Most Important Element in life Most important Element in Engineering Most important element in...

84
Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C

Transcript of Most Important Element in life Most important Element in Engineering Most important element in...

Page 1: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Most Important Element in life

Most important Element in Engineering

Most important element in Nanotechnology

C

Page 2: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Graphite Diamond

Buckminster Fullerene1985

Carbon Nanotubes1991

Graphene2004

Allotropes of C

Page 3: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Contents

Crystal, Lattice and Motif

Unit cells, Lattice Parameters and Projections

Miller Indices & Miller-Bravais IndicesDirections and Planes

Classification of Lattices:7 crystal systems14 Bravais lattices

Reciprocal lattice

Page 4: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

A 3D translationaly periodic arrangement of atoms in space is called a crystal.

Crystal, Lattice and Motif

Crystal ?

Page 5: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Lattice?

A 3D translationally periodic arrangement of points in space is called a lattice.

Crystal, Lattice and Motif

Page 6: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

A 3D translationally periodic arrangement of atoms

Crystal

A 3D translationally periodic arrangement of points

Lattice

Crystal, Lattice and Motif

Page 7: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Motif?

Crystal = Lattice + Motif

Motif or basis: an atom or a group of atoms associated with each lattice point

Crystal, Lattice and Motif

Page 8: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

“Nothing that is worth knowing can be taught.”

Oscar Wilde

Page 9: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

+

Love Pattern(Crystal)

Love Lattice

+ Heart (Motif)

=

=

Lattice + Motif = Crystal

Love Pattern

Page 10: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Air, Water and Earth

Maurits Cornelis Escher 1898-1972

Dutch Graphic Artist

Page 11: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Every periodic pattern (and hence a crystal) has a unique lattice associated with it

Page 12: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.
Page 13: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Cu Crystal NaCl Crystal

Lattice

FCC FCC

Motif 1 Cu+ ion

1 Na+ ion + 1 Cl- ion

Crystal

Crystal, Lattice and Motif

Page 14: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Contents

Crystal, Lattice and Motif

Unit cells, Lattice Parameters and Projections

Miller Indices & Miller-Bravais IndicesDirections and Planes

Classification of Lattices:7 crystal systems14 Bravais lattices

Page 15: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Translational Periodicity

One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)

Unit Cell

Unit cell description : 1

Page 16: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

The most common shape of a unit cell is a parallelopiped

with lattice points at corners.

UNIT CELL:

Primitive Unit Cell: Lattice Points only at corners

Non-Primitive Unit cell: Lattice Point at corners as well as other some points

Page 17: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Lattice Parameters:

1. A corner as origin

2. Three edge vectors {a, b, c} from the origin define

a CRSYTALLOGRAPHIC COORDINATE

SYSTEM

3. The three lengths a, b, c and the three interaxial angles , , are called the LATTICE PARAMETERS

a

b

c

Unit cell description : 4

Page 18: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Wigner-Seitz Unit Cells

FCC

BCC

Rhombic Dodcahedron

Tetrakaidecahedron

Page 19: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

The six lattice parameters a, b, c, , ,

The cell of the lattice

lattice

crystal

+ Motif

Page 20: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Contents

Crystal, Lattice and Motif

Unit cells, Lattice Parameters and Projections

Miller Indices & Miller-Bravais IndicesDirections and Planes

Classification of Lattices:7 crystal systems14 Bravais lattices

Page 21: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

7 crystal SystemsCrystal System Conventional Unit Cell

1. Cubic a=b=c, ===90

2. Tetragonal a=bc, ===90

3. Orthorhombic abc, ===90

4. Hexagonal a=bc, == 90, =120

5. Rhombohedral a=b=c, ==90 OR Trigonal

6. Monoclinic abc, ==90

7. Triclinic abc,

Unit cell description : 5

Page 22: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Crystal System Bravais Lattices

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

?

Why so many empty boxes?

E.g. Why cubic C is absent?

P: Simple; I: body-centred;F: Face-centred; C: End-centred

Page 23: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

The three cubic Bravais lattices

Crystal system Bravais lattices

1. Cubic P I F

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

Page 24: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

Page 25: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Monatomic Body-Centred Cubic (BCC) crystal

Lattice: bcc

CsCl crystal

Lattice: simple cubic

BCC Feynman!

Corner and body-centres have the same neighbourhood

Corner and body-centred atoms do not have the same neighbourhood

Motif: 1 atom 000Motif: two atoms Cl 000; Cs ½ ½ ½

Cs

Cl

Page 26: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

½

½

½

½

Lattice: Simple hexagonal

hcp lattice hcp crystal

Example: Hexagonal close-packed (HCP) crystal

x

y

z

Corner and inside atoms do not have the same neighbourhood

Motif: Two atoms: 000; 2/3 1/3 1/2

Page 27: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Crystal System Bravais Lattices

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

?

Why so many empty boxes?

E.g. Why cubic C is absent?

P: Simple; I: body-centred;F: Face-centred; C: End-centred

Page 28: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2

a2

a

Page 29: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

Page 30: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Face-centred cubic in the Bravais list ?

Cubic F = Tetragonal I ?!!!

Page 31: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

Page 32: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Couldn’t find his photo on the net

1811-1863

Auguste Bravais

1850: 14 lattices1835: 15 lattices

ML Frankenheim 1801-1869

06 Aug 2009: 13 lattices !!!

AML750

IIT-D

X

1856: 14 lattices

History:

Page 33: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Why can’t the Face-Centred Cubic lattice

(Cubic F) be considered as a Body-

Centred Tetragonal lattice (Tetragonal I) ?

Page 34: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Primitivecell

Primitivecell

Non-primitive cell

A unit cell of a lattice is NOT unique.

UNIT CELLS OF A LATTICE

Unit cell shape CANNOT be the basis for classification of Lattices

Page 35: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

What is the basis for classification of lattices

into 7 crystal systems

and 14 Bravais lattices?

Page 36: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Lattices are classified on the

basis of their symmetry

Page 37: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

What is symmetry?

Page 38: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

Symmetry

Page 39: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

NOW NO SWIMS ON MON

Page 40: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Lattices also have translational symmetry

Translational symmetry

In fact this is the defining symmetry of a lattice

Page 41: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an n-fold rotation axis where

0360n

=180

=90

Rotation Axis

n=2 2-fold rotation axis

n=4 4-fold rotation axis

Page 42: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Rotational Symmetries

Z180 120 90 72 60

2 3 4 5 6

45

8

Angles:

Fold:

Graphic symbols

Page 43: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Crsytallographic Restriction

5-fold symmetry or Pentagonal symmetry is not possible for Periodic TilingsSymmetries higher than 6-fold also not possible

Only possible rotational symmetries for periodic tilings

2 3 4 5 6 7 8 9…

Page 44: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Proof of The Crystallographic Restriction

A rotation can be represented by a matrix

333231

232221

131211

ttt

ttt

ttt

T

1cos2][ 332211 tttTTrace

If T is a rotational symmetry of a lattice then all its elements must be integers (wrt primitive basis vectors)

N 1cos2 31 N

N -1 0 1 2 3

180° 120° 90° 60° 0°

n-fold 2 3 4 6 1

Page 45: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4

“Fig. 1-4 is an invented arrangement for ice, and although it contains many of the correct features of the ice, it is not the true arrangement. One of the correct features is that there is a part of the symmetry that is hexagonal. You can see that if we turn the picture around an axis by 120°, the picture returns to itself.”

Hexagonal symmetry

o606

360

Page 46: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Correction: Shift the box

Michael Gottlieb’s correction:

But gives H:O = 1.5 : 1

Page 47: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

QUASICRYSTALS (1984)Icosahedral symmetry (5-fold symmetry)

Lack strict translational periodicity -> Quasiperiodic

Icosahedron

Penrose Tiling

Diffraction PatternExternal

Morphology

Page 48: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Reflection (or mirror symmetry)

Page 49: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

Page 50: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

The group of all symmetry elements of a crystal except translations (e.g. rotation, reflection etc.) is called its

POINT GROUP.

The complete group of all symmetry elements including translations of a crystal is called its SPACE GROUP

Point Group and Space Group

Page 51: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Classification of lattices

Based on the space group symmetry, i.e., rotational, reflection and translational symmetry

14 types of lattices 14 Bravais lattices

Based on the point group symmetry alone (i.e. excluding translational symmetry 7 types of lattices

7 crystal systems

Crystal systems and Bravais Lattices

Classification of Lattices

Page 52: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

7 crystal SystemsSystem Required symmetry

• Cubic Three 4-fold axis

• Tetragonal one 4-fold axis

• Orthorhombic three 2-fold axis

• Hexagonal one 6-fold axis

• Rhombohedral one 3-fold axis

• Monoclinic one 2-fold axis

• Triclinic none

Page 53: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Tetragonal symmetry Cubic symmetry

Cubic C = Tetragonal P Cubic F Tetragonal I

Page 54: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

The three Bravais lattices in the cubic crystal system have the same rotational symmetry but different translational symmetry.

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

Page 55: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Neumann’s Principle

Symmetry elements of a physical property of a crystal must include all the symmetry elements of its point group (i.e., all its rotational axes and mirror planes).

Electrical resistance of a cubic crystal is isotropic (spherical symmetry)

All properties that can be represented by tensors of rank up to 2 are isotropic for cubic crystals

Page 56: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Jian-Min Zhanga,Yan Zhanga, Ke-Wei Xub and Vincent Ji

J Phys. Chem. Solids, 68 (2007) 503-510

Elasic modulus (4th. Rank Tensor) is not isotropic

Representation surfaces of Young’s modulus of fcc metals

Ag

Au

Ni

Pb

Cu

Al

Page 57: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

QUESTIONS?

Page 58: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Contents

Crystal, Lattice and Motif

Unit cells, Lattice Parameters and Projections

Miller Indices & Miller-Bravais IndicesDirections and Planes

Classification of Lattices:7 crystal systems14 Bravais lattices

Page 59: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

1. Choose a point on the direction as the origin.

2. Choose a coordinate system with axes parallel to the unit cell edges.

x

y 3. Find the coordinates of another point on the direction in terms of a, b and c

4. Reduce the coordinates to smallest integers. 5. Put in square brackets

Miller Indices of Directions

[100]

1a+0b+0c

z

1, 0, 0

1, 0, 0

Miller Indices 2

ab

c

Page 60: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

y

zMiller indices of a direction:

only the orientation not its position or sense

All parallel directions have the same Miller indices

[100]x

Miller Indices 3

Page 61: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

x

y

z

O

A

1/2, 1/2, 1

[1 1 2]

OA=1/2 a + 1/2 b + 1 c

P

Q

x

y

z

PQ = -1 a -1 b + 1 c-1, -1, 1

Miller Indices of Directions (contd.)

[ 1 1 1 ]

_ _

-ve steps are shown as bar over the number

Direction OA

Direction PQ

Page 62: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller indices of a family of symmetry related directions

[100]

[001]

[010]

uvw = [uvw] and all other directions related to [uvw] by the symmetry of the crystal

cubic100 = [100], [010],

[001]tetragonal

100 = [100], [010]

CubicTetragonal

[010][100]

Miller Indices 4

Page 63: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller indices of slip directions in CCP

y

z

[110]

[110]

[101] [ 101]

[011]

[101]

Slip directions = close-packed directions = face diagonals

x

Six slip directions:[110

][101]

[011]

[110]

[101]

[101]

All six slip directions in

ccp:110

Miller Indices 5

Page 64: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Vectors vs Directions:

Miller Indices of a direction

Exception:

The Burger’s vector

[110] is a direction along the face diagonal of a unit cell. It is not a vector of fixed length

FCC 1102

1b

BCC 1112

1b A vector equal to half body

diagonal

A vector equal to half face diagonal

Page 65: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

5. Enclose in parenthesis

Miller Indices for planes

3. Take reciprocal

2. Find intercepts along axes

1. Select a crystallographic

coordinate system with

origin not on the plane

4. Convert to smallest

integers in the same ratio

1 1 1

1 1 1

1 1 1

(111)

x

y

z

O

Page 66: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller Indices for planes (contd.)

origin

intercepts

reciprocalsMiller Indices

AB

CD

O

ABCD

O

1 ∞ ∞

1 0 0

(1 0 0)

OCBE

O*

1 -1 ∞

1 -1 0

(1 1 0)_

Plane

x

z

y

O*

x

z

E

Zero represents

that the plane is parallel to

the corresponding

axis

Bar represents a negative intercept

Page 67: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller indices of a plane specifies only its orientation in space not its position

All parallel planes have the same Miller Indices

AB

CD

O

x

z

y

E

(100)

(h k l ) (h k l )_ _ _

(100) (100)

_

Page 68: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller indices of a family of symmetry related planes

= (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal

{hkl }

All the faces of the cube are equivalent to each other by symmetry

Front & back faces: (100)Left and right faces: (010)

Top and bottom faces: (001)

{100} = (100), (010), (001)

Page 69: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Slip planes in a ccp crystalSlip planes in ccp are the close-packed planes

AEG (111)

A D

FE

D

G

x

y

z

BC

y

ACE (111)

A

E

D

x

z

C

ACG (111)

A

G

x

y

z

C

A D

FE

DG

x

y

z

BC

All four slip planes of a ccp crystal:

{111}

D

E

G

x

y

z

C

CEG (111)

Page 70: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

{100}cubic = (100), (010),

(001)

{100}tetragonal = (100), (010)

(001)

Cubic

Tetragonal

Miller indices of a family of symmetry related planes

x

z

y

z

x

y

Page 71: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

[100]

[010]

Symmetry related directions in the hexagonal crystal system

cubic100 = [100], [010],

[001]

hexagonal100

Not permutations

Permutations

[110]

x

y

z= [100],

[010],

[110]

Page 72: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

(010)(100

)

(110)

x

{100}hexagonal = (100), (010),

{100}cubic = (100), (010), (001)

Not permutations

Permutations

Symmetry related planes in the hexagonal crystal system

(110)

y

z

Page 73: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Problem:

In hexagonal system symmetry related planes and directions do NOT have Miller indices which are permutations

Solution:

Use the four-index Miller-Bravais Indices instead

Page 74: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

x1

x2

x3 (1010)

(0110)

(1100)

(hkl)=>(hkil) with i=-(h+k)

Introduce a fourth axis in the basal plane

x1

x3

Miller-Bravais Indices of Planes

x2

Prismatic planes:{1100}

= (1100)

(1010)

(0110)

z

Page 75: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Miller-Bravais Indices of Directions in hexagonal crystals

x1

x2

x3

Basal plane=slip plane=(0001)

[uvw]=>[UVTW]

wWvuTuvVvuU );();2(3

1);2(

3

1

Require that:

0 TVU

Vectorially 0aaa 21 3

caa

caaa

wvu

WTVU

21

321

a1

a2

a2

Page 76: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

a1

a1-a2

-a3

]0112[

a2a3

x1

x2

x3

wWvuTuvVvuU );();2(3

1);2(

3

1

]0112[0]100[ 31

31

32

]0121[0]010[ 31

32

31

]2011[0]011[ 32

31

31

x1:

x2:

x3:

Slip directions in hcp

0112

Miller-Bravais indices of slip directions in hcp crystal:

Page 77: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Some IMPORTANT Results

Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl):

h u + k v + l w = 0

Weiss zone law

True for ALL crystal systems

h U + k V + i T +l W = 0

Page 78: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

CUBIC CRYSTALS

[hkl] (hkl)

Angle between two directions [h1k1l1] and [h2k2l2]:

C

[111]

(111)

22

22

22

21

21

21

212121coslkhlkh

llkkhh

Page 79: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

dhkl

Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell

222 lkh

acubichkld

O

x(100)

ad 100

BO

x

z

E

2011

ad

Page 80: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Orientation Relationships

In solid state phase-transformation, the new crystalline phase has a particular orientation relationship with the parent phase.

When proeutectioid ferrite forms from austenite in steels the following orientation relationship, known as Kurdjumov-Sachs relationship is observed:

110||111

}111{||}110{

Page 81: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Orientation relationship between TiC and B2 phase in as-cast and heat-treated NiTi shape memory alloys

Z. Zhang et al.Mat. Sci. Engg. A, 2006

Page 82: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Dendrite Growth Directions

FCC

100

BCC

100

HCP

0110

Page 83: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

[uvw] Miller indices of a direction (i.e. a set of parallel directions)

<uvw> Miller indices of a family of symmetry related directions

(hkl) Miller Indices of a plane (i.e. a set of parallel planes)

{hkl} Miller indices of a family of symmetry related planes

[uvtw] Miller-Bravais indices of a direction, (hkil) plane in a hexagonal system

Summary of Notation convention for Indices

Page 84: Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C.

Thank You