In this lesson we're going to address the question, why the Bravais lattice? Remember, in reviewing what we talked about previously, one of the seven crystal systems, the cubic system, has a total of three Bravais lattices, what we refer to as a primitive cell. And again, what we mean by the term primitive is that each one of those lattice points, each contribute an eighth. And since there are eight corners on a cube, we have a total one lattice point associated with that simple cubic structure. If we move to this center figure, what we have is the face-centered cubic structure. And again, we have those corner positions, as well as the six faces. And that gives us then a total of four lattice points in that unit cell, making it non-primitive. If we go over to the body-centered cubic, what we're seeing here is, again, the eight positions at the corner and the one that sits wholly in the center, so gives us a total of two. But in this case, each one of these cells are cubic, and therefore, we should be able to use the basic ideas of vector analysis when we are describing the crystallography of the primitive face and body-centered cubic cells. We'll talk about these by looking at them individually. First, what we can thus see is that when we talk about the face-centered cubic, we have the lattice positions that are sitting at each one of the corners and each one of the faces. And if I were to take that basic unit, and I were to translate it in the three orthogonal directions, I would wind up building up a unit that looks like what I have on the right-hand side. So I have my faces covered, I have my corners all covered and I will wind up then, having a total of four lattice points in that unit. So, let's return to this, FCC structure again. If you look at each one of those particular points what you find is that each point turns out to be surrounded by a total of 12 equivalent lattice points. That is, they have 12 nearest neighbors. So if you identify any one of those points, the closest distance between any one of those points gives you a total of 12. And sometimes, as we've talked about perviously, that would then refer to the coordination number of those lattice points. And it turns out that when we look at this figure carefully, we can use three alternative vectors to define the unit cell. And I've described them on here is the vectors a1, a2 and a3, and those are all equal, the magnitude of a1, a2 and a3. The interaxial angles associated with a1, a2, and a3 are not 90 degrees. But what I can do is, if I go through and do my vector additions, I can produce all the other points inside of that cubic face and or cubic structure by simply vector addition. So, let's see how that all works. So here are my three vectors a1, a2 and a3, and they're going to those particular points that are located on the face positions of the FCC unit. Now what I can do is I can create one of faces by simply adding a1 and a2. I can create another by adding a2 and a3. I can add a1 and a2 and produce that additional one. And by suitable combinations of a1, a2, and a3, I can produce that point that I have on the upper corner of the cube. Then, by continued additions of combinations of a1, a2, and a3, I can fill in all of the positions that were given by the face-centered cubic unit cell. So what I've done then is to take the a1, a2, a3 and made it a primitive rhombohedral cell. And it turns out that when we look at that unit over here, what we have is a rhombohedral structure, which has a total of eight corners, and each one is surrounded by eight unit cells. And what we will see then, it has one lattice point associated with it. So if I take those vectors, a1, a2, and a3, rather than describing the unit as a cube, I can describe it alternatively as a primitive rhombohedral unit. Here's our picture over here to the right, so this is now our primitive unit cell associated with a rhombohedral structure. Now the question then becomes why not let it be the rhombohedral structure? Well as it turns out, because we do not have interaxial angles of 90 degrees, it's far more convenient to change the number of lattice points through the incorporation of the addition of those vectors and produce the face-centered cubic array. So the particular interaxial angles that are associated with the a1, a2, a3 will wind up giving you a face-centered cubic distribution of points, which now we can look at from the point of view of the three orthogonal vectors x, y, and z. Before we continue with this, we need to briefly review a few basic points. We need to go back and review what we mean by the term cross product, and of course we also need to understand the concept of a triple scalar product. When we talk about the cross product of two vectors, the cross product turns out to be a vector, and the vector is normal to the plane. So when we do b cross c we get the vector normal. Now when we go through the operation where we take the normal and we dot it with a which is a vector, so we take those two vectors and form the dot product between a and N. And what that provides us is the volume of the unit cell. So the triple scalar product, a dot b cross c, winds up providing the volume of the unit cell. So, now what we'll do is we'll make some comparisons between the FCC unit cell, which is non-primitive and the rhombohedral cell, which is primitive. So here we are looking at our FCC cell, and I've included the face positions and the corner position. And what I'm going to do is to determine what the triple scalar product of that is, using the vectors along the x, y, and z-axis. What I'll do is write those in terms of my vectors. And I have a, b, and c, and I will wind up with a triple scalar product, which is nothing more than the value of a0 cubed. So that's giving us the volume of the cube, and you would expect that because x, y and z are all the same dimensions, and when we look at the cube we'll see that it should be just a0 cubed. Now, if we look at the primitive, so here's our primitive cell over here, and now what I'm doing is I'm using the vectors that describe those three points. And those are related to my FCC unit cell that I described previously but now I have included the magnitudes of those vectors. And when I'm going to take the cross product and I take the dot product and come up with a triple scalar product, what I wind up with is a value of a0, which is the edge of the cube divided by 4. Now it is not a coincidence that the value of the non-primitive FCC cell happens to be four times larger than the associated cell that we refer to the primitive rhombohedral. It's useful for you to consider looking at the body-centered cubic. And coming up with a very similar relationship between the volume of the non-primitive BCC and the primitive rhombohedral cell that's described by the vectors associated with the body-centered cubic positions. We'll address those in subsequent lessons. Thank you.