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Introduction to Miller Indices

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While studying crystallography, understanding crystal planes are of high importance. Miller indices are the mathematical representation of the crystal planes. The concept of Miller indices was introduced in the early 1839s by the British mineralogist and physicist William Hallowes Miller. This method was also historically known as the Millerian system and the indices as Millerian or the miller indices. 


The orientation and direction of a surface or a crystal plane may be defined by considering how the crystal plane intersects the main crystallographic axes of the solid. The use of a set of rules leads to the assignment of the Miller Indices (hkl) a set of integers that quantify the intercepts and thus may be used to uniquely identify the plane or surface. In this article, miller indices are explained in detail along with some solved examples for a better understanding.


Crystallographic Planes

We know that crystal lattices are the infinite array of points arranged periodically in space. These points can be joined together by drawing a straight line and by extending these lines in the three-dimension we notice that they appear to be a set of crystal planes or crystallographic planes. The crystal lattices are constructed by the set of parallel lines known as the crystallographic planes.


The lattice points have different mechanical, electrical, or optical properties in different directions, this will make the study of crystal structure difficult. To overcome this difficulty, we will choose a set of crystal planes such that the properties of the crystal lattice remain unchanged in the direction of the crystal plane. In order to choose specific crystal planes, a famous mineralogist William Hallowes Miller introduced a method known as the miller indices. Basically, miller indices are the mathematical representation of the set of parallel crystallographic planes.

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Miller Indices Definition

After joining the crystal lattice points by straight lines and those straight lines were assumed to be the set of parallel crystal planes extending them in three-dimensional geometry. The problem that arose of explanation of the orientation and direction of these planes. 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.


Construction of Miller Planes

Let us understand the steps involved in the construction of miller planes one by one. To construct the Miller indices and the miller plane we follow the following method:

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Step 1:

Consider a point or an atom as the origin, construct a three-coordinate axis and find the intercepts of the planes along the coordinate axis. 


Step 2:

Measure the distance or the length of the intercepts from the origin in multiples of the lattice constant.


Step 3:

Consider reciprocal of the intercepts. Reduce the reciprocals of the intercepts into the smallest set of integers in the same ratio by multiplying with their LCM. 


Step 4:

Enclose the smallest set of integers in parentheses and hence we found the miller indices that explain the crystal plane mathematically.


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.

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 Miler 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.

Examples

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

Sol: 

Given that we have a plane [1, ∞, ∞], our aim is to determine the Miller indices for the given set of the plane. We know that we have a set of rules for determining the miller indices and they are as follows:


Step 1:

Consider the given plane [1, ∞, ∞].


Step 2:

Take reciprocals of the intercepts,

\[\frac{1}{1}\], \[\frac{1}{\infty}\], \[\frac{1}{\infty}\] 


Step 3:

Take LCM of these fractions to reduce them into the smallest set of integers.

[1, 0, 0]

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Therefore, the miller indices for the given plane is [ 1, 0, 0].


2. Determine the Miller Indices for the Plane [1, ∞, 1] 

Sol:

Given that we have a plane [1, ∞, 1], our aim is to determine the Miller indices for the given set of the plane. We know that we have a set of rules for determining the miller indices and they are as follows:


Step 1:

Consider the given plane [1, ∞, 1].


Step 2:

Take reciprocals of the intercepts,

\[\frac{1}{1}\], \[\frac{1}{\infty}\], \[\frac{1}{1}\] 


Step 3:

Take LCM of these fractions to reduce them into the smallest set of integers.

[1, 0, 1]

Therefore, the miller indices for the given plane is [ 1 0 1].


3. Determine the Miller Indices for the Plane [½, 1, ∞]  

Sol:

Given that we have a plane [½, 1, ∞], our aim is to determine the Miller indices for the given set of the plane. We know that we have a set of rules for determining the miller indices and they are as follows:


Step 1:

Consider the given plane [½, 1, ∞].


Step 2:

Take reciprocals of the intercepts,

\[\frac{1}{\frac{1}{2}}\], \[\frac{1}{1}\], \[\frac{1}{\infty}\] 


Step 3:

Take LCM of these fractions to reduce them into the smallest set of integers.

[2, 1, 0]

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Therefore, the miller indices for the given plane is [ 2, 1, 0].


4. Determine the Miller Indices for the Plane [-1, ∞, ½]

Sol:

Given that we have a plane [-1, ∞, ½], our aim is to determine the Miller indices for the given set of the plane. We know that we have a set of rules for determining the miller indices and they are as follows:


Step 1:

Consider the given plane [-1, ∞, ½].


Step 2:

Take reciprocals of the intercepts,

\[\frac{1}{-1}\], \[\frac{1}{\infty}\], \[\frac{1}{\frac{1}{2}}\] 


Step 3:

Take LCM of these fractions to reduce them into the smallest set of integers.

[-1, 0, 2]

Therefore, the miller indices for the given plane is [1, 0, 2]

FAQ (Frequently Asked Questions)

1. Explain Miller Indices?

Ans: 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 hkl miller indices are also known as the hkl planes.

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

Ans: 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 a hkl plane has a negative intercept, the negative number is denoted by a bar (  ̅) above the number.