Geos 306, Fall 2004, Lecture 6, Pauling's Rules

Geos 306, Fall 2004, Lecture 6
Pauling's Rules Reading:
Klein: p 64-80
Nesse: Chapter 4

Recall that the closest-packing models, bcc and diamond structures all involved single element atoms that were all the same size. We also know that each element has its own unique size. Therefore, each structure will have its own unique volume.
It is also possible to consider ionic crystals as constructed of packed anions with cations filling the voids. This results in 2, 3, 4, or 6 coordinated cations. Furthermore, if a cation had the same size as an anion then it could substitute for the anion and be 12 coordinated.
The first step towards understanding the role of atom sizes is seeing the voids in the closest packed array, and realizing that small atoms can fit into these voids.
The question that naturally arises is which atoms can fit into which holes? Only the smallest cations can be 2 or 3 coordinated, and the larger the cation, the higher its coordination number.
Linus Pauling addressed this question with his radius ratio rule. It is part of a set of rules that he developed to understand crystal structures that are now called Pauling's Rules.
Another important question addressed by Pauling in his rules deals with determining what the possible neighbours (or configuration) that atoms can have, given that we know their coordinations. For example, we never see Al-O-Al linkages when the Al atoms are 4-coordinated with O atoms.
The history behind these rules is that in the 1920's-1930's a lot of crystal structures were being examined using X-ray diffraction techniques. The methods to solve these structures were not developed like they are today, so rules of thumb were devised to help solve the structures.

The "Radius Ratio" Concept

This is a scheme for understanding the packing of atoms of unequal sizes.

Premises:

Large radius atoms pack around a small radius atom in as tight a configuration as possible such that the small atom never rattles around in the space. The large radius atoms are always in contact with the small radius atom. For instance, both of these arrangements are okay.

The next image is not okay because the cation is too small.

This means that the number of large radius atoms that can pack around a small radius atom must be a function of the ratio of their radii. We define the number of large atoms that are in contact with the small atom to be the coordination number (CN) of the small atom.

Do a couple of examples, say for CN=3,or CN=4

Radius Ratio Rules

r_C/r_A	Coord. #	Angle	Config.	Image	Example
0.000-0.155	2	180°	Linear		(HF₂)^-1
0.155-0.225	3	120°	Trigonal planar		(CO₃)^-2
0.225-0.414	4	109.47°	Tetrahedral		(SiO₄)^-4
0.414-0.732	4	90°	Square planar		(CuO₄)^-6
0.414-0.732	6	90°	Octahedral		(NaCl₆)^-5
0.732-1.000	8	70.53°	Square -bipyramid		(CsCl₈)^-7
1.000	12	60°	Closest-packed		(KO₁₂)^-23

Now, let's look at Pauling's rules.

Pauling's Rules

1. A coordination polyhedron of anions is formed about each cation, the cation-anion distance equaling the sum of their characteristic packing radii and their radius ratio determining both the nature of the coordination polyhedron and therefore the coordination number of the cation.

2. An ionic structure will be stable to the extent that the sum of the strengths, S, of the electrostatic bonds that reach an anion from adjacent cations equals the charge, Z, on that anion.

Z_A= S S_i

(The strength of an electrostatic bond is defined as the cations's valence divided by its coordination number, S = Z_C/CN).

3. The sharing of edges and particularly of faces by two anion polyhedra decreases the stability of an ionic crystal structure.

4. In a crystal structure containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with each other.

5. The number of essentially different kinds of constituents in a crystal tends to be small.

From Bloss, Crystallography and Crystal Chemistry

Prewitt's addendum: Given that the chemical formula for a crystal is charge balanced, then the sum of the coordination numbers of the cations must equal the sum of the coordination numbers of the anions.

Rule 1 has already been covered. This gives us the coordination number of the cation. It is of use whether the bonding is ionic, covalent, or metallic. It tells us what can fit in the closest-packing voids and so on.

Rule 2 is a very useful rule, the one that I use the most.
Let us focus on the concept of bond strength first.

S=Z_C/CN If the charge on a cation is Z_C and the coordination number is CN, then the number of electrons per bond is what Pauling considered to be the bond strength. It seems obvious that Si with 4 valence electrons can make a stronger bond than Mg with 2 valence electrons.

Eg. MgO₆: S_Mg = 2/6 = 1/3
Eg. SiO₄: S_Si = 4/4 = 1
Eg. NaO₅: S_Na = 1/5.
Note that rule#1 give the coordination number of the cation, rule#2 gives the coordination number of the anion. Sometimes there are several possible coordination numbers that can be chosen for the anion. Prewitt's addendum helps us choose between them.
Eg Topaz. Al₂SiO₄(F,OH)₂

r(Al) = .57 Angstroms
r(Si) = .39 Angstroms
r(O) = 1.32 Angstroms

then r(Si)/r(O) = .3 ÞCN_Si = 4

and r(Al)/r(O) = .43 ÞCN_Al= 6

and so S(Al) = 3/6=1/2, S(Si) = 4/4=1 and each O is coordinated to 2Al+1Si.

Note that by Prewitt's addendum we can obtain the coordination number of the (F,OH) atoms to be 2:

2 ´ CN(Al) + CN(Si) = 2 ´ 6 + 4 = 16 = 4 ´ CN(O) + 2 ´ CN(F,OH) = 4 ´ 3 + 2 ´ 2

Question: If you were to melt topaz, which bonds would break first?

Rule 3 discusses the sharing of edges and faces

In general, the more corners that are shared between two polyhedrons, then the closer together the cations are. This destabilizes the structure because of cation-cation repulsion.

Rule 4 (In a crystal structure containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with each other.) is a generalization of rule 3
Rule 5 (The number of essentially different kinds of constituents in a crystal tends to be small.) means that we will usually see coordinated polyhedra in crystals.