Close-packed Spheres Units cells: point and space symmetry Crystalline Solids Close-packed Spheres Units cells: point and space symmetry

Building Up Solid Structures From Close-Packed Spheres

Close Packed Circles?

Close Packed Circles!

Close Packed Circles! What is percent area filled for each case?? ½ area of circle Area of triangle % Area filled = Area of circle Area of square % Area filled = pr2/(2r)2 = 78.5 Area of circle = pr2 Area of triangle = bh/2 Area of square = l2

Three-Dimensional Packing Empty Hollows Filled hollows At start all sites equivalent After placing first atom in second layer, two sites now present

Three-Dimensional Packing At start all sites equivalent Empty Hollows Filled hollows After placing first atom in second layer, two sites now present

Three-Dimensional Packing A site B C site When placing atoms in third layer, we have two choices Similar to forming second layer, we can only choose 1 site.

Three-Dimensional Packing A site B Filling the A site gives an ABABABAB packing pattern Resulting in hexagonal close packing (hcp)

Three-Dimensional Packing B C C site Filling the C site gives an ABCABCABC packing pattern Resulting in cubic close packing (ccp)

FCC Unit Cell Each corner atom 1/8 in cell Each face atom ½ in cell Derived from ABC packing of spheres, ccp

Hexagonal Unit Cell Derived from Hexagonal Close Packing (hcp) Two views of the Hexagonal Unit Cell with Close-Packed Planes indicated in Blue and Green Side view Top view Derived from AB packing of spheres

All Solids Contain Empty Space. Empty Space Can Be Filled All Solids Contain Empty Space! Empty Space Can Be Filled! (and it is energetically favorable to do so)

Occupation of Octahedral Holes three blue atoms on bottom three purple atoms on top Typically, close-packed spheres are anions and species filling tetrahedral and octahedral holes are cations Occupation of Tetrahedral Holes one blue atom on bottom three purple atoms on top

Tetrahedral and Octahedral Holes Two views of octahedral hole Two views of tetrahedral hole

Rock Salt Structure Filling of octahedral holes

Rock Salt Structure Highlighting the close-packed planes B C A

Rock Salt Structure highlighting the two interpenetrating fcc lattices

Zinc Blende (ccp lattice, abc) Filling the tetrahedral holes Note adamantane-like structure

Diamond a = 3.56 Å Can be considered as filling of tetrahedral holes

All of these Group 14 Elements Have Diamond Structure silicon Carbon - diamond tin germanium

Perovskite - An Important Class of Cubic Mineral Strontium Titanate SrTiO3 Sr in cell center: 1 Ti+4 O-2 Sr+2 Titanium on cell Corners: 8 x 1/8 = 1 Oxygen on cell Edges: 12 x 1/4 = 3

Perovskite - An Important Class of Cubic Mineral Strontium Titanate SrTiO3 Sr on cell Corners: 8 x 1/8 = 1 Ti+4 O-2 Sr+2 Titanium in cell center: 1 Oxygen on cell faces: 6 x 1/2 = 3

1987 Nobel Prize in Physics                                                                                        Age 37 Age 60 "for their important break-through in the discovery of superconductivity in ceramic materials"

Discovery of the 1-2-3 Class of High Temperature Superconductor Maw-Kuen Wu Paul Chu Director, Texas Center for Superconductivity University of Houston

1-2-3 Superconductors A perovskite-like structure Use simpler structures to understand more complex structures

1-2-3 Superconductors One Yttrium in cell center: 1 Two Bariums in upper and lower sections: 2 Eight Cu on cell vertices: 8 x 1/8 = 1 Eight Cu on cell edges: 8 x 1/4 = 2 Total = 3 Twelve O on cell edges: 12 x 1/4 = 3 Eight O on cell faces: 8 x 1/2 = 4 Total = 7

1-2-3 Superconductors YBa2Cu3O7-x ( x < 0.1) These structure of these materials is related to Perovskite

The Materials Minute Brought to you today by John Henssler

Quantitative Assessment of the Spherical Packing Model For the following problems, consider a close-packed, three-dimensional structure made up of hard spheres all of radius a: Show that the interlayer separation between planes is equal to 1.633a b) Show that the largest sphere that can be inscribed inside the triangle formed by 3 spheres in the plane of a layer has a radius of 0.154a c) Show that the radius of the tetrahedral holes between the close-packed layers is 0.225a d) Show that the radius of the octahedral holes between close-packed layers is 0.414a e) Show that the volume fraction of space occupied by the spheres is 0.741 s