In this lesson, we're going to continue our discussion of the face centered cubic model, but what we're going to do, is to add to it the concept of a hard sphere. When we look at the figure on the left, when we described the lattice, so we have each one of those points that represent the FCC structure. And if we move to the figure to the right, what we're seeing are the spheres that are sitting at each one of those lattice points and they're developing the hard sphere FCC structure. Now, it turns out that this is a very useful way of describing certain materials, for example, copper crystallizes into a structure that's referred to as face center cubic. That is, each one of the atoms is represented as a sphere, and it touches in particular directions, and it produces a dense structure called the FCC structure. Now it turns out that regardless of the dimensions of the sphere, the structures all will look the same and describe what we consider to be the packing factor and see that regardless of the radius of the sphere. When we're describing a face center cubic structure, the packing is always the same in terms of the packing fracture. So, if we now look at a section through the FCC structure where it having the hard sphere, we see that on the lower left. And we can see how those corner positions and face positions are cut, developing the face center cubic structure. Now what is important for you to remember is that in order for us to develop this hard sphere structure of face center cubic, the spheres must touch along a particular direction. And the direction where they must touch is along one of the diagonals of the face. We can relate the diagonal of the face to the edge of the unit cell by recognizing the simple geometry that the length of the direction on the phase is A0 which is the edge of the lattice times the square root of two. So we can relate then the dimension where all of those spheres touch, that length as a zero onto the square root of two. Now what we can do is, we can describe something we refer to as the packing factor. And the packing factor is going to be the number of spheres and the volume of the spheres divided by the volume of the unit cell. And we know already that there is a relationship in the hard sphere approximation that we can relate a0 and r to one another and what we and then, enable to do is develop a non-dimensional packing factor. Here is our FCC structure, we have six faces, each sharing with a half, eight corners and we turned out with a total of force pierce in the volume. Now, what we're going to do is to represent the packing factor as the numbers of spheres times the volume of the spheres divided by the volume of the cubic unit. And we know the relationship that exists between the edge of the unit cell and the radii. So we know for example that a onto the square root of 2 = 4r. And therefore, we know the relationship between a and r for this face centered cubic structure. So all we then have to do is to substitute so that we can cancel out the terms of r, and we wind up with a packing factor of 0.74. It turns out that this is very important because the packing factor for a distribution of hard spheres in the face center cubic geometry is the highest packing factor that we can get. These are all the same dimension spheres and they pack 74%. We'll develop a similar relationship in the next lesson where we describe the hard sphere packing in the body centered cubic arrangement. Thank you.