[MUSIC] In the previous lecture, we looked at plutonic solids. And the plutonic solids were created by folding up two-dimensional networks. Those of you who are paying attention, will also have noticed some different symmetry symbols appearing in the animations. Some of the symmetry symbols contained a square in which there was an open circle. This is an indication of an inversion center. And it was operating in both the octahedra and tetrahedra. In this lecture, you will learn what an inversion center is. In addition, you'll discover what all the additional three dimensional symmetry operators are that can be derived from the two dimensional operations. The sequence in which we'll go through these additional symmetry operators begin with screw axis, or improper rotations. These are to be differentiated from plain rotation, or proper rotation. In addition, we'll look at the way in which we can develop double glide and axial glide. So far we've only met plane glide, in other words, it occurs in one dimension. But in double glide and axial glide, we'll see the way in which we can move up or down steps. This is an example of axial glide. We'll also look at how to create these centers of inversion. This is perhaps one of the most difficult of the three-dimensional operations to understand. And finally, we'll look at how we can use inversion centers to create roto-reflection or roto-inversion. Let's recap where we are so far. In plane symmetry, we have rotation, reflection, translation, and glide. Once we move to three dimensions, we have to introduce a screw axis. Now the screw axis is also a compound operation. It involves rotation in a plane, followed by translation in a direction at right angles to that plane. A real life example of a screw axis in operation would be a spiral staircase. If we look at the diagram shown in this slide, what can we see in that staircase? The first thing is that there must be a vertical translation. In other words, there will be a point at which one of the lower steps is reproduced exactly higher up the staircase. That will be your translation distance. Obviously, there is rotation taking place. In this case the rotation is 15 fold. There are 15 stairs in the translational repeat. This is not something which we would meet on a regular bases. There also must be a periodic repeat. In other words, the translational distance. And finally there should be an asymmetric unit. The asymmetric unit will be just a single stair. So that single asymmetric unit is rotated slightly and lifted, rotated slightly and lifted, and we continue that operation until we have the total repeat occurring. This of course is common to all symmetry operations. The replication of the symmetry operation until we return to our starting point. Let's look at this a little more formally. We begin by looking at two fold rotation. The diagram on the right shows what's happened, in plain, twofold, or dyad rotation. We start with an object, shown in the bottom disc, we rotate through minus 180 degrees, and there is no vertical translation taking place. We repeat that rotation, over and over again to create the repeating object. When we move to two fold-screw axes though, we begin at the same point but now we introduce some translation vertical to that disc. In this case, the translation is half the full repeat. So we rotate, translate, and rotate again, to get back to where we started. You'll notice the symbol for two fold-screw axes, is two subscript one. The two subscript one, is telling us the repeat is one half. Let's now look at three fold-screw axes. The diagram on the left shows a normal proper rotation. We begin with an object and we rotate it sequentially in a counter-clockwise direction in increments of -120 degrees. In a screw axis, we do the same thing. We rotate counterclockwise in increments of -120 degrees, but between the rotations we're going to either lift by 1/3 or 2/3 along the screw axis. So if we start with an object at 0.1, we rotate and lift to 0.2, rotate and lift to 0.3. You'll notice that in this example, the lifting is through 1/3, therefore the screw axis label is 3 subscript 1. In the second example shown on right, we have a 3 2 screw axis. So we begin with an object at position 1. We rotate and lift, but now were lifting through 2/3. We do that operation again and we return to a point in the middle and finally another application of the operation puts us back to where we started at position one. There are three types of four fold-screw axes. in the first type, four one, we lift the object through a distance of 1/4 along a translational axis. We repeat the operation until we return to the disk at the top, where we have the object lying directly above the disk in the bottom. In the 4,2 c screw axis we lift by half the translational difference. If you like you take the symbol 4 subscript 2 right at it's two on four, divide by two and that's your translation distance. When we start with an object in the lower disc at position 1, we rotate 3 minus 90 degrees lift by a half we get to position two. We repeat that operation and we get to the top disc, but you'll notice that at this point we still haven't got back to where we began on disc one so we have to repeat the operation twice more to return to our original starting point. You'll also notice that in a four two screw-axes, we don't have to use all of the discs. The disc at 1/4 and 3/4 is not utilized. Finally, we turn the four-three screw-axis, where the lift is through 3/4 and we end up with the arrangement of objects shown. You'll notice that the sense of the screw in four-one and the four-three axes are opposite. In this slide, I show all of the six fold-screw axis. They come in five different varieties. I won't go through this slide in detail, because the principles are exactly the same as we've already seen for the two fold, three fold, and four fold axis. What I will highlight, however, is that not every disc needs to be used. If we look at the case of the 6 2 screw axis, in other words we're lifting by 1/3, then what you will see is the disc at 1/6, the disc at 1/2, and the disc at 5/6 is not needed. Similarly, if we look at the 6 3 screw axis, we only use a disk at zero and a disk at half. You will also notice that the sense of the 6 1 screw axis is opposite to the sense of the 6 5 screw axis. Let's recap what we've learned about the screw axis. The first point is that they are compound operations. They involve rotation in a negative, or counter clockwise direction, followed by translation at right angles to the plane of the rotation. The symbols to describe these screw axes are very specific. They are derived from the symbols used for proper rotation. But in addition, they have little wings attached to them to show the sense of the rotation and also which of the disks an object lies on when we apply that operation. If we take a simple example, say the 4 1 and the 4 2 screw axes, in 4 1's screw axis, the object lies at 1/4, 1/2, 3/4. But in the 4 2 case, it only lies at naught and a half. The disks that we find at 1/4 and 3/4 are not used. Therefore, the symbol only has two wings attached to it. Let's now look at axial glide. And this is the first case where we really have to carefully consider the relationship between two dimensional, or plane symmetry operation, and three dimensional operations. The easiest way to think about axial glide is that it's like climbing a ladder. When you climb a ladder, your feet represent the chiral objects. There's also a compound operation, because it involves reflection followed by translation along the ladder direction. Consequently, the relationships between the objects must be chiral. So if you think about your feet as chiral objects, then as you climb the ladder, they represent the chiral pairs. In the diagram, I've shown the feet as chiral pairs. If we look carefully, we can see there is a vertical translation. So as we move from one chiral object to the next, It is the distance between the rungs on the ladder. There is evidently reflection taking place, hence the chirality of the objects. And there will be an overall periodic repeat. This periodic repeat will be equal to the distance of two rungs on the ladder. There must, of course, be an asymmetric unit which can be your feet or one of the chiral objects. To fully understand axial glide, we need to label the three directions, x, y, and zed. If we look along the x direction, which in other words is looking directly at the ladder. We see the chiralobjects separated by a plane glide line, in other words, a dashed line. If we look at the relationship between objects along the y direction, we see a straight line of chiral objects. And finally, if we look down on top of the ladder, in other words, along the zed direction, we see the pairs of chiral objects that now rerepresent the axial glide through the dotted line. So you'll notice the difference between an axial glide representation, the dotted line, and the plain glide representation, which is a dashed line. And depending how we look at this operation taking place, we will either see the axial glide, or just the plain glide. This again, is a compound operation. It involves reflection plus translation, and the axial glides are defined according to the direction upon which the translation takes place. In this case, I've labelled the ladder direction as zed, therefore this would be called a c-glide. Now let's look at double glide. In this case we have a mirror, but it's combined with two orthogonal translation vectors. So it's like a staircase. Your feet represent the chiral pairs and as you climb the staircase, you're lifting your feet vertically and moving them forward. Again, we find that we have translation taking place. There is a vertical translation, and also a horizontal translation. Hence, it's called double glide. There is reflection, therefore we will be able to see chiral objects and there must be a periodic repeat, and also an asymmetric unit, as before. If we introduce the vectors x, y and z, then in this particular instance, we can see that by looking along x, all looking along y we will see a two-fold screw axis. If we want the view to double glide we have to look along z, in other words look down on the staircase. And the formal representation for double glide is a dot dash line. In general, the translation vector is a half of the total periodic repeat. But as we'll find in crystallography, N glide, or double glide, can also involve translation along a quarter of the unit cell, or a quarter of the translational repeat. Now we turn our attention to inversion. Inversion is also a compound operator. It involves rotation and reflection. In this example, I've borrowed from The Matrix, and if we look at Neo, starting at position one. He would rotate through 90 degrees in a counter-clockwise direction. He would then reflect when you would get to position three. If we carry out that operation again, rotation and reflection, we get back to where we started. If we want to represent this operation more formally, we could say that Neo began at position z, y, z. And through the action of rotation and reflection, he moved to position -x, -y, and -z. What we have done through this combination of rotation and reflection is create an inversion center. It's also known as a center of symmetry and in the crystallographic symbols, it's represented by an open circle. Now that we understand what a circle of symmetry is, we can combine the center of symmetry with reflection. Looking at the right hand diagram, we begin with Neo at position one. We then rotate through minus 180 degrees and invert. We're still not back to where we started, so we rotate again, invert again, and we did get back to our starting position. What you will notice though is that the sequence from one to five is different. And this difference in sequence will lead to distinct symmetry symbols as we'll soon see. We're now in a position to describe Roto-Reflection and Roto-Inversion in a formal sense. To do this we're going to use some of the same symbols that we've met already. In other words, the representation of the chiral objects. To understand this we have to take a cone. The cone can point up, which is given a positive symbol, or it can point down, which is given a negative symbol. Because reflection is involved, we'll create a chiral pair. And for this example, we'll use rotation of -180 degrees, in other words, a two-fold rotation. First we begin, by that rotation, we move the cone to the other side of the central axis. We then reflect, rotate, reflect again, and we get back to where we started. But what you'll notice is we have a cone pointing up and a cone pointing down. So we have objects which are chiral objects. So we have an open circle and a circle with a comma. But we also have a sense of direction, the positive or negative sign. In carrying out the rotation and reflection, we've created the inversion center, which is shown by the open circle in the projection. The crystallographic symbol for the rotation center is bar one. Now we move to the next step. In other words, we create roto inversion. We start again, by looking at a cone pointing upwards, we rotate and invert. Now we have the cone pointing down. We carry out the operation again, through rotation and inversion, and now we've created two cones, but they're not separated from each other, one pointing up, one pointing down. And you'll notice that the symbol now involves one circle, cut in half. 1/2 has a comma, 1/2 is absent, so we have the chiral pair. And there is still a positive and negative sense. The symbol roto-inversion is 2/m. Now of course, there are great many combinations of rotation and inversion which can be used. It's not the purpose of this course to enumerate all the combinations of rotation and inversion and reflection which are possible. I simply list a few of these in the bottom of the slide. I'm sure after this lecture you'll be much relieved to know that there's not more to be taught about symmetry operators. We've learned that in plane symmetry there is reflection, there is rotation and glide. There is also translation. These symmetry operators are captured within the ten point groups, the five Bravais lattices, and the 17 plane groups. Once we moved to three dimensions, then we increase the number of point groups from 10 to 32. There are also 14 Bravais lattices, and these will be discussed in detail in the next lecture. We will also learn that all of the three dimensional symmetry operations can be combined to create 230 space groups.