AnnouncementDate Change Date Change

10/13/10 Nick Heinz 11/05/10 8:30am start

10/18/10 8:30am start 11/08/10 8:30am start

10/20/10 8:30am start 11/10/10 no lecture

10/22/10 no lecture 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 ABCABC….C

What are the unit cell dimensions?

face diagonal is close-packed direction

a

2 4a R

2 2a R

|a1| = |a2| = |a3| 1 = 2 = 3 = 90°

Cubic Close-packed Structure

only one cell parameter to be specified

2 2R|a1| = |a2| = |a3|

1 = 2 = 3

atoms per unit cell?

coordination number?

lattice points per unit cell?

a unit cell with more than one lattice point is a non-primitive cell

12

atoms per lattice point?

4

4

1

CCP structure is often simply called the FCC structure (misleading)

lattice type of CCP is called “face-centered cubic”

CCP

Cubic “Loose-packed” StructureBody-centered cubic (BCC)

body diagonal is closest-packed direction

a

3 4a R4

3a R

|a1| = |a2| = |a3|

1 = 2 = 3 = 90°

atoms per unit cell?

coordination number?

lattice points per unit cell?

8

atoms per lattice point?

2

2

1

another example of a non-primitive cell

no common name that distinguishes lattice type from structure type

lattice type of ‘CLP’ is “body-centered cubic”

Summary: Common Metal Structures

Unit Cell

a

b

c

hcp ccp (fcc) bcc

ABABABABCABC not close-packed

• space filling• defined by 3 vectors• parallelipiped• arbitrary coord system• lattice pts at corners +

The Crystalline State• Crystalline

– 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

Crystal system Lattices

triclinic

simple base-centered

monoclinic = 90°

Convention: = 90° instead of

simple base-centered body-centered face-centered

orthorhombic = = = 90°

hexagonal = 120° c

a

a

rhombohedral (trigonal)

= =

simple body-centered

tetragonal = = = 90°

a = b

simple body-centered face-centered cubic

(isometric) = = = 90°

a = b = c

6 or 7 crystal systems

14 lattices

Ionic Bonding & Structures

Which is more stable?

+ –

–

––

–

–

+ –

–

––

–

–

Isotropic bonding; alternate anions and cations

––

–

– –

–+

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

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture3/Lec3.html#anchor4

central atom drawn smaller than available space for clarity

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

2RA

rc + RA

2 2A c AR r R

22

2c A

A

r R

R

2 1 0.414c

A

r

R

rc + RA

2RA

a

Cubic Coordination: CN = 8

2RA

2(rc + RA)

2 AR a

3c A

A

r R

R

3 1 0.732c

A

r

R

a

2( ) 3A cR r a

Cuboctahedral: CN = 12

rc + RA = 2RA

rc = RA rc/RA = 1

2RA

rc + RA

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