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Linus Pauling (1929) established five rules, now called Pauling's rules, which describe cation–anion relationships in ionically bonded substances and are paraphrased below:

 Rule 1: A polyhedron of anions is formed about each cation, with the distance between a cation and an anion determined by the sum of their radii (radius sum). The number of coordinated anions in the polyhedron is determined by the cation : anion radius ratio.

 Rule 2: An ionic structure is stable when the sum of the strengths of all the bonds that join the cation to the anions in the polyhedron equals (balances) the charge on the cations and on the anions. This rule is called the electrostatic valency rule.

 Rule 3: The sharing of edges and particularly faces by adjacent anion polyhedral elements decreases the stability of an ionic structure. Similar charges tend to repel. If they share components, adjacent polyhedra tend to share corners, rather than edges, to maximize cation spacing.

 Rule 4: Cations with high valence charges and small coordination number tend not to share polyhedral elements. Their large positive charges tend to repel.

 Rule 5: The number of different cations and anions in a crystal structure tends to be small. This is called the rule of parsimony.

Pauling's rules provide important tools for understanding crystal structures. Especially important is the rule concerning radius ratio and coordination polyhedra. Coordination polyhedra provide a powerful means for visualizing crystal structures and their relationship to crystal chemistry. In fact, they provide a fundamental link between the two. When atoms and ions combine to form crystals, they bond together into geometric patterns in which each atom or ion is bonded to a number of nearest neighbors. The number of nearest neighbor ions or atoms is called the coordination number (CN). Clusters of atoms or ions bonded to other coordinating atoms produce coordination polyhedron structures. Polyhedrons include triangles, cubes, octahedra, tetrahedra, and other geometric forms.

When ions of opposite charge combine to form minerals, each cation attracts as many nearest neighbor anions as can fit around it as approximate “spheres in contact”. In this way, the basic units of crystal structure are formed which grow into crystals as multiples of such units are added to the existing structure. One can visualize crystal structures in terms of different coordinating cations and coordinated anions that together define a simple three‐dimensional polyhedron structure. As detailed in Chapters 3 and 4, complex polyhedral structures develop by linking of multiple coordination polyhedra.

The number of nearest neighbor anions that can be coordinated with a single cation “as spheres in contact” depends on the radius ratio (RR = Rc/Ra) which is the radius of the smaller cation (Rc) divided by the radius of the larger anion (Ra). For very small, highly charged cations coordinated with much larger, highly charged anions, the radius ratio (RR) and the coordination number (CN) are small. This is analogous to fitting basketballs as spheres in contact around a small marble. Only two basketballs can fit as spheres in contact with the marble. For cations of smaller charge coordinated with anions of smaller charge, the coordination number is larger. This is analogous to fitting golf balls around a larger marble. One can fit a larger number of golf balls around a large marble as spheres in contact because the radius ratio is larger.

Table 2.5 Relationship between radius ratio, coordination number, and coordination polyhedra.

Radius ratio (Rc/Ra)	Coordination number	Coordination type	Coordination polyhedron
<0.155	2	Linear	Line
0.155–0.225	3	Triangular	Triangle
0.225–0.414	4	Tetrahedral	Tetrahedron
0.414–0.732	6	Octahedral	Octahedron
0.732–1.00	8	Cubic	Cube
>1.00	12	Cubic or hexagonal closest packed	Cubeoctahedron complex

The general relationship between radius ratio, coordination number and the type of coordination polyhedron that results is summarized in Table 2.5. For radius ratios less than 0.155, the coordination number is 2 and the “polyhedron” is a line. The appearance of these coordination polyhedra is summarized in Figure 2.19.

When predicting coordination number using radius ratios, several caveats must be kept in mind.

1 The ionic radius and coordination number are not independent. As illustrated by Table 2.6, effective ionic radius increases as coordination number increases.

2 Since bonds are never truly ionic, models based on spheres in contact are only approximations. As bonds become more covalent and more highly polarized, radius ratios become increasing less effective in predicting coordination numbers.

3 Radius ratios do not successfully predict coordination numbers for metallically bonded substances.

The great value of the concept of coordination polyhedra is that it yields insights into the fundamental patterns in which atoms bond during the formation of crystalline materials. These patterns most commonly involve threefold (triangular), fourfold (tetrahedral), sixfold (octahedral), eightfold (cubic) and, to a lesser extent, 12‐fold coordination polyhedra or small variations of such basic patterns. Other coordination numbers and polyhedron types exist, but are rare in inorganic Earth materials.

Another advantage of using spherical ions to model coordination polyhedra is that it allows one to calculate the size or volume of the resulting polyhedron. In a coordination polyhedron of anions, the cation–anion distance is determined by the radius sum (R _∑ ). The radius sum is simply the sum of the radii of the two ions (Rc + Ra); that is, the distance between their respective centers. Once this is known, the size of any polyhedron can be calculated using the principles of geometry. Such calculations are beyond the scope of this book but are discussed in Wenk and Bulakh (2016) and Klein and Dutrow (2007).