[MUSIC] Now it's time to bring together everything we've learned about plane symmetry and collect this in a very formal way. In this lecture, you will gain practice in reading the Plane Group Tables. You'll also learn again how to specify the general and the special positions. And for this exercise we'll look at both hexagonal and rectangular lattices. We will also consider again the derivation of the fractional coordinates or the positions of objects that are generated through the action of the symmetry operators. We come again to the primitive and centered cells, but now we look at the way in which these are represented in the plane groups tables and how the plane groups and how the centering condition is specified. And finally, we collate and examine the 17 plane group symbols. There are only 17 of these, and these are sometimes also known as the wallpaper symbols. Now, let's see how this information is presented in the Plane Group Tables of the International Tables of Crystallography. In the top left hand part of the information, you can see the lattice type. It's a rectangular lattice. On the right hand side, you have the plane group symbol p2mm. The lower case p indicating a primitive cell. The 2 indicating the operation of the diad and the two mirror planes, vertical and horizontal. This part of the symbol is reflecting the point group symmetry of this particular plane group. We can also look at the two diagrams. The diagram on the left is the arrangement of symmetry elements in a unit cell. The general position diagram shows the arrangement of each of the objects in the unit cell. The origin of the unit cell is also specified. Up to this point, we've always said that the unit cell can be placed anywhere on top of a pattern, provided it is the same size and shape. But now we're considering these symmetries more mathematically, the origin is specified. And in general, the origin will always be placed where there is some form of rotation. So in this case, the origin is at the rotation point which has the point symmetry 2mm. We can also specify where the asymmetric unit lies. Here, what's shown is that the asymmetric unit has to lie from zero to a half in x and zero to a half in y. Now let's return to the question of centering. For this we'll examine two of the plane groups. Number three was symbol pm and number five with symbol cm. You'll notice that if we start to look at the pm, plane group, there are two general positions inside the unit cell and they are chiral positions and they occur at the coordinates x, y and bar x, y. And when you count across you can see the multiplicity must be two. However, if you look at the other example, plane group cm, you can see that if you count up the number of objects inside of the unit cell, it is full. But then, when you look at the coordinates of the general position, you'll see only two of them. Position x, y and position minus x, y. What's happened to the other two coordinates? In this case, the information is contained at the top of the coordinate set where it shows you the centering condition. What that is telling you, is that at every position naught, naught, the origin, there is an identical position at the center, at a half, a half. So the total number of positions is four and it would be position one, x, y, together with position three. x plus a half, and y plus a half. Position two would be position minus x, y and the centering condition produces minus x plus y and y plus a half. To get this centering condition we've used a glide line. And remember the glide is a combination of mirroring and translation. So if we begin with the general position in the top left-hand corner, we reflect across the glide line, translate by half the unit cell. We then carry out that operation again to return to the point where we began. We should now come back to the question of multiplicity and object coordinates. Up to now we've only looked at rectangular and square lattices. Let's now take an example from a hexagonal lattice. In this case we will use the plane group p3. So a primitive cell with threefold rotation. The threefold rotation point would be at the origin. And we can show here four of the unit cells with one of those three-fold rotation points illustrated through the blue triangle. If we have a general position, xy, we generate two additional positions. In other words, the multiplicity of a general position will be three. Once we have done that do we find any other threefold rotation points? The answer is yes. We can also find a rotation point as shown here, with the three objects lying on the blue circle. There is also an additional threefold rotation point, as shown with the objects lying on the orange circle. So in that hexagonal lattice, through the imposition of a single threefold rotation point. we actually generate an additional two rotation points inside the unit cell. We can also generate special positions. To do this, we have to place the object on top of the symmetry operation, the three-fold rotation point. If we put a red circle on top of the threefold rotation at the origin of the unit cell, we generate no additional objects. In other words, the multiplicity is one. We can do the same thing by placing a green and a blue circle on the other threefold rotation points. All of these maintain the multiplicity of one. The asymmetric unit is shown in outline. As you can see, the assymetric unit again contains only one general position from which the other two positions can be generated. Now, let's look at how p3 symmetry is represented in the plane group tables. There are to begin the two diagrams, the symmetry element diagram, and the general position diagram. The symmetry elements are all threefold rotation points. In the general position diagram, we can see that there are three objects. Therefore, the multiplicity is 3. The position of those objects, or their coordinates, is also given. Because these objects do not lie on top of any symmetry operation, the site symmetry is 1. If, however, we place an object on any of threefold rotation points, we create a site with multiplicity of only one and the positions and fractional coordinates as shown. You'll notice that the location of the special site positions in this case is completely fixed. They have to be at naught, naught, a third or two-thirds, two-thirds or one-third. We can now assign the Wyckoff Letters. In terms of site symmetry, we are showing what the rotation points are. The rotation is always coming out of the page and it's always the first point in terms of the site symmetry description. As there are no mirror lines or glide lines, we leave the other two positions, indicated by the full stops, blank. So now, let's consolidate everything that we need to know about the 17 Plane Groups. All of them must conform to one of the five bravais lattices. They can be oblique lattices, rectangular, square or hexagonal lattices. And in the case of the rectangular lattice, it may be primitive or centered. The point group for each of these lattices is also specified. We can also describe the 17 plane groups in terms of the so called full symbol, or short symbol. The short symbol is what we have been using so far. The first part of the short symbol describes whether the cell is primitive or centered, as always represented by a lowercase p or c. In addition, the symbol indicates the point group symmetry. Where there is a glide plane which incorporates the mirror, the m is sometimes replaced by a g. What have we learned in this lecture? We now know that this position of the symmetry operators inside of the unit cell determine the size of the asymmetric unit. Recall that the asymmetric unit contains only one object. And it is the operators which create the rest of the pattern on that single object. We also known that special positions appear when the object lies directly on top of a symmetry operator. In this process, it changes multiplicity. A general position lies away from any symmetry operators, and consequently has the highest multiplicity. But the special position is less abundant. They are less abundant because some of the operators have no effect. Consequently, they have lower multiplicity. We now understand the difference between centered and primitive unit cells. In a primitive cell, we have objects dispersed around the origin, but there is no similar dispersion around the center of the unit cell. In a centered cell, the exact opposite occurs. The same disposition of objects occurs both at the center and the origin. And finally, we must remember that there are only 17 Plane Groups. These are also known as the wallpaper groups. And with these 17 Plane Groups, it's possible to create all of the patterns which are known in plane symmetry.
