1 00:00:06,610 --> 00:00:10,870 PROFESSOR: OK, let us resume. 2 00:00:10,870 --> 00:00:15,040 I had no idea how many people would be here today, and I 3 00:00:15,040 --> 00:00:18,770 think I made 25 copies of the handout. 4 00:00:18,770 --> 00:00:22,800 And I see 25 names on the list. 5 00:00:22,800 --> 00:00:25,490 And that means that two people did not get 6 00:00:25,490 --> 00:00:26,970 a copy of the syllabus. 7 00:00:26,970 --> 00:00:28,220 Does anybody need a copy? 8 00:00:30,760 --> 00:00:33,180 That's strange. 9 00:00:33,180 --> 00:00:34,430 OK. 10 00:00:36,280 --> 00:00:36,760 All right. 11 00:00:36,760 --> 00:00:39,740 We covered some introductory material, and I think we've 12 00:00:39,740 --> 00:00:42,080 covered enough that you can do a problem set. 13 00:00:42,080 --> 00:00:44,270 So it gives me great pleasure to hand out 14 00:00:44,270 --> 00:00:45,670 problem set number one. 15 00:01:00,570 --> 00:01:03,330 OK, you can think about that. 16 00:01:03,330 --> 00:01:05,570 It is the sort of problem that will either take you two 17 00:01:05,570 --> 00:01:09,690 minutes or two hours or infinity. 18 00:01:09,690 --> 00:01:14,310 So don't spend too much time on it, but I would like you to 19 00:01:14,310 --> 00:01:17,290 put your name on it and turn it in either at the end of the 20 00:01:17,290 --> 00:01:20,960 hour or next time so I can make comments if there's 21 00:01:20,960 --> 00:01:22,970 something that's mostly right but not quite right. 22 00:01:33,880 --> 00:01:40,430 Let's return to these three simple patterns that we put on 23 00:01:40,430 --> 00:01:41,450 the blackboard. 24 00:01:41,450 --> 00:01:44,700 And let me make another point about symmetry. 25 00:01:47,720 --> 00:01:53,430 The people who sensed that this pattern and the one on 26 00:01:53,430 --> 00:01:57,960 the bottom were the same because they had the same 27 00:01:57,960 --> 00:02:01,070 motif in them, that they had the same rectangle with one 28 00:02:01,070 --> 00:02:02,530 concave side. 29 00:02:02,530 --> 00:02:08,830 And I drew a mirror line in here because that locus, when 30 00:02:08,830 --> 00:02:12,590 I review this is a reflection from left to right, left the 31 00:02:12,590 --> 00:02:18,720 motif, as well as the entire pattern, unchanged upon making 32 00:02:18,720 --> 00:02:20,610 that transformation. 33 00:02:20,610 --> 00:02:25,860 Is there not also a mirror line there? 34 00:02:25,860 --> 00:02:27,760 Worked for this motif. 35 00:02:27,760 --> 00:02:30,370 Why not for this motif? 36 00:02:30,370 --> 00:02:35,710 Well, the answer is no, that this is not a mirror line 37 00:02:35,710 --> 00:02:39,550 because the symmetry transformations acts on 38 00:02:39,550 --> 00:02:43,850 everything, and not just one little bit of space. 39 00:02:43,850 --> 00:02:46,980 And if I would take this chain of objects that's 40 00:02:46,980 --> 00:02:49,880 translationally periodic with a translation running this 41 00:02:49,880 --> 00:02:53,970 way, and I reflected that, I should have another chain 42 00:02:53,970 --> 00:02:55,930 running like this. 43 00:02:55,930 --> 00:02:59,560 So the direction of the translation vector is not left 44 00:02:59,560 --> 00:03:01,920 invariant by this reflection. 45 00:03:01,920 --> 00:03:08,040 So the conclusion here, and it's a subtle one, matter of 46 00:03:08,040 --> 00:03:12,580 definition, almost, is that the transformation, the 47 00:03:12,580 --> 00:03:15,260 symmetry transformation, if it's to be a symmetry 48 00:03:15,260 --> 00:03:35,170 transformation, acts on all of the space, and not just on one 49 00:03:35,170 --> 00:03:37,940 local domain. 50 00:03:37,940 --> 00:03:43,930 So let me give you an example of a pattern that doesn't 51 00:03:43,930 --> 00:03:45,960 involve translational periodicity. 52 00:03:45,960 --> 00:03:51,630 So let me try to make a star as carefully as I can. 53 00:03:55,320 --> 00:03:57,740 What sort of symmetric does that have, or would it have 54 00:03:57,740 --> 00:03:59,010 had I drawn it more perfectly? 55 00:03:59,010 --> 00:04:02,900 Well, that would be a five-fold rotation axis in the 56 00:04:02,900 --> 00:04:05,180 middle because I could rotate through one 57 00:04:05,180 --> 00:04:06,430 fifth of the circle. 58 00:04:08,580 --> 00:04:12,960 And any of those rotations twice or three times, or just 59 00:04:12,960 --> 00:04:16,959 2 pi over five, would be something that maps the 60 00:04:16,959 --> 00:04:19,420 pattern into congruence with itself. 61 00:04:19,420 --> 00:04:23,720 There are also mirror lines that go from one tip of the 62 00:04:23,720 --> 00:04:26,640 star to the other end. 63 00:04:26,640 --> 00:04:30,280 So that is an example of a pattern, non-periodic, but one 64 00:04:30,280 --> 00:04:34,410 that has five-fold rotational symmetry and mirror symmetry. 65 00:04:34,410 --> 00:04:40,040 Now, if I put that star in a box and ask, what is the 66 00:04:40,040 --> 00:04:42,420 symmetry of that space? 67 00:04:42,420 --> 00:04:45,520 There's only one operation which is common to the star 68 00:04:45,520 --> 00:04:47,990 and to the enclosing rectangle, and that's this 69 00:04:47,990 --> 00:04:49,240 mirror plane. 70 00:04:52,640 --> 00:04:56,880 So the symmetry acts not just on one little part of the 71 00:04:56,880 --> 00:05:00,580 space, but it has to leave everything invariant. 72 00:05:00,580 --> 00:05:03,990 So in that sense, going to this pattern here, this is not 73 00:05:03,990 --> 00:05:06,800 a mirror plane because it doesn't leave the entire 74 00:05:06,800 --> 00:05:08,760 pattern invariant. 75 00:05:08,760 --> 00:05:11,690 That plane would reflect this one up to here, and we don't 76 00:05:11,690 --> 00:05:12,830 have anything there. 77 00:05:12,830 --> 00:05:14,620 So the space is not left invariant. 78 00:05:23,340 --> 00:05:25,510 One further definition. 79 00:05:25,510 --> 00:05:29,960 We defined what we mean when we say a space or an object 80 00:05:29,960 --> 00:05:31,220 has symmetry. 81 00:05:31,220 --> 00:05:35,120 We said an object or a space possesses symmetry when there 82 00:05:35,120 --> 00:05:38,540 is an operation, or set of operations, that maps the 83 00:05:38,540 --> 00:05:41,900 space or the object into congruence with itself. 84 00:05:41,900 --> 00:05:44,950 Let me make another definition, and that a 85 00:05:44,950 --> 00:05:46,200 symmetry element-- 86 00:05:54,550 --> 00:05:56,060 another bit of terminology-- 87 00:05:56,060 --> 00:05:59,440 is the locus of points that's left unmoved by the operation. 88 00:06:07,090 --> 00:06:08,620 Left unmoved or left invariant. 89 00:06:24,820 --> 00:06:32,670 So for some specific examples, this vertical line is a locus 90 00:06:32,670 --> 00:06:36,760 which is left invariant by either the five-fold rotation 91 00:06:36,760 --> 00:06:46,900 or any of the mirror planes passing through the points 92 00:06:46,900 --> 00:06:49,460 that would be true of the star. 93 00:06:49,460 --> 00:06:53,060 So for the star, these are all symmetry elements. 94 00:06:53,060 --> 00:06:57,340 For the net combination of the star and the enclosing 95 00:06:57,340 --> 00:07:01,630 rectangle, the only thing that leaves a space 96 00:07:01,630 --> 00:07:03,050 invariant is this line. 97 00:07:03,050 --> 00:07:05,670 The locus that's left and moved is this line, so we 98 00:07:05,670 --> 00:07:07,350 refer to that as a mirror plane. 99 00:07:12,560 --> 00:07:20,680 Now, these may be seeming kind of definitions. 100 00:07:20,680 --> 00:07:23,230 Nice to have, but what use are they? 101 00:07:23,230 --> 00:07:26,100 We will use some of these definitions to answer a 102 00:07:26,100 --> 00:07:29,110 question which may seem tricky. 103 00:07:29,110 --> 00:07:32,780 If we do a couple of things in sequence, for example, what is 104 00:07:32,780 --> 00:07:36,360 the net consequence of doing, let's say, a rotation combined 105 00:07:36,360 --> 00:07:38,270 with a reflection? 106 00:07:38,270 --> 00:07:40,480 You can answer that question by saying, what 107 00:07:40,480 --> 00:07:42,720 has been left unmoved? 108 00:07:42,720 --> 00:07:46,220 And that is the locuses of whatever net transformation 109 00:07:46,220 --> 00:07:48,720 results from a combination of two or more. 110 00:07:51,710 --> 00:07:55,510 So that, again, is abstract, but we'll use that later on. 111 00:07:55,510 --> 00:07:57,420 Any question on this? 112 00:07:57,420 --> 00:08:00,370 Let me summarize very quickly what we have found for two 113 00:08:00,370 --> 00:08:02,550 dimensions. 114 00:08:02,550 --> 00:08:05,350 We found that there are, in a two-dimensional space, three 115 00:08:05,350 --> 00:08:07,240 kinds of operations. 116 00:08:07,240 --> 00:08:14,720 There is the operation of translation, which we'll call 117 00:08:14,720 --> 00:08:16,890 by the vector, T, corresponding to that 118 00:08:16,890 --> 00:08:18,310 transformation. 119 00:08:18,310 --> 00:08:20,990 And there is an operation of reflection. 120 00:08:23,790 --> 00:08:26,880 And the locus of the plane, in which the reflection occurs, 121 00:08:26,880 --> 00:08:28,730 we'll call a mirror plane. 122 00:08:28,730 --> 00:08:30,260 And that's a linear locus. 123 00:08:30,260 --> 00:08:33,650 And then, we've seen in these two patterns here an operation 124 00:08:33,650 --> 00:08:35,309 of rotation. 125 00:08:35,309 --> 00:08:37,990 In particular, in these two-dimensional patterns, we 126 00:08:37,990 --> 00:08:41,027 saw the rotation operation A pi. 127 00:08:44,230 --> 00:08:50,880 Now, I would put forth for your consideration something 128 00:08:50,880 --> 00:08:54,330 that is a profound conclusion. 129 00:08:54,330 --> 00:08:58,850 These are the only single-step transformations that can exist 130 00:08:58,850 --> 00:09:02,090 in a two-dimensional space. 131 00:09:02,090 --> 00:09:04,850 These are the only ones that can exist as single-step 132 00:09:04,850 --> 00:09:07,720 operations. 133 00:09:07,720 --> 00:09:11,400 We can view these as operations that result in a 134 00:09:11,400 --> 00:09:15,280 transformation of coordinates. 135 00:09:15,280 --> 00:09:19,910 And in two dimensions, if we have some position, x, y, in 136 00:09:19,910 --> 00:09:23,160 the space, what the transformation of a 137 00:09:23,160 --> 00:09:29,310 translation does is to take x and add a constant to it. 138 00:09:29,310 --> 00:09:34,680 It takes y and adds, perhaps, a different constant to it. 139 00:09:34,680 --> 00:09:38,680 Do the operation a second time, and we'll go to x plus 140 00:09:38,680 --> 00:09:40,740 2a and y plus 2b. 141 00:09:44,480 --> 00:09:50,320 So analytically, we can look at these symmetry operations 142 00:09:50,320 --> 00:09:54,210 in terms of the transformation of a representative 143 00:09:54,210 --> 00:09:55,840 coordinate. 144 00:09:55,840 --> 00:09:59,140 If we have a reflection plane, and let's set up a coordinate 145 00:09:59,140 --> 00:10:03,820 system where this is y and this is x. 146 00:10:03,820 --> 00:10:08,730 We have an object here at the location x, y, and we reflect 147 00:10:08,730 --> 00:10:11,180 it across this locus. 148 00:10:11,180 --> 00:10:17,640 It goes to minus x, y, if the mirror line runs through the 149 00:10:17,640 --> 00:10:20,260 origin and is perpendicular to x. 150 00:10:20,260 --> 00:10:23,850 So one example of a transformation by reflection 151 00:10:23,850 --> 00:10:29,340 is that x, y goes to minus x, y. 152 00:10:29,340 --> 00:10:33,420 And this is a case where the mirror plane is perpendicular 153 00:10:33,420 --> 00:10:36,340 to x and passes through the origin. 154 00:10:45,470 --> 00:10:49,890 If we do the operation a second time, minus x, y would 155 00:10:49,890 --> 00:10:53,170 get mapped back into x, y again. 156 00:10:53,170 --> 00:10:56,090 It comes back to where it started from. 157 00:10:56,090 --> 00:10:57,740 So this would be the first reflection. 158 00:11:00,540 --> 00:11:02,155 This would be the second reflection. 159 00:11:08,410 --> 00:11:11,960 And we saw in the patterns an example of one other 160 00:11:11,960 --> 00:11:14,150 transformation. 161 00:11:14,150 --> 00:11:17,970 Let's suppose there was an operation, A pi, at the 162 00:11:17,970 --> 00:11:21,850 origin, and this was x, and this was y. 163 00:11:21,850 --> 00:11:25,090 We started out with a motif here at x, y. 164 00:11:25,090 --> 00:11:29,620 If we rotated that by 180 degrees, it would go down to a 165 00:11:29,620 --> 00:11:36,040 location minus x, minus y. 166 00:11:36,040 --> 00:11:42,780 So the operation of a 180-degree rotation is going 167 00:11:42,780 --> 00:11:46,180 to analytically correspond to a transformation of going from 168 00:11:46,180 --> 00:11:49,890 x, y to minus x, minus y. 169 00:11:49,890 --> 00:11:53,170 If we perform it again, it would go back to x, y. 170 00:11:55,890 --> 00:11:59,070 OK, let's look at this in more general terms. 171 00:11:59,070 --> 00:12:01,950 In a two-dimensional space, we've got two dimensions to 172 00:12:01,950 --> 00:12:03,730 diddle with. 173 00:12:03,730 --> 00:12:06,520 We can change the sense of no coordinate. 174 00:12:06,520 --> 00:12:08,190 That's translation. 175 00:12:08,190 --> 00:12:11,280 We can change the sense of one coordinate. 176 00:12:11,280 --> 00:12:13,280 That's going to be reflection. 177 00:12:13,280 --> 00:12:16,370 We can change the sense of both coordinates. 178 00:12:16,370 --> 00:12:19,740 That's going to be a rotation. 179 00:12:19,740 --> 00:12:21,070 That's all we can do. 180 00:12:21,070 --> 00:12:23,900 So these are the three basic operations in a 181 00:12:23,900 --> 00:12:25,150 two-dimensional space. 182 00:12:29,060 --> 00:12:34,870 I gave you special cases to make things easy, but 183 00:12:34,870 --> 00:12:38,900 regardless of where the mirror plane is, parallel to or 184 00:12:38,900 --> 00:12:42,160 perpendicular to an axis or not, and whether it passes 185 00:12:42,160 --> 00:12:46,930 through the origin or not, a mirror plane has the operation 186 00:12:46,930 --> 00:12:50,570 of reversing the sense of one direction. 187 00:12:50,570 --> 00:12:53,320 Just the sense of one direction that is reversed. 188 00:12:53,320 --> 00:12:56,450 And the rotation, be it a rotation through 60 degrees or 189 00:12:56,450 --> 00:13:00,700 90 degrees or 180 degrees, is always taking both 190 00:13:00,700 --> 00:13:01,470 coordinates. 191 00:13:01,470 --> 00:13:04,590 It's making a transformation of both coordinates. 192 00:13:04,590 --> 00:13:06,770 If you only have two coordinates with which to 193 00:13:06,770 --> 00:13:08,750 play, that's all you can do. 194 00:13:12,760 --> 00:13:15,120 Let's do some giant extrapolations. 195 00:13:15,120 --> 00:13:18,510 If we have a strictly one-dimensional pattern where 196 00:13:18,510 --> 00:13:21,560 there's x and nothing else, than they're only going to be 197 00:13:21,560 --> 00:13:22,860 two coordinates. 198 00:13:22,860 --> 00:13:25,920 And there are going to be only two ways we can transform 199 00:13:25,920 --> 00:13:26,520 coordinates. 200 00:13:26,520 --> 00:13:36,570 So in a one-dimensional space, we can change the sense of no 201 00:13:36,570 --> 00:13:43,410 coordinate, and that's going to be the operation of 202 00:13:43,410 --> 00:13:52,170 translation, or we can change the sense of one coordinate, 203 00:13:52,170 --> 00:13:56,220 and that's going to be the operation of reflection. 204 00:13:56,220 --> 00:13:58,460 No rotation in a one-dimensional space. 205 00:14:01,960 --> 00:14:03,930 Now, let's extrapolate in the other direction. 206 00:14:03,930 --> 00:14:06,750 In a three-dimensional space, the sort that we're going to 207 00:14:06,750 --> 00:14:10,190 be concerned with when we want to describe the symmetry of 208 00:14:10,190 --> 00:14:16,220 real crystals, you've got three coordinates to permute. 209 00:14:16,220 --> 00:14:22,820 So it follows then, without saying what they are, in 3D, 210 00:14:22,820 --> 00:14:29,715 there are going to be four distinct one-step operations. 211 00:14:40,880 --> 00:14:43,086 And then five dimensions? 212 00:14:43,086 --> 00:14:45,060 Hey, that's a nice thing about mathematics. 213 00:14:45,060 --> 00:14:47,030 You could play any game you like. 214 00:14:47,030 --> 00:14:49,680 Not only that, but you make up the rules. 215 00:14:49,680 --> 00:14:51,900 In a five-dimensional space, there's going to be six 216 00:14:51,900 --> 00:14:54,050 transformations. 217 00:14:54,050 --> 00:14:56,850 Would we ever want to worry about five-dimensional 218 00:14:56,850 --> 00:14:59,090 crystallography? 219 00:14:59,090 --> 00:15:02,850 Well, let me hang out a teaser and not answer the question. 220 00:15:02,850 --> 00:15:06,980 Yeah, there are crystals for which as many as 221 00:15:06,980 --> 00:15:12,200 six-dimensional symmetries are necessary. 222 00:15:12,200 --> 00:15:12,870 Wow. 223 00:15:12,870 --> 00:15:14,130 Doesn't that blow the mind? 224 00:15:14,130 --> 00:15:18,190 We'll return to that, and I'll explain why later on. 225 00:15:24,860 --> 00:15:27,770 Another thing you might ask, why did I sneak this in? 226 00:15:27,770 --> 00:15:31,150 Why did I say one-step operation? 227 00:15:31,150 --> 00:15:33,190 Well, it's something we should worry about, and 228 00:15:33,190 --> 00:15:34,750 unfortunately, we will. 229 00:15:34,750 --> 00:15:39,880 What if you take a motif, translate it, rotate it around 230 00:15:39,880 --> 00:15:43,550 a couple of times, reflect it, bounce it up and down three 231 00:15:43,550 --> 00:15:46,880 times, and then put it down? 232 00:15:46,880 --> 00:15:50,780 How do you get from the first motif to the final one there? 233 00:15:50,780 --> 00:15:54,520 Is there an infinite number of operations? 234 00:15:54,520 --> 00:15:55,650 Mercifully, no. 235 00:15:55,650 --> 00:15:59,570 The number is small and very finite. 236 00:15:59,570 --> 00:16:03,460 And we will systematically, in another week's time, examine 237 00:16:03,460 --> 00:16:06,470 specifically two-step operations. 238 00:16:06,470 --> 00:16:10,630 And as with many things that we'll encounter, we might not 239 00:16:10,630 --> 00:16:12,610 be clever enough to think them up. 240 00:16:12,610 --> 00:16:15,180 But when we start putting things together into a 241 00:16:15,180 --> 00:16:17,980 synthesis, suddenly we're going to stumble over 242 00:16:17,980 --> 00:16:21,440 something we don't know how to explain, and we will have 243 00:16:21,440 --> 00:16:25,690 arrived, like it or not, at a new feature which we perhaps 244 00:16:25,690 --> 00:16:26,940 hadn't anticipated. 245 00:16:29,430 --> 00:16:31,080 OK, any question at this point? 246 00:16:35,370 --> 00:16:38,730 All this has been in a way of general introduction. 247 00:16:38,730 --> 00:16:45,260 We're going to now take things more slowly and proceed one 248 00:16:45,260 --> 00:16:46,510 step at a time. 249 00:16:57,570 --> 00:17:06,790 I would like to confine our attention for the moment on 250 00:17:06,790 --> 00:17:10,920 two-dimensional symmetries and examine the sorts of patterns 251 00:17:10,920 --> 00:17:13,990 that can exist in two dimensions, fabric patterns, 252 00:17:13,990 --> 00:17:17,900 floor tile, grillwork, and so on. 253 00:17:17,900 --> 00:17:20,880 And we've seen that, basically, there seem to be 254 00:17:20,880 --> 00:17:24,150 three operations, three kinds of operations, translation, 255 00:17:24,150 --> 00:17:26,750 reflection, and rotation. 256 00:17:26,750 --> 00:17:30,330 That's an infinite number of operations because we are not 257 00:17:30,330 --> 00:17:34,140 specifying whether or not the rotation angle is restricted 258 00:17:34,140 --> 00:17:35,940 to any particular value. 259 00:17:35,940 --> 00:17:38,290 No reason why it should be. 260 00:17:38,290 --> 00:17:40,155 There are lots of rotational symmetries that 261 00:17:40,155 --> 00:17:42,590 are absolutely lovely. 262 00:17:42,590 --> 00:17:44,580 But let's build things up. 263 00:17:44,580 --> 00:17:48,190 And I would like to first look at the operation of 264 00:17:48,190 --> 00:17:53,410 translation, which we've said a great deal 265 00:17:53,410 --> 00:17:56,430 about to this point. 266 00:17:56,430 --> 00:18:00,090 Translation has magnitude. 267 00:18:00,090 --> 00:18:01,060 It has direction. 268 00:18:01,060 --> 00:18:02,590 So it acts like a vector. 269 00:18:02,590 --> 00:18:06,650 But just like a vector, it has no unique origin. 270 00:18:06,650 --> 00:18:11,660 Perform the operation twice, and you have a position that 271 00:18:11,660 --> 00:18:14,740 is two translations removed from the origin. 272 00:18:14,740 --> 00:18:17,220 Do it three times, you have a translation 273 00:18:17,220 --> 00:18:21,090 that's three times out. 274 00:18:21,090 --> 00:18:25,310 If a motif sits here, the motif must sit at the end of 275 00:18:25,310 --> 00:18:27,830 this translation in the same 276 00:18:27,830 --> 00:18:30,190 orientation parallel to itself. 277 00:18:30,190 --> 00:18:33,200 It must exist at the end of two translations. 278 00:18:33,200 --> 00:18:37,750 And if the operation acts on all of the space, if we say a 279 00:18:37,750 --> 00:18:42,130 translation is present, we really imply that there's an 280 00:18:42,130 --> 00:18:44,710 infinite row going to plus infinity and 281 00:18:44,710 --> 00:18:46,490 back to minus infinity. 282 00:18:46,490 --> 00:18:49,570 And there is a motif hanging at the terminal 283 00:18:49,570 --> 00:18:51,135 point of every vector. 284 00:18:56,320 --> 00:19:03,350 Now, we can summarize this periodicity with 285 00:19:03,350 --> 00:19:06,050 a convenient device. 286 00:19:06,050 --> 00:19:11,940 Let's take some fiducial point and summarize the 287 00:19:11,940 --> 00:19:18,150 translational periodicity by saying that something that is 288 00:19:18,150 --> 00:19:28,580 hung at one point, either here, or maybe hung also off 289 00:19:28,580 --> 00:19:32,360 in some other direction relative to the translation, 290 00:19:32,360 --> 00:19:36,840 that something hung on one of these points is automatically 291 00:19:36,840 --> 00:19:39,910 reproduced for us at every point. 292 00:19:39,910 --> 00:19:43,510 So what we have done through this device is defined 293 00:19:43,510 --> 00:19:45,810 something that is called a lattice point. 294 00:19:48,940 --> 00:19:54,480 And this is an abstraction of the translational periodicity. 295 00:19:54,480 --> 00:19:58,690 There is an array of points, geometric fictions, which we 296 00:19:58,690 --> 00:20:01,150 have constructed. 297 00:20:01,150 --> 00:20:04,730 And we ascribe to this geometric fiction the property 298 00:20:04,730 --> 00:20:09,190 that anything hung at one of these points, be it a benzene 299 00:20:09,190 --> 00:20:13,910 ring or be it a Santa Claus on Christmas wrapping paper, is 300 00:20:13,910 --> 00:20:17,960 understood to be automatically reproduced at every other one 301 00:20:17,960 --> 00:20:21,240 of these points. 302 00:20:21,240 --> 00:20:27,720 It is this array of fictitious points that is the proper 303 00:20:27,720 --> 00:20:33,110 designation of what we refer to as a lattice. 304 00:20:33,110 --> 00:20:37,460 So a lattice is an array of fictitious points that 305 00:20:37,460 --> 00:20:41,990 summarizes the translational periodicity of the crystal. 306 00:20:41,990 --> 00:20:45,150 It has a property to repeat that something hung at a 307 00:20:45,150 --> 00:20:48,430 particular disposition relative to that point and 308 00:20:48,430 --> 00:20:52,590 with a particular orientation is understood to be hung at 309 00:20:52,590 --> 00:20:56,750 every other lattice point in exactly the same way. 310 00:20:56,750 --> 00:20:58,040 So that is a lattice. 311 00:20:58,040 --> 00:21:01,100 And this is one of the most abused terms in 312 00:21:01,100 --> 00:21:02,880 crystallography. 313 00:21:02,880 --> 00:21:06,290 We talk about the sodium chloride lattice. 314 00:21:06,290 --> 00:21:10,970 The sodium chloride lattice is a set of points that are 315 00:21:10,970 --> 00:21:14,740 placed at the corners of a cube and in the 316 00:21:14,740 --> 00:21:16,820 middle of all the faces. 317 00:21:16,820 --> 00:21:18,460 This is the NaCl lattice. 318 00:21:21,400 --> 00:21:26,420 If I choose to decorate that lattice with one sodium and 319 00:21:26,420 --> 00:21:29,180 one chlorine, then I have atoms sitting at 320 00:21:29,180 --> 00:21:31,240 these lattice points. 321 00:21:31,240 --> 00:21:34,810 And that is the sodium chloride structure. 322 00:21:34,810 --> 00:21:38,540 That is the proper term for the atomic configuration. 323 00:21:41,430 --> 00:21:45,170 So lattice is a geometrical term, and it's an abstraction. 324 00:21:45,170 --> 00:21:49,320 Structure is the actual atomic arrangement. 325 00:21:49,320 --> 00:21:53,990 Now, since I realize already that I am among friends, I can 326 00:21:53,990 --> 00:21:57,580 confess that I very often recklessly 327 00:21:57,580 --> 00:21:58,820 abuse the term lattice. 328 00:21:58,820 --> 00:22:03,140 If I talk about lattice energy, lattice diffusion, 329 00:22:03,140 --> 00:22:07,140 lattice vibration, I'm not talking about abstract points 330 00:22:07,140 --> 00:22:11,380 bobbling around or something going through this array of 331 00:22:11,380 --> 00:22:12,710 little points. 332 00:22:12,710 --> 00:22:16,060 I mean, I should talk about structure diffusion, structure 333 00:22:16,060 --> 00:22:18,490 energy, structure vibration. 334 00:22:18,490 --> 00:22:22,190 But man, that just doesn't have the established 335 00:22:22,190 --> 00:22:26,000 terminology, and it doesn't have the zing and music of 336 00:22:26,000 --> 00:22:28,060 something like lattice vibrations. 337 00:22:28,060 --> 00:22:30,240 So I do it all the time. 338 00:22:30,240 --> 00:22:33,740 Don't tell anybody else that I said this to you, frankly. 339 00:22:33,740 --> 00:22:36,550 But it's never going to be stamped out. 340 00:22:36,550 --> 00:22:41,450 But now you perhaps are informed enough to at least 341 00:22:41,450 --> 00:22:44,750 blush slightly when you talk about lattice energy or 342 00:22:44,750 --> 00:22:48,330 lattice diffusion, realizing you're using the term 343 00:22:48,330 --> 00:22:50,990 incorrectly and that you know better, but everybody else 344 00:22:50,990 --> 00:22:53,430 does it, so you do the same thing. 345 00:22:53,430 --> 00:22:55,210 So that is the definition of lattice. 346 00:22:58,520 --> 00:23:04,980 Now, suppose I take this space, to which I've added a 347 00:23:04,980 --> 00:23:13,630 first translation, and I'll call it T1, implying that I'm 348 00:23:13,630 --> 00:23:16,030 going to add something else to this space, 349 00:23:16,030 --> 00:23:17,870 which I'm free to do. 350 00:23:17,870 --> 00:23:21,590 I can put in a second translational periodicity 351 00:23:21,590 --> 00:23:24,260 because this is a two-dimensional space. 352 00:23:24,260 --> 00:23:25,250 How do I do this? 353 00:23:25,250 --> 00:23:29,590 And the answer is very carefully because the second 354 00:23:29,590 --> 00:23:34,400 translation could not go in the space parallel to the 355 00:23:34,400 --> 00:23:38,890 first one if I put in a second translation, T2, which is 356 00:23:38,890 --> 00:23:42,301 totally incommensurate with T1. 357 00:23:42,301 --> 00:23:44,890 The things blow up in my face. 358 00:23:44,890 --> 00:23:46,550 I don't have a lattice. 359 00:23:46,550 --> 00:23:49,540 I will get lattice points all over the place. 360 00:23:49,540 --> 00:23:51,100 So this is impossible. 361 00:23:51,100 --> 00:23:59,125 So if T1 is not equal to T2, this space self destructs. 362 00:24:02,090 --> 00:24:07,070 If T1 is a multiple of T2, then if I say a translation 363 00:24:07,070 --> 00:24:12,380 exists of length T1, and I add a second translation twice T1. 364 00:24:12,380 --> 00:24:13,990 I've already got those lattice points. 365 00:24:13,990 --> 00:24:15,880 And that's nothing new. 366 00:24:15,880 --> 00:24:18,400 So if I want to say there's a second translational 367 00:24:18,400 --> 00:24:22,820 periodicity in the space, the only thing I can do is pick a 368 00:24:22,820 --> 00:24:27,200 T2 which is not parallel to T1. 369 00:24:27,200 --> 00:24:30,120 And then this T2 will pick up everything in the space. 370 00:24:30,120 --> 00:24:33,770 It's going to take these lattice points and generate 371 00:24:33,770 --> 00:24:35,890 them at equal intervals, T2. 372 00:24:35,890 --> 00:24:39,360 But for that matter, it acts on everything in the space. 373 00:24:39,360 --> 00:24:44,350 So we could think of this translation, T2, as moving 374 00:24:44,350 --> 00:24:48,610 this entire infinite string of lattice points separated by T1 375 00:24:48,610 --> 00:24:55,330 and giving me a whole string of lattice points. 376 00:24:57,990 --> 00:25:09,690 So now, having taken two noncollinear translations, 377 00:25:09,690 --> 00:25:16,100 those translations will imply a two-dimensional space 378 00:25:16,100 --> 00:25:32,990 lattice in which motifs will be hung at translations nT1 379 00:25:32,990 --> 00:25:40,390 plus mT2 where n, m are integers that go from minus 380 00:25:40,390 --> 00:25:42,980 infinity to plus infinity. 381 00:25:45,660 --> 00:25:48,500 OK, so this is a two-dimensional space lattice, 382 00:25:48,500 --> 00:25:52,730 or sometimes it's referred to by the term a lattice net. 383 00:25:55,230 --> 00:25:55,625 Good term. 384 00:25:55,625 --> 00:25:57,485 It looks like what fishermen throw in the 385 00:25:57,485 --> 00:25:58,720 water to snag fish. 386 00:25:58,720 --> 00:26:03,110 So it is a net, in terms of something that we're familiar 387 00:26:03,110 --> 00:26:04,640 with in everyday life. 388 00:26:09,260 --> 00:26:09,500 All right. 389 00:26:09,500 --> 00:26:13,040 So we've specified a space lattice, but it is a highly 390 00:26:13,040 --> 00:26:15,020 redundant pattern. 391 00:26:15,020 --> 00:26:19,150 We've got a doubly infinite set of lattice points. 392 00:26:19,150 --> 00:26:23,750 And the unique nature of the pattern, the structure, is 393 00:26:23,750 --> 00:26:26,630 going to be whatever is associated 394 00:26:26,630 --> 00:26:29,660 with one lattice point. 395 00:26:29,660 --> 00:26:33,050 So if we specify what's going on in the vicinity of one 396 00:26:33,050 --> 00:26:36,050 lattice point and establish that at every other lattice 397 00:26:36,050 --> 00:26:38,010 point, we have the entire infinite 398 00:26:38,010 --> 00:26:40,150 two-dimensional structure. 399 00:26:40,150 --> 00:26:45,600 So let's ask now, how we can define the area that is unique 400 00:26:45,600 --> 00:26:46,850 to one lattice point. 401 00:26:49,380 --> 00:26:51,155 And there are several ways of doing this. 402 00:26:54,850 --> 00:27:00,120 We can specify T1, and then specify T2. 403 00:27:05,830 --> 00:27:08,580 We'll repeat T1 up to here. 404 00:27:08,580 --> 00:27:15,980 T1 will repeat to T2 over to here, and we will have defined 405 00:27:15,980 --> 00:27:19,330 the area that is uniquely associated 406 00:27:19,330 --> 00:27:21,540 with one lattice point. 407 00:27:21,540 --> 00:27:25,190 So if I can tell you what's going on within the confines 408 00:27:25,190 --> 00:27:31,940 of this parallelogram, then I have given you the unique part 409 00:27:31,940 --> 00:27:35,470 of what is hung at a lattice point, and which is reproduced 410 00:27:35,470 --> 00:27:37,630 only by translation. 411 00:27:37,630 --> 00:27:40,130 And this is a very important construct. 412 00:27:40,130 --> 00:27:46,100 It is something that is referred to as the unit cell, 413 00:27:46,100 --> 00:27:48,220 or sometimes just cell for short. 414 00:27:53,840 --> 00:28:02,840 And now we encounter a curious ambiguity. 415 00:28:02,840 --> 00:28:07,915 T1 and T2 imply an array of lattice points. 416 00:28:15,870 --> 00:28:20,480 And this particular choice of T1 and T2 define a cell. 417 00:28:25,240 --> 00:28:28,270 But the reverse is not true. 418 00:28:28,270 --> 00:28:29,695 If I give you-- 419 00:28:36,290 --> 00:28:37,080 and what do I want to say? 420 00:28:37,080 --> 00:28:43,810 That a particular lattice does not specify 421 00:28:43,810 --> 00:28:45,960 a unique unit cell. 422 00:28:45,960 --> 00:28:49,220 Or, stated another way, there are many different choices for 423 00:28:49,220 --> 00:28:56,170 T1 and T2 that would specify the same unique area. 424 00:28:56,170 --> 00:29:00,210 I could take this as a T1 prime, and then I would have a 425 00:29:00,210 --> 00:29:03,380 cell that looks like this. 426 00:29:03,380 --> 00:29:06,600 And that would also define the area associated with one 427 00:29:06,600 --> 00:29:07,770 lattice point. 428 00:29:07,770 --> 00:29:11,600 It's not clear this oblique thing with one very long T1 429 00:29:11,600 --> 00:29:16,400 prime would have very much to commend it, but there are many 430 00:29:16,400 --> 00:29:28,780 ways, many choices, for T1 and T2, to find 431 00:29:28,780 --> 00:29:30,150 exactly the same lattice. 432 00:29:45,260 --> 00:29:48,950 We could take this as T1, this as T2, same lattice, same 433 00:29:48,950 --> 00:29:50,010 array of lattice points. 434 00:29:50,010 --> 00:29:53,060 Take this as T1, this as T2, same array of lattice points. 435 00:29:53,060 --> 00:29:57,450 Take this as T1, this as T2, that's yet another choice. 436 00:29:57,450 --> 00:30:01,500 So there are an infinite number of translations. 437 00:30:01,500 --> 00:30:06,620 Special name for this, to introduce a bit of jargon 438 00:30:06,620 --> 00:30:09,550 again, these are very often called conjugate translations. 439 00:30:25,060 --> 00:30:27,920 So all this is still nothing more than simple geometry, but 440 00:30:27,920 --> 00:30:32,520 if you invent some fancy words, you really have to do 441 00:30:32,520 --> 00:30:33,840 that to impress your friends. 442 00:30:33,840 --> 00:30:35,360 Yeah, you had a question here? 443 00:30:35,360 --> 00:30:36,340 AUDIENCE: Yeah. 444 00:30:36,340 --> 00:30:38,300 So you can define magnitude for T1 and 445 00:30:38,300 --> 00:30:39,770 T2, all those constants. 446 00:30:39,770 --> 00:30:41,975 But you're changing the directions of T1 and T2, and 447 00:30:41,975 --> 00:30:43,690 you're saying, even though you're changing those 448 00:30:43,690 --> 00:30:45,100 directions, it's still the same unit cell? 449 00:30:45,100 --> 00:30:49,580 PROFESSOR: Yeah, provided I have some new translation like 450 00:30:49,580 --> 00:30:54,630 this one here, which is really this T1 plus this T2, this 451 00:30:54,630 --> 00:30:57,200 would define a very, very oblique cell 452 00:30:57,200 --> 00:30:59,140 that looks like this. 453 00:30:59,140 --> 00:31:04,983 But yet, the terminal points of T1 prime and-- 454 00:31:04,983 --> 00:31:06,280 I need a term for this. 455 00:31:06,280 --> 00:31:08,590 I'll call this T2 prime. 456 00:31:08,590 --> 00:31:11,510 The terminal points here are going to be exactly the same 457 00:31:11,510 --> 00:31:14,140 as the nodes that are defined here. 458 00:31:14,140 --> 00:31:19,880 So they are two choices for one in the same lattice. 459 00:31:19,880 --> 00:31:25,020 OK, so the implication of this is we're going to 460 00:31:25,020 --> 00:31:26,270 have to have rules. 461 00:31:29,950 --> 00:31:34,980 And some of these make common sense. 462 00:31:34,980 --> 00:31:38,540 You could pick, in a two-dimensional lattice, some 463 00:31:38,540 --> 00:31:44,170 absolutely ridiculous unit cells defined in terms of very 464 00:31:44,170 --> 00:31:55,080 long vectors that define a cell that is a very, very 465 00:31:55,080 --> 00:31:56,910 oblique cell. 466 00:31:56,910 --> 00:31:59,590 So it's the lattice that's defined by 467 00:31:59,590 --> 00:32:02,230 this translation here. 468 00:32:02,230 --> 00:32:06,140 And the next translation parallel to this one would go 469 00:32:06,140 --> 00:32:08,905 way up to something like this. 470 00:32:13,540 --> 00:32:14,450 So there's a T1. 471 00:32:14,450 --> 00:32:15,360 There's a T2. 472 00:32:15,360 --> 00:32:18,070 This crazy cell here works. 473 00:32:18,070 --> 00:32:19,850 That's the area that's associated 474 00:32:19,850 --> 00:32:20,780 with one lattice point. 475 00:32:20,780 --> 00:32:23,860 But clearly, it has absolutely nothing to 476 00:32:23,860 --> 00:32:25,630 commend this choice. 477 00:32:25,630 --> 00:32:27,940 There's nothing to be gained by using these long 478 00:32:27,940 --> 00:32:30,640 translations that make very extreme 479 00:32:30,640 --> 00:32:33,010 intertranslation angles. 480 00:32:33,010 --> 00:32:35,030 Your intuition would say, why would you want to 481 00:32:35,030 --> 00:32:36,290 do that, you dummy? 482 00:32:36,290 --> 00:32:39,880 Let's take these as the translations, which is 483 00:32:39,880 --> 00:32:42,620 something I sort of naturally did all along. 484 00:32:42,620 --> 00:32:43,740 And what are we doing? 485 00:32:43,740 --> 00:32:46,435 We're picking the shortest translations. 486 00:32:57,230 --> 00:32:59,330 So there's one very common sense rule. 487 00:33:04,080 --> 00:33:07,940 Another rule, getting a little bit ahead of the game, but 488 00:33:07,940 --> 00:33:11,700 suppose I examine the lattice that describes the arrangement 489 00:33:11,700 --> 00:33:15,400 of four floor tiles. 490 00:33:15,400 --> 00:33:17,910 If I take a lattice point right at the point of 491 00:33:17,910 --> 00:33:23,620 intersection of the joins between the tiles, that is a 492 00:33:23,620 --> 00:33:27,310 cell that is exactly square. 493 00:33:27,310 --> 00:33:30,420 And it's exactly square because there's a four-fold 494 00:33:30,420 --> 00:33:35,460 axis in that pattern that leaves things invariant after 495 00:33:35,460 --> 00:33:39,070 a 90-degree rotation. 496 00:33:39,070 --> 00:33:42,310 So if that's the nature of a lattice, if it in fact is 497 00:33:42,310 --> 00:33:45,970 constrained because of the symmetry that is there to have 498 00:33:45,970 --> 00:33:48,790 two translations identical in length, in fact, 499 00:33:48,790 --> 00:33:50,170 identical in every way. 500 00:33:52,760 --> 00:33:56,270 Pick those as the choice of the cell to emphasize that 501 00:33:56,270 --> 00:33:59,380 special key feature of the lattice. 502 00:33:59,380 --> 00:34:07,470 So a second row, which is a second and final one, is to 503 00:34:07,470 --> 00:34:16,100 pick a T1 and T2 that displays the symmetry, 504 00:34:16,100 --> 00:34:31,334 if any, of the lattice. 505 00:34:35,670 --> 00:34:38,800 Which introduces us to a feature which we'll elaborate 506 00:34:38,800 --> 00:34:42,409 much more later on, that the translational periodicity and 507 00:34:42,409 --> 00:34:44,110 the symmetry of the lattice are two 508 00:34:44,110 --> 00:34:46,159 things that go together. 509 00:34:46,159 --> 00:34:49,780 That the fact that there is translational symmetry 510 00:34:49,780 --> 00:34:53,620 drastically reduces the number of symmetries that you could 511 00:34:53,620 --> 00:34:57,410 have, the fact that there are symmetries possible for 512 00:34:57,410 --> 00:34:59,800 presence in a lattice restricts the number of 513 00:34:59,800 --> 00:35:02,000 different kinds of cells. 514 00:35:02,000 --> 00:35:04,960 So these are two aspects of the pattern, the symmetry 515 00:35:04,960 --> 00:35:08,600 that's in it and its periodicity. 516 00:35:08,600 --> 00:35:13,770 OK, but these are the only two rules that we really need to 517 00:35:13,770 --> 00:35:15,920 pick what's called the standard cell. 518 00:35:22,140 --> 00:35:25,660 Take the shortest translations that are available to you, and 519 00:35:25,660 --> 00:35:31,520 pick translations that display the symmetry that may be 520 00:35:31,520 --> 00:35:32,770 present in the lattice. 521 00:35:47,320 --> 00:35:48,670 Any questions or comments? 522 00:36:10,840 --> 00:36:13,410 Any comments? 523 00:36:13,410 --> 00:36:21,240 OK, I think I have time for one last major point of 524 00:36:21,240 --> 00:36:22,460 discussion. 525 00:36:22,460 --> 00:36:27,220 And what we are going to embark on now is a process of 526 00:36:27,220 --> 00:36:31,170 synthesis, which will occupy us for a couple of weeks. 527 00:36:36,966 --> 00:36:43,630 What I'm going to do is start with a translation. 528 00:36:43,630 --> 00:36:49,180 And this defines an infinite string of lattice points. 529 00:36:49,180 --> 00:36:54,830 Now, I know that in two dimensions, I have two kinds 530 00:36:54,830 --> 00:36:58,630 of symmetry operations that are present, either rotation 531 00:36:58,630 --> 00:37:00,930 or translation. 532 00:37:00,930 --> 00:37:04,160 So now, I'm going to ask the question, what happens if I 533 00:37:04,160 --> 00:37:07,610 define a lattice, or at least one translation in a lattice, 534 00:37:07,610 --> 00:37:14,600 and now I add to that lattice an operation of rotation, OK? 535 00:37:14,600 --> 00:37:15,050 I can do that. 536 00:37:15,050 --> 00:37:18,170 We've seen examples of translationally periodic 537 00:37:18,170 --> 00:37:21,390 patterns that have rotational symmetry. 538 00:37:21,390 --> 00:37:27,190 So let me suppose I add to this space a rotation 539 00:37:27,190 --> 00:37:30,730 operation, A alpha. 540 00:37:30,730 --> 00:37:34,490 And there's no unique origin to the translation. 541 00:37:34,490 --> 00:37:37,330 There is no unique location for a lattice point. 542 00:37:37,330 --> 00:37:42,460 So I can put the operation A alpha in at my designated 543 00:37:42,460 --> 00:37:45,380 lattice point. 544 00:37:45,380 --> 00:37:50,440 Now, if I do that, all hell breaks loose because now I 545 00:37:50,440 --> 00:37:53,150 have a rotation operation A alpha. 546 00:37:53,150 --> 00:37:55,700 This has a translation coming out of it. 547 00:37:55,700 --> 00:37:59,270 That translation will be repeated up here, an angle 548 00:37:59,270 --> 00:38:00,520 alpha away. 549 00:38:00,520 --> 00:38:03,160 A alpha acts on everything, so it's going to take this 550 00:38:03,160 --> 00:38:09,360 translation and move it over here to a location for another 551 00:38:09,360 --> 00:38:10,210 translation. 552 00:38:10,210 --> 00:38:11,250 This is a lattice point. 553 00:38:11,250 --> 00:38:12,540 This is a lattice point. 554 00:38:12,540 --> 00:38:16,340 And this business is going to go on until it comes around 555 00:38:16,340 --> 00:38:19,090 full circle. 556 00:38:19,090 --> 00:38:22,040 Let me focus my attention on just one of these 557 00:38:22,040 --> 00:38:26,170 translations, and this will be this one up here, the one that 558 00:38:26,170 --> 00:38:28,960 is alpha away from the first in a 559 00:38:28,960 --> 00:38:31,190 counterclockwise direction. 560 00:38:31,190 --> 00:38:35,080 So here sits another translation, and that means 561 00:38:35,080 --> 00:38:36,330 this is a lattice point. 562 00:38:41,220 --> 00:38:43,750 At this end of the translation, the same thing is 563 00:38:43,750 --> 00:38:44,760 going to happen. 564 00:38:44,760 --> 00:38:48,970 The operation A alpha is moved to this location at the end of 565 00:38:48,970 --> 00:38:50,100 the translation. 566 00:38:50,100 --> 00:38:52,520 That means that anything coming out of this lattice 567 00:38:52,520 --> 00:38:57,790 point must also be repeated at angular intervals, alpha. 568 00:38:57,790 --> 00:39:00,550 And now I'm going to focus my attention on 569 00:39:00,550 --> 00:39:02,900 this translation here. 570 00:39:02,900 --> 00:39:05,960 And there will be a translation that goes up like 571 00:39:05,960 --> 00:39:09,140 this, and this is a lattice point. 572 00:39:14,390 --> 00:39:17,790 And now, in the words of that famous musical, there's big 573 00:39:17,790 --> 00:39:20,960 trouble in River City. 574 00:39:20,960 --> 00:39:24,150 Because we started out by saying that everything in the 575 00:39:24,150 --> 00:39:28,090 space was periodic at an interval, T, a translational 576 00:39:28,090 --> 00:39:29,320 interval, T. 577 00:39:29,320 --> 00:39:33,540 This is T. This is T. This is T. Here we 578 00:39:33,540 --> 00:39:35,000 have a lattice point. 579 00:39:35,000 --> 00:39:41,270 This jolly well has to be T as well, or we've contradicted 580 00:39:41,270 --> 00:39:44,250 the basic assumption of our construction. 581 00:39:44,250 --> 00:39:46,840 Well, that's over restrictive. 582 00:39:46,840 --> 00:39:53,165 This doesn't have to be T, but it has to be some multiple, p, 583 00:39:53,165 --> 00:39:54,600 of that translation. 584 00:39:54,600 --> 00:39:55,950 p could be 0. 585 00:39:55,950 --> 00:39:57,180 p could be 5. 586 00:39:57,180 --> 00:40:00,640 But it has to be an integral number of translations because 587 00:40:00,640 --> 00:40:03,620 this translational periodicity has to work everywhere, 588 00:40:03,620 --> 00:40:06,620 including up on the top of this trapezohedron. 589 00:40:09,610 --> 00:40:12,450 So that's a constraint. 590 00:40:12,450 --> 00:40:14,080 This angle is alpha. 591 00:40:14,080 --> 00:40:17,620 We cannot let alpha be arbitrary because the only way 592 00:40:17,620 --> 00:40:21,010 we can add a rotation operation A alpha to a lattice 593 00:40:21,010 --> 00:40:25,040 is for a value of alpha which makes this translation be a 594 00:40:25,040 --> 00:40:28,270 multiple of the original one. 595 00:40:28,270 --> 00:40:30,990 Now, let me take this geometry, and I'm going to 596 00:40:30,990 --> 00:40:33,980 extract the basic constraint from it. 597 00:40:33,980 --> 00:40:37,745 This is some integer, p times T. This is T. This is T. This 598 00:40:37,745 --> 00:40:39,920 is T. This is alpha. 599 00:40:39,920 --> 00:40:44,770 Let me lickety split drop down a perpendicular to the 600 00:40:44,770 --> 00:40:46,770 original translation. 601 00:40:46,770 --> 00:40:52,730 This is T times the cosine of alpha. 602 00:40:52,730 --> 00:40:56,450 This is T times the cosine of alpha. 603 00:40:56,450 --> 00:41:03,490 This total length is T. This length in here is p times T. 604 00:41:03,490 --> 00:41:07,060 And now I can go away from the geometry to an equation, 605 00:41:07,060 --> 00:41:09,650 something you probably prefer to deal with. 606 00:41:09,650 --> 00:41:14,490 And what this constraint is expressed analytically is that 607 00:41:14,490 --> 00:41:26,690 my original translation, T, minus twice T times the cosine 608 00:41:26,690 --> 00:41:36,090 of alpha has to come out equal to an integer, p times T. And 609 00:41:36,090 --> 00:41:39,390 there's my constraint. 610 00:41:39,390 --> 00:41:41,980 Alpha has to satisfy that condition. 611 00:41:41,980 --> 00:41:46,330 Well, I can immediately cancel the T and write this as one 612 00:41:46,330 --> 00:41:52,331 minus 2 cosine of alpha is equal to an integer, p. 613 00:41:52,331 --> 00:41:55,470 And it figures that that has to be the case because none of 614 00:41:55,470 --> 00:41:58,350 this construction depends on the size of the original 615 00:41:58,350 --> 00:42:00,070 translation that I took. 616 00:42:00,070 --> 00:42:03,980 And now, let me solve for the values of alpha which are 617 00:42:03,980 --> 00:42:06,360 compatible with a lattice. 618 00:42:06,360 --> 00:42:11,270 This says that cosine of alpha is 1 minus p over 2. 619 00:42:14,400 --> 00:42:19,290 And unless that condition holds, my combination is 620 00:42:19,290 --> 00:42:20,540 incompatible. 621 00:42:24,880 --> 00:42:29,060 So I'm going to let that stew with you until next time. 622 00:42:29,060 --> 00:42:33,480 But what we've set up is something where we can just 623 00:42:33,480 --> 00:42:37,280 plug and chug, put in different values of p. 624 00:42:37,280 --> 00:42:43,380 And if I start out with a value of p, and let's let p be 625 00:42:43,380 --> 00:42:52,620 equal to 4, and then find one minus p over 2, which is 626 00:42:52,620 --> 00:42:55,790 supposedly the cosine of an angle, alpha. 627 00:42:55,790 --> 00:42:59,230 If that's 4, I will have minus 3/2. 628 00:42:59,230 --> 00:43:02,550 And the value of alpha obviously does not exist. 629 00:43:02,550 --> 00:43:06,560 Cosine of alpha cannot get greater than 1. 630 00:43:06,560 --> 00:43:15,210 If p is equal to 3, then 1 minus 3 over 2 is minus 2 over 631 00:43:15,210 --> 00:43:17,850 2, or minus 1. 632 00:43:17,850 --> 00:43:19,720 You like the way I do that arithmetic in my 633 00:43:19,720 --> 00:43:22,370 head just like that? 634 00:43:22,370 --> 00:43:30,030 And the angle whose cosine is minus 1 is 180 degrees. 635 00:43:30,030 --> 00:43:33,930 And what that says is that a two-fold axis works. 636 00:43:40,290 --> 00:43:44,580 So I can drop a two-fold axis into a net. 637 00:43:44,580 --> 00:43:47,920 And what that's going to do is take my original translation, 638 00:43:47,920 --> 00:43:53,710 rotate it 180 degrees, and the second translation is 639 00:43:53,710 --> 00:43:54,960 going to sit here. 640 00:43:57,260 --> 00:44:01,250 Rotate it 180 degrees in the reverse direction, and then 641 00:44:01,250 --> 00:44:03,930 the second lattice point sits here. 642 00:44:03,930 --> 00:44:07,360 And lo and behold, just as advertised, the distance 643 00:44:07,360 --> 00:44:10,500 between the first lattice point and the final lattice 644 00:44:10,500 --> 00:44:11,750 point is three translations. 645 00:44:14,570 --> 00:44:18,740 So I can put a two-fold axis in any lattice whatsoever 646 00:44:18,740 --> 00:44:23,730 because this is compatible simply with a lattice row. 647 00:44:23,730 --> 00:44:27,520 So one possible combination of rotation in a lattice is going 648 00:44:27,520 --> 00:44:33,140 to be any lattice whatsoever, and what we can add to this is 649 00:44:33,140 --> 00:44:35,020 a rotation operation A pi. 650 00:44:41,140 --> 00:44:44,630 And we'll have two full rotation operations which are 651 00:44:44,630 --> 00:44:46,600 translationally equivalent. 652 00:44:49,312 --> 00:44:49,770 All right. 653 00:44:49,770 --> 00:44:50,970 Several integers to go. 654 00:44:50,970 --> 00:44:53,230 We would want to try p equals 2. 655 00:44:53,230 --> 00:44:54,450 That's going to work. 656 00:44:54,450 --> 00:44:56,900 p equals plus 1 is going to work. 657 00:44:56,900 --> 00:44:59,500 p equals 0 is going to work. 658 00:44:59,500 --> 00:45:03,700 And we will find a very limited number of rotational 659 00:45:03,700 --> 00:45:07,560 operations that are compatible with a lattice. 660 00:45:07,560 --> 00:45:12,580 And this is going to give us a small number of the possible 661 00:45:12,580 --> 00:45:16,660 combinations of lattice and rotational symmetry in two 662 00:45:16,660 --> 00:45:17,910 dimensions. 663 00:45:19,950 --> 00:45:22,010 So we'll pick up from there next time, and we'll very 664 00:45:22,010 --> 00:45:26,750 quickly determine the remaining possibilities and 665 00:45:26,750 --> 00:45:29,400 take a look at what the arrangement of symmetry 666 00:45:29,400 --> 00:45:30,925 elements look like in these lattices. 667 00:45:36,510 --> 00:45:39,620 OK, once again, I have some extra copies of the syllabus 668 00:45:39,620 --> 00:45:42,240 if somebody did not get one. 669 00:45:42,240 --> 00:45:45,910 And I'll have extra copies of the problem set.