In this lesson we're going to describe something we refer to as the Miller Indices in terms of planes. We're going to be interested ultimately in talking about planes for example of densest packing. What are are those planes and how can we go ahead and describe them simply? So we'll start out again with a cube. And this time we're going to be talking about planes. So first we're going to identify the coordinate system. And we have out coordinates system x, y, and z. Now there's a point here that I want to make. And that is I'm using a left-handed system, meaning that when I look at x and I look at y, if I take x and I cross it into y, I get the direction z. So, when you describe your position with respect to your identified coordinates, what you want to do is use a right-handed system. Now let's say that we're interested in the plane at the top. In order for us to do that what we have to do is to identify the intercepts of that plane. And what we can see is that, that plane is parallel to x and y, and it intersects the z-axis at Unit one. So that tells me that my intercepts are going to wind up being infinity, infinity one. In other words, it never touches x or y, but it intersects the z axis at unity. Now what we're going to do is to take the reciprocal of the infinity, infinity one individually and what will happen is we will wind up then having 001. So what we've done now is to describe the plane of interest, which is the blue plane. We've described it and it now becomes the 001 plane. Now if we look at this additional plane that I've indicated in the visual. It actually intersects the z axis at a value of a half, so it's half way up the z axis and what it does it it never intersects x or y. And so what we have in terms of our intercepts is infinity, infinity, one half. Now, what we're going to do is to take the individual reciprocals of each one of those and we'll have that as one over infinity, one over infinity, one over one-half. And when we evaluate that becomes the 002 so the plane that I've indicated here is now called the zero zero two and I want to point out here that when we're describing planes The way we describe the planes is, there are no commas that are between the integers. And in addition to that what we have are the parenthesis. So the parenthesis describe for us the planes whereas the brackets wind up describing the directions. So, that's now the 002 plane. So, let's take a look at describing another plane. So here is our cube. What we have been doing actually is to place our origin back in that left hand corner. But because the plane of interest actually passes through there, what we're going to do is to come up with an alternative origin. So the plane does not pass through the origin. So we choose an alternative origin and that's our origin here. And again, It's the right handed system of x coming out the board y, and z. And so that then becomes the zero, comma, zero, comma zero so that's our origin. Now when we look at that plane, we see that that plane now is parallel to the z axis. It intersects the y axis at negative one and it never intersects the x axis. And so when we look at this, we have infinity -1 infinity. And now if we take the reciprocal of that what we get is the 01bar0. So again, the fact that our intercept is negative we actually delineate that by putting the bar over tops and when we see a plane that's written in this way We immediately know that this is the 0 1 bar 0. And so, our negative sign then is put up at the top and we read it as a bar. In review, what we can do is, we can take a look at the plane that I've indicated here and that is a dihedral plane. That is, it passes through the cube and when we start looking at our axes, we position the axes at the point in the rear and then, as a result, this becomes the one, one bar, zero plane. As we did in the case of the directions where we talked about specific directions and directions which are all of the same type, we can do the same thing when we talk about planes. So, for example, let's describe again all the faces of the cube. What I've done is to indicate the origin. Remember, x, y, and z in a right hand way. And that first visual that we see is the plane 010. The 01 bar 0 represents the negative and pay attention to the fact that I have changed the origin with respect to figure on the left and the figure on the right. I've done that because when I'm looking at a specific plane, I do not want to have that plane passing through my selected origin. And now when we look at the plane that is associated with the third figure what we see is that's going to be the 001 plane, never intersecting X or Y, but intersect Z at 1. Now when we talk about the bottom face of that particular cube. We have to redirect our position with respect to the origin of our coordinate system and I've moved it to the top. And now we don't intersect x, y but we do intersect z. But this time we're talking about intersecting it at negative one. And when we look at the 100 plane, the corresponding 1 bar 00 again I have translated the origin. And we have thus described all the six faces of the cube and what we have is specific planes, and now what I can do is lump them all together, and I can talk about the family of planes as written in red. So I’ve described then the planes that make up the family of 100. Thank you.
