Now what I'd like to do is to introduce the concept of linear and planar densities. So, we'll begin first then by looking at what we mean by the idea of a linear density. And this is describing for us the directions in which we have the most number of spheres that are along a line. So for example, if we look at our two structures, the BCC and the FCC, again on the left for the FCC structure. What we have is a line that's indicated in that bottom plane, and the direction would therefore be 1, 1, 0 direction. So that's that specific direction in the FCC. When we look at the BCC structure we're going to be talking about a specific direction for example. Let's describe the direction that moves along the body diagonal, and that body diagonal direction would be the associated 1 1 1 direction. Now when we talk about calculating a linear density, there are a couple of different ways that we can do that, and certainly one of the fairly simple ways to do that is to talk about the line, and then talk about how many centers that that line passes through. So, for example, when we're looking at the touching of spheres along the 110 direction, what we see is that we have a half a center, one full center in the center of that square and then another one-half. So when we add those two, what we see are those three positions. We have one-half plus one plus a half. So that's going to give us a value of two. Now the other thing that we want to pay attention to is, we want to know how long that line is, and we know that that line is going to be a 0 or the edge of the cube times the square root of two. So what we'll have then when we talk about.a direction and we talk about a linear density, then what we will have is the number divided by the length and so we have a total of two intercepts that are going to be divided by A0 onto the square root of two. When we talk about the density along the 111 in this particular case for the body centered cubic case. We have the direction is along that 111 direction and what we see is that's going to be a zero onto the square root of three. And when we talk about the number of intercepts again it's one half, one and one half and so in order to describe that particular packing or linear density what we will have is one-half plus one plus a half, so that's two, and that's now going to be divided by a0 onto the square root of three. So, that describes for us a linear density. Now when we talk about a planar density, for example, let's look at the 1 1 1 plane in the FCC lattice. We see that in terms of that particular plane we will have the figure if we replace the lattice points with spheres. Now what we have is when we're describing that triangular unit that describes the 1 1 1 plane in the unit cube. What we will see is the circles have an area. One of them is the sixth of the circle, one of them is a half of a circle, the other half of a circle, a half of a circle, and then those two at the bottom corner represent a sixth of a circle. And so what we would then do is to add up all of those circles. The three that are associated with the corners, so that would be three times the area, divided by six. And then with respect to the ones that are along the edges, they would be three times the area divided by two. And so that would give us then the ability for us to calculate the planer density that's associated with the 1 1 1 plane. Now, when we look at the 1 1 0 plane, again in the FCC lattice, we look at the positions of the hard spheres on top of those lattice points. And we can calculate what the total area is associated with all of those circles that lie on the plane. And so we have one quarter at each of the corners, and then we have a half along those two edges of the face diagonal and we see that the area would be a0 times a0 onto the square root of two. And what that tells us is that's the area of that particular rectangle of interest to us. And then we divide that into one half, one half plus the four one quarters and that will tell us then what the packing density is associated with those circles that are lying on that plane. One point that I want to bring up here is that when we're talking about determining the fractions we want to make sure for the particular plane of interest that we're making our calculations, that that plane actually passes through the center of those circles. And so that describes then the portion of the sphere that lies on in this case the 1 1 0 plane. I'm bringing these two up at this particular point in time because what we'll see is, when we describe the FCC and the BCC structures, we're going to come up with a very important concept. And that concept is the planes of densest packing and the directions of densest packing. And when we look at the FCC system, what we see is that all of the [111] planes in the FCC structure wind up being bound by the [110] type directions and that describes the plane of densest packing for us. And we would include the plane and we would include the direction. When we describe the body center cube, now that plane of densest packing is along the dihedral plane and that's the 1 1 0 type plane. And then what we have is on that plane two directions of closest approach with respect to the directions in which the spheres are touching. And so we have the combination of the densely packed 1 1 0 and the directions 1 1 1 in the body centered cubic. We'll use this concept a little bit later when we begin to talk about the effect of particular orientations on the mechanical behavior in the FCC and BCC materials. Thank you.