1 00:00:08,940 --> 00:00:13,840 PROFESSOR: Today, we are going to change gears after the 2 00:00:13,840 --> 00:00:17,870 first hour to a very different set of topics. 3 00:00:17,870 --> 00:00:23,620 What I wanted to do today was to take a look at notation for 4 00:00:23,620 --> 00:00:24,920 space groups. 5 00:00:24,920 --> 00:00:28,050 And this is a question that is far from trivial. 6 00:00:28,050 --> 00:00:31,280 There are conventions, and we'll see the reasons for them 7 00:00:31,280 --> 00:00:33,090 with a few examples. 8 00:00:33,090 --> 00:00:37,480 And then I would like to pass around representations of some 9 00:00:37,480 --> 00:00:39,290 non-trivial space groups. 10 00:00:39,290 --> 00:00:43,000 First, as they appear in the old International Tables for 11 00:00:43,000 --> 00:00:45,680 X-Ray Crystallography, Volume 1. 12 00:00:45,680 --> 00:00:50,670 And then second, to give you a look at the larger, heavier, 13 00:00:50,670 --> 00:00:54,890 and much more expensive Volume A, which gives much more 14 00:00:54,890 --> 00:01:01,050 information which is both its liability and its advantage. 15 00:01:01,050 --> 00:01:02,720 There are things in there that are not in the 16 00:01:02,720 --> 00:01:03,690 international tables. 17 00:01:03,690 --> 00:01:06,560 But there's so much information and so cluttered 18 00:01:06,560 --> 00:01:07,890 that it's really overwhelming. 19 00:01:07,890 --> 00:01:10,900 But I'll give you a chance of what their representations of 20 00:01:10,900 --> 00:01:13,780 the space group look like. 21 00:01:13,780 --> 00:01:19,640 Let me begin by passing around an example of the information 22 00:01:19,640 --> 00:01:24,270 in the International Tables, Volume 1 for one of the 23 00:01:24,270 --> 00:01:27,910 orthorhombic space groups. 24 00:01:27,910 --> 00:01:30,540 And it all fits on one page. 25 00:01:30,540 --> 00:01:32,560 It's not terribly complicated. 26 00:01:32,560 --> 00:01:35,790 But it is considerably more complex both in the 27 00:01:35,790 --> 00:01:40,050 arrangement of atoms and in the cluster of symmetry 28 00:01:40,050 --> 00:01:45,530 elements, compared to the very simple monoclinic space groups 29 00:01:45,530 --> 00:01:48,200 which we looked at in their entirety and in 30 00:01:48,200 --> 00:01:49,450 considerable detail. 31 00:01:52,040 --> 00:01:56,030 So first of all, some indication of what's on the 32 00:01:56,030 --> 00:01:57,870 lower part of the page. 33 00:01:57,870 --> 00:02:02,050 This is exactly analogous to what you've seen and 34 00:02:02,050 --> 00:02:04,390 presumably are familiar with for the two 35 00:02:04,390 --> 00:02:07,280 dimensional plane groups. 36 00:02:07,280 --> 00:02:10,490 There are many possible origins that could be selected 37 00:02:10,490 --> 00:02:16,440 for the 0, 0, 0, position relative to which the atomic 38 00:02:16,440 --> 00:02:18,040 locations are expressed. 39 00:02:18,040 --> 00:02:20,870 So the first thing, just below the picture of the arrangement 40 00:02:20,870 --> 00:02:24,360 of atoms and the depiction of the arrangement of symmetry 41 00:02:24,360 --> 00:02:27,600 elements is a statement of where the origin is. 42 00:02:27,600 --> 00:02:33,760 So it says, origin at the inversion center 2/m. 43 00:02:33,760 --> 00:02:36,070 And if you look at the arrangement of symmetry 44 00:02:36,070 --> 00:02:44,570 elements, there is nothing but the representation for screw 45 00:02:44,570 --> 00:02:46,230 axes normal to the page. 46 00:02:46,230 --> 00:02:49,120 But there is, off on the right-hand side, a solid 47 00:02:49,120 --> 00:02:53,270 chevron, and that is the indication for a mirror plane. 48 00:02:53,270 --> 00:02:56,620 Something again that is nowhere stated, but always 49 00:02:56,620 --> 00:03:07,560 assumed is that the two variables x and y or the axes 50 00:03:07,560 --> 00:03:15,080 a and b are assumed to run from upper left to lower left 51 00:03:15,080 --> 00:03:20,210 for a and the coordinate x and from upper left to upper right 52 00:03:20,210 --> 00:03:21,980 for the axis b. 53 00:03:21,980 --> 00:03:26,230 And therefore, plus c comes out normal to the plane of the 54 00:03:26,230 --> 00:03:27,480 projection. 55 00:03:29,350 --> 00:03:33,150 Here, for the first time, we're seeing a complex space 56 00:03:33,150 --> 00:03:38,530 group that really requires the full range of symbols for 57 00:03:38,530 --> 00:03:42,450 representing the atomic locations. 58 00:03:42,450 --> 00:03:46,230 You will see, for each circle, a vertical line 59 00:03:46,230 --> 00:03:47,500 dividing it in half. 60 00:03:50,250 --> 00:03:53,870 These are two atoms then that are superposed in projection. 61 00:03:53,870 --> 00:03:58,530 One of them, corresponding to the little comma inside the 62 00:03:58,530 --> 00:04:03,780 circle, is the enantiomorph of those circles which are 63 00:04:03,780 --> 00:04:05,950 unadorned with that little comma. 64 00:04:05,950 --> 00:04:10,340 And they are, therefore, of opposite handedness. 65 00:04:10,340 --> 00:04:14,310 Alongside of each circle, you will see two symbols giving 66 00:04:14,310 --> 00:04:17,579 the elevation of that atom within the cell. 67 00:04:17,579 --> 00:04:23,070 The one at the furthest upper left corner says 1/2 plus. 68 00:04:23,070 --> 00:04:26,380 So this says that the coordinate is z, whatever that 69 00:04:26,380 --> 00:04:30,310 is for the representative atom, plus 1/2. 70 00:04:30,310 --> 00:04:35,280 The half of the circle that has the comma inside has a 71 00:04:35,280 --> 00:04:36,620 minus sign next do it. 72 00:04:36,620 --> 00:04:41,200 That means that it is at an elevation minus z. 73 00:04:41,200 --> 00:04:43,140 Well, where is the representative atom? 74 00:04:43,140 --> 00:04:47,080 This is tucked just inside the corner of the cell where the 75 00:04:47,080 --> 00:04:50,770 vertical line divides the circle into 1/2. 76 00:04:50,770 --> 00:04:53,750 That is a plus, so that's an atom at plus z. 77 00:04:53,750 --> 00:04:56,810 The right-hand side of that circle has a comma. 78 00:04:56,810 --> 00:05:02,270 It's an enantiomorphic motif, and that exist at 1/2 minus z. 79 00:05:02,270 --> 00:05:05,570 So you find labels on all of the atoms in the cell. 80 00:05:05,570 --> 00:05:10,810 There are a total of four clusters of eight. 81 00:05:10,810 --> 00:05:15,660 So are a total of 16 atoms within the unit cell. 82 00:05:19,360 --> 00:05:22,790 If you look at this pattern and reflect on it for a 83 00:05:22,790 --> 00:05:27,640 moment, so to speak, you can see that the same cluster of 84 00:05:27,640 --> 00:05:31,370 eight sits at the corners of the cell as in the 85 00:05:31,370 --> 00:05:32,300 middle of the cell. 86 00:05:32,300 --> 00:05:33,570 Exactly the same. 87 00:05:33,570 --> 00:05:38,050 So therefore, this is a lattice that is side-centered. 88 00:05:38,050 --> 00:05:40,830 There's a lattice point at the corner of the cell and one 89 00:05:40,830 --> 00:05:43,810 lattice point, in addition, at 1/2, 1/2, 0, in the 90 00:05:43,810 --> 00:05:45,870 center of the cell. 91 00:05:45,870 --> 00:05:49,660 The c-axis comes out of the plane of the paper. 92 00:05:49,660 --> 00:05:54,320 This then is a side-centered c-lattice. 93 00:05:54,320 --> 00:05:57,680 And that is the capital letter symbol that appears first in 94 00:05:57,680 --> 00:06:00,300 the symbol for the space group. 95 00:06:00,300 --> 00:06:04,330 It is a side-centered lattice, and the extra lattice point is 96 00:06:04,330 --> 00:06:06,660 in the middle of the face out of which c comes. 97 00:06:09,520 --> 00:06:14,180 Then come, in the space group symbol at upper left-- this is 98 00:06:14,180 --> 00:06:15,800 the abbreviated form-- 99 00:06:15,800 --> 00:06:17,050 mcm. 100 00:06:18,840 --> 00:06:23,840 And what I wanted to go over this example specifically for, 101 00:06:23,840 --> 00:06:29,570 there are three directions in an orthorhombic symmetry, no 102 00:06:29,570 --> 00:06:33,440 one any more or less special than any other. 103 00:06:33,440 --> 00:06:35,540 So if you look at the arrangement of symmetry 104 00:06:35,540 --> 00:06:39,490 elements, there's a 2 sub 1 screw axis coming out in the 105 00:06:39,490 --> 00:06:42,430 direction of c. 106 00:06:42,430 --> 00:06:46,960 The symbols running left to right, arrows and barbs, the 107 00:06:46,960 --> 00:06:48,880 arrows stand for two-fold axes. 108 00:06:48,880 --> 00:06:52,200 The barbs stand for 2 sub 1 screw axes. 109 00:06:52,200 --> 00:06:56,080 And running up and down parallel to the a-axis, again, 110 00:06:56,080 --> 00:07:00,430 arrows that indicate two-fold axes and barbs that indicate 2 111 00:07:00,430 --> 00:07:01,870 sub 1 screw axes. 112 00:07:01,870 --> 00:07:04,440 So which is the special direction? 113 00:07:04,440 --> 00:07:07,510 Which one should come first? 114 00:07:07,510 --> 00:07:10,790 It's not like a cubic symmetry, where the direction 115 00:07:10,790 --> 00:07:13,860 of the four-fold axis is always along the edge of the 116 00:07:13,860 --> 00:07:19,200 unit cell, or a tetragonal symmetry, where the four-fold 117 00:07:19,200 --> 00:07:22,120 axis is always taken to be the direction of c. 118 00:07:22,120 --> 00:07:24,430 Here, that's not specified. 119 00:07:24,430 --> 00:07:31,880 So again, the first convention is that the magnitudes of the 120 00:07:31,880 --> 00:07:51,030 translations determine a, b, and c in orthorhombic. 121 00:07:56,420 --> 00:07:59,640 Because there's nothing more or less special about the 122 00:07:59,640 --> 00:08:02,890 symmetry of these three orthogonal directions. 123 00:08:02,890 --> 00:08:07,010 And the convention which we've mentioned earlier is that the 124 00:08:07,010 --> 00:08:10,470 magnitude of b is greater than the magnitude of a. 125 00:08:10,470 --> 00:08:14,810 And then seemingly logically, c is the smallest. 126 00:08:14,810 --> 00:08:17,790 So if this is the direction of a and this is the direction of 127 00:08:17,790 --> 00:08:21,220 b, then c comes up. 128 00:08:21,220 --> 00:08:29,290 And this is the intermediate length translation. 129 00:08:29,290 --> 00:08:33,380 This is the maximum translation. 130 00:08:33,380 --> 00:08:36,120 And normal to the blackboard is the shortest translation. 131 00:08:40,850 --> 00:08:43,909 What symmetry goes first? 132 00:08:43,909 --> 00:08:47,010 And here again, for orthorhombic, we need a 133 00:08:47,010 --> 00:08:48,180 convention. 134 00:08:48,180 --> 00:08:52,460 And the space group symbol has-- 135 00:08:52,460 --> 00:08:53,340 I [INAUDIBLE] 136 00:08:53,340 --> 00:08:56,660 I use lambda for this, but-- a symbol for the lattice type. 137 00:08:56,660 --> 00:08:59,980 And we've seen that the capital C that comes first 138 00:08:59,980 --> 00:09:03,230 indicates that there is a c-centered lattice. 139 00:09:03,230 --> 00:09:09,080 And then what one does for orthorhombic is give the 140 00:09:09,080 --> 00:09:12,340 symmetry elements in the order abc. 141 00:09:12,340 --> 00:09:16,550 So the axis that is parallel to a over the plane that is 142 00:09:16,550 --> 00:09:19,940 perpendicular to a. 143 00:09:19,940 --> 00:09:24,450 Next, the axis that's parallel to b over the plane that is 144 00:09:24,450 --> 00:09:27,560 perpendicular to b. 145 00:09:27,560 --> 00:09:32,180 And then the axis that is parallel to c over the plane, 146 00:09:32,180 --> 00:09:34,680 if any, that is perpendicular to c. 147 00:09:38,320 --> 00:09:40,910 So then this is a convention for orthorhombic. 148 00:09:40,910 --> 00:09:45,000 And again, this is the only crystal system where 149 00:09:45,000 --> 00:09:47,090 conventions this elaborate are necessary. 150 00:09:47,090 --> 00:09:51,380 If this were tetragonal, 4 or 4/m would always come first 151 00:09:51,380 --> 00:09:53,040 because that's the direction of c. 152 00:09:53,040 --> 00:09:54,740 So even the order of the 153 00:09:54,740 --> 00:09:58,300 symmetry elements is different. 154 00:09:58,300 --> 00:09:59,860 But we've got another problem. 155 00:09:59,860 --> 00:10:04,010 If we look at the axes, if any, that are along a, we've 156 00:10:04,010 --> 00:10:07,660 got this symbol that stands for a two-fold axis and this 157 00:10:07,660 --> 00:10:11,590 single barbed arrow that stands for a 2 sub 1 screw. 158 00:10:11,590 --> 00:10:13,970 So what kind of axis is along a? 159 00:10:13,970 --> 00:10:15,980 There are two different kinds of axes. 160 00:10:15,980 --> 00:10:22,380 So in here, we could put either a 2 or a 2 sub 1. 161 00:10:22,380 --> 00:10:26,720 And if we look at the planes that are perpendicular to a, 162 00:10:26,720 --> 00:10:27,930 there's two of them. 163 00:10:27,930 --> 00:10:31,720 There's a dashed line, and this is a glide where the 164 00:10:31,720 --> 00:10:34,770 direction of tau is in the plane of the paper so that 165 00:10:34,770 --> 00:10:38,420 we're looking edge on in the glide plane and 166 00:10:38,420 --> 00:10:40,050 perpendicular to tau. 167 00:10:40,050 --> 00:10:43,290 And then there is a solid line. 168 00:10:43,290 --> 00:10:45,890 That's the symbol for a mirror plane. 169 00:10:45,890 --> 00:10:50,890 This would be a glide in which tau is a long b, 170 00:10:50,890 --> 00:10:54,330 so this is a b-glide. 171 00:10:54,330 --> 00:10:57,150 So those are the two sorts of symmetry planes that are 172 00:10:57,150 --> 00:10:59,000 perpendicular to a. 173 00:10:59,000 --> 00:11:00,230 Again, there are two of them. 174 00:11:00,230 --> 00:11:06,140 So we could put in either m, or we could put in a for the 175 00:11:06,140 --> 00:11:08,730 plane that is perpendicular to a. 176 00:11:08,730 --> 00:11:11,100 So already, there are four different possibilities for 177 00:11:11,100 --> 00:11:12,400 the first character. 178 00:11:12,400 --> 00:11:14,620 So obviously, what I'm leading up to is the fact that we're 179 00:11:14,620 --> 00:11:18,320 going to have to have conventions of necessity to 180 00:11:18,320 --> 00:11:22,560 come to a symbol on which everyone agrees. 181 00:11:22,560 --> 00:11:25,400 I think you get the idea now, so let me move along quickly 182 00:11:25,400 --> 00:11:30,400 along the direction of the arrow for a two-fold axis, a 183 00:11:30,400 --> 00:11:32,400 barb for 2 sub 1 screw axis. 184 00:11:32,400 --> 00:11:36,070 So again, there are two kinds of symmetry axes that could go 185 00:11:36,070 --> 00:11:38,720 in here for the axes that are parallel to b, 186 00:11:38,720 --> 00:11:41,310 either 2 or 2 sub 1. 187 00:11:41,310 --> 00:11:44,820 If we look at the planes that are perpendicular to b, 188 00:11:44,820 --> 00:11:46,790 there's a dotted line. 189 00:11:46,790 --> 00:11:54,510 And this is a glide for which tau is directed upwards. 190 00:11:54,510 --> 00:11:58,690 We're looking directly along the orientation of tau. 191 00:11:58,690 --> 00:12:02,960 The axis that is normal to the board is the c-axis. 192 00:12:02,960 --> 00:12:04,270 So this is a c-glide. 193 00:12:07,980 --> 00:12:11,600 And then appears a dash-dot line alongside of it. 194 00:12:11,600 --> 00:12:14,800 The dash-dot line is a glide plane where you're looking 195 00:12:14,800 --> 00:12:18,800 neither perpendicular to tau nor parallel to tau, but 196 00:12:18,800 --> 00:12:20,280 halfway in between. 197 00:12:20,280 --> 00:12:24,710 So this is a glide plane where tau is equal to 1/2 of 198 00:12:24,710 --> 00:12:26,390 a plus 1/2 of b. 199 00:12:29,370 --> 00:12:32,680 And that's a diagonal glide represented, just to confuse 200 00:12:32,680 --> 00:12:35,780 the innocent, by the symbol n. 201 00:12:35,780 --> 00:12:38,820 So two choices for axes along b. 202 00:12:38,820 --> 00:12:44,960 Two choices for the glide plane, either c or n. 203 00:12:44,960 --> 00:12:47,910 And finally, mercifully, along the direction of the c-axis, 204 00:12:47,910 --> 00:12:53,460 which is easiest to visualize, because things are either 205 00:12:53,460 --> 00:12:56,280 along it or parallel to it, we've got only a 206 00:12:56,280 --> 00:12:57,670 2 sub 1 screw axis. 207 00:12:57,670 --> 00:13:02,230 So this is the only symbol which is not ambiguous. 208 00:13:02,230 --> 00:13:05,020 And then perpendicular to that, we have the symbol for a 209 00:13:05,020 --> 00:13:08,740 mirror plane, the solid chevron off to one side. 210 00:13:08,740 --> 00:13:13,090 And then the chevron adorned with a diagonal arrow and that 211 00:13:13,090 --> 00:13:19,250 is, again, a diagonal glide n. 212 00:13:19,250 --> 00:13:25,600 We notice that the center of the cell at z equals 0 is the 213 00:13:25,600 --> 00:13:28,070 same as the grouping of atoms at the corner. 214 00:13:28,070 --> 00:13:31,250 So that is a side-centered lattice. 215 00:13:31,250 --> 00:13:38,670 The centered lattice point is coming out of the face which c 216 00:13:38,670 --> 00:13:40,330 emerges from. 217 00:13:40,330 --> 00:13:42,265 And so this is a c-lattice. 218 00:13:42,265 --> 00:13:43,448 You have a question? 219 00:13:43,448 --> 00:13:47,432 AUDIENCE: Professor, for the first term [? x of the ?] 220 00:13:47,432 --> 00:13:53,090 plane perpendicular to a, why is it a and not b? 221 00:13:53,090 --> 00:13:53,630 PROFESSOR: I'm sorry. 222 00:13:53,630 --> 00:13:55,130 You're absolutely right. 223 00:13:55,130 --> 00:13:56,550 I wrote it down as a b-glide. 224 00:13:56,550 --> 00:13:59,963 Perpendicular to a is a b-glide. 225 00:13:59,963 --> 00:14:01,440 Good. 226 00:14:01,440 --> 00:14:02,690 Congratulations. 227 00:14:06,530 --> 00:14:09,510 So we need a convention. 228 00:14:09,510 --> 00:14:13,220 And the convention for establishing a hierarchy of 229 00:14:13,220 --> 00:14:15,430 symbols is easy to remember. 230 00:14:15,430 --> 00:14:19,480 It's if you have both a two-fold screw axis and a 231 00:14:19,480 --> 00:14:23,370 two-fold axis, the order of preference is that the 2 is 232 00:14:23,370 --> 00:14:26,700 chosen in preference to the 2 sub 1 screw symbol. 233 00:14:26,700 --> 00:14:30,010 So by this greater than, I mean the preference for 234 00:14:30,010 --> 00:14:32,820 choosing it is greater than the symbol to the right. 235 00:14:32,820 --> 00:14:34,840 So we get rid of the 2 sub 1 here. 236 00:14:34,840 --> 00:14:36,200 We pick the 2. 237 00:14:36,200 --> 00:14:41,310 Get rid of the 2 sub 1 here, and that stays at 2 sub 1. 238 00:14:41,310 --> 00:14:46,780 For planes, you pick m in preference to an a-glide, in 239 00:14:46,780 --> 00:14:50,670 preference to a b-glide, in preference to a c-glide, in 240 00:14:50,670 --> 00:14:54,200 preference to a diagonal glide, in preference to a 241 00:14:54,200 --> 00:14:56,950 diamond glide, assuming you have a choice of 242 00:14:56,950 --> 00:14:59,330 two different planes. 243 00:14:59,330 --> 00:15:03,210 So in this case, we picked the m in preference to b. 244 00:15:03,210 --> 00:15:05,510 Here, we've got a c-glide and an n-glide. 245 00:15:05,510 --> 00:15:08,470 We picked the c, and it's in preference to the n. 246 00:15:08,470 --> 00:15:11,920 And here again, for the plane that is perpendicular to c, we 247 00:15:11,920 --> 00:15:14,030 picked the m in preference to the n. 248 00:15:14,030 --> 00:15:22,890 So the full symbol would be C, 2 over m, 2 over c, and 249 00:15:22,890 --> 00:15:26,360 2 sub 1 over m. 250 00:15:26,360 --> 00:15:28,590 Now I'll peek and see if I got this right. 251 00:15:28,590 --> 00:15:29,220 Yes, I did. 252 00:15:29,220 --> 00:15:31,380 How about that? 253 00:15:31,380 --> 00:15:39,780 And the short symbol would be simply Cmcm, which indeed is 254 00:15:39,780 --> 00:15:43,810 what we see listed in the upper left-hand 255 00:15:43,810 --> 00:15:46,100 corner of the page. 256 00:15:46,100 --> 00:15:56,320 This is a space group based on the point group 2/m, 2/m, 2/m, 257 00:15:56,320 --> 00:15:58,530 which is D2h. 258 00:15:58,530 --> 00:16:02,260 And there is the large number 17 in the superscript, which 259 00:16:02,260 --> 00:16:06,450 is the 17th orthorhombic space group, which Schoenflies was 260 00:16:06,450 --> 00:16:09,443 able to derive in the order in which he did them. 261 00:16:13,790 --> 00:16:15,390 Any questions? 262 00:16:15,390 --> 00:16:16,146 Yes. 263 00:16:16,146 --> 00:16:18,376 AUDIENCE: If you look at the other symbols to describe the 264 00:16:18,376 --> 00:16:21,754 space group, would we construct the same space group 265 00:16:21,754 --> 00:16:24,070 using that function [INAUDIBLE]? 266 00:16:24,070 --> 00:16:26,570 PROFESSOR: Yes, you could. 267 00:16:26,570 --> 00:16:29,990 And let me say, yes, it's possible. 268 00:16:29,990 --> 00:16:34,450 But also to answer a question, you might have about the short 269 00:16:34,450 --> 00:16:38,690 form of the symbol, we're throwing away information. 270 00:16:38,690 --> 00:16:42,050 We've thrown away the fact that there are two-fold axes, 271 00:16:42,050 --> 00:16:44,200 and a pair of directions, and 2 sub 1 only 272 00:16:44,200 --> 00:16:45,490 in the third direction. 273 00:16:45,490 --> 00:16:51,090 Is it possible that two space groups could have exactly the 274 00:16:51,090 --> 00:16:54,030 same three symbols in the short form? 275 00:16:54,030 --> 00:16:55,900 And the answer is no. 276 00:16:55,900 --> 00:17:01,740 Because if you think about it, you start with a lattice type. 277 00:17:01,740 --> 00:17:05,060 If you put a mirror plane in one orientation, a mirror 278 00:17:05,060 --> 00:17:09,089 plane in the other orientation, that really is 279 00:17:09,089 --> 00:17:14,270 all you have to specify in order to determine every other 280 00:17:14,270 --> 00:17:16,740 symmetry element that's present. 281 00:17:16,740 --> 00:17:20,690 You simply combine those operations, and you find that 282 00:17:20,690 --> 00:17:25,619 when you've taken these symmetries and rotations and 283 00:17:25,619 --> 00:17:28,480 combine them with the lattice translation, it comes out to 284 00:17:28,480 --> 00:17:32,660 one and the same unique result. 285 00:17:32,660 --> 00:17:35,510 So your question was what happens if you take another 286 00:17:35,510 --> 00:17:37,470 three symbols. 287 00:17:37,470 --> 00:17:39,810 You end up with the same result. 288 00:17:39,810 --> 00:17:43,420 Three of them, really, are enough to specify what the 289 00:17:43,420 --> 00:17:44,670 group will turn out to be. 290 00:17:48,320 --> 00:17:52,580 Now the other thing I wanted to do was to show you how 291 00:17:52,580 --> 00:17:57,410 remarkably the symbol for the space group will change if you 292 00:17:57,410 --> 00:18:02,150 take a different set of axes simply because the relative 293 00:18:02,150 --> 00:18:05,340 lengths of the cell edge for this arrangement of symmetry 294 00:18:05,340 --> 00:18:08,050 elements have changed. 295 00:18:08,050 --> 00:18:09,430 This takes dominance. 296 00:18:09,430 --> 00:18:12,040 This determines the labels that go to the axes. 297 00:18:12,040 --> 00:18:15,020 And then the symbol for the plane group, and what is a 298 00:18:15,020 --> 00:18:18,010 b-glide, and what is a c-glide follows from 299 00:18:18,010 --> 00:18:20,350 this choice of axes. 300 00:18:20,350 --> 00:18:21,650 So let me take another example. 301 00:18:21,650 --> 00:18:24,400 And I think if you follow what I've done so far, I can zip 302 00:18:24,400 --> 00:18:27,710 through this a little more quickly. 303 00:18:27,710 --> 00:18:31,070 Somebody pick a new direction for this axis. 304 00:18:31,070 --> 00:18:31,960 Let's not pick a. 305 00:18:31,960 --> 00:18:34,160 We have used that for this. 306 00:18:34,160 --> 00:18:36,380 So somebody pick a b or a c. 307 00:18:36,380 --> 00:18:37,170 Do you want to pick a b or a c? 308 00:18:37,170 --> 00:18:37,460 AUDIENCE: [INAUDIBLE]. 309 00:18:37,460 --> 00:18:38,340 PROFESSOR: b. 310 00:18:38,340 --> 00:18:42,660 So we're going to change this direction to b. 311 00:18:42,660 --> 00:18:45,092 Somebody want to pick a label for this axis? 312 00:18:45,092 --> 00:18:46,070 AUDIENCE: c. 313 00:18:46,070 --> 00:18:47,320 PROFESSOR: c. 314 00:18:49,410 --> 00:18:52,200 And nobody gets to pick this axis. 315 00:18:52,200 --> 00:18:53,500 If you think you can, you're out of here. 316 00:18:53,500 --> 00:18:55,580 Because that now has determined 317 00:18:55,580 --> 00:18:56,940 the coordinate system. 318 00:18:56,940 --> 00:18:58,930 So we've got b this way, c this way. 319 00:18:58,930 --> 00:19:02,300 We want a right-handed abc convention. 320 00:19:02,300 --> 00:19:03,550 So a goes up. 321 00:19:07,840 --> 00:19:12,190 So abc right-handed system. 322 00:19:12,190 --> 00:19:14,270 So exactly the same arrangement of symmetry 323 00:19:14,270 --> 00:19:17,100 elements except that we've now changed the 324 00:19:17,100 --> 00:19:18,280 labels on the axes. 325 00:19:18,280 --> 00:19:22,380 So we have changed the name that we have to apply to the 326 00:19:22,380 --> 00:19:24,310 glide planes. 327 00:19:24,310 --> 00:19:31,320 So let me now proceed to write down here the new collection 328 00:19:31,320 --> 00:19:33,890 of symbols which would appear for this setting of exactly 329 00:19:33,890 --> 00:19:35,480 the same space group. 330 00:19:35,480 --> 00:19:40,580 First of all, the centered lattice point is now in the 331 00:19:40,580 --> 00:19:43,460 middle of the face out of which a comes. 332 00:19:43,460 --> 00:19:48,720 So this is an a-lattice, side-centered a. 333 00:19:48,720 --> 00:19:54,100 If we look along the direction of a, all that we have for an 334 00:19:54,100 --> 00:19:56,840 axis is 2 sub 1. 335 00:19:56,840 --> 00:19:58,360 Then we have these two planes. 336 00:19:58,360 --> 00:19:59,620 One is an m. 337 00:19:59,620 --> 00:20:02,050 One is a diagonal glide. 338 00:20:02,050 --> 00:20:08,260 So that is m or n. 339 00:20:08,260 --> 00:20:12,470 If I look along b, I have the choice of either 2 or 2 sub 1 340 00:20:12,470 --> 00:20:14,820 as both are present. 341 00:20:14,820 --> 00:20:22,720 Perpendicular to b, however, is a glide for which tau is 342 00:20:22,720 --> 00:20:24,740 along the direction of c. 343 00:20:24,740 --> 00:20:27,400 So that is a c-glide. 344 00:20:27,400 --> 00:20:31,240 And then the other symbol for an axis that's perpendicular 345 00:20:31,240 --> 00:20:33,220 to b is a mirror plane. 346 00:20:33,220 --> 00:20:36,010 So it's either c or m. 347 00:20:36,010 --> 00:20:39,500 Then finally, along the direction of c, I once more 348 00:20:39,500 --> 00:20:44,390 have two-fold axes and 2 sub 1 screw axes. 349 00:20:44,390 --> 00:20:49,080 And then perpendicular to c is a diagonal glide. 350 00:20:49,080 --> 00:20:50,200 And this is a glide. 351 00:20:50,200 --> 00:20:53,320 The dotted line indicates that the direction of 352 00:20:53,320 --> 00:20:54,930 tau is along a. 353 00:20:54,930 --> 00:21:00,570 So I have a choice of either a or n in here. 354 00:21:00,570 --> 00:21:04,320 So putting down the symbols in the order of preference, this 355 00:21:04,320 --> 00:21:14,070 would be A, 2 sub 1 over m, 2 over m, 2 over a. 356 00:21:14,070 --> 00:21:20,590 Or the short form of the symbol would be Amma, which I 357 00:21:20,590 --> 00:21:24,240 submit doesn't look anything like Cmcm. 358 00:21:28,470 --> 00:21:34,040 And when you start out in diffraction, many people have 359 00:21:34,040 --> 00:21:36,480 a first disconcerting experience. 360 00:21:36,480 --> 00:21:40,390 You spend probably the better part of a month taking single 361 00:21:40,390 --> 00:21:42,090 crystal diffraction patterns. 362 00:21:42,090 --> 00:21:45,250 You come up with a trial space group for 363 00:21:45,250 --> 00:21:47,100 your particular material. 364 00:21:47,100 --> 00:21:49,730 And then you go to the International Tables, and you 365 00:21:49,730 --> 00:21:52,930 say, oh, it's not in there, what did I do wrong? 366 00:21:52,930 --> 00:21:55,530 And you go back, and you check all your calculations. 367 00:21:55,530 --> 00:21:58,220 You look once more at all of the systematic absence of 368 00:21:58,220 --> 00:21:58,780 reflection. 369 00:21:58,780 --> 00:21:59,960 See, I have it right. 370 00:21:59,960 --> 00:22:02,400 Why is it not in there? 371 00:22:02,400 --> 00:22:05,250 Well, the answer is more than likely that you have a 372 00:22:05,250 --> 00:22:07,440 different setting for the labels that you 373 00:22:07,440 --> 00:22:08,910 applied to the axes. 374 00:22:08,910 --> 00:22:12,820 And you have, therefore, metamorphosed the symbol for 375 00:22:12,820 --> 00:22:15,660 the space group from one form to a symbol that 376 00:22:15,660 --> 00:22:16,910 is very, very different. 377 00:22:23,070 --> 00:22:27,440 How do you tell, when you have a particular space group, what 378 00:22:27,440 --> 00:22:30,170 all of the possible symbols for one and the same 379 00:22:30,170 --> 00:22:31,600 symmetry might be? 380 00:22:31,600 --> 00:22:34,060 So let me now pass out two things. 381 00:22:34,060 --> 00:22:40,050 If you have a question like that, you can bet your bottom 382 00:22:40,050 --> 00:22:42,700 bippy that it's in the International Tables if you 383 00:22:42,700 --> 00:22:43,970 look in the right place. 384 00:22:43,970 --> 00:22:46,600 So first of all, let me do something I should 385 00:22:46,600 --> 00:22:48,660 have done early on. 386 00:22:48,660 --> 00:22:51,240 I meant to hand out a copy of the space group 387 00:22:51,240 --> 00:22:53,480 that I drew up here. 388 00:22:53,480 --> 00:22:54,730 If I can find it again. 389 00:22:57,690 --> 00:23:00,660 It's buried with all of the stuff that I brought in. 390 00:23:00,660 --> 00:23:07,780 So I do have a copy for you, but it's gone at the moment. 391 00:23:07,780 --> 00:23:14,460 Let me pass out a listing of all of the possible variants 392 00:23:14,460 --> 00:23:17,582 of the space groups for the 230 393 00:23:17,582 --> 00:23:21,100 three-dimensional space groups. 394 00:23:21,100 --> 00:23:26,740 And it starts out simple for triclinic symmetries. 395 00:23:26,740 --> 00:23:33,150 There's only two possible symbols period, P1 and P1 bar. 396 00:23:33,150 --> 00:23:41,260 And then when you come to monoclinic, the Schoenflies 397 00:23:41,260 --> 00:23:47,350 symbol has the marvelous property that it is 398 00:23:47,350 --> 00:23:53,350 independent of the orientation and labelling of axes. 399 00:23:53,350 --> 00:23:56,530 It depends only on the point group that the crystal has. 400 00:23:56,530 --> 00:23:59,890 For monoclinic, either C2 or Cs-- 401 00:23:59,890 --> 00:24:01,230 that's a mirror plane-- 402 00:24:01,230 --> 00:24:03,910 or C2h-- 403 00:24:03,910 --> 00:24:06,790 that's 2 over a horizontal mirror plane. 404 00:24:06,790 --> 00:24:11,400 But the labels can still change depending whether in 405 00:24:11,400 --> 00:24:17,700 the first setting either a, b, or the diagonal of the oblique 406 00:24:17,700 --> 00:24:21,980 net has to be labeled a and c. 407 00:24:21,980 --> 00:24:25,450 So at the top of the column, you'll see the permutation of 408 00:24:25,450 --> 00:24:28,940 the two symbols a and b or b and a. 409 00:24:28,940 --> 00:24:33,780 And again, the change is significant. 410 00:24:33,780 --> 00:24:37,090 Number eight, as indicated in the left-hand column, 411 00:24:37,090 --> 00:24:40,700 can be Am or Bm. 412 00:24:45,600 --> 00:24:49,230 And for the groups that are based on 2 over m, you can 413 00:24:49,230 --> 00:24:52,160 see, again, there are different symbols. 414 00:24:52,160 --> 00:24:58,070 A, 2 over b, A, 2 over a. 415 00:24:58,070 --> 00:25:00,730 Again, not at all similar. 416 00:25:00,730 --> 00:25:02,950 Next, come the orthorhombic space groups. 417 00:25:02,950 --> 00:25:05,470 And you can amuse yourself by looking through those. 418 00:25:09,130 --> 00:25:16,640 Again, what is called the standard setting under the 419 00:25:16,640 --> 00:25:19,640 heading symbols for various settings, this is the one 420 00:25:19,640 --> 00:25:21,180 that's listed in the tables. 421 00:25:21,180 --> 00:25:24,160 But then for various permutations of a, b, and c, 422 00:25:24,160 --> 00:25:27,210 you can have five other possibilities. 423 00:25:27,210 --> 00:25:30,210 And you can marvel at how the symbol for the space group 424 00:25:30,210 --> 00:25:33,970 changes, as you do nothing but change the labels a, b, and c 425 00:25:33,970 --> 00:25:37,500 that goes onto the axes. 426 00:25:37,500 --> 00:25:42,400 So there are tons of orthorhombic space groups 427 00:25:42,400 --> 00:25:44,840 because you have three different axes to play with, 428 00:25:44,840 --> 00:25:46,900 and three different planes, and four 429 00:25:46,900 --> 00:25:48,430 different lattice types. 430 00:25:48,430 --> 00:25:51,630 So there is an absolutely mind boggling collection of 431 00:25:51,630 --> 00:25:53,060 orthorhombic space groups. 432 00:25:53,060 --> 00:25:59,090 It is the most densely populated crystal system. 433 00:25:59,090 --> 00:26:06,660 For tetragonal, no real alternative. 434 00:26:15,100 --> 00:26:19,640 Something like P4, 4 is in the direction of the c-axis. 435 00:26:19,640 --> 00:26:22,300 There are two-fold axes parallel to that, but the 436 00:26:22,300 --> 00:26:25,030 symbol that is used for the symmetry part of the space 437 00:26:25,030 --> 00:26:28,440 group symbol looks very, very much like the symbol for the 438 00:26:28,440 --> 00:26:29,830 point group. 439 00:26:29,830 --> 00:26:32,430 So these are pretty much self-explanatory. 440 00:26:32,430 --> 00:26:39,630 Notice that there are two kinds of lattices for 441 00:26:39,630 --> 00:26:42,710 tetragonal crystals, either primitive or body-centered. 442 00:26:42,710 --> 00:26:49,120 So you see families with P for primitive or I for 443 00:26:49,120 --> 00:26:50,490 body-centered. 444 00:26:50,490 --> 00:26:54,470 But notice how many you can get out of a single type of 445 00:26:54,470 --> 00:26:56,080 axis and a single lattice. 446 00:26:56,080 --> 00:27:01,110 Numbers 75, 76, 77, and 78 are a four-fold axis and a 447 00:27:01,110 --> 00:27:04,430 primitive lattice, a 4 sub 1 screw, a 4 sub 2 screw, or a 4 448 00:27:04,430 --> 00:27:06,180 sub 3 screw. 449 00:27:06,180 --> 00:27:14,140 Then you do the same thing with I. And so it goes. 450 00:27:14,140 --> 00:27:16,580 Then it picks up at the top of the page again. 451 00:27:16,580 --> 00:27:25,160 And you see, again, different possibilities for the symbols. 452 00:27:28,470 --> 00:27:31,400 Continuing still on-- and I think you have the idea now-- 453 00:27:31,400 --> 00:27:34,170 same for trigonal crystals. 454 00:27:34,170 --> 00:27:37,150 They chose to list those separate from hexagonal. 455 00:27:37,150 --> 00:27:39,190 And then hexagonal symmetries. 456 00:27:39,190 --> 00:27:42,140 The number is fairly large here because when you take an 457 00:27:42,140 --> 00:27:45,230 axis and add it to a primitive hexagonal lattice-- 458 00:27:45,230 --> 00:27:48,160 for number 168, for example-- 459 00:27:48,160 --> 00:27:49,270 you get P6. 460 00:27:49,270 --> 00:27:53,830 Change the six-fold axis to a screw axis. 461 00:27:53,830 --> 00:28:01,690 There are five different types of six-fold screw axes. 462 00:28:01,690 --> 00:28:06,935 So there's a P6 sub 1, a P6 sub 2, a P6 sub 3, a P6 sub 4, 463 00:28:06,935 --> 00:28:08,520 and a P6 sub 5. 464 00:28:12,950 --> 00:28:15,210 And then finally, cubic. 465 00:28:15,210 --> 00:28:17,180 A fair number of cubic symmetries. 466 00:28:17,180 --> 00:28:24,330 But the edge of the cell is always along the direction of 467 00:28:24,330 --> 00:28:27,410 the four-fold axis or the two-fold axis, depending on 468 00:28:27,410 --> 00:28:30,880 whether the symmetry is based on 2, 3 or 4, 3, 2. 469 00:28:30,880 --> 00:28:35,280 So there are really no alternative symbols for 470 00:28:35,280 --> 00:28:37,300 different settings of the symmetry elements 471 00:28:37,300 --> 00:28:39,990 relative to the edges. 472 00:28:39,990 --> 00:28:41,470 So there you have them. 473 00:28:41,470 --> 00:28:46,330 All 230 of our cast of characters. 474 00:28:46,330 --> 00:28:51,490 But in forms of the very different mantles in which 475 00:28:51,490 --> 00:28:52,740 they can be cloaked. 476 00:29:02,410 --> 00:29:05,220 What else did I want to do? 477 00:29:05,220 --> 00:29:12,550 I want to contrast what is done for us in the old Volume 478 00:29:12,550 --> 00:29:15,640 1 of the International Tables. 479 00:29:15,640 --> 00:29:28,720 And I will pass out an example for one of the tetragonal 480 00:29:28,720 --> 00:29:30,350 space groups. 481 00:29:30,350 --> 00:29:33,400 This is the one with maximum symmetry 4 over m, 2 482 00:29:33,400 --> 00:29:34,600 over m, 2 over m. 483 00:29:34,600 --> 00:29:35,850 Sorry. 484 00:29:37,550 --> 00:29:38,800 Can you pass those back? 485 00:29:47,354 --> 00:29:56,740 And this is something that is analogous to P4mm in two 486 00:29:56,740 --> 00:30:02,100 dimensions with the four-fold and two-fold axes extended 487 00:30:02,100 --> 00:30:06,470 parallel to the edge of what is now a tetragonal cell. 488 00:30:06,470 --> 00:30:10,360 And then there are two-fold axes perpendicular to the 489 00:30:10,360 --> 00:30:11,620 four-fold axis. 490 00:30:11,620 --> 00:30:15,220 So actually 4 over m, 2 over m, 2 over m has been dropped 491 00:30:15,220 --> 00:30:20,190 in at a lattice point of a primitive tetragonal lattice. 492 00:30:20,190 --> 00:30:24,550 The chevron in the upper right of the arrangement of symmetry 493 00:30:24,550 --> 00:30:26,980 elements is the mirror plane that's perpendicular to the 494 00:30:26,980 --> 00:30:28,060 four-fold axis. 495 00:30:28,060 --> 00:30:29,870 The other mirror planes are the same as in 496 00:30:29,870 --> 00:30:31,120 the plane group P4mm. 497 00:30:33,380 --> 00:30:42,100 And the glide plane, which now would be called a diagonal 498 00:30:42,100 --> 00:30:44,810 glide, is exactly the same as in the 499 00:30:44,810 --> 00:30:47,490 two-dimensional symmetry. 500 00:30:47,490 --> 00:30:52,920 Notice, however, the exquisite number of general and special 501 00:30:52,920 --> 00:30:55,880 positions that are present in the space group. 502 00:30:55,880 --> 00:31:01,140 The general position has the letter U. You've gone through 503 00:31:01,140 --> 00:31:04,470 almost the entire alphabet before you've labeled all of 504 00:31:04,470 --> 00:31:06,070 these positions. 505 00:31:06,070 --> 00:31:10,830 And again, either on the mirror planes, but now there's 506 00:31:10,830 --> 00:31:13,850 a vertical mirror plane, a diagonal vertical mirror 507 00:31:13,850 --> 00:31:17,460 plane, a horizontal mirror plane, mirror planes halfway 508 00:31:17,460 --> 00:31:20,420 along each of the axes, mirror planes halfway along the 509 00:31:20,420 --> 00:31:21,990 diagonal translation. 510 00:31:21,990 --> 00:31:26,030 So there are lots of positions of point group m. 511 00:31:26,030 --> 00:31:29,010 Lots of positions with point group 2mm. 512 00:31:29,010 --> 00:31:32,750 And then finally, 2 over m, 2 over m, 2 over m. 513 00:31:32,750 --> 00:31:36,440 And the highest symmetry, 4 over m, 2 over m, 2 over m, 514 00:31:36,440 --> 00:31:44,190 which occurs at the origin, at the center of the cell, and at 515 00:31:44,190 --> 00:31:48,710 the positions 0, 0, 1/2 and 1/2, 1/2, 1/2. 516 00:31:53,290 --> 00:31:57,090 Then I'd like to pass around one more example of a space 517 00:31:57,090 --> 00:32:00,260 group from the International Tables. 518 00:32:00,260 --> 00:32:05,250 And I know I copied it, and I don't have it with me. 519 00:32:09,370 --> 00:32:11,710 I left some stuff behind, which we'll get 520 00:32:11,710 --> 00:32:12,850 after we take our break. 521 00:32:12,850 --> 00:32:15,760 No, here it is. 522 00:32:15,760 --> 00:32:20,870 This is an example from Volume 1, the older edition, for one 523 00:32:20,870 --> 00:32:22,600 of the cubic point groups. 524 00:32:25,180 --> 00:32:26,840 And this is a very high symmetry. 525 00:32:26,840 --> 00:32:31,860 This is symmetry 4 over m, 3 bar 2 over m, the highest 526 00:32:31,860 --> 00:32:34,400 symmetry cubic point group dropped into a 527 00:32:34,400 --> 00:32:36,460 body-centered lattice. 528 00:32:36,460 --> 00:32:41,350 So this turns out to be, in the long form, I, 4 over m, 3 529 00:32:41,350 --> 00:32:42,720 bar, 2 over m. 530 00:32:50,870 --> 00:32:52,120 Let me pass these back. 531 00:32:55,490 --> 00:33:00,560 So the first thing you'll notice is no picture of 532 00:33:00,560 --> 00:33:02,870 symmetry elements. 533 00:33:02,870 --> 00:33:05,830 How do you draw something where the symmetry elements 534 00:33:05,830 --> 00:33:10,350 are not merely parallel to the plane of the depiction or 535 00:33:10,350 --> 00:33:11,700 perpendicular to it? 536 00:33:11,700 --> 00:33:14,090 Here, you've got mirror planes that are inclined to the 537 00:33:14,090 --> 00:33:16,600 paper, axes that are inclined to the paper. 538 00:33:16,600 --> 00:33:19,520 How do you represent them? 539 00:33:19,520 --> 00:33:23,830 You'll notice also the enormous number of atoms in 540 00:33:23,830 --> 00:33:25,200 the general position. 541 00:33:25,200 --> 00:33:26,890 96 atoms. 542 00:33:26,890 --> 00:33:31,550 Drop in one atom at xyz, and all hell breaks loose. 543 00:33:31,550 --> 00:33:34,590 You get 95 other atoms. 544 00:33:34,590 --> 00:33:39,630 And again, when the lattice is centered, they do not list all 545 00:33:39,630 --> 00:33:41,630 96 sets of coordinates. 546 00:33:41,630 --> 00:33:49,310 They denote at the top of the page 0, 0, 0, 1/2, 1/2, plus. 547 00:33:49,310 --> 00:33:53,770 That means because the lattice is body-centered to those 48 548 00:33:53,770 --> 00:33:57,180 positions that are listed, you add 0, 0, 0, which is easy to 549 00:33:57,180 --> 00:34:01,830 do, and then add 1/2, 1/2, 1/2 to x, y, and z. 550 00:34:01,830 --> 00:34:03,490 And these are the atoms that hang at the 551 00:34:03,490 --> 00:34:04,740 centered lattice point. 552 00:34:07,780 --> 00:34:10,710 Again, because the symmetry is so high, surprisingly, there 553 00:34:10,710 --> 00:34:12,550 are not that many special positions. 554 00:34:12,550 --> 00:34:16,100 Because all of the symmetry elements can serve as special 555 00:34:16,100 --> 00:34:20,150 positions are related by the symmetry that's there. 556 00:34:20,150 --> 00:34:24,949 So this is all that you see for the body-centered lattice 557 00:34:24,949 --> 00:34:29,330 with 4 over m, 3 bar, 2 over m dropped into it. 558 00:34:29,330 --> 00:34:32,580 And again, you can describe some very, very complicated 559 00:34:32,580 --> 00:34:36,110 crystal structures with a very brief set of notes, give the 560 00:34:36,110 --> 00:34:39,110 value of the single lattice constant a. 561 00:34:39,110 --> 00:34:42,100 And then say, you've got an atom in the general position 562 00:34:42,100 --> 00:34:45,960 96l with x equals something, y equals something, z equals 563 00:34:45,960 --> 00:34:46,989 some number. 564 00:34:46,989 --> 00:34:51,710 And another atom in position 16f x, x, x, and just the 565 00:34:51,710 --> 00:34:53,409 value of x would be given. 566 00:34:53,409 --> 00:35:00,970 So you've described a structure that has roughly 125 567 00:35:00,970 --> 00:35:03,350 atoms in it with just that modest 568 00:35:03,350 --> 00:35:05,485 specification of atomic positions. 569 00:35:12,930 --> 00:35:18,030 Let me now pass out for you some samples of what the space 570 00:35:18,030 --> 00:35:23,000 groups look like in Volume A of the new International 571 00:35:23,000 --> 00:35:25,220 Tables for Crystallography. 572 00:35:25,220 --> 00:35:27,420 Not X-ray crystallography, but just 573 00:35:27,420 --> 00:35:28,670 crystallography in general. 574 00:35:31,870 --> 00:35:38,160 And I think you will be blown away because the amount of 575 00:35:38,160 --> 00:35:42,420 information that's there is almost suffocating in its 576 00:35:42,420 --> 00:35:47,200 detail and its density. 577 00:35:47,200 --> 00:35:52,830 The first sample space group that is given here is, not by 578 00:35:52,830 --> 00:35:58,970 coincidence, one of the tetragonal space groups, the 579 00:35:58,970 --> 00:36:03,840 one that we just discussed, P, 4 over m, 2 over m, 2 over m. 580 00:36:06,640 --> 00:36:08,560 They've done some very useful things. 581 00:36:08,560 --> 00:36:14,420 They have specified the asymmetric unit within which 582 00:36:14,420 --> 00:36:17,900 you need specify coordinates of atoms. 583 00:36:17,900 --> 00:36:21,260 And here, they say that you have to tell what's in the 584 00:36:21,260 --> 00:36:27,310 range x equals 0 to x equal 1/2, y equals 0 to y equals 585 00:36:27,310 --> 00:36:36,230 1/2, and z equal to 1/2, but greater or equal to 0. 586 00:36:36,230 --> 00:36:40,130 Then they give a representative set of symmetry 587 00:36:40,130 --> 00:36:44,390 operations which generate all of the atoms 588 00:36:44,390 --> 00:36:47,960 within the unit cell. 589 00:36:47,960 --> 00:36:54,230 The symmetry operations are not independent, but they list 590 00:36:54,230 --> 00:36:56,310 16 symmetry elements. 591 00:36:56,310 --> 00:37:03,880 And now if you look to the next page on the list of 16 592 00:37:03,880 --> 00:37:06,850 atomic coordinates in the general position, you will see 593 00:37:06,850 --> 00:37:10,120 a number in parentheses in front of each one. 594 00:37:10,120 --> 00:37:15,780 The 15, for example, tells you that you get this atom from 595 00:37:15,780 --> 00:37:20,820 the atom at xyz by the operation of symmetry element 596 00:37:20,820 --> 00:37:25,560 15, which turns out to be a mirror plane parallel to the x 597 00:37:25,560 --> 00:37:28,150 minus xz direction. 598 00:37:28,150 --> 00:37:31,260 And this is very, very useful if you want to describe a 599 00:37:31,260 --> 00:37:39,870 structure and you've labeled your atoms silicon 1 and 600 00:37:39,870 --> 00:37:43,710 silicon 2 if they're two different kinds of silicon ion 601 00:37:43,710 --> 00:37:46,290 in the structure. 602 00:37:46,290 --> 00:37:49,230 And then you're talking about a silicon, oxygen, silicon 603 00:37:49,230 --> 00:37:53,340 bond, and it's not at all clear which of the symmetry 604 00:37:53,340 --> 00:37:56,020 related atoms are involved in that bond. 605 00:37:56,020 --> 00:38:02,860 Well, the presentation of a standard numbering of atoms 606 00:38:02,860 --> 00:38:06,980 related by symmetry lets you say that, for example, this is 607 00:38:06,980 --> 00:38:13,300 the bond between silicon superscript 5, oxygen, silicon 608 00:38:13,300 --> 00:38:14,520 superscript 14. 609 00:38:14,520 --> 00:38:18,340 And you can identify the coordinates of the atoms that 610 00:38:18,340 --> 00:38:22,250 went into that particular bond angle or bond distance. 611 00:38:22,250 --> 00:38:24,570 And that's very nice. 612 00:38:24,570 --> 00:38:28,270 All sorts of information a la group theory. 613 00:38:28,270 --> 00:38:33,220 The maximal non-isomorphic subgroups, the maximal 614 00:38:33,220 --> 00:38:37,120 isomorphic subgroups of lowest index, the minimal 615 00:38:37,120 --> 00:38:40,420 non-isomorphic supergroups. 616 00:38:40,420 --> 00:38:44,230 At one time, I may have known what all that meant, and I've 617 00:38:44,230 --> 00:38:46,920 long since forgotten and have never 618 00:38:46,920 --> 00:38:48,840 regretted having forgotten. 619 00:38:48,840 --> 00:38:56,860 So this is very exotic, higher-level information about 620 00:38:56,860 --> 00:38:59,980 the subgroups that exist for the space group. 621 00:38:59,980 --> 00:39:03,470 Notice how much more information is present though, 622 00:39:03,470 --> 00:39:10,070 compared to the depiction of the same space group in the 623 00:39:10,070 --> 00:39:14,240 earlier International Tables for X-Ray Crystallography. 624 00:39:14,240 --> 00:39:16,340 Well, more or less at random, I picked out 625 00:39:16,340 --> 00:39:20,590 some other space groups. 626 00:39:20,590 --> 00:39:24,020 I, 4 sub 1, amd. 627 00:39:24,020 --> 00:39:29,590 So this is 4 over m, 2 over m, 2 over m in which the 4 has 628 00:39:29,590 --> 00:39:32,390 been replaced by a 4 sub 1 screw axis. 629 00:39:32,390 --> 00:39:35,490 The mirror plane perpendicular to the edge of the cell 630 00:39:35,490 --> 00:39:36,780 replaced b an a-glide. 631 00:39:43,610 --> 00:39:46,690 The mirror plane for the second mirrors that's 632 00:39:46,690 --> 00:39:50,660 perpendicular to the cell edge, the a-glide is diagonal. 633 00:39:50,660 --> 00:39:55,120 And then perpendicular to the diagonal, 634 00:39:55,120 --> 00:39:56,620 two-fold axis is a d-glide. 635 00:39:59,890 --> 00:40:06,840 Again, the different operations are identified by a 636 00:40:06,840 --> 00:40:11,070 number, the glide plane, and the inversion centers, and 637 00:40:11,070 --> 00:40:13,450 two-fold axes that are present. 638 00:40:13,450 --> 00:40:17,790 And then in the coordinates of the general position, you are 639 00:40:17,790 --> 00:40:23,250 given the nature of the symmetry element that produces 640 00:40:23,250 --> 00:40:25,800 the atom at a particular set of coordinates from the atom 641 00:40:25,800 --> 00:40:27,820 that's in the general position. 642 00:40:27,820 --> 00:40:30,250 Now again, the maximal non-isomorphic this, the 643 00:40:30,250 --> 00:40:33,790 minimal non-isomorphic supergroup, and so on. 644 00:40:33,790 --> 00:40:35,040 Lots of information. 645 00:40:37,100 --> 00:40:41,660 Then some examples of still higher symmetries. 646 00:40:41,660 --> 00:40:47,840 P6 sub 3 over mmc based on 6 over m, 2 over m, 2 over m 647 00:40:47,840 --> 00:40:52,110 with the 6 replaced by a 6 sub 3 screw axis and one of the 648 00:40:52,110 --> 00:40:54,340 mirror planes replaced by a c-glide. 649 00:40:58,036 --> 00:41:01,310 If you go on, here comes the really exciting part. 650 00:41:01,310 --> 00:41:04,470 The next page gives you a depiction of 651 00:41:04,470 --> 00:41:07,450 a cubic space group. 652 00:41:07,450 --> 00:41:08,530 Heroic. 653 00:41:08,530 --> 00:41:13,360 What they do is show stereographic projections at 654 00:41:13,360 --> 00:41:25,090 locations, such as 0, 0, 0 and 0, 1/2, 1/2, I believe it is. 655 00:41:25,090 --> 00:41:29,000 And then the best part of all, down at the bottom of that 656 00:41:29,000 --> 00:41:34,050 page, you see the arrangement of atoms in stereo. 657 00:41:34,050 --> 00:41:36,600 And if you're one of these people who can stare at the 658 00:41:36,600 --> 00:41:40,350 thing cross-eyed and let your eyes sort of blonk out, you 659 00:41:40,350 --> 00:41:42,510 can watch the two halves merge. 660 00:41:42,510 --> 00:41:45,900 And all of a sudden, zing, the thing leaps out of the page at 661 00:41:45,900 --> 00:41:47,620 you in three dimensions. 662 00:41:47,620 --> 00:41:50,570 If you don't have that ability to cross your eyes and see 663 00:41:50,570 --> 00:41:53,440 stereographic projections, you have to get a little viewer. 664 00:41:53,440 --> 00:41:56,340 But nevertheless, if you want to see it in three dimensions 665 00:41:56,340 --> 00:41:58,370 badly enough, you can do it. 666 00:41:58,370 --> 00:41:59,060 So there you are. 667 00:41:59,060 --> 00:42:02,260 For the first time, pictures of the general position in 668 00:42:02,260 --> 00:42:06,320 cubic space groups. 669 00:42:06,320 --> 00:42:09,370 I think that's kind of fun. 670 00:42:09,370 --> 00:42:12,490 Sometimes I gaze at it for so long I'm afraid my eyes are 671 00:42:12,490 --> 00:42:13,410 going to get stuck. 672 00:42:13,410 --> 00:42:16,120 And then I'll be in real trouble. 673 00:42:16,120 --> 00:42:17,380 Here's another one. 674 00:42:17,380 --> 00:42:22,260 P 4 over m, 3 bar, 2 over m. 675 00:42:22,260 --> 00:42:25,340 You notice the problems that they have in indicating the 676 00:42:25,340 --> 00:42:28,770 orientation of three-fold axes which are not parallel to or 677 00:42:28,770 --> 00:42:30,660 perpendicular to the paper. 678 00:42:30,660 --> 00:42:32,350 You have to do that with a little stereographic 679 00:42:32,350 --> 00:42:33,600 projection. 680 00:42:35,840 --> 00:42:42,610 And then finally, after we've done a couple of more cubic 681 00:42:42,610 --> 00:42:50,090 space groups, you come to the one that I gave you the 682 00:42:50,090 --> 00:42:53,340 handout from the International Tables for X-Ray 683 00:42:53,340 --> 00:42:55,200 Crystallography. 684 00:42:55,200 --> 00:42:57,320 I, 4 over m, 3 bar, 2 over m. 685 00:42:57,320 --> 00:43:00,430 And look at all the information that is there that 686 00:43:00,430 --> 00:43:04,760 no attempt is made to give you in the earlier tables. 687 00:43:04,760 --> 00:43:08,530 So this is for better or for worse what space 688 00:43:08,530 --> 00:43:11,060 group tables look like. 689 00:43:11,060 --> 00:43:13,980 This is the information that hopefully you'll 690 00:43:13,980 --> 00:43:16,030 be equipped to use. 691 00:43:16,030 --> 00:43:18,960 If nothing else, if you can't derive these things or 692 00:43:18,960 --> 00:43:21,950 reproduce the arrangement of symmetries from the symbol, if 693 00:43:21,950 --> 00:43:26,050 you ever have to construct the atomic arrangement from a 694 00:43:26,050 --> 00:43:29,820 material from the crystallographic notation in 695 00:43:29,820 --> 00:43:33,410 which the atomic arrangement is provided, you can hopefully 696 00:43:33,410 --> 00:43:36,260 go to the International Tables, know where to look, 697 00:43:36,260 --> 00:43:38,830 and how to go about identifying the coordinates of 698 00:43:38,830 --> 00:43:42,550 all of the atoms within the unit cell. 699 00:43:42,550 --> 00:43:45,840 Timed myself to finish just at five of the hour. 700 00:43:45,840 --> 00:43:55,210 So let me, before you leave, give you one final problem set 701 00:43:55,210 --> 00:43:58,660 on symmetry, problem set number 10. 702 00:43:58,660 --> 00:44:04,500 And this is related to identifying what the symbols 703 00:44:04,500 --> 00:44:08,030 used to give atomic locations represent. 704 00:44:08,030 --> 00:44:12,770 First problem asks you to look at a pair of atoms and 705 00:44:12,770 --> 00:44:17,280 determine the symmetry element which has related them. 706 00:44:17,280 --> 00:44:22,550 The second problem on the second sheet asks you to do 707 00:44:22,550 --> 00:44:25,760 what we did here for this orthorhombic space group. 708 00:44:25,760 --> 00:44:32,230 Determine the symbol when a, b, and c are particular 709 00:44:32,230 --> 00:44:35,100 translations of the three that are unique. 710 00:44:35,100 --> 00:44:38,740 And then change the orientation, and see what the 711 00:44:38,740 --> 00:44:41,070 space group symbol morphs into. 712 00:44:48,420 --> 00:44:53,160 Let me finish with one final handout. 713 00:44:53,160 --> 00:44:58,660 And that is could you please summarize what we spent the 714 00:44:58,660 --> 00:45:00,970 last month and a half doing? 715 00:45:00,970 --> 00:45:02,240 Gladly. 716 00:45:02,240 --> 00:45:06,550 I have summarized everything that we did and the theorems 717 00:45:06,550 --> 00:45:11,590 that we used to derive it on one piece of paper. 718 00:45:11,590 --> 00:45:15,710 So all of symmetry theory is here admittedly written in a 719 00:45:15,710 --> 00:45:17,560 rather tight hand. 720 00:45:17,560 --> 00:45:23,030 But here is an indication of everything we did to get from 721 00:45:23,030 --> 00:45:26,150 a definition of basic operations, a flowsheet that 722 00:45:26,150 --> 00:45:29,480 gets us down to 230 three-dimensional 723 00:45:29,480 --> 00:45:32,705 crystallographic space groups. 724 00:45:32,705 --> 00:45:37,500 So I'll pass that around for your awe and amazement. 725 00:45:37,500 --> 00:45:40,170 Sorry I gave a big pack out on the 726 00:45:40,170 --> 00:45:41,360 right-hand side of the room. 727 00:45:41,360 --> 00:45:42,610 They'll come around to you. 728 00:45:47,250 --> 00:45:50,710 So let us take our usual 10 minute break.