# Miller Indices

Miller introduced a set of three number of designate a plane in a crystal.This set of three numbers are known as Miller indices of the concerned plane.

**Definition: M**iller indices is defined as the reciprocal of the intercepts made by the plane on the crystallographic axes which is reduced to smallest numbers.

or

Miller indices are the three smallest possible integers, which have the same ratio as the reciprocals of the intercepts of the plane concerned along the 3 axes.

**Step 1** : *Identify the intercepts on the x- , y- and z- axes.*

In this case the intercept on the x-axis is at *x* = *a* ( at the point (*a*,0,0) ), but the surface is parallel to the *y*– and *z*-axes – strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at infinity ( ∞ ) for the special case where the plane is parallel to an axis. The intercepts on the *x*– , *y*– and *z*-axes are thus

Intercepts : *a* , ∞ , ∞

**Step 2** : Specify the intercepts in fractional co-ordinates

Co-ordinates are converted to fractional co-ordinates by dividing by the respective cell-dimension – for example, a point (*x*,*y*,*z*) in a unit cell of dimensions *a* x *b* x *c* has fractional co-ordinates of ( *x*/*a* , *y*/*b* , *z*/*c* ). In the case of a cubic unit cell each co-ordinate will simply be divided by the cubic cell constant , *a* . This gives

Fractional Intercepts : ** a/a , ∞/a, ∞/a** i.e.

**1 , ∞ , ∞**

**Step 3** : Take the reciprocals of the fractional intercepts

This final manipulation generates the Miller Indices which (by convention) should then be specified without being separated by any commas or other symbols. The Miller Indices are also enclosed within standard brackets (….) when one is specifying a unique surface such as that being considered here.

The reciprocals of 1 and ∞ are 1 and 0 respectively, thus yielding

Miller Indices : **(100)**

So the surface/plane illustrated is the (100) plane of the cubic crystal.

**Other Examples**

1. The (110) surface

Intercepts : Fractional intercepts : 1 , 1 ,∞ Miller Indices : |