1 00:00:07,240 --> 00:00:09,750 PROFESSOR: That I know of that derives two-dimensional 2 00:00:09,750 --> 00:00:10,810 symmetries. 3 00:00:10,810 --> 00:00:15,340 And so, what I will give you instead is a set of notes-- 4 00:00:15,340 --> 00:00:17,140 nothing fancy, just handwritten-- 5 00:00:17,140 --> 00:00:22,680 but which will carry out each of the derivations in detail, 6 00:00:22,680 --> 00:00:25,500 so you'll have something to fall back on if it seems 7 00:00:25,500 --> 00:00:27,330 overly complicated. 8 00:00:27,330 --> 00:00:29,280 Any questions on what we've done to this point? 9 00:00:29,280 --> 00:00:33,740 We've really set forth the ground rules and components of 10 00:00:33,740 --> 00:00:35,970 our final step. 11 00:00:35,970 --> 00:00:41,700 And that, as I mentioned earlier, is to combine one or 12 00:00:41,700 --> 00:00:44,860 more of these collections of symmetry elements around a 13 00:00:44,860 --> 00:00:49,770 fixed point in space, with one of the five two-dimensional 14 00:00:49,770 --> 00:00:53,030 lattices that can accommodate them. 15 00:00:53,030 --> 00:01:00,400 And when we find a successful, feasible combination, we will 16 00:01:00,400 --> 00:01:08,060 have derived something that we could call the two-dimensional 17 00:01:08,060 --> 00:01:15,515 crystallographic space groups. 18 00:01:21,570 --> 00:01:24,120 These are collections of operations which really pick 19 00:01:24,120 --> 00:01:25,870 up everything-- the whole space-- 20 00:01:25,870 --> 00:01:28,810 move it, tumble it end over end, and throw it back down 21 00:01:28,810 --> 00:01:30,350 into coincidence with itself. 22 00:01:30,350 --> 00:01:33,450 So hence the term space group, rather than point group, where 23 00:01:33,450 --> 00:01:37,265 it's only a point that's left invariant at worst. 24 00:01:41,520 --> 00:01:44,590 OK, so what we're going to do then is to take each of the 25 00:01:44,590 --> 00:01:48,460 point groups in turn, and combine each of them, when 26 00:01:48,460 --> 00:01:51,370 possible, with one of the five lattice types. 27 00:01:51,370 --> 00:01:57,520 And I remind you that those are the oblique lattice or 28 00:01:57,520 --> 00:01:58,770 parallelogram lattice. 29 00:02:03,100 --> 00:02:10,740 And this was what was required by two and one, 30 00:02:10,740 --> 00:02:11,990 no symmetry at all. 31 00:02:14,270 --> 00:02:28,970 Then there was the primitive rectangular net, and the 32 00:02:28,970 --> 00:02:33,160 symmetry element that required that was a mirror plane, or we 33 00:02:33,160 --> 00:02:36,820 could have, depending on how the mirror plane was aligned 34 00:02:36,820 --> 00:02:40,670 relative to the translation, we could also have a lattice 35 00:02:40,670 --> 00:02:42,790 that was non-primitive, a double cell. 36 00:02:42,790 --> 00:02:45,300 And that was a centered rectangular net. 37 00:02:50,230 --> 00:02:52,900 And either of those was compatible 38 00:02:52,900 --> 00:02:54,590 with a mirror plane. 39 00:02:54,590 --> 00:02:56,400 Then we had a square cell-- 40 00:03:00,440 --> 00:03:03,770 square net or square cell-- and that was what was required 41 00:03:03,770 --> 00:03:05,440 by a 4-fold axis. 42 00:03:05,440 --> 00:03:12,320 And then the hexagonal net, and that could be required by 43 00:03:12,320 --> 00:03:16,360 either a 3-fold or a 6-fold. 44 00:03:16,360 --> 00:03:19,260 Well, if you add up the number of point groups that we've 45 00:03:19,260 --> 00:03:23,980 listed here, there are only six, but there are 10 46 00:03:23,980 --> 00:03:28,120 crystallographic point groups. 47 00:03:28,120 --> 00:03:31,850 So what about the ones that we have not yet considered adding 48 00:03:31,850 --> 00:03:37,676 to a lattice, and these are the nets of the form Cnv. 49 00:03:40,800 --> 00:03:45,130 So let's consider, in turn, each of those point groups and 50 00:03:45,130 --> 00:03:48,680 see what they would require of the lattice that could 51 00:03:48,680 --> 00:03:50,340 accommodate them. 52 00:03:50,340 --> 00:03:53,900 But let's do 2MM first. 53 00:03:53,900 --> 00:03:58,610 2MM, the 2 is compatible with an oblique net. 54 00:03:58,610 --> 00:04:01,990 The mirror plane wants either a rectangular net or a 55 00:04:01,990 --> 00:04:03,860 centered rectangular net. 56 00:04:03,860 --> 00:04:10,450 So all we can do is to add this to a rectangular or 57 00:04:10,450 --> 00:04:12,930 centered rectangular net. 58 00:04:12,930 --> 00:04:16,600 Let's put a 2-fold axis at the lattice point. 59 00:04:16,600 --> 00:04:23,120 The mirror plane has to be aligned along the edge of a 60 00:04:23,120 --> 00:04:24,100 rectangular net. 61 00:04:24,100 --> 00:04:26,710 If there are two of them, you just put one along each of the 62 00:04:26,710 --> 00:04:29,960 edges of the rectangular net. 63 00:04:29,960 --> 00:04:34,850 A mirror plane in a net could either go in like this, or 64 00:04:34,850 --> 00:04:38,490 could go, for the centered net, in like this. 65 00:04:38,490 --> 00:04:42,040 So what we're saying, 2-fold doesn't care, add a mirror 66 00:04:42,040 --> 00:04:44,420 plane, it means that the parallelogram 67 00:04:44,420 --> 00:04:46,270 has to become a rectangle. 68 00:04:46,270 --> 00:04:49,470 You can put the two mirror planes of 2MM in along the two 69 00:04:49,470 --> 00:04:53,120 edges of the rectangle, or do this equally well with a 70 00:04:53,120 --> 00:04:57,840 centered rectangular net, put the mirror plane in parallel 71 00:04:57,840 --> 00:05:06,410 to the edges, just like we did in CM, put the 2-fold axis in 72 00:05:06,410 --> 00:05:08,760 at the lattice points. 73 00:05:08,760 --> 00:05:11,430 So there's a lattice point at the corner, and also a lattice 74 00:05:11,430 --> 00:05:14,340 point in the middle of the cell. 75 00:05:14,340 --> 00:05:22,040 So for the rectangular net, we could put in either M or 2MM 76 00:05:22,040 --> 00:05:24,570 in both of these rectangular nets, the 77 00:05:24,570 --> 00:05:25,820 primitive and the centered. 78 00:05:28,270 --> 00:05:31,650 4-fold access requires that the net become still more 79 00:05:31,650 --> 00:05:35,454 specialized and have the shape of a square. 80 00:05:35,454 --> 00:05:35,922 AUDIENCE: Professor? 81 00:05:35,922 --> 00:05:37,800 PROFESSOR: Hmm? 82 00:05:37,800 --> 00:05:38,670 AUDIENCE: We can't see on this side. 83 00:05:38,670 --> 00:05:39,800 PROFESSOR: Oh, I'm sorry. 84 00:05:39,800 --> 00:05:41,790 I'll pace back and forth, which I usually do. 85 00:05:41,790 --> 00:05:46,380 Then everybody will get equal time. 86 00:05:46,380 --> 00:05:50,470 OK, so I'm glad you spoke up. 87 00:05:50,470 --> 00:05:56,270 4-fold axis requires a square net. 88 00:05:56,270 --> 00:06:02,470 There are two mirror planes, 45 degrees apart, and those 89 00:06:02,470 --> 00:06:04,610 mirror planes have to be along the edges of 90 00:06:04,610 --> 00:06:06,060 a rectangular net. 91 00:06:06,060 --> 00:06:10,260 Put one of them in this orientation, and that surely 92 00:06:10,260 --> 00:06:14,060 is along the edge of a rectangular net, along the 93 00:06:14,060 --> 00:06:16,800 edge of a net that's not only rectangular but square. 94 00:06:16,800 --> 00:06:19,334 But it has a higher symmetry, and that's fine. 95 00:06:19,334 --> 00:06:22,510 Looks as though this diagonal mirror plane might cause 96 00:06:22,510 --> 00:06:27,490 troubles, but let's note that if we look at the diagonal 97 00:06:27,490 --> 00:06:32,230 translations in this lattice, that they will define a 98 00:06:32,230 --> 00:06:35,940 centered rectangular net, so putting the other mirror plane 99 00:06:35,940 --> 00:06:38,100 in this way puts it along the edge of a 100 00:06:38,100 --> 00:06:39,550 centered rectangular net. 101 00:06:39,550 --> 00:06:42,200 So this mirror plane is perfectly happy as well. 102 00:06:42,200 --> 00:06:44,110 So in the square lattice we could pop either 103 00:06:44,110 --> 00:06:45,960 symmetry 4 or 4MM. 104 00:06:53,760 --> 00:06:57,080 3-fold is an interesting one. 105 00:06:57,080 --> 00:07:03,290 That's the odd symmetry that does strange things. 106 00:07:03,290 --> 00:07:08,810 Here is a hexagonal net, and that's what 107 00:07:08,810 --> 00:07:12,030 a 3-fold axis requires. 108 00:07:12,030 --> 00:07:15,600 Two translations, identical in magnitude, and in terms of 109 00:07:15,600 --> 00:07:19,195 what goes on along them, in exactly 120 degrees apart. 110 00:07:21,740 --> 00:07:28,300 Now, I don't readily see any rectangle along the edge of 111 00:07:28,300 --> 00:07:31,875 which I could align the mirror plane in 3M. 112 00:07:31,875 --> 00:07:35,830 But it's the oblique shape of this net that throws you off. 113 00:07:35,830 --> 00:07:40,130 Let's notice that if I draw several unit cells side by 114 00:07:40,130 --> 00:07:51,510 side, here is a double cell that has a rectangular shape. 115 00:07:51,510 --> 00:07:53,940 OK? 116 00:07:53,940 --> 00:07:59,590 Take 2T1 plus T2 as one edge of the cell, and leave the 117 00:07:59,590 --> 00:08:05,610 other one as T2, and we have defined a cell where that is 118 00:08:05,610 --> 00:08:08,680 identically a 90-degree angle. 119 00:08:08,680 --> 00:08:12,000 So one of the mirror planes of 3M could go in this 120 00:08:12,000 --> 00:08:18,440 orientation, and the other one would be 30 degrees away. 121 00:08:18,440 --> 00:08:21,570 So this would be the orientation of 122 00:08:21,570 --> 00:08:23,200 the two mirror planes. 123 00:08:23,200 --> 00:08:30,970 So we're putting in 3M with M perpendicular to the 124 00:08:30,970 --> 00:08:33,630 translations that define the primitive hexagonal 125 00:08:33,630 --> 00:08:36,270 cell, T1 and T2. 126 00:08:41,220 --> 00:08:44,540 But there's another way you can do it. 127 00:08:44,540 --> 00:08:47,810 Let me draw the same net, the same hexagonal net, and look 128 00:08:47,810 --> 00:08:49,861 at a different kind of double cell. 129 00:08:57,080 --> 00:09:02,590 OK, here's my hexagonal net, two translations, T1 and T2, 130 00:09:02,590 --> 00:09:07,730 with a 3-fold axis at the lattice point. 131 00:09:07,730 --> 00:09:13,880 And now I will again draw a couple of extra cells-- 132 00:09:18,670 --> 00:09:21,650 double T1, and let's double T2. 133 00:09:31,110 --> 00:09:35,420 And then let me point out that I can put in the mirror plane 134 00:09:35,420 --> 00:09:44,570 this way and along this way, and here is a double cell, and 135 00:09:44,570 --> 00:09:49,180 I can put the mirror plane along this 136 00:09:49,180 --> 00:09:51,370 direction and this direction. 137 00:09:55,700 --> 00:09:57,970 So this is crazy. 138 00:09:57,970 --> 00:10:04,510 This is the same lattice and the same point group, but 139 00:10:04,510 --> 00:10:08,120 there are two different ways I can align the point group in 140 00:10:08,120 --> 00:10:09,670 the lattice. 141 00:10:09,670 --> 00:10:13,320 I can put the mirror plane in such that it's perpendicular 142 00:10:13,320 --> 00:10:16,790 to a cell edge, or put the mirror plane in 30 degrees 143 00:10:16,790 --> 00:10:20,600 away from that orientation such that it is along the edge 144 00:10:20,600 --> 00:10:26,370 of a centric rectangular net. 145 00:10:26,370 --> 00:10:33,960 So I could add 3M as well with the mirror plane aligned 146 00:10:33,960 --> 00:10:38,800 parallel to the translations in the net. 147 00:10:38,800 --> 00:10:41,600 So lots of funny permutations here. 148 00:10:41,600 --> 00:10:45,350 Here I have one symmetry element, two different 149 00:10:45,350 --> 00:10:48,300 lattices that I can combine with it. 150 00:10:48,300 --> 00:10:51,920 Here I have one point group, 3M, but two different ways in 151 00:10:51,920 --> 00:10:54,990 which I can set it relative to the edges of the cell. 152 00:10:54,990 --> 00:10:55,670 Yes? 153 00:10:55,670 --> 00:10:57,446 AUDIENCE: Why are you putting two mirror planes in each of 154 00:10:57,446 --> 00:10:58,514 those figures when you really only could 155 00:10:58,514 --> 00:11:00,550 do one mirror plane? 156 00:11:00,550 --> 00:11:02,640 PROFESSOR: Actually, I need-- 157 00:11:02,640 --> 00:11:05,990 well, there are going to be a radiating sheaf of mirror 158 00:11:05,990 --> 00:11:08,040 lines coming out of that 3-fold axis. 159 00:11:08,040 --> 00:11:09,310 AUDIENCE: An angle of 60 degrees. 160 00:11:09,310 --> 00:11:10,960 PROFESSOR: They are at an angle of 60 degrees. 161 00:11:10,960 --> 00:11:12,710 It has to be 60 degrees, and that is 162 00:11:12,710 --> 00:11:15,000 indeed what I have here. 163 00:11:15,000 --> 00:11:18,780 I've just drawn part of 3M, and if I draw the remaining 164 00:11:18,780 --> 00:11:20,530 mirror planes, they would do this. 165 00:11:32,266 --> 00:11:34,710 OK, I think that's fairly intelligible. 166 00:11:34,710 --> 00:11:38,800 Another reason for notes is to draw these diagrams when 167 00:11:38,800 --> 00:11:41,900 they're reasonably complex. 168 00:11:41,900 --> 00:11:46,510 On a blackboard, when you're rushing to get through it, is 169 00:11:46,510 --> 00:11:49,030 something that's easier said than done, even though you may 170 00:11:49,030 --> 00:11:52,380 have done it a number of times. 171 00:11:52,380 --> 00:11:56,950 So there is the job that we've laid out for ourselves. 172 00:11:56,950 --> 00:12:02,000 Five lattices, 10 point groups, and we can drop each 173 00:12:02,000 --> 00:12:07,380 of the point groups into one or more of the lattices of the 174 00:12:07,380 --> 00:12:08,630 five that we have determined. 175 00:12:11,620 --> 00:12:16,830 As soon as we combine things, we're going to have the 176 00:12:16,830 --> 00:12:19,290 situation that we've encountered before. 177 00:12:19,290 --> 00:12:22,950 In general, if I have an operation number one that 178 00:12:22,950 --> 00:12:26,800 produces one transformation of coordinates, and combine it 179 00:12:26,800 --> 00:12:32,370 with an operation number two which has another change of 180 00:12:32,370 --> 00:12:34,640 coordinates, there must automatically 181 00:12:34,640 --> 00:12:37,340 arise some third operation. 182 00:12:37,340 --> 00:12:41,040 And we can write it in terms of the language of operators 183 00:12:41,040 --> 00:12:45,230 by saying operation number one followed by operation number 184 00:12:45,230 --> 00:12:48,840 two is equivalent to an operation number three. 185 00:12:48,840 --> 00:12:51,970 And in terms of the pattern, this says that whenever you 186 00:12:51,970 --> 00:12:54,250 throw two symmetry transformations together, a 187 00:12:54,250 --> 00:12:56,500 third one pops up. 188 00:12:56,500 --> 00:12:57,490 You can't push it down. 189 00:12:57,490 --> 00:13:00,350 It's going to be there once you add the first two 190 00:13:00,350 --> 00:13:02,510 combinations. 191 00:13:02,510 --> 00:13:05,080 You may have to be really clever to see where it is and 192 00:13:05,080 --> 00:13:07,060 what it is, but it's going to be there. 193 00:13:07,060 --> 00:13:08,020 It's going to be there. 194 00:13:08,020 --> 00:13:11,390 It comes in automatically. 195 00:13:11,390 --> 00:13:15,680 So this general class of statement, this equality in 196 00:13:15,680 --> 00:13:18,710 operations, is something that I like to call, for short, a 197 00:13:18,710 --> 00:13:19,960 combination theorem. 198 00:13:22,840 --> 00:13:25,140 And we've seen several examples of this already, 199 00:13:25,140 --> 00:13:28,390 notably the one in the last hour that said whenever you 200 00:13:28,390 --> 00:13:33,080 combine two mirror lines at an angle mu, you automatically 201 00:13:33,080 --> 00:13:37,640 create, like it or not, a rotation operation of a 2 mu. 202 00:13:40,570 --> 00:13:43,810 So as soon as we start making these combinations, we're 203 00:13:43,810 --> 00:13:47,610 going to find a new operation arising. 204 00:13:47,610 --> 00:13:50,560 So let's start with the simplest one. 205 00:13:50,560 --> 00:13:54,300 Add a 1-fold axis to a general oblique net. 206 00:13:54,300 --> 00:13:58,800 So there is T1, there is T2, we don't put anything in it, 207 00:13:58,800 --> 00:14:03,580 it's just a simple array of translations with lattice 208 00:14:03,580 --> 00:14:05,890 points that we can put at the terminal point of these 209 00:14:05,890 --> 00:14:07,160 translations. 210 00:14:07,160 --> 00:14:08,470 There's no symmetry. 211 00:14:08,470 --> 00:14:11,860 If we put in one atom, that atom hangs on every lattice 212 00:14:11,860 --> 00:14:17,035 point, and that's all there is to it. 213 00:14:17,035 --> 00:14:20,960 It's so simple it's not only uninteresting, it's ugly. 214 00:14:20,960 --> 00:14:25,740 It's just a simple atom arranged in a periodic array. 215 00:14:25,740 --> 00:14:29,000 As with everything we've done so far, it's convenient to 216 00:14:29,000 --> 00:14:36,320 have a notation to indicate which particular plane group 217 00:14:36,320 --> 00:14:43,370 one has, and the symbol here is to give side by side the 218 00:14:43,370 --> 00:14:49,630 type of lattice as the first part of the symbol, and then 219 00:14:49,630 --> 00:14:51,793 the point group that you have added. 220 00:14:57,710 --> 00:14:58,830 So what do we have here? 221 00:14:58,830 --> 00:15:03,980 We've taken a primitive oblique lattice, and we've 222 00:15:03,980 --> 00:15:07,380 added no symmetry at all, a 1-fold axis. 223 00:15:07,380 --> 00:15:12,390 And here people who have derived symmetry theory assume 224 00:15:12,390 --> 00:15:14,090 that the reader knows something. 225 00:15:14,090 --> 00:15:16,990 And so you're going to get a lot of respect when you show 226 00:15:16,990 --> 00:15:21,250 your colleagues that you know what this notation means. 227 00:15:21,250 --> 00:15:24,560 All you have to do, if you know the symmetry, is to state 228 00:15:24,560 --> 00:15:27,760 whether the lattice is primitive, as it has to be for 229 00:15:27,760 --> 00:15:30,880 an oblique net, or whether it's centered, in the case of 230 00:15:30,880 --> 00:15:33,150 the rectangular net. 231 00:15:33,150 --> 00:15:35,020 Three dimensions is a lot more complicated. 232 00:15:35,020 --> 00:15:37,320 If the lattice is cubic, it could be primitive, it could 233 00:15:37,320 --> 00:15:39,760 be face-centered, it could be body-centered. 234 00:15:39,760 --> 00:15:42,820 So you need a symbol that tells the type of lattice. 235 00:15:42,820 --> 00:15:46,010 In this case it's primitive, and the point group that we've 236 00:15:46,010 --> 00:15:49,820 added is 1, so this rather boring ugly thing 237 00:15:49,820 --> 00:15:51,070 here is called P1. 238 00:15:54,120 --> 00:15:57,150 Another bit of code. 239 00:15:57,150 --> 00:15:59,190 We're going to do eventually the same thing in three 240 00:15:59,190 --> 00:15:59,980 dimensions. 241 00:15:59,980 --> 00:16:02,020 We're going to take just a general oblique 242 00:16:02,020 --> 00:16:05,000 three-dimensional lattice, and not adorn it with any sort of 243 00:16:05,000 --> 00:16:09,200 symmetry at all, and that will also be called P1. 244 00:16:09,200 --> 00:16:14,470 But to distinguish the two, capital P1 means a 245 00:16:14,470 --> 00:16:18,640 three-dimensional lattice, and we won't be working with those 246 00:16:18,640 --> 00:16:19,120 for a while. 247 00:16:19,120 --> 00:16:21,360 We haven't even enumerated them. 248 00:16:21,360 --> 00:16:28,450 A lowercase p, that is the symbol for a 249 00:16:28,450 --> 00:16:32,320 two-dimensional net. 250 00:16:32,320 --> 00:16:35,200 In other words, a plane group rather than a space group. 251 00:16:35,200 --> 00:16:42,980 So here's the first, P1, and it really doesn't give shivers 252 00:16:42,980 --> 00:16:46,060 running up and down my spine to look at it. 253 00:16:46,060 --> 00:16:48,850 So let's do one that's a little more interesting. 254 00:16:48,850 --> 00:16:54,830 Let's take a 2-fold axis and combine it first with a 255 00:16:54,830 --> 00:16:56,545 primitive rectangular net. 256 00:17:05,290 --> 00:17:06,190 I'm sorry, what am I saying? 257 00:17:06,190 --> 00:17:07,440 I want an oblique net. 258 00:17:14,760 --> 00:17:16,359 So the symbol-- 259 00:17:16,359 --> 00:17:18,960 and I'm assuming this is going to be unique, and we haven't 260 00:17:18,960 --> 00:17:22,060 seen anything with a 2-fold axis in it, so it is. 261 00:17:22,060 --> 00:17:26,280 So this would be P for a primitive lattice, 2 for the 262 00:17:26,280 --> 00:17:28,790 symmetry that you've added to the lattice. 263 00:17:28,790 --> 00:17:31,140 And since you are all cognoscenti, I don't have to 264 00:17:31,140 --> 00:17:34,050 tell you it's an oblique lattice, because you know, 265 00:17:34,050 --> 00:17:37,030 don't you, deep down in your heart, that a 2-fold axis 266 00:17:37,030 --> 00:17:40,460 requires only that the lattice be an oblique lattice. 267 00:17:40,460 --> 00:17:41,710 Nothing special is required. 268 00:17:44,460 --> 00:17:50,090 So what we will do is to take this general oblique lattice 269 00:17:50,090 --> 00:17:54,910 with two translations, T1 and T2, and to this-- remember, we 270 00:17:54,910 --> 00:17:57,040 don't deal with symmetry elements. 271 00:17:57,040 --> 00:18:00,300 We have to do this operation by operation. 272 00:18:00,300 --> 00:18:03,780 So what we're doing is taking the operation A pi, the only 273 00:18:03,780 --> 00:18:07,160 non-trivial operation contained in a 2-fold axis, 274 00:18:07,160 --> 00:18:11,090 and putting that in at a lattice point. 275 00:18:11,090 --> 00:18:13,550 OK. 276 00:18:13,550 --> 00:18:16,190 Bells ring and whistles go off. 277 00:18:16,190 --> 00:18:20,350 We don't know what happens when we make this combination. 278 00:18:20,350 --> 00:18:23,840 And let me do this in more general terms, so we can 279 00:18:23,840 --> 00:18:25,730 derive the theorem once and for all. 280 00:18:25,730 --> 00:18:30,960 Here's a translation, and let me add to the end of that 281 00:18:30,960 --> 00:18:34,730 translation a rotation operation A alpha. 282 00:18:34,730 --> 00:18:36,730 In this case, it would be the operation A pi. 283 00:18:41,760 --> 00:18:47,450 What's the net result of rotating and then translating? 284 00:18:47,450 --> 00:18:50,100 That's a non-trivial question. 285 00:18:50,100 --> 00:18:55,680 So if I'm doing this for a general rotation of alpha, let 286 00:18:55,680 --> 00:19:11,790 me take a translation, and I'm going to make my construction 287 00:19:11,790 --> 00:19:15,090 in the following way, because I'm really clever. 288 00:19:15,090 --> 00:19:20,680 And I'm going to put this translation T at one half of 289 00:19:20,680 --> 00:19:25,180 alpha on one side of the perpendicular to T. 290 00:19:25,180 --> 00:19:29,650 And then I'll let that rotation go to work. 291 00:19:29,650 --> 00:19:35,180 And it'll take the translation and move it to alpha over 2 on 292 00:19:35,180 --> 00:19:36,885 the other side of the perpendicular. 293 00:19:40,130 --> 00:19:44,330 And now I'll let the translation move to this 294 00:19:44,330 --> 00:19:49,330 rotated translation, and if it does so, it's going to move 295 00:19:49,330 --> 00:19:50,970 the translation over to here. 296 00:19:54,450 --> 00:19:57,480 Here's a motif hanging on this translation. 297 00:19:57,480 --> 00:20:04,020 It's rotated over to here, and then gets slid over to here. 298 00:20:04,020 --> 00:20:06,970 This one is right-handed, this one is right-handed, this one 299 00:20:06,970 --> 00:20:08,470 is right-handed. 300 00:20:08,470 --> 00:20:11,890 How do I get from this one to this one? 301 00:20:11,890 --> 00:20:14,780 Only two ways, translation or rotation. 302 00:20:17,710 --> 00:20:22,960 If we rotate, about what point are we rotating? 303 00:20:22,960 --> 00:20:24,440 Now I'm going to use a definition 304 00:20:24,440 --> 00:20:25,530 which seemed trivial. 305 00:20:25,530 --> 00:20:30,480 I said, a symmetry element is the locus of points that is 306 00:20:30,480 --> 00:20:33,730 unmoved by the operation. 307 00:20:33,730 --> 00:20:37,530 And if we decided, because these two motifs are not 308 00:20:37,530 --> 00:20:40,720 parallel to one another and have the same corrality, that 309 00:20:40,720 --> 00:20:41,970 the rotation-- 310 00:20:43,780 --> 00:20:46,560 has to be a rotation that relates the two. 311 00:20:46,560 --> 00:20:48,650 When asked finally the question, what 312 00:20:48,650 --> 00:20:49,950 point is left unmoved? 313 00:20:52,620 --> 00:20:53,470 Ha. 314 00:20:53,470 --> 00:20:57,120 It's this point where these two translations come together 315 00:20:57,120 --> 00:20:58,700 at a single point. 316 00:20:58,700 --> 00:21:02,150 That's the point that's left unmoved by rotating from here 317 00:21:02,150 --> 00:21:06,350 to here and then sliding over by the translation T. 318 00:21:06,350 --> 00:21:10,760 And I can say exactly where that rotation is. 319 00:21:10,760 --> 00:21:15,050 It's going to be along the perpendicular bisector of my 320 00:21:15,050 --> 00:21:17,450 original translation. 321 00:21:17,450 --> 00:21:21,110 And where is it going to be located? 322 00:21:21,110 --> 00:21:27,980 This line makes an angle of alpha over 2 with respect to 323 00:21:27,980 --> 00:21:30,740 the perpendicular to T. So that angle is 324 00:21:30,740 --> 00:21:33,090 also alpha over 2. 325 00:21:33,090 --> 00:21:37,620 This line and this line both make an angle of alpha over 2 326 00:21:37,620 --> 00:21:42,210 relative to the normal to the translation, so this angle is 327 00:21:42,210 --> 00:21:44,480 also alpha over 2. 328 00:21:44,480 --> 00:21:48,550 And look how this has turned out. 329 00:21:48,550 --> 00:21:54,310 The combination of a rotation with a translation is another 330 00:21:54,310 --> 00:22:00,530 rotation, B, about a different point but through the same net 331 00:22:00,530 --> 00:22:05,070 amount alpha as the original translation. 332 00:22:05,070 --> 00:22:08,710 On top of that, really to nail this down, 333 00:22:08,710 --> 00:22:11,210 where should we look? 334 00:22:11,210 --> 00:22:23,130 This distance in here is T over 2, and the tangent of 335 00:22:23,130 --> 00:22:26,233 alpha is T over 2 over this distance x. 336 00:22:31,800 --> 00:22:38,900 And if I just solve for that distance, x is 337 00:22:38,900 --> 00:22:42,960 equal to T over 2-- 338 00:22:42,960 --> 00:22:44,370 I'm sorry. 339 00:22:44,370 --> 00:22:47,560 I got this garbled. 340 00:22:47,560 --> 00:22:57,280 T over 2 over x is equal to the tangent of alpha. 341 00:22:57,280 --> 00:22:58,550 That's what I wanted to do. 342 00:22:58,550 --> 00:23:03,350 So if I solve for x, that distance x is T over 2 over 343 00:23:03,350 --> 00:23:07,580 the tangent of alpha, and I can write that as T over 2 344 00:23:07,580 --> 00:23:10,840 times the cotangent of alpha over 2. 345 00:23:15,940 --> 00:23:18,000 So here it is, done properly at last. 346 00:23:18,000 --> 00:23:23,140 Go up a distance x along the perpendicular bisector of T, 347 00:23:23,140 --> 00:23:27,130 and you go up a distance T over 2 times the cotangent of 348 00:23:27,130 --> 00:23:28,330 alpha over 2. 349 00:23:28,330 --> 00:23:34,220 So this is a general result for any rotation operation. 350 00:23:34,220 --> 00:23:41,500 Combine a rotation operation A alpha with a perpendicular 351 00:23:41,500 --> 00:23:42,130 translation-- 352 00:23:42,130 --> 00:23:45,000 perpendicular to the rotation axis A-- 353 00:23:45,000 --> 00:23:48,900 and the result is a new rotation operation through the 354 00:23:48,900 --> 00:23:52,810 same angle, but at a different location. 355 00:23:52,810 --> 00:23:58,330 It's at a distance x, which is equal to T over 2 times the 356 00:23:58,330 --> 00:24:02,360 cotangent of alpha over 2 along the perpendicular 357 00:24:02,360 --> 00:24:08,690 bisector of T. 358 00:24:08,690 --> 00:24:10,050 So this is a general theorem. 359 00:24:10,050 --> 00:24:13,530 This is a fact of two-dimensional life that is 360 00:24:13,530 --> 00:24:16,560 always going to be true regardless of alpha. 361 00:24:16,560 --> 00:24:20,960 And the only constraint is that the translation has to be 362 00:24:20,960 --> 00:24:26,690 added perpendicular to the locus of the rotation axis, 363 00:24:26,690 --> 00:24:29,360 and that has to be the case in two dimensions, because the 364 00:24:29,360 --> 00:24:33,950 rotation axis has to always be normal to the space, the 365 00:24:33,950 --> 00:24:34,940 two-dimensional space. 366 00:24:34,940 --> 00:24:38,190 In three dimensions, we'll have to generalize this to 367 00:24:38,190 --> 00:24:41,400 make the translation have arbitrary orientation relative 368 00:24:41,400 --> 00:24:45,140 to the rotation axis, but we've got enough to chew on at 369 00:24:45,140 --> 00:24:46,390 the moment. 370 00:24:51,670 --> 00:24:52,090 Yes. 371 00:24:52,090 --> 00:24:54,680 AUDIENCE: [INAUDIBLE] 372 00:24:54,680 --> 00:24:58,050 PROFESSOR: The distance x is the distance up along the 373 00:24:58,050 --> 00:24:59,200 perpendicular bisector. 374 00:24:59,200 --> 00:25:00,850 I'm sorry. 375 00:25:00,850 --> 00:25:03,076 I said it but I didn't write it in. 376 00:25:03,076 --> 00:25:04,890 OK? 377 00:25:04,890 --> 00:25:06,870 Let me go through it again since we've got 378 00:25:06,870 --> 00:25:07,630 everything on the board. 379 00:25:07,630 --> 00:25:11,470 We start with the original object, rotate it by alpha, 380 00:25:11,470 --> 00:25:15,070 and then we then further map it to a new location by the 381 00:25:15,070 --> 00:25:17,090 translation T. 382 00:25:17,090 --> 00:25:20,260 Both motifs are right-handed, but they're not parallel, so 383 00:25:20,260 --> 00:25:23,930 therefore, they have to be related by a rotation. 384 00:25:23,930 --> 00:25:28,290 If we ask what is the locus about which this location has 385 00:25:28,290 --> 00:25:31,900 occurred, we use the fact that the locus of a rotation axis 386 00:25:31,900 --> 00:25:34,330 is the only point that's left unmoved by the net 387 00:25:34,330 --> 00:25:38,210 transformation, and the place where this translation 388 00:25:38,210 --> 00:25:41,360 intersects the translation that has been shifted over by 389 00:25:41,360 --> 00:25:45,760 T is a point that sits up along the perpendicular 390 00:25:45,760 --> 00:25:50,520 bisector of the translation by a distance T over 2 times the 391 00:25:50,520 --> 00:25:53,440 cotangent of alpha over 2. 392 00:25:53,440 --> 00:25:54,960 So that's in the abstract. 393 00:25:54,960 --> 00:26:00,380 Let's now look at a particular application of this to our 394 00:26:00,380 --> 00:26:05,360 addition of a 2-fold rotation to a translation. 395 00:26:05,360 --> 00:26:11,900 So here's the translation T. We're going to put a 2-fold 396 00:26:11,900 --> 00:26:15,280 axis and the only operation which it 397 00:26:15,280 --> 00:26:18,660 possesses, namely A pi. 398 00:26:18,660 --> 00:26:22,800 And according to our theorem, up along the perpendicular 399 00:26:22,800 --> 00:26:29,290 bisector of T by distance x equals T over 2 times the 400 00:26:29,290 --> 00:26:32,700 cotangent of alpha over 2 should be a 401 00:26:32,700 --> 00:26:35,630 new operation B pi. 402 00:26:35,630 --> 00:26:40,280 So in this case, this would be T over 2 times the cotangent 403 00:26:40,280 --> 00:26:42,810 of pi over 2. 404 00:26:42,810 --> 00:26:48,560 Cotangent of 90 degrees is 0. 405 00:26:48,560 --> 00:26:51,910 So this says that you go 0 of the way along the 406 00:26:51,910 --> 00:26:56,920 perpendicular bisector, which means you stay right at the 407 00:26:56,920 --> 00:27:01,400 midpoint of T. So this says that if you rotate through 180 408 00:27:01,400 --> 00:27:05,230 degrees, follow that by this translation, you should find 409 00:27:05,230 --> 00:27:08,160 that the net effect is a rotation of pi over 2 about 410 00:27:08,160 --> 00:27:09,410 this point. 411 00:27:11,250 --> 00:27:14,750 Boggles the mind, but let's show that it works. 412 00:27:14,750 --> 00:27:20,350 Here is an initial point, initial motif number 1, and it 413 00:27:20,350 --> 00:27:22,110 is right-handed. 414 00:27:22,110 --> 00:27:25,320 We rotate by 180 degrees. 415 00:27:25,320 --> 00:27:30,050 Here's number 2, it stays right-handed, and I pick it up 416 00:27:30,050 --> 00:27:33,980 and I move it by this translation, and it moves from 417 00:27:33,980 --> 00:27:39,180 here to here. 418 00:27:39,180 --> 00:27:42,610 Here's number 3, it stayed right-handed, and the way I 419 00:27:42,610 --> 00:27:47,600 get from 1 to 3 in one shot is to rotate 180 degrees about 420 00:27:47,600 --> 00:27:48,850 the midpoint of the translation. 421 00:27:53,320 --> 00:27:54,570 How about that? 422 00:27:57,340 --> 00:28:00,300 People are so astounded that they're stunned, except for 423 00:28:00,300 --> 00:28:02,870 the people who couldn't see what I was doing, and they're 424 00:28:02,870 --> 00:28:04,570 moving back and forth frantically trying 425 00:28:04,570 --> 00:28:05,820 to see what I wrote. 426 00:28:13,306 --> 00:28:16,680 I'll let you catch your breath, and then let's do 427 00:28:16,680 --> 00:28:22,740 this, use this to derive the two-dimensional space group, 428 00:28:22,740 --> 00:28:26,930 the plane group that results when we add a 2-fold access to 429 00:28:26,930 --> 00:28:28,738 an oblique net. 430 00:28:28,738 --> 00:28:32,420 And it's turned out very simply, because if I've got 431 00:28:32,420 --> 00:28:35,900 the operation B pi here, that implies that I've got a 2-fold 432 00:28:35,900 --> 00:28:39,840 axis, because that's the only operation that exists within a 433 00:28:39,840 --> 00:28:41,040 2-fold axis. 434 00:28:41,040 --> 00:28:44,710 So I'll start by putting in a 2-fold axis here. 435 00:28:44,710 --> 00:28:48,840 I'll combine A pi with this translation T1, I get a new 436 00:28:48,840 --> 00:28:51,350 rotation operation, B pi, in the middle of this 437 00:28:51,350 --> 00:28:55,580 translation, so there's a new 2-fold axis there. 438 00:28:55,580 --> 00:28:59,200 The 2-fold axis here hanging at this lattice point as well, 439 00:28:59,200 --> 00:29:01,910 and at these other lattice points. 440 00:29:01,910 --> 00:29:06,270 Combine the rotation A pi with this translation T2, I get a 441 00:29:06,270 --> 00:29:09,980 new operation, B pi, sitting in the middle of this 442 00:29:09,980 --> 00:29:13,940 translation, so I've got all I need to say that there's a 443 00:29:13,940 --> 00:29:15,490 2-fold axis there. 444 00:29:15,490 --> 00:29:19,410 That 2-fold axis will get translated down here by T1. 445 00:29:19,410 --> 00:29:23,620 This 2-fold axis will be translated over to here by T2. 446 00:29:23,620 --> 00:29:26,810 Then the only other thing that I have to combine this 447 00:29:26,810 --> 00:29:32,990 translation with is the translation T1 plus T2, and 448 00:29:32,990 --> 00:29:36,540 that combined with A pi says I should have B pi in the middle 449 00:29:36,540 --> 00:29:39,290 of that translation as well, so I'll get another 2-fold 450 00:29:39,290 --> 00:29:41,610 axis in the middle of the cell. 451 00:29:41,610 --> 00:29:42,860 And that's it. 452 00:29:42,860 --> 00:29:47,790 We've got our first nontrivial two-dimensional space group. 453 00:29:47,790 --> 00:29:50,740 And what do we call this? 454 00:29:50,740 --> 00:29:53,540 Confusing as hell, some might say, but crystallographers 455 00:29:53,540 --> 00:29:55,980 would say that it's a primitive lattice combined 456 00:29:55,980 --> 00:29:59,710 with a 2-fold axis, this is P2. 457 00:29:59,710 --> 00:30:02,170 I don't have to tell you that it's an oblique net, because 458 00:30:02,170 --> 00:30:05,790 you know that that is all that a 2-fold axis requires, that 459 00:30:05,790 --> 00:30:07,040 the net be oblique. 460 00:30:09,440 --> 00:30:12,440 What does the pattern look like? 461 00:30:12,440 --> 00:30:14,930 I'll say another thing that's so simple, it takes a while 462 00:30:14,930 --> 00:30:15,800 for it to sink in. 463 00:30:15,800 --> 00:30:19,720 The pattern of a plane group is the pattern that is 464 00:30:19,720 --> 00:30:24,760 generated by the point group merely hung at every lattice 465 00:30:24,760 --> 00:30:26,870 point in the net. 466 00:30:26,870 --> 00:30:31,190 So if this is what a 2-fold axis does, relates a pair of 467 00:30:31,190 --> 00:30:37,510 motifs like this, a pattern for P2 is this pair hung at 468 00:30:37,510 --> 00:30:38,760 every lattice point. 469 00:30:44,130 --> 00:30:48,360 So all of the objects are of the same pirality, all either 470 00:30:48,360 --> 00:30:49,440 left-handed or right-handed. 471 00:30:49,440 --> 00:30:52,930 I've got the same pair at every corner of the cell. 472 00:30:52,930 --> 00:30:58,230 And notice that these new symmetry elements that popped 473 00:30:58,230 --> 00:31:02,900 up just like what we found in the point groups, the new 474 00:31:02,900 --> 00:31:09,150 mirror planes that came arose as a consequence of the 475 00:31:09,150 --> 00:31:11,390 combination that we had made. 476 00:31:11,390 --> 00:31:16,300 And they were independent in most cases, 3M being the one 477 00:31:16,300 --> 00:31:17,870 case where they weren't. 478 00:31:17,870 --> 00:31:21,610 These additional three 2-fold axes are 479 00:31:21,610 --> 00:31:23,520 independent 2-fold axes. 480 00:31:23,520 --> 00:31:26,820 They're different kinds of 2-fold axes than the ones at 481 00:31:26,820 --> 00:31:27,720 the corner. 482 00:31:27,720 --> 00:31:30,910 Not only are they not equivalent by translation, but 483 00:31:30,910 --> 00:31:33,782 they do different things relative to the motifs that 484 00:31:33,782 --> 00:31:34,890 are in the pattern. 485 00:31:34,890 --> 00:31:39,160 This 2-fold axis has a different relation relative to 486 00:31:39,160 --> 00:31:43,210 the two closest motifs than does this one, 487 00:31:43,210 --> 00:31:45,660 than does this one. 488 00:31:45,660 --> 00:31:50,440 They relate different objects pairwise in the pattern. 489 00:31:50,440 --> 00:31:57,810 And no two of these 2-fold axes describe the same 490 00:31:57,810 --> 00:31:59,470 pairwise relationship. 491 00:31:59,470 --> 00:32:02,720 So these are independent 2-fold axes. 492 00:32:16,380 --> 00:32:19,110 Independent in the sense that they do different things 493 00:32:19,110 --> 00:32:22,250 relative to the motifs in the pattern, different in the 494 00:32:22,250 --> 00:32:26,740 sense that there is no other operation that throws this one 495 00:32:26,740 --> 00:32:29,920 into this one, and therefore, of necessity, would require 496 00:32:29,920 --> 00:32:33,180 that the pair about this 2-fold axis be the same as the 497 00:32:33,180 --> 00:32:34,565 pair about this 2-fold axis. 498 00:32:47,175 --> 00:32:47,670 OK. 499 00:32:47,670 --> 00:32:55,230 So there is a first nontrivial two-dimensional space group. 500 00:32:55,230 --> 00:32:57,460 I should mention-- 501 00:32:57,460 --> 00:32:59,240 probably an obvious fact-- 502 00:32:59,240 --> 00:33:04,420 that this also has a cousin in three dimensions. 503 00:33:04,420 --> 00:33:08,440 If I just imagine all of these 2-fold axes extending out 504 00:33:08,440 --> 00:33:11,820 through space perpendicular to the plane of the blackboard, 505 00:33:11,820 --> 00:33:15,810 and take another translation that is also perpendicular to 506 00:33:15,810 --> 00:33:18,930 the blackboard, I would have the three-dimensional space 507 00:33:18,930 --> 00:33:22,830 group which has the same symbol except capital P, 508 00:33:22,830 --> 00:33:25,770 indicating that it's a space lattice. 509 00:33:25,770 --> 00:33:28,590 So almost at no extra charge, we've done all the work to 510 00:33:28,590 --> 00:33:30,440 determine one of the 511 00:33:30,440 --> 00:33:33,119 three-dimensional space groups. 512 00:33:33,119 --> 00:33:34,005 Yes? 513 00:33:34,005 --> 00:33:36,850 AUDIENCE: If you change the order of the combination of 514 00:33:36,850 --> 00:33:40,420 the translation rotation, do we get the same operation? 515 00:33:40,420 --> 00:33:40,900 PROFESSOR: OK. 516 00:33:40,900 --> 00:33:44,135 That is a good question. 517 00:33:46,865 --> 00:33:49,260 I really meant to mention it earlier in connection with the 518 00:33:49,260 --> 00:33:50,950 point groups. 519 00:33:50,950 --> 00:33:55,820 Does interchange of the order of the operations in a 520 00:33:55,820 --> 00:34:00,630 combination change the result, or change, in other words, the 521 00:34:00,630 --> 00:34:03,750 symmetry element that is equivalent to the net 522 00:34:03,750 --> 00:34:06,320 combination of the two? 523 00:34:06,320 --> 00:34:10,600 How many think yes, it does change? 524 00:34:10,600 --> 00:34:11,790 AUDIENCE: Could you say that again? 525 00:34:11,790 --> 00:34:15,070 PROFESSOR: OK, if we have two operations A and B, is 526 00:34:15,070 --> 00:34:19,840 operation A followed by B the same result as the operation B 527 00:34:19,840 --> 00:34:20,780 followed by A? 528 00:34:20,780 --> 00:34:23,370 Does the order make a difference? 529 00:34:23,370 --> 00:34:26,066 And how many say yes-- 530 00:34:26,066 --> 00:34:27,060 AUDIENCE: The order makes a difference? 531 00:34:27,060 --> 00:34:29,179 PROFESSOR: The order does make a difference. 532 00:34:29,179 --> 00:34:32,260 How many say no? 533 00:34:32,260 --> 00:34:34,739 Well, this is one of these happy occasions where I can 534 00:34:34,739 --> 00:34:37,719 say you're all right. 535 00:34:37,719 --> 00:34:40,949 Sometimes yes, sometimes no. 536 00:34:40,949 --> 00:34:43,560 So let's ask when does it make a difference and 537 00:34:43,560 --> 00:34:44,810 when does it not. 538 00:34:47,409 --> 00:34:50,989 Let's look at one of our point groups. 539 00:34:50,989 --> 00:34:55,420 Let me take a non-trivial point group. 540 00:34:55,420 --> 00:34:59,260 This would be point group 4MM. 541 00:34:59,260 --> 00:34:59,470 OK? 542 00:34:59,470 --> 00:35:02,780 So those are all mirror planes of two different kinds. 543 00:35:02,780 --> 00:35:06,050 And let me quickly sketch in the pattern. 544 00:35:06,050 --> 00:35:09,865 The pattern is just a pair of things hanging on the mirror 545 00:35:09,865 --> 00:35:13,805 plane, and that pair is rotated by 90 degrees. 546 00:35:18,094 --> 00:35:18,590 OK. 547 00:35:18,590 --> 00:35:22,710 Suppose we look at the operation A pi over 2, a 548 00:35:22,710 --> 00:35:27,140 90-degree rotation in a counterclockwise direction, 549 00:35:27,140 --> 00:35:30,090 and let the other operation be sigma 1, this 550 00:35:30,090 --> 00:35:31,670 reflection plane here. 551 00:35:31,670 --> 00:35:39,470 Suppose we reflect and follow that by A pi over 2. 552 00:35:39,470 --> 00:35:46,900 We reflect to here and then we rotate by 90 degrees, and 553 00:35:46,900 --> 00:35:49,870 that's going to give us the number 3 sitting up here. 554 00:35:49,870 --> 00:35:52,030 So we reflect it and then rotate it. 555 00:35:54,810 --> 00:35:59,240 How about if we first rotate that same object by A pi over 556 00:35:59,240 --> 00:36:03,660 2 and follow it by sigma 1. 557 00:36:03,660 --> 00:36:12,280 So we'll rotate number 1 to here, so this is 2 prime, and 558 00:36:12,280 --> 00:36:14,190 you can see we're already in trouble. 559 00:36:14,190 --> 00:36:17,920 If we now reflect along sigma 1, this is 3 prime. 560 00:36:17,920 --> 00:36:21,080 So in one case we ended up here, in the other case we 561 00:36:21,080 --> 00:36:22,560 ended up here. 562 00:36:22,560 --> 00:36:23,850 So the order did make a difference. 563 00:36:28,540 --> 00:36:30,970 Let us look at another example. 564 00:36:30,970 --> 00:36:32,470 Let's look at 2MM. 565 00:36:37,420 --> 00:36:41,010 So here's an operation sigma 1, here's an operation sigma 566 00:36:41,010 --> 00:36:46,950 2, so let's first do sigma 1 followed by A pi. 567 00:36:46,950 --> 00:36:50,980 So here's object 1, reflect to here, and then we rotate by 568 00:36:50,980 --> 00:36:54,320 180 degrees, and this is number 3. 569 00:36:57,050 --> 00:37:02,010 Now let's do the operation A pi and follow it by sigma 1. 570 00:37:02,010 --> 00:37:08,430 So we'll take number 1 and rotate it up to here. 571 00:37:08,430 --> 00:37:10,410 So here's number 1 prime. 572 00:37:10,410 --> 00:37:14,070 And then we reflect by sigma 1, and 3 and 3 prime are 573 00:37:14,070 --> 00:37:15,590 exactly the same. 574 00:37:15,590 --> 00:37:18,590 So it didn't make a difference. 575 00:37:18,590 --> 00:37:22,450 So the answer is sometimes yes, sometimes no, which is 576 00:37:22,450 --> 00:37:23,830 interesting, but even more interesting 577 00:37:23,830 --> 00:37:25,555 is how can you tell? 578 00:37:25,555 --> 00:37:26,545 Yeah, you have a question? 579 00:37:26,545 --> 00:37:28,525 AUDIENCE: In that first identity you drew-- 580 00:37:28,525 --> 00:37:29,020 PROFESSOR: Yep. 581 00:37:29,020 --> 00:37:34,217 AUDIENCE: The right side, it seems like your commas are 582 00:37:34,217 --> 00:37:36,940 drawn backwards and that just affects-- 583 00:37:36,940 --> 00:37:37,930 PROFESSOR: Could be. 584 00:37:37,930 --> 00:37:39,540 Mm, no, they're-- 585 00:37:39,540 --> 00:37:40,880 ah, these are drawn backwards. 586 00:37:40,880 --> 00:37:42,690 They should all have their tails pointing in. 587 00:37:42,690 --> 00:37:48,030 No, it doesn't work, actually, because even if I screwed up 588 00:37:48,030 --> 00:37:50,710 on the orientation, the positions are different. 589 00:37:50,710 --> 00:37:54,680 Again, first time I reflected, and then 590 00:37:54,680 --> 00:37:57,670 rotated, so this is 3. 591 00:37:57,670 --> 00:38:02,966 If I rotate and then reflect, I'm way over here. 592 00:38:02,966 --> 00:38:08,110 So this is one location, this is another location, and even 593 00:38:08,110 --> 00:38:10,140 if I screwed up by having the tails pointed the wrong 594 00:38:10,140 --> 00:38:14,290 orientation, still not the same. 595 00:38:14,290 --> 00:38:18,580 But here, clearly, reflecting and rotating is the same as 596 00:38:18,580 --> 00:38:19,645 rotating and reflecting. 597 00:38:19,645 --> 00:38:22,780 I end up in the same place. 598 00:38:22,780 --> 00:38:25,120 So how can you tell? 599 00:38:25,120 --> 00:38:28,010 It's a very difficult thing to prove that this is the case, 600 00:38:28,010 --> 00:38:33,540 but you can interchange the order of operations when the 601 00:38:33,540 --> 00:38:38,700 two symmetry operations that are involved leave each other 602 00:38:38,700 --> 00:38:39,950 untransformed. 603 00:38:41,820 --> 00:38:46,200 And if the two symmetry operations move one another, 604 00:38:46,200 --> 00:38:49,875 then changing the order changes the net result. 605 00:38:49,875 --> 00:38:52,690 It doesn't change the nature of the operation, but it does 606 00:38:52,690 --> 00:38:54,910 change the locus about which it operates. 607 00:38:54,910 --> 00:38:56,220 Let me show you what I mean. 608 00:38:56,220 --> 00:38:58,220 Here's 2MM. 609 00:38:58,220 --> 00:39:01,730 The 2-fold axis takes this mirror plane, 610 00:39:01,730 --> 00:39:03,020 turns it upside down. 611 00:39:03,020 --> 00:39:05,740 It takes this mirror plane and flips it over. 612 00:39:05,740 --> 00:39:09,490 The mirror plane is left unchanged in both cases. 613 00:39:09,490 --> 00:39:12,070 The mirror plane has the 2-fold axis sitting right on 614 00:39:12,070 --> 00:39:14,990 it, so it doesn't change the location of the 2-fold axis 615 00:39:14,990 --> 00:39:16,270 when it acts on it. 616 00:39:16,270 --> 00:39:19,540 And similarly, this mirror plane takes this mirror plane 617 00:39:19,540 --> 00:39:22,110 and reflects it, and it doesn't change anything. 618 00:39:22,110 --> 00:39:23,730 So all of these operations-- 619 00:39:23,730 --> 00:39:26,410 the three independent operations, sigma 1, 620 00:39:26,410 --> 00:39:28,320 sigma 2, and A pi-- 621 00:39:28,320 --> 00:39:32,600 all leave their locus of operation unchanged. 622 00:39:32,600 --> 00:39:36,350 Whereas that, clearly, is not the same for a 4-fold axis and 623 00:39:36,350 --> 00:39:37,880 a mirror plane. 624 00:39:37,880 --> 00:39:41,020 The mirror plane, when it acts on the 4-fold axis, leaves it 625 00:39:41,020 --> 00:39:43,720 unchanged, but the 4-fold axis, when it acts on the 626 00:39:43,720 --> 00:39:45,560 mirror plane, moves it into a new, 627 00:39:45,560 --> 00:39:47,290 entirely different location. 628 00:39:47,290 --> 00:39:49,640 And when that is the case, the order does make a difference. 629 00:39:58,430 --> 00:40:01,840 So I can take that corny joke of the mathematicians, what is 630 00:40:01,840 --> 00:40:03,590 purple and commutes? 631 00:40:03,590 --> 00:40:04,970 An Abelian grape. 632 00:40:04,970 --> 00:40:07,380 I can say, what is purple and commutes? 633 00:40:07,380 --> 00:40:13,135 It's a purple 4-fold axis in a mirror plane, a purple 4MM. 634 00:40:19,150 --> 00:40:23,010 OK, what other aspect of the plane groups and three 635 00:40:23,010 --> 00:40:30,090 dimensional space groups is an analytic representation of the 636 00:40:30,090 --> 00:40:37,350 way the plane group maps atoms around? 637 00:40:37,350 --> 00:40:43,170 And I see that it is five of the hour, so let me just make 638 00:40:43,170 --> 00:40:48,920 one final statement to indicate what I'm going to 639 00:40:48,920 --> 00:40:50,760 talk about next time. 640 00:40:50,760 --> 00:40:58,830 Here's the arrangement of 2-fold axes, and if you have 641 00:40:58,830 --> 00:41:04,600 an atom sitting in here, and this is T1 and T2, and we use 642 00:41:04,600 --> 00:41:07,140 this as the basis of a coordinate system, so this 643 00:41:07,140 --> 00:41:12,555 atom sits at a location x along T1 and y along T2. 644 00:41:12,555 --> 00:41:15,800 And then they ask, what other atoms are related to it? 645 00:41:15,800 --> 00:41:20,810 Well, that 2-fold axis will move x to minus x, and move y 646 00:41:20,810 --> 00:41:26,500 to minus y, and those are the only two atoms I get. 647 00:41:26,500 --> 00:41:31,600 So a characteristic of this plane group, if I were 648 00:41:31,600 --> 00:41:34,150 describing to you a crystal structure which had this 649 00:41:34,150 --> 00:41:38,260 symmetry, I might say something like a lithium in a 650 00:41:38,260 --> 00:41:44,120 position with coordinates xy, and therefore minus x minus y 651 00:41:44,120 --> 00:41:45,700 would also have to occur. 652 00:41:45,700 --> 00:41:49,070 I have an oxygen at a different number, x prime y 653 00:41:49,070 --> 00:41:51,910 prime, and that must then also occur at minus x 654 00:41:51,910 --> 00:41:54,370 prime minus y prime. 655 00:41:54,370 --> 00:41:57,410 And it looks as though any time you throw in an atom at a 656 00:41:57,410 --> 00:42:00,790 location xy, you get a symmetry related companion to 657 00:42:00,790 --> 00:42:04,040 it with negative coordinates. 658 00:42:04,040 --> 00:42:06,620 And this is what is called the general position 659 00:42:06,620 --> 00:42:07,870 of the space group. 660 00:42:11,560 --> 00:42:14,960 There's nothing special about it, and you put in an atom 661 00:42:14,960 --> 00:42:17,190 here, you get another atom here. 662 00:42:17,190 --> 00:42:20,300 This is going to be a nice economical way of describing 663 00:42:20,300 --> 00:42:23,550 an atomic arrangement, particularly if you realize 664 00:42:23,550 --> 00:42:27,280 that there are almost 150 atoms that appear in the unit 665 00:42:27,280 --> 00:42:29,280 cell for some of the cubic symmetries. 666 00:42:29,280 --> 00:42:33,820 Drop in one, wow, all hell breaks loose, 150 other ones. 667 00:42:33,820 --> 00:42:36,440 But what you can do is codify the coordinates of all these 668 00:42:36,440 --> 00:42:37,920 symmetry-related atoms. 669 00:42:37,920 --> 00:42:41,420 And that is a great utility in describing crystal structures, 670 00:42:41,420 --> 00:42:44,320 because I only have to give you the coordinates of one 671 00:42:44,320 --> 00:42:46,450 representative atom. 672 00:42:46,450 --> 00:42:49,140 But this is not the only thing that happens. 673 00:42:49,140 --> 00:42:52,760 This is the general position because it's general relative 674 00:42:52,760 --> 00:42:57,180 to its location with respect to the symmetry elements. 675 00:42:57,180 --> 00:43:02,930 If x and y were both 0, I would have one atom moving to 676 00:43:02,930 --> 00:43:06,312 the origin to coincide with the second one. 677 00:43:06,312 --> 00:43:09,740 And if I put one in at 0, 0, that's all I'm going to get. 678 00:43:09,740 --> 00:43:12,960 The 2-fold axis is just going to twirl it around, and the 679 00:43:12,960 --> 00:43:16,020 translation will repeat it to the other lattice points. 680 00:43:16,020 --> 00:43:19,230 So if I put in an atom at 0, 0, that's all 681 00:43:19,230 --> 00:43:21,230 I'm going to get. 682 00:43:21,230 --> 00:43:24,630 And the same thing would occur for any of these four 683 00:43:24,630 --> 00:43:26,630 independent 2-fold axes. 684 00:43:26,630 --> 00:43:32,300 If I let x and y migrate to 0, 1/2, this atom will join 685 00:43:32,300 --> 00:43:35,820 together with this atom, and I would get just one atom 686 00:43:35,820 --> 00:43:37,440 sitting there. 687 00:43:37,440 --> 00:43:41,950 So there's a position also of the form 0, 1/2, and a 688 00:43:41,950 --> 00:43:46,570 position 1/2, 0, and a position 1/2, 1/2 that are the 689 00:43:46,570 --> 00:43:49,070 location of these four distinct 2-fold axes. 690 00:43:52,590 --> 00:43:57,590 OK, to have a shorthand way of referring to the general 691 00:43:57,590 --> 00:44:02,170 position in these positions, which are called special 692 00:44:02,170 --> 00:44:03,420 positions-- 693 00:44:05,210 --> 00:44:09,085 what's special about them is that they're right on top of a 694 00:44:09,085 --> 00:44:19,400 symmetry element, and whenever that happens, the number per 695 00:44:19,400 --> 00:44:37,920 cell is a sub-multiple of the number in the general 696 00:44:37,920 --> 00:44:43,665 position, because coalescence has occurred. 697 00:44:50,680 --> 00:44:53,130 And so they're three independent 2-fold axes, so 698 00:44:53,130 --> 00:44:57,330 there are four locations where you get only one atom per cell 699 00:44:57,330 --> 00:44:58,930 rather than two. 700 00:44:58,930 --> 00:45:03,060 And then to give you a way of referring to these positions 701 00:45:03,060 --> 00:45:08,620 readily, they are labeled starting with the most 702 00:45:08,620 --> 00:45:11,420 specialized, just going through the alphabet, position 703 00:45:11,420 --> 00:45:16,450 a, position b, position c, position d, position e, which 704 00:45:16,450 --> 00:45:17,700 is the general position. 705 00:45:21,350 --> 00:45:24,160 And then another piece of information that's almost 706 00:45:24,160 --> 00:45:28,820 trivial here, but is not for more complicated symmetries, 707 00:45:28,820 --> 00:45:34,620 what's called the rank of the position, and that is the 708 00:45:34,620 --> 00:45:39,500 number that you get per cell, one per cell, one per cell, 709 00:45:39,500 --> 00:45:42,150 and for the general position you get two. 710 00:45:42,150 --> 00:45:46,470 So the rank of the position is just the number per cell. 711 00:45:46,470 --> 00:45:49,580 And then, if it's a special position, the atom has to sit 712 00:45:49,580 --> 00:45:57,320 at a site of some symmetry, and in this case, the atoms 713 00:45:57,320 --> 00:46:00,160 all sit at 2-fold axes for the special position. 714 00:46:00,160 --> 00:46:04,485 The general position always has site symmetry 1 always. 715 00:46:12,400 --> 00:46:14,940 So that's the characteristics of the way in which this 716 00:46:14,940 --> 00:46:16,910 symmetry can move atoms around. 717 00:46:16,910 --> 00:46:22,730 And this is of great utility in interpreting structures. 718 00:46:22,730 --> 00:46:26,740 For example, suppose our motif was not an individual atom, 719 00:46:26,740 --> 00:46:30,000 but was a molecule. 720 00:46:30,000 --> 00:46:36,150 And suppose you had one molecule per cell. 721 00:46:36,150 --> 00:46:38,200 That would tell you the molecule has to fit in a 722 00:46:38,200 --> 00:46:39,370 special position. 723 00:46:39,370 --> 00:46:42,320 And the special positions are all 2-fold axes. 724 00:46:42,320 --> 00:46:45,300 So if the molecule has to sit on a 2-fold axis, the 725 00:46:45,300 --> 00:46:48,790 configuration of that molecule has to have at least a 2-fold 726 00:46:48,790 --> 00:46:52,860 symmetry, or else you can't fit in one per cell in this 727 00:46:52,860 --> 00:46:55,050 particular plane group. 728 00:46:55,050 --> 00:46:58,120 So you can use this information to determine 729 00:46:58,120 --> 00:46:58,900 structures. 730 00:46:58,900 --> 00:47:01,680 I'm going to give you a problem a little bit later on 731 00:47:01,680 --> 00:47:02,940 when we get to three-dimensional 732 00:47:02,940 --> 00:47:07,620 crystallography, and I'm going to give you the magnitude of 733 00:47:07,620 --> 00:47:11,320 the cell edge and the density of rock salt. 734 00:47:11,320 --> 00:47:14,840 And from that, in 10 minutes, you can find out that there's 735 00:47:14,840 --> 00:47:18,250 only one possible structure for rock salt. 736 00:47:18,250 --> 00:47:21,170 The Braggs fiddled around for the better part of a year 737 00:47:21,170 --> 00:47:25,750 doing x-ray experiments and trying to calculate what the 738 00:47:25,750 --> 00:47:28,480 intensities would be for different atomic positions. 739 00:47:28,480 --> 00:47:32,110 You, in another two weeks, can get the same answer in 10 740 00:47:32,110 --> 00:47:34,660 minutes knowing only the lattice constant and the 741 00:47:34,660 --> 00:47:38,882 density and the space group. 742 00:47:38,882 --> 00:47:42,440 So a lot of physical consequences of the nature of 743 00:47:42,440 --> 00:47:45,050 the structure, the confirmation of the molecule, 744 00:47:45,050 --> 00:47:48,780 the extent or absence of solid substitution of one species 745 00:47:48,780 --> 00:47:51,360 for another, can be dredged out of the properties of the 746 00:47:51,360 --> 00:47:53,580 space group. 747 00:47:53,580 --> 00:47:57,250 OK, I've run 2 minutes over, but we took our break two 748 00:47:57,250 --> 00:48:00,980 minutes later, so let's call it a day and I shall see you 749 00:48:00,980 --> 00:48:03,980 on Thursday, hopefully armed with pictures so that I can 750 00:48:03,980 --> 00:48:06,460 personally present each one of you with your corrected 751 00:48:06,460 --> 00:48:07,710 problem sets.