Introduction to Mineralogy Dr. Tark Hamilton Chapter 4: Lecture 10 The Chemical Basis of Minerals (coordination polyhedra & lattice energy) Camosun College GEOS 250 Lectures: 9:30-10:20 M T Th F300 Lab: 9:30-12:20 W F300

Principles of Mineral Lattices SPACE is used most efficiently SYMMETRY is the greatest possible CONNECTION is maximized (greatest CN)

CsCl Formula unit 4 NaCl Formula unit 4 ZnS Formula unit

Radius RatioCoordination no. Binary (AB) Structure- type r + /r - = 1 12none known 1 > r + /r - > CsCl > r + /r - > NaCl > r + /r - > ZnS Limiting Radius Ratios – Coordinating anions contact each other & cation (Pauling’s 1 st rule)

Goldschmidt’s Ionic Model Ions are charged Incompressible Non-polarizable spheres Modifications: –Most charge resides in a central hard core –Soft outer sphere has low electron density

Pauling’s 1 st Rule Coordination Polyhedra –"A coordination polyhedron of anions is formed around every cation (and vice-versa) - it will only be stable if the cation is in contact with each of its neighbours. –Ionic crystals may thus be considered as sets of linked polyhedra. –The cation-anion distance is regarded as the sum of the ionic radii.

Shape & Coordination No. Molecular Materials Absolute coordination numbers are controlled by valency (VSEPR): H 2 O, (CO 3 ) -2, (SO 4 ) -2 etc. Non-Molecular Materials Valency has only an indirect bearing on coordination number e.g. Na I Cl, Mg II O, Sc III N, Ti IV C all have the Rock Salt (6:6) Structure despite change in valency and from predominantly ionic to covalent bonding Ionic Size does influence coordination number The Coordination Number of the Cation will be Maximized subject to maintaining Cation-Anion Contact, according to the ratio of the ionic radii, r+/r- & geometry.

What is the Numerical Value of ionic radius?

What's the Numerical Value of a specific Ionic Radius? Ionic Radii in most scales do not generally meet at experimental electron density minima, because of polarization of the anion by the cation The various scales are designed to be self-consistent in reproducing r o = r+ + r- Ionic radii change with coordination number –r 8 > r 6 > r 4 {use the appropriate one!} Use the same scale for cation and anion

Test with Structures of Alkali Halides Radius Ratio Test for Alkali Halides

Do the Radius Ratio Rules Work? Graph compares structures (CsCl / NaCl) with predictions by radius ratio rules from r+/r- {r-/r+ if cation is larger} For Li+ and Na+ salts, ratios calculated from both r 6 and r 4 are indicated Radius ratios suggest adoption of CsCl structure more than is observed in reality NaCl structure is observed more than is predicted Radius ratios are only correct ca. 50% of the time, not very good for a family of archetypal ionic solids - random choice might be just as successful as radius ratio rules and saying that all adopt the NaCl structure more so! Is the Goldschmidt cation-anion Contact Criterion all there is to it? - No!

Lattice Energy Qu. Why is highest possible Coordination adopted? Ans. Greater Madelung Potentials!Madelung Structure TypeMadelung Constant (A) CsCl1.763 NaCl1.748 ZnS (Wurtzite Hi-T°C)Wurtzite1.641 ZnS (Sphalerite Lo-T°C)1.638 u = (A q 1 q 2 )/r

Plots of Coulombic Madelung Energy vs Radius Ratio Tetrahedral Octahedral Cubic <0.414 <0.732

Lattice Energy and Madelung Constants: Madelung energy rises as r + /r - falls until the structure can no longer support cation-anion contact. At the geometric limiting radius ratio, the Madelung energy remains constant as r + /r - falls, because r o ­ r + + r - but instead is limited by "anion-anion contact". Plot indicates structural transitions do not occur at limiting radius ratios NaCl CsCl transition at r + /r - Å 0.71 ZnS NaCl transition at r + /r - Å 0.32 Taking into account that r 8 > r 6 indicates that the CsCl structure is never favourable (dotted line) CsCl structure is only adopted where 8:8 coordination maximizes Dispersion Forces NaCl structure is highly favoured by Covalency best utilization of Cl - p 3 orbitals

Compression of ions under pressure:- Expect CsCl structure to be favoured at high pressuresCsCl structure e.g. RbCl undergoes NaCl CsCl structural transition at 5~20 kbar

Structure Maps Plots of r A versus r B with structure-type indicated Mg 2 SiO 4 Be 2 SiO 4 Thenardite