[MUSIC] We've looked at how we can use symmetry to arrange objects in simple and complicated fashions. Now we're going to turn our attention to the application of these rules to the creation of crystal structures. And to do this, we need to learn how to read the international tables of crystallography. These tables contain all the information that is required to plot atomic positions. From these atomic positions we can then look at the arrangement of atoms and how they control the external form of crystals. In addition to this we will need to learn about the relationship between Plane Groups and Space Groups. So now let's begin by looking at some pages from the International Tables of Crystallography. These pages always come in pairs. One page, which you can see now, contains information about the space group symbol. And also the fractional coordinates of atoms that were generated using this space symmetry. There is a second page which we'll look at momentarily which includes the symmetry diagrams. And these are important for seeing the location of the symmetry operators. First of all, there is a space group number. We've already learned that there are 230 space groups. For this particular space group which we're observing now, space group P2, it is number three of those 230 space groups. The space group symbol P2 is telling us first of all that it is a primitive unit cell that's the capital P. And it also contains a two fold or dyad rotation operation, that's the two. Below that we can see information that relates the playing group to the space group. And if you look in detail what it's telling you is that as you view a p2 type cell down the 0 0 1 c axis, 0 1 0 b axis or 1 0 0 a axis, you have particular plane groups. In the middle of this page, we see the very key information that we'll be focusing on in this lecture. This is where we see the fractional coordinates that atoms will adopt in P2. There is other information also on this page which is not key to this course, but I'll explain what it's for. To the right-hand side, there is information concerning diffraction. We'll see in a few lectures how the diffraction is used to determine crystal structures. At the bottom of the page, relationships between the different 230 space groups are given. As you would expect, there is a mathematical hierarchy that changes as you subtract or add symmetry operations. Now let's look at the second page for the P2 space group. Here we have the symmetry diagrams. There is also some duplication of the information on the first page. Again, you can see the space group symbol P2. But the full symbol is also given, which is P121. What the full symbol is telling us is that the diad rotation axis, the two part, is along the b axis. There is no symmetry operators along the a or the c axis. You also see a different way to represent the symmetry the schoenflies symbol. These symbols are used, not so much in crystallography which is the concern of this course but more so in spectroscopy. In addition, we see that it is a monoclinic system, and we also see the point symmetry of this space group. When we look at the positional diagrams, we find that in the bottom right hand corner, there is a general position diagram. And you'll notice the standard symbol which is used to convey the sense of the object, the open circle. This particular general position diagram is conventionally looking down the b axis, so this is why the angles are not 90 degrees. Because we have rotation but no mirror, all of those circles are open, none of them contain the comma which indicates chiral positions. The other three diagrams show what operations are working when we look down and a and the b and the c axis. So the top left-hand diagram is looking down the b axis and corresponds in direction to the general position diagram shown lower right. At the bottom is shown where the origin lies and also the constraint on the choice in the position of the atom x,y. This is actually indicating the size of the asymmetrical unit. Perhaps the easiest way to understand how to use the space group tables, is to work with a real example. And for this, we will take lead germanium oxide. In fact, it's trilead germinate. This particular structure also conforms to the space group P2 in its short form or P121 in its long form. Underneath that we're showing where all of the atom lie with respect to the symmetry operators. So we have three types of lead atoms, Pb(1), Pb(2), and Pb(3). We have a germanium atom, and we have three oxygen atoms, 1, 2, and 3. If we look at oxygen 2, position 2e, that is a general position. And the oxygen lies at 0.19, 0.27, 0.69 with respect to the x, y, and z axes. What we would like to do now is to create a complete list of atoms by referring to the space group table. We begin by looking at the symmetry diagrams and these are extracted directly from the space group tables. The general position diagram we've already discussed and that shown in the bottom left. The general position diagram for a monoclinic system is always the projection along the unique axis. In other words a long the b direction which is also the o one o direction. The other three diagrams will show what symmetry operators are acting when we look along [010], [100], and [001]. In other words, along the b direction, the a direction, and the c direction. You'll notice specifically how the origin is defined in these projections. That's where the zero is shown. And the a and the c directions are also given. We begin by looking at the full symbol P121. The 1, and this is a identity symbol means there is no symmetry operation along the a axis. The 2, which is two-fold rotation or diad is what is operating along the b axis. And finally the one at the end of the symbol refers to the fact there is an identity or no symmetry operation along the c axis. When we look at the positions 2 e 1 a b c d, what are we referring to? We are talking about how many atoms will be generated due to this certain symmetry and in this case the multiplicity is one or two and the a, b, c, and d are the Wyckoff positions. If we look at the positions with respect to the general position diagram then we can see that we have atom at x y z. These are the atoms that are located at the top left-hand corner, because the origin is the top left hand corner, and we also generate an atom at -x,y, -z. Remember, we are looking down the b axis in the general position diagram so the y symbol refers to the height of the atom in this particular projection. There are two fold rotation axises and the symmetry diagram shows where these lie. If you look back at the previous page, it said that the origin was at the two fold rotation position two. There are also two fold rotation axis found in the center of the unit cell. And also in the edges or faces of the unit cell. When we have an object, the atom, lying directly on top of these two fold rotation axes, then we generate the special positions. In this case, there are four special positions, a, b, c, d, as showing. Also we can illustrate where the asymmetric unit is. In this case, it lies as shown in the shaded region of the general position diagram. You're now ready to generate the positions of the atoms in lead trigermanate. The slide that you are looking at now is quite busy, and you will need to study it in your own time. You also need to look at it in conjunction with the earlier slide that contained the table of the atom positions. What we'll do though, is just speak briefly about what you should be working with in this particular slide. You can see that we have given the fractional coordinates, x, y, z, for all of the atoms inside the unit cell. For those that are on the special position and have only the multiplicity one, you don't have to do anything more. There is only one atom of that type available in the unit cell. However, for oxygen 1 and oxygen 2, that lie on the general position, there is a multiplicity 2, which means that you have to apply the symmetry operation accordingly. You do this by looking at the x, y, z fractional coordinates for the 2 e position. For every x, y, z atom, you can generate a -x, y, -z atom, and that is what is shown in this table. Now we have generated all of the atoms of lead, germanium and oxygen. Inside the unit cell, we're ready to plot them. What I've drawn on the right-hand side of this slide are four unit cells projected down the b-axis. So we want to do a b-axis projection of the structure. We draw four unit cells rather than one because we need to see the complete pattern of atoms inside this structure. If we only drew one unit cell, it wouldn't be quite so easy to appreciate the relative positioning of the atoms. I've placed the origin of the unit cell in the top left hand corner. Of course for these four unit cells, the origin is always exactly the same. You'll also notice when we do the plot, it sometimes easier to work with positive values of the fractional coordinates. To convert a negative value to a positive value, all you have to do is add one. If we begin with the lead atom at the origin, naught, naught, naught. Then we can plot it shown as a green circle. It's not in x, not in z, and its height is also naught. We write the zero next to it, to indicate its height. Every origin is equivalent, so we can also place lead atoms at all of the origins. We then carry out the same exercise for lead two. In this case, its position is a half no a half. So it's a half in x, a half in z, and it's height is again, naught. Every unit cell is equivalent, so we can place three equivalent atoms in the adjacent unit cells. We do the same thing with lead three which is at height .52. We would do the same thing for the germanium which is shown by a yellow circle. And we do the same thing for oxygen1, oxygen2, and oxygen3. A final point to make is when you look very closely at this plot of the atom positions, you will notice that the oxygen one and oxygen two positions, the general positions for the oxygen conform to the twofold rotation axis, as you would expect. And you can see this by looking at their position in the x zed plane, and also their heights. So for example, the oxygen, which is at height 0.34, conforms to having a two-fold rotation relationship around the origin, and also around the center of the unit cell. In this lecture, we have looked at two things. First, we have considered how we use space group information to generate the positions of atoms. Secondly, we have looked at the plotting of those atoms, to better understand the patterning and tessellation of the different atomic species. When we generate the fractional co-ordinates, particularly the fractional co-ordinates of the general position atoms, we will often find that you end up with negative co-ordinates. The negative co-ordinates are perfectly fine, and you can plot them directly on top of the unit cell. However, for some of you, it might be easier to always work with positive co-ordinates. In this case simply add one and you convert to a positive figure which is exactly equivalent to the negative figure. Secondly we know that when we plot the atoms it is in accord with the space group symmetry, therefore if you were to superimpose the general position diagram. And the symmetry elements associated with that diagram, on top of the atomic plot, the atoms must lie in accordance with those positions and those symmetry operations. And this is what we just saw for the oxygen one and the oxygen two in the lead germinate. You're now ready to start to look at some real crystal structures. And this is what we will do when we start work on part three.
