Graphite Diamond

Buckminster Fullerene1985

Carbon Nanotubes1991

Graphene2004

Allotropes of C

GraphiteSp2 hybridization 3 covalent bonds

Hexagonal sheets

x ya b=a=120

a = 2 d cos 30°

= √3 dd = 1.42 Åa = 2.46 Å

Graphite

x y

a = 2.46 Å c = 6.70 Å

B

A

A

www.scifun.ed.ac.uk/

c

Lattice: Simple HexagonalMotif: 4 carbon atoms

Graphite Highly Anisotropic:

Properties are very different in the a and c directions

www.sciencemuseum.org.uk/

Uses:Solid lubricantPencils (clay + graphite, hardness

depends on fraction of clay)carbon fibre

DiamondSp3 hybridization 4 covalent bonds

Location of atoms:8 Corners6 face centres4 one on each of the 4 body diagonals

Tetrahedral bonding

Diamond Cubic Crystal: Lattice & motif?

AA BB

C

CD

D

x

y

P

P

QQ

RR

S

S

T

T

KK

L

L

MM

N

N

0,1

0,1

0,1

0,1

0,1

41

41

43

43

Diamond Cubic Crystal= FCC lattice + motif:

x

y

21

21

21

21

Projection of the unit cell on the bottom face of the cube

000; ¼¼¼

Diamond

Effective number of atoms in the unit cell = 881

Corners

Relaton between lattice parameter and atomic radius

ra 243

38ra

Packing efficiency

34.01633

483

3

a

r

Coordination number 4

8621 41

InsideFace

Diamond Cubic Crystal StructuresC Si Ge Gray Sn

a (Å) 3.57 5.43 5.65 6.46

0,1 0,1

21

IV-IV compound: SiCIII-V compound:

AlP, AlAs, AlSb, GaP, GaAs, GaSb,

InP, InAs, InSbII-VI compound:

ZnO, ZnS,CdS, CdSe, CdTe

I-VII compound:CuCl, AgI

y

S

0,1 0,1

0,1

41

41

43

43

21

21

21

Equiatomic binary AB compounds having diamond cubic like structure

USES:

DiamondAbrasive in polishing and grindingwire drawing dies

Si, Ge, compounds: semiconducting devices

SiCabrasives, heating elements of furnaces

Graphite Diamond

Buckminster Fullerene1985

Carbon Nanotubes1991

Graphene2004

Allotropes of C

C60 BuckminsterfullereneH.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. SmalleyNature 318 (1985) 162-163

1996 Nobel Prize

Long-chain carbon molecules in interstellar space

A carbon atom at each vertex

American architect, author, designer, futurist, inventor, and visionary.

He was expelled from Harvard twice: 1. first for spending all his money partying with

a Vaudeville troupe, 2. for his "irresponsibility and lack of interest".

what he, as an individual, could do to improve humanity's condition, which large organizations, governments, and private enterprises inherently could not do.

Montreal Biosphere in Montreal, Canada

Truncated Icosahedron

Icosahedron: A Platonic solid (a regular solid)Truncated Icosahedron: An Archimedean solid

A regular polygon

A polygon with all sides equal and all angles equal

Square regular

Rectangle unequal sides not regular

Rhombusunequal angles not regular

Regular Polygons: All sides equal all angles equal

A regular n-gon with any n >= 3 is possible

3 4 5 6…

There are infinitely many regular polygons

Triangle square pentagon hexagon…

3D: Regular Polyhedra or Platonic SolidsAll faces regular congruent polygons, all corners identical.

Cube

How many regular solids?

Tetrahedron

There are 5 and only 5 Platonic or regular solids !

Icosahedron

Octahedron

Tetrahedron

Cube

Dodecahedron

1. Tetrahedron 4 64

2. Octahedron6 12 83. Cube 8 12 64. Icosahedron 12 30

205. Dodecahedron 20 30

12

Duals

Duals

Euler’s Polyhedron Formula

V-E+F=2

Duality

Tetrahedron Self-DualOctahedron-CubeIcosahedron-Dodecahedron

Proof of Five Platonic Solids

At any vertex at least three faces should meetThe sum of polygonal angles at any vertex should be less the 360

Triangles (60) 3 Tetrahedron4 Octahedron5 Icosahedron6 or more: not possible

Square (90) 3 Cube4 or more: not possible

Pentagon (108) 3 Dodecahedron

Dense packings of the Platonic and ArchimedeansolidsS. Torquato & Y. JiaoNature, Aug 13, 2009

Truncated Icosahedron: V=60, E=90, F=32

Synthesis of Fullerene

Arc Evaporation of graphite in inert atmosphere

M. CARAMAN, G. LAZAR, M. STAMATE, I. LAZAR

Nature 391, 59-62 (1 January 1998)Electronic structure of atomically resolved carbon nanotubesJeroen W. G. Wilder, Liesbeth C. Venema, Andrew G. Rinzler, Richard E. Smalley & Cees Dekker

zigzig (n,0)

armchair (n,n)

(n,m)=(6,3)

a1a

2

wrapping vector

Structural features of carbon nanotubes

=chiral angle

Material Young's Modulus (TPa)

Tensile Strength

(GPa)

Elongation at Break (%)

SWNT ~1 (from 1 to 5) 13-53E 16Armchair SWNT

0.94T 126.2T 23.1

Zigzag SWNT

0.94T 94.5T 15.6-17.5

Chiral SWNT

0.92

MWNT 0.8-0.9E 150

Stainless Steel

~0.2 ~0.65-1 15-50

Kevlar ~0.15 ~3.5 ~2KevlarT 0.25 29.6 Source: wiki

ElectricalFor a given (n,m) nanotube, if n = m, the nanotube is metallic;

if n − m is a multiple of 3, then the nanotube is semiconducting with a very small band gap,

otherwise the nanotube is a moderate semiconductor.

Thus all armchair (n=m) nanotubes are metallic,

and nanotubes (5,0), (6,4), (9,1), etc. are semiconducting.

In theory, metallic nanotubes can carry an electrical current density of 4×109 A/cm2 which is more than 1,000 times greater than metals such as copper[23].

While the fantasy of a space elevator has been around for about 100 years, the idea became slightly more realistic by the 1991 discovery of "carbon nanotubes,"

The Space Engineering and Science Institute 

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  Thursday, August 13 through Sunday, August 16,

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