1. 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

2. 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

3. AB C
ABCABC….
Cubic???

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

5. 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)

6. 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°

7. 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

8. 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

9. 6 or 7 crystal systems
14 lattices

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

11. 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

12. central atom drawn smaller than available space for clarity

13. 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

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

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

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

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

18. 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