This document discusses directions, planes, and Miller indices in crystal structures. It begins by introducing lattice planes and how Miller indices are used to describe planes and directions within crystal structures. It then provides general rules and conventions for Miller indices, including how they are determined for both directions and planes. Specific examples are given to illustrate how to calculate Miller indices. Important directions and planes within crystal structures are also highlighted. The document emphasizes that Miller indices allow for the standardized description of orientations within crystalline materials.

1. DIRECTIONS, PLANES AND
MILLER INDICES
By Srilakshmi B
UCST, Tumkur

2. CONTENTS
• INTRODUCTION
• NEED OF DIRECTIONS AND PLANES
• GENERAL RULES AND CONVENTION
• MILLER INDICES FOR DIRECTIONS
• MILLER INDICES FOR PLANES
• IMPORTANT FEATURES OF MILLER INDICES

3. INTRODUCTION
The crystal lattice may be regarded as made
up of an infinite set of parallel equidistant
planes passing through the lattice points
which are known as lattice planes.
In simple terms, the planes passing through
lattice points are called ‘lattice planes’.
For a given lattice, the lattice planes can be
chosen in a different number of ways.

4. • The orientation of planes or faces in a crystal can
be described in terms of their intercepts on the
three axes.
• Miller introduced a system to designate a plane
in a crystal.
• He introduced a set of three numbers to specify
a plane in a crystal.
• This set of three numbers is known as ‘Miller
Indices’ of the concerned plane.

5. NEED OF DIRECTIONS AND PLANES
• 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. Other properties of materials (electrical
conductivity, thermal conductivity, elastic
modulus) can vary in a crystal with orientation.

7. GENERAL RULES FOR LATTICE
DIRECTIONS, PLANES AND MILLER
INDICES
• Miller indices used to express lattice planes and
directions
• x, y, z are the axes (on arbitrarily positioned
origin)
• a, b, c are lattice parameters (length of unit cell
along a side) h, k, l are the Miller indices for
planes and directions -
expressed as planes: (hkl) and directions: [hkl]

8. CONVENTION FOR NAMING
• There are NO COMMAS between numbers
• Negative values are expressed with a bar over
the number
Example: -5 is expressed 5

9. MILLER INDICES FOR DIRECTIONS
• Draw vector, and find the coordinates of the
head, h1,k1,l1 and the tail h2,k2,l2.
• subtract coordinates of tail from coordinates
of head
• Remove fractions by multiplying by smallest
possible factor
• Enclose in square brackets

11. The direction can also be
determined by giving
the coordinates of the
first whole numbered
point (x,y) through
which each of the
direction passes.
In this figure direction of
OA is[110] and OB is
[520]

12. IMPORTANT DIRECTIONS IN A CRYSTAL

13. MILLER INDICES FOR PLANES
• If the plane passes through the origin, select
an equivalent plane or move the origin
• Determine the intersection of the plane with
the axes in terms of a,b, and c
• Take the reciprocal (1/∞ = 0)
• Convert to smallest integers
• Enclose by parentheses

14. EXAMPLE
• Here x,y and z
intercepts are 1,1,1.
• Therefore (111) is the
miller indices of the
plane

15. • DETERMINATION OF ‘MILLER
INDICES
• Step 1:The intercepts are 2,3 and
2 on the three axes.
• Step 2:The reciprocals are 1/2,
1/3 and 1/2.
• Step 3:The least common
denominator is ‘6’. Multiplying
each reciprocal by lcd, we get, 3,2
and 3.
• Step 4:Hence Miller indices for
the plane ABC is (3 2 3)

16. IMPORTANT FEATURES OF MILLER
INDICES
• A plane passing through the origin is defined in
terms of a parallel plane having non zero
intercepts.
• All equally spaced parallel planes have same
‘Miller indices’ i.e. The Miller indices do not only
define a particular plane but also a set of parallel
planes. Thus the planes whose intercepts are 1,
1,1; 2,2,2; -3,-3,-3 etc., are all represented by the
same set of Miller indices.

17. • It is only the ratio of the indices which is
important in this notation. The (6 2 2) planes
are the same as (3 1 1) planes.
• If a plane cuts an axis on the negative side of
the origin, corresponding index is negative. It
_
is represented by a bar, like (1 0 0). i.e. It
indicates that the plane has an intercept in
the –ve X –axis.

18. Planes and their negatives are
equivalent

19. In the cubic system, a plane and a
direction with the same indices are
orthogonal
Here the direction is
[001]
and plane is (001)

21. SOME IMPORTANT PLANES

22. BIBLIOGRAPHY
• SOLID STATE PHYSICS BY S.O.PILLAI, FIFTH
EDITION BY NEW AGE INTERNATIONAL
PUBLICATION