< Engineering Physics - I >
Can you guess the figure?
Introduction
Miller Indices are a method of describing the orientation of a plane or set of planes within a lattice in relation to the unit cell. They were developed by William Hallowes Miller. These indices are useful in understanding many phenomena in materials science, such as explaining the shapes of single crystals, the form of some materials' microstructure, the interpretation of X-ray diffraction patterns, and the movement of a dislocation, which may determine the mechanical properties of the material.
Learning Objectives On completion of this session you will be able to: 1. Understand the concept of a lattice plane; 2. Be able to determine the Miller indices of a plane from its intercepts with the edges of the unit cell; 3. Be able to visualise and draw a plane when given its Miller indices; 4. Be aware of how knowledge of lattice planes and their Miller indices can help to understand other concepts in materials science.
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< Engineering Physics - I >
Lattice planes The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. The crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal. Orientations of planes or faces in a crystal can be described in terms of their intercepts on the three axes. Miller gave a method of indicating the orientation of a plane, by the reciprocal of its numerical parameters. These reciprocals have to be converted into whole numbers in order to get Miller indices.
Examples of determining indices for a plane using intercepts with axes; left (111), right (221)
The general form of indices of a plane is (h k l). The planes which are parallel have the same Miller indices.
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Procedure of finding Miller indices 1. The intercepts of the desired plane on the three coordinate axes has to be determined. 2. These intercepts has to be expressed as multiples of the unit cell dimensions or lattice parameters i.e. (p q r). 3. The ratio of reciprocals of these numbers has to be taken (1/p 1/q 1/r). 4. The reciprocals have to be converted into whole numbers by multiplying each with their L.C.M. to get the smallest whole number. 5. This gives the Miller indices (h k l) of the plane.
Miller indices of some important crystal planes
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< Engineering Physics - I >
The (100), (010), (001), (100), (010) and (001) planes form the faces of the unit cell. Here, they are shown as the faces of a triclinic (a ≠ b ≠ c, α ≠ β ≠ γ) unit cell. Although in this image, the (100) and (100) planes are shown as the front and back of the unit cell, both indices refer to the same family of planes. It should be noted that these six planes are not all symmetrically related, as they are in the cubic system. The (101), (110), (011), (101), (110) and (011) planes form the sections through the diagonals of the unit cell, along with those planes whose indices are the negative of these. In the image the planes are shown in a different triclinic unit cell.
The (111) type planes in a face centered cubic lattice are the close packed planes. • When a plane is parallel to any axis, the intercept of the plane on that axis is infinity. So the Miller index for that axis is zero.
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< Engineering Physics - I >
• When the intercept of a plane on any axis is negative, a bar is drawn on the corresponding Miller index. • When the plane passes through the origin, it is indexed in terms of a parallel plane having non-zero intercept. • Equally spaced parallel planes have the same Miller indices (h k l). Interplanar spacing d in a cubic lattice
Interplanar spacing in a cubic crystal is given by
d = inter-planar spacing between planes with Miller indices h, k and l. a = lattice constant (edge of the cube). h, k, l = Miller indices of cubic planes being considered.
Check your understanding 1. Draw the 14 Bravais lattices. 2. State if the following statement is true or false? All unit cells are primitive cells. a) True
b) False
3. Fill in the blank with the right answer. When a plane is parallel to any axis, the intercept of the plane on that axis is --------------------. Check the correct answers on page _6_.
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Summary On completion of this chapter you have learned:
Miller Indices are the convention used to label lattice planes. This mathematical description allows us to define accurately, planes within a crystal, and quantitatively analyze many problems in materials science.
Activity Draw the crystal planes with Miller indices (1 0 1) and (0 2 0).
Suggested Reading 1. ‘Engineering Physics’ by P.K. Palanisamy. 2. ‘Introduction to Solid State Physics’ by C. Kittel, John Wiley and Sons, 2004. Chapter 1 covers crystallography. The book then goes on to cover a wide range of more advanced solid state science.
Answers to CYU 2. b 3. Infinity
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