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

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

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

Crystal, Lattice and Motif

Crystal ?

Lattice?

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

Crystal, Lattice and Motif

A 3D translationally periodic arrangement of atoms

Crystal

A 3D translationally periodic arrangement of points

Lattice

Crystal, Lattice and Motif

Motif?

Crystal = Lattice + Motif

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

Crystal, Lattice and Motif

“Nothing that is worth knowing can be taught.”

Oscar Wilde

+

Love Pattern(Crystal)

Love Lattice

+ Heart (Motif)

=

=

Lattice + Motif = Crystal

Love Pattern

Air, Water and Earth

Maurits Cornelis Escher 1898-1972

Dutch Graphic Artist

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

Cu Crystal NaCl Crystal

Lattice

FCC FCC

Motif 1 Cu+ ion

1 Na+ ion + 1 Cl- ion

Crystal

Crystal, Lattice and Motif

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

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

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

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

Wigner-Seitz Unit Cells

FCC

BCC

Rhombic Dodcahedron

Tetrakaidecahedron

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

The cell of the lattice

lattice

crystal

+ Motif

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

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

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

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

Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

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

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

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

End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2

a2

a

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

Face-centred cubic in the Bravais list ?

Cubic F = Tetragonal I ?!!!

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

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:

Why can’t the Face-Centred Cubic lattice

(Cubic F) be considered as a Body-

Centred Tetragonal lattice (Tetragonal I) ?

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

What is the basis for classification of lattices

into 7 crystal systems

and 14 Bravais lattices?

Lattices are classified on the

basis of their symmetry

What is symmetry?

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

Symmetry

NOW NO SWIMS ON MON

Lattices also have translational symmetry

Translational symmetry

In fact this is the defining symmetry of a lattice

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

Rotational Symmetries

Z180 120 90 72 60

2 3 4 5 6

45

8

Angles:

Fold:

Graphic symbols

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…

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

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

Correction: Shift the box

Michael Gottlieb’s correction:

But gives H:O = 1.5 : 1

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

Lack strict translational periodicity -> Quasiperiodic

Icosahedron

Penrose Tiling

Diffraction PatternExternal

Morphology

Reflection (or mirror symmetry)

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

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

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

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

Tetragonal symmetry Cubic symmetry

Cubic C = Tetragonal P Cubic F Tetragonal I

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

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

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

QUESTIONS?

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

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

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

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

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

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

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

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

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

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)

_

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)

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)

{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

[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]

(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

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

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

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

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:

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

CUBIC CRYSTALS

[hkl] (hkl)

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

C

[111]

(111)

22

22

22

21

21

21

212121coslkhlkh

llkkhh

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

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{

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

Dendrite Growth Directions

FCC

100

BCC

100

HCP

0110

[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

Thank You