Closest Packing
Atoms can be treated approximately as hard spheres for
many types of bonding.
–
+
d
Covalent bond
d
Ionic bond
d
Metallic bond
Different types of bonding
The absence of preferred directional bonding leads to closest packing.
packing.
Closest Packing
CloseClose-packing – Most efficient way of packing equalequal-sized
spheres
Primitive packing
# near neighbors = 4
Close packing
# near neighbors = 6
Which packing is most efficient?
1
Closest Packing
Packing spheres in 3D Space
B
A
t+
t
o +
t–
t–
t+
t+
o
o
t–
t+
t+
o
t+
t+
t–
o
t–
t+
t+
What is the ratio of tetrahedral
to octahedral sites?
2(t) to 1(o) per packing sphere
(2t is 1 t+ + 1 t–)
Tetrahedral
cavity
Octahedral
cavity
Closest Packing
Packing spheres in 3D Space – 3rd layer
B
c
a
a
c
c
a
a
c
A
c
a
The next layer can pack on top of either the a or c sites
If a, the ABAB pattern is called Hexagonal Close Packed
(hcp)
hcp)
If c, the ABCABC pattern is called Cubic Close Packed
(ccp)
ccp)
2
Closest Packing
The Unit Cell of Cubic Close Packing (ABCABC)
[111]
[110]
a [001]
A [100] c
C
B
Face centered cubic
[010] b
The close packed layers stack along the body diagonal of the
cubic unit cell - [111] direction
Closest Packing
The Unit Cell of Hexagonal Close Packing (ABAB)
A
B
c [010]
[001]
b
A
Hexagonal
[100] a
The close packed layers stack along the [001] direction
3
In-Class Problem Set
Calculate the shortest UU-U distance for the 2D projection of
the structure of U3Si2 (a = b = 7.33Å):
y
x
Positions: (x, y)
U = 0, 0
0.5, 0.5
0.182, 0.682
0.818, 0.318
0.318, 0.182
0.682, 0.818
The last four positional coordinates are related via a 44-fold
rotational axis. Give the operation upon the general position
(x,y) Æ (±x,±
x,±y) which leads to each coordinate.
coordinate.
Structures With CCP Packing
The Unit Cell of Cubic Close Packing (ABCABC)
Octahedral sites in the faceface-centered cubic (ccp
(ccp)) unit cell are
located in the center and on every edge
What is the # of octahedral sites per unit cell?
1 + ¼(12) = 4
What is the coordination number of each atom?
Na = 6, Cl = 6
4
Structures With CCP Packing
The Unit Cell of Cubic Close Packing (ABCABC)
(1,1,1)
(0,0,1)
(½,½, ½)
(½,½,0)
c
b
(0,0,0)
a
(½,0,0)
(1,0,0)
Atomic positions are described by fractional coordinates.
If all octahedral sites are filled, the structure is NaClNaCl-type.
type.
If every other layer of octahedral sites are filled, the structure
structure is
CdCl2-type.
type.
Structures With CCP Packing
Some compounds with the NaCl structure type, a (Angstroms)
MgO
CaO
SrO
BaO
4.213
4.8105
5.160
5.539
MgS
CaS
SrS
BaS
5.200 LiF
5.6948 LiCl
6.020
LiBr
6.386
LiI
4.0270
5.1396
5.5013
6.00
KF
KCl
KBr
KI
5.347
6.2931
6.5966
7.0655
Unit cell increases with: 1) Increasing cation size
2) Increasing anion size
size
5
Structures With CCP Packing
The Unit Cell of Cubic Close Packing (ABCABC)
Polyhedral drawing
Tetrahedral sites in the faceface-centered cubic (ccp
(ccp)) unit cell are
located in each of the eight octants.
What is the # of tetrahedral sites per unit cell?
4 T+, 4 T–
What is the coordination number of each atom?
Yellow, Blue = 4; Red = 8
Structures With CCP Packing
The Unit Cell of Cubic Close Packing (ABCABC)
Polyhedral drawing
What ½ of the tetrahedral sites (either all T+ or all T–) are filled,
the structure is ZnS (sphalerite)
sphalerite)-type.
type.
When all of the tetrahedral sites are filled, the structure is K2Otype (or CaF2; antianti-type)
6
Structures With CCP Packing
Some compounds with the ZnS (sphalerite)
sphalerite) structure type, a
(Angstroms)
CuF 4.255
CuCl 5.416
BeS
BeSe
4.862
5.070
CdS 5.818
CdSe 6.077
BN 3.616
GaAs 5.653
Some compounds with the K2O structure type, a (Angstroms)
SrF2 5.800
SrCl2 6.977
PbO2 5.349 Li2O 4.611 K2O 6.449
CeO2 5.411 Li2S 5.710 K2S 7.406
C (diamond) is fcc
Structures With HCP Packing
Octahedral sites of Hexagonal Close Packing (ABAB)
A
B
t+
B
t+
c
a
b
o
t–
A
t+
t+
A
View from above
Unit cell
Octahedral sites (2) are located in each unit cell
-- They form faceface-sharing chains
What is the coordination number of the packing sphere?
7
Structures With HCP Packing
Octahedral sites of Hexagonal Close Packing (ABAB)
A
B
c
b
a
A
Polyhedral drawing
Unit cell
If all octahedral sites are filled, the structure is NiAs structurestructure-type
If ½ octahedral sites are filled, the structure is CdI2 structurestructure-type
What is the coordination geometry of the anions? Cations?
Cations?
Structures With HCP Packing
Tetrahedral sites of Hexagonal Close Packing (ABAB)
A
B
t+
B
t+
a
b
o
t–
A
t+
t+
A
View from above
Unit cell
Tetrahedral sites are located within the unit cell (1T+, 1T–) and
on each edge (1T+, 1T– per edge).
-- A face/vertex sharing chain is formed when all tetrahedral
sites are filled
8
Structures With HCP Packing
Tetrahedral sites of Hexagonal Close Packing (ABAB)
A
B
b
a
A
Polyhedral drawing
Unit cell
Filling ½ the tetrahedral sites (either T+ or 1T–) gives the ZnS
(wurtzite)
wurtzite) structure type.
Fill all tetrahedral sites results in unfavorable cation/
cation/cation
repulsion Æ No analogue to ccp K2O type.
Structures With HCP Packing
Some compounds with the NiAsNiAs-type, a c (Angstroms)
NiS
NiSn
FeSe
FeTe
3.439 5.348
4.048 5.123
3.637 5.958
3.800 5.651
PtB
PtSn
CoSb
MnSb
3.358
4.103
3.866
4.120
4.058
5.428
5.188
5.784
Solids with this structure type tend to be metallic
Ideally c/a = 1.633; A smaller ratio will have shorter metalmetalmetal bonds
Some compounds with the ZnS (wurtzite)
wurtzite) structure type, a
ZnO
ZnTe
BeO
MnS
3.250
4.270
2.698
3.976
5.207
6.990
4.380
6.432
AgI 4.580
AlN 3.111
GaN 3.180
SiC 3.076
7.494
4.978
5.166
5.048
9
Closest Packing (Summary)
Common Structures and Types
Anion
Packing
ccp
hcp
Interstitial Sites
T+
T–
O
Coordination
Cation Anion
Examples
0
0
1
6
6
NaCl
1
0
0
4
4
ZnS (sphalerite)
1
1
0
4
8
K2O, CaF2
0
0
1/2
6
3
CdCl2
0
0
1
6
6
NiAs
1
0
0
4
4
ZnS (Wurtzite)
0
0
1/2
6
3
CdI2
10