crystalline solids 1.close-packed spheres 2.units cells: point and space symmetry

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