1 00:00:07,840 --> 00:00:08,240 PROFESSOR: All right. 2 00:00:08,240 --> 00:00:11,940 The end of this lecture generated so much excitement 3 00:00:11,940 --> 00:00:15,410 that we've continued on into the start of the next segment. 4 00:00:15,410 --> 00:00:19,220 So I think we'd better begin. 5 00:00:19,220 --> 00:00:23,030 What I told you, as several people were clever enough to 6 00:00:23,030 --> 00:00:33,630 observe, that the number of planes between the origin and 7 00:00:33,630 --> 00:00:43,030 intercept plane is equal to A times B times C, where the 8 00:00:43,030 --> 00:00:50,010 intercept is A T1 out plus B T2 out equals C T3 out. 9 00:00:54,740 --> 00:00:58,440 That is true only if the numbers A, B, and C are 10 00:00:58,440 --> 00:01:00,980 mutually prime. 11 00:01:00,980 --> 00:01:04,239 If it's something like 4, 2, 1, you won't get 12 00:01:04,239 --> 00:01:05,390 that number of planes. 13 00:01:05,390 --> 00:01:07,270 You'll get a submultiple. 14 00:01:07,270 --> 00:01:09,960 And the reason for that is that some of the planes, when 15 00:01:09,960 --> 00:01:13,700 you generate them by these three different translations 16 00:01:13,700 --> 00:01:17,000 in different directions, some of them are mapped on top of 17 00:01:17,000 --> 00:01:18,760 existing planes. 18 00:01:18,760 --> 00:01:21,210 And I'm not going to attempt to prove this, but let me just 19 00:01:21,210 --> 00:01:26,670 tell you the result, that if A and B contain some common 20 00:01:26,670 --> 00:01:32,670 factor, p, and B and C contain some common factor, q, and in 21 00:01:32,670 --> 00:01:37,250 the worst possible case, A and C contains some common factor, 22 00:01:37,250 --> 00:01:46,290 r, then the number of intervals is equal to A times 23 00:01:46,290 --> 00:01:52,310 B times C divided by p times q times r. 24 00:01:52,310 --> 00:01:55,960 And probably the best way to show that is let you show, on 25 00:01:55,960 --> 00:02:00,070 a problem set in the very near future, to actually map out 26 00:02:00,070 --> 00:02:02,440 the planes in the two-dimensional situation, for 27 00:02:02,440 --> 00:02:05,240 a and b that are mutually prime, and then do the same 28 00:02:05,240 --> 00:02:10,880 thing for a and b that contain some common factor. 29 00:02:10,880 --> 00:02:13,790 So this is the so-called Miller indices, the 30 00:02:13,790 --> 00:02:17,000 Miller-Bravais indices, and everybody hears about them, 31 00:02:17,000 --> 00:02:22,280 but it is not at all clear why one takes this very indirect 32 00:02:22,280 --> 00:02:25,630 and non-intuitive way of defining the 33 00:02:25,630 --> 00:02:27,700 orientation of a plane. 34 00:02:27,700 --> 00:02:34,680 And the reason for this, ultimately, is that in x-ray 35 00:02:34,680 --> 00:02:38,870 diffraction, you want to integrate up the scattering of 36 00:02:38,870 --> 00:02:40,970 all atoms in a crystal. 37 00:02:40,970 --> 00:02:45,350 So you want to integrate over a plane that contains the 38 00:02:45,350 --> 00:02:49,890 atoms, and this locus has a convenient form only if you 39 00:02:49,890 --> 00:02:54,480 choose these indices to find the orientation of the plane. 40 00:02:54,480 --> 00:02:57,450 And what comes out of this, as I said to a couple of people 41 00:02:57,450 --> 00:03:01,150 during intermission, is that the amplitude of the scattered 42 00:03:01,150 --> 00:03:07,990 wave has some summation over all of the atoms. 43 00:03:07,990 --> 00:03:19,290 And in that phase of that ray, that is e to the plus 2 pi i, 44 00:03:19,290 --> 00:03:21,620 hx plus ky plus lz. 45 00:03:24,960 --> 00:03:26,790 Very, very simple form. 46 00:03:26,790 --> 00:03:32,550 And it has that form only because the locus, the way 47 00:03:32,550 --> 00:03:35,810 this plane is defined was done very, very carefully. 48 00:03:35,810 --> 00:03:38,720 So it's something beyond just simple geometry that forms a 49 00:03:38,720 --> 00:03:39,970 reason for it. 50 00:03:42,100 --> 00:03:44,840 Let me say a couple more things in general about the 51 00:03:44,840 --> 00:03:47,240 Miller-Bravais indices. 52 00:03:47,240 --> 00:03:52,330 They are very, very intuitive, counter-intuitive. 53 00:03:52,330 --> 00:03:57,340 Let's suppose that this is a, and this is b, and this is c. 54 00:03:57,340 --> 00:04:01,620 And I put down here, again, the equations of the entire 55 00:04:01,620 --> 00:04:03,900 set of planes in the stack. 56 00:04:03,900 --> 00:04:06,460 And let's look at the first planes and ask what the 57 00:04:06,460 --> 00:04:07,710 intercepts are. 58 00:04:10,850 --> 00:04:16,680 The intercepts on x, y, and z are going to be 1 over h on x, 59 00:04:16,680 --> 00:04:21,140 1 over k on y, and 1 over l times z. 60 00:04:21,140 --> 00:04:26,500 So the first plane from the origin is going to be 1 over h 61 00:04:26,500 --> 00:04:33,590 of the a translation, 1 over k of the b translation, and 1 62 00:04:33,590 --> 00:04:37,890 over l of the c translation. 63 00:04:37,890 --> 00:04:44,060 So this gives us one way of determining, given a 64 00:04:44,060 --> 00:04:49,080 particular rational plane, show it relative to the three 65 00:04:49,080 --> 00:04:50,710 translations. 66 00:04:50,710 --> 00:04:55,510 From the first plane from the origin, h is going to be 1 67 00:04:55,510 --> 00:04:58,120 over the intercept on T1. 68 00:05:02,310 --> 00:05:04,330 I'll call it a, which is what we usually do. 69 00:05:04,330 --> 00:05:13,090 k is going to be 1 over the intercept on b, and l is going 70 00:05:13,090 --> 00:05:17,930 to be 1 over the intercept on c for the first 71 00:05:17,930 --> 00:05:19,180 plane from the origin. 72 00:05:25,750 --> 00:05:29,100 So if you have a drawing of the set of planes relative to 73 00:05:29,100 --> 00:05:32,510 the translation, such as the one that I obliterated from 74 00:05:32,510 --> 00:05:35,960 just a moment ago, reciprocals of the intercepts of the first 75 00:05:35,960 --> 00:05:38,795 plane from the origin give you h, k, and l very easily. 76 00:05:41,880 --> 00:05:46,310 The thing that is counter-intuitive is that, if 77 00:05:46,310 --> 00:05:52,310 you look out of context at the three indices, h, k, and l, 78 00:05:52,310 --> 00:05:59,160 you have the feeling that as l gets larger, the intercept of 79 00:05:59,160 --> 00:06:02,160 the plane on c should gradually creep up from here, 80 00:06:02,160 --> 00:06:06,250 to here, to here, and so on, as l gets larger. 81 00:06:06,250 --> 00:06:08,410 Actually, the reverse is true. 82 00:06:08,410 --> 00:06:13,480 As l increases, the intercept on c gets smaller and smaller 83 00:06:13,480 --> 00:06:19,590 and smaller and smaller, until finally, when l is infinity, 84 00:06:19,590 --> 00:06:23,390 the intercept is 0. 85 00:06:23,390 --> 00:06:26,140 So it's counter-intuitive. 86 00:06:26,140 --> 00:06:29,190 The intercept on the axis gets larger as the Miller index 87 00:06:29,190 --> 00:06:30,440 gets larger. 88 00:06:32,700 --> 00:06:39,290 Let me give you a proprietary Uncle Bernie's secret method 89 00:06:39,290 --> 00:06:42,260 for determining Miller indices without stress and strain. 90 00:06:45,440 --> 00:06:48,510 This works every time. 91 00:06:48,510 --> 00:06:50,670 Find the intercepts of any plane. 92 00:07:01,130 --> 00:07:03,100 And we know from these equations what those 93 00:07:03,100 --> 00:07:03,780 are going to be. 94 00:07:03,780 --> 00:07:12,940 For the nth plane, the intercepts are going to be n 95 00:07:12,940 --> 00:07:23,310 over h, n over k on b, and n over l on c. 96 00:07:23,310 --> 00:07:24,560 Take the reciprocals. 97 00:07:30,630 --> 00:07:34,680 And the reciprocals are going to be h over n, k over 98 00:07:34,680 --> 00:07:39,050 n, and l over n. 99 00:07:39,050 --> 00:07:46,140 And then find whatever integer, n, you must multiply 100 00:07:46,140 --> 00:07:49,650 those reciprocals by in order to convert them to mutually 101 00:07:49,650 --> 00:07:54,700 prime intercepts, and you've got the Miller indices in a 102 00:07:54,700 --> 00:07:56,475 no-brainer, automatic fashion. 103 00:08:04,010 --> 00:08:08,290 Another bit of jargon. 104 00:08:08,290 --> 00:08:13,330 We will also use the basis vectors to define locations 105 00:08:13,330 --> 00:08:16,360 and to define directions in a lattice. 106 00:08:20,440 --> 00:08:27,470 To distinguish three integers that indicate a plane, and you 107 00:08:27,470 --> 00:08:30,780 all know this from previous experience, the parentheses 108 00:08:30,780 --> 00:08:33,994 says plane, an individual plane. 109 00:08:45,340 --> 00:08:53,500 There are other times when you would like to use the indices 110 00:08:53,500 --> 00:08:58,780 of one representative plane to indicate the entire set of 111 00:08:58,780 --> 00:09:03,490 planes that are related by the symmetry of the crystal. 112 00:09:03,490 --> 00:09:05,580 So let me give one specific example. 113 00:09:05,580 --> 00:09:12,670 Suppose we had a cubic crystal, where these are the 114 00:09:12,670 --> 00:09:17,130 three translations that form the edge of the unit cell. 115 00:09:17,130 --> 00:09:21,610 And let's say that this is a1, this is a2, and this is a3. 116 00:09:21,610 --> 00:09:24,680 I'm using a1, a2, and a3, not a, b, and c. 117 00:09:24,680 --> 00:09:26,700 But let's not go there. 118 00:09:26,700 --> 00:09:29,590 OK, these are lattice points. 119 00:09:29,590 --> 00:09:33,350 And if I look at this plane that's perpendicular to a1 and 120 00:09:33,350 --> 00:09:36,060 cuts it at one lattice point, l, this would 121 00:09:36,060 --> 00:09:38,900 be the 1, 0, 0 plane. 122 00:09:38,900 --> 00:09:44,390 And I would indicate that that is an individual plane that 123 00:09:44,390 --> 00:09:46,190 we're talking about. 124 00:09:46,190 --> 00:09:53,010 But if this crystal is cubic, all six faces of the set, 1, 125 00:09:53,010 --> 00:10:04,930 0, 0; 0, 1, 0; 0, 0, 1; and then the plane that's in the 126 00:10:04,930 --> 00:10:11,450 opposite direction, minus 1, 0, 0; 0, minus 1, 0; 127 00:10:11,450 --> 00:10:13,910 and 0, 0, minus 1. 128 00:10:13,910 --> 00:10:20,390 These six planes define the surfaces of the cube. 129 00:10:20,390 --> 00:10:23,470 So there are times, since these are equivalent to planes 130 00:10:23,470 --> 00:10:26,090 and will have the same properties, the same hardness, 131 00:10:26,090 --> 00:10:28,270 the same [? edge ?] bits and everything else, there are 132 00:10:28,270 --> 00:10:31,440 times when we may want to refer to the entire 133 00:10:31,440 --> 00:10:34,190 symmetrical set. 134 00:10:34,190 --> 00:10:36,180 And we do that. 135 00:10:36,180 --> 00:10:40,000 We call that set a form. 136 00:10:40,000 --> 00:10:42,570 It is a symmetry-related set. 137 00:10:50,830 --> 00:10:54,590 And you can pick any one you like. 138 00:10:54,590 --> 00:10:57,900 Generally, you pick one with simple indices, like the one 139 00:10:57,900 --> 00:11:01,820 that is perpendicular to the translation a. 140 00:11:01,820 --> 00:11:07,210 And then you indicate by braces, which is 141 00:11:07,210 --> 00:11:11,810 the symbol for set. 142 00:11:11,810 --> 00:11:15,520 This means the set 1, 0, 0, and that would correspond to 143 00:11:15,520 --> 00:11:16,770 all six of these faces. 144 00:11:19,470 --> 00:11:26,350 The number, if you do this in reverse, and I'm going to give 145 00:11:26,350 --> 00:11:28,750 you the indices, what are the separate faces? 146 00:11:28,750 --> 00:11:30,540 That depends on the symmetry. 147 00:11:30,540 --> 00:11:34,980 So to give you an example, if I have a unit cell that has 148 00:11:34,980 --> 00:11:40,380 three orthogonal translations, and this is a, and this is b, 149 00:11:40,380 --> 00:11:44,310 and this is c, so that there is no cubic symmetry to it. 150 00:11:44,310 --> 00:11:52,310 This face, which would be 1, 0, 0, and the one that cuts 151 00:11:52,310 --> 00:11:57,680 the axes at minus a, minus 1, 0 0, those are going to be the 152 00:11:57,680 --> 00:12:03,920 only faces with a single 1 and a pair of 0's which are 153 00:12:03,920 --> 00:12:05,270 equivalent by symmetry. 154 00:12:05,270 --> 00:12:08,572 So in this case, for this brick shaped unit cell, 1, 0, 155 00:12:08,572 --> 00:12:14,720 0 would mean this pair of two faces, rather than six. 156 00:12:14,720 --> 00:12:18,830 So to go from a representative face to the set of planes, you 157 00:12:18,830 --> 00:12:20,180 have to know the symmetry. 158 00:12:20,180 --> 00:12:23,060 If you can write the planes on the basis of symmetry, take 159 00:12:23,060 --> 00:12:25,180 one representative and let that be the form. 160 00:12:33,370 --> 00:12:38,370 So it leads one to observe that crystals undergoing 161 00:12:38,370 --> 00:12:44,075 plastic deformation have bad form. 162 00:12:44,075 --> 00:12:45,037 Yeah? 163 00:12:45,037 --> 00:12:47,442 AUDIENCE: Is that different from when you 164 00:12:47,442 --> 00:12:48,885 talk about a direction? 165 00:12:48,885 --> 00:12:50,330 PROFESSOR: Yes, yes. 166 00:12:50,330 --> 00:12:53,830 You will see in just a moment that we use the same system of 167 00:12:53,830 --> 00:12:57,200 three integers, but in order to distinguish which one he's 168 00:12:57,200 --> 00:13:00,184 talking about when one's thinking about integers with a 169 00:13:00,184 --> 00:13:01,434 different set of numbers. 170 00:13:14,940 --> 00:13:17,760 Again, if we use the lattice as the basis of a coordinate 171 00:13:17,760 --> 00:13:28,570 system, another thing we might want to specify is the 172 00:13:28,570 --> 00:13:30,850 orientation of a rational direction. 173 00:13:30,850 --> 00:13:34,660 And let me go to a two-dimensional case. 174 00:13:34,660 --> 00:13:38,980 Assume that c goes straight up. 175 00:13:38,980 --> 00:13:45,570 Directions of easy plastic deformation will very often 176 00:13:45,570 --> 00:13:47,960 involve integral number of translations. 177 00:13:47,960 --> 00:13:51,620 For example, this direction here is obtained by going one 178 00:13:51,620 --> 00:13:54,090 translation in the a direction plus one 179 00:13:54,090 --> 00:13:55,750 translation in the b direction. 180 00:13:55,750 --> 00:13:59,680 So we could use the pair of integers 1, 1, and if we are 181 00:13:59,680 --> 00:14:04,540 normal to see we'd use a 0 for the third integer. 182 00:14:04,540 --> 00:14:09,920 But how are we going to tell whether we're talking about a 183 00:14:09,920 --> 00:14:13,450 direction or a plane? 184 00:14:13,450 --> 00:14:19,410 And the way one does that is to use square braces to 185 00:14:19,410 --> 00:14:21,175 indicate one particular direction. 186 00:14:35,230 --> 00:14:41,720 You may want to also indicate the location of a particular 187 00:14:41,720 --> 00:14:45,350 atom within the unit cell. 188 00:14:45,350 --> 00:14:51,730 In order to do that, one could really use the location of a 189 00:14:51,730 --> 00:14:57,570 vector, the components of a vector, xa plus yb plus zc, 190 00:14:57,570 --> 00:15:01,340 the vector from the origin to that atom as coefficients 191 00:15:01,340 --> 00:15:03,880 which correspond to the atom location. 192 00:15:03,880 --> 00:15:08,050 So if that is the case, the numbers 193 00:15:08,050 --> 00:15:10,480 are usually not integers. 194 00:15:10,480 --> 00:15:13,290 And one simply uses the three integers in 195 00:15:13,290 --> 00:15:14,540 an unadorned fashion. 196 00:15:20,890 --> 00:15:24,360 So the lattice translations give you the basis of a 197 00:15:24,360 --> 00:15:28,520 coordinate system, and we use those to specify features, 198 00:15:28,520 --> 00:15:33,856 rational or not, of planes and directions and coordinates. 199 00:15:40,870 --> 00:15:42,120 OK, any questions here? 200 00:15:52,900 --> 00:15:57,460 OK, we are next going to take the first small step in a 201 00:15:57,460 --> 00:16:00,420 process of synthesis. 202 00:16:00,420 --> 00:16:05,000 We have found that there are, in two dimensions, five 203 00:16:05,000 --> 00:16:06,790 distinct kinds of lattices. 204 00:16:06,790 --> 00:16:26,660 The oblique, the rectangular, the centric rectangular, the 205 00:16:26,660 --> 00:16:32,880 square, and the hexagonal. 206 00:16:32,880 --> 00:16:34,130 Five kinds of lattices. 207 00:16:37,010 --> 00:16:40,430 We've shown, to this point, that each of these lattices 208 00:16:40,430 --> 00:16:44,430 can accommodate a certain rotational symmetry. 209 00:16:44,430 --> 00:16:47,390 1 or 2 here. 210 00:16:47,390 --> 00:16:51,770 m and m for the rectangular and centric rectangular. 211 00:16:51,770 --> 00:16:54,670 4, the fourfold axis for a square. 212 00:16:54,670 --> 00:16:56,870 3 or 6 for the hexagonal lattice. 213 00:17:01,660 --> 00:17:05,839 So what I would like to do next, after we make a brief 214 00:17:05,839 --> 00:17:11,550 direct diversion, is to combine with each of these 215 00:17:11,550 --> 00:17:20,790 nets one of the two symmetries which can be 216 00:17:20,790 --> 00:17:24,400 accommodated in that net. 217 00:17:24,400 --> 00:17:27,359 And what we will have determined then are the 218 00:17:27,359 --> 00:17:30,630 patterns' lattice and symmetry, which can exist in 219 00:17:30,630 --> 00:17:31,670 two dimensions. 220 00:17:31,670 --> 00:17:34,510 The sort of things we look at actually, like the pattern of 221 00:17:34,510 --> 00:17:38,680 square tiles on the floor, or the two-dimensional pattern 222 00:17:38,680 --> 00:17:41,110 that is on a fabric design. 223 00:17:41,110 --> 00:17:46,850 So we'll derive those exhaustively. 224 00:17:46,850 --> 00:17:50,160 The thing that results is something that is called a 225 00:17:50,160 --> 00:17:51,410 plane group. 226 00:17:55,430 --> 00:17:59,100 And I'm going to take this through in detail because 227 00:17:59,100 --> 00:18:03,950 there are not that many combinations that can be made. 228 00:18:03,950 --> 00:18:06,490 And the number is enough to count on 229 00:18:06,490 --> 00:18:07,720 your fingers and toes. 230 00:18:07,720 --> 00:18:10,400 You've got to use both, so they're a fair number, but not 231 00:18:10,400 --> 00:18:12,480 a staggering number. 232 00:18:12,480 --> 00:18:15,330 If we were to do this exercise in three dimensions, there 233 00:18:15,330 --> 00:18:20,050 would be 230 distinct combinations. 234 00:18:20,050 --> 00:18:23,812 And that's too much for anybody to go through. 235 00:18:23,812 --> 00:18:27,430 I know of very few people who work professionally in this 236 00:18:27,430 --> 00:18:31,910 area who've actually sat down and derived every single one. 237 00:18:31,910 --> 00:18:34,950 If you know the principles of the derivation and know how 238 00:18:34,950 --> 00:18:39,780 you could go about doing it if you were forced to, that's 239 00:18:39,780 --> 00:18:41,050 sufficient. 240 00:18:41,050 --> 00:18:45,450 The results are tabulated in an admirable compilation. 241 00:18:45,450 --> 00:18:46,850 We mentioned that at the beginning. 242 00:18:46,850 --> 00:18:49,270 This is the International Tables for X-ray 243 00:18:49,270 --> 00:18:51,530 Crystallography, Volume 1. 244 00:18:51,530 --> 00:18:55,610 So what we should end up with is a way for you to go to this 245 00:18:55,610 --> 00:18:57,890 reference data and know how to interpret it 246 00:18:57,890 --> 00:18:59,250 and make use of it. 247 00:18:59,250 --> 00:19:01,030 But this is a nice microcosm. 248 00:19:01,030 --> 00:19:03,620 There are not many plane groups, and to do this 249 00:19:03,620 --> 00:19:07,660 exhaustively, will let us see the sort of reasoning that 250 00:19:07,660 --> 00:19:09,470 goes into it. 251 00:19:09,470 --> 00:19:12,690 I have to tell you that this is not in Buerger's book. 252 00:19:12,690 --> 00:19:14,020 We're going to do two-dimensional 253 00:19:14,020 --> 00:19:15,890 crystallography first, and then extend 254 00:19:15,890 --> 00:19:17,340 it in a third direction. 255 00:19:17,340 --> 00:19:20,680 In fact, I know of no book that does this. 256 00:19:20,680 --> 00:19:23,840 And so you're going to get a unique, handwritten document 257 00:19:23,840 --> 00:19:27,620 by none other than yours truly that will do this in detail 258 00:19:27,620 --> 00:19:28,850 every step of the way. 259 00:19:28,850 --> 00:19:32,360 And I don't know of any other place where this is done. 260 00:19:32,360 --> 00:19:35,240 Now, what is the origin of this name? 261 00:19:35,240 --> 00:19:37,290 Plane, well, these are symmetries 262 00:19:37,290 --> 00:19:38,810 confined to a plane. 263 00:19:38,810 --> 00:19:42,480 And then there's this mysterious term, group. 264 00:19:42,480 --> 00:19:45,660 And the corresponding thing in 3D is something 265 00:19:45,660 --> 00:19:46,910 called a space group. 266 00:19:54,890 --> 00:19:57,990 This is a language derived from a branch of mathematics 267 00:19:57,990 --> 00:20:06,230 that is called group theory, and it is exactly the language 268 00:20:06,230 --> 00:20:09,690 that describes things that we've mentioned 269 00:20:09,690 --> 00:20:14,430 conversationally, such as if you take two transformations 270 00:20:14,430 --> 00:20:17,070 and perform them in succession, they're going to 271 00:20:17,070 --> 00:20:19,880 give you a third that has to be present. 272 00:20:19,880 --> 00:20:22,510 If you've got two present, there must be a third. 273 00:20:22,510 --> 00:20:26,260 We talk about symmetries being self consistent. 274 00:20:26,260 --> 00:20:29,070 If you have a rotation operation that is not an 275 00:20:29,070 --> 00:20:33,850 integral submultiple of 2 pi, it's not going to be self 276 00:20:33,850 --> 00:20:36,010 consistent because you go around, and around, and 277 00:20:36,010 --> 00:20:39,280 around, and you will never come exactly full circle. 278 00:20:39,280 --> 00:20:44,000 So group theory is a branch of mathematics that deals with 279 00:20:44,000 --> 00:20:45,300 concepts of this sort. 280 00:20:45,300 --> 00:20:50,150 So it's not very hard to develop, and it gives its 281 00:20:50,150 --> 00:20:54,050 ideas to the notation that is used in crystallography. 282 00:20:54,050 --> 00:20:57,730 So let me say a few words about group theory. 283 00:20:57,730 --> 00:20:58,980 What is a group? 284 00:21:01,410 --> 00:21:03,440 A group is a set of elements-- 285 00:21:08,510 --> 00:21:10,220 And what are elements? 286 00:21:10,220 --> 00:21:11,470 Elements are things-- 287 00:21:18,230 --> 00:21:22,930 For which a law of combination is defined. 288 00:21:35,790 --> 00:21:38,150 And the nature of the elements can be very, 289 00:21:38,150 --> 00:21:39,140 very different things. 290 00:21:39,140 --> 00:21:41,135 They could be numbers. 291 00:21:45,410 --> 00:21:55,060 They could be complex numbers, and the law of combination 292 00:21:55,060 --> 00:21:57,790 here might be multiplication. 293 00:21:57,790 --> 00:22:03,900 They could be matrices, for which the law of combination 294 00:22:03,900 --> 00:22:07,990 is defined as matrix multiplication. 295 00:22:07,990 --> 00:22:11,280 Or they could be pots of different color paint, for 296 00:22:11,280 --> 00:22:14,360 which the law of combination is defined as mixing the two 297 00:22:14,360 --> 00:22:16,160 colors 50-50. 298 00:22:16,160 --> 00:22:18,910 I don't know any practical application of that sort of 299 00:22:18,910 --> 00:22:22,390 group, but nevertheless, it follows the definition, a set 300 00:22:22,390 --> 00:22:27,000 of things for which a law of combination is defined. 301 00:22:27,000 --> 00:22:40,615 And in addition, which satisfies three so-called 302 00:22:40,615 --> 00:22:41,865 group postulates. 303 00:22:49,930 --> 00:22:56,220 And the group postulates are the combination of any two 304 00:22:56,220 --> 00:23:06,705 elements is also a member of the group. 305 00:23:23,530 --> 00:23:28,800 The second group postulate is that the identity-- 306 00:23:28,800 --> 00:23:30,050 I'm going to call it just identity-- 307 00:23:36,250 --> 00:23:39,100 and let me define, identity is doing nothing-- 308 00:23:45,710 --> 00:23:47,610 Is also a member of the group. 309 00:23:53,890 --> 00:24:01,290 And if we define the identity operation as I, this means 310 00:24:01,290 --> 00:24:08,770 that for every element, I followed by a or a followed by 311 00:24:08,770 --> 00:24:12,490 I, is the same as a by itself. 312 00:24:12,490 --> 00:24:15,360 If the law of multiplication were just arithmetic 313 00:24:15,360 --> 00:24:18,570 multiplication, then number 1 would be the identity 314 00:24:18,570 --> 00:24:22,300 operation because 1 times any number gives you the number. 315 00:24:22,300 --> 00:24:26,340 Any number times 1 gives you the same number back again. 316 00:24:26,340 --> 00:24:35,400 And the third postulate is that, for every element, an 317 00:24:35,400 --> 00:24:36,650 inverse exists-- 318 00:24:45,000 --> 00:24:50,670 let's call the elements a and a minus 1-- 319 00:24:50,670 --> 00:24:56,000 such that a followed by a minus 1 is equal to the 320 00:24:56,000 --> 00:25:00,530 identity operation, and the inverse element followed by 321 00:25:00,530 --> 00:25:03,825 the original element, a, is also the identity operation. 322 00:25:07,880 --> 00:25:10,950 So if the law of combination were defined as 323 00:25:10,950 --> 00:25:17,750 multiplication, you would have to find, for every number, 324 00:25:17,750 --> 00:25:20,660 another number, which multiplied by it, gives you 325 00:25:20,660 --> 00:25:21,910 the number 1. 326 00:25:25,810 --> 00:25:29,060 OK, let me now illustrate with a couple of simple examples. 327 00:25:35,670 --> 00:25:42,540 Let's consider the set of numbers 1, n minus 1. 328 00:25:46,180 --> 00:25:51,070 The number of elements in the group is said to be the rank 329 00:25:51,070 --> 00:25:54,675 of the group, or some people like to use the term order. 330 00:26:00,020 --> 00:26:04,470 And I don't like that term because second order or fourth 331 00:26:04,470 --> 00:26:08,940 order is a term applied to quantities which are not 332 00:26:08,940 --> 00:26:09,960 terribly important. 333 00:26:09,960 --> 00:26:13,500 So we expand this except for higher order terms, which are 334 00:26:13,500 --> 00:26:14,880 negligible. 335 00:26:14,880 --> 00:26:17,500 And I don't like that connotation. 336 00:26:17,500 --> 00:26:20,190 So I will not use order, although it's sometimes used 337 00:26:20,190 --> 00:26:21,900 to talk about the rank of a group. 338 00:26:21,900 --> 00:26:24,260 The rank or order of the group is simply the number of 339 00:26:24,260 --> 00:26:25,560 elements contained in a set. 340 00:26:44,420 --> 00:26:50,050 So let's show that the numbers 1 and minus 1 constitute a 341 00:26:50,050 --> 00:26:52,270 group of rank two. 342 00:26:52,270 --> 00:26:55,460 Is the combination of any pair of elements also a 343 00:26:55,460 --> 00:26:56,560 member of the group? 344 00:26:56,560 --> 00:26:59,970 Well, to do this, it's convenient to set up an array 345 00:26:59,970 --> 00:27:01,910 called the group multiplication table. 346 00:27:09,230 --> 00:27:13,730 And what you do is you simply put the elements of the group 347 00:27:13,730 --> 00:27:15,230 along a row. 348 00:27:15,230 --> 00:27:17,680 And in this case, it's 1 and minus 1. 349 00:27:17,680 --> 00:27:20,630 And then you put the same elements along a column, and 350 00:27:20,630 --> 00:27:22,380 then you combine them. 351 00:27:22,380 --> 00:27:24,080 1 times 1 is 1. 352 00:27:24,080 --> 00:27:26,590 Minus 1 times 1 is minus 1. 353 00:27:26,590 --> 00:27:28,650 1 times minus 1 is minus 1. 354 00:27:28,650 --> 00:27:31,770 Minus 1 times minus 1 is plus 1. 355 00:27:31,770 --> 00:27:34,860 Sometimes, as I'm giving this part of the lecture, somebody 356 00:27:34,860 --> 00:27:37,880 passes down the hallway, and they hear 1 times 1 is 1. 357 00:27:37,880 --> 00:27:39,080 1 times minus 1-- 358 00:27:39,080 --> 00:27:41,530 and they go in reverse to see what in the world could be 359 00:27:41,530 --> 00:27:46,170 going on in this high-powered mathematical lecture. 360 00:27:46,170 --> 00:27:48,240 That's why I close the door. 361 00:27:48,240 --> 00:27:50,580 OK, so any combination of elements is also a 362 00:27:50,580 --> 00:27:52,710 member of the group. 363 00:27:52,710 --> 00:27:55,750 Is the identity operation present? 364 00:27:55,750 --> 00:28:02,170 Yes because minus 1 times plus 1 or plus 1 times minus 1 365 00:28:02,170 --> 00:28:04,080 gives you the same element back again. 366 00:28:04,080 --> 00:28:08,570 So for every element in the group, the combination of any 367 00:28:08,570 --> 00:28:11,620 two elements is present. 368 00:28:11,620 --> 00:28:17,240 For each operation, we've shown that an inverse exists 369 00:28:17,240 --> 00:28:21,840 and for every element there's an identity operation, and the 370 00:28:21,840 --> 00:28:22,940 inverse exists. 371 00:28:22,940 --> 00:28:26,160 So the pair of numbers, not terribly exciting, 1 and minus 372 00:28:26,160 --> 00:28:32,380 1 where the law of combination is defined is a 373 00:28:32,380 --> 00:28:34,930 group of rank 2. 374 00:28:41,090 --> 00:28:43,100 I'll give you another example. 375 00:28:43,100 --> 00:28:46,180 I won't bother to carry out all the terms, but the numbers 376 00:28:46,180 --> 00:28:53,220 1, minus 1, i, and minus i, where i is equal to the square 377 00:28:53,220 --> 00:28:57,690 root of minus 1, and where the law of combination is defined 378 00:28:57,690 --> 00:29:03,020 as multiplication, constitute a group of rank 4. 379 00:29:14,380 --> 00:29:21,320 With this simple introduction, I think one can see that 380 00:29:21,320 --> 00:29:26,380 symmetry transformations can be regarded as operations that 381 00:29:26,380 --> 00:29:28,690 are elements in a group. 382 00:29:28,690 --> 00:29:34,870 And let me give you a simple example. 383 00:29:34,870 --> 00:29:36,590 Let's look at a combination. 384 00:29:36,590 --> 00:29:40,790 We still haven't considered the mind-boggling possibility 385 00:29:40,790 --> 00:29:43,890 of having more than one symmetry element present in a 386 00:29:43,890 --> 00:29:45,730 space at the same time. 387 00:29:45,730 --> 00:29:48,210 But let's suppose we have two mirror planes that are 388 00:29:48,210 --> 00:29:52,850 completely orthogonal, and see what sort of pattern they 389 00:29:52,850 --> 00:29:55,290 would generate. 390 00:29:55,290 --> 00:29:57,560 Here's a first motif. 391 00:29:57,560 --> 00:30:03,300 If I call this mirror plane m1, and this mirror plane m2. 392 00:30:03,300 --> 00:30:07,950 m1 is going to take this object and reflect 393 00:30:07,950 --> 00:30:09,380 it across to here. 394 00:30:09,380 --> 00:30:15,174 m2 will take this object and reflect it down to here. 395 00:30:15,174 --> 00:30:17,720 Take this one, and reflect it down to here. 396 00:30:17,720 --> 00:30:20,440 And now I know how this is related to this, how this is 397 00:30:20,440 --> 00:30:23,310 related to this, how this is related to this. 398 00:30:23,310 --> 00:30:28,410 But this one and this one are exactly the same thing. 399 00:30:28,410 --> 00:30:35,360 And those two objects are not related by reflection. 400 00:30:35,360 --> 00:30:40,040 If this is a right-handed one, this is a left-handed one, 401 00:30:40,040 --> 00:30:43,940 this is a left-handed one, and this is a right-handed one. 402 00:30:43,940 --> 00:30:47,830 So if something relates these two, it has to be something 403 00:30:47,830 --> 00:30:50,780 that does not produce an enantiomorph. 404 00:30:50,780 --> 00:30:54,760 The only thing that's possible then would be translation. 405 00:30:54,760 --> 00:30:57,080 And these two guys are not parallel to one another. 406 00:30:57,080 --> 00:30:59,060 You can't get this one by just sliding 407 00:30:59,060 --> 00:31:00,420 this parallel to itself. 408 00:31:00,420 --> 00:31:02,750 The only thing that's left is rotation. 409 00:31:02,750 --> 00:31:08,870 And, lo and behold, we can get from this one to this one by a 410 00:31:08,870 --> 00:31:10,530 180-degree rotation. 411 00:31:10,530 --> 00:31:12,695 And we can get from this one to this one by 412 00:31:12,695 --> 00:31:15,850 a 180-degree rotation. 413 00:31:15,850 --> 00:31:18,860 So here's an example of a way in which we've combined 414 00:31:18,860 --> 00:31:22,920 without showing why this is possible, or how I got there. 415 00:31:22,920 --> 00:31:25,590 Here's a combination of symmetry elements that gives 416 00:31:25,590 --> 00:31:28,025 us a possible arrangement of objects in space. 417 00:31:30,760 --> 00:31:35,890 It's going to be convenient to have a notation to indicate 418 00:31:35,890 --> 00:31:38,150 such combinations. 419 00:31:38,150 --> 00:31:43,260 And we call them Harry, George, and Sam, or some 420 00:31:43,260 --> 00:31:44,690 popular, affectionate name. 421 00:31:44,690 --> 00:31:47,330 Or call them number 1, number 2, number 3. 422 00:31:47,330 --> 00:31:50,730 But a nice notation is going to be a descriptive one which 423 00:31:50,730 --> 00:31:52,280 tells you what you have. 424 00:31:52,280 --> 00:31:56,510 And what is done in crystallography is to indicate 425 00:31:56,510 --> 00:32:00,470 these possible combinations by a running list of the 426 00:32:00,470 --> 00:32:03,100 different kinds of symmetry elements that are present. 427 00:32:03,100 --> 00:32:05,875 And here, we've got a twofold axis, for which the symbol is 428 00:32:05,875 --> 00:32:09,680 2, and two mirror planes that are different and function in 429 00:32:09,680 --> 00:32:11,140 different ways in the pattern. 430 00:32:11,140 --> 00:32:12,940 So this combination is called 2mm. 431 00:32:16,520 --> 00:32:19,610 OK, that's getting ahead of our story. 432 00:32:19,610 --> 00:32:25,140 But what I wanted to do now is to show you that this set of 433 00:32:25,140 --> 00:32:28,790 symmetry elements contains four operations that 434 00:32:28,790 --> 00:32:30,700 constitute a group. 435 00:32:30,700 --> 00:32:33,710 There's the operation of a one-fold axis. 436 00:32:33,710 --> 00:32:35,980 That's the identity operation. 437 00:32:35,980 --> 00:32:38,670 There is a reflection across the first 438 00:32:38,670 --> 00:32:40,370 type of mirror plane. 439 00:32:40,370 --> 00:32:44,170 And notice, now, the utility of having a symbol that 440 00:32:44,170 --> 00:32:46,500 indicates a specific operation rather 441 00:32:46,500 --> 00:32:48,280 than a symmetry element. 442 00:32:48,280 --> 00:32:51,970 There's a second reflection operation, sigma 2, and there 443 00:32:51,970 --> 00:32:55,860 is a rotation operation through a 180 degrees, A pi. 444 00:32:58,520 --> 00:33:03,340 So I'd like to show that these four operations constitute a 445 00:33:03,340 --> 00:33:05,210 group of rank 4. 446 00:33:05,210 --> 00:33:06,820 So how do we show this? 447 00:33:06,820 --> 00:33:09,340 We first set up the group multiplication table. 448 00:33:16,810 --> 00:33:21,590 And then we simply combine these 449 00:33:21,590 --> 00:33:24,640 four operations pairwise. 450 00:33:24,640 --> 00:33:29,440 Put the same 1, sigma 1, sigma 2, and A pi down 451 00:33:29,440 --> 00:33:32,110 this side of the array. 452 00:33:34,800 --> 00:33:38,470 Doing nothing, doing nothing, is the same as a one-fold axis 453 00:33:38,470 --> 00:33:41,260 doing nothing. 454 00:33:41,260 --> 00:33:44,170 Doing a reflection sigma 1 followed by doing nothing 455 00:33:44,170 --> 00:33:46,230 gives you sigma 1 back again. 456 00:33:46,230 --> 00:33:49,420 Doing a reflection sigma 2 followed by doing nothing 457 00:33:49,420 --> 00:33:51,350 gives you sigma 2 back again. 458 00:33:51,350 --> 00:33:54,210 And similarly, rotating A pi followed by doing 459 00:33:54,210 --> 00:33:55,730 nothing is the same. 460 00:33:55,730 --> 00:33:58,060 If I go along the columns, doing nothing followed by 461 00:33:58,060 --> 00:34:02,300 sigma 1 is sigma 1. 462 00:34:02,300 --> 00:34:05,850 Sigma 1 and 1 combined is sigma 2. 463 00:34:05,850 --> 00:34:10,060 A pi combined with sigma 1 is A pi. 464 00:34:10,060 --> 00:34:13,739 So far so good, but not surprising. 465 00:34:13,739 --> 00:34:19,159 Doing sigma 1 and following it by sigma 1 is reflecting left 466 00:34:19,159 --> 00:34:23,290 to right across mirror plane 1, and then reflecting right 467 00:34:23,290 --> 00:34:24,409 back again. 468 00:34:24,409 --> 00:34:29,870 So that is going to be a one-fold axis doing nothing. 469 00:34:29,870 --> 00:34:34,719 Doing sigma 1 following up by sigma 2 would reflect across, 470 00:34:34,719 --> 00:34:37,080 and then follow up by reflection in the second 471 00:34:37,080 --> 00:34:38,270 mirror plane. 472 00:34:38,270 --> 00:34:41,040 I get from the first one to the final one by 473 00:34:41,040 --> 00:34:43,690 the rotation A pi. 474 00:34:43,690 --> 00:34:47,060 So this is A pi. 475 00:34:47,060 --> 00:34:50,420 And if I do the first reflection, sigma 1, and 476 00:34:50,420 --> 00:34:58,270 follow that by A pi, that gets me from number 1 to number 2 477 00:34:58,270 --> 00:34:59,120 to number 3. 478 00:34:59,120 --> 00:35:01,930 And I get from number 1 to number 3 directly by the 479 00:35:01,930 --> 00:35:05,430 reflection operation, sigma 2. 480 00:35:05,430 --> 00:35:06,550 OK, you notice what's happening? 481 00:35:06,550 --> 00:35:09,050 I get the same four elements back again in 482 00:35:09,050 --> 00:35:12,020 every column or row. 483 00:35:12,020 --> 00:35:14,360 Sigma 1, 1, A pi, sigma 2. 484 00:35:14,360 --> 00:35:17,550 And if I rattle these off quickly, this as A pi. 485 00:35:17,550 --> 00:35:22,290 This is identity, and this is sigma 1. 486 00:35:22,290 --> 00:35:25,410 And this will be sigma 2. 487 00:35:25,410 --> 00:35:28,480 This will be sigma 1. 488 00:35:28,480 --> 00:35:31,610 And this will be the identity operation. 489 00:35:31,610 --> 00:35:35,470 So the first group postulate is satisfied. 490 00:35:35,470 --> 00:35:38,610 These four elements combined pairwise always give you 491 00:35:38,610 --> 00:35:43,285 nothing but one of these elements back. 492 00:35:43,285 --> 00:35:46,600 An identity operation is present because the one-fold 493 00:35:46,600 --> 00:35:50,320 rotation axis is the same as leaving the thing alone. 494 00:35:50,320 --> 00:35:54,540 And we've seen that for every operation, an inverse exists 495 00:35:54,540 --> 00:35:58,040 because 1 occurs once in each row and column. 496 00:35:58,040 --> 00:36:05,230 So sigma 2 is its own inverse, and for sigma 1, sigma 1 is 497 00:36:05,230 --> 00:36:06,140 its own inverse. 498 00:36:06,140 --> 00:36:10,260 For A pi, A pi is its own inverse. 499 00:36:10,260 --> 00:36:11,510 So an inverse exists. 500 00:36:15,400 --> 00:36:17,590 So we've got a group. 501 00:36:17,590 --> 00:36:23,350 And this is another way of saying, in general terms, if 502 00:36:23,350 --> 00:36:27,600 we put two mirror planes and a twofold axis together in this 503 00:36:27,600 --> 00:36:32,050 specific fashion, these operations, when they go to 504 00:36:32,050 --> 00:36:37,280 work on an initial motif, reproduce a finite set of 505 00:36:37,280 --> 00:36:40,990 objects and not a clutter that just fills space and never 506 00:36:40,990 --> 00:36:42,510 closes upon itself. 507 00:36:42,510 --> 00:36:43,395 Yes, sir? 508 00:36:43,395 --> 00:36:45,335 AUDIENCE: I see why you have sigma 1, sigma 2, A pi in that 509 00:36:45,335 --> 00:36:48,460 group, but why do you have the 1, which is a conversion? 510 00:36:48,460 --> 00:36:48,760 PROFESSOR: OK. 511 00:36:48,760 --> 00:36:51,620 If I didn't put it in-- no, that's the identity operation. 512 00:36:51,620 --> 00:36:52,900 This is a one-fold axis. 513 00:36:55,910 --> 00:36:58,050 Just this 2 was a twofold axis. 514 00:36:58,050 --> 00:37:01,670 So one-fold axis is a nice symbol for identity because it 515 00:37:01,670 --> 00:37:02,500 doesn't do anything. 516 00:37:02,500 --> 00:37:05,020 It picks something up and puts it right back down 517 00:37:05,020 --> 00:37:05,820 where it came from. 518 00:37:05,820 --> 00:37:08,120 OK. 519 00:37:08,120 --> 00:37:08,560 Yeah. 520 00:37:08,560 --> 00:37:11,240 That's deceptive because 1, when we're talking about 521 00:37:11,240 --> 00:37:14,010 multiplication of numbers, also functions as 522 00:37:14,010 --> 00:37:16,506 the identity operation. 523 00:37:16,506 --> 00:37:16,988 Yeah? 524 00:37:16,988 --> 00:37:19,880 AUDIENCE: If all these symmetries are commutative, 525 00:37:19,880 --> 00:37:21,808 [? that is, ?] 526 00:37:21,808 --> 00:37:24,459 if they all commute, do you really [INAUDIBLE] bottom 527 00:37:24,459 --> 00:37:25,182 diagonal [INAUDIBLE]? 528 00:37:25,182 --> 00:37:26,146 PROFESSOR: In general, yes. 529 00:37:26,146 --> 00:37:27,592 AUDIENCE: Generally? 530 00:37:27,592 --> 00:37:28,080 PROFESSOR: Yeah. 531 00:37:28,080 --> 00:37:32,450 Actually, this is a special group because if I have two 532 00:37:32,450 --> 00:37:38,530 elements, a, and take b and follow it by a, that is the 533 00:37:38,530 --> 00:37:42,700 same result as a followed by b. 534 00:37:42,700 --> 00:37:46,340 And a group that has this character 535 00:37:46,340 --> 00:37:47,590 is said to be abelian. 536 00:37:53,680 --> 00:37:57,590 There's a standing rotten joke in mathematics that asks 537 00:37:57,590 --> 00:38:02,700 rhetorically, what is purple and commutes? 538 00:38:02,700 --> 00:38:04,440 The answer is an abelian grape. 539 00:38:07,242 --> 00:38:09,690 I don't think it's terribly funny either, but if you had a 540 00:38:09,690 --> 00:38:12,990 gaggle of mathematicians here, they would chortle and double 541 00:38:12,990 --> 00:38:14,546 over in laughter. 542 00:38:14,546 --> 00:38:17,380 Ha, ha, ha, abelian grape. 543 00:38:17,380 --> 00:38:18,530 Well, I don't feel badly. 544 00:38:18,530 --> 00:38:20,200 You react about the same way to some of my 545 00:38:20,200 --> 00:38:21,310 funny things, too. 546 00:38:21,310 --> 00:38:23,360 But you get used to that sort of thing. 547 00:38:30,690 --> 00:38:31,010 All right. 548 00:38:31,010 --> 00:38:35,860 The feeling I want to leave you with at this point is the 549 00:38:35,860 --> 00:38:41,220 idea that group theory and some of its concepts are 550 00:38:41,220 --> 00:38:45,830 exactly what we're meaning when we grope for a definition 551 00:38:45,830 --> 00:38:49,040 of the fact that a certain collection of symmetry 552 00:38:49,040 --> 00:38:51,190 operation should be finite. 553 00:38:51,190 --> 00:38:52,940 That is to say, it's finite. 554 00:38:52,940 --> 00:38:56,450 The operations combined on each other have to give a 555 00:38:56,450 --> 00:39:01,250 finite number, must constitute a group. 556 00:39:01,250 --> 00:39:06,120 And the main reason for being familiar with this is that 557 00:39:06,120 --> 00:39:13,900 group theory forms the basis of a lot of the words that are 558 00:39:13,900 --> 00:39:15,080 used to describe the 559 00:39:15,080 --> 00:39:19,070 combinations of symmetry elements. 560 00:39:19,070 --> 00:39:26,120 For example, what we have derived here, the symmetry 561 00:39:26,120 --> 00:39:29,050 that's called 2mm, is something that's called a 562 00:39:29,050 --> 00:39:30,300 point group. 563 00:39:35,070 --> 00:39:38,330 Point because there is at least one point in space 564 00:39:38,330 --> 00:39:40,440 that's left unchanged, and that's this point of 565 00:39:40,440 --> 00:39:43,040 intersection of all three symmetry elements. 566 00:39:43,040 --> 00:39:45,380 And it's called a group because the individual 567 00:39:45,380 --> 00:39:49,610 operations of these symmetry elements, when combined 568 00:39:49,610 --> 00:39:51,520 according to group theory, can be shown to 569 00:39:51,520 --> 00:39:54,860 constitute a group. 570 00:39:54,860 --> 00:40:00,520 The patterns that we get when we add symmetry elements to a 571 00:40:00,520 --> 00:40:10,190 lattice are groups in the sense, if you regard the 572 00:40:10,190 --> 00:40:14,220 pattern as extending infinitely in all directions, 573 00:40:14,220 --> 00:40:16,750 that the combination of any two elements gives you 574 00:40:16,750 --> 00:40:18,700 something that's also in the group. 575 00:40:18,700 --> 00:40:22,740 But the number of elements is infinite. 576 00:40:22,740 --> 00:40:25,420 For example, in the lattice, here's T1. 577 00:40:25,420 --> 00:40:26,870 Here's T2. 578 00:40:26,870 --> 00:40:29,660 Every other lattice point can be described as some 579 00:40:29,660 --> 00:40:33,450 combination of T1 followed by T2. 580 00:40:33,450 --> 00:40:35,500 But the numbers in front of those 581 00:40:35,500 --> 00:40:37,690 translations can be infinite. 582 00:40:37,690 --> 00:40:42,660 If we put a symmetry element in this lattice, like a 583 00:40:42,660 --> 00:40:46,370 twofold axis, for example, the translations reproduce this 584 00:40:46,370 --> 00:40:49,120 twofold axis to an infinite number of locations. 585 00:40:49,120 --> 00:40:52,790 But yet, all of the group postulates are satisfied. 586 00:40:52,790 --> 00:40:56,210 The number of elements in the group does, however, not have 587 00:40:56,210 --> 00:40:57,560 to be finite. 588 00:40:57,560 --> 00:41:01,260 So that is another distinction that's worth making, that 589 00:41:01,260 --> 00:41:05,530 there are infinite groups and there are finite groups. 590 00:41:14,400 --> 00:41:16,630 And the number of elements can be infinite. 591 00:41:21,930 --> 00:41:25,840 Collection of symmetry elements that leave one point 592 00:41:25,840 --> 00:41:28,130 in space unmoved is called a point group. 593 00:41:28,130 --> 00:41:32,000 If we do something like this, put a twofold axis in 594 00:41:32,000 --> 00:41:36,310 combination with a pair of translations, that is an 595 00:41:36,310 --> 00:41:41,140 infinite set of operations that acts on all of space. 596 00:41:41,140 --> 00:41:43,610 This is something that's called a space group. 597 00:41:53,600 --> 00:41:56,390 Another designation, distinction, that's sometimes 598 00:41:56,390 --> 00:42:09,310 useful to make are crystallographic point groups, 599 00:42:09,310 --> 00:42:15,890 such as this one, 2mm, as opposed to combinations of 600 00:42:15,890 --> 00:42:18,950 symmetry elements, which are perfectly lovely and which are 601 00:42:18,950 --> 00:42:24,910 valid groups, such as a combination of rotations of 2 602 00:42:24,910 --> 00:42:29,590 pi over 5, or a combination of a fivefold axis 603 00:42:29,590 --> 00:42:31,500 with a mirror plane. 604 00:42:31,500 --> 00:42:34,920 Perfectly lovely symbols, constitute groups, but these 605 00:42:34,920 --> 00:42:37,180 are non-crystallographic 606 00:42:37,180 --> 00:42:37,790 These would be 607 00:42:37,790 --> 00:42:40,080 non-crystallographic point groups. 608 00:42:40,080 --> 00:42:43,140 Satisfy all the requirements of the group, but if it's to 609 00:42:43,140 --> 00:42:46,430 be in a crystal, the fivefold axis must be combined with 610 00:42:46,430 --> 00:42:49,840 translation, and that's impossible. 611 00:42:49,840 --> 00:42:50,480 So you can have 612 00:42:50,480 --> 00:42:52,370 non-crystallographic point groups. 613 00:42:52,370 --> 00:42:54,720 There are no non-crystallographic space 614 00:42:54,720 --> 00:42:57,150 groups because they are, by definition, something that 615 00:42:57,150 --> 00:42:59,775 involve translation yet constitute groups. 616 00:43:04,710 --> 00:43:06,190 All right. 617 00:43:06,190 --> 00:43:09,590 Set the stage for next time, so you all come back excited. 618 00:43:09,590 --> 00:43:18,800 We have shown that there are a limited number of symmetry 619 00:43:18,800 --> 00:43:24,472 elements that are possible in a lattice, rotation 1, 2, 3, 620 00:43:24,472 --> 00:43:28,300 4, and 6 in a mirror plane. 621 00:43:28,300 --> 00:43:31,790 And these are now examples of what we would call 622 00:43:31,790 --> 00:43:33,530 crystallographic point groups. 623 00:43:38,520 --> 00:43:43,870 These would be the sort of symmetry elements that are 624 00:43:43,870 --> 00:43:46,635 candidates for two-dimensional patterns. 625 00:43:58,310 --> 00:44:02,210 But are these the only symmetries that can be added 626 00:44:02,210 --> 00:44:04,510 to lattices? 627 00:44:04,510 --> 00:44:09,100 Could we not combine a mirror plane with a twofold axis? 628 00:44:09,100 --> 00:44:11,300 The answer I say to that is, sure. 629 00:44:11,300 --> 00:44:12,040 You bet you can. 630 00:44:12,040 --> 00:44:13,840 You just did it for us. 631 00:44:13,840 --> 00:44:17,120 But could you combine a mirror plane with a threefold axis, a 632 00:44:17,120 --> 00:44:19,450 mirror plane with a fourfold axis, a mirror plane with a 633 00:44:19,450 --> 00:44:20,670 sixfold axis? 634 00:44:20,670 --> 00:44:23,160 The answer to all of those questions are yes. 635 00:44:23,160 --> 00:44:25,700 So there are going to be spaces here, where there will 636 00:44:25,700 --> 00:44:29,270 be additional groups. 637 00:44:29,270 --> 00:44:33,030 And when we're done, we will have the two-dimensional 638 00:44:33,030 --> 00:44:35,340 crystallographic point groups. 639 00:44:35,340 --> 00:44:38,480 But now I have to be able to complete something. 640 00:44:38,480 --> 00:44:44,020 If I take A pi and combine it with a mirror operation sigma 641 00:44:44,020 --> 00:44:46,200 1, what do I get? 642 00:44:46,200 --> 00:44:51,520 If I take A 2 pi over 3, a threefold rotation, combine 643 00:44:51,520 --> 00:44:53,460 that with a mirror plane that passes through 644 00:44:53,460 --> 00:44:55,310 it, what do I get? 645 00:44:55,310 --> 00:44:58,890 Again, a characteristic of symmetry theory is that 646 00:44:58,890 --> 00:45:04,410 whenever I take a motif and repeat it by operation number 647 00:45:04,410 --> 00:45:10,780 1 to get a second motif, and then I repeat that by 648 00:45:10,780 --> 00:45:13,840 operation number 2, I have three 649 00:45:13,840 --> 00:45:15,980 things that are identical. 650 00:45:15,980 --> 00:45:19,940 And there must be some way, some operation, that 651 00:45:19,940 --> 00:45:23,320 automatically arises that tells me how number 1 is 652 00:45:23,320 --> 00:45:25,910 related to number 2. 653 00:45:25,910 --> 00:45:28,880 Number 1 is related to number 3. 654 00:45:28,880 --> 00:45:32,700 Tied in with group theory, this says that if you combine 655 00:45:32,700 --> 00:45:36,180 operation 1 in a space, if it's to be a group, if 656 00:45:36,180 --> 00:45:40,030 operation number 2 is present, you must combine those two 657 00:45:40,030 --> 00:45:44,390 steps, and whatever pops up must be a member of the set of 658 00:45:44,390 --> 00:45:46,600 operations that are present. 659 00:45:46,600 --> 00:45:51,000 So we're going to have to come up with something that I call 660 00:45:51,000 --> 00:45:52,250 combination theorems. 661 00:45:58,050 --> 00:46:05,650 And this is simply expressing the result of a combination of 662 00:46:05,650 --> 00:46:09,360 two elements in a group multiplication table. 663 00:46:09,360 --> 00:46:10,720 It goes with a caveat. 664 00:46:10,720 --> 00:46:12,560 If you're so excited about this stuff, you're going to 665 00:46:12,560 --> 00:46:14,760 run home and start reading Buerger's book as 666 00:46:14,760 --> 00:46:16,010 soon as you get there. 667 00:46:18,640 --> 00:46:23,490 You can regard these symmetry transformations as operators. 668 00:46:23,490 --> 00:46:27,300 And you're familiar with other sorts of operators like d by 669 00:46:27,300 --> 00:46:30,640 dx, d by dy. 670 00:46:30,640 --> 00:46:33,660 And when you apply them in succession, you actually write 671 00:46:33,660 --> 00:46:35,860 it as a product. 672 00:46:35,860 --> 00:46:42,900 d by dy followed by d by dx you write as d squared dxdy. 673 00:46:42,900 --> 00:46:45,750 Or you write it as d squared dx squared when you 674 00:46:45,750 --> 00:46:46,700 differentiate twice. 675 00:46:46,700 --> 00:46:49,330 You don't mean you take the differential of the function 676 00:46:49,330 --> 00:46:51,560 with respect to x and then square it. 677 00:46:51,560 --> 00:46:53,600 You mean you do this operation twice. 678 00:46:53,600 --> 00:46:55,180 So these are all operators. 679 00:46:59,150 --> 00:47:07,450 And what we usually understand is that if you do d by dy 680 00:47:07,450 --> 00:47:15,800 first and then follow that with d by dx, the sequence of 681 00:47:15,800 --> 00:47:17,755 operation goes from right to left. 682 00:47:25,810 --> 00:47:30,320 And I think that will come right quite naturally to you. 683 00:47:30,320 --> 00:47:32,820 You do that with differentials and other sort of operators. 684 00:47:32,820 --> 00:47:36,110 That is what we'll do. 685 00:47:36,110 --> 00:47:37,445 Not Martin Buerger. 686 00:47:37,445 --> 00:47:40,760 A very strong-minded person, and he did what 687 00:47:40,760 --> 00:47:42,460 he thought was sensible. 688 00:47:42,460 --> 00:47:47,490 And he said we read things from left to right. 689 00:47:47,490 --> 00:47:49,920 We read everything from left to right. 690 00:47:49,920 --> 00:47:54,400 So I will be damned if I'm going to write d by dx, d by 691 00:47:54,400 --> 00:47:57,390 dy to be a sequence of operations that goes from 692 00:47:57,390 --> 00:47:59,600 right to left. 693 00:47:59,600 --> 00:48:02,100 So all throughout Buerger's book-- and it's easy to get 694 00:48:02,100 --> 00:48:08,020 adjusted to, if he writes A dot B, he means B followed by 695 00:48:08,020 --> 00:48:19,840 A. Whereas normally, in the calculus of operations, we 696 00:48:19,840 --> 00:48:27,700 will use, as the rest of the world does, this is A followed 697 00:48:27,700 --> 00:48:38,290 by B. Trivial point, but one that can throw you for a loop 698 00:48:38,290 --> 00:48:40,100 the first time you see it in Buerger's book 699 00:48:40,100 --> 00:48:41,350 if you follow it. 700 00:48:48,870 --> 00:48:55,410 Distinction does not matter if the group is abelian. 701 00:48:55,410 --> 00:48:57,970 So that the order of operations doesn't make any 702 00:48:57,970 --> 00:48:59,895 difference. 703 00:48:59,895 --> 00:49:02,070 OK, that's enough for one day. 704 00:49:02,070 --> 00:49:04,870 Beginning next time, know we will start to begin this 705 00:49:04,870 --> 00:49:07,220 process of synthesis. 706 00:49:07,220 --> 00:49:09,530 And the first thing we'll ask is, how do you combine a 707 00:49:09,530 --> 00:49:13,390 rotation operation with a reflection operation? 708 00:49:13,390 --> 00:49:14,640 See you next week.