Lecture 5

Structure of Materials

23 August 2022

1
Symmetry and the 14 Space (Bravais) Lattices
Based on the symmetry elements we just discussed,
crystallographers have shown that ALL space lattices can be
classified ONLY into the 14 types we had listed.

7 crystal systems : 7 different point groups (rotational and

reflection symmetries)
14 Bravais lattices: 14 different types of complete symmetry.

Thus the unit cells listed are based on the inherent

symmetry of the lattice itself.

2
Now we can answer those two questions about this table
• What is the basis of classification of lattices into these 14 types?
• Why are certain Bravais lattices NOT found, e.g. face-centered tetragonal?

No. Crystal System Example Conventional Unit Cell Bravais Lattice

1 Cubic (c) Cu, Al, Fe, NaCl a = b = c; = = = 90o P I F --
2 Tetragonal (t) β-Sn, TiO2 a = b ≠ c; = = = 90o P I -- --
3 Orthorhombic (o) -S, Ga, Fe3C a ≠ b ≠ c; = = = 90o P I F C
4 Hexagonal (h) Zn, Mg, NiAs a = b ≠ c; = = 90o, = 120o P -- -- --
5 Rhombohedral As, Sb, Bi a = b = c; = = ≠ 90o P -- -- --
(R)
6 Monoclinic (m) β-S, CaSO4.2H2O a ≠ b ≠ c; = = 90o ≠ P -- -- C
7 Triclinic (a) K2CrO7 a ≠ b ≠ c; ≠ ≠ ≠ 90o P -- -- --

The Answer is SYMMETRY!

3
End-centered cubic lattices – do they exist?

4
End-centered cubic lattices – do they exist?

5
What about a face-centered tetragonal (FCT) Lattice?

a
• Face-centered a
tetragonal is in fact
body-centered
tetragonal (BCT).
c

a/√2

6
Exercise Problem: Why don’t Edge-Centered Lattices Exist?

A a
7
• This summarizes our discussion on crystal systems, lattices and unit cells.

• Hope it is clear that the classification of lattices is not based just on

drawing any random unit cell, but on principles of symmetry – the
geometry of these 3D objects!

• The unit cell should

• reflect the intrinsic symmetry of the lattice and

• be the simplest one out of the infinite possible cells!!

8
Topic II
Crystallographic Points
Miller Indices for Directions and Planes

9
An example: why are these important?
Materials with oriented crystals show better properties, e.g., easy magnetisation!

a b c

100 µm 200 µm

10
Indexing of points, directions and planes in a crystal

• In dealing with crystal structure of materials, it is essential to

quantify the point coordinates, crystallographic directions and
planes so that they can be correlated with certain material
properties.
• How is it done?
• A convention is followed, where a corner of the unit cell is taken
as the origin and its 3 sides are called the x, y and z axes.
• This applies to ALL unit cells, irrespective of whether the axes
are mutually perpendicular or not, i.e. all crystal systems and
not just cubic.

11
Point Coordinates
z
• O is the origin 0, 0, 0.
• The three axes are x, y and z and
the corresponding vectors are a, b
and c respectively (the edge
lengths of the unit cell). D??
c
• Point coordinates are fractions/
multiples of the a, b, c unit cell C
edge lengths. O y
b
• Thus, point coordinates of a
A: 1a, 0b, 0c = 1, 0, 0
B: 1a,1b,0c = 1, 1, 0 x A B
C: 0.5a, 0.5b, 0.5c = 0.5, 0.5, 0.5

12
Miller Indices for Directions

A vector r passing from the origin to a lattice point can be written as:

r = r1 a + r2 b + r3 c z

r3 r c
O r2 y
r1 a
x b

13
Procedure for Miller Indices of Directions
• A crystallographic direction is a line directed between two points
(e.g. OA). It is a vector with a Head & a Tail.
z
• Point coordinates
Tail: x1, y1, z1 and Head: x2, y2, z2
• Subtract the coordinates C
x2 -x1, y2 -y1, z2 -z1 c
• Convert them into fractions/multiples of the respective lattice y
parameters.
O b
𝑥 −𝑥 𝑧 −𝑧
a
𝑦 −𝑦
𝑎 𝑏 𝑐
• Multiply/divide (if necessary) by n to get integers! x A B
• The Direction Indices are

𝑥 −𝑥 𝑦 −𝑦 𝑧 −𝑧
𝑢 = 𝑛. 𝑣 = 𝑛. 𝑤 = 𝑛.
𝑎 𝑏 𝑐
• Enclose them in square brackets [uvw].
14
Examples of Directions

• Point Coordinates are z

O: 0, 0, 0; A: 1, 0, 0; B: 1, 1, 0; C: 1, 1, 1
• Hence OA is
(1-0)a+(0-0)b+(0-0)c = 1a+0b+0c C
• u = 1; v = 0; w = 0. c
• Direction OA is O b y
[uvw] = [100]. a
• Likewise, OB is [110] and OC is [111]. x B
A

15
Parallel Directions
z
• Point E is 0, 1, 0 and B is 1, 1, 0. G F
• Hence EB = 1a+0b+0c = [100] like OA!

D c C
• In fact, OA, EB, FC and GD are all [100].
O b E y
a
• So, the [uvw] indices tell us the
orientation of the direction vector, not x A B
its actual position.

16
Another example

z
• O is 0, 0, 0 A
• A is ½, ½, 1
• OA = 0.5a+0.5b+1c = [1/2 1/2 1]
c
• Convert to integers by multiplying O
with 2
b y
a
• Hence OA = [112].
x

17
What if the origin does not lie on the direction vector? We want to index
CA!
z z
A
A
c
c O
O b
b y a y
a
C x C
x
• Shifting the origin (O) to C essentially does the
• Can of course use the same
same operation as on the left.
method.
• C is 1a,1b, 0c; A is 0.5a, 0.5b, 1c • C is 0a, 0b, 0c; A is −0.5a, −0.5b, 1c
• CA is −0.5a−0.5b+1c • CA is −0.5a−0.5b+1c
• CA = [−1 −1 2] = [ ] • CA = [−1 −1 2] = [ ]
18
Family of Directions
• Any direction [ ] is essentially z
the same line as ].
[001]
• For the cubic lattice, [100], [010]
and [001] are equivalent due to [100]
symmetry. [010] y
[010]

• A family of crystallographically [100]

equivalent directions is clubbed x
together as <u v w>
• e.g. <100> includes [100],
[010], [001], [100], [010] and [001]
[001].

19
Miller Indices of Crystallographic Planes

20
Miller Indices for Planes

• Pick an origin that does NOT lie on z

the plane!
• Note the intercepts made by the
plane on the coordinate axes.
• Express intercepts in terms of c
multiples of the lattice vectors, a, b O
and c. b y
• Take the reciprocals of the a
intercepts.
x
• Convert into smallest integers with
the same ratio (if required).
• Enclose in parentheses, e.g. (hkl)
21