1 00:00:06,410 --> 00:00:09,330 PROFESSOR: All right, we've been slogging our way through 2 00:00:09,330 --> 00:00:11,460 derivation of the plane groups. 3 00:00:11,460 --> 00:00:15,400 And I think I'll do a few more, because we'll stumble 4 00:00:15,400 --> 00:00:20,620 across some major tricks in deriving a subfamily of them. 5 00:00:20,620 --> 00:00:26,440 But to not get lost in the forest because of all the 6 00:00:26,440 --> 00:00:30,090 trees, I have a set of notes. 7 00:00:30,090 --> 00:00:34,200 They are handwritten because my secretary would resign if 8 00:00:34,200 --> 00:00:38,200 she had to fit in all these figures and subscripts and 9 00:00:38,200 --> 00:00:39,810 strange symbols. 10 00:00:39,810 --> 00:00:43,880 So, they are as neat as I could make them. 11 00:00:43,880 --> 00:00:47,350 Sorry to say that, in running them through the Xerox machine 12 00:00:47,350 --> 00:00:50,530 in an attempt to get everything on one sheet, some 13 00:00:50,530 --> 00:00:53,100 of the last lines got clipped. 14 00:00:53,100 --> 00:00:59,050 So I'll run these through again and give you a copy 15 00:00:59,050 --> 00:01:01,180 that's minus those truncations. 16 00:01:14,210 --> 00:01:14,500 All right. 17 00:01:14,500 --> 00:01:19,860 What we've been doing so far, to have a brief reprise, was 18 00:01:19,860 --> 00:01:23,640 to take the symmetries, the 10 two-dimensional plane group 19 00:01:23,640 --> 00:01:24,150 symmetries. 20 00:01:24,150 --> 00:01:28,410 And they were one-, two-, three-, four-, or sixfold 21 00:01:28,410 --> 00:01:37,850 axes, a mirror plane, 2mm, 3m, 4mm, 6mm. 22 00:01:37,850 --> 00:01:39,100 So there are 10 of them. 23 00:01:41,330 --> 00:01:45,240 And these are the so-called crystallographic point groups. 24 00:01:45,240 --> 00:01:50,330 Crystallographic because we considered only those rotation 25 00:01:50,330 --> 00:01:52,790 axes that are compatible with a lattice. 26 00:01:52,790 --> 00:01:55,360 And they are point groups because they leave at least 27 00:01:55,360 --> 00:01:59,910 one point in space, invariant-- 28 00:01:59,910 --> 00:02:02,780 stays there rigidly fixed. 29 00:02:02,780 --> 00:02:07,510 And they are groups because the collection of operations 30 00:02:07,510 --> 00:02:11,130 that are present satisfies the postulates of the mathematical 31 00:02:11,130 --> 00:02:14,020 entity called a group. 32 00:02:14,020 --> 00:02:18,190 We've then are in the process of taking these 10 symmetries 33 00:02:18,190 --> 00:02:23,670 and adding them to the 5 two-dimensional lattices. 34 00:02:23,670 --> 00:02:34,480 The parallelogram net, the rectangular, the centered 35 00:02:34,480 --> 00:02:50,605 rectangular, and the square, and the hexagonal. 36 00:02:53,230 --> 00:02:56,950 And clearly, we can't put each of those point groups in every 37 00:02:56,950 --> 00:02:58,380 one of the lattices. 38 00:02:58,380 --> 00:03:02,550 For example, the lattice has to be square for either 4 or 39 00:03:02,550 --> 00:03:08,110 4mm, so we would attempt to place only those two of the 10 40 00:03:08,110 --> 00:03:11,480 point groups into a square net. 41 00:03:11,480 --> 00:03:13,760 We've gone through quite a few of them. 42 00:03:13,760 --> 00:03:15,900 I won't bother to draw the pattern of 43 00:03:15,900 --> 00:03:16,980 the symmetry elements. 44 00:03:16,980 --> 00:03:19,680 But we put no symmetry at all in the general 45 00:03:19,680 --> 00:03:21,250 parallelogram net. 46 00:03:21,250 --> 00:03:24,240 And that is plane group P1. 47 00:03:24,240 --> 00:03:26,310 Maybe I will draw the figures, after all. 48 00:03:26,310 --> 00:03:32,340 P2 was a twofold axis dropped into a net, which had to have 49 00:03:32,340 --> 00:03:36,315 no specialized shape, simply because a twofold [INAUDIBLE] 50 00:03:36,315 --> 00:03:41,150 axis requires nothing but a lattice row if one translation 51 00:03:41,150 --> 00:03:42,400 is combined with it. 52 00:03:44,922 --> 00:03:52,060 The threefold axis could fit into an equilateral net. 53 00:03:52,060 --> 00:03:59,590 And there's the threefold axis we added to a lattice point. 54 00:03:59,590 --> 00:04:02,740 And we have two additional threefold axes in the centers 55 00:04:02,740 --> 00:04:06,440 of those 2 equilateral triangles, which each make up 56 00:04:06,440 --> 00:04:08,350 half the cell. 57 00:04:08,350 --> 00:04:13,750 P4 was the square net. 58 00:04:13,750 --> 00:04:16,589 We put a fourfold axis at the corner of the cell. 59 00:04:16,589 --> 00:04:18,680 We've got another one in the middle. 60 00:04:18,680 --> 00:04:21,945 Two folds in the middle of the edges because of the 61 00:04:21,945 --> 00:04:27,520 180-degree rotation that is built into the fourfold axis. 62 00:04:27,520 --> 00:04:30,430 And P6. 63 00:04:30,430 --> 00:04:33,130 We put that into a hexagonal net. 64 00:04:33,130 --> 00:04:36,900 We've got sixfold axes that we dropped in at the lattice 65 00:04:36,900 --> 00:04:39,430 point only, nothing else. 66 00:04:42,970 --> 00:04:46,430 There's a 120-degree rotation at the lattice point, and that 67 00:04:46,430 --> 00:04:48,510 gives us the threefold axes that are 68 00:04:48,510 --> 00:04:50,530 present in P3, as well. 69 00:04:50,530 --> 00:04:54,510 There's a 180-degree rotation contained in a sixfold axis, 70 00:04:54,510 --> 00:04:59,180 and that gives us the twofold axes in all of the 71 00:04:59,180 --> 00:05:00,430 locations of P2. 72 00:05:04,790 --> 00:05:08,190 OK, I haven't drawn in any representative patterns, but 73 00:05:08,190 --> 00:05:09,360 let me remind you again. 74 00:05:09,360 --> 00:05:13,550 The pattern that is characteristic of every one of 75 00:05:13,550 --> 00:05:17,610 the plane groups is just the pattern that the point group 76 00:05:17,610 --> 00:05:20,460 that you've placed at the lattice point would produce. 77 00:05:20,460 --> 00:05:23,810 And that pattern is, in turn, hung at every lattice point of 78 00:05:23,810 --> 00:05:25,580 the two-dimensional cell. 79 00:05:25,580 --> 00:05:29,880 So even though the huge number of symmetry elements that's 80 00:05:29,880 --> 00:05:32,710 present in the higher symmetry point groups is rather 81 00:05:32,710 --> 00:05:33,820 intimidating-- 82 00:05:33,820 --> 00:05:37,560 you say, wow, how would one draw a pattern for that-- 83 00:05:37,560 --> 00:05:40,840 the pattern is nothing more complicated than the pattern 84 00:05:40,840 --> 00:05:43,530 of the symmetry that you've placed at the lattice point. 85 00:05:43,530 --> 00:05:47,290 And all of these other symmetry elements arise to 86 00:05:47,290 --> 00:05:51,280 express relations between the motifs that you've placed at 87 00:05:51,280 --> 00:05:54,590 the initial representative lattice point. 88 00:05:54,590 --> 00:05:57,220 Deriving these, we used one theorem, which it's well to 89 00:05:57,220 --> 00:05:58,150 remind you of. 90 00:05:58,150 --> 00:06:04,050 And that is that if I have a rotation operation A alpha, 91 00:06:04,050 --> 00:06:07,290 follow that by a translation that's perpendicular to the 92 00:06:07,290 --> 00:06:10,730 rotation axis, what I get is a new rotation 93 00:06:10,730 --> 00:06:12,790 operation is B alpha. 94 00:06:12,790 --> 00:06:15,880 And it's located in a very specific location. 95 00:06:15,880 --> 00:06:20,650 And that is at a location that's a distance x equal half 96 00:06:20,650 --> 00:06:23,050 the magnitude of the translation times the 97 00:06:23,050 --> 00:06:26,570 cotangent of alpha over 2. 98 00:06:26,570 --> 00:06:31,320 So this combination term, if nothing else, reminds us that 99 00:06:31,320 --> 00:06:36,250 these combination theorems, as I call them, are not equations 100 00:06:36,250 --> 00:06:37,890 in symmetry elements. 101 00:06:37,890 --> 00:06:41,680 They are equations in individual operations. 102 00:06:41,680 --> 00:06:46,160 So, for example, if I combine the fourfold axis with the 103 00:06:46,160 --> 00:06:50,750 translation, the 180-degree operation that's present in a 104 00:06:50,750 --> 00:06:53,610 fourfold axis puts B pi here. 105 00:06:53,610 --> 00:06:58,650 The 90-degree rotation puts B pi over 2 here. 106 00:06:58,650 --> 00:07:01,270 And the 270-degree rotation-- 107 00:07:01,270 --> 00:07:04,750 which I can define just as well as A minus pi over 2-- 108 00:07:04,750 --> 00:07:07,950 puts the operation A minus 2 pi over 2 down here. 109 00:07:07,950 --> 00:07:10,480 So it's not an equation in symmetry elements. 110 00:07:10,480 --> 00:07:12,960 It's an equation in individual operations. 111 00:07:16,986 --> 00:07:20,680 Then some peculiarities started to arise, which we 112 00:07:20,680 --> 00:07:23,930 perhaps might not think of. 113 00:07:23,930 --> 00:07:28,400 If we put a mirror plane in a primitive rectangular net, 114 00:07:28,400 --> 00:07:31,980 that gave a group that we called Pm. 115 00:07:31,980 --> 00:07:36,180 If we combine that with a translation, we need a theorem 116 00:07:36,180 --> 00:07:42,590 that says what happens if you combine a mirror reflection 117 00:07:42,590 --> 00:07:45,610 operation with a translation that's perpendicular to it. 118 00:07:45,610 --> 00:07:48,370 We sketch that out, and once and for all could decide that 119 00:07:48,370 --> 00:07:51,670 it's going to be a new reflection operation that's 120 00:07:51,670 --> 00:07:58,010 located halfway along this perpendicular translation. 121 00:07:58,010 --> 00:08:02,670 So this reflection operation sigma, combined with this 122 00:08:02,670 --> 00:08:06,830 translation, gave us a new reflection operation, sigma 123 00:08:06,830 --> 00:08:08,710 prime, halfway along the cell. 124 00:08:08,710 --> 00:08:10,440 And, of course, we have this one hanging at a 125 00:08:10,440 --> 00:08:13,340 lattice point as well. 126 00:08:13,340 --> 00:08:14,730 And then came the interesting one. 127 00:08:14,730 --> 00:08:17,920 When we combined a mirror plane now with the centered 128 00:08:17,920 --> 00:08:25,500 rectangular net, we have all of the mirror planes that we 129 00:08:25,500 --> 00:08:28,590 have here, because this primitive rectangular lattice 130 00:08:28,590 --> 00:08:30,745 is a subgroup of the centered lattice. 131 00:08:30,745 --> 00:08:34,049 Then the interesting thing happened when we combined the 132 00:08:34,049 --> 00:08:39,850 reflection operation sigma with this translation that 133 00:08:39,850 --> 00:08:42,340 went to the center of the cell. 134 00:08:42,340 --> 00:08:45,600 And that gave us a transformation that was 135 00:08:45,600 --> 00:08:49,150 something we had not encountered before. 136 00:08:49,150 --> 00:08:53,940 We took an object, reflected it in this plane, and then 137 00:08:53,940 --> 00:08:56,030 translated it down to a centered 138 00:08:56,030 --> 00:08:58,760 lattice point over here-- 139 00:08:58,760 --> 00:09:01,440 to give us one that sat here-- 140 00:09:01,440 --> 00:09:07,005 and then asked, how did I get from the first one, a 141 00:09:07,005 --> 00:09:10,020 right-handed one perhaps, to a second one that's a 142 00:09:10,020 --> 00:09:13,425 left-handed one, and then a third left-handed one that 143 00:09:13,425 --> 00:09:14,890 sits down here? 144 00:09:14,890 --> 00:09:18,220 The answer is that we found there was no way we could 145 00:09:18,220 --> 00:09:23,230 specify getting from number 1 to number 3 in a single shot. 146 00:09:23,230 --> 00:09:25,880 We had to take two steps to do it. 147 00:09:25,880 --> 00:09:28,890 And there was nothing more simple than that. 148 00:09:28,890 --> 00:09:34,020 We had to first translate down by the part of the 149 00:09:34,020 --> 00:09:34,880 translation-- 150 00:09:34,880 --> 00:09:35,920 let me call it tau-- 151 00:09:35,920 --> 00:09:39,870 which is equal to that part of the translation t, which is 152 00:09:39,870 --> 00:09:42,160 parallel to the reflection operation. 153 00:09:42,160 --> 00:09:46,430 And then we had to reflect across, and that would get us 154 00:09:46,430 --> 00:09:47,920 from the first to the third. 155 00:09:47,920 --> 00:09:50,380 That was a new sort of operation. 156 00:09:50,380 --> 00:09:53,670 We'd indicate its locus by a dashed line to distinguish it 157 00:09:53,670 --> 00:09:54,860 from a mirror plane. 158 00:09:54,860 --> 00:09:58,760 And it has a translation part and a reflection part. 159 00:09:58,760 --> 00:10:00,650 Doesn't matter what order in which we do them. 160 00:10:00,650 --> 00:10:03,787 We get to the same location if we reflect first and then 161 00:10:03,787 --> 00:10:06,880 slide or slide first and then reflect. 162 00:10:06,880 --> 00:10:09,180 So this was a new operation that I'll 163 00:10:09,180 --> 00:10:12,360 represent by sigma tau-- 164 00:10:12,360 --> 00:10:14,790 looks sort of like a symbol for reflection, but the 165 00:10:14,790 --> 00:10:17,990 subscript reminds us you've got to translate by an amount 166 00:10:17,990 --> 00:10:22,050 tau that is parallel to the initial mirror plane. 167 00:10:22,050 --> 00:10:26,010 And this gave us a new theorem that a general translation 168 00:10:26,010 --> 00:10:29,020 that had a part t perpendicular and a component 169 00:10:29,020 --> 00:10:33,520 t parallel, when it followed a reflection operation, was 170 00:10:33,520 --> 00:10:39,125 equal to a net effect of reproducing the object by a 171 00:10:39,125 --> 00:10:41,220 glide plane, sigma tau prime. 172 00:10:41,220 --> 00:10:44,840 It had tau equal to the part of the translation that was 173 00:10:44,840 --> 00:10:48,540 parallel to the reflection part of the locus. 174 00:10:48,540 --> 00:10:52,145 And it was located always at one-half the perpendicular 175 00:10:52,145 --> 00:10:53,395 part of the translation. 176 00:10:56,910 --> 00:11:02,180 So using that theorem and completing the mirror planes 177 00:11:02,180 --> 00:11:08,360 that hang at the lattice points, we have a very 178 00:11:08,360 --> 00:11:11,410 interesting group that consists of a centered 179 00:11:11,410 --> 00:11:16,610 lattice, mirror plane hung at the corner lattice point-- 180 00:11:16,610 --> 00:11:19,965 this is also a lattice point, so we automatically get the 181 00:11:19,965 --> 00:11:22,040 mirror plane in the middle of the cell-- 182 00:11:22,040 --> 00:11:27,260 and then in between, we get this new 2-step operation, the 183 00:11:27,260 --> 00:11:29,050 glide plane. 184 00:11:29,050 --> 00:11:31,400 And the pattern that's representative of this plane 185 00:11:31,400 --> 00:11:35,960 group is, as advertised, just what a mirror plane does. 186 00:11:35,960 --> 00:11:40,340 And that is hung at every lattice point of the centered 187 00:11:40,340 --> 00:11:42,350 rectangular net. 188 00:11:42,350 --> 00:11:46,310 So the glide plane, this new operation that popped up, does 189 00:11:46,310 --> 00:11:50,190 what the new operations did in all of the 190 00:11:50,190 --> 00:11:51,950 other preceding groups. 191 00:11:51,950 --> 00:11:55,220 It tells you how do you get from the pair that you've hung 192 00:11:55,220 --> 00:11:58,920 at the lattice point to these that sit in the 193 00:11:58,920 --> 00:12:00,340 center of the cell. 194 00:12:00,340 --> 00:12:02,660 They're related by translation, but the relation 195 00:12:02,660 --> 00:12:05,910 of one to another of these motifs is by the glide plane. 196 00:12:05,910 --> 00:12:10,270 So that is an operation which has arisen in the group. 197 00:12:10,270 --> 00:12:14,840 And this, then, is a group that we would call Cm. 198 00:12:14,840 --> 00:12:18,210 C for a centered rectangular net, as opposed to the 199 00:12:18,210 --> 00:12:21,255 primitive net, which is the one we got immediately before. 200 00:12:24,450 --> 00:12:27,330 Again, I sound like a cracked record sometimes, but let me 201 00:12:27,330 --> 00:12:30,160 emphasize the simplicity of these patterns. 202 00:12:30,160 --> 00:12:35,140 Again, the pattern consists of what you would get when you 203 00:12:35,140 --> 00:12:39,730 hung a motif on one lattice point, and then that is 204 00:12:39,730 --> 00:12:42,470 repeated by m, which is the symmetry you've placed at a 205 00:12:42,470 --> 00:12:43,280 lattice point. 206 00:12:43,280 --> 00:12:45,500 And that's hung at every lattice point of the centered 207 00:12:45,500 --> 00:12:46,420 rectangular net. 208 00:12:46,420 --> 00:12:50,540 The glide planes just express relations between things that 209 00:12:50,540 --> 00:12:53,580 you already have when you've hung the motifs 210 00:12:53,580 --> 00:12:55,760 on the lattice point. 211 00:12:55,760 --> 00:13:00,740 Any questions after this brief reprieve? 212 00:13:00,740 --> 00:13:01,980 Comments? 213 00:13:01,980 --> 00:13:02,545 Yes, sir. 214 00:13:02,545 --> 00:13:06,790 AUDIENCE: You called those [INAUDIBLE] glides, 215 00:13:06,790 --> 00:13:08,990 that's a sigma tau? 216 00:13:08,990 --> 00:13:10,150 PROFESSOR: Yeah. 217 00:13:10,150 --> 00:13:12,220 The relation between this one and this 218 00:13:12,220 --> 00:13:14,180 one, I've called sigma. 219 00:13:14,180 --> 00:13:16,050 That's the reflection operation. 220 00:13:16,050 --> 00:13:19,290 The relation between this one and this one down here would 221 00:13:19,290 --> 00:13:22,290 be the operation that has a reflection part and a 222 00:13:22,290 --> 00:13:23,540 translation part, tau. 223 00:13:31,730 --> 00:13:36,630 Let me point out something that's worth observing when we 224 00:13:36,630 --> 00:13:39,020 start making some more complex additions. 225 00:13:44,320 --> 00:13:50,220 We said early on that we only have to consider translations 226 00:13:50,220 --> 00:13:52,920 that terminate within the unit cell. 227 00:13:52,920 --> 00:13:57,570 Because everything is translationally periodic-- 228 00:13:57,570 --> 00:14:00,120 not only the atoms in the motifs but the symmetry 229 00:14:00,120 --> 00:14:01,730 elements as well. 230 00:14:01,730 --> 00:14:07,760 But the observation that I want to make is that if we put 231 00:14:07,760 --> 00:14:08,780 a glide plane in. 232 00:14:08,780 --> 00:14:13,340 And let's do that for the direct addition of a glide 233 00:14:13,340 --> 00:14:16,070 plane to a lattice point. 234 00:14:16,070 --> 00:14:21,940 And having done that, we have the potential of possibly 235 00:14:21,940 --> 00:14:24,941 having derived a group that we would call Cg. 236 00:14:31,194 --> 00:14:31,680 OK. 237 00:14:31,680 --> 00:14:36,570 This diagonal translation-- 238 00:14:36,570 --> 00:14:39,030 this is T1, and this is T2. 239 00:14:39,030 --> 00:14:41,850 We might ask, what is the reflect [INAUDIBLE] glide 240 00:14:41,850 --> 00:14:44,980 operation that sits at the origin lattice point, followed 241 00:14:44,980 --> 00:14:46,540 by T1 plus T2. 242 00:14:50,950 --> 00:14:54,480 Well, what does our theorem tell us? 243 00:14:54,480 --> 00:14:58,250 It says that we should get a new glide plane. 244 00:14:58,250 --> 00:15:03,250 It should be at one-half of the perpendicular part of the 245 00:15:03,250 --> 00:15:04,950 translation. 246 00:15:04,950 --> 00:15:07,720 And the perpendicular part of the translation is T2. 247 00:15:11,250 --> 00:15:16,710 So this is at one-half of T2, and it should have a glide 248 00:15:16,710 --> 00:15:18,880 component equal to the parallel part of the 249 00:15:18,880 --> 00:15:21,280 translation, and that is T1. 250 00:15:27,570 --> 00:15:31,780 T1 plus the original glide component, tau. 251 00:15:31,780 --> 00:15:32,510 What is this? 252 00:15:32,510 --> 00:15:41,350 A glide component of half of T1 plus the entire T1. 253 00:15:41,350 --> 00:15:45,060 If we ask, does that make sense? 254 00:15:45,060 --> 00:15:51,360 Yeah, we would glide down to here and then we would 255 00:15:51,360 --> 00:15:53,240 translate down to here. 256 00:15:53,240 --> 00:16:03,960 And that would give us sitting at half of a path of T1, a new 257 00:16:03,960 --> 00:16:05,280 object that sat here. 258 00:16:09,300 --> 00:16:13,935 Is that a glide operation? 259 00:16:16,445 --> 00:16:17,560 Yeah, it is. 260 00:16:17,560 --> 00:16:21,700 But it is one that is not really distinct from the glide 261 00:16:21,700 --> 00:16:23,210 plane that sits here. 262 00:16:23,210 --> 00:16:29,130 Because if we have a glide operation with tau equal to 263 00:16:29,130 --> 00:16:34,310 one-half of T1, and if this sits in a lattice, then 264 00:16:34,310 --> 00:16:39,020 there's going to be a glide plane has tau equal to 1 plus 265 00:16:39,020 --> 00:16:41,410 one-half of T1. 266 00:16:41,410 --> 00:16:43,910 And there will have to be another glide operation that 267 00:16:43,910 --> 00:16:47,380 consists of 2 plus one-half of T1. 268 00:16:47,380 --> 00:16:49,920 And the reason is that everything is periodic at an 269 00:16:49,920 --> 00:16:51,290 interval T1. 270 00:16:51,290 --> 00:16:56,660 So the moral that I'm trying to draw here is that one can 271 00:16:56,660 --> 00:17:04,069 add or subtract to identify the 272 00:17:04,069 --> 00:17:06,589 actual nature of a symmetry. 273 00:17:06,589 --> 00:17:09,440 One can add or subtract an integral number of 274 00:17:09,440 --> 00:17:10,690 translations. 275 00:17:19,560 --> 00:17:33,960 And that permits one to reduce any tau to a sigma tau prime, 276 00:17:33,960 --> 00:17:38,170 such that tau prime is always less than the translation 277 00:17:38,170 --> 00:17:39,470 that's parallel to the tau. 278 00:17:39,470 --> 00:17:43,190 In other words, lop off this translation of an entire T1, 279 00:17:43,190 --> 00:17:47,790 this translation of the entire T2, and you have identified 280 00:17:47,790 --> 00:17:51,610 the basic nature of the glide operation that sits here as 281 00:17:51,610 --> 00:17:54,830 something with a translation that's half of T1. 282 00:17:54,830 --> 00:17:58,750 The translations that move motifs out of the cell may be 283 00:17:58,750 --> 00:18:02,530 related by a glide operation that involves an integral 284 00:18:02,530 --> 00:18:04,720 number of T1s plus half of T1. 285 00:18:04,720 --> 00:18:08,290 But it doesn't change the basic nature of the simplest 286 00:18:08,290 --> 00:18:10,450 glide step that's in there. 287 00:18:10,450 --> 00:18:12,950 That's a very obscure explanation of probably 288 00:18:12,950 --> 00:18:15,510 something that didn't puzzle you in the first place, but 289 00:18:15,510 --> 00:18:18,630 it's worth saying when we make some of these additions. 290 00:18:18,630 --> 00:18:20,000 Let's finish this off. 291 00:18:20,000 --> 00:18:22,340 We have another translation in here, and that's the 292 00:18:22,340 --> 00:18:30,240 translation T1 plus T2 over 2. 293 00:18:30,240 --> 00:18:36,840 So, what would we get if we took the glide operation sigma 294 00:18:36,840 --> 00:18:43,740 one-half of T1 and followed that by a translation T1 295 00:18:43,740 --> 00:18:45,920 plus T2 over 2. 296 00:18:45,920 --> 00:18:50,520 And that operation, again, would be equal to a glide 297 00:18:50,520 --> 00:18:55,250 plane sigma prime with the tau equal to the original glide 298 00:18:55,250 --> 00:18:59,500 component plus the part of the translation that is parallel 299 00:18:59,500 --> 00:19:03,040 to the glide plane. 300 00:19:03,040 --> 00:19:06,980 And it'd be located at one-half of the perpendicular 301 00:19:06,980 --> 00:19:10,330 part of the translation, which is T2. 302 00:19:10,330 --> 00:19:14,820 So it'd be at one-half of one-half of T2. 303 00:19:14,820 --> 00:19:18,050 So this says that the combination of the glide 304 00:19:18,050 --> 00:19:22,060 operation with the centered translation is a glide plane 305 00:19:22,060 --> 00:19:25,900 with a glide component T1. 306 00:19:25,900 --> 00:19:30,920 And it's located at one-quarter of T2. 307 00:19:35,500 --> 00:19:42,390 And that, indeed, is what you would do if you reflected and 308 00:19:42,390 --> 00:19:46,160 reproduced the object by a glide down to here and then 309 00:19:46,160 --> 00:19:50,900 translated by T2 over to here. 310 00:19:54,570 --> 00:19:54,810 OK. 311 00:19:54,810 --> 00:19:55,270 No, I'm sorry. 312 00:19:55,270 --> 00:19:57,920 We glided and then we added on the parallel part of the 313 00:19:57,920 --> 00:19:58,810 translation. 314 00:19:58,810 --> 00:20:01,840 So we would end up down here. 315 00:20:05,660 --> 00:20:07,760 And if one sits here, there has to be one 316 00:20:07,760 --> 00:20:11,130 repeated by T1 up here. 317 00:20:11,130 --> 00:20:15,270 And this is exactly the same thing as a reflection plane 318 00:20:15,270 --> 00:20:16,550 that's been introduced. 319 00:20:16,550 --> 00:20:21,260 So here is a case where we could subtract off the entire 320 00:20:21,260 --> 00:20:25,640 translation T1 and say this is identical to a mirror plane 321 00:20:25,640 --> 00:20:30,090 passing through the origin, a pure mirror plane sigma that 322 00:20:30,090 --> 00:20:35,010 is at one-quarter of T2. 323 00:20:35,010 --> 00:20:42,260 So, let me clean this up and show you what we have. 324 00:20:42,260 --> 00:20:48,350 Completing the operations, we would have a pair of objects 325 00:20:48,350 --> 00:20:51,820 related by glide, like this, that is hung at 326 00:20:51,820 --> 00:20:54,070 every lattice point. 327 00:20:54,070 --> 00:20:56,420 It's also hung at the centered lattice point. 328 00:20:56,420 --> 00:21:00,330 And what that is going to give us is a pair of objects that 329 00:21:00,330 --> 00:21:03,500 sit like this. 330 00:21:03,500 --> 00:21:07,730 And what has come in as a result of those combinations 331 00:21:07,730 --> 00:21:11,245 is a mirror plane interleaved between the glide planes. 332 00:21:16,870 --> 00:21:23,020 And this is exactly what we have in Cm, which was also 333 00:21:23,020 --> 00:21:27,100 interleaved mirror planes and glide planes with the origin 334 00:21:27,100 --> 00:21:40,820 shifted by one-half one-quarter of T2. 335 00:21:40,820 --> 00:21:43,660 So this is not in a group. 336 00:21:43,660 --> 00:21:47,020 Proceeding logically, we'd take Cm and replace the mirror 337 00:21:47,020 --> 00:21:48,450 plane by a glide. 338 00:21:48,450 --> 00:21:51,400 When we do that, we have a consistent group. 339 00:21:51,400 --> 00:21:53,950 But it turns out to be exactly the same arrangement of 340 00:21:53,950 --> 00:21:57,830 symmetry elements and exactly the same pattern as Cm, but 341 00:21:57,830 --> 00:22:01,500 with a little nudge over along T2 by one-quarter of that 342 00:22:01,500 --> 00:22:02,750 translation. 343 00:22:04,740 --> 00:22:05,070 Yes. 344 00:22:05,070 --> 00:22:07,284 AUDIENCE: But in Cm, didn't we have mirror planes going 345 00:22:07,284 --> 00:22:09,010 through the lattice points? 346 00:22:09,010 --> 00:22:09,400 PROFESSOR: Yeah. 347 00:22:09,400 --> 00:22:11,750 And what I'm saying is here there are glide planes going 348 00:22:11,750 --> 00:22:14,780 through, but the lattice point is arbitrary. 349 00:22:14,780 --> 00:22:16,040 I can put it anywhere I like. 350 00:22:16,040 --> 00:22:20,550 And if I decide to put the lattice point here, that turns 351 00:22:20,550 --> 00:22:21,800 it into Cm. 352 00:22:24,136 --> 00:22:25,520 OK? 353 00:22:25,520 --> 00:22:31,130 So, let's put the two of them side by side. 354 00:22:31,130 --> 00:22:37,140 Cm was this mirror plane, mirror plane, mirror plane 355 00:22:37,140 --> 00:22:42,685 glides, and the atoms motifs did this. 356 00:22:45,570 --> 00:22:47,340 A lattice point here in the center. 357 00:22:47,340 --> 00:22:55,010 And I'll deliberately, to make my case, offset the cell by 358 00:22:55,010 --> 00:22:56,130 one-quarter of T1. 359 00:22:56,130 --> 00:22:59,870 Now I've got a glide plane here, same as this one. 360 00:22:59,870 --> 00:23:01,970 Mirror plane at a quarter of T1. 361 00:23:01,970 --> 00:23:05,210 Glide plane here at the centered lattice point. 362 00:23:05,210 --> 00:23:09,610 Mirror plane here, glide point T2 away. 363 00:23:09,610 --> 00:23:11,640 And here are the lattice points. 364 00:23:11,640 --> 00:23:21,670 But the pattern of objects looks like this. 365 00:23:25,000 --> 00:23:27,920 Armed in advance on what the thing has to look like. 366 00:23:27,920 --> 00:23:28,710 Looks like this. 367 00:23:28,710 --> 00:23:31,240 And this is exactly the same pattern of motifs. 368 00:23:40,510 --> 00:23:40,840 OK. 369 00:23:40,840 --> 00:23:46,410 So, coming out of this consideration, we have with 370 00:23:46,410 --> 00:23:56,400 the rectangular nets Pm, Cm, Pg. 371 00:23:56,400 --> 00:24:04,090 But Cg, if we try to construct it, was the same as Cm. 372 00:24:04,090 --> 00:24:07,120 And that exhausts the possibilities for a single 373 00:24:07,120 --> 00:24:10,555 symmetry plane and the rectangular nets. 374 00:24:24,160 --> 00:24:24,490 OK. 375 00:24:24,490 --> 00:24:27,510 Let me move on then to the next step. 376 00:24:27,510 --> 00:24:30,170 And I'm going to skip over a threefold axis. 377 00:24:30,170 --> 00:24:36,190 And I'm going to look at the square net combined with the 378 00:24:36,190 --> 00:24:38,900 other symmetry that would require a net with this 379 00:24:38,900 --> 00:24:39,770 dimensionality. 380 00:24:39,770 --> 00:24:47,280 And this would be a primitive square lattice plus 4 mm. 381 00:24:53,130 --> 00:24:55,030 We've already done almost all the work. 382 00:24:55,030 --> 00:24:59,130 So as we derive the symmetries that are subgroups of these 383 00:24:59,130 --> 00:25:02,840 higher symmetries, we've done P4. 384 00:25:02,840 --> 00:25:06,525 And that says that if we put a fourfold access at the origin, 385 00:25:06,525 --> 00:25:09,090 that fourfold axis, we'll get another one in the 386 00:25:09,090 --> 00:25:10,400 center of the cell. 387 00:25:10,400 --> 00:25:13,790 And we'll get twofold axes in the midpoints of all the edges 388 00:25:13,790 --> 00:25:16,770 of the cell. 389 00:25:16,770 --> 00:25:21,200 And then 4 mm has one kind of mirror plane. 390 00:25:21,200 --> 00:25:24,230 Two Ms because there are two kinds of mirror planes. 391 00:25:24,230 --> 00:25:28,000 The mirror plane says, hey, if I'm in a lattice, I want to be 392 00:25:28,000 --> 00:25:32,060 at right angles to a rectangular or a centered 393 00:25:32,060 --> 00:25:33,080 rectangular net. 394 00:25:33,080 --> 00:25:34,010 Well, OK. 395 00:25:34,010 --> 00:25:38,320 This one is happy, because a square net is a special case 396 00:25:38,320 --> 00:25:39,490 of a rectangular net. 397 00:25:39,490 --> 00:25:41,820 The two translations are merely equal. 398 00:25:41,820 --> 00:25:43,620 So he is happy. 399 00:25:43,620 --> 00:25:46,540 Combine that with T2, and we'll get another mirror plane 400 00:25:46,540 --> 00:25:49,930 like this, and another mirror plane like this. 401 00:25:49,930 --> 00:25:53,000 Fourfold axis is going to rotate that mirror plane, so 402 00:25:53,000 --> 00:25:56,230 we'll have mirror planes running this way, 403 00:25:56,230 --> 00:25:57,480 and this way as well. 404 00:26:02,570 --> 00:26:08,110 And now we have a different kind of mirror plane that we 405 00:26:08,110 --> 00:26:10,290 tried to put in in this location. 406 00:26:10,290 --> 00:26:14,190 This mirror plane says, hey, I have to be parallel to the 407 00:26:14,190 --> 00:26:19,520 edge of a rectangular net or a centered rectangular net. 408 00:26:19,520 --> 00:26:24,530 So if we look at the translations that are parallel 409 00:26:24,530 --> 00:26:33,060 to and perpendicular to this translation, lo and behold, 410 00:26:33,060 --> 00:26:41,080 this mirror plane is aligned along the edge of a centered 411 00:26:41,080 --> 00:26:44,830 rectangular net that has the additional specialization of 412 00:26:44,830 --> 00:26:48,480 being a centered square. 413 00:26:48,480 --> 00:26:51,040 But that mirror plane now is perfectly happy. 414 00:26:51,040 --> 00:26:52,520 And he says, OK, I'll hold my piece. 415 00:26:52,520 --> 00:26:56,820 I have the arrangements of translations relative to my 416 00:26:56,820 --> 00:26:58,560 orientation that makes me happy. 417 00:26:58,560 --> 00:27:01,160 So I can say now that there is a mirror plane 418 00:27:01,160 --> 00:27:02,330 running this way. 419 00:27:02,330 --> 00:27:05,070 There has to be another mirror plane 90 degrees away from 420 00:27:05,070 --> 00:27:06,320 those orientations. 421 00:27:09,410 --> 00:27:13,192 And this, then, is going to be the location of all the mirror 422 00:27:13,192 --> 00:27:16,150 planes in that net. 423 00:27:16,150 --> 00:27:19,840 And at no time did the chalk ever leave my fingers. 424 00:27:19,840 --> 00:27:22,190 Just doing what we said we're doing. 425 00:27:22,190 --> 00:27:26,830 Putting in 4 mm, making sure the requirements imposed on 426 00:27:26,830 --> 00:27:29,950 the lattice are those that that symmetry element demands. 427 00:27:29,950 --> 00:27:31,200 And here we go. 428 00:27:33,710 --> 00:27:39,080 Except that there is now one other combination that we have 429 00:27:39,080 --> 00:27:41,410 not considered. 430 00:27:41,410 --> 00:27:43,940 Here is a mirror plane. 431 00:27:43,940 --> 00:27:46,870 And now this mirror plane is diagonal to 432 00:27:46,870 --> 00:27:49,360 our translation T2. 433 00:27:49,360 --> 00:27:50,970 And here is a mirror plane. 434 00:27:50,970 --> 00:27:53,130 And that mirror plane is diagonal to 435 00:27:53,130 --> 00:27:56,440 our translation T1. 436 00:27:56,440 --> 00:27:58,810 So what is that going to require? 437 00:27:58,810 --> 00:28:02,450 Here is the mirror plane, that's the operation sigma. 438 00:28:02,450 --> 00:28:08,240 And here is our translation, T1, down to here. 439 00:28:08,240 --> 00:28:12,770 We have a theorem that says that a reflection followed by 440 00:28:12,770 --> 00:28:16,550 a translation that has a general orientation is going 441 00:28:16,550 --> 00:28:20,560 to give me a new reflection operation, sigma prime, that's 442 00:28:20,560 --> 00:28:23,420 located at one-half the perpendicular part of the 443 00:28:23,420 --> 00:28:24,910 translation. 444 00:28:24,910 --> 00:28:30,450 And it's going to have a glide component that is equal to 445 00:28:30,450 --> 00:28:34,690 one-half of the parallel part of the translation. 446 00:28:34,690 --> 00:28:36,550 So what has that told us? 447 00:28:39,100 --> 00:28:45,860 To get from this mirror plane down to here, we've gone this 448 00:28:45,860 --> 00:28:53,210 far in a sense that is perpendicular to the mirror 449 00:28:53,210 --> 00:28:55,880 plane, so this is T perpendicular. 450 00:28:55,880 --> 00:28:59,830 And we have a part of the translation that is parallel 451 00:28:59,830 --> 00:29:00,670 to the mirror plane. 452 00:29:00,670 --> 00:29:02,840 This is T parallel. 453 00:29:02,840 --> 00:29:07,040 So the plane that results is going to be a 454 00:29:07,040 --> 00:29:11,010 glide plane in here. 455 00:29:11,010 --> 00:29:16,500 It's located at one-half of the perpendicular part of the 456 00:29:16,500 --> 00:29:17,400 translation. 457 00:29:17,400 --> 00:29:29,740 And that is one-half of one-half of T1 plus T2. 458 00:29:35,620 --> 00:29:39,440 And it has a glide component equal to T parallel, which is 459 00:29:39,440 --> 00:29:44,640 equal to one-half of T1 plus T2. 460 00:29:44,640 --> 00:29:47,200 So we get a new glide plane in here. 461 00:29:47,200 --> 00:29:51,280 And that will require glide planes through similar 462 00:29:51,280 --> 00:29:53,480 arguments that go down the diagonals of 463 00:29:53,480 --> 00:29:56,780 the cell, like this. 464 00:29:56,780 --> 00:30:01,170 And let me convince you that that, in fact, is an operation 465 00:30:01,170 --> 00:30:09,400 that must arise if I place, at the origin of the cell, a set 466 00:30:09,400 --> 00:30:14,950 of objects that have 4 mm symmetry. 467 00:30:14,950 --> 00:30:19,350 We have one like this, one like this, one like this. 468 00:30:19,350 --> 00:30:21,530 Another pair hanging here. 469 00:30:21,530 --> 00:30:24,080 Another pair hanging here. 470 00:30:24,080 --> 00:30:29,330 And if we do a reflection operation, let's say this pair 471 00:30:29,330 --> 00:30:33,290 up to this pair, and then slide it down by the diagonal 472 00:30:33,290 --> 00:30:34,540 translation-- 473 00:30:37,760 --> 00:30:41,320 and we have the same set, again, hanging down here at 474 00:30:41,320 --> 00:30:44,470 the diagonally opposed lattice point. 475 00:30:49,530 --> 00:30:52,820 Reflect across, slide down to here. 476 00:30:52,820 --> 00:31:00,440 The way in which you get that is, believe it or not, to 477 00:31:00,440 --> 00:31:03,250 reflect across this glide plane. 478 00:31:07,194 --> 00:31:11,020 How do we get there? 479 00:31:11,020 --> 00:31:13,810 We reflect across. 480 00:31:13,810 --> 00:31:16,150 We translate down to here. 481 00:31:16,150 --> 00:31:20,180 And the way I do that is by reflecting across this 482 00:31:20,180 --> 00:31:22,170 diagonal glide. 483 00:31:22,170 --> 00:31:24,020 That probably has convinced no one. 484 00:31:24,020 --> 00:31:27,780 But map it out with your own pattern, and I 485 00:31:27,780 --> 00:31:30,970 think you'll agree. 486 00:31:30,970 --> 00:31:35,840 Let me observe, while you're considering that. 487 00:31:35,840 --> 00:31:40,770 When we derive groups based on 3m or 6mm, we're going to have 488 00:31:40,770 --> 00:31:44,210 other cases where a translation along the edge of 489 00:31:44,210 --> 00:31:49,560 the cell is inclined to a mirror plane. 490 00:31:49,560 --> 00:31:52,550 And the effect of combining a translation with a mirror 491 00:31:52,550 --> 00:31:56,120 plane that's inclined to that translation always has the 492 00:31:56,120 --> 00:32:00,710 effect of interposing a glide plane halfway in between the 493 00:32:00,710 --> 00:32:05,160 mirror planes that are related by translation. 494 00:32:05,160 --> 00:32:08,500 So, let me say that in general, because that will be 495 00:32:08,500 --> 00:32:13,690 an observation that'll let us identify glide planes quickly. 496 00:32:13,690 --> 00:32:25,790 So the general resolve is if we have a translation inclined 497 00:32:25,790 --> 00:32:36,180 to a mirror plane, which of necessity is repeated by 498 00:32:36,180 --> 00:32:54,450 translation, a quite general result is that a glide plane 499 00:32:54,450 --> 00:33:06,590 is interleaved always between the two. 500 00:33:06,590 --> 00:33:09,360 And they'll be parallel because they're related by 501 00:33:09,360 --> 00:33:10,610 translation. 502 00:33:20,130 --> 00:33:22,780 So we'll see some more cases where we can immediately state 503 00:33:22,780 --> 00:33:24,134 that without further thought. 504 00:33:46,170 --> 00:33:46,460 OK. 505 00:33:46,460 --> 00:33:49,160 There is something that we might consider doing. 506 00:33:49,160 --> 00:33:50,900 I'd like to put it off, though. 507 00:33:50,900 --> 00:33:53,230 Here we start with a mirror plane. 508 00:33:53,230 --> 00:33:58,470 Can we replace the mirror plane with a glide plane? 509 00:33:58,470 --> 00:34:01,400 The answer is yes, we can. 510 00:34:01,400 --> 00:34:05,740 But it's not at all clear if I take a fourfold axis and put a 511 00:34:05,740 --> 00:34:08,840 glide plane through it, what this plane has to be. 512 00:34:08,840 --> 00:34:11,710 So I'd like to leave this one for now and come back to that. 513 00:34:18,400 --> 00:34:18,670 OK. 514 00:34:18,670 --> 00:34:24,560 In the notes, I've tried to do all of these derivations 515 00:34:24,560 --> 00:34:26,520 thoroughly and logically. 516 00:34:26,520 --> 00:34:30,489 So if this is a bit fast or me waving my hands, saying a 517 00:34:30,489 --> 00:34:32,540 glide plane is here, and this, that, and the other thing, 518 00:34:32,540 --> 00:34:34,980 when you can't really see what I'm pointing at, I think if 519 00:34:34,980 --> 00:34:36,570 you refer to the notes that'll be clear. 520 00:34:36,570 --> 00:34:37,699 But yes, you had a question? 521 00:34:37,699 --> 00:34:40,184 AUDIENCE: So you said that this glide plane [INAUDIBLE] 522 00:34:40,184 --> 00:34:41,675 one-half T perpendicular. 523 00:34:41,675 --> 00:34:43,663 What did you write underneath that? 524 00:34:43,663 --> 00:34:48,384 I guess that maybe is where your T and one-half T 525 00:34:48,384 --> 00:34:49,850 perpendicular is where the glide plane is. 526 00:34:49,850 --> 00:34:50,190 PROFESSOR: OK. 527 00:34:50,190 --> 00:34:54,380 I was just demonstrating that, in fact, for this mirror plane 528 00:34:54,380 --> 00:34:58,760 with the diagonal translation, the glide plane would come in 529 00:34:58,760 --> 00:35:02,570 at one-half of the perpendicular part of the 530 00:35:02,570 --> 00:35:04,560 translation. 531 00:35:04,560 --> 00:35:09,000 And the part of the translation that-- 532 00:35:09,000 --> 00:35:11,610 we were combining this fourfold access with this 533 00:35:11,610 --> 00:35:12,880 translation. 534 00:35:12,880 --> 00:35:16,310 So the perpendicular part of the translation is half of the 535 00:35:16,310 --> 00:35:17,770 body diagonal. 536 00:35:17,770 --> 00:35:20,840 And if a glide plane comes in one-half of the perpendicular 537 00:35:20,840 --> 00:35:24,335 part of the translation, that put it parallel to the mirror 538 00:35:24,335 --> 00:35:27,760 plane and at one-quarter of the way of the translation. 539 00:35:27,760 --> 00:35:31,640 So this little scrawl here says it one-half of T 540 00:35:31,640 --> 00:35:40,570 perpendicular and one-half of one-half of the body diagonal. 541 00:35:40,570 --> 00:35:42,490 So it's one-quarter. 542 00:35:42,490 --> 00:35:45,170 All this equals one-quarter of T1 plus T2. 543 00:35:58,690 --> 00:35:58,960 OK. 544 00:35:58,960 --> 00:36:02,480 Now something unexpected happens. 545 00:36:02,480 --> 00:36:08,770 If we would move along, I skipped over 3m. 546 00:36:08,770 --> 00:36:11,690 We know what P3 looks like. 547 00:36:11,690 --> 00:36:20,420 And that is a hexagonal net, an equilateral net. 548 00:36:20,420 --> 00:36:23,750 And we have a threefold axis that we've added to the 549 00:36:23,750 --> 00:36:25,270 lattice point. 550 00:36:25,270 --> 00:36:29,020 And then we found additional threefold axes in the center 551 00:36:29,020 --> 00:36:31,700 of these triangles. 552 00:36:31,700 --> 00:36:35,410 And, once again, the pattern that has this symmetry is just 553 00:36:35,410 --> 00:36:40,310 a triangle of objects hung at every lattice point. 554 00:36:40,310 --> 00:36:46,560 And now we want to add a mirror plane to 555 00:36:46,560 --> 00:36:49,010 that threefold axis. 556 00:36:49,010 --> 00:36:57,240 So we have a primitive equilateral net plus the 557 00:36:57,240 --> 00:37:05,560 two-dimensional plane point group 3m. 558 00:37:05,560 --> 00:37:08,930 The question is, which way should we 559 00:37:08,930 --> 00:37:10,220 orient the mirror plane? 560 00:37:13,890 --> 00:37:16,950 Well, a mirror plane says, I want to be along the edge of 561 00:37:16,950 --> 00:37:19,110 either a centered rectangular net or a 562 00:37:19,110 --> 00:37:20,760 primitive rectangular net. 563 00:37:20,760 --> 00:37:23,250 There's nothing about this lattice that looks 564 00:37:23,250 --> 00:37:25,920 rectangular. 565 00:37:25,920 --> 00:37:27,346 Yeah, there is. 566 00:37:27,346 --> 00:37:30,020 I see a couple of people doing this, and they 567 00:37:30,020 --> 00:37:32,460 have the right idea. 568 00:37:32,460 --> 00:37:33,560 Here's a lattice point. 569 00:37:33,560 --> 00:37:39,340 If we go more than one unit cell in this diagram, here's a 570 00:37:39,340 --> 00:37:40,140 lattice point. 571 00:37:40,140 --> 00:37:41,120 Here's a lattice point. 572 00:37:41,120 --> 00:37:42,270 Here's a lattice point. 573 00:37:42,270 --> 00:37:43,740 Here's a lattice point. 574 00:37:43,740 --> 00:37:51,090 Go down to here and go over to here. 575 00:37:51,090 --> 00:37:55,350 And lo and behold, here is a centered rectangular net, 576 00:37:55,350 --> 00:37:58,320 hiding incognito in a hexagonal net. 577 00:38:03,170 --> 00:38:04,010 Well, that helps. 578 00:38:04,010 --> 00:38:09,800 I can put in the mirror plane. 579 00:38:09,800 --> 00:38:12,680 But let me point out-- this is getting very messy, so I'll 580 00:38:12,680 --> 00:38:14,090 redraw it on a smaller scale. 581 00:38:25,740 --> 00:38:30,825 This will be worth doing so that it's perfectly obvious. 582 00:38:30,825 --> 00:38:35,155 I'm going to draw a number of equilateral nets. 583 00:38:41,590 --> 00:38:43,880 The reason I'm drawing more than one cell, I would like to 584 00:38:43,880 --> 00:38:49,640 point out that here is one centered rectangular net. 585 00:38:49,640 --> 00:38:55,240 And it has its edge perpendicular to the 586 00:38:55,240 --> 00:38:57,830 translations in the hexagonal net. 587 00:38:57,830 --> 00:39:03,290 But there is also a rectangular net that 588 00:39:03,290 --> 00:39:05,835 goes down like this. 589 00:39:10,870 --> 00:39:14,740 That's a centered rectangular net that has its edge along 590 00:39:14,740 --> 00:39:18,790 one of edges of the unit cell. 591 00:39:18,790 --> 00:39:20,310 So I have two possibilities. 592 00:39:20,310 --> 00:39:23,340 One is to draw the rectangular net like this. 593 00:39:23,340 --> 00:39:27,470 The other one is to draw the rectangular net like this. 594 00:39:27,470 --> 00:39:31,500 And having confused you thoroughly, let me simply draw 595 00:39:31,500 --> 00:39:33,415 this larger cell again. 596 00:39:37,550 --> 00:39:41,430 And say that I could, if I wanted to, put the mirror 597 00:39:41,430 --> 00:39:47,210 planes in in these directions, along the edges of the cell. 598 00:39:52,900 --> 00:39:53,470 OK. 599 00:39:53,470 --> 00:39:56,470 And in this case, if you're convinced that there's a 600 00:39:56,470 --> 00:39:59,285 centered rectangular net sitting there. 601 00:39:59,285 --> 00:40:02,220 Let me clean this up. 602 00:40:02,220 --> 00:40:04,970 And here's one mirror plane. 603 00:40:04,970 --> 00:40:07,300 60 degrees away is another mirror plane. 604 00:40:07,300 --> 00:40:10,580 And there'll be another mirror plane at this lattice point 605 00:40:10,580 --> 00:40:11,830 doing this. 606 00:40:15,130 --> 00:40:21,540 So now I've got 3m at every one of these threefold axes. 607 00:40:21,540 --> 00:40:27,175 And I got all those simply by repeating these mirror planes 608 00:40:27,175 --> 00:40:28,380 by translation. 609 00:40:28,380 --> 00:40:31,940 But in one case, the mirror planes are perpendicular to 610 00:40:31,940 --> 00:40:33,120 the cell edge. 611 00:40:33,120 --> 00:40:37,350 This being T1, let's say, and this being T2. 612 00:40:37,350 --> 00:40:39,600 In this case, the mirror planes are along the 613 00:40:39,600 --> 00:40:41,350 edges of the cell. 614 00:40:41,350 --> 00:40:46,000 This being T1, and this being T2. 615 00:40:46,000 --> 00:40:46,880 So, holy mackerel! 616 00:40:46,880 --> 00:40:49,220 One point group. 617 00:40:49,220 --> 00:40:53,220 Two different space groups, which differ in the way in 618 00:40:53,220 --> 00:40:57,845 which the symmetry is oriented relative to the lattice. 619 00:41:00,470 --> 00:41:03,370 And if you think back a little bit, this particular net, with 620 00:41:03,370 --> 00:41:06,790 this dimensional specialization, was happy with 621 00:41:06,790 --> 00:41:11,840 a sixfold access as well, and will be compatible with 6mm. 622 00:41:11,840 --> 00:41:14,825 So, really, these two different orientations for the 623 00:41:14,825 --> 00:41:17,600 mirror planes, even though we haven't got there yet, are the 624 00:41:17,600 --> 00:41:20,120 two different orientations which are both present 625 00:41:20,120 --> 00:41:24,970 simultaneously if we were to put 6mm into 626 00:41:24,970 --> 00:41:27,720 this hexagonal net. 627 00:41:27,720 --> 00:41:30,090 So, same point group. 628 00:41:30,090 --> 00:41:31,530 Same lattice. 629 00:41:31,530 --> 00:41:35,430 Two different plane groups that depend on the orientation 630 00:41:35,430 --> 00:41:37,490 of the symmetry relative to the lattice. 631 00:41:40,680 --> 00:41:43,250 So there isn't even a one-to-one correspondence 632 00:41:43,250 --> 00:41:46,045 between the point groups and the plane groups. 633 00:41:49,280 --> 00:41:54,340 The pattern for either of these is, once again, going to 634 00:41:54,340 --> 00:41:58,870 be just what the pattern of the point group does. 635 00:41:58,870 --> 00:42:03,120 If we start with a motif here and repeat that by the 636 00:42:03,120 --> 00:42:07,640 threefold axis and the mirror planes, we would 637 00:42:07,640 --> 00:42:14,250 have motifs like this. 638 00:42:14,250 --> 00:42:21,560 And here the translation points out in between the 639 00:42:21,560 --> 00:42:22,890 mirror planes. 640 00:42:22,890 --> 00:42:27,060 In this case, the mirror planes would repeat the 641 00:42:27,060 --> 00:42:29,495 objects in two locations, like this. 642 00:42:35,890 --> 00:42:38,290 So, indeed, both patterns have 3mm symmetry. 643 00:42:38,290 --> 00:42:41,600 And what's different is the way the translations come out 644 00:42:41,600 --> 00:42:44,780 in between those mirror planes. 645 00:42:44,780 --> 00:42:44,990 OK. 646 00:42:44,990 --> 00:42:49,300 Now our notation has come up against an impasse. 647 00:42:49,300 --> 00:42:55,040 We said that the notation for the plane group should be the 648 00:42:55,040 --> 00:42:58,250 symbol for the lattice type, which the initiated know has 649 00:42:58,250 --> 00:43:01,300 to be a hexagonal net. 650 00:43:01,300 --> 00:43:04,350 The point group that we've added, which is 3m, but now 651 00:43:04,350 --> 00:43:08,873 how do we distinguish the two different orientations of the 652 00:43:08,873 --> 00:43:10,150 mirror planes? 653 00:43:10,150 --> 00:43:16,630 And the way around that is to actually use three symbols. 654 00:43:16,630 --> 00:43:22,420 And the second symbol will be what is 655 00:43:22,420 --> 00:43:24,320 perpendicular to the cell edge. 656 00:43:28,870 --> 00:43:30,560 And the third symbol will be what's 657 00:43:30,560 --> 00:43:32,180 parallel to the cell edge. 658 00:43:36,610 --> 00:43:39,325 So this one would be called P3M1. 659 00:43:42,250 --> 00:43:45,330 And this one, just to distinguish it, would be 660 00:43:45,330 --> 00:43:48,060 called P31M. 661 00:43:48,060 --> 00:43:50,900 So if this weren't bad enough, now you've got almost the same 662 00:43:50,900 --> 00:43:55,590 symbol with a permutation of two of the terms in it. 663 00:43:55,590 --> 00:43:59,920 In this case, the mirror plane is along the cell edge along 664 00:43:59,920 --> 00:44:01,270 the translations. 665 00:44:01,270 --> 00:44:02,720 In this case, it's perpendicular. 666 00:44:09,100 --> 00:44:14,042 But we're not quite yet done, because we've got translations 667 00:44:14,042 --> 00:44:17,460 that are inclined to mirror planes. 668 00:44:17,460 --> 00:44:21,580 And now you can see how prudent I was in saying quite 669 00:44:21,580 --> 00:44:24,090 generally, without trying to figure out what is the 670 00:44:24,090 --> 00:44:27,220 perpendicular part and what is the parallel part, here's a 671 00:44:27,220 --> 00:44:31,920 translation and it's at an angle to a mirror plane. 672 00:44:31,920 --> 00:44:38,580 And therefore we get a glide that has to be halfway between 673 00:44:38,580 --> 00:44:40,030 these mirror planes. 674 00:44:40,030 --> 00:44:42,490 And there'll have to be a glide halfway between these 675 00:44:42,490 --> 00:44:45,750 mirror planes, and a glide that's halfway between these 676 00:44:45,750 --> 00:44:47,150 mirror planes. 677 00:44:47,150 --> 00:44:50,060 In this case, the mirror planes are this way. 678 00:44:50,060 --> 00:44:52,980 Again, a translation that's inclined to the mirror plane. 679 00:44:52,980 --> 00:44:56,630 We have to have a glide that is midway between 680 00:44:56,630 --> 00:44:58,070 these mirror planes. 681 00:44:58,070 --> 00:45:02,850 In this case, they're a little easier to identify. 682 00:45:02,850 --> 00:45:06,450 They sort of make a triangular box that surrounds the 683 00:45:06,450 --> 00:45:09,230 threefold axis. 684 00:45:09,230 --> 00:45:16,390 So now we see another difference in the plane groups 685 00:45:16,390 --> 00:45:18,390 and the reason why they're so very distinct. 686 00:45:18,390 --> 00:45:22,010 And that is the rearrangement of the glides relative to the 687 00:45:22,010 --> 00:45:24,560 threefold axes are quite different in these two cases. 688 00:45:28,570 --> 00:45:30,500 So these are two quite different symmetries. 689 00:45:30,500 --> 00:45:35,560 Notice that when we start identifying special locations 690 00:45:35,560 --> 00:45:38,910 in these two plane groups, here there is a location only 691 00:45:38,910 --> 00:45:43,520 of symmetry 3mm for a point group, 3m for a point group. 692 00:45:43,520 --> 00:45:46,560 Here there are two different locations, one that's symmetry 693 00:45:46,560 --> 00:45:50,240 3m, the other one is just symmetry 3. 694 00:45:50,240 --> 00:45:54,720 So this is location of point group 3 and a location of 695 00:45:54,720 --> 00:45:56,940 point group 3m. 696 00:45:56,940 --> 00:45:59,960 So the sort of special positions that exist in these 697 00:45:59,960 --> 00:46:02,610 two plane groups will be very, very different. 698 00:46:20,720 --> 00:46:24,010 One more addition, and then I think we can leave with a 699 00:46:24,010 --> 00:46:25,950 light heart because we should be done. 700 00:46:30,350 --> 00:46:35,190 What about a primitive hexagonal net 701 00:46:35,190 --> 00:46:38,980 combined with 6mm? 702 00:46:38,980 --> 00:46:40,880 And if you think I'm going to try to draw 703 00:46:40,880 --> 00:46:42,730 that, you're crazy. 704 00:46:42,730 --> 00:46:46,410 But I can say quite simply and hopefully convince you that 705 00:46:46,410 --> 00:46:57,410 this looks like P31M plus P3M1 right on top of one another, 706 00:46:57,410 --> 00:47:01,370 because we've got mirror planes that 707 00:47:01,370 --> 00:47:03,830 are 30 degrees apart. 708 00:47:03,830 --> 00:47:09,840 So, as a schematic direction and an invitation to do this 709 00:47:09,840 --> 00:47:15,030 at home in your spare time, is to take what P6 looks like. 710 00:47:21,080 --> 00:47:23,130 And that's sixfold axes at the lattice points. 711 00:47:23,130 --> 00:47:24,660 Threefolds here. 712 00:47:24,660 --> 00:47:28,010 Twofolds here. 713 00:47:28,010 --> 00:47:31,060 And then, directly on top of this, place the mirrors and 714 00:47:31,060 --> 00:47:33,910 glides in this orientation, and the mirrors and glides in 715 00:47:33,910 --> 00:47:35,090 this orientation. 716 00:47:35,090 --> 00:47:42,200 So we will have mirror planes coming out like that, and 717 00:47:42,200 --> 00:47:47,230 glide planes in between all the parallel mirror planes. 718 00:47:47,230 --> 00:47:52,910 As I say, I'm heroic but not foolish. 719 00:47:52,910 --> 00:47:53,305 Yes. 720 00:47:53,305 --> 00:47:55,110 AUDIENCE: Can you just explain that derivation again? 721 00:47:55,110 --> 00:47:56,990 Your P3 [INAUDIBLE]. 722 00:47:56,990 --> 00:47:58,880 PROFESSOR: OK. 723 00:47:58,880 --> 00:48:04,380 The object of this additional bit of confusion is to put in 724 00:48:04,380 --> 00:48:08,180 the symbol the point group, which is 3m, and then tell you 725 00:48:08,180 --> 00:48:11,600 how the mirror plane in a point group is oriented 726 00:48:11,600 --> 00:48:15,090 relative to the edges of the cell. 727 00:48:15,090 --> 00:48:18,410 And we could call them anything we wanted to. 728 00:48:18,410 --> 00:48:22,590 We could call them P3M perpendicular, subscript 729 00:48:22,590 --> 00:48:23,580 perpendicular. 730 00:48:23,580 --> 00:48:26,970 Or P3M subscript parallel to indicate mirror plane parallel 731 00:48:26,970 --> 00:48:30,010 to the cell edge. 732 00:48:30,010 --> 00:48:32,050 And mirror plane perpendicular to the cell edge. 733 00:48:32,050 --> 00:48:35,840 Or we could call them some obscenity, which would be very 734 00:48:35,840 --> 00:48:37,580 descriptive, as well. 735 00:48:37,580 --> 00:48:41,660 But this is just a way of keeping track of what's 736 00:48:41,660 --> 00:48:43,210 parallel to the cell edge and what's 737 00:48:43,210 --> 00:48:44,460 perpendicular to the cell edge. 738 00:48:47,100 --> 00:48:51,510 So, this middle symbol is what's oriented perpendicular 739 00:48:51,510 --> 00:48:52,600 to the cell edge. 740 00:48:52,600 --> 00:48:55,910 And in this case, this is the mirror plane and that is 741 00:48:55,910 --> 00:48:56,820 perpendicular. 742 00:48:56,820 --> 00:48:59,460 All the mirror planes are perpendicular to the 743 00:48:59,460 --> 00:49:01,920 edges of the cell. 744 00:49:01,920 --> 00:49:04,770 And then parallel to the cell edge, there's 745 00:49:04,770 --> 00:49:07,350 no symmetry at all. 746 00:49:07,350 --> 00:49:08,500 So that's 1. 747 00:49:08,500 --> 00:49:10,020 1 stands for no symmetry. 748 00:49:10,020 --> 00:49:14,050 In this case, the mirror plane is along the cell edge, so 749 00:49:14,050 --> 00:49:16,520 there's an M in the third position. 750 00:49:16,520 --> 00:49:19,360 And perpendicular to the cell edge, there's 751 00:49:19,360 --> 00:49:21,130 no symmetry at all-- 752 00:49:21,130 --> 00:49:22,380 no symmetry plane at all. 753 00:49:25,500 --> 00:49:31,690 So it's purely arbitrary, but we need some mechanism for 754 00:49:31,690 --> 00:49:34,380 distinguishing which one we're talking about. 755 00:49:34,380 --> 00:49:36,750 That pretty much works as well as anything. 756 00:49:36,750 --> 00:49:39,570 And it's a minor perturbation of what we've done for the 757 00:49:39,570 --> 00:49:41,520 additional plane group symbols. 758 00:49:46,900 --> 00:49:47,100 All right. 759 00:49:47,100 --> 00:49:51,956 It's time for our midafternoon break. 760 00:49:51,956 --> 00:49:55,020 It's a nice place to quit, because you're probably 761 00:49:55,020 --> 00:49:55,680 feeling good. 762 00:49:55,680 --> 00:49:59,530 We've done this, and we can move on to something new. 763 00:49:59,530 --> 00:50:01,460 Actually, there's one more trick. 764 00:50:01,460 --> 00:50:04,810 And unless you were really clever, you wouldn't have 765 00:50:04,810 --> 00:50:05,730 thought of it. 766 00:50:05,730 --> 00:50:09,510 But there's another small family of plane groups that we 767 00:50:09,510 --> 00:50:12,510 can get through one particular device. 768 00:50:12,510 --> 00:50:14,420 I shouldn't have told you that, because you'll be less 769 00:50:14,420 --> 00:50:15,860 inclined to come back now. 770 00:50:15,860 --> 00:50:17,730 But we're almost through. 771 00:50:17,730 --> 00:50:20,630 And I think it'll be very clear what these additional 772 00:50:20,630 --> 00:50:21,880 possibilities are.