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

2. Building Up Solid Structures From Close-Packed Spheres

3. Close Packed Circles?

4. Close Packed Circles!

5. 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

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

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

8. 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.

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

10. Three-Dimensional PackingB C
C site
Filling the C site gives an ABCABCABC packing pattern
Resulting in cubic close packing (ccp)

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

12. 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

14. 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)

15. 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

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

17. Rock Salt Structure
Filling of octahedral holes

18. Rock Salt Structure Highlighting the close-packed planesB C A

19. Rock Salt Structure highlighting the two interpenetrating fcc lattices

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

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

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

23. 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

24. 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

25. 1987 Nobel Prize in Physics
                               
                                
                     
Age 37
Age 60
"for their important break-through in the discovery of superconductivity in ceramic materials"

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

27. 1-2-3 Superconductors A perovskite-like structure
Use simpler structures to understand more complex structures

28. 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

29. 1-2-3 Superconductors YBa2Cu3O7-x ( x < 0.1)
These structure of these materials is related to Perovskite

30. The Materials Minute
Brought to you today by John Henssler

31. 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