Hello today we’ll learn about an important concept in material science or chemistry, even electronics, which is miller indices now for finding water miller indices. We first have to draw a simple unit cubic cell. We draw the x axis y axis and the z axis?
It’s a right handed coordinate system, as you can understand that if you take a cross product of x, – y, it will give z! Now we draw a unit cube a unit cube is a cube whose each side is of unit length in any unit dimensions! Now, let’s define a plane which intersects the x-axis at this point, 1 0 and 0 the plane could be defined as something like by this blue color. We find the intercept made by this plane on x axis to be 1 on y axis, to be infinity the reason, because this plane is parallel to y axis, it does not intersect at all. Similarly, the plane is parallel to c axis, so the intercept is infinity. Next, we take reciprocal of the planes or the intercepts. The reciprocal of x intercept is 1 by 1, which is 1. Reciprocal of y intercept is 1 by infinity, is equal to 0, and the reciprocal of z intercept is again 1 by infinity is equal to 0, hence miller indices, which are denoted as h k al with, in parentheses, when h refers to this number, the reciprocal of intercept on x-axis k refers to the reciprocal of the intercept on y-axis and l represents the reciprocal of intercept on z-axis will become 1, 0 0 within parenthesis! This is cede allowed as 1 0 0 plane rather than 100. Next, let’s try to find something a little more challenging. We first start out by drawing again a cubic unit cell. We take a plane which is a little different than parallel to any axis. Suppose the plane is intercepting the x-axis at a point: 1/3 comma 0, omar 0, which is this point because this is 1? It is intercepting the y-axis at a point: 0, comma 2, third, comma 0. This is what x this is y.
This is z. I should have leveled them before and suppose the plane is intercepting the point on z axis as 0 0 1. Now, let’s draw this plane? It is a triangular looking clean as we see here and we can hatch it. The procedure to be followed here is similar. The x intercept is 1/3? On y, the intercept is 2/3 on z, the intercept is 1. The reciprocal on 4 x is reciprocal of 1/3 or 3/4. Why reciprocal of 2 by 3, which is 3 by 2 and for z? it is 1? If we write these three numbers together as we did, it will become 3 3 by 2 and 1. However, miller indices have to be integers, so we multiply throughout by 2, which is the denominator, and we get 6 3 & 2 as the three numbers as the reciprocals of intercepts.
We put them inside parentheses? To see this plane is denoted as 6 3, 2 clean. We got the example of 1 0 0 plane, as we saw earlier! However, what would happen if a plane was to become such that it intercepts? the y-axis at y is equal to 1 or point is 0 1 0, and it is parallel to x axis. This is the plane and it is parallel to z axis? Similarly, the x intercept here will be infinity? The y intercept here will be 1 and the z intercept here will be again infinity!
Taking the reciprocal? We have h as 0 k as 1 and l as 0? The plane, as you can see from above, will become 0 1 0 plane as the miller indices notation. However, 1 0 0 planes, 0, 1 0 planes and similarly 0:01 planes all could be written as a family of planes, as within this curly bracket as 1 0 0 planes, the miller indices are important for finding many important concepts in material science and chemistry! One of them is the d spacing or inter planar, spacing between the parallel planes, which is defined as e, which is the side of cube which differs for different material and which is generally in angstrom, divided by root over h square plus k square plus l square, where h k l refer to the miller indices as an example, if we have a crystal structure whose side of the cube is 0. 36, one nanometer the d spacing for 2 to 0 plane would become 0. 36, 1/2, squared plus 2, squared plus 1 0 squared, which will be 0. 128 nanometer? Thank you for watching the video! We could go into more details in later videos. .