1 00:00:07,010 --> 00:00:10,190 PROFESSOR: I think it's about five after the hour so we 2 00:00:10,190 --> 00:00:11,670 ought to get started. 3 00:00:11,670 --> 00:00:16,610 Before we forge bravely ahead, I'd like to make sure that 4 00:00:16,610 --> 00:00:19,490 everybody has a copy of the things that we've 5 00:00:19,490 --> 00:00:20,710 been handing out. 6 00:00:20,710 --> 00:00:23,710 There is a copy of the handwritten notes on the 7 00:00:23,710 --> 00:00:26,893 derivation of the 17 plane groups. 8 00:00:26,893 --> 00:00:28,150 Need a copy of that? 9 00:00:30,900 --> 00:00:32,200 There you go. 10 00:00:32,200 --> 00:00:43,220 And then there is, in addition, a set of diagrams 11 00:00:43,220 --> 00:00:47,030 from the international tables that give very nice pictures 12 00:00:47,030 --> 00:00:48,870 of all of the plane groups in the 13 00:00:48,870 --> 00:00:50,120 arrangement of symmetry elements. 14 00:00:52,580 --> 00:00:54,550 That's the only other thing that was handed 15 00:00:54,550 --> 00:00:55,800 out up to this point. 16 00:00:58,150 --> 00:01:03,090 For your continued edification and amusement, I have another 17 00:01:03,090 --> 00:01:04,459 problem set. 18 00:01:04,459 --> 00:01:06,720 And let me say something about this problem set. 19 00:01:06,720 --> 00:01:11,650 The first problem gives you some angles between crystal 20 00:01:11,650 --> 00:01:19,010 faces and asks you to deduce a possible set of lattice 21 00:01:19,010 --> 00:01:22,410 translations which are consistent with those angles. 22 00:01:22,410 --> 00:01:25,890 And in the days before diffraction, that was what 23 00:01:25,890 --> 00:01:27,660 crystallography was all about. 24 00:01:27,660 --> 00:01:31,190 It really was mapping the geometry of crystals. 25 00:01:31,190 --> 00:01:33,560 And the interesting thing is, you could measure these angles 26 00:01:33,560 --> 00:01:37,330 very precisely with a device called a reflecting 27 00:01:37,330 --> 00:01:41,140 goniometer, Provided you had a crystal with nice, 28 00:01:41,140 --> 00:01:42,690 shiny faces on it. 29 00:01:42,690 --> 00:01:45,960 You had a two circle instrument, and you could 30 00:01:45,960 --> 00:01:49,040 adjust the crystal so that a light beam, a very finely 31 00:01:49,040 --> 00:01:53,140 collimated beam of light, was focused into an eyepiece. 32 00:01:53,140 --> 00:01:57,745 And you could measure angles to within not one minute, but 33 00:01:57,745 --> 00:01:59,900 one second of arc. 34 00:01:59,900 --> 00:02:03,940 And from this you could deduce if you could assign Miller 35 00:02:03,940 --> 00:02:05,130 Indices to the faces. 36 00:02:05,130 --> 00:02:09,389 You could deduce not the absolute magnitudes of 37 00:02:09,389 --> 00:02:13,310 translations, but you could deduce the ratio of them. 38 00:02:13,310 --> 00:02:15,690 And along came x-ray diffraction and shook 39 00:02:15,690 --> 00:02:16,680 everything up. 40 00:02:16,680 --> 00:02:20,020 Turns out, sometimes you were right and other times you were 41 00:02:20,020 --> 00:02:23,720 wrong, because you had determined a self consistent 42 00:02:23,720 --> 00:02:26,480 set of indices, but not the ones that were correct. 43 00:02:26,480 --> 00:02:29,520 So this is to give you a little look at the early days 44 00:02:29,520 --> 00:02:34,050 of crystallography, and to see if using the way Miller 45 00:02:34,050 --> 00:02:38,220 Indices are defined you can calculate the ratio of axes. 46 00:02:38,220 --> 00:02:41,510 Not terribly demanding, but worth doing. 47 00:02:41,510 --> 00:02:44,360 And we had talked earlier about these combinations of 48 00:02:44,360 --> 00:02:47,280 symmetry elements as constituting the 49 00:02:47,280 --> 00:02:49,040 elements of a group. 50 00:02:49,040 --> 00:02:52,430 And in the second problem I ask you to take a not 51 00:02:52,430 --> 00:02:56,100 terrifically simple, but a fairly complex point group, 52 00:02:56,100 --> 00:02:59,530 4mm, and show that, indeed, the operations that are 53 00:02:59,530 --> 00:03:02,820 present satisfy the group postulates. 54 00:03:02,820 --> 00:03:05,600 And that is worth doing once so that you convince yourself 55 00:03:05,600 --> 00:03:08,090 these really are groups. 56 00:03:08,090 --> 00:03:12,740 And then the third problem is easy to state and it is 57 00:03:12,740 --> 00:03:14,350 diabolically tricky. 58 00:03:14,350 --> 00:03:17,690 So I'll let you have a go at it, but don't beat your brains 59 00:03:17,690 --> 00:03:20,210 against your desktop for an entire evening over it. 60 00:03:20,210 --> 00:03:20,870 Try it. 61 00:03:20,870 --> 00:03:24,740 If you don't see the solution to it, all will be revealed 62 00:03:24,740 --> 00:03:26,220 later on in class. 63 00:03:26,220 --> 00:03:30,650 But it's based on the fact that the plane groups, which 64 00:03:30,650 --> 00:03:35,380 we've now derived, 17 of them, these can be viewed as the 65 00:03:35,380 --> 00:03:38,970 base level of a three dimensional space group. 66 00:03:38,970 --> 00:03:41,370 And for each of them, you could take another 67 00:03:41,370 --> 00:03:45,360 translation, t3, that was perpendicular to the plane of 68 00:03:45,360 --> 00:03:49,200 the group, and just imagine all of those rotation axes and 69 00:03:49,200 --> 00:03:51,960 all of those mirror planes not being two dimensional 70 00:03:51,960 --> 00:03:54,930 operations, but three dimensional operations. 71 00:03:54,930 --> 00:03:57,420 The only one where you could pick the translation, 72 00:03:57,420 --> 00:04:01,330 generally, would be the oblique lattice, p1, no 73 00:04:01,330 --> 00:04:02,380 symmetry at all. 74 00:04:02,380 --> 00:04:05,180 And there you could pick the translation in any orientation 75 00:04:05,180 --> 00:04:06,130 you wished. 76 00:04:06,130 --> 00:04:11,580 But for each one of the 17 two dimensional plane groups, 77 00:04:11,580 --> 00:04:16,040 there is a corresponding three dimensional space group. 78 00:04:16,040 --> 00:04:19,769 So we got 17 three dimensional symmetries for free without 79 00:04:19,769 --> 00:04:21,930 really doing any additional work. 80 00:04:21,930 --> 00:04:24,120 However, there's one thing that one 81 00:04:24,120 --> 00:04:25,400 should not gloss over. 82 00:04:25,400 --> 00:04:31,150 We found that a rotation point in a two dimensional plane 83 00:04:31,150 --> 00:04:36,300 allowed only a very limited number of lattices. 84 00:04:36,300 --> 00:04:40,660 But we were not thinking of things in terms of three 85 00:04:40,660 --> 00:04:41,500 dimensions. 86 00:04:41,500 --> 00:04:44,660 So what I'm inviting you to do in the third problem is to 87 00:04:44,660 --> 00:04:47,610 consider, now, that construction that we did where 88 00:04:47,610 --> 00:04:52,890 we showed that rotation angles were restricted to values of 89 00:04:52,890 --> 00:04:57,650 cosine of alpha equals 1 minus p over 2. 90 00:04:57,650 --> 00:05:00,910 Now generalize that to a three dimensional situation, where 91 00:05:00,910 --> 00:05:04,340 the translation does not have to be perpendicular to the 92 00:05:04,340 --> 00:05:07,790 rotation axis, but can be inclined to it. 93 00:05:07,790 --> 00:05:11,420 Are there additional rotation axes allowed? 94 00:05:11,420 --> 00:05:13,350 Are their fewer? 95 00:05:13,350 --> 00:05:15,280 Something we ought to examine. 96 00:05:15,280 --> 00:05:19,450 But there's a little bit of ingenuity that you have to use 97 00:05:19,450 --> 00:05:20,210 in that proof. 98 00:05:20,210 --> 00:05:23,250 But I invite you to have a go at it on your own. 99 00:05:23,250 --> 00:05:26,460 So with that tantalizing introduction that will want to 100 00:05:26,460 --> 00:05:29,390 send you running home and start it as soon as we finish 101 00:05:29,390 --> 00:05:32,580 our class this afternoon, let me hand out, without any 102 00:05:32,580 --> 00:05:37,040 further blather, problem set number five. 103 00:05:37,040 --> 00:05:38,290 Pass that in the back corner. 104 00:05:45,220 --> 00:05:49,390 Last time, without saying much about them, I handed out this 105 00:05:49,390 --> 00:05:56,150 extract from the international tables that summarized the 17 106 00:05:56,150 --> 00:05:59,840 two dimensional plane groups and their properties. 107 00:05:59,840 --> 00:06:03,770 And I'd like to spend a few minutes going over the 108 00:06:03,770 --> 00:06:05,300 considerable information that's 109 00:06:05,300 --> 00:06:07,750 contained on these pages. 110 00:06:07,750 --> 00:06:11,940 Each one is treated in the same way, and let me start 111 00:06:11,940 --> 00:06:14,600 with one that's almost trivially simple, and that's 112 00:06:14,600 --> 00:06:17,190 the two-fold axis in the oblique net. 113 00:06:17,190 --> 00:06:21,880 Across the top of the page you find, in boldface, the symbol 114 00:06:21,880 --> 00:06:24,040 for the plane group. 115 00:06:24,040 --> 00:06:28,430 And then, just counting in order of increasing symmetry, 116 00:06:28,430 --> 00:06:33,620 the number of that particular plane group in the set. 117 00:06:33,620 --> 00:06:39,130 And then a symbol that really has its full meaning only in 118 00:06:39,130 --> 00:06:42,760 three dimensions, but it gives the symbol for the lattice, 119 00:06:42,760 --> 00:06:48,230 again, 2 and then a 1 and a 1, which says that if this were a 120 00:06:48,230 --> 00:06:53,710 three dimensional space group, there would be a two-fold axis 121 00:06:53,710 --> 00:06:56,090 in one direction and no symmetry at all in the other 122 00:06:56,090 --> 00:06:58,420 two directions. 123 00:06:58,420 --> 00:07:02,080 And then, next comes the symbol for the point group of 124 00:07:02,080 --> 00:07:03,780 the crystal. 125 00:07:03,780 --> 00:07:07,422 And we derived this particular group by dropping point group 126 00:07:07,422 --> 00:07:09,320 2 into a lattice. 127 00:07:09,320 --> 00:07:16,270 And then the coordinate system that is necessary to describe 128 00:07:16,270 --> 00:07:20,020 the features of this plane group, if you take the edges 129 00:07:20,020 --> 00:07:23,660 of the unit cell as the basis of a coordinate system. 130 00:07:23,660 --> 00:07:25,720 So this is something called a crystal system. 131 00:07:30,930 --> 00:07:33,300 So for each of the plane groups, you have this 132 00:07:33,300 --> 00:07:36,690 information spread across the top of the page. 133 00:07:36,690 --> 00:07:42,950 Then underneath that is a diagram that, by means of open 134 00:07:42,950 --> 00:07:48,690 circles, shows the way in which that particular symmetry 135 00:07:48,690 --> 00:07:53,810 moves a motif where atoms, in the case of a crystal, moves 136 00:07:53,810 --> 00:07:55,310 atoms around. 137 00:07:55,310 --> 00:08:01,010 And for p2 we have the pair of motifs of the same chirality 138 00:08:01,010 --> 00:08:03,130 related by a two-fold axis. 139 00:08:03,130 --> 00:08:05,994 And that, then, is hung at every lattice point of an 140 00:08:05,994 --> 00:08:07,790 oblique net. 141 00:08:07,790 --> 00:08:14,490 And then, immediately to the right, is the arrangement of 142 00:08:14,490 --> 00:08:18,360 symmetry elements in the group. 143 00:08:18,360 --> 00:08:22,200 Then, the way in which that particular plane group can 144 00:08:22,200 --> 00:08:25,120 move atoms around to generate a structure is 145 00:08:25,120 --> 00:08:26,880 summarized for you. 146 00:08:26,880 --> 00:08:30,470 And the first thing they have to state is where you're going 147 00:08:30,470 --> 00:08:31,720 to choose the origin. 148 00:08:34,280 --> 00:08:36,120 There is no unique lattice point. 149 00:08:36,120 --> 00:08:39,460 It's convenient to take the origin at a location of high 150 00:08:39,460 --> 00:08:43,270 symmetry, and in this case the smart thing to do is to take 151 00:08:43,270 --> 00:08:47,710 the origin at one of the two-fold axes. 152 00:08:47,710 --> 00:08:51,060 That's a non trivial question, because there are some groups 153 00:08:51,060 --> 00:08:54,600 that have different kinds of rotation axes. 154 00:08:54,600 --> 00:08:57,350 P4 has a four-fold and the two-fold. 155 00:08:57,350 --> 00:09:01,960 Taking the origin of the coordinate system at a 156 00:09:01,960 --> 00:09:04,370 location of high symmetry, then, gives you the scope of 157 00:09:04,370 --> 00:09:07,000 picking either a two-fold or a four-fold axes. 158 00:09:07,000 --> 00:09:07,706 Yes,sir? 159 00:09:07,706 --> 00:09:10,562 AUDIENCE: On the left hand figure, what's with the-- 160 00:09:10,562 --> 00:09:13,710 those aren't mirror planes [INAUDIBLE] are they? 161 00:09:13,710 --> 00:09:15,015 PROFESSOR: On the-- 162 00:09:15,015 --> 00:09:17,360 AUDIENCE: [INAUDIBLE]. 163 00:09:17,360 --> 00:09:18,620 PROFESSOR: Here they have this. 164 00:09:18,620 --> 00:09:23,570 That's just to split things up into quadrants, so you have a 165 00:09:23,570 --> 00:09:28,390 feel for how the atoms hung at the lattice points split up 166 00:09:28,390 --> 00:09:30,510 into different quadrants. 167 00:09:30,510 --> 00:09:32,440 That is a very good point, though. 168 00:09:32,440 --> 00:09:34,910 Mirror planes are shown as bold lines. 169 00:09:34,910 --> 00:09:37,600 Sometimes the outlines of the cell look pretty bold 170 00:09:37,600 --> 00:09:38,580 themselves. 171 00:09:38,580 --> 00:09:43,400 So if you turn the page to pm, if you have the notes with 172 00:09:43,400 --> 00:09:46,670 you, can see the lines that are reference lines for the 173 00:09:46,670 --> 00:09:51,365 cells are lighter in their weight than the symbols for 174 00:09:51,365 --> 00:09:52,290 the mirror planes. 175 00:09:52,290 --> 00:09:54,270 But it's a very subtle difference. 176 00:09:54,270 --> 00:09:57,450 And when you Xerox it a couple of times it becomes almost 177 00:09:57,450 --> 00:09:59,486 indistinguishable 178 00:09:59,486 --> 00:10:05,410 AUDIENCE: Can you just say, one more time, [INAUDIBLE]? 179 00:10:05,410 --> 00:10:10,530 PROFESSOR: That means that this is the location of 0,0. 180 00:10:10,530 --> 00:10:15,600 And it is always assumed, but nowhere stated specifically, 181 00:10:15,600 --> 00:10:19,230 that the origin is in the upper left hand corner. 182 00:10:27,980 --> 00:10:32,555 And what is also assumed, but never stated, is that the 183 00:10:32,555 --> 00:10:40,160 x-coordinate in the a-axis goes down and the y-coordinate 184 00:10:40,160 --> 00:10:44,960 in the b-axis goes from upper left to the right. 185 00:10:44,960 --> 00:10:47,380 That is nowhere stated anywhere in the international 186 00:10:47,380 --> 00:10:50,720 tables, but that is the direction that is assumed for 187 00:10:50,720 --> 00:10:54,010 the reference axes. 188 00:10:54,010 --> 00:10:56,430 So we'll take our origin of the two-fold axis. 189 00:10:56,430 --> 00:10:58,470 That's a sensible thing to do. 190 00:10:58,470 --> 00:11:02,210 And then there are two ways you can look at how the 191 00:11:02,210 --> 00:11:04,720 particular plane group generates a pattern. 192 00:11:04,720 --> 00:11:07,670 You can do it with little circles or little commas, or 193 00:11:07,670 --> 00:11:08,590 something like that. 194 00:11:08,590 --> 00:11:10,970 In other words, do it graphically. 195 00:11:10,970 --> 00:11:14,260 But a way to communicate the atomic arrangement in a 196 00:11:14,260 --> 00:11:19,700 structure, which is going to be the most free of ambiguity 197 00:11:19,700 --> 00:11:22,950 and rigorous, is to do it analytically. 198 00:11:22,950 --> 00:11:26,500 And that's what the tables do for you next. 199 00:11:26,500 --> 00:11:29,760 They give an analytic description of the way in 200 00:11:29,760 --> 00:11:33,500 which the symmetry elements will move an atom around. 201 00:11:33,500 --> 00:11:37,620 For this particular plane group, things are very simple. 202 00:11:37,620 --> 00:11:42,270 This is the direction of x, this is the direction of y. 203 00:11:42,270 --> 00:11:46,390 And if we plop one atom in here, it will be at a 204 00:11:46,390 --> 00:11:49,820 location x and y. 205 00:11:49,820 --> 00:11:55,270 So that's what happens when you drop one atom in it. 206 00:11:55,270 --> 00:11:59,930 That atom gets hung at every lattice point, and it gets 207 00:11:59,930 --> 00:12:01,330 rotated by the two-fold axis. 208 00:12:08,550 --> 00:12:13,270 So you're going to get two of them per lattice point. 209 00:12:13,270 --> 00:12:16,300 Somewhat arbitrary which pair you take. 210 00:12:16,300 --> 00:12:19,714 The coordinates of this atom are x and y. 211 00:12:19,714 --> 00:12:24,850 The coordinates of this atom inside of the box is 1, minus 212 00:12:24,850 --> 00:12:26,700 x and 1, minus y. 213 00:12:29,740 --> 00:12:35,200 That would do it, but it makes sense, esthetically, and to 214 00:12:35,200 --> 00:12:41,000 see that the atoms are related by symmetry, if, instead, you 215 00:12:41,000 --> 00:12:45,200 specify as the two atoms per cell the two that are hanging 216 00:12:45,200 --> 00:12:46,800 at the lattice point. 217 00:12:46,800 --> 00:12:49,130 And in that case, the coordinates of the first, if 218 00:12:49,130 --> 00:12:54,080 they are x and y, would be minus x and minus y. 219 00:12:54,080 --> 00:12:59,780 If you have a pair of numbers, 0.283 and 0.456, then minus 220 00:12:59,780 --> 00:13:05,280 0.233 and minus 0.456 leaves no doubt that these are 221 00:13:05,280 --> 00:13:08,550 positions that are related by symmetry. 222 00:13:08,550 --> 00:13:13,990 If you have coordinates, like 0.2, 0.3 and then 0.8, 0.7, 223 00:13:13,990 --> 00:13:17,490 without doing some arithmetic in your head, it's not clear 224 00:13:17,490 --> 00:13:20,560 that those are going to be atoms related by symmetry. 225 00:13:20,560 --> 00:13:23,510 So the coordinates that will be stated for you, then, will 226 00:13:23,510 --> 00:13:28,560 be the coordinate of the pair of atoms at the lattice point, 227 00:13:28,560 --> 00:13:30,450 and some of the coordinates would be negative. 228 00:13:33,760 --> 00:13:37,220 This is something called the general position. 229 00:13:42,230 --> 00:13:45,210 And that transformation of coordinates is different for 230 00:13:45,210 --> 00:13:47,080 every one of the plane groups. 231 00:13:47,080 --> 00:13:48,560 That's what makes them different. 232 00:13:48,560 --> 00:13:51,170 They're different in the way they move around an atom in a 233 00:13:51,170 --> 00:13:54,910 general location and fill space with it. 234 00:13:54,910 --> 00:13:58,030 So that's the general position, and it is unique for 235 00:13:58,030 --> 00:13:59,280 each plane group. 236 00:14:14,360 --> 00:14:17,830 And there are several ways in which this information is 237 00:14:17,830 --> 00:14:19,600 conveyed to you. 238 00:14:19,600 --> 00:14:22,740 Move this over a little bit; x, y and minus x, y. 239 00:14:26,930 --> 00:14:31,510 The first thing that's a characteristic of the position 240 00:14:31,510 --> 00:14:35,150 is the number per cell. 241 00:14:35,150 --> 00:14:41,660 So you get the number 2, or this is sometimes referred to 242 00:14:41,660 --> 00:14:43,100 as the rank of the position. 243 00:14:46,980 --> 00:14:52,810 Drop in 1 and you get a second one out related by symmetry. 244 00:14:52,810 --> 00:14:56,910 The next is, and I'll go to the last next, this is the 245 00:14:56,910 --> 00:14:58,160 site symmetry. 246 00:15:03,910 --> 00:15:09,930 And by definition, this is always 1, no symmetry at all, 247 00:15:09,930 --> 00:15:11,180 for a general position. 248 00:15:21,530 --> 00:15:26,410 So why make a big deal about all of these characteristics 249 00:15:26,410 --> 00:15:28,290 of the general position? 250 00:15:28,290 --> 00:15:32,130 The reason is that there are locations that are termed 251 00:15:32,130 --> 00:15:34,070 special positions. 252 00:15:34,070 --> 00:15:39,870 And let me clean up this right hand part of my diagram. 253 00:15:39,870 --> 00:15:44,880 And say that, for plane group p2 you will always get two 254 00:15:44,880 --> 00:15:48,010 atoms per cell if you place an atom in an 255 00:15:48,010 --> 00:15:49,260 unspecialized location. 256 00:15:51,820 --> 00:15:54,260 What would be a specialized location? 257 00:15:54,260 --> 00:15:58,490 Suppose we would drop the atom down right smack on top of 258 00:15:58,490 --> 00:15:59,740 that two-fold axis? 259 00:16:01,690 --> 00:16:05,610 Then that two-fold axis is just going to twirl the atom 260 00:16:05,610 --> 00:16:09,810 around on its axis and it's not going to map it into a 261 00:16:09,810 --> 00:16:11,940 second location. 262 00:16:11,940 --> 00:16:15,190 So following the general position is always provided to 263 00:16:15,190 --> 00:16:17,120 you a set of special positions. 264 00:16:23,720 --> 00:16:25,500 And what's special about them? 265 00:16:25,500 --> 00:16:26,900 They are on a symmetry element. 266 00:16:37,460 --> 00:16:41,250 You can look at the characteristics of a special 267 00:16:41,250 --> 00:16:45,090 position in two ways, either geometrically or you can do it 268 00:16:45,090 --> 00:16:47,700 analytically, as I'll show you in just a moment. 269 00:16:47,700 --> 00:16:56,800 Imagine that the atom is to sit here, x and y migrate 270 00:16:56,800 --> 00:16:59,800 progressively towards the location 0,0. 271 00:16:59,800 --> 00:17:03,010 And then the other atom related to it by two-fold 272 00:17:03,010 --> 00:17:06,770 symmetry will move towards it until finally the two of them 273 00:17:06,770 --> 00:17:08,639 will merge into just one single motif. 274 00:17:12,130 --> 00:17:16,450 So when that happens you get not two per cell, you get only 275 00:17:16,450 --> 00:17:18,180 one per cell. 276 00:17:18,180 --> 00:17:21,540 And the reason for that is the site symmetry is 2. 277 00:17:21,540 --> 00:17:22,790 It's on a two-fold axis. 278 00:17:26,690 --> 00:17:28,890 But there are other two-fold axes as well. 279 00:17:28,890 --> 00:17:36,500 What if the element migrated to the position 0, 1/2? 280 00:17:36,500 --> 00:17:40,570 If that were the case, if the representative atom moves to 281 00:17:40,570 --> 00:17:44,630 here, this one up here will migrate down and the two will 282 00:17:44,630 --> 00:17:48,360 merge into one atom that sits at just 0, 1/2. 283 00:17:51,160 --> 00:17:55,072 So there'll be another type of special position, one per 284 00:17:55,072 --> 00:17:58,400 cell, also, on a two-fold axis. 285 00:17:58,400 --> 00:18:05,930 This one was at 0,0, and this one is at 0, 1/2. 286 00:18:05,930 --> 00:18:08,610 You remember, we made a point of saying that there are four 287 00:18:08,610 --> 00:18:13,280 different kinds of two-fold axes within this plane group. 288 00:18:13,280 --> 00:18:16,160 Different in how they're positioned relative to a 289 00:18:16,160 --> 00:18:19,490 pattern, different in that they are not mapped into one 290 00:18:19,490 --> 00:18:23,440 another by some other symmetry element which is present. 291 00:18:23,440 --> 00:18:27,720 So each of these four locations is a location of 292 00:18:27,720 --> 00:18:30,870 another general position of rank 2, sitting 293 00:18:30,870 --> 00:18:32,250 on a two-fold axis. 294 00:18:32,250 --> 00:18:36,600 x and y migrate down to 1/2, 0, and this one and this one 295 00:18:36,600 --> 00:18:37,840 will come together. 296 00:18:37,840 --> 00:18:42,340 So there's another one, giving you one per cell, sits on a 297 00:18:42,340 --> 00:18:46,250 two-fold axis, and this would be at the location 1/2, 0. 298 00:18:46,250 --> 00:18:48,930 And finally there's another one in the center of the cell, 299 00:18:48,930 --> 00:18:53,890 and if you let the atom migrate to that location, 300 00:18:53,890 --> 00:18:57,480 again, the atoms will coalesce pair wise into a single one. 301 00:18:57,480 --> 00:19:03,120 So there'll be another one, one per cell, and also on a 302 00:19:03,120 --> 00:19:06,330 two-fold axis, and this would sit at the location 1/2, 1/2. 303 00:19:10,670 --> 00:19:13,710 And those, then, are the characteristics of this 304 00:19:13,710 --> 00:19:15,320 particular simple plane group. 305 00:19:17,880 --> 00:19:21,240 There's another symbol that's added, and this is something 306 00:19:21,240 --> 00:19:22,750 called the Wyckoff symbol. 307 00:19:32,780 --> 00:19:37,390 Wyckoff was a crystal chemist who attempted valiantly, back 308 00:19:37,390 --> 00:19:40,270 in the early days of the century, when there were a 309 00:19:40,270 --> 00:19:43,440 couple of dozen structures determined per year because it 310 00:19:43,440 --> 00:19:47,070 was such a grand, new adventure, and few people knew 311 00:19:47,070 --> 00:19:48,180 how to do it. 312 00:19:48,180 --> 00:19:54,620 But Wyckoff tried to summarize, between one pair of 313 00:19:54,620 --> 00:19:57,970 hard covers, all of the crystal structure 314 00:19:57,970 --> 00:20:01,400 determinations that had been performed in a single year. 315 00:20:01,400 --> 00:20:05,250 And he went at this courageously for perhaps a 316 00:20:05,250 --> 00:20:08,950 dozen years, and then results began to accumulate so rapidly 317 00:20:08,950 --> 00:20:11,890 he just said, the heck with it, clapped covers on it, and 318 00:20:11,890 --> 00:20:13,520 that was the end of the series of books. 319 00:20:13,520 --> 00:20:16,670 So they don't get very far, but it was a real useful 320 00:20:16,670 --> 00:20:18,500 contribution at the time. 321 00:20:18,500 --> 00:20:25,920 This is just a shorthand way of referring to the position. 322 00:20:25,920 --> 00:20:29,640 And he starts with a for the most specialized, runs the way 323 00:20:29,640 --> 00:20:33,160 up through the alphabet until you've assigned a letter to 324 00:20:33,160 --> 00:20:34,650 each of the positions. 325 00:20:34,650 --> 00:20:35,740 This is a nicety. 326 00:20:35,740 --> 00:20:39,730 All these special positions, for example, sit on a two-fold 327 00:20:39,730 --> 00:20:44,430 axis, and you don't really have to specify the 328 00:20:44,430 --> 00:20:48,430 coordinates, because they have to be either 0 or 1/2. 329 00:20:48,430 --> 00:20:53,125 So this gives us a way of saying you have an osmium atom 330 00:20:53,125 --> 00:20:57,860 in position 1a and you have an oxygen atom in position 1c. 331 00:20:57,860 --> 00:21:01,990 And it's a nice shorthand way of avoiding mentioning things 332 00:21:01,990 --> 00:21:04,640 which could be calculated once and for all and not stated 333 00:21:04,640 --> 00:21:05,890 explicitly. 334 00:21:10,160 --> 00:21:13,470 So this, my friends, is the language in which you will see 335 00:21:13,470 --> 00:21:17,880 structural datas cited, when people have determined, using 336 00:21:17,880 --> 00:21:21,430 diffraction methods, the locations of all of the atoms 337 00:21:21,430 --> 00:21:24,500 within the units, some of a particular material. 338 00:21:24,500 --> 00:21:29,940 If the coordinate is variable, with today's techniques and 339 00:21:29,940 --> 00:21:34,140 high speed computation, you can get x as a fraction of a 340 00:21:34,140 --> 00:21:39,330 cell to usually at least plus or minus 1 341 00:21:39,330 --> 00:21:41,150 in the fourth place. 342 00:21:41,150 --> 00:21:42,690 So it's data that can be 343 00:21:42,690 --> 00:21:44,170 determined extremely precisely. 344 00:21:44,170 --> 00:21:45,826 Yes sir? 345 00:21:45,826 --> 00:21:49,242 AUDIENCE: I didn't quite get the exact definition of the 346 00:21:49,242 --> 00:21:50,218 general position. 347 00:21:50,218 --> 00:21:52,660 It's a set of points related by-- 348 00:21:52,660 --> 00:21:54,500 PROFESSOR: The general position, it's a set of 349 00:21:54,500 --> 00:21:58,476 equivalent positions that are related by symmetry. 350 00:21:58,476 --> 00:21:59,726 Yeah. 351 00:22:01,460 --> 00:22:04,800 And the general position, yeah-- 352 00:22:04,800 --> 00:22:09,900 Each of these is called a set of equivalent positions, 353 00:22:09,900 --> 00:22:12,460 equivalent in the sense of being related to one another 354 00:22:12,460 --> 00:22:15,540 by symmetry. 355 00:22:15,540 --> 00:22:20,390 And the entire set is sometimes referred to as an, 356 00:22:20,390 --> 00:22:21,920 singular, equipoint. 357 00:22:24,930 --> 00:22:29,500 Sort of a condensed jargon for a set of equivalent positions. 358 00:22:29,500 --> 00:22:32,730 So when one speaks about an equipoint of rank 2, in this 359 00:22:32,730 --> 00:22:33,980 case, or rank 1. 360 00:22:37,010 --> 00:22:39,660 Let me just, even for this trivial two dimensional 361 00:22:39,660 --> 00:22:45,780 symmetry, give you an example of some information that falls 362 00:22:45,780 --> 00:22:47,030 out of this immediately. 363 00:22:50,780 --> 00:22:53,210 First of all, you can use these special 364 00:22:53,210 --> 00:22:56,320 positions only once. 365 00:22:56,320 --> 00:23:00,000 If you put an atom in position 1c at 1/2, 0, it's used up. 366 00:23:00,000 --> 00:23:04,010 You can't put another atom in there in a given structure. 367 00:23:04,010 --> 00:23:07,030 Secondly, if you think more globally, not in terms of 368 00:23:07,030 --> 00:23:09,400 atoms, which, to a good approximation, 369 00:23:09,400 --> 00:23:10,490 are spherically symmetric. 370 00:23:10,490 --> 00:23:13,480 But think in terms of a molecule, if you're a polymer 371 00:23:13,480 --> 00:23:15,610 scientist or an organic chemist. 372 00:23:15,610 --> 00:23:17,670 and not place an individual atom, but 373 00:23:17,670 --> 00:23:20,690 place an entire molecule. 374 00:23:20,690 --> 00:23:23,760 If there's one molecule per cell in this particular 375 00:23:23,760 --> 00:23:30,800 symmetry, that molecule has to sit on a two-fold axis. 376 00:23:30,800 --> 00:23:33,840 And that means the configuration of that molecule 377 00:23:33,840 --> 00:23:38,470 is going to be limited to having to conform to a 378 00:23:38,470 --> 00:23:40,460 two-fold rotational symmetric. 379 00:23:40,460 --> 00:23:43,060 So just from the density in the lattice constitute, if you 380 00:23:43,060 --> 00:23:46,310 find there's one molecule per cell, you know that molecule 381 00:23:46,310 --> 00:23:48,870 has to have two-fold rotational symmetry. 382 00:23:48,870 --> 00:23:52,380 So you can say something about the structure of the molecule. 383 00:23:52,380 --> 00:23:55,000 So there are lots of ways in which the symmetry information 384 00:23:55,000 --> 00:23:58,050 has something to say about the arrangement of 385 00:23:58,050 --> 00:23:59,300 atoms in the structure. 386 00:24:04,730 --> 00:24:06,543 Any further questions? 387 00:24:06,543 --> 00:24:07,529 Yes? 388 00:24:07,529 --> 00:24:10,158 AUDIENCE: How did they decide, between b, c, and d, which was 389 00:24:10,158 --> 00:24:12,459 the most special form? 390 00:24:12,459 --> 00:24:14,020 PROFESSOR: That is a very good question. 391 00:24:14,020 --> 00:24:17,970 And the way it's decided is by you getting there first before 392 00:24:17,970 --> 00:24:19,170 anybody else did it. 393 00:24:19,170 --> 00:24:23,340 So everybody follows, now, the conventions that are given in 394 00:24:23,340 --> 00:24:24,640 the international tables. 395 00:24:24,640 --> 00:24:29,320 That's sort of the dictionary of all these terms. 396 00:24:29,320 --> 00:24:30,440 But you're absolutely right. 397 00:24:30,440 --> 00:24:34,560 What is more special about this one than this one? 398 00:24:34,560 --> 00:24:39,320 Well, they're all locations of the same symmetry. 399 00:24:39,320 --> 00:24:43,960 You can turn one into another by changing the origin of the 400 00:24:43,960 --> 00:24:48,090 cell, where you're going to define a lattice point. 401 00:24:48,090 --> 00:24:49,810 So they are thoroughly arbitrary. 402 00:24:49,810 --> 00:24:54,890 What is generally done is to take 0, 0 for your particular 403 00:24:54,890 --> 00:24:57,770 choice of origin as position a. 404 00:24:57,770 --> 00:25:03,470 And then what should come next, 0, 1/2 or 1/2, 0? 405 00:25:03,470 --> 00:25:07,360 The truly observant among you will notice I snuck a quick 406 00:25:07,360 --> 00:25:10,280 peek at the international tables before I wrote one of 407 00:25:10,280 --> 00:25:11,210 these down. 408 00:25:11,210 --> 00:25:14,750 I can never remember which is which. 409 00:25:14,750 --> 00:25:16,440 And it is arbitrary. 410 00:25:16,440 --> 00:25:21,130 Usually you end up with the one that has both coordinates 411 00:25:21,130 --> 00:25:24,930 non-zero, but that is code that's embodied in the 412 00:25:24,930 --> 00:25:25,840 international tables. 413 00:25:25,840 --> 00:25:27,160 But that's a good question. 414 00:25:27,160 --> 00:25:27,863 Yes, sir? 415 00:25:27,863 --> 00:25:34,765 AUDIENCE: How did you decide to make the motifs [INAUDIBLE] 416 00:25:34,765 --> 00:25:36,260 special decisions? 417 00:25:36,260 --> 00:25:39,590 PROFESSOR: Oh, I just changed the value of x and y. 418 00:25:39,590 --> 00:25:42,540 Those are numbers that can be, for the representative atom, 419 00:25:42,540 --> 00:25:46,410 each of them can be between 0 and 1. 420 00:25:46,410 --> 00:25:51,400 So what I just said was, what will happen if we let x and y 421 00:25:51,400 --> 00:25:55,420 take on not general values, but very special values? 422 00:25:55,420 --> 00:25:57,970 And the special values are going to be things like 0 and 423 00:25:57,970 --> 00:26:01,800 1/2, if I have picked my origin at a two-fold axis. 424 00:26:04,540 --> 00:26:07,350 It's another reason for picking your origin at a 425 00:26:07,350 --> 00:26:08,440 location of high symmetry. 426 00:26:08,440 --> 00:26:11,700 AUDIENCE: But how about the other non-origins of these 427 00:26:11,700 --> 00:26:13,680 special decisions [INAUDIBLE]? 428 00:26:17,670 --> 00:26:18,330 PROFESSOR: There are two ways you can do it. 429 00:26:18,330 --> 00:26:19,480 One of them we just did. 430 00:26:19,480 --> 00:26:22,310 We just looked at the geometry and we said, if this 431 00:26:22,310 --> 00:26:25,270 representative atom, this is the one we define as being at 432 00:26:25,270 --> 00:26:26,670 location x, y. 433 00:26:26,670 --> 00:26:30,720 If x and y both approach 0, this atom and the one over 434 00:26:30,720 --> 00:26:33,450 here are going to migrate towards the origin lattice 435 00:26:33,450 --> 00:26:34,905 point and eventually just coalesce. 436 00:26:39,010 --> 00:26:44,120 If I let x and y and migrate, x go to 0 and y go to 1/2, 437 00:26:44,120 --> 00:26:45,710 then this one and this one will come 438 00:26:45,710 --> 00:26:47,720 together and coalesce. 439 00:26:47,720 --> 00:26:50,740 So that's why I did that, because I knew that the number 440 00:26:50,740 --> 00:26:55,840 I would generate would be smaller than the number that I 441 00:26:55,840 --> 00:26:59,120 would get for a set of coordinates for an atom that 442 00:26:59,120 --> 00:27:01,140 was off the symmetry element. 443 00:27:01,140 --> 00:27:02,650 I said there's another way of doing it. 444 00:27:02,650 --> 00:27:09,490 For something this simple it is not really that profound, 445 00:27:09,490 --> 00:27:11,990 but it is a way of checking whether you've counted the 446 00:27:11,990 --> 00:27:14,030 same position twice. 447 00:27:14,030 --> 00:27:18,820 I said you can imagine these special positions arising when 448 00:27:18,820 --> 00:27:21,310 x and y assume special values. 449 00:27:21,310 --> 00:27:27,640 So let's let, for example, x be 1/2 and y be 0 and plug 450 00:27:27,640 --> 00:27:31,140 those coordinates into my equation 451 00:27:31,140 --> 00:27:32,620 for the general positions. 452 00:27:32,620 --> 00:27:37,570 So I'll put in 1/2 for x, 0 for y, and then I'll put in 453 00:27:37,570 --> 00:27:41,850 minus 1/2 for x and minus 0 for y. 454 00:27:41,850 --> 00:27:45,750 But what goes on at minus 1/2 has to be the same as what 455 00:27:45,750 --> 00:27:49,550 goes on at plus 1/2, if the structure is based on a 456 00:27:49,550 --> 00:27:50,920 lattice and periodic. 457 00:27:50,920 --> 00:27:56,690 So what I'm really getting is 1/2, 0, 1/2, 0 twice, which is 458 00:27:56,690 --> 00:27:59,190 the same as saying, if x is 1/2 and y equals 0, I'm 459 00:27:59,190 --> 00:28:02,700 putting two atoms together right on top of one another. 460 00:28:02,700 --> 00:28:04,360 So you can do it analytically. 461 00:28:04,360 --> 00:28:06,910 And for the more complicated symmetries, where there might 462 00:28:06,910 --> 00:28:12,750 be for 20 or 48 atoms, that's a good way to see if you've 463 00:28:12,750 --> 00:28:16,650 looked at the same position twice. 464 00:28:16,650 --> 00:28:20,660 So we're going to do a couple more before we move on, and 465 00:28:20,660 --> 00:28:23,000 this one is almost trivially simple. 466 00:28:23,000 --> 00:28:25,750 Any other questions before we go forward? 467 00:28:30,370 --> 00:28:36,070 Let's take one that is slightly more complex. 468 00:28:36,070 --> 00:28:38,180 And I'll go to one of the rectangular groups. 469 00:28:41,640 --> 00:28:51,240 And let me look at, let's take plane group p2mm, which has a 470 00:28:51,240 --> 00:28:53,110 fair amount of symmetry to it. 471 00:28:57,790 --> 00:29:02,970 So this is the group that we got by putting 2mm into a 472 00:29:02,970 --> 00:29:06,840 lattice that had to be rectangular, then. 473 00:29:06,840 --> 00:29:10,180 And across the top of the page, this is number six, 474 00:29:10,180 --> 00:29:11,720 you'll see that it is rectangular. 475 00:29:14,350 --> 00:29:16,820 It has to have a lattice in a rectangular shape because of 476 00:29:16,820 --> 00:29:18,050 the symmetry. 477 00:29:18,050 --> 00:29:19,785 Next comes the short hand symbol. 478 00:29:24,900 --> 00:29:27,660 And I always feel a little bit apologetic when I have to 479 00:29:27,660 --> 00:29:32,090 point out the existence of this, and the reason is that 480 00:29:32,090 --> 00:29:35,910 people who work with symmetry are no more or less lazy than 481 00:29:35,910 --> 00:29:37,510 any other human being. 482 00:29:37,510 --> 00:29:41,270 And some of the symbols become really ungainly when we go to 483 00:29:41,270 --> 00:29:43,230 three dimensional symmetry. 484 00:29:43,230 --> 00:29:52,770 And something like a 2 sub 1 over a 2 over m 2 sub 1 over d 485 00:29:52,770 --> 00:29:54,180 is a mouthful. 486 00:29:54,180 --> 00:29:57,790 And actually what's done is to just specify the minimum 487 00:29:57,790 --> 00:30:01,290 amount of information, which defines the symmetry. 488 00:30:01,290 --> 00:30:05,610 So here, two m's that intersect have to give you 489 00:30:05,610 --> 00:30:07,900 point group 2mm. 490 00:30:07,900 --> 00:30:13,530 Then comes the full symbol of the plane group, p2mm. 491 00:30:13,530 --> 00:30:15,650 And for those who are counting, this is number six 492 00:30:15,650 --> 00:30:20,690 and the shorthand symbol is pmm, where the short symbol 493 00:30:20,690 --> 00:30:23,730 for the point group, along with the lattice, is 494 00:30:23,730 --> 00:30:28,600 sufficient to tell you that this is 2mm placed at the 495 00:30:28,600 --> 00:30:30,130 nodes of a rectangular lattice. 496 00:30:33,770 --> 00:30:37,620 Then, underneath, comes the representative pattern. 497 00:30:37,620 --> 00:30:39,960 Again, there's a cross in the middle of this just to split 498 00:30:39,960 --> 00:30:41,210 it up into quadrants. 499 00:30:43,970 --> 00:30:47,240 The symmetry is one that we've already encountered and was 500 00:30:47,240 --> 00:30:50,980 fairly easy to come to terms with. 501 00:30:50,980 --> 00:30:55,850 Two-fold axes at the corners of the cell, and in the center 502 00:30:55,850 --> 00:30:59,040 and in the midpoint of the edges. 503 00:30:59,040 --> 00:31:02,270 That's just like p2, except the mirror planes that are 504 00:31:02,270 --> 00:31:07,630 present require that that parallelogram in p2 straighten 505 00:31:07,630 --> 00:31:09,560 out into a rectangle. 506 00:31:09,560 --> 00:31:13,145 And then we put 2mm at the lattice points so the mirror 507 00:31:13,145 --> 00:31:19,210 planes run down through the cell like this, up and down 508 00:31:19,210 --> 00:31:22,430 and left to right. 509 00:31:22,430 --> 00:31:25,930 So here is all the symmetry here. 510 00:31:25,930 --> 00:31:27,990 Pattern, I'll say it again. 511 00:31:27,990 --> 00:31:29,300 It's so easy to forget. 512 00:31:29,300 --> 00:31:32,610 The pattern is nothing more than the pattern that 2mm 513 00:31:32,610 --> 00:31:39,150 produces hung at every lattice point of a rectangular net. 514 00:31:39,150 --> 00:31:43,330 And all of the symmetry planes and symmetry axes that arise 515 00:31:43,330 --> 00:31:48,400 are just ways of defining all of the relations that exist 516 00:31:48,400 --> 00:31:52,710 between these things when you perform 517 00:31:52,710 --> 00:31:55,290 that process of addition. 518 00:31:55,290 --> 00:31:58,520 So this is, then, the arrangement of all of the 519 00:31:58,520 --> 00:32:03,570 atoms in the cell and then some. 520 00:32:03,570 --> 00:32:05,820 And there's a new symbol that's introduced, and 521 00:32:05,820 --> 00:32:08,590 somebody asked about that last time. 522 00:32:08,590 --> 00:32:13,141 To me it looks like a little tadpole inside of a frog egg. 523 00:32:13,141 --> 00:32:15,372 Has anybody seen a frog egg? 524 00:32:15,372 --> 00:32:16,710 That's just what it looks like. 525 00:32:16,710 --> 00:32:19,320 It's a little tiny tadpole in there, waiting to hatch out. 526 00:32:19,320 --> 00:32:21,620 So here is our little tadpole. 527 00:32:21,620 --> 00:32:24,845 This is used to indicate an enantiomorph. 528 00:32:27,980 --> 00:32:31,470 If this were a motif with chirality, this one would be 529 00:32:31,470 --> 00:32:34,340 right handed, this one would be left handed. 530 00:32:34,340 --> 00:32:39,180 So that indicates all of the atoms or molecules that are 531 00:32:39,180 --> 00:32:40,430 the same chirality. 532 00:32:42,710 --> 00:32:48,000 The purists among you will notice that the dummy that 533 00:32:48,000 --> 00:32:52,160 produced this diagram did not show the tadpoles conforming 534 00:32:52,160 --> 00:32:53,430 to the two-fold symmetry. 535 00:32:53,430 --> 00:32:55,730 Both of the commas point in the same direction. 536 00:32:55,730 --> 00:32:56,680 Tsk, tsk. 537 00:32:56,680 --> 00:32:57,930 Little oversight. 538 00:33:00,050 --> 00:33:03,340 So this, then, is the arrangement of 539 00:33:03,340 --> 00:33:06,166 motifs in the pattern. 540 00:33:06,166 --> 00:33:09,830 And now let's proceed to analyze that, according to the 541 00:33:09,830 --> 00:33:13,375 nature of the positions that are available 542 00:33:13,375 --> 00:33:15,240 of different sorts. 543 00:33:15,240 --> 00:33:18,020 Again, coordinates have meaning only if 544 00:33:18,020 --> 00:33:21,260 you define the origin. 545 00:33:21,260 --> 00:33:24,930 And nobody in their right mind would want to pick an origin 546 00:33:24,930 --> 00:33:27,660 that is off, at least, the symmetry plane. 547 00:33:27,660 --> 00:33:31,190 But the thing to do is to take the origin at 2mm. 548 00:33:35,270 --> 00:33:37,510 Again, I'll remind you that x goes down this 549 00:33:37,510 --> 00:33:40,180 way, y goes this way. 550 00:33:40,180 --> 00:33:45,030 So if I have a representative atom at x, y, I'll have 551 00:33:45,030 --> 00:33:50,280 another one at minus x, plus x, another one at minus x, 552 00:33:50,280 --> 00:33:55,700 minus y, and one at plus x, minus y. 553 00:33:55,700 --> 00:33:57,280 So there are the coordinates of all the 554 00:33:57,280 --> 00:33:58,580 symmetry of related atoms. 555 00:34:01,130 --> 00:34:06,780 So putting down the general position, I get four of them 556 00:34:06,780 --> 00:34:10,239 if I place one in the cell. 557 00:34:10,239 --> 00:34:14,889 By definition, the general position is always at a site 558 00:34:14,889 --> 00:34:16,929 of symmetry 1. 559 00:34:16,929 --> 00:34:18,179 So this is the site symmetry. 560 00:34:21,540 --> 00:34:24,250 Then we just rattled of the coordinates. 561 00:34:24,250 --> 00:34:31,110 They're x,y, minus x, y, x, minus y, and minus x, minus y. 562 00:34:40,360 --> 00:34:43,010 There are lots of special positions here, now, because 563 00:34:43,010 --> 00:34:47,790 we have not only two-fold axes with mirror planes 564 00:34:47,790 --> 00:34:51,190 intersecting at them, but they're also locations of just 565 00:34:51,190 --> 00:34:52,549 a mirror plane alone. 566 00:34:55,800 --> 00:34:59,160 And there are a total of four distinct, independent mirror 567 00:34:59,160 --> 00:35:04,810 planes, this one, this one, this one, and this one. 568 00:35:04,810 --> 00:35:10,360 So we're going to have four positions where the atom sits 569 00:35:10,360 --> 00:35:12,210 at a location of symmetry 2mm. 570 00:35:16,540 --> 00:35:18,880 And those are going to be analogous to the positions 571 00:35:18,880 --> 00:35:21,990 that we found for the two-fold axes in p2. 572 00:35:21,990 --> 00:35:24,230 And, not surprisingly, the coordinates look 573 00:35:24,230 --> 00:35:26,202 very much the same. 574 00:35:26,202 --> 00:35:31,005 0,0 puts us on a location of symmetry 2mm. 575 00:35:31,005 --> 00:35:37,050 0, 1/2 does the same, 1/2, 0 and 1/2, 1/2. 576 00:35:40,950 --> 00:35:45,400 So there are four positions of symmetry to 2mm. 577 00:35:45,400 --> 00:35:48,970 And then there are four positions just on a mirror 578 00:35:48,970 --> 00:35:50,820 plane, but not on a two-fold axis. 579 00:35:50,820 --> 00:35:53,460 And we could do that for the mirror plane that runs along 580 00:35:53,460 --> 00:35:56,770 the x-axis, or the mirror plane that runs along the 581 00:35:56,770 --> 00:35:59,530 direction of x at y equals 2. 582 00:35:59,530 --> 00:36:02,710 And the order in which you pick them is rather arbitrary, 583 00:36:02,710 --> 00:36:12,140 but there will be a pair of positions at x, 0 and minus x, 584 00:36:12,140 --> 00:36:18,520 0 that would happen if the y-coordinate was exactly-- 585 00:36:18,520 --> 00:36:23,020 let me draw what the position is, just off here to the right 586 00:36:23,020 --> 00:36:24,320 in a small diagram. 587 00:36:24,320 --> 00:36:27,000 This would be where you put the atom on the mirror plane 588 00:36:27,000 --> 00:36:30,060 running through the origin, then these things would give 589 00:36:30,060 --> 00:36:32,410 you only a pair. 590 00:36:32,410 --> 00:36:38,560 The next one is at a position x, 1/2, and that's where the 591 00:36:38,560 --> 00:36:48,320 mirror plane would be the mirror plane that is along the 592 00:36:48,320 --> 00:36:50,980 position y equals 1/2. 593 00:36:50,980 --> 00:36:57,130 And here we would get atoms coalescing like this. 594 00:36:57,130 --> 00:37:01,890 x is general, but y is exactly 1/2. 595 00:37:01,890 --> 00:37:06,970 So we'd have x, 1/2, minus x, 1/2, and I better put these 596 00:37:06,970 --> 00:37:07,720 off to the right, here. 597 00:37:07,720 --> 00:37:13,630 The remaining two are 0, y and 0, minus y. 598 00:37:13,630 --> 00:37:16,440 That is also a position of site symmetry m. 599 00:37:25,180 --> 00:37:33,260 And that would be a pair of objects that has x equal to 0. 600 00:37:33,260 --> 00:37:37,360 y is anything, so that's this mirror plane here. 601 00:37:37,360 --> 00:37:42,220 And in that case we'd have a pair of atoms, left to right, 602 00:37:42,220 --> 00:37:46,040 reflected by the mirror plane that goes through the origin. 603 00:37:46,040 --> 00:37:49,280 And then last one, thank goodness, is one at a location 604 00:37:49,280 --> 00:37:54,130 1/2, y and 1/2, minus y. 605 00:37:54,130 --> 00:37:59,860 And this would be the mirror plane exactly at 1/2. 606 00:37:59,860 --> 00:38:01,110 y can be anything. 607 00:38:07,740 --> 00:38:09,050 x is 1/2. 608 00:38:09,050 --> 00:38:13,720 So that would be a pair that sits somewhere on either side 609 00:38:13,720 --> 00:38:14,940 of the mirror plane running through the 610 00:38:14,940 --> 00:38:16,520 center of the cell. 611 00:38:16,520 --> 00:38:19,700 That's also a site of symmetry m. 612 00:38:19,700 --> 00:38:23,240 Now we assign the Wyckoff symbol by working our way up 613 00:38:23,240 --> 00:38:23,930 the alphabet. 614 00:38:23,930 --> 00:38:30,610 So this is a, this is b, this is c, this is d, this is e, f, 615 00:38:30,610 --> 00:38:34,350 this is g, this is h, and finally we end 616 00:38:34,350 --> 00:38:38,040 up at position i. 617 00:38:38,040 --> 00:38:43,210 For all of these, we get two atoms per cell. 618 00:38:43,210 --> 00:38:45,720 These have rank 2. 619 00:38:45,720 --> 00:38:48,258 For each of these we get just a single atom. 620 00:38:52,082 --> 00:38:53,994 AUDIENCE: Can you explain the two atom per cell thing? 621 00:38:53,994 --> 00:38:55,244 How do you [INAUDIBLE]? 622 00:38:58,740 --> 00:39:00,960 PROFESSOR: Which one would you like me to do, that last one? 623 00:39:00,960 --> 00:39:01,630 1/2, y? 624 00:39:01,630 --> 00:39:02,880 AUDIENCE:[INAUDIBLE]. 625 00:39:05,220 --> 00:39:07,570 PROFESSOR: So if x is 1/2 and y is anything, that's this 626 00:39:07,570 --> 00:39:10,040 locus here. 627 00:39:10,040 --> 00:39:13,050 And that is a mirror plane. 628 00:39:13,050 --> 00:39:18,710 And if we let this atom move from its location, x,y, down 629 00:39:18,710 --> 00:39:23,530 to this location, the one in the lower part of the cell is 630 00:39:23,530 --> 00:39:27,890 going to move up and these two will coalesce to a blob that 631 00:39:27,890 --> 00:39:28,816 sits there. 632 00:39:28,816 --> 00:39:30,440 AUDIENCE: That would be 2m? 633 00:39:30,440 --> 00:39:33,110 PROFESSOR: So this would be site symmetry m. 634 00:39:33,110 --> 00:39:35,942 And instead of getting four per cell I would get two per 635 00:39:35,942 --> 00:39:38,515 cell; this one and the one that's reflected across. 636 00:39:42,710 --> 00:39:47,220 Now the other way of doing it is to not think about it all. 637 00:39:47,220 --> 00:39:55,920 Say here is x,y, minus x, y, x, minus y, minus x, minus y. 638 00:39:55,920 --> 00:39:58,900 And then let's make one of the coordinates specialized. 639 00:39:58,900 --> 00:40:01,370 Let's let x be exactly 1/2. 640 00:40:01,370 --> 00:40:09,150 So I'll get 1/2, y, minus 1/2, y, and then I'll get 1/2, 641 00:40:09,150 --> 00:40:14,370 minus y and minus 1/2, minus y. 642 00:40:14,370 --> 00:40:19,090 Looks like four atoms, except if I have one at plus 1/2, the 643 00:40:19,090 --> 00:40:21,640 one in the neighbor unit cell is at minus 1/2. 644 00:40:21,640 --> 00:40:25,800 So actually I've had the atoms coalesce pairwise, because 645 00:40:25,800 --> 00:40:33,200 these coordinates are actually identical to saying 1/2, y 646 00:40:33,200 --> 00:40:39,480 plus 1/2, y, 1/2, minus y, and minus 1/2 is the same as plus 647 00:40:39,480 --> 00:40:41,650 1/2, minus y. 648 00:40:41,650 --> 00:40:42,790 So you can do it that way. 649 00:40:42,790 --> 00:40:46,730 Just put in a special value for x or y, turn the crank on 650 00:40:46,730 --> 00:40:49,140 the general position, and you'll find you're going to 651 00:40:49,140 --> 00:40:50,390 get the same thing twice. 652 00:41:00,840 --> 00:41:05,130 And if we look at position a, which is 0,0. 653 00:41:05,130 --> 00:41:09,990 put in 0 for x, 0 for y, then put 0,0 for all these other 654 00:41:09,990 --> 00:41:11,750 four positions, you're going to get 0,0 655 00:41:11,750 --> 00:41:13,000 four different times. 656 00:41:24,310 --> 00:41:28,820 I'm going to do just one more, and the results are here for 657 00:41:28,820 --> 00:41:31,290 all of the remaining 14. 658 00:41:31,290 --> 00:41:35,230 So I don't think there's any need to do them all. 659 00:41:35,230 --> 00:41:39,940 One of the things I would like to examine with you is a group 660 00:41:39,940 --> 00:41:42,750 that has a glide point in it, and see how 661 00:41:42,750 --> 00:41:45,700 that changes things. 662 00:41:45,700 --> 00:41:46,950 Any other questions on this? 663 00:41:49,748 --> 00:41:53,024 AUDIENCE: Can you just go over the rank of fours? 664 00:41:53,024 --> 00:41:53,970 Four-- 665 00:41:53,970 --> 00:41:57,130 PROFESSOR: Four means four per cell. 666 00:41:57,130 --> 00:41:57,920 AUDIENCE: Four per cell. 667 00:41:57,920 --> 00:42:00,620 PROFESSOR: And if you count them around the 668 00:42:00,620 --> 00:42:02,340 origin as 1, 2, 3, 4. 669 00:42:02,340 --> 00:42:04,467 If you count the number that's caught within the box 670 00:42:04,467 --> 00:42:05,940 it's 1, 2, 3, 4. 671 00:42:09,110 --> 00:42:10,280 AUDIENCE: And site symmetry of 1? 672 00:42:10,280 --> 00:42:13,250 PROFESSOR:That's site symmetry 1 because 1 is no symmetry at 673 00:42:13,250 --> 00:42:15,770 all, and that is what's general about it. 674 00:42:15,770 --> 00:42:18,900 It's not sitting on a symmetry element which would then fail 675 00:42:18,900 --> 00:42:20,320 to reproduce it. 676 00:42:27,200 --> 00:42:29,810 And again, if the motif was not an atom but was a 677 00:42:29,810 --> 00:42:33,320 molecule, if you find only one molecule per cell from the 678 00:42:33,320 --> 00:42:36,070 density in the cell dimensions, that molecule has 679 00:42:36,070 --> 00:42:39,430 to have symmetry 2mm, because it has to sit. 680 00:42:39,430 --> 00:42:42,990 If there's only one per cell, that set of atoms has to sit 681 00:42:42,990 --> 00:42:47,360 at a location that conforms to symmetry 2mm. 682 00:42:47,360 --> 00:42:51,170 Let us go to one more. 683 00:42:51,170 --> 00:42:58,850 And I'm going to take p2mg, because it's one that has a 684 00:42:58,850 --> 00:43:00,190 glide plane in it. 685 00:43:00,190 --> 00:43:03,340 This is pmg for short. 686 00:43:03,340 --> 00:43:06,600 This is number seven. 687 00:43:06,600 --> 00:43:13,520 P2mg is the proper symbol, and that's point group 2mm. 688 00:43:13,520 --> 00:43:15,736 And, again, the coordinate system is rectangular. 689 00:43:19,340 --> 00:43:21,390 This is one that we derived last time. 690 00:43:21,390 --> 00:43:26,360 The arrangements of elements in this is, once again, a 691 00:43:26,360 --> 00:43:29,600 rectangular lattice, two-fold axes. 692 00:43:29,600 --> 00:43:34,820 If we take the origin at 2, would be at the corner of the 693 00:43:34,820 --> 00:43:38,080 cell and in the midpoint of the edges. 694 00:43:38,080 --> 00:43:40,400 Then there's a glide plane passing through 695 00:43:40,400 --> 00:43:41,650 the two-fold axes. 696 00:43:45,590 --> 00:43:49,460 And there is a mirror plane passing in between the 697 00:43:49,460 --> 00:43:51,480 two-fold axes. 698 00:43:51,480 --> 00:43:56,540 And here's a good example of an arrangement of symmetry 699 00:43:56,540 --> 00:43:59,850 elements that has a mirror plane in it, and then a line 700 00:43:59,850 --> 00:44:03,280 that just indicates the edges of the unit cell. 701 00:44:03,280 --> 00:44:07,870 And can you distinguish the bold line from the light line? 702 00:44:07,870 --> 00:44:10,330 Barely. 703 00:44:10,330 --> 00:44:12,520 But the bold line is the mirror plane. 704 00:44:12,520 --> 00:44:15,880 And what does the arrangement of atoms look like? 705 00:44:24,150 --> 00:44:26,720 This is a tricky one, because there are two different 706 00:44:26,720 --> 00:44:30,010 symmetry elements; a two-fold axis, and that's going to take 707 00:44:30,010 --> 00:44:37,350 an atom at x, y and repeat it to minus x, minus y. 708 00:44:37,350 --> 00:44:41,540 But here, now, is a case where there's another symmetry 709 00:44:41,540 --> 00:44:44,950 element that does not intersect the first one. 710 00:44:44,950 --> 00:44:49,350 So that mirror plane is going to reflect this atom down here 711 00:44:49,350 --> 00:44:53,920 to an enantiomorph and reflect this atom down here to another 712 00:44:53,920 --> 00:44:55,980 enantiomorph. 713 00:44:55,980 --> 00:45:02,080 So if this is x then this is y, this one sits at minus x 714 00:45:02,080 --> 00:45:04,960 and minus y. 715 00:45:04,960 --> 00:45:12,600 For this atom here, y is the same as before. 716 00:45:12,600 --> 00:45:17,070 But if this distance is x, the distance up from the center 717 00:45:17,070 --> 00:45:23,670 line will be x, so this second coordinate is 1/2, minus x. 718 00:45:23,670 --> 00:45:26,240 I'll do that again since this is pretty small and tight. 719 00:45:26,240 --> 00:45:27,820 This distance is x. 720 00:45:27,820 --> 00:45:33,800 We reflect the atom across a mirror line at 1/4, and 721 00:45:33,800 --> 00:45:37,930 therefore the one towards the center of the cell sits up 722 00:45:37,930 --> 00:45:42,330 above the location 1/2 by the same number, x. 723 00:45:42,330 --> 00:45:45,690 So this is 1/2 minus x, y. 724 00:45:45,690 --> 00:45:52,140 And by the same argument, this one here is at minus y, and it 725 00:45:52,140 --> 00:45:56,660 sticks out beyond the position x equals 1/2 by the 726 00:45:56,660 --> 00:45:58,110 same amount, x. 727 00:45:58,110 --> 00:46:06,170 So this one is 1/2 plus x minus y. 728 00:46:06,170 --> 00:46:07,420 Nontrivial. 729 00:46:09,850 --> 00:46:11,730 So if I began to tabulate these-- 730 00:46:22,480 --> 00:46:27,680 We've picked the origin at 2. 731 00:46:27,680 --> 00:46:28,870 Didn't have to do that. 732 00:46:28,870 --> 00:46:31,390 If you wanted to, you could take the origin at m if you 733 00:46:31,390 --> 00:46:36,920 wanted to, and just switch everything by 1/4 of the 734 00:46:36,920 --> 00:46:40,090 coordinate x. 735 00:46:40,090 --> 00:46:41,340 We found four per cell. 736 00:46:44,680 --> 00:46:50,630 The site symmetry is 1, and the atom locations were at x, 737 00:46:50,630 --> 00:46:54,950 y, minus x, minus y. 738 00:46:54,950 --> 00:47:08,630 And then we had 1/2, minus x, y and 1/2, plus x, minus y. 739 00:47:11,330 --> 00:47:17,650 So the coordinates are getting permuted around in a very more 740 00:47:17,650 --> 00:47:19,950 complex fashion than just changing sign. 741 00:47:23,570 --> 00:47:24,820 Special positions. 742 00:47:29,040 --> 00:47:35,280 Just as before, any of these two-fold axes is going to be a 743 00:47:35,280 --> 00:47:38,610 site for a special position. 744 00:47:38,610 --> 00:47:40,310 Let x and y go to 0. 745 00:47:40,310 --> 00:47:45,200 This pair at the origin will coalesce and this pair will 746 00:47:45,200 --> 00:47:49,620 coalesce at a point that's halfway along the cell. 747 00:47:49,620 --> 00:47:57,050 So there will be two of those, one at the origin, one in the 748 00:47:57,050 --> 00:48:00,270 middle of this edge, and their locations would be 749 00:48:00,270 --> 00:48:05,376 0,0 and 1/2, 0. 750 00:48:05,376 --> 00:48:09,720 And this is a site on a two-fold axis and we have only 751 00:48:09,720 --> 00:48:10,970 two per cell. 752 00:48:19,450 --> 00:48:23,750 Would this be a location for a special position, this 753 00:48:23,750 --> 00:48:25,665 two-fold axis? 754 00:48:25,665 --> 00:48:26,915 It was before. 755 00:48:29,660 --> 00:48:31,680 Is it now? 756 00:48:31,680 --> 00:48:35,850 Key point is independent sites of symmetry. 757 00:48:35,850 --> 00:48:37,900 There's a mirror plane here. 758 00:48:37,900 --> 00:48:41,620 What goes on here has to be exactly the same as 759 00:48:41,620 --> 00:48:44,110 what goes on here. 760 00:48:44,110 --> 00:48:48,265 So this is not an independent special position. 761 00:48:59,660 --> 00:49:02,780 In the same way, there's a mirror plane. 762 00:49:02,780 --> 00:49:07,280 And if I let the atom migrate down to the mirror plane I 763 00:49:07,280 --> 00:49:15,805 will have the mirror plane at a location 1/4, y. 764 00:49:18,810 --> 00:49:23,740 And these will be on a mirror plane. 765 00:49:26,430 --> 00:49:32,670 1/4, y would be a first, 3/4, minus y would be the second 766 00:49:32,670 --> 00:49:35,345 one that I get, and they would coalesce pairwise. 767 00:49:42,710 --> 00:49:50,450 Is this is another mirror plane that could be regarded 768 00:49:50,450 --> 00:49:52,450 as an independent position? 769 00:49:52,450 --> 00:49:59,330 This mirror plane, again, is not independent, because it's 770 00:49:59,330 --> 00:50:02,800 related to the one we just considered by the two-fold 771 00:50:02,800 --> 00:50:05,080 rotation about the center of the cell. 772 00:50:05,080 --> 00:50:08,850 So there's only one kind of mirror plane in the structure. 773 00:50:08,850 --> 00:50:12,160 But, going back to the two-fold axes, there is a 774 00:50:12,160 --> 00:50:13,240 two-fold axis here. 775 00:50:13,240 --> 00:50:16,970 There's another one that is halfway along y. 776 00:50:16,970 --> 00:50:21,110 So there's another two-fold axis that is independent at 0, 777 00:50:21,110 --> 00:50:24,060 1/2, 1/2, 1/2. 778 00:50:24,060 --> 00:50:27,570 And this is the entire set. 779 00:50:27,570 --> 00:50:31,550 So there are three special positions of rank 2, 780 00:50:31,550 --> 00:50:32,840 two atoms per cell. 781 00:50:32,840 --> 00:50:36,920 We make the most specialized one be named a, work our way 782 00:50:36,920 --> 00:50:39,280 up to c, and the general position, then, is 783 00:50:39,280 --> 00:50:40,530 referred to as d. 784 00:50:43,370 --> 00:50:47,890 Again, so what I did was to say that this mirror plane is 785 00:50:47,890 --> 00:50:50,600 related to this one by symmetry, and if there's 786 00:50:50,600 --> 00:50:53,390 something on this mirror plane as a special position, I'm 787 00:50:53,390 --> 00:50:57,220 automatically going to get it on the lower mirror plane. 788 00:50:57,220 --> 00:51:00,210 This two-fold axis is the same as this one. 789 00:51:00,210 --> 00:51:03,500 So whatever is going on as a result of specializing the 790 00:51:03,500 --> 00:51:06,870 location on this two-fold axis is going to be provided for me 791 00:51:06,870 --> 00:51:09,940 automatically at x equals 1/2 by the action 792 00:51:09,940 --> 00:51:10,900 of this mirror plane. 793 00:51:10,900 --> 00:51:13,850 There's nothing that throws this two-fold axis into this 794 00:51:13,850 --> 00:51:15,400 one, though, so they're both independent. 795 00:51:18,950 --> 00:51:22,070 Now, one of the symmetry elements that I steadfastly 796 00:51:22,070 --> 00:51:26,310 ignored was the glide plane. 797 00:51:26,310 --> 00:51:31,050 What happens if I place an atom right on the glide plane? 798 00:51:31,050 --> 00:51:36,950 A glide plane, if you use a motif that has some handedness 799 00:51:36,950 --> 00:51:41,050 to it, we repeat the atom by translating half of a 800 00:51:41,050 --> 00:51:45,620 translation, not yet putting it down, first 801 00:51:45,620 --> 00:51:47,110 reflecting it across. 802 00:51:47,110 --> 00:51:48,880 This is an enantiomorph. 803 00:51:48,880 --> 00:51:51,750 Doing it again gives me an atom that's related to the 804 00:51:51,750 --> 00:51:53,420 first by a translation. 805 00:51:56,510 --> 00:52:02,135 What happens if I let the atom migrate onto the glide plane? 806 00:52:02,135 --> 00:52:03,800 Let this atom move down to here. 807 00:52:03,800 --> 00:52:05,730 Now it sits here. 808 00:52:05,730 --> 00:52:09,610 This one, repeated from it by glide, will 809 00:52:09,610 --> 00:52:12,470 migrate up to here. 810 00:52:12,470 --> 00:52:14,390 Nothing happens. 811 00:52:14,390 --> 00:52:17,090 It's just that the amount that the atom is displaced from the 812 00:52:17,090 --> 00:52:20,330 locus of the glide plane is different in the two cases. 813 00:52:20,330 --> 00:52:25,490 But there is no coalescence of the atoms, because of this 814 00:52:25,490 --> 00:52:29,080 translation component to the glide plane. 815 00:52:29,080 --> 00:52:37,890 So a conclusion, then, is that glide is never a candidate 816 00:52:37,890 --> 00:52:39,935 location for a special position. 817 00:52:58,570 --> 00:53:03,130 This is a good example of a case where I can check what 818 00:53:03,130 --> 00:53:05,430 I've done, and to see if I counted the 819 00:53:05,430 --> 00:53:07,540 same position twice. 820 00:53:07,540 --> 00:53:10,290 I said that both of these locations 821 00:53:10,290 --> 00:53:11,820 are not special position. 822 00:53:11,820 --> 00:53:13,860 The mirror planes are related by symmetry. 823 00:53:13,860 --> 00:53:19,150 Suppose I were not clever enough to notice that, and I 824 00:53:19,150 --> 00:53:25,910 said, OK, there's going to be a special position, 1/4, x, 825 00:53:25,910 --> 00:53:28,970 3/4, minus y. 826 00:53:28,970 --> 00:53:34,490 That's what happened if I use this mirror plane 827 00:53:34,490 --> 00:53:36,880 as a special position. 828 00:53:36,880 --> 00:53:50,900 What would I get if I took 3/4, y and 1/4, minus y? 829 00:53:54,910 --> 00:53:58,140 Looks like a different specialized position. 830 00:53:58,140 --> 00:54:01,960 But let me put this number into my expression that is the 831 00:54:01,960 --> 00:54:03,350 formula for the general position. 832 00:54:03,350 --> 00:54:09,940 If x is 1/4 and y is y, then this position here would be 833 00:54:09,940 --> 00:54:19,520 minus 1/4 and plus y, 1/2, minus x would be 834 00:54:19,520 --> 00:54:23,880 1/4, and y is y. 835 00:54:23,880 --> 00:54:27,160 Over here, I got the same thing back again. 836 00:54:27,160 --> 00:54:34,630 And 1/2, plus x would be 3/4, and minus y is minus y, minus 837 00:54:34,630 --> 00:54:37,940 1/2 and 3/4, plus 3/4 are the same thing. 838 00:54:37,940 --> 00:54:42,000 So what I've gotten is the position 1/4, y, and 3/4, 839 00:54:42,000 --> 00:54:43,250 minus y twice. 840 00:54:50,330 --> 00:54:53,915 So what this is telling me, analytically, if I just plug 841 00:54:53,915 --> 00:54:57,650 in the coordinates of what seem to be a distinct, special 842 00:54:57,650 --> 00:55:02,190 position is, I get the same atom on top of itself twice. 843 00:55:02,190 --> 00:55:04,650 And it looks exactly like choosing the 844 00:55:04,650 --> 00:55:06,520 first mirror plane. 845 00:55:06,520 --> 00:55:08,630 So that's a good way of checking, particularly in a 846 00:55:08,630 --> 00:55:12,010 very high symmetry where there are all sorts of sites of 847 00:55:12,010 --> 00:55:15,500 possible point groups that could cause coalescence, to 848 00:55:15,500 --> 00:55:18,530 just crank out the coordinates and see if, in fact, you have 849 00:55:18,530 --> 00:55:23,010 exactly the same pair with the same coordinates that 850 00:55:23,010 --> 00:55:27,220 correspond exactly to the x and y of some previous 851 00:55:27,220 --> 00:55:28,710 location that you identified. 852 00:55:31,880 --> 00:55:40,340 I think, hopefully, you have a feeling for how it works. 853 00:55:40,340 --> 00:55:43,980 If you look at some of the higher symmetries, the number 854 00:55:43,980 --> 00:55:51,030 of atoms that constitute the full equipoint set of 855 00:55:51,030 --> 00:55:52,180 equivalent positions. 856 00:55:52,180 --> 00:55:57,510 For p6mm there are 12 locations in the general 857 00:55:57,510 --> 00:56:01,830 position, and because the coordinate system is oblique, 858 00:56:01,830 --> 00:56:04,260 x and y get permuted into linear 859 00:56:04,260 --> 00:56:05,930 combinations of one another. 860 00:56:05,930 --> 00:56:10,100 So that's really, extremely complicated. 861 00:56:10,100 --> 00:56:13,230 If you look at some of the square symmetries, the number 862 00:56:13,230 --> 00:56:17,650 of the special positions is very, very high. 863 00:56:17,650 --> 00:56:22,350 There are two positions in p4mm, there are two 864 00:56:22,350 --> 00:56:24,580 positions of rank 1. 865 00:56:24,580 --> 00:56:29,810 There is a position of rank 2 that sits on 2mm. 866 00:56:29,810 --> 00:56:33,770 Three different mirror planes, and the general position with 867 00:56:33,770 --> 00:56:37,490 rank 8 requires working your way all the way through g in 868 00:56:37,490 --> 00:56:38,460 the alphabet. 869 00:56:38,460 --> 00:56:39,792 Yes? 870 00:56:39,792 --> 00:56:42,272 AUDIENCE: I had a question about the p2gg [INAUDIBLE] 871 00:56:45,744 --> 00:56:51,210 whether they're at the entrance of the [INAUDIBLE]? 872 00:56:51,210 --> 00:56:59,360 PROFESSOR: For p2gg, the only site of point symmetry is a 873 00:56:59,360 --> 00:57:00,610 two-fold axis. 874 00:57:02,360 --> 00:57:06,390 So the only thing that you could use for a special 875 00:57:06,390 --> 00:57:11,550 position in p2gg is a two-fold axis. 876 00:57:11,550 --> 00:57:16,630 And there are two different two-fold axes. 877 00:57:16,630 --> 00:57:21,950 They relate in a curious way because of a glide plane. 878 00:57:21,950 --> 00:57:25,145 And this is a case where the site of a special position is 879 00:57:25,145 --> 00:57:28,990 not related by rotation or reflection, but by a glide. 880 00:57:28,990 --> 00:57:35,074 P2gg looks like this. 881 00:57:35,074 --> 00:57:41,800 Two-fold axes in the old familiar places, 0, 1/2. 882 00:57:41,800 --> 00:57:49,000 The glide planes are here, here, here, and here. 883 00:57:49,000 --> 00:57:55,550 So which two-fold axes are equivalent? 884 00:57:55,550 --> 00:57:58,920 The glide would take this two-fold axis, reflect it 885 00:57:58,920 --> 00:58:01,370 down, and slide it over to here. 886 00:58:01,370 --> 00:58:06,260 So whatever goes on here goes on at this two-fold axis. 887 00:58:06,260 --> 00:58:06,742 AUDIENCE: Sorry. 888 00:58:06,742 --> 00:58:10,598 I guess my question was, can you put the glide planes at 889 00:58:10,598 --> 00:58:11,562 any edges of this sites? 890 00:58:11,562 --> 00:58:14,940 [INAUDIBLE], or-- 891 00:58:14,940 --> 00:58:15,440 PROFESSOR: You could. 892 00:58:15,440 --> 00:58:21,490 You could, but you would lose, then, the apparent similarity 893 00:58:21,490 --> 00:58:24,011 in coordinates in that, if you put the glide plane at the 894 00:58:24,011 --> 00:58:29,150 two-fold axis, if this is x, y, this one over here is at 895 00:58:29,150 --> 00:58:32,000 minus x, minus y. 896 00:58:32,000 --> 00:58:36,070 And then if you repeat it by glide, that would slide over 897 00:58:36,070 --> 00:58:39,020 to here and then be reflected down here. 898 00:58:39,020 --> 00:58:42,140 So there'd be one pair up here and one pair down here of 899 00:58:42,140 --> 00:58:50,270 opposite handedness, so you would not see the similarity. 900 00:58:50,270 --> 00:58:53,150 No, that's right. 901 00:58:55,980 --> 00:59:05,750 If you took the origin at the glide plane, you would not see 902 00:59:05,750 --> 00:59:09,420 the simple relation between the numbers that you do when 903 00:59:09,420 --> 00:59:12,680 there is that a rotation axis of a mirror plane. 904 00:59:12,680 --> 00:59:16,840 But you can take the origin anywhere you want. 905 00:59:16,840 --> 00:59:20,150 And the advantage of doing it at a symmetry element is then 906 00:59:20,150 --> 00:59:24,990 the numbers that describe where the atoms sit are simple 907 00:59:24,990 --> 00:59:28,780 permutations of sign, or adding 1/2 to the number of 908 00:59:28,780 --> 00:59:32,120 the preceding atom that was just mapped to a new location. 909 00:59:39,700 --> 00:59:43,400 I think you're more than ready for a break. 910 00:59:43,400 --> 00:59:48,190 Let's resume in 10 minutes, and we'll move on to something 911 00:59:48,190 --> 00:59:49,440 completely different.