1 00:00:00,000 --> 00:00:12,250 PROFESSOR: I think it's time that we got started. 2 00:00:12,250 --> 00:00:15,340 I haven't given a problem set out for a couple of days. 3 00:00:15,340 --> 00:00:19,300 So I have one that will cover some material that we'll go 4 00:00:19,300 --> 00:00:23,720 over with and develop today and some new material that 5 00:00:23,720 --> 00:00:25,880 we're not going to introduce for a while. 6 00:00:25,880 --> 00:00:28,868 AUDIENCE: [INAUDIBLE]? 7 00:00:28,868 --> 00:00:30,100 PROFESSOR: Yes, this is Thursday. 8 00:00:30,100 --> 00:00:34,160 And this is for the near future. 9 00:00:34,160 --> 00:00:38,340 I've given you two-- two a week before. 10 00:00:38,340 --> 00:00:41,315 So it's the adjective you're objecting to not the problem 11 00:00:41,315 --> 00:00:42,565 set, of course. 12 00:00:44,800 --> 00:00:48,630 Today we're going undertake another major step in our 13 00:00:48,630 --> 00:00:50,930 development of three dimensional crystallographic 14 00:00:50,930 --> 00:00:52,220 symmetries. 15 00:00:52,220 --> 00:00:56,340 We have just completed derivation of the 16 00:00:56,340 --> 00:01:00,030 crystallographic point groups, so these are the macroscopic 17 00:01:00,030 --> 00:01:04,080 symmetries that crystals can have. 18 00:01:04,080 --> 00:01:07,940 And then, by analogy, to what we did in two dimensions, the 19 00:01:07,940 --> 00:01:12,970 next step would be to take each of these point groups and 20 00:01:12,970 --> 00:01:17,270 hang it at a lattice point of some three dimensional lattice 21 00:01:17,270 --> 00:01:18,890 that can accommodate them. 22 00:01:18,890 --> 00:01:21,590 We have not as yet developed the three dimensional 23 00:01:21,590 --> 00:01:24,400 lattices, and that will be our agenda for today. 24 00:01:27,590 --> 00:01:29,890 Seems like we've already done that, haven't we? 25 00:01:29,890 --> 00:01:34,560 We showed that in two dimensions, a cell can be 26 00:01:34,560 --> 00:01:38,210 oblique, it can be rectangular, it can be 27 00:01:38,210 --> 00:01:40,480 centered rectangular, it can be square, 28 00:01:40,480 --> 00:01:42,100 or it can be hexagonal. 29 00:01:42,100 --> 00:01:44,480 So these are the different kinds of bases that one can 30 00:01:44,480 --> 00:01:45,750 have for the cell, right? 31 00:01:45,750 --> 00:01:50,150 So there should be the same number of space lattices. 32 00:01:50,150 --> 00:01:52,920 Actually it's more complicated than that. 33 00:01:52,920 --> 00:02:00,770 And let me remind you by another handout what the two 34 00:02:00,770 --> 00:02:04,080 dimensional space groups, the plane groups, look like 35 00:02:04,080 --> 00:02:08,320 assembled in all of their glory on one sheet. 36 00:02:08,320 --> 00:02:11,630 And there you see the different ways symmetry can be 37 00:02:11,630 --> 00:02:15,630 placed in these five different kinds of lattices-- 38 00:02:15,630 --> 00:02:17,850 oblique, rectangular, centered 39 00:02:17,850 --> 00:02:21,840 rectangular, square, and hexagonal. 40 00:02:21,840 --> 00:02:25,550 Now a lattice in two dimensions, a potential base 41 00:02:25,550 --> 00:02:32,110 for a unit cell, can be rectangular if and only if 42 00:02:32,110 --> 00:02:34,790 there is symmetry within that net. 43 00:02:34,790 --> 00:02:38,860 That demands that it be exactly rectangular. 44 00:02:38,860 --> 00:02:42,940 In other words, if a lattice is to be rectangular in the 45 00:02:42,940 --> 00:02:45,750 base of the cell, it has to have in it one of the 46 00:02:45,750 --> 00:02:49,610 symmetries that correspond to those in the plane groups that 47 00:02:49,610 --> 00:02:54,830 give and require, indeed demand, a rectangular lattice. 48 00:02:54,830 --> 00:02:58,490 Similarly a lattice can be truly square, identically 49 00:02:58,490 --> 00:03:01,880 square, only if there's a fourfold axis in it. 50 00:03:01,880 --> 00:03:03,330 That demands that it be square. 51 00:03:03,330 --> 00:03:05,350 The same could be said for either 52 00:03:05,350 --> 00:03:07,270 threefold or sixfold symmetry. 53 00:03:07,270 --> 00:03:11,050 One of the groups, P3, [? P3, ?] 54 00:03:11,050 --> 00:03:17,700 mP3, 1m, P6, and P6mm, has to be in the base of the cell if 55 00:03:17,700 --> 00:03:21,430 that cell is to have the specialization. 56 00:03:21,430 --> 00:03:24,900 So let's give an example with a twofold axis. 57 00:03:24,900 --> 00:03:29,230 Let's suppose we have a base of the cell that has an 58 00:03:29,230 --> 00:03:31,060 unspecialized shape-- 59 00:03:31,060 --> 00:03:34,740 two translations that are unequal in length and an angle 60 00:03:34,740 --> 00:03:38,410 between them which is a general obtuse angle. 61 00:03:38,410 --> 00:03:40,260 That's about as general as you can get. 62 00:03:40,260 --> 00:03:48,380 But suppose that this net has as well a two dimensional-- a 63 00:03:48,380 --> 00:03:51,220 twofold axis in it. 64 00:03:51,220 --> 00:03:57,150 And suppose we pick our third translation such that it picks 65 00:03:57,150 --> 00:04:00,460 up this net which is going to be the base of our three 66 00:04:00,460 --> 00:04:04,260 dimensional lattice and translates it over to this 67 00:04:04,260 --> 00:04:06,850 point in the base. 68 00:04:06,850 --> 00:04:08,540 Well that translation picks up everything. 69 00:04:08,540 --> 00:04:11,560 It picks up not only the two translations that we've been 70 00:04:11,560 --> 00:04:15,470 calling T1 and T2 to this point, but it also picks up 71 00:04:15,470 --> 00:04:18,019 all the twofold axes that are hanging on 72 00:04:18,019 --> 00:04:19,610 these lattice points. 73 00:04:19,610 --> 00:04:25,190 So if this is the terminus of T3 and projection, it will 74 00:04:25,190 --> 00:04:28,810 have picked up a twofold axis and plopped down a twofold 75 00:04:28,810 --> 00:04:33,490 axis here where no twofold axis exists. 76 00:04:33,490 --> 00:04:36,130 And we can take one of the theorems that we've seen, 77 00:04:36,130 --> 00:04:40,590 namely that the operation A pi, followed by translation, 78 00:04:40,590 --> 00:04:45,630 is equal to a new rotation operation through pi located 79 00:04:45,630 --> 00:04:47,600 halfway along the translation. 80 00:04:47,600 --> 00:04:52,540 And turn this around and ask what happens when we combine 81 00:04:52,540 --> 00:04:55,340 two rotations through 180 degrees 82 00:04:55,340 --> 00:04:56,500 about different points. 83 00:04:56,500 --> 00:04:59,160 And let's just draw it out to see what happens. 84 00:04:59,160 --> 00:05:01,000 Let's say there's a twofold axis here. 85 00:05:01,000 --> 00:05:07,480 It takes motif number 1, moves it to number 2. 86 00:05:07,480 --> 00:05:10,470 And then here's a second twofold axis and this takes 87 00:05:10,470 --> 00:05:15,130 number 2, rotates it over here to number 3. 88 00:05:15,130 --> 00:05:17,380 I is number 1 related to 3? 89 00:05:17,380 --> 00:05:22,730 But in terms of this first theorem, not surprisingly, we 90 00:05:22,730 --> 00:05:28,160 have introduced a translation into the space which is equal 91 00:05:28,160 --> 00:05:32,020 to twice the spacing between the twofold axes, which I"ll 92 00:05:32,020 --> 00:05:34,490 label as delta. 93 00:05:34,490 --> 00:05:37,710 So in other words, if we put a twofold axis down in this 94 00:05:37,710 --> 00:05:41,090 place in addition to the ones that we already have, we have 95 00:05:41,090 --> 00:05:45,150 created, in the base of the cell, a new twofold axis in 96 00:05:45,150 --> 00:05:46,340 this location. 97 00:05:46,340 --> 00:05:49,740 That just mucks everything up because it's going to rotate 98 00:05:49,740 --> 00:05:52,050 the translations that we have, it's going to rotate the 99 00:05:52,050 --> 00:05:53,710 twofold axes that we have. 100 00:05:53,710 --> 00:05:56,110 And we will not any longer have a group. 101 00:05:56,110 --> 00:05:58,310 We will not have a set of operations 102 00:05:58,310 --> 00:06:01,560 that closes upon itself. 103 00:06:01,560 --> 00:06:06,560 So the conclusion then is that if we're going to safely pick 104 00:06:06,560 --> 00:06:11,930 a third translation, T3, and combine it with a net that has 105 00:06:11,930 --> 00:06:17,210 twofold axes in it, in order to get a group, we are going 106 00:06:17,210 --> 00:06:22,100 to have to ensure that the twofold axes line up in every 107 00:06:22,100 --> 00:06:24,270 net in the stack. 108 00:06:24,270 --> 00:06:26,410 And there could be several ways of doing that. 109 00:06:26,410 --> 00:06:31,650 We could pick T3 so that it goes straight up. 110 00:06:31,650 --> 00:06:34,950 And then the twofold axes are just all slit up parallel to 111 00:06:34,950 --> 00:06:38,280 themselves, and they remain in coincidence. 112 00:06:38,280 --> 00:06:43,780 Or we could pick T3 so that it terminates over the point 113 00:06:43,780 --> 00:06:46,020 that's halfway along T2. 114 00:06:46,020 --> 00:06:48,730 And then this twofold axis gets picked up and put down 115 00:06:48,730 --> 00:06:50,290 over one that's already there. 116 00:06:50,290 --> 00:06:52,740 This one gets picked up and put down over one that's 117 00:06:52,740 --> 00:06:53,940 already there and so on. 118 00:06:53,940 --> 00:06:55,710 So that's an OK choice. 119 00:06:55,710 --> 00:06:58,270 That's a safe choice as well. 120 00:06:58,270 --> 00:07:01,870 Similarly we could pick T3 such that, within the plane of 121 00:07:01,870 --> 00:07:06,260 the net, it had a component that was one half of T1, so 122 00:07:06,260 --> 00:07:09,790 that it picked up this net and slid it over to this location. 123 00:07:09,790 --> 00:07:13,892 So once again all the twofold axes lined up. 124 00:07:13,892 --> 00:07:17,150 Or, in the same fashion, let it terminate over the center 125 00:07:17,150 --> 00:07:19,840 of the cell. 126 00:07:19,840 --> 00:07:23,060 So those are the four ways we can combine an oblique net 127 00:07:23,060 --> 00:07:28,450 that contains a twofold axis in such a way that the lattice 128 00:07:28,450 --> 00:07:32,870 points in the existing twofold axes remain invariant and are 129 00:07:32,870 --> 00:07:38,700 not moved into locations that create new twofold axes. 130 00:07:38,700 --> 00:07:41,650 OK so let's back off a little bit and start 131 00:07:41,650 --> 00:07:43,850 with plane group P1-- 132 00:07:43,850 --> 00:07:46,370 no symmetry at all. 133 00:07:46,370 --> 00:07:48,400 So this is T1 and T2. 134 00:07:48,400 --> 00:07:51,050 If there's no symmetry whatsoever in the base of the 135 00:07:51,050 --> 00:07:54,080 cell, we are free to pick the third translation in any 136 00:07:54,080 --> 00:07:56,420 orientation that we choose. 137 00:07:56,420 --> 00:08:02,280 So trying to sketch it in three dimensions. 138 00:08:02,280 --> 00:08:09,520 If this is T1, and this is T2, T3 can be in any orientation 139 00:08:09,520 --> 00:08:12,930 such that not only is this angle general, but this angle 140 00:08:12,930 --> 00:08:15,680 is general, and this angle general as well. 141 00:08:15,680 --> 00:08:19,440 And if I would try to carefully complete the 142 00:08:19,440 --> 00:08:22,900 parallelepiped so that its geometry looks reasonable. 143 00:08:25,520 --> 00:08:29,900 This is going to be the general oblique parallelepiped 144 00:08:29,900 --> 00:08:37,960 with three translations that are unequal in magnitude, and 145 00:08:37,960 --> 00:08:43,780 three angles between these translations. 146 00:08:43,780 --> 00:08:48,000 Let me write them as the angle between T1 and T2-- 147 00:08:48,000 --> 00:08:54,220 not equal to the angle between T2 and T3 necessarily, not 148 00:08:54,220 --> 00:08:59,790 equal to the angle between T3 and T1. 149 00:08:59,790 --> 00:09:02,452 And all of these angles are completely general and can 150 00:09:02,452 --> 00:09:05,760 assume any values that they like. 151 00:09:05,760 --> 00:09:07,830 So this then is the lowest common denominator. 152 00:09:07,830 --> 00:09:10,760 This is a totally unspecialized space lattice. 153 00:09:10,760 --> 00:09:14,020 It has the shape of the general parallelepiped-- 154 00:09:14,020 --> 00:09:16,960 no special relations between any of the translations 155 00:09:16,960 --> 00:09:19,129 whatsoever. 156 00:09:19,129 --> 00:09:22,580 AUDIENCE: [INAUDIBLE] 157 00:09:22,580 --> 00:09:24,890 PROFESSOR: I say they're-- 158 00:09:24,890 --> 00:09:26,920 Oh I'm sorry, magnitude are not equal. 159 00:09:26,920 --> 00:09:27,690 [INAUDIBLE] 160 00:09:27,690 --> 00:09:29,130 I meant to put not equal signs there. 161 00:09:29,130 --> 00:09:31,890 Yeah, they can have any length they wish. 162 00:09:31,890 --> 00:09:33,140 They're not equivalent. 163 00:09:41,760 --> 00:09:43,010 Any other questions? 164 00:09:46,680 --> 00:09:54,950 OK let's go back then to the case of an oblique net to 165 00:09:54,950 --> 00:09:57,400 which we've decided to add a twofold axis. 166 00:09:57,400 --> 00:10:00,740 And that gives twofold axes in the locations that we've 167 00:10:00,740 --> 00:10:03,460 become familiar with in the plane group. 168 00:10:03,460 --> 00:10:09,490 And if I pick the translation so that it goes directly 169 00:10:09,490 --> 00:10:12,880 normal to the base of the cell, I'm going to have a new 170 00:10:12,880 --> 00:10:14,580 kind of lattice. 171 00:10:14,580 --> 00:10:19,140 The lattice is going to have two angles between axes that 172 00:10:19,140 --> 00:10:23,880 are exactly 90 degrees, not approximately so, but exactly 173 00:10:23,880 --> 00:10:28,350 so, because only if that translation, T3, is exactly 174 00:10:28,350 --> 00:10:32,880 normal to the base of the cell will the twofold axes in all 175 00:10:32,880 --> 00:10:34,865 of these nets lineup exactly. 176 00:10:39,500 --> 00:10:42,370 So here we have a degree of specialization. 177 00:10:42,370 --> 00:10:46,080 The translations remain unequal in magnitude. 178 00:10:46,080 --> 00:10:50,420 They can be any length they wish and not change things, 179 00:10:50,420 --> 00:10:52,580 but two of the angles-- 180 00:10:52,580 --> 00:11:01,270 axial angles, T1 to T2, is identically 90 degrees. 181 00:11:01,270 --> 00:11:06,940 The angle between T2 and T3 is identically 90 degrees. 182 00:11:06,940 --> 00:11:08,880 And the third angle, the angle-- 183 00:11:08,880 --> 00:11:11,590 I'm sorry, T1 and T2 is my general angle. 184 00:11:11,590 --> 00:11:12,865 So this can be anything. 185 00:11:18,710 --> 00:11:20,730 So that can be any general angle. 186 00:11:20,730 --> 00:11:27,200 T2 and T3 and T1 and T3, want to be exactly 90 degrees. 187 00:11:33,100 --> 00:11:37,590 So we're getting space lattices that have a degree of 188 00:11:37,590 --> 00:11:39,860 precise specialization. 189 00:11:39,860 --> 00:11:43,140 And the reason is that the geometry of the base of the 190 00:11:43,140 --> 00:11:47,110 cell is inseparable from symmetry in the base of the 191 00:11:47,110 --> 00:11:51,220 cell that demands the degree of specialization that makes 192 00:11:51,220 --> 00:11:52,470 the space lattice unique. 193 00:11:56,820 --> 00:12:02,100 So let us look at a couple of the other choices for the 194 00:12:02,100 --> 00:12:04,930 third translation that will leave the twofold axes in 195 00:12:04,930 --> 00:12:06,180 coincidence. 196 00:12:08,030 --> 00:12:13,450 Another choice would be to pick the third translation 197 00:12:13,450 --> 00:12:16,420 such that it terminated exactly over 198 00:12:16,420 --> 00:12:20,420 the midpoint of T2. 199 00:12:20,420 --> 00:12:21,430 So here's T1. 200 00:12:21,430 --> 00:12:22,680 Here's T2. 201 00:12:26,590 --> 00:12:29,090 Now what I'm going to do is something I have not done to 202 00:12:29,090 --> 00:12:31,010 this point. 203 00:12:31,010 --> 00:12:35,260 Having this translation T3 terminate exactly over the 204 00:12:35,260 --> 00:12:37,830 midpoint of a translation [INAUDIBLE] was clearly a 205 00:12:37,830 --> 00:12:39,840 specialization. 206 00:12:39,840 --> 00:12:43,740 If I were to go up two translations, T3, I would be 207 00:12:43,740 --> 00:12:49,620 directly over the end of T2, and that is a much more 208 00:12:49,620 --> 00:12:52,290 convenient way of demonstrating 209 00:12:52,290 --> 00:12:53,970 that special feature. 210 00:12:53,970 --> 00:12:56,460 So let me pick a new T3 prime. 211 00:12:59,530 --> 00:13:04,480 It goes up exactly normal to the base of the cell, and in 212 00:13:04,480 --> 00:13:08,780 so doing, I will have defined a cell which is a double cell. 213 00:13:14,840 --> 00:13:18,400 And the catch is an extra lattice point in the middle of 214 00:13:18,400 --> 00:13:21,870 one of the pair of faces. 215 00:13:21,870 --> 00:13:25,750 So I'll refer to this general sort of situation-- 216 00:13:25,750 --> 00:13:27,700 this is a double cell. 217 00:13:27,700 --> 00:13:32,695 And I'll call this, in words, a side-centered cell. 218 00:13:40,260 --> 00:13:44,030 And it has, with this redefinition of translations, 219 00:13:44,030 --> 00:13:48,370 exactly the same degree of specialization as before-- 220 00:13:48,370 --> 00:13:52,440 magnitude of T1 not equal to the magnitude of T2, not equal 221 00:13:52,440 --> 00:13:54,620 to the magnitude of T3. 222 00:13:54,620 --> 00:13:58,220 And it has the same specialization of the angles. 223 00:13:58,220 --> 00:14:04,250 The angle between T1 and T3 is exactly 90 degrees. 224 00:14:04,250 --> 00:14:11,470 The angle between T2 and T3 is exactly 90 degrees. 225 00:14:11,470 --> 00:14:18,050 And the angle between T1 and T2 is general. 226 00:14:18,050 --> 00:14:22,750 General, but by convention we will assume the obtuse angle 227 00:14:22,750 --> 00:14:24,000 rather than the acute angle. 228 00:14:26,770 --> 00:14:30,660 OK so by picking this double cell, we've defined a cell 229 00:14:30,660 --> 00:14:33,520 that has a twofold redundancy. 230 00:14:33,520 --> 00:14:40,130 But in doing so, we have gained the advantage of 231 00:14:40,130 --> 00:14:45,510 showing that this specialized geometry is exactly the same 232 00:14:45,510 --> 00:14:46,610 as this one-- 233 00:14:46,610 --> 00:14:49,530 in the same sense that in two dimensions, the primitive 234 00:14:49,530 --> 00:14:53,020 rectangular cell and the centered rectangular cell were 235 00:14:53,020 --> 00:14:54,820 both cousins. 236 00:14:54,820 --> 00:14:57,610 They were both had orthogonal translations. 237 00:14:57,610 --> 00:15:00,830 But the other thing that we gain is that even though we 238 00:15:00,830 --> 00:15:03,900 can't have an orthogonal coordinate system, if we use 239 00:15:03,900 --> 00:15:07,110 the three translations as the basis of a coordinate system, 240 00:15:07,110 --> 00:15:10,960 at least we have two 90 degree angles on 241 00:15:10,960 --> 00:15:11,950 our coordinate system. 242 00:15:11,950 --> 00:15:13,620 And operationally, that is going to 243 00:15:13,620 --> 00:15:15,400 be an enormous advantage. 244 00:15:15,400 --> 00:15:15,730 Yes, sir. 245 00:15:15,730 --> 00:15:17,104 AUDIENCE: So is T3 [INAUDIBLE] 246 00:15:20,090 --> 00:15:27,220 PROFESSOR: T3 was selected so that we picked it as 2 T2. 247 00:15:30,876 --> 00:15:35,050 I'm sorry, 2 T3 minus T2. 248 00:15:35,050 --> 00:15:38,150 That was my new T3 prime. 249 00:15:38,150 --> 00:15:42,630 I went up two translations in T2 and then back T2, and that 250 00:15:42,630 --> 00:15:45,660 put me directly over the lattice point from which my 251 00:15:45,660 --> 00:15:46,730 translations emanate. 252 00:15:46,730 --> 00:15:50,410 AUDIENCE: So your T3 is actually [INAUDIBLE] 253 00:15:50,410 --> 00:15:55,990 PROFESSOR: It goes to the next level up in the stack of nets. 254 00:15:59,880 --> 00:16:02,690 If I would draw this in projection, which is not 255 00:16:02,690 --> 00:16:05,630 nearly as clear in some respects, this would be the 256 00:16:05,630 --> 00:16:06,880 base of the cell. 257 00:16:06,880 --> 00:16:10,870 Let me use x's for the next level up. 258 00:16:10,870 --> 00:16:13,300 So here's the net offset by half of T2. 259 00:16:16,130 --> 00:16:19,820 And then if I go up two of these T3's and then back by 260 00:16:19,820 --> 00:16:24,020 T2, my next layer up is directly over the first one-- 261 00:16:24,020 --> 00:16:26,330 the third layer up is directly over the first. 262 00:16:26,330 --> 00:16:28,530 AUDIENCE: Shouldn't it be T3 prime then? 263 00:16:28,530 --> 00:16:33,010 PROFESSOR: Yes if I were to be consistent, I should call this 264 00:16:33,010 --> 00:16:36,040 T3 prime, which I did here. 265 00:16:36,040 --> 00:16:40,520 But now I should call that T3 as well. 266 00:16:40,520 --> 00:16:43,160 I got a little bit careless because we're going to forget 267 00:16:43,160 --> 00:16:46,640 that we ever had this T3-- to find this selected as a 268 00:16:46,640 --> 00:16:47,890 stacking vector. 269 00:16:51,520 --> 00:16:54,730 OK well in P2 there are four different 270 00:16:54,730 --> 00:16:57,430 kinds of twofold axes. 271 00:16:57,430 --> 00:17:01,320 So we could have the third translation terminate over 272 00:17:01,320 --> 00:17:04,980 either of the remaining pair of twofold axes as well. 273 00:17:04,980 --> 00:17:10,700 I think that it's apparent that if I pick T3, the 274 00:17:10,700 --> 00:17:17,930 original T3, as one half of T1, so that the next layer up 275 00:17:17,930 --> 00:17:23,030 terminates in lattice points in this orientation. 276 00:17:23,030 --> 00:17:30,060 And then I pick a T3 prime, which is equal to 2 T1 minus-- 277 00:17:30,060 --> 00:17:35,490 2 T3 minus T1, I will have gotten again a 278 00:17:35,490 --> 00:17:36,710 side-centered lattice. 279 00:17:36,710 --> 00:17:39,690 The only difference is that it's going to be a different 280 00:17:39,690 --> 00:17:40,500 pair of faces. 281 00:17:40,500 --> 00:17:44,080 It's going to be the O1Oa faces that have the extra 282 00:17:44,080 --> 00:17:45,270 lattice point. 283 00:17:45,270 --> 00:17:47,990 So that is not fundamentally a different sort of lattice. 284 00:17:47,990 --> 00:17:50,250 It is also side-centered. 285 00:17:50,250 --> 00:17:55,170 And differs from our second result only in whether it is 286 00:17:55,170 --> 00:17:59,490 the side face with the shortest translation or the 287 00:17:59,490 --> 00:18:02,430 longest translations of the base that bears the extra 288 00:18:02,430 --> 00:18:04,080 lattice point. 289 00:18:04,080 --> 00:18:11,910 One final choice though does result in a new lattice with 290 00:18:11,910 --> 00:18:13,180 special features. 291 00:18:13,180 --> 00:18:18,420 And this would be one where I pick the third translation 292 00:18:18,420 --> 00:18:24,810 such that it terminates over the center of the net below. 293 00:18:29,020 --> 00:18:36,070 So let's let this be T1, this be T2, and pick T3 so that it 294 00:18:36,070 --> 00:18:41,380 moves the origin lattice point directly over the center of 295 00:18:41,380 --> 00:18:44,490 the parallelogram face below. 296 00:18:44,490 --> 00:18:51,420 And then if I go up two translations, T3, and then 297 00:18:51,420 --> 00:19:01,520 subtract off T1, and subtract off T2, I will again have a T3 298 00:19:01,520 --> 00:19:04,810 prime, which is exactly normal to the base of the cell. 299 00:19:08,150 --> 00:19:10,920 OK this is a type of lattice that you've 300 00:19:10,920 --> 00:19:12,180 come to know and love. 301 00:19:12,180 --> 00:19:15,040 In the cubic system, this is referred to as a 302 00:19:15,040 --> 00:19:16,290 body-centered lattice. 303 00:19:28,116 --> 00:19:31,574 AUDIENCE: So then, T3 is just [INAUDIBLE] 304 00:19:31,574 --> 00:19:38,980 PROFESSOR: T3, the original choice, would pick this 305 00:19:38,980 --> 00:19:47,740 original net up and move it so that that net was moved-- 306 00:19:47,740 --> 00:19:50,750 such that it's corner was over the center of 307 00:19:50,750 --> 00:19:53,040 the original net. 308 00:19:53,040 --> 00:19:55,930 So if we go up two translations, we have lattice 309 00:19:55,930 --> 00:20:00,922 point over lattice point, and we redefine T3 by going back 1 310 00:20:00,922 --> 00:20:03,390 T1 and minus 1 T2. 311 00:20:03,390 --> 00:20:05,420 And that makes it exactly normal to the base. 312 00:20:05,420 --> 00:20:07,036 AUDIENCE: Not on the sides? 313 00:20:07,036 --> 00:20:09,046 PROFESSOR: Nothing on the sides . 314 00:20:09,046 --> 00:20:11,810 Now in fact, if we try to do that, we would have something 315 00:20:11,810 --> 00:20:12,700 that's not a lattice. 316 00:20:12,700 --> 00:20:16,360 If we tried to have some additional lattice points in 317 00:20:16,360 --> 00:20:19,290 the center and the edges of this upper level, we would 318 00:20:19,290 --> 00:20:22,420 have destroyed the original T1 and T2 in 319 00:20:22,420 --> 00:20:23,670 the base of the cell. 320 00:20:36,100 --> 00:20:39,920 Let me do one or two more and then I think the way in which 321 00:20:39,920 --> 00:20:43,310 one proceeds should be quite clear. 322 00:20:43,310 --> 00:20:50,630 If you take this list of all of the plane groups, if we 323 00:20:50,630 --> 00:20:55,980 would like to have a three dimensional cell that has a 324 00:20:55,980 --> 00:21:03,030 rectangular base, the base can be rectangular only if there 325 00:21:03,030 --> 00:21:09,440 is a mirror plane in that net or a twofold axis with the 326 00:21:09,440 --> 00:21:10,920 mirror plane. 327 00:21:10,920 --> 00:21:12,730 The two situations are different. 328 00:21:12,730 --> 00:21:24,990 So let's examine first the case of a rectangular net that 329 00:21:24,990 --> 00:21:28,825 has either a mirror plane or a glide plane in it. 330 00:21:37,570 --> 00:21:42,270 And they're both going to have the same sort of constraint. 331 00:21:42,270 --> 00:21:49,270 If I have a mirror plane, that-- let me use a slightly 332 00:21:49,270 --> 00:21:55,200 wiggly line so the mirror line is not confused with the edges 333 00:21:55,200 --> 00:21:55,870 of the cell. 334 00:21:55,870 --> 00:22:00,430 So here are two translations, T1 and T2. 335 00:22:00,430 --> 00:22:05,630 And if I pick a T3 that moves this rectangular net up and 336 00:22:05,630 --> 00:22:09,630 over itself, we have to do so in such a way that the mirror 337 00:22:09,630 --> 00:22:11,510 planes coincide. 338 00:22:11,510 --> 00:22:17,440 Because if I have a mirror plane in space that passes in 339 00:22:17,440 --> 00:22:22,800 proximity to a lattice point, I can turn my theorem around 340 00:22:22,800 --> 00:22:29,350 and say that a translation, T1, combined with a mirror 341 00:22:29,350 --> 00:22:32,710 plane that is a reflection operation that's removed from 342 00:22:32,710 --> 00:22:39,910 the lattice point, I would have to introduce a new 343 00:22:39,910 --> 00:22:42,130 translation to delta which is going to be 344 00:22:42,130 --> 00:22:44,620 incommensurate with T1. 345 00:22:44,620 --> 00:22:46,850 So I have to pick my third translation 346 00:22:46,850 --> 00:22:48,120 in one of two ways. 347 00:22:48,120 --> 00:22:50,810 Let me indicate that by drawing a circle. 348 00:22:50,810 --> 00:22:55,600 T3 can terminate anywhere over this mirror plane. 349 00:22:55,600 --> 00:22:59,430 I could pick this net up, slide it parallel to T2, and 350 00:22:59,430 --> 00:23:02,790 as long as I put it down so that this is the lattice point 351 00:23:02,790 --> 00:23:08,780 in the next layer up, that will be a legitimate placement 352 00:23:08,780 --> 00:23:09,470 of the net. 353 00:23:09,470 --> 00:23:11,590 Because all the mirror planes are simply slid into 354 00:23:11,590 --> 00:23:14,130 coincidence with one another. 355 00:23:14,130 --> 00:23:17,405 So if I sketch that thing in three dimensions-- 356 00:23:20,480 --> 00:23:22,950 this is the base of the cell and it has 357 00:23:22,950 --> 00:23:25,030 a rectangular shape. 358 00:23:25,030 --> 00:23:27,350 And the third translation goes up at an angle. 359 00:23:33,570 --> 00:23:35,790 That's going to be exactly something 360 00:23:35,790 --> 00:23:37,810 that I've made before. 361 00:23:37,810 --> 00:23:42,940 And the only difference is that the oblique angle is now 362 00:23:42,940 --> 00:23:46,150 on the vertical plane but that's a right angle. 363 00:23:46,150 --> 00:23:50,650 And this stays a right angle. 364 00:23:50,650 --> 00:23:53,860 So that's exactly a lattice of the shape that we had last 365 00:23:53,860 --> 00:23:57,440 time with twofold axes, except this cell is sitting on one of 366 00:23:57,440 --> 00:24:00,000 its rectangular sides rather than sitting 367 00:24:00,000 --> 00:24:02,400 on its oblique base. 368 00:24:02,400 --> 00:24:04,710 So that is something that's not new. 369 00:24:07,410 --> 00:24:12,220 And if I would pick the translations so that T3 had a 370 00:24:12,220 --> 00:24:18,540 component that was one half of T1, plus some amount z that is 371 00:24:18,540 --> 00:24:23,650 perpendicular to the base, I can redefine that in terms of 372 00:24:23,650 --> 00:24:31,350 a cell that has one pair of faces inclined to one another. 373 00:24:31,350 --> 00:24:33,740 And the difference is going to be that the lattice point 374 00:24:33,740 --> 00:24:37,320 would be caught in the center of the cell. 375 00:24:37,320 --> 00:24:40,710 This would be my original translation T3 and I'll 376 00:24:40,710 --> 00:24:43,760 redefine that as a T3 prime. 377 00:24:43,760 --> 00:24:48,870 And that gives me two right angles and one general angle. 378 00:24:48,870 --> 00:24:53,650 So those are exactly the same lattices that I obtained from 379 00:24:53,650 --> 00:24:55,920 stacking a net with a twofold axis in it. 380 00:25:02,290 --> 00:25:04,060 Let me introduce now another piece of 381 00:25:04,060 --> 00:25:05,310 jargon as we go along. 382 00:25:10,170 --> 00:25:15,820 The relative angles between the translations are going to 383 00:25:15,820 --> 00:25:21,350 determine the shape of the coordinate system that we pick 384 00:25:21,350 --> 00:25:25,570 along those translations to specify the geometry of 385 00:25:25,570 --> 00:25:29,270 features within the lattice. 386 00:25:29,270 --> 00:25:34,990 So we've had a first case, and this was just a general 387 00:25:34,990 --> 00:25:36,940 oblique net. 388 00:25:36,940 --> 00:25:42,690 So we had a T1 not equal to a T2, not equal to T3, and all 389 00:25:42,690 --> 00:25:45,290 three angles were general. 390 00:25:51,690 --> 00:25:53,760 All three pair of axes are inclined. 391 00:26:04,330 --> 00:26:07,780 And this coordinate system, if we want to refer to it in 392 00:26:07,780 --> 00:26:12,386 terms of words, is called a triclinic. 393 00:26:12,386 --> 00:26:14,500 This is the triclinic system. 394 00:26:18,920 --> 00:26:22,560 And the triclinic system is short for coordinate system. 395 00:26:33,010 --> 00:26:34,890 So this is as general as things get. 396 00:26:34,890 --> 00:26:36,870 And if this is the shape of the 397 00:26:36,870 --> 00:26:38,795 lattice there is no symmetry. 398 00:26:42,920 --> 00:26:46,370 Which is the same as saying that the point group, the only 399 00:26:46,370 --> 00:26:48,750 point group that can fit into such a lattice, 400 00:26:48,750 --> 00:26:50,301 is point group 1. 401 00:26:59,440 --> 00:27:05,900 And then we hit another sort of geometry. 402 00:27:05,900 --> 00:27:08,980 And that is where the three translations 403 00:27:08,980 --> 00:27:12,992 had a general angle-- 404 00:27:12,992 --> 00:27:17,660 a third translation that was exactly normal to the plane of 405 00:27:17,660 --> 00:27:19,850 the first two. 406 00:27:19,850 --> 00:27:24,410 And one pair of axes is inclined. 407 00:27:24,410 --> 00:27:28,950 And you can see that this notation is a little strange, 408 00:27:28,950 --> 00:27:32,170 perhaps, but consistent. 409 00:27:32,170 --> 00:27:34,810 And it says something about the nature of the arrangement 410 00:27:34,810 --> 00:27:36,400 of cell edges. 411 00:27:36,400 --> 00:27:44,290 So one pair of axes inclined is called 412 00:27:44,290 --> 00:27:47,715 the monoclinic system. 413 00:27:56,140 --> 00:28:05,060 And this can contain lattices that are either primitive, 414 00:28:05,060 --> 00:28:06,310 side-centered-- 415 00:28:11,020 --> 00:28:13,950 with either the longest or shortest translation in the 416 00:28:13,950 --> 00:28:17,140 parallelogram that's coming out of 417 00:28:17,140 --> 00:28:18,270 the face that's centered. 418 00:28:18,270 --> 00:28:21,190 Or we saw there was one final possibility. 419 00:28:21,190 --> 00:28:22,440 And that was body-centered. 420 00:28:25,400 --> 00:28:28,950 So we'll refer to these lattices subsequently as 421 00:28:28,950 --> 00:28:32,290 monoclinic primitive, monoclinic side-centered, or 422 00:28:32,290 --> 00:28:35,030 monoclinic body-centered. 423 00:28:35,030 --> 00:28:44,160 These lattices were compatible with point group 2 424 00:28:44,160 --> 00:28:47,720 or point group m. 425 00:28:47,720 --> 00:28:51,220 We obtained a lattice of the same shape with point group m 426 00:28:51,220 --> 00:28:53,170 or with point group 2. 427 00:28:53,170 --> 00:28:56,680 If it works for 2 and it works for m, why not 428 00:28:56,680 --> 00:28:59,050 both at the same time. 429 00:28:59,050 --> 00:29:02,290 It would also work for 2 over m-- 430 00:29:05,170 --> 00:29:07,470 twofold axis perpendicular to a mirror plane. 431 00:29:16,700 --> 00:29:19,900 Any question at this point? 432 00:29:19,900 --> 00:29:20,215 Yeah? 433 00:29:20,215 --> 00:29:21,406 AUDIENCE: Also 2m, right? 434 00:29:21,406 --> 00:29:22,720 [INAUDIBLE] 435 00:29:22,720 --> 00:29:25,862 Yeah 436 00:29:25,862 --> 00:29:28,560 PROFESSOR: No,because as soon as you got a mirror plane, 437 00:29:28,560 --> 00:29:32,770 then you have to have a rectangular net. 438 00:29:35,790 --> 00:29:37,230 So it's going to be-- it's going to 439 00:29:37,230 --> 00:29:40,100 fall into this family. 440 00:29:40,100 --> 00:29:42,140 You're right, a mirror plane by itself and 441 00:29:42,140 --> 00:29:43,400 we've got that here. 442 00:29:43,400 --> 00:29:45,550 That's where the mirror plane falls in 443 00:29:45,550 --> 00:29:47,844 the side of the thing. 444 00:29:47,844 --> 00:29:49,094 You're right. 445 00:29:51,280 --> 00:29:53,790 But if it has a rectangular shape, then we have to have 446 00:29:53,790 --> 00:29:56,510 either orthogonal mirror planes and a twofold axis. 447 00:29:59,040 --> 00:30:00,522 The base, if the base-- yeah. 448 00:30:04,140 --> 00:30:05,690 OK. 449 00:30:05,690 --> 00:30:10,300 Part of the amusing part of this game is that you can, 450 00:30:10,300 --> 00:30:16,900 upon appropriate redefinition of cells, come up with a given 451 00:30:16,900 --> 00:30:22,180 coordinate system that defines more than one lattice. 452 00:30:25,400 --> 00:30:28,710 And I'll give you-- 453 00:30:28,710 --> 00:30:31,410 because you get a little space doing this-- 454 00:30:31,410 --> 00:30:34,600 I'll give you a two dimensional example. 455 00:30:34,600 --> 00:30:36,430 Here is one cell. 456 00:30:36,430 --> 00:30:39,560 This is T1, this is T2. 457 00:30:39,560 --> 00:30:40,820 This is a primitive cell. 458 00:30:45,320 --> 00:30:47,223 Nobody should bother to write this down. 459 00:30:50,640 --> 00:30:51,890 Here's a T1, here's a T2. 460 00:30:57,570 --> 00:30:59,330 So anybody know what that is? 461 00:31:02,080 --> 00:31:03,720 This is a primitive cell. 462 00:31:03,720 --> 00:31:04,970 This is a jail cell. 463 00:31:10,900 --> 00:31:12,180 Couple sniggers, thank you. 464 00:31:14,960 --> 00:31:18,260 Still another one-- 465 00:31:18,260 --> 00:31:20,530 this is T1, this is T2. 466 00:31:27,040 --> 00:31:28,330 Anybody guess what that is? 467 00:31:31,100 --> 00:31:32,350 That's a soft sell. 468 00:31:35,950 --> 00:31:37,200 I got a million of them. 469 00:31:42,826 --> 00:31:44,220 Know what that is? 470 00:31:47,070 --> 00:31:48,370 It's an underground cell. 471 00:31:52,840 --> 00:31:54,220 Here's one that's kind of dated-- 472 00:31:54,220 --> 00:31:55,920 I don't think anybody will get this one. 473 00:32:06,030 --> 00:32:08,200 Does anybody know what that is? 474 00:32:08,200 --> 00:32:11,430 That was a misadventure of the Ford Motor Company that was 475 00:32:11,430 --> 00:32:12,870 named the Ed cell. 476 00:32:17,410 --> 00:32:19,880 Well, I'll give you equal time. 477 00:32:19,880 --> 00:32:22,420 Anybody who, after intermission, wants to respond 478 00:32:22,420 --> 00:32:26,470 in kind, I'll let you have an opportunity to do so. 479 00:32:26,470 --> 00:32:29,570 But obviously these are not standard crystallographic 480 00:32:29,570 --> 00:32:31,530 cells, but they make for a smile in an 481 00:32:31,530 --> 00:32:32,910 otherwise dreary business. 482 00:32:36,130 --> 00:32:39,380 I don't know how to proceed from here. 483 00:32:39,380 --> 00:32:45,050 I have a set of notes for you which carries all this out in 484 00:32:45,050 --> 00:32:48,450 incredible thoroughness. 485 00:32:48,450 --> 00:32:53,370 So let me pass this around. 486 00:32:53,370 --> 00:32:56,080 Let you take a copy of this and we won't go over every 487 00:32:56,080 --> 00:33:01,460 single thing, but I think just to outline quickly the sort of 488 00:33:01,460 --> 00:33:03,760 choices that we have. 489 00:33:03,760 --> 00:33:08,630 We haven't yet finished with the rectangular cell, but if 490 00:33:08,630 --> 00:33:11,290 it's got something like 2mm in it. 491 00:33:14,330 --> 00:33:17,060 with twofold axes here and mirror planes 492 00:33:17,060 --> 00:33:18,310 running this way-- 493 00:33:24,725 --> 00:33:42,360 P2mm in the base, P2gm, P2gg, can be stacked only in such a 494 00:33:42,360 --> 00:33:47,650 fashion that T3 brings twofold axes and mirror planes into 495 00:33:47,650 --> 00:33:49,950 coincidence with one another. 496 00:33:49,950 --> 00:33:59,560 So T3 could be 0T1 plus 0T2 plus some amount z which is 497 00:33:59,560 --> 00:34:01,930 straight up. 498 00:34:01,930 --> 00:34:11,400 Or it could be equal to one half of T1, 0T2, plus z. 499 00:34:11,400 --> 00:34:16,020 That would be making the origin twofold axis fall over 500 00:34:16,020 --> 00:34:17,770 this location. 501 00:34:17,770 --> 00:34:26,989 Or T3 could be equal to 0 plus one half of T2 plus z. 502 00:34:26,989 --> 00:34:29,520 And they're both going to, upon redefinition, give the 503 00:34:29,520 --> 00:34:30,780 same result. 504 00:34:30,780 --> 00:34:39,480 Or T2 could be equal to one half of T1 plus one half of T2 505 00:34:39,480 --> 00:34:42,030 plus z straight up. 506 00:34:42,030 --> 00:34:47,030 And upon redefinition, the first one that would be a 507 00:34:47,030 --> 00:34:48,630 brick-shaped unit cell. 508 00:34:51,630 --> 00:34:53,210 That's going to be a primitive cell. 509 00:34:55,830 --> 00:35:00,220 Picking T3 with a component of either one half of T1 or one 510 00:35:00,220 --> 00:35:04,570 half of T2 is going to be something that gives us a 511 00:35:04,570 --> 00:35:10,466 cell, which can again be redefined. 512 00:35:16,000 --> 00:35:21,200 This was the original T3 in terms of a T3 prime, which is 513 00:35:21,200 --> 00:35:31,320 equal to 2T1 or 2T2 minus [? 2T3 ?] 514 00:35:31,320 --> 00:35:33,610 minus T1 or T2. 515 00:35:33,610 --> 00:35:34,860 This is going to be side-centered. 516 00:35:41,920 --> 00:35:46,900 And the final choice where it is a T3 that's one half of T1 517 00:35:46,900 --> 00:35:51,600 plus one half of T2 can once again be defined in terms of-- 518 00:35:51,600 --> 00:35:54,370 redefined in terms of a body-centered lattice. 519 00:36:04,720 --> 00:36:08,490 OK this is a coordinate system where all interaxial angles 520 00:36:08,490 --> 00:36:09,940 are 90 degrees. 521 00:36:09,940 --> 00:36:14,680 The translations, however, have arbitrary magnitude with 522 00:36:14,680 --> 00:36:18,100 respect to one another. 523 00:36:18,100 --> 00:36:23,180 All of these cells could be described as rhombuses. 524 00:36:23,180 --> 00:36:27,870 In this case all angles in the rhombus are 90 degrees, so 525 00:36:27,870 --> 00:36:32,630 this is called the orthorhombic system-- 526 00:36:32,630 --> 00:36:35,850 orthorhombic coordinate system. 527 00:36:35,850 --> 00:36:42,620 All the angles between the axes are 90 degrees. 528 00:36:45,540 --> 00:36:51,410 So we can get that out of any one of the rectangular groups 529 00:36:51,410 --> 00:36:55,570 that has twofold axes and symmetry planes in it. 530 00:36:55,570 --> 00:37:00,930 And conversely, take one of these lattices and let the 531 00:37:00,930 --> 00:37:03,290 symmetry elements that we found in the plane group 532 00:37:03,290 --> 00:37:05,370 extend through them and you've got a three 533 00:37:05,370 --> 00:37:08,210 dimensional space group. 534 00:37:08,210 --> 00:37:11,170 So without emphasizing the fact is we go along, we're 535 00:37:11,170 --> 00:37:14,350 picking up a healthy number of three dimensional space groups 536 00:37:14,350 --> 00:37:15,600 as we go along. 537 00:37:29,258 --> 00:37:31,738 AUDIENCE: [INAUDIBLE]? 538 00:37:31,738 --> 00:37:34,010 PROFESSOR: This is primitive. 539 00:37:34,010 --> 00:37:35,590 It's also orthorhombic -- 540 00:37:35,590 --> 00:37:39,550 same dimensionality, three translations, and they're all 541 00:37:39,550 --> 00:37:42,230 mutually orthogonal. 542 00:37:42,230 --> 00:37:45,340 That's true of these other cells except they're double 543 00:37:45,340 --> 00:37:48,520 cells in the case of the side-centered and double as 544 00:37:48,520 --> 00:37:49,770 well for the case of the body-centered. 545 00:38:04,410 --> 00:38:09,210 It's time we introduced some other conventions for notation 546 00:38:09,210 --> 00:38:11,610 in describing lattices. 547 00:38:11,610 --> 00:38:15,920 We've been using T1, T2, and T3 to designate the three 548 00:38:15,920 --> 00:38:17,940 translations that are used to define the cell. 549 00:38:23,600 --> 00:38:27,430 One has to have some sort of rules for picking a standard 550 00:38:27,430 --> 00:38:31,330 unit cell so that two people can do an x-ray diffraction 551 00:38:31,330 --> 00:38:35,140 experiment, let's say, on a particular material and end up 552 00:38:35,140 --> 00:38:38,600 defining the lattice in exactly the same way. 553 00:38:38,600 --> 00:38:42,210 So let me now list, so that we can use these labels as we go 554 00:38:42,210 --> 00:38:51,805 along further, conventions for selecting cell edges. 555 00:39:02,400 --> 00:39:07,080 OK one convention is that one should select the shortest 556 00:39:07,080 --> 00:39:08,330 translations in the lattice. 557 00:39:21,630 --> 00:39:26,340 And these shortest translations, if there's no 558 00:39:26,340 --> 00:39:29,920 symmetry, will be [? dignified ?] 559 00:39:29,920 --> 00:39:33,700 by calling them a, b and, c-- 560 00:39:33,700 --> 00:39:36,060 standing for the x, y, and z directions. 561 00:39:40,970 --> 00:39:45,280 One picks always, if there's nothing special about the 562 00:39:45,280 --> 00:39:50,130 translations in the triclinic system, the magnitude of b 563 00:39:50,130 --> 00:39:51,380 greater than a. 564 00:39:54,060 --> 00:39:55,965 And in a rational world-- 565 00:39:59,170 --> 00:40:01,560 every so often you come to a point like this and it feels 566 00:40:01,560 --> 00:40:04,860 almost the silliest talking about underground cells and 567 00:40:04,860 --> 00:40:06,740 jail cells-- 568 00:40:06,740 --> 00:40:11,540 in a logical, rational thinking world, 569 00:40:11,540 --> 00:40:12,790 you would do this. 570 00:40:18,000 --> 00:40:20,340 This is not what one does-- 571 00:40:20,340 --> 00:40:22,410 for reasons that have to do with the 572 00:40:22,410 --> 00:40:24,760 perverse nature of crystals. 573 00:40:24,760 --> 00:40:31,410 Take b greater than a, one does that always, but you pick 574 00:40:31,410 --> 00:40:34,160 c as the shortest translation. 575 00:40:38,930 --> 00:40:40,180 Why? 576 00:40:41,970 --> 00:40:44,340 It has something to do with the 577 00:40:44,340 --> 00:40:47,260 peculiar behavior of crystals. 578 00:40:47,260 --> 00:40:51,630 If you were working with a crystal that had the shape of 579 00:40:51,630 --> 00:40:54,960 a needle, in other words, greatly elongated in one 580 00:40:54,960 --> 00:40:59,480 direction so that the crystal looks something like this. 581 00:40:59,480 --> 00:41:02,870 How would you draw a picture of that? 582 00:41:02,870 --> 00:41:05,910 Would you put it straight up and down like this or would 583 00:41:05,910 --> 00:41:07,160 you put it on its side. 584 00:41:10,690 --> 00:41:12,110 Takes up a lot of room this way. 585 00:41:12,110 --> 00:41:15,410 You'd almost certainly instinctively put it this way. 586 00:41:18,030 --> 00:41:20,250 Now if you didn't know anything about the 587 00:41:20,250 --> 00:41:23,250 translations down inside the innards of that crystal-- and 588 00:41:23,250 --> 00:41:26,890 this was the position that the people who did crystallography 589 00:41:26,890 --> 00:41:30,910 solely on the basis of crystal shape and morphology-- 590 00:41:30,910 --> 00:41:33,850 and had no idea if there was even a lattice inside of the 591 00:41:33,850 --> 00:41:36,850 crystal that caused this regular shape. 592 00:41:36,850 --> 00:41:46,180 When you draw a coordinate system x, y, z, 593 00:41:46,180 --> 00:41:49,620 z always goes up. 594 00:41:49,620 --> 00:41:54,160 Who draws a Cartesian coordinate system with z going 595 00:41:54,160 --> 00:41:58,530 this way, x going this way, and y going this way? 596 00:41:58,530 --> 00:41:59,160 It's obscene. 597 00:41:59,160 --> 00:42:02,460 We always put z going straight up and down. 598 00:42:02,460 --> 00:42:05,020 You do the same if you don't know what the lattice 599 00:42:05,020 --> 00:42:09,720 translations are with the labels on 600 00:42:09,720 --> 00:42:11,460 the axes of the crystal. 601 00:42:11,460 --> 00:42:16,550 So this is a, this is b, and this is c. 602 00:42:16,550 --> 00:42:22,190 And there were tons of pages published with drawings of 603 00:42:22,190 --> 00:42:25,820 crystal morphology on them. 604 00:42:25,820 --> 00:42:28,500 And then along came x-rays. 605 00:42:28,500 --> 00:42:32,415 And when they begin to determine lattice constants of 606 00:42:32,415 --> 00:42:36,520 a the material that looked like this, almost invariably 607 00:42:36,520 --> 00:42:39,370 they found the translations inside of a needle-like 608 00:42:39,370 --> 00:42:43,020 crystal corresponded to a unit cell in the 609 00:42:43,020 --> 00:42:44,270 shape of the pancake. 610 00:42:47,660 --> 00:42:52,150 Crystals, when they have a very, very short lattice 611 00:42:52,150 --> 00:42:55,740 vector, almost always grow most 612 00:42:55,740 --> 00:42:57,400 rapidly in that direction. 613 00:43:00,330 --> 00:43:03,400 So this is a terrible pickle. 614 00:43:03,400 --> 00:43:06,630 Here are reams of books that published descriptions of 615 00:43:06,630 --> 00:43:09,390 crystals that had a needle-like-- 616 00:43:09,390 --> 00:43:13,670 an acicular shape, if you want to use a fancy term, which had 617 00:43:13,670 --> 00:43:17,470 c as a direction which corresponded to the shortest 618 00:43:17,470 --> 00:43:18,620 translation in the crystal. 619 00:43:18,620 --> 00:43:19,760 So what do you do? 620 00:43:19,760 --> 00:43:22,420 You don't want to have to redraw all these figures and 621 00:43:22,420 --> 00:43:25,570 redefine all these directions in the crystal. 622 00:43:25,570 --> 00:43:26,935 So what you do is cop out. 623 00:43:26,935 --> 00:43:30,940 And you make c the shortest direction in the crystal. 624 00:43:34,640 --> 00:43:37,350 You think this field is rigorous but people wimp out 625 00:43:37,350 --> 00:43:40,150 and they do something that's contrary to the 626 00:43:40,150 --> 00:43:42,280 logical thing to do. 627 00:43:42,280 --> 00:43:45,030 Interesting question is why crystals should grow most 628 00:43:45,030 --> 00:43:49,290 rapidly in the direction of the shortest lattice vector. 629 00:43:49,290 --> 00:43:51,240 And I think I know the answer to that-- 630 00:43:51,240 --> 00:43:54,085 I've never seen anybody express it in the writing. 631 00:43:56,640 --> 00:44:02,720 The clue comes from some early work that looked at the growth 632 00:44:02,720 --> 00:44:05,600 of the crystal from solution. 633 00:44:05,600 --> 00:44:10,940 And this was done first with-- 634 00:44:10,940 --> 00:44:15,120 I think it was some sort of iodide. 635 00:44:15,120 --> 00:44:18,160 I can look it up-- something like cadmium iodide. 636 00:44:18,160 --> 00:44:24,300 And this material in solution grew very, very slowly in the 637 00:44:24,300 --> 00:44:25,900 form of hexagonal plates. 638 00:44:28,710 --> 00:44:33,280 And continued to grow most rapidly in this direction. 639 00:44:33,280 --> 00:44:36,830 This was the direction of the shortest lattice translation. 640 00:44:36,830 --> 00:44:39,170 Then all a sudden something happened. 641 00:44:39,170 --> 00:44:42,670 And suddenly the rate of growth and the nature of the 642 00:44:42,670 --> 00:44:44,430 morphology changed. 643 00:44:44,430 --> 00:44:47,360 The crystal took off like a bat in these directions and 644 00:44:47,360 --> 00:44:52,680 finally ended up in a needle-like fashion like this. 645 00:44:56,530 --> 00:45:02,220 This was an observation that led to the dislocation theory 646 00:45:02,220 --> 00:45:04,770 of crystal growth. 647 00:45:04,770 --> 00:45:09,790 The interpretation was that in the early stages of growth, 648 00:45:09,790 --> 00:45:13,500 the crystal is growing by accretion of atoms on to these 649 00:45:13,500 --> 00:45:15,190 planar surfaces. 650 00:45:15,190 --> 00:45:21,660 And the probability of an atom sticking is less high when the 651 00:45:21,660 --> 00:45:24,860 atom is on the flat surface than when an atom is on the 652 00:45:24,860 --> 00:45:28,440 surface that might have a little crevice in it so it can 653 00:45:28,440 --> 00:45:31,130 bind to two planes at once. 654 00:45:31,130 --> 00:45:33,410 So the original postulate-- 655 00:45:33,410 --> 00:45:35,350 and now we all take this for granted-- 656 00:45:35,350 --> 00:45:39,200 is that if there was a flaw such that the crystal 657 00:45:39,200 --> 00:45:44,080 developed a dislocation on this surface, and an exposed 658 00:45:44,080 --> 00:45:49,940 ledge, atoms would accrete and stick more tenaciously to this 659 00:45:49,940 --> 00:45:52,610 exposed ledge of the dislocation. 660 00:45:52,610 --> 00:45:54,990 And this is the so-called dislocation-- 661 00:45:54,990 --> 00:45:56,540 screw dislocation mechanism-- 662 00:45:56,540 --> 00:45:58,050 for crystal growth. 663 00:45:58,050 --> 00:46:02,170 And after the crystal has grown for a while, you can see 664 00:46:02,170 --> 00:46:06,790 spiral steps on the surface of the crystal that correspond to 665 00:46:06,790 --> 00:46:11,730 this dislocation winding around. 666 00:46:11,730 --> 00:46:16,650 OK so why should crystals grow most rapidly along the 667 00:46:16,650 --> 00:46:19,290 direction of the shortest translation? 668 00:46:19,290 --> 00:46:23,580 It's going to be a lot easier to create a screw dislocation 669 00:46:23,580 --> 00:46:28,320 with a Burgers vector for a translation that is short, 670 00:46:28,320 --> 00:46:31,250 rather than the translation that's 10 or 20 angstroms that 671 00:46:31,250 --> 00:46:33,230 the Burgers vector would have to involve-- 672 00:46:33,230 --> 00:46:34,960 many, many layers of atoms. 673 00:46:34,960 --> 00:46:37,370 If the translation is very small, just two or three 674 00:46:37,370 --> 00:46:40,250 atoms, it doesn't cost you much work to make a screw 675 00:46:40,250 --> 00:46:41,380 dislocation. 676 00:46:41,380 --> 00:46:45,090 So I think that is the reason why crystals with very short 677 00:46:45,090 --> 00:46:48,930 lattice translations grow most rapidly in that direction, 678 00:46:48,930 --> 00:46:52,180 because it's easier to make the screw dislocation with a 679 00:46:52,180 --> 00:46:54,220 Burgers vector that's parallel to that translation. 680 00:47:00,490 --> 00:47:04,560 So here's the first flagrant bit of logic that 681 00:47:04,560 --> 00:47:05,810 flies in our face. 682 00:47:08,430 --> 00:47:12,050 Let me give a few more definitions and then we'll 683 00:47:12,050 --> 00:47:15,280 return to this. 684 00:47:15,280 --> 00:47:20,020 If we have a crystal in which two translations are 685 00:47:20,020 --> 00:47:22,050 equivalent by symmetry-- 686 00:47:22,050 --> 00:47:26,540 and we would have such a situation if we picked a 687 00:47:26,540 --> 00:47:34,290 translation that was perpendicular to a square net. 688 00:47:34,290 --> 00:47:40,380 And would we call the base of the cell something defined by 689 00:47:40,380 --> 00:47:43,050 translations a and b? 690 00:47:43,050 --> 00:47:47,240 If this is a fourfold axis, what goes on in this direction 691 00:47:47,240 --> 00:47:51,510 is identical to what goes on in this direction by symmetry. 692 00:47:51,510 --> 00:47:54,500 And if these two directions are identical, why call one of 693 00:47:54,500 --> 00:47:56,890 them a and one of them b? 694 00:47:56,890 --> 00:48:01,480 So if two directions are equivalent, and this is true 695 00:48:01,480 --> 00:48:07,300 of the square net, what you do is call one a1 and 696 00:48:07,300 --> 00:48:08,550 you call one a2-- 697 00:48:11,095 --> 00:48:12,460 because they're the same thing. 698 00:48:12,460 --> 00:48:15,060 So let's call them both a. 699 00:48:15,060 --> 00:48:18,740 And the third direction then is c. 700 00:48:18,740 --> 00:48:23,740 And in such a situation here, c is always defined as the 701 00:48:23,740 --> 00:48:24,990 unique direction. 702 00:48:34,550 --> 00:48:37,600 What do I mean by a unique direction? 703 00:48:37,600 --> 00:48:42,590 This is the direction of high symmetry in crystals that are 704 00:48:42,590 --> 00:48:48,220 based on square bases and hexagonal bases. 705 00:48:56,160 --> 00:48:56,320 Yes sir? 706 00:48:56,320 --> 00:48:58,860 AUDIENCE: You pretty much contradict yourself if c is 707 00:48:58,860 --> 00:49:00,016 the longest [INAUDIBLE]. 708 00:49:00,016 --> 00:49:02,380 PROFESSOR: Not always, not always. 709 00:49:02,380 --> 00:49:03,134 AUDIENCE: Well, I mean in that picture. 710 00:49:03,134 --> 00:49:04,930 PROFESSOR: Well I drew it that way. 711 00:49:04,930 --> 00:49:05,430 You know why? 712 00:49:05,430 --> 00:49:08,170 Because if I draw it like this, the top of my cell 713 00:49:08,170 --> 00:49:11,480 doesn't get in the way of the bottom of my cell. 714 00:49:11,480 --> 00:49:17,810 And if I want to draw a small c translation, then I get into 715 00:49:17,810 --> 00:49:21,966 geometrical complications that doesn't look nearly as nice. 716 00:49:21,966 --> 00:49:26,124 AUDIENCE: Well, then I'm just saying two [INAUDIBLE]. 717 00:49:26,124 --> 00:49:27,260 PROFESSOR: No, no, no. 718 00:49:27,260 --> 00:49:28,570 It's different for different systems. 719 00:49:28,570 --> 00:49:31,590 This is what is true for a triclinic system. 720 00:49:31,590 --> 00:49:35,370 And in the case of a brick-shaped unit cell-- 721 00:49:35,370 --> 00:49:36,490 orthorhombic. 722 00:49:36,490 --> 00:49:39,830 So this is done for triclinic. 723 00:49:39,830 --> 00:49:41,080 It's also done for orthorhombic. 724 00:49:49,180 --> 00:49:51,170 Not for monoclinic. 725 00:49:51,170 --> 00:49:54,630 For monoclinic, there is a special direction because 726 00:49:54,630 --> 00:49:59,350 there is a direction of a twofold axis or maybe a 727 00:49:59,350 --> 00:50:01,960 twofold axis perpendicular to a mirror plane. 728 00:50:01,960 --> 00:50:04,310 In that case, if we were rational, we'd call that 1c. 729 00:50:04,310 --> 00:50:08,290 OK 730 00:50:08,290 --> 00:50:17,190 I'll close with an opinion poll. 731 00:50:17,190 --> 00:50:21,060 I will invariably, and I'm sure most people do, draw a 732 00:50:21,060 --> 00:50:25,760 crystal with a square base, with the c axis the longest. 733 00:50:25,760 --> 00:50:33,480 And if we have someone drawing a sketch of a cell in which 734 00:50:33,480 --> 00:50:37,160 the base of the cell is a hexagonal net, we'd again call 735 00:50:37,160 --> 00:50:39,240 one a1, one a2. 736 00:50:39,240 --> 00:50:41,670 And perpendicular to that would be the same direction. 737 00:50:41,670 --> 00:50:43,040 Say look, there I did it again. 738 00:50:43,040 --> 00:50:45,320 I made c longer than a1 and a2. 739 00:50:49,130 --> 00:50:49,700 All right. 740 00:50:49,700 --> 00:50:51,065 Let's look at hexagonal crystals. 741 00:50:53,680 --> 00:50:55,000 Let me ask for a poll. 742 00:50:55,000 --> 00:51:00,960 What percent of all crystals with threefold and sixfold 743 00:51:00,960 --> 00:51:04,540 axes do in fact have the length of c greater than the 744 00:51:04,540 --> 00:51:05,790 length of a? 745 00:51:07,800 --> 00:51:09,050 50/50? 746 00:51:11,930 --> 00:51:14,460 How many would say equally probable to have it the 747 00:51:14,460 --> 00:51:15,710 longest or the shortest? 748 00:51:18,260 --> 00:51:19,480 No opinion. 749 00:51:19,480 --> 00:51:22,950 Let me tell you the astounding fact. 750 00:51:22,950 --> 00:51:31,930 Seventy percent of all hexagonal crystals have c 751 00:51:31,930 --> 00:51:33,180 greater than a. 752 00:51:43,140 --> 00:51:44,390 Astounding. 753 00:51:47,530 --> 00:51:50,200 Let's look at the next-- 754 00:51:50,200 --> 00:51:54,210 haven't really defined these terms-- tetragonal crystals. 755 00:51:54,210 --> 00:51:58,900 These are crystals that have a fourfold axis in them. 756 00:51:58,900 --> 00:52:02,950 What percentage have c greater than a? 757 00:52:02,950 --> 00:52:06,360 This is almost a first cousin of the cubic crystal or all 758 00:52:06,360 --> 00:52:09,110 three translations are equal. 759 00:52:09,110 --> 00:52:14,110 How many would say more have c greater than a, fewer have c 760 00:52:14,110 --> 00:52:16,570 greater than a? 761 00:52:16,570 --> 00:52:17,980 One vote. 762 00:52:17,980 --> 00:52:19,780 More? 763 00:52:19,780 --> 00:52:23,640 That's a good answer, but it's only about 60 some percent. 764 00:52:27,130 --> 00:52:29,450 So let's go down to orthorhombic crystals. 765 00:52:35,220 --> 00:52:39,950 And orthorhombic crystals are the ones that have a 766 00:52:39,950 --> 00:52:43,380 brick-shaped unit cell. 767 00:52:43,380 --> 00:52:48,426 How many have c greater than a? 768 00:52:48,426 --> 00:52:50,340 It's declining. 769 00:52:50,340 --> 00:52:55,475 So if you extrapolate 50/50? 770 00:52:55,475 --> 00:52:56,357 AUDIENCE: Zero? 771 00:52:56,357 --> 00:52:58,500 PROFESSOR: Who said that? 772 00:52:58,500 --> 00:53:00,620 Are you a wise guy? 773 00:53:00,620 --> 00:53:03,880 You weren't supposed to give the right answer. 774 00:53:03,880 --> 00:53:05,690 It's 0 percent. 775 00:53:05,690 --> 00:53:09,200 It's 0 percent because there's nothing that makes one 776 00:53:09,200 --> 00:53:11,180 direction more special than any other. 777 00:53:11,180 --> 00:53:14,980 So you decide on the labelling of axes on the basis of their 778 00:53:14,980 --> 00:53:16,940 relative lengths. 779 00:53:16,940 --> 00:53:22,110 And you, by definition, make c the smallest axis. 780 00:53:22,110 --> 00:53:22,930 Congratulations. 781 00:53:22,930 --> 00:53:26,230 To the rest of you, gotcha. 782 00:53:26,230 --> 00:53:28,400 So that's a good note on which to quit. 783 00:53:28,400 --> 00:53:32,930 And a few remaining mysteries of coordinate systems and 784 00:53:32,930 --> 00:53:36,410 lattices will be expounded in the 785 00:53:36,410 --> 00:53:37,660 second half of our lecture.