Crystalline Solids
1. Close-packed Spheres2. Units cells: point and space symmetry
Building Up Solid StructuresFrom Close-Packed Spheres
Close Packed Circles?
Close Packed Circles!
Close Packed Circles!
% Area filled = r2/(2r)2 = 78.5Area of circleArea of square
½ area of circleArea of triangle
% Area filled = 7.9032
2)3)(2(
21 2
rr
r
Area of circle = r2 Area of triangle = bh/2 Area of square = l2
What is percent area filled for each case??
Three-Dimensional Packing
At start all sites equivalent
After placing first atom in second layer, two sites now present
Filled hollows Empty HollowsEmpty Hollows
Three-Dimensional Packing
At start all sites equivalent
After placing first atom in second layer, two sites now present
Filled hollows Empty HollowsEmpty Hollows
Three-Dimensional Packing
When placing atoms in third layer, we have two choicesSimilar to forming second layer, we can only choose 1 site.
A site C site
AB
Three-Dimensional Packing
Filling the A site gives an ABABABAB packing patternResulting in hexagonal close packing (hcp)
A site
AB
A
Three-Dimensional Packing
Filling the C site gives an ABCABCABC packing patternResulting in cubic close packing (ccp)
C site
AB
C
FCC Unit Cell
Each corner atom 1/8 in cellEach 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 withClose-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!
(and it is energetically favorable to do so)
Occupation of Octahedral Holes
Occupation of Tetrahedral Holesone blue atom on bottomthree purple atoms on top
three blue atoms on bottomthree purple atoms on top
Typically, close-packed spheres are anions and species fillingtetrahedral and octahedral holes are cations
Tetrahedral and Octahedral Holes
Two views of tetrahedral hole Two views of octahedral hole
Rock Salt Structure
Filling of octahedral holes
Rock Salt StructureHighlighting the close-packed planes
B
C
A
AB C
Rock Salt Structurehighlighting the two
interpenetrating fcc lattices
Zinc Blende (ccp lattice, abc)
Filling the tetrahedral holes
Note adamantane-likestructure
Diamond
Can be considered as filling of tetrahedral holes
a = 3.56 Å
All of these Group 14 Elements Have Diamond Structure
Carbon - diamond silicon
germanium tin
Perovskite - An Important Class of Cubic Mineral
Strontium Titanate SrTiO3
Titanium on cellCorners: 8 x 1/8 = 1
Oxygen on cellEdges: 12 x 1/4 = 3
Sr in cell center: 1
Ti+4
O-2
Sr+2
Perovskite - An Important Class of Cubic Mineral
Strontium Titanate SrTiO3
Titanium in cell center: 1
Oxygen on cell faces: 6 x 1/2 = 3
Sr on cellCorners: 8 x 1/8 = 1
Ti+4
O-2
Sr+2
1987 Nobel Prize in Physics
"for their important break-through in the discovery of superconductivity in ceramic materials"
Age 37 Age 60
Discovery of the 1-2-3 Class of High Temperature Superconductor
Paul ChuDirector, Texas Center for Superconductivity
University of Houston
Maw-Kuen Wu
A perovskite-like structure
1-2-3 Superconductors
Use simpler structures to understand more complex structures
1-2-3 Superconductors
One Yttrium in cell center: 1
Two Bariums in upper andlower sections: 2
Eight Cu on cell vertices: 8 x 1/8 = 1Eight Cu on cell edges: 8 x 1/4 = 2Total = 3
Twelve O on cell edges: 12 x 1/4 = 3Eight O on cell faces: 8 x 1/2 = 4Total = 7
1-2-3 SuperconductorsYBa2Cu3O7-x ( x < 0.1)
These structure of these materials is related to Perovskite
The Materials Minute
Brought to you today by John HensslerChromophore Fluorination Enhances Crystallization and Stability of Soluble Anthradithiophene Semiconductors
Sankar Subramanian, Sung kyu Park, Sean R. Parkin, Vitaly Podzorov, Thomas N. Jackson, and John E. Anthony
J. Am. Chem. Soc. 2008, 130, 2706
powerlight.com/newsletters/news_issue/3/newsletter.htm; Rogers, J. A.; Bao, Z.; J. Polym. Sci., Part A 2002, 40, 3327.; http://www.powerlight.com/newsletters/news_issue/3/newsletter_industry.htm
SS S
S SS
pentacene sexithiophene
SiR3
SiR3
S
SF F
SF
SiR3
SiR3
S
S
S
Applications and highly studied organic semiconducting materials:
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:
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