directions, planes and miller indices

DIRECTIONS, PLANES AND MILLER INDICES

By Srilakshmi BUCST, Tumkur

CONTENTS

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

INTRODUCTIONThe 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.

• 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.

NEED OF DIRECTIONS AND PLANES

• Deformation under loading (slip) occurs on certaincrystalline 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.

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]

CONVENTION FOR NAMING

• There are NO COMMAS between numbers• Negative values are expressed with a bar over

the number Example: -5 is expressed 5

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

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]

IMPORTANT DIRECTIONS IN A CRYSTAL

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

EXAMPLE

• Here x,y and z intercepts are 1,1,1.

• Therefore (111) is the miller indices of the plane

• 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)

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.

• 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.

Planes and their negatives areequivalent

In the cubic system, a plane and adirection with the same indices are

orthogonal

Here the direction is [001]

and plane is (001)

SOME IMPORTANT PLANES

BIBLIOGRAPHY

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