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Common crystal structures
•Simple close packed structures
atoms hard spheres problem of structure most efficient packing
Donuts
*
*Proposition made by Goldschmidt 1926. Useful approach for metals, where the chemical bond does not provide geometrical constrains like in diamond for instance
hexagonal layers packed in 2 different ways
Hexagonal Close-Packed Structure
hexagonal close-packed cubic close packed
layers stack according to ABAB
hexagonal layers are stacked ABCCubic Close-Packed Structure
FCC
(close packed
structures
- unit cells )
* more complicated packing sequence such as ABAC, ABCB, etc
*
= FCC
Computer proof:
hcp and fcc have both 74% packing ratio :=Volume of the spheres in the unit cell
Volume the unit cell 100%
all others including bcc have less packing ratio
Johannes Kepler asserted in the early 1600's that
no packing can improve the Face-Centered Cubic packing
Proof took nearly 400 years
1998 Thomas Hales (presently University of Pittsburgh) announced to have a proof of the Kepler conjecture
250 pages of notes and 3 gigabytes of computer programs, data and results
minimizing a function with 150 variables
!
(click to see an e-mail of T.Hales
announcing the proof)
Lattices which can be considered as 2 interpenetrating fcc lattices
diamond lattice: fcc lattice with basis
diamond lattice: not packing but symmetrically placed valence bonds
),,(4
1
4
1
4
1= two identical atoms at (0,0,0) and
(0,0,0)),,(4
1
4
1
4
1
determine the structure
What happens if atoms of the basis are different ?
ZnS (zincblende), or GaAs
Four neighbors
all of opposite chemical species
(click here for animations)
NaCl: fcc translational symmetry with basis
(0,0,0))
21
,21
,21
(
CsCl: Simple cubic space lattice with basis
(0,0,0)
)21
,21
,21
(
The most advantageous crystal structure for ionic solids*
NaCl versus CsCl structure
Competition between packing and avoiding of e.g. anion-anion contact
r0
20r
R+
R-
Radius Ratio Coordination no. Binary (AB) Structure-type
r+/r- = 1 12 none known
1 > r+/r- > 0.732 8 CsCl
0.732 > r+/r- > 0.414 6 NaCl
0.414 > r+/r- > 0.225 4 ZnS
To avoid - contact in NaCl structure
0 2 2r R
2R R R 2 1R
R
(*the explanation in Blakemore page 15 is misleading )