1 00:00:12,500 --> 00:00:14,450 PROFESSOR: But you all knew that. 2 00:00:14,450 --> 00:00:16,219 That is quite a mouthful. 3 00:00:16,219 --> 00:00:20,700 So if you like, you can refer to it as SST. 4 00:00:20,700 --> 00:00:23,430 And today, we're taking off. 5 00:00:23,430 --> 00:00:26,780 So my name, for those of you who don't know 6 00:00:26,780 --> 00:00:28,340 me, is Bernardt Wuensch. 7 00:00:38,160 --> 00:00:45,210 My room number is 13-4037, The office that's become a legend 8 00:00:45,210 --> 00:00:47,440 in its own time. 9 00:00:47,440 --> 00:00:58,530 And my extension number is 3-6889. 10 00:00:58,530 --> 00:01:02,630 And hey, just to get you sensitized and thinking about 11 00:01:02,630 --> 00:01:07,090 the right things, let me point out that my extension number 12 00:01:07,090 --> 00:01:12,820 has a point of 180 degree rotational symmetry right in 13 00:01:12,820 --> 00:01:13,720 the middle. 14 00:01:13,720 --> 00:01:16,020 You can pick it up, turn it head over 15 00:01:16,020 --> 00:01:17,990 heels by 180 degrees. 16 00:01:17,990 --> 00:01:21,790 And it's mapped into coincidence with itself. 17 00:01:21,790 --> 00:01:22,920 Just happened to get it. 18 00:01:22,920 --> 00:01:24,990 You might think I would have had to have fought for years 19 00:01:24,990 --> 00:01:27,560 to get an extension number like that. 20 00:01:27,560 --> 00:01:29,290 But no, it just happened to come my way. 21 00:01:32,260 --> 00:01:35,195 OK, some words about the formalities of the subject. 22 00:01:38,050 --> 00:01:42,090 First of all, the format of the class is unusual. 23 00:01:42,090 --> 00:01:47,155 We meet four hours a week, but because most if not all of you 24 00:01:47,155 --> 00:01:50,260 are graduate students anxious to get some work done in the 25 00:01:50,260 --> 00:01:54,610 laboratory, we do this in two two-hour chunks. 26 00:01:54,610 --> 00:01:57,580 So we meet Tuesday and Thursdays for two hours. 27 00:01:57,580 --> 00:02:01,900 Two hours is a lot of time for anything, however good. 28 00:02:01,900 --> 00:02:05,710 So what we do is to take a long intermission halfway 29 00:02:05,710 --> 00:02:09,449 through and let you go out and enjoy what's left of the 30 00:02:09,449 --> 00:02:12,690 lingering summer for 10 or 12 minutes. 31 00:02:12,690 --> 00:02:16,590 And then, come back refreshed and we will resume. 32 00:02:16,590 --> 00:02:21,330 Most graduate students like this arrangement because it 33 00:02:21,330 --> 00:02:24,180 gives them a chance to duck out and make a setting on a 34 00:02:24,180 --> 00:02:27,670 furnace or turn something off in the laboratory. 35 00:02:27,670 --> 00:02:32,240 And it works out better for them than having a one hour 36 00:02:32,240 --> 00:02:35,900 time chunk every day of the week or four of the five days 37 00:02:35,900 --> 00:02:36,990 of the week. 38 00:02:36,990 --> 00:02:39,200 In any case, nobody's complained about it. 39 00:02:39,200 --> 00:02:40,660 So I assume that will work 40 00:02:40,660 --> 00:02:42,280 satisfactorily for you as well. 41 00:02:45,460 --> 00:02:48,080 The other question that one immediately asked at the 42 00:02:48,080 --> 00:02:51,305 beginning of the term, how many quizzes? 43 00:02:51,305 --> 00:02:54,290 And we're supposed to tell you that straight up. 44 00:02:54,290 --> 00:02:57,575 There will be three quizzes. 45 00:03:01,440 --> 00:03:03,780 No final examination-- 46 00:03:03,780 --> 00:03:07,060 do I look like the kind of Scrooge that would prevent you 47 00:03:07,060 --> 00:03:11,000 from getting a good flight home at Christmas time or have 48 00:03:11,000 --> 00:03:14,570 you working, cramming for a final examination a few days 49 00:03:14,570 --> 00:03:16,070 before Christmas? 50 00:03:16,070 --> 00:03:18,720 And for my part, I can remember the good old days-- 51 00:03:18,720 --> 00:03:21,150 or not so good old days-- when I did give a final. 52 00:03:21,150 --> 00:03:24,530 And there I would be, lying down on my stomach under the 53 00:03:24,530 --> 00:03:27,260 Christmas tree grading final examinations. 54 00:03:27,260 --> 00:03:30,025 And every time one of my little kids would come near, 55 00:03:30,025 --> 00:03:33,250 I'd lash out with my foot and say, get out of here, kid! 56 00:03:33,250 --> 00:03:35,840 Can't you see Daddy's got papers to correct? 57 00:03:35,840 --> 00:03:37,100 Well, no Scrooge. 58 00:03:37,100 --> 00:03:38,770 No final examination. 59 00:03:38,770 --> 00:03:41,360 We'll have three quizzes. 60 00:03:41,360 --> 00:03:45,990 The quizzes will also be a little bit unique. 61 00:03:45,990 --> 00:03:50,250 Since we have two hour chunks of time, I found by experience 62 00:03:50,250 --> 00:03:54,960 that if I give the quiz during the first hour everybody is 63 00:03:54,960 --> 00:03:58,870 sitting around glassy-eyed, absolutely brain dead and pay 64 00:03:58,870 --> 00:04:01,470 no attention to the lecture that follows. 65 00:04:01,470 --> 00:04:06,040 If I give the quiz in the second hour, everybody is 66 00:04:06,040 --> 00:04:08,080 pretending to pay attention and then sneaking 67 00:04:08,080 --> 00:04:10,670 surreptitious looks at their notes just so they can have 68 00:04:10,670 --> 00:04:14,490 everything packed away before they have to write on paper. 69 00:04:14,490 --> 00:04:20,480 So my quizzes are two hour quizzes which lets you not 70 00:04:20,480 --> 00:04:23,840 work for two hours, but gives you all the time you could 71 00:04:23,840 --> 00:04:25,700 possibly want. 72 00:04:25,700 --> 00:04:28,780 And people start leaving after about an hour and a quarter. 73 00:04:28,780 --> 00:04:32,820 But you can stay for the entire two hours if you want. 74 00:04:32,820 --> 00:04:37,260 Even at that, I found by experience that if I give you 75 00:04:37,260 --> 00:04:40,390 two hours for the quiz, after about an hour and a half, 76 00:04:40,390 --> 00:04:44,130 everybody's looking out the window, looking back at the 77 00:04:44,130 --> 00:04:47,000 ceiling, not a single pencil is moving. 78 00:04:47,000 --> 00:04:49,720 Then I will say, OK, you all done? 79 00:04:49,720 --> 00:04:51,450 Everybody starts writing again and going through 80 00:04:51,450 --> 00:04:52,660 their papers once more. 81 00:04:52,660 --> 00:04:55,450 And even after two hours I find that in order to get the 82 00:04:55,450 --> 00:04:58,850 quiz papers, I have to plant one foot on the edge of your 83 00:04:58,850 --> 00:05:03,690 table, grab hold of your quiz with both hands and drag it 84 00:05:03,690 --> 00:05:05,470 out of your clutching fingers. 85 00:05:05,470 --> 00:05:07,390 So you can take the full two hours. 86 00:05:07,390 --> 00:05:09,730 But it should not be necessary for you to 87 00:05:09,730 --> 00:05:11,740 consume that much time. 88 00:05:11,740 --> 00:05:15,390 The quizzes will come-- and this is something else we're 89 00:05:15,390 --> 00:05:16,800 supposed to explain to you-- 90 00:05:16,800 --> 00:05:20,925 they will come at one third of the way through the term, two 91 00:05:20,925 --> 00:05:27,280 thirds of the way through the term, and 2.983/3 of the way 92 00:05:27,280 --> 00:05:28,510 through the term. 93 00:05:28,510 --> 00:05:33,070 And if you're wondering where that number comes from, this 94 00:05:33,070 --> 00:05:35,970 lets me put the quiz just before the final week of the 95 00:05:35,970 --> 00:05:38,860 term when we're not supposed to give examinations. 96 00:05:38,860 --> 00:05:41,230 So there'll be three quizzes. 97 00:05:41,230 --> 00:05:43,600 You will have opportunity for lots of 98 00:05:43,600 --> 00:05:45,370 practice with problems. 99 00:05:45,370 --> 00:05:50,640 We will have on the order of 15 problem sets. 100 00:05:50,640 --> 00:05:55,530 And for the most part, they will be very short, or 101 00:05:55,530 --> 00:05:59,960 modestly short, and designed to give you some practice in 102 00:05:59,960 --> 00:06:01,530 working with the material. 103 00:06:01,530 --> 00:06:05,350 Because as the nature of this subject begins to unfold, 104 00:06:05,350 --> 00:06:08,300 you'll see that it involves a type of mathematics that 105 00:06:08,300 --> 00:06:11,260 you've really perhaps not had much practice with. 106 00:06:11,260 --> 00:06:13,750 It involves geometrical relations. 107 00:06:13,750 --> 00:06:17,220 And to really master it, you have to work with the material 108 00:06:17,220 --> 00:06:18,470 and get some practice. 109 00:06:20,600 --> 00:06:24,050 Another question that's perennially asked if not 110 00:06:24,050 --> 00:06:28,580 outright raised in private, what do the quizzes count? 111 00:06:28,580 --> 00:06:31,640 What do the problem sets count? 112 00:06:31,640 --> 00:06:35,590 The answer to that is that the problem sets will count a lot 113 00:06:35,590 --> 00:06:38,080 towards your understanding of the material. 114 00:06:38,080 --> 00:06:39,850 But I'm not going to grade them. 115 00:06:39,850 --> 00:06:42,360 And I'm not going to factor them in along with the quizzes 116 00:06:42,360 --> 00:06:47,550 to decide your final eventual fortunes in this class. 117 00:06:47,550 --> 00:06:50,460 The problem sets, moreover, there are a lot of them. 118 00:06:50,460 --> 00:06:55,030 But they will be optional in the sense that if you do them, 119 00:06:55,030 --> 00:06:59,320 I will carefully correct them, add words of inspiration and 120 00:06:59,320 --> 00:07:04,910 advice, correct things where you've gone wrong and then 121 00:07:04,910 --> 00:07:07,460 return them to you as quickly as possible. 122 00:07:07,460 --> 00:07:12,080 But if you how to do the problem, you say, ah! 123 00:07:12,080 --> 00:07:16,110 Why does he wants us to do this and waste our time with a 124 00:07:16,110 --> 00:07:17,650 silly problem like this? 125 00:07:17,650 --> 00:07:18,810 Don't do it. 126 00:07:18,810 --> 00:07:21,510 Don't do it, because if you know how to do the problem 127 00:07:21,510 --> 00:07:22,680 set, that's fine. 128 00:07:22,680 --> 00:07:25,870 And you've got better things to do with your time. 129 00:07:25,870 --> 00:07:29,410 And I've got better things to do with my time if the 130 00:07:29,410 --> 00:07:31,950 feedback is not going to be of benefit to you. 131 00:07:31,950 --> 00:07:33,370 So I hope you will do them. 132 00:07:33,370 --> 00:07:37,410 And as I said, if you do them, I will correct them promptly 133 00:07:37,410 --> 00:07:38,080 and thoroughly. 134 00:07:38,080 --> 00:07:41,450 And if you haven't got the foggiest idea how to do the 135 00:07:41,450 --> 00:07:44,230 problem, do what you can. 136 00:07:44,230 --> 00:07:46,530 And then, write down a plea of help-- 137 00:07:46,530 --> 00:07:49,000 I don't understand what's going on here! 138 00:07:49,000 --> 00:07:51,770 OK, and then I will take the time to write out what's going 139 00:07:51,770 --> 00:07:57,370 on, hopefully to your benefit and use. 140 00:07:57,370 --> 00:08:00,160 Will I turn out solutions? 141 00:08:00,160 --> 00:08:02,710 Only on an individual basis in the fashion 142 00:08:02,710 --> 00:08:04,610 that I've just described. 143 00:08:04,610 --> 00:08:08,460 I find that if I write out a solution to each of these 144 00:08:08,460 --> 00:08:10,160 problems, you don't do them. 145 00:08:10,160 --> 00:08:12,560 And you say, oh, so that's how you do that, throw it in a 146 00:08:12,560 --> 00:08:15,850 file, and not look at it until the night before the quiz. 147 00:08:15,850 --> 00:08:18,960 So there will not be solutions handed out other than 148 00:08:18,960 --> 00:08:23,280 correction on an individual basis on your papers. 149 00:08:23,280 --> 00:08:25,650 Is that all that I wanted to say? 150 00:08:25,650 --> 00:08:28,510 I think that's about all for the formalities. 151 00:08:32,260 --> 00:08:39,340 Actually, I should add one postscript to say the problem 152 00:08:39,340 --> 00:08:41,390 sets don't count anything toward your final grade. 153 00:08:44,159 --> 00:08:46,510 They do in one minor sense. 154 00:08:46,510 --> 00:08:50,010 When you have a large class and you plot up the grades and 155 00:08:50,010 --> 00:08:54,080 there are no lumps with gaps in between, there comes a 156 00:08:54,080 --> 00:08:56,780 point where you have to separate 157 00:08:56,780 --> 00:08:58,690 one grade from another. 158 00:08:58,690 --> 00:09:01,420 And if you've done well on the quizzes and you've done well 159 00:09:01,420 --> 00:09:04,260 on the problem sets and there's just one quiz that's a 160 00:09:04,260 --> 00:09:06,660 little bit, I say, OK, he or she had a 161 00:09:06,660 --> 00:09:08,610 bad day that afternoon. 162 00:09:08,610 --> 00:09:10,880 And I'll give you the benefit of the doubt. 163 00:09:10,880 --> 00:09:13,420 And even though I'm not a vindictive sort, if you're 164 00:09:13,420 --> 00:09:15,780 right on the fence and you haven't done any of the 165 00:09:15,780 --> 00:09:19,810 problems, then without malice I say, gotcha! 166 00:09:19,810 --> 00:09:21,350 And you go down [INAUDIBLE] 167 00:09:21,350 --> 00:09:23,090 on the low side of the barricade. 168 00:09:23,090 --> 00:09:25,930 And I think that's only a natural indication because 169 00:09:25,930 --> 00:09:29,490 there are some cases where, with Solomonic judgement, you 170 00:09:29,490 --> 00:09:33,020 have to decide who gets what grade. 171 00:09:36,360 --> 00:09:42,570 OK, let me say a little bit about the texts. 172 00:09:42,570 --> 00:09:49,030 There are a number of books that deal with 173 00:09:49,030 --> 00:09:51,360 crystallography. 174 00:09:51,360 --> 00:09:56,060 For the most part, though, they consists of an 175 00:09:56,060 --> 00:09:57,550 introductory chapter. 176 00:09:57,550 --> 00:10:02,040 Every single book on the solid state feels compelled to write 177 00:10:02,040 --> 00:10:06,450 some sort of half baked chapter on crystal structure 178 00:10:06,450 --> 00:10:08,320 or crystallography. 179 00:10:08,320 --> 00:10:11,550 And usually, these chapters consist of big tables. 180 00:10:11,550 --> 00:10:15,720 And they say, there are 14 of these. 181 00:10:15,720 --> 00:10:17,430 There are 17 of these. 182 00:10:17,430 --> 00:10:19,190 There are 32 of these. 183 00:10:19,190 --> 00:10:21,190 There are like 230 of these. 184 00:10:21,190 --> 00:10:25,310 There are 1,170 of these. 185 00:10:25,310 --> 00:10:29,050 And then, that has all the excitement and stimulation of 186 00:10:29,050 --> 00:10:31,850 reading the telephone directory. 187 00:10:31,850 --> 00:10:34,450 It's a crazy cast of characters, but it's awfully 188 00:10:34,450 --> 00:10:36,950 hard to see the plot. 189 00:10:36,950 --> 00:10:41,320 So what we will do all the way through is derive everything. 190 00:10:41,320 --> 00:10:46,080 So you can not only see how it turns out, but why it has to 191 00:10:46,080 --> 00:10:47,380 be that way. 192 00:10:47,380 --> 00:10:50,380 And that's the way, in my opinion, one really learns 193 00:10:50,380 --> 00:10:51,630 this material. 194 00:10:53,570 --> 00:10:55,830 A couple of other very pedantic 195 00:10:55,830 --> 00:10:57,695 comments about the material. 196 00:10:57,695 --> 00:11:00,980 In the early part of the term, the first half in fact, we're 197 00:11:00,980 --> 00:11:03,940 going to use plain old geometry. 198 00:11:03,940 --> 00:11:07,210 Now, geometry really doesn't cut much 199 00:11:07,210 --> 00:11:09,160 mustard around the Institute. 200 00:11:09,160 --> 00:11:12,680 If you can't integrate it or take its Fourier transform, 201 00:11:12,680 --> 00:11:15,916 that's a mathematics you don't have to take seriously. 202 00:11:15,916 --> 00:11:20,530 Well, geometry is a perfectly valid branch of mathematics. 203 00:11:20,530 --> 00:11:24,280 And one can do what we're going to do in more complex 204 00:11:24,280 --> 00:11:26,740 terms using the language of group theory. 205 00:11:26,740 --> 00:11:29,520 And we will, in fact, use a little bit of that later on. 206 00:11:29,520 --> 00:11:33,790 But for the most part, just diagrams with simple geometry 207 00:11:33,790 --> 00:11:37,260 are going to be one of the principle tools in the initial 208 00:11:37,260 --> 00:11:39,470 part of the class. 209 00:11:39,470 --> 00:11:42,130 About halfway through, we'll switch over to something that 210 00:11:42,130 --> 00:11:45,000 is much more mathematical in the traditional sense. 211 00:11:45,000 --> 00:11:49,180 Here, a little bit of linear algebra and matrix algebra 212 00:11:49,180 --> 00:11:50,120 will help you. 213 00:11:50,120 --> 00:11:52,440 If you haven't had that or haven't looked at it for a 214 00:11:52,440 --> 00:11:56,250 while, we'll build it up from ground zero so that you'll be 215 00:11:56,250 --> 00:11:58,410 able to fully understand it. 216 00:11:58,410 --> 00:12:01,360 We'll hit a few eigenvalue problems towards 217 00:12:01,360 --> 00:12:02,190 the end of the term. 218 00:12:02,190 --> 00:12:05,650 If that doesn't get your adrenaline pumping, that will 219 00:12:05,650 --> 00:12:08,600 be developed in a physical context so that you're doing 220 00:12:08,600 --> 00:12:11,480 the sort of problem before you even know what it's called. 221 00:12:11,480 --> 00:12:13,800 So it's going to be a user friendly course that doesn't 222 00:12:13,800 --> 00:12:15,630 rely on something that you may have had two 223 00:12:15,630 --> 00:12:16,880 or three years ago. 224 00:12:19,280 --> 00:12:24,310 OK, but the other thing that I wanted to say was that this 225 00:12:24,310 --> 00:12:27,920 class is not like many classes in that you talk about 226 00:12:27,920 --> 00:12:29,470 something for one week. 227 00:12:29,470 --> 00:12:31,490 And then, you put it aside and you talk about something 228 00:12:31,490 --> 00:12:33,730 completely different the next week. 229 00:12:33,730 --> 00:12:36,800 Our first half of the course will be one 230 00:12:36,800 --> 00:12:40,510 long process of synthesis. 231 00:12:40,510 --> 00:12:44,070 We're going to start out very, very simply with little 232 00:12:44,070 --> 00:12:45,930 mapping transformations. 233 00:12:45,930 --> 00:12:49,220 This is picked up and rotated and slid over to here. 234 00:12:49,220 --> 00:12:52,530 And you'll say, ho-hum, let's get on with it. 235 00:12:52,530 --> 00:12:53,850 Come on, go faster. 236 00:12:53,850 --> 00:12:57,990 But we'll build on this and then build on what we've just 237 00:12:57,990 --> 00:12:59,850 done to what comes next. 238 00:12:59,850 --> 00:13:03,960 And unlike most of the classes in science that you take where 239 00:13:03,960 --> 00:13:09,280 you start with general terms and you zero in on some little 240 00:13:09,280 --> 00:13:16,690 nugget like f equals ma, e equals mc squared, lambda 241 00:13:16,690 --> 00:13:20,220 equals 2d sin (theta), a little nugget like a bullion 242 00:13:20,220 --> 00:13:22,280 cube that you can drop in your pocket. 243 00:13:22,280 --> 00:13:24,910 And then when you need it later on, you pull it out and 244 00:13:24,910 --> 00:13:26,740 add hot water. 245 00:13:26,740 --> 00:13:30,260 And then, you have a tool that you can use. 246 00:13:30,260 --> 00:13:32,610 We will do something that's completely 247 00:13:32,610 --> 00:13:33,700 different in its structure. 248 00:13:33,700 --> 00:13:35,160 It will start out simple. 249 00:13:35,160 --> 00:13:36,040 It will grow. 250 00:13:36,040 --> 00:13:39,090 It will blossom like an elegant [? Filigree ?] 251 00:13:39,090 --> 00:13:43,280 structure that gets more and more complicated and diverges 252 00:13:43,280 --> 00:13:46,900 rather than converging to a nice, tight, little nugget. 253 00:13:46,900 --> 00:13:49,010 It's going to get very, very complicated. 254 00:13:49,010 --> 00:13:51,700 And the reason for doing this gradually and thoroughly is so 255 00:13:51,700 --> 00:13:54,430 that you can understand the complexity and 256 00:13:54,430 --> 00:13:56,360 where it comes from. 257 00:13:56,360 --> 00:13:58,920 OK, so my moral here is keep up. 258 00:13:58,920 --> 00:14:00,670 It may seem easy when you start. 259 00:14:00,670 --> 00:14:03,450 But we're going to assume that you've got that down cold 260 00:14:03,450 --> 00:14:05,560 before we go on to the next step. 261 00:14:08,550 --> 00:14:09,800 OK, texts. 262 00:14:12,910 --> 00:14:16,120 Apart from these half baked treatments which I just keep 263 00:14:16,120 --> 00:14:24,580 [INAUDIBLE] on, one of the very best books is by an old 264 00:14:24,580 --> 00:14:32,610 MIT guy, Martin Buerger, who was one of MIT's most 265 00:14:32,610 --> 00:14:33,730 distinguished faculty. 266 00:14:33,730 --> 00:14:37,560 He was the very first faculty member to be honored with the 267 00:14:37,560 --> 00:14:40,290 title Institute Professor, the very first one. 268 00:14:40,290 --> 00:14:43,740 Chairman of the Faculty, all sorts of awards from 269 00:14:43,740 --> 00:14:45,270 professional societies-- 270 00:14:45,270 --> 00:14:48,260 he has a book called Elementary Crystallography. 271 00:15:00,510 --> 00:15:02,235 This is published by Wiley. 272 00:15:05,820 --> 00:15:08,810 There's some who dispute the term "elementary." But he 273 00:15:08,810 --> 00:15:12,780 really has a book which uses, at the outset, nothing more 274 00:15:12,780 --> 00:15:14,610 than geometry. 275 00:15:14,610 --> 00:15:18,290 He doesn't throw in comments like, "It can be shown that," 276 00:15:18,290 --> 00:15:21,080 or "By further work, it turns out--." He does 277 00:15:21,080 --> 00:15:22,610 everything for you. 278 00:15:22,610 --> 00:15:26,020 Everything is down there so you can see how it's done and 279 00:15:26,020 --> 00:15:27,270 what the results are. 280 00:15:29,810 --> 00:15:32,820 To me, it is the best book on the subject. 281 00:15:32,820 --> 00:15:34,360 That's the good part. 282 00:15:34,360 --> 00:15:36,750 The bad part is that it's been out of print 283 00:15:36,750 --> 00:15:39,750 for about 15 years. 284 00:15:39,750 --> 00:15:43,980 So what I am going to do is to make-- 285 00:15:43,980 --> 00:15:46,890 now that I know how many of you are going to be present-- 286 00:15:46,890 --> 00:15:49,340 I'm going to make a Xerox copy for you of the 287 00:15:49,340 --> 00:15:52,480 first half of the book. 288 00:15:52,480 --> 00:15:54,110 What a department! 289 00:15:54,110 --> 00:15:55,600 What a class! 290 00:15:55,600 --> 00:15:58,780 You're going to get a classic text, 50% of it, without 291 00:15:58,780 --> 00:16:00,360 spending a nickel. 292 00:16:00,360 --> 00:16:05,310 And that'll be the text for the first part of the class. 293 00:16:05,310 --> 00:16:07,760 We will be doing some derivation that 294 00:16:07,760 --> 00:16:09,060 are not in this book. 295 00:16:09,060 --> 00:16:13,130 And for that, I will have notes that I have written out. 296 00:16:13,130 --> 00:16:14,990 And you'll get Xerox copies of that. 297 00:16:14,990 --> 00:16:18,120 So we'll have lots and lots of handouts during the course of 298 00:16:18,120 --> 00:16:19,370 this semester. 299 00:16:21,410 --> 00:16:23,560 I'd like to call your attention, though, to two 300 00:16:23,560 --> 00:16:25,630 other books. 301 00:16:25,630 --> 00:16:26,970 These are not textbooks. 302 00:16:26,970 --> 00:16:28,920 These are reference books. 303 00:16:28,920 --> 00:16:31,600 And you can see from the shape of this one that this is one 304 00:16:31,600 --> 00:16:32,880 of my favorite volumes. 305 00:16:32,880 --> 00:16:35,490 It's thoroughly worn out. 306 00:16:35,490 --> 00:16:38,360 This is something that is called The International 307 00:16:38,360 --> 00:16:41,533 Tables for X-ray Crystallography. 308 00:17:02,660 --> 00:17:05,760 And it is published by an organization called the 309 00:17:05,760 --> 00:17:07,715 International Union for Crystallography. 310 00:17:16,150 --> 00:17:18,660 The funny sounding term, "International Union for 311 00:17:18,660 --> 00:17:22,640 Crystallography," sounds like an organization under which 312 00:17:22,640 --> 00:17:25,430 diffractionists go out and strike for higher pay. 313 00:17:25,430 --> 00:17:29,410 But no, this is actually a federation of all of the 314 00:17:29,410 --> 00:17:31,970 national societies of crystallography from 315 00:17:31,970 --> 00:17:33,810 all over the world. 316 00:17:33,810 --> 00:17:36,490 And among the useful things that they do, besides having a 317 00:17:36,490 --> 00:17:41,010 splendid conference every couple of years, is to publish 318 00:17:41,010 --> 00:17:42,670 these tables. 319 00:17:42,670 --> 00:17:48,440 And volume one is called Symmetry Tables. 320 00:17:53,650 --> 00:17:57,880 And everything that we will derive and all of its 321 00:17:57,880 --> 00:17:58,680 properties-- 322 00:17:58,680 --> 00:18:00,500 physical and geometrical-- 323 00:18:00,500 --> 00:18:02,900 are tabulated in this book. 324 00:18:02,900 --> 00:18:06,790 It is, however, a reference book and not a textbook. 325 00:18:06,790 --> 00:18:10,110 You don't learn it for the first time from this book. 326 00:18:10,110 --> 00:18:13,760 But in terms of generating atomic arrangements from the 327 00:18:13,760 --> 00:18:17,320 data that's present in the literature, looking at the 328 00:18:17,320 --> 00:18:21,460 arrangement of symmetry elements in space, and how 329 00:18:21,460 --> 00:18:26,050 they move atoms around, it is the code book that tells you 330 00:18:26,050 --> 00:18:30,610 how to crack the arcane language in which diffraction 331 00:18:30,610 --> 00:18:34,060 and structural results are recorded and find out how to 332 00:18:34,060 --> 00:18:36,330 unravel it. 333 00:18:36,330 --> 00:18:39,030 I call also to your attention, although it will not be 334 00:18:39,030 --> 00:18:41,960 germane to this class, there are four other volumes. 335 00:18:46,420 --> 00:18:49,030 Volume two is called Mathematical Tables. 336 00:18:56,760 --> 00:18:58,810 And this has all sorts of useful stuff. 337 00:18:58,810 --> 00:19:02,860 If you've ever done diffraction, you know that 338 00:19:02,860 --> 00:19:05,690 depending on the symmetry of the crystal, there are some 339 00:19:05,690 --> 00:19:11,120 planes for which h squared plus k squared plus l squared 340 00:19:11,120 --> 00:19:14,890 divided by 2 pi is not a reflection [INAUDIBLE] 341 00:19:14,890 --> 00:19:18,450 if the crystal is green, and other arcane rules like that. 342 00:19:18,450 --> 00:19:21,040 All of these are summarized in these books. 343 00:19:21,040 --> 00:19:24,680 There are quantities that you need to calculate, things like 344 00:19:24,680 --> 00:19:26,820 interplanar spacings. 345 00:19:26,820 --> 00:19:28,350 Tables are available there. 346 00:19:28,350 --> 00:19:32,120 So this is a handy thing primarily for diffraction. 347 00:19:32,120 --> 00:19:35,645 Volume three is called Physical Tables. 348 00:19:41,460 --> 00:19:44,090 And this is where you find things like absorption 349 00:19:44,090 --> 00:19:47,030 coefficients for x-rays and for neutrons. 350 00:19:47,030 --> 00:19:51,580 It's where you find the latest values of absorption 351 00:19:51,580 --> 00:19:54,320 coefficients, neutron scattering length. 352 00:19:54,320 --> 00:19:57,030 And since these things are derived experimentally, the 353 00:19:57,030 --> 00:19:59,820 values improve and change from time to time. 354 00:19:59,820 --> 00:20:04,220 So this is where you find the most up to date values of 355 00:20:04,220 --> 00:20:08,880 physical constants and items that are necessary for 356 00:20:08,880 --> 00:20:10,130 diffraction. 357 00:20:13,250 --> 00:20:17,120 It never ceases to amaze me how somebody who has the good 358 00:20:17,120 --> 00:20:21,040 fortune of having to use the diffraction for a thesis will 359 00:20:21,040 --> 00:20:24,260 labor carefully over making the measurements and 360 00:20:24,260 --> 00:20:25,620 reducing the data. 361 00:20:25,620 --> 00:20:29,670 And then when it comes to using a wavelength, which is 362 00:20:29,670 --> 00:20:33,200 how the final numbers will be determined, goes to an 363 00:20:33,200 --> 00:20:36,190 appendix of a book on diffraction that was published 364 00:20:36,190 --> 00:20:37,530 20 years ago. 365 00:20:37,530 --> 00:20:39,740 And that's not the most up to date value. 366 00:20:39,740 --> 00:20:44,560 Scattering powers of x-rays by the electrons on the atoms are 367 00:20:44,560 --> 00:20:47,320 calculated from wave functions, which constantly 368 00:20:47,320 --> 00:20:48,950 get better from year to year. 369 00:20:48,950 --> 00:20:51,100 And the value of the scattering powers of the 370 00:20:51,100 --> 00:20:54,460 function of angle gets better from year to year. 371 00:20:54,460 --> 00:20:56,770 So this is where you want to go if you need any of that 372 00:20:56,770 --> 00:20:57,780 physical data. 373 00:20:57,780 --> 00:21:04,980 And finally, volume four is-- 374 00:21:04,980 --> 00:21:07,700 it's not its title, but it's essentially an update of the 375 00:21:07,700 --> 00:21:12,230 Physical Tables, giving later values which came out about 10 376 00:21:12,230 --> 00:21:13,480 years later. 377 00:21:15,450 --> 00:21:18,970 OK, this series was getting out of hand. 378 00:21:18,970 --> 00:21:23,770 So I have to bend my knees and use two hands when I 379 00:21:23,770 --> 00:21:25,030 pick up this one. 380 00:21:25,030 --> 00:21:28,770 This is a continuation of the series, essentially. 381 00:21:28,770 --> 00:21:31,770 But this one is called International Tables for 382 00:21:31,770 --> 00:21:37,020 Crystallography, period, no x-rays because neutrons and 383 00:21:37,020 --> 00:21:39,320 electrons are just as important today for doing 384 00:21:39,320 --> 00:21:41,140 scattering experiments. 385 00:21:41,140 --> 00:21:48,025 And this is International Tables for Crystallography. 386 00:21:55,110 --> 00:21:57,990 No x-ray in there. 387 00:21:57,990 --> 00:22:01,660 And there are now something like six volumes out. 388 00:22:01,660 --> 00:22:04,340 They're not called one, two, three, and four, but they're 389 00:22:04,340 --> 00:22:08,200 called A, B, and C to avoid confusion. 390 00:22:08,200 --> 00:22:13,465 And volume A is one called Space Group Symmetry. 391 00:22:21,740 --> 00:22:23,590 And then, there are a whole series of other ones. 392 00:22:23,590 --> 00:22:25,880 As I say, I think there's six of them that give physical 393 00:22:25,880 --> 00:22:28,480 data and all sorts of useful guides. 394 00:22:31,400 --> 00:22:35,580 I have mixed feelings about the new series. 395 00:22:35,580 --> 00:22:45,610 You will see that it is about three times as large and three 396 00:22:45,610 --> 00:22:51,040 times as heavy, which means it's nine times as expensive. 397 00:22:51,040 --> 00:22:56,180 And to me, it's almost the case for most people of a 398 00:22:56,180 --> 00:23:01,030 situation where if it wasn't broke, you shouldn't fix it. 399 00:23:01,030 --> 00:23:04,020 And what they've done is that they've put in all sorts of 400 00:23:04,020 --> 00:23:09,690 esoteric theory which probably is going to be of interest and 401 00:23:09,690 --> 00:23:12,960 use to perhaps 5% of the readers. 402 00:23:12,960 --> 00:23:15,090 But nevertheless, if you wanted, you'll find it there, 403 00:23:15,090 --> 00:23:18,000 which is something that could not be said before. 404 00:23:18,000 --> 00:23:23,130 They've added a few things which are useful, but a lot of 405 00:23:23,130 --> 00:23:25,750 additional information which you don't really need. 406 00:23:25,750 --> 00:23:28,430 And you pay for that whether you want it or not. 407 00:23:28,430 --> 00:23:29,740 Nevertheless, it's been done. 408 00:23:29,740 --> 00:23:33,490 You can't buy the old volumes any longer. 409 00:23:33,490 --> 00:23:37,350 You have to buy the new volumes. 410 00:23:37,350 --> 00:23:39,550 So anyway, this is what you'll find in the library now. 411 00:23:39,550 --> 00:23:45,020 Maybe they do still have the old volumes, one through four. 412 00:23:45,020 --> 00:23:48,220 This, we will make reference to in the course of the term. 413 00:23:48,220 --> 00:23:52,050 I will give you some copies of certain pages in here as 414 00:23:52,050 --> 00:23:54,530 handouts when we need them for purposes of 415 00:23:54,530 --> 00:23:57,070 illustration or for use. 416 00:23:57,070 --> 00:24:00,990 But I spent the last five minutes just to make you aware 417 00:24:00,990 --> 00:24:02,860 of the existence of these books. 418 00:24:02,860 --> 00:24:05,830 And these are really the penultimate source of 419 00:24:05,830 --> 00:24:12,470 information and numerical quantities that will be used 420 00:24:12,470 --> 00:24:15,200 in diffraction, one of the principle applications of 421 00:24:15,200 --> 00:24:16,960 crystallography. 422 00:24:16,960 --> 00:24:19,170 I think I have just enough enough-- 423 00:24:19,170 --> 00:24:24,630 to start things off, I have a syllabus for the course that 424 00:24:24,630 --> 00:24:28,500 is, in very dense form, exactly what we will be 425 00:24:28,500 --> 00:24:29,750 covering this term. 426 00:24:32,500 --> 00:24:36,070 And I'd like to lead you by the hand through this. 427 00:24:52,480 --> 00:24:56,710 All right, what we will be doing in the first half of the 428 00:24:56,710 --> 00:25:00,503 term is something that is known as crystallography. 429 00:25:06,910 --> 00:25:09,870 OK, the meaning of the word is almost self-explanatory. 430 00:25:09,870 --> 00:25:11,700 The first part is crystal. 431 00:25:11,700 --> 00:25:12,870 We're going to be dealing with the 432 00:25:12,870 --> 00:25:15,600 crystalline state of matter. 433 00:25:15,600 --> 00:25:19,240 To me, amorphous materials, although they may be 434 00:25:19,240 --> 00:25:23,130 important, have all the interest of a piece of steak 435 00:25:23,130 --> 00:25:24,610 before it's been cooked. 436 00:25:24,610 --> 00:25:26,740 The atoms in amorphous materials are fine. 437 00:25:26,740 --> 00:25:29,320 But they really get interesting when they organize 438 00:25:29,320 --> 00:25:31,910 themselves into an ordered fashion. 439 00:25:31,910 --> 00:25:33,540 So the name is self-explanatory. 440 00:25:33,540 --> 00:25:36,680 The first part, crystal, means we're going to deal with the 441 00:25:36,680 --> 00:25:38,210 crystalline state. 442 00:25:38,210 --> 00:25:40,350 What does the graphy mean? 443 00:25:40,350 --> 00:25:43,410 That means mapping or geometry. 444 00:25:43,410 --> 00:25:47,950 And let me give you an example of a few other words that have 445 00:25:47,950 --> 00:25:49,950 the same sort of structure. 446 00:25:49,950 --> 00:25:50,870 Geo-- 447 00:25:50,870 --> 00:25:52,400 the Earth-- 448 00:25:52,400 --> 00:25:58,290 followed by graph, geography, is the mapping of the Earth. 449 00:25:58,290 --> 00:26:00,910 And there are many other terms that involve these two 450 00:26:00,910 --> 00:26:02,960 separate parts. 451 00:26:02,960 --> 00:26:08,760 Crystallography, though, is very often subdivided into 452 00:26:08,760 --> 00:26:10,660 different flavors. 453 00:26:10,660 --> 00:26:13,210 There is something well defined called x-ray 454 00:26:13,210 --> 00:26:14,460 crystallography. 455 00:26:21,600 --> 00:26:27,540 And this is the experimental determination of the 456 00:26:27,540 --> 00:26:32,620 crystallography of a material using diffraction, usually 457 00:26:32,620 --> 00:26:36,500 x-rays because they're relatively inexpensive and 458 00:26:36,500 --> 00:26:37,970 they're widely available. 459 00:26:37,970 --> 00:26:42,080 But increasingly, neutron scattering or electron 460 00:26:42,080 --> 00:26:44,740 scattering is used for this purpose. 461 00:26:44,740 --> 00:26:49,960 And there are a number of very powerful, very exciting 462 00:26:49,960 --> 00:26:54,780 sources of neutrons, either from reactor sources of 463 00:26:54,780 --> 00:27:00,810 unprecedented intensity or from what's called a 464 00:27:00,810 --> 00:27:07,340 spallation source, where an entire synchrotron is built 465 00:27:07,340 --> 00:27:10,020 just to direct a beam of particles onto 466 00:27:10,020 --> 00:27:11,640 a heavy metal target. 467 00:27:11,640 --> 00:27:16,390 And those high energy particles split off neutrons 468 00:27:16,390 --> 00:27:18,420 from the nuclei of the target material. 469 00:27:18,420 --> 00:27:20,240 Doesn't really matter what the material is. 470 00:27:20,240 --> 00:27:22,820 It helps if it's a heavy metal. 471 00:27:22,820 --> 00:27:26,800 The nice thing about these sources of neutron radiation 472 00:27:26,800 --> 00:27:30,660 is that they're so expensive they are all national 473 00:27:30,660 --> 00:27:32,640 facilities. 474 00:27:32,640 --> 00:27:37,770 And the consequence of that is that anybody with a good idea 475 00:27:37,770 --> 00:27:42,800 and a project worth doing can apply for beam time. 476 00:27:42,800 --> 00:27:44,470 And if it's a good problem, you get it. 477 00:27:44,470 --> 00:27:48,550 So you're using a facility that cost $1 billion. 478 00:27:48,550 --> 00:27:51,380 You have people whose sole function in life is to help 479 00:27:51,380 --> 00:27:54,630 you do the experiment and make sure you're doing it properly. 480 00:27:54,630 --> 00:27:59,400 And this is a very, very exciting time to be somebody 481 00:27:59,400 --> 00:28:03,840 working with diffraction using these neutron sources. 482 00:28:03,840 --> 00:28:06,170 There's another branch of crystallography which is 483 00:28:06,170 --> 00:28:07,470 called optical crystallography. 484 00:28:14,980 --> 00:28:17,900 And this is the characterization and study of 485 00:28:17,900 --> 00:28:21,150 crystalline materials using polarized light. 486 00:28:21,150 --> 00:28:26,020 You can identify unknowns using their optical properties 487 00:28:26,020 --> 00:28:29,410 if they're transparent about 10 times faster than you can 488 00:28:29,410 --> 00:28:32,000 do with x-ray diffraction. 489 00:28:32,000 --> 00:28:34,590 It's a technique that today is little used. 490 00:28:34,590 --> 00:28:36,690 But it's a very powerful technique. 491 00:28:36,690 --> 00:28:39,430 And all it takes is a microscope, and 492 00:28:39,430 --> 00:28:41,310 you're off and running. 493 00:28:41,310 --> 00:28:46,380 Some other flavors of crystallography, well, I'll 494 00:28:46,380 --> 00:28:48,440 mention the one that we're going to use. 495 00:28:48,440 --> 00:28:50,730 What we're going to talk about is something called 496 00:28:50,730 --> 00:28:55,080 geometrical crystallography, to distinguish it from these 497 00:28:55,080 --> 00:28:56,330 other branches. 498 00:29:00,850 --> 00:29:04,920 And this is synonymous with symmetry theory. 499 00:29:15,030 --> 00:29:17,860 So that's what we'll do for the first month 500 00:29:17,860 --> 00:29:19,110 and a half or so. 501 00:29:24,130 --> 00:29:29,700 All right, let me introduce now some basic concepts. 502 00:29:29,700 --> 00:29:34,190 Geometrical crystallography is the study of patterns and 503 00:29:34,190 --> 00:29:35,450 their symmetry. 504 00:29:35,450 --> 00:29:42,370 So let me give you an example of some very simple patterns 505 00:29:42,370 --> 00:29:45,130 that extent in one dimension. 506 00:29:45,130 --> 00:29:49,450 And let me put in a figure. 507 00:29:49,450 --> 00:29:51,980 The thing that is in the pattern is something that's 508 00:29:51,980 --> 00:29:54,780 called the motif. 509 00:29:54,780 --> 00:29:58,210 And let me use a plump, little fat comma. 510 00:30:02,620 --> 00:30:04,310 And I'll make a chain of these things 511 00:30:04,310 --> 00:30:07,200 extending in one dimension. 512 00:30:07,200 --> 00:30:10,430 The nice thing about this fat little comma is that it is a 513 00:30:10,430 --> 00:30:14,220 figure which, in itself, has no inherent symmetry. 514 00:30:14,220 --> 00:30:17,990 So it is asymmetric, without symmetry. 515 00:30:17,990 --> 00:30:22,440 And imagine this is being repeated without limit in both 516 00:30:22,440 --> 00:30:27,710 directions, both to the left and to the right. 517 00:30:27,710 --> 00:30:29,400 We then draw another pattern with a 518 00:30:29,400 --> 00:30:31,810 different sort of motif. 519 00:30:31,810 --> 00:30:36,190 And let me use a rectangle with one concave side. 520 00:30:47,950 --> 00:30:51,690 OK, and I think you get the picture of this one. 521 00:30:51,690 --> 00:30:55,130 And imagine that as extending without limit indefinitely to 522 00:30:55,130 --> 00:30:58,260 the left and to the right. 523 00:30:58,260 --> 00:31:00,390 Then, I'm getting tired of inventing new motifs. 524 00:31:00,390 --> 00:31:03,980 So let me use the same motif the second time, but arrange 525 00:31:03,980 --> 00:31:05,780 it in a slightly different way. 526 00:31:16,610 --> 00:31:18,605 And again, imagine that as extended indefinitely. 527 00:31:21,390 --> 00:31:30,700 OK, having now generated these three patterns in two 528 00:31:30,700 --> 00:31:33,790 dimensions but extending periodically in two 529 00:31:33,790 --> 00:31:34,380 dimensions. 530 00:31:34,380 --> 00:31:36,370 Let me ask the question now. 531 00:31:36,370 --> 00:31:39,830 And even if you have not the foggiest idea, you have a 50% 532 00:31:39,830 --> 00:31:41,260 chance of being right. 533 00:31:41,260 --> 00:31:44,240 Are any of these patterns the same? 534 00:31:44,240 --> 00:31:45,900 Or are they all different? 535 00:31:49,140 --> 00:31:50,670 Are any of the patterns the same? 536 00:31:50,670 --> 00:31:51,920 Or are they different? 537 00:31:55,120 --> 00:31:56,030 Well, that's a-- yeah? 538 00:31:56,030 --> 00:31:57,910 AUDIENCE: [INAUDIBLE]. 539 00:31:57,910 --> 00:32:03,650 PROFESSOR: OK, that is an answer that's right because 540 00:32:03,650 --> 00:32:07,196 the bottom two involve the same sort of figure. 541 00:32:07,196 --> 00:32:10,900 They have the same sort of the motif. 542 00:32:10,900 --> 00:32:12,150 They both have the same rectangle 543 00:32:12,150 --> 00:32:13,490 with one concave side. 544 00:32:13,490 --> 00:32:15,422 And that's a valid answer. 545 00:32:15,422 --> 00:32:16,540 Do you have a different answer? 546 00:32:16,540 --> 00:32:17,940 AUDIENCE: The first and third are the same. 547 00:32:17,940 --> 00:32:19,330 PROFESSOR: First and third are the same. 548 00:32:19,330 --> 00:32:20,530 Why do you say that? 549 00:32:20,530 --> 00:32:22,720 AUDIENCE: They both have [? rotational symmetry. ?] 550 00:32:22,720 --> 00:32:24,040 PROFESSOR: OK. 551 00:32:24,040 --> 00:32:26,190 This is the point I was trying to introduce. 552 00:32:26,190 --> 00:32:29,060 And that is your choice of answering the question, one is 553 00:32:29,060 --> 00:32:30,400 the nature the motif. 554 00:32:30,400 --> 00:32:31,780 And you're absolutely correct. 555 00:32:31,780 --> 00:32:33,990 This pattern and this pattern are both 556 00:32:33,990 --> 00:32:35,900 based on the same motif. 557 00:32:35,900 --> 00:32:43,130 But in patterns, we are less concerned with the motif that 558 00:32:43,130 --> 00:32:48,750 is in the pattern than we are with the relations between one 559 00:32:48,750 --> 00:32:52,340 motif and all of the others. 560 00:32:52,340 --> 00:32:57,800 And in that context, the first and the third pattern, 561 00:32:57,800 --> 00:33:01,290 although they look entirely different, are really exactly 562 00:33:01,290 --> 00:33:02,780 the same sort of pattern. 563 00:33:02,780 --> 00:33:08,100 So let's begin to analyze what sort of operations are in 564 00:33:08,100 --> 00:33:10,350 these patterns that take one motif-- 565 00:33:10,350 --> 00:33:12,190 and obviously, they're all the same-- 566 00:33:12,190 --> 00:33:16,400 and relate it to all of the others. 567 00:33:16,400 --> 00:33:21,890 First of all, there is an operation which I'll call 568 00:33:21,890 --> 00:33:24,100 translation for obvious reasons. 569 00:33:24,100 --> 00:33:28,010 And I'll represent that by a vector, T, since a translation 570 00:33:28,010 --> 00:33:34,700 has magnitude and direction but no unique origin. 571 00:33:34,700 --> 00:33:39,000 I could take this pair of objects sitting nose to nose, 572 00:33:39,000 --> 00:33:43,420 pick them up, slide them over by T, put them down again. 573 00:33:43,420 --> 00:33:46,410 And I have the relation that gives me 574 00:33:46,410 --> 00:33:48,570 this neighboring pair. 575 00:33:48,570 --> 00:33:51,570 Pick it up again, move it to the right by the same 576 00:33:51,570 --> 00:33:54,370 translation in the same direction, put it down again. 577 00:33:54,370 --> 00:33:56,600 And I've got this pair. 578 00:33:56,600 --> 00:34:00,410 So that is one operation that can exist in patterns. 579 00:34:00,410 --> 00:34:03,930 This is the operation of translation. 580 00:34:03,930 --> 00:34:07,470 So let me call that by a vector relation. 581 00:34:07,470 --> 00:34:08,749 And it has magnitude. 582 00:34:12,520 --> 00:34:23,920 It has direction, but no unique origin, just like a 583 00:34:23,920 --> 00:34:25,350 plain old vector. 584 00:34:25,350 --> 00:34:29,750 So in other words, I can't say that the translation moves us 585 00:34:29,750 --> 00:34:32,949 from here to here or from here to here. 586 00:34:32,949 --> 00:34:34,360 It's all the same thing-- 587 00:34:34,360 --> 00:34:36,889 magnitude and direction, no unique origin. 588 00:34:41,360 --> 00:34:46,429 In fact, all of these patterns have translational 589 00:34:46,429 --> 00:34:48,900 periodicity. 590 00:34:48,900 --> 00:34:52,030 There's a translation in this bottom pattern and another 591 00:34:52,030 --> 00:34:54,800 translation from here to here in the middle pattern. 592 00:34:57,620 --> 00:35:03,510 The thing that makes a crystal a crystal is that it is an 593 00:35:03,510 --> 00:35:08,320 arrangement of atoms or molecules which is related one 594 00:35:08,320 --> 00:35:12,560 part to another by the operation of translation. 595 00:35:12,560 --> 00:35:15,730 If you don't have translational periodicity, you 596 00:35:15,730 --> 00:35:17,250 do not have a crystal. 597 00:35:17,250 --> 00:35:19,670 So that comes to the essence of what 598 00:35:19,670 --> 00:35:22,040 crystallography is about. 599 00:35:22,040 --> 00:35:27,470 You can imagine, in one sense, the generation of this pattern 600 00:35:27,470 --> 00:35:31,530 by a rubber stamp sort of operation. 601 00:35:31,530 --> 00:35:33,600 Suppose I have a rubber stamp. 602 00:35:33,600 --> 00:35:39,880 And I put on the rubber stamp the pair of motifs like this. 603 00:35:39,880 --> 00:35:41,920 Pick it up, move it over, chunk. 604 00:35:41,920 --> 00:35:43,560 Pick it up, move it over, chunk. 605 00:35:43,560 --> 00:35:47,830 And I can stamp out the pattern in that fashion. 606 00:35:47,830 --> 00:35:51,400 Notice that my statement about no unique origin in these 607 00:35:51,400 --> 00:35:55,210 terms can be stated that it doesn't matter where the two 608 00:35:55,210 --> 00:35:57,690 motifs are on the stamp. 609 00:35:57,690 --> 00:35:59,860 they could be up in the upper left hand corner, right in the 610 00:35:59,860 --> 00:36:01,320 middle, down in the bottom. 611 00:36:01,320 --> 00:36:05,100 As long as I move the stamp through the same distance and 612 00:36:05,100 --> 00:36:07,430 the same direction, I get the same pattern. 613 00:36:09,940 --> 00:36:11,630 Now, that's not bad for an introduction. 614 00:36:11,630 --> 00:36:17,740 But I want to be more general than this because when I deal 615 00:36:17,740 --> 00:36:20,940 in terms of a rubber stamp operation, that is a 616 00:36:20,940 --> 00:36:25,830 transformation that involves taking one little chunk of a 617 00:36:25,830 --> 00:36:30,300 two dimensional space, picking it up, and putting it down in 618 00:36:30,300 --> 00:36:37,850 another location to another unique location in space. 619 00:36:37,850 --> 00:36:42,270 So I'm going to now make another generalization that 620 00:36:42,270 --> 00:36:45,760 operations, which we've begun to define, 621 00:36:45,760 --> 00:36:47,470 act on all of space. 622 00:36:57,130 --> 00:36:59,660 So I don't want you to think of this repetition in terms of 623 00:36:59,660 --> 00:37:02,550 a rubber stamp, although we could get the pattern that way 624 00:37:02,550 --> 00:37:04,780 and it's conceptually appealing. 625 00:37:04,780 --> 00:37:10,130 But I'm going to say now that this string of motifs has 626 00:37:10,130 --> 00:37:16,610 translational periodicity if, when I pick it up, move it by 627 00:37:16,610 --> 00:37:20,200 T in a particular direction, and drop the whole infinite 628 00:37:20,200 --> 00:37:23,310 chain back down again, it is mapped into 629 00:37:23,310 --> 00:37:26,558 congruence with itself. 630 00:37:26,558 --> 00:37:30,940 Which leads me to another definition-- 631 00:37:30,940 --> 00:37:54,670 an object or a space possesses symmetry when there is an 632 00:37:54,670 --> 00:38:19,230 operation or a set of operations that maps it into 633 00:38:19,230 --> 00:38:20,480 congruence with itself. 634 00:38:39,010 --> 00:38:41,790 In other words, in plain words, you can't tell that 635 00:38:41,790 --> 00:38:43,040 it's been moved. 636 00:39:01,480 --> 00:39:05,660 OK, is there anything else that is a transformation which 637 00:39:05,660 --> 00:39:09,470 leaves the set invariant? 638 00:39:09,470 --> 00:39:13,930 OK, if we look at the first pattern, there are [? rho ?] 639 00:39:13,930 --> 00:39:19,030 sides such as this one here, or this one here, or this one 640 00:39:19,030 --> 00:39:26,630 here, about which I can rotate one motif into its neighbor 641 00:39:26,630 --> 00:39:30,660 or, for that matter, pick up the entire chain and flip it 642 00:39:30,660 --> 00:39:34,370 end over elbow through 180 degrees. 643 00:39:34,370 --> 00:39:39,060 And it will be mapped into coincidence with itself. 644 00:39:39,060 --> 00:39:42,410 And that is an operation, and another sort of distinct 645 00:39:42,410 --> 00:39:44,770 operation of transformation. 646 00:39:44,770 --> 00:39:46,520 And this is one that I could call 647 00:39:46,520 --> 00:39:49,530 rotation for obvious reasons. 648 00:39:55,110 --> 00:39:57,230 And there are two things I have to tell you about a 649 00:39:57,230 --> 00:39:59,050 rotation operation. 650 00:39:59,050 --> 00:40:03,230 The first one is the point about which the rotation takes 651 00:40:03,230 --> 00:40:06,080 place, and that's going to be some point. 652 00:40:06,080 --> 00:40:18,590 And let me call this point here A. So this will be some 653 00:40:18,590 --> 00:40:23,015 labelled point that is the location of the rotation axis. 654 00:40:27,070 --> 00:40:29,160 But then, the other thing that I have to tell you is the 655 00:40:29,160 --> 00:40:31,730 angle through which I'm going to rotate. 656 00:40:31,730 --> 00:40:36,700 And I'll append to the A as a subscript 657 00:40:36,700 --> 00:40:37,950 the angle of rotation. 658 00:40:45,010 --> 00:40:49,070 So this particular operation, called a twofold rotation 659 00:40:49,070 --> 00:40:52,120 because it rotates through half of a circle, would be the 660 00:40:52,120 --> 00:40:55,520 operation A pi. 661 00:40:55,520 --> 00:41:00,690 This point is A. We rotate through an angle pi. 662 00:41:00,690 --> 00:41:07,390 This pattern here has also rotational symmetry. 663 00:41:07,390 --> 00:41:09,400 In addition to the translation, there is a 664 00:41:09,400 --> 00:41:12,860 rotation operation, A pi, in the lower pattern. 665 00:41:12,860 --> 00:41:18,710 So the follow who is unfortunate enough not to have 666 00:41:18,710 --> 00:41:20,270 a seat-- and I should have given you this 667 00:41:20,270 --> 00:41:21,520 one a long time ago. 668 00:41:21,520 --> 00:41:23,260 I'll give that to you as your reward for 669 00:41:23,260 --> 00:41:24,970 giving the best answer. 670 00:41:24,970 --> 00:41:29,800 And you get a seat wherever you would like to place it. 671 00:41:29,800 --> 00:41:34,130 The first and the final pattern are the same in the 672 00:41:34,130 --> 00:41:36,240 sense that they contain two operations, 673 00:41:36,240 --> 00:41:38,160 translation and rotation. 674 00:41:38,160 --> 00:41:40,810 This pattern is a much more interesting one. 675 00:41:40,810 --> 00:41:44,370 This also has a rotational symmetry, A pi. 676 00:41:47,180 --> 00:41:50,160 It also is based on a translation. 677 00:41:50,160 --> 00:41:54,240 But now, there's another operation that we can do to 678 00:41:54,240 --> 00:41:55,960 leave the pattern invariant. 679 00:41:55,960 --> 00:41:59,580 There exists [? rho sides ?] 680 00:41:59,580 --> 00:42:04,510 that pass through the center of this rectangular figure 681 00:42:04,510 --> 00:42:10,380 across which I could flip an individual motif, or for that 682 00:42:10,380 --> 00:42:13,810 matter the entire pattern, from left to right. 683 00:42:13,810 --> 00:42:18,280 It's a reflection sort of operation. 684 00:42:18,280 --> 00:42:21,606 So this is a new type of transformation. 685 00:42:28,890 --> 00:42:30,500 So we'll add that to our list. 686 00:42:34,790 --> 00:42:40,300 And the symbol that's usually used to indicate the locus of 687 00:42:40,300 --> 00:42:44,070 this operation is m, standing from mirror. 688 00:42:48,590 --> 00:42:51,770 And that does it for these particular patterns. 689 00:42:51,770 --> 00:42:53,210 Three sorts of operations-- 690 00:42:53,210 --> 00:42:57,700 translation, rotation, and reflection. 691 00:42:57,700 --> 00:43:02,840 And in fact, that is all you can have in a 692 00:43:02,840 --> 00:43:04,610 two dimensional space-- 693 00:43:04,610 --> 00:43:06,570 not necessarily a rotation that's 694 00:43:06,570 --> 00:43:08,740 restricted to 180 degrees. 695 00:43:08,740 --> 00:43:12,920 If these patterns are translational periodic in more 696 00:43:12,920 --> 00:43:16,780 than one direction, you can have higher symmetries. 697 00:43:19,660 --> 00:43:24,390 One of the things I would like to suggest to you is that you 698 00:43:24,390 --> 00:43:29,770 look around you in everyday life at the sort of patterns 699 00:43:29,770 --> 00:43:32,330 that enrich your environment. 700 00:43:32,330 --> 00:43:35,850 I see a one dimensionally periodic pattern there, the 701 00:43:35,850 --> 00:43:37,630 black and white stripes. 702 00:43:37,630 --> 00:43:40,770 It's translationally periodic, going up and down. 703 00:43:40,770 --> 00:43:44,640 It also has mirror planes running through the black 704 00:43:44,640 --> 00:43:47,366 stripes and the white stripes. 705 00:43:47,366 --> 00:43:51,740 I see another two dimensional pattern back there. 706 00:43:51,740 --> 00:43:53,700 That has translation. 707 00:43:53,700 --> 00:43:56,920 But you could rotate-- 708 00:43:56,920 --> 00:43:57,970 no, you can't do anything. 709 00:43:57,970 --> 00:43:59,710 That just has translation, nothing else. 710 00:43:59,710 --> 00:44:00,650 Get a new shirt. 711 00:44:00,650 --> 00:44:03,420 That's not terribly interesting. 712 00:44:03,420 --> 00:44:05,670 There's another one there that's so complex I don't 713 00:44:05,670 --> 00:44:09,390 think I can look at it without climbing all over him and 714 00:44:09,390 --> 00:44:13,050 drawing some translational vectors and things like that. 715 00:44:13,050 --> 00:44:14,610 But that's a nice periodic pattern. 716 00:44:14,610 --> 00:44:16,210 That's a good one. 717 00:44:16,210 --> 00:44:18,410 But there's lots of stuff like that. 718 00:44:18,410 --> 00:44:21,350 Look at the grills in the ventilators. 719 00:44:21,350 --> 00:44:22,820 They have mirror planes. 720 00:44:22,820 --> 00:44:26,960 They are translationally periodic in one direction. 721 00:44:26,960 --> 00:44:29,170 We've got floor tiles. 722 00:44:29,170 --> 00:44:33,450 These are lovely because these have examples of 90 degree 723 00:44:33,450 --> 00:44:35,500 rotational symmetry. 724 00:44:35,500 --> 00:44:38,830 Same is true of the tiles up in the ceiling, 725 00:44:38,830 --> 00:44:40,330 same sort of pattern. 726 00:44:40,330 --> 00:44:42,780 And there's another pattern with four-fold symmetry in the 727 00:44:42,780 --> 00:44:50,360 grills that are underneath the fluorescent fixtures. 728 00:44:50,360 --> 00:44:51,495 So symmetry is everywhere. 729 00:44:51,495 --> 00:44:52,860 It surrounds us. 730 00:44:52,860 --> 00:44:54,090 We wear it. 731 00:44:54,090 --> 00:44:55,030 We walk on it. 732 00:44:55,030 --> 00:44:56,400 We sit on it. 733 00:44:56,400 --> 00:44:59,660 And think how much richer your life will be when you can 734 00:44:59,660 --> 00:45:01,575 understand this part of your environment. 735 00:45:01,575 --> 00:45:05,070 Hey, that's a good, chauvinistic note, overstated, 736 00:45:05,070 --> 00:45:06,140 on which to end. 737 00:45:06,140 --> 00:45:08,060 So why don't we take our break? 738 00:45:08,060 --> 00:45:10,800 I'll hang around if you have any questions or get you a 739 00:45:10,800 --> 00:45:12,680 copy of anything that came around that you 740 00:45:12,680 --> 00:45:14,100 missed getting one of. 741 00:45:14,100 --> 00:45:19,100 And it is now, according to my Timex watch, about three 742 00:45:19,100 --> 00:45:20,420 minutes before the hour. 743 00:45:20,420 --> 00:45:24,630 So let's take a break and stretch for 10 minutes. 744 00:45:24,630 --> 00:45:27,020 Ya'll come back because I've got your name's on a list.