 66% of coordination environments agree with Pauling's first rule.
[6] For a given cation, Pauling defined[2] the electrostatic bond strength to each coordinated anion as
A stable ionic structure is arranged to preserve local electroneutrality, so that the sum of the strengths of the electrostatic bonds to an anion equals the charge on that anion.
Pauling showed that this rule is useful in limiting the possible structures to consider for more complex crystals such as the aluminosilicate mineral orthoclase, KAlSi3O8, with three different cations.
[2] However, from data analysis of oxides from the Inorganic Crystal Structure Database (ICSD), the result showed that only 20% of all oxygen atoms matched with the prediction from second rule (using a cutoff of 0.01).
[6] The sharing of edges and particularly faces by two anion polyhedra decreases the stability of an ionic structure.
[2]: 559 The decrease in stability is due to the fact that sharing edges and faces places cations in closer proximity to each other, so that cation-cation electrostatic repulsion is increased.
[6] As one example, Pauling considered the three mineral forms of titanium dioxide, each with a coordination number of 6 for the Ti4+ cations.
The other two, less stable, forms are brookite and anatase, in which each octahedron shares three and four edges respectively with adjoining octahedra.
[2]: 559 In a crystal containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with one another.
[2]: 561 This rule tends to increase the distance between highly charged cations, so as to reduce the electrostatic repulsion between them.
The structure contains distinct SiO4 tetrahedra which do not share any oxygens (at corners, edges or faces) with each other.
The lower-valence Mg2+ and Fe2+ cations are surrounded by polyhedra which do share oxygens.
The number of essentially different kinds of constituents in a crystal tends to be small.
[2] The repeating units will tend to be identical because each atom in the structure is most stable in a specific environment.
