Transcript Chemical Bonding - California Institute of Technology

Announcement
Date
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
ABCABCABC….
What are the unit cell dimensions?
a
face diagonal is
close-packed
direction
2a  4R
a  2 2R
|a1| = |a2| = |a3|
a1 = a2 = a3 = 90°
Cubic Close-packed Structure
|a1| = |a2| = |a3|  2 2R
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)
a
coordination number?
8
atoms per unit cell?
2
lattice points per unit cell?
2
atoms per lattice point?
1
another example of a non-primitive cell
body diagonal is closestpacked direction
4

a

R
3a  4R
3
|a1| = |a2| = |a3|
a1 = a2 = a3 = 90°
lattice type of ‘CLP’ is “body-centered cubic”
no common name that distinguishes
lattice type from structure type
Summary: Common Metal Structures
hcp
bcc
ccp (fcc)
ABCABC
ABABAB
c
Unit Cell
b
a
a
g
b
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
b
a
– Points of identical environment
– Related by translational symmetry
– Lattice = array of lattice points
c
a
g
b
Crystal system
Lattices
triclinic
simple
base-centered
monoclinic
Convention:
b = 90°
instead of a
a = 90°
simple
base-centered body-centered face-centered
orthorhombic
a = b = g = 90°
6 or 7
crystal
systems
hexagonal
g = 120°
c
14 lattices
a
a
rhombohedral
(trigonal)
a = b = ga
simple
body-centered
simple
body-centered face-centered
tetragonal
a = b = g = 90°
a=b
cubic
(isometric)
a = b = g = 90°
a=b=c
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)
2
Geometry
min rc/RA
none
(linear)
3
0.155
(trigonal planar)
4
0.225
(tetrahedral)
Consider: CN = 6, 8 12
Octahedral Coordination: CN=6
rc + R A
2RA
a
rc + RA
2RA
2RA  2  rc  RA 
 rc  RA  
RA
2
 2
2
rc
 2  1  0.414
RA
Cubic Coordination: CN = 8
2RA  a
2(RA  rc )  3a
2(rc + RA)
a
 rc  RA  
RA
3
rc
 3  1  0.732
RA
2RA
Cuboctahedral: CN = 12
2RA
rc + RA
rc + RA = 2RA
rc = RA  rc/RA = 1
CN
6
Geometry
min rc/RA
0.414
(octahedral)
8
0.732
(cubic)
12
1
(cuboctahedral)
Radius Ratio Rules
CN (cation)
2
Geometry
linear
min rc/RA (f)
3
trigonal planar
0.155
4
tetrahedral
0.225
6
octahedral
0.414
8
cubic
0.732
12
cubo-octahedral
1
none
if rc is smaller than fRA, then the space is too big and the structure is unstable