Common crystal structures

1. Common crystal structures • Simple close packed structures * most efficient packing problem of structure atoms hard spheres Donuts *Proposition made by Goldschmidt 1926. Useful approach for metals, where the chemical bond does not provide geometrical constrains like in diamond for instance

2. hexagonal close-packed cubic close packed hexagonal layers packed in 2 different ways Hexagonal Close-Packed Structure layers stack according to ABAB

3. Cubic Close-Packed Structure FCC hexagonal layers are stacked ABC  (close packed structures - unit cells )

4. * = FCC * more complicated packing sequence such as ABAC, ABCB, etc

5. no packing can improve the Face-Centered Cubic packing Computer proof: Volume of the spheres in the unit cell 100% hcp and fcc have both 74% packing ratio := Volume the unit cell all others including bcc have less packing ratio Johannes Kepler asserted in the early 1600's that ! Proof took nearly 400 years 1998 Thomas Hales (presently University of Pittsburgh) announced to have a proof of the Kepler conjecture (click to see an e-mail of T.Hales announcing the proof) minimizing a function with 150 variables 250 pages of notes and 3 gigabytes of computer programs, data and results

6. (0,0,0) Lattices which can be considered as 2 interpenetrating fcc lattices diamond lattice: not packing but symmetrically placed valence bonds determine the structure diamond lattice: fcc lattice with basis = two identical atoms at (0,0,0) and

7. ? What happens if atoms of the basis are different ZnS (zincblende), or GaAs Four neighbors all of opposite chemical species (click here for animations)

8. (0,0,0) (0,0,0) NaCl: fcc translational symmetry with basis CsCl: Simple cubic space lattice with basis

9. To avoid - contact in NaCl structure The most advantageous crystal structure for ionic solids* CsCl structure versus NaCl Competition between packing and avoiding of e.g. anion-anion contact R- R+ r0 (*the explanation in Blakemore page 15 is misleading )