Announcement Date Change 10/13/10 Nick Heinz 11/05/10 8:30am start 10/18/10 11/08/10 10/20/10 11/10/10 no lecture 10/22/10 11/29/10 TBA Subject to change

Close-packed Structures Metallic materials have isotropic bonding In 2-D close-packed spheres generate a hexagonal array In 3-D, the close-packed layers can be stacked in all sorts of sequences Most common are ABABAB.. ABCABCABC… Hexagonal close-packed Cubic close-packed

AB C ABCABC…. Cubic???

a What are the unit cell dimensions? face diagonal is close-packed direction |a1| = |a2| = |a3| a1 = a2 = a3 = 90°

Cubic Close-packed Structure |a1| = |a2| = |a3| a1 = a2 = a3 only one cell parameter to be specified coordination number? 12 atoms per unit cell? 4 lattice points per unit cell? 4 atoms per lattice point? 1 CCP a unit cell with more than one lattice point is a non-primitive cell lattice type of CCP is called “face-centered cubic” CCP structure is often simply called the FCC structure (misleading)

Cubic “Loose-packed” Structure Body-centered cubic (BCC) coordination number? 8 atoms per unit cell? 2 a lattice points per unit cell? 2 atoms per lattice point? 1 another example of a non-primitive cell body diagonal is closest-packed direction lattice type of ‘CLP’ is “body-centered cubic” no common name that distinguishes lattice type from structure type |a1| = |a2| = |a3| a1 = a2 = a3 = 90°

Summary: Common Metal Structures hcp ccp (fcc) bcc ABCABC not close-packed ABABAB c space filling defined by 3 vectors parallelipiped arbitrary coord system lattice pts at corners + Unit Cell a b b g a

The Crystalline State Crystalline Three translation vectors define: Periodic arrangement of atoms Pattern is repeated by translation Three translation vectors define: Coordinate system Crystal system Unit cell shape Lattice points Points of identical environment Related by translational symmetry Lattice = array of lattice points a b c g

6 or 7 crystal systems 14 lattices

Ionic Bonding & Structures Isotropic bonding; alternate anions and cations Which is more stable?  + – + – – + Just barely stable

Ionic Bonding & Structures Isotropic bonding Maximize # of bonds, subject to constraints Maintain stoichiometry Alternate anions and cations Like atoms should not touch ‘Radius Ratio Rules’ – rather, guidelines Develop assuming rc < RA But inverse considerations also apply n-fold coordinated atom must be at least some size

central atom drawn smaller than available space for clarity http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture3/Lec3.html#anchor4

Radius Ratio Rules CN (cation) Geometry min rc/RA 2 none (linear) 3 0.155 (trigonal planar) 4 0.225 (tetrahedral) Consider: CN = 6, 8 12

Octahedral Coordination: CN=6 rc + RA 2RA rc + RA a 2RA

Cubic Coordination: CN = 8 2(rc + RA) a 2RA

Cuboctahedral: CN = 12 2RA rc + RA rc + RA = 2RA rc = RA  rc/RA = 1

CN Geometry min rc/RA 6 0.414 (octahedral) 8 0.732 (cubic) 12 1 (cuboctahedral)

Radius Ratio Rules CN (cation) Geometry min rc/RA (f) 2 linear none 3 trigonal planar 0.155 4 tetrahedral 0.225 6 octahedral 0.414 8 cubic 0.732 12 cubo-octahedral 1 if rc is smaller than fRA, then the space is too big and the structure is unstable