1 00:00:06,958 --> 00:00:11,680 PROFESSOR: All right, my trusty \$5 Timex watch says 2 00:00:11,680 --> 00:00:12,920 it's five after the hour. 3 00:00:12,920 --> 00:00:17,180 So I guess we might as well begin. 4 00:00:17,180 --> 00:00:22,460 I have corrected all the problems that were turned in-- 5 00:00:22,460 --> 00:00:24,710 the puzzles, not problems, puzzles. 6 00:00:24,710 --> 00:00:28,110 And I just yesterday got hold of the 7 00:00:28,110 --> 00:00:29,720 photographs of you few people. 8 00:00:29,720 --> 00:00:32,570 And I'd like to hand out the papers individually so I can 9 00:00:32,570 --> 00:00:35,120 start to put names together with faces. 10 00:00:35,120 --> 00:00:39,040 So I'll turn them back individually next time. 11 00:00:39,040 --> 00:00:41,600 I have one additional problem set that for you. 12 00:00:41,600 --> 00:00:46,210 And again, this is in the form of a puzzle. 13 00:00:46,210 --> 00:00:49,790 Be assured that this is the last entertaining problem set. 14 00:00:49,790 --> 00:00:53,800 From here on, we move into drudgery-- 15 00:00:53,800 --> 00:00:56,760 tedious, drawn out things that hopefully 16 00:00:56,760 --> 00:00:57,620 will teach you something. 17 00:00:57,620 --> 00:00:59,880 But this is the last one that's done with a little bit 18 00:00:59,880 --> 00:01:02,750 of tongue in cheek humor. 19 00:01:02,750 --> 00:01:05,820 This one is a little more subtle than the other two. 20 00:01:05,820 --> 00:01:09,200 And for this one, I will hand out a solution because I would 21 00:01:09,200 --> 00:01:13,230 like to use these problems to make a couple of basic points 22 00:01:13,230 --> 00:01:16,310 about the nature of symmetry theory in crystals. 23 00:01:16,310 --> 00:01:18,070 So see how you do on this one. 24 00:01:22,620 --> 00:01:31,190 OK, also before we begin, he gets a little bit uneasy 25 00:01:31,190 --> 00:01:32,190 before a crowd. 26 00:01:32,190 --> 00:01:35,110 But I brought a little friend in with me today. 27 00:01:35,110 --> 00:01:38,090 And could someone hold the box, please, so I 28 00:01:38,090 --> 00:01:39,340 could take him out? 29 00:01:49,090 --> 00:01:50,570 Sorry. 30 00:01:50,570 --> 00:01:53,020 He gets a little bit nervous before a crowd. 31 00:01:53,020 --> 00:01:54,230 Come on, take it easy. 32 00:01:54,230 --> 00:01:55,480 Take it easy. 33 00:01:59,490 --> 00:02:00,820 This is Rodney. 34 00:02:03,910 --> 00:02:06,070 He's an old friend of mine. 35 00:02:06,070 --> 00:02:10,820 Rodney is a crystallographic abomination. 36 00:02:10,820 --> 00:02:12,535 Rodney has five-fold symmetry-- 37 00:02:16,040 --> 00:02:20,330 impossible in crystallography, an abomination. 38 00:02:20,330 --> 00:02:24,130 Why did Rodney decide to do this? 39 00:02:24,130 --> 00:02:27,470 Anybody have any idea? 40 00:02:27,470 --> 00:02:29,740 Why not six fold symmetry? 41 00:02:29,740 --> 00:02:31,490 He's got five-fold symmetry. 42 00:02:31,490 --> 00:02:32,400 Yuck! 43 00:02:32,400 --> 00:02:33,480 But why? 44 00:02:33,480 --> 00:02:37,420 There's always a reason for things if you look carefully 45 00:02:37,420 --> 00:02:40,176 enough and know what questions to ask. 46 00:02:40,176 --> 00:02:40,634 Yeah? 47 00:02:40,634 --> 00:02:42,930 AUDIENCE: He had narrow symmetry. 48 00:02:42,930 --> 00:02:44,440 PROFESSOR: Yeah, he had narrow symmetry. 49 00:02:44,440 --> 00:02:46,190 But he could do that-- what if you were square? 50 00:02:46,190 --> 00:02:47,440 Why not a square starfish? 51 00:02:51,230 --> 00:02:52,750 Why five? 52 00:02:52,750 --> 00:02:54,940 He did this for a reason. 53 00:02:54,940 --> 00:02:57,340 He's been doing this for probably a couple of million 54 00:02:57,340 --> 00:02:59,985 years, he and his ancestors. 55 00:02:59,985 --> 00:03:01,930 There has to be a reason. 56 00:03:01,930 --> 00:03:03,650 OK, let me turn him over. 57 00:03:03,650 --> 00:03:06,900 And maybe that will give you a clue. 58 00:03:06,900 --> 00:03:11,210 The backside is not nearly as nice as the front side. 59 00:03:11,210 --> 00:03:18,230 And in fact, if you look at Rodney carefully, Rodney has 60 00:03:18,230 --> 00:03:20,650 sutures between these five arms. 61 00:03:26,856 --> 00:03:29,400 OK, this is what he looks like from the top. 62 00:03:29,400 --> 00:03:31,890 But if you look at him from the bottom, there are sutures 63 00:03:31,890 --> 00:03:36,280 between these arms that do this. 64 00:03:36,280 --> 00:03:39,970 And these sutures are weak points in his structure. 65 00:03:39,970 --> 00:03:43,500 And by having five-fold symmetry, he puts the suture 66 00:03:43,500 --> 00:03:48,030 between this pair of arms opposite the midpoint of the 67 00:03:48,030 --> 00:03:49,290 opposite rib. 68 00:03:49,290 --> 00:03:51,150 And that gives him a great deal 69 00:03:51,150 --> 00:03:54,480 more structural integrity. 70 00:03:54,480 --> 00:03:57,580 So when a large fish comes along and gloms onto one of 71 00:03:57,580 --> 00:04:01,655 his arms and shakes, he doesn't split in half and end 72 00:04:01,655 --> 00:04:03,530 his career prematurely. 73 00:04:03,530 --> 00:04:06,250 What happens instead, at very worst, the tip of the arm 74 00:04:06,250 --> 00:04:07,920 breaks off. 75 00:04:07,920 --> 00:04:12,000 And then, Rodney gets to scamper away and make more 76 00:04:12,000 --> 00:04:16,899 starfish or do whatever his purpose in our environment is. 77 00:04:16,899 --> 00:04:18,550 If he had an even-fold symmetry, 78 00:04:18,550 --> 00:04:20,070 that wouldn't happen. 79 00:04:20,070 --> 00:04:23,070 He would really be done in by a fish 80 00:04:23,070 --> 00:04:25,652 grabbing hold and shaking. 81 00:04:25,652 --> 00:04:28,530 AUDIENCE: What if [INAUDIBLE]? 82 00:04:28,530 --> 00:04:30,730 PROFESSOR: That's a very good question. 83 00:04:30,730 --> 00:04:34,650 If he had only three arms and a fish glomed on and one arm 84 00:04:34,650 --> 00:04:38,300 broke off, his mobility would be greatly impaired. 85 00:04:38,300 --> 00:04:40,060 In a way, he would go hip, hop, flop. 86 00:04:40,060 --> 00:04:40,550 Hip, hip, flop. 87 00:04:40,550 --> 00:04:41,500 Hip, hip, flop. 88 00:04:41,500 --> 00:04:42,640 Hip, hip, flop. 89 00:04:42,640 --> 00:04:45,590 It'd be a pathetic thing. 90 00:04:45,590 --> 00:04:49,387 So five is much more advantageous than three. 91 00:04:49,387 --> 00:04:50,201 AUDIENCE: Why not seven? 92 00:04:50,201 --> 00:04:52,130 PROFESSOR: Why not seven? 93 00:04:52,130 --> 00:04:53,520 That's a good question, too. 94 00:04:53,520 --> 00:04:57,965 Would you believe that on the Australian Barrier Reef, there 95 00:04:57,965 --> 00:05:00,936 are starfish with nine-fold symmetry? 96 00:05:00,936 --> 00:05:03,070 Quite odd. 97 00:05:03,070 --> 00:05:05,730 That's really a creepy thing with all those 98 00:05:05,730 --> 00:05:06,740 arms flopping around. 99 00:05:06,740 --> 00:05:11,430 But as many things in our environment in our natural 100 00:05:11,430 --> 00:05:15,130 world, there's a reason for things if you can only think 101 00:05:15,130 --> 00:05:16,610 what it might be. 102 00:05:16,610 --> 00:05:19,455 So we gave you another example of a non-crystallographic 103 00:05:19,455 --> 00:05:25,130 symmetry, the large cacti of the Southwest, the saguaros. 104 00:05:25,130 --> 00:05:27,160 And there are lots of other ribbed casts. 105 00:05:27,160 --> 00:05:32,270 They have symmetries that range between 19 and 23 fold. 106 00:05:32,270 --> 00:05:35,480 Why do you think they have this rib structure? 107 00:05:35,480 --> 00:05:37,160 Why can't they have a nice, smooth trunk 108 00:05:37,160 --> 00:05:39,370 like a birch tree? 109 00:05:39,370 --> 00:05:40,430 Any idea there? 110 00:05:40,430 --> 00:05:41,180 I'll give you a hint. 111 00:05:41,180 --> 00:05:43,760 They live out in the desert. 112 00:05:43,760 --> 00:05:44,840 There's not much water. 113 00:05:44,840 --> 00:05:45,794 AUDIENCE: [INAUDIBLE]. 114 00:05:45,794 --> 00:05:47,225 PROFESSOR: Hm? 115 00:05:47,225 --> 00:05:50,087 AUDIENCE: More surface area for the [INAUDIBLE]. 116 00:05:50,087 --> 00:05:51,800 PROFESSOR: Yeah, but that's bad. 117 00:05:51,800 --> 00:05:55,170 That means they would lose their water. 118 00:05:55,170 --> 00:05:58,730 So things that don't want to lose water 119 00:05:58,730 --> 00:05:59,500 through their surface-- 120 00:05:59,500 --> 00:06:02,510 you're right-- try to minimize their surface. 121 00:06:02,510 --> 00:06:03,690 But you're onto something. 122 00:06:03,690 --> 00:06:06,070 And it is exactly the reverse of that. 123 00:06:06,070 --> 00:06:07,370 Water is scarce. 124 00:06:07,370 --> 00:06:11,260 So when it rains, this cactus wants to soak up the moisture 125 00:06:11,260 --> 00:06:12,540 and store it. 126 00:06:12,540 --> 00:06:15,090 If he has this pleated structure, 127 00:06:15,090 --> 00:06:17,300 he's able to expand. 128 00:06:17,300 --> 00:06:20,680 If he had a smooth surface with minimal area, 129 00:06:20,680 --> 00:06:21,630 he would just pop. 130 00:06:21,630 --> 00:06:24,560 And that would be the end of him or her, as the case may 131 00:06:24,560 --> 00:06:27,590 be. "It" is probably the appropriate term. 132 00:06:27,590 --> 00:06:31,380 But anyway, there's the reason for these many, many ribs. 133 00:06:31,380 --> 00:06:34,280 It gives them the ability to expand, store water, and then 134 00:06:34,280 --> 00:06:37,800 contract when the water evaporates. 135 00:06:37,800 --> 00:06:42,140 So again, it's always interesting to look at the 136 00:06:42,140 --> 00:06:44,610 world around us and see examples of things that are 137 00:06:44,610 --> 00:06:47,860 strange because it's the strange things that very often 138 00:06:47,860 --> 00:06:51,760 can lead to penetrating conclusions. 139 00:06:51,760 --> 00:06:56,650 All right, last time we had talked about chirality in 140 00:06:56,650 --> 00:06:59,790 finite atomic structures, molecules. 141 00:06:59,790 --> 00:07:03,340 And I had pointed out-- but did not make the obvious 142 00:07:03,340 --> 00:07:04,510 observation-- 143 00:07:04,510 --> 00:07:10,260 I cannot think of a more dramatic connection between 144 00:07:10,260 --> 00:07:13,240 structure and properties, which is what material science 145 00:07:13,240 --> 00:07:18,240 is all about, than the very, very different properties of 146 00:07:18,240 --> 00:07:21,490 molecules that are of opposite handedness. 147 00:07:21,490 --> 00:07:26,700 We pointed out, for example, that many pharmaceuticals are 148 00:07:26,700 --> 00:07:31,230 of chiral molecules that exist in right handed and left 149 00:07:31,230 --> 00:07:32,450 handed forms. 150 00:07:32,450 --> 00:07:36,570 And very often, one form of one handedness has a very 151 00:07:36,570 --> 00:07:39,870 different set of properties, and in particular 152 00:07:39,870 --> 00:07:44,180 pharmacological action, then it it's enan enantiomorph. 153 00:07:44,180 --> 00:07:47,860 So in the case of thalidomide, one handedness of the molecule 154 00:07:47,860 --> 00:07:49,200 acts as a tranquilizer. 155 00:07:49,200 --> 00:07:54,390 The other handedness acts as a source of birth defects. 156 00:07:54,390 --> 00:07:59,460 One of the other ones less familiar to me, but this is 157 00:07:59,460 --> 00:08:00,885 [? ethambutol. ?] 158 00:08:00,885 --> 00:08:04,510 It's used as a drug to treat tuberculosis. 159 00:08:04,510 --> 00:08:09,400 The molecule of opposite handedness creates blindness. 160 00:08:09,400 --> 00:08:17,010 The drug Ritalin that is used to treat hyperactivity, one 161 00:08:17,010 --> 00:08:20,150 handedness of the molecule is absolutely useless. 162 00:08:20,150 --> 00:08:23,710 So you only use half of what you pay for. 163 00:08:23,710 --> 00:08:27,610 So it's of great interest to pharmaceutical companies to be 164 00:08:27,610 --> 00:08:32,370 able to synthesize a molecule of one particular handedness 165 00:08:32,370 --> 00:08:35,200 with little if none of the other one. 166 00:08:35,200 --> 00:08:39,030 And I looked up a number since we met last time. 167 00:08:39,030 --> 00:08:45,010 It turns out that in 1997, enantiomorph pure drugs were a 168 00:08:45,010 --> 00:08:49,610 \$400 billion dollar industry. 169 00:08:49,610 --> 00:08:55,180 So again, this is structure property relations in action. 170 00:08:55,180 --> 00:08:56,740 So let me ask you another question. 171 00:08:56,740 --> 00:09:00,460 It turns out that products that are derived from nature-- 172 00:09:00,460 --> 00:09:04,240 a good example being sugar, which is produced by sugar 173 00:09:04,240 --> 00:09:06,440 beets or sugar cane-- 174 00:09:06,440 --> 00:09:10,300 produces a molecule only of one handedness. 175 00:09:10,300 --> 00:09:11,570 They are [? enantiomer ?] 176 00:09:11,570 --> 00:09:12,650 pure. 177 00:09:12,650 --> 00:09:16,120 But every synthetic compound that can exist in right handed 178 00:09:16,120 --> 00:09:18,890 and left handed forms occurs with equal probability. 179 00:09:21,760 --> 00:09:27,430 So how do you make a pharmaceutical of one 180 00:09:27,430 --> 00:09:30,600 particular handedness? 181 00:09:30,600 --> 00:09:32,900 And this is an interesting question because I think it 182 00:09:32,900 --> 00:09:36,940 was two years ago the Nobel Prize in chemistry was given 183 00:09:36,940 --> 00:09:41,250 to three people for their work in being able to synthesize 184 00:09:41,250 --> 00:09:42,960 molecules of a given handedness. 185 00:09:42,960 --> 00:09:44,020 Anybody an idea how you do it? 186 00:09:44,020 --> 00:09:46,756 AUDIENCE: I don't have an idea how you do it, but what do you 187 00:09:46,756 --> 00:09:48,421 mean by one handedness versus--? 188 00:09:48,421 --> 00:09:50,377 PROFESSOR: OK, left handed or right handed. 189 00:09:53,320 --> 00:09:57,460 So my left hand is exactly the same as my right hand, but I 190 00:09:57,460 --> 00:10:00,250 cannot match them into congruence with one another. 191 00:10:00,250 --> 00:10:03,080 And we don't know how to describe this other than we 192 00:10:03,080 --> 00:10:07,490 used the term derived from our physiology. 193 00:10:07,490 --> 00:10:10,240 We say left handed and right handed. 194 00:10:10,240 --> 00:10:13,190 So when I was referring to a molecule of one handedness-- 195 00:10:13,190 --> 00:10:16,860 and I'll bring in a Xerox copy of a couple of examples. 196 00:10:16,860 --> 00:10:19,340 There are many examples of fairly common molecules which 197 00:10:19,340 --> 00:10:24,300 can exist in one form in which the hydroxyl group points out 198 00:10:24,300 --> 00:10:26,870 this way, then another form where the hydroxyl groups 199 00:10:26,870 --> 00:10:28,350 points out this way. 200 00:10:28,350 --> 00:10:29,420 Sorry, you should have asked that earlier. 201 00:10:29,420 --> 00:10:32,230 AUDIENCE: So if it's not the same molecule, it would be 202 00:10:32,230 --> 00:10:33,200 that molecule but--? 203 00:10:33,200 --> 00:10:38,000 PROFESSOR: Chemically, it is exactly the same concept. 204 00:10:38,000 --> 00:10:42,245 But the way in which those component models are 205 00:10:42,245 --> 00:10:44,806 configured has the same number of side groups, the same 206 00:10:44,806 --> 00:10:45,984 number of appendages. 207 00:10:45,984 --> 00:10:49,760 But they, don't have a better word. 208 00:10:49,760 --> 00:10:50,310 One is left handed. 209 00:10:50,310 --> 00:10:52,240 One is right handed. 210 00:10:52,240 --> 00:10:54,420 I'll give you some examples next time. 211 00:10:54,420 --> 00:10:55,560 Somebody else, you had a question? 212 00:10:55,560 --> 00:10:57,528 AUDIENCE: I was just going to see if I could venture a guess 213 00:10:57,528 --> 00:10:58,512 how they do it. 214 00:10:58,512 --> 00:11:01,464 I was going to guess they use an electric field that has its 215 00:11:01,464 --> 00:11:04,088 bearings to set the right handed course [INAUDIBLE] that 216 00:11:04,088 --> 00:11:05,400 would make [? enantiomorphs ?] 217 00:11:05,400 --> 00:11:10,070 or more advantageous than [INAUDIBLE]. 218 00:11:10,070 --> 00:11:12,320 PROFESSOR: That's not a bad idea. 219 00:11:12,320 --> 00:11:16,750 Anything that could bias the energy of one configuration 220 00:11:16,750 --> 00:11:18,810 over the other would do it. 221 00:11:18,810 --> 00:11:21,410 But actually, no reason why you should know how to do it. 222 00:11:21,410 --> 00:11:23,210 If you knew how to do it, maybe you could have gotten 223 00:11:23,210 --> 00:11:25,350 the Nobel Prize instead of these other guys. 224 00:11:25,350 --> 00:11:28,720 But actually, you do it through catalysis. 225 00:11:28,720 --> 00:11:33,850 And you use a catalyst which itself has chirality. 226 00:11:33,850 --> 00:11:37,920 And that chirality favors the formation of one molecule over 227 00:11:37,920 --> 00:11:40,590 the other almost exclusively. 228 00:11:40,590 --> 00:11:43,980 And that's why living plants make-- 229 00:11:43,980 --> 00:11:47,140 or animals-- make chiral molecules in just one 230 00:11:47,140 --> 00:11:48,990 handedness as well. 231 00:11:48,990 --> 00:11:52,300 So it's using a catalyst that also has some chirality. 232 00:11:52,300 --> 00:11:55,288 AUDIENCE: How do you get the one hundred catalysts 233 00:11:55,288 --> 00:11:56,782 [INAUDIBLE]? 234 00:11:56,782 --> 00:11:58,032 PROFESSOR: How do you have it--? 235 00:12:02,260 --> 00:12:04,730 OK, there are lots of materials. 236 00:12:04,730 --> 00:12:09,330 Any crystalline material that does not have a mirror plane 237 00:12:09,330 --> 00:12:13,870 in it can exist in chiral forms. 238 00:12:13,870 --> 00:12:18,790 And one of the classic examples of this is quartz. 239 00:12:18,790 --> 00:12:21,280 Quartz is SiO2. 240 00:12:21,280 --> 00:12:25,430 And quartz has actually only a threefold symmetry. 241 00:12:25,430 --> 00:12:29,480 But it forms crystals that look prismatic, as though they 242 00:12:29,480 --> 00:12:32,460 were hexagonal prisms. 243 00:12:32,460 --> 00:12:36,190 And if I draw two of these prisms side by side, you could 244 00:12:36,190 --> 00:12:38,980 not tell which prism was left handed and 245 00:12:38,980 --> 00:12:40,150 which was right handed. 246 00:12:40,150 --> 00:12:44,330 But quartz ends up growing with a couple of little facets 247 00:12:44,330 --> 00:12:46,640 on it that look like this. 248 00:12:46,640 --> 00:12:50,850 They spiral up this way in a crystal of one handedness and 249 00:12:50,850 --> 00:12:54,515 they spiral up in the opposite direction for a crystal of 250 00:12:54,515 --> 00:12:55,900 opposite handedness. 251 00:12:55,900 --> 00:13:00,170 So the external faces on the crystal show you that this 252 00:13:00,170 --> 00:13:03,150 crystal shape cannot be mapped one into another. 253 00:13:03,150 --> 00:13:05,390 They are of opposite handedness. 254 00:13:05,390 --> 00:13:07,810 And they have very different properties. 255 00:13:07,810 --> 00:13:09,300 And so here's an example. 256 00:13:09,300 --> 00:13:10,330 I'm not answer your question. 257 00:13:10,330 --> 00:13:14,310 I'm talking around it very adroitly in the hope that if I 258 00:13:14,310 --> 00:13:16,480 talk enough you will forget what you asked. 259 00:13:16,480 --> 00:13:18,760 But how do you make a chiral catalyst? 260 00:13:18,760 --> 00:13:21,280 Well, there are some materials if you prepare them in a 261 00:13:21,280 --> 00:13:25,510 single crystal form, once you nucleate one enantiomorph, 262 00:13:25,510 --> 00:13:27,710 that will continue to grow. 263 00:13:27,710 --> 00:13:29,980 So I would presume one could do exactly 264 00:13:29,980 --> 00:13:32,840 the same with a catalyst. 265 00:13:32,840 --> 00:13:35,940 Actually, the properties of quartz are strikingly 266 00:13:35,940 --> 00:13:38,990 different for these two different forms. 267 00:13:38,990 --> 00:13:44,680 If you would subject the crystal to compression, this 268 00:13:44,680 --> 00:13:46,520 crystal would develop a positive 269 00:13:46,520 --> 00:13:48,810 charge on the surface. 270 00:13:48,810 --> 00:13:50,860 This crystal would develop a negative 271 00:13:50,860 --> 00:13:52,720 charge on the surface. 272 00:13:52,720 --> 00:13:55,310 That's a property that we'll discuss in detail later on. 273 00:13:55,310 --> 00:13:58,240 That's something called piezoelectricity, literally 274 00:13:58,240 --> 00:14:00,310 pressure electricity. 275 00:14:00,310 --> 00:14:03,440 So the nature of the charge induced on the crystal is 276 00:14:03,440 --> 00:14:06,945 different for the crystals of the two chiralities. 277 00:14:10,830 --> 00:14:12,940 OK, any other comment or question? 278 00:14:17,400 --> 00:14:25,510 OK, last time we had taken the first small steps in what will 279 00:14:25,510 --> 00:14:26,740 turn out to be a long and 280 00:14:26,740 --> 00:14:31,310 protracted process of synthesis. 281 00:14:31,310 --> 00:14:35,330 We've covered by now some of the general notions of 282 00:14:35,330 --> 00:14:39,290 operations that can exist in a pattern, two dimensional 283 00:14:39,290 --> 00:14:43,630 pattern of wallpaper or fabric or a three dimensional pattern 284 00:14:43,630 --> 00:14:46,260 that constitutes a crystal. 285 00:14:46,260 --> 00:14:49,960 And we said that in three dimensions, there are four 286 00:14:49,960 --> 00:14:53,800 basic different operations that can exist that tell you 287 00:14:53,800 --> 00:14:56,640 how one part of the system is related to another. 288 00:14:56,640 --> 00:14:58,370 There is an operation of translation. 289 00:15:01,840 --> 00:15:04,780 And this has all the characteristics of a vector, 290 00:15:04,780 --> 00:15:07,360 magnitude and direction. 291 00:15:07,360 --> 00:15:10,590 We summarize that translational periodicity by a 292 00:15:10,590 --> 00:15:14,815 set of fiducial markers that are called lattice points. 293 00:15:20,320 --> 00:15:23,060 And analytically, the transformation takes the 294 00:15:23,060 --> 00:15:24,596 coordinate x, y, z. 295 00:15:24,596 --> 00:15:29,560 And if you do the operation once, maps this to x plus a, y 296 00:15:29,560 --> 00:15:35,078 plus b, z plus c where the translation has components a, 297 00:15:35,078 --> 00:15:38,570 b, and c in x, y, and z directions. 298 00:15:38,570 --> 00:15:41,270 Another operation is the operation of reflection. 299 00:15:45,390 --> 00:15:52,230 And what this operation does is to take the coordinates of 300 00:15:52,230 --> 00:15:57,400 an initial part of the space atom or location and maps it 301 00:15:57,400 --> 00:16:03,780 into some location where one direction is reversed in sign. 302 00:16:03,780 --> 00:16:05,920 Which direction that is or whether it's a pair of 303 00:16:05,920 --> 00:16:10,010 directions will depend on how the reflection locus is 304 00:16:10,010 --> 00:16:13,830 located relative to the coordinate system. 305 00:16:13,830 --> 00:16:16,120 Next, there was an operation of rotation. 306 00:16:20,550 --> 00:16:25,440 And this involves repeating things at angular intervals. 307 00:16:25,440 --> 00:16:29,070 Take the space or the location of the atom and rotate it 308 00:16:29,070 --> 00:16:33,240 through some angle, alpha, about some rotation, a. 309 00:16:33,240 --> 00:16:37,440 And in a sense, this is a periodicity which is 310 00:16:37,440 --> 00:16:41,650 reminiscent of translation except translation strings 311 00:16:41,650 --> 00:16:44,490 things out in a line. 312 00:16:44,490 --> 00:16:47,110 Rotation wraps up the directions about 313 00:16:47,110 --> 00:16:48,150 some central axis. 314 00:16:48,150 --> 00:16:50,370 But it's also periodic. 315 00:16:50,370 --> 00:16:54,150 Then finally, in three dimensions, there is another 316 00:16:54,150 --> 00:16:55,400 operation called inversion. 317 00:16:57,830 --> 00:16:59,580 And what this does is-- 318 00:16:59,580 --> 00:17:03,870 to could describe it as such in literal terms-- 319 00:17:03,870 --> 00:17:07,300 take something and turns it inside out to a point. 320 00:17:07,300 --> 00:17:10,650 So if the inversion point were at the origin, it would take 321 00:17:10,650 --> 00:17:16,930 x, y, z, and map it into minus x, minus y, minus z. 322 00:17:16,930 --> 00:17:22,180 And then, we concluded last time with a definition of a 323 00:17:22,180 --> 00:17:27,880 vocabulary for indicating analytically a single 324 00:17:27,880 --> 00:17:37,150 operation, the set of operations that constitutes 325 00:17:37,150 --> 00:17:45,800 the symmetry element, and then a geometric symbol for the 326 00:17:45,800 --> 00:17:51,050 locus of that operation. 327 00:17:55,570 --> 00:17:58,790 And for translation, we could represent a single 328 00:17:58,790 --> 00:18:00,830 operation by t. 329 00:18:00,830 --> 00:18:03,820 The set of operations could be-- 330 00:18:03,820 --> 00:18:07,430 and I'll use this customary set of braces to 331 00:18:07,430 --> 00:18:09,240 indicate the set-- 332 00:18:09,240 --> 00:18:13,270 this would be a one dimensional lattice, a set of 333 00:18:13,270 --> 00:18:17,990 operations, p, where p is an integer times t. 334 00:18:17,990 --> 00:18:21,950 The geometric symbol, it's convenient to use an arrow 335 00:18:21,950 --> 00:18:24,800 extending from one lattice point to another. 336 00:18:24,800 --> 00:18:29,630 But I emphasize that there's no unique origin to the 337 00:18:29,630 --> 00:18:30,730 translation. 338 00:18:30,730 --> 00:18:34,910 We cannot be entitled to say it sits here as opposed to 339 00:18:34,910 --> 00:18:36,780 down here or down here. 340 00:18:36,780 --> 00:18:40,420 There are an infinite number of parallel directions that 341 00:18:40,420 --> 00:18:45,020 could specify a given interval between lattice points. 342 00:18:45,020 --> 00:18:49,080 For rotation operations, the notation for a single 343 00:18:49,080 --> 00:18:53,800 operation in as much as we must specify two things, the 344 00:18:53,800 --> 00:18:57,730 point about which we rotate and then the angular interval 345 00:18:57,730 --> 00:19:00,750 about which we perform the rotation. 346 00:19:00,750 --> 00:19:03,420 And so we need to specify two things, the location of the 347 00:19:03,420 --> 00:19:06,160 axis, a point a, and the angle, alpha, 348 00:19:06,160 --> 00:19:08,870 through which we rotate. 349 00:19:08,870 --> 00:19:14,240 The set of operations, the rotation axis, is indicated by 350 00:19:14,240 --> 00:19:20,230 an integer, n, where the rotation part of the operation 351 00:19:20,230 --> 00:19:23,810 is 2 pi divided by the integer, n. 352 00:19:23,810 --> 00:19:28,420 So we would use, for example, 2 to represent a rotation axis 353 00:19:28,420 --> 00:19:34,130 where things were moved through 180 degrees, 3 for a 354 00:19:34,130 --> 00:19:37,810 symmetry where things were mapped through intervals of 355 00:19:37,810 --> 00:19:41,890 120 degrees, and 22 for my good friend, the Saguaro 356 00:19:41,890 --> 00:19:44,950 cactus, which is left invariant perhaps by a 357 00:19:44,950 --> 00:19:51,290 rotation through 1/22 of 2 pi. 358 00:19:51,290 --> 00:19:54,920 For the geometric symbol for a rotation axis, we will use a 359 00:19:54,920 --> 00:19:56,170 little polyhedron. 360 00:19:58,160 --> 00:20:00,970 Triangle for a threefold axis. 361 00:20:00,970 --> 00:20:03,340 Square for a fourfold axis. 362 00:20:03,340 --> 00:20:06,940 For a twofold axis, the polyhedron that had that 363 00:20:06,940 --> 00:20:08,790 symmetry would be a line segment. 364 00:20:08,790 --> 00:20:11,350 So we'll take a little artistic license and let the 365 00:20:11,350 --> 00:20:15,720 middle of the line segment bulge out a little bit. 366 00:20:15,720 --> 00:20:18,990 Finally for inversion, for reasons that will not become 367 00:20:18,990 --> 00:20:24,470 clear for another meeting or two, the single operation is 368 00:20:24,470 --> 00:20:25,780 denoted one bar. 369 00:20:25,780 --> 00:20:28,960 The set of operations of which there are only two, inverting 370 00:20:28,960 --> 00:20:31,360 and coming back again, also indicated by 371 00:20:31,360 --> 00:20:32,840 the symbol one bar. 372 00:20:32,840 --> 00:20:36,210 And the geometric symbol for the locus is a little, tiny 373 00:20:36,210 --> 00:20:40,220 circle that's open large enough to be noticed, but not 374 00:20:40,220 --> 00:20:46,100 so large that it looks like an atom in an atomic arrangement. 375 00:20:46,100 --> 00:20:48,470 So there's our basic bag of tricks. 376 00:20:48,470 --> 00:20:52,250 And last time, we had already used some simple geometry to 377 00:20:52,250 --> 00:20:55,270 come to a rather astonishing conclusion. 378 00:20:55,270 --> 00:20:55,710 Yes, sir? 379 00:20:55,710 --> 00:20:57,128 AUDIENCE: Reflection? 380 00:20:57,128 --> 00:21:00,551 PROFESSOR: Reflection, that's this middle one. 381 00:21:00,551 --> 00:21:02,507 Whoops, not going to help. 382 00:21:02,507 --> 00:21:04,463 Thank you very much. 383 00:21:04,463 --> 00:21:06,419 I get excited and carried away. 384 00:21:06,419 --> 00:21:08,375 OK, reflection, the second operation. 385 00:21:12,310 --> 00:21:18,090 Single operation, although crystallographers don't often 386 00:21:18,090 --> 00:21:21,890 use it in condensed matter physics, a single operation is 387 00:21:21,890 --> 00:21:26,660 represented by the symbol sigma. 388 00:21:26,660 --> 00:21:30,820 Symbol for a mirror plane is m, easily appreciated. 389 00:21:30,820 --> 00:21:34,520 And the geometric locus across which things are reflected 390 00:21:34,520 --> 00:21:38,570 left to right and vice versa is done with a bold, solid 391 00:21:38,570 --> 00:21:43,580 line when you're looking down parallel to the surface of the 392 00:21:43,580 --> 00:21:44,830 reflection locus. 393 00:21:48,210 --> 00:21:50,220 Then, I will now be caught up. 394 00:21:50,220 --> 00:21:53,165 And I can set off on something new. 395 00:21:56,110 --> 00:22:01,480 We had said that in a pattern, very often more than one of 396 00:22:01,480 --> 00:22:05,490 these operations are combined in a single space. 397 00:22:05,490 --> 00:22:07,460 And one of the things that must be present-- 398 00:22:07,460 --> 00:22:09,720 if the symmetry of the entity we are 399 00:22:09,720 --> 00:22:11,950 discussing is a crystal-- 400 00:22:11,950 --> 00:22:15,710 one of the things that must be present is a translation. 401 00:22:15,710 --> 00:22:19,940 And so I asked last time the question, what if we want to 402 00:22:19,940 --> 00:22:24,120 say that our space has a translational periodicity 403 00:22:24,120 --> 00:22:26,280 described by t. 404 00:22:26,280 --> 00:22:30,440 And then, I add to one lattice point a rotation operation, A 405 00:22:30,440 --> 00:22:35,100 alpha, that has to get translated to another location 406 00:22:35,100 --> 00:22:38,180 where that operation, A alpha, exists. 407 00:22:38,180 --> 00:22:45,190 And if A alpha exists and I repeat that translation by a 408 00:22:45,190 --> 00:22:49,810 rotation alpha, I'm going to have a sheath of translations 409 00:22:49,810 --> 00:22:52,810 all separated by a distance alpha. 410 00:22:52,810 --> 00:22:59,095 If I single out the one that is removed from the first by 411 00:22:59,095 --> 00:23:03,200 an operation of rotation A alpha in a counterclockwise 412 00:23:03,200 --> 00:23:07,540 direction and single out from that sheath another one that 413 00:23:07,540 --> 00:23:12,640 is alpha away in the clockwise direction. 414 00:23:12,640 --> 00:23:14,960 Then, I have lattice points at the end of all of these 415 00:23:14,960 --> 00:23:16,040 translations. 416 00:23:16,040 --> 00:23:19,680 But these two guys up here are also lattice points. 417 00:23:19,680 --> 00:23:24,510 And therefore, I have mucked up to the entire construction 418 00:23:24,510 --> 00:23:27,830 unless I can claim that this distance between these two 419 00:23:27,830 --> 00:23:33,430 lattice points is either T or some multiple P times T, where 420 00:23:33,430 --> 00:23:35,540 P is an integer. 421 00:23:35,540 --> 00:23:38,350 And then lickety split, before you could catch your breath in 422 00:23:38,350 --> 00:23:40,580 wonder, I dropped a perpendicular down to the 423 00:23:40,580 --> 00:23:42,260 original translation. 424 00:23:42,260 --> 00:23:44,490 This distance in here was PT. 425 00:23:44,490 --> 00:23:47,230 This distance on either side was T times 426 00:23:47,230 --> 00:23:49,120 the cosine of alpha. 427 00:23:49,120 --> 00:23:53,600 And that led me to a constraint that alpha, that 428 00:23:53,600 --> 00:23:57,260 rotation operation A alpha, if it were to be combined with 429 00:23:57,260 --> 00:24:01,080 translation, would be restricted to values such that 430 00:24:01,080 --> 00:24:05,750 the cosine of alpha was equal to 1 minus an integer P 431 00:24:05,750 --> 00:24:07,000 divided by 2. 432 00:24:09,690 --> 00:24:13,690 Wow, simple, but so incredibly profound. 433 00:24:13,690 --> 00:24:16,390 What came out of this was that alpha could 434 00:24:16,390 --> 00:24:20,870 either be 0 or 360 degrees. 435 00:24:20,870 --> 00:24:23,170 That was a one-fold axis. 436 00:24:23,170 --> 00:24:26,420 Alpha could be 180 degrees. 437 00:24:26,420 --> 00:24:29,460 That was a two-fold axis. 438 00:24:29,460 --> 00:24:32,750 120, which would be the angular rotation of a 439 00:24:32,750 --> 00:24:35,210 three-fold axis. 440 00:24:35,210 --> 00:24:40,150 90, that would be the operation of a four-fold axis. 441 00:24:40,150 --> 00:24:45,090 Or 60, that would be the operation of a six-fold axis. 442 00:24:45,090 --> 00:24:46,940 It's incredible. 443 00:24:46,940 --> 00:24:51,760 Anything, any pattern, any crystal that is based on a 444 00:24:51,760 --> 00:24:56,610 lattice can contain only these five rotational symmetries 445 00:24:56,610 --> 00:25:00,630 including no symmetry at all. 446 00:25:00,630 --> 00:25:05,630 So this says a lot about what the morphology of crystalline 447 00:25:05,630 --> 00:25:07,450 materials are permitted to be. 448 00:25:10,270 --> 00:25:13,790 Then, we looked at the nature of the lattices that were 449 00:25:13,790 --> 00:25:18,430 described by these angular rotations. 450 00:25:18,430 --> 00:25:22,640 And we found that there were only three types of lattices 451 00:25:22,640 --> 00:25:24,095 that were required by rotation. 452 00:25:26,710 --> 00:25:28,915 These were two dimensional lattice nets. 453 00:25:33,250 --> 00:25:37,170 One was a general oblique net which had two different 454 00:25:37,170 --> 00:25:41,730 translations, some arbitrary angle between them, two 455 00:25:41,730 --> 00:25:45,730 translations, T1 and T2, which were not equal to one another. 456 00:25:45,730 --> 00:25:48,100 We would call this the oblique net. 457 00:25:50,860 --> 00:25:55,340 Then, there was another net that was square in the sense 458 00:25:55,340 --> 00:25:58,770 that two translations, T1 and T2, were 459 00:25:58,770 --> 00:26:02,410 identical to one another-- 460 00:26:02,410 --> 00:26:04,480 not close, but identical. 461 00:26:04,480 --> 00:26:07,940 And the angle between them was identically 90 degrees. 462 00:26:07,940 --> 00:26:11,450 This oblique net could accept either a one-fold axis or a 463 00:26:11,450 --> 00:26:12,470 two-fold axis. 464 00:26:12,470 --> 00:26:14,910 This was the specialized shape that was required by a 465 00:26:14,910 --> 00:26:16,340 four-fold axis. 466 00:26:16,340 --> 00:26:19,330 And then finally, there was another net that looked 467 00:26:19,330 --> 00:26:22,740 oblique except it was a specialized oblique net. 468 00:26:22,740 --> 00:26:26,690 It had two translations, T1 and T2, which were identical 469 00:26:26,690 --> 00:26:33,130 just as in the square net and an angle between them, which 470 00:26:33,130 --> 00:26:35,460 was exactly 120 degrees. 471 00:26:35,460 --> 00:26:38,400 And this same sort of net could be compatible with 472 00:26:38,400 --> 00:26:40,935 either a three-fold axis or a six-fold axis. 473 00:26:43,780 --> 00:26:46,450 Then, we took the only other operation that could be 474 00:26:46,450 --> 00:26:49,210 present in a two dimensional pattern, and this was the 475 00:26:49,210 --> 00:26:53,750 operation of reflection, a mirror point, m. 476 00:26:53,750 --> 00:26:58,250 And we saw that this by itself could require two different 477 00:26:58,250 --> 00:27:01,170 sorts of specializations in a lattice, a lattice in which 478 00:27:01,170 --> 00:27:05,450 two directions were exactly orthogonal but the lengths of 479 00:27:05,450 --> 00:27:07,790 the translations were unequal. 480 00:27:07,790 --> 00:27:11,881 And then, a diamond shaped net-- 481 00:27:11,881 --> 00:27:14,020 and this was a new wrinkle. 482 00:27:14,020 --> 00:27:18,350 This was a net that had two translations that were 483 00:27:18,350 --> 00:27:21,890 Identical in length and an arbitrary angle between them. 484 00:27:21,890 --> 00:27:26,080 Or alternatively, if we chose to pick a redundant lattice, 485 00:27:26,080 --> 00:27:30,330 one that had two translations, T1 and T2 prime which were 486 00:27:30,330 --> 00:27:33,770 different but an angle between them that was exactly 90 487 00:27:33,770 --> 00:27:37,970 degrees, so this one we would refer to as a rectangular 488 00:27:37,970 --> 00:27:42,860 lattice, a rectangular net, and this one as a centered 489 00:27:42,860 --> 00:27:43,810 rectangular net-- 490 00:27:43,810 --> 00:27:46,620 same shape, but an extra lattice point in the middle. 491 00:27:57,050 --> 00:28:00,050 Why pick a double cell, which is redundant, takes up more, 492 00:28:00,050 --> 00:28:02,370 and doesn't convey any more information? 493 00:28:02,370 --> 00:28:07,080 And the advantage comes when you want to analytically 494 00:28:07,080 --> 00:28:11,050 describe the relation between different directions or the 495 00:28:11,050 --> 00:28:13,350 positions of atoms within the cell. 496 00:28:13,350 --> 00:28:16,900 In that case, an orthogonal coordinate system, even if 497 00:28:16,900 --> 00:28:21,120 it's not Cartesian, is of immense advantage as opposed 498 00:28:21,120 --> 00:28:22,540 to an oblique system. 499 00:28:25,100 --> 00:28:28,530 OK, slightly out of breath, but that is everything we've 500 00:28:28,530 --> 00:28:29,870 covered to this point. 501 00:28:29,870 --> 00:28:34,500 And you can find this material discussed in the first 502 00:28:34,500 --> 00:28:38,340 chapter, in the introduction, and then half of the second 503 00:28:38,340 --> 00:28:41,420 chapter of Berger's book if you want to read over it. 504 00:28:44,730 --> 00:28:46,110 OK, any questions or comments? 505 00:28:55,940 --> 00:29:00,810 All right, I wanted to say a little bit more about lattices 506 00:29:00,810 --> 00:29:07,040 and something with which you are no doubt familiar, but 507 00:29:07,040 --> 00:29:10,630 perhaps have some questions about why one does it in such 508 00:29:10,630 --> 00:29:13,140 an obscure, difficult way. 509 00:29:13,140 --> 00:29:18,360 And let me introduce what's done by an analogy which we 510 00:29:18,360 --> 00:29:20,460 have around us in everyday life. 511 00:29:20,460 --> 00:29:27,240 Let me draw a system of streets and avenues for a city 512 00:29:27,240 --> 00:29:31,410 that might be one like Boston where the streets were laid 513 00:29:31,410 --> 00:29:39,960 out by cows wandering around in search of forage. 514 00:29:39,960 --> 00:29:44,080 And suppose somebody met you on the street and said, may I 515 00:29:44,080 --> 00:29:46,390 stop you here at point A and ask you a question? 516 00:29:46,390 --> 00:29:52,760 How do I get from point A to point B? 517 00:29:52,760 --> 00:29:53,410 That's not a vector. 518 00:29:53,410 --> 00:29:55,190 That's a point. 519 00:29:55,190 --> 00:29:58,330 How do I do that? 520 00:29:58,330 --> 00:30:07,560 You would not say, go 342 meters to the east and 26.4 521 00:30:07,560 --> 00:30:09,670 meters to the south. 522 00:30:09,670 --> 00:30:13,410 That's perfectly logical that is admirably Cartesian but you 523 00:30:13,410 --> 00:30:17,830 wouldn't do it that way what we just said you would say go 524 00:30:17,830 --> 00:30:21,620 one block straight ahead, and then turn one block to your 525 00:30:21,620 --> 00:30:25,700 left, and then go down the street to your right and turn 526 00:30:25,700 --> 00:30:28,700 to your right again, and you'll be right there. 527 00:30:28,700 --> 00:30:30,550 You can't miss it. 528 00:30:30,550 --> 00:30:32,620 Of course, the person in Boston would miss it and would 529 00:30:32,620 --> 00:30:36,340 have to stop a second time and ask again and by a method of 530 00:30:36,340 --> 00:30:40,020 successive approximations eventually get to where he or 531 00:30:40,020 --> 00:30:43,640 she would like to be. 532 00:30:43,640 --> 00:30:49,120 I submit if a system gives you a grid, it makes sense to use 533 00:30:49,120 --> 00:30:55,290 that grid as the basis vectors of coordinate system even if 534 00:30:55,290 --> 00:30:59,150 the intervals are different, in two different directions, 535 00:30:59,150 --> 00:31:03,500 and even though the intervals might be not vectors that are 536 00:31:03,500 --> 00:31:05,710 orthogonal to one another. 537 00:31:05,710 --> 00:31:09,545 If the system is periodic and has this grid work to it, it 538 00:31:09,545 --> 00:31:13,740 makes sense to use that as the basis of a coordinate system. 539 00:31:13,740 --> 00:31:20,020 So if we have a lattice that is oblique and there are 540 00:31:20,020 --> 00:31:27,570 lattice points defined by these two translations, T1 and 541 00:31:27,570 --> 00:31:37,820 T2, it makes great sense if you want to specify the vector 542 00:31:37,820 --> 00:31:42,510 that runs from this lattice point to this lattice point to 543 00:31:42,510 --> 00:31:47,740 do that by saying that this is a lattice point that is at the 544 00:31:47,740 --> 00:31:55,370 end of the vector 1 T1 plus 2 T2. 545 00:31:58,290 --> 00:32:02,000 And that, just as a system of streets and avenues, is a lot 546 00:32:02,000 --> 00:32:07,290 more efficient than saying go 12.2 angstroms in the x 547 00:32:07,290 --> 00:32:12,900 direction and go 15.3 angstroms in the y direction. 548 00:32:12,900 --> 00:32:16,370 What we have defined here is something that is called a 549 00:32:16,370 --> 00:32:17,620 rational direction. 550 00:32:20,960 --> 00:32:26,240 And a rational direction is something that is going to be 551 00:32:26,240 --> 00:32:29,280 a direction that extends between lattice points. 552 00:32:29,280 --> 00:32:33,110 And what's rational about it is that the coefficients in 553 00:32:33,110 --> 00:32:37,340 front of T1 and T2 that would appear would be integers. 554 00:32:37,340 --> 00:32:39,450 And an integers is sometimes referred to 555 00:32:39,450 --> 00:32:42,840 as a rational number. 556 00:32:42,840 --> 00:32:46,680 That would be quite different from a vector that extended 557 00:32:46,680 --> 00:32:51,210 from a lattice point to, let's say, some atom that is within 558 00:32:51,210 --> 00:32:53,770 the boundaries of this cell. 559 00:32:53,770 --> 00:32:56,310 And this might be a radial vector from the 560 00:32:56,310 --> 00:32:57,650 origin to one lattice. 561 00:32:57,650 --> 00:32:59,880 And that could be used to specify the atomic 562 00:32:59,880 --> 00:33:01,350 coordinates. 563 00:33:01,350 --> 00:33:05,910 This is going to complete the dichotomy, a feature that is 564 00:33:05,910 --> 00:33:08,190 referred to as irrational. 565 00:33:08,190 --> 00:33:08,590 Why? 566 00:33:08,590 --> 00:33:15,400 Because the components of the steps along T1, x, and along 567 00:33:15,400 --> 00:33:19,110 T2, y, are going to be fractional numbers. 568 00:33:19,110 --> 00:33:22,620 So you'll hit irrational notation for directions that 569 00:33:22,620 --> 00:33:25,760 go to different locations with a unit cell. 570 00:33:25,760 --> 00:33:30,350 You'll use rational vectors in directions that extend between 571 00:33:30,350 --> 00:33:32,190 integral numbers of lattice points. 572 00:33:39,700 --> 00:33:44,880 These directions occur all the time when one discusses the 573 00:33:44,880 --> 00:33:47,270 physical properties of crystals. 574 00:33:47,270 --> 00:33:50,160 I'm sure in any class that you've had that's discussed 575 00:33:50,160 --> 00:33:54,610 mechanical properties, one of the favorite topics is to 576 00:33:54,610 --> 00:33:57,340 discuss the ways in which close 577 00:33:57,340 --> 00:33:59,930 packed metals can deform. 578 00:33:59,930 --> 00:34:03,810 And this can be discussed easily in terms of one layer 579 00:34:03,810 --> 00:34:06,980 of close packed spheres sliding over another. 580 00:34:06,980 --> 00:34:11,620 And one of the ways that the layer can slide to go from one 581 00:34:11,620 --> 00:34:15,070 set of close packed hollows to another is in a 582 00:34:15,070 --> 00:34:16,780 direction like this. 583 00:34:16,780 --> 00:34:19,219 And that turns out to be a rational direction. 584 00:34:19,219 --> 00:34:22,170 And these are directions in which a close packed metal 585 00:34:22,170 --> 00:34:25,190 will easily deform. 586 00:34:25,190 --> 00:34:30,449 So directions of easy plastic deformations almost always 587 00:34:30,449 --> 00:34:33,590 involve rational directions. 588 00:34:33,590 --> 00:34:39,110 So there is a unique application of these sorts of 589 00:34:39,110 --> 00:34:42,590 features in the discussion of mechanical properties. 590 00:34:42,590 --> 00:34:47,960 Another thing that can happen is that a particular 591 00:34:47,960 --> 00:34:52,580 structure, if it is weakly bonded or relatively weakly 592 00:34:52,580 --> 00:34:57,965 bonded in one direction, if you come down with a chisel on 593 00:34:57,965 --> 00:35:03,490 that plane and give it a little tunk with a hammer, the 594 00:35:03,490 --> 00:35:07,230 crystal will fall exactly in half and give you two very 595 00:35:07,230 --> 00:35:10,620 smooth surfaces on either side of some plane. 596 00:35:10,620 --> 00:35:12,920 And the reason is if you look down into the guts of the 597 00:35:12,920 --> 00:35:15,150 crystal and look at the atoms and how they're bonded 598 00:35:15,150 --> 00:35:18,480 together, there may be some direction of weak bonding 599 00:35:18,480 --> 00:35:22,710 between atoms that is very easily separated by some sort 600 00:35:22,710 --> 00:35:24,770 of mechanical force. 601 00:35:24,770 --> 00:35:28,280 This is true even for crystals that are very strongly bonded. 602 00:35:28,280 --> 00:35:31,620 The way in which people begin to facet diamonds, for 603 00:35:31,620 --> 00:35:35,570 example, is to sit for a month and study the diamond and then 604 00:35:35,570 --> 00:35:38,360 decide how you're going to put a straight edge to it and give 605 00:35:38,360 --> 00:35:41,660 it a little tunk so that it falls neatly into two pieces 606 00:35:41,660 --> 00:35:45,890 at the sides that you want to facet. 607 00:35:45,890 --> 00:35:48,960 Doesn't always work, so you sit for a long time to figure 608 00:35:48,960 --> 00:35:51,990 out just what you're going to do. 609 00:35:51,990 --> 00:35:57,110 I think diamond cutters burn out early and have a useful 610 00:35:57,110 --> 00:35:59,580 career that last perhaps 3 and 1/2 years. 611 00:35:59,580 --> 00:36:00,700 And then, they're burned out. 612 00:36:00,700 --> 00:36:01,800 I don't know that for sure. 613 00:36:01,800 --> 00:36:03,460 But it must be a nerve wracking way 614 00:36:03,460 --> 00:36:06,660 of making an existence. 615 00:36:06,660 --> 00:36:08,020 Not a far-fetched story. 616 00:36:08,020 --> 00:36:12,010 At one time, I was doing a diffraction study on a 617 00:36:12,010 --> 00:36:15,090 material which was a low temperature face, very 618 00:36:15,090 --> 00:36:19,070 interesting, but formed stable only at such low temperatures 619 00:36:19,070 --> 00:36:21,950 that you could not make it by cooking the components. 620 00:36:21,950 --> 00:36:27,240 But it had been found as a mineral in the Swiss Alps, all 621 00:36:27,240 --> 00:36:32,200 sorts of exotica attached to this particular material. 622 00:36:32,200 --> 00:36:34,670 There was a particular place up in the Alps which had a 623 00:36:34,670 --> 00:36:40,040 very unusual chemistry involving lead and arsenic and 624 00:36:40,040 --> 00:36:42,180 thallium and sulfur-- 625 00:36:42,180 --> 00:36:45,430 not what you find in your usual backyard rock pile. 626 00:36:45,430 --> 00:36:48,730 And there was something like 37 unique minerals that were 627 00:36:48,730 --> 00:36:52,570 found in this little spot that were known nowhere else on the 628 00:36:52,570 --> 00:36:53,970 face of the Earth. 629 00:36:53,970 --> 00:36:55,910 And I was very interested in the atomic 630 00:36:55,910 --> 00:36:56,940 arrangements in one of them. 631 00:36:56,940 --> 00:37:01,130 So I wrote to the British Museum that had found the sole 632 00:37:01,130 --> 00:37:04,510 supplier of this material that-- 633 00:37:04,510 --> 00:37:06,590 back in about 1901 or so-- 634 00:37:06,590 --> 00:37:07,900 I wrote and I said, do you have any 635 00:37:07,900 --> 00:37:09,360 crystals of this stuff? 636 00:37:09,360 --> 00:37:10,450 Yes. 637 00:37:10,450 --> 00:37:12,580 Would you like to study it? 638 00:37:12,580 --> 00:37:13,430 And I said yes, I would. 639 00:37:13,430 --> 00:37:14,930 Well, we'd be happy to send it. 640 00:37:14,930 --> 00:37:19,170 But please be careful not to damage the morphology because 641 00:37:19,170 --> 00:37:22,460 it is a very rare material. 642 00:37:22,460 --> 00:37:22,980 So I said fine. 643 00:37:22,980 --> 00:37:26,150 I'm as careful as the next meticulous guy. 644 00:37:26,150 --> 00:37:27,760 They sent two crystals. 645 00:37:27,760 --> 00:37:30,320 The crystals measured-- 646 00:37:30,320 --> 00:37:33,480 one of them was 2/10 of a millimeter in diameter. 647 00:37:33,480 --> 00:37:38,770 One of them was about 0.15 millimeters in diameter. 648 00:37:38,770 --> 00:37:42,230 And I was supposed to take off a piece for study without 649 00:37:42,230 --> 00:37:44,230 damaging the morphology. 650 00:37:44,230 --> 00:37:48,240 Man, I felt just like one of these Dutch diamond cutters. 651 00:37:48,240 --> 00:37:51,210 I sat and I looked at those two little suckers under the 652 00:37:51,210 --> 00:37:53,380 microscope for, I think, a week. 653 00:37:53,380 --> 00:37:56,680 And then finally, the moment of truth, and I took a needle 654 00:37:56,680 --> 00:37:58,785 and popped off a corner. 655 00:37:58,785 --> 00:38:00,640 And it went into two pieces. 656 00:38:00,640 --> 00:38:03,350 One went for a microprobe analysis to determine the 657 00:38:03,350 --> 00:38:04,240 composition. 658 00:38:04,240 --> 00:38:06,740 One of them went to a single crystal x-ray diffraction 659 00:38:06,740 --> 00:38:09,070 study to determine the atomic arrangement. 660 00:38:09,070 --> 00:38:12,820 Both analyses can be done on a very small piece of material. 661 00:38:12,820 --> 00:38:17,400 So my story about the diamond cutter agonizing before 662 00:38:17,400 --> 00:38:21,040 looking for a cleavage or a fracture surface is not just 663 00:38:21,040 --> 00:38:21,960 made up fiction. 664 00:38:21,960 --> 00:38:25,430 It is something that, when you work with crystals very often, 665 00:38:25,430 --> 00:38:27,090 is encountered. 666 00:38:27,090 --> 00:38:31,220 OK, cleavage planes are also rational directions. 667 00:38:31,220 --> 00:38:34,580 The cleavage plane of rock salt is legendary. 668 00:38:34,580 --> 00:38:37,030 And it's nice, really, to spend an afternoon splitting 669 00:38:37,030 --> 00:38:39,560 it up just to see how neat it falls apart. 670 00:38:43,440 --> 00:38:48,520 OK, how are we going to denote rational planes? 671 00:38:48,520 --> 00:38:51,590 Talked about rational and irrational directions, and 672 00:38:51,590 --> 00:38:53,520 that's easy. 673 00:38:53,520 --> 00:38:59,815 How are we going to describe rational planes in crystals? 674 00:39:03,050 --> 00:39:06,250 And this is rather bizarre. 675 00:39:06,250 --> 00:39:11,240 So let me tell you what you do, which is something you've 676 00:39:11,240 --> 00:39:14,860 probably heard before, and then explain why one 677 00:39:14,860 --> 00:39:18,010 does it this way. 678 00:39:18,010 --> 00:39:24,020 Let's suppose we have three vectors that are the vectors 679 00:39:24,020 --> 00:39:25,560 we will use to define the lattice. 680 00:39:29,840 --> 00:39:34,300 And if there is a plane hanging on one of these 681 00:39:34,300 --> 00:39:38,990 lattice points, everything is translationally periodic, so 682 00:39:38,990 --> 00:39:42,610 there must be a similar plane in the same orientation that 683 00:39:42,610 --> 00:39:45,930 passes through all the lattice points. 684 00:39:45,930 --> 00:39:50,220 So let me look at a very special plane, namely one that 685 00:39:50,220 --> 00:39:54,280 cuts a lattice point at an integral number of 686 00:39:54,280 --> 00:39:57,840 translations, T3, and an integral number of 687 00:39:57,840 --> 00:40:01,330 translations, T2, and an integral number of 688 00:40:01,330 --> 00:40:04,650 translations, T1. 689 00:40:04,650 --> 00:40:07,930 And I don't want any common factor here, so let's look at 690 00:40:07,930 --> 00:40:09,610 this point here. 691 00:40:09,610 --> 00:40:14,700 And now, let's connect these lattice points together. 692 00:40:14,700 --> 00:40:20,900 And that will define a plane uniquely. 693 00:40:20,900 --> 00:40:24,260 And what is special about this point is that it hits a 694 00:40:24,260 --> 00:40:27,510 lattice point on each of these three axes. 695 00:40:27,510 --> 00:40:30,185 And this is a plane that I'll call the intercept plane. 696 00:40:37,390 --> 00:40:40,580 There will be a similar plane hanging on all of these other 697 00:40:40,580 --> 00:40:42,060 lattice points. 698 00:40:42,060 --> 00:40:47,400 But those planes will not hit a lattice point on the 699 00:40:47,400 --> 00:40:49,990 directions of T1, T2, and T3. 700 00:40:49,990 --> 00:40:52,540 This is the first one out from the origin that does that. 701 00:40:55,426 --> 00:41:02,080 Now, we will use this oblique set of vectors, T1, T2, and 702 00:41:02,080 --> 00:41:05,890 T3, as the coordinate system for specifying 703 00:41:05,890 --> 00:41:07,470 direction and angles. 704 00:41:07,470 --> 00:41:10,920 So let me ask a question now that's going to blow you away. 705 00:41:10,920 --> 00:41:16,630 What is the equation for the locus of this plane If I use 706 00:41:16,630 --> 00:41:23,010 this oblique coordinate system based on T1, T2, and T3 as my 707 00:41:23,010 --> 00:41:24,260 coordinate system? 708 00:41:26,440 --> 00:41:28,050 Holy mackerel. 709 00:41:28,050 --> 00:41:31,370 What a way to wrap up a Tuesday. 710 00:41:31,370 --> 00:41:33,570 Actually, it's very, very easy. 711 00:41:33,570 --> 00:41:37,500 First of all, it has to be a linear equation. 712 00:41:37,500 --> 00:41:40,100 So it's going to be linear in x, y, and z. 713 00:41:40,100 --> 00:41:45,270 So something times x, something times y, something 714 00:41:45,270 --> 00:41:49,690 times z equals a constant. 715 00:41:49,690 --> 00:41:53,430 That's going to define, if I use the proper coefficients, 716 00:41:53,430 --> 00:41:56,980 the equation for this plane. 717 00:41:56,980 --> 00:42:01,970 And let me determine quite easily what those 718 00:42:01,970 --> 00:42:04,170 coefficients are. 719 00:42:04,170 --> 00:42:07,920 What is the coordinate of this point, which sits on the plane 720 00:42:07,920 --> 00:42:10,280 and therefore must satisfy this equation? 721 00:42:10,280 --> 00:42:12,860 Well, y is 0. z is zero. 722 00:42:12,860 --> 00:42:14,270 So this is the point-- 723 00:42:14,270 --> 00:42:20,620 if I am a translations out from the origin, this is the 724 00:42:20,620 --> 00:42:23,340 point x equals A. 725 00:42:23,340 --> 00:42:28,120 So if I'm three translations out from the origin, that 726 00:42:28,120 --> 00:42:32,610 integer is the coefficient in front of x because when y and 727 00:42:32,610 --> 00:42:36,200 z are 0, if I make the constants on the right hand 728 00:42:36,200 --> 00:42:43,040 side equal to 1, this is the point x equals a. 729 00:42:43,040 --> 00:42:45,220 And it's nice to have one on the right hand side. 730 00:42:45,220 --> 00:42:48,270 That's about as nice and neat and tidy a constant as you 731 00:42:48,270 --> 00:42:49,900 could have. 732 00:42:49,900 --> 00:42:52,870 We do the same thing for this point at the end of two 733 00:42:52,870 --> 00:42:54,540 translations, T2. 734 00:42:54,540 --> 00:42:58,050 let's say in general that for this intercept plane, we are 735 00:42:58,050 --> 00:43:01,350 two translations out along T2. 736 00:43:01,350 --> 00:43:04,810 So for the point 0-- 737 00:43:04,810 --> 00:43:07,700 this point, 0-- 738 00:43:07,700 --> 00:43:10,500 if I put the number of translations, b, in front of 739 00:43:10,500 --> 00:43:20,620 y, This is the point y equals b when x and z are 0. 740 00:43:20,620 --> 00:43:26,100 And in the same way, if we are C translations out along T3, C 741 00:43:26,100 --> 00:43:29,504 times z-- that integer-- is equal to 1. 742 00:43:29,504 --> 00:43:30,754 AUDIENCE: Can you explain it again? 743 00:43:33,157 --> 00:43:34,011 [INAUDIBLE] 744 00:43:34,011 --> 00:43:35,040 PROFESSOR: I'm sorry. 745 00:43:35,040 --> 00:43:36,823 What? 746 00:43:36,823 --> 00:43:39,522 AUDIENCE: [INAUDIBLE]. 747 00:43:39,522 --> 00:43:41,480 PROFESSOR: Yeah, I'm saying I'm going to [INAUDIBLE] my 748 00:43:41,480 --> 00:43:47,520 definition at a plane which cuts the direction of T1, 749 00:43:47,520 --> 00:43:50,030 which is what I called the variable x, at a 750 00:43:50,030 --> 00:43:50,980 translations out. 751 00:43:50,980 --> 00:43:53,880 In this case, this is 3. 752 00:43:53,880 --> 00:43:55,150 This is 2. 753 00:43:55,150 --> 00:43:56,320 And this is 1. 754 00:43:56,320 --> 00:44:01,890 So for this particular example, I would have 3x plus 755 00:44:01,890 --> 00:44:09,350 2y plus 1z equals 1 because when y is 0 and z is 0, the 756 00:44:09,350 --> 00:44:13,410 point on the surface is-- 757 00:44:13,410 --> 00:44:15,830 1/3. 758 00:44:15,830 --> 00:44:17,080 Sorry. 759 00:44:19,430 --> 00:44:20,680 Good question. 760 00:44:26,150 --> 00:44:28,230 OK, that's a lot better. 761 00:44:28,230 --> 00:44:30,750 I would've gotten in much deeper trouble if you had let 762 00:44:30,750 --> 00:44:31,960 me persist in that. 763 00:44:31,960 --> 00:44:34,110 So thank you for the correction. 764 00:44:34,110 --> 00:44:36,370 But now, I would like to have integers out in front here. 765 00:44:36,370 --> 00:44:41,750 So let me multiply both sides through by A times B times C. 766 00:44:41,750 --> 00:44:46,820 And now, I do get something with integers in front of x 767 00:44:46,820 --> 00:44:49,815 and integers in front of y. 768 00:44:49,815 --> 00:44:53,890 And this would be A times C, and integers in front of z. 769 00:44:53,890 --> 00:44:56,650 And that's A times B equals-- 770 00:44:56,650 --> 00:45:01,560 and now on the right side, I would have a times B times C. 771 00:45:01,560 --> 00:45:03,710 So everything, every coefficient and the term on 772 00:45:03,710 --> 00:45:05,550 the right hand side are products of integers. 773 00:45:05,550 --> 00:45:07,200 So they are all integers themselves. 774 00:45:16,320 --> 00:45:23,470 OK, let me next ask the question these planes passing 775 00:45:23,470 --> 00:45:26,200 through all of the other lattice points that are 776 00:45:26,200 --> 00:45:30,540 contained within this little tetrahedron of volume are 777 00:45:30,540 --> 00:45:31,790 going to be equally spaced. 778 00:45:35,830 --> 00:45:39,710 How will I claim that these planes are all equally spaced? 779 00:45:39,710 --> 00:45:41,960 Well, that's easy just in general terms. 780 00:45:41,960 --> 00:45:43,540 Here is a lattice point. 781 00:45:43,540 --> 00:45:45,180 There's a plane passing through it. 782 00:45:45,180 --> 00:45:47,500 There's some lattice point neighboring it that's out at 783 00:45:47,500 --> 00:45:54,860 distance, D. Here is another lattice point. 784 00:45:54,860 --> 00:45:57,000 It has a plane passing through it. 785 00:45:57,000 --> 00:46:01,290 There must be a plane out here on D. And I'm not going to be 786 00:46:01,290 --> 00:46:04,850 able to have the same environment for every lattice 787 00:46:04,850 --> 00:46:07,920 point unless all of these planes have the same spacing 788 00:46:07,920 --> 00:46:09,950 from one another. 789 00:46:09,950 --> 00:46:12,190 So to say that they pass in lattice points in the 790 00:46:12,190 --> 00:46:15,530 environment of everybody lattice point is by definition 791 00:46:15,530 --> 00:46:19,585 identical is possible only if the planes are equally spaced. 792 00:46:22,630 --> 00:46:23,830 Next question-- 793 00:46:23,830 --> 00:46:27,040 and I'm not going to be gutsy enough to try to demonstrate 794 00:46:27,040 --> 00:46:31,340 this in three dimensions, so let me do it in two. 795 00:46:31,340 --> 00:46:36,770 Let us ask the question, how many planes are there hanging 796 00:46:36,770 --> 00:46:40,350 on all the lattice points between the origin and the 797 00:46:40,350 --> 00:46:41,600 intercept plane? 798 00:46:44,940 --> 00:46:48,980 And let me demonstrate that for a two dimensional case. 799 00:46:48,980 --> 00:46:55,080 And I will refer you to a drawing in Berger's book 800 00:46:55,080 --> 00:46:58,740 because this is not convincing unless you do 801 00:46:58,740 --> 00:47:01,630 it absolutely exactly. 802 00:47:01,630 --> 00:47:06,390 And to do that in front of a live audience with chalk is 803 00:47:06,390 --> 00:47:08,830 not always the easiest thing in the world. 804 00:47:08,830 --> 00:47:10,230 So here is T1. 805 00:47:10,230 --> 00:47:11,640 Here is T2. 806 00:47:11,640 --> 00:47:13,910 And let me look at this plane. 807 00:47:13,910 --> 00:47:15,800 This is the intercept plane. 808 00:47:15,800 --> 00:47:21,170 A is equal to 3 and B is equal to 2. 809 00:47:21,170 --> 00:47:24,680 Now, what I'm going to do is to use plus and minus the 810 00:47:24,680 --> 00:47:28,370 translation T1 to repeat this intercept plane. 811 00:47:34,740 --> 00:47:36,520 OK, I go minus T1. 812 00:47:36,520 --> 00:47:38,720 And that's going to give me a plane in here. 813 00:47:38,720 --> 00:47:40,370 I'm going to go minus T1 again. 814 00:47:40,370 --> 00:47:42,880 And that's going to give me a plane in here. 815 00:47:42,880 --> 00:47:46,510 So I have split the interval between the origin and the 816 00:47:46,510 --> 00:47:49,535 intercept plane into A parts. 817 00:47:54,830 --> 00:47:57,620 OK so far? 818 00:47:57,620 --> 00:48:00,220 Let me now use the translation T2. 819 00:48:00,220 --> 00:48:05,010 And I will use the plus and minus the translation T2 to 820 00:48:05,010 --> 00:48:09,660 take this stack of A planes and repeat it B times. 821 00:48:09,660 --> 00:48:12,400 And that's going to take the plane at the origin and move 822 00:48:12,400 --> 00:48:13,850 it up to here. 823 00:48:13,850 --> 00:48:18,520 It's going to move the planes into stacks like this. 824 00:48:18,520 --> 00:48:26,180 I will map the stack of A planes B times. 825 00:48:30,800 --> 00:48:39,520 And that is going to give me A times B intervals except if a 826 00:48:39,520 --> 00:48:41,880 special condition applies. 827 00:48:41,880 --> 00:48:43,690 And then, I will go back to it in a moment. 828 00:48:47,020 --> 00:48:49,715 This is the equation of the intercept plane. 829 00:49:00,230 --> 00:49:04,730 I will argue that the three dimensional analogy of what 830 00:49:04,730 --> 00:49:09,440 I've done is that there will be, if I let the translations 831 00:49:09,440 --> 00:49:14,020 T1, T2, and T3 go to work, there will be A times B times 832 00:49:14,020 --> 00:49:20,800 C intervals between the origin and the intercept plane. 833 00:49:36,090 --> 00:49:39,740 So if this is the intercept plane and this is A times B 834 00:49:39,740 --> 00:49:43,930 times C times the spacing between the planes out from 835 00:49:43,930 --> 00:49:50,250 the origin, then the first plane from the origin is going 836 00:49:50,250 --> 00:49:54,836 to happen to intercept which is 1ABC to the first. 837 00:49:54,836 --> 00:49:59,880 And ABC divided by ABC turns out to be 1 for any value of 838 00:49:59,880 --> 00:50:00,940 A, B, and C. 839 00:50:00,940 --> 00:50:03,880 So the first plane from the origin is going to have the 840 00:50:03,880 --> 00:50:12,110 simple equation be BC times x plus AC times y plus AB times 841 00:50:12,110 --> 00:50:14,710 z equals 1. 842 00:50:14,710 --> 00:50:17,850 And the second plane out from the origin will have the same 843 00:50:17,850 --> 00:50:21,350 coefficients except the term on the left will be 2. 844 00:50:21,350 --> 00:50:25,020 Third plane will have 3 on the right and finally the integer 845 00:50:25,020 --> 00:50:27,170 ABC when we get to the intercept plane. 846 00:50:29,930 --> 00:50:33,510 OK, now I'm ready to make a momentous definition just as 847 00:50:33,510 --> 00:50:35,710 we get five of the hour. 848 00:50:35,710 --> 00:50:37,650 B times C is an integer. 849 00:50:37,650 --> 00:50:40,940 Let me define that integer by a single integer. 850 00:50:40,940 --> 00:50:45,330 And just for the heck of it, I'll call it H. A times C is 851 00:50:45,330 --> 00:50:45,880 an integer. 852 00:50:45,880 --> 00:50:51,800 Let me call that integer K. And AB is an integer. 853 00:50:51,800 --> 00:50:55,310 Let me call that integer L. 854 00:50:55,310 --> 00:51:01,720 So using T1 and T2 and T3 as the basis vectors of my 855 00:51:01,720 --> 00:51:05,670 coordinate system, the first plane from the origin has the 856 00:51:05,670 --> 00:51:10,000 equation Hx plus Ky plus Lz equals 1. 857 00:51:10,000 --> 00:51:15,000 And these three integers, H, K, and L, are said to be the 858 00:51:15,000 --> 00:51:18,110 Miller indices of this plane. 859 00:51:24,956 --> 00:51:25,928 PROFESSOR: Yes, sir? 860 00:51:25,928 --> 00:51:29,168 AUDIENCE: Just on that figure, [INAUDIBLE] connect to the 861 00:51:29,168 --> 00:51:29,816 lattice points. 862 00:51:29,816 --> 00:51:32,489 I'm trying to make sense of why did you put the planes in 863 00:51:32,489 --> 00:51:33,720 between them? 864 00:51:33,720 --> 00:51:36,490 PROFESSOR: OK, what I did was I started with just these 865 00:51:36,490 --> 00:51:41,200 three hanging on the lattice point separated by T1. 866 00:51:41,200 --> 00:51:44,710 And then, I move this by T2. 867 00:51:44,710 --> 00:51:49,360 OK, so T2 has a plane in the middle of it. 868 00:51:49,360 --> 00:51:52,590 And it's going to move that up to here and 869 00:51:52,590 --> 00:51:54,110 give me another plane. 870 00:51:54,110 --> 00:51:55,720 If you just draw this out-- and you're going to get a 871 00:51:55,720 --> 00:51:58,140 problem that asks you to do this-- 872 00:51:58,140 --> 00:52:02,300 since these integers are mutually prime, the second 873 00:52:02,300 --> 00:52:05,430 translation is going to interleave planes 874 00:52:05,430 --> 00:52:06,850 between the first set. 875 00:52:06,850 --> 00:52:09,660 And it gets even worse in three dimensions because the 876 00:52:09,660 --> 00:52:16,840 third translation, which is the axis C translations out, 877 00:52:16,840 --> 00:52:21,010 is going to interleave that set of AB planes C times. 878 00:52:21,010 --> 00:52:23,180 And they have to be at equal distances. 879 00:52:23,180 --> 00:52:27,370 Otherwise, the definition of the lattice point having 880 00:52:27,370 --> 00:52:29,830 identical environment is violated. 881 00:52:29,830 --> 00:52:35,560 OK, these are the infamous Miller indices that are the 882 00:52:35,560 --> 00:52:40,080 downfall of all freshman who take 309.1. 883 00:52:40,080 --> 00:52:46,320 It is not necessarily a very straightforward definition. 884 00:52:46,320 --> 00:52:51,520 But the advantage of it is that it lets you analytically 885 00:52:51,520 --> 00:52:56,360 describe the locus of the lattice planes in a very, very 886 00:52:56,360 --> 00:52:58,070 convenient way. 887 00:52:58,070 --> 00:53:01,180 Actually, let me tell you something maybe you have not 888 00:53:01,180 --> 00:53:02,500 heard before. 889 00:53:02,500 --> 00:53:07,300 These indices were not invented by a fellow named 890 00:53:07,300 --> 00:53:10,540 Miller who was an Englishman who wrote a book, an early 891 00:53:10,540 --> 00:53:13,920 book on crystallography that incorporated this notation. 892 00:53:13,920 --> 00:53:18,220 The guy who invented them and first proposed them was 893 00:53:18,220 --> 00:53:22,990 Auguste Bravais, a very famous French crystallographer-- 894 00:53:22,990 --> 00:53:24,440 mathematician, actually-- 895 00:53:24,440 --> 00:53:28,780 who gave his name to the three dimensional space lattices. 896 00:53:28,780 --> 00:53:31,830 These are universally known as the Bravais 897 00:53:31,830 --> 00:53:33,110 lattices, 14 of them. 898 00:53:33,110 --> 00:53:33,980 It's a great name. 899 00:53:33,980 --> 00:53:36,280 It's music. it rolls of the tongue-- 900 00:53:36,280 --> 00:53:39,100 Bravis, Bravais. 901 00:53:39,100 --> 00:53:41,080 Actually, Bravias wasn't the first guy to 902 00:53:41,080 --> 00:53:42,300 try to derive them. 903 00:53:42,300 --> 00:53:46,700 The first guy who tried to drive the lattices was 904 00:53:46,700 --> 00:53:47,950 somebody named Frankenheim. 905 00:53:50,290 --> 00:53:51,890 He blew it. 906 00:53:51,890 --> 00:53:52,750 He didn't get 14. 907 00:53:52,750 --> 00:53:56,040 He only found 13. 908 00:53:56,040 --> 00:53:57,575 So there's a moral here. 909 00:53:57,575 --> 00:54:01,050 If immortality is your game, get it right the first time 910 00:54:01,050 --> 00:54:04,400 because everybody has heard of Bravais, Bravais. 911 00:54:04,400 --> 00:54:07,520 Nobody has heard of Frankenheim. 912 00:54:07,520 --> 00:54:08,960 And for that, I say thank god. 913 00:54:08,960 --> 00:54:09,500 What a name! 914 00:54:09,500 --> 00:54:11,960 It makes you think of somebody who's got plugs in his neck 915 00:54:11,960 --> 00:54:14,610 and walks around like this and has a green face. 916 00:54:14,610 --> 00:54:15,390 Frankenheim-- 917 00:54:15,390 --> 00:54:16,520 bah! 918 00:54:16,520 --> 00:54:19,760 Bravais, Bravais, it really sings to you. 919 00:54:19,760 --> 00:54:25,590 OK, I'm getting silly so it's time to quit and take a break. 920 00:54:25,590 --> 00:54:27,000 Come up and say hello to Rodney.