Structure of Materials::Miller Indices (Planes and Directions)

Miller Indices (Planes and Directions)

In this lecture, we will discuss the Miller Indices, Planes, and Directions. In the previous lectures we saw, we understood the fundamentals of crystallography, lattices, crystal systems, Bravais lattices, and symmetry and their correlations. Now, we will try to understand how we can quantify the crystals in terms of the directions and various phases because this is the knowledge of this is very important to understand the correlation of anisotropy, directional response of various properties in crystals. So, if you measure certain properties in one direction, it is different from other directions, and this is true about mechanical properties, electrical properties, thermal properties, and other magnetic properties.

 

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So, to correlate the properties with directions, you need to have a method to quantify the directions and planes of the crystal, and that is where this concept of miller indices comes into the picture.

 I draw a simple parallelepiped here. So, you can see that the separation or spacing between these atoms is something else it is let us say a by root 2. So, given that different atoms have different spacings with respect to each other. The properties also change in different directions. So, if I measure some response in this direction, it is different from the response that is measured in another direction. So, that is why we need to understand what is this direction. Similarly, you can see that different faces of crystal have different atomic density, for example, this face has these four atoms located at certain distances, if I make this face, this has a different density it again has the same number of atoms, but it has different density. It will change when you go to FCC and BCC structures. For example, this face has a different density. As a result, they will have a different response because they have a different spacing of atoms between them, and there are packed differently in these directions. So, that is why it is necessary to evolve a system to quantify these things.

Miller indices are in the name of a person called William Hallowes Miller, who coined the term, and that is why these are called miller indices. Crystallographic planes are nothing but the faces of facets of crystals you can say facets of crystals they are defined as, so, this is to identify they are defined as (h k l) for one plane. However, it could be for a set of identical planes dependent upon one crystal structure, whether it is just cubic or whether it is tetragonal. So, you may not be able to write the same connotation for tetragonal. The second thing is crystallographic directions in various atomic directions in crystal, and these are depicted as [u v w], and <u v w> represents a set of directions. Again just like planes, it is dependent upon the crystal structure, and here h, k, l and u, v, w are integers, and they could be positive or negative.

where h/a is intercept on the intercept of the plane on the x-axis, h/b will be intercepted on the y-axis, l/c will be intercepted on the z-axis. And a, b, c are unit cell lengths of lattice parameters. And h, k, l are miller indices.

Let us say, I have a parallelogram like this, let us say this is the origin and I define this as in some multiple of A, 2A, 3A, and so on. So, this will be 4A, 6A, and this 2A. So, I have a body which is something like that, and if I connect these so, this is my plane. So, I can see my unit cell parameters are so, a is equal to 4A, b is equal to 8A, and c is equal to 3A. What are the fractional intercepts?.

So, fractional intercept along x is 2A by this is along x, y it is 6A divided by 8A, and along z, it is 3A divided by 3A. So, this is 1 over 2, this is 3 over 4, and this is 1. So, now, you need to convert this into reciprocals. The plane indices, (h k l), have to be integers. So, you need to convert this into the smallest set of integers. So, if you convert this in the smallest set of integers, what do you get? You get 6, 4, and 3. So, this plane is (6 4 3). That is how you determine the Miller indices of a given plane. Similarly, So, let us do the same exercise for let us say this plane.

The process of determining a plane of the unit cell includes defining the origin, determine the intercepts, take the reciprocal, and then convert to the smallest set of integers. Why the smallest set of integers? Because if you look at (0 1 0) and (0 2 0) are parallel planes, one is at the half spacing of the unit cell; another is at the full spacing of the unit cell. So, h k l and 2h, 2k, 2l, and 3h, 3k, 3l are the same set of planes, and they are parallel to each other, it is just that the spacing between them is different.

I ask you to draw (1 2 3) and (210) , I draw a unit cell, and the choice of origin is very important. So, if you have (1 2 3) and (210), how do you choose the origin? You can see that h intercept is in positive x direction when you do not have any negative is typically determined. So, if you have (h k l) with a negative sign, then it will be ¿). So, what it means is that if you have 1, then you are moving your intercept is along the positive x-direction, 2 means half the intercept is along the positive y-direction, and 3 means one-third of the intercept along the positive z-direction.

        So, the origin which satisfied all the 3 directions is this origin. So, if I choose this as the origin o, then my intercept along the x-axis is 1. So 1, 2, 3 should be so you write 1, 2, 3 take the reciprocal that is 1, 1/2 and 1/3. So, these are the reciprocals and then put them as an intercept in the unit cell.

Every successive (2 4 6) plane will be spaced closer as compared to every successive (1 2 3) plane. So, it is nothing but the family of planes or set of planes which are parallel to each other. Now, let us say I want to draw a plane which is which has a negative interest indices. So, I draw a unit cell let us say I want to draw (1´1 0). So I shift my origin here, I can go in positive x, and if I go in that direction, I go in negative y, and this is important to be within the unit cell. So, if I do that this is the intercept on x, this is the intercept on minus y, and there is no intercept on z, which means it is parallel to z, it is infinity. So, the plane would be this and that so, this would be the (1´1 0). So, you can do now the exercise at home as to. So, I will just now do one last exercise in this case.

I Draw a unit cell just one last demonstration and to help you with how to find out, now let us say I draw a random plane let me choose one. So, I connect this point, this point and this point is it a legitimate plane, have intercept along x, intercept along y, and intercept along z.

So, how do you choose the origin now? So, you can see that the trick here is let us say so, this is that half right this is that half. So, you can see that if you choose this is this point is located at half of minus half x from here, this is located at -1/2 along x, -1/2 along y, and this is a 1/2 along z. If you take the reciprocal, this becomes -2, -2, 2, or this is nothing but (´1´1 1).

So, what is 1, bar 1, 1 plane? Basically 1 bar 1 plane will be this and that if I put these together nothing, but a parallel plane, but if you have to determine this red one, this 1 is a legitimate plane as well you might have atom sitting here. So, one way to do it is you do it in this fashion, or another way to do it could be by drawing a parallel plane so that you can choose an origin which is located on one of the corners, which means you have to draw the
parallel planes one more parallel plane. So, that instead of ending in this fashion, it ends here. So, this will go out of the system. So, this is something you do to determine the planes and need to know about the spacing between the planes.

For cubic system spacing between the planes is given as, dhkl, shown in the figure. Where a is the lattice parameter, and h, k, l are the miller indices. So, what is the spacing between these planes that is or you can have this is 1 0 0 plane and 0 1 0 plane, the spacing between these two is a. (1 00) dhkl=a

So, that is how you can determine the plane spacing, and you can also find that different planes are at different angles. If I want to calculate between the angle between the two of them, the angle is given as cosθ,

This is called as interplanar angle. These are only for cubic systems. By the way, for the tetragonal and orthorhombic systems, the relations are different. In the next lecture, we will now discuss the Miller indices for directions.

For Next Lecture Click below

Structure of Materials  

 Structure of Materials : Bonding in Materials

Structure of Materials : Correlation between bond and physical properties

 Crystal Structure: Lattice and Basis 

Primitive and Non-primitive Lattices  

Crystal Systems and Bravais Lattice 

Bravais Lattices Symmetry in Crystals 

Symmetry in Crystals  

Symmetry and Correlations with the Bravais Lattices  

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