# Miller Indices

In this lecture, we are going to learn about the Miller Indices, their definition of it, the Procedure for finding Miller Indices, and some important features of Miller Indices of crystal plane. So let’s start with the introduction of Miller Indices.

## Introduction of Miller Indices

In a crystal, there exists direction and planes which contain a large concentration of atoms. Therefore it is necessary to locate these directions and planes for a crystal to analyze. The problem is how to identify the direction and to designate ( to choose) a plane in a crystal.

Here, let us discuss briefly the method of designating a plane in crystal. This method was suggested by Miller.

Miller Indices

Miller introduces a set of three numbers to designate a plane in a crystal. This set of three numbers in known as Miller indices of the concerned plane.

## Procedure for Finding Miller Indices

The steps in the determination of Miller indices of a plane are illustrated with the aid of Figure. Consider the plane ABC which cuts 1 unit along the X-axis, 2 units along the Y-axis, and three units along the Z-axis.

Step 1:

• Find the intercept of the plane ABC along the three axes X, Y, and Z. Let it be OA, OB, and OC. Express the intercepts in terms of multiples of axial lengths, i.e., lattice parameters. Let them be OA = pa, OB = qb, and OC = rc where p, q, and r are the intercept numerical values along the three-axis.
• This example shows:

p=1, q=2 and r = 3

Hence, OA : OB: OC = pa:qb: rc = 1a: 2b : 3c

• Therefore, the intercepts are 1a, 2b, and 3c along the three axes.

Step 2:

• Find the reciprocal of the numerical intercept values.

i.e., \frac{1}{p}\frac{1}{q}\frac{1}{r}

• For the example shown the reciprocal of the numerical intercept values are:

\frac{1}{1}\frac{1}{2}\frac{1}{3}

Step 3:

• Convert these reciprocals into whole numbers by multiplying each with their least common multiple (LCM). In this example, the LCM is 6. Therefore,

6\times \frac{1}{1}\;,6\times \frac{1}{2}\;, 6\times\frac{1}{3}
6 \;3\; 2

Step 4:

• Enclose these numbers in the bracket. This represents the indices of the given plane and is called the Miller Indices of the plane.
• For example, as shown, the Miller indices are (6 3 2).
• It is generally denoted by ( h k l). It can also be noticed that,

h:k:l=\frac{1}{1}:\frac{1}{2}:\frac{1}{3}

Definition -1: Thus, Miller indices may be defined as the reciprocal of the intercepts made by the plane along the three crystallographic axes which are reduced to the smallest numbers.

Definition-2: 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 three axes.

## Important Features of Miller Indices of Crystal Plane

When it comes to Miller indices, several important features come into play, providing valuable insights into crystallography. Let’s delve into these essential aspects:

1. Infinite Intercepts:

• Any plane that runs parallel to at least one coordinate axis exhibits an infinite intercept (∞). Consequently, the Miller index for that particular axis becomes zero.

2. Parallel Planes:

• Equally spaced parallel planes with a specific alignment share the same index numbers (h k l). It’s important to note that Miller indices represent not just a single plane, but rather a combination of multiple parallel planes.

3. The ratio of Indices:

• The ratio between the indices holds significance above all else. The specific planes themselves are not of primary importance.

4. Origin and Non-Zero Intercept:

• A plane passing through the origin is defined in comparison to a parallel plane with non-zero intercepts.

5. Equally Distant Planes:

• Parallel planes that are equally spaced possess identical Miller indices. Hence, Miller indices are utilized to represent a set of parallel planes.

6. Parallelism through Ratio:

• If two planes have the same ratio of Miller indices, such as 844 and 422 or 211, they can be deemed parallel to each other.

7. Dividing the Axes:

• If the Miller indices for a plane are denoted as h k l, the plane will divide the axes into equivalent sections of a/h, b/k, and c/l.

8. Precision in Multi-Digit Indices:

• When the integers in Miller indices consist of more than one digit, it is essential to separate them by commas for clarity, for example, (3, 11, 12).

9. Crystal Directions and Planes:

• In a crystal family, the directions of the crystals are not necessarily parallel to each other. Similarly, not all planes within a family are guaranteed to be parallel.

10. Antiparallel Directions and Planes:

• By changing the signs of all indices in a crystal direction, an antiparallel or conflicting direction is obtained. Similarly, altering the signs of all indices in a plane leads to a plane situated at an equivalent distance on the opposite side of its origin.

## Frequently Asked Questions on Miller Indices

1. ### Explain Miller Indices.

The Miller indices definition can be stated as the mathematical representation of the crystallographic planes in three dimensions. Miller evolved a method to designate the orientation and direction of the set of parallel planes with respect to the coordinate system by numbers h, k, and l (integers) known as the Miller indices. The planes represented by the h k l miller indices are also known as the h k l planes.

2. ### State Any Two Rules to Determine the Miller Indices.

Following are the rules to be followed for determining the miller indices:

->Determine the intercepts (a,b,c) of the planes along the crystallographic axes, in terms of unit cell dimensions.
->Consider the reciprocal of the intercepts measured.
->Clear the fractions, and reduce them to the lowest terms in the same ratio by considering the LCM.
->If the h k l plane has a negative intercept, the negative number is denoted by a bar (  ̅) above the number.

3. ### Are miller indices at all times positive in value or they can be negative as well?

According to Miller indices, two or more parallel planes can have similar indices which can ultimately be a negative value, zero, or a positive value. This completely depends on the intercept on the axes and nothing else. This leads us to the conclusion that Miller indices do not necessarily always have to be positive.

4. ### What is the meaning of the line that is displayed over a number in Miller indices?

A Miller index having a 0 value simply means that the plane is in a parallel position to the corresponding axis. Negative indices, on the other hand, are indicated with a bar drawn over the integer number. In cubic crystal systems, the Miller indices of a plane are exactly similar to those of the way vertical to the plane.