1. DST-SERC School on Texture and Microstructure
BASIC CRYSTALLOGRAPHY
Rajesh Prasad
Department of Applied Mechanics
Indian Institute of Technology
New Delhi 110016
rajesh@am.iitd.ac.in
2. 2
Contents
Crystal, Lattice and Motif
Unit cells, Lattice Parameters and Projections
Miller Indices & Miller-Bravais Indices
Directions and Planes
Classification of Lattices:
7 crystal systems
14 Bravais lattices
Reciprocal lattice
13. 13
Contents
Crystal, Lattice and Motif
Unit cells, Lattice Parameters and Projections
Miller Indices & Miller-Bravais Indices
Directions and Planes
Classification of Lattices:
7 crystal systems
14 Bravais lattices
Reciprocal lattice
14. 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. 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. 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
COORDINATE
SYSTEM
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. 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. 18
The description of a unit cell
requires:
1. Its Size and shape (the six lattice parameters)
2. Its atomic content
(fractional
coordinates):
Coordinates
of atoms as
fractions of
the respective
lattice parameters
Unit cell description : 6
19. 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. 20
The six lattice parameters a, b, c, α, β, γ
The cell of the lattice
lattice
crystal
+ Motif
21. 21
Contents
Crystal, Lattice and Motif
Unit cells, Lattice Parameters and Projections
Miller Indices & Miller-Bravais Indices
Directions and Planes
Classification of Lattices:
7 crystal systems
14 Bravais lattices
Reciprocal lattice
22. 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. 23
The three cubic Bravais lattices
Crystal system Bravais lattices
1. Cubic P I F
Simple cubic
Primitive cubic
Cubic P
Body-centred cubic
Cubic I
Face-centred cubic
Cubic F
25. 25
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 000
Motif: two atoms
Cl 000; Cs ½ ½ ½
Cs
Cl
26. 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. 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
?
29. 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
31. 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
32. 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. 33
Why can’t the Face-
Centred Cubic lattice
(Cubic F) be considered
as a Body-Centred
Tetragonal lattice
(Tetragonal I) ?
41. 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
0
360
=n
θ=180°
θ=90°
Rotation Axis
n=2 2-fold rotation axis
n=4 4-fold rotation axis
45. 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. 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. 47
7 crystal Systems
System 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
49. 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
51. 51
Contents
Crystal, Lattice and Motif
Unit cells, Lattice Parameters and Projections
Miller Indices & Miller-Bravais Indices
Directions and Planes
Classification of Lattices:
7 crystal systems
14 Bravais lattices
Reciprocal lattice
52. 52
Miller Indices of directions and
planes
William Hallowes Miller
(1801 – 1880)
University of Cambridge
Miller Indices 1
53. 53
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
a
b
c
54. 54
y
z
Miller 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
55. 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. 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
cubic
100 = [100], [010],
[001]
tetragonal
100 = [100], [010]
Cubic
Tetragonal
[010]
[100]
Miller Indices 4
57. 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. 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. 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. 60
Miller Indices for planes (contd.)
origin
intercepts
reciprocals
Miller
Indices
A
B
C
D
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. 61
Miller indices of a plane
specifies only its orientation
in space not its position
All parallel planes
have the same Miller
Indices
A
B
C
D
O
x
z
y
E
(100)
(h k l ) ≡≡≡≡ (h k l )
_ _ _
(100) ≡≡≡≡ (100)
_
62. 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. 63
Slip planes in a ccp crystal
Slip planes in ccp are the close-packed planes
AEG (111)
A
D
FE
D
G
x
y
z
B
C
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
B
C
All four slip
planes of a
ccp crystal:
{111}
D
E
G
x
y
z
C
CEG (111)
64. 64
{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
65. 65
[100]
[010]
Symmetry related directions in the
hexagonal crystal system
cubic
100 = [100], [010], [001]
hexagonal
100
Not permutations
Permutations
[110]
x
y
z
= [100], [010], [110]
67. 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
71. 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. 72
CUBIC CRYSTALS
[hkl] ⊥⊥⊥⊥ (hkl)
Angle between two directions [h1k1l1] and [h2k2l2]:
C
[111]
(111)
2
2
2
2
2
2
2
1
2
1
2
1
212121
cos
lkhlkh
llkkhh
++++
++
=θ
74. 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. 75
Contents
Crystal, Lattice and Motif
Unit cells, Lattice Parameters and Projections
Miller Indices & Miller-Bravais Indices
Directions and Planes
Classification of Lattices:
7 crystal systems
14 Bravais lattices
Reciprocal lattice
76. 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)(a
c
a)(c
b
c)(b
a
×
=
×
=
×
= ∗∗∗
;;
V= volume of the unit cell
77. 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. 78
hkl
hkl
d
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. 79
Full significance of the
concept of the reciprocal
lattice will be appreciated
in the discussion of the x-
ray and electron
diffraction.
81. 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
218
8
1
=+×
corners Body
centre
000
½ ½ ½