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 T = tetrahedral sites (4-fold) and O = octahedral sites (6-fold) 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
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 = R c /R a 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 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. The maximum number is limited by the requirement that cations maintain contact with their coordinating anions 6 possible geometries
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 Fe 2+, and Mg 2+ can be in 6-fold and 8-fold coordination w/o 2- ; Al 3+ may be in 4-fold and 6-fold coordination Si 4+ is only found in 4-fold coordination -> NB that Si-O bonds are ~50% ionic and 50% covalent, so RR and sp 3 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 HCPtype 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 CaCO 3. Also observed in sulphates and phosphates with SO 4 2- and PO 4 3- 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 O 2- 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 Edge Sharing Face Sharing Least Stable Corner Sharing Most 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 CaSO 4 (anhydrite): Isolated SO 4 2- groups are linked together through the Ca 2+ cations
Simple Examples of Pauling s Rules NB that Si 4+ is in a 4-fold or tetrahedral site, while Mg 2+ (Fe 2+ ) is in an 6-fold or octahedral site. Mesodesmic Bonding Silicate structures. In this example we see the olivine structure. Note different ways in which the structure may be depicted.
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.