WEBVTT 00:00:06.740 --> 00:00:08.850 PROFESSOR: Good afternoon and welcome back. 00:00:08.850 --> 00:00:11.630 I'm glad you all came back even though there was a 00:00:11.630 --> 00:00:15.790 problem in finding a seat for every one last time. 00:00:15.790 --> 00:00:21.860 I would like you to turn in the problem set if you've been 00:00:21.860 --> 00:00:22.785 able to do it. 00:00:22.785 --> 00:00:25.960 If you haven't been able to do it, that's no great problem, 00:00:25.960 --> 00:00:30.260 but I hope you find it mildly amusing. 00:00:30.260 --> 00:00:33.350 I have given that problem set out a couple of times over the 00:00:33.350 --> 00:00:39.930 years, and I can recall one very pale student who appeared 00:00:39.930 --> 00:00:44.400 at my door the next morning saying it's the first problem 00:00:44.400 --> 00:00:47.380 set, you've only talked an hour and I can't do it. 00:00:47.380 --> 00:00:49.940 And then there was another group of three who formed a 00:00:49.940 --> 00:00:55.350 consortium to attempt to solve the code using a computer. 00:00:55.350 --> 00:00:57.800 They didn't get very far either. 00:00:57.800 --> 00:01:01.230 This is an interesting sort of problem because it requires 00:01:01.230 --> 00:01:04.269 you to think in a slightly different direction than 00:01:04.269 --> 00:01:07.190 you're accustomed to thinking, and therefore it's a little 00:01:07.190 --> 00:01:08.700 bit amusing. 00:01:08.700 --> 00:01:13.020 This is not my creation, it came from a book by a man 00:01:13.020 --> 00:01:15.730 named Polya, the title of the book is Mathematics and 00:01:15.730 --> 00:01:17.640 Plausible Reasoning. 00:01:17.640 --> 00:01:21.360 And he gives this as an example of a problem that can 00:01:21.360 --> 00:01:22.780 be solved only if you think in a 00:01:22.780 --> 00:01:25.100 slightly different direction. 00:01:25.100 --> 00:01:28.450 I'll give you another example of a problem from his book. 00:01:28.450 --> 00:01:33.520 Suppose, not that the problem arises in this era when you do 00:01:33.520 --> 00:01:37.550 all of your graphics on a computer console, but suppose 00:01:37.550 --> 00:01:41.160 you had to, in solving a problem in short notice, draw 00:01:41.160 --> 00:01:47.800 a circle that had a diameter of 4 inches. 00:01:47.800 --> 00:01:51.300 And you went looking for your pair of compasses which you 00:01:51.300 --> 00:01:54.820 never use very often, and when you found them, they were 00:01:54.820 --> 00:01:58.380 rusted solid, and they were open to a 00:01:58.380 --> 00:02:01.300 distance of 5 inches. 00:02:01.300 --> 00:02:02.370 OK, so there's the problem. 00:02:02.370 --> 00:02:04.840 You have a pair of compasses that can only draw a circle 00:02:04.840 --> 00:02:09.180 that's 5 inches in diameter, you must draw a circle that's 00:02:09.180 --> 00:02:11.130 4 inches in diameter. 00:02:11.130 --> 00:02:13.890 What sort of construction, what sort of mapping out of 00:02:13.890 --> 00:02:17.650 arcs that you connected together could you do to 00:02:17.650 --> 00:02:21.910 create the circle of smaller diameter? 00:02:21.910 --> 00:02:23.160 Anybody have an idea? 00:02:25.850 --> 00:02:28.960 It seems impossible doesn't it? 00:02:28.960 --> 00:02:33.600 Well suppose you got yourself a little block of wood that 00:02:33.600 --> 00:02:39.480 had a height that was equal to the square root of 5 squared 00:02:39.480 --> 00:02:42.460 minus 4 squared. 00:02:42.460 --> 00:02:45.170 And so you have one end of the compass is up on top of the 00:02:45.170 --> 00:02:49.580 block of wood the other end of the compass traces out a 00:02:49.580 --> 00:02:52.450 circle that has a smaller diameter. 00:02:52.450 --> 00:02:53.960 It's fairly obvious. 00:02:53.960 --> 00:02:57.530 But what you have to do is to think of a problem that has 00:02:57.530 --> 00:03:00.630 poked your nose into two-dimensions and think of it 00:03:00.630 --> 00:03:02.380 in terms of a three-dimensional problem, and 00:03:02.380 --> 00:03:03.830 then the answer is easy. 00:03:03.830 --> 00:03:06.140 So that was the sort of thought provoking thing that 00:03:06.140 --> 00:03:10.870 Polya presented in one portion of this book. 00:03:10.870 --> 00:03:15.200 OK, if you enjoyed that problem, I have another one 00:03:15.200 --> 00:03:17.540 for you in a similar vein. 00:03:17.540 --> 00:03:19.830 And perhaps you'll enjoy this one as well. 00:03:19.830 --> 00:03:22.700 So while I'm talking, let me pass this around, I think 00:03:22.700 --> 00:03:23.950 there's enough for everyone. 00:03:28.890 --> 00:03:33.580 Last time I got so caught up with the displaying the heft 00:03:33.580 --> 00:03:37.150 of the International Tables for X-Ray Crystallography that 00:03:37.150 --> 00:03:40.320 I forgot to mention entirely that there is another text 00:03:40.320 --> 00:03:43.510 that we will use in the class. 00:03:43.510 --> 00:03:47.070 And this we will not need until halfway through the 00:03:47.070 --> 00:03:51.450 semester, and therefore I did not feel terribly remiss in 00:03:51.450 --> 00:03:52.600 not mentioning it. 00:03:52.600 --> 00:03:59.570 It's a book by somebody named Nye, and it's called the 00:03:59.570 --> 00:04:02.530 Physical Properties of Crystals. 00:04:18.350 --> 00:04:23.680 And this is published by Oxford University Press and 00:04:23.680 --> 00:04:37.050 the publication date of the original addition was in 1967. 00:04:37.050 --> 00:04:42.760 This is a book that is really, I don't think I'm being 00:04:42.760 --> 00:04:46.590 extravagant in calling it a classic. 00:04:46.590 --> 00:04:48.550 It is a beautifully written book. 00:04:48.550 --> 00:04:53.340 The first 2/3 of it deal systematically with tensor 00:04:53.340 --> 00:04:58.130 properties, the particular sort of mathematics that is 00:04:58.130 --> 00:05:01.450 used to set them up, transformation of axes, and 00:05:01.450 --> 00:05:04.930 then looks at specific physical properties and 00:05:04.930 --> 00:05:09.780 numbers that have to be described in terms of tensors. 00:05:09.780 --> 00:05:14.210 This is actually the third book in a sequence. 00:05:14.210 --> 00:05:23.200 It was a book by Wooster that covered things very similarly, 00:05:23.200 --> 00:05:29.620 but the notation that was used for the tensors was not the 00:05:29.620 --> 00:05:31.350 modern current notation. 00:05:37.220 --> 00:05:43.790 And then the subject started in the form of a third book, 00:05:43.790 --> 00:05:45.380 Woldemar Voigt. 00:05:47.900 --> 00:05:54.060 And the title of this book is Lehrbuch der Kristallphysik. 00:06:10.000 --> 00:06:14.820 And this was published back in 1910. 00:06:14.820 --> 00:06:16.970 This is the first time anybody had anything 00:06:16.970 --> 00:06:19.926 to say on the matter. 00:06:19.926 --> 00:06:24.460 It in fact is a big fat book that contains some topics that 00:06:24.460 --> 00:06:27.030 are not covered in Wooster and Nye. 00:06:29.680 --> 00:06:31.940 Nye's book is beautifully written. 00:06:31.940 --> 00:06:36.410 The first 2/3 of it concerns tensor formalism and the last 00:06:36.410 --> 00:06:39.580 1/3, which we will not touch at all, deals with 00:06:39.580 --> 00:06:42.470 thermodynamic relations between different properties 00:06:42.470 --> 00:06:45.330 that are represented by tensors. 00:06:45.330 --> 00:06:52.120 So I don't recommend you go out and buy this book until 00:06:52.120 --> 00:06:54.680 you determine whether or not you need it because I'll try 00:06:54.680 --> 00:06:57.180 to make the lectures self-contained and I'll have 00:06:57.180 --> 00:06:59.570 lots of notes and handouts. 00:06:59.570 --> 00:07:03.610 The problem with Nye's book, and any book that is intensely 00:07:03.610 --> 00:07:09.000 mathematical, is that you can't jump in on page 73 to 00:07:09.000 --> 00:07:11.330 find the answer to a specific question. 00:07:11.330 --> 00:07:14.150 Because when you go there, it will say as we showed back in 00:07:14.150 --> 00:07:16.540 Chapter 4, now what is he talking about? 00:07:16.540 --> 00:07:20.520 So you go back to Chapter 4, and Chapter 4 says, starting 00:07:20.520 --> 00:07:23.720 with our definition of Chapter 2, and you have to go back and 00:07:23.720 --> 00:07:24.630 read Chapter 2. 00:07:24.630 --> 00:07:27.920 So it's awfully hard to pick something out to answer a 00:07:27.920 --> 00:07:28.810 specific question. 00:07:28.810 --> 00:07:32.230 You have to really go all the way through it. 00:07:32.230 --> 00:07:36.200 The good news is that Nye's book has been 00:07:36.200 --> 00:07:38.960 published in paperback. 00:07:38.960 --> 00:07:42.650 And it is available at the COOP and paperback means 00:07:42.650 --> 00:07:46.000 cheap, cheap, or relatively inexpensive. 00:07:46.000 --> 00:07:49.230 No books are really cheap these days. 00:07:49.230 --> 00:07:53.000 So we will cover material that's in there, we'll have a 00:07:53.000 --> 00:07:55.950 slightly different emphasis, but the notation and the 00:07:55.950 --> 00:08:01.380 general mathematics that's involved is in Nye's book. 00:08:01.380 --> 00:08:05.240 The second thing that I mentioned last time is that 00:08:05.240 --> 00:08:10.510 there is a very, very nice and thorough and geometric 00:08:10.510 --> 00:08:12.950 treatment of crystal symmetry. 00:08:12.950 --> 00:08:14.390 And I said that's the good news. 00:08:14.390 --> 00:08:18.470 The bad news is that it's out of print, so I promised, what 00:08:18.470 --> 00:08:21.260 a guy, that I give you a Xerox copy of the 00:08:21.260 --> 00:08:22.900 first half of the book. 00:08:22.900 --> 00:08:25.630 So here is the text that we'll use in the 00:08:25.630 --> 00:08:26.880 first part of the term. 00:08:33.450 --> 00:08:37.260 I included at the beginning, the table of contents, so you 00:08:37.260 --> 00:08:42.840 can see each other topics that are covered in the book. 00:08:46.310 --> 00:08:52.190 We will not go through all of the material that's covered. 00:08:52.190 --> 00:08:59.420 There are a lot of different symmetries to be derived, and 00:08:59.420 --> 00:09:02.610 it turns out that if you get the general idea and you can 00:09:02.610 --> 00:09:06.940 summarize the results, there's no need to derive every single 00:09:06.940 --> 00:09:09.560 one of them. 00:09:09.560 --> 00:09:12.840 It's nice to know that there's a place where you can find out 00:09:12.840 --> 00:09:16.105 how it is done if you really have a particular question. 00:09:16.105 --> 00:09:19.940 Did everybody get a copy or are there a few who did not? 00:09:19.940 --> 00:09:23.540 I made extras, OK, nobody in need of one, great. 00:09:27.420 --> 00:09:36.250 OK, let me now start with a general rhetorical question. 00:09:36.250 --> 00:09:40.740 Crystallography, as we mentioned last time, is the 00:09:40.740 --> 00:09:42.740 geometry of crystals. 00:09:42.740 --> 00:09:46.100 It's the geometry of patterns and the sorts of symmetries 00:09:46.100 --> 00:09:48.280 that are in those patterns. 00:09:48.280 --> 00:09:50.950 Now you might ask yourself, why should I as a material 00:09:50.950 --> 00:09:53.940 scientist or a physical scientist of some sort, worry 00:09:53.940 --> 00:09:55.070 about this stuff? 00:09:55.070 --> 00:09:58.810 I'm not training to be a wallpaper designer, I'm going 00:09:58.810 --> 00:10:00.040 to do physical things. 00:10:00.040 --> 00:10:04.250 I'm going to heat things and measure properties, and that 00:10:04.250 --> 00:10:06.290 sort of thing. 00:10:06.290 --> 00:10:11.550 Well there are at least three answers to that question. 00:10:11.550 --> 00:10:15.330 First of all, whether you like it or not, the arcane language 00:10:15.330 --> 00:10:17.830 of symmetry is the language that's 00:10:17.830 --> 00:10:20.620 used to describe crystals. 00:10:20.620 --> 00:10:24.460 It's the language that's used to describe structures. 00:10:24.460 --> 00:10:27.000 The normal thing that you do when you're trying to describe 00:10:27.000 --> 00:10:30.410 verbally a ball and pin model of the geometrical arrangement 00:10:30.410 --> 00:10:33.940 of atoms in a crystal is to say the red balls are at the 00:10:33.940 --> 00:10:37.410 corners of the cube, the green balls are in the middle of the 00:10:37.410 --> 00:10:41.520 edges, and the chartreuse balls are sort of tucked up 00:10:41.520 --> 00:10:45.210 inside one of the corners, but slightly closer to 1 face than 00:10:45.210 --> 00:10:48.090 to the other 2 faces. 00:10:48.090 --> 00:10:51.840 The point I'm trying to make is that is a language that has 00:10:51.840 --> 00:10:54.350 limited utility. 00:10:54.350 --> 00:10:58.340 It deals, it's capable of dealing only with the simplest 00:10:58.340 --> 00:11:00.560 sort of atomic configurations. 00:11:00.560 --> 00:11:03.580 So there is a general language based on symmetry theory, 00:11:03.580 --> 00:11:08.270 based on group theory, that is universally used to describe 00:11:08.270 --> 00:11:10.060 atomic arrangements. 00:11:10.060 --> 00:11:13.300 So instead of saying red balls at the corners of the cube and 00:11:13.300 --> 00:11:17.400 green balls in the middle of the faces, I can say space 00:11:17.400 --> 00:11:26.600 group 4 over m3 bar 2 over m, atom a in position for b, m3m, 00:11:26.600 --> 00:11:32.090 atom b in position for c, m3m. 00:11:32.090 --> 00:11:34.100 That's what rock salt it. 00:11:34.100 --> 00:11:38.400 And that is the way, not only it, but especially more 00:11:38.400 --> 00:11:42.540 complicated structural arrangements are described in 00:11:42.540 --> 00:11:44.210 the literature. 00:11:44.210 --> 00:11:46.780 So this is the language of describing such arrangements. 00:11:46.780 --> 00:11:51.640 And finally, sooner or later, I bet you that every one of 00:11:51.640 --> 00:11:55.950 you will be involved with some crystalline material, and the 00:11:55.950 --> 00:12:00.590 first question you will answer is what is its structure, what 00:12:00.590 --> 00:12:01.870 is the atomic arrangement. 00:12:01.870 --> 00:12:04.050 That's where properties start. 00:12:04.050 --> 00:12:09.080 And you'll go a book or a set of volumes that describe 00:12:09.080 --> 00:12:10.530 structural data. 00:12:10.530 --> 00:12:14.020 There's a big long compendium of books that fill about that 00:12:14.020 --> 00:12:17.520 much of a bookshelf which are called structure reports. 00:12:17.520 --> 00:12:22.720 They started a number of years ago to compile all of the 00:12:22.720 --> 00:12:24.970 structures that had been determined within a given 00:12:24.970 --> 00:12:26.290 calendar year. 00:12:26.290 --> 00:12:29.710 They did a pretty good job of staying caught up 00:12:29.710 --> 00:12:32.680 back in 1915 and 1920. 00:12:32.680 --> 00:12:37.260 And then as it became easier to obtain such results, partly 00:12:37.260 --> 00:12:42.730 due to the advent of rapid large computers, they fell 00:12:42.730 --> 00:12:45.270 further and further behind and I think now they are about 5 00:12:45.270 --> 00:12:45.970 or 10 miles-- 00:12:45.970 --> 00:12:49.350 5 or 10 years, miles as well. 00:12:49.350 --> 00:12:54.400 But this is one of the places to go to look up, without 00:12:54.400 --> 00:12:57.080 going to the original literature, whether the 00:12:57.080 --> 00:12:59.260 material that you're interested in has had its 00:12:59.260 --> 00:13:02.270 atomic arrangement determined. 00:13:02.270 --> 00:13:04.600 When you go there you're going to find the atomic 00:13:04.600 --> 00:13:08.140 arrangement, not in terms of red balls at one position on 00:13:08.140 --> 00:13:10.690 the cell, but you're going to find it in terms of the 00:13:10.690 --> 00:13:12.590 language of symmetry theory. 00:13:12.590 --> 00:13:15.370 So one of the things I hope you'll be able to do by the 00:13:15.370 --> 00:13:18.860 time we finish this time together, is to be able to go 00:13:18.860 --> 00:13:22.070 to such literature and know exactly what to do and where 00:13:22.070 --> 00:13:25.190 to go to reconstruct the geometrical 00:13:25.190 --> 00:13:26.440 arrangement of the atoms. 00:13:28.880 --> 00:13:32.880 OK, so hopefully you're at least mildly convinced that 00:13:32.880 --> 00:13:37.410 this exercise is going to be worthwhile. 00:13:37.410 --> 00:13:40.800 Before we continue where we left off last time, I would 00:13:40.800 --> 00:13:47.340 like to say a little bit about the language in which these 00:13:47.340 --> 00:13:50.490 geometries are described. 00:13:50.490 --> 00:13:54.090 And we mentioned last time without thoroughly 00:13:54.090 --> 00:13:58.130 demonstrating why that in a 3-dimensional space there are 00:13:58.130 --> 00:14:01.410 4 basically different kinds of operations. 00:14:01.410 --> 00:14:06.430 And one of these is something that all crystals must by 00:14:06.430 --> 00:14:10.690 definition display, and this is the operation of 00:14:10.690 --> 00:14:13.070 translation. 00:14:13.070 --> 00:14:17.310 Analytically it can be described as a mapping in 00:14:17.310 --> 00:14:22.500 which every coordinate in a space xyz is mapped to a 00:14:22.500 --> 00:14:27.460 location x plus some constant, y plus some constant, z plus 00:14:27.460 --> 00:14:28.870 some constant. 00:14:28.870 --> 00:14:32.850 And if you do the operation again, this would go to a 00:14:32.850 --> 00:14:38.620 location x plus 2a y plus 2b, z plus 2c. 00:14:42.240 --> 00:14:46.430 A feature of translation that is unique to this particular 00:14:46.430 --> 00:14:51.390 symmetry transformation is that it has no origin. 00:14:51.390 --> 00:14:55.700 If I have a pair of motifs that are related by 00:14:55.700 --> 00:15:01.290 translation, we could think of them as being related by a 00:15:01.290 --> 00:15:04.570 vector, magnitude and direction, that takes this 00:15:04.570 --> 00:15:06.970 motif and moves it to this location. 00:15:06.970 --> 00:15:10.130 We said that more generally we should view these operations, 00:15:10.130 --> 00:15:13.900 not just acting on one little domain and space, but acting 00:15:13.900 --> 00:15:14.980 on everything. 00:15:14.980 --> 00:15:21.360 So this implies that there be a infinite chain of motifs if 00:15:21.360 --> 00:15:24.340 the operation of translation is to be present. 00:15:24.340 --> 00:15:27.740 Because only that infinite, doubly infinite string is 00:15:27.740 --> 00:15:32.100 consistent with all of space being mapped into itself. 00:15:32.100 --> 00:15:34.900 Like any vector, there's no unique origin. 00:15:34.900 --> 00:15:37.910 You could say it extends from here to here, or from here to 00:15:37.910 --> 00:15:43.720 here, or any other choice of translation, provided the 00:15:43.720 --> 00:15:49.140 direction and the magnitude are the same in every choice. 00:15:49.140 --> 00:15:56.110 As a result, it's not really possible to specify the locus 00:15:56.110 --> 00:15:57.495 of this particular operation. 00:15:57.495 --> 00:15:59.310 It has magnitude and direction, 00:15:59.310 --> 00:16:00.875 but no unique origin. 00:16:03.860 --> 00:16:08.480 What we can do through a very neat device is to 00:16:08.480 --> 00:16:19.130 nevertheless, take some reference point and have each 00:16:19.130 --> 00:16:22.830 of these reference points separated by T, and have each 00:16:22.830 --> 00:16:27.470 motif lurking off in space in exactly the same location and 00:16:27.470 --> 00:16:30.650 distance from this point that we've constructed. 00:16:30.650 --> 00:16:33.640 And this array of abstractions, so these 00:16:33.640 --> 00:16:37.210 geometrical abstractions, these points, are what are 00:16:37.210 --> 00:16:48.550 called lattice points, and they are a very neat summary 00:16:48.550 --> 00:16:52.150 of the translational periodicity of the crystal. 00:16:52.150 --> 00:16:57.030 It is absolutely essential not to mix up these lattice points 00:16:57.030 --> 00:17:00.910 which are a construct that we have created, and the atoms 00:17:00.910 --> 00:17:04.230 themselves that are present in a crystal. 00:17:04.230 --> 00:17:07.200 The atoms are atoms and they're not necessarily the 00:17:07.200 --> 00:17:08.520 lattice points. 00:17:08.520 --> 00:17:11.470 Another way of saying that not all atoms of the same chemical 00:17:11.470 --> 00:17:14.440 species need be translation equivalent. 00:17:14.440 --> 00:17:18.650 We'll see some examples of this later on, so do not mix 00:17:18.650 --> 00:17:21.310 up the atoms and the lattice points. 00:17:21.310 --> 00:17:25.339 When I talk about the sodium chloride lattice, I mean an 00:17:25.339 --> 00:17:31.430 array of points in space that are located at the corners of 00:17:31.430 --> 00:17:34.510 a cube and in the middle of the faces of the cube. 00:17:34.510 --> 00:17:37.860 If I talk about the arrangement of sodium ions and 00:17:37.860 --> 00:17:41.930 chlorine ions, that is the sodium chloride structure and 00:17:41.930 --> 00:17:45.160 not the sodium chloride lattice. 00:17:45.160 --> 00:17:48.190 And then last time I apologized for usage so as not 00:17:48.190 --> 00:17:50.160 to appear hypocritical. 00:17:50.160 --> 00:17:53.730 Everybody talks about lattice vibration, lattice energy, 00:17:53.730 --> 00:17:56.190 lattice dynamics, and so on, but that's a 00:17:56.190 --> 00:17:57.650 misuse of the term. 00:17:57.650 --> 00:18:01.100 But nevertheless, it is much more musical than saying 00:18:01.100 --> 00:18:03.390 structure energy, structure vibration. 00:18:03.390 --> 00:18:09.150 So we'll go on misusing the term lattice I'm afraid. 00:18:09.150 --> 00:18:12.420 OK, let's look at another operation. 00:18:12.420 --> 00:18:15.860 Here we change the sense of no coordinate. 00:18:15.860 --> 00:18:21.570 Let's next look at an operation that might take xyz, 00:18:21.570 --> 00:18:25.838 and map it into minus xyz. 00:18:29.830 --> 00:18:33.520 This would be a situation where if I set up a coordinate 00:18:33.520 --> 00:18:41.900 system, here's x, here's y, here's z. 00:18:41.900 --> 00:18:46.170 What I've done is to take an object that sits off here, at 00:18:46.170 --> 00:18:51.260 a coordinate plus x, and I've changed the sign of x so that 00:18:51.260 --> 00:18:53.725 this object now sits off here. 00:18:59.410 --> 00:19:04.020 This is exactly what happens when I take something and 00:19:04.020 --> 00:19:07.010 reflect it in a mirror. 00:19:07.010 --> 00:19:10.270 And if that's not immediately obvious, it just so happens, 00:19:10.270 --> 00:19:12.170 not at all by accident, I brought 00:19:12.170 --> 00:19:15.280 along with me a mirror. 00:19:15.280 --> 00:19:20.590 OK, here is one hand, and if you look in the mirror, there 00:19:20.590 --> 00:19:21.480 is the other hand. 00:19:21.480 --> 00:19:26.410 It's the same distance behind the plane of the mirror, two 00:19:26.410 --> 00:19:30.230 coordinates have been left unchanged. 00:19:30.230 --> 00:19:33.620 The two coordinates within the plane of the mirror, if that 00:19:33.620 --> 00:19:36.220 is my choice of the reference system, and one of them has 00:19:36.220 --> 00:19:38.670 been reversed. 00:19:38.670 --> 00:19:42.650 Now I'll take the second motif out of the geometric 00:19:42.650 --> 00:19:45.800 construct, and I'd like to point out one very curious 00:19:45.800 --> 00:19:49.220 feature of the pair of motifs that's generated by this 00:19:49.220 --> 00:19:51.110 transformation. 00:19:51.110 --> 00:19:54.820 They're both the same thing, clearly. 00:19:54.820 --> 00:19:59.820 But no matter how I try, I cannot move one so that it 00:19:59.820 --> 00:20:02.740 coincides with the other. 00:20:02.740 --> 00:20:06.100 And we intuitively appreciate this difference by saying we 00:20:06.100 --> 00:20:08.850 actually use our hands by analogy. 00:20:08.850 --> 00:20:14.930 We say one is left-handed and one is right-handed, and they 00:20:14.930 --> 00:20:16.380 are not congruent. 00:20:16.380 --> 00:20:19.740 The fancy name that's used to describe this relation is to 00:20:19.740 --> 00:20:21.420 say that they are enantiomorph. 00:20:27.770 --> 00:20:31.740 Another term that's used, particularly in chemistry, is 00:20:31.740 --> 00:20:33.470 to say that they are chiral. 00:20:39.880 --> 00:20:41.750 Which one is the left-handed one, which is a 00:20:41.750 --> 00:20:44.150 right-handed one? 00:20:44.150 --> 00:20:46.620 This is what I call my left hand, this is what I call my 00:20:46.620 --> 00:20:47.230 right hand. 00:20:47.230 --> 00:20:52.010 But can we distinguish them physically, any other way? 00:20:52.010 --> 00:20:57.040 No, these are terms that have come in to regular use in both 00:20:57.040 --> 00:21:01.800 our everyday language and also in science because we use our 00:21:01.800 --> 00:21:07.620 hands instinctively as readily available examples of 00:21:07.620 --> 00:21:12.410 enantiomorphs, readily at hand, I might say to almost 00:21:12.410 --> 00:21:15.280 make a pun. 00:21:15.280 --> 00:21:20.690 One of the really brilliant figures in physics was a man 00:21:20.690 --> 00:21:22.790 named Richard Feynman. 00:21:22.790 --> 00:21:28.160 Recently deceased, Feynman gave a very famous series of 00:21:28.160 --> 00:21:31.220 lectures on science at Cornell University. 00:21:31.220 --> 00:21:35.640 And he has one entire vector that was devoted to the 00:21:35.640 --> 00:21:39.600 difference between right and left. 00:21:39.600 --> 00:21:43.220 And he comes up with a funny story, he pretends that the 00:21:43.220 --> 00:21:46.750 hero of the story is someone who's trying to communicate 00:21:46.750 --> 00:21:51.540 with beings in outer space and suddenly he gets lucky and he 00:21:51.540 --> 00:21:54.130 gets a response to his message. 00:21:54.130 --> 00:21:56.830 And they work out a way to communicate and eventually 00:21:56.830 --> 00:22:02.100 they try to describe each other to the other individual. 00:22:02.100 --> 00:22:03.390 Well what they look like? 00:22:03.390 --> 00:22:08.780 Well we're bipedal, and we have two organs related by 00:22:08.780 --> 00:22:12.340 reflection that let us sense light and form images. 00:22:12.340 --> 00:22:15.940 And we have an aperture through which we ingest things 00:22:15.940 --> 00:22:17.820 that can be metabolized. 00:22:17.820 --> 00:22:21.750 And our circulation and body works because we have a pump 00:22:21.750 --> 00:22:24.790 on the left hand side that circulates 00:22:24.790 --> 00:22:26.100 fluids through our-- 00:22:26.100 --> 00:22:29.900 wait, I don't understand, what's left? 00:22:29.900 --> 00:22:34.590 So then there follows along this course on how to define 00:22:34.590 --> 00:22:37.690 left in an absolute sense. 00:22:37.690 --> 00:22:40.620 How do you describe to someone what makes your left hand 00:22:40.620 --> 00:22:43.450 left, and your right hand right without being 00:22:43.450 --> 00:22:45.080 anthropomorphic about it. 00:22:45.080 --> 00:22:48.220 So it goes on and on and on and he gets into physical 00:22:48.220 --> 00:22:49.810 phenomena which are objective and 00:22:49.810 --> 00:22:51.530 independent of a human being. 00:22:51.530 --> 00:22:58.360 And finally he comes to the anisotropic emission beta 00:22:58.360 --> 00:23:01.050 particles in radioactive decay. 00:23:01.050 --> 00:23:04.360 And that depends on direction relative to the magnetic 00:23:04.360 --> 00:23:07.350 moment and that defines an absolute sense 00:23:07.350 --> 00:23:08.880 of right and left. 00:23:08.880 --> 00:23:11.460 So that's something physical, it doesn't depend on the 00:23:11.460 --> 00:23:13.500 nature of the human being. 00:23:13.500 --> 00:23:17.970 And then Feynman wraps up his story by saying, if finally 00:23:17.970 --> 00:23:22.140 our extraterrestrial being travels to space, gets out of 00:23:22.140 --> 00:23:25.320 his spaceship and he walks forward to greet you and you 00:23:25.320 --> 00:23:28.810 put out your right hand, and he puts out his left hand, get 00:23:28.810 --> 00:23:32.490 out of their fast because it means he is made out of 00:23:32.490 --> 00:23:34.710 anti-matter. 00:23:34.710 --> 00:23:38.810 And nuclei of matter in this anisotropic emission of data 00:23:38.810 --> 00:23:42.020 rays shoot out the beta particle in one sense, 00:23:42.020 --> 00:23:44.220 anti-matter shoots out the beta particle 00:23:44.220 --> 00:23:48.160 in the chiral sense. 00:23:48.160 --> 00:23:53.640 So it's a cute little story that emphasizes the problem in 00:23:53.640 --> 00:23:57.700 defining absolutely left from right. 00:23:57.700 --> 00:23:59.930 But they're of opposite handedness that we can say. 00:24:06.540 --> 00:24:15.320 Mirrors are interesting things and I brought along a couple 00:24:15.320 --> 00:24:24.000 of mirrors and they have very, very peculiar characteristics. 00:24:24.000 --> 00:24:28.260 And I would invite you to come up and look at these in 00:24:28.260 --> 00:24:30.900 private because if I hold them up in front of you, you're not 00:24:30.900 --> 00:24:33.670 going to be able to see what I'm doing at all, although I 00:24:33.670 --> 00:24:37.440 kid myself that you can, and I walk around and show 00:24:37.440 --> 00:24:38.800 you what I'm doing. 00:24:38.800 --> 00:24:42.750 Here is something scientific, it's a chemical compound 00:24:42.750 --> 00:24:45.890 carbon dioxide, and-- 00:24:55.850 --> 00:25:01.830 OK, and if I hand this down you can see carbon dioxide in 00:25:01.830 --> 00:25:03.646 the mirror. 00:25:03.646 --> 00:25:06.930 Can you see that? 00:25:06.930 --> 00:25:08.850 Uh oh, it's not reflecting-- 00:25:08.850 --> 00:25:10.560 sorry, I didn't turn it on. 00:25:14.180 --> 00:25:16.960 Now I think we can get it. 00:25:16.960 --> 00:25:19.200 And is it working now? 00:25:32.580 --> 00:25:35.720 OK, now it's working. 00:25:35.720 --> 00:25:40.200 You can see why this is a very special kind of mirror because 00:25:40.200 --> 00:25:45.180 it reflects only red letters and it leaves the black 00:25:45.180 --> 00:25:46.430 letters unchanged. 00:25:56.342 --> 00:25:58.460 You're going to have to come up, I see some of you 00:25:58.460 --> 00:26:00.450 straining your necks, you'll have to come up and look at 00:26:00.450 --> 00:26:01.860 that in person. 00:26:01.860 --> 00:26:07.160 But the black letters are completely unchanged, the red 00:26:07.160 --> 00:26:09.130 letters are reflected into letters 00:26:09.130 --> 00:26:12.140 of an opposite chirality. 00:26:12.140 --> 00:26:16.290 It's a very special kind of mirror. 00:26:16.290 --> 00:26:20.010 I've got another kind of mirror that works in a 00:26:20.010 --> 00:26:21.260 different way. 00:26:26.940 --> 00:26:29.450 Where Is my other piece of paper? 00:26:29.450 --> 00:26:30.700 OK. 00:26:35.520 --> 00:26:42.620 This is an interesting mirror because it reflects only male 00:26:42.620 --> 00:26:48.300 names are not female names. 00:26:48.300 --> 00:26:51.880 This was a very topical sort of mirror a few years ago when 00:26:51.880 --> 00:26:52.840 there was a lawsuit. 00:26:52.840 --> 00:26:56.210 There was a college down South called the Citadel which would 00:26:56.210 --> 00:27:01.380 only admit male applicants and not female applicants. 00:27:01.380 --> 00:27:03.830 So I claim that this was a mirror that I got from the 00:27:03.830 --> 00:27:09.590 Citadel because it doesn't change the male names, but 00:27:09.590 --> 00:27:15.160 does change, does reject or reflect, female names. 00:27:15.160 --> 00:27:18.480 So you can play with this during our break. 00:27:18.480 --> 00:27:23.120 But if I look at myself in a mirror, I 00:27:23.120 --> 00:27:25.870 take a look at myself. 00:27:25.870 --> 00:27:29.520 If I wink my left eye, the in there winks his 00:27:29.520 --> 00:27:32.880 right eye back at me. 00:27:32.880 --> 00:27:36.540 So I'm not really seeing myself, what I'm seeing is my 00:27:36.540 --> 00:27:38.390 enantiomorph. 00:27:38.390 --> 00:27:40.620 Doesn't that shake you up? 00:27:40.620 --> 00:27:45.330 You have never ever seen yourself in exactly the same 00:27:45.330 --> 00:27:49.510 way as other people see you. 00:27:49.510 --> 00:27:51.740 You are only familiar with your enantiomorph. 00:27:56.020 --> 00:27:57.330 Does that make a difference? 00:27:57.330 --> 00:27:59.450 Well I'll bring in something that I put together and I 00:27:59.450 --> 00:28:01.850 couldn't put my hands on. 00:28:01.850 --> 00:28:06.470 We are very, very sensitive to the symmetry in our faces. 00:28:06.470 --> 00:28:09.060 And if they are reflected left to right, you surely are going 00:28:09.060 --> 00:28:11.320 to look different to the other person. 00:28:11.320 --> 00:28:14.780 And the way to see that is to take a photograph of somebody 00:28:14.780 --> 00:28:17.940 and cut it down the middle, and put the two different 00:28:17.940 --> 00:28:20.750 sides reflected left to right. 00:28:20.750 --> 00:28:24.460 And the expression on the person's face, and the general 00:28:24.460 --> 00:28:29.670 spirit that that image conveys is entirely different if you 00:28:29.670 --> 00:28:33.020 use the one half of the face reflected left to right, and 00:28:33.020 --> 00:28:36.040 the other half reflected left to right. 00:28:36.040 --> 00:28:39.810 So think of this, when you look in the mirror, you see 00:28:39.810 --> 00:28:44.260 your enantiomorph and other people see you differently. 00:28:44.260 --> 00:28:46.070 Let me ask you to scratch your head now. 00:28:46.070 --> 00:28:49.640 Is there any time when, in point of fact, you may have 00:28:49.640 --> 00:28:54.930 seen yourself without being reflected into the 00:28:54.930 --> 00:28:55.490 enantiomorph? 00:28:55.490 --> 00:28:55.870 Yeah. 00:28:55.870 --> 00:28:57.430 AUDIENCE: Picture? 00:28:57.430 --> 00:28:57.710 PROFESSOR: Absolutely. 00:28:57.710 --> 00:29:01.170 A Photograph or a TV monitor. 00:29:01.170 --> 00:29:06.220 When you look at a picture on television, you can read all 00:29:06.220 --> 00:29:09.540 the signs, and they didn't make up special signs in 00:29:09.540 --> 00:29:11.290 reflection so that they'd look right when they 00:29:11.290 --> 00:29:12.400 photographed you. 00:29:12.400 --> 00:29:15.760 So photography or a video camera does 00:29:15.760 --> 00:29:18.120 not change the chirality. 00:29:18.120 --> 00:29:22.920 But I've got another way in which I can see myself in the 00:29:22.920 --> 00:29:24.650 exact same chirality. 00:29:24.650 --> 00:29:26.950 And this I can't really convince you of, 00:29:26.950 --> 00:29:28.200 you'll have to try it. 00:29:28.200 --> 00:29:32.730 If I put two mirrors together at 90 degrees, and then adjust 00:29:32.730 --> 00:29:37.450 them so that I am looking right the point of 00:29:37.450 --> 00:29:42.690 intersection so that my two images coincide, then I see 00:29:42.690 --> 00:29:46.270 myself in the normal way. 00:29:46.270 --> 00:29:51.130 If I now blink my left eye, this guy blinks his 00:29:51.130 --> 00:29:53.020 left eye at me too. 00:29:53.020 --> 00:29:56.710 This is really astounding, two mirrors at 90 degrees, if you 00:29:56.710 --> 00:29:59.260 look at yourself right at their point of intersection, 00:29:59.260 --> 00:30:03.970 give you a non-chiral image of yourself. 00:30:03.970 --> 00:30:06.350 So I invite you to come up and try that, that's truly 00:30:06.350 --> 00:30:07.600 astounding. 00:30:09.550 --> 00:30:12.980 So why should somebody in material science or chemistry 00:30:12.980 --> 00:30:14.950 care about chirality? 00:30:14.950 --> 00:30:16.270 Does it really make any difference? 00:30:18.815 --> 00:30:23.350 Let me give you a little experiment that you can try. 00:30:23.350 --> 00:30:25.840 Suppose you have a little cell on a couple pieces of 00:30:25.840 --> 00:30:28.950 Polaroid, and the cell has a glass front and a glass back 00:30:28.950 --> 00:30:31.690 and you fill it with sugar solution. 00:30:31.690 --> 00:30:35.570 And then you pass a beam of polarized light through the 00:30:35.570 --> 00:30:41.250 sugar solution, and what happens is that the sugar 00:30:41.250 --> 00:30:48.240 solution rotates the direction of polarization in proportion 00:30:48.240 --> 00:30:50.620 to the thickness of solution at the light is passed 00:30:50.620 --> 00:30:54.690 through, in proportion to the concentration of sugar. 00:30:54.690 --> 00:30:57.600 The point of polarization gets rotated. 00:30:57.600 --> 00:31:00.440 Now that's pretty curious, so you scratch 00:31:00.440 --> 00:31:01.340 your head about that. 00:31:01.340 --> 00:31:03.332 Why does that happen? 00:31:03.332 --> 00:31:05.810 Well, maybe I'd better go back and try it again. 00:31:05.810 --> 00:31:08.390 And a day or two later, you go back and you repeat the 00:31:08.390 --> 00:31:09.410 experiment. 00:31:09.410 --> 00:31:16.430 And once again, the point of polarization rotates, but it 00:31:16.430 --> 00:31:18.620 rotates in the opposite direction. 00:31:23.500 --> 00:31:27.400 The reason for this is that if I was not careful in 00:31:27.400 --> 00:31:30.410 cleanliness and there were some little bugs lurking in 00:31:30.410 --> 00:31:33.440 the corners of that cell and when they sense the sugar 00:31:33.440 --> 00:31:36.165 solution, they said wow, free lunch, and they crawled out 00:31:36.165 --> 00:31:37.050 And gobbled it up. 00:31:37.050 --> 00:31:40.650 It turns out, those guys can gobble up just the sugar of 00:31:40.650 --> 00:31:42.160 one chirality. 00:31:42.160 --> 00:31:45.650 Sugar is a chiral molecule. 00:31:45.650 --> 00:31:50.710 And in fact there is a product that's called invert sugar and 00:31:50.710 --> 00:31:54.210 this is sugar that is all of one handedness. 00:31:54.210 --> 00:31:57.520 But everything in the world around us, everything from 00:31:57.520 --> 00:32:00.980 sugar beats to sugar cane to other things that make 00:32:00.980 --> 00:32:08.830 sucrose, manufacture sugar of one chirality, not mixed. 00:32:08.830 --> 00:32:12.880 All chiral molecules that are produced by living organisms 00:32:12.880 --> 00:32:15.575 are all of the same kind chirality. 00:32:15.575 --> 00:32:19.910 If we make them synthetically, there's no reason to favor 00:32:19.910 --> 00:32:23.820 synthesis of one molecule or the opposite handedness, so 00:32:23.820 --> 00:32:28.830 synthesized molecules are of equal proportion in the 00:32:28.830 --> 00:32:33.170 left-handed chirality and the right-hand chirality. 00:32:33.170 --> 00:32:38.130 This means that in the case of pharmaceuticals at the very 00:32:38.130 --> 00:32:44.030 best, you are going to use only half of the product that 00:32:44.030 --> 00:32:45.220 you've made. 00:32:45.220 --> 00:32:49.950 There is a pharmaceutical product that is prescribed for 00:32:49.950 --> 00:32:55.600 attention deficit disorder, this is called Ritalin, and 00:32:55.600 --> 00:33:00.480 only one chirality of the Ritalin molecule does 00:33:00.480 --> 00:33:01.610 anything for you. 00:33:01.610 --> 00:33:03.020 The other part is just metabolized 00:33:03.020 --> 00:33:05.360 and doesn't do anything. 00:33:05.360 --> 00:33:08.990 But there are other much more sinister cases. 00:33:08.990 --> 00:33:13.750 There was a serious problem about 20 years ago, primarily 00:33:13.750 --> 00:33:20.670 in Europe, where a particular pharmaceutical thalidomide was 00:33:20.670 --> 00:33:25.650 prescribed for pregnant women, it was to act as a sedative. 00:33:25.650 --> 00:33:29.530 Only one chirality of the molecule did this, the other 00:33:29.530 --> 00:33:34.590 chirality tragically caused birth defects. 00:33:34.590 --> 00:33:38.980 So you have to be very careful about the chirality of the 00:33:38.980 --> 00:33:42.630 pharmaceutical molecule that you synthesize. 00:33:42.630 --> 00:33:45.800 Another example, there is-- 00:33:48.340 --> 00:33:50.405 I don't remember the name of it. 00:33:52.950 --> 00:33:57.095 This is something that is taken to-- 00:34:02.760 --> 00:34:06.990 this is something called Ethambutal which is used to 00:34:06.990 --> 00:34:09.340 treat tuberculosis. 00:34:09.340 --> 00:34:11.949 Only the molecule of one handedness does this, the 00:34:11.949 --> 00:34:14.090 other one causes blindness. 00:34:14.090 --> 00:34:16.889 That's really a sinister and antiomorph. 00:34:16.889 --> 00:34:20.380 Then there's some even crazier examples. 00:34:20.380 --> 00:34:25.260 Ibuprofen is a chiral molecule, and this in a most 00:34:25.260 --> 00:34:29.190 remarkable situation is a molecule which you're body 00:34:29.190 --> 00:34:34.090 converts to the molecule of the chirality that has the 00:34:34.090 --> 00:34:35.710 intended purpose. 00:34:35.710 --> 00:34:39.800 So here your body is clever enough to change ibuprofen 00:34:39.800 --> 00:34:43.139 into the molecule which is the one that you need for its 00:34:43.139 --> 00:34:44.389 pharmaceutical effect. 00:34:46.820 --> 00:34:50.530 OK, so mirrors are interesting things. 00:34:50.530 --> 00:34:53.979 I would invite you to come up and play with the special 00:34:53.979 --> 00:34:56.920 mirrors that do strange things and see yourself 00:34:56.920 --> 00:35:00.770 as others see you. 00:35:00.770 --> 00:35:04.370 And now I would like to continue on in this discussion 00:35:04.370 --> 00:35:09.710 to mention the ways in which we can represent a mirror 00:35:09.710 --> 00:35:13.040 plane in a graphic language. 00:35:13.040 --> 00:35:15.390 This is what a mirror plane does, it changes the sense of 00:35:15.390 --> 00:35:16.600 one coordinate. 00:35:16.600 --> 00:35:20.720 If there is a locus across which that transformation is 00:35:20.720 --> 00:35:26.240 performed, we would like first of all, an analytic symbol. 00:35:31.680 --> 00:35:35.310 Some way of indicating the presence of that particular 00:35:35.310 --> 00:35:40.430 operation in the pattern, and a mirror is very descriptive 00:35:40.430 --> 00:35:44.360 so the symbol m is used to represent the presence of a 00:35:44.360 --> 00:35:48.810 mirror plane in a particular symbol. 00:35:48.810 --> 00:35:55.610 We might want to indicate a specific operation. 00:35:55.610 --> 00:35:59.480 There are only two operations in the case of a mirror plane 00:35:59.480 --> 00:36:02.620 reflecting left to right and reflecting right to left. 00:36:02.620 --> 00:36:05.590 But there are other operations such as rotation. 00:36:05.590 --> 00:36:10.500 If we have a 16-fold rotation axis, there is one operation 00:36:10.500 --> 00:36:15.170 that consists of rotating 1/16 of 2 pi, another operation 00:36:15.170 --> 00:36:18.750 that will also leave the space invariant that's rotating 2/16 00:36:18.750 --> 00:36:20.190 of 2 pi, and so on. 00:36:20.190 --> 00:36:23.080 So an individual operation is something that we 00:36:23.080 --> 00:36:27.280 will want to designate. 00:36:27.280 --> 00:36:30.960 And for a mirror plane, something that is used 00:36:30.960 --> 00:36:34.960 commonly in physics is to use an operation sigma for a 00:36:34.960 --> 00:36:38.150 particular reflection. 00:36:38.150 --> 00:36:42.120 This is not done in Buerger. 00:36:42.120 --> 00:36:48.260 If you get into reading it, he uses m for both. 00:36:48.260 --> 00:36:53.500 And then finally, it's going to be convenient when we have 00:36:53.500 --> 00:36:59.550 a pattern before us to use a geometric symbol to indicate 00:36:59.550 --> 00:37:03.650 in the pattern the locus of this particular operation. 00:37:03.650 --> 00:37:09.710 And what we use in the case of a mirror plane is a bold line. 00:37:09.710 --> 00:37:12.700 And that if this were the pattern and we wanted to 00:37:12.700 --> 00:37:16.280 indicate where the mirror plane was, or the mirror line 00:37:16.280 --> 00:37:18.830 in 2-dimensions that relates those two motifs, 00:37:18.830 --> 00:37:20.080 we draw it in thusly. 00:37:24.980 --> 00:37:28.910 We began last time to examine the properties of rotation, 00:37:28.910 --> 00:37:34.080 but that's another sort of symmetry and that is a 00:37:34.080 --> 00:37:37.165 rotation which takes place about a rotation axis. 00:37:40.920 --> 00:37:48.760 The symbol that is used to represent the collection of 00:37:48.760 --> 00:37:52.550 operations, the analytic symbol, is based on the fact 00:37:52.550 --> 00:37:58.590 that the angular rotation, alpha, has to be equal to some 00:37:58.590 --> 00:38:01.020 sub-multiple of 2 pi. 00:38:01.020 --> 00:38:03.200 2 pi over n, where n is some integer. 00:38:08.600 --> 00:38:12.050 And the reason for that I think is quite clear, if I 00:38:12.050 --> 00:38:19.860 take a particular motif and rotate through an angle alpha, 00:38:19.860 --> 00:38:24.220 if I am not rotating by some sub-multiple of 2 pi, I'll 00:38:24.220 --> 00:38:26.950 just go round and round and round and I will never get a 00:38:26.950 --> 00:38:33.160 finite set of objects that is separated from its neighbor by 00:38:33.160 --> 00:38:35.760 the same angular interval alpha. 00:38:35.760 --> 00:38:38.950 This will only happen if alpha is an integral 00:38:38.950 --> 00:38:41.220 sub-multiple of 2 pi. 00:38:41.220 --> 00:38:46.170 And the symbol that is used for the collection of 00:38:46.170 --> 00:38:49.780 operations that is usually embodied in a rotation axis is 00:38:49.780 --> 00:38:53.520 n, the same n that is in the denominator. 00:38:53.520 --> 00:38:58.180 The symbol for individual rotation, so we mentioned last 00:38:58.180 --> 00:39:02.080 time we have to specify the location of the point about 00:39:02.080 --> 00:39:07.570 which we rotate, and we have to indicate the angle alpha 00:39:07.570 --> 00:39:09.190 through which we've rotated. 00:39:09.190 --> 00:39:13.890 So A alpha will be an individual operation, and the 00:39:13.890 --> 00:39:22.010 geometric symbol will be an n-Gon which has the symmetry 00:39:22.010 --> 00:39:23.500 of the rotation axis. 00:39:23.500 --> 00:39:28.050 So for a sixfold axis we would use a hexagon. 00:39:28.050 --> 00:39:33.170 For a fivefold axis we will use a pentagon, for a 00:39:33.170 --> 00:39:37.950 fourfold, a square, for a threefold, a triangle. 00:39:37.950 --> 00:39:44.350 Now an n-Gon with 180 degree rotation is a line segment. 00:39:44.350 --> 00:39:46.800 And that would be easily overlooked and it's not very 00:39:46.800 --> 00:39:50.970 aesthetic, so here we indulge in a little bit of artistic 00:39:50.970 --> 00:39:55.640 license and fatten out the middle of the line segment to 00:39:55.640 --> 00:39:58.380 get an oval with pointed ends. 00:39:58.380 --> 00:40:02.090 And that's the symbol for a twofold axis. 00:40:02.090 --> 00:40:03.990 What about a one-fold axis? 00:40:03.990 --> 00:40:06.910 One-fold axes exist anywhere, so you can sprinkle them 00:40:06.910 --> 00:40:09.030 around with reckless abandon. 00:40:09.030 --> 00:40:13.170 A one-fold axis has no symmetry at all, but that is a 00:40:13.170 --> 00:40:16.990 very nice symbol to use for no symmetry at all. 00:40:16.990 --> 00:40:21.860 So symmetry 1 is the absence of symmetry. 00:40:21.860 --> 00:40:27.030 So it does come up occasionally in notation. 00:40:27.030 --> 00:40:32.290 Now if you look at what we've done so far, we have a 00:40:32.290 --> 00:40:36.040 transformation that changes the sense of no coordinate. 00:40:36.040 --> 00:40:40.230 We have a transformation that changes the sense of two 00:40:40.230 --> 00:40:44.470 coordinates, one coordinate, no coordinate, one coordinate. 00:40:44.470 --> 00:40:48.220 Rotation is interchanging the sense of two coordinates in a 00:40:48.220 --> 00:40:52.350 plane, and in a 2-dimensional pattern, that's all there is. 00:40:52.350 --> 00:41:01.660 But for a 3-dimensional space, we have the option of changing 00:41:01.660 --> 00:41:07.470 the sense of no coordinate, the sense 1, the sense of 2, 00:41:07.470 --> 00:41:10.750 or change the sense of all 3 coordinates. 00:41:10.750 --> 00:41:16.310 So if this is x and this is y and this is z, and up here in 00:41:16.310 --> 00:41:22.640 space lurks my initial motif, if I change the sense of x, 00:41:22.640 --> 00:41:29.520 the sense of y, and the sense of z, namely take xyz and map 00:41:29.520 --> 00:41:34.800 it to minus x, minus y, minus z, what I'm going to do is to 00:41:34.800 --> 00:41:38.510 essentially turn the object inside out. 00:41:38.510 --> 00:41:44.050 And if my initial one was right-handed, I will produce a 00:41:44.050 --> 00:41:48.340 chiral object, a left-handed object. 00:41:48.340 --> 00:41:52.280 This is an operation which is called inversion. 00:41:55.760 --> 00:41:59.870 In this operation of turning the object inside out if you 00:41:59.870 --> 00:42:03.050 will, is inverting it to a new location. 00:42:03.050 --> 00:42:06.490 And this analytically is the exchange in coordinates 00:42:06.490 --> 00:42:11.330 provided the point of inversion is at the center. 00:42:11.330 --> 00:42:18.090 The analytic symbol for inversion is 1 with a bar over 00:42:18.090 --> 00:42:19.830 the top, pronounced 1 bar. 00:42:23.020 --> 00:42:30.660 And I'll have to leave to later indication of exactly 00:42:30.660 --> 00:42:34.010 where that notation comes from. 00:42:34.010 --> 00:42:40.810 The individual operation is also called 1 bar, and the 00:42:40.810 --> 00:42:44.100 geometric symbol that is used to indicate the location of an 00:42:44.100 --> 00:42:49.840 inversion center is a tiny little open circle large 00:42:49.840 --> 00:42:53.070 enough so that you don't miss it, but not so large that it 00:42:53.070 --> 00:42:56.910 might be confused with an atom in a drawing of an atomic 00:42:56.910 --> 00:42:57.950 arrangement. 00:42:57.950 --> 00:43:02.550 So in this case, we would adorn our sketch was a little 00:43:02.550 --> 00:43:05.470 circle at the origin if that was the point through which 00:43:05.470 --> 00:43:06.865 the space was being inverted. 00:43:09.600 --> 00:43:14.870 So that, ladies and gentlemen, is our basic bag of tricks in 00:43:14.870 --> 00:43:17.430 3-dimensions. 00:43:17.430 --> 00:43:23.410 Let me point out that inversion can exist in 00:43:23.410 --> 00:43:30.970 3-dimensions only because I have to have 3 coordinates to 00:43:30.970 --> 00:43:35.570 play with or else I cannot define the operation. 00:43:35.570 --> 00:43:40.530 Suppose I have a mapping operation xyz that goes to 00:43:40.530 --> 00:43:46.820 minus x, minus y, minus z, and I get rid of z to make it 00:43:46.820 --> 00:43:48.560 2-dimensional. 00:43:48.560 --> 00:43:53.220 Then my transformation is xy going to minus x, minus y and 00:43:53.220 --> 00:43:54.760 that's exactly what a twofold axis does. 00:44:01.510 --> 00:44:03.990 So inversion, when you throw out the third coordinate, 00:44:03.990 --> 00:44:07.260 looks like a 180 degree rotation. 00:44:07.260 --> 00:44:09.990 So you need 3 dimensions in order to define that 00:44:09.990 --> 00:44:11.240 transformation. 00:44:17.540 --> 00:44:20.000 If we really wanted to go crazy, we could go on to say 00:44:20.000 --> 00:44:23.030 what happens in 4-dimensions? 00:44:26.260 --> 00:44:30.140 There should in principle be five different operations and 00:44:30.140 --> 00:44:32.250 yes, mathematically you can define them. 00:44:32.250 --> 00:44:36.060 They're very difficult to draw because we have to have some 00:44:36.060 --> 00:44:39.290 sort of operation that take something and pulls it out of 00:44:39.290 --> 00:44:40.640 our 3-dimensional world. 00:44:40.640 --> 00:44:43.310 We have no idea where it went and then all of sudden, [POP], 00:44:43.310 --> 00:44:46.050 it pops back into our space. 00:44:46.050 --> 00:44:50.000 But mathematically there are cases when you need a fourth 00:44:50.000 --> 00:44:55.230 variable to describe the symmetry of an arrangement. 00:44:55.230 --> 00:45:00.650 And this generally occurs in something called a modulated 00:45:00.650 --> 00:45:04.430 structure where there's a periodic change in some 00:45:04.430 --> 00:45:07.650 variable other than the atomic positions. 00:45:07.650 --> 00:45:11.670 And let me give you two quick examples without going into it 00:45:11.670 --> 00:45:12.920 exhaustively. 00:45:15.480 --> 00:45:19.910 One characteristic of an atom besides its location and its 00:45:19.910 --> 00:45:26.670 atomic mass and things like that, is perhaps a magnetic 00:45:26.670 --> 00:45:29.780 atom that has a magnetic moment attached to it. 00:45:32.560 --> 00:45:39.430 There are magnetically ordered structures. 00:45:39.430 --> 00:45:44.250 One of them looks exactly like rock salt. 00:45:44.250 --> 00:45:47.010 And I'll draw just the magnetic cations which sit in 00:45:47.010 --> 00:45:48.280 locations like this. 00:45:50.870 --> 00:45:56.120 And the magnetic moment here is up, the magnetic moment 00:45:56.120 --> 00:45:59.040 here is up, the magnetic moment here is down, the 00:45:59.040 --> 00:46:02.190 magnetic moment here is down. 00:46:02.190 --> 00:46:07.810 So what I've drawn here is no longer the lattice and in 00:46:07.810 --> 00:46:11.570 fact, the lattice constant of this material looks like a 00:46:11.570 --> 00:46:15.190 rock salt as far as the atomic positions are concerned, but 00:46:15.190 --> 00:46:21.100 the magnetic moments have to be continued on in another 00:46:21.100 --> 00:46:28.180 direction and some extra distance. 00:46:28.180 --> 00:46:32.500 Actually some examples of this sort of behavior is FeO, 00:46:32.500 --> 00:46:35.110 cobalt oxide, nickel oxide. 00:46:35.110 --> 00:46:38.420 All of these cations are magnetic, they have magnetic 00:46:38.420 --> 00:46:45.060 moments which are ordered and the unit cell turns out to be 00:46:45.060 --> 00:46:51.610 when you take magnetic moment into account, a larger cell, a 00:46:51.610 --> 00:46:53.180 super cell. 00:46:53.180 --> 00:46:57.320 There's a more interesting type of magnetic structure 00:46:57.320 --> 00:47:03.400 though in which the magnetic moment is inclined relative to 00:47:03.400 --> 00:47:06.660 some translation in the structure. 00:47:06.660 --> 00:47:09.730 And the magnetic moments all lie on the generators of 00:47:09.730 --> 00:47:17.940 cones, but as you walk along the chain of atoms, the 00:47:17.940 --> 00:47:21.180 orientation of the moment rotates to different 00:47:21.180 --> 00:47:22.430 orientations. 00:47:26.560 --> 00:47:32.840 There is a family of materials that are said to have cubicle 00:47:32.840 --> 00:47:45.060 spin structures in which the periodicity of the march of 00:47:45.060 --> 00:47:49.270 the magnetic moment around the surface of the cone occurs 00:47:49.270 --> 00:47:52.060 with a period that is in commensurate with the spacing 00:47:52.060 --> 00:47:54.550 of the chain of atoms. 00:47:54.550 --> 00:47:57.850 So strictly speaking, this material does not have a 00:47:57.850 --> 00:48:01.570 lattice in this direction, so it's not a crystal unless you 00:48:01.570 --> 00:48:08.200 use a fourth variable to describe the periodicity of 00:48:08.200 --> 00:48:09.790 the orientation of the moment. 00:48:09.790 --> 00:48:13.950 And one final one at the risk of carrying this too far, 00:48:13.950 --> 00:48:17.010 here's a pattern that is based on a square lattice. 00:48:17.010 --> 00:48:21.490 It has a fourfold axis in it unless I make the pattern out 00:48:21.490 --> 00:48:25.270 of squares that are black and white, make a checkerboard. 00:48:33.440 --> 00:48:36.480 This is now no longer a fourfold axis because I can't 00:48:36.480 --> 00:48:39.740 rotate 90 degrees and leave the pattern invariant. 00:48:39.740 --> 00:48:42.280 So this is an example of something called a black-white 00:48:42.280 --> 00:48:44.590 symmetry, or a color symmetry. 00:48:44.590 --> 00:48:50.610 And it requires more than just 4 operations to describe the 00:48:50.610 --> 00:48:52.670 relation between one motif and another. 00:48:52.670 --> 00:48:55.810 We need a fourth operation, switching of a color from 00:48:55.810 --> 00:48:59.030 black to white, or switching it from white to black, that's 00:48:59.030 --> 00:49:00.210 a forth operation. 00:49:00.210 --> 00:49:02.750 Again, within the confines of a pattern that 00:49:02.750 --> 00:49:03.820 exists in our space. 00:49:03.820 --> 00:49:04.317 Yes? 00:49:04.317 --> 00:49:06.429 AUDIENCE: Does that actually have-- so you're saying it 00:49:06.429 --> 00:49:08.520 doesn't add value to the [INAUDIBLE]? 00:49:08.520 --> 00:49:11.950 PROFESSOR: I did say no rotational symmetry if it has 00:49:11.950 --> 00:49:16.940 a fourfold axis here but this used to be a fourfold axis, 00:49:16.940 --> 00:49:21.430 and that now changes into a twofold axis. 00:49:21.430 --> 00:49:24.470 And then I have the problem of describing how this square is 00:49:24.470 --> 00:49:28.710 a square exactly like this square except for its color. 00:49:28.710 --> 00:49:32.120 So I need then an operation which rotates 90 degrees and 00:49:32.120 --> 00:49:34.070 switches from white to black. 00:49:34.070 --> 00:49:37.190 And then rotates 90 degrees again and switches 00:49:37.190 --> 00:49:40.130 from black to white. 00:49:40.130 --> 00:49:45.970 So there's a fifth operation, a color change that is 00:49:45.970 --> 00:49:49.990 necessary in a 3-dimensional space or a forth operation, a 00:49:49.990 --> 00:49:51.480 color change in a 2-dimensional 00:49:51.480 --> 00:49:53.820 checkerboard for example. 00:49:53.820 --> 00:49:57.660 So there are lots of nuances to symmetry theory, it's 00:49:57.660 --> 00:50:00.960 mathematics and the nice thing about mathematics is it's your 00:50:00.960 --> 00:50:03.360 ballgame, you could make up the rules and as long as you 00:50:03.360 --> 00:50:06.780 play according to those rules consistently, then you've got 00:50:06.780 --> 00:50:10.770 something that people can't quarrel with. 00:50:10.770 --> 00:50:15.320 OK, I think my internal clock has just told me that it's 00:50:15.320 --> 00:50:17.720 five minutes of the hour and it's time to take our break. 00:50:20.360 --> 00:50:23.120 Come up by all means and play with the mirrors if you'd like 00:50:23.120 --> 00:50:26.800 and we'll resume the lecture part of our 00:50:26.800 --> 00:50:28.170 discussion in 10 minutes.