Transcript Slide 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 hexagonal layers packed in 2 different ways hexagonal close-packed Hexagonal Close-Packed Structure cubic close packed layers stack according to ABAB hexagonal layers are stacked ABC Cubic Close-Packed Structure FCC (close packed structures - unit cells ) * = FCC * more complicated packing sequence such as ABAC, ABCB, etc hcp and fcc have both 74% packing ratio := Volume of the spheres in the unit cell 100% Volume the unit cell 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 (click to see an e-mail of T.Hales Kepler conjecture announcing the proof) Computer proof: minimizing a function with 150 variables 250 pages of notes and 3 gigabytes of computer programs, data and results Lattices which can be considered as 2 interpenetrating fcc lattices diamond lattice: not packing but symmetrically placed valence bonds determine the structure (0,0,0) 1 1 1 ( , , ) 4 4 4 diamond lattice: fcc lattice with basis 1 1 1 4 4 4 = two identical atoms at (0,0,0) and ( , , ) What happens if atoms of the basis are different ZnS (zincblende), or GaAs Four neighbors all of opposite chemical species (click here for animations) ? 1 1 1 ( , , ) 2 2 2 (0,0,0) 1 1 1 ( , , ) 2 2 2 (0,0,0) NaCl: fcc translational symmetry with basis CsCl: Simple cubic space lattice with basis The most advantageous crystal structure for ionic solids* NaCl CsCl versus structure Competition between packing and avoiding of e.g. anion-anion contact RR+ To avoid r0 r0 2 - contact in NaCl structure 2 2R R R 2 R R 2 1 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 r0 (*the explanation in Blakemore page 15 is misleading )