· Controls of Crystal Structure
- largely from bonding
- leads to convenient organization of crystal types:
- Metallic, Covalent, Molecular, Ionic
 

Metallic Bonding

· Atoms pack in regular arrangement to minimize open space
 - Hexagonal closest packing
 - Cubic closest packing

· Closest way to pack spheres of equal size
 - Assume spheres are in layers
 - Difference is the way the layers are stacked
 - In both cases, each sphere has 12 nearest neighbors
 

·  Hexagonal closest packing
- Every second layer aligned
- ABABAB structure
- Hexagonal symmetry

· Cubic Closest packing
- Every third layer aligned
- ABCABC structure
- face-centered cubic lattice
- Native metals good example
- if same size and chemistry can form alloys, e.g. gold and silver
 

· Body centered cubic packing
 - e.g. Fe
- atoms at nodes of body-centered cubic lattice
- Arrangement leaves only 8 rather than 12 neighbors

· Resulting physical properties
- metals often high density relative to ionic-bonded crystals
- conduct electricity and malleable
 

Covalent bonding

· Atoms need to be in specific arrangements
 - orbitals must overlap
 - Assume close packed arrangement
- Typically hard and high melting points

Molecular Crystals

· Discrete molecules packed in systematic way
- molecules bonded internally with covalent bonds
- molecules held together with van der Waals or Hydrogen bonds
- e.g Graphite and Ice
 

Ionic bonding

· Oxygen most abundant anion
· Oxygen bonds mostly ionic
- 50% with Si
- higher per cent with other cations (Al, Fe, Mg, Ca, Na, K)
· Assume most mineral bonds are ionic
- ionic bonds not directions
- don’t care about overlapped orbitals
- only consider geometric arrangements
 

· Set of rules developed to describe the arrangements: Pauling’s Rules:

(1) Coordination Principle
(2) Electrostatic Valency Principle
(3) Sharing of Polyhedral elements I
(4) Sharing of Polyhedral elements II
(5) Principle of Parsimony
 

Coordination Principle (rule 1)

· Packing arrangements depend on ionic size

· Coordination number = number of ions surrounding central ion.
 - usually cation surrounded by anions
 - common numbers: 12, 8, 6, 4, 3, 2
 

· Coordination polyhedra = shape defined by anions coordinating around cation
- depends on coordination number
- 8 = cube
- 6 = octahedron
- 4 = tetrahedron
- 3 = triangle
- 2 = line

· Coordination number can be calculated by radius ratio, RR

- RR = R_c/R_a
 

· Irregular numbers and shapes

- cations can coordinate with 5, 7, 9, 10, or 11 anions
- Shapes of regular coordination polyhedra may be distorted

· Irregularities caused by many factors

- often related to bonds being covalent (i.e. directional)
- also if there are strongly bonded anion group
 CO₃^2-, SO₄^2-, SiO₄^4-
 

· Most coordination is between O and common cations
 - effective radii are known for ions
- possible to predict coordination between ions
- Some cations fit in more than one coordination, e.g.
Al³⁺ - 4 or 6 fold
Fe²⁺ and Mg²⁺ - 6 or 8 fold
Na⁺ and Ca²⁺ - 8 or 12 fold
 

Electrostatic Valency Principle (rule 2)

· Bonding capacity is proportional to oxidation state (charge) and coordination number:
 - called electrostatic valence bonds (evb)

evb = ion charge/CN

Two broad types: Uniform and non-uniform bond strengths
 

Uniform bond strength - Isodesmic

• All bonds between cations and anions have same strength

• Anions tend to pack into highly symmetrical arrangements
 - typically isometric, tetragonal or hexagonal
 
 

Non-uniform bond strength

· Where certain bonds stronger than others

• Form in anionic groups

· Commonly, forms in small cations with high charge
 - e.g. C4+, S6+, P5+, Si4+

· Commonly bonds with O2-
 -e.g. CO₃^2-, SO₄^2-, PO₄^3-, SiO₄^2-
 

· Anisodesmic
- some anion-cation bonds take more than half of the anion charge
- soluble into cations and cation groups
 - e.g. calcite
- also Sulfates (SO₄^2-) and Phosphates (PO₄^3-)

· Mesodesmic
- some anion-cation bonds take exactly half the anion charge
- e.g. Silicon bonds with 4 oxygen
- Silica tetrahedron
- Arrangement of tetrahedron is basis of silicate mineral classification
 

Sharing of polyhedral elements I (rule 3)

- cations generally share only single anions (points)
- occasionally will share two anions (edge)
- never share 3 (face)
- Reason is that the cations have to be separated by sufficient difference

Sharing of polyhedral elements II (rule 4)

• highly charged cations are not placed near each other in a structure

- like charges repel, so highly charged ions are separated by large distances
- small high charged cations have low coordination number
 * use up more than half of anion charge
 * other bonded cations have to have low charge and thus are large
 

Principle of Parsimony (rule 5)

• Number of fundamentally different sites for a mineral is small
- typically fewer than 4 different polyhedron for cations
- means there are small integer ratios of elements in mineral formulas
 

Examples of applications of Pauling’s rules

• Halite:

- Uniform Isodesmic Bonding (rule 2)
- Cubic closest packing
- RR = Cl/Na = 1.81Å/1.16Å = 0.64
 * Coordination number = 6 (rule 1)
- Filling of octahedrons mean that Na⁺ polyhedron has to share edges (rule 3)
 * Ok because Na small charge (+1)
- Very simple structure (e.g. rule 5)

• Anhydrite (CaSO4)

- Nonuniform anisodesmic bonding
- S⁶⁺ is tetrahedrally coordinated with 4 O^2-
- Rule 2: evb = 1.5 on S-O bonds, so there is –1/2 charge for bonding with Ca²⁺
- Ca-O coordination usually 8 fold (rule 1)
- Rule 2: evb = 0.25 on Ca-O bonds
 * must arrange 8 O around cations
- Insufficient space for cubic coordination
- Ca coordination polyhedron distorted cube