Lecture 4: Crystal Chemistry

Atoms, Periodic Table

Bohr Atom - electron shells and orbits; electron transitions absorb or emit characteristic amounts of energy

Schrodinger Atom - probability of finding an electron at a spot; s, p, d, f subshells

Ions - cations and anions; periodic relationships; electronegativity

Atomic and Ionic Radii

It should be apparent that an isolated atom has no definite size, because the probability of finding an electron is zero only at a radius of infinity. In a crystal, however, the electrons of any atom are constrained by its neighboring atoms. It is therefore possible to assign to each ion a "radius" such that the sum of the radii of two adjacent atoms equals the observed inter-atomic separation/distance.

Ionic radii depend strongly upon the valence state of the ion. We would intuitively predict, for example that the radius of Na+ should be less than that of Na0. Conversely, if we add electrons to a neutral atom, the radius of the electron cloud should increase. This concept is well illustrated by sulfur which can range in size from 0.2 A in the +6 valence, to 1.04 A for neutral S, and 1.7A for S in the -2 valence state.

Ionic radii also depend on the configuration of the ion relative to its nearest neighbors, chiefly in terms of the number of surrounding ions. The number of nearest neighbors bonded to an ion is referred to as the coordination number. Your text lists the radii of common ions in their most common coordination numbers (p.188). Examination of this table (4.8) reveals a wide range in ionic radii. Silicate minerals, for example, are composed of small cations (e.g., Si4+ in 4-fold coordination, r = 0.26 A), intermediate cations (e.g., Mg2+ in 6-fold coordination, r = 0.72 A), and large cations (e.g., K+, r = 1.38 - 1.64 A) bonded to large anions of oxygen (r ~ 1.4 A).

Radius Ratio

Paulings Rules

How do coordination polyhedra or environments relate to real crystals?
How do we build crystals?

The following is essentially independent of whether the bonds are ionic or covalent although we will find it easiest to think about these properties as being ionic.

Closest Packing: HCP and CCP

cation size
anion size
radius ratio
coord. #
geometric shape
1.0
1.0
1.0
12
cuboctahedron
1.0
1.3
0.7320
8
cube
1.0
2.5
0.414
6
octahedron
1.0
4.5
0.225
4
tetrahedron
1.0
6.5
0.155
3
triangular
1.0
>6.5
<0.155
2
linear

5, 7, 9, and 10 fold coordination are possible in non-closest packed structures

Site size is proportional to CN environment, oxidation state:

Oxygen in CN=2 = 1.35 Å
Oxygen in CN=8 = 1.42 Å


Fe = 1.26 Å
Fe ++ = 0.78 Å
Fe +++ = 0.65 Å

Pauling noticed that size determines CN (Rule #1).

In the coordination polyhedron of anions about each cation, the cation-anion distance is constrained by the radius sum and the coordination number of the cation is controlled by the radius ratio.
Ex: Mg:O .72/1.36 = .53 therefore 6 C.N.

Pauling Rule #2: Electrostatic Valency Principle
Sum of the bond strengths reaching a cation from the anions must equal the charge on the cation.
Pauling Rule #3: Sharing of edges and especially faces decreases the stability of a structure.
Pauling Rule #4: When stacking polyhedra in crystals using different cations, there is a tendency for those individual atoms with high valence and small coordination number NOT to share edges and faces.
Rules 1-4 maximize cation-anion attraction and minimize cation-cation and anion- anion repulsion forces.
Pauling Rule #5: Principle of Parsimony
Nature is stingy. The # of essentially different types of constituents is small (due to limited # of different sites in close packed atoms).
Problems with the Rules:

 

 

 

Quick Bonding Review

Before getting going here, lets quickly review the different bonding types we need to be concerned with in minerals (should have covered this on day one):