DST-SERC School on Texture and Microstructure
BASIC CRYSTALLOGRAPHY
Rajesh PrasadDepartment of Applied MechanicsIndian Institute of Technology
New Delhi 110016
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
12
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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
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
Unit cell description : 4
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
Unit cell description : 5
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
Unit cell description : 6
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
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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
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
Crystal system Bravais lattices
1. Cubic P I F
Simple cubicPrimitive cubicCubic P
Body-centred cubicCubic I
Face-centred cubicCubic F
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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
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
?
28
End-centred cubic not in the Bravais list ?
End-centred cubic = Simple Tetragonal
2
a
2
a
29
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
30
Face-centred cubic in the Bravais list ?
Cubic F = Tetragonal I ?!!!
31
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
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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
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
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
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
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
Unit cell description : 8
Effective no. of atoms in the unit cell
2188
1=+×
corners Body centre
000
½ ½ ½