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 )