1 00:00:06,740 --> 00:00:08,850 PROFESSOR: Good afternoon and welcome back. 2 00:00:08,850 --> 00:00:11,630 I'm glad you all came back even though there was a 3 00:00:11,630 --> 00:00:15,790 problem in finding a seat for every one last time. 4 00:00:15,790 --> 00:00:21,860 I would like you to turn in the problem set if you've been 5 00:00:21,860 --> 00:00:22,785 able to do it. 6 00:00:22,785 --> 00:00:25,960 If you haven't been able to do it, that's no great problem, 7 00:00:25,960 --> 00:00:30,260 but I hope you find it mildly amusing. 8 00:00:30,260 --> 00:00:33,350 I have given that problem set out a couple of times over the 9 00:00:33,350 --> 00:00:39,930 years, and I can recall one very pale student who appeared 10 00:00:39,930 --> 00:00:44,400 at my door the next morning saying it's the first problem 11 00:00:44,400 --> 00:00:47,380 set, you've only talked an hour and I can't do it. 12 00:00:47,380 --> 00:00:49,940 And then there was another group of three who formed a 13 00:00:49,940 --> 00:00:55,350 consortium to attempt to solve the code using a computer. 14 00:00:55,350 --> 00:00:57,800 They didn't get very far either. 15 00:00:57,800 --> 00:01:01,230 This is an interesting sort of problem because it requires 16 00:01:01,230 --> 00:01:04,269 you to think in a slightly different direction than 17 00:01:04,269 --> 00:01:07,190 you're accustomed to thinking, and therefore it's a little 18 00:01:07,190 --> 00:01:08,700 bit amusing. 19 00:01:08,700 --> 00:01:13,020 This is not my creation, it came from a book by a man 20 00:01:13,020 --> 00:01:15,730 named Polya, the title of the book is Mathematics and 21 00:01:15,730 --> 00:01:17,640 Plausible Reasoning. 22 00:01:17,640 --> 00:01:21,360 And he gives this as an example of a problem that can 23 00:01:21,360 --> 00:01:22,780 be solved only if you think in a 24 00:01:22,780 --> 00:01:25,100 slightly different direction. 25 00:01:25,100 --> 00:01:28,450 I'll give you another example of a problem from his book. 26 00:01:28,450 --> 00:01:33,520 Suppose, not that the problem arises in this era when you do 27 00:01:33,520 --> 00:01:37,550 all of your graphics on a computer console, but suppose 28 00:01:37,550 --> 00:01:41,160 you had to, in solving a problem in short notice, draw 29 00:01:41,160 --> 00:01:47,800 a circle that had a diameter of 4 inches. 30 00:01:47,800 --> 00:01:51,300 And you went looking for your pair of compasses which you 31 00:01:51,300 --> 00:01:54,820 never use very often, and when you found them, they were 32 00:01:54,820 --> 00:01:58,380 rusted solid, and they were open to a 33 00:01:58,380 --> 00:02:01,300 distance of 5 inches. 34 00:02:01,300 --> 00:02:02,370 OK, so there's the problem. 35 00:02:02,370 --> 00:02:04,840 You have a pair of compasses that can only draw a circle 36 00:02:04,840 --> 00:02:09,180 that's 5 inches in diameter, you must draw a circle that's 37 00:02:09,180 --> 00:02:11,130 4 inches in diameter. 38 00:02:11,130 --> 00:02:13,890 What sort of construction, what sort of mapping out of 39 00:02:13,890 --> 00:02:17,650 arcs that you connected together could you do to 40 00:02:17,650 --> 00:02:21,910 create the circle of smaller diameter? 41 00:02:21,910 --> 00:02:23,160 Anybody have an idea? 42 00:02:25,850 --> 00:02:28,960 It seems impossible doesn't it? 43 00:02:28,960 --> 00:02:33,600 Well suppose you got yourself a little block of wood that 44 00:02:33,600 --> 00:02:39,480 had a height that was equal to the square root of 5 squared 45 00:02:39,480 --> 00:02:42,460 minus 4 squared. 46 00:02:42,460 --> 00:02:45,170 And so you have one end of the compass is up on top of the 47 00:02:45,170 --> 00:02:49,580 block of wood the other end of the compass traces out a 48 00:02:49,580 --> 00:02:52,450 circle that has a smaller diameter. 49 00:02:52,450 --> 00:02:53,960 It's fairly obvious. 50 00:02:53,960 --> 00:02:57,530 But what you have to do is to think of a problem that has 51 00:02:57,530 --> 00:03:00,630 poked your nose into two-dimensions and think of it 52 00:03:00,630 --> 00:03:02,380 in terms of a three-dimensional problem, and 53 00:03:02,380 --> 00:03:03,830 then the answer is easy. 54 00:03:03,830 --> 00:03:06,140 So that was the sort of thought provoking thing that 55 00:03:06,140 --> 00:03:10,870 Polya presented in one portion of this book. 56 00:03:10,870 --> 00:03:15,200 OK, if you enjoyed that problem, I have another one 57 00:03:15,200 --> 00:03:17,540 for you in a similar vein. 58 00:03:17,540 --> 00:03:19,830 And perhaps you'll enjoy this one as well. 59 00:03:19,830 --> 00:03:22,700 So while I'm talking, let me pass this around, I think 60 00:03:22,700 --> 00:03:23,950 there's enough for everyone. 61 00:03:28,890 --> 00:03:33,580 Last time I got so caught up with the displaying the heft 62 00:03:33,580 --> 00:03:37,150 of the International Tables for X-Ray Crystallography that 63 00:03:37,150 --> 00:03:40,320 I forgot to mention entirely that there is another text 64 00:03:40,320 --> 00:03:43,510 that we will use in the class. 65 00:03:43,510 --> 00:03:47,070 And this we will not need until halfway through the 66 00:03:47,070 --> 00:03:51,450 semester, and therefore I did not feel terribly remiss in 67 00:03:51,450 --> 00:03:52,600 not mentioning it. 68 00:03:52,600 --> 00:03:59,570 It's a book by somebody named Nye, and it's called the 69 00:03:59,570 --> 00:04:02,530 Physical Properties of Crystals. 70 00:04:18,350 --> 00:04:23,680 And this is published by Oxford University Press and 71 00:04:23,680 --> 00:04:37,050 the publication date of the original addition was in 1967. 72 00:04:37,050 --> 00:04:42,760 This is a book that is really, I don't think I'm being 73 00:04:42,760 --> 00:04:46,590 extravagant in calling it a classic. 74 00:04:46,590 --> 00:04:48,550 It is a beautifully written book. 75 00:04:48,550 --> 00:04:53,340 The first 2/3 of it deal systematically with tensor 76 00:04:53,340 --> 00:04:58,130 properties, the particular sort of mathematics that is 77 00:04:58,130 --> 00:05:01,450 used to set them up, transformation of axes, and 78 00:05:01,450 --> 00:05:04,930 then looks at specific physical properties and 79 00:05:04,930 --> 00:05:09,780 numbers that have to be described in terms of tensors. 80 00:05:09,780 --> 00:05:14,210 This is actually the third book in a sequence. 81 00:05:14,210 --> 00:05:23,200 It was a book by Wooster that covered things very similarly, 82 00:05:23,200 --> 00:05:29,620 but the notation that was used for the tensors was not the 83 00:05:29,620 --> 00:05:31,350 modern current notation. 84 00:05:37,220 --> 00:05:43,790 And then the subject started in the form of a third book, 85 00:05:43,790 --> 00:05:45,380 Woldemar Voigt. 86 00:05:47,900 --> 00:05:54,060 And the title of this book is Lehrbuch der Kristallphysik. 87 00:06:10,000 --> 00:06:14,820 And this was published back in 1910. 88 00:06:14,820 --> 00:06:16,970 This is the first time anybody had anything 89 00:06:16,970 --> 00:06:19,926 to say on the matter. 90 00:06:19,926 --> 00:06:24,460 It in fact is a big fat book that contains some topics that 91 00:06:24,460 --> 00:06:27,030 are not covered in Wooster and Nye. 92 00:06:29,680 --> 00:06:31,940 Nye's book is beautifully written. 93 00:06:31,940 --> 00:06:36,410 The first 2/3 of it concerns tensor formalism and the last 94 00:06:36,410 --> 00:06:39,580 1/3, which we will not touch at all, deals with 95 00:06:39,580 --> 00:06:42,470 thermodynamic relations between different properties 96 00:06:42,470 --> 00:06:45,330 that are represented by tensors. 97 00:06:45,330 --> 00:06:52,120 So I don't recommend you go out and buy this book until 98 00:06:52,120 --> 00:06:54,680 you determine whether or not you need it because I'll try 99 00:06:54,680 --> 00:06:57,180 to make the lectures self-contained and I'll have 100 00:06:57,180 --> 00:06:59,570 lots of notes and handouts. 101 00:06:59,570 --> 00:07:03,610 The problem with Nye's book, and any book that is intensely 102 00:07:03,610 --> 00:07:09,000 mathematical, is that you can't jump in on page 73 to 103 00:07:09,000 --> 00:07:11,330 find the answer to a specific question. 104 00:07:11,330 --> 00:07:14,150 Because when you go there, it will say as we showed back in 105 00:07:14,150 --> 00:07:16,540 Chapter 4, now what is he talking about? 106 00:07:16,540 --> 00:07:20,520 So you go back to Chapter 4, and Chapter 4 says, starting 107 00:07:20,520 --> 00:07:23,720 with our definition of Chapter 2, and you have to go back and 108 00:07:23,720 --> 00:07:24,630 read Chapter 2. 109 00:07:24,630 --> 00:07:27,920 So it's awfully hard to pick something out to answer a 110 00:07:27,920 --> 00:07:28,810 specific question. 111 00:07:28,810 --> 00:07:32,230 You have to really go all the way through it. 112 00:07:32,230 --> 00:07:36,200 The good news is that Nye's book has been 113 00:07:36,200 --> 00:07:38,960 published in paperback. 114 00:07:38,960 --> 00:07:42,650 And it is available at the COOP and paperback means 115 00:07:42,650 --> 00:07:46,000 cheap, cheap, or relatively inexpensive. 116 00:07:46,000 --> 00:07:49,230 No books are really cheap these days. 117 00:07:49,230 --> 00:07:53,000 So we will cover material that's in there, we'll have a 118 00:07:53,000 --> 00:07:55,950 slightly different emphasis, but the notation and the 119 00:07:55,950 --> 00:08:01,380 general mathematics that's involved is in Nye's book. 120 00:08:01,380 --> 00:08:05,240 The second thing that I mentioned last time is that 121 00:08:05,240 --> 00:08:10,510 there is a very, very nice and thorough and geometric 122 00:08:10,510 --> 00:08:12,950 treatment of crystal symmetry. 123 00:08:12,950 --> 00:08:14,390 And I said that's the good news. 124 00:08:14,390 --> 00:08:18,470 The bad news is that it's out of print, so I promised, what 125 00:08:18,470 --> 00:08:21,260 a guy, that I give you a Xerox copy of the 126 00:08:21,260 --> 00:08:22,900 first half of the book. 127 00:08:22,900 --> 00:08:25,630 So here is the text that we'll use in the 128 00:08:25,630 --> 00:08:26,880 first part of the term. 129 00:08:33,450 --> 00:08:37,260 I included at the beginning, the table of contents, so you 130 00:08:37,260 --> 00:08:42,840 can see each other topics that are covered in the book. 131 00:08:46,310 --> 00:08:52,190 We will not go through all of the material that's covered. 132 00:08:52,190 --> 00:08:59,420 There are a lot of different symmetries to be derived, and 133 00:08:59,420 --> 00:09:02,610 it turns out that if you get the general idea and you can 134 00:09:02,610 --> 00:09:06,940 summarize the results, there's no need to derive every single 135 00:09:06,940 --> 00:09:09,560 one of them. 136 00:09:09,560 --> 00:09:12,840 It's nice to know that there's a place where you can find out 137 00:09:12,840 --> 00:09:16,105 how it is done if you really have a particular question. 138 00:09:16,105 --> 00:09:19,940 Did everybody get a copy or are there a few who did not? 139 00:09:19,940 --> 00:09:23,540 I made extras, OK, nobody in need of one, great. 140 00:09:27,420 --> 00:09:36,250 OK, let me now start with a general rhetorical question. 141 00:09:36,250 --> 00:09:40,740 Crystallography, as we mentioned last time, is the 142 00:09:40,740 --> 00:09:42,740 geometry of crystals. 143 00:09:42,740 --> 00:09:46,100 It's the geometry of patterns and the sorts of symmetries 144 00:09:46,100 --> 00:09:48,280 that are in those patterns. 145 00:09:48,280 --> 00:09:50,950 Now you might ask yourself, why should I as a material 146 00:09:50,950 --> 00:09:53,940 scientist or a physical scientist of some sort, worry 147 00:09:53,940 --> 00:09:55,070 about this stuff? 148 00:09:55,070 --> 00:09:58,810 I'm not training to be a wallpaper designer, I'm going 149 00:09:58,810 --> 00:10:00,040 to do physical things. 150 00:10:00,040 --> 00:10:04,250 I'm going to heat things and measure properties, and that 151 00:10:04,250 --> 00:10:06,290 sort of thing. 152 00:10:06,290 --> 00:10:11,550 Well there are at least three answers to that question. 153 00:10:11,550 --> 00:10:15,330 First of all, whether you like it or not, the arcane language 154 00:10:15,330 --> 00:10:17,830 of symmetry is the language that's 155 00:10:17,830 --> 00:10:20,620 used to describe crystals. 156 00:10:20,620 --> 00:10:24,460 It's the language that's used to describe structures. 157 00:10:24,460 --> 00:10:27,000 The normal thing that you do when you're trying to describe 158 00:10:27,000 --> 00:10:30,410 verbally a ball and pin model of the geometrical arrangement 159 00:10:30,410 --> 00:10:33,940 of atoms in a crystal is to say the red balls are at the 160 00:10:33,940 --> 00:10:37,410 corners of the cube, the green balls are in the middle of the 161 00:10:37,410 --> 00:10:41,520 edges, and the chartreuse balls are sort of tucked up 162 00:10:41,520 --> 00:10:45,210 inside one of the corners, but slightly closer to 1 face than 163 00:10:45,210 --> 00:10:48,090 to the other 2 faces. 164 00:10:48,090 --> 00:10:51,840 The point I'm trying to make is that is a language that has 165 00:10:51,840 --> 00:10:54,350 limited utility. 166 00:10:54,350 --> 00:10:58,340 It deals, it's capable of dealing only with the simplest 167 00:10:58,340 --> 00:11:00,560 sort of atomic configurations. 168 00:11:00,560 --> 00:11:03,580 So there is a general language based on symmetry theory, 169 00:11:03,580 --> 00:11:08,270 based on group theory, that is universally used to describe 170 00:11:08,270 --> 00:11:10,060 atomic arrangements. 171 00:11:10,060 --> 00:11:13,300 So instead of saying red balls at the corners of the cube and 172 00:11:13,300 --> 00:11:17,400 green balls in the middle of the faces, I can say space 173 00:11:17,400 --> 00:11:26,600 group 4 over m3 bar 2 over m, atom a in position for b, m3m, 174 00:11:26,600 --> 00:11:32,090 atom b in position for c, m3m. 175 00:11:32,090 --> 00:11:34,100 That's what rock salt it. 176 00:11:34,100 --> 00:11:38,400 And that is the way, not only it, but especially more 177 00:11:38,400 --> 00:11:42,540 complicated structural arrangements are described in 178 00:11:42,540 --> 00:11:44,210 the literature. 179 00:11:44,210 --> 00:11:46,780 So this is the language of describing such arrangements. 180 00:11:46,780 --> 00:11:51,640 And finally, sooner or later, I bet you that every one of 181 00:11:51,640 --> 00:11:55,950 you will be involved with some crystalline material, and the 182 00:11:55,950 --> 00:12:00,590 first question you will answer is what is its structure, what 183 00:12:00,590 --> 00:12:01,870 is the atomic arrangement. 184 00:12:01,870 --> 00:12:04,050 That's where properties start. 185 00:12:04,050 --> 00:12:09,080 And you'll go a book or a set of volumes that describe 186 00:12:09,080 --> 00:12:10,530 structural data. 187 00:12:10,530 --> 00:12:14,020 There's a big long compendium of books that fill about that 188 00:12:14,020 --> 00:12:17,520 much of a bookshelf which are called structure reports. 189 00:12:17,520 --> 00:12:22,720 They started a number of years ago to compile all of the 190 00:12:22,720 --> 00:12:24,970 structures that had been determined within a given 191 00:12:24,970 --> 00:12:26,290 calendar year. 192 00:12:26,290 --> 00:12:29,710 They did a pretty good job of staying caught up 193 00:12:29,710 --> 00:12:32,680 back in 1915 and 1920. 194 00:12:32,680 --> 00:12:37,260 And then as it became easier to obtain such results, partly 195 00:12:37,260 --> 00:12:42,730 due to the advent of rapid large computers, they fell 196 00:12:42,730 --> 00:12:45,270 further and further behind and I think now they are about 5 197 00:12:45,270 --> 00:12:45,970 or 10 miles-- 198 00:12:45,970 --> 00:12:49,350 5 or 10 years, miles as well. 199 00:12:49,350 --> 00:12:54,400 But this is one of the places to go to look up, without 200 00:12:54,400 --> 00:12:57,080 going to the original literature, whether the 201 00:12:57,080 --> 00:12:59,260 material that you're interested in has had its 202 00:12:59,260 --> 00:13:02,270 atomic arrangement determined. 203 00:13:02,270 --> 00:13:04,600 When you go there you're going to find the atomic 204 00:13:04,600 --> 00:13:08,140 arrangement, not in terms of red balls at one position on 205 00:13:08,140 --> 00:13:10,690 the cell, but you're going to find it in terms of the 206 00:13:10,690 --> 00:13:12,590 language of symmetry theory. 207 00:13:12,590 --> 00:13:15,370 So one of the things I hope you'll be able to do by the 208 00:13:15,370 --> 00:13:18,860 time we finish this time together, is to be able to go 209 00:13:18,860 --> 00:13:22,070 to such literature and know exactly what to do and where 210 00:13:22,070 --> 00:13:25,190 to go to reconstruct the geometrical 211 00:13:25,190 --> 00:13:26,440 arrangement of the atoms. 212 00:13:28,880 --> 00:13:32,880 OK, so hopefully you're at least mildly convinced that 213 00:13:32,880 --> 00:13:37,410 this exercise is going to be worthwhile. 214 00:13:37,410 --> 00:13:40,800 Before we continue where we left off last time, I would 215 00:13:40,800 --> 00:13:47,340 like to say a little bit about the language in which these 216 00:13:47,340 --> 00:13:50,490 geometries are described. 217 00:13:50,490 --> 00:13:54,090 And we mentioned last time without thoroughly 218 00:13:54,090 --> 00:13:58,130 demonstrating why that in a 3-dimensional space there are 219 00:13:58,130 --> 00:14:01,410 4 basically different kinds of operations. 220 00:14:01,410 --> 00:14:06,430 And one of these is something that all crystals must by 221 00:14:06,430 --> 00:14:10,690 definition display, and this is the operation of 222 00:14:10,690 --> 00:14:13,070 translation. 223 00:14:13,070 --> 00:14:17,310 Analytically it can be described as a mapping in 224 00:14:17,310 --> 00:14:22,500 which every coordinate in a space xyz is mapped to a 225 00:14:22,500 --> 00:14:27,460 location x plus some constant, y plus some constant, z plus 226 00:14:27,460 --> 00:14:28,870 some constant. 227 00:14:28,870 --> 00:14:32,850 And if you do the operation again, this would go to a 228 00:14:32,850 --> 00:14:38,620 location x plus 2a y plus 2b, z plus 2c. 229 00:14:42,240 --> 00:14:46,430 A feature of translation that is unique to this particular 230 00:14:46,430 --> 00:14:51,390 symmetry transformation is that it has no origin. 231 00:14:51,390 --> 00:14:55,700 If I have a pair of motifs that are related by 232 00:14:55,700 --> 00:15:01,290 translation, we could think of them as being related by a 233 00:15:01,290 --> 00:15:04,570 vector, magnitude and direction, that takes this 234 00:15:04,570 --> 00:15:06,970 motif and moves it to this location. 235 00:15:06,970 --> 00:15:10,130 We said that more generally we should view these operations, 236 00:15:10,130 --> 00:15:13,900 not just acting on one little domain and space, but acting 237 00:15:13,900 --> 00:15:14,980 on everything. 238 00:15:14,980 --> 00:15:21,360 So this implies that there be a infinite chain of motifs if 239 00:15:21,360 --> 00:15:24,340 the operation of translation is to be present. 240 00:15:24,340 --> 00:15:27,740 Because only that infinite, doubly infinite string is 241 00:15:27,740 --> 00:15:32,100 consistent with all of space being mapped into itself. 242 00:15:32,100 --> 00:15:34,900 Like any vector, there's no unique origin. 243 00:15:34,900 --> 00:15:37,910 You could say it extends from here to here, or from here to 244 00:15:37,910 --> 00:15:43,720 here, or any other choice of translation, provided the 245 00:15:43,720 --> 00:15:49,140 direction and the magnitude are the same in every choice. 246 00:15:49,140 --> 00:15:56,110 As a result, it's not really possible to specify the locus 247 00:15:56,110 --> 00:15:57,495 of this particular operation. 248 00:15:57,495 --> 00:15:59,310 It has magnitude and direction, 249 00:15:59,310 --> 00:16:00,875 but no unique origin. 250 00:16:03,860 --> 00:16:08,480 What we can do through a very neat device is to 251 00:16:08,480 --> 00:16:19,130 nevertheless, take some reference point and have each 252 00:16:19,130 --> 00:16:22,830 of these reference points separated by T, and have each 253 00:16:22,830 --> 00:16:27,470 motif lurking off in space in exactly the same location and 254 00:16:27,470 --> 00:16:30,650 distance from this point that we've constructed. 255 00:16:30,650 --> 00:16:33,640 And this array of abstractions, so these 256 00:16:33,640 --> 00:16:37,210 geometrical abstractions, these points, are what are 257 00:16:37,210 --> 00:16:48,550 called lattice points, and they are a very neat summary 258 00:16:48,550 --> 00:16:52,150 of the translational periodicity of the crystal. 259 00:16:52,150 --> 00:16:57,030 It is absolutely essential not to mix up these lattice points 260 00:16:57,030 --> 00:17:00,910 which are a construct that we have created, and the atoms 261 00:17:00,910 --> 00:17:04,230 themselves that are present in a crystal. 262 00:17:04,230 --> 00:17:07,200 The atoms are atoms and they're not necessarily the 263 00:17:07,200 --> 00:17:08,520 lattice points. 264 00:17:08,520 --> 00:17:11,470 Another way of saying that not all atoms of the same chemical 265 00:17:11,470 --> 00:17:14,440 species need be translation equivalent. 266 00:17:14,440 --> 00:17:18,650 We'll see some examples of this later on, so do not mix 267 00:17:18,650 --> 00:17:21,310 up the atoms and the lattice points. 268 00:17:21,310 --> 00:17:25,339 When I talk about the sodium chloride lattice, I mean an 269 00:17:25,339 --> 00:17:31,430 array of points in space that are located at the corners of 270 00:17:31,430 --> 00:17:34,510 a cube and in the middle of the faces of the cube. 271 00:17:34,510 --> 00:17:37,860 If I talk about the arrangement of sodium ions and 272 00:17:37,860 --> 00:17:41,930 chlorine ions, that is the sodium chloride structure and 273 00:17:41,930 --> 00:17:45,160 not the sodium chloride lattice. 274 00:17:45,160 --> 00:17:48,190 And then last time I apologized for usage so as not 275 00:17:48,190 --> 00:17:50,160 to appear hypocritical. 276 00:17:50,160 --> 00:17:53,730 Everybody talks about lattice vibration, lattice energy, 277 00:17:53,730 --> 00:17:56,190 lattice dynamics, and so on, but that's a 278 00:17:56,190 --> 00:17:57,650 misuse of the term. 279 00:17:57,650 --> 00:18:01,100 But nevertheless, it is much more musical than saying 280 00:18:01,100 --> 00:18:03,390 structure energy, structure vibration. 281 00:18:03,390 --> 00:18:09,150 So we'll go on misusing the term lattice I'm afraid. 282 00:18:09,150 --> 00:18:12,420 OK, let's look at another operation. 283 00:18:12,420 --> 00:18:15,860 Here we change the sense of no coordinate. 284 00:18:15,860 --> 00:18:21,570 Let's next look at an operation that might take xyz, 285 00:18:21,570 --> 00:18:25,838 and map it into minus xyz. 286 00:18:29,830 --> 00:18:33,520 This would be a situation where if I set up a coordinate 287 00:18:33,520 --> 00:18:41,900 system, here's x, here's y, here's z. 288 00:18:41,900 --> 00:18:46,170 What I've done is to take an object that sits off here, at 289 00:18:46,170 --> 00:18:51,260 a coordinate plus x, and I've changed the sign of x so that 290 00:18:51,260 --> 00:18:53,725 this object now sits off here. 291 00:18:59,410 --> 00:19:04,020 This is exactly what happens when I take something and 292 00:19:04,020 --> 00:19:07,010 reflect it in a mirror. 293 00:19:07,010 --> 00:19:10,270 And if that's not immediately obvious, it just so happens, 294 00:19:10,270 --> 00:19:12,170 not at all by accident, I brought 295 00:19:12,170 --> 00:19:15,280 along with me a mirror. 296 00:19:15,280 --> 00:19:20,590 OK, here is one hand, and if you look in the mirror, there 297 00:19:20,590 --> 00:19:21,480 is the other hand. 298 00:19:21,480 --> 00:19:26,410 It's the same distance behind the plane of the mirror, two 299 00:19:26,410 --> 00:19:30,230 coordinates have been left unchanged. 300 00:19:30,230 --> 00:19:33,620 The two coordinates within the plane of the mirror, if that 301 00:19:33,620 --> 00:19:36,220 is my choice of the reference system, and one of them has 302 00:19:36,220 --> 00:19:38,670 been reversed. 303 00:19:38,670 --> 00:19:42,650 Now I'll take the second motif out of the geometric 304 00:19:42,650 --> 00:19:45,800 construct, and I'd like to point out one very curious 305 00:19:45,800 --> 00:19:49,220 feature of the pair of motifs that's generated by this 306 00:19:49,220 --> 00:19:51,110 transformation. 307 00:19:51,110 --> 00:19:54,820 They're both the same thing, clearly. 308 00:19:54,820 --> 00:19:59,820 But no matter how I try, I cannot move one so that it 309 00:19:59,820 --> 00:20:02,740 coincides with the other. 310 00:20:02,740 --> 00:20:06,100 And we intuitively appreciate this difference by saying we 311 00:20:06,100 --> 00:20:08,850 actually use our hands by analogy. 312 00:20:08,850 --> 00:20:14,930 We say one is left-handed and one is right-handed, and they 313 00:20:14,930 --> 00:20:16,380 are not congruent. 314 00:20:16,380 --> 00:20:19,740 The fancy name that's used to describe this relation is to 315 00:20:19,740 --> 00:20:21,420 say that they are enantiomorph. 316 00:20:27,770 --> 00:20:31,740 Another term that's used, particularly in chemistry, is 317 00:20:31,740 --> 00:20:33,470 to say that they are chiral. 318 00:20:39,880 --> 00:20:41,750 Which one is the left-handed one, which is a 319 00:20:41,750 --> 00:20:44,150 right-handed one? 320 00:20:44,150 --> 00:20:46,620 This is what I call my left hand, this is what I call my 321 00:20:46,620 --> 00:20:47,230 right hand. 322 00:20:47,230 --> 00:20:52,010 But can we distinguish them physically, any other way? 323 00:20:52,010 --> 00:20:57,040 No, these are terms that have come in to regular use in both 324 00:20:57,040 --> 00:21:01,800 our everyday language and also in science because we use our 325 00:21:01,800 --> 00:21:07,620 hands instinctively as readily available examples of 326 00:21:07,620 --> 00:21:12,410 enantiomorphs, readily at hand, I might say to almost 327 00:21:12,410 --> 00:21:15,280 make a pun. 328 00:21:15,280 --> 00:21:20,690 One of the really brilliant figures in physics was a man 329 00:21:20,690 --> 00:21:22,790 named Richard Feynman. 330 00:21:22,790 --> 00:21:28,160 Recently deceased, Feynman gave a very famous series of 331 00:21:28,160 --> 00:21:31,220 lectures on science at Cornell University. 332 00:21:31,220 --> 00:21:35,640 And he has one entire vector that was devoted to the 333 00:21:35,640 --> 00:21:39,600 difference between right and left. 334 00:21:39,600 --> 00:21:43,220 And he comes up with a funny story, he pretends that the 335 00:21:43,220 --> 00:21:46,750 hero of the story is someone who's trying to communicate 336 00:21:46,750 --> 00:21:51,540 with beings in outer space and suddenly he gets lucky and he 337 00:21:51,540 --> 00:21:54,130 gets a response to his message. 338 00:21:54,130 --> 00:21:56,830 And they work out a way to communicate and eventually 339 00:21:56,830 --> 00:22:02,100 they try to describe each other to the other individual. 340 00:22:02,100 --> 00:22:03,390 Well what they look like? 341 00:22:03,390 --> 00:22:08,780 Well we're bipedal, and we have two organs related by 342 00:22:08,780 --> 00:22:12,340 reflection that let us sense light and form images. 343 00:22:12,340 --> 00:22:15,940 And we have an aperture through which we ingest things 344 00:22:15,940 --> 00:22:17,820 that can be metabolized. 345 00:22:17,820 --> 00:22:21,750 And our circulation and body works because we have a pump 346 00:22:21,750 --> 00:22:24,790 on the left hand side that circulates 347 00:22:24,790 --> 00:22:26,100 fluids through our-- 348 00:22:26,100 --> 00:22:29,900 wait, I don't understand, what's left? 349 00:22:29,900 --> 00:22:34,590 So then there follows along this course on how to define 350 00:22:34,590 --> 00:22:37,690 left in an absolute sense. 351 00:22:37,690 --> 00:22:40,620 How do you describe to someone what makes your left hand 352 00:22:40,620 --> 00:22:43,450 left, and your right hand right without being 353 00:22:43,450 --> 00:22:45,080 anthropomorphic about it. 354 00:22:45,080 --> 00:22:48,220 So it goes on and on and on and he gets into physical 355 00:22:48,220 --> 00:22:49,810 phenomena which are objective and 356 00:22:49,810 --> 00:22:51,530 independent of a human being. 357 00:22:51,530 --> 00:22:58,360 And finally he comes to the anisotropic emission beta 358 00:22:58,360 --> 00:23:01,050 particles in radioactive decay. 359 00:23:01,050 --> 00:23:04,360 And that depends on direction relative to the magnetic 360 00:23:04,360 --> 00:23:07,350 moment and that defines an absolute sense 361 00:23:07,350 --> 00:23:08,880 of right and left. 362 00:23:08,880 --> 00:23:11,460 So that's something physical, it doesn't depend on the 363 00:23:11,460 --> 00:23:13,500 nature of the human being. 364 00:23:13,500 --> 00:23:17,970 And then Feynman wraps up his story by saying, if finally 365 00:23:17,970 --> 00:23:22,140 our extraterrestrial being travels to space, gets out of 366 00:23:22,140 --> 00:23:25,320 his spaceship and he walks forward to greet you and you 367 00:23:25,320 --> 00:23:28,810 put out your right hand, and he puts out his left hand, get 368 00:23:28,810 --> 00:23:32,490 out of their fast because it means he is made out of 369 00:23:32,490 --> 00:23:34,710 anti-matter. 370 00:23:34,710 --> 00:23:38,810 And nuclei of matter in this anisotropic emission of data 371 00:23:38,810 --> 00:23:42,020 rays shoot out the beta particle in one sense, 372 00:23:42,020 --> 00:23:44,220 anti-matter shoots out the beta particle 373 00:23:44,220 --> 00:23:48,160 in the chiral sense. 374 00:23:48,160 --> 00:23:53,640 So it's a cute little story that emphasizes the problem in 375 00:23:53,640 --> 00:23:57,700 defining absolutely left from right. 376 00:23:57,700 --> 00:23:59,930 But they're of opposite handedness that we can say. 377 00:24:06,540 --> 00:24:15,320 Mirrors are interesting things and I brought along a couple 378 00:24:15,320 --> 00:24:24,000 of mirrors and they have very, very peculiar characteristics. 379 00:24:24,000 --> 00:24:28,260 And I would invite you to come up and look at these in 380 00:24:28,260 --> 00:24:30,900 private because if I hold them up in front of you, you're not 381 00:24:30,900 --> 00:24:33,670 going to be able to see what I'm doing at all, although I 382 00:24:33,670 --> 00:24:37,440 kid myself that you can, and I walk around and show 383 00:24:37,440 --> 00:24:38,800 you what I'm doing. 384 00:24:38,800 --> 00:24:42,750 Here is something scientific, it's a chemical compound 385 00:24:42,750 --> 00:24:45,890 carbon dioxide, and-- 386 00:24:55,850 --> 00:25:01,830 OK, and if I hand this down you can see carbon dioxide in 387 00:25:01,830 --> 00:25:03,646 the mirror. 388 00:25:03,646 --> 00:25:06,930 Can you see that? 389 00:25:06,930 --> 00:25:08,850 Uh oh, it's not reflecting-- 390 00:25:08,850 --> 00:25:10,560 sorry, I didn't turn it on. 391 00:25:14,180 --> 00:25:16,960 Now I think we can get it. 392 00:25:16,960 --> 00:25:19,200 And is it working now? 393 00:25:32,580 --> 00:25:35,720 OK, now it's working. 394 00:25:35,720 --> 00:25:40,200 You can see why this is a very special kind of mirror because 395 00:25:40,200 --> 00:25:45,180 it reflects only red letters and it leaves the black 396 00:25:45,180 --> 00:25:46,430 letters unchanged. 397 00:25:56,342 --> 00:25:58,460 You're going to have to come up, I see some of you 398 00:25:58,460 --> 00:26:00,450 straining your necks, you'll have to come up and look at 399 00:26:00,450 --> 00:26:01,860 that in person. 400 00:26:01,860 --> 00:26:07,160 But the black letters are completely unchanged, the red 401 00:26:07,160 --> 00:26:09,130 letters are reflected into letters 402 00:26:09,130 --> 00:26:12,140 of an opposite chirality. 403 00:26:12,140 --> 00:26:16,290 It's a very special kind of mirror. 404 00:26:16,290 --> 00:26:20,010 I've got another kind of mirror that works in a 405 00:26:20,010 --> 00:26:21,260 different way. 406 00:26:26,940 --> 00:26:29,450 Where Is my other piece of paper? 407 00:26:29,450 --> 00:26:30,700 OK. 408 00:26:35,520 --> 00:26:42,620 This is an interesting mirror because it reflects only male 409 00:26:42,620 --> 00:26:48,300 names are not female names. 410 00:26:48,300 --> 00:26:51,880 This was a very topical sort of mirror a few years ago when 411 00:26:51,880 --> 00:26:52,840 there was a lawsuit. 412 00:26:52,840 --> 00:26:56,210 There was a college down South called the Citadel which would 413 00:26:56,210 --> 00:27:01,380 only admit male applicants and not female applicants. 414 00:27:01,380 --> 00:27:03,830 So I claim that this was a mirror that I got from the 415 00:27:03,830 --> 00:27:09,590 Citadel because it doesn't change the male names, but 416 00:27:09,590 --> 00:27:15,160 does change, does reject or reflect, female names. 417 00:27:15,160 --> 00:27:18,480 So you can play with this during our break. 418 00:27:18,480 --> 00:27:23,120 But if I look at myself in a mirror, I 419 00:27:23,120 --> 00:27:25,870 take a look at myself. 420 00:27:25,870 --> 00:27:29,520 If I wink my left eye, the in there winks his 421 00:27:29,520 --> 00:27:32,880 right eye back at me. 422 00:27:32,880 --> 00:27:36,540 So I'm not really seeing myself, what I'm seeing is my 423 00:27:36,540 --> 00:27:38,390 enantiomorph. 424 00:27:38,390 --> 00:27:40,620 Doesn't that shake you up? 425 00:27:40,620 --> 00:27:45,330 You have never ever seen yourself in exactly the same 426 00:27:45,330 --> 00:27:49,510 way as other people see you. 427 00:27:49,510 --> 00:27:51,740 You are only familiar with your enantiomorph. 428 00:27:56,020 --> 00:27:57,330 Does that make a difference? 429 00:27:57,330 --> 00:27:59,450 Well I'll bring in something that I put together and I 430 00:27:59,450 --> 00:28:01,850 couldn't put my hands on. 431 00:28:01,850 --> 00:28:06,470 We are very, very sensitive to the symmetry in our faces. 432 00:28:06,470 --> 00:28:09,060 And if they are reflected left to right, you surely are going 433 00:28:09,060 --> 00:28:11,320 to look different to the other person. 434 00:28:11,320 --> 00:28:14,780 And the way to see that is to take a photograph of somebody 435 00:28:14,780 --> 00:28:17,940 and cut it down the middle, and put the two different 436 00:28:17,940 --> 00:28:20,750 sides reflected left to right. 437 00:28:20,750 --> 00:28:24,460 And the expression on the person's face, and the general 438 00:28:24,460 --> 00:28:29,670 spirit that that image conveys is entirely different if you 439 00:28:29,670 --> 00:28:33,020 use the one half of the face reflected left to right, and 440 00:28:33,020 --> 00:28:36,040 the other half reflected left to right. 441 00:28:36,040 --> 00:28:39,810 So think of this, when you look in the mirror, you see 442 00:28:39,810 --> 00:28:44,260 your enantiomorph and other people see you differently. 443 00:28:44,260 --> 00:28:46,070 Let me ask you to scratch your head now. 444 00:28:46,070 --> 00:28:49,640 Is there any time when, in point of fact, you may have 445 00:28:49,640 --> 00:28:54,930 seen yourself without being reflected into the 446 00:28:54,930 --> 00:28:55,490 enantiomorph? 447 00:28:55,490 --> 00:28:55,870 Yeah. 448 00:28:55,870 --> 00:28:57,430 AUDIENCE: Picture? 449 00:28:57,430 --> 00:28:57,710 PROFESSOR: Absolutely. 450 00:28:57,710 --> 00:29:01,170 A Photograph or a TV monitor. 451 00:29:01,170 --> 00:29:06,220 When you look at a picture on television, you can read all 452 00:29:06,220 --> 00:29:09,540 the signs, and they didn't make up special signs in 453 00:29:09,540 --> 00:29:11,290 reflection so that they'd look right when they 454 00:29:11,290 --> 00:29:12,400 photographed you. 455 00:29:12,400 --> 00:29:15,760 So photography or a video camera does 456 00:29:15,760 --> 00:29:18,120 not change the chirality. 457 00:29:18,120 --> 00:29:22,920 But I've got another way in which I can see myself in the 458 00:29:22,920 --> 00:29:24,650 exact same chirality. 459 00:29:24,650 --> 00:29:26,950 And this I can't really convince you of, 460 00:29:26,950 --> 00:29:28,200 you'll have to try it. 461 00:29:28,200 --> 00:29:32,730 If I put two mirrors together at 90 degrees, and then adjust 462 00:29:32,730 --> 00:29:37,450 them so that I am looking right the point of 463 00:29:37,450 --> 00:29:42,690 intersection so that my two images coincide, then I see 464 00:29:42,690 --> 00:29:46,270 myself in the normal way. 465 00:29:46,270 --> 00:29:51,130 If I now blink my left eye, this guy blinks his 466 00:29:51,130 --> 00:29:53,020 left eye at me too. 467 00:29:53,020 --> 00:29:56,710 This is really astounding, two mirrors at 90 degrees, if you 468 00:29:56,710 --> 00:29:59,260 look at yourself right at their point of intersection, 469 00:29:59,260 --> 00:30:03,970 give you a non-chiral image of yourself. 470 00:30:03,970 --> 00:30:06,350 So I invite you to come up and try that, that's truly 471 00:30:06,350 --> 00:30:07,600 astounding. 472 00:30:09,550 --> 00:30:12,980 So why should somebody in material science or chemistry 473 00:30:12,980 --> 00:30:14,950 care about chirality? 474 00:30:14,950 --> 00:30:16,270 Does it really make any difference? 475 00:30:18,815 --> 00:30:23,350 Let me give you a little experiment that you can try. 476 00:30:23,350 --> 00:30:25,840 Suppose you have a little cell on a couple pieces of 477 00:30:25,840 --> 00:30:28,950 Polaroid, and the cell has a glass front and a glass back 478 00:30:28,950 --> 00:30:31,690 and you fill it with sugar solution. 479 00:30:31,690 --> 00:30:35,570 And then you pass a beam of polarized light through the 480 00:30:35,570 --> 00:30:41,250 sugar solution, and what happens is that the sugar 481 00:30:41,250 --> 00:30:48,240 solution rotates the direction of polarization in proportion 482 00:30:48,240 --> 00:30:50,620 to the thickness of solution at the light is passed 483 00:30:50,620 --> 00:30:54,690 through, in proportion to the concentration of sugar. 484 00:30:54,690 --> 00:30:57,600 The point of polarization gets rotated. 485 00:30:57,600 --> 00:31:00,440 Now that's pretty curious, so you scratch 486 00:31:00,440 --> 00:31:01,340 your head about that. 487 00:31:01,340 --> 00:31:03,332 Why does that happen? 488 00:31:03,332 --> 00:31:05,810 Well, maybe I'd better go back and try it again. 489 00:31:05,810 --> 00:31:08,390 And a day or two later, you go back and you repeat the 490 00:31:08,390 --> 00:31:09,410 experiment. 491 00:31:09,410 --> 00:31:16,430 And once again, the point of polarization rotates, but it 492 00:31:16,430 --> 00:31:18,620 rotates in the opposite direction. 493 00:31:23,500 --> 00:31:27,400 The reason for this is that if I was not careful in 494 00:31:27,400 --> 00:31:30,410 cleanliness and there were some little bugs lurking in 495 00:31:30,410 --> 00:31:33,440 the corners of that cell and when they sense the sugar 496 00:31:33,440 --> 00:31:36,165 solution, they said wow, free lunch, and they crawled out 497 00:31:36,165 --> 00:31:37,050 And gobbled it up. 498 00:31:37,050 --> 00:31:40,650 It turns out, those guys can gobble up just the sugar of 499 00:31:40,650 --> 00:31:42,160 one chirality. 500 00:31:42,160 --> 00:31:45,650 Sugar is a chiral molecule. 501 00:31:45,650 --> 00:31:50,710 And in fact there is a product that's called invert sugar and 502 00:31:50,710 --> 00:31:54,210 this is sugar that is all of one handedness. 503 00:31:54,210 --> 00:31:57,520 But everything in the world around us, everything from 504 00:31:57,520 --> 00:32:00,980 sugar beats to sugar cane to other things that make 505 00:32:00,980 --> 00:32:08,830 sucrose, manufacture sugar of one chirality, not mixed. 506 00:32:08,830 --> 00:32:12,880 All chiral molecules that are produced by living organisms 507 00:32:12,880 --> 00:32:15,575 are all of the same kind chirality. 508 00:32:15,575 --> 00:32:19,910 If we make them synthetically, there's no reason to favor 509 00:32:19,910 --> 00:32:23,820 synthesis of one molecule or the opposite handedness, so 510 00:32:23,820 --> 00:32:28,830 synthesized molecules are of equal proportion in the 511 00:32:28,830 --> 00:32:33,170 left-handed chirality and the right-hand chirality. 512 00:32:33,170 --> 00:32:38,130 This means that in the case of pharmaceuticals at the very 513 00:32:38,130 --> 00:32:44,030 best, you are going to use only half of the product that 514 00:32:44,030 --> 00:32:45,220 you've made. 515 00:32:45,220 --> 00:32:49,950 There is a pharmaceutical product that is prescribed for 516 00:32:49,950 --> 00:32:55,600 attention deficit disorder, this is called Ritalin, and 517 00:32:55,600 --> 00:33:00,480 only one chirality of the Ritalin molecule does 518 00:33:00,480 --> 00:33:01,610 anything for you. 519 00:33:01,610 --> 00:33:03,020 The other part is just metabolized 520 00:33:03,020 --> 00:33:05,360 and doesn't do anything. 521 00:33:05,360 --> 00:33:08,990 But there are other much more sinister cases. 522 00:33:08,990 --> 00:33:13,750 There was a serious problem about 20 years ago, primarily 523 00:33:13,750 --> 00:33:20,670 in Europe, where a particular pharmaceutical thalidomide was 524 00:33:20,670 --> 00:33:25,650 prescribed for pregnant women, it was to act as a sedative. 525 00:33:25,650 --> 00:33:29,530 Only one chirality of the molecule did this, the other 526 00:33:29,530 --> 00:33:34,590 chirality tragically caused birth defects. 527 00:33:34,590 --> 00:33:38,980 So you have to be very careful about the chirality of the 528 00:33:38,980 --> 00:33:42,630 pharmaceutical molecule that you synthesize. 529 00:33:42,630 --> 00:33:45,800 Another example, there is-- 530 00:33:48,340 --> 00:33:50,405 I don't remember the name of it. 531 00:33:52,950 --> 00:33:57,095 This is something that is taken to-- 532 00:34:02,760 --> 00:34:06,990 this is something called Ethambutal which is used to 533 00:34:06,990 --> 00:34:09,340 treat tuberculosis. 534 00:34:09,340 --> 00:34:11,949 Only the molecule of one handedness does this, the 535 00:34:11,949 --> 00:34:14,090 other one causes blindness. 536 00:34:14,090 --> 00:34:16,889 That's really a sinister and antiomorph. 537 00:34:16,889 --> 00:34:20,380 Then there's some even crazier examples. 538 00:34:20,380 --> 00:34:25,260 Ibuprofen is a chiral molecule, and this in a most 539 00:34:25,260 --> 00:34:29,190 remarkable situation is a molecule which you're body 540 00:34:29,190 --> 00:34:34,090 converts to the molecule of the chirality that has the 541 00:34:34,090 --> 00:34:35,710 intended purpose. 542 00:34:35,710 --> 00:34:39,800 So here your body is clever enough to change ibuprofen 543 00:34:39,800 --> 00:34:43,139 into the molecule which is the one that you need for its 544 00:34:43,139 --> 00:34:44,389 pharmaceutical effect. 545 00:34:46,820 --> 00:34:50,530 OK, so mirrors are interesting things. 546 00:34:50,530 --> 00:34:53,979 I would invite you to come up and play with the special 547 00:34:53,979 --> 00:34:56,920 mirrors that do strange things and see yourself 548 00:34:56,920 --> 00:35:00,770 as others see you. 549 00:35:00,770 --> 00:35:04,370 And now I would like to continue on in this discussion 550 00:35:04,370 --> 00:35:09,710 to mention the ways in which we can represent a mirror 551 00:35:09,710 --> 00:35:13,040 plane in a graphic language. 552 00:35:13,040 --> 00:35:15,390 This is what a mirror plane does, it changes the sense of 553 00:35:15,390 --> 00:35:16,600 one coordinate. 554 00:35:16,600 --> 00:35:20,720 If there is a locus across which that transformation is 555 00:35:20,720 --> 00:35:26,240 performed, we would like first of all, an analytic symbol. 556 00:35:31,680 --> 00:35:35,310 Some way of indicating the presence of that particular 557 00:35:35,310 --> 00:35:40,430 operation in the pattern, and a mirror is very descriptive 558 00:35:40,430 --> 00:35:44,360 so the symbol m is used to represent the presence of a 559 00:35:44,360 --> 00:35:48,810 mirror plane in a particular symbol. 560 00:35:48,810 --> 00:35:55,610 We might want to indicate a specific operation. 561 00:35:55,610 --> 00:35:59,480 There are only two operations in the case of a mirror plane 562 00:35:59,480 --> 00:36:02,620 reflecting left to right and reflecting right to left. 563 00:36:02,620 --> 00:36:05,590 But there are other operations such as rotation. 564 00:36:05,590 --> 00:36:10,500 If we have a 16-fold rotation axis, there is one operation 565 00:36:10,500 --> 00:36:15,170 that consists of rotating 1/16 of 2 pi, another operation 566 00:36:15,170 --> 00:36:18,750 that will also leave the space invariant that's rotating 2/16 567 00:36:18,750 --> 00:36:20,190 of 2 pi, and so on. 568 00:36:20,190 --> 00:36:23,080 So an individual operation is something that we 569 00:36:23,080 --> 00:36:27,280 will want to designate. 570 00:36:27,280 --> 00:36:30,960 And for a mirror plane, something that is used 571 00:36:30,960 --> 00:36:34,960 commonly in physics is to use an operation sigma for a 572 00:36:34,960 --> 00:36:38,150 particular reflection. 573 00:36:38,150 --> 00:36:42,120 This is not done in Buerger. 574 00:36:42,120 --> 00:36:48,260 If you get into reading it, he uses m for both. 575 00:36:48,260 --> 00:36:53,500 And then finally, it's going to be convenient when we have 576 00:36:53,500 --> 00:36:59,550 a pattern before us to use a geometric symbol to indicate 577 00:36:59,550 --> 00:37:03,650 in the pattern the locus of this particular operation. 578 00:37:03,650 --> 00:37:09,710 And what we use in the case of a mirror plane is a bold line. 579 00:37:09,710 --> 00:37:12,700 And that if this were the pattern and we wanted to 580 00:37:12,700 --> 00:37:16,280 indicate where the mirror plane was, or the mirror line 581 00:37:16,280 --> 00:37:18,830 in 2-dimensions that relates those two motifs, 582 00:37:18,830 --> 00:37:20,080 we draw it in thusly. 583 00:37:24,980 --> 00:37:28,910 We began last time to examine the properties of rotation, 584 00:37:28,910 --> 00:37:34,080 but that's another sort of symmetry and that is a 585 00:37:34,080 --> 00:37:37,165 rotation which takes place about a rotation axis. 586 00:37:40,920 --> 00:37:48,760 The symbol that is used to represent the collection of 587 00:37:48,760 --> 00:37:52,550 operations, the analytic symbol, is based on the fact 588 00:37:52,550 --> 00:37:58,590 that the angular rotation, alpha, has to be equal to some 589 00:37:58,590 --> 00:38:01,020 sub-multiple of 2 pi. 590 00:38:01,020 --> 00:38:03,200 2 pi over n, where n is some integer. 591 00:38:08,600 --> 00:38:12,050 And the reason for that I think is quite clear, if I 592 00:38:12,050 --> 00:38:19,860 take a particular motif and rotate through an angle alpha, 593 00:38:19,860 --> 00:38:24,220 if I am not rotating by some sub-multiple of 2 pi, I'll 594 00:38:24,220 --> 00:38:26,950 just go round and round and round and I will never get a 595 00:38:26,950 --> 00:38:33,160 finite set of objects that is separated from its neighbor by 596 00:38:33,160 --> 00:38:35,760 the same angular interval alpha. 597 00:38:35,760 --> 00:38:38,950 This will only happen if alpha is an integral 598 00:38:38,950 --> 00:38:41,220 sub-multiple of 2 pi. 599 00:38:41,220 --> 00:38:46,170 And the symbol that is used for the collection of 600 00:38:46,170 --> 00:38:49,780 operations that is usually embodied in a rotation axis is 601 00:38:49,780 --> 00:38:53,520 n, the same n that is in the denominator. 602 00:38:53,520 --> 00:38:58,180 The symbol for individual rotation, so we mentioned last 603 00:38:58,180 --> 00:39:02,080 time we have to specify the location of the point about 604 00:39:02,080 --> 00:39:07,570 which we rotate, and we have to indicate the angle alpha 605 00:39:07,570 --> 00:39:09,190 through which we've rotated. 606 00:39:09,190 --> 00:39:13,890 So A alpha will be an individual operation, and the 607 00:39:13,890 --> 00:39:22,010 geometric symbol will be an n-Gon which has the symmetry 608 00:39:22,010 --> 00:39:23,500 of the rotation axis. 609 00:39:23,500 --> 00:39:28,050 So for a sixfold axis we would use a hexagon. 610 00:39:28,050 --> 00:39:33,170 For a fivefold axis we will use a pentagon, for a 611 00:39:33,170 --> 00:39:37,950 fourfold, a square, for a threefold, a triangle. 612 00:39:37,950 --> 00:39:44,350 Now an n-Gon with 180 degree rotation is a line segment. 613 00:39:44,350 --> 00:39:46,800 And that would be easily overlooked and it's not very 614 00:39:46,800 --> 00:39:50,970 aesthetic, so here we indulge in a little bit of artistic 615 00:39:50,970 --> 00:39:55,640 license and fatten out the middle of the line segment to 616 00:39:55,640 --> 00:39:58,380 get an oval with pointed ends. 617 00:39:58,380 --> 00:40:02,090 And that's the symbol for a twofold axis. 618 00:40:02,090 --> 00:40:03,990 What about a one-fold axis? 619 00:40:03,990 --> 00:40:06,910 One-fold axes exist anywhere, so you can sprinkle them 620 00:40:06,910 --> 00:40:09,030 around with reckless abandon. 621 00:40:09,030 --> 00:40:13,170 A one-fold axis has no symmetry at all, but that is a 622 00:40:13,170 --> 00:40:16,990 very nice symbol to use for no symmetry at all. 623 00:40:16,990 --> 00:40:21,860 So symmetry 1 is the absence of symmetry. 624 00:40:21,860 --> 00:40:27,030 So it does come up occasionally in notation. 625 00:40:27,030 --> 00:40:32,290 Now if you look at what we've done so far, we have a 626 00:40:32,290 --> 00:40:36,040 transformation that changes the sense of no coordinate. 627 00:40:36,040 --> 00:40:40,230 We have a transformation that changes the sense of two 628 00:40:40,230 --> 00:40:44,470 coordinates, one coordinate, no coordinate, one coordinate. 629 00:40:44,470 --> 00:40:48,220 Rotation is interchanging the sense of two coordinates in a 630 00:40:48,220 --> 00:40:52,350 plane, and in a 2-dimensional pattern, that's all there is. 631 00:40:52,350 --> 00:41:01,660 But for a 3-dimensional space, we have the option of changing 632 00:41:01,660 --> 00:41:07,470 the sense of no coordinate, the sense 1, the sense of 2, 633 00:41:07,470 --> 00:41:10,750 or change the sense of all 3 coordinates. 634 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 635 00:41:16,310 --> 00:41:22,640 space lurks my initial motif, if I change the sense of x, 636 00:41:22,640 --> 00:41:29,520 the sense of y, and the sense of z, namely take xyz and map 637 00:41:29,520 --> 00:41:34,800 it to minus x, minus y, minus z, what I'm going to do is to 638 00:41:34,800 --> 00:41:38,510 essentially turn the object inside out. 639 00:41:38,510 --> 00:41:44,050 And if my initial one was right-handed, I will produce a 640 00:41:44,050 --> 00:41:48,340 chiral object, a left-handed object. 641 00:41:48,340 --> 00:41:52,280 This is an operation which is called inversion. 642 00:41:55,760 --> 00:41:59,870 In this operation of turning the object inside out if you 643 00:41:59,870 --> 00:42:03,050 will, is inverting it to a new location. 644 00:42:03,050 --> 00:42:06,490 And this analytically is the exchange in coordinates 645 00:42:06,490 --> 00:42:11,330 provided the point of inversion is at the center. 646 00:42:11,330 --> 00:42:18,090 The analytic symbol for inversion is 1 with a bar over 647 00:42:18,090 --> 00:42:19,830 the top, pronounced 1 bar. 648 00:42:23,020 --> 00:42:30,660 And I'll have to leave to later indication of exactly 649 00:42:30,660 --> 00:42:34,010 where that notation comes from. 650 00:42:34,010 --> 00:42:40,810 The individual operation is also called 1 bar, and the 651 00:42:40,810 --> 00:42:44,100 geometric symbol that is used to indicate the location of an 652 00:42:44,100 --> 00:42:49,840 inversion center is a tiny little open circle large 653 00:42:49,840 --> 00:42:53,070 enough so that you don't miss it, but not so large that it 654 00:42:53,070 --> 00:42:56,910 might be confused with an atom in a drawing of an atomic 655 00:42:56,910 --> 00:42:57,950 arrangement. 656 00:42:57,950 --> 00:43:02,550 So in this case, we would adorn our sketch was a little 657 00:43:02,550 --> 00:43:05,470 circle at the origin if that was the point through which 658 00:43:05,470 --> 00:43:06,865 the space was being inverted. 659 00:43:09,600 --> 00:43:14,870 So that, ladies and gentlemen, is our basic bag of tricks in 660 00:43:14,870 --> 00:43:17,430 3-dimensions. 661 00:43:17,430 --> 00:43:23,410 Let me point out that inversion can exist in 662 00:43:23,410 --> 00:43:30,970 3-dimensions only because I have to have 3 coordinates to 663 00:43:30,970 --> 00:43:35,570 play with or else I cannot define the operation. 664 00:43:35,570 --> 00:43:40,530 Suppose I have a mapping operation xyz that goes to 665 00:43:40,530 --> 00:43:46,820 minus x, minus y, minus z, and I get rid of z to make it 666 00:43:46,820 --> 00:43:48,560 2-dimensional. 667 00:43:48,560 --> 00:43:53,220 Then my transformation is xy going to minus x, minus y and 668 00:43:53,220 --> 00:43:54,760 that's exactly what a twofold axis does. 669 00:44:01,510 --> 00:44:03,990 So inversion, when you throw out the third coordinate, 670 00:44:03,990 --> 00:44:07,260 looks like a 180 degree rotation. 671 00:44:07,260 --> 00:44:09,990 So you need 3 dimensions in order to define that 672 00:44:09,990 --> 00:44:11,240 transformation. 673 00:44:17,540 --> 00:44:20,000 If we really wanted to go crazy, we could go on to say 674 00:44:20,000 --> 00:44:23,030 what happens in 4-dimensions? 675 00:44:26,260 --> 00:44:30,140 There should in principle be five different operations and 676 00:44:30,140 --> 00:44:32,250 yes, mathematically you can define them. 677 00:44:32,250 --> 00:44:36,060 They're very difficult to draw because we have to have some 678 00:44:36,060 --> 00:44:39,290 sort of operation that take something and pulls it out of 679 00:44:39,290 --> 00:44:40,640 our 3-dimensional world. 680 00:44:40,640 --> 00:44:43,310 We have no idea where it went and then all of sudden, [POP], 681 00:44:43,310 --> 00:44:46,050 it pops back into our space. 682 00:44:46,050 --> 00:44:50,000 But mathematically there are cases when you need a fourth 683 00:44:50,000 --> 00:44:55,230 variable to describe the symmetry of an arrangement. 684 00:44:55,230 --> 00:45:00,650 And this generally occurs in something called a modulated 685 00:45:00,650 --> 00:45:04,430 structure where there's a periodic change in some 686 00:45:04,430 --> 00:45:07,650 variable other than the atomic positions. 687 00:45:07,650 --> 00:45:11,670 And let me give you two quick examples without going into it 688 00:45:11,670 --> 00:45:12,920 exhaustively. 689 00:45:15,480 --> 00:45:19,910 One characteristic of an atom besides its location and its 690 00:45:19,910 --> 00:45:26,670 atomic mass and things like that, is perhaps a magnetic 691 00:45:26,670 --> 00:45:29,780 atom that has a magnetic moment attached to it. 692 00:45:32,560 --> 00:45:39,430 There are magnetically ordered structures. 693 00:45:39,430 --> 00:45:44,250 One of them looks exactly like rock salt. 694 00:45:44,250 --> 00:45:47,010 And I'll draw just the magnetic cations which sit in 695 00:45:47,010 --> 00:45:48,280 locations like this. 696 00:45:50,870 --> 00:45:56,120 And the magnetic moment here is up, the magnetic moment 697 00:45:56,120 --> 00:45:59,040 here is up, the magnetic moment here is down, the 698 00:45:59,040 --> 00:46:02,190 magnetic moment here is down. 699 00:46:02,190 --> 00:46:07,810 So what I've drawn here is no longer the lattice and in 700 00:46:07,810 --> 00:46:11,570 fact, the lattice constant of this material looks like a 701 00:46:11,570 --> 00:46:15,190 rock salt as far as the atomic positions are concerned, but 702 00:46:15,190 --> 00:46:21,100 the magnetic moments have to be continued on in another 703 00:46:21,100 --> 00:46:28,180 direction and some extra distance. 704 00:46:28,180 --> 00:46:32,500 Actually some examples of this sort of behavior is FeO, 705 00:46:32,500 --> 00:46:35,110 cobalt oxide, nickel oxide. 706 00:46:35,110 --> 00:46:38,420 All of these cations are magnetic, they have magnetic 707 00:46:38,420 --> 00:46:45,060 moments which are ordered and the unit cell turns out to be 708 00:46:45,060 --> 00:46:51,610 when you take magnetic moment into account, a larger cell, a 709 00:46:51,610 --> 00:46:53,180 super cell. 710 00:46:53,180 --> 00:46:57,320 There's a more interesting type of magnetic structure 711 00:46:57,320 --> 00:47:03,400 though in which the magnetic moment is inclined relative to 712 00:47:03,400 --> 00:47:06,660 some translation in the structure. 713 00:47:06,660 --> 00:47:09,730 And the magnetic moments all lie on the generators of 714 00:47:09,730 --> 00:47:17,940 cones, but as you walk along the chain of atoms, the 715 00:47:17,940 --> 00:47:21,180 orientation of the moment rotates to different 716 00:47:21,180 --> 00:47:22,430 orientations. 717 00:47:26,560 --> 00:47:32,840 There is a family of materials that are said to have cubicle 718 00:47:32,840 --> 00:47:45,060 spin structures in which the periodicity of the march of 719 00:47:45,060 --> 00:47:49,270 the magnetic moment around the surface of the cone occurs 720 00:47:49,270 --> 00:47:52,060 with a period that is in commensurate with the spacing 721 00:47:52,060 --> 00:47:54,550 of the chain of atoms. 722 00:47:54,550 --> 00:47:57,850 So strictly speaking, this material does not have a 723 00:47:57,850 --> 00:48:01,570 lattice in this direction, so it's not a crystal unless you 724 00:48:01,570 --> 00:48:08,200 use a fourth variable to describe the periodicity of 725 00:48:08,200 --> 00:48:09,790 the orientation of the moment. 726 00:48:09,790 --> 00:48:13,950 And one final one at the risk of carrying this too far, 727 00:48:13,950 --> 00:48:17,010 here's a pattern that is based on a square lattice. 728 00:48:17,010 --> 00:48:21,490 It has a fourfold axis in it unless I make the pattern out 729 00:48:21,490 --> 00:48:25,270 of squares that are black and white, make a checkerboard. 730 00:48:33,440 --> 00:48:36,480 This is now no longer a fourfold axis because I can't 731 00:48:36,480 --> 00:48:39,740 rotate 90 degrees and leave the pattern invariant. 732 00:48:39,740 --> 00:48:42,280 So this is an example of something called a black-white 733 00:48:42,280 --> 00:48:44,590 symmetry, or a color symmetry. 734 00:48:44,590 --> 00:48:50,610 And it requires more than just 4 operations to describe the 735 00:48:50,610 --> 00:48:52,670 relation between one motif and another. 736 00:48:52,670 --> 00:48:55,810 We need a fourth operation, switching of a color from 737 00:48:55,810 --> 00:48:59,030 black to white, or switching it from white to black, that's 738 00:48:59,030 --> 00:49:00,210 a forth operation. 739 00:49:00,210 --> 00:49:02,750 Again, within the confines of a pattern that 740 00:49:02,750 --> 00:49:03,820 exists in our space. 741 00:49:03,820 --> 00:49:04,317 Yes? 742 00:49:04,317 --> 00:49:06,429 AUDIENCE: Does that actually have-- so you're saying it 743 00:49:06,429 --> 00:49:08,520 doesn't add value to the [INAUDIBLE]? 744 00:49:08,520 --> 00:49:11,950 PROFESSOR: I did say no rotational symmetry if it has 745 00:49:11,950 --> 00:49:16,940 a fourfold axis here but this used to be a fourfold axis, 746 00:49:16,940 --> 00:49:21,430 and that now changes into a twofold axis. 747 00:49:21,430 --> 00:49:24,470 And then I have the problem of describing how this square is 748 00:49:24,470 --> 00:49:28,710 a square exactly like this square except for its color. 749 00:49:28,710 --> 00:49:32,120 So I need then an operation which rotates 90 degrees and 750 00:49:32,120 --> 00:49:34,070 switches from white to black. 751 00:49:34,070 --> 00:49:37,190 And then rotates 90 degrees again and switches 752 00:49:37,190 --> 00:49:40,130 from black to white. 753 00:49:40,130 --> 00:49:45,970 So there's a fifth operation, a color change that is 754 00:49:45,970 --> 00:49:49,990 necessary in a 3-dimensional space or a forth operation, a 755 00:49:49,990 --> 00:49:51,480 color change in a 2-dimensional 756 00:49:51,480 --> 00:49:53,820 checkerboard for example. 757 00:49:53,820 --> 00:49:57,660 So there are lots of nuances to symmetry theory, it's 758 00:49:57,660 --> 00:50:00,960 mathematics and the nice thing about mathematics is it's your 759 00:50:00,960 --> 00:50:03,360 ballgame, you could make up the rules and as long as you 760 00:50:03,360 --> 00:50:06,780 play according to those rules consistently, then you've got 761 00:50:06,780 --> 00:50:10,770 something that people can't quarrel with. 762 00:50:10,770 --> 00:50:15,320 OK, I think my internal clock has just told me that it's 763 00:50:15,320 --> 00:50:17,720 five minutes of the hour and it's time to take our break. 764 00:50:20,360 --> 00:50:23,120 Come up by all means and play with the mirrors if you'd like 765 00:50:23,120 --> 00:50:26,800 and we'll resume the lecture part of our 766 00:50:26,800 --> 00:50:28,170 discussion in 10 minutes.