Miller Indices

What is Miller Indices?

What is Miller Indices?

Miller Indices

The concept of Miller Indices was introduced in the early 1839s by the British mineralogist and physicist William Hallowes Miller.
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 hkl Miller Indices are also known as the hkl planes. Therefore, the Miller Indices definition can be stated as the mathematical representation of the crystallographic planes in three dimensions.

General Principles of Miller Indices

If a Miller index is zero, then it indicates that the given plane is parallel to that axis.
The smaller a Miller index is, it will be more nearly parallel to the plane of the axis.
The larger a Miller index, it will be more nearly perpendicular to the plane of that axis.
Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane.
When the integers used in the Miller indices contain more than one digit, the indices must be separated by commas to avoid confusions. E.g. (3,10,13)
By changing the signs of the indices 3 planes, we obtain a plane located at the same distance on the other side of the origin.

Rules for 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 a hkl plane has a negative intercept, the negative number is denoted by a bar ( ̅) above the number.
Never alter or change the negative numbers. For example, do not divide -3,-3, -3 by -1 to get 3,3,3.
If the crystal plane is parallel to an axis, its intercept is zero and they will meet each other at infinity.
The three indices are enclosed in parenthesis, hkl and known as the hkl indices. A family of planes is represented by hkl and this is the Miller index notation.

Q. Determine the Miller Indices of Simple Cubic Unit Cell Plane 1,∞,∞.

Given that,
Plane 1,∞,∞
Miller Indices
Step 1:
Consider the given plane 1,∞,∞.
Step 2:
Take reciprocals of the intercepts,
1/1, 1/∞, 1/∞
Step 3:
Take LCM of these fractions to reduce them into the smallest set of integers.
Therefore, the miller indices for the given plane is 1,0,0.

Weiss Indices