Structure of Materials::Miller Indices (contd)...

Miller Indices (contd)...

In the last lecture, we talked about the planes, family of planes in different systems, and how the interplanar spacing, then we will move on to the hexagonal system. We will come back to cube and distinction with other systems a little later.

You can see that in the cubic system, if this is x, y, and z, this direction is depicted as what would this direction be if I take this as origin? Would this direction be? [1 0 0] Because you have an intercept of 1 along x-axis, so, this is [1 0 0]. What is this direction passing between these two is?

[1 1 1]. Also, you have [1 1 0]. Now in directions, you also have a concept of negative, so let us say if I want to draw [1 1̅ 0] and again, I need to shift the origin, shift the origin appropriately. So, I can see that one translation along the y-axis requires to shift the origin. So, I can move along the negative y-axis.

So, I can now put my origin here, which is O prime. So, if I need to go one along x, and one along -y and if I connect these two points, this would be? [1 1̅ 0] . And in the directions, you can see that [1 0 0], [2 0 0], [3 0 0] all are same directions. In the case of planes, (2 0 0), (3 0 0) because they have different spacing. In case of directions, when it is [3 0 0], [4 0 0], you do not write it is [4 0 0], you write it as [1 0 0] because it is a vector.

So, they are all represented as [1 0 0], in terms of the smallest set of integers. So, let us say this is AB direction, what is the indices of AB direction? Coordinates of this point are [0 1 1]. The common origin is O which is [0 0 0] and this point; the one in the center this is ½ of x and ½ of z. So, this is [12 0 12]. So, one way to determine the direction is half minus 1, minus half; so, this is nothing but [1 2̅ 1̅] another way to determine is you just move along the XY-directions. So, I can see that I need to go if I take this as a ½ in positive x-direction. So, I have moved x-intercept as ½, -1 along y-direction, and ½ along z-direction. So, the [u v w] is [1 2̅ 1̅]. For the cubic system, <u v w> means a family of direction.

Similarly, for <1 2 3> you can have multiple possibilities like [1 2 3], [1 3 2], [3 2 1], [3 1 2] and so on. The total possibilities you will reach 48. Similarly in tetragonal system, for example <1 0 0> will mean only [1 0 0], [0 1 0], [1̅ 0 0], [ 0 1̅ 0], and <0 0 1> would mean [0 0 1] and [0 0 1̅] because these two are the translation along those two is different. So, they are two different vectors. So, this is the difference between tetragonal and orthorhombic. So, again you can work out the multiplicities by yourself. So, for example, for the cubic system, <1 0 0> will have a multiplicity of 6, <1 1 0> will have a multiplicity of 12, <1 1 1> will have a multiplicity of 8. So, that is how you draw various directions. So, the direction is simpler as compared to the plane in the sense of understanding it.

Now, there is a special case, hexagonal system. Another thing that I want to draw to your attention towards says that in some systems we take, let us say we have a plane (h k l) perpendicular to the plane that is a vector all direction. So, for a cubic system, the (h k l) plane is normal to the direction [h k l].

So, if you have a plane (1 2 3), the plane normal will also be [1 2 3]. So, this is specifically for a cubic system, it applies to all the (h k l) planes, in case of other systems it may apply to certain planes it may not apply to certain planes. For example, in the case of tetragonal, you can see that (1 0 0) plane and [1 0 0] direction is perpendicular. Similarly, (0 1 0) and [0 1 0], (0 0 1) and [0 0 1], (1 1 0) and [1 1 0] are perpendicular, but (1 0 1) is not perpendicular to [1 0 1], which means the plane normal does not have same Miller indices as the plane. In the case of a hexagonal system,

It can be represented by this individual unit cell, which is here. So, this is one-third of the hexagonal unit cell, which is a single unit cell. Now, this single unit cell can be represented very well by a small unit cell off by a three-axis system by {h k l} <u v w>. However, what we can do here is the following if I take this as an axis a1 and this is an axis a2 then there is a third axis in this direction, which is a3. So, this can be, and of course, the fourth axis remains this one.

So, the (h k l) plane in this case if I take this four-axis system a1, a2, a3, and c typically in the hexagonal system, we call this as a-axis and c-axis because a1 and a2 are equal and they are symmetric. We call both of them as a-axis, and the perpendicular one is basal perpendicular to the basal plane is called as c axis. So, this plane is called a basal plane. Then we can conveniently choose as a four-axis system where a1, a2, a3 are three-axis within the basal plane, and the c-axis is the axis that is out of the basal plane. And you can see that by symmetry a1 and a2, a3 three are related to each other.

You can see above Fig. that a1 + a2 + a3 = 0, so, a3=-(a1+a2) because they close to each other. So, we define a new system which is called, so, instead of the three-axis (h k l) system, we take (h k i l). However, the direction [u v w], can be written as [u v t w], where one can prove it vectorially, U = 2u+v, V=2v+u and W=w. So, you can see that [u v w]. So, the plane (h k l) can be represented as (h k i l). So, plane (h k l) can be represented as (h k i l) in the four coordinate systems where (h + k) = -i. So, this is one simple representation of the hexagonal system.

Planes (h k l) becomes (h k i l), and directions [U V W] becomes [u v t w]. Where, U = 2u + v, V = 2v + u, and W = w.

If I take these planes, if you remember, there are two types of unit cells; one is the smaller set of the unit cell, and the other is the bigger unit cell. So, if I look at these unit cells, these planes marked in red. The first one is (1 0 0), it is parallel to a2 axis. So, this face on the right will be (0 1 0) system. So, in some sense, this is the face, which is just like (1 0 0), (0 1 0) face, but basically, this face is parallel to this plane in the middle. And if it is parallel to this plane in the middle, this becomes (1 1̅ 0).

So, there is a discrepancy if you rotate the hexagon this (1 1̅ 0) will become (1 0 0). So, why is it that it is noted differently, so that is where the four-axis system helps four-axis system will allow us to determine all of these planes in an identical fashion. Because now, if I convert this into the four-axis system, this (1 0 0) will become (h k i l). So, h and k will remain the same, and i will be -(h+k), and l will remain the same. So, this will become (1 0 1̅ 0). Similarly, this will become (0 1 1̅ 0), and if you now look at this, what will this become? This will become (1 1̅ 0 0). So, basically, they are the same family of planes (1 1̅ 0 0), (1 0 1̅ 0), (0 1 1̅ 0).

So, they remain the same family of planes in four coordinate systems, and they look as if they are the same. They are the same family of planes. However, in three coordinate systems, you end up having an index that looks different, which feels different in a hexagonal system. That is why to make it easier for one to understand the four-axis system is invented.

Now, let us look at this problem in a slightly different manner. So, you know how to draw the plane. If you wanted to draw, for example, (1 0 1̅ 0), you have one intercept along this direction, one intercept along this direction, and the other ones are parallel. So, this is parallel to the a2 axis and c-axis.

So, now let us try to draw the direction. So, in this case, let us say I want to draw a direction (1 2 0) this is a hexagonal basal plane and I am drawing (1 2 0), is simple I go one along a1 axis and two along a2 axis and then join with the origin. So, (1 2 0) can be converted into (0 1 1̅ 0), because you know that,

So, this is how you draw the direction in four coordinate system is hexagonal. So, in the previous case, I showed you how you draw the plane the four coordinate system. So, try doing these conversions and drawing these directions and planes in three and four coordinate system at home. Follow the same routine which I have told you in this lecture. So, we finish it here. We can describe the unit cells with the use of Miller indices; planes and directions are nothing but vectors and can be drawn very easily and determined very easily in the hexagonal system because of the symmetry of the unit cell.

If you want to represent the unit cell in the hexagonal form, you can conveniently choose a system which is you can choose conveniently choose a notation system that is easier to comprehend for a hexagonal system than for a normal unit cell. So, I can take up a little bit more problems in the next class on the hexagonal system to make it a little bit more clear.

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Miller Indices (Planes and Directions)

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