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

Miller Indices (Planes and Directions) contd....

In this lecture, we will discuss Miller indices, and we will continue some part of the previous ones before we move on to the Miller indices for directions.

In the last lecture, we are talking about Miller indices for planes and directions. Now, this is needed to complete the description of the unit cell. So, for plane Miller indices are usually described as (h k l), and for the direction, you would describe the Miller indices as [u v w].

As we saw that for a plane, the procedure is ( h k l) are nothing but the reciprocal of the intercepts on respective unit cell lengths and which are converted to the smallest set of integers. So, h k l all three are integers; they can be positive as well as negative.

I will go through some more descriptions of (h k l) in a unit cell. Now, let us say you want to draw a plane, which is (1 2 3). So, on the x-axis, your intercept is 1, y-axis intercept is 1/2, and on the z-axis, the intercept is 1/3, and you can choose this as an origin "O" because all three are positive. So, if you go in this direction, it is plus x, in this direction plus y, and if you go in this direction is plus z. And if you connect all three of them, this is your plane (1 2 3). Now suppose you wanted to draw a plane which is (1 2̅ 3).

So, what this means is that you have intercept on the x-axis as 1, along with minus y-axis you have half, and along plus z-axis, you have one-third. The problem is that you cannot choose this origin and the and the thing Miller indices is that that the planes and direction should be represented within one unit cell. So, in this case, what you need to do is that you need to shift your origin.

If you have a plane drawn within a unit cell, which is in green color, so, what is the origin you are going to choose? So, you need to choose an origin from which you can count the intercepts. So, I choose an intercept, which is the origin, o' let us say. So, I have an intercept, which is along the x-axis is 1, -1 along the y-axis, and 1/2 along the z-axis. So the plane is (1 1̅ 2).

Family of planes, for example, consider a cube. So, in a cube (0 0 1), (0 1 0) and (0 0 1) planes are similar planes because of symmetry.

So, I can represent these planes by these curled brackets, {1 0 0}, which means the family of (1 0 0) type planes, but only valid for a cubic system. So, {1 0 0} implies (1 0 0), (0 1 0) and (0 0 1). So, likewise, you can see that there is a multiplicity, in case of cube 1 0 0 implies; you also have (1̅ 0 0), (0 1̅ 0), and (0 0 1̅). So, you have depending on how you look at it, there are three different types of which are identical. Similarly, if you look at {1 1 0}, there are 12 different types of which are identical.

Similary, {1 1 1} implies,

You have in total of 8. So, you have 8 multiplicity. Similarly, for a cube, for {h k l} type, there will be 48 possible planes. And for {h h l} type, it will be 24. In the case of the tetragon, a is equal to b is not equal to c. So, you have two lattice parameters a and c.

In tetragonal symmetry, this plane is different as compared to the other two planes. Which has a 4-fold, and the other two have 2-fold symmetry, and also, in terms of atomic density and spacing, they are different. So, in case of tetragonal system, (1 0 0) and (0 1 0) are equivalent, but (0 0 1) is not. So, in the case of tetragonal, (h 0 0) multiplicity will be 4, and they are (h 0 0), (0 h 0), and their negative planes. Similarly, (0 0 l) will have a multiplicity of 2. Similarly, there are other planes, as well. So, when you have (h k l) or (h k 0), (h k 0) will not be equivalent to (h 0 k) or (0 h k) because of tetragonal symmetry, where the third indices is along the c axis have different length as compared to a and b. So, the multiplicity will go down in the tetragonal system, and you will also see that d100 and d010 are the same, but d001 is not same as d100 and d010.

So, the formula, dhkl is valid only for cubic, and that is,

For the tetragonal system, dhkl is,

For orthorhombic system, dhkl is,

The formula looks very similar to these three are orthogonal systems. For non-orthogonal systems, different dhkl are different. So, if you look at interplanar angles, they will also be different, so how to find the interplanar spacing and how to find the interplanar angles?.

So, interplanar spacing d100, d010, d001 are all the same for cubic. In the case of tetragonal, d100 is the same as d010, but it is not the same as d001. In the case of orthorhombic and things are even different for other planes other systems. For the interplanar angle,

Now, directions are nothing but vectors, and they are denoted as [u v w], called crystallographic directions. So, resolve the vector components and then along the crystallographic axis and then reduce them to the smallest set of integers, let us see how we work along with this.

The representation (h k l) is independent of the crystal system. The multiplicity may change, but how you determine a plane, how you draw a plane that is independent of the crystal system. So, that is applied to all the crystal systems as long as you have a unit cell.

Let me draw a cubic unit cell, and I want to represent direction [1 2 3]. So, we have 1 intercept along the x-axis, intercepting along the y-axis at 2, and along the z-axis, it is 3. That is how you will represent a vector. So, this is the origin (O); you will have one step along the x-axis, two steps along the y-axis, and three steps along the z-axis. So, you will go out of the unit cell, which is not desirable. What we want to do instead is, convert the intercept values by dividing with the largest integer. So as per our previous example, which is 1/3, 2/3, and 1. Now you connect with the endpoint. So, this is the direction OA, nothing but [1 2 3]. So, a single direction is determined by these square brackets, and family of direction is given as <u v w>.

We will finish this lecture here, and in the next lecture, we look at some more examples of how do we draw the directions. Moreover, we will also look at a little different system, which is a hexagonal system. In a hexagonal system, the directions and planes are can be drawn in a different manner because the depiction of them can be done in four digits rather than three digits. Because as we will see hexagonal system can also be characterized by four-axis, and which can be reduced to three axes, but the fourth axis is drawn just for the sake of convenience, which is related to the two axes. So, there is a third axis in the basal plane of the hexagonal system.

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

Miller Indices (Planes and Directions)

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