1 00:00:10,143 --> 00:00:15,380 PROFESSOR: Since z, that's straight up. 2 00:00:15,380 --> 00:00:23,150 And that is going to provide for you a cell that has the 3 00:00:23,150 --> 00:00:25,380 shape of a square prism. 4 00:00:25,380 --> 00:00:26,960 This would be a1. 5 00:00:26,960 --> 00:00:28,360 This would be a2. 6 00:00:28,360 --> 00:00:30,330 And this would be a3-- 7 00:00:30,330 --> 00:00:33,960 I'm sorry, this would be c. 8 00:00:33,960 --> 00:00:38,820 And if you took the choice for the third translation as 1/2 9 00:00:38,820 --> 00:00:48,050 half of a1 plus 1/2 of a2 plus some amount z up above the 10 00:00:48,050 --> 00:00:53,340 base of the cell, and then redefined a T3 prime that 11 00:00:53,340 --> 00:00:54,590 would be equal to-- 12 00:00:57,936 --> 00:00:58,390 sorry. 13 00:00:58,390 --> 00:00:59,640 What am I doing here? 14 00:01:03,324 --> 00:01:03,830 Yeah. 15 00:01:03,830 --> 00:01:04,300 This is right. 16 00:01:04,300 --> 00:01:07,760 1/2 of a1 1/2 of a2, and z straight up. 17 00:01:07,760 --> 00:01:18,200 Define a T3 prime as 2T3 minus a1 minus a2. 18 00:01:18,200 --> 00:01:22,510 And that will define for you, again, a cell in the shape of 19 00:01:22,510 --> 00:01:28,420 a square prism with a1 and a2. 20 00:01:28,420 --> 00:01:31,620 And now the translation that went up directly over the 21 00:01:31,620 --> 00:01:34,940 center of the base of the net below. 22 00:01:34,940 --> 00:01:41,890 Twice that minus a2 minus a1 brings you back to a third 23 00:01:41,890 --> 00:01:45,740 translation T3 prime that's normal to the base. 24 00:01:45,740 --> 00:01:47,890 So both cells have the same shape. 25 00:01:47,890 --> 00:01:49,790 This is a primitive. 26 00:01:49,790 --> 00:01:53,180 This is a body-centered tetragonal lattice. 27 00:01:53,180 --> 00:01:56,540 And the symbol that's used to represent the body-centered 28 00:01:56,540 --> 00:02:00,620 lattice is I. One of the few cases, again, where one has to 29 00:02:00,620 --> 00:02:02,150 be bilingual. 30 00:02:02,150 --> 00:02:08,860 The German word for body-centered is 31 00:02:08,860 --> 00:02:10,110 innenzentriert. 32 00:02:12,890 --> 00:02:15,620 So there's Schoenflies at work again. 33 00:02:15,620 --> 00:02:20,600 So primitive body-centered represented by I. 34 00:02:20,600 --> 00:02:25,040 And that brings us to one final case, and that is the 35 00:02:25,040 --> 00:02:28,933 plane groups that have a hexagonal shape. 36 00:02:34,310 --> 00:02:42,690 And here we have a funny situation in that the same 37 00:02:42,690 --> 00:02:47,070 choices of the third translation do not work for 38 00:02:47,070 --> 00:02:51,075 all of the plane groups. 39 00:02:53,990 --> 00:02:58,610 If you have a base to the cell-- 40 00:03:02,260 --> 00:03:04,690 and again, the two edges of that net are 41 00:03:04,690 --> 00:03:06,340 identical by symmetry-- 42 00:03:06,340 --> 00:03:12,820 if it has a three-fold axis in it and nothing else, than 43 00:03:12,820 --> 00:03:17,020 there are two choices. 44 00:03:17,020 --> 00:03:27,520 You can either have T3 be equal to 0a1 plus 0a2 and an 45 00:03:27,520 --> 00:03:28,860 amount z straight up. 46 00:03:31,480 --> 00:03:46,130 And that will give you a cell that is a primitive cell. 47 00:03:46,130 --> 00:03:47,650 This would be a1. 48 00:03:47,650 --> 00:03:49,570 This would be a2. 49 00:03:49,570 --> 00:03:52,040 And just as we did for tetragonal, we'd call the 50 00:03:52,040 --> 00:03:59,080 third axis c, This will work for a three-fold axis. 51 00:03:59,080 --> 00:04:03,630 So there is a space group P3. 52 00:04:03,630 --> 00:04:07,970 This will also work for all of the other groups. 53 00:04:07,970 --> 00:04:12,280 So there's P3M1 and a P31M. 54 00:04:15,820 --> 00:04:18,490 And a P6 will work. 55 00:04:18,490 --> 00:04:23,530 And a P6mm will work. 56 00:04:23,530 --> 00:04:24,340 So look it here. 57 00:04:24,340 --> 00:04:28,490 We've got five additional space groups in addition to 58 00:04:28,490 --> 00:04:30,200 the lattice type. 59 00:04:30,200 --> 00:04:34,430 This is the primitive hexagonal lattice. 60 00:04:34,430 --> 00:04:38,930 For a three-fold axis, we have another choice. 61 00:04:38,930 --> 00:04:42,420 We could pick T3 so that it took the origin three-fold 62 00:04:42,420 --> 00:04:45,200 axis and put it directly over this one. 63 00:04:45,200 --> 00:04:50,060 Looks as though you could have a different distinct lattice 64 00:04:50,060 --> 00:04:54,230 if you move the origin lattice point over the three-fold axis 65 00:04:54,230 --> 00:04:57,450 that sits at a location 1/3, 2/3. 66 00:04:57,450 --> 00:04:59,680 This one is at 2/3, 1/3. 67 00:05:02,330 --> 00:05:05,930 Let me demonstrate, I think convincingly and clearly, that 68 00:05:05,930 --> 00:05:09,600 those two choices are exactly the same thing. 69 00:05:09,600 --> 00:05:13,380 And the way that I can show that is, once again, to draw 70 00:05:13,380 --> 00:05:21,180 this net a1, a2. 71 00:05:21,180 --> 00:05:27,730 And suppose I pick the part of the offset of T3 that is 72 00:05:27,730 --> 00:05:32,190 within the plane of the net as going from this lattice point 73 00:05:32,190 --> 00:05:35,364 to this one here. 74 00:05:35,364 --> 00:05:40,410 Then if I go twice that translation T3, that's gonna 75 00:05:40,410 --> 00:05:43,270 put me directly over this three-fold axis. 76 00:05:43,270 --> 00:05:48,460 And if I go three translations T3, I'll be directly over the 77 00:05:48,460 --> 00:05:51,430 lattice point on the ground floor. 78 00:05:51,430 --> 00:05:55,030 So putting the original lattice point over this 79 00:05:55,030 --> 00:05:58,760 three-fold axis or this three-fold axis is exactly the 80 00:05:58,760 --> 00:05:59,650 same thing. 81 00:05:59,650 --> 00:06:02,500 And I can change one into the other just by flipping this 82 00:06:02,500 --> 00:06:04,120 thing 180 degrees. 83 00:06:04,120 --> 00:06:07,090 So it's obviously the same choice. 84 00:06:07,090 --> 00:06:13,760 This description of T3, if I define the direction of c now 85 00:06:13,760 --> 00:06:25,040 as 3T3 defined in this fashion, minus a1 minus a2, is 86 00:06:25,040 --> 00:06:28,860 gonna be a funny situation. 87 00:06:28,860 --> 00:06:35,220 It's gonna be a peculiar sort of double body-centered cell. 88 00:06:35,220 --> 00:06:38,230 Of course, along the long diagonal of the hexagonal 89 00:06:38,230 --> 00:06:41,810 cell, I'll have one lattice point that's 1/3 of the way 90 00:06:41,810 --> 00:06:45,510 along that long diagonal, and another interior lattice point 91 00:06:45,510 --> 00:06:48,405 that's 2/3 of the way along that long diagonal. 92 00:06:53,090 --> 00:06:54,340 OK. 93 00:06:59,350 --> 00:07:06,680 This double body-centered cell, so to speak, actually 94 00:07:06,680 --> 00:07:10,860 can be redefined in terms of a primitive cell, not a 95 00:07:10,860 --> 00:07:15,900 primitive cell that is this particular description of a 96 00:07:15,900 --> 00:07:17,170 primitive hexagonal lattice. 97 00:07:17,170 --> 00:07:19,420 This is not a primitive hexagonal lattice. 98 00:07:19,420 --> 00:07:25,060 But suppose I go from this interior lattice point along 99 00:07:25,060 --> 00:07:28,370 the long diagonal of the cell to this location 100 00:07:28,370 --> 00:07:30,660 and call that a1. 101 00:07:30,660 --> 00:07:34,590 Remember that there is a three-fold axis sneaking down 102 00:07:34,590 --> 00:07:37,410 through the center of this triangle, goes through the 103 00:07:37,410 --> 00:07:39,190 lattice point, and comes out the center of 104 00:07:39,190 --> 00:07:40,440 the triangle below. 105 00:07:43,040 --> 00:07:48,070 If I go up to this lattice point and up to this lattice 106 00:07:48,070 --> 00:07:52,440 point, I have three translations that are 107 00:07:52,440 --> 00:07:55,270 straddling the three-fold axis. 108 00:07:55,270 --> 00:07:59,880 And that three-fold axis rotates one into the other. 109 00:07:59,880 --> 00:08:03,150 So this new definition of a cell that is in fact 110 00:08:03,150 --> 00:08:05,050 primitive a1, a2-- 111 00:08:05,050 --> 00:08:06,445 and they're all equivalent by symmetry. 112 00:08:11,780 --> 00:08:15,620 I always get in trouble when I try to draw it, but that 113 00:08:15,620 --> 00:08:18,070 actually defines a cell with a shape that 114 00:08:18,070 --> 00:08:19,580 we've not seen before. 115 00:08:19,580 --> 00:08:23,420 That is a cell that has the shape of a rhombohedron. 116 00:08:23,420 --> 00:08:31,200 And as I say, I always get in trouble when I try to draw it. 117 00:08:31,200 --> 00:08:33,520 It's got three translations like this and three 118 00:08:33,520 --> 00:08:37,380 translations skewed by 60 degree sitting on top. 119 00:08:37,380 --> 00:08:39,720 Everybody can see that's a rhombohedron, can't you? 120 00:08:39,720 --> 00:08:40,370 Say yes. 121 00:08:40,370 --> 00:08:42,530 Be nice. 122 00:08:42,530 --> 00:08:45,890 So this is another primitive cell that's a choice for this 123 00:08:45,890 --> 00:08:47,160 triple cell. 124 00:08:47,160 --> 00:08:51,060 And this consequently gives its name to this lattice. 125 00:08:51,060 --> 00:08:55,680 This is called a rhombohedral lattice, regardless of whether 126 00:08:55,680 --> 00:08:58,220 you pick the primitive cell in the shape of the rhombohedron 127 00:08:58,220 --> 00:09:01,110 or this peculiar double body-centered cell. 128 00:09:01,110 --> 00:09:03,230 So this is a rhombohedral lattice. 129 00:09:05,970 --> 00:09:09,350 And this is represented in a space group symbol by the 130 00:09:09,350 --> 00:09:10,950 letter R, standing for rhombohedral. 131 00:09:13,470 --> 00:09:15,870 So just the three-fold axis with this choice of 132 00:09:15,870 --> 00:09:18,200 translations, that would be space group R3. 133 00:09:22,132 --> 00:09:23,382 AUDIENCE: [INAUDIBLE]? 134 00:09:26,100 --> 00:09:26,890 PROFESSOR: Yeah. 135 00:09:26,890 --> 00:09:30,360 a1 like this. a2 like this. a3, I'm taking a right-handed 136 00:09:30,360 --> 00:09:32,150 system, comes from the centered lattice point. 137 00:09:32,150 --> 00:09:33,950 So let me draw it in projection. 138 00:09:33,950 --> 00:09:35,770 It's easier to see. 139 00:09:35,770 --> 00:09:39,320 Here's the outline of the hexagonal net. 140 00:09:39,320 --> 00:09:41,410 Here's the three-fold axis. 141 00:09:41,410 --> 00:09:44,366 And I've taken a1 coming up like this. 142 00:09:47,150 --> 00:09:52,130 I've taken a2 coming up out of the board like this. 143 00:09:52,130 --> 00:09:56,402 And I've taken a3 coming out like this. 144 00:09:56,402 --> 00:09:56,770 OK. 145 00:09:56,770 --> 00:09:58,850 So these are the three edges of the rhombohedron. 146 00:10:03,750 --> 00:10:09,830 Notice that I cannot have that lattice with a lot of the 147 00:10:09,830 --> 00:10:14,190 hexagonal plane groups. 148 00:10:14,190 --> 00:10:17,020 I cannot do it for six-fold axis. 149 00:10:17,020 --> 00:10:19,280 Because a six-fold axis-- 150 00:10:19,280 --> 00:10:23,630 P6 and P6MM have a six-fold axis here and three-fold axes 151 00:10:23,630 --> 00:10:25,280 in the middle of the cell. 152 00:10:25,280 --> 00:10:28,870 So I cannot have a rhombohedral lattice for P6 or 153 00:10:28,870 --> 00:10:31,800 P6 00:10:34,280 only thing that works. 155 00:10:34,280 --> 00:10:37,430 And then we have this curious situation with the two 156 00:10:37,430 --> 00:10:43,100 alternative settings of point group 3M in a hexagonal net, 157 00:10:43,100 --> 00:10:45,530 P3M1 and P31M. 158 00:10:48,565 --> 00:10:58,050 For P3M1, I have symmetry 3M at both the origin lattice 159 00:10:58,050 --> 00:11:01,600 point and in the center of these two triangles. 160 00:11:01,600 --> 00:11:10,430 So for P3M1, in addition to a three-fold axis, I could put 161 00:11:10,430 --> 00:11:17,560 in 3M1, stack up plane group P3M1. 162 00:11:17,560 --> 00:11:23,810 But I cannot do this for 31M. 163 00:11:23,810 --> 00:11:25,310 That's impossible. 164 00:11:25,310 --> 00:11:30,190 Because if you look at P31M, there is symmetry 3M at the 165 00:11:30,190 --> 00:11:35,360 origin and only symmetry 3 in the center of the triangles. 166 00:11:35,360 --> 00:11:39,650 So if I try to pick this as a third translation for plane 167 00:11:39,650 --> 00:11:44,670 group P31M, I'm taking symmetry 3M and plopping it 168 00:11:44,670 --> 00:11:48,790 down on top of symmetry 3, and that wrecks the plane group. 169 00:11:48,790 --> 00:11:50,630 So for the rhombohedral lattice, I can 170 00:11:50,630 --> 00:11:53,520 only have 3M1 or P3. 171 00:11:56,290 --> 00:11:59,750 P6 has a six-fold axis at the origin lattice point. 172 00:11:59,750 --> 00:12:03,370 The two centers of the triangles are three-fold axes. 173 00:12:03,370 --> 00:12:09,020 So primitive for the six-fold axis, either P6 or P6MM, is 174 00:12:09,020 --> 00:12:10,270 the only choice. 175 00:12:24,130 --> 00:12:26,470 So if anybody's counting-- and I wasn't-- 176 00:12:26,470 --> 00:12:35,320 we've got 11 lattices so far. 177 00:12:35,320 --> 00:12:39,020 And the only ones that we haven't been able to get by 178 00:12:39,020 --> 00:12:45,990 simple stacking of the plane groups are the systems that 179 00:12:45,990 --> 00:12:49,640 you all know as your favorites. 180 00:12:49,640 --> 00:12:55,720 The cubic or isometric stems for the fact that the 181 00:12:55,720 --> 00:12:58,390 translations are identical in all three directions. 182 00:13:02,650 --> 00:13:12,660 Let me quickly dispose of those by starting with a 183 00:13:12,660 --> 00:13:21,010 four-fold axis coming out of one face of a tetragonal cell 184 00:13:21,010 --> 00:13:25,040 and saying now if this is a cubic symmetry, I have a 185 00:13:25,040 --> 00:13:28,540 three-fold access that comes out, sort of, in the direction 186 00:13:28,540 --> 00:13:31,450 of a body diagonal. 187 00:13:31,450 --> 00:13:34,690 But this three-fold axis is gonna rotate that four-fold 188 00:13:34,690 --> 00:13:40,220 access to this direction and to this direction. 189 00:13:40,220 --> 00:13:45,660 We've seen that if I look down along the diagonal of a 190 00:13:45,660 --> 00:13:51,710 equally axed tetragonal prism, the three-fold axis takes this 191 00:13:51,710 --> 00:13:56,480 face, rotates it into this face, and rotates 192 00:13:56,480 --> 00:13:58,360 it into this face. 193 00:13:58,360 --> 00:14:01,470 So I have to have a four-fold axis coming out of every one 194 00:14:01,470 --> 00:14:02,520 of the faces. 195 00:14:02,520 --> 00:14:06,200 And having that arrangement of four-fold axes means that not 196 00:14:06,200 --> 00:14:11,400 only will the two translations be identical in the base of 197 00:14:11,400 --> 00:14:14,750 the cell, as is the case for tetragonal, but if I put in 198 00:14:14,750 --> 00:14:18,230 that three-fold axis as an extender, all three edges of 199 00:14:18,230 --> 00:14:25,070 the cell have to be identical by symmetry. 200 00:14:25,070 --> 00:14:29,550 So I can do this for a primitive tetragonal lattice. 201 00:14:29,550 --> 00:14:32,480 And that's gonna give me a primitive isometric lattice. 202 00:14:36,050 --> 00:14:37,320 I can do this for a 203 00:14:37,320 --> 00:14:40,920 body-centered tetragonal lattice. 204 00:14:40,920 --> 00:14:44,870 So that would be indicated by the symbol I. So it's a 205 00:14:44,870 --> 00:14:46,815 body-centered cubic lattice. 206 00:14:46,815 --> 00:14:51,030 And if you add up everything we've done to this point, 207 00:14:51,030 --> 00:14:59,400 there are 13 space lattices, a very unlucky number. 208 00:14:59,400 --> 00:15:01,310 And as you all know, there are 14. 209 00:15:01,310 --> 00:15:02,950 We've missed one. 210 00:15:02,950 --> 00:15:04,490 So what have we not done? 211 00:15:04,490 --> 00:15:07,480 We stacked up all the plane groups in all possible ways. 212 00:15:10,570 --> 00:15:17,220 We've taken the four-fold axes out of the face normals, all 213 00:15:17,220 --> 00:15:19,680 three face normals of a tetragonal cell. 214 00:15:19,680 --> 00:15:23,670 And that forced it to become a cubic lattice. 215 00:15:23,670 --> 00:15:25,219 What have we done wrong? 216 00:15:30,710 --> 00:15:39,680 Actually, we can get the fourteenth space lattice by a 217 00:15:39,680 --> 00:15:42,680 little bit of sleight of hand that is not obvious. 218 00:15:42,680 --> 00:15:46,190 Let's say that this is the base of a tetragonal lattice 219 00:15:46,190 --> 00:15:52,910 that we used as the parent for our cubic lattice. 220 00:15:52,910 --> 00:15:57,790 Let me change that primitive square in the base of the 221 00:15:57,790 --> 00:16:04,330 tetragonal cell and change it into a centered square. 222 00:16:04,330 --> 00:16:07,670 There'd be no reason for doing that for a plane group. 223 00:16:07,670 --> 00:16:11,260 But let me notice now that if this is the direction of a 224 00:16:11,260 --> 00:16:16,620 four-fold axis, rather than having the three-fold axis 225 00:16:16,620 --> 00:16:20,830 come out of this direction relative to this super double 226 00:16:20,830 --> 00:16:25,120 square, I can make the three-fold axis come out in 227 00:16:25,120 --> 00:16:26,370 this direction. 228 00:16:28,265 --> 00:16:29,660 Ah-ha. 229 00:16:29,660 --> 00:16:34,740 So what I have now is a cell that is centered, has a center 230 00:16:34,740 --> 00:16:40,030 in the base and has translations that are 231 00:16:40,030 --> 00:16:42,155 perpendicular to the base like this. 232 00:16:42,155 --> 00:16:46,040 And if this is the direction of the four-fold axis and this 233 00:16:46,040 --> 00:16:48,780 is the direction of the three-fold axis, that 234 00:16:48,780 --> 00:16:52,950 three-fold axis is going to put a centered lattice point 235 00:16:52,950 --> 00:16:59,480 on the remaining faces of the cube. 236 00:16:59,480 --> 00:17:04,810 So it's only if I notice I can make a double square base to 237 00:17:04,810 --> 00:17:08,819 the cell again, put the three-fold axis of a cubic 238 00:17:08,819 --> 00:17:12,819 symmetry along the direction of the body diagonal, I 239 00:17:12,819 --> 00:17:16,470 generate lattice points in the middle of the other faces as 240 00:17:16,470 --> 00:17:19,579 well, and this gives me the lattice known as F, the 241 00:17:19,579 --> 00:17:21,090 face-centered cubic lattice. 242 00:17:29,830 --> 00:17:34,880 And the edges of the cell here are all identical, equivalent 243 00:17:34,880 --> 00:17:35,510 by symmetry. 244 00:17:35,510 --> 00:17:37,720 So they're labeled a1, a2, a3. 245 00:17:42,660 --> 00:17:45,900 Have you ever seen a lattice constant for a 246 00:17:45,900 --> 00:17:49,160 cubic crystal a0? 247 00:17:53,010 --> 00:17:56,270 Has anybody ever seen that? 248 00:17:56,270 --> 00:17:58,440 A lot of people do it. 249 00:17:58,440 --> 00:18:03,110 And if you see that notation in a publication, you will be 250 00:18:03,110 --> 00:18:05,016 entitled to sneer at it knowingly. 251 00:18:07,610 --> 00:18:09,780 It's logical, three, two, one, zero. 252 00:18:09,780 --> 00:18:12,710 By extrapolation, if you've got an a3, an a2, a1, you 253 00:18:12,710 --> 00:18:14,630 gotta have an a0, right? 254 00:18:14,630 --> 00:18:16,730 Well, not necessarily. 255 00:18:16,730 --> 00:18:18,810 That sort of algebra doesn't apply to 256 00:18:18,810 --> 00:18:22,280 labeling of cell edges. 257 00:18:22,280 --> 00:18:30,900 It comes from a curious historic precedent. 258 00:18:30,900 --> 00:18:38,660 Come back, again, to the fact that for literally more than 259 00:18:38,660 --> 00:18:43,240 100 years, people developed symmetry theory and had a 260 00:18:43,240 --> 00:18:48,150 pretty good idea that crystals were based on packing of 261 00:18:48,150 --> 00:18:51,520 units, molecules, whatever they were, that were on the 262 00:18:51,520 --> 00:18:53,490 nodes of a lattice. 263 00:18:53,490 --> 00:18:56,300 But they couldn't measure the size and shape of the lattice. 264 00:18:56,300 --> 00:19:01,070 And that was the brilliance of the famous experiment by Max 265 00:19:01,070 --> 00:19:07,740 von Laue because that, in one fell swoop, show that the 266 00:19:07,740 --> 00:19:11,780 mysterious radiation x-rays were just electromagnetic 267 00:19:11,780 --> 00:19:14,120 radiation of a very short wavelength. 268 00:19:14,120 --> 00:19:18,670 And that's what he really gained notoriety for. 269 00:19:18,670 --> 00:19:21,470 But at the same time, he proved for the first time, 270 00:19:21,470 --> 00:19:25,940 unequivocally, that crystals were based on a lattice. 271 00:19:25,940 --> 00:19:29,230 And that was the other side of his double-edged 272 00:19:29,230 --> 00:19:30,480 accomplishment. 273 00:19:33,570 --> 00:19:36,660 Up until that time, people learned about crystals by 274 00:19:36,660 --> 00:19:39,950 studying the morphology and measuring the angles between 275 00:19:39,950 --> 00:19:47,130 crystal faces and recording different faces on a crystal. 276 00:19:47,130 --> 00:19:49,500 I mean, I can just see one of these old guys working a 277 00:19:49,500 --> 00:19:53,060 little bit before his wife made him come down and sit for 278 00:19:53,060 --> 00:19:54,880 supper because it was getting cold. 279 00:19:54,880 --> 00:19:57,900 And he'd let out a, whoop, huh? 280 00:19:57,900 --> 00:20:00,015 A 13 27 face. 281 00:20:00,015 --> 00:20:00,800 What the hey. 282 00:20:00,800 --> 00:20:04,950 It's a new face on potassium tartrate, something like that. 283 00:20:04,950 --> 00:20:07,560 This was the way these guys got their jollies. 284 00:20:07,560 --> 00:20:10,380 And on the basis of the angles between faces, as I said on 285 00:20:10,380 --> 00:20:15,980 several occasions, they could determine a ratio of axes, a 286 00:20:15,980 --> 00:20:22,340 ratio of a to b to c, if their assignment of indices to faces 287 00:20:22,340 --> 00:20:23,830 were correct. 288 00:20:23,830 --> 00:20:25,565 And how did you know? 289 00:20:25,565 --> 00:20:28,840 Well, along came x-rays, a and you could measure for sure. 290 00:20:28,840 --> 00:20:34,620 So what people started doing to distinguish a to b to c 291 00:20:34,620 --> 00:20:43,790 ratios determined from crystal morphology, they put the zeros 292 00:20:43,790 --> 00:20:53,750 on it to indicate real values from diffraction. 293 00:21:01,420 --> 00:21:03,400 And when little people came afterwards and they didn't 294 00:21:03,400 --> 00:21:06,830 realize the reason for doing that-- nobody's done this for 295 00:21:06,830 --> 00:21:09,510 over 100 years because everybody determines lattice 296 00:21:09,510 --> 00:21:10,910 constants using diffraction-- 297 00:21:10,910 --> 00:21:12,810 but some people have seen these zeros. 298 00:21:12,810 --> 00:21:15,120 So they say, I'm not doing proper notation. 299 00:21:15,120 --> 00:21:17,720 And you know how furious crystallographers get if you 300 00:21:17,720 --> 00:21:19,380 don't use proper notation. 301 00:21:19,380 --> 00:21:21,760 So let me write a0 as the lattice 302 00:21:21,760 --> 00:21:23,690 constant of my cubic crystal. 303 00:21:23,690 --> 00:21:25,200 You see this commonly. 304 00:21:25,200 --> 00:21:28,580 And there's been no need for it for 100 years. 305 00:21:28,580 --> 00:21:31,662 So anyway, that's the origin of the subscript zero. 306 00:21:31,662 --> 00:21:35,080 There are a lot of dumb things that take place among normally 307 00:21:35,080 --> 00:21:36,330 rational people. 308 00:21:39,910 --> 00:21:42,230 I said a moment ago-- and I crossed my 309 00:21:42,230 --> 00:21:43,650 fingers when I said it-- 310 00:21:43,650 --> 00:21:52,460 that c is the label that is assigned to the unique axis, 311 00:21:52,460 --> 00:21:55,120 the axis that is unique because of symmetry. 312 00:22:02,490 --> 00:22:09,530 Sometimes you see monoclinic crystals, which have a 313 00:22:09,530 --> 00:22:13,780 two-fold axis and/or a mirror plane perpendicular to the 314 00:22:13,780 --> 00:22:24,770 two-fold axis, labelled with this as b, this is a, and this 315 00:22:24,770 --> 00:22:28,460 as c, and the angle between them-- that's the general 316 00:22:28,460 --> 00:22:29,680 obtuse angle-- 317 00:22:29,680 --> 00:22:30,930 as beta. 318 00:22:34,910 --> 00:22:39,060 There is a story behind this lapse of 319 00:22:39,060 --> 00:22:43,250 standardness in notation. 320 00:22:43,250 --> 00:22:48,660 How do people decide on what proper notation is? 321 00:22:48,660 --> 00:22:52,510 Well, there is an organization called the International Union 322 00:22:52,510 --> 00:22:55,630 of Crystallography, which is the equivalent of the 323 00:22:55,630 --> 00:22:58,510 International Union of Pure and Applied Physics and the 324 00:22:58,510 --> 00:23:02,050 International Union of Pure and Applied Chemistry. 325 00:23:02,050 --> 00:23:04,440 And they meet every three years. 326 00:23:04,440 --> 00:23:06,950 And there are special commissions for people who 327 00:23:06,950 --> 00:23:11,730 like to fight and argue out conventions and nomenclature. 328 00:23:11,730 --> 00:23:15,100 And this is such a silly abomination that there was a 329 00:23:15,100 --> 00:23:17,920 move made when they came out with their new edition of 330 00:23:17,920 --> 00:23:22,330 international tables to change this and do it the way you do 331 00:23:22,330 --> 00:23:27,410 for all of the other crystal systems, namely a, b with 332 00:23:27,410 --> 00:23:30,850 gamma between them, and c coming out this way as the 333 00:23:30,850 --> 00:23:32,340 direction of the two-fold axis. 334 00:23:32,340 --> 00:23:33,775 And this would be the mirror plane. 335 00:23:33,775 --> 00:23:36,510 And that would put this in accord with tetragonal and 336 00:23:36,510 --> 00:23:37,920 hexagonal and everything else. 337 00:23:37,920 --> 00:23:41,570 And it was logical and seemed the right thing to do. 338 00:23:41,570 --> 00:23:45,040 So how do people decide on nomenclature? 339 00:23:45,040 --> 00:23:48,650 This is something that people fight with with more passion 340 00:23:48,650 --> 00:23:50,810 than anything else in science. 341 00:23:50,810 --> 00:23:52,310 So you make a proposal. 342 00:23:52,310 --> 00:23:55,100 The proposal goes to the national committees of 343 00:23:55,100 --> 00:23:57,720 crystallography in countries over the world. 344 00:23:57,720 --> 00:23:58,620 They discuss it. 345 00:23:58,620 --> 00:23:59,600 They fight about it. 346 00:23:59,600 --> 00:24:00,650 They vote on it. 347 00:24:00,650 --> 00:24:03,560 Then it comes back to the International Union. 348 00:24:03,560 --> 00:24:06,760 And finally, if everybody seems agreed, they present it 349 00:24:06,760 --> 00:24:09,330 at the next meeting of the International Union Of 350 00:24:09,330 --> 00:24:11,530 crystallography for a vote. 351 00:24:11,530 --> 00:24:14,830 And that's exactly what happened with the proposal to 352 00:24:14,830 --> 00:24:16,370 standardized notation. 353 00:24:16,370 --> 00:24:22,400 This took place maybe, I don't know, in the 1950s. 354 00:24:22,400 --> 00:24:24,380 Everybody was in favor. 355 00:24:24,380 --> 00:24:27,740 The discussion proceeded rationally. 356 00:24:27,740 --> 00:24:31,850 Then this little guy jumps up in the front row, short of 357 00:24:31,850 --> 00:24:38,140 stature, balding of head, with a very piercing nasal voice. 358 00:24:38,140 --> 00:24:41,710 And he gets up and he enters into this diatribe. 359 00:24:41,710 --> 00:24:47,310 b-axis unique was good enough for Curie. 360 00:24:47,310 --> 00:24:51,930 b-axis unique was good enough for Mauguin. 361 00:24:51,930 --> 00:24:55,080 B-axis unique was-- and all the names were French. 362 00:24:55,080 --> 00:24:55,960 Actually, he wasn't French. 363 00:24:55,960 --> 00:24:56,535 He was Belgian. 364 00:24:56,535 --> 00:24:58,810 So he went on and on and on. 365 00:24:58,810 --> 00:25:05,790 And he so cowed this group of 300 rational delegates, that 366 00:25:05,790 --> 00:25:09,300 they didn't cave in entirely. 367 00:25:09,300 --> 00:25:14,160 What they did was they entered every entry for monoclinic 368 00:25:14,160 --> 00:25:20,060 crystals in the international tables twice, once with this 369 00:25:20,060 --> 00:25:23,390 set of labels, once with this set of labels. 370 00:25:23,390 --> 00:25:26,440 And to indicate that this was the preferred notation, this 371 00:25:26,440 --> 00:25:29,720 is labeled the first setting. 372 00:25:29,720 --> 00:25:33,590 And to indicate second-class citizenship no matter what 373 00:25:33,590 --> 00:25:38,500 Curie, Mauguin, Bravais, and all of these other guys felt, 374 00:25:38,500 --> 00:25:42,090 this was called the second setting. 375 00:25:42,090 --> 00:25:43,090 And I'm not kidding. 376 00:25:43,090 --> 00:25:46,390 I'll bring in a copy of the international tables for x-ray 377 00:25:46,390 --> 00:25:47,560 crystallography. 378 00:25:47,560 --> 00:25:55,034 Every entry for a monoclinic crystals is in there twice. 379 00:25:55,034 --> 00:25:57,364 AUDIENCE: And no one in the past 50 years has tried to get 380 00:25:57,364 --> 00:25:58,770 rid of the second set? 381 00:25:58,770 --> 00:25:59,010 PROFESSOR: No. 382 00:25:59,010 --> 00:26:01,530 Actually, the reason is it is expensive to 383 00:26:01,530 --> 00:26:03,590 revise these tables. 384 00:26:03,590 --> 00:26:07,000 And I brought in on the first day, you don't remember, my 385 00:26:07,000 --> 00:26:10,340 copy of international tables for x-ray crystallography, 386 00:26:10,340 --> 00:26:12,730 which is about this big and cost $100. 387 00:26:12,730 --> 00:26:17,440 And then the new version which is twice as large 388 00:26:17,440 --> 00:26:23,500 dimensionally, four times as heavy in terms of weight, and 389 00:26:23,500 --> 00:26:25,760 I think about eight times as expensive in 390 00:26:25,760 --> 00:26:27,220 terms of actual cost. 391 00:26:27,220 --> 00:26:29,900 So you don't do this casually. 392 00:26:29,900 --> 00:26:31,880 If there's something actually new comes out, you make a 393 00:26:31,880 --> 00:26:33,590 little supplement that people can stick 394 00:26:33,590 --> 00:26:35,965 in the pages somewhere. 395 00:26:35,965 --> 00:26:36,935 AUDIENCE: Don't they still have 396 00:26:36,935 --> 00:26:38,390 PDFs nowadays or something? 397 00:26:41,810 --> 00:26:45,350 PROFESSOR: The last volume came out before PDF, before 398 00:26:45,350 --> 00:26:50,690 very intensive use of computers for such data. 399 00:26:50,690 --> 00:26:52,540 The journals now are all done online. 400 00:26:52,540 --> 00:26:56,300 You submit papers online and they're archived online as 401 00:26:56,300 --> 00:26:58,270 well in addition to hard copy. 402 00:26:58,270 --> 00:27:02,490 But that's something that was before the time of the 403 00:27:02,490 --> 00:27:04,105 publication of the last set of volumes. 404 00:27:06,700 --> 00:27:10,550 Well, I am almost done with talking about lattices. 405 00:27:10,550 --> 00:27:12,780 But I have a thought question. 406 00:27:15,740 --> 00:27:21,780 You've all heard of this castle in Ireland that has 407 00:27:21,780 --> 00:27:24,330 this famous rock in the side of the castle called the 408 00:27:24,330 --> 00:27:26,600 blarney stone? 409 00:27:26,600 --> 00:27:31,420 And you, to get the gift of gab, have to lean over this 410 00:27:31,420 --> 00:27:36,770 rather dangerous wall and somebody holds your ankles and 411 00:27:36,770 --> 00:27:41,260 you go down and you kiss the blarney stone. 412 00:27:41,260 --> 00:27:43,190 And that gives you supposedly the gift of gab. 413 00:27:45,760 --> 00:27:48,460 But kissing is a dangerous occupation. 414 00:27:48,460 --> 00:27:53,465 You've all heard about kissing disease mononucleosis? 415 00:27:53,465 --> 00:27:56,960 And with all those people smooching the blarney stone, 416 00:27:56,960 --> 00:27:59,590 I've always been afraid the blarney stone might 417 00:27:59,590 --> 00:28:01,700 come down with mono. 418 00:28:01,700 --> 00:28:04,500 And if the blarney stone became ill in this session, 419 00:28:04,500 --> 00:28:05,260 would it have to be 420 00:28:05,260 --> 00:28:07,230 transferred to the Mono Clinic? 421 00:28:12,858 --> 00:28:14,108 I don't know. 422 00:28:17,250 --> 00:28:19,560 All right. 423 00:28:19,560 --> 00:28:23,260 Here in all their glory are the 14 space lattices, or 424 00:28:23,260 --> 00:28:27,000 Bravais lattices as they're finally called. 425 00:28:27,000 --> 00:28:31,420 The labels a, b, c, et cetera, are not on them, but the c 426 00:28:31,420 --> 00:28:33,950 axis is vertical. 427 00:28:33,950 --> 00:28:38,000 You'll see that the hexagonal rhombohedral lattice is shown 428 00:28:38,000 --> 00:28:42,120 referred to hexagonal axes. 429 00:28:42,120 --> 00:28:44,140 In the international tables, you will find all the 430 00:28:44,140 --> 00:28:49,890 information both referred to rhombohedral lattices and to 431 00:28:49,890 --> 00:28:51,950 the triple hexagonal cell as well. 432 00:28:54,770 --> 00:29:01,730 Now, into these lattices one should drop those of the 32 433 00:29:01,730 --> 00:29:04,730 point groups that the lattices can accommodate. 434 00:29:07,300 --> 00:29:08,915 That's gonna be a lot of additions. 435 00:29:13,260 --> 00:29:18,050 Besides that, there is the option of having the symmetry 436 00:29:18,050 --> 00:29:21,390 elements of the point group not intersect at a common 437 00:29:21,390 --> 00:29:26,230 point, but interleave them as we did, for example, with the 438 00:29:26,230 --> 00:29:35,380 glide planes in P2GG, and in some of the other 439 00:29:35,380 --> 00:29:39,310 two-dimensional plane groups as well. 440 00:29:39,310 --> 00:29:42,710 Then we would do what we also did in two dimensions, take 441 00:29:42,710 --> 00:29:44,910 the mirror planes and replace them by glide 442 00:29:44,910 --> 00:29:47,260 planes, one or the both. 443 00:29:47,260 --> 00:29:50,650 And you might think that we really have an enormous amount 444 00:29:50,650 --> 00:29:52,360 of work to do. 445 00:29:52,360 --> 00:29:56,710 We haven't considered any symmetry combinations that 446 00:29:56,710 --> 00:29:59,370 involve a horizontal mirror plane. 447 00:29:59,370 --> 00:30:03,270 For example, we look at what symmetry 2 requires and we 448 00:30:03,270 --> 00:30:10,560 found the primitive monoclinic lattice, and the side-centered 449 00:30:10,560 --> 00:30:12,140 monoclinic lattice, and the 450 00:30:12,140 --> 00:30:14,200 body-centered monoclinic lattice. 451 00:30:14,200 --> 00:30:16,870 So there's more than one lattice type into which we can 452 00:30:16,870 --> 00:30:19,170 drop a given point group. 453 00:30:19,170 --> 00:30:24,480 But what would you have if you had a mirror plane 454 00:30:24,480 --> 00:30:27,130 perpendicular to the two-fold axis? 455 00:30:27,130 --> 00:30:31,420 What kind of lattice will that require? 456 00:30:31,420 --> 00:30:31,720 OK. 457 00:30:31,720 --> 00:30:37,600 Let me, in one masterful swoop, convince you that we 458 00:30:37,600 --> 00:30:43,380 have done almost the better part of the work. 459 00:30:43,380 --> 00:30:51,220 Let us ask what does inversion require of a lattice? 460 00:30:51,220 --> 00:30:53,670 What does the symmetry of inversion, which didn't appear 461 00:30:53,670 --> 00:30:56,020 in any of the plane groups at all because that's inherently 462 00:30:56,020 --> 00:30:59,270 a three-dimensional point group operation? 463 00:30:59,270 --> 00:31:00,040 Well, think about it. 464 00:31:00,040 --> 00:31:03,080 All inversion says is that, if, hey, there's a translation 465 00:31:03,080 --> 00:31:05,860 that goes up this way, there should be another translation 466 00:31:05,860 --> 00:31:08,830 minus T in the opposite direction. 467 00:31:08,830 --> 00:31:13,010 And any lattice whatsoever can make that claim and satisfy 468 00:31:13,010 --> 00:31:15,520 that requirement. 469 00:31:15,520 --> 00:31:20,510 So let me now just quickly say that if we have derived the 470 00:31:20,510 --> 00:31:23,950 lattice types, and by implication the space groups 471 00:31:23,950 --> 00:31:28,460 as well, for a two-fold axis, for a four-fold axis, for a 472 00:31:28,460 --> 00:31:34,160 six-fold axis, and so on, we have automatically satisfied 473 00:31:34,160 --> 00:31:40,040 the requirements for any point group that results when you 474 00:31:40,040 --> 00:31:43,130 add inversion to a two-fold axis, when you add inversion 475 00:31:43,130 --> 00:31:47,730 to a four-fold axis, or add inversion to a six-fold axis. 476 00:31:47,730 --> 00:31:53,350 So it turns out that the requirements of a six-fold 477 00:31:53,350 --> 00:31:56,770 axis, which gave us the primitive hexagonal lattice, 478 00:31:56,770 --> 00:32:01,830 are also met by 6 over M. The requirements of a four-fold 479 00:32:01,830 --> 00:32:05,835 axis, which gave us the lattices primitive tetragonal 480 00:32:05,835 --> 00:32:12,640 and body-centered tetragonal, are also satisfied for 4 over 481 00:32:12,640 --> 00:32:13,910 M. 482 00:32:13,910 --> 00:32:15,890 So, any one of the point groups that 483 00:32:15,890 --> 00:32:21,480 differs from the 11-- 484 00:32:21,480 --> 00:32:25,860 from the 10 two-dimensional symmetries-- 485 00:32:25,860 --> 00:32:32,650 namely 1, 2, 3, 4, 6, M, 2MM, 3M, 4MM, and 6MM-- 486 00:32:32,650 --> 00:32:36,850 any symmetry that you get by adding inversion to those 487 00:32:36,850 --> 00:32:40,410 two-dimensional point groups is automatically going to be 488 00:32:40,410 --> 00:32:42,640 happy in the space lattices that we've 489 00:32:42,640 --> 00:32:44,140 derived to this point. 490 00:32:44,140 --> 00:32:46,960 So we have done an enormous fraction of the work that's 491 00:32:46,960 --> 00:32:52,540 required to derive space lattices, and the 492 00:32:52,540 --> 00:32:55,520 space groups as well. 493 00:32:55,520 --> 00:32:59,730 So there is not too much more to be done. 494 00:32:59,730 --> 00:33:04,500 And what I will do for probably the next meeting is 495 00:33:04,500 --> 00:33:09,820 to derive a few of the space groups for the lower 496 00:33:09,820 --> 00:33:12,600 symmetries which are not too complicated, just to show you 497 00:33:12,600 --> 00:33:14,730 how the game plays out. 498 00:33:14,730 --> 00:33:18,410 And then I will mention a little bit about the special 499 00:33:18,410 --> 00:33:22,440 idiosyncrasies of the orthorhombic brick-shaped 500 00:33:22,440 --> 00:33:26,070 cells where there is nothing more distinct about one 501 00:33:26,070 --> 00:33:27,120 direction and the other. 502 00:33:27,120 --> 00:33:29,470 All the inter-axial angles are 90 degrees. 503 00:33:29,470 --> 00:33:31,180 The translations are general. 504 00:33:31,180 --> 00:33:37,190 So there are peculiarities in notation there that are unique 505 00:33:37,190 --> 00:33:38,440 to the orthorhombic system. 506 00:33:41,200 --> 00:33:46,760 And then I want to spend, perhaps, one class talking a 507 00:33:46,760 --> 00:33:49,350 little bit about crystal chemistry. 508 00:33:49,350 --> 00:33:52,210 Have any of you had a class in solid state chemistry or 509 00:33:52,210 --> 00:33:53,620 crystal chemistry? 510 00:33:53,620 --> 00:33:54,100 OK. 511 00:33:54,100 --> 00:33:56,980 One. 512 00:33:56,980 --> 00:34:01,260 One of the main uses of space group theory is the 513 00:34:01,260 --> 00:34:03,480 description of crystal structures. 514 00:34:03,480 --> 00:34:07,010 So I think I might, as a minor digression, take an hour to 515 00:34:07,010 --> 00:34:10,719 talk about packing and some of the concepts used to describe 516 00:34:10,719 --> 00:34:15,620 the close-packed structures and ionic structures and give 517 00:34:15,620 --> 00:34:21,120 you a problem that ask you, as I already did, to generate a 518 00:34:21,120 --> 00:34:23,834 structure, simple-minded two-dimensional structure, and 519 00:34:23,834 --> 00:34:25,980 do some interpretation of it. 520 00:34:25,980 --> 00:34:31,710 So, I think this will be worth saying a little bit about just 521 00:34:31,710 --> 00:34:33,900 so that one of the main applications of symmetry 522 00:34:33,900 --> 00:34:39,590 theory does not escape without any discussion at all. 523 00:34:39,590 --> 00:34:46,310 We will, after about two more lectures, have a complete 524 00:34:46,310 --> 00:34:49,440 change of direction and start talking about properties of 525 00:34:49,440 --> 00:34:53,770 crystals and how these properties are impacted by the 526 00:34:53,770 --> 00:34:56,929 fact that crystals have symmetry. 527 00:34:56,929 --> 00:34:59,730 So we're gonna use some of our results in symmetry, but not 528 00:34:59,730 --> 00:35:07,540 all that much, but develop remarkable mileage in 529 00:35:07,540 --> 00:35:11,950 predicting anisotropy in crystals just on the basis of 530 00:35:11,950 --> 00:35:14,550 the symmetry which that crystal possesses, that 531 00:35:14,550 --> 00:35:17,250 certain physical properties as a function of direction are 532 00:35:17,250 --> 00:35:21,910 weird surfaces that have lumps and noses and bulges in them. 533 00:35:21,910 --> 00:35:28,060 And certain properties have this anisotropy universally, 534 00:35:28,060 --> 00:35:31,670 regardless of the chemistry or composition of the crystal. 535 00:35:31,670 --> 00:35:34,940 Given the symmetry of the crystal, the anisotropy has to 536 00:35:34,940 --> 00:35:36,310 be in this fashion. 537 00:35:36,310 --> 00:35:38,650 We'll be able to show that some properties simply cannot 538 00:35:38,650 --> 00:35:42,860 exist, period, regardless of composition and structure in 539 00:35:42,860 --> 00:35:44,890 materials that have certain symmetries. 540 00:35:44,890 --> 00:35:48,070 So that'll be the direction we take off in. 541 00:35:48,070 --> 00:35:50,830 I mentioned at the beginning of the term-- you've probably 542 00:35:50,830 --> 00:35:51,740 forgotten-- 543 00:35:51,740 --> 00:35:55,870 there is a lovely book on tensors and physical 544 00:35:55,870 --> 00:36:01,320 properties by a gentleman named J. F. Nye, N-Y-E. And 545 00:36:01,320 --> 00:36:03,000 the title of the book is Physical 546 00:36:03,000 --> 00:36:04,250 Properties of Crystals. 547 00:36:06,870 --> 00:36:12,330 This is a nice place to go for further reading. 548 00:36:12,330 --> 00:36:16,950 We will go through roughly half of Nye's book with a 549 00:36:16,950 --> 00:36:20,610 little more emphasis on the consequences of symmetry than 550 00:36:20,610 --> 00:36:21,870 what he does. 551 00:36:21,870 --> 00:36:25,630 But if you want to do some extra reading, I asked the 552 00:36:25,630 --> 00:36:27,240 Coop to stock this book. 553 00:36:27,240 --> 00:36:31,550 But I suggest that you not rush out and buy it. 554 00:36:31,550 --> 00:36:34,910 It's available in paperback, which means it's cheap, but 555 00:36:34,910 --> 00:36:36,150 only relatively so. 556 00:36:36,150 --> 00:36:39,030 There's no such thing as a cheap technical book, 557 00:36:39,030 --> 00:36:41,340 paperbacked or not, these days. 558 00:36:41,340 --> 00:36:43,120 It's also on reserve in the library. 559 00:36:43,120 --> 00:36:45,590 So, see if you need the extra help. 560 00:36:45,590 --> 00:36:48,220 See if you would like to do some reading in Nye. 561 00:36:48,220 --> 00:36:51,460 And make sure you want to before you go out and put your 562 00:36:51,460 --> 00:36:53,640 money down at the Coop. 563 00:36:53,640 --> 00:36:56,540 The last part of Nye is an elegant, 564 00:36:56,540 --> 00:36:59,310 beautiful part of the book. 565 00:36:59,310 --> 00:37:02,580 It deals with the thermodynamic relations 566 00:37:02,580 --> 00:37:04,230 between different properties. 567 00:37:04,230 --> 00:37:09,140 And it's a unique treatment that is developed by Nye. 568 00:37:09,140 --> 00:37:12,680 We won't do any of that, but it's a nice thing to have. 569 00:37:12,680 --> 00:37:14,330 So don't rush out and spend your money. 570 00:37:14,330 --> 00:37:17,980 But if you want additional reading and just going to the 571 00:37:17,980 --> 00:37:21,890 reserved book collection is not enough, then Nye is not at 572 00:37:21,890 --> 00:37:26,750 all a bad copy of a book to have. 573 00:37:26,750 --> 00:37:30,810 I think most of you have probably seen Nye, not the 574 00:37:30,810 --> 00:37:33,890 book but the man. 575 00:37:33,890 --> 00:37:38,360 How many of you have seen the famous Bragg bubble raft movie 576 00:37:38,360 --> 00:37:42,780 on disk locations, where he floats sheets of bundles and 577 00:37:42,780 --> 00:37:46,900 sheers them and you see his dislocation go zipping across? 578 00:37:46,900 --> 00:37:50,000 At one stage in that movie, there's an assistant who 579 00:37:50,000 --> 00:37:52,940 tiptoes, trying to be unobtrusive, across the 580 00:37:52,940 --> 00:37:53,470 background. 581 00:37:53,470 --> 00:37:56,730 But the camera catches him scurrying behind Bragg. 582 00:37:56,730 --> 00:37:58,330 That's Nye. 583 00:37:58,330 --> 00:37:59,730 So if you've seen the movie, you've seen 584 00:37:59,730 --> 00:38:01,800 what Nye looks like. 585 00:38:01,800 --> 00:38:02,150 OK. 586 00:38:02,150 --> 00:38:04,040 I'll call it quits there for today. 587 00:38:04,040 --> 00:38:06,610 And we'll see you next week on Tuesday.