1. 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

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

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

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

5. 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 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 )

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

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

9. The most advantageous crystal structure for ionic solids* NaCl versus CsCl structure Competition between packing and avoiding of e.g. anion-anion contact r0r0 R+R+ R-R- Radius RatioCoordination no.Binary (AB) Structure-type r + /r - = 112 none known 1 > r + /r - > 0.7328CsCl 0.732 > r + /r - > 0.4146NaCl 0.414 > r + /r - > 0.2254ZnS To avoid - contact in NaCl structure (* the explanation in Blakemore page 15 is misleading )