In this lesson, we're going to be talking about more complicated structures where rather than just having one or two, we now have several atoms per lattice point. So we can begin with a structure that is of the type Mx2. And when we look at a structure like this Mx2 what I have done is to illustrate one of the ions as being blue and the other two as being black filled in circles. And I've coupled them with dotted lines to indicate how those structures are coupled together or how those ions are coupled together in the structure. So when we look at this, what we're seeing here is again a face centered cubic lattice and this time it has a total of three atoms per lattice point so that gives us then a total of 12 ions in the unit cell. And because of the stoichiometry, what we have to have is two of the X type ions and one of the M type so that the charge is balanced out in the unit cell. We can look at the structure of a material which is based upon the Silica tetrahedron, where we have silicons that sits in the center of the tetrahedron and is surrounded by a total of four oxygen ions. And one of the structures that can develop as a consequence of this tetrahedral arrangement of silicon and oxygen is one of the forms of silica, which is referred to a crystobalite. There's several different crystal structures that are associated with SiO2 but this happens to be one of them and this one in particular is a variation again on the face centered cubic structure. So let's examine this in a bit more detail. In terms of the FCC structure, we're going to see that they're a total of six atoms per lattice point. As we go through this, because we're dealing with the face centered cubic structure, and we have six of the atoms associated with each lattice point, we therefore must have a total of 24 of these atoms in the unit cell. So let's take a look at this. So here is our structure, and what I'm going to do is to identify the number of the various types of species that are inside of the structure. So the first thing we're looking at are all those indicated by the number 1 and that 1 indicates the total number of silicon atoms that are at those eight corner points. Remember each one of those corner points has a total of eight unit cells around it, so since we have eight corners, we there for have one silicon atom that is associated with a corner. Now what we're going to do is we're going to take a look at the silicon atoms that occupy the face centering positions. So, when we look at those, those are designated as 3s, meaning we have six faces. Each one of the faces shares with an adjacent above or below, in front or back and since we have six faces in every share we have then a total of three silicon atoms that are associated with those faces. Now we still have some silicons that are not labelled. And we're going to go through and label those, and those there indicated as four. And in this particular case when we look at the four, we see that there are a total of four of those silicons that lie wholly inside of the unit cell. And therefore, we must have a total of four silicon atoms that are inside and associated with those particular atoms in space. Now we're going to sum up all of the atoms that appear in the unit cell in terms of the structure of silicon dioxide. Remember that we had a total of 8 silicon atoms and because the structure is one to two, we therefore must have a total of 16 oxygen atoms per unit cell. And as a consequence, we have a total of 24 atoms that are in this unit cell of crystobalite, SiO2. Let's look at another structure. This happens to be once again a simple cubic structure. And when you look at the structure, and what I'll do is indicate what those various points mean by recognizing that I have assigned the center position as titanium, and I've put calciums on the corners. And then what I've done, is to put the oxygens on the faces of the unit cell. Now it's clear that the corner positions are different than the face positions, which are different than the center position which means this structure must be simple cubic. And it turns out, it's referred to as calcium titanate, and it has the stoichiometry of one titanium, one calcium and three oxygens. And that is consistent with the picture that is up on the screen and this particular structure, because of the importance and the number of different compounds that crystallize in this form, it's referred to as the Perovskite Structure. What we can do is we can look at an alternative Perovskite structure of titanium, calcium, oxygen material. And this time what we're going to do is reposition the atoms. So the atoms now are located at the titanium at the corners. And when we start looking at the calcium, it lies in the center. And those two, then, are just simply interchanged and consequently, we maintain the one to one relationship between titanium and calcium. All right, now let's take a look at the oxygens. Because of our new origin, the oxygens are not located on the faces, but now they're located on the edges. So let's do a little bit of counting here. It turns out that we're dealing with a queue, which means that we're going to have a total of 12 edges. When we look at each one of those edges wherein oxygen is occupied. Then what we find is, each one of those oxygens is contributing a total of one quarter. So hence, when we sum up all the edges and the one quarter contribution, we'll get a total then of 12 divided by three or four which is ultimately going to give us three. So now we go back and we see the structure of calcium titanate. Again, one titanium, one calcium and three oxygen. So everything is consistent here. If we were to go down the column of the periodic chart and replace calcium with barium, we would come up with a structure barium titanate. When we look at the structure of barium titanate, it's similar to what we saw with calcium titanate with the exception that the titanium is now displaced in the center of the unit cell. As a result of that displacement, what we see is on the diagram to the right, we see that the titanium has been moved up relative to the oxygen. And then what we find is as a consequence of that charge displacement, we develop a dipole that's associated with the structure now. So we have a dipole. And what does that mean? Well, what happens in this particular case. If we were to take this unit so and compress it along the C direction or the Z direction, what we could do is to move the titanium back into the center of the cell and the dipole then winds up the superion. Well it turns out that there is a very fascinating material that can be developed as a consequence of utilizing this structure, and that is we can produce a material which can take electrical energy and mechanical energy, or mechanical and electrical, and go back and forth between those two. And this then becomes the basis for something that we refer to as a transducer. The last structure that I want to introduce in this lesson is that associated with a crystalline polyethylene. And remember we can begin to think about these polymer chains as spaghetti. And what we've done in this particular case is to align each one of those change in such a way that we produce this unit cell that happens to be orthogonal. So the a, b and c have different dimensions but again all of those interaxial angles are 90 degrees. So you can begin to see as we move away from some of these simple structures and we come in to these more complicated structures that we can still begin to analyze the individual crystal structures of the various materials. It just takes a lot more detail in order to do that. Thank you.
