- Crystallographic points, directions and planes are specified using indexing schemes like Miller indices. - Materials can be single crystals or polycrystalline aggregates of randomly oriented grains, leading to anisotropic or isotropic properties respectively. - A crystal's diffraction pattern in reciprocal space is determined by its real space lattice and atomic structure. The reciprocal lattice is constructed geometrically from the real lattice and maps planes in real space to points in reciprocal space.

1. Lec. (4,5)
Z
X
Y
(100)
1

2. 2
Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume that contains
the complete lattice pattern of a crystal.
a, b and c are the lattice constants

3. Unit Cells Types
Primitive Face-Centered
Body-Centered End-Centered
A unit cell is the smallest component of the crystal that reproduces the whole
crystal when stacked together.
• Primitive (P) unit cells contain only a single lattice point.
• Internal (I) unit cell contains an atom in the body center.
• Face (F) unit cell contains atoms in the all faces of the planes composing the cell.
• Centered (C) unit cell contains atoms centered on the sides of the unit cell.
Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic,
trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice.
3

4. (c) 2003 Brooks/Cole Publishing / Thomson Learning™
Lattice parameters in cubic, orthorhombic and hexagonal
crystal systems.
4

5. Define basic terms and give examples of each:
• Points (atomic positions)
• Vectors (defines a particular direction - plane normal)
• Miller Indices (defines a particular plane)
• relation to diffraction
• 3-index for cubic and 4-index notation for HCP
Basic definitions – Lattice sites
5

6.  Miller indices - A shorthand notation to describe certain
crystallographic directions and planes in a material.
Denoted by [ ], <>, ( ) brackets. A negative number is
represented by a bar over the number.
Points, Directions and Planes in the
Unit Cell
6

7. • Coordinates of selected points in the unit cell.
• The number refers to the distance from the origin in terms of lattice parameters.
Point Coordinates
7

8. 8
Point Coordinates
Point coordinates for unit cell
center are
a/2, b/2, c/2 ½½½
Point coordinates for unit cell
corner are 111
Translation: integer multiple of
lattice constants  identical
position in another unit cell
z
x
y
a b
c
000
111
y
z

2c



b
b

9. 9
Crystallographic Directions
1. Vector repositioned (if necessary) to
pass through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvw]
ex: 1, 0, ½ => 2, 0, 1 => [201 ]
-1, 1, 1
z
x
where overbar represents a negative
index
[ 111]=>
y
[ 111]
[ 201]
Algorithm

10. Procedure:
1. Any line (or vector direction) is specified by 2 points.
• The first point is, typically, at the origin (000).
2. Determine length of vector projection in each of 3 axes in
units (or fractions) of a, b, and c.
• X (a), Y(b), Z(c)
1 1 0
3. Multiply or divide by a common factor to reduce the
lengths to the smallest integer values, u v w.
4. Enclose in square brackets: [u v w]: [110] direction.
a
b
c
DIRECTIONS will help define PLANES (Miller Indices or plane normal).

[ 1 10]5. Designate negative numbers by a bar
• Pronounced “bar 1”, “bar 1”, “zero” direction.
6. “Family” of [110] directions is designated as <110>.
Directions in a Crystal
10

11. Examples
210
X = ½ , Y = ½ , Z = 1
[½ ½ 1] [1 1 2]
X = 1 , Y = ½ , Z = 0
[1 ½ 0] [2 1 0]
11

12. 12
• When we write the direction
[n1n2n3] depending on the origin,
negative directions are written as
R = n1a1 + n2a2 + n3a3
To specify the direction, the
smallest possible integers must
be used.
Y direction
(origin) O
- Y direction
X direction
- X direction
Z direction
- Z direction
][ 321 nnn
Negative Directions
12

13. 13
X = -1 , Y = -1 , Z = 0 [110]
X = 1 , Y = 0 , Z = 0 [1 0 0]
Examples of Crystal Directions
13

14. 14
X =-1 , Y = 1 , Z = -1/6
[-1 1 -1/6] [6 6 1]
A vector can be moved to the origin.
Examples
14

15. 15
• Within a crystal lattice it is possible to identify sets of
equally spaced parallel planes.
These are called lattice planes.
• In the figure, the density of lattice points on each plane of a set
is the same & all lattice points are contained on each set of
planes.
b
a
b
a
The set of
planes for a
2D lattice.
Crystal Planes
15

16. • If the plane passes through origin, either:
– Construct another plane, or
– Create a new origin
– Then, for each axis, decide whether plane intersects
or parallels the axis.
• Algorithm for Miller indices
1. Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no commas.
16
Crystallographic Planes

17. 17
Crystallographic Planes
• Crystallographic planes are specified by 3 Miller
Indices (h k l). All parallel planes have same Miller
indices.

18. 18
• Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice & are defined
as the reciprocals of the fractional intercepts which the plane
makes with the crystallographic axes.
• To find the Miller indices of a plane, take the following steps:
1. Determine the intercepts of the plane along each of the three
crystallographic directions.
2. Take the reciprocals of the intercepts.
3. If fractions result, multiply each by the denominator of the
smallest fraction.
18

19. 19
Crystallographic Planesz
x
y
a b
c
4. Miller Indices (110)
example a b c
z
x
y
a b
c
4. Miller Indices (200)
1. Intercepts 1 1 
2. Reciprocals 1/1 1/1 1/
1 1 0
3. Reduction 1 1 0
1. Intercepts 1/2  
2. Reciprocals 1/½ 1/ 1/
2 0 0
3. Reduction 2 0 0
example a b c

20. 20
Crystallographic Planes
z
x
y
a b
c



4. Miller Indices (634)
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4

21. x y z
Intercepts
Intercept in terms of lattice parameters
Reciprocals
Reductions
Enclosure
a -b c/2
 -1 1/2
0 -1 2
N/A
(012)
Determine the Miller indices for the plane shown in the sketch
21

22. 22
Family of Planes
• Planes that are crystallographically
equivalent have the same atomic packing.
• Also, in cubic systems only, planes having
the same indices, regardless of order and
sign, are equivalent.
• Ex: {111}
= (111), (111), (111), (111), (111), (111), (111), (111)
(001)(010), (100), (010),(001),Ex: {100} = (100),
_ __ __ _ __ _ _ _ _

23. Family of planes{110}
23

24. FCC Unit Cell with (110) plane
24

25. BCC Unit Cell with (110) plane
25

26. 26
Axis X Y Z
Intercept
points 1 ∞ ∞
Reciprocals 1/1 1/ ∞ 1/ ∞
Smallest
Ratio 1 0 0
Miller İndices (100)
Example-1
(1,0,0)

27. 27
Axis X Y Z
Intercept
points 1 1 ∞
Reciprocals 1/1 1/ 1 1/ ∞
Smallest
Ratio 1 1 0
Miller İndices (110)
Example-2
(1,0,0)
(0,1,0)

28. 28
Axis X Y Z
Intercept
points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1
Smallest
Ratio 1 1 1
Miller İndices (111)(1,0,0)
(0,1,0)
(0,0,1)
Example-3

29. 29
Axis X Y Z
Intercept
points 1/2 1 ∞
Reciprocals 1/(½) 1/ 1 1/ ∞
Smallest
Ratio 2 1 0
Miller İndices (210)
(1/2, 0, 0)
(0,1,0)
Example-4

30. 30
Miller Indices
Reciprocal numbers are:
2
1
,
2
1
,
3
1
Plane intercepts axes at cba 2,2,3
Indices of the plane (Miller): (2,3,3)
(200) (100)
Indices of the direction: [2,3,3]a
3
2
2
b
c
[2,3,3]
Z
X
Y
(100)
Z
X
Y
(110)
Z
X
Y
(111)

31. 31
Example 7
31

32. SUMMARY
• Crystallographic points, directions and planes are
specified in terms of indexing schemes.
• Materials can be single crystals or polycrystalline.
• Material properties generally vary with single crystal
orientation (anisotropic), but are generally non-
directional (isotropic) in polycrystals with randomly
oriented grains.
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
32

33.  A crystal resides in real space. The diffraction pattern of the crystal in Fraunhofer
diffraction geometry resides in Reciprocal Space. In a diffraction experiment
(powder diffraction using X-rays, selected area diffraction in a TEM), a part of
this reciprocal space is usually sampled.
 From the real lattice the reciprocal lattice can be geometrically constructed. The
properties of the reciprocal lattice are ‘inverse’ of the real lattice → planes ‘far
away’ in the real crystal are closer to the origin in the reciprocal lattice.
 As a real crystal can be thought of as decoration of a lattice with motif; a
reciprocal crystal can be visualized as a Reciprocal Lattice decorated with a
motif* of Intensities.
 Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif*
 The reciprocal of the ‘reciprocal lattice’ is nothing but the real lattice!
 Planes in real lattice become points in reciprocal lattice and vice-versa.
Reciprocal Lattice

34.  In diffraction patterns (Fraunhofer geometry) (e.g. SAD), planes are mapped as spots (ideally points).
Hence, we would like to have a construction which maps planes in a real crystal as points.
 Apart from the use in ‘diffraction studies’ we will see that it makes sense to use reciprocal lattice
when we are dealing with planes.
Motivation for constructing reciprocal lattices
 As the index of the plane increases
→ the interplanar spacing decreases
→ and ‘planes start to crowd’ around
the origin in the real lattice (refer figure).
 Hence, we work in reciprocal lattice
when dealing with planes.
As seen in the figure the diagonal is divided
into (h + k) parts.

35. Let us start with a one dimensional lattice and construct the reciprocal lattice
Reciprocal Lattice
Real Lattice
How is this reciprocal lattice constructed
Note in 1D planes are points and
have Miller indices of single digit
(they have been extended into the
second dimension (as lines) for
better visibility
 The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc.
 As the index of the plane increases it gets closer to the origin (there is
a crowding towards the origin)
One unit cell
Reciprocal Lattice
Each one of these points
correspond to a set of
‘planes’ in real space
Note that in reciprocal space index
has NO brackets
Real Lattice

36. Reciprocal Lattice
(01)
(10)
(11)
(21)
10 20
11
221202
01 21
00
The reciprocal lattice has an origin!
1a

2a

1a
1
1
a
*
11g *
21g*
b2

*
b1

Now let us construct some 2D reciprocal lattices
Example-1
Each one of these points correspond to
a set of ‘planes’ in real space
2
1
a
Note that vectors in reciprocal
space are perpendicular to planes
in real space (as constructed!)
But do not measure distances from the figure!
Overlay of real and
reciprocal lattices
g vectors connect
origin to reciprocal
lattice points