Pauling's rules are five rules published by Linus Pauling

Linus Pauling

Linus Carl Pauling was an American chemist, biochemist, peace activist, author, and educator. He was one of the most influential chemists in history and ranks among the most important scientists of the 20th century...

 in 1929 for determining the crystal structure

Crystal structure

In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...

s of complex ionic crystal

Ionic crystal

An ionic crystal is a crystal consisting of ions bound together by their electrostatic attraction. Examples of such crystals are the alkali halides, including potassium fluoride, potassium chloride, potassium bromide, potassium iodide, sodium fluoride, and other combinations of sodium, caesium,...

s.

First rule

A coordinated polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 of anions is formed about each cation, the cation-anion distance determined by the sum of ionic radii

Ionic radius

Ionic radius, rion, is the radius of an atom's ion. Although neither atoms nor ions have sharp boundaries, it is important to treat them as if they are hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice...

 and the coordination number

Coordination number

In chemistry and crystallography, the coordination number of a central atom in a molecule or crystal is the number of its nearest neighbours. This number is determined somewhat differently for molecules and for crystals....

 (C.N.) by the radius ratio.

The numbers in the table are mathematically derived minimum radius ratios for that geometry. An octahedron may form with a radius ratio greater than or equal to .414, but as the ratio rises above .732, a cubic conformation becomes more stable. These mathematically derived ratios are deviated from in practice; thus, octahedral salt structures with a radius ratio of less than .414 or more than .732 have been observed. If the radius ratio falls below the minimum calculated for ions modeled as spherical balls, it is presumed that the ions are compressed into oblong balls that are not perfect spheres.

Examples of radius ratios and related polyhedra

Radius ratio	C.N.	polyhedron
.225	4	tetrahedron Tetrahedral molecular geometry In a tetrahedral molecular geometry a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are cos−1 ≈ 109.5° when all four substituents are the same, as in CH4. This molecular geometry is common throughout the first...
.414	6	octahedron Octahedral molecular geometry In chemistry, octahedral molecular geometry describes the shape of compounds where in six atoms or groups of atoms or ligands are symmetrically arranged around a central atom, defining the vertices of an octahedron...
.592	7	capped octahedron
.645	8	square antiprism Square antiprism In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps...  (anticube)
.732	8	cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
.732	9	triaugmented triangular prism Triaugmented triangular prism In geometry, the triaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by attaching square pyramids to each of the three equatorial faces of the triangular prism...
1.00	12	cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

Second rule: the electrostatic valence rule

An ionic structure will be stable to the extent that the sum of the strengths of the electrostatic bonds that reach an anion equal the charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

 on that anion.

i.e. A stable ionic structure must be arranged to preserve Local Electroneutrality.

This is expressed mathematically as:

where is the charge of the anion and the summation is over the adjacent cations.

Some examples are:

Cations with oxide O²⁻ ion

Cation	Radius ratio	CN	Electrostatic bond strength
Li+	0.34	4	0.25
Mg2+	0.47	6	0.33
Sc3+	0.60	6	0.5

Third rule

The sharing of edges and particularly faces by two anion polyhedra decreases the stability of an ionic structure. Sharing of corners does not decrease stability as much, so (for example) octahedra may share corners with one another.

Effect is largest for cations with high charge and low C.N. (especially when r+/r- approaches the lower limit of the polyhedral stability).

Fourth rule

In a crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

 containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with one another.

Fifth rule: the rule of parsimony

The number of essentially different kinds of constituents in a crystal tends to be small. The repeating units will tend to be identical because each atom in the structure is most stable in a specific environment. There may be two or three types of polyhedra, such as tetrahedra or octahedra, but there will not be many different types.