Crystal Structure and Chemistry

•  Controls on Crystal Structure –  Metallic bonding – closest packing –  Covalent bonding – depends on orbital overlap and geometry –  Ionic bonding

•  Pauling’s Rules –  Coordination Principle and Radius Ratios –  Electrostatic Valency Principle –  Sharing of Polyhedral Elements –  Principle of Parsimony

•  Applications of Pauling’s Rules •  Isostructural Minerals •  Polymorphism •  Mineral Classification and Compositional Variation

Metallic Bonding and Closest Packing •  Metals tend to pack

together closely due to free movement of valence electrons

•  Two Forms –  Hexagonal Closest

Packing (HCP) -> AB layers

–  Cubic Closest Packing (CCP) -> ABC layers. Along {111} plane FCC lattice

–  Each sphere (atom) in contact with 12 others

•  Au and Ag are examples T = tetrahedral sites (4-fold) and O = octahedral sites (6-fold)

Fe metal and Body-Centered Cubic

•  Body-Centered Cubic Packing is another form of closest packing

•  Lower density than HCP or CCP

•  Fe metal is found in this form

•  Each sphere in contact with 8 rather than 12 other atoms in structure

Structural Controls on Ionic Bonding •  Pauling’s Rules

–  Rule #1: Coordination Principle and Radius Ratios •  Each cation is surrounded by a coordination polyhedron (CP). The form of the

CP is defined by the cation and anion radii and the number of anions in the CP is fixed by the relative size of the cation and anion

•  The CP and the coordination number (CN) are related and yield well-defined geometric relationships

–  Rule# 2: Electrostatic Valency Principle •  The total strength of the valence bonds that reach an anion from all nearest

neighbor cations is equal to the charge of the anion

–  Rule#3: Sharing of Polyhedral Elements I •  Face and edge sharing of individual CP within a crystal structure tends to

decrease its overall stability

–  Rule#4: Sharing of Polyhedral Elements II •  In a crystal with different cations, those with large valence (high positive charge)

and small coordination number tend to not share CP elements (edges and faces)

–  Rule#5: Principle of Parsimony •  The number of different constituents in a crystal structure tends to be small

Radius Ratios and Coordination

•  Radius ratio (RR) is simply defined as the ratio of the cation radius to the anion radius:

–  RR = Rc/Ra

•  The minimum number of anions that will coordinate a specific cation is NOT strictly limited, but cations tend to bond with as many anions as possible

•  The maximum number is limited by the requirement that cations maintain contact with their coordinating anions

•  6 possible geometries

CN and CP are related. Note that a “cubic” coordination polyhedron has a coordination number of 8, while a “octahedral” CP has a CN of 6.

Coordination Polyhedra – 6 forms

12-fold coordination occurs when atoms are approximately the same size (metals), while decreasing CN is associated with an decrease in the RR (larger anion relative to cation)

Expected CN’s for common elements

•  Oxygen is the most common element in the crust and therefore the most common anion in all minerals

•  Note that common cations of Fe2+, and Mg2+ can be in 6-fold and 8-fold coordination w/O2-; Al3+ may be in 4-fold and 6-fold coordination

•  Si4+ is only found in 4-fold coordination -> NB that Si-O bonds are ~50% ionic and 50% covalent, so RR and sp3 hybrid orbitals both work together to yield stable silicon tetrahedral geometry

Electrostatic Valency Principle For an ionic bond, the bonding capacity is proportional to is oxidation or valence state (that is charge).

Electrostatic bond strength (also known as electrostatic valence bonds or evb) is calculated as:

Bond Strength (evb) = ionic charge / coordination number

Uniform bond strength, called isodesmic. Common to ionic minerals with a single cation and anion and some with multiple cations and/or anions. Yield highly symmetrical crystal structures since all bonds are of uniform strength, CCP or HCP-type structures in the isometric, tetragonal, and hexagonal crystal systems.

Nonuniform bond strength – two types: -  Anisodesmic: some cation-anion bonds take more than 50% of the anion charge; commonly found in minerals with highly charged cations, such as CaCO3. Also observed in sulphates and phosphates with SO4

2- and PO43- anionic groups

-  Mesodesmic: some cation-anion bonds exactly 50% of the anion charge; best example is the silicon tetrahedron, with each Si-O evb = 1

Anisodesmic and Mesodesmic Examples

Anisodesmic: Carbonate group Note that the C-O bonds are much stronger than the Ca-O bonds at an evb of 1.33 vs. 0.33 Mesodesmic: Silicon Tetrahedron

Note that this allows each O2- to bond with other oxygen anions.

Sharing of Polyhedral Elements Shared anion at corner keeps other anions and two cations farthest away to minimize repulsion

Corner Sharing Most Stable

Edge Sharing Face Sharing Least Stable

Corner -> Edge -> Face sharing progression brings cations, with high positive charge density into closer proximity. This in turn tends to lower the overall stability of the crystal structure.

Simple Examples of Pauling’s Rules

Uniform Isodesmic Bonding in NaCl: Shared edge of Cl-

octahedra keeps like charges maximum distance away

Anisodesmic Bonding in CaSO4 (anhydrite): Isolated SO4

2- groups are linked together through the Ca2+ cations

Simple Examples of Pauling’s Rules

Mesodesmic Bonding Silicate structures. In this example we see the olivine structure. Note different ways in which the structure may be depicted.

NB that Si4+ is in a 4-fold or tetrahedral site, while Mg2+ (Fe2+) is in an 6-fold or octahedral site.

Isomorphism vs. Polymorphism

Isomorphism is when two different minerals have the same crystal structure. For example Halite, NaCl and Gelena, PbS. May also have isostructural groups that are related by a common anion or anion group, such as the carbonates, sulphates, and phosphates.

Polymorphism is when a chemical compound may crystallize in more than one structure. Common examples include calcite and aragonite and diamond and graphite.