1 00:00:06,692 --> 00:00:09,260 PROFESSOR: OK, ready for more? 2 00:00:13,670 --> 00:00:15,070 I defined a problem for you. 3 00:00:15,070 --> 00:00:17,110 Now let's address it. 4 00:00:17,110 --> 00:00:20,980 Said that if we have one coordinate system, and if we 5 00:00:20,980 --> 00:00:27,560 have some vector, q, that's defined as a second-rank 6 00:00:27,560 --> 00:00:35,390 tensor, aij times some other vector p sub j-- 7 00:00:35,390 --> 00:00:38,360 and let me digress in passing. 8 00:00:38,360 --> 00:00:45,760 I am very careful to say "a second-rank tensor" and not a 9 00:00:45,760 --> 00:00:51,820 "second-order tensor," because higher-order order terms means 10 00:00:51,820 --> 00:00:54,260 negligible and non-important. 11 00:00:54,260 --> 00:00:58,140 And when I say "second-order tensor," I don't mean to say 12 00:00:58,140 --> 00:01:00,000 it's not important and negligible. 13 00:01:00,000 --> 00:01:03,430 It's very important, so I say "rank," which has some sort of 14 00:01:03,430 --> 00:01:04,390 dignity to it. 15 00:01:04,390 --> 00:01:06,160 So I don't like the term "order," because it has 16 00:01:06,160 --> 00:01:07,500 another meaning. 17 00:01:07,500 --> 00:01:15,500 OK, so here is a tensor that relates a vector pj to give us 18 00:01:15,500 --> 00:01:20,020 the components of a vector qi. 19 00:01:20,020 --> 00:01:24,860 If we change coordinate system, the components of p, 20 00:01:24,860 --> 00:01:28,290 representing exactly one in the same vector, wink on and 21 00:01:28,290 --> 00:01:30,100 off and take different values. 22 00:01:30,100 --> 00:01:33,470 The values for q take on different values, and 23 00:01:33,470 --> 00:01:37,490 therefore, of necessity, the three by three array of 24 00:01:37,490 --> 00:01:40,450 coefficients, which relates these different numbers, must 25 00:01:40,450 --> 00:01:44,350 also change its numerical values. 26 00:01:44,350 --> 00:01:47,150 And I hopefully convinced you at the end of last hour that 27 00:01:47,150 --> 00:01:50,870 there's times when you might actually want to do this when 28 00:01:50,870 --> 00:01:53,750 you're cutting out a particular sample from a 29 00:01:53,750 --> 00:01:55,000 single-crystal specimen. 30 00:01:57,780 --> 00:02:02,990 How do we get the new tensor in terms of the direction 31 00:02:02,990 --> 00:02:07,890 cosine scheme that specifies the change of axes and the 32 00:02:07,890 --> 00:02:09,849 original tensor? 33 00:02:09,849 --> 00:02:13,520 So I am going to refer to my notes quite closely here, 34 00:02:13,520 --> 00:02:17,270 because I want it to come out pretty and not have to 35 00:02:17,270 --> 00:02:19,350 redefine variables when I'm done. 36 00:02:19,350 --> 00:02:22,870 So let's start with the original tensor relation, q 37 00:02:22,870 --> 00:02:26,040 sub i equals aij times p sub j. 38 00:02:32,880 --> 00:02:37,780 Now, what we want is something of the form q sub i prime 39 00:02:37,780 --> 00:02:42,780 equals aij prime times p sub j prime. 40 00:02:42,780 --> 00:02:46,050 And we know the relations forward and reverse in terms 41 00:02:46,050 --> 00:02:52,680 of the direction cosine scheme cij So let's begin by writing 42 00:02:52,680 --> 00:02:58,640 qi prime in terms of qi And we know how 43 00:02:58,640 --> 00:03:00,090 that is going to transform. 44 00:03:00,090 --> 00:03:03,990 It's going to be elements of the direction cosine scheme 45 00:03:03,990 --> 00:03:10,430 cim times q sub m. 46 00:03:10,430 --> 00:03:15,190 So that will give me the i-th component of q in the new 47 00:03:15,190 --> 00:03:18,090 coordinate system. 48 00:03:18,090 --> 00:03:22,010 I know how q arises from the applied vector p. 49 00:03:22,010 --> 00:03:27,110 So let me write qm in terms of the applied pj. 50 00:03:27,110 --> 00:03:31,320 And this is going to be aij times-- 51 00:03:37,900 --> 00:03:39,750 be careful of my variables here-- 52 00:03:39,750 --> 00:03:45,295 this is going to be a ml times p sub l. 53 00:03:48,630 --> 00:03:53,510 And that is going to, from my definition of a second-rank 54 00:03:53,510 --> 00:04:01,890 tensor, give me the m-th component of q. 55 00:04:01,890 --> 00:04:04,245 So far, so good. 56 00:04:04,245 --> 00:04:08,170 I've got two different repeated subscripts here, so 57 00:04:08,170 --> 00:04:10,590 this is a double summation. 58 00:04:10,590 --> 00:04:14,890 Now, I'll have what I want to have, namely a q sub i prime 59 00:04:14,890 --> 00:04:19,829 on the left-hand side and a p sub j prime on the right-hand 60 00:04:19,829 --> 00:04:25,540 side if I can express the original components of p in 61 00:04:25,540 --> 00:04:27,790 terms of the new components of p. 62 00:04:27,790 --> 00:04:31,330 And I do that by the reverse transformation. 63 00:04:31,330 --> 00:04:35,160 So let me now write a ml. 64 00:04:35,160 --> 00:04:42,600 And then in place of p sub l, I will write cjl 65 00:04:42,600 --> 00:04:46,370 times p sub j prime. 66 00:04:46,370 --> 00:04:48,780 Notice the inverted order of the subscripts. 67 00:04:48,780 --> 00:04:50,850 That is the reverse transformation that's going to 68 00:04:50,850 --> 00:04:52,100 give me p sub l. 69 00:04:55,480 --> 00:04:57,660 So you really have now what I'm after. 70 00:05:00,420 --> 00:05:05,870 This is a triple summation in m, l, and j. 71 00:05:05,870 --> 00:05:12,540 I can write the terms that are in what is going to be a 72 00:05:12,540 --> 00:05:15,110 triple summation over a product of terms. 73 00:05:15,110 --> 00:05:17,810 I could write these terms in any order. 74 00:05:17,810 --> 00:05:22,280 So to simplify it, let me write q sub i prime is equal 75 00:05:22,280 --> 00:05:34,590 to cim cjl times a ml, times p sub j prime. 76 00:05:37,150 --> 00:05:42,530 And now, hotcha, I've got an expression that has q prime on 77 00:05:42,530 --> 00:05:46,440 the left and p prime on the right, and paying close 78 00:05:46,440 --> 00:05:49,640 attention to my notes so that the subscripts all came out 79 00:05:49,640 --> 00:05:51,260 the way I would like them to. 80 00:05:51,260 --> 00:06:01,030 So what this says is that the transform tensor aij prime is 81 00:06:01,030 --> 00:06:04,535 going to be equal to, by definition-- 82 00:06:07,100 --> 00:06:09,640 or what we've shown here, it's going to be equal-- 83 00:06:09,640 --> 00:06:10,980 just picking off terms-- it's going to be 84 00:06:10,980 --> 00:06:14,700 cim cjl times a ml. 85 00:06:19,820 --> 00:06:23,330 Just picking off these terms. 86 00:06:23,330 --> 00:06:28,810 m has no physical meaning, because m simply is an index 87 00:06:28,810 --> 00:06:30,020 of summation. 88 00:06:30,020 --> 00:06:32,240 l has no specific meaning. 89 00:06:32,240 --> 00:06:34,170 That is just an index of summation. 90 00:06:34,170 --> 00:06:38,150 But the i and the j do have meaning. 91 00:06:38,150 --> 00:06:42,370 They go with the i and j on the particular tensor element 92 00:06:42,370 --> 00:06:45,610 that we were attempting to evaluate. 93 00:06:45,610 --> 00:06:48,580 So in other words, to be specific, if we want the new 94 00:06:48,580 --> 00:06:53,116 value of the tensor element a1 2 prime, it's going to be c1 95 00:06:53,116 --> 00:07:01,560 something, c2 something, and those somethings m and l would 96 00:07:01,560 --> 00:07:04,160 vary from 1 to 3. 97 00:07:04,160 --> 00:07:07,410 So if I write this out not in the reduced subscript notation 98 00:07:07,410 --> 00:07:11,650 but put a summation sign in there, so a12 prime is going 99 00:07:11,650 --> 00:07:18,605 to be the sum over m and the sum over l of terms c1m-- 100 00:07:18,605 --> 00:07:20,750 the first index is always 1-- 101 00:07:20,750 --> 00:07:21,910 c2 something-- 102 00:07:21,910 --> 00:07:23,600 first index is always 2-- 103 00:07:23,600 --> 00:07:26,085 and then m and l take on all possible values. 104 00:07:33,270 --> 00:07:35,290 OK, we've got two results. 105 00:07:35,290 --> 00:07:38,950 We learned how to transform a vector. 106 00:07:38,950 --> 00:07:43,890 And a vector, if you will, is simply a tensor of first rank. 107 00:07:43,890 --> 00:07:47,700 And transforming the vector, we summed over all three 108 00:07:47,700 --> 00:07:49,940 components of the original vector. 109 00:07:49,940 --> 00:07:53,080 And the coefficients in that summation were 110 00:07:53,080 --> 00:07:56,100 one direction cosine. 111 00:07:56,100 --> 00:07:58,810 Now we're transforming a second-rank tensor. 112 00:07:58,810 --> 00:08:04,010 Again, each new element is a linear combination of all nine 113 00:08:04,010 --> 00:08:07,200 of the elements in the original tensor, and the 114 00:08:07,200 --> 00:08:11,640 coefficients are a product of two direction cosines. 115 00:08:11,640 --> 00:08:13,150 So we've got two points. 116 00:08:13,150 --> 00:08:16,020 Let's draw a line through them, and we can say that, in 117 00:08:16,020 --> 00:08:19,280 general, any new tensor element of any rank-- 118 00:08:19,280 --> 00:08:23,990 and you could prove it through exactly this method by going 119 00:08:23,990 --> 00:08:27,180 up, now, to the third rank, fourth rank, and so on, and 120 00:08:27,180 --> 00:08:29,380 writing substitutions of this form-- 121 00:08:29,380 --> 00:08:34,919 it turns out that a new tensor element aijkl however far you 122 00:08:34,919 --> 00:08:38,970 want to go, is going to be given by a linear combination 123 00:08:38,970 --> 00:08:48,010 of direction cosine element ciI, cj capital J, ck capital 124 00:08:48,010 --> 00:08:54,630 K, cl capital L, times however far you have to go times 125 00:08:54,630 --> 00:08:59,200 aijkl, and so on. 126 00:08:59,200 --> 00:09:02,690 So these are the true indices. 127 00:09:02,690 --> 00:09:04,510 These have physical meaning. 128 00:09:04,510 --> 00:09:11,200 These have relevance to how a particular 129 00:09:11,200 --> 00:09:15,360 property will behave. 130 00:09:15,360 --> 00:09:20,870 The capital I, capital J, capital L, and so on, are what 131 00:09:20,870 --> 00:09:24,980 we referred to last time as dummy indices. 132 00:09:24,980 --> 00:09:26,295 These are indices of summation. 133 00:09:33,040 --> 00:09:37,020 But the first index on the direction cosines 134 00:09:37,020 --> 00:09:38,940 has specific meaning. 135 00:09:38,940 --> 00:09:43,280 They are tied to the indices on the subscript of the term 136 00:09:43,280 --> 00:09:44,530 that you would like to evaluate. 137 00:09:52,040 --> 00:09:56,370 So we've got now a very profound relation for a tensor 138 00:09:56,370 --> 00:09:57,350 of any rank. 139 00:09:57,350 --> 00:10:01,590 And really, it just involves substitutions using the 140 00:10:01,590 --> 00:10:05,580 reverse or the forward transformation until you get 141 00:10:05,580 --> 00:10:11,200 one element on the one side related to another element on 142 00:10:11,200 --> 00:10:12,400 the right-hand side. 143 00:10:12,400 --> 00:10:15,960 And this is a specific element in the new tensor, in our 144 00:10:15,960 --> 00:10:17,725 case, of the second-rank tensor, aml. 145 00:10:31,090 --> 00:10:34,290 This is all very abstract. 146 00:10:34,290 --> 00:10:37,290 It is something that we'll have to do a couple of times 147 00:10:37,290 --> 00:10:41,670 for a real problem before you see how it works out. 148 00:10:41,670 --> 00:10:47,010 The number of elements that figure into these 149 00:10:47,010 --> 00:10:51,800 transformations is really astronomical. 150 00:10:51,800 --> 00:10:56,120 Suppose, for example, the tensor involved were something 151 00:10:56,120 --> 00:11:01,700 like the elastic stiffness tensor, which is 152 00:11:01,700 --> 00:11:04,030 represented by c. 153 00:11:04,030 --> 00:11:07,350 And we need four subscripts. 154 00:11:07,350 --> 00:11:11,140 This is a tensor or fourth rank. 155 00:11:11,140 --> 00:11:17,410 If we wanted to transform a particular stiffness to a new 156 00:11:17,410 --> 00:11:22,840 coordinate system, we would need a summation ci capital I, 157 00:11:22,840 --> 00:11:35,920 cj capital J, ck capital K, cl capital L, times all of the 158 00:11:35,920 --> 00:11:37,510 elements in the original tensor, aijkl. 159 00:11:40,010 --> 00:11:45,810 A fourth-rank tensor consists of an array of 9 by 9 terms. 160 00:11:45,810 --> 00:11:49,390 So there are 81 of these. 161 00:11:49,390 --> 00:11:53,890 We'd have four direction cosines out in front, and 162 00:11:53,890 --> 00:11:59,860 there would be a total of 81 times 5 characters that we 163 00:11:59,860 --> 00:12:00,770 would have to write. 164 00:12:00,770 --> 00:12:06,780 So to do the complete tensor transformation, we would have 165 00:12:06,780 --> 00:12:13,910 to write on the order of 400 quantities to get just one of 166 00:12:13,910 --> 00:12:19,070 the 81 elements in the new transform tensor. 167 00:12:19,070 --> 00:12:21,430 So the total number of elements we'd have to write to 168 00:12:21,430 --> 00:12:24,810 do this would be 81 squared times 5. 169 00:12:24,810 --> 00:12:26,260 That's a lot of elements. 170 00:12:30,610 --> 00:12:33,920 We'll do a few of these transformations directly, but 171 00:12:33,920 --> 00:12:37,700 let me assure you that if we do a transformation that is 172 00:12:37,700 --> 00:12:42,860 going to involve symmetry, a lot of the direction cosines, 173 00:12:42,860 --> 00:12:44,870 if we're lucky, will be 0. 174 00:12:44,870 --> 00:12:47,580 So it's not quite as onerous as it seems. 175 00:12:56,660 --> 00:13:00,300 So we would make use of this sort of formalism if we wanted 176 00:13:00,300 --> 00:13:04,300 to go from one set of reference axes to a new set 177 00:13:04,300 --> 00:13:06,470 that might represent a special specimen that we 178 00:13:06,470 --> 00:13:07,690 cut out of a crystal. 179 00:13:07,690 --> 00:13:12,460 But there's another formal way in which we could make very 180 00:13:12,460 --> 00:13:15,045 profound and non-intuitive use of these relations. 181 00:13:17,750 --> 00:13:21,530 Crystals, except for the abominable triclinic crystals, 182 00:13:21,530 --> 00:13:23,740 have symmetry. 183 00:13:23,740 --> 00:13:28,880 If a crystal has symmetry, you can transform the solid 184 00:13:28,880 --> 00:13:32,000 physically by that symmetry operation. 185 00:13:32,000 --> 00:13:33,750 And you have to measure the same 186 00:13:33,750 --> 00:13:36,370 property before and after. 187 00:13:36,370 --> 00:13:40,795 So suppose we have a crystal that has a twofold axis. 188 00:13:43,570 --> 00:13:48,095 And this crystal is something that looks like this. 189 00:13:53,740 --> 00:14:00,010 So this is side A, and this is side B. We could move the 190 00:14:00,010 --> 00:14:03,030 crystal by a 180-degree rotation. 191 00:14:03,030 --> 00:14:03,800 Put it down. 192 00:14:03,800 --> 00:14:05,960 I won't draw it, because it's going to look exactly the 193 00:14:05,960 --> 00:14:09,860 same, except now this thing-- 194 00:14:09,860 --> 00:14:11,730 I won't draw, and I do draw it-- 195 00:14:11,730 --> 00:14:15,230 this is face A, and this is face B. If we had electrodes 196 00:14:15,230 --> 00:14:18,860 on the crystal before and after that transformation, we 197 00:14:18,860 --> 00:14:21,390 have to measure, let's say, the same electrical 198 00:14:21,390 --> 00:14:24,350 conductivity for both orientations of the crystal. 199 00:14:29,980 --> 00:14:34,430 Now, moving a crystal relative to some coordinate system, 200 00:14:34,430 --> 00:14:37,430 relative to a pair of electrodes that we're 201 00:14:37,430 --> 00:14:40,710 fastening onto the crystal, is exactly the same thing as 202 00:14:40,710 --> 00:14:43,390 doing the reverse transformation of the 203 00:14:43,390 --> 00:14:46,080 coordinate system. 204 00:14:46,080 --> 00:14:48,540 That's a vague, strange-sounding term. 205 00:14:48,540 --> 00:14:52,410 So suppose we have a crystal with a fourfold axis with four 206 00:14:52,410 --> 00:14:59,980 faces, A, B, C, D. And here are our electrodes. 207 00:14:59,980 --> 00:15:03,015 To move the crystal relative to the electrodes by a 208 00:15:03,015 --> 00:15:09,390 90-degree rotation would involve rotating face D up to 209 00:15:09,390 --> 00:15:11,580 this location. 210 00:15:11,580 --> 00:15:13,710 A would move to this location. 211 00:15:13,710 --> 00:15:15,520 C would move to this location. 212 00:15:15,520 --> 00:15:18,950 B would move to this location, and we'd fasten electrodes on 213 00:15:18,950 --> 00:15:20,530 the crystal again. 214 00:15:20,530 --> 00:15:24,340 If the crystal originally had a coordinate system such that 215 00:15:24,340 --> 00:15:32,410 this were X1 and this is X2, moving the electrodes onto a 216 00:15:32,410 --> 00:15:35,570 different direction on the crystal is the same as moving 217 00:15:35,570 --> 00:15:37,980 the crystal in the opposite sense. 218 00:15:37,980 --> 00:15:43,010 So we could either envision moving the crystal relative to 219 00:15:43,010 --> 00:15:48,280 the electrodes like this, or we could move the electrodes 220 00:15:48,280 --> 00:15:52,960 relative to the crystal by the reverse transformation. 221 00:15:52,960 --> 00:15:56,700 And the result is the same. 222 00:15:56,700 --> 00:16:01,590 So what I'm saying is that if a crystal has symmetry-- 223 00:16:01,590 --> 00:16:08,260 and let me be specific and suppose that our crystal has a 224 00:16:08,260 --> 00:16:12,435 twofold rotation axis along X3. 225 00:16:20,130 --> 00:16:24,300 Let's ask how that twofold access would change the 226 00:16:24,300 --> 00:16:26,780 coordinate system relative to the crystal. 227 00:16:26,780 --> 00:16:29,060 That's the same as moving the crystal relative to the 228 00:16:29,060 --> 00:16:30,380 coordinate system. 229 00:16:30,380 --> 00:16:36,930 It's going to take X1 and move it to this location X1 prime. 230 00:16:36,930 --> 00:16:39,610 It's going to take X2 and move it 231 00:16:39,610 --> 00:16:41,600 through 180-degree rotation. 232 00:16:41,600 --> 00:16:43,810 This is going to be X2 prime. 233 00:16:43,810 --> 00:16:47,620 And if the twofold axis is along X3, X3 prime 234 00:16:47,620 --> 00:16:51,480 is the same as X3. 235 00:16:51,480 --> 00:16:54,060 So what is the direction cosine scheme for 236 00:16:54,060 --> 00:16:56,070 this change of axes? 237 00:16:56,070 --> 00:17:00,200 You might immediately start working and saying, well, C11 238 00:17:00,200 --> 00:17:03,940 is the cosine of the angle between X1 prime and X1. 239 00:17:03,940 --> 00:17:05,430 That's 180 degrees. 240 00:17:05,430 --> 00:17:08,230 Cosine of 180 degrees is minus 1. 241 00:17:08,230 --> 00:17:12,359 But let me remind you that the direction cosine scheme, c ij, 242 00:17:12,359 --> 00:17:16,660 simply gives us the relation between the new axes x sub i 243 00:17:16,660 --> 00:17:19,200 prime and the old axes, x sub j. 244 00:17:19,200 --> 00:17:22,900 So let me, just by inspection, write down the relation 245 00:17:22,900 --> 00:17:26,599 between these two sets of coordinate systems. 246 00:17:26,599 --> 00:17:29,755 So X1 prime is equal to minus X1. 247 00:17:29,755 --> 00:17:33,610 X2 prime is equal to minus X2. 248 00:17:33,610 --> 00:17:37,200 X3 prime is equal to X3. 249 00:17:37,200 --> 00:17:41,350 So the direction cosine scheme for this particular 250 00:17:41,350 --> 00:17:49,170 transformation is simply minus 1 0 0, 0 minus 1 0, 0 0 1. 251 00:17:49,170 --> 00:17:52,070 So I just evaluated a nine-element direction cosine 252 00:17:52,070 --> 00:17:55,540 scheme by inspection, if I can write the relation between the 253 00:17:55,540 --> 00:17:57,790 coordinate system before and after the transformation. 254 00:18:02,750 --> 00:18:04,000 OK? 255 00:18:06,400 --> 00:18:09,630 And as we examine higher symmetry, the same is going to 256 00:18:09,630 --> 00:18:13,330 be true for the threefold axis, let's say, along the 1, 257 00:18:13,330 --> 00:18:16,880 1, 1 direction of a cubic crystal, for a sixfold 258 00:18:16,880 --> 00:18:20,130 axis, and so on. 259 00:18:20,130 --> 00:18:22,810 So we'll be able to write the direction cosine schemes for a 260 00:18:22,810 --> 00:18:25,455 symmetry transformation simply by inspection. 261 00:18:29,500 --> 00:18:34,950 So for a twofold axis along X3, this is the form of the 262 00:18:34,950 --> 00:18:36,490 direction cosine scheme. 263 00:18:47,670 --> 00:18:52,640 So now let me transform the elements of a second-rank 264 00:18:52,640 --> 00:18:58,590 tensor term by term and see what we get. 265 00:18:58,590 --> 00:18:59,840 Suppose I want-- 266 00:19:02,590 --> 00:19:05,840 let's stick with conductivity as an example. 267 00:19:05,840 --> 00:19:11,140 Suppose I want the value for the conductivity element 268 00:19:11,140 --> 00:19:13,640 sigma1 1 prime. 269 00:19:13,640 --> 00:19:18,140 That's going to be c1 something, c1 something, 270 00:19:18,140 --> 00:19:21,760 because these are the elements that go in here, times every 271 00:19:21,760 --> 00:19:26,180 element of a conductivity tensor sigma lm. 272 00:19:26,180 --> 00:19:31,230 The only element of the form c1 something that is non-zero 273 00:19:31,230 --> 00:19:37,540 in this row c11, c12, c13, is the term c11. 274 00:19:37,540 --> 00:19:40,830 In the same way in the same row, the only term of the form 275 00:19:40,830 --> 00:19:42,990 c1 something is c11. 276 00:19:42,990 --> 00:19:47,850 So this is going to be simply c11 times c11 times sigma11. 277 00:19:47,850 --> 00:19:50,550 That's the only term that survives. 278 00:19:50,550 --> 00:19:58,400 c11 has a numerical value of minus 1, and that says that 279 00:19:58,400 --> 00:20:00,860 sigma11 prime is equal to sigma11. 280 00:20:04,730 --> 00:20:09,180 So is there any constraint, any restriction on sigma 11? 281 00:20:09,180 --> 00:20:14,540 No, sigma11 could be anything it likes. 282 00:20:14,540 --> 00:20:15,790 So there's no constraint. 283 00:20:21,290 --> 00:20:22,750 Let's do another element. 284 00:20:22,750 --> 00:20:25,590 Let's see what sigma12 prime would be. 285 00:20:25,590 --> 00:20:32,430 This will be c1 something times c2 something times sigma 286 00:20:32,430 --> 00:20:35,380 something something. 287 00:20:35,380 --> 00:20:38,070 The only element of the form c1 something that 288 00:20:38,070 --> 00:20:42,360 is non-zero is c11. 289 00:20:42,360 --> 00:20:46,790 So I'll put in a 1 for the l. 290 00:20:46,790 --> 00:20:49,430 The only direction cosine element of the form c2 291 00:20:49,430 --> 00:20:52,810 something which is non-zero is c22. 292 00:20:52,810 --> 00:20:58,630 So I'll put in c22 and let n be equal to 2. 293 00:20:58,630 --> 00:21:02,840 c11 is minus 1, c22 is minus 1. 294 00:21:02,840 --> 00:21:04,950 So again, this gives us something not terribly 295 00:21:04,950 --> 00:21:06,010 interesting. 296 00:21:06,010 --> 00:21:10,690 Sigma 12 prime is equal to sigma 12. 297 00:21:10,690 --> 00:21:17,900 So there's no constraint, at which point you're probably 298 00:21:17,900 --> 00:21:20,310 getting very restive, say, this is 299 00:21:20,310 --> 00:21:22,580 not telling us anything. 300 00:21:22,580 --> 00:21:25,930 So let me shake you up by doing one further 301 00:21:25,930 --> 00:21:31,940 transformation, and that is to find the value for c13 prime. 302 00:21:31,940 --> 00:21:34,880 And that would be c1 something, c3 303 00:21:34,880 --> 00:21:39,050 something times sigma lm. 304 00:21:39,050 --> 00:21:42,330 The only form of this term of the form c1 something, that's 305 00:21:42,330 --> 00:21:44,720 non-zero is c11, as we've seen. 306 00:21:44,720 --> 00:21:46,730 So I'll put in just that single term and 307 00:21:46,730 --> 00:21:49,400 replace l by 1. 308 00:21:49,400 --> 00:21:52,880 The term of the form c3 something that 309 00:21:52,880 --> 00:21:56,581 is non-zero is c33. 310 00:21:56,581 --> 00:22:02,130 And I'll put in a 3 for m, and this is then c11 times c33 311 00:22:02,130 --> 00:22:02,790 times sigma13. 312 00:22:02,790 --> 00:22:09,520 1, And c11 is minus 1. 313 00:22:09,520 --> 00:22:12,630 c33 is plus 1. 314 00:22:12,630 --> 00:22:18,950 And that says that sigma13 prime is minus sigma13. 315 00:22:18,950 --> 00:22:22,620 But if this is a symmetry transformation, the tensor has 316 00:22:22,620 --> 00:22:25,020 to remain invariant. 317 00:22:25,020 --> 00:22:30,630 And if we're to have sigma13 equals minus sigma13, there's 318 00:22:30,630 --> 00:22:35,960 only one number that can make that claim, and that's 0. 319 00:22:35,960 --> 00:22:41,800 So sigma13 is identically 0. 320 00:22:41,800 --> 00:22:45,850 And that places a rather severe constraint on the way 321 00:22:45,850 --> 00:22:50,150 in which the crystal is going to relate an applied electric 322 00:22:50,150 --> 00:22:52,100 field to a current flow. 323 00:22:52,100 --> 00:22:53,470 Sigma11 is anything. 324 00:22:53,470 --> 00:22:55,890 Sigma12 is anything. 325 00:22:55,890 --> 00:22:59,470 Sigma13 has to be identically 0. 326 00:22:59,470 --> 00:23:01,330 Now, let's cut to the bottom line. 327 00:23:01,330 --> 00:23:06,450 The direction cosine scheme is diagonal, so we can say that 328 00:23:06,450 --> 00:23:14,630 for any element that we pick to transform sigma ij prime is 329 00:23:14,630 --> 00:23:20,360 going to be cii, the diagonal term which has the second 330 00:23:20,360 --> 00:23:25,030 subscript equal to the first, times cjj times sigma ij. 331 00:23:29,710 --> 00:23:39,410 And this says that if we have i or j is equal to 3, then 332 00:23:39,410 --> 00:23:43,530 sigma ij has to be 0, because we're going to have a minus 1 333 00:23:43,530 --> 00:23:45,250 times a plus 1. 334 00:23:45,250 --> 00:23:57,280 If neither i or j is equal to 3, then again, we will have a 335 00:23:57,280 --> 00:23:59,330 minus 1 times a minus 1. 336 00:23:59,330 --> 00:24:00,580 There will be no constraint. 337 00:24:04,250 --> 00:24:10,420 And the only final possibility is that both i and j 338 00:24:10,420 --> 00:24:11,320 are equal to 3. 339 00:24:11,320 --> 00:24:14,040 That would be the single element c33. 340 00:24:14,040 --> 00:24:17,810 Then we would have plus 1 times plus 1 as the product of 341 00:24:17,810 --> 00:24:20,070 direction cosines, and there will be no constraint. 342 00:24:25,460 --> 00:24:30,980 So for a crystal that has a twofold axis, and in which 343 00:24:30,980 --> 00:24:35,960 that twofold axis is along the direction of x3, the form of 344 00:24:35,960 --> 00:24:44,660 the tensor will be sigma11, sigma12, 0, sigma21, sigma22, 345 00:24:44,660 --> 00:25:00,900 0, sigma31, that's going to be equal to 0, and sigma33 has no 346 00:25:00,900 --> 00:25:02,150 constraints. 347 00:25:04,370 --> 00:25:13,100 So rather than having nine elements, there are only five 348 00:25:13,100 --> 00:25:25,485 independent elements rather than nine. 349 00:25:28,100 --> 00:25:34,400 And there is now another relation that can occur in a 350 00:25:34,400 --> 00:25:36,390 second-rank tensor. 351 00:25:36,390 --> 00:25:44,370 The off-diagonal terms sigma12 and sigma21 do 352 00:25:44,370 --> 00:25:48,200 not have to be related. 353 00:25:48,200 --> 00:25:49,823 But for most-- 354 00:25:52,570 --> 00:25:53,820 but not all-- 355 00:25:58,200 --> 00:26:07,035 most second-rank tensor properties happily have sigma 356 00:26:07,035 --> 00:26:12,440 ij identical to sigma ji. 357 00:26:12,440 --> 00:26:15,710 In other words, the tensor is symmetric across 358 00:26:15,710 --> 00:26:19,490 its principal diagonal. 359 00:26:19,490 --> 00:26:26,470 That is a condition that does not arise from symmetry That 360 00:26:26,470 --> 00:26:29,420 depends on the specific physical property. 361 00:26:32,400 --> 00:26:34,780 So let me emphasize that this depends 362 00:26:34,780 --> 00:26:36,030 on the tensor property. 363 00:26:46,120 --> 00:26:48,970 And for a great many physical properties-- 364 00:26:48,970 --> 00:26:53,230 conductivity, diffusivity, permeability, susceptibility-- 365 00:26:53,230 --> 00:26:58,270 you can show that the tensor has to be symmetric. 366 00:26:58,270 --> 00:27:01,310 But there are a lot of tensors, particularly for the 367 00:27:01,310 --> 00:27:03,800 more obscure physical properties, where to my 368 00:27:03,800 --> 00:27:08,980 knowledge, this proof has never been given. 369 00:27:08,980 --> 00:27:12,350 And along the same lines, it is well known that there is 370 00:27:12,350 --> 00:27:16,020 one physical property for which this is not true. 371 00:27:16,020 --> 00:27:19,730 This is the thermal electricity tensor. 372 00:27:19,730 --> 00:27:24,300 So for at least one property, you can show for sure that the 373 00:27:24,300 --> 00:27:29,260 tensor does not have to be symmetric and that for a 374 00:27:29,260 --> 00:27:33,550 crystal of symmetry 2, this term and this term are 375 00:27:33,550 --> 00:27:34,860 definitely not equal. 376 00:27:42,060 --> 00:27:45,840 All right, let us do another transformation for another 377 00:27:45,840 --> 00:27:49,390 symmetry, and we can see that it goes fast when the 378 00:27:49,390 --> 00:27:52,370 direction cosine scheme is relatively sparse. 379 00:27:52,370 --> 00:27:55,360 Let's ask the restrictions, if any, that 380 00:27:55,360 --> 00:27:56,730 are imposed by inversion. 381 00:28:08,690 --> 00:28:12,880 So what is the direction cosine scheme here? 382 00:28:12,880 --> 00:28:18,880 Here's x1, here's x2, here's x3. 383 00:28:18,880 --> 00:28:22,380 Then operation of inversion at the intersection of these axes 384 00:28:22,380 --> 00:28:26,035 will invert the direction of x1 prime to here. 385 00:28:26,035 --> 00:28:29,500 It'll invert the direction of x2 prime here. 386 00:28:29,500 --> 00:28:33,440 It will invert the direction of x3 prime to here. 387 00:28:33,440 --> 00:28:36,770 So the relation between the reference axes is that x1 388 00:28:36,770 --> 00:28:39,870 prime is equal to minus x1. 389 00:28:39,870 --> 00:28:42,990 x2 prime is equal to minus x2. 390 00:28:42,990 --> 00:28:48,090 x3 prime is equal to minus x3, so that the form of the 391 00:28:48,090 --> 00:28:53,940 direction cosine scheme crj is minus 1, 00, 0 minus 392 00:28:53,940 --> 00:28:56,300 1 0, 00 minus 1. 393 00:29:04,080 --> 00:29:06,900 Slightly different from that for a two-fold axis for which 394 00:29:06,900 --> 00:29:09,460 the first two diagonal elements were minus 1, the 395 00:29:09,460 --> 00:29:11,110 third one was 0. 396 00:29:11,110 --> 00:29:13,350 Well, let's jump right to it and see if we 397 00:29:13,350 --> 00:29:14,440 can generalize this. 398 00:29:14,440 --> 00:29:19,180 It's a diagonal direction cosine scheme once again. 399 00:29:19,180 --> 00:29:23,080 And this says that if we transform a particular element 400 00:29:23,080 --> 00:29:31,840 sigma ij, it's going to be given by cil, cjm, sigma lm, 401 00:29:31,840 --> 00:29:35,070 where l and m are variables of summation. 402 00:29:35,070 --> 00:29:39,530 The only ones that survive are the ones for which i equals l 403 00:29:39,530 --> 00:29:41,590 and for which j equals m. 404 00:29:41,590 --> 00:29:47,330 So it's going to be cii, cjj times sigma ij. 405 00:29:47,330 --> 00:29:51,530 Regardless of the values of i and j, the diagonal terms are 406 00:29:51,530 --> 00:29:53,330 always minus 1. 407 00:29:53,330 --> 00:29:57,590 And therefore, sigma ij prime is always going to turn out to 408 00:29:57,590 --> 00:30:00,460 be equal to sigma ij, so there's going to be no 409 00:30:00,460 --> 00:30:04,070 constraint on any element. 410 00:30:08,170 --> 00:30:13,470 And this shortens the job that's facing us immeasurably. 411 00:30:13,470 --> 00:30:18,820 So let me write that down, because that's important. 412 00:30:18,820 --> 00:30:38,070 Inversion imposes no constraint on any second-rank 413 00:30:38,070 --> 00:30:39,320 tensor property. 414 00:30:46,460 --> 00:30:51,240 So if we stay with monoclinic crystals, we looked at 415 00:30:51,240 --> 00:30:53,920 symmetry 2. 416 00:30:53,920 --> 00:30:59,910 Symmetry 2 over m is equal to 2 with an inversion 417 00:30:59,910 --> 00:31:01,290 center put on it. 418 00:31:01,290 --> 00:31:05,270 But inversion doesn't require anything, so the symmetry 419 00:31:05,270 --> 00:31:10,620 constraints for 2 over m have to be the same as 420 00:31:10,620 --> 00:31:14,640 for symmetry 2. 421 00:31:14,640 --> 00:31:18,600 We look at the constraints that might be imposed by a 422 00:31:18,600 --> 00:31:20,870 mirror plane. 423 00:31:20,870 --> 00:31:26,555 A mirror plane plus inversion is 2 over m. 424 00:31:26,555 --> 00:31:29,310 2 over m has to be the same as 2. 425 00:31:29,310 --> 00:31:31,760 So this will be the same as 2. 426 00:31:31,760 --> 00:31:35,240 And now we've shown that for any monoclinic crystal, 427 00:31:35,240 --> 00:31:37,840 regardless of whether the symmetry, the point group, is 428 00:31:37,840 --> 00:31:42,050 2m or 2 over m, the form of the tensor has to 429 00:31:42,050 --> 00:31:43,840 be exactly the same. 430 00:31:43,840 --> 00:31:56,760 So for any monoclinic crystal, namely 2m or 2 over m, this is 431 00:31:56,760 --> 00:32:00,380 the form of the tensor where the twofold axis, again, is 432 00:32:00,380 --> 00:32:03,420 along x3, the mirror plane would have to be perpendicular 433 00:32:03,420 --> 00:32:07,945 to x3, and for 2 over m, both of the preceding conditions. 434 00:32:11,420 --> 00:32:12,670 So how about that? 435 00:32:17,500 --> 00:32:22,860 Let me issue a caveat, because we're almost out of time. 436 00:32:22,860 --> 00:32:25,570 There are five independent elements. 437 00:32:25,570 --> 00:32:27,810 And that's true. 438 00:32:27,810 --> 00:32:45,570 But elements sigma12, sigma21, sigma13, and sigma31 are 0 439 00:32:45,570 --> 00:32:53,150 only for this arrangement of axes relative to 440 00:32:53,150 --> 00:32:54,400 the symmetry elements. 441 00:33:06,980 --> 00:33:14,086 If you wanted to take a different set of axes, you 442 00:33:14,086 --> 00:33:17,400 know how to get the tensor for that set of axes. 443 00:33:17,400 --> 00:33:20,310 Each tensor element is going to be given by a linear 444 00:33:20,310 --> 00:33:23,470 combination of these five non-zero elements. 445 00:33:23,470 --> 00:33:27,520 And if the orientation of the axes relative to the symmetry 446 00:33:27,520 --> 00:33:35,080 elements is quite general, all nine elements of the tensor 447 00:33:35,080 --> 00:33:37,790 will be non-zero. 448 00:33:37,790 --> 00:33:40,770 There will be only five independent numbers, which 449 00:33:40,770 --> 00:33:44,430 composes each of those nine elements, and they will be 450 00:33:44,430 --> 00:33:47,800 given by a product of two direction cosines, sometimes 451 00:33:47,800 --> 00:33:50,860 each of these five non-zero elements. 452 00:33:50,860 --> 00:33:53,750 But there will be no zeros in this array at all for an 453 00:33:53,750 --> 00:33:55,500 arbitrary set of coordinate systems. 454 00:33:58,620 --> 00:34:02,460 Something that I think I'll ask you to do as a problem, 455 00:34:02,460 --> 00:34:06,230 because it's really easy to do, if the tensor is 456 00:34:06,230 --> 00:34:10,100 symmetric, which most of them are, one thing that you can 457 00:34:10,100 --> 00:34:20,210 show quite directly is that a symmetric tensor remains 458 00:34:20,210 --> 00:34:30,375 symmetric for any arbitrary change of axes. 459 00:34:43,780 --> 00:34:46,900 And that, again, is something that's fairly easy to prove, 460 00:34:46,900 --> 00:34:49,750 and I'll let you have the fun and exhilaration of doing that 461 00:34:49,750 --> 00:34:51,000 for yourself. 462 00:34:52,980 --> 00:34:56,980 OK, so this means that for everything except thermal 463 00:34:56,980 --> 00:35:01,010 electricity, you really have to transform, at most, only 464 00:35:01,010 --> 00:35:02,800 six elements if you go from one 465 00:35:02,800 --> 00:35:05,040 coordinate system to another. 466 00:35:05,040 --> 00:35:09,270 That's still a lot, but it's considerably better than 467 00:35:09,270 --> 00:35:11,110 transforming all nine. 468 00:35:11,110 --> 00:35:14,070 So if the tensor originally is symmetric in one coordinate 469 00:35:14,070 --> 00:35:17,370 system, it stays symmetric in any other coordinate system. 470 00:35:20,720 --> 00:35:22,380 Now, one thing that I should mention-- 471 00:35:22,380 --> 00:35:24,230 I passed over it rather quickly-- 472 00:35:27,350 --> 00:35:33,780 we said that a property of a direction cosine scheme is 473 00:35:33,780 --> 00:35:37,860 that it is what's called a unitary transformation. 474 00:35:37,860 --> 00:35:42,680 And it has the property that the determinant of the 475 00:35:42,680 --> 00:35:46,030 coefficients is plus 1 if the axis 476 00:35:46,030 --> 00:35:49,580 retains the same chirality. 477 00:35:49,580 --> 00:35:53,140 The determinant is minus 1 if you change the handedness. 478 00:35:53,140 --> 00:35:59,520 That is only true for what is called a measure-preserving 479 00:35:59,520 --> 00:36:02,160 transformation. 480 00:36:02,160 --> 00:36:02,920 That's what it's called. 481 00:36:02,920 --> 00:36:05,490 And when it's a measure-preserving 482 00:36:05,490 --> 00:36:09,690 transformation, then the direction cosine scheme is a 483 00:36:09,690 --> 00:36:11,440 unitary matrix. 484 00:36:11,440 --> 00:36:17,210 What is measure-preserving transformation? 485 00:36:17,210 --> 00:36:19,070 If it's a right-handed system 486 00:36:19,070 --> 00:36:21,560 beforehand, there's no squishing. 487 00:36:21,560 --> 00:36:23,740 It doesn't go to an oblique coordinate system. 488 00:36:23,740 --> 00:36:26,100 Cartesian stays Cartesian. 489 00:36:26,100 --> 00:36:30,470 If the reference axes are of equal lengths, they don't 490 00:36:30,470 --> 00:36:33,440 stretch upon the transformation. 491 00:36:33,440 --> 00:36:36,150 You can define transformations like that if you like, where 492 00:36:36,150 --> 00:36:40,950 the angles between them go from orthogonal to oblique 493 00:36:40,950 --> 00:36:43,670 after the transformation, and the units of length along the 494 00:36:43,670 --> 00:36:45,900 three axes change dimension. 495 00:36:45,900 --> 00:36:49,920 But then the determinant of the coefficients is not unity, 496 00:36:49,920 --> 00:36:52,170 and a lot of the nice, convenient properties that 497 00:36:52,170 --> 00:36:53,420 we've seen here do not hold. 498 00:36:57,205 --> 00:37:01,620 All right, that is a good place to stop, I think. 499 00:37:01,620 --> 00:37:02,970 Next week, no quiz. 500 00:37:02,970 --> 00:37:04,650 If you came in late, we're going to postpone 501 00:37:04,650 --> 00:37:06,070 the quiz for a week. 502 00:37:06,070 --> 00:37:09,710 And let's see, look at all I can ask you just after one 503 00:37:09,710 --> 00:37:12,930 lecture on tensors. 504 00:37:12,930 --> 00:37:16,700 What we will do next is explore the form of the 505 00:37:16,700 --> 00:37:18,830 tensors for other crystal symmetries. 506 00:37:18,830 --> 00:37:21,600 It goes fairly quickly. 507 00:37:21,600 --> 00:37:25,340 And then having done all that, I'll show you how you can 508 00:37:25,340 --> 00:37:30,970 determine the symmetry constraints by inspection for 509 00:37:30,970 --> 00:37:31,990 a tensor of any rank. 510 00:37:31,990 --> 00:37:35,240 And you're going to despise me for that, but this was useful, 511 00:37:35,240 --> 00:37:39,090 because we can get used to manipulating the notation. 512 00:37:39,090 --> 00:37:40,760 But there is a method called-- 513 00:37:40,760 --> 00:37:43,380 appropriately enough-- the method of direct inspection 514 00:37:43,380 --> 00:37:47,170 where you can very quickly and very easily do the symmetry 515 00:37:47,170 --> 00:37:48,800 transformations. 516 00:37:48,800 --> 00:37:52,470 So all this and more will be revealed next time, which is 517 00:37:52,470 --> 00:37:55,120 going to be a lot more beneficial than taking a quiz.