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1. Directions
 Direction:
 A line between two points and it is a
vector .

2. For more help contact me
Muhammad Umair Bukhari
Engr.umair.bukhari@gmail.com
www.bzuiam.webs.com
03136050151

3. Process
 Steps utilized in determination of three
directional indices are :
 A vector is positioned such that it pass through
the origin of coordinate system.
 The length of vector projection on each of the
three axis is determined; these are measured in
term of unit cell dimensions a, b, and c.
 These three vectors are multiplied or divided by
common factor to reduce them to smallest
integer values.
 The three indices not separated by commas.
They are enclosed by square brackets: [uvw].

4. Process (contd.)
 If more than one direction for a particular
crystal structure is required, it is
necessary for maintaining the
consistency that the positive-negative
once established not be changed
 Changing the sign of the indices
produce an anti-parallel direction.
 For each of the three axis there will exist
both +ive and –ive coordinates. Thus
negative coordinates are represented by
a bar over the index.

5. Directions

6. Important Notes
 Because the directions are vectors, a
direction and its negative are not
identical; [100] is not equal to [-100].
They represent the same line, but
opposite directions.
 A direction and its multiple are identical;
[100] is the same direction as [200].

7. What is plane ?
 A flat surface determine the position of 3
point in space.

8. Planes
 Miller indices are used to identify planes.
 As described in following procedure:
 Identify the points at which the plane intercepts
the x, y and z coordinates in terms of the
number of the lattice parameters.
 If the plane passes through the origin, the origin
of the coordinate system must be moved.
 Take reciprocals of these intercepts.
 Clear fractions but do not reduce to lowest
integers.
 Enclose the resulting numbers in brackets “()”.
 Negative numbers should be written with the bar
over the number.

9. Planes

10. Important aspects of Miller
indices for Planes
 Planes and their negatives are identical.
Therefore (020)=(0-20)
 Planes and their multiples are not
identical.
 In cube systems, a direction that has
same indices as a plane, is
perpendicular to that plane.

11. Why we need Directions & Planes
 we need a way to identify directions and
planes of atoms
 Why ????????????
 Deformation under loading (slip) occurs
on certain crystalline planes and in
certain crystallographic directions.
 Before we can predict how materials fail,
we need to know what modes of failure
are more likely to occur.

12. Contd.
 Other properties of materials (electrical
conductivity, thermal conductivity, elastic
modulus) can vary in a crystal with
orientation.

13. Simple Cubic

14. Volume of Simple Cubic
Volume of SC=a*a*a
vol. of SC=a3
as a=2R
So,
Volume of SC= (2R)3
Volume of SC= 8R3

15. Face-centered Cubic

16. Volume of FCC
AC=4R
According to Pythagoras Theorem
(AC)2=(AB)2+(BC)2
(4R)2=(a)2+(a)2
16R2=2a2
a2=8R2
a=2√2 R
vol. of FCC=V3=(2√2 R)3 =16√2 R3

17. Body-Centered Cubic

18. Vol. of BCC
 BC=√2a
(AC)2= (AB)2 +(BC)2
(4R)2= (a)2 +(√2a)2
16R2= 3a2
a2 =16/3 R2
a=4R/ √3
 vol. of BCC= a3
vol. of BCC= 64R3
3√3