14 - Lecture notes for GEOL3010

Pauling's Rules a set of "rules" (i.e., generalizations) used to determine the nature of crystalline structures.

In effect, the rules are design to:


1. Coordination Rule. Coordination is defined by the polyhedron of anions that form about a cation. The size of the polyhedron (i.e., cation-anion distance) is determined by sum of the atomic radii and the coordination number (C.N.) of the cation by the radius ratio (R =rcation/ranion).

Coordination type

C.N.

Radius ratio

 Example
HCP 12 1.00  
 CCP  12  1.00  
Cubic Coordination 8 1.00 - 0.73  
Octahedral 6 0.73 - 0.42  
Tetrahedral 4 0.42 - 0.23  
Triangular 3 0.23 - 0.16  
Linear 2 < 0.16  


2. Electrostatic valence Rule The total strength of valencey bonds that reach a cation from all neighboring anions must equal the charge of the cation.

The strength of an electrostatic bond, known as the electrostatic valence (e.v.)

where:

e.v. = | valence/C.N.|

Example: Na+ is octahedrally coordinated. Therefore e.v. = 1/6. Cl- also has an e.v. of 1/6.

When all bonds are of equal strength then this is termed isodesmic.

In many cases the bond strengths are not all equal. If you have a small compact group comprising a highly charged cation and less strongly charged anions then their e.v.'s will be large.


B. C4+ is 3-coordinated with O2- into a carbonate group. e.v. = 4/3.

The 1 1/3 value is greater than one-half the oxygen ion (1)
Therefore, a radical carbonate group exists (i.e., CO32-)
These functional groups typically bonds more weakly with another cation (e.g., Calcite CaCO3).

When there is a disparity in the bond strengths (e.v.'s) then these structures are said to be anisodesmic.


C. When the bond strength is exactly one half the bonding energy of the anion bonds are termed mesodesmic.

The most common example is the silicate tetrahedra, where e.v. = 1, which is half of the oxygen ion.

In the case the anion groups may bond together to form chains, sheets, and boxwork polymers (e.g. quartz).


3. Polyhedra Sharing Rule. The sharing of anion polyhedra edges and faces tend to decrease the stability of the structure. (face sharing less stable than edge sharing). 

polyshare


4. High valence - Small CN Avoidance Rule. In crystals containing different cations, those cations of high valence and small coordination number tend not to share polyhedral elements. (examples are carbonate and sulfate groups). If there is sharing of a polyhedral element, then the polyhedra are usually distorted.


5. Principle of Parsimony (parsimony = excessive or extreme frugality or stinginess) The number of different type of constituents in a crystal tends to be small. Therefore, even in a crystal with very complex structure and composition, a particular anion will occupy only a few sites. All ions (although they may be different) that occupy the same site are considered constituents of that site.

The amphibole structure above has one of the most complex site assemblages.