WEBVTT 00:00:06.610 --> 00:00:10.870 PROFESSOR: OK, let us resume. 00:00:10.870 --> 00:00:15.040 I had no idea how many people would be here today, and I 00:00:15.040 --> 00:00:18.770 think I made 25 copies of the handout. 00:00:18.770 --> 00:00:22.800 And I see 25 names on the list. 00:00:22.800 --> 00:00:25.490 And that means that two people did not get 00:00:25.490 --> 00:00:26.970 a copy of the syllabus. 00:00:26.970 --> 00:00:28.220 Does anybody need a copy? 00:00:30.760 --> 00:00:33.180 That's strange. 00:00:33.180 --> 00:00:34.430 OK. 00:00:36.280 --> 00:00:36.760 All right. 00:00:36.760 --> 00:00:39.740 We covered some introductory material, and I think we've 00:00:39.740 --> 00:00:42.080 covered enough that you can do a problem set. 00:00:42.080 --> 00:00:44.270 So it gives me great pleasure to hand out 00:00:44.270 --> 00:00:45.670 problem set number one. 00:01:00.570 --> 00:01:03.330 OK, you can think about that. 00:01:03.330 --> 00:01:05.570 It is the sort of problem that will either take you two 00:01:05.570 --> 00:01:09.690 minutes or two hours or infinity. 00:01:09.690 --> 00:01:14.310 So don't spend too much time on it, but I would like you to 00:01:14.310 --> 00:01:17.290 put your name on it and turn it in either at the end of the 00:01:17.290 --> 00:01:20.960 hour or next time so I can make comments if there's 00:01:20.960 --> 00:01:22.970 something that's mostly right but not quite right. 00:01:33.880 --> 00:01:40.430 Let's return to these three simple patterns that we put on 00:01:40.430 --> 00:01:41.450 the blackboard. 00:01:41.450 --> 00:01:44.700 And let me make another point about symmetry. 00:01:47.720 --> 00:01:53.430 The people who sensed that this pattern and the one on 00:01:53.430 --> 00:01:57.960 the bottom were the same because they had the same 00:01:57.960 --> 00:02:01.070 motif in them, that they had the same rectangle with one 00:02:01.070 --> 00:02:02.530 concave side. 00:02:02.530 --> 00:02:08.830 And I drew a mirror line in here because that locus, when 00:02:08.830 --> 00:02:12.590 I review this is a reflection from left to right, left the 00:02:12.590 --> 00:02:18.720 motif, as well as the entire pattern, unchanged upon making 00:02:18.720 --> 00:02:20.610 that transformation. 00:02:20.610 --> 00:02:25.860 Is there not also a mirror line there? 00:02:25.860 --> 00:02:27.760 Worked for this motif. 00:02:27.760 --> 00:02:30.370 Why not for this motif? 00:02:30.370 --> 00:02:35.710 Well, the answer is no, that this is not a mirror line 00:02:35.710 --> 00:02:39.550 because the symmetry transformations acts on 00:02:39.550 --> 00:02:43.850 everything, and not just one little bit of space. 00:02:43.850 --> 00:02:46.980 And if I would take this chain of objects that's 00:02:46.980 --> 00:02:49.880 translationally periodic with a translation running this 00:02:49.880 --> 00:02:53.970 way, and I reflected that, I should have another chain 00:02:53.970 --> 00:02:55.930 running like this. 00:02:55.930 --> 00:02:59.560 So the direction of the translation vector is not left 00:02:59.560 --> 00:03:01.920 invariant by this reflection. 00:03:01.920 --> 00:03:08.040 So the conclusion here, and it's a subtle one, matter of 00:03:08.040 --> 00:03:12.580 definition, almost, is that the transformation, the 00:03:12.580 --> 00:03:15.260 symmetry transformation, if it's to be a symmetry 00:03:15.260 --> 00:03:35.170 transformation, acts on all of the space, and not just on one 00:03:35.170 --> 00:03:37.940 local domain. 00:03:37.940 --> 00:03:43.930 So let me give you an example of a pattern that doesn't 00:03:43.930 --> 00:03:45.960 involve translational periodicity. 00:03:45.960 --> 00:03:51.630 So let me try to make a star as carefully as I can. 00:03:55.320 --> 00:03:57.740 What sort of symmetric does that have, or would it have 00:03:57.740 --> 00:03:59.010 had I drawn it more perfectly? 00:03:59.010 --> 00:04:02.900 Well, that would be a five-fold rotation axis in the 00:04:02.900 --> 00:04:05.180 middle because I could rotate through one 00:04:05.180 --> 00:04:06.430 fifth of the circle. 00:04:08.580 --> 00:04:12.960 And any of those rotations twice or three times, or just 00:04:12.960 --> 00:04:16.959 2 pi over five, would be something that maps the 00:04:16.959 --> 00:04:19.420 pattern into congruence with itself. 00:04:19.420 --> 00:04:23.720 There are also mirror lines that go from one tip of the 00:04:23.720 --> 00:04:26.640 star to the other end. 00:04:26.640 --> 00:04:30.280 So that is an example of a pattern, non-periodic, but one 00:04:30.280 --> 00:04:34.410 that has five-fold rotational symmetry and mirror symmetry. 00:04:34.410 --> 00:04:40.040 Now, if I put that star in a box and ask, what is the 00:04:40.040 --> 00:04:42.420 symmetry of that space? 00:04:42.420 --> 00:04:45.520 There's only one operation which is common to the star 00:04:45.520 --> 00:04:47.990 and to the enclosing rectangle, and that's this 00:04:47.990 --> 00:04:49.240 mirror plane. 00:04:52.640 --> 00:04:56.880 So the symmetry acts not just on one little part of the 00:04:56.880 --> 00:05:00.580 space, but it has to leave everything invariant. 00:05:00.580 --> 00:05:03.990 So in that sense, going to this pattern here, this is not 00:05:03.990 --> 00:05:06.800 a mirror plane because it doesn't leave the entire 00:05:06.800 --> 00:05:08.760 pattern invariant. 00:05:08.760 --> 00:05:11.690 That plane would reflect this one up to here, and we don't 00:05:11.690 --> 00:05:12.830 have anything there. 00:05:12.830 --> 00:05:14.620 So the space is not left invariant. 00:05:23.340 --> 00:05:25.510 One further definition. 00:05:25.510 --> 00:05:29.960 We defined what we mean when we say a space or an object 00:05:29.960 --> 00:05:31.220 has symmetry. 00:05:31.220 --> 00:05:35.120 We said an object or a space possesses symmetry when there 00:05:35.120 --> 00:05:38.540 is an operation, or set of operations, that maps the 00:05:38.540 --> 00:05:41.900 space or the object into congruence with itself. 00:05:41.900 --> 00:05:44.950 Let me make another definition, and that a 00:05:44.950 --> 00:05:46.200 symmetry element-- 00:05:54.550 --> 00:05:56.060 another bit of terminology-- 00:05:56.060 --> 00:05:59.440 is the locus of points that's left unmoved by the operation. 00:06:07.090 --> 00:06:08.620 Left unmoved or left invariant. 00:06:24.820 --> 00:06:32.670 So for some specific examples, this vertical line is a locus 00:06:32.670 --> 00:06:36.760 which is left invariant by either the five-fold rotation 00:06:36.760 --> 00:06:46.900 or any of the mirror planes passing through the points 00:06:46.900 --> 00:06:49.460 that would be true of the star. 00:06:49.460 --> 00:06:53.060 So for the star, these are all symmetry elements. 00:06:53.060 --> 00:06:57.340 For the net combination of the star and the enclosing 00:06:57.340 --> 00:07:01.630 rectangle, the only thing that leaves a space 00:07:01.630 --> 00:07:03.050 invariant is this line. 00:07:03.050 --> 00:07:05.670 The locus that's left and moved is this line, so we 00:07:05.670 --> 00:07:07.350 refer to that as a mirror plane. 00:07:12.560 --> 00:07:20.680 Now, these may be seeming kind of definitions. 00:07:20.680 --> 00:07:23.230 Nice to have, but what use are they? 00:07:23.230 --> 00:07:26.100 We will use some of these definitions to answer a 00:07:26.100 --> 00:07:29.110 question which may seem tricky. 00:07:29.110 --> 00:07:32.780 If we do a couple of things in sequence, for example, what is 00:07:32.780 --> 00:07:36.360 the net consequence of doing, let's say, a rotation combined 00:07:36.360 --> 00:07:38.270 with a reflection? 00:07:38.270 --> 00:07:40.480 You can answer that question by saying, what 00:07:40.480 --> 00:07:42.720 has been left unmoved? 00:07:42.720 --> 00:07:46.220 And that is the locuses of whatever net transformation 00:07:46.220 --> 00:07:48.720 results from a combination of two or more. 00:07:51.710 --> 00:07:55.510 So that, again, is abstract, but we'll use that later on. 00:07:55.510 --> 00:07:57.420 Any question on this? 00:07:57.420 --> 00:08:00.370 Let me summarize very quickly what we have found for two 00:08:00.370 --> 00:08:02.550 dimensions. 00:08:02.550 --> 00:08:05.350 We found that there are, in a two-dimensional space, three 00:08:05.350 --> 00:08:07.240 kinds of operations. 00:08:07.240 --> 00:08:14.720 There is the operation of translation, which we'll call 00:08:14.720 --> 00:08:16.890 by the vector, T, corresponding to that 00:08:16.890 --> 00:08:18.310 transformation. 00:08:18.310 --> 00:08:20.990 And there is an operation of reflection. 00:08:23.790 --> 00:08:26.880 And the locus of the plane, in which the reflection occurs, 00:08:26.880 --> 00:08:28.730 we'll call a mirror plane. 00:08:28.730 --> 00:08:30.260 And that's a linear locus. 00:08:30.260 --> 00:08:33.650 And then, we've seen in these two patterns here an operation 00:08:33.650 --> 00:08:35.309 of rotation. 00:08:35.309 --> 00:08:37.990 In particular, in these two-dimensional patterns, we 00:08:37.990 --> 00:08:41.027 saw the rotation operation A pi. 00:08:44.230 --> 00:08:50.880 Now, I would put forth for your consideration something 00:08:50.880 --> 00:08:54.330 that is a profound conclusion. 00:08:54.330 --> 00:08:58.850 These are the only single-step transformations that can exist 00:08:58.850 --> 00:09:02.090 in a two-dimensional space. 00:09:02.090 --> 00:09:04.850 These are the only ones that can exist as single-step 00:09:04.850 --> 00:09:07.720 operations. 00:09:07.720 --> 00:09:11.400 We can view these as operations that result in a 00:09:11.400 --> 00:09:15.280 transformation of coordinates. 00:09:15.280 --> 00:09:19.910 And in two dimensions, if we have some position, x, y, in 00:09:19.910 --> 00:09:23.160 the space, what the transformation of a 00:09:23.160 --> 00:09:29.310 translation does is to take x and add a constant to it. 00:09:29.310 --> 00:09:34.680 It takes y and adds, perhaps, a different constant to it. 00:09:34.680 --> 00:09:38.680 Do the operation a second time, and we'll go to x plus 00:09:38.680 --> 00:09:40.740 2a and y plus 2b. 00:09:44.480 --> 00:09:50.320 So analytically, we can look at these symmetry operations 00:09:50.320 --> 00:09:54.210 in terms of the transformation of a representative 00:09:54.210 --> 00:09:55.840 coordinate. 00:09:55.840 --> 00:09:59.140 If we have a reflection plane, and let's set up a coordinate 00:09:59.140 --> 00:10:03.820 system where this is y and this is x. 00:10:03.820 --> 00:10:08.730 We have an object here at the location x, y, and we reflect 00:10:08.730 --> 00:10:11.180 it across this locus. 00:10:11.180 --> 00:10:17.640 It goes to minus x, y, if the mirror line runs through the 00:10:17.640 --> 00:10:20.260 origin and is perpendicular to x. 00:10:20.260 --> 00:10:23.850 So one example of a transformation by reflection 00:10:23.850 --> 00:10:29.340 is that x, y goes to minus x, y. 00:10:29.340 --> 00:10:33.420 And this is a case where the mirror plane is perpendicular 00:10:33.420 --> 00:10:36.340 to x and passes through the origin. 00:10:45.470 --> 00:10:49.890 If we do the operation a second time, minus x, y would 00:10:49.890 --> 00:10:53.170 get mapped back into x, y again. 00:10:53.170 --> 00:10:56.090 It comes back to where it started from. 00:10:56.090 --> 00:10:57.740 So this would be the first reflection. 00:11:00.540 --> 00:11:02.155 This would be the second reflection. 00:11:08.410 --> 00:11:11.960 And we saw in the patterns an example of one other 00:11:11.960 --> 00:11:14.150 transformation. 00:11:14.150 --> 00:11:17.970 Let's suppose there was an operation, A pi, at the 00:11:17.970 --> 00:11:21.850 origin, and this was x, and this was y. 00:11:21.850 --> 00:11:25.090 We started out with a motif here at x, y. 00:11:25.090 --> 00:11:29.620 If we rotated that by 180 degrees, it would go down to a 00:11:29.620 --> 00:11:36.040 location minus x, minus y. 00:11:36.040 --> 00:11:42.780 So the operation of a 180-degree rotation is going 00:11:42.780 --> 00:11:46.180 to analytically correspond to a transformation of going from 00:11:46.180 --> 00:11:49.890 x, y to minus x, minus y. 00:11:49.890 --> 00:11:53.170 If we perform it again, it would go back to x, y. 00:11:55.890 --> 00:11:59.070 OK, let's look at this in more general terms. 00:11:59.070 --> 00:12:01.950 In a two-dimensional space, we've got two dimensions to 00:12:01.950 --> 00:12:03.730 diddle with. 00:12:03.730 --> 00:12:06.520 We can change the sense of no coordinate. 00:12:06.520 --> 00:12:08.190 That's translation. 00:12:08.190 --> 00:12:11.280 We can change the sense of one coordinate. 00:12:11.280 --> 00:12:13.280 That's going to be reflection. 00:12:13.280 --> 00:12:16.370 We can change the sense of both coordinates. 00:12:16.370 --> 00:12:19.740 That's going to be a rotation. 00:12:19.740 --> 00:12:21.070 That's all we can do. 00:12:21.070 --> 00:12:23.900 So these are the three basic operations in a 00:12:23.900 --> 00:12:25.150 two-dimensional space. 00:12:29.060 --> 00:12:34.870 I gave you special cases to make things easy, but 00:12:34.870 --> 00:12:38.900 regardless of where the mirror plane is, parallel to or 00:12:38.900 --> 00:12:42.160 perpendicular to an axis or not, and whether it passes 00:12:42.160 --> 00:12:46.930 through the origin or not, a mirror plane has the operation 00:12:46.930 --> 00:12:50.570 of reversing the sense of one direction. 00:12:50.570 --> 00:12:53.320 Just the sense of one direction that is reversed. 00:12:53.320 --> 00:12:56.450 And the rotation, be it a rotation through 60 degrees or 00:12:56.450 --> 00:13:00.700 90 degrees or 180 degrees, is always taking both 00:13:00.700 --> 00:13:01.470 coordinates. 00:13:01.470 --> 00:13:04.590 It's making a transformation of both coordinates. 00:13:04.590 --> 00:13:06.770 If you only have two coordinates with which to 00:13:06.770 --> 00:13:08.750 play, that's all you can do. 00:13:12.760 --> 00:13:15.120 Let's do some giant extrapolations. 00:13:15.120 --> 00:13:18.510 If we have a strictly one-dimensional pattern where 00:13:18.510 --> 00:13:21.560 there's x and nothing else, than they're only going to be 00:13:21.560 --> 00:13:22.860 two coordinates. 00:13:22.860 --> 00:13:25.920 And there are going to be only two ways we can transform 00:13:25.920 --> 00:13:26.520 coordinates. 00:13:26.520 --> 00:13:36.570 So in a one-dimensional space, we can change the sense of no 00:13:36.570 --> 00:13:43.410 coordinate, and that's going to be the operation of 00:13:43.410 --> 00:13:52.170 translation, or we can change the sense of one coordinate, 00:13:52.170 --> 00:13:56.220 and that's going to be the operation of reflection. 00:13:56.220 --> 00:13:58.460 No rotation in a one-dimensional space. 00:14:01.960 --> 00:14:03.930 Now, let's extrapolate in the other direction. 00:14:03.930 --> 00:14:06.750 In a three-dimensional space, the sort that we're going to 00:14:06.750 --> 00:14:10.190 be concerned with when we want to describe the symmetry of 00:14:10.190 --> 00:14:16.220 real crystals, you've got three coordinates to permute. 00:14:16.220 --> 00:14:22.820 So it follows then, without saying what they are, in 3D, 00:14:22.820 --> 00:14:29.715 there are going to be four distinct one-step operations. 00:14:40.880 --> 00:14:43.086 And then five dimensions? 00:14:43.086 --> 00:14:45.060 Hey, that's a nice thing about mathematics. 00:14:45.060 --> 00:14:47.030 You could play any game you like. 00:14:47.030 --> 00:14:49.680 Not only that, but you make up the rules. 00:14:49.680 --> 00:14:51.900 In a five-dimensional space, there's going to be six 00:14:51.900 --> 00:14:54.050 transformations. 00:14:54.050 --> 00:14:56.850 Would we ever want to worry about five-dimensional 00:14:56.850 --> 00:14:59.090 crystallography? 00:14:59.090 --> 00:15:02.850 Well, let me hang out a teaser and not answer the question. 00:15:02.850 --> 00:15:06.980 Yeah, there are crystals for which as many as 00:15:06.980 --> 00:15:12.200 six-dimensional symmetries are necessary. 00:15:12.200 --> 00:15:12.870 Wow. 00:15:12.870 --> 00:15:14.130 Doesn't that blow the mind? 00:15:14.130 --> 00:15:18.190 We'll return to that, and I'll explain why later on. 00:15:24.860 --> 00:15:27.770 Another thing you might ask, why did I sneak this in? 00:15:27.770 --> 00:15:31.150 Why did I say one-step operation? 00:15:31.150 --> 00:15:33.190 Well, it's something we should worry about, and 00:15:33.190 --> 00:15:34.750 unfortunately, we will. 00:15:34.750 --> 00:15:39.880 What if you take a motif, translate it, rotate it around 00:15:39.880 --> 00:15:43.550 a couple of times, reflect it, bounce it up and down three 00:15:43.550 --> 00:15:46.880 times, and then put it down? 00:15:46.880 --> 00:15:50.780 How do you get from the first motif to the final one there? 00:15:50.780 --> 00:15:54.520 Is there an infinite number of operations? 00:15:54.520 --> 00:15:55.650 Mercifully, no. 00:15:55.650 --> 00:15:59.570 The number is small and very finite. 00:15:59.570 --> 00:16:03.460 And we will systematically, in another week's time, examine 00:16:03.460 --> 00:16:06.470 specifically two-step operations. 00:16:06.470 --> 00:16:10.630 And as with many things that we'll encounter, we might not 00:16:10.630 --> 00:16:12.610 be clever enough to think them up. 00:16:12.610 --> 00:16:15.180 But when we start putting things together into a 00:16:15.180 --> 00:16:17.980 synthesis, suddenly we're going to stumble over 00:16:17.980 --> 00:16:21.440 something we don't know how to explain, and we will have 00:16:21.440 --> 00:16:25.690 arrived, like it or not, at a new feature which we perhaps 00:16:25.690 --> 00:16:26.940 hadn't anticipated. 00:16:29.430 --> 00:16:31.080 OK, any question at this point? 00:16:35.370 --> 00:16:38.730 All this has been in a way of general introduction. 00:16:38.730 --> 00:16:45.260 We're going to now take things more slowly and proceed one 00:16:45.260 --> 00:16:46.510 step at a time. 00:16:57.570 --> 00:17:06.790 I would like to confine our attention for the moment on 00:17:06.790 --> 00:17:10.920 two-dimensional symmetries and examine the sorts of patterns 00:17:10.920 --> 00:17:13.990 that can exist in two dimensions, fabric patterns, 00:17:13.990 --> 00:17:17.900 floor tile, grillwork, and so on. 00:17:17.900 --> 00:17:20.880 And we've seen that, basically, there seem to be 00:17:20.880 --> 00:17:24.150 three operations, three kinds of operations, translation, 00:17:24.150 --> 00:17:26.750 reflection, and rotation. 00:17:26.750 --> 00:17:30.330 That's an infinite number of operations because we are not 00:17:30.330 --> 00:17:34.140 specifying whether or not the rotation angle is restricted 00:17:34.140 --> 00:17:35.940 to any particular value. 00:17:35.940 --> 00:17:38.290 No reason why it should be. 00:17:38.290 --> 00:17:40.155 There are lots of rotational symmetries that 00:17:40.155 --> 00:17:42.590 are absolutely lovely. 00:17:42.590 --> 00:17:44.580 But let's build things up. 00:17:44.580 --> 00:17:48.190 And I would like to first look at the operation of 00:17:48.190 --> 00:17:53.410 translation, which we've said a great deal 00:17:53.410 --> 00:17:56.430 about to this point. 00:17:56.430 --> 00:18:00.090 Translation has magnitude. 00:18:00.090 --> 00:18:01.060 It has direction. 00:18:01.060 --> 00:18:02.590 So it acts like a vector. 00:18:02.590 --> 00:18:06.650 But just like a vector, it has no unique origin. 00:18:06.650 --> 00:18:11.660 Perform the operation twice, and you have a position that 00:18:11.660 --> 00:18:14.740 is two translations removed from the origin. 00:18:14.740 --> 00:18:17.220 Do it three times, you have a translation 00:18:17.220 --> 00:18:21.090 that's three times out. 00:18:21.090 --> 00:18:25.310 If a motif sits here, the motif must sit at the end of 00:18:25.310 --> 00:18:27.830 this translation in the same 00:18:27.830 --> 00:18:30.190 orientation parallel to itself. 00:18:30.190 --> 00:18:33.200 It must exist at the end of two translations. 00:18:33.200 --> 00:18:37.750 And if the operation acts on all of the space, if we say a 00:18:37.750 --> 00:18:42.130 translation is present, we really imply that there's an 00:18:42.130 --> 00:18:44.710 infinite row going to plus infinity and 00:18:44.710 --> 00:18:46.490 back to minus infinity. 00:18:46.490 --> 00:18:49.570 And there is a motif hanging at the terminal 00:18:49.570 --> 00:18:51.135 point of every vector. 00:18:56.320 --> 00:19:03.350 Now, we can summarize this periodicity with 00:19:03.350 --> 00:19:06.050 a convenient device. 00:19:06.050 --> 00:19:11.940 Let's take some fiducial point and summarize the 00:19:11.940 --> 00:19:18.150 translational periodicity by saying that something that is 00:19:18.150 --> 00:19:28.580 hung at one point, either here, or maybe hung also off 00:19:28.580 --> 00:19:32.360 in some other direction relative to the translation, 00:19:32.360 --> 00:19:36.840 that something hung on one of these points is automatically 00:19:36.840 --> 00:19:39.910 reproduced for us at every point. 00:19:39.910 --> 00:19:43.510 So what we have done through this device is defined 00:19:43.510 --> 00:19:45.810 something that is called a lattice point. 00:19:48.940 --> 00:19:54.480 And this is an abstraction of the translational periodicity. 00:19:54.480 --> 00:19:58.690 There is an array of points, geometric fictions, which we 00:19:58.690 --> 00:20:01.150 have constructed. 00:20:01.150 --> 00:20:04.730 And we ascribe to this geometric fiction the property 00:20:04.730 --> 00:20:09.190 that anything hung at one of these points, be it a benzene 00:20:09.190 --> 00:20:13.910 ring or be it a Santa Claus on Christmas wrapping paper, is 00:20:13.910 --> 00:20:17.960 understood to be automatically reproduced at every other one 00:20:17.960 --> 00:20:21.240 of these points. 00:20:21.240 --> 00:20:27.720 It is this array of fictitious points that is the proper 00:20:27.720 --> 00:20:33.110 designation of what we refer to as a lattice. 00:20:33.110 --> 00:20:37.460 So a lattice is an array of fictitious points that 00:20:37.460 --> 00:20:41.990 summarizes the translational periodicity of the crystal. 00:20:41.990 --> 00:20:45.150 It has a property to repeat that something hung at a 00:20:45.150 --> 00:20:48.430 particular disposition relative to that point and 00:20:48.430 --> 00:20:52.590 with a particular orientation is understood to be hung at 00:20:52.590 --> 00:20:56.750 every other lattice point in exactly the same way. 00:20:56.750 --> 00:20:58.040 So that is a lattice. 00:20:58.040 --> 00:21:01.100 And this is one of the most abused terms in 00:21:01.100 --> 00:21:02.880 crystallography. 00:21:02.880 --> 00:21:06.290 We talk about the sodium chloride lattice. 00:21:06.290 --> 00:21:10.970 The sodium chloride lattice is a set of points that are 00:21:10.970 --> 00:21:14.740 placed at the corners of a cube and in the 00:21:14.740 --> 00:21:16.820 middle of all the faces. 00:21:16.820 --> 00:21:18.460 This is the NaCl lattice. 00:21:21.400 --> 00:21:26.420 If I choose to decorate that lattice with one sodium and 00:21:26.420 --> 00:21:29.180 one chlorine, then I have atoms sitting at 00:21:29.180 --> 00:21:31.240 these lattice points. 00:21:31.240 --> 00:21:34.810 And that is the sodium chloride structure. 00:21:34.810 --> 00:21:38.540 That is the proper term for the atomic configuration. 00:21:41.430 --> 00:21:45.170 So lattice is a geometrical term, and it's an abstraction. 00:21:45.170 --> 00:21:49.320 Structure is the actual atomic arrangement. 00:21:49.320 --> 00:21:53.990 Now, since I realize already that I am among friends, I can 00:21:53.990 --> 00:21:57.580 confess that I very often recklessly 00:21:57.580 --> 00:21:58.820 abuse the term lattice. 00:21:58.820 --> 00:22:03.140 If I talk about lattice energy, lattice diffusion, 00:22:03.140 --> 00:22:07.140 lattice vibration, I'm not talking about abstract points 00:22:07.140 --> 00:22:11.380 bobbling around or something going through this array of 00:22:11.380 --> 00:22:12.710 little points. 00:22:12.710 --> 00:22:16.060 I mean, I should talk about structure diffusion, structure 00:22:16.060 --> 00:22:18.490 energy, structure vibration. 00:22:18.490 --> 00:22:22.190 But man, that just doesn't have the established 00:22:22.190 --> 00:22:26.000 terminology, and it doesn't have the zing and music of 00:22:26.000 --> 00:22:28.060 something like lattice vibrations. 00:22:28.060 --> 00:22:30.240 So I do it all the time. 00:22:30.240 --> 00:22:33.740 Don't tell anybody else that I said this to you, frankly. 00:22:33.740 --> 00:22:36.550 But it's never going to be stamped out. 00:22:36.550 --> 00:22:41.450 But now you perhaps are informed enough to at least 00:22:41.450 --> 00:22:44.750 blush slightly when you talk about lattice energy or 00:22:44.750 --> 00:22:48.330 lattice diffusion, realizing you're using the term 00:22:48.330 --> 00:22:50.990 incorrectly and that you know better, but everybody else 00:22:50.990 --> 00:22:53.430 does it, so you do the same thing. 00:22:53.430 --> 00:22:55.210 So that is the definition of lattice. 00:22:58.520 --> 00:23:04.980 Now, suppose I take this space, to which I've added a 00:23:04.980 --> 00:23:13.630 first translation, and I'll call it T1, implying that I'm 00:23:13.630 --> 00:23:16.030 going to add something else to this space, 00:23:16.030 --> 00:23:17.870 which I'm free to do. 00:23:17.870 --> 00:23:21.590 I can put in a second translational periodicity 00:23:21.590 --> 00:23:24.260 because this is a two-dimensional space. 00:23:24.260 --> 00:23:25.250 How do I do this? 00:23:25.250 --> 00:23:29.590 And the answer is very carefully because the second 00:23:29.590 --> 00:23:34.400 translation could not go in the space parallel to the 00:23:34.400 --> 00:23:38.890 first one if I put in a second translation, T2, which is 00:23:38.890 --> 00:23:42.301 totally incommensurate with T1. 00:23:42.301 --> 00:23:44.890 The things blow up in my face. 00:23:44.890 --> 00:23:46.550 I don't have a lattice. 00:23:46.550 --> 00:23:49.540 I will get lattice points all over the place. 00:23:49.540 --> 00:23:51.100 So this is impossible. 00:23:51.100 --> 00:23:59.125 So if T1 is not equal to T2, this space self destructs. 00:24:02.090 --> 00:24:07.070 If T1 is a multiple of T2, then if I say a translation 00:24:07.070 --> 00:24:12.380 exists of length T1, and I add a second translation twice T1. 00:24:12.380 --> 00:24:13.990 I've already got those lattice points. 00:24:13.990 --> 00:24:15.880 And that's nothing new. 00:24:15.880 --> 00:24:18.400 So if I want to say there's a second translational 00:24:18.400 --> 00:24:22.820 periodicity in the space, the only thing I can do is pick a 00:24:22.820 --> 00:24:27.200 T2 which is not parallel to T1. 00:24:27.200 --> 00:24:30.120 And then this T2 will pick up everything in the space. 00:24:30.120 --> 00:24:33.770 It's going to take these lattice points and generate 00:24:33.770 --> 00:24:35.890 them at equal intervals, T2. 00:24:35.890 --> 00:24:39.360 But for that matter, it acts on everything in the space. 00:24:39.360 --> 00:24:44.350 So we could think of this translation, T2, as moving 00:24:44.350 --> 00:24:48.610 this entire infinite string of lattice points separated by T1 00:24:48.610 --> 00:24:55.330 and giving me a whole string of lattice points. 00:24:57.990 --> 00:25:09.690 So now, having taken two noncollinear translations, 00:25:09.690 --> 00:25:16.100 those translations will imply a two-dimensional space 00:25:16.100 --> 00:25:32.990 lattice in which motifs will be hung at translations nT1 00:25:32.990 --> 00:25:40.390 plus mT2 where n, m are integers that go from minus 00:25:40.390 --> 00:25:42.980 infinity to plus infinity. 00:25:45.660 --> 00:25:48.500 OK, so this is a two-dimensional space lattice, 00:25:48.500 --> 00:25:52.730 or sometimes it's referred to by the term a lattice net. 00:25:55.230 --> 00:25:55.625 Good term. 00:25:55.625 --> 00:25:57.485 It looks like what fishermen throw in the 00:25:57.485 --> 00:25:58.720 water to snag fish. 00:25:58.720 --> 00:26:03.110 So it is a net, in terms of something that we're familiar 00:26:03.110 --> 00:26:04.640 with in everyday life. 00:26:09.260 --> 00:26:09.500 All right. 00:26:09.500 --> 00:26:13.040 So we've specified a space lattice, but it is a highly 00:26:13.040 --> 00:26:15.020 redundant pattern. 00:26:15.020 --> 00:26:19.150 We've got a doubly infinite set of lattice points. 00:26:19.150 --> 00:26:23.750 And the unique nature of the pattern, the structure, is 00:26:23.750 --> 00:26:26.630 going to be whatever is associated 00:26:26.630 --> 00:26:29.660 with one lattice point. 00:26:29.660 --> 00:26:33.050 So if we specify what's going on in the vicinity of one 00:26:33.050 --> 00:26:36.050 lattice point and establish that at every other lattice 00:26:36.050 --> 00:26:38.010 point, we have the entire infinite 00:26:38.010 --> 00:26:40.150 two-dimensional structure. 00:26:40.150 --> 00:26:45.600 So let's ask now, how we can define the area that is unique 00:26:45.600 --> 00:26:46.850 to one lattice point. 00:26:49.380 --> 00:26:51.155 And there are several ways of doing this. 00:26:54.850 --> 00:27:00.120 We can specify T1, and then specify T2. 00:27:05.830 --> 00:27:08.580 We'll repeat T1 up to here. 00:27:08.580 --> 00:27:15.980 T1 will repeat to T2 over to here, and we will have defined 00:27:15.980 --> 00:27:19.330 the area that is uniquely associated 00:27:19.330 --> 00:27:21.540 with one lattice point. 00:27:21.540 --> 00:27:25.190 So if I can tell you what's going on within the confines 00:27:25.190 --> 00:27:31.940 of this parallelogram, then I have given you the unique part 00:27:31.940 --> 00:27:35.470 of what is hung at a lattice point, and which is reproduced 00:27:35.470 --> 00:27:37.630 only by translation. 00:27:37.630 --> 00:27:40.130 And this is a very important construct. 00:27:40.130 --> 00:27:46.100 It is something that is referred to as the unit cell, 00:27:46.100 --> 00:27:48.220 or sometimes just cell for short. 00:27:53.840 --> 00:28:02.840 And now we encounter a curious ambiguity. 00:28:02.840 --> 00:28:07.915 T1 and T2 imply an array of lattice points. 00:28:15.870 --> 00:28:20.480 And this particular choice of T1 and T2 define a cell. 00:28:25.240 --> 00:28:28.270 But the reverse is not true. 00:28:28.270 --> 00:28:29.695 If I give you-- 00:28:36.290 --> 00:28:37.080 and what do I want to say? 00:28:37.080 --> 00:28:43.810 That a particular lattice does not specify 00:28:43.810 --> 00:28:45.960 a unique unit cell. 00:28:45.960 --> 00:28:49.220 Or, stated another way, there are many different choices for 00:28:49.220 --> 00:28:56.170 T1 and T2 that would specify the same unique area. 00:28:56.170 --> 00:29:00.210 I could take this as a T1 prime, and then I would have a 00:29:00.210 --> 00:29:03.380 cell that looks like this. 00:29:03.380 --> 00:29:06.600 And that would also define the area associated with one 00:29:06.600 --> 00:29:07.770 lattice point. 00:29:07.770 --> 00:29:11.600 It's not clear this oblique thing with one very long T1 00:29:11.600 --> 00:29:16.400 prime would have very much to commend it, but there are many 00:29:16.400 --> 00:29:28.780 ways, many choices, for T1 and T2, to find 00:29:28.780 --> 00:29:30.150 exactly the same lattice. 00:29:45.260 --> 00:29:48.950 We could take this as T1, this as T2, same lattice, same 00:29:48.950 --> 00:29:50.010 array of lattice points. 00:29:50.010 --> 00:29:53.060 Take this as T1, this as T2, same array of lattice points. 00:29:53.060 --> 00:29:57.450 Take this as T1, this as T2, that's yet another choice. 00:29:57.450 --> 00:30:01.500 So there are an infinite number of translations. 00:30:01.500 --> 00:30:06.620 Special name for this, to introduce a bit of jargon 00:30:06.620 --> 00:30:09.550 again, these are very often called conjugate translations. 00:30:25.060 --> 00:30:27.920 So all this is still nothing more than simple geometry, but 00:30:27.920 --> 00:30:32.520 if you invent some fancy words, you really have to do 00:30:32.520 --> 00:30:33.840 that to impress your friends. 00:30:33.840 --> 00:30:35.360 Yeah, you had a question here? 00:30:35.360 --> 00:30:36.340 AUDIENCE: Yeah. 00:30:36.340 --> 00:30:38.300 So you can define magnitude for T1 and 00:30:38.300 --> 00:30:39.770 T2, all those constants. 00:30:39.770 --> 00:30:41.975 But you're changing the directions of T1 and T2, and 00:30:41.975 --> 00:30:43.690 you're saying, even though you're changing those 00:30:43.690 --> 00:30:45.100 directions, it's still the same unit cell? 00:30:45.100 --> 00:30:49.580 PROFESSOR: Yeah, provided I have some new translation like 00:30:49.580 --> 00:30:54.630 this one here, which is really this T1 plus this T2, this 00:30:54.630 --> 00:30:57.200 would define a very, very oblique cell 00:30:57.200 --> 00:30:59.140 that looks like this. 00:30:59.140 --> 00:31:04.983 But yet, the terminal points of T1 prime and-- 00:31:04.983 --> 00:31:06.280 I need a term for this. 00:31:06.280 --> 00:31:08.590 I'll call this T2 prime. 00:31:08.590 --> 00:31:11.510 The terminal points here are going to be exactly the same 00:31:11.510 --> 00:31:14.140 as the nodes that are defined here. 00:31:14.140 --> 00:31:19.880 So they are two choices for one in the same lattice. 00:31:19.880 --> 00:31:25.020 OK, so the implication of this is we're going to 00:31:25.020 --> 00:31:26.270 have to have rules. 00:31:29.950 --> 00:31:34.980 And some of these make common sense. 00:31:34.980 --> 00:31:38.540 You could pick, in a two-dimensional lattice, some 00:31:38.540 --> 00:31:44.170 absolutely ridiculous unit cells defined in terms of very 00:31:44.170 --> 00:31:55.080 long vectors that define a cell that is a very, very 00:31:55.080 --> 00:31:56.910 oblique cell. 00:31:56.910 --> 00:31:59.590 So it's the lattice that's defined by 00:31:59.590 --> 00:32:02.230 this translation here. 00:32:02.230 --> 00:32:06.140 And the next translation parallel to this one would go 00:32:06.140 --> 00:32:08.905 way up to something like this. 00:32:13.540 --> 00:32:14.450 So there's a T1. 00:32:14.450 --> 00:32:15.360 There's a T2. 00:32:15.360 --> 00:32:18.070 This crazy cell here works. 00:32:18.070 --> 00:32:19.850 That's the area that's associated 00:32:19.850 --> 00:32:20.780 with one lattice point. 00:32:20.780 --> 00:32:23.860 But clearly, it has absolutely nothing to 00:32:23.860 --> 00:32:25.630 commend this choice. 00:32:25.630 --> 00:32:27.940 There's nothing to be gained by using these long 00:32:27.940 --> 00:32:30.640 translations that make very extreme 00:32:30.640 --> 00:32:33.010 intertranslation angles. 00:32:33.010 --> 00:32:35.030 Your intuition would say, why would you want to 00:32:35.030 --> 00:32:36.290 do that, you dummy? 00:32:36.290 --> 00:32:39.880 Let's take these as the translations, which is 00:32:39.880 --> 00:32:42.620 something I sort of naturally did all along. 00:32:42.620 --> 00:32:43.740 And what are we doing? 00:32:43.740 --> 00:32:46.435 We're picking the shortest translations. 00:32:57.230 --> 00:32:59.330 So there's one very common sense rule. 00:33:04.080 --> 00:33:07.940 Another rule, getting a little bit ahead of the game, but 00:33:07.940 --> 00:33:11.700 suppose I examine the lattice that describes the arrangement 00:33:11.700 --> 00:33:15.400 of four floor tiles. 00:33:15.400 --> 00:33:17.910 If I take a lattice point right at the point of 00:33:17.910 --> 00:33:23.620 intersection of the joins between the tiles, that is a 00:33:23.620 --> 00:33:27.310 cell that is exactly square. 00:33:27.310 --> 00:33:30.420 And it's exactly square because there's a four-fold 00:33:30.420 --> 00:33:35.460 axis in that pattern that leaves things invariant after 00:33:35.460 --> 00:33:39.070 a 90-degree rotation. 00:33:39.070 --> 00:33:42.310 So if that's the nature of a lattice, if it in fact is 00:33:42.310 --> 00:33:45.970 constrained because of the symmetry that is there to have 00:33:45.970 --> 00:33:48.790 two translations identical in length, in fact, 00:33:48.790 --> 00:33:50.170 identical in every way. 00:33:52.760 --> 00:33:56.270 Pick those as the choice of the cell to emphasize that 00:33:56.270 --> 00:33:59.380 special key feature of the lattice. 00:33:59.380 --> 00:34:07.470 So a second row, which is a second and final one, is to 00:34:07.470 --> 00:34:16.100 pick a T1 and T2 that displays the symmetry, 00:34:16.100 --> 00:34:31.334 if any, of the lattice. 00:34:35.670 --> 00:34:38.800 Which introduces us to a feature which we'll elaborate 00:34:38.800 --> 00:34:42.409 much more later on, that the translational periodicity and 00:34:42.409 --> 00:34:44.110 the symmetry of the lattice are two 00:34:44.110 --> 00:34:46.159 things that go together. 00:34:46.159 --> 00:34:49.780 That the fact that there is translational symmetry 00:34:49.780 --> 00:34:53.620 drastically reduces the number of symmetries that you could 00:34:53.620 --> 00:34:57.410 have, the fact that there are symmetries possible for 00:34:57.410 --> 00:34:59.800 presence in a lattice restricts the number of 00:34:59.800 --> 00:35:02.000 different kinds of cells. 00:35:02.000 --> 00:35:04.960 So these are two aspects of the pattern, the symmetry 00:35:04.960 --> 00:35:08.600 that's in it and its periodicity. 00:35:08.600 --> 00:35:13.770 OK, but these are the only two rules that we really need to 00:35:13.770 --> 00:35:15.920 pick what's called the standard cell. 00:35:22.140 --> 00:35:25.660 Take the shortest translations that are available to you, and 00:35:25.660 --> 00:35:31.520 pick translations that display the symmetry that may be 00:35:31.520 --> 00:35:32.770 present in the lattice. 00:35:47.320 --> 00:35:48.670 Any questions or comments? 00:36:10.840 --> 00:36:13.410 Any comments? 00:36:13.410 --> 00:36:21.240 OK, I think I have time for one last major point of 00:36:21.240 --> 00:36:22.460 discussion. 00:36:22.460 --> 00:36:27.220 And what we are going to embark on now is a process of 00:36:27.220 --> 00:36:31.170 synthesis, which will occupy us for a couple of weeks. 00:36:36.966 --> 00:36:43.630 What I'm going to do is start with a translation. 00:36:43.630 --> 00:36:49.180 And this defines an infinite string of lattice points. 00:36:49.180 --> 00:36:54.830 Now, I know that in two dimensions, I have two kinds 00:36:54.830 --> 00:36:58.630 of symmetry operations that are present, either rotation 00:36:58.630 --> 00:37:00.930 or translation. 00:37:00.930 --> 00:37:04.160 So now, I'm going to ask the question, what happens if I 00:37:04.160 --> 00:37:07.610 define a lattice, or at least one translation in a lattice, 00:37:07.610 --> 00:37:14.600 and now I add to that lattice an operation of rotation, OK? 00:37:14.600 --> 00:37:15.050 I can do that. 00:37:15.050 --> 00:37:18.170 We've seen examples of translationally periodic 00:37:18.170 --> 00:37:21.390 patterns that have rotational symmetry. 00:37:21.390 --> 00:37:27.190 So let me suppose I add to this space a rotation 00:37:27.190 --> 00:37:30.730 operation, A alpha. 00:37:30.730 --> 00:37:34.490 And there's no unique origin to the translation. 00:37:34.490 --> 00:37:37.330 There is no unique location for a lattice point. 00:37:37.330 --> 00:37:42.460 So I can put the operation A alpha in at my designated 00:37:42.460 --> 00:37:45.380 lattice point. 00:37:45.380 --> 00:37:50.440 Now, if I do that, all hell breaks loose because now I 00:37:50.440 --> 00:37:53.150 have a rotation operation A alpha. 00:37:53.150 --> 00:37:55.700 This has a translation coming out of it. 00:37:55.700 --> 00:37:59.270 That translation will be repeated up here, an angle 00:37:59.270 --> 00:38:00.520 alpha away. 00:38:00.520 --> 00:38:03.160 A alpha acts on everything, so it's going to take this 00:38:03.160 --> 00:38:09.360 translation and move it over here to a location for another 00:38:09.360 --> 00:38:10.210 translation. 00:38:10.210 --> 00:38:11.250 This is a lattice point. 00:38:11.250 --> 00:38:12.540 This is a lattice point. 00:38:12.540 --> 00:38:16.340 And this business is going to go on until it comes around 00:38:16.340 --> 00:38:19.090 full circle. 00:38:19.090 --> 00:38:22.040 Let me focus my attention on just one of these 00:38:22.040 --> 00:38:26.170 translations, and this will be this one up here, the one that 00:38:26.170 --> 00:38:28.960 is alpha away from the first in a 00:38:28.960 --> 00:38:31.190 counterclockwise direction. 00:38:31.190 --> 00:38:35.080 So here sits another translation, and that means 00:38:35.080 --> 00:38:36.330 this is a lattice point. 00:38:41.220 --> 00:38:43.750 At this end of the translation, the same thing is 00:38:43.750 --> 00:38:44.760 going to happen. 00:38:44.760 --> 00:38:48.970 The operation A alpha is moved to this location at the end of 00:38:48.970 --> 00:38:50.100 the translation. 00:38:50.100 --> 00:38:52.520 That means that anything coming out of this lattice 00:38:52.520 --> 00:38:57.790 point must also be repeated at angular intervals, alpha. 00:38:57.790 --> 00:39:00.550 And now I'm going to focus my attention on 00:39:00.550 --> 00:39:02.900 this translation here. 00:39:02.900 --> 00:39:05.960 And there will be a translation that goes up like 00:39:05.960 --> 00:39:09.140 this, and this is a lattice point. 00:39:14.390 --> 00:39:17.790 And now, in the words of that famous musical, there's big 00:39:17.790 --> 00:39:20.960 trouble in River City. 00:39:20.960 --> 00:39:24.150 Because we started out by saying that everything in the 00:39:24.150 --> 00:39:28.090 space was periodic at an interval, T, a translational 00:39:28.090 --> 00:39:29.320 interval, T. 00:39:29.320 --> 00:39:33.540 This is T. This is T. This is T. Here we 00:39:33.540 --> 00:39:35.000 have a lattice point. 00:39:35.000 --> 00:39:41.270 This jolly well has to be T as well, or we've contradicted 00:39:41.270 --> 00:39:44.250 the basic assumption of our construction. 00:39:44.250 --> 00:39:46.840 Well, that's over restrictive. 00:39:46.840 --> 00:39:53.165 This doesn't have to be T, but it has to be some multiple, p, 00:39:53.165 --> 00:39:54.600 of that translation. 00:39:54.600 --> 00:39:55.950 p could be 0. 00:39:55.950 --> 00:39:57.180 p could be 5. 00:39:57.180 --> 00:40:00.640 But it has to be an integral number of translations because 00:40:00.640 --> 00:40:03.620 this translational periodicity has to work everywhere, 00:40:03.620 --> 00:40:06.620 including up on the top of this trapezohedron. 00:40:09.610 --> 00:40:12.450 So that's a constraint. 00:40:12.450 --> 00:40:14.080 This angle is alpha. 00:40:14.080 --> 00:40:17.620 We cannot let alpha be arbitrary because the only way 00:40:17.620 --> 00:40:21.010 we can add a rotation operation A alpha to a lattice 00:40:21.010 --> 00:40:25.040 is for a value of alpha which makes this translation be a 00:40:25.040 --> 00:40:28.270 multiple of the original one. 00:40:28.270 --> 00:40:30.990 Now, let me take this geometry, and I'm going to 00:40:30.990 --> 00:40:33.980 extract the basic constraint from it. 00:40:33.980 --> 00:40:37.745 This is some integer, p times T. This is T. This is T. This 00:40:37.745 --> 00:40:39.920 is T. This is alpha. 00:40:39.920 --> 00:40:44.770 Let me lickety split drop down a perpendicular to the 00:40:44.770 --> 00:40:46.770 original translation. 00:40:46.770 --> 00:40:52.730 This is T times the cosine of alpha. 00:40:52.730 --> 00:40:56.450 This is T times the cosine of alpha. 00:40:56.450 --> 00:41:03.490 This total length is T. This length in here is p times T. 00:41:03.490 --> 00:41:07.060 And now I can go away from the geometry to an equation, 00:41:07.060 --> 00:41:09.650 something you probably prefer to deal with. 00:41:09.650 --> 00:41:14.490 And what this constraint is expressed analytically is that 00:41:14.490 --> 00:41:26.690 my original translation, T, minus twice T times the cosine 00:41:26.690 --> 00:41:36.090 of alpha has to come out equal to an integer, p times T. And 00:41:36.090 --> 00:41:39.390 there's my constraint. 00:41:39.390 --> 00:41:41.980 Alpha has to satisfy that condition. 00:41:41.980 --> 00:41:46.330 Well, I can immediately cancel the T and write this as one 00:41:46.330 --> 00:41:52.331 minus 2 cosine of alpha is equal to an integer, p. 00:41:52.331 --> 00:41:55.470 And it figures that that has to be the case because none of 00:41:55.470 --> 00:41:58.350 this construction depends on the size of the original 00:41:58.350 --> 00:42:00.070 translation that I took. 00:42:00.070 --> 00:42:03.980 And now, let me solve for the values of alpha which are 00:42:03.980 --> 00:42:06.360 compatible with a lattice. 00:42:06.360 --> 00:42:11.270 This says that cosine of alpha is 1 minus p over 2. 00:42:14.400 --> 00:42:19.290 And unless that condition holds, my combination is 00:42:19.290 --> 00:42:20.540 incompatible. 00:42:24.880 --> 00:42:29.060 So I'm going to let that stew with you until next time. 00:42:29.060 --> 00:42:33.480 But what we've set up is something where we can just 00:42:33.480 --> 00:42:37.280 plug and chug, put in different values of p. 00:42:37.280 --> 00:42:43.380 And if I start out with a value of p, and let's let p be 00:42:43.380 --> 00:42:52.620 equal to 4, and then find one minus p over 2, which is 00:42:52.620 --> 00:42:55.790 supposedly the cosine of an angle, alpha. 00:42:55.790 --> 00:42:59.230 If that's 4, I will have minus 3/2. 00:42:59.230 --> 00:43:02.550 And the value of alpha obviously does not exist. 00:43:02.550 --> 00:43:06.560 Cosine of alpha cannot get greater than 1. 00:43:06.560 --> 00:43:15.210 If p is equal to 3, then 1 minus 3 over 2 is minus 2 over 00:43:15.210 --> 00:43:17.850 2, or minus 1. 00:43:17.850 --> 00:43:19.720 You like the way I do that arithmetic in my 00:43:19.720 --> 00:43:22.370 head just like that? 00:43:22.370 --> 00:43:30.030 And the angle whose cosine is minus 1 is 180 degrees. 00:43:30.030 --> 00:43:33.930 And what that says is that a two-fold axis works. 00:43:40.290 --> 00:43:44.580 So I can drop a two-fold axis into a net. 00:43:44.580 --> 00:43:47.920 And what that's going to do is take my original translation, 00:43:47.920 --> 00:43:53.710 rotate it 180 degrees, and the second translation is 00:43:53.710 --> 00:43:54.960 going to sit here. 00:43:57.260 --> 00:44:01.250 Rotate it 180 degrees in the reverse direction, and then 00:44:01.250 --> 00:44:03.930 the second lattice point sits here. 00:44:03.930 --> 00:44:07.360 And lo and behold, just as advertised, the distance 00:44:07.360 --> 00:44:10.500 between the first lattice point and the final lattice 00:44:10.500 --> 00:44:11.750 point is three translations. 00:44:14.570 --> 00:44:18.740 So I can put a two-fold axis in any lattice whatsoever 00:44:18.740 --> 00:44:23.730 because this is compatible simply with a lattice row. 00:44:23.730 --> 00:44:27.520 So one possible combination of rotation in a lattice is going 00:44:27.520 --> 00:44:33.140 to be any lattice whatsoever, and what we can add to this is 00:44:33.140 --> 00:44:35.020 a rotation operation A pi. 00:44:41.140 --> 00:44:44.630 And we'll have two full rotation operations which are 00:44:44.630 --> 00:44:46.600 translationally equivalent. 00:44:49.312 --> 00:44:49.770 All right. 00:44:49.770 --> 00:44:50.970 Several integers to go. 00:44:50.970 --> 00:44:53.230 We would want to try p equals 2. 00:44:53.230 --> 00:44:54.450 That's going to work. 00:44:54.450 --> 00:44:56.900 p equals plus 1 is going to work. 00:44:56.900 --> 00:44:59.500 p equals 0 is going to work. 00:44:59.500 --> 00:45:03.700 And we will find a very limited number of rotational 00:45:03.700 --> 00:45:07.560 operations that are compatible with a lattice. 00:45:07.560 --> 00:45:12.580 And this is going to give us a small number of the possible 00:45:12.580 --> 00:45:16.660 combinations of lattice and rotational symmetry in two 00:45:16.660 --> 00:45:17.910 dimensions. 00:45:19.950 --> 00:45:22.010 So we'll pick up from there next time, and we'll very 00:45:22.010 --> 00:45:26.750 quickly determine the remaining possibilities and 00:45:26.750 --> 00:45:29.400 take a look at what the arrangement of symmetry 00:45:29.400 --> 00:45:30.925 elements look like in these lattices. 00:45:36.510 --> 00:45:39.620 OK, once again, I have some extra copies of the syllabus 00:45:39.620 --> 00:45:42.240 if somebody did not get one. 00:45:42.240 --> 00:45:45.910 And I'll have extra copies of the problem set.