Hello today we will start a new topic till now we have been discussing ah bravais lattices, fourteen bravais, lattices and seven crystal systems. We looked how symmetry helps in classification of crystals into these schemes!
We will now start a new topic, the miller indices of directions and planes? These are techniques or tools to specify a various directions. As when we work with crystals, we need to specify or name different directions and planes in a crystal so miller, indexing is an standard method which has been which is being used for this purpose. So let us look, we will first in this video we will look at miller, indices of directions and in the next one we will take miller indices of planes? So let us look at miller indices of direction. Suppose we have this crystal. A unit cell is shown a face centred cubic lattice. So all these large circle are the lattice points and we want to specify a certain direction. Let us say this blue direction edge of the cube? Now, of course, the cube has this edge. It has this another edge, two horizontal edges and one vertical edge. So i have picked out one of them, this blue one, and i want to give a name to it? Of course, in common language i can say that it is the edge of the cube on its bottom face coming out of the screen, so to say, and this one is an edge lying in the screen, and this is a vertical edge and so on. However, miller indexing will give us a specific notation, a specific system, to name this blue line. So let us look at how we do that.
We will go step by step. The first step in miller in indexing is to choose an origin on the direction, so i have chosen this ah back corner, as my origin pointed out in red, so the first step is always to choose the origin, and i have highlighted here that on the direction, so the origin always has to be on the direction it should lie on the direction or vice versa. The direction should pass through the origin or origin should be so chosen that it lies on the direction. This freedom of choice exists in crystallography in the crystallographic coordinate system. We are free to choose the origin anywhere we wish!
So if i want to index this blue direction, i choose the origin on the blue direction and i took this point as the origin? The next step is to choose a coordinate system, crystallographic coordinate system with axis parallel to the unit cell edges. So in this case i have chosen x, y and z, with the three x y z directions parallel to the unit cell edges.
Here i have red for illustration purpose! I have taken a cube even in a non cubic crystal.
Even if the angle between x and y is not ninety degree and even if z is not perpendicular to x and y, we will always choose our x y and z parallel to the unit cell edges. This is what is called the crystallographic coordinate system, so we we will be using the crystallographic coordinate system with unit cell edges as our axis, so we have done that for this direction.
Now we have taken this red origin and red axis?
The next step is to find the coordinates of another point on the direction in terms of a b and c! A b and c are the three lattice parameters.
So in this case they are the edge lengths. So a is the edge length of the unit cell along the x axis b. Is the edge length of the unit cell along y axis and c is the edge length along the z axis. So in terms of these three vectors, the a b and c vectors, i i will now try to express the blue vector, which is the vector of my choice, direction of my choice ad in terms of these three vectors. So here it is very simple: it is one times a because the direction is along the x axis and it is of the length equal to a so it is one times a zero times b and zero times c! So i just take these coefficients one o o to represent this direction, so i find the coordinates of another vector in terms of a b and c.
So the first one means one times a the second zero means zero times b and then zero times c, the next step, which in this case a redundant, but we will write it out because we will use it in. The next example is to reduce the coordinates to smallest integers, and this can be done either by dividing by a common factor or multiplying by common factor, so suppose we had fractions, then we will multiply by some common factor such that the fractions get cancelled or suppose. If we had a common factor in all these three, then we will divide by that common factor to cancel out the common factor. So this is a step of reducing the coordinates to smallest integers in this case, and nothing is required. One zero. Zero is already smallest integers, so we carry on with that and then the final step is to just put these three numbers in a square bracket?
There is an important step!
A square bracket is not my choice in this presentation or this slide. It is an internationally agreed upon convention. The directions will always be represented by numbers inside square bracket. So we will follow this convention, so one zero zero is the direction which is represented by this blue line, so one zero zero.
There is a slide difference between vector terminology and the miller indices of a direction!
Although i picked up one vector this blue vector too along this line- and you use that to calculate my miller indice once i have found the miller indices, one zero zero, its not representing just this blue vector, but this entire x axis, so the entire x axis, as well as the negative x axis, can be represented?
This full line is represented by the number one: zero zero another peculiarity of convention here when i wrote the ah components separately. I am writing it with commas, but in the miller indices, i am not using any commas.
So this is a useful convention? Unless and until we have a two digit miller indices for one of the component, if it is only three numbers, we write them without any commas, and it is understood that the first number is with respect to the x axis, the second one, with respect to the y and third one with respect to z. So let us look at some more examples now before looking at those examples, one more point so miller, indices of a direction represents only the orientation of the line, not its particular position in space or also its sense. So i already told that not only the positive x axis, but the negative x axis will also be represented by one zero zero. If we do not really want to distinguish positive and negative, that is to say if we are only interested in the line, not in the sense and in miller indices, that is usually the case. Then one zero zero represents the entire x axis. Not only that, because what we said about freedom of choosing the origin. If we had parallel lines somewhere here, then i can again choose my origin here, and this will become my x axis or i can choose my origin there, and this will become my x axis! So all parallel directions have the same miller indices! Let us now look at some more examples. So again we take this face centred cubic unit cell and we want to indices a direction which is starting from this corner and passing through the top face centre?
So this direction is the direction of my choice! So i choose this as a coordinate first step to choose a coordinate origin and a coordinate system.
So i chose this point as my origin and axis along the unit cell edges. With respect to this coordinate system, i now write the vector o a so you can see to reach o a! I have to go a by two steps along x, a by two steps along y and then c, step along z, so o a the vector o a is half a plus half b and one times c! So the coordinates which we will use in the miller indices in terms of a b and c are half half and one now i will use the cancelling of fractions step, which we did not require in the previous one. In the case of one zero zero, we did not require that, but now, in the case of half half one, we will not call this direction, half half one, but will simply multiply by two all these three numbers to get one one. Two and of course i put them in the square bracket which we have agreed upon to use as a convention for directions. So the o a direction not just the o, a vector, but the entire o a direction will be represented by this miller indices, one one, two one more example: let us look at this black direction now, of course, i have to choose the origin on the black line and that freedom is there. So i shift the origin to this point p on the black line. You can note that when i have shifted the origin, i have kept the axis parallel, so we have the freedom to choose. Our origin anywhere, but in a given problem, once we have specified the orientation of the axis, the that orientation cannot change! So the x y z in the new with the new origin is exactly parallel to the x y z, before the black x y z is parallel to the blue x y z. So now let us try to indices this direction along p q, which is one of the body diagonals of this q. So if we want to look at this p q, we will start with p, and you can see that now i have to take a minus one step along x, one step along y and one step along z to reach q.
So the p q vector is minus one, a minus one b and one c, so the components are minus one minus one one in terms of a b and c now i write this in a square bracket with one additional convention that the negatives are written as bars over the number!
Instead of on the side, as in useful mathematics in the miller, indice indexing notation a bar above, the number represents negative quantity so, and it is red also as bar instead of minus one. So we will call this direction. P q, as my bar one bar one one! Let us now take so the negative steps are shown as bar over the number! Let us now look at another convention which is used many times we. We are not interested in just one direction because we have talked about the symmetry in the crystal and crystals can have symmetry and symmetry relates many directions, so many directions become equivalent because of the existing symmetry of the crystal. So, for example, if you take a cubic crystal so all the edges of the cube are equivalent by the cubic symmetry?
So if we, if we indices the edge along x axis, it will be one zero zero. If we indices along the y axis zero one, zero and indices along z axis zero, zero one, but suppose i am not interested in a specific direction? I just want to talk about the cube edges for all directions along the cube edge, which are equivalent by symmetry? Then there is a new notation that you can put the miller indices of any one of them in an angular bracket, so an angular bracket.
U v! w means the specific direction, u v w and all other direction related to? u v w by the symmetry of the crystal, and it is important that when we are using this notation, we have to know which crystal system we are talking about, because different crystals will have different symmetry and the symbol will mean different things. We will show you this, ah with the help of cubic and tetragonal examples! So let us look at first, the cubic so in the miller indices of cubic crystal the one zero zero direction is equivalent to zero one zero zero, zero one as well as, if you take the negatives minus bar one: zero, zero, zero bar one zero and zero zero bar one. So the all six direction are equivalent by the cubic symmetry. So if we simply write one zero zero, if i pick any one of them, i have picked up one zero zero. You could have picked up zero one zero or zero zero one any of these six, so any member of this direction.
Ah, this is a family of six members? This is a family of symmetry related directions, and i pick up any member of the family to represent the entire family. So when i say one zero, zero in angular bracket- and i know that it is for cubic- then ill mean all these six directions? But now let us look at the tetragonal crystals in tetragonal, you know x and y are equivalent by symmetry, but not the z.
So if i say one zero zero for tetragonal, it will only mean these four directions: one: zero, zero, zero one zero and their negative. The third direction: zero zero one is absent from here in this list, because tetragonal symmetry does not make zero zero one equivalent to one zero zero! With this we will end this video in the next video we will take. Ah the discussion on miller indices of planes! .