1 00:00:09,348 --> 00:00:12,880 PROFESSOR: Let the minutes of the proceeding show that I 2 00:00:12,880 --> 00:00:15,540 re-entered the room at 3:00 and 12 3 00:00:15,540 --> 00:00:19,090 seconds, true to my word. 4 00:00:19,090 --> 00:00:24,980 OK, I wanted to give you one example of a screw axis that 5 00:00:24,980 --> 00:00:27,570 you're probably familiar with in everyday life. 6 00:00:27,570 --> 00:00:32,150 That was a telephone pole that I was very familiar with when 7 00:00:32,150 --> 00:00:35,050 I was a little kid because we used to love to wait until it 8 00:00:35,050 --> 00:00:37,220 got dark and our parents couldn't see us. 9 00:00:37,220 --> 00:00:38,660 And then, we'd climb up the thing. 10 00:00:38,660 --> 00:00:41,770 And wow, you could see all over the whole neighborhood. 11 00:00:41,770 --> 00:00:43,280 I'm surprised nobody said, what? 12 00:00:43,280 --> 00:00:45,640 They had electricity when you were a little kid? 13 00:00:45,640 --> 00:00:46,850 Yes, they did. 14 00:00:46,850 --> 00:00:48,530 But now it's all underground. 15 00:00:48,530 --> 00:00:49,880 So kids, I don't know what they do for 16 00:00:49,880 --> 00:00:53,090 recreation these days. 17 00:00:53,090 --> 00:00:57,750 OK, let me pursue this question of screw axes. 18 00:00:57,750 --> 00:01:01,390 We've seen what a 2 sub 1 screw access looks like. 19 00:01:01,390 --> 00:01:03,960 I just erased it. 20 00:01:03,960 --> 00:01:09,920 But it consists of objects of the same chirality extending 21 00:01:09,920 --> 00:01:13,180 left and right on either side of the principal axis. 22 00:01:13,180 --> 00:01:17,160 The symbol for a twofold axis is this. 23 00:01:17,160 --> 00:01:22,300 The symbol for a twofold screw axis is the symbol for a 24 00:01:22,300 --> 00:01:26,030 twofold access with alternate sides extended like a 25 00:01:26,030 --> 00:01:29,720 propeller, a very descriptive symbol. 26 00:01:29,720 --> 00:01:31,380 The thing is rotating around. 27 00:01:31,380 --> 00:01:34,180 And you can think of these as the little ribbons that you 28 00:01:34,180 --> 00:01:36,646 used to have on the handle bars of your bike when you 29 00:01:36,646 --> 00:01:37,320 were a little kid. 30 00:01:37,320 --> 00:01:39,310 And as you pedaled along, these things fluttered out 31 00:01:39,310 --> 00:01:42,280 behind, and it was really cool. 32 00:01:42,280 --> 00:01:47,350 OK, let's move on to the next family of screw axes. 33 00:01:47,350 --> 00:01:56,880 And let me look at an operation A2 pi over 3 with a 34 00:01:56,880 --> 00:02:00,210 translation component, tau. 35 00:02:00,210 --> 00:02:01,870 And these things are going to get awfully 36 00:02:01,870 --> 00:02:04,380 cumbersome to draw. 37 00:02:04,380 --> 00:02:07,120 So let me use a device-- 38 00:02:07,120 --> 00:02:09,199 I wish I could claim credit for it. 39 00:02:09,199 --> 00:02:12,605 But actually, a student in my class one year said, hey, I've 40 00:02:12,605 --> 00:02:15,560 got a really cool way to draw these things for you. 41 00:02:15,560 --> 00:02:17,720 Let's imagine that this is a screw axis. 42 00:02:17,720 --> 00:02:22,580 We put a cylinder of paper around the screw. 43 00:02:22,580 --> 00:02:29,150 And then, let's divide up the surface of the paper in terms 44 00:02:29,150 --> 00:02:31,660 of end segments. 45 00:02:31,660 --> 00:02:35,200 So if this were a threefold screw axis, these would be 120 46 00:02:35,200 --> 00:02:37,760 degrees segments. 47 00:02:37,760 --> 00:02:41,960 And then, let's draw horizontal lines on the sides 48 00:02:41,960 --> 00:02:44,220 of the cylinder. 49 00:02:44,220 --> 00:02:48,750 Then, if we want to see how the screw axis reproduces a 50 00:02:48,750 --> 00:02:52,450 pattern from a given motif, you just fill in these boxes. 51 00:02:52,450 --> 00:02:57,200 And then when you're all done, cut the cylinder and you can 52 00:02:57,200 --> 00:03:02,810 draw the pattern quite nicely on a two dimensional surface. 53 00:03:02,810 --> 00:03:07,220 So this is what I call the unrolled cylinder device. 54 00:03:07,220 --> 00:03:09,620 Somebody very imaginative invented it. 55 00:03:09,620 --> 00:03:11,960 His name, unfortunately, is lost to history. 56 00:03:11,960 --> 00:03:15,360 But I didn't want to claim credit for it myself. 57 00:03:15,360 --> 00:03:19,935 So what is the pattern of a threefold screw 58 00:03:19,935 --> 00:03:21,110 axis going to be? 59 00:03:21,110 --> 00:03:23,550 Let's let the difference between boxes be the 60 00:03:23,550 --> 00:03:25,610 translation component, tau. 61 00:03:25,610 --> 00:03:30,520 And so what we will do is take an initial motif, rotate it 62 00:03:30,520 --> 00:03:36,150 120 degrees, slide it up by tau relative to the axis about 63 00:03:36,150 --> 00:03:37,630 which we're rotating. 64 00:03:37,630 --> 00:03:40,650 Performing the operation again would involve rotating 120 65 00:03:40,650 --> 00:03:42,580 degrees, sliding up by tau. 66 00:03:42,580 --> 00:03:46,340 Doing it yet get again brings us back down full circle. 67 00:03:46,340 --> 00:03:49,480 So this would be the translational periodicity 68 00:03:49,480 --> 00:03:52,260 along the direction of the screw axis. 69 00:03:52,260 --> 00:03:56,540 And this would be the value of tau equal to 1/3 of the 70 00:03:56,540 --> 00:03:57,790 translation. 71 00:03:59,640 --> 00:04:03,030 So that is what a threefold screw access would look like. 72 00:04:03,030 --> 00:04:07,040 And the pattern obviously, if we would keep repeating it, 73 00:04:07,040 --> 00:04:08,760 would do something like this-- 74 00:04:12,360 --> 00:04:15,670 so perfectly interpretable, easy to draw. 75 00:04:19,070 --> 00:04:24,540 But things are actually more interesting and more 76 00:04:24,540 --> 00:04:25,790 complicated than this. 77 00:04:29,950 --> 00:04:33,790 We are stating two things about this operation. 78 00:04:33,790 --> 00:04:35,510 We're specifying a translation. 79 00:04:40,970 --> 00:04:44,170 But then the other thing that we're specifying is the 80 00:04:44,170 --> 00:04:48,200 translation component tau. 81 00:04:48,200 --> 00:04:54,180 And why should we be constrained to say that tau 82 00:04:54,180 --> 00:04:56,275 can only be one nth of the translation? 83 00:05:00,260 --> 00:05:06,560 If we do the operation n times, then doing the 84 00:05:06,560 --> 00:05:10,400 operation n times would give us a total 85 00:05:10,400 --> 00:05:13,780 translation of n tau. 86 00:05:13,780 --> 00:05:19,150 But there's no reason why that has to be one translation. 87 00:05:19,150 --> 00:05:20,534 Why not two? 88 00:05:20,534 --> 00:05:22,500 Why not three? 89 00:05:22,500 --> 00:05:23,740 Why not four? 90 00:05:23,740 --> 00:05:27,750 So the only real constraint is that n tau has to be some 91 00:05:27,750 --> 00:05:32,220 integer, m, times the translation that is parallel 92 00:05:32,220 --> 00:05:34,370 to the screw axis. 93 00:05:34,370 --> 00:05:37,190 And this means that the value of tau is not just 94 00:05:37,190 --> 00:05:38,710 equal to 1 nth of-- 95 00:05:38,710 --> 00:05:40,390 restricted to 1 nth-- 96 00:05:40,390 --> 00:05:40,770 of t. 97 00:05:40,770 --> 00:05:47,370 Tau can be m/n times T. 98 00:05:47,370 --> 00:05:50,590 And that is perfectly compatible. 99 00:05:50,590 --> 00:05:52,200 You do the operation n times. 100 00:05:52,200 --> 00:05:54,580 You're going to be directly above where you started. 101 00:05:54,580 --> 00:05:57,120 That's going to be a translation vector. 102 00:05:57,120 --> 00:06:00,350 But why not two translations, or three translations, or four 103 00:06:00,350 --> 00:06:02,130 translations? 104 00:06:02,130 --> 00:06:07,900 So there are infinitely more screw axes than just the n 105 00:06:07,900 --> 00:06:10,800 subscript something translations. 106 00:06:10,800 --> 00:06:16,300 So let's look at some of the possibilities. 107 00:06:16,300 --> 00:06:24,220 For a threefold screw axis, tau could be equal to 0T Tau 108 00:06:24,220 --> 00:06:30,810 could be equal to 1/3 of T. Tau could be equal to 2/3 of a 109 00:06:30,810 --> 00:06:32,380 translation. 110 00:06:32,380 --> 00:06:38,050 Tau could be equal to 3/3 of the translation, 4/3, 5/3, and 111 00:06:38,050 --> 00:06:39,710 so on, on and on and on. 112 00:06:39,710 --> 00:06:43,080 We could fill the whole board with possible screw axes 113 00:06:43,080 --> 00:06:44,550 having different values of tau. 114 00:06:50,030 --> 00:06:57,940 Now, let me convince you, hopefully easily, that if tau 115 00:06:57,940 --> 00:07:02,030 is equal to 3/3 t, tau would be an integral number of 116 00:07:02,030 --> 00:07:04,060 translations-- 117 00:07:04,060 --> 00:07:05,330 in this case, one. 118 00:07:05,330 --> 00:07:09,540 Down here, 6/3 t would be two translations. 119 00:07:09,540 --> 00:07:12,880 We're already going to have those operations in the 120 00:07:12,880 --> 00:07:17,830 pattern when tau was equal to 0T. 121 00:07:17,830 --> 00:07:20,090 So let me show you what I mean if it's not clear. 122 00:07:20,090 --> 00:07:30,750 Here's a trio of objects on either side of the rotation 123 00:07:30,750 --> 00:07:32,000 part of the operation. 124 00:07:35,280 --> 00:07:38,750 So what I'm saying is tau would be equal to two 125 00:07:38,750 --> 00:07:40,000 translations. 126 00:07:42,840 --> 00:07:47,450 That would, for some perverse reason, defining this as the 127 00:07:47,450 --> 00:07:48,700 pitch of the screw. 128 00:07:53,740 --> 00:07:56,970 Is it clear that that is going to be an operation that I 129 00:07:56,970 --> 00:08:01,170 already have when I say there's a threefold axis and a 130 00:08:01,170 --> 00:08:04,910 translation, T, parallel to that threefold axis? 131 00:08:04,910 --> 00:08:09,400 That's going to give all sorts of screw operations. 132 00:08:09,400 --> 00:08:12,300 But they are going to be integral multiples of the 133 00:08:12,300 --> 00:08:16,290 translation that's parallel to the axis. 134 00:08:16,290 --> 00:08:19,730 So the rule is that we can always subtract off one 135 00:08:19,730 --> 00:08:24,480 translation from any of these definitions of tau and reduce 136 00:08:24,480 --> 00:08:26,430 it to something smaller. 137 00:08:26,430 --> 00:08:34,970 So we can always define tau less than or equal to T 138 00:08:34,970 --> 00:08:39,870 without any change or artificial restrictedness on 139 00:08:39,870 --> 00:08:42,730 the nature of the pattern. 140 00:08:42,730 --> 00:08:52,290 So that says that for a threefold screw, for a 141 00:08:52,290 --> 00:08:55,380 threefold screw axis, there are only three 142 00:08:55,380 --> 00:08:56,530 that we should consider. 143 00:08:56,530 --> 00:09:01,560 Tau equal to 0, and that's a pure threefold rotation axis. 144 00:09:01,560 --> 00:09:07,380 tau could be equal to 1/3 of a translation. 145 00:09:07,380 --> 00:09:09,830 And now, we introduce the notation used 146 00:09:09,830 --> 00:09:13,200 to designate screws. 147 00:09:13,200 --> 00:09:20,360 If n tau was equal to mT, the symbol for the screw axis is 148 00:09:20,360 --> 00:09:25,435 n, the rank of the axis, with m as a subscript. 149 00:09:36,280 --> 00:09:41,420 So the pattern that we just drew here would be designated 150 00:09:41,420 --> 00:09:44,080 3 subscript 1. 151 00:09:44,080 --> 00:09:47,350 And that's says automatically that the value of tau was 152 00:09:47,350 --> 00:09:49,380 equal to 1/3 of the translation that's 153 00:09:49,380 --> 00:09:50,930 parallel to the axis. 154 00:09:50,930 --> 00:09:54,650 So the only other ones we have to consider besides 3 and 3 155 00:09:54,650 --> 00:10:03,520 sub 1 is 3 sub 2, where tau would be equal to 2/3 of T. 156 00:10:03,520 --> 00:10:07,850 Let's see what that looks like by quickly drawing it out 157 00:10:07,850 --> 00:10:11,686 again using this dandy little unrolled cylinder device. 158 00:10:18,570 --> 00:10:21,570 Let's let this be my first motif. 159 00:10:21,570 --> 00:10:26,430 And I'll take this as the value of tau, two boxes up. 160 00:10:26,430 --> 00:10:32,720 So I rotate, slide up by tau, rotate, slide up by tau, 161 00:10:32,720 --> 00:10:36,610 rotate, slide up by tau, and I'm back up here. 162 00:10:36,610 --> 00:10:43,050 And here is my translation. 163 00:10:43,050 --> 00:10:46,850 tau is equal to 2/3 of a translation. 164 00:10:46,850 --> 00:10:49,090 And my translation, then, should be equal 165 00:10:49,090 --> 00:10:52,560 to 3/2 times tau. 166 00:10:52,560 --> 00:10:54,790 But this is supposed to be the translation. 167 00:10:57,310 --> 00:10:58,630 And there's nothing in this box. 168 00:11:02,250 --> 00:11:06,640 OK, this introduces another very important aspect of the 169 00:11:06,640 --> 00:11:09,850 patterns produced by screw axes. 170 00:11:09,850 --> 00:11:17,260 And it's sufficiently important we must use first 171 00:11:17,260 --> 00:11:30,290 the basic screw operation, A alpha tau and the translation 172 00:11:30,290 --> 00:11:34,240 in terms of which we have defined tau. 173 00:11:47,490 --> 00:11:54,200 So use the spiral that you have defined by stating A 174 00:11:54,200 --> 00:11:56,160 alpha tau is the basic operation. 175 00:11:56,160 --> 00:11:57,720 But then, don't quit. 176 00:11:57,720 --> 00:12:01,270 You're saying another thing, that tau is a certain fraction 177 00:12:01,270 --> 00:12:02,670 of a translation. 178 00:12:02,670 --> 00:12:05,970 And you have to use that translation to generate 179 00:12:05,970 --> 00:12:08,640 additional objects in the pattern. 180 00:12:08,640 --> 00:12:11,990 So in this pattern that we've drawn here, this is supposed 181 00:12:11,990 --> 00:12:13,350 to be a translation. 182 00:12:13,350 --> 00:12:18,520 So I have to take this one and slide it up by T. I have to 183 00:12:18,520 --> 00:12:22,730 take this one and slide it up by T. I have to take this one 184 00:12:22,730 --> 00:12:27,200 and slide it down by T. So I'm using a different kind of 185 00:12:27,200 --> 00:12:33,110 shading to indicate the ones I produced by A alpha tau and 186 00:12:33,110 --> 00:12:36,550 the ones that I produced from that helix by using the 187 00:12:36,550 --> 00:12:39,710 translation, T. 188 00:12:39,710 --> 00:12:41,260 So this is the final pattern. 189 00:12:41,260 --> 00:12:44,910 It has a basic screw operation that's equal to 2/3 of the 190 00:12:44,910 --> 00:12:45,910 translation. 191 00:12:45,910 --> 00:12:49,620 And it has a translation that's 3/2 of tau. 192 00:12:49,620 --> 00:12:52,100 So everything's fine. 193 00:12:52,100 --> 00:12:52,868 Yeah? 194 00:12:52,868 --> 00:12:55,856 AUDIENCE: For the very top one where you have it colored in, 195 00:12:55,856 --> 00:12:56,860 [? should that stay ?] outside the box? 196 00:12:56,860 --> 00:13:00,900 PROFESSOR: Yeah, well, it's supposed to be another row. 197 00:13:00,900 --> 00:13:04,240 AUDIENCE: Shouldn't that be a box lower [INAUDIBLE]? 198 00:13:04,240 --> 00:13:05,800 PROFESSOR: Yes, it should indeed. 199 00:13:05,800 --> 00:13:07,050 Thank you. 200 00:13:09,580 --> 00:13:12,530 When your nose is right in the middle of the thing, sometimes 201 00:13:12,530 --> 00:13:13,950 you don't notice that as readily as 202 00:13:13,950 --> 00:13:15,870 somebody in the audience. 203 00:13:19,730 --> 00:13:22,850 OK, so that's the pattern of 3 sub 2. 204 00:13:22,850 --> 00:13:26,300 And now, if we realize that we must use both the basic 205 00:13:26,300 --> 00:13:30,050 operation, A alpha tau, and translation to fill things in, 206 00:13:30,050 --> 00:13:33,690 it's clear that if we try to have tau greater than the 207 00:13:33,690 --> 00:13:37,540 translation, the translations, when we apply them, would give 208 00:13:37,540 --> 00:13:42,100 us a screw that had the possibility of being redefined 209 00:13:42,100 --> 00:13:43,480 in terms of a shorter tau. 210 00:13:46,710 --> 00:13:58,120 OK, one thing that is apparent if you look at this pattern is 211 00:13:58,120 --> 00:14:06,290 that 3 sub 2 produces the same sort of helix but in an 212 00:14:06,290 --> 00:14:08,690 enantiomorphic sense. 213 00:14:08,690 --> 00:14:12,580 This one is a spiral that goes this way. 214 00:14:12,580 --> 00:14:18,140 And this one increases when we rotate in a clockwise fashion. 215 00:14:18,140 --> 00:14:19,390 So are they distinct? 216 00:14:21,240 --> 00:14:25,880 Yeah, they really are because you can have both the left 217 00:14:25,880 --> 00:14:29,860 handed spiral and the right handed spiral together in the 218 00:14:29,860 --> 00:14:31,270 same space group. 219 00:14:31,270 --> 00:14:33,600 I don't remember whether that's true for the threefold 220 00:14:33,600 --> 00:14:37,110 screw axes or not. 221 00:14:37,110 --> 00:14:38,210 I'm not sure. 222 00:14:38,210 --> 00:14:39,030 I have to check that. 223 00:14:39,030 --> 00:14:39,340 Yes? 224 00:14:39,340 --> 00:14:41,770 AUDIENCE: So really [INAUDIBLE] 225 00:14:41,770 --> 00:14:44,410 enantiomorphic because they're all left handed. 226 00:14:44,410 --> 00:14:45,646 So it's just minus 1, isn't it? 227 00:14:45,646 --> 00:14:46,730 PROFESSOR: Good for you. 228 00:14:46,730 --> 00:14:48,430 I was about to make that point. 229 00:14:48,430 --> 00:14:53,320 The sense of the spiral is enantiomorphic in the sense 230 00:14:53,320 --> 00:14:57,810 that this one is a right handed spiral. 231 00:14:57,810 --> 00:14:59,960 This one is a left handed spiral. 232 00:14:59,960 --> 00:15:02,890 But they're not truly enantiomorphic patterns 233 00:15:02,890 --> 00:15:05,800 because if we start with a right handed motif, this one 234 00:15:05,800 --> 00:15:08,060 also has a right handed motif. 235 00:15:08,060 --> 00:15:09,820 So I should qualify that statement, which I 236 00:15:09,820 --> 00:15:10,730 was about to do. 237 00:15:10,730 --> 00:15:13,560 It's the sense of the spiral which is enantiomorphic. 238 00:15:13,560 --> 00:15:18,810 This is not to say that one has motifs of one chirality 239 00:15:18,810 --> 00:15:22,024 and the other one has to have the opposite chirality. 240 00:15:22,024 --> 00:15:24,016 AUDIENCE: Couldn't you just [INAUDIBLE] 241 00:15:24,016 --> 00:15:26,506 being minus 1? 242 00:15:26,506 --> 00:15:29,510 Just [INAUDIBLE] minus 1? 243 00:15:29,510 --> 00:15:34,140 PROFESSOR: Yeah, I could do that, I could do that. 244 00:15:34,140 --> 00:15:37,990 And remember that we can add or subtract a translation at 245 00:15:37,990 --> 00:15:39,490 will from tau. 246 00:15:39,490 --> 00:15:41,830 And if I took a full translation and subtracted it 247 00:15:41,830 --> 00:15:45,330 from 3 sub 1, I would get-- 248 00:15:50,330 --> 00:15:53,370 I get tau equal to minus 2. 249 00:15:56,500 --> 00:15:58,164 Nope, doesn't work that way. 250 00:15:58,164 --> 00:16:00,910 AUDIENCE: [INAUDIBLE]? 251 00:16:00,910 --> 00:16:03,030 PROFESSOR: Yeah, it's the same tau but in the 252 00:16:03,030 --> 00:16:05,668 opposite sense, yeah. 253 00:16:08,910 --> 00:16:12,470 OK, so what comes out of this is that there are three kinds 254 00:16:12,470 --> 00:16:16,710 of axes that involve a 120 degree rotation. 255 00:16:16,710 --> 00:16:21,280 Three, which we can view as 3 sub 0, 3 sub 1, 3 sub 2. 256 00:16:21,280 --> 00:16:25,560 The symbols that are used for them now, we know and love the 257 00:16:25,560 --> 00:16:29,410 triangle that represents the locus of a threefold axis. 258 00:16:29,410 --> 00:16:35,590 Four 3 sub 1 and 3 sub 2, what one does is to extend the 259 00:16:35,590 --> 00:16:38,950 edges of the triangle. 260 00:16:38,950 --> 00:16:41,700 There's the streamers on the bike handle again. 261 00:16:41,700 --> 00:16:46,670 And if you look down on the pattern and use a right handed 262 00:16:46,670 --> 00:16:51,850 spiral, then you extend the streamers, the edges of the 263 00:16:51,850 --> 00:16:54,940 triangle, that goes down into the board this way. 264 00:16:54,940 --> 00:16:57,592 And 3 sub 2 would-- 265 00:16:57,592 --> 00:17:00,080 if it was in the same pattern-- you would indicate 266 00:17:00,080 --> 00:17:04,400 by extending the opposite pair of the opposite set of edges. 267 00:17:09,900 --> 00:17:13,900 AUDIENCE: So would they be different space groups? 268 00:17:13,900 --> 00:17:16,470 [INAUDIBLE]? 269 00:17:16,470 --> 00:17:21,819 PROFESSOR: Yes, there is a P3 sub 1. 270 00:17:21,819 --> 00:17:23,970 And there is a P3 sub 2. 271 00:17:26,900 --> 00:17:30,211 And they are distinct space groups. 272 00:17:30,211 --> 00:17:33,995 AUDIENCE: So how can you distinguish [INAUDIBLE]? 273 00:17:37,970 --> 00:17:41,200 PROFESSOR: You would find out where the atoms sit. 274 00:17:41,200 --> 00:17:46,170 And in one case, the spiral would occur in a 275 00:17:46,170 --> 00:17:47,300 right handed fashion. 276 00:17:47,300 --> 00:17:50,290 And in the other one, from a left handed fashion. 277 00:17:50,290 --> 00:17:52,615 You can determine the position of the atoms unambiguously. 278 00:17:56,670 --> 00:18:00,650 OK, let's do one that's more interesting where the rank of 279 00:18:00,650 --> 00:18:03,575 the rotation part of the operation is higher. 280 00:18:10,310 --> 00:18:13,190 And let's look at fourfold screw axes. 281 00:18:20,920 --> 00:18:28,270 So here, we would consider tau equal 0, tau equals 1/4 of a 282 00:18:28,270 --> 00:18:34,350 translation, tau equal to 2/4 of a translation, and tau 283 00:18:34,350 --> 00:18:36,200 equal to 3/4 of a translation. 284 00:18:40,280 --> 00:18:45,990 So let's take our cylinder that was formerly wrapped 285 00:18:45,990 --> 00:18:50,835 around the rotation access, straighten it out. 286 00:18:54,820 --> 00:18:58,480 And I won't bother to draw the pattern for a fourfold access. 287 00:18:58,480 --> 00:19:03,290 But for a 4 sub1 1 screw axis, this would be tau. 288 00:19:03,290 --> 00:19:07,360 And that would be one, two, three, one quarter of a 289 00:19:07,360 --> 00:19:09,630 translation that's parallel to tau. 290 00:19:13,010 --> 00:19:19,040 Or conversely, so T is equal to 4 tau. 291 00:19:19,040 --> 00:19:23,860 Or conversely, how is equal to one quarter of the 292 00:19:23,860 --> 00:19:26,000 translation. 293 00:19:26,000 --> 00:19:33,240 So start with a first motif, rotate 90 degrees, slide up by 294 00:19:33,240 --> 00:19:34,760 one quarter of the translation. 295 00:19:34,760 --> 00:19:37,620 Rotate and slide, rotate and slide. 296 00:19:37,620 --> 00:19:39,450 Now I've come full circle. 297 00:19:39,450 --> 00:19:42,420 Rotate and slide. 298 00:19:42,420 --> 00:19:47,760 So this, then, is the pattern for the screw axis that I'll 299 00:19:47,760 --> 00:19:53,540 label as 4 sub 1 by analogy to what I've done with threefold 300 00:19:53,540 --> 00:19:54,790 screw axes. 301 00:19:57,050 --> 00:20:00,220 Over here, let me immediately jump to 4 sub 3. 302 00:20:04,460 --> 00:20:06,800 I'm going to have to be a little more stingy with the 303 00:20:06,800 --> 00:20:09,560 spacing of my boxes or I'm going to run out of room. 304 00:20:14,720 --> 00:20:22,120 So if this is 4 sub 3, turns out that tau should be equal 305 00:20:22,120 --> 00:20:25,720 to 3/4 of a translation. 306 00:20:25,720 --> 00:20:28,920 And conversely, the translation should equal to 307 00:20:28,920 --> 00:20:32,520 4/3 of tau. 308 00:20:32,520 --> 00:20:38,540 So that's the situation that I would obtain if one, two, 309 00:20:38,540 --> 00:20:46,510 three boxes gives me the length of tau and four boxes 310 00:20:46,510 --> 00:20:48,650 give me the length of the translation. 311 00:20:52,700 --> 00:20:56,750 OK, so again I'll use different sorts of shading to 312 00:20:56,750 --> 00:20:58,500 indicate which way I have gotten this. 313 00:20:58,500 --> 00:21:00,290 I would rotate 90 degrees. 314 00:21:00,290 --> 00:21:02,650 I would jump up one, two, three boxes. 315 00:21:02,650 --> 00:21:03,870 So here's the next one. 316 00:21:03,870 --> 00:21:07,530 Rotate, one, two, three boxes up brings me to here. 317 00:21:07,530 --> 00:21:12,770 Rotate, one, two, three boxes brings me up to here. 318 00:21:12,770 --> 00:21:16,450 Rotating again, sliding up three boxes brings me to here. 319 00:21:19,950 --> 00:21:24,920 OK, so there's the basic helix. 320 00:21:24,920 --> 00:21:34,536 And how is indeed equal to 4/3 of the translation. 321 00:21:34,536 --> 00:21:36,040 But I can't quit yet. 322 00:21:36,040 --> 00:21:39,000 I've used the basic operation, A alpha tau. 323 00:21:39,000 --> 00:21:42,020 Now, I've got to fill in with translation. 324 00:21:42,020 --> 00:21:46,950 And according to my rule, the translation should be 325 00:21:46,950 --> 00:21:51,450 equal to 4/3 of-- 326 00:21:51,450 --> 00:21:56,130 I'm sorry, this should be the translation is 4/3 of tau. 327 00:21:56,130 --> 00:21:59,640 So this should be the translation running up one, 328 00:21:59,640 --> 00:22:02,710 two, three, four boxes. 329 00:22:02,710 --> 00:22:03,690 So that's-- 330 00:22:03,690 --> 00:22:05,050 I'll use a different shading here. 331 00:22:05,050 --> 00:22:07,000 That would fill in one here. 332 00:22:07,000 --> 00:22:10,390 One, two, three, four boxes up brings me to here. 333 00:22:10,390 --> 00:22:22,630 One, two, three, four boxes up would bring me to here. 334 00:22:22,630 --> 00:22:24,750 Go down by four-- one, two, three, four-- 335 00:22:24,750 --> 00:22:26,280 brings me to here. 336 00:22:26,280 --> 00:22:29,520 And let me use still a different shading here. 337 00:22:29,520 --> 00:22:32,280 One, two, three, four brings me down to here. 338 00:22:35,070 --> 00:22:38,910 Filling in quickly, one, two, three, four. 339 00:22:38,910 --> 00:22:40,400 Another one would sit here. 340 00:22:44,280 --> 00:22:46,140 One, two, three, four, another one would sit here. 341 00:22:46,140 --> 00:22:49,570 One, two, three, four, another one would sit here. 342 00:22:49,570 --> 00:23:00,040 And you can see what is happening here, that 4 sub 3 343 00:23:00,040 --> 00:23:07,540 looks exactly like 4 sub 1 but in the opposite sense. 344 00:23:07,540 --> 00:23:14,640 So it's the same relation as between 3 sub 1 and 3 sub 2. 345 00:23:14,640 --> 00:23:17,590 What do we use as symbols? 346 00:23:17,590 --> 00:23:19,950 This is a 4 sub 1 screw. 347 00:23:19,950 --> 00:23:25,125 And again, if we look down on the pattern from the top, use 348 00:23:25,125 --> 00:23:33,750 a right handed rule and extend every edge of the square, that 349 00:23:33,750 --> 00:23:36,030 would be the symbol for 4 sub 1. 350 00:23:36,030 --> 00:23:40,210 4 sub 3 is the same sort of spiral, but in 351 00:23:40,210 --> 00:23:41,390 the opposite direction. 352 00:23:41,390 --> 00:23:46,430 So we will, with a right handed rule and going between 353 00:23:46,430 --> 00:23:51,060 neighboring closest motifs, we would extend this, extend 354 00:23:51,060 --> 00:23:52,310 this, and extend this. 355 00:23:56,270 --> 00:23:57,860 So that's the symbol for 4 sub 3. 356 00:24:02,470 --> 00:24:03,740 With that, we come back over to here. 357 00:24:03,740 --> 00:24:09,000 And that is everything except 4 sub 2. 358 00:24:19,370 --> 00:24:22,470 And again, I'll mark up a cylinder that's been split 359 00:24:22,470 --> 00:24:28,225 into quadrants, draw reference horizontal lines. 360 00:24:32,460 --> 00:24:35,500 This is 4 sub 2. 361 00:24:35,500 --> 00:24:37,365 And we define this as tau. 362 00:24:40,550 --> 00:24:45,730 Then, the translation, this should be equal to 1/4 of the 363 00:24:45,730 --> 00:24:48,790 translation parallel to the axis. 364 00:24:48,790 --> 00:24:51,550 And conversely, the translation 365 00:24:51,550 --> 00:24:55,846 should be four halves. 366 00:25:03,100 --> 00:25:05,720 So let's draw the pattern. 367 00:25:05,720 --> 00:25:07,000 This was the first one. 368 00:25:07,000 --> 00:25:11,060 Rotate, slide up by tau. 369 00:25:11,060 --> 00:25:12,430 This is the next one. 370 00:25:12,430 --> 00:25:16,440 Rotate and slide, rotate and slide. 371 00:25:16,440 --> 00:25:23,340 Rotate and slide begin, rotate and slide. 372 00:25:23,340 --> 00:25:27,520 So this turns out to be the nature of the helix that is 373 00:25:27,520 --> 00:25:32,854 produced by A pi over 2 tau. 374 00:25:35,810 --> 00:25:40,160 But this is not yet a pattern that has the translational 375 00:25:40,160 --> 00:25:44,380 periodicity that I have claimed. 376 00:25:44,380 --> 00:25:46,340 That's supposed to be the translation, 377 00:25:46,340 --> 00:25:49,280 twice tau, up to here. 378 00:25:49,280 --> 00:25:54,220 So I would have two slide this one up to here. 379 00:25:54,220 --> 00:25:57,380 I'd have to slide this one up to here. 380 00:25:57,380 --> 00:26:06,302 I'd have to slide this one up to here. 381 00:26:06,302 --> 00:26:10,270 And something's wrong here-- one, two, three, four. 382 00:26:13,040 --> 00:26:17,960 I think I went too far up for one of them. 383 00:26:17,960 --> 00:26:19,305 This one was one, two, three. 384 00:26:23,120 --> 00:26:25,040 This one, I went up too far. 385 00:26:25,040 --> 00:26:26,290 I skipped one. 386 00:26:30,070 --> 00:26:33,090 What did I do wrong? 387 00:26:33,090 --> 00:26:34,572 AUDIENCE: You did the rotation wrong. 388 00:26:34,572 --> 00:26:36,054 PROFESSOR: Hm? 389 00:26:36,054 --> 00:26:37,536 AUDIENCE: The two [INAUDIBLE]. 390 00:26:42,476 --> 00:26:45,165 PROFESSOR: OK, let me do this all over again. 391 00:26:45,165 --> 00:26:47,420 Again, I can't see standing right on top of it. 392 00:26:57,220 --> 00:27:01,740 It's not a very good attempt to make it look easy. 393 00:27:01,740 --> 00:27:03,230 Let's split this up into four quadrants. 394 00:27:08,090 --> 00:27:10,300 If this one comes out quite differently, I'll make sure I 395 00:27:10,300 --> 00:27:11,420 have it right. 396 00:27:11,420 --> 00:27:15,515 So this is my translation, one, two, three. 397 00:27:19,000 --> 00:27:21,640 This is the translation. 398 00:27:21,640 --> 00:27:29,010 And that should be equal to four halves of tau. 399 00:27:29,010 --> 00:27:32,390 So this, then, is my value of tau. 400 00:27:32,390 --> 00:27:34,490 It should go up two boxes to here. 401 00:27:39,150 --> 00:27:48,390 And that should be equal to 1/2 of T. So I want to rotate, 402 00:27:48,390 --> 00:27:55,300 slide up two boxes, rotate, slide up two boxes, rotate, 403 00:27:55,300 --> 00:28:00,570 slide up two boxes, rotate, slide up two boxes, rotate, 404 00:28:00,570 --> 00:28:03,300 and slide up two boxes. 405 00:28:03,300 --> 00:28:05,310 And now, I have to fill in with translation. 406 00:28:05,310 --> 00:28:11,240 So I will take this one and go up one, two, three, four 407 00:28:11,240 --> 00:28:11,990 translations. 408 00:28:11,990 --> 00:28:13,870 That's what I did wrong. 409 00:28:13,870 --> 00:28:16,190 I'll take this one and go down one, two, three, four 410 00:28:16,190 --> 00:28:17,620 translations. 411 00:28:17,620 --> 00:28:21,030 Take this one and go up one, two, three, four translations. 412 00:28:21,030 --> 00:28:24,690 Take this one and go one, two, three, four translations. 413 00:28:24,690 --> 00:28:28,090 Take this one and go one, two, three, four translations. 414 00:28:28,090 --> 00:28:35,680 And now, lo and behold, what I have is one, two, three, four. 415 00:28:35,680 --> 00:28:37,190 Curious sort of thing-- 416 00:28:37,190 --> 00:28:41,090 I have two motifs at every level. 417 00:28:41,090 --> 00:28:44,970 And they are 180 degrees apart. 418 00:28:44,970 --> 00:28:50,380 So a 4 sub 2 screw axis produces a pattern 419 00:28:50,380 --> 00:28:51,440 that looks like this. 420 00:28:51,440 --> 00:28:56,350 Pair of motifs like this 180 degrees apart, pair of motifs 421 00:28:56,350 --> 00:29:00,990 skewed by 90 degrees to that pair, another pair 422 00:29:00,990 --> 00:29:02,950 translationally equivalent to the first. 423 00:29:06,810 --> 00:29:12,120 But what is the chirality of this spiral? 424 00:29:12,120 --> 00:29:14,570 3 sub 1 was right handed. 425 00:29:14,570 --> 00:29:16,260 4 sub 1 was right handed. 426 00:29:16,260 --> 00:29:18,290 4 sub 3 was left handed. 427 00:29:18,290 --> 00:29:20,670 4 sub 2 was right in the middle. 428 00:29:20,670 --> 00:29:22,110 Is that right handed or left handed? 429 00:29:22,110 --> 00:29:24,420 The answer is it's both. 430 00:29:24,420 --> 00:29:25,390 It's both. 431 00:29:25,390 --> 00:29:29,170 There's one spiral that goes up this way. 432 00:29:29,170 --> 00:29:31,020 And there's another spiral that goes up in 433 00:29:31,020 --> 00:29:33,120 the opposite sense. 434 00:29:33,120 --> 00:29:35,280 So it's both. 435 00:29:35,280 --> 00:29:38,270 It's a left handed spiral and a right handed spiral together 436 00:29:38,270 --> 00:29:39,520 in the same pattern. 437 00:29:42,410 --> 00:29:45,800 And the pattern for this screw axis, the symbol for this 438 00:29:45,800 --> 00:29:52,230 screw axis if it occurs in a pattern, is every other edge 439 00:29:52,230 --> 00:29:55,350 of the square extended. 440 00:29:55,350 --> 00:29:56,360 Which do you extend? 441 00:29:56,360 --> 00:29:58,800 Well, it really doesn't matter because it's both left handed 442 00:29:58,800 --> 00:30:00,480 and right handed simultaneously. 443 00:30:00,480 --> 00:30:03,030 So you can use either one. 444 00:30:03,030 --> 00:30:04,748 Yes, a couple questions? 445 00:30:04,748 --> 00:30:06,212 AUDIENCE: Can you explain why you explain 446 00:30:06,212 --> 00:30:07,676 you did that shading? 447 00:30:07,676 --> 00:30:09,930 PROFESSOR: I just wanted to differentiate 448 00:30:09,930 --> 00:30:11,010 the ones that I used. 449 00:30:11,010 --> 00:30:12,630 They're all the same. 450 00:30:12,630 --> 00:30:15,650 And where they come from doesn't affect what the basic 451 00:30:15,650 --> 00:30:16,800 pattern looks like. 452 00:30:16,800 --> 00:30:20,990 But I tried to make clear which ones I got by using the 453 00:30:20,990 --> 00:30:23,200 operation A alpha tau. 454 00:30:23,200 --> 00:30:27,480 And for 4 sub 2, which I just did twice, once correctly and 455 00:30:27,480 --> 00:30:31,190 once incorrectly, these with shading were produced by the 456 00:30:31,190 --> 00:30:34,660 operation of rotating and sliding up by 1/2 of the 457 00:30:34,660 --> 00:30:35,700 translation. 458 00:30:35,700 --> 00:30:39,780 The other ones, the ones for which I used this shading, I 459 00:30:39,780 --> 00:30:42,690 filled in by using what the translation was, 460 00:30:42,690 --> 00:30:44,950 namely twice tau. 461 00:30:44,950 --> 00:30:48,740 So it's just a way of keeping track of what operations I 462 00:30:48,740 --> 00:30:51,090 used to produce everything that's in the pattern. 463 00:30:54,510 --> 00:30:58,740 And for 4 sub 3, I had to use three kinds of shading. 464 00:30:58,740 --> 00:30:59,535 Yes, sir? 465 00:30:59,535 --> 00:31:01,960 AUDIENCE: Should I [INAUDIBLE] the fact that this was 466 00:31:01,960 --> 00:31:03,900 actually two superimposed twofold axis 467 00:31:03,900 --> 00:31:07,780 separated by 1/2 T? 468 00:31:07,780 --> 00:31:10,040 PROFESSOR: Yes, you could read that into it. 469 00:31:10,040 --> 00:31:19,160 And you would be very observant because a 4 sub 2 470 00:31:19,160 --> 00:31:24,700 screw axis contains a twofold rotation as a subgroup. 471 00:31:24,700 --> 00:31:27,050 AUDIENCE: [INAUDIBLE]. 472 00:31:27,050 --> 00:31:32,190 PROFESSOR: And since you were shrewd enough to deduce this-- 473 00:31:32,190 --> 00:31:36,620 we ask the question with glide planes when we're dealing with 474 00:31:36,620 --> 00:31:39,240 two dimensional plane groups. 475 00:31:39,240 --> 00:31:42,400 Is a glide plane a candidate location 476 00:31:42,400 --> 00:31:45,170 for a special position? 477 00:31:45,170 --> 00:31:48,495 In other words, if I move the atom directly onto a glide 478 00:31:48,495 --> 00:31:50,030 plane, is there coalescence? 479 00:31:50,030 --> 00:31:52,040 The answer was no. 480 00:31:52,040 --> 00:31:55,730 Suppose I asked the same question now of a screw axis. 481 00:31:55,730 --> 00:31:59,150 Is a screw axis a candidate location 482 00:31:59,150 --> 00:32:02,720 for a special position? 483 00:32:02,720 --> 00:32:05,827 The answer is sometimes yes, sometimes no. 484 00:32:05,827 --> 00:32:09,810 AUDIENCE: Well, only if you have [INAUDIBLE]. 485 00:32:09,810 --> 00:32:13,030 PROFESSOR: Yeah, so actually if I move the atom, the first 486 00:32:13,030 --> 00:32:17,070 atom, onto the locus of the rotation part of the 487 00:32:17,070 --> 00:32:20,280 operation, I get pairwise coalescence. 488 00:32:20,280 --> 00:32:24,150 And instead of getting all of these, I'll have just a string 489 00:32:24,150 --> 00:32:27,610 of single atoms separated by half a 490 00:32:27,610 --> 00:32:30,650 translation, half as many. 491 00:32:30,650 --> 00:32:32,100 So that's a special position. 492 00:32:32,100 --> 00:32:36,140 I don't get the full number out when I throw one into the 493 00:32:36,140 --> 00:32:37,390 space group. 494 00:32:40,980 --> 00:32:46,120 Now, there are an infinite number of screw axes. 495 00:32:46,120 --> 00:32:52,050 My old friend, the saguaro cactus, which has a 19 to 496 00:32:52,050 --> 00:32:57,090 maybe 23-fold rotational component to its symmetry, 497 00:32:57,090 --> 00:33:01,620 does not, if I look at it carefully, have the same thing 498 00:33:01,620 --> 00:33:08,180 on every one of these ribs because the tufts of spines 499 00:33:08,180 --> 00:33:10,500 spiral up like this. 500 00:33:10,500 --> 00:33:14,380 So a saguaro cactus, if I take the spines into account, 501 00:33:14,380 --> 00:33:16,840 doesn't have 22-fold rotational symmetry. 502 00:33:16,840 --> 00:33:20,480 It has some sort of 22-fold screw axis. 503 00:33:20,480 --> 00:33:23,510 And I've never been determined enough to risk puncturing my 504 00:33:23,510 --> 00:33:27,180 finger by going down around a saguaro and seeing if there's 505 00:33:27,180 --> 00:33:30,670 some other tuft at the same level and whether it's a 22 506 00:33:30,670 --> 00:33:34,670 subscript 3 or 22 subscript 2 screw axis. 507 00:33:34,670 --> 00:33:39,410 But a saguaro cactus does have a screw axis as 508 00:33:39,410 --> 00:33:40,660 its symmetry element. 509 00:33:43,030 --> 00:33:45,690 OK, let us generalize on the basis of what 510 00:33:45,690 --> 00:33:46,940 we've done so far. 511 00:33:53,810 --> 00:34:00,740 For any screw axis whatsoever, crystallographic or not, we 512 00:34:00,740 --> 00:34:07,390 will have an n sub 1 screw, an n sub 2 screw, where tau is 513 00:34:07,390 --> 00:34:18,790 equal to 1 nth of a translation, tau equal to 2 514 00:34:18,790 --> 00:34:20,760 nths of a translation. 515 00:34:20,760 --> 00:34:27,300 And we will go all the way up n sub n minus 1, 516 00:34:27,300 --> 00:34:31,610 n sub n minus 2. 517 00:34:31,610 --> 00:34:38,020 And always, regardless of n, the ones at either end tend to 518 00:34:38,020 --> 00:34:41,590 be spirals in an enantiomorphic sense. 519 00:34:41,590 --> 00:34:45,485 And n sub 2 and an n sub n minus 2 are spirals. 520 00:34:48,310 --> 00:34:53,830 If you end up with an odd one in the middle, this has no 521 00:34:53,830 --> 00:34:55,540 chirality to the spiral. 522 00:34:55,540 --> 00:34:59,320 There's a left handed one and a right handed one. 523 00:34:59,320 --> 00:35:02,880 On the other hand, for something like two and four 524 00:35:02,880 --> 00:35:12,580 and six, the ones at opposite ends of the series are 525 00:35:12,580 --> 00:35:14,790 enantiomorphs in the sense of the rotation. 526 00:35:18,610 --> 00:35:24,220 For any six-fold screw axis-- 527 00:35:24,220 --> 00:35:27,250 and I will pass out the nature of the pattern without taking 528 00:35:27,250 --> 00:35:29,080 time to draw it-- 529 00:35:29,080 --> 00:35:32,660 6 sub 1 and 6 sub 5 are enantiomorphous. 530 00:35:32,660 --> 00:35:36,620 6 sub 2 and 6 sub 4 are enantiomorphous. 531 00:35:36,620 --> 00:35:39,020 And 6 sub 3 stands alone. 532 00:35:39,020 --> 00:35:45,610 6 sub 3 consists of a pattern of three objects on a triangle 533 00:35:45,610 --> 00:35:49,860 pointing in one sense, three on a triangle pointing in the 534 00:35:49,860 --> 00:35:51,460 opposite sense. 535 00:35:51,460 --> 00:35:54,430 This is the value of tau. 536 00:35:54,430 --> 00:35:58,030 And that is equal to 3/6 of the translation. 537 00:35:58,030 --> 00:36:00,970 So no left handed or right handed sense. 538 00:36:00,970 --> 00:36:05,270 6 sub 2 consist of pairs of objects 539 00:36:05,270 --> 00:36:06,850 separated by 60 degrees. 540 00:36:13,230 --> 00:36:20,880 6 sub 4, exactly the same pairs, but they go in the 541 00:36:20,880 --> 00:36:22,130 opposite sense. 542 00:36:30,660 --> 00:36:35,390 6 sub 1 is just a single spiral going up with atoms at 543 00:36:35,390 --> 00:36:37,680 intervals of 1/6 of the translation. 544 00:36:37,680 --> 00:36:41,340 6 sub 5, a spiral in the opposite sense. 545 00:36:48,020 --> 00:36:53,090 OK, with that, let me pass out a sheet that has patterns done 546 00:36:53,090 --> 00:36:55,440 in a decent fashion for all of the 547 00:36:55,440 --> 00:36:57,880 crystallographic screw axes. 548 00:36:57,880 --> 00:37:01,360 Bear in mind that there are fascinating, interesting, and 549 00:37:01,360 --> 00:37:02,690 intriguing patterns for 550 00:37:02,690 --> 00:37:04,570 non-crystallographic screw axes. 551 00:37:10,280 --> 00:37:16,220 And let me say a few words about the symbol used in the 552 00:37:16,220 --> 00:37:18,910 space group for glide planes. 553 00:37:25,450 --> 00:37:27,470 Any questions on this, by the way? 554 00:37:27,470 --> 00:37:30,730 I don't want to have beat it into the ground. 555 00:37:30,730 --> 00:37:31,950 Yes? 556 00:37:31,950 --> 00:37:34,375 AUDIENCE: I have a question on symbols. 557 00:37:34,375 --> 00:37:37,770 How do you know which way the arrows point on the triangle 558 00:37:37,770 --> 00:37:41,435 or the squares? 559 00:37:41,435 --> 00:37:42,140 PROFESSOR: Oh, here? 560 00:37:42,140 --> 00:37:42,580 AUDIENCE: Yeah. 561 00:37:42,580 --> 00:37:44,170 PROFESSOR: OK. 562 00:37:44,170 --> 00:37:47,910 Probably the best thing to do is to look at the patterns 563 00:37:47,910 --> 00:37:50,100 that I just passed around if you got one yet. 564 00:37:50,100 --> 00:37:56,210 If you look at 3 sub 1 and look down on it from the top 565 00:37:56,210 --> 00:38:02,140 and indicate the sense of rotation that gets you from 566 00:38:02,140 --> 00:38:06,180 the one above to the one that's directly below, your 567 00:38:06,180 --> 00:38:10,040 hand will curl around in a clockwise fashion. 568 00:38:10,040 --> 00:38:14,440 And the little tails on the symbol for the threefold axis 569 00:38:14,440 --> 00:38:18,770 trail out behind the way in which you're rotating. 570 00:38:18,770 --> 00:38:23,230 If you look at 3 sub 2 and look down on it, you would 571 00:38:23,230 --> 00:38:26,330 have to, using your right hand, rotate in a 572 00:38:26,330 --> 00:38:27,820 counterclockwise sense. 573 00:38:27,820 --> 00:38:31,650 And therefore, the streamers go out again in 574 00:38:31,650 --> 00:38:33,950 the opposite direction. 575 00:38:33,950 --> 00:38:37,440 Same for 4 sub 1. 576 00:38:37,440 --> 00:38:40,390 There, every edge of the square is enclosed. 577 00:38:40,390 --> 00:38:44,360 If you look at how you have to rotate to get from the one on 578 00:38:44,360 --> 00:38:47,510 top to the one immediately below using your right hand, 579 00:38:47,510 --> 00:38:50,830 again, you have to rotate in a clockwise sense. 580 00:38:50,830 --> 00:38:56,770 And therefore, the edges that are elongated trail out in the 581 00:38:56,770 --> 00:38:58,780 opposite direction. 582 00:38:58,780 --> 00:39:03,770 And finally, drawn in a decent fashion are 6 sub 1, 6 sub 5, 583 00:39:03,770 --> 00:39:06,240 which are spirals in the opposite sense. 584 00:39:06,240 --> 00:39:11,160 6 sub 2, 6 sub 4, pairs of things at each level. 585 00:39:11,160 --> 00:39:13,340 And the level is 1/3 of the translation. 586 00:39:13,340 --> 00:39:17,700 6 sub 3, triangles of objects at intervals that are equal to 587 00:39:17,700 --> 00:39:26,920 1/2 of T, 3/6 of T. OK, does that answer? 588 00:39:26,920 --> 00:39:29,650 That is really a convention of some higher order. 589 00:39:29,650 --> 00:39:31,630 But nevertheless, it is a convention and it is 590 00:39:31,630 --> 00:39:32,880 consistent. 591 00:39:35,290 --> 00:39:37,481 Other questions about screw axes? 592 00:39:37,481 --> 00:39:37,968 Yeah? 593 00:39:37,968 --> 00:39:40,403 AUDIENCE: So in nature, is there a difference between the 594 00:39:40,403 --> 00:39:41,864 right handed screw [INAUDIBLE]? 595 00:39:45,290 --> 00:39:48,940 PROFESSOR: That's like asking is there something like a 596 00:39:48,940 --> 00:39:51,660 Coriolis effect in crystals. 597 00:39:51,660 --> 00:39:54,000 You know the Coriolis effect that if you're in the southern 598 00:39:54,000 --> 00:39:57,640 hemisphere, the water gurgles down the drain one direction. 599 00:39:57,640 --> 00:39:58,870 If you're in the northern hemisphere, 600 00:39:58,870 --> 00:40:00,120 the opposite direction? 601 00:40:03,960 --> 00:40:09,080 There was actually a very famous scientist, 602 00:40:09,080 --> 00:40:10,320 what was his name? 603 00:40:10,320 --> 00:40:12,370 I think it was Serret. 604 00:40:12,370 --> 00:40:14,200 Serret, so famous I can't think of his name. 605 00:40:14,200 --> 00:40:22,660 I remembers his name, Serret He developed a very famous 606 00:40:22,660 --> 00:40:26,150 theorem in kinetics, and that was thermal migration, 607 00:40:26,150 --> 00:40:29,250 migration driven not by a chemical potential gradient 608 00:40:29,250 --> 00:40:32,120 but by a temperature gradient. 609 00:40:32,120 --> 00:40:34,820 And it's very difficult to measure. 610 00:40:34,820 --> 00:40:37,620 And only recently have people been testing that theory and 611 00:40:37,620 --> 00:40:39,710 making measurements. 612 00:40:39,710 --> 00:40:41,590 He asked this very same question. 613 00:40:41,590 --> 00:40:46,900 And he said, I think it was potassium chlorate that he 614 00:40:46,900 --> 00:40:50,670 looked at which has no mirror plane or inversion in it. 615 00:40:50,670 --> 00:40:53,690 So it occurs in left handed and right handed forms. 616 00:40:53,690 --> 00:40:57,250 He asked, is there something like a Coriolis effect that 617 00:40:57,250 --> 00:41:00,660 would give you more right handed crystals in the 618 00:41:00,660 --> 00:41:03,130 northern hemisphere and more left handed crystals in the 619 00:41:03,130 --> 00:41:04,260 southern hemisphere? 620 00:41:04,260 --> 00:41:06,350 So what he'd do, he drew a lot of crystals. 621 00:41:06,350 --> 00:41:07,560 And he started counting. 622 00:41:07,560 --> 00:41:09,320 And he counted, and he counted, and counted. 623 00:41:09,320 --> 00:41:10,990 And he couldn't tell if there was a difference. 624 00:41:10,990 --> 00:41:13,170 And counted, counted, counted. 625 00:41:13,170 --> 00:41:17,770 And still, after lots and lots of counting, still could not 626 00:41:17,770 --> 00:41:20,150 get a result that was statistically significant. 627 00:41:20,150 --> 00:41:21,445 And he went nuts. 628 00:41:21,445 --> 00:41:23,590 Sorry to tell you this. 629 00:41:23,590 --> 00:41:27,190 So I think it remains to be seen whether there is indeed a 630 00:41:27,190 --> 00:41:29,335 Coriolis effect in crystal shapes. 631 00:41:29,335 --> 00:41:29,970 And that's true. 632 00:41:29,970 --> 00:41:30,760 It's tragic. 633 00:41:30,760 --> 00:41:33,410 We're all laughing at the poor guy. 634 00:41:33,410 --> 00:41:36,060 But he actually just couldn't stand it. 635 00:41:36,060 --> 00:41:39,470 He counted so many crystals, poof! 636 00:41:42,320 --> 00:41:45,820 That's a true, unfortunate story. 637 00:41:45,820 --> 00:41:49,400 No, I don't think there is any difference between left handed 638 00:41:49,400 --> 00:41:53,000 crystals or right handed crystals. 639 00:41:53,000 --> 00:41:57,640 Remember my story about sugar and its chirality and the 640 00:41:57,640 --> 00:42:01,610 little bugs that can only eat one chirality. 641 00:42:01,610 --> 00:42:04,980 Other questions of the less frivolous nature? 642 00:42:04,980 --> 00:42:05,420 That was not frivolous. 643 00:42:05,420 --> 00:42:06,670 That was a good question. 644 00:42:09,240 --> 00:42:11,490 OK, well let's say a little bit about glide planes and how 645 00:42:11,490 --> 00:42:13,500 they appear in space groups. 646 00:42:16,330 --> 00:42:20,140 Things were very easy in two dimensions because the glide 647 00:42:20,140 --> 00:42:22,930 plane was a glide line. 648 00:42:22,930 --> 00:42:26,470 And everything had to be confined to a 649 00:42:26,470 --> 00:42:28,150 two dimensional space. 650 00:42:28,150 --> 00:42:32,970 So we used a solid line to indicate a mirror plane, the 651 00:42:32,970 --> 00:42:34,740 locus of the operation, sigma. 652 00:42:34,740 --> 00:42:37,980 And we use a dashed line to indicate the locus of an 653 00:42:37,980 --> 00:42:39,900 operation, sigma tau. 654 00:42:39,900 --> 00:42:44,510 And tau was in the direction of the dashes. 655 00:42:44,510 --> 00:42:46,930 Well, I've given it all away with the little diagram that I 656 00:42:46,930 --> 00:42:47,550 passed around. 657 00:42:47,550 --> 00:42:49,900 Things are much more complicated in three 658 00:42:49,900 --> 00:42:50,540 dimensions. 659 00:42:50,540 --> 00:42:53,670 If this is the locus of a glide plane and this is the 660 00:42:53,670 --> 00:42:58,500 direction of tau in a space group, we could be looking at 661 00:42:58,500 --> 00:43:04,570 that glide plane in a total of four different ways. 662 00:43:04,570 --> 00:43:07,780 We could be looking at the glide plane from the side, 663 00:43:07,780 --> 00:43:11,090 edge on, and perpendicular to tau. 664 00:43:11,090 --> 00:43:15,460 We could be looking at the glide plane edge on and along 665 00:43:15,460 --> 00:43:17,230 the direction of tau. 666 00:43:17,230 --> 00:43:21,170 Or we could be between these two orientations and looking 667 00:43:21,170 --> 00:43:25,290 at the glide plane edge on but in between normal to tau and 668 00:43:25,290 --> 00:43:27,690 parallel to tau. 669 00:43:27,690 --> 00:43:30,310 And finally, we could look at the glide plane 670 00:43:30,310 --> 00:43:31,560 down from the top. 671 00:43:34,560 --> 00:43:38,270 If we view the glide plane from above, for example 672 00:43:38,270 --> 00:43:42,040 suppose we have a monoclinic crystal and we put a glide 673 00:43:42,040 --> 00:43:45,040 plane in the base of the cell which has tau in this 674 00:43:45,040 --> 00:43:48,250 direction, the way you indicate that is with a little 675 00:43:48,250 --> 00:43:50,610 chevron off to the side. 676 00:43:50,610 --> 00:43:54,560 And if this is the direction of tau, you put a little barb 677 00:43:54,560 --> 00:43:58,830 on the chevron that indicates the direction of tau. 678 00:43:58,830 --> 00:44:03,200 So the three possibilities for a monoclinic crystal is that 679 00:44:03,200 --> 00:44:06,020 how could be this way, in which case you use a 680 00:44:06,020 --> 00:44:07,610 chevron like this. 681 00:44:07,610 --> 00:44:11,660 Or how could be in this direction. 682 00:44:11,660 --> 00:44:13,817 And then, you use a chevron with a barb 683 00:44:13,817 --> 00:44:15,660 attached like this. 684 00:44:15,660 --> 00:44:21,310 Or how could be a diagonal glide, in which case the atoms 685 00:44:21,310 --> 00:44:25,040 would be up and opposite chirality and 686 00:44:25,040 --> 00:44:28,010 down and back up again. 687 00:44:28,010 --> 00:44:33,000 In that case, you add a little barb in between the two 688 00:44:33,000 --> 00:44:39,550 directions to indicate the direction of tau. 689 00:44:39,550 --> 00:44:44,280 We put some labels on this, this a and this b. 690 00:44:44,280 --> 00:44:49,250 This situation would be a tau that is equal to 1/2 of b. 691 00:44:49,250 --> 00:44:54,300 In this case, how would be equal to 1/2 of a. 692 00:44:54,300 --> 00:45:00,580 In this case, how would be equal to 1/2 of a plus b. 693 00:45:04,180 --> 00:45:10,140 This is designated as a b glide. 694 00:45:10,140 --> 00:45:14,060 And rather than having the symbol n appear in the symbol 695 00:45:14,060 --> 00:45:16,800 for the space group, a b would appear. 696 00:45:16,800 --> 00:45:18,260 This is an a glide. 697 00:45:21,100 --> 00:45:24,510 And the a would appear in the symbol. 698 00:45:24,510 --> 00:45:26,695 And this is a diagonal glide. 699 00:45:32,620 --> 00:45:36,780 And that, for reasons that I have never found anyone able 700 00:45:36,780 --> 00:45:40,600 to explain to me, is called an n glide. 701 00:45:40,600 --> 00:45:44,065 I know of no language in which the word for diagonal 702 00:45:44,065 --> 00:45:45,315 begins with an n. 703 00:45:47,460 --> 00:45:48,710 But that's what it's called. 704 00:45:50,850 --> 00:45:54,120 Now, those same glides can be viewed from directions that 705 00:45:54,120 --> 00:45:57,110 are parallel to the plane of the glide plane. 706 00:45:57,110 --> 00:46:01,020 Looking normal to tau is exactly the same situation we 707 00:46:01,020 --> 00:46:03,190 had in plane groups. 708 00:46:03,190 --> 00:46:05,710 So one uses the dash. 709 00:46:05,710 --> 00:46:08,620 If you're looking along the glide plane end on but in the 710 00:46:08,620 --> 00:46:13,140 direction of tau, you use a dotted line. 711 00:46:13,140 --> 00:46:14,850 It's easy to keep straight which is which. 712 00:46:14,850 --> 00:46:16,800 If there's a little arrow in there and you look at it from 713 00:46:16,800 --> 00:46:18,300 the side, you see a line segment. 714 00:46:18,300 --> 00:46:20,780 If you look at it from the end, you see a dot. 715 00:46:20,780 --> 00:46:24,340 So it's easy to remember this convention. 716 00:46:24,340 --> 00:46:26,900 If you're neither perpendicular to nor parallel 717 00:46:26,900 --> 00:46:30,080 to the glide plane, you just mix the two symbols. 718 00:46:30,080 --> 00:46:31,690 So you use a dashed dot line. 719 00:46:35,540 --> 00:46:39,830 So those are the symbols for glide. 720 00:46:39,830 --> 00:46:42,460 Glide plane is something like a piece of wood. 721 00:46:42,460 --> 00:46:44,240 It's got grain to it. 722 00:46:44,240 --> 00:46:46,950 The direction of the grain is the direction of 723 00:46:46,950 --> 00:46:48,200 tau, if you'd like. 724 00:46:52,460 --> 00:46:59,050 All right, in the hand out, I give you some examples of the 725 00:46:59,050 --> 00:47:02,620 way in which a pattern of objects that has glide in it 726 00:47:02,620 --> 00:47:07,780 would be interpreted in terms of a geometric symbol. 727 00:47:07,780 --> 00:47:11,040 At the bottom right of the page, the left hand diagram 728 00:47:11,040 --> 00:47:15,550 has atoms related by a glide plane in the base of the cell 729 00:47:15,550 --> 00:47:18,920 where tau is running left to right. 730 00:47:18,920 --> 00:47:21,800 And so you have a chevron with an arrow on 731 00:47:21,800 --> 00:47:24,400 the right hand branch. 732 00:47:24,400 --> 00:47:29,400 If you have atoms alternately left handed, right handed at a 733 00:47:29,400 --> 00:47:33,140 level and 1/2 plus that elevation, then you're looking 734 00:47:33,140 --> 00:47:35,840 at a glide plane edge on and down along the 735 00:47:35,840 --> 00:47:37,060 direction of tau. 736 00:47:37,060 --> 00:47:40,200 So you would indicate the locus of that glide plane by a 737 00:47:40,200 --> 00:47:41,500 series of dots. 738 00:47:49,320 --> 00:47:53,150 What other type of glide plane is possible, not for 739 00:47:53,150 --> 00:48:08,330 monoclinic, but for lattices that have a centered lattice 740 00:48:08,330 --> 00:48:10,480 point in the middle of the cell? 741 00:48:16,890 --> 00:48:19,480 So this takes a special type of Bravais lattice. 742 00:48:23,610 --> 00:48:29,900 And this would be a lattice, for example, that had lattice 743 00:48:29,900 --> 00:48:33,210 points at the corners of the cell and another lattice point 744 00:48:33,210 --> 00:48:35,080 in the middle of the cell. 745 00:48:35,080 --> 00:48:36,960 One face of the cell is centered. 746 00:48:36,960 --> 00:48:41,860 Then, you have not only translations a, b, and a plus 747 00:48:41,860 --> 00:48:47,080 b, which can serve as directions for glide. 748 00:48:47,080 --> 00:48:49,890 But you have another translation that is not 749 00:48:49,890 --> 00:48:51,540 present in the primitive lattice. 750 00:48:51,540 --> 00:48:58,090 And this is 1/2 of a plus 1/2 of b. 751 00:48:58,090 --> 00:49:01,500 And if that's a translation, you could have a glide plane 752 00:49:01,500 --> 00:49:06,545 parallel to the base of the cell and have tau equal to 1/2 753 00:49:06,545 --> 00:49:10,955 of 1/2 of a plus b, namely have this little vector in 754 00:49:10,955 --> 00:49:13,830 here be tau. 755 00:49:13,830 --> 00:49:17,340 And the symbol for that kind of glide is d. 756 00:49:17,340 --> 00:49:19,270 And I do know what that stands for. 757 00:49:19,270 --> 00:49:24,160 That stands for a diamond glide because diamond, one of 758 00:49:24,160 --> 00:49:29,030 the very simple structures for an element or any compound and 759 00:49:29,030 --> 00:49:33,990 one of the early structures to be determined experimentally, 760 00:49:33,990 --> 00:49:36,110 does have a d glide in it. 761 00:49:36,110 --> 00:49:39,310 So the name of the compound, diamond, gave its name to the 762 00:49:39,310 --> 00:49:40,560 type of glide plane. 763 00:49:51,540 --> 00:49:53,770 OK, I've got still 10 more minutes. 764 00:49:53,770 --> 00:49:58,610 What I would like to do next is to look at some of the 765 00:49:58,610 --> 00:50:03,810 other monoclinic space groups and also examine the way in 766 00:50:03,810 --> 00:50:06,730 which this information is presented for you in the 767 00:50:06,730 --> 00:50:09,320 international tables. 768 00:50:09,320 --> 00:50:14,580 So I've got a big, fat pack of material that shows you all of 769 00:50:14,580 --> 00:50:17,210 the monoclinic space group, that is the space groups 770 00:50:17,210 --> 00:50:21,510 derived from point group two, the space groups with point 771 00:50:21,510 --> 00:50:24,970 group m, and the space groups with point group 2/m. 772 00:50:30,400 --> 00:50:31,620 And we've done a few of these. 773 00:50:31,620 --> 00:50:36,020 But what we haven't looked at is the Shcoenflies notation 774 00:50:36,020 --> 00:50:41,610 for the result or the way in which this information is 775 00:50:41,610 --> 00:50:45,005 presented in the international tables. 776 00:50:48,520 --> 00:50:52,990 To introduce you to these gradually, this is the 777 00:50:52,990 --> 00:50:54,570 monoclinic space group-- 778 00:50:54,570 --> 00:50:57,940 these are the monoclinic space groups as presented in the old 779 00:50:57,940 --> 00:51:01,430 international tables for x-ray crystallography. 780 00:51:01,430 --> 00:51:05,360 A little bit later, I will pass out to you a few higher 781 00:51:05,360 --> 00:51:11,690 symmetry space groups that are represented in both the old 782 00:51:11,690 --> 00:51:15,100 international tables and the new international tables. 783 00:51:15,100 --> 00:51:18,780 And as I indicated somewhat disparagingly earlier on, 784 00:51:18,780 --> 00:51:21,920 there is so much information in the new version that 785 00:51:21,920 --> 00:51:24,860 they're very, very hard to use. 786 00:51:24,860 --> 00:51:33,070 But OK, if you open the hand out, you find space group P2. 787 00:51:33,070 --> 00:51:35,160 So that's a twofold axis added to a 788 00:51:35,160 --> 00:51:36,600 primitive monoclinic axis. 789 00:51:39,850 --> 00:51:44,440 You see on the left hand page this situation, this 790 00:51:44,440 --> 00:51:47,790 ridiculous situation, that exists only for monoclinic 791 00:51:47,790 --> 00:51:53,920 crystals the first setting, where the axes are a on the 792 00:51:53,920 --> 00:51:57,020 left hand side running down, and b on the top running from 793 00:51:57,020 --> 00:51:59,970 left to right, and c coming straight up out of the page. 794 00:51:59,970 --> 00:52:08,110 And the second setting, where the unique axis is b, and that 795 00:52:08,110 --> 00:52:12,880 means if you're going to draw the space group with c coming 796 00:52:12,880 --> 00:52:18,460 up, a down, and b to the right, b now is the direction 797 00:52:18,460 --> 00:52:19,450 of the twofold axis. 798 00:52:19,450 --> 00:52:22,100 So the diagram looks completely different. 799 00:52:22,100 --> 00:52:27,770 But it's the same space group except that the first setting 800 00:52:27,770 --> 00:52:33,050 is tilted on its side so that a comes down, b is from left 801 00:52:33,050 --> 00:52:37,390 to right, and c, which is now not a direction of the twofold 802 00:52:37,390 --> 00:52:39,010 axis, comes out. 803 00:52:39,010 --> 00:52:44,060 So there's some jargon here, that geometric jargon. 804 00:52:44,060 --> 00:52:47,300 The atoms occur at different elevations. 805 00:52:47,300 --> 00:52:52,190 So in these two cases, the atoms occur only at elevations 806 00:52:52,190 --> 00:52:54,490 plus z or minus z. 807 00:52:54,490 --> 00:52:57,970 And you see next to the little circles that represent the 808 00:52:57,970 --> 00:52:59,880 symmetry equivalent positions for the 809 00:52:59,880 --> 00:53:01,980 general position a plus. 810 00:53:01,980 --> 00:53:05,360 And that means elevation plus z. 811 00:53:05,360 --> 00:53:08,980 And for the first setting, all the atoms occur at an 812 00:53:08,980 --> 00:53:12,170 elevation plus z for the general position. 813 00:53:12,170 --> 00:53:14,880 For the second setting in the right hand diagram, you're 814 00:53:14,880 --> 00:53:18,090 looking at the twofold axis from the side. 815 00:53:18,090 --> 00:53:20,470 And therefore, one atom is at plus z. 816 00:53:20,470 --> 00:53:22,940 The one that's related by symmetry gets rotated 817 00:53:22,940 --> 00:53:24,080 down to minus z. 818 00:53:24,080 --> 00:53:27,750 So you see a little plus next to one atom, a little minus 819 00:53:27,750 --> 00:53:31,550 next to the next to it. 820 00:53:31,550 --> 00:53:36,670 We need a symbol for a view down along the locus of a 821 00:53:36,670 --> 00:53:37,650 twofold axis. 822 00:53:37,650 --> 00:53:40,520 And that's the little pointed oval that we became familiar 823 00:53:40,520 --> 00:53:42,710 with with the plane groups. 824 00:53:42,710 --> 00:53:45,800 When you're looking at the twofold axis from the side, 825 00:53:45,800 --> 00:53:47,300 how are you going to indicate that? 826 00:53:47,300 --> 00:53:50,420 Well, the convention is to use an arrow. 827 00:53:50,420 --> 00:53:52,950 And that's what you see on the right hand side for the second 828 00:53:52,950 --> 00:53:55,910 setting of P2. 829 00:53:55,910 --> 00:53:59,990 If this were not a twofold axis but a 2 sub 1 screw axis, 830 00:53:59,990 --> 00:54:02,720 it would be a one-sided barb. 831 00:54:02,720 --> 00:54:09,090 So looking down at these two axes with rotations of 180 832 00:54:09,090 --> 00:54:13,020 degrees, this is the symbol for two viewed from the side. 833 00:54:13,020 --> 00:54:17,030 And a one-sided barb would be the symbol for a 834 00:54:17,030 --> 00:54:19,030 2 sub 1 screw axis. 835 00:54:19,030 --> 00:54:23,510 Mercifully, these are the only screw axes that need to be 836 00:54:23,510 --> 00:54:25,810 depicted in a view from the side. 837 00:54:25,810 --> 00:54:30,520 So there's no standard symbol for a sixfold access or a 838 00:54:30,520 --> 00:54:32,650 fourfold axis looked at from the side. 839 00:54:36,570 --> 00:54:40,240 In boldface, in the outer corner of these pages, you see 840 00:54:40,240 --> 00:54:44,590 the international symbol, which is what we used for the 841 00:54:44,590 --> 00:54:49,610 plane groups, the symbol for the lattice type, primitive, 842 00:54:49,610 --> 00:54:52,970 except now we use uppercase symbols for the lattice type 843 00:54:52,970 --> 00:54:56,890 to distinguish the space groups from the plane groups. 844 00:54:56,890 --> 00:55:02,590 Underneath it, you see a C2 superscript 1. 845 00:55:02,590 --> 00:55:04,560 This is the Schoenflies symbol. 846 00:55:04,560 --> 00:55:08,700 You'll recognize C2 as the Schoenflies symbol for a 847 00:55:08,700 --> 00:55:10,470 twofold axis. 848 00:55:10,470 --> 00:55:13,580 Schoenflies' symbol for the space group was to add a 849 00:55:13,580 --> 00:55:17,240 superscript, namely the first one that old [? Artur ?] 850 00:55:17,240 --> 00:55:20,740 got, the second one that [? Artur ?] got from this 851 00:55:20,740 --> 00:55:22,720 point group, the third one he got from the point 852 00:55:22,720 --> 00:55:23,880 group, and so on. 853 00:55:23,880 --> 00:55:27,560 So it's pretty much an arbitrary order, except he 854 00:55:27,560 --> 00:55:32,140 starts with, as you'll see, a twofold access, then replaces 855 00:55:32,140 --> 00:55:36,130 the twofold axis with a screw access. 856 00:55:36,130 --> 00:55:40,790 So if you turn to the next page, you see symbols for not 857 00:55:40,790 --> 00:55:44,810 twofold axes, but 2 sub 1 screw axes along the 858 00:55:44,810 --> 00:55:47,040 edges of the cell. 859 00:55:47,040 --> 00:55:51,420 You see now a different sort of symbol alongside the atoms. 860 00:55:51,420 --> 00:55:52,720 There's one at plus z. 861 00:55:52,720 --> 00:55:55,200 That's the representative one at x, y, z. 862 00:55:55,200 --> 00:55:58,620 If you repeat it by a screw rotation, you rotate it 180 863 00:55:58,620 --> 00:56:02,030 degrees and slide it up by 1/2 of [? c. ?] 864 00:56:02,030 --> 00:56:05,320 So you see the symbol 1/2 plus. 865 00:56:05,320 --> 00:56:06,770 So plus is plus z. 866 00:56:06,770 --> 00:56:11,580 1/2 plus is 1/2 plus z. 867 00:56:11,580 --> 00:56:16,630 Over to the second settings, trying not to sneer too 868 00:56:16,630 --> 00:56:20,910 vigorously, you'll see pointed barbs. 869 00:56:20,910 --> 00:56:22,740 That's the symbol for a twofold axis 870 00:56:22,740 --> 00:56:24,100 viewed from the side. 871 00:56:24,100 --> 00:56:27,950 And then, one of them is plus. 872 00:56:27,950 --> 00:56:30,990 The one that's a little long parallel to the 2 sub 1 screw 873 00:56:30,990 --> 00:56:34,150 axis goes to the other side and goes from plus z down to 874 00:56:34,150 --> 00:56:35,400 the minus z. 875 00:56:38,670 --> 00:56:41,470 Let me ask you a question which people usually don't 876 00:56:41,470 --> 00:56:41,950 think about. 877 00:56:41,950 --> 00:56:44,760 If you look at the first setting and ask, what are the 878 00:56:44,760 --> 00:56:48,020 lengths of the translations? 879 00:56:48,020 --> 00:56:48,505 That's a. 880 00:56:48,505 --> 00:56:50,100 And that's so many angstroms. 881 00:56:50,100 --> 00:56:50,780 And that's b. 882 00:56:50,780 --> 00:56:53,170 That's so many angstroms. 883 00:56:53,170 --> 00:56:57,180 If I ask you that for the second setting, the answer is 884 00:56:57,180 --> 00:56:59,720 a little trickier. 885 00:56:59,720 --> 00:57:02,170 What is the length of this translation? 886 00:57:02,170 --> 00:57:04,540 That's b. 887 00:57:04,540 --> 00:57:06,960 What is the length of this translation? 888 00:57:06,960 --> 00:57:08,120 I'm sorry, that's c. 889 00:57:08,120 --> 00:57:11,430 And this translation is a, right? 890 00:57:11,430 --> 00:57:12,890 Wrong. 891 00:57:12,890 --> 00:57:15,060 That is no longer a. 892 00:57:15,060 --> 00:57:21,815 And the reason for that is that when you have a cell with 893 00:57:21,815 --> 00:57:27,260 non-orthogonal angles in it, let me indicate for the most 894 00:57:27,260 --> 00:57:32,560 general case for a triclinic crystal. 895 00:57:32,560 --> 00:57:35,310 So this is a. 896 00:57:35,310 --> 00:57:37,060 This is b. 897 00:57:37,060 --> 00:57:41,915 And then the third translation normal to that plane is c. 898 00:57:46,040 --> 00:57:51,610 And if we have a structure with atoms in these locations 899 00:57:51,610 --> 00:57:56,050 and we want to project the structure along c, you do 900 00:57:56,050 --> 00:57:57,340 exactly that. 901 00:57:57,340 --> 00:58:01,170 You don't plop the atoms down onto the base of the cell when 902 00:58:01,170 --> 00:58:07,820 you project it because if you did so, the atom that is up in 903 00:58:07,820 --> 00:58:10,470 the neighboring unit cell would come down 904 00:58:10,470 --> 00:58:12,980 to a different location. 905 00:58:12,980 --> 00:58:16,750 And you would not have a pattern that was periodic 906 00:58:16,750 --> 00:58:18,640 based on a lattice. 907 00:58:18,640 --> 00:58:22,580 You get that only if, when you're projecting in an 908 00:58:22,580 --> 00:58:27,750 oblique lattice, you project along the translation. 909 00:58:27,750 --> 00:58:32,080 And then, and only then, can you end up with a pattern that 910 00:58:32,080 --> 00:58:35,320 looks like it has translational periodicity and 911 00:58:35,320 --> 00:58:37,640 really does. 912 00:58:37,640 --> 00:58:40,190 So if a cell is oblique, you do not project just by 913 00:58:40,190 --> 00:58:42,680 plopping the atoms down on the base of the cell. 914 00:58:42,680 --> 00:58:46,450 You project them parallel to the translation that extends 915 00:58:46,450 --> 00:58:50,140 up above the direction onto which you're projecting. 916 00:58:50,140 --> 00:59:01,730 So if our cell in the second setting is monoclinic and this 917 00:59:01,730 --> 00:59:09,000 is a and this b and c comes up, if we look at that cell 918 00:59:09,000 --> 00:59:12,960 from the side as is done in the depiction on the right 919 00:59:12,960 --> 00:59:17,590 hand, then we let this be the direction of b. 920 00:59:17,590 --> 00:59:20,080 So that is b. 921 00:59:20,080 --> 00:59:29,980 But the thing that is at right angles to b is going to be a 922 00:59:29,980 --> 00:59:36,300 times the cosine of beta. 923 00:59:36,300 --> 00:59:37,696 And it's not a itself. 924 00:59:41,000 --> 00:59:42,470 So this is a right angle. 925 00:59:42,470 --> 00:59:45,900 But this is not the lattice translation. 926 00:59:45,900 --> 00:59:49,510 It is the lattice translation times a 927 00:59:49,510 --> 00:59:51,570 trigonometric function. 928 00:59:51,570 --> 00:59:55,350 The triclinic case is a hairy beast. 929 00:59:58,150 --> 01:00:01,830 To calculate the length of these axes and projection, 930 01:00:01,830 --> 01:00:03,910 it's clear what they are. 931 01:00:03,910 --> 01:00:07,210 But to get these values involves alpha 932 01:00:07,210 --> 01:00:09,040 and beta and gamma. 933 01:00:09,040 --> 01:00:11,950 And it's a very, very complicated function. 934 01:00:14,842 --> 01:00:17,975 AUDIENCE: So the angle between a and c in general is not 90 935 01:00:17,975 --> 01:00:19,670 degrees in monoclinics? 936 01:00:19,670 --> 01:00:21,850 PROFESSOR: The angle between depends on how 937 01:00:21,850 --> 01:00:22,970 you label your axes. 938 01:00:22,970 --> 01:00:25,970 For monoclinic, if you do the first setting, then by 939 01:00:25,970 --> 01:00:30,620 definition this is a, and this is b, and this is c. 940 01:00:30,620 --> 01:00:33,910 So these are always by definition 90 degrees. 941 01:00:33,910 --> 01:00:36,530 If you use the second setting, then this is the 942 01:00:36,530 --> 01:00:38,030 direction of b. 943 01:00:38,030 --> 01:00:40,220 And this is the direction of a. 944 01:00:40,220 --> 01:00:42,940 And this is the direction of c. 945 01:00:42,940 --> 01:00:44,800 That keeps the system right handed. 946 01:00:44,800 --> 01:00:47,748 Then, this is a right angle and this is a general angle. 947 01:00:50,556 --> 01:00:52,428 AUDIENCE: But all you're doing is taking that top one and 948 01:00:52,428 --> 01:00:53,840 just [? tilting it over ?]. 949 01:00:53,840 --> 01:00:58,100 PROFESSOR: No, I'm doing more than that because if there 950 01:00:58,100 --> 01:01:00,330 were a twofold axis, that would be the direction of the 951 01:01:00,330 --> 01:01:01,490 twofold axis. 952 01:01:01,490 --> 01:01:03,610 Now, this is the direction of the twofold axis. 953 01:01:06,330 --> 01:01:08,600 The other thing that happens is the symbols 954 01:01:08,600 --> 01:01:10,640 for the glides change. 955 01:01:10,640 --> 01:01:15,950 So if you look at space group number seven, which is a case 956 01:01:15,950 --> 01:01:20,830 where the mirror plane has been replaced by a b glide in 957 01:01:20,830 --> 01:01:23,760 the first setting, and that means the glide is 958 01:01:23,760 --> 01:01:25,650 perpendicular to the twofold axis. 959 01:01:25,650 --> 01:01:31,050 If the twofold axis is now the direction of b, the glide 960 01:01:31,050 --> 01:01:32,500 plane turns into a b glide. 961 01:01:38,410 --> 01:01:40,820 But let's just thumb through them quickly. 962 01:01:40,820 --> 01:01:43,780 And then, we're actually two minutes past my promised 963 01:01:43,780 --> 01:01:45,370 quitting hour. 964 01:01:45,370 --> 01:01:47,340 First one, P2. 965 01:01:47,340 --> 01:01:51,485 Then, replace the twofold access by a 2 sub 1 screw. 966 01:01:51,485 --> 01:01:57,190 So that's Schoenflies symbol C2 superscript 2. 967 01:01:57,190 --> 01:02:00,275 Then, do this with centered lattices. 968 01:02:00,275 --> 01:02:03,170 The international table chooses to use a 969 01:02:03,170 --> 01:02:04,900 side centered cell. 970 01:02:04,900 --> 01:02:07,230 So that's the third one you can get from a twofold axis. 971 01:02:07,230 --> 01:02:13,950 So Schoenflies calls it C subscript 2 superscript 3. 972 01:02:13,950 --> 01:02:17,860 And if you add that to a side centered lattice, you get 973 01:02:17,860 --> 01:02:22,950 screw axes interleaved between the twofold axes. 974 01:02:22,950 --> 01:02:27,300 Put it on its side, it stays C2 superscript 3 in the 975 01:02:27,300 --> 01:02:29,630 Schoenflies notation. 976 01:02:29,630 --> 01:02:36,720 But the side centered d become side centered c because it's b 977 01:02:36,720 --> 01:02:39,330 that comes out of the oblique net. 978 01:02:39,330 --> 01:02:42,470 So the international symbol tells you 979 01:02:42,470 --> 01:02:45,140 exactly what you have. 980 01:02:45,140 --> 01:02:48,080 The Schoenflies symbol is arbitrary. 981 01:02:48,080 --> 01:02:50,520 It tells you the point group and it tells you the order in 982 01:02:50,520 --> 01:02:52,170 which Schoenflies derived them. 983 01:02:52,170 --> 01:02:56,990 But that non-informative nature to the 984 01:02:56,990 --> 01:02:58,750 symbol has a blessing. 985 01:02:58,750 --> 01:03:02,520 And that is it doesn't change with different labellings of 986 01:03:02,520 --> 01:03:04,170 a, b, and c. 987 01:03:04,170 --> 01:03:09,075 It doesn't change with a being shortest, b being shortest. 988 01:03:11,760 --> 01:03:14,580 It just sits there and its dumb, uninformative fashion. 989 01:03:14,580 --> 01:03:15,770 And that is nice. 990 01:03:15,770 --> 01:03:17,450 So both of them have survived. 991 01:03:17,450 --> 01:03:21,910 OK, so could we get another space group out of this by 992 01:03:21,910 --> 01:03:24,540 replacing the twofold axis by a 2 sub 1? 993 01:03:24,540 --> 01:03:29,060 No, because we've already got 2 sub 1 screw axes in C2 994 01:03:29,060 --> 01:03:30,420 superscript 3. 995 01:03:30,420 --> 01:03:35,470 So then, Schoenflies moves on and begins to work with 996 01:03:35,470 --> 01:03:36,700 symmetry planes. 997 01:03:36,700 --> 01:03:40,010 So number six puts a mirror plane in the primitive 998 01:03:40,010 --> 01:03:42,690 monoclinic lattice. 999 01:03:42,690 --> 01:03:44,330 That becomes CS. 1000 01:03:44,330 --> 01:03:47,470 That's the Schoenflies symbol for point group m. 1001 01:03:47,470 --> 01:03:49,790 And it's the first thing you can get. 1002 01:03:49,790 --> 01:03:53,820 Another piece of convention for displaying the 1003 01:03:53,820 --> 01:03:57,650 representative pattern of atoms, in the first setting, 1004 01:03:57,650 --> 01:04:01,070 the two atoms sit directly above one another. 1005 01:04:01,070 --> 01:04:02,960 So there are two new things here. 1006 01:04:02,960 --> 01:04:07,210 A vertical line through the atom indicates that there are 1007 01:04:07,210 --> 01:04:10,350 two of them that are superimposed in projection. 1008 01:04:10,350 --> 01:04:14,640 And moreover, the little tadpole has appeared inside 1009 01:04:14,640 --> 01:04:17,230 the frog eggs in the left hand part. 1010 01:04:17,230 --> 01:04:20,740 So this says on the right hand side of that split circle, the 1011 01:04:20,740 --> 01:04:22,900 atom is at plus z. 1012 01:04:22,900 --> 01:04:25,490 On the left hand side with the comment to indicate an 1013 01:04:25,490 --> 01:04:29,810 enantiomorph is 1 at minus z that sits directly 1014 01:04:29,810 --> 01:04:31,360 below the first one. 1015 01:04:31,360 --> 01:04:33,210 And you'll notice in the diagram of the symmetry 1016 01:04:33,210 --> 01:04:36,230 elements is a chevron working off the lower right hand 1017 01:04:36,230 --> 01:04:37,150 corner of the cell. 1018 01:04:37,150 --> 01:04:39,330 And that's the symbol for a mirror plane that's being 1019 01:04:39,330 --> 01:04:42,110 viewed directly from above. 1020 01:04:42,110 --> 01:04:43,880 And then I'll just go a couple more. 1021 01:04:43,880 --> 01:04:48,400 You could replace the mirror plane with a glide plane. 1022 01:04:48,400 --> 01:04:50,810 And that's a b glide. 1023 01:04:50,810 --> 01:04:54,340 Now, you can see the atoms going up at plus z, slide 1024 01:04:54,340 --> 01:04:58,550 along by 1/2 z, and popped down to minus z, going down to 1025 01:04:58,550 --> 01:05:02,630 an enantiomorph, so there's a comma inside the circle. 1026 01:05:02,630 --> 01:05:06,890 The chevron in the lower right hand corner of the diagram of 1027 01:05:06,890 --> 01:05:09,990 symmetry elements now has acquired an arrow, a barb, 1028 01:05:09,990 --> 01:05:14,060 that indicates the direction of tau. 1029 01:05:14,060 --> 01:05:19,430 The second setting, again, flops the thing on its side so 1030 01:05:19,430 --> 01:05:23,660 that you're looking now down along the b glide. 1031 01:05:23,660 --> 01:05:27,240 And so you see a dotted line that indicates a glide plane 1032 01:05:27,240 --> 01:05:31,070 viewed edge on in the direction of tau. 1033 01:05:31,070 --> 01:05:34,230 Then, you can put a mirror plane for number eight into a 1034 01:05:34,230 --> 01:05:36,960 side centered b lattice. 1035 01:05:36,960 --> 01:05:41,870 And that gives you another space group. 1036 01:05:41,870 --> 01:05:46,520 And then, you can put a glide plane into the b lattice. 1037 01:05:46,520 --> 01:05:51,180 And that, interestingly, gives you a different space group. 1038 01:05:53,790 --> 01:05:57,650 Number eight has an alternating mirror plane with 1039 01:05:57,650 --> 01:06:00,120 an axial glide. 1040 01:06:00,120 --> 01:06:05,680 Number nine has an alternating axial glide alternating with a 1041 01:06:05,680 --> 01:06:07,690 diagonal glide-- two glide planes. 1042 01:06:07,690 --> 01:06:09,820 Not a mirror plane and a glide, but two different kinds 1043 01:06:09,820 --> 01:06:11,180 of glide planes. 1044 01:06:11,180 --> 01:06:14,200 And then, you've used up all the tricks you can pull there. 1045 01:06:14,200 --> 01:06:18,260 So now, we take the last monoclinic point group, 2/m, 1046 01:06:18,260 --> 01:06:20,430 drop it into a primitive lattice. 1047 01:06:20,430 --> 01:06:25,260 Replace the twofold axis with a 2 sub 1 screw axis. 1048 01:06:25,260 --> 01:06:27,650 That gives you P2 sub 1 over m. 1049 01:06:27,650 --> 01:06:31,450 Then, leave the twofold axis alone and leave the plane 1050 01:06:31,450 --> 01:06:34,830 alone and put 2/m in a side centered cell. 1051 01:06:34,830 --> 01:06:39,290 And then, replace the mirror plane by a glide. 1052 01:06:39,290 --> 01:06:41,610 That's number 13. 1053 01:06:41,610 --> 01:06:48,930 And then finally, the summa cum ultra, replace the twofold 1054 01:06:48,930 --> 01:06:52,576 axis y a 2 sub 1 screw and replace the mirror plane with 1055 01:06:52,576 --> 01:06:55,280 a glide, number 14. 1056 01:06:55,280 --> 01:06:59,410 And then, do the same thing in a centered lattice. 1057 01:06:59,410 --> 01:07:04,600 So we get a total of 13 different space groups from 1058 01:07:04,600 --> 01:07:07,890 two lousy kinds of lattices and three 1059 01:07:07,890 --> 01:07:09,310 possible point groups. 1060 01:07:09,310 --> 01:07:14,940 You get 13 different distinct space groups. 1061 01:07:14,940 --> 01:07:16,275 OK, that's a quick overview. 1062 01:07:19,930 --> 01:07:23,140 Notice that just as was the case for the plane groups, the 1063 01:07:23,140 --> 01:07:28,880 tables go on to list the way in which atoms are related by 1064 01:07:28,880 --> 01:07:34,370 symmetry and the number per cell and the site symmetry 1065 01:07:34,370 --> 01:07:36,100 just as in the plane groups. 1066 01:07:36,100 --> 01:07:38,140 And the way a three dimensional structure is going 1067 01:07:38,140 --> 01:07:41,450 to be described to you is exactly analogous to what we 1068 01:07:41,450 --> 01:07:43,370 did with two dimensions. 1069 01:07:43,370 --> 01:07:53,930 Magnesium in position 3b, symmetry 6/m, oxygen in 1070 01:07:53,930 --> 01:07:58,710 position 12d, symmetry 1. 1071 01:07:58,710 --> 01:08:01,730 And then, the tables give you the coordinates of all the 1072 01:08:01,730 --> 01:08:03,540 symmetry S atoms. 1073 01:08:03,540 --> 01:08:05,550 OK, thank you for your patience. 1074 01:08:05,550 --> 01:08:10,010 That was a long stretch in one shot. 1075 01:08:10,010 --> 01:08:15,990 I'll say a little bit, very little, next Tuesday about the 1076 01:08:15,990 --> 01:08:22,029 convention for orthorhombic space groups because there, no 1077 01:08:22,029 --> 01:08:25,080 one direction is any more or less special than any other. 1078 01:08:25,080 --> 01:08:28,290 So the symbol for the space group changes all over the 1079 01:08:28,290 --> 01:08:31,130 place when the axes take on different 1080 01:08:31,130 --> 01:08:32,660 lengths, relative lengths. 1081 01:08:32,660 --> 01:08:35,200 But the symmetry is exactly the same. 1082 01:08:35,200 --> 01:08:36,450 So we'll see you--