1.  Atomic and Ionic Radii

• Atomic radius- radius of the maximum distance that an e- (charge density) can exist in the outermost shell. Referred to as metallic radii, based on ½ bond length in pure metals.
• Ionic radius- "effective" radius; ions of the same element have variable radii depending on (1) charge, (2) type and # of surrounding ions. For example: Mg²⁺ = 0.88Å ionic; = 1.36Å covalent; = 1.60Å metallic.
• Ionic radii of cations are always smaller than atomic (metallic) radii because ions have lost e^- (i.e. e^- cloud is smaller). Opposite for radii of anions.
• Ionic radii increase downward in columns (groups) on Periodic Chart with increasing atomic # (Z); decrease across rows (periods) until reaching Group VI cations. Oxidized species of same element generally have smaller radii due to greater attractive force per unit e^- from excess protons in the nucleus.

• Crystals can be envisioned as 3D networks of anions with cations in the spaces between anions (interstices) at regularly spaced intervals.
• Coordination Number (C.N.):  # of anions that a particular cation bonds to.

Example 1: Si tetrahedron: SiO4^4-, Si has C.N. = 4 or is said to be in 4-fold coordination (Si⁴⁺ is bonded to four O^2-).
Example 2: halite (NaCl): Na+ is surrounded by six Cl- (and vice versa), Na has C.N. = 6 or 6-fold coordination (octahedral coordination).
• Chemical Formulas are sometimes written to indicate cation C.N. Example: magnetite Fe^IVFe₂^VIO₄; where roman numerals signify C.N.
• Coordination Polyhedron:  geometrical shape that describes 3D arrangement of anions surrounding a central cation.

• Pauling’s Rule #1 (Radius Ratio Concept)- ratio of cation radius to anion radius (R_cation/R_anion or R_A+/R_X-). Used as a 1^st approximation in predicting the C.N. of a cation. Basic point is that as cation radius increases, so does C.N.

• Pauling’s Rule #3- coordination polyhedra become increasingly unstable when they share of anions along corners Þ edges Þ faces.
• Tetrahedrons: corner sharing is common, stable; edge sharing is possible, but not common, tetrahedra are distorted; face sharing is not observed, cation distances are too close, unstable.
• Octahedrons: corner, edge, and face sharing is common, stable. Why? Because cation distances are larger in octahedral sharing due to lower charge on cation.

 

• Visualize all mineral structures as essentially a closest packed array (either HCP or CCP) of anions with cations regularly spaced throughout in the voids (interstices) between cations.
• These interstitial positions (or sites) are usually either in 4-, 6-, or 8-fold coordination (depending on the radius ratio) and may or may not be completely filled with cations (i.e. some may be empty). In general, B voids are in 4-fold coordination (tetrahedral); C voids are in 6-fold (octahedral).

• Structural types depend on (1) radius ratio or ‘effective’ radius; (2) charge; (3) vacancies (empty sites) and/or ion substitutions.

• Based on silica tetrahedron (SiO₄)^4-, it is the fundamental "packing" unit of all silicates. Minerals must be electrically neutral (charge balanced). Silica tetrahedra are balanced by: (1) bonding to cations; and/or (2) sharing oxygens between tetrahedra (polymerization).
• Polymerization is the basis for silicate subclasses.
• Isolated (nesosilicates)- (SiO₄)^4-; no polymerization; cations link tetrahedra,         (olivine(Fe,Mg)₂SiO₄; garnet).
• Paired (sorosilicates)- (Si₂O₇)^6-, one O^2- is shared on corner of two tetrahedra; (hemimorphite Zn₄(Si₂O₇)(OH₂)· H₂O).
• Ring (cyclosilicates)- (Si_xO_3x)^2x- where x ³ 3, two O^2- are shared on each tetrahedron that link in rings (beryl Be₃Al₂Si₆O₁₈).
• Single-chains (inosilicates)- (Si₂O₆)^4- or (SiO₃)^2-, two O^2- are shared on each tetrahedron that link in infinite single chains (pyroxenes).
• Double-chains (inosilicates)- (Si₄O₁₁)^6-, two O^2- are shared on some tetrahedron and some share three O^2- that link in infinite double chains (amphiboles).
• Sheets (phyllosilicates)- (Si₂O₅)^2-, three O^2- are shared on all tetrahedron that link in planes (micas, clays).
• 3D networks (tectosilicates)- (SiO₂)⁰, all four O^2- are shared on all tetrahedron that link in infinite 3D networks (quartz, feldspars).

• Complexity in Silicate Structures (and chemical formulas)
• presence of OH- or other ions/molecules (i.e. muscovite KAl₃Si₃O₁₀(OH)₂).
• O^2- that is not associated with the silica tetrahedron (i.e. kyanite Al₂(SiO₄)O; normally written out as Al₂SiO₅).
• Some minerals don’t fit nicely into silicate subclasses. For example, minerals that have paired and isolated tetrahedra; (i.e. clinozoisite Ca₂Al₃O(SiO₄)(Si₂O₇)(OH)).
• Elemental substitutions, for example Fe²⁺ and Mg²⁺ are about the same size (and charge), therefore they can substitute for each other in the octahedral site in the olivine structure (Fe,Mg)₂SiO₄.

• Elemental Substitutions (or solid solution)- ions (both anions and cations) of similar size and charge may occupy similar sites in crystal structures. There 3 main types (substitutional, interstitial, and omission).
• Substitutional:  there are simple and coupled substitutions.
• Simple: simple 1:1 exchange of ions that have like size and charge. Examples, Ca²⁺, Mn²⁺, Fe²⁺, Mg²⁺ in garnet, pyroxene; K⁺, Na⁺ in feldspar, amphibole; Fe³⁺, Al³⁺ in garnet, spinel; Fe²⁺, Mg²⁺ in olivine. Usually a complete solid solution (substitutions occur over the entire compositional range). For example, the full compositional range between 100% pure Mg-olivine (forsterite Mg₂SiO₄) and 100% pure Fe-olivine (fayalite Fe₂SiO₄) is possible.
• Coupled: this involves the exchange of ions with different charges. For example, feldspars have complete solid solution between Ca- and Na-rich varieties (CaAl₂Si₂O₈ Û NaAlSi₃O₈) where Ca²⁺ substitutes for Na⁺. In order to maintain total charge balance, other ions must also be exchanged. Thus, Al³⁺ Þ Si⁴⁺. The total exchange is Ca²⁺Al³⁺ Þ Na⁺Si⁴⁺.
 
• Interstitial:  large molecules (H₂O, CO₂) or ions (K⁺, OH^-, Rb⁺, Cs⁺, etc.) will sometimes occupy large voids (interstices) in crystal structures. Common for ring silicates and zeolite minerals (tectosilicates).

• Omission:  where a highly charged cation replaces two or more cations of lower charge. In order to maintain charge balance, a site vacancy in the crystal structure. For example, 2K⁺ Þ Pb²⁺ +  in feldspars;  Si⁴⁺ = K⁺Al³⁺ in amphiboles ( represents the vacancy).