basic crystallography

DST-SERC School on Texture and Microstructure

BASIC CRYSTALLOGRAPHY

Rajesh PrasadDepartment of Applied MechanicsIndian Institute of Technology

New Delhi 110016

[email protected]

2

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

3

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

Crystal ?

4

Lattice?

A 3D translationally

periodic arrangement

of points in space is

called a lattice.

5

A 3D

translationally

periodic

arrangement

of atoms

Crystal

A 3D

translationally

periodic

arrangement of

points

Lattice

6

What is the relation between

the two?

Crystal = Lattice + Motif

Motif or basis: an atom or

a group of atoms associated

with each lattice point

7

Crystal=lattice+basis

Lattice: the underlying periodicity of

the crystal,

Basis: atom or group of atoms

associated with each lattice points

Lattice: how to repeat

Motif: what to repeat

8

A 3D translationally periodic arrangement of points

Each lattice point in a lattice has identical neighbourhood of

other lattice points.

Lattice

9

+

Love Pattern(Crystal)

Love Lattice + Heart(Motif)

=

=

Lattice + Motif = Crystal

Love Pattern

10

Air,

Water

and

Earth

by

M.C.

Esher

11

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

13

Contents





Reciprocal lattice

14

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

15

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

16

Size and shape of the unit cell:

1. A corner as origin

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

COORDINATESYSTEM

3. The three lengths a, b, c and the three

interaxial angles α, β, γ are called the

LATTICE PARAMETERS

α

β

γ

a

b

c


17

7 crystal Systems

Crystal System Conventional Unit Cell

1. Cubic a=b=c, α=β=γ=90°

2. Tetragonal a=b≠c, α=β=γ=90°

3. Orthorhombic a≠b≠c, α=β=γ=90°

4. Hexagonal a=b≠c, α=β= 90°, γ=120°

5. Rhombohedral a=b=c, α=β=γ≠90°OR Trigonal

6. Monoclinic a≠b≠c, α=β=90°≠γ

7. Triclinic a≠b≠c, α≠β≠γ


18

The description of a unit cell requires: 1. Its Size and shape (the six lattice parameters)

2. Its atomic content (fractionalcoordinates):

Coordinatesof atoms asfractions of

the respectivelattice parameters


19x

y

Projection/plan view of unit cells

½ ½ 0

Example 1: Cubic close-packed (CCP) crystal

e.g. Cu, Ni, Au, Ag, Pt, Pb etc.

2

1

2

1

2

1

2

1

x

y

z

Plan description : 1

20

The six lattice parameters a, b, c, α, β, γ

The cell of the lattice

lattice

crystal

+ Motif

21

Contents





Reciprocal lattice

22

14 Bravais lattices divided into seven crystal systems

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

P: Primitive

(lattice points only at

the 8 corners of the

unit cell)

I: Body-centred (lattice

points at the corners + one

lattice point at the centre of

the unit cell)

F: Face-centred

(lattice points at the

corners + lattice

points at centres of

all faces of the unit

cell)

C: End-centred or

base-centred

(lattice points at the

corners + lattice

points at the centres of

a pair of opposite

faces)

23

The three cubic Bravais lattices


1. Cubic P I F

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

24

Orthorhombic C

End-centred orthorhombic

Base-centred orthorhombic

25

Monatomic Body-CentredCubic (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

26

½

½½

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

27

14 Bravais lattices divided into seven crystal

systems


1. Cubic P I F

2. Tetragonal P I


4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

?

28

End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2

a

2

a

29


systems


1. Cubic P I F C

2. Tetragonal P I


4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

30

Face-centred cubic in the Bravais list ?

Cubic F = Tetragonal I ?!!!

31


systems


1. Cubic P I F C

2. Tetragonal P I


4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

32

Couldn’t

find his

photo on

the net

1811-1863

Auguste Bravais

1850: 14 lattices1835: 15 lattices

ML Frankenheim

1801-1869

24 March 2008:

13 lattices !!!

DST-SERC

School

X

1856: 14 lattices

History:

33

Why can’t the Face-Centred Cubic lattice (Cubic F) be considered as a Body-CentredTetragonal lattice (Tetragonal I) ?

34

Primitive

cell

Primitive

cell

Non-

primitive cellA unit cell of a lattice is NOT unique.

UNIT CELLS OF A LATTICE

Unit cell shape CANNOT be the basis for classification of Lattices

35

What is the basis for

classification of lattices

into

7 crystal systems

and

14 Bravais lattices?

36

Lattices are

classified on the

basis of their

symmetry

37

What is

symmetry?

38

If an object is brought into self-

coincidence after some

operation it said to possess

symmetry with respect to that

operation.

Symmetry

39

NOW NO SWIMS ON MON

40

Rotational symmetry

http://fab.cba.mit.edu/

41

If an object come into self-coincidence through smallest

non-zero rotation angle of θ then it is said to have an n-

fold rotation axis where

θ

0360=n

θ=180°

θ=90°

Rotation Axis

n=2 2-fold rotation axis

n=4 4-fold rotation axis

42

Reflection (or mirror symmetry)

43

Lattices also have

translational

symmetry

Translational symmetry

In fact this is the

defining symmetry of

a lattice

44

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

45

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 of a crystal including translations

is called its SPACE GROUP

Point Group and Space Group

46

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⇒ 7 types of lattices

⇒ 7 crystal systems

47

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

48

Tetragonal symmetry Cubic symmetry

Cubic C = Tetragonal P Cubic F ≠≠≠≠ Tetragonal I

49

The three Bravais lattices in the cubic crystal

system have the same rotational symmetry but

different translational symmetry.

Simple cubic

Primitive cubic

Cubic P

Body-centred cubic

Cubic I

Face-centred cubic

Cubic F

50

QUESTIONS?

51

Contents





Reciprocal lattice

52

Miller Indices of directions and planes

William Hallowes Miller(1801 – 1880)

University of Cambridge

Miller Indices 1

53

1. Choose a point on the directionas 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

54

y

zMiller indices of a direction:

only the orientationnot its position or sense

All parallel directions have the same Miller indices

[100]x

Miller Indices 3

55

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

56

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

57

Slip System

A common microscopic mechanism of plastic deformation involves slip on some crystallographic planes known as slip planes in certain crystallographic directions called slip directions.

A combination of slip direction lying in a slip plane is called a slip system

58

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

59

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

60

Miller Indices for planes (contd.)

origin

intercepts

reciprocals

Miller 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

61

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)_

62

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)

63

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

D

G

x

y

z

BC

All four slip planes of a ccp crystal:

{111}

D

E

G

x

y

z

C

CEG (111)

64

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

(001)

Cubic

TetragonalMiller indices of a family of symmetry related planes

x

z

y

z

x

y

65[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]

66

(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

67

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

68x1

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

69

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

70

a1

a1-a2

-a3]0112[

a2a3

x1

x2

x3wWvuTuvVvuU =+−=−=−= );();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 hcpcrystal:

71

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

72

CUBIC CRYSTALS

[hkl] ⊥⊥⊥⊥ (hkl)

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

C

[111]

(111)

22

22

22

21

21

21

212121coslkhlkh

llkkhh

++++

++=θ

73

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

222lkh

acubichkld

++=

O

x(100)

ad =100

BO

x

z

E

2011

ad =

74

[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

75

Contents





Reciprocal lattice

76

Reciprocal Lattice

A lattice can be defined in terms of three vectors a, b and c along the edges of the unit cell

We now define another triplet of vectors a*, b* and c* satisfying the following property:

The triplet a*, b* and c* define another unit cell and thus a lattice called the reciprocal lattice. The original lattice in relation to the reciprocal lattice is called the direct lattice.

VVV

b)(ac

a)(cb

c)(ba

×=

×=

×= ∗∗∗ ; ;

V= volume of the unit cell

77

Every crystal has a unique Direct Lattice and a corresponding Reciprocal Lattice.

The reciprocal lattice vector ghkl has the following interesting and useful properties:

A RECIPROCAL LATTICE VECTOR ghkl is a vector with integer components h, k, l in the reciprocal space:

∗∗∗ ++= cbag lkhhkl

Reciprocal Lattice (contd.)

78

hkl

hkld

1=g

1. The reciprocal lattice vector ghkl with integer components h, k, l is perpendicular to the direct lattice plane with Miller indices (hkl):

)(hklhkl ⊥g

Properties of Reciprocal Lattice Vector:

2. The length of the reciprocal lattice vector ghkl with integer components h, k, l is equal to the reciprocal of the interplanar spacing of direct lattice plane (hkl):

79

Full significance of the concept of the reciprocal lattice will be appreciated in the discussion of the x-ray and electron diffraction.

80

Thank You

81

Example: Monatomic Body-Centred Cubic (BCC) crystal

e.g., Fe, Ti, Cr, W, Mo etc.

Atomic content of a unit cell: fractional coordinates:

Fractional coordinates of the two atoms

000; ½ ½ ½


Effective no. of atoms in the unit cell

2188

1=+×

corners Body centre

000

½ ½ ½

basic crystallography

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