Lecture 4 – Radius ratios and Pauling's Rules

As we have discussed, the outward symmetry of crystals is an expression of internal ordering of atoms and ions in the crystal structure. This in turn reflects the intrinsic symmetry of the packing of atoms, and their interaction with neighboring atoms...

The ultimate reason for a particular arrangement of atoms in a mineral structure lies in the nature of the cohesive forces that hold atoms together. In theory, we should be able to predict a mineral structure from the chemical composition, but in reality the problem rapidly becomes too complex to solve.

We'll be discussing the subject of crystal chemistry for the next few weeks - defined as the elucidation of the relationship between chemical composition, internal structure and physical properties of crystalline material.

A reminder: the chemical composition of the Earth's crust - 8 elements make up ~99 wt% of the crust ("major elements") ... O and Si are most abundant, thus most common minerals are silicates and oxides.

Ionic radii

Size of atoms difficult to define, let alone measure. Determined be maximum charge density, which itself is a function of the type and number of nearest neighbor atoms. Therefore it is possible to assign each ion a radius such that the sum of the radii of two adjacent ions is each to the interatomic (separation) distance. Thus we can determine effective radii by measuring bond lengths in crystals.

Within a given period (say, the alkalis), the radius increases with atomic number. (Table 13.1)

Radii also vary systematically across a row, being smaller at the center (cation charge increases) and largest to the right (the anions; Table 13.2).

Ionic radii depend strongly upon the valence state of the ion, with larger sizes for negative ions and smaller sizes for positive ions (Table 13.3, 13.4).

EX:

S⁺⁶: radius = 0.6 A

S: radius = 1.04 A

S^-2 radius = 1.7A

Finally, the size of an ion is dependent on its coordination number.

Coordination number

Many simple mineral structures can be viewed as close packing of large anions, with smaller cations in interstitial sites. The anions are packed in a regular structure, while the cations fit in between. The number of anions to which a particular cation bonds is the cation’s coordination number. EX: Si⁺⁴ typically bonds to 4 O atoms, and therefore has a coordination number of 4.

The size of the interstices depends how the anions are packed – different in 2- and 3-dimensions.

EXAMPLES

We give coordination arrangements geometrical names:

2-fold linear

3-fold triangular

4-fold tetrahedral

6-fold octahedral

8-fold cubic

12-fold dodecahedral

Closest packing

What is the most economical way to pack spheres?

2D: If spheres of equal size are packed together as close as possible in a plane, they arrange themselves as follows:

Center of spheres are at the corners of equilateral triangles; each sphere is in contact with 6 others. Note that within this layer there are 3 close-packed directions, each at 60o. Unit cell is hexagonal, with lattice parameter a = 2r. Packing is 90.7% efficient. Hexagonal (closest) packing

Here the center of spheres are at the corner of squares, each sphere is in contact with 4 others. There are only two close-packed directions and the unit cell is square. Packing is 78.5% efficient. Tetragonal packing

Closest packing in three dimensions

Metals have structures that are typically formed by close-packing of atoms, around which electrons pass freely. This type of structure is highly ordered so as to minimize void space.

Build 3-D structure by placing these layers one on top of the other. Most economical way of doing this is such that spheres in one layer rest in hollows of layer below. Two different positions are possible, B or C. (choice equivalent to rotating 180o). By doing so, create a layered sequence of AB.

Let's add a third layer. Again, we have two choices. If third layer goes above A position: ABABABABA

simplest form of close packing - hexagonal closest packing (has underlying hexagonal lattice) … this is true for Na metal

If third layer goes in the C position, stacking sequence would be ABCABCABC.

cubic closest packing

(has underlying cubic lattice). In both of these closest packing sequences, each atom has twelve equidistant nearest neighbors, six in its own plane, and three each in the layer above and the layer below. Examples include Au (shown to the left), Ag, and Cu.

This simple structure means that metal atoms of similar size can easily substitute for each other, thus allowing for alloys of metals like silver and gold. Because of the close packing, metals are dense; they are also malleable and good electrical conductors.

Other minerals that have a cubic closest packed structure are

sphalerite halite

ZnS NaCl

Most minerals are not formed by metallic bonds, and thus do not have this simple structure. For example, the covalent bonds of diamond are strongly directional, which prevents the atoms from adopting a close-packed structure. As a consequence, diamond has a lower specific gravity than a typical metal.

Ionic Bonding

One of the most successful models for predicting crystal structure is to treat crystals as packing of anions and cations as different sized spheres… these rules are collectively known as Paulings Rule and can be summarized as follows:

1. The Coordination (radius ratio) Principle – a coordination polyhedron of anions surrounds each cation. The cation-anion distance is determined by the sum of the cation and anion radii and the number of anions coordinating with the cation is determined by the relative size of the cation and anion.

2. Electrostatic Valency Principle – in a stable ionic structure, the total strength of the valency bonds that reach an anion from all neighboring cations is equal to the charge of the anion.

3. Sharing of Polyhedral Elements I – the existence of edges (and particularly faces) common to coordination polyhedra decreases the stability of ionic structures

4. Sharing of Polyhedral Elements II – in a crystal containing different cations, those with large valence and small coordination number tend not to share polyhedral elements with each other.

5. Principle of Parsimony – the number of essentially different kinds of constituents in a crystal tends to be small.

We’ll look at each rule separately.

Coordination (radius ratio) Principle:

This principle states that the number of anions with which a cation coordinates is determined by the ratio of their radii r_c/r_a.

Bottom line: most stable configuration is achieved when oppositely charged ions (e.g. Na+ and Cl-) are as close together as possible without overlapping. Inter-ionic distance determined by the balance of electrostatic attractive forces between outer electron charges, and repulsive forces between nuclei. Thus in 3 dimensions, ions with positions that follow principles of ionic bonding form highly symmetric polyhedra (coordination polyhedra) that have same inter-ionic distances - will control where certain cations fit into crystal structures. Tetrahedra and octahedra are most common structural types, but triangles, cubes, and other forms important. These coordination polyhedra link together in various ways to form the polyhedral-frame structures. Include all of rock-forming silicates, as well as many borates, sulfates, phosphates, tungstates, oxides, hydroxides.

To reiterate, a coordination polyhedron of anions is formed about each cation, the cation-anion distance being determined by the radius sum and the coordination number of the cation by radius ratio. Thus when bonding dominantly ionic, each cation in the structure will tend to attract, or coordinate, as many anions as will fit around it.

NaCl

Appropriate radii:

Na⁺ = 0.097nm

Cl^- = 0.181nm (almost twice as large)

r_c/r_a = 0.54

If we imagine these as rigid spheres, how closely can we pack them?

First let’s look at 2 dimensions. If the radius of the cation is very small relative to the cation, the cation can fit into small space between three close-packed anions. As the cation becomes larger, the anions move farther apart. At some ratio of ionic radii, closest packing switches to one in which the cation is surrounded by 4 anions (this is the case for NaCl)

From trigonometry:

rc / ra = .414

Thus the radius ratio between anions and cations tells us how the spheres can be packed. For smaller ratios, all 4 anions would not touch the cations, and distances would not be minimized. For larger ratio, distance between anions > 2ra, and eventually a new configuration becomes stable.

What about the third dimension? In order to maintain r_c/r_a = .414 (minimum separation = closest packing), we must add two additional anions, one above and one below. Thus each cation is in 6-fold (octahedral) coordination.

In general:

Rc/Ra	Expected coordination of cation	C.N.
<0.15	2-fold coordination	2
0.15 0.15-0.22	ideal triangular triangular	3
0.22 0.22-0.41	ideal tetrahedral tetrahedral	4
0.41 0.41-0.73	ideal octahedral octahedral	6
0.73 0.73-1.0	ideal cubic cubic	8
1.0 > 1.0	ideal dodecahedral dodecahedral	12

Let’s return to our model of close-packed spheres. As you determined in lab, stacking of close-packed layers of spheres generates two kinds of interstices:

tertrahedral site between 4 close-packed atoms. Thus any small atom occupying this site will be tetrahedrally-coordinated with its neighbors. Tetrahedral sites form in two distinct orientations - apex pointing up or apex pointing down. For this reason, there are twice as many tetrahedral sites as there are close-packed ions (one above and one below).

octahedral site is larger - has 3 atoms below and 3 above.

NOTE: when one close-packed layer is placed on top of another, both types of sites are created. Specifically, there are two 4-fold sites and one 6-fold site per sphere.

Let's look at elements in rock-forming minerals.

O ranion = 0.13 nm

Radius ratios with oxygen:

Si4+ = 0.30nm

Al3+ = 0.47nm

Si fits into range for tetrahedral coordination - usually found tetrahedrally coordinated in silicate minerals. However, Al in boundary region between tetrahedral and octahedral coordination... In natural minerals it is found in either coordination.

Most other common cations in Earth's crust fall in range of octahedral coordination.

Electrostatic valency principle: we can calculate the strength of a bond (its electrostatic valence) by dividing valence by coordination number (CN). As a consequence, in a stable crystal structure the total strength of the valency bonds that reach an anion from all neighboring cations is equal to the charge of the anion. The rule is a direct consequence of ionic bonding... total bonding capacity of a cation is proportional to its charge (z): z/CN

Ex. rutile TiO2 Ti+4 is in octahedral coordination with oxygen. Each Ti-O bond has a strength Z/CN = 4/6 = 2/3. Each oxygen has three neighboring Ti+4 cations such that their collective bond strength (3 x 2/3) equals the oxygen charge of -2. For this reason, anions tend to be locally charge-balanced.

SOME CONSEQUENCES:

A. Geometrical and electrical stability - (Ex. fluorite, CaF₂). Each Ca has 8 fluorine neighbors, while each fluorine has only four Ca neighbors.

rCa = .99A rF = 1.33A rc/ra = .74

Even though relative sizes would allow closest packing, charge balance requires the 2:1 ratio, and thus determines the structure.

Fluorite and halite illustrate another consequence of rule 2, which is that when all ionic bonds have the same strength, anions pack together in a highly symmetrical arrangement, thus these minerals are highly symmetric. Minerals with uniform bond strengths include the oxides, fluorides, chlorides, etc.

In contrast, when there are nonuniform bond strengths, crystal structures have lower symmetry. This is true when structures include small cations of high charge (C⁴⁺, S⁶⁺, P⁵⁺, Si⁴⁺). Additionally, this rule means that the number and kinds of coordination polyhedra that can meet together at a point are severely limited. For example, no more than 2 Si⁴⁺ tetrahedra can share a common oxygen, even though the radius ratio considerations alone would permit three, four or more ... each Si-O bond contributes an electrostatic strength of 4/4 = 1, so that two Si-O bonds will just satisfy the -2 charge of the oxygen. Similarly, exactly three divalent cation octahedra will share a common oxygen with a Si⁴⁺ tetrahedron. Mineral groups included in this category are the carbonates, sulfates, phosphates and silicates.

3) Sharing of polyhedral elements. I. The existence of edges, and particularly of faces, common to two anion polyhedra in a coordinated structure decreases its stability. Direct outgrowth of electrostatic forces... Most stable configuration is when two polyhedra share only a corner, because then the two central cations are as far apart as possible.

The figure above shows that the more anions shared between polyhedra, the closer the positively charged cations. This reduces stability, particularly when the cations are highly charged (e.g., Si⁴⁺).

4) Sharing of polyhedral elements II. In a crystal containing different cations, those of high valency and small coordination number tend not to share polyhedral elements with each other. Corollary of rule three - emphasizes the fact that highly charged cations will be as far apart from each other as possible. Effect stronger if coordination number is low. Ex. - no silicate minerals have edge-sharing or face-sharing Si tetrahedra. However, edge-shared octahedra are common (TiO₂, or, as shown in the diagram below, NaCl), and even face-shared octahedra are found (Fe₂O₃).

5) The principle of parsimony. Number of essentially different kinds of constituents in a crystal tends to be small because, there are only a few types of cation and anion sites. No more than two or three different types of coordination polyhedra in a mineral. The number of crystallographically different sites is thus small - fundamental reason why various cations and anions in chemical fromulas are generally in small integer ratios to each other. Relative abundance controlled by availability of sites in a structure. Thus in structures of complex compositions, a number of different ions may occupy the same structural position (site).

Summary:

It is possible to regard a crystal as being made up of AXn groups that are joined together by sharing corners, edges or faces of coordination polyhedra rather than as individual ions... we'll see a lot more of these. Coordination polyhedra commonly distorted.

1. polyhedral framework structures

Most of rock-forming minerals in this category, especially silicates. All structures are direct consequence of predominantly ionic bonds between constituent ions. As result of bonding, anions tend to group around cations in highly symmetric manner to define coordination polyhedra.

Ex: silica tetrahedron (SiO4)-4

divalent cation octahedra (MgO6)-10

By sharing apical oxygens, polyhedra link together to define a structural frame that possesses at least half of the total bonding energy of the mineral - resulting frame is relatively strong and has important influence on most physical and chemical properties.

2. Symmetrically packed structures

Either bonds between atoms are nondirectional or bond directions are highly symmetrical.

Ex. metallic bond, also many examples of covalent and ionic.

Atoms form highly symmetrical structures in which atoms packed together in symmetrical ways:

a) monatomic (native metals) - if atoms are in contact in and between sheets - highly efficient packing called closest packing. If atoms lose contact within sheets but retain contact between sheets - close-packed.

b) mulitatomic - both cations and anions... many oxides, sulfides, halides and most of important silicates considered as framework are in this category. Anions are in symmetrically packed sites, and cation soccupy voids between. symmetry of anion packing is basic characteristic.

3. Molecular structures

Composed of atoms characterized by strongly directional and low symmetry bonds. Asymmetric bonds form strong clusters, chains and layers of atoms that behave as discrete units connnected by much weaker bonds to for 3-D networks. Ex: ice