1 00:00:00,000 --> 00:00:12,320 2 00:00:12,320 --> 00:00:16,340 PROFESSOR: My watch says five after 2:00, so why don't we 3 00:00:16,340 --> 00:00:18,710 get started? 4 00:00:18,710 --> 00:00:22,740 This is going to be a strange lecture. 5 00:00:22,740 --> 00:00:25,260 I'm expecting some wise guy in the back to say, all your 6 00:00:25,260 --> 00:00:27,020 lectures are rather peculiar! 7 00:00:27,020 --> 00:00:30,990 But I have a phone call from Europe coming in sometime 8 00:00:30,990 --> 00:00:33,340 around 2:15, 2:30. 9 00:00:33,340 --> 00:00:36,950 Somebody's coming to visit us next week, and we're going to 10 00:00:36,950 --> 00:00:38,340 set up a seminar. 11 00:00:38,340 --> 00:00:40,550 So I don't know exactly when it's going to come in, but I 12 00:00:40,550 --> 00:00:41,730 have to be in my office. 13 00:00:41,730 --> 00:00:46,090 So what I will do is adjourn at quarter 14 00:00:46,090 --> 00:00:47,970 after, 20 past the hour. 15 00:00:47,970 --> 00:00:50,730 And then so I don't keep you here sitting restively 16 00:00:50,730 --> 00:00:52,580 wondering when I'm going to come back, we'll 17 00:00:52,580 --> 00:00:54,460 adjourn until 3 o'clock. 18 00:00:54,460 --> 00:00:57,580 So you seem to have been taken longer and longer breaks 19 00:00:57,580 --> 00:00:58,680 during intermission. 20 00:00:58,680 --> 00:01:04,010 So I'll give you a full 3/4 of an hour so you can get it out 21 00:01:04,010 --> 00:01:04,849 of your systems. 22 00:01:04,849 --> 00:01:08,680 And then, we'll take shorter breaks from here on in. 23 00:01:08,680 --> 00:01:11,910 OK this is going to be pretty much our 24 00:01:11,910 --> 00:01:14,310 last lecture on symmetry. 25 00:01:14,310 --> 00:01:18,640 And we will begin, at least in half of the next class on 26 00:01:18,640 --> 00:01:21,700 Tuesday, to begin talking about physical properties, and 27 00:01:21,700 --> 00:01:24,190 in particular tensor properties. 28 00:01:24,190 --> 00:01:28,650 But I'd like to say a little bit about the derivation of 29 00:01:28,650 --> 00:01:34,030 space groups and take a look at how this information is 30 00:01:34,030 --> 00:01:37,940 tabulated for you by the kind folks who prepare the 31 00:01:37,940 --> 00:01:39,700 international tables. 32 00:01:39,700 --> 00:01:44,700 And there are really very strict parallels between what 33 00:01:44,700 --> 00:01:45,970 we did in two dimensions. 34 00:01:45,970 --> 00:01:48,970 And that's the reason why we did it so thoroughly and 35 00:01:48,970 --> 00:01:52,220 systematically, except that there are three dimensions. 36 00:01:52,220 --> 00:01:56,920 And because we are taking 32 point groups and putting them 37 00:01:56,920 --> 00:02:01,150 into 14 lattices, there are a lot more combinations to be 38 00:02:01,150 --> 00:02:01,760 considered. 39 00:02:01,760 --> 00:02:04,900 And there are a lot more of them that are unique. 40 00:02:04,900 --> 00:02:09,080 Also, they're a little bit more difficult to visualize 41 00:02:09,080 --> 00:02:09,850 and depict. 42 00:02:09,850 --> 00:02:13,440 So there are some new conventions in preparing a 43 00:02:13,440 --> 00:02:16,720 representation of three dimensional symmetries on a 44 00:02:16,720 --> 00:02:19,740 sheet of paper or page of a book which, of necessity, has 45 00:02:19,740 --> 00:02:20,970 to be two dimensional. 46 00:02:20,970 --> 00:02:23,620 So we'll go over some of those conventions. 47 00:02:23,620 --> 00:02:26,970 But the main reason for going a little bit further is that 48 00:02:26,970 --> 00:02:31,070 there is another surprise lurking in there for us as 49 00:02:31,070 --> 00:02:36,250 soon as we begin to combine translation in three 50 00:02:36,250 --> 00:02:38,930 dimensions with a symmetry element. 51 00:02:38,930 --> 00:02:44,880 But let me go through in some detail the 52 00:02:44,880 --> 00:02:46,650 monoclinic space groups. 53 00:02:46,650 --> 00:02:52,710 54 00:02:52,710 --> 00:02:57,550 So we have at our disposal to decorate with symmetry 55 00:02:57,550 --> 00:02:59,305 elements two lattices. 56 00:02:59,305 --> 00:03:02,940 57 00:03:02,940 --> 00:03:14,190 And they are a primitive monoclinic lattice with two 58 00:03:14,190 --> 00:03:18,445 translations, a and b, that define an oblique net, and a 59 00:03:18,445 --> 00:03:21,560 third translation, c, at right angles to that net. 60 00:03:21,560 --> 00:03:25,990 And then, either two choices for a double cell-- 61 00:03:25,990 --> 00:03:30,790 62 00:03:30,790 --> 00:03:33,720 and I'll draw these in projection because I can do 63 00:03:33,720 --> 00:03:36,040 both of them with one diagram. 64 00:03:36,040 --> 00:03:44,720 Either this is a and b and the extra lattice point is in the 65 00:03:44,720 --> 00:03:47,460 center of the cell halfway up from the base. 66 00:03:47,460 --> 00:03:48,775 This would be body centered. 67 00:03:48,775 --> 00:03:54,800 68 00:03:54,800 --> 00:04:00,640 And that's represented by I for innenzentriert. 69 00:04:00,640 --> 00:04:03,660 In German, that's German for body centered. 70 00:04:03,660 --> 00:04:09,350 Or the alternative would be to pick a cell like this, in 71 00:04:09,350 --> 00:04:12,970 which case the extra lattice point gets caught in the 72 00:04:12,970 --> 00:04:14,800 center of one of the faces. 73 00:04:14,800 --> 00:04:19,120 So this depiction here would be a side centered lattice. 74 00:04:19,120 --> 00:04:24,420 75 00:04:24,420 --> 00:04:27,800 And it could have the extra lattice point in the middle of 76 00:04:27,800 --> 00:04:30,370 the face out of which b comes. 77 00:04:30,370 --> 00:04:34,450 And in that case, the symbol for the lattice is B. Or 78 00:04:34,450 --> 00:04:37,590 alternatively, without changing any of the nature of 79 00:04:37,590 --> 00:04:40,100 the specialness of the lattice, the extra lattice 80 00:04:40,100 --> 00:04:41,580 point could be in the middle of the face 81 00:04:41,580 --> 00:04:43,250 out of which a comes. 82 00:04:43,250 --> 00:04:46,770 And that would be an A lattice. 83 00:04:46,770 --> 00:04:51,960 With the first, setting with the c axis unique, there is no 84 00:04:51,960 --> 00:04:56,310 c lattice because the oblique base of the cell would not 85 00:04:56,310 --> 00:04:59,890 give us anything new if it were centered. 86 00:04:59,890 --> 00:05:03,530 All right, so there are those two lattices. 87 00:05:03,530 --> 00:05:05,815 And then, there are three point groups. 88 00:05:05,815 --> 00:05:09,460 89 00:05:09,460 --> 00:05:15,180 And they were 2, m, and then a combination of a twofold axis 90 00:05:15,180 --> 00:05:18,200 perpendicular to a mirror plane. 91 00:05:18,200 --> 00:05:22,060 So it looks as though in principle there are going to 92 00:05:22,060 --> 00:05:24,286 be 6 combinations. 93 00:05:24,286 --> 00:05:26,600 In point of fact, there are a lot more because of the 94 00:05:26,600 --> 00:05:28,790 surprise that we have in store for us. 95 00:05:28,790 --> 00:05:31,680 96 00:05:31,680 --> 00:05:37,260 Before proceeding further, though, let me ask 97 00:05:37,260 --> 00:05:38,930 rhetorically the question. 98 00:05:38,930 --> 00:05:45,340 Which lattice type would you pick as the standard one? 99 00:05:45,340 --> 00:05:49,550 And here, there's a major decision that has to be made. 100 00:05:49,550 --> 00:05:54,760 101 00:05:54,760 --> 00:06:04,750 Should the labels on an axes give a unique 102 00:06:04,750 --> 00:06:06,000 symbol for the lattice? 103 00:06:06,000 --> 00:06:13,210 104 00:06:13,210 --> 00:06:15,860 In other words, if you have the double cell for a 105 00:06:15,860 --> 00:06:20,340 monoclinic crystal, should you define the cell to make it 106 00:06:20,340 --> 00:06:23,160 body centered always, or to make it A centered or B 107 00:06:23,160 --> 00:06:24,410 centered always? 108 00:06:24,410 --> 00:06:26,950 109 00:06:26,950 --> 00:06:33,350 Or should the labels be defined by the relative 110 00:06:33,350 --> 00:06:35,030 lengths of the axes? 111 00:06:35,030 --> 00:06:54,490 112 00:06:54,490 --> 00:06:56,460 So those are the two possibilities. 113 00:06:56,460 --> 00:06:59,710 And there's no right or wrong way. 114 00:06:59,710 --> 00:07:02,970 You pays your money and you makes your choice. 115 00:07:02,970 --> 00:07:09,230 And on pragmatic grounds, this is the 116 00:07:09,230 --> 00:07:12,000 convention that's followed. 117 00:07:12,000 --> 00:07:15,790 And it's purely a pragmatic decision because lattice 118 00:07:15,790 --> 00:07:19,530 constants have been determined for literally tens of 119 00:07:19,530 --> 00:07:21,340 thousands of materials. 120 00:07:21,340 --> 00:07:23,530 Sometimes, the lattice constants are known and the 121 00:07:23,530 --> 00:07:25,160 space group is not. 122 00:07:25,160 --> 00:07:30,470 But it would be terrible if the label that you applied to 123 00:07:30,470 --> 00:07:38,710 the axis could be C attached to the longest one, the 124 00:07:38,710 --> 00:07:40,500 shortest one, or the intermediate one. 125 00:07:40,500 --> 00:07:44,710 It would be impossible to calculate any database for 126 00:07:44,710 --> 00:07:48,910 lattice constants displayed by particular materials. 127 00:07:48,910 --> 00:07:53,790 So there's all this-- this is what the decision was made. 128 00:07:53,790 --> 00:07:56,110 This is the decision that was made many years ago. 129 00:07:56,110 --> 00:08:02,490 And for monoclinic crystals, the labeling of the axes is 130 00:08:02,490 --> 00:08:07,050 determined by the c axis being the unique axis. 131 00:08:07,050 --> 00:08:10,630 That is, the axis that is along the twofold axis or 132 00:08:10,630 --> 00:08:13,210 perpendicular to the mirror plane. 133 00:08:13,210 --> 00:08:20,190 And then, the labels b and c are defined by magnitude of b 134 00:08:20,190 --> 00:08:22,570 greater than the magnitude of a. 135 00:08:22,570 --> 00:08:27,690 So I inherently drew this is in the proper fashion. 136 00:08:27,690 --> 00:08:30,110 So the labels are applied to monoclinic 137 00:08:30,110 --> 00:08:31,360 crystals in that fashion. 138 00:08:31,360 --> 00:08:34,270 139 00:08:34,270 --> 00:08:37,799 The price you pay for that is that the symbol for the 140 00:08:37,799 --> 00:08:41,039 lattice for a monoclinic crystal that has the double 141 00:08:41,039 --> 00:08:46,630 cell could either be A, B, or I depending on the relative 142 00:08:46,630 --> 00:08:47,690 lengths of the axes. 143 00:08:47,690 --> 00:08:50,610 If these were the two shortest, it would be a body 144 00:08:50,610 --> 00:08:51,890 centered lattice. 145 00:08:51,890 --> 00:08:54,910 On the other hand, if this and this were the shortest pair, 146 00:08:54,910 --> 00:08:57,030 then it would be a side centered lattice. 147 00:08:57,030 --> 00:08:59,730 So the symbol for the lattice type will bounce around 148 00:08:59,730 --> 00:09:01,710 depending on the dimensions of the translations. 149 00:09:01,710 --> 00:09:04,210 150 00:09:04,210 --> 00:09:08,080 So that's a complication in the space groups. 151 00:09:08,080 --> 00:09:10,960 In the two dimensional plane group, that 152 00:09:10,960 --> 00:09:12,240 issue never came up. 153 00:09:12,240 --> 00:09:16,730 We just took B greater than A for the rectangular and the 154 00:09:16,730 --> 00:09:18,775 oblique nets. 155 00:09:18,775 --> 00:09:22,210 156 00:09:22,210 --> 00:09:24,940 OK, so let me do quickly a couple of the 157 00:09:24,940 --> 00:09:26,620 monoclinic space groups. 158 00:09:26,620 --> 00:09:31,720 Let's take a twofold axis and put it into a primitive 159 00:09:31,720 --> 00:09:34,210 monoclinic net. 160 00:09:34,210 --> 00:09:41,870 And this is exactly the same as our plane group, P2, with 161 00:09:41,870 --> 00:09:49,720 the twofold axes extended parallel to the c translation. 162 00:09:49,720 --> 00:09:56,130 So we would have twofold axes in all of these orientations. 163 00:09:56,130 --> 00:10:03,090 And that's simply P2, little p in two dimensions, with the 164 00:10:03,090 --> 00:10:05,200 twofold axes extended normal to the plane 165 00:10:05,200 --> 00:10:05,880 of the plane group. 166 00:10:05,880 --> 00:10:07,270 So this is p2. 167 00:10:07,270 --> 00:10:09,560 This is capital P2. 168 00:10:09,560 --> 00:10:11,550 And that is way we distinguish plane 169 00:10:11,550 --> 00:10:14,710 groups from space groups. 170 00:10:14,710 --> 00:10:18,510 So the symbol for the lattice is a capital symbol for the 171 00:10:18,510 --> 00:10:19,760 space groups. 172 00:10:19,760 --> 00:10:23,410 173 00:10:23,410 --> 00:10:29,866 So I didn't have to use any new combinations theorems. 174 00:10:29,866 --> 00:10:32,140 I simply say there's a third translation 175 00:10:32,140 --> 00:10:33,370 perpendicular to the net. 176 00:10:33,370 --> 00:10:35,470 So everything protrudes out of the plane group 177 00:10:35,470 --> 00:10:36,720 into a third dimension. 178 00:10:36,720 --> 00:10:42,549 179 00:10:42,549 --> 00:10:57,400 Let me now combine a twofold axis this a and this b and 180 00:10:57,400 --> 00:11:02,020 this c with a lattice that is not the primitive lattice but 181 00:11:02,020 --> 00:11:03,250 one of the double cells. 182 00:11:03,250 --> 00:11:05,720 And I'll take it, for convenience, to be the body 183 00:11:05,720 --> 00:11:08,530 centered flavor because it's easier to draw. 184 00:11:08,530 --> 00:11:11,330 185 00:11:11,330 --> 00:11:17,760 And the operation that we've added to the lattice, if this 186 00:11:17,760 --> 00:11:23,230 is a combination of a twofold axis, is the symbol A pi. 187 00:11:23,230 --> 00:11:26,590 And I know what happens if I combine A pi with a 188 00:11:26,590 --> 00:11:28,760 translation that is perpendicular to 189 00:11:28,760 --> 00:11:30,420 the rotation axis. 190 00:11:30,420 --> 00:11:34,740 I get a new rotation axis, B pi, that's halfway along the 191 00:11:34,740 --> 00:11:37,030 perpendicular part of the translation. 192 00:11:37,030 --> 00:11:38,780 And that's where all of these additional 193 00:11:38,780 --> 00:11:40,800 twofold axes came in. 194 00:11:40,800 --> 00:11:46,150 But if a lattice is a body centered lattice, there is now 195 00:11:46,150 --> 00:11:49,750 a translation that goes up to the centered lattice point. 196 00:11:49,750 --> 00:11:52,300 197 00:11:52,300 --> 00:11:53,460 So what's that going to be? 198 00:11:53,460 --> 00:12:01,080 What is T followed by A pi followed by T, where T is 1/2 199 00:12:01,080 --> 00:12:04,340 A plus 1/2 of B? 200 00:12:04,340 --> 00:12:09,180 And that is a part that is perpendicular to the axis, and 201 00:12:09,180 --> 00:12:15,820 then a third component, c, which is parallel to the axis. 202 00:12:15,820 --> 00:12:18,470 As with all these theorems, what you do is you draw it out 203 00:12:18,470 --> 00:12:20,790 once and for all and see what it turns out to be. 204 00:12:20,790 --> 00:12:28,070 205 00:12:28,070 --> 00:12:31,110 So let's do that. 206 00:12:31,110 --> 00:12:34,820 Let's say that this is the translation that goes up to 207 00:12:34,820 --> 00:12:38,360 the point 1/2, 1/2, 1/2. 208 00:12:38,360 --> 00:12:41,560 Here is the parallel part-- 209 00:12:41,560 --> 00:12:42,870 the perpendicular part, rather. 210 00:12:42,870 --> 00:12:49,420 This is 1/2 of A plus 1/2 of B. 211 00:12:49,420 --> 00:12:53,390 And then, we have a part that's parallel to the 212 00:12:53,390 --> 00:12:55,910 rotation operation, A pi. 213 00:12:55,910 --> 00:12:59,330 And that is 1/2 of c. 214 00:12:59,330 --> 00:13:01,190 So let's just do it. 215 00:13:01,190 --> 00:13:03,030 Here's my first object number one. 216 00:13:03,030 --> 00:13:04,260 It's right handed. 217 00:13:04,260 --> 00:13:09,520 I'll rotate 180 degrees to get a second one 218 00:13:09,520 --> 00:13:10,720 that's right handed. 219 00:13:10,720 --> 00:13:15,220 And then, I will translate it up to here. 220 00:13:15,220 --> 00:13:17,030 And here sits the third one. 221 00:13:17,030 --> 00:13:18,280 It's also right handed. 222 00:13:18,280 --> 00:13:22,120 223 00:13:22,120 --> 00:13:24,580 How do I get from one to three? 224 00:13:24,580 --> 00:13:29,530 225 00:13:29,530 --> 00:13:32,180 It's not really clear how I do that. 226 00:13:32,180 --> 00:13:34,640 And again, I'll draw it in projection because this is 227 00:13:34,640 --> 00:13:37,000 looking pretty messy in three dimensions. 228 00:13:37,000 --> 00:13:40,150 So here's my first one. 229 00:13:40,150 --> 00:13:43,615 I rotate 180 degrees to get a second one. 230 00:13:43,615 --> 00:13:45,460 The chirality is all the same. 231 00:13:45,460 --> 00:13:48,190 And then, I translate up to this point. 232 00:13:48,190 --> 00:13:51,610 So if this one is at z and this one is at z, this one 233 00:13:51,610 --> 00:13:59,800 here will sit at z plus 1/2. 234 00:13:59,800 --> 00:14:02,860 And this is the third one. 235 00:14:02,860 --> 00:14:04,110 Anybody got any idea? 236 00:14:04,110 --> 00:14:07,760 237 00:14:07,760 --> 00:14:08,800 Can't be reflection. 238 00:14:08,800 --> 00:14:10,410 They're of the same chirality. 239 00:14:10,410 --> 00:14:12,640 It can't be translation because the 240 00:14:12,640 --> 00:14:16,325 orientation is different. 241 00:14:16,325 --> 00:14:16,814 Yes, sir? 242 00:14:16,814 --> 00:14:19,750 AUDIENCE: Then it has to be some form of rotation. 243 00:14:19,750 --> 00:14:21,680 PROFESSOR: Right, has to be. 244 00:14:21,680 --> 00:14:28,040 But how do we get the object up to a different elevation? 245 00:14:28,040 --> 00:14:35,220 Well, we've stumbled headlong over a new type of operation, 246 00:14:35,220 --> 00:14:37,780 just as we discovered the glide plane when we started 247 00:14:37,780 --> 00:14:40,330 putting mirror planes into a lattice in combination with 248 00:14:40,330 --> 00:14:41,670 translation. 249 00:14:41,670 --> 00:14:45,950 The only way I can get from the first to the third is to 250 00:14:45,950 --> 00:14:50,990 rotate 180 degrees about exactly the same point that I 251 00:14:50,990 --> 00:14:54,050 would have before, namely at 1/2 of the part of the 252 00:14:54,050 --> 00:14:55,820 translation that is 253 00:14:55,820 --> 00:14:59,200 perpendicular to the operation. 254 00:14:59,200 --> 00:15:01,680 But then before putting it down, I've got to take a 255 00:15:01,680 --> 00:15:02,910 second step. 256 00:15:02,910 --> 00:15:08,470 I've got to slide it up by a translation component that is 257 00:15:08,470 --> 00:15:11,550 parallel to the axis about which I've rotated. 258 00:15:11,550 --> 00:15:16,320 So to complete my theorem, A alpha followed by a 259 00:15:16,320 --> 00:15:22,070 translation that has a perpendicular part plus a 260 00:15:22,070 --> 00:15:26,210 parallel part is a new type of operation that involves 261 00:15:26,210 --> 00:15:30,020 rotating through 180 degrees. 262 00:15:30,020 --> 00:15:36,450 And this is located at 1/2 of T perpendicular. 263 00:15:36,450 --> 00:15:39,360 But it has a translation component which I'll call 264 00:15:39,360 --> 00:15:40,970 generically tau. 265 00:15:40,970 --> 00:15:45,540 And tau is equal to the parallel part of the 266 00:15:45,540 --> 00:15:48,120 translation. 267 00:15:48,120 --> 00:15:53,060 So if I separate out this new sort of operation in all of 268 00:15:53,060 --> 00:15:59,180 its grandeur, what it does is it takes a first object, 269 00:15:59,180 --> 00:16:02,920 rotates us 180 degrees, does not yet put it down, first 270 00:16:02,920 --> 00:16:06,160 slides up by the amount tau. 271 00:16:06,160 --> 00:16:09,200 Doing the operation, again rotates it 180 degrees, 272 00:16:09,200 --> 00:16:13,320 doesn't yet put it down, slides it up by tau. 273 00:16:13,320 --> 00:16:18,700 Doing it again moves us to another one up here. 274 00:16:18,700 --> 00:16:24,950 So what we've generated is a helical spiral of objects 275 00:16:24,950 --> 00:16:29,450 about the central axis about which we're rotating. 276 00:16:29,450 --> 00:16:32,140 And this is a new two step operation that's called a 277 00:16:32,140 --> 00:16:34,310 screw axis. 278 00:16:34,310 --> 00:16:38,730 There is a persistent rumor that it derives its name from 279 00:16:38,730 --> 00:16:41,860 the effect that it has on 360 quizzes. 280 00:16:41,860 --> 00:16:43,460 But that is just an ugly rumor. 281 00:16:43,460 --> 00:16:45,310 There's nothing to that at all. 282 00:16:45,310 --> 00:16:48,930 So this is a new two step operation that we'll call, in 283 00:16:48,930 --> 00:16:53,910 general, A alpha tau. 284 00:16:53,910 --> 00:16:57,900 And what it will do is to generate a screw like 285 00:16:57,900 --> 00:16:59,680 distribution of objects. 286 00:16:59,680 --> 00:17:04,700 For those of you who have come in late, since you've been 287 00:17:04,700 --> 00:17:07,290 demonstrating a predilection towards longer and longer 288 00:17:07,290 --> 00:17:11,890 breaks, we're going to adjourn now and have a 40 minute break 289 00:17:11,890 --> 00:17:13,849 so you can get it out of your system. 290 00:17:13,849 --> 00:17:18,980 And we will resume at exactly 3 o'clock sharp. 291 00:17:18,980 --> 00:17:27,849 And we will launch into an elucidation of the nature of 292 00:17:27,849 --> 00:17:29,099 screw axes. 293 00:17:29,099 --> 00:17:40,050 294 00:17:40,050 --> 00:17:43,960 And again, for those of you who were not here at the very 295 00:17:43,960 --> 00:17:48,190 start of class, the reason is that I have a phone call that 296 00:17:48,190 --> 00:17:52,110 had to come in about this time of day and no other time. 297 00:17:52,110 --> 00:17:55,300 So rather than keep you shifting, wondering whether 298 00:17:55,300 --> 00:17:57,950 I'm coming back, we'll start at three when I should be done 299 00:17:57,950 --> 00:17:59,050 with my business. 300 00:17:59,050 --> 00:18:00,810 OK, so that's it. 301 00:18:00,810 --> 00:18:02,910 You missed all the important discussion of what's going to 302 00:18:02,910 --> 00:18:06,320 be on the next quiz, and nobody will tell you because 303 00:18:06,320 --> 00:18:09,420 they want to keep class average down. 304 00:18:09,420 --> 00:18:12,710 All right, so sorry for the longer than usual interlude, 305 00:18:12,710 --> 00:18:14,310 but we'll start again promptly at 3:00. 306 00:18:14,310 --> 00:18:16,540