In this example we’re gonna, look at how we determine the miller indices when we have a hexagonal unit cell we use a different notation from a cubic cell. We use a notation with four indices, so there’s “h”,, “k”,, “i” and “l”, which is different from the cubic structure, and we do this by having three vectors that represent direction of the unit cell distances are the same so a_1 a_2. This distance is the distance from here to here in each case. And having three vectors in the same plane is redundant. They are each at 120 degrees to each other, but it is a convenient way to do it and then the fourth in the z-direction, this distance from here to here is “c”. So!
We want to determine the miller indices by first determining the intersections the plane with the axes and? So the first is: where are intersections? so with a_1? it’s a distance. Here it is “a”. A_2. This plane is parallel to the a_2 axis, so it never intersects. So the intersection is infinity.
With a_3. We have to extent the axis in the other direction, so it’s minus “a”. And in the z direction.
The intersection here is “c”, so remembers? “c” is not equal to “a” in this structure, and then we write these indices first, the determined indices,. We write these in terms of the multiples of the unit cell dimensions, so 1, infinity,, -1 and 1! And? Then we take the inverse so 1, 0,, -1 and 1! And.
Then, if we want the final miller indices, we write them as (1 0 and -1 is written as 1 with the line over it.
So? This is our answer for this green plane. These are the miller indices in the hexagonal structure. I mentioned that there’s redundancy, which means that “h” plus “k” is equal to minus “i”. So we can always just determined “i”. You, see in this case 1 plus 0 equals minus “i” is -1 so. We can always determine the third indices from the other. Two. We can also look at another plane. We can look at this plane that i colored in here and we can see the miller indices,. The intersections are infinity, infinity,, infinity and “c” so, using the same idea, taking inverses and so ( 0, 0, 0, 1) would be the miller indices for this top plane. That i have indicated.
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