1 00:00:07,430 --> 00:00:13,700 PROFESSOR: All right, an announcement that I must make, 2 00:00:13,700 --> 00:00:16,690 after developing all these friendships and working 3 00:00:16,690 --> 00:00:20,350 together, we will, next Tuesday, once again, enter a 4 00:00:20,350 --> 00:00:22,570 brief adversarial relationship. 5 00:00:22,570 --> 00:00:26,340 And we will have quiz number two. 6 00:00:26,340 --> 00:00:31,970 What we will cover on that quiz will be about 50%, 7 00:00:31,970 --> 00:00:35,330 perhaps a little bit more, on symmetry. 8 00:00:35,330 --> 00:00:37,630 And I'll remind you in just a minute what we've covered 9 00:00:37,630 --> 00:00:39,220 since the last quiz. 10 00:00:39,220 --> 00:00:45,180 And the second half or 40% of the quiz will be on the 11 00:00:45,180 --> 00:00:48,550 formalism of tensors and tensor manipulation. 12 00:00:48,550 --> 00:00:50,390 And obviously, you've had a chance only to 13 00:00:50,390 --> 00:00:52,090 do a very few problems. 14 00:00:52,090 --> 00:00:55,620 So these will be pretty basic questions and nothing tricky 15 00:00:55,620 --> 00:00:57,280 or very difficult. 16 00:00:57,280 --> 00:01:00,460 So let me remind you very, very quickly just by topics 17 00:01:00,460 --> 00:01:01,810 what we've covered. 18 00:01:01,810 --> 00:01:06,950 We began the period following the first quiz discussing how 19 00:01:06,950 --> 00:01:10,440 we can take the plane groups, two-dimensional symmetries 20 00:01:10,440 --> 00:01:13,720 that are translationally periodic, and defining a third 21 00:01:13,720 --> 00:01:18,590 translation to stack up the plane groups to give us a 22 00:01:18,590 --> 00:01:22,140 three-dimensional periodic 23 00:01:22,140 --> 00:01:24,030 arrangement of symmetry elements. 24 00:01:24,030 --> 00:01:28,580 In doing so, we found that we had to pick that third 25 00:01:28,580 --> 00:01:32,360 translation such that the symmetry elements that were in 26 00:01:32,360 --> 00:01:35,080 the plane group, and those were the things that made the 27 00:01:35,080 --> 00:01:38,780 lattices of the plane groups special, made them square or 28 00:01:38,780 --> 00:01:42,340 hexagonal or rectangular, that was true because there was 29 00:01:42,340 --> 00:01:43,260 symmetry there. 30 00:01:43,260 --> 00:01:47,210 Once again, symmetry and the specialness of a lattice are 31 00:01:47,210 --> 00:01:49,870 two inseparable parts of the story. 32 00:01:49,870 --> 00:01:52,160 But if we did this, the symmetry 33 00:01:52,160 --> 00:01:53,960 elements had to coincide. 34 00:01:53,960 --> 00:01:57,380 Otherwise, we would reproduce new symmetry elements in the 35 00:01:57,380 --> 00:01:59,740 base of the cell and wreck everything. 36 00:01:59,740 --> 00:02:02,160 So when we did that, there were a very limited number of 37 00:02:02,160 --> 00:02:03,410 ways of doing it. 38 00:02:07,020 --> 00:02:11,420 In the course of doing that, we fell headlong over screw 39 00:02:11,420 --> 00:02:15,400 axis in the same way that we fell over glide planes as soon 40 00:02:15,400 --> 00:02:20,390 as we put mirror lines in a two-dimensional lattice. 41 00:02:20,390 --> 00:02:24,910 The interesting thing was that that exercise of stacking the 42 00:02:24,910 --> 00:02:28,580 two-dimensional plane groups gave us almost all of the 43 00:02:28,580 --> 00:02:31,610 lattice types that exist in three dimensions. 44 00:02:31,610 --> 00:02:34,080 And the reason for that is we showed very directly that 45 00:02:34,080 --> 00:02:36,020 inversion requires nothing. 46 00:02:36,020 --> 00:02:39,100 So therefore, any of the point groups that were used in 47 00:02:39,100 --> 00:02:42,210 deriving the plane groups, when augmented by inversion to 48 00:02:42,210 --> 00:02:45,080 give a three-dimensional point group, are going to require 49 00:02:45,080 --> 00:02:48,000 nothing more than the lattices that we derived by stacking 50 00:02:48,000 --> 00:02:49,240 the plane groups. 51 00:02:49,240 --> 00:02:52,800 And that took care of just about everything except for 52 00:02:52,800 --> 00:02:54,650 the cubic symmetries. 53 00:02:54,650 --> 00:02:57,350 And we disposed of those in very short order. 54 00:02:57,350 --> 00:02:59,560 So that led us to the 14 space lattices, 55 00:02:59,560 --> 00:03:02,070 so-called bravais lattices. 56 00:03:02,070 --> 00:03:06,840 And then we proceeded to work with those lattices and put in 57 00:03:06,840 --> 00:03:09,370 the point groups that could accommodate them. 58 00:03:09,370 --> 00:03:12,370 Having discovered glide planes, we then, if we had 59 00:03:12,370 --> 00:03:14,700 done the process systematically, would have 60 00:03:14,700 --> 00:03:17,620 replaced mirror planes by glide planes. 61 00:03:17,620 --> 00:03:22,810 And having discovered screw axes, we replace pure rotation 62 00:03:22,810 --> 00:03:26,940 by screw axis rotations. 63 00:03:26,940 --> 00:03:29,780 And then doing what we did but didn't have to do very often, 64 00:03:29,780 --> 00:03:33,130 mercifully, for the plane groups, we said one could also 65 00:03:33,130 --> 00:03:35,350 interweave symmetry elements. 66 00:03:35,350 --> 00:03:39,230 And that gave a few additional space groups. 67 00:03:39,230 --> 00:03:41,590 We did not derive very many of them. 68 00:03:41,590 --> 00:03:44,320 What is important at this stage is simply to realize 69 00:03:44,320 --> 00:03:48,420 that their properties and applications are very similar 70 00:03:48,420 --> 00:03:51,850 to what we discovered for the two-dimensional plane groups. 71 00:03:51,850 --> 00:03:53,730 The notation is a little bit different. 72 00:03:53,730 --> 00:03:57,460 Because there can be symmetry elements parallel to the plane 73 00:03:57,460 --> 00:04:00,410 of the paper so that in depicting the arrangement of 74 00:04:00,410 --> 00:04:03,390 symmetry elements you need a little [? chevron-type ?] 75 00:04:03,390 --> 00:04:06,830 device to indicate whether you've got a mirror plane or a 76 00:04:06,830 --> 00:04:09,340 glide plane in the base of the cell. 77 00:04:09,340 --> 00:04:12,360 And similarly with two-fold axes, and that's about the 78 00:04:12,360 --> 00:04:15,460 only thing that would be in the plane of the paper if the 79 00:04:15,460 --> 00:04:21,459 principal symmetry is normal to the paper, we need a device 80 00:04:21,459 --> 00:04:24,380 for indicating the elevation and whether the axis is a 81 00:04:24,380 --> 00:04:26,490 two-fold rotation axis or a screw. 82 00:04:26,490 --> 00:04:29,280 So there's a geometrical language for describing the 83 00:04:29,280 --> 00:04:30,930 arrangement of the space groups. 84 00:04:30,930 --> 00:04:34,320 We had a few problems in determining space group 85 00:04:34,320 --> 00:04:37,500 symbols and interpreting what these mean. 86 00:04:37,500 --> 00:04:41,420 But again, I don't think I would be sadistic enough to 87 00:04:41,420 --> 00:04:46,000 ask you to derive a complicated three-dimensional 88 00:04:46,000 --> 00:04:47,380 space group. 89 00:04:47,380 --> 00:04:47,710 I don't know. 90 00:04:47,710 --> 00:04:50,300 Maybe over the weekend, if I get to feeling mean, I might 91 00:04:50,300 --> 00:04:51,740 ask you something in those directions. 92 00:04:51,740 --> 00:04:55,150 But it will not be anything terribly formidable. 93 00:04:55,150 --> 00:04:58,620 Then finally, we looked at the way in which space groups can 94 00:04:58,620 --> 00:05:02,890 be used to describe three-dimensional periodic 95 00:05:02,890 --> 00:05:04,100 arrays of atoms. 96 00:05:04,100 --> 00:05:06,560 And that again was strictly analogous to what we did in 97 00:05:06,560 --> 00:05:09,070 two dimensions. 98 00:05:09,070 --> 00:05:13,290 The description of the three-dimensional arrangement 99 00:05:13,290 --> 00:05:17,820 of atoms, particularly if it is too complicated to be 100 00:05:17,820 --> 00:05:21,230 readily appreciated in a projection, would have to be 101 00:05:21,230 --> 00:05:24,710 given in terms of the space group, the type of position 102 00:05:24,710 --> 00:05:27,600 that is occupied, either special or the general 103 00:05:27,600 --> 00:05:31,260 position, and then depending on how many degrees of freedom 104 00:05:31,260 --> 00:05:34,410 are available in those positions to give you the 105 00:05:34,410 --> 00:05:37,480 coordinates of the element or just the x-coordinate or the x 106 00:05:37,480 --> 00:05:38,760 and y-coordinate. 107 00:05:38,760 --> 00:05:40,840 And then you run to the international tables. 108 00:05:40,840 --> 00:05:42,840 You don't have to work this out yourself. 109 00:05:42,840 --> 00:05:46,510 And you find the list of coordinates that are related 110 00:05:46,510 --> 00:05:49,590 by symmetry for that particular special position 111 00:05:49,590 --> 00:05:54,310 and then plug in x and y to get all the coordinates of the 112 00:05:54,310 --> 00:05:55,560 atoms in the unit cell. 113 00:05:58,180 --> 00:06:03,380 To go to the practical aspects of this, usually only simple 114 00:06:03,380 --> 00:06:10,520 structures can be projected with any clarity that lets 115 00:06:10,520 --> 00:06:12,410 them be interpreted. 116 00:06:12,410 --> 00:06:16,640 And even with a computer these days one can show an 117 00:06:16,640 --> 00:06:20,210 orthographic projection of the structure and then in real 118 00:06:20,210 --> 00:06:23,850 time manipulate it so you can see this beautiful 119 00:06:23,850 --> 00:06:26,770 construction with colored shiny spheres 120 00:06:26,770 --> 00:06:28,550 in it rotating around. 121 00:06:28,550 --> 00:06:31,670 But if it's a complicated structure, even that doesn't 122 00:06:31,670 --> 00:06:33,340 do you very much. 123 00:06:33,340 --> 00:06:36,320 It works for NaCO and zinc sulfide. 124 00:06:36,320 --> 00:06:37,840 But you know what those look like. 125 00:06:37,840 --> 00:06:41,450 And so you don't really need this very attractive, real 126 00:06:41,450 --> 00:06:42,680 time interaction. 127 00:06:42,680 --> 00:06:47,167 What I found works well is to take the structure and slice 128 00:06:47,167 --> 00:06:52,020 it apart in layers like a layer cake and look at a 129 00:06:52,020 --> 00:06:57,590 limited range of the one translation, generally, one of 130 00:06:57,590 --> 00:07:01,440 the shorter translations, and examine the coordination of 131 00:07:01,440 --> 00:07:03,040 atoms within that layer. 132 00:07:03,040 --> 00:07:06,540 And so by slicing the structure away and looking at 133 00:07:06,540 --> 00:07:09,760 it in several layers that are stacked on top of one another, 134 00:07:09,760 --> 00:07:12,910 the way I work between the ears, usually lets me gain an 135 00:07:12,910 --> 00:07:15,080 appreciation of what's going on in that structure. 136 00:07:15,080 --> 00:07:17,530 Something different might work for you. 137 00:07:17,530 --> 00:07:20,180 Matter of fact, you might be a gifted person who can see the 138 00:07:20,180 --> 00:07:23,080 things spinning around on a computer screen and understand 139 00:07:23,080 --> 00:07:24,040 exactly what it is. 140 00:07:24,040 --> 00:07:27,980 But everybody's mind works in a slightly different way. 141 00:07:27,980 --> 00:07:31,740 OK, that is roughly what we'll be covering on symmetry. 142 00:07:31,740 --> 00:07:34,770 Then we got into the connection between symmetry 143 00:07:34,770 --> 00:07:36,760 and physical properties. 144 00:07:36,760 --> 00:07:43,755 And we defined what we meant by a tensor. 145 00:07:43,755 --> 00:07:46,460 A tensor of the second rank is something 146 00:07:46,460 --> 00:07:48,240 that relates to vectors. 147 00:07:48,240 --> 00:07:52,700 We got into a notation involving the assumption that 148 00:07:52,700 --> 00:07:55,800 any repeated subscript, and there's subscripts all over 149 00:07:55,800 --> 00:07:57,680 the place in tensor properties. 150 00:07:57,680 --> 00:08:02,890 Any repeated subscript is automatically implied to be 151 00:08:02,890 --> 00:08:04,525 summed over from 1 to 3. 152 00:08:07,580 --> 00:08:14,690 We then derived the laws for transformation of a tensor. 153 00:08:14,690 --> 00:08:17,970 Namely that, if you specify the change of coordinate 154 00:08:17,970 --> 00:08:24,890 system by a direction cosine scheme, cij, where cij is the 155 00:08:24,890 --> 00:08:30,810 cosine of the angle between x of i prime and x sub j, that, 156 00:08:30,810 --> 00:08:34,140 since these direction cosines constitute unit vectors, there 157 00:08:34,140 --> 00:08:36,370 had to be relations between them. 158 00:08:36,370 --> 00:08:39,830 Namely the sum of the squares of the terms in any row or 159 00:08:39,830 --> 00:08:42,750 column had to be unity. 160 00:08:42,750 --> 00:08:46,960 The sum of corresponding terms in any row or column 161 00:08:46,960 --> 00:08:48,140 had to add to 0. 162 00:08:48,140 --> 00:08:51,860 Because these represented dot products of unit vectors in a 163 00:08:51,860 --> 00:08:54,280 Cartesian coordinate system. 164 00:08:54,280 --> 00:08:59,060 And then using this array of direction cosines, we found 165 00:08:59,060 --> 00:09:03,970 that the way in which a second ranked tensor transformed was 166 00:09:03,970 --> 00:09:09,560 that each tensor element was given by a linear combination 167 00:09:09,560 --> 00:09:13,300 of all of the elements of the original tensor. 168 00:09:13,300 --> 00:09:18,550 And out in front of the tensor was a product of two direction 169 00:09:18,550 --> 00:09:24,920 cosines, the subscripts of which were determined by the 170 00:09:24,920 --> 00:09:27,210 indices of that element. 171 00:09:27,210 --> 00:09:30,300 A vector could be regarded as a first ranked tensor. 172 00:09:30,300 --> 00:09:32,430 And everybody understands the law for 173 00:09:32,430 --> 00:09:34,000 transformation of a vector. 174 00:09:34,000 --> 00:09:36,440 Namely that each component of the vector is a linear 175 00:09:36,440 --> 00:09:40,270 combination of every component of the original vector with 176 00:09:40,270 --> 00:09:42,370 one direction cosine out in front. 177 00:09:42,370 --> 00:09:45,590 Tensor of second rank has a product of two direction 178 00:09:45,590 --> 00:09:49,080 cosines summed over all the elements of 179 00:09:49,080 --> 00:09:50,330 the original tensor. 180 00:09:50,330 --> 00:09:52,710 Third ranked tensors, which, believe it or not, we'll get 181 00:09:52,710 --> 00:09:56,340 into after the quiz in considerable detail, and even 182 00:09:56,340 --> 00:09:59,310 shudder fourth ranked tensors are going to involve 183 00:09:59,310 --> 00:10:04,090 summations that involve scads of terms and products of three 184 00:10:04,090 --> 00:10:08,120 and four direction cosines respectively. 185 00:10:08,120 --> 00:10:10,890 So we won't get very far into tensors. 186 00:10:10,890 --> 00:10:16,060 We will perhaps have a couple of questions on the nature of 187 00:10:16,060 --> 00:10:19,710 anisotropy that's implied by these relations. 188 00:10:19,710 --> 00:10:22,940 And I think, if you looked at the problem sets to this 189 00:10:22,940 --> 00:10:25,520 point, that's as far as we'll go on the quiz. 190 00:10:25,520 --> 00:10:27,830 And we will not throw something at you that you've 191 00:10:27,830 --> 00:10:28,850 not seen before. 192 00:10:28,850 --> 00:10:30,710 So it'll be merciful. 193 00:10:30,710 --> 00:10:33,750 We'll pull out all the stops in the last quiz. 194 00:10:33,750 --> 00:10:35,290 Because then you'll be leaving. 195 00:10:35,290 --> 00:10:36,270 And you can hate me. 196 00:10:36,270 --> 00:10:38,190 But I never have to deal with you again. 197 00:10:38,190 --> 00:10:42,455 So that will not make any difference. 198 00:10:42,455 --> 00:10:48,140 All right, I have handed back all of the problem sets except 199 00:10:48,140 --> 00:10:50,360 two, which are partly done. 200 00:10:50,360 --> 00:10:54,190 I never again will ask anybody to draw out patterns of 201 00:10:54,190 --> 00:10:56,480 rotoinversion and rotoreflection axes. 202 00:10:56,480 --> 00:10:58,840 Because they go on and on and on. 203 00:10:58,840 --> 00:11:01,580 And I am so tired of staring at little circles and trying 204 00:11:01,580 --> 00:11:04,340 to tell which ones are right and which one are wrong. 205 00:11:04,340 --> 00:11:07,010 But that's something else we have in common. 206 00:11:07,010 --> 00:11:09,770 You probably hated looking at them when you were drawing 207 00:11:09,770 --> 00:11:11,330 them out for yourself. 208 00:11:11,330 --> 00:11:14,610 So let me tell you what's going to go on for the next 209 00:11:14,610 --> 00:11:15,600 several days. 210 00:11:15,600 --> 00:11:19,680 Tomorrow is a holiday, not for instructors who are making up 211 00:11:19,680 --> 00:11:20,530 a quiz, however. 212 00:11:20,530 --> 00:11:25,630 So I will be in my office from early morning on for the rest 213 00:11:25,630 --> 00:11:26,090 of the day. 214 00:11:26,090 --> 00:11:29,430 So if you want to come by to ask questions, what I will be 215 00:11:29,430 --> 00:11:32,180 doing primarily is finishing up grading the problem sets. 216 00:11:32,180 --> 00:11:34,170 And if you want to pick up your problem sets and ask a 217 00:11:34,170 --> 00:11:37,430 question about those, I am there at your disposal 218 00:11:37,430 --> 00:11:40,220 hopefully with no ringing phone calls and no committee 219 00:11:40,220 --> 00:11:43,180 meetings and nothing of that sort. 220 00:11:43,180 --> 00:11:47,340 If I have finished the problem sets and you haven't gotten 221 00:11:47,340 --> 00:11:52,390 yours yet, what I will do is tape an envelope to my door 222 00:11:52,390 --> 00:11:53,760 and have them tucked in there. 223 00:11:53,760 --> 00:11:56,990 So you can come by any time over the weekend and pick them 224 00:11:56,990 --> 00:11:59,390 up if you want to get hold of them and look them over at 225 00:11:59,390 --> 00:12:01,280 your leisure. 226 00:12:01,280 --> 00:12:03,650 I have to tell you, looking ahead in the schedule, I, 227 00:12:03,650 --> 00:12:08,420 unfortunately, bad timing, I have to be away on Monday. 228 00:12:08,420 --> 00:12:13,530 So if you have a last minute question or you need to be 229 00:12:13,530 --> 00:12:17,660 calmed down from a last minute panic, you can catch me 230 00:12:17,660 --> 00:12:18,630 Tuesday morning. 231 00:12:18,630 --> 00:12:22,050 I'll be in all of the earlier part of the day. 232 00:12:22,050 --> 00:12:27,450 But if you do have questions, please come in on Friday. 233 00:12:27,450 --> 00:12:30,440 Because I'll have lots of time to spend with you. 234 00:12:30,440 --> 00:12:34,170 So your problem sets, anything turned in to this point and 235 00:12:34,170 --> 00:12:39,190 turned in today, should be on my door, if you haven't called 236 00:12:39,190 --> 00:12:42,950 for it no, later than the beginning of the week. 237 00:12:42,950 --> 00:12:44,430 OK, any questions? 238 00:12:44,430 --> 00:12:47,620 Any puzzles that you want to go over before we launch into 239 00:12:47,620 --> 00:12:48,870 new material? 240 00:12:52,940 --> 00:12:57,300 OK, what we have done with second ranked tensor 241 00:12:57,300 --> 00:13:01,900 properties is to derive a whole set of symmetry 242 00:13:01,900 --> 00:13:06,380 constraints on something aij that represents a physical 243 00:13:06,380 --> 00:13:08,640 property of a crystal. 244 00:13:08,640 --> 00:13:11,200 And one of the things that we've seen, let's look at 245 00:13:11,200 --> 00:13:14,300 conductivity once more since that's something we're all 246 00:13:14,300 --> 00:13:20,090 familiar with and can readily appreciate, that the 247 00:13:20,090 --> 00:13:25,910 components of the current density vector depend on all 248 00:13:25,910 --> 00:13:30,340 of the components of the applied electric field, 249 00:13:30,340 --> 00:13:33,490 voltage per unit length. 250 00:13:33,490 --> 00:13:41,260 And in as much as for anything other than a crystal of cubic 251 00:13:41,260 --> 00:13:44,860 symmetry, the numbers in this array, at least some of them, 252 00:13:44,860 --> 00:13:46,240 are different. 253 00:13:46,240 --> 00:13:49,530 And that gives two surprising results. 254 00:13:49,530 --> 00:13:54,420 One is that the direction of J relative to the applied 255 00:13:54,420 --> 00:13:58,000 electric field will change as you apply the field in 256 00:13:58,000 --> 00:14:00,520 different directions in the crystal. 257 00:14:00,520 --> 00:14:04,380 And moreover, the magnitude of the current is going to change 258 00:14:04,380 --> 00:14:07,390 as you change the orientation of field. 259 00:14:07,390 --> 00:14:11,310 That's true of every crystal in a crystal 260 00:14:11,310 --> 00:14:12,560 system other than cubic. 261 00:14:15,080 --> 00:14:18,680 So the thing that I would like to address today is the 262 00:14:18,680 --> 00:14:24,280 interesting matter of what is the nature of this anisotropy 263 00:14:24,280 --> 00:14:28,100 and if a consequence of a second ranked tensor relation 264 00:14:28,100 --> 00:14:31,990 is that, if we apply an electric field, the current 265 00:14:31,990 --> 00:14:34,290 doesn't go in the same direction 266 00:14:34,290 --> 00:14:35,360 as the applied field. 267 00:14:35,360 --> 00:14:39,380 It runs off in some other direction as non-intuitive as 268 00:14:39,380 --> 00:14:42,440 that might seem. 269 00:14:42,440 --> 00:14:47,250 First of all, there are two directions. 270 00:14:47,250 --> 00:14:50,760 If we talk about a property varying with direction, which 271 00:14:50,760 --> 00:14:53,320 direction are we talking about, the direction of the 272 00:14:53,320 --> 00:14:58,350 applied field, the direction of the resulting current flux, 273 00:14:58,350 --> 00:15:03,420 or something halfway in between, split the difference? 274 00:15:03,420 --> 00:15:06,560 Well, we can answer that question by doing a couple of 275 00:15:06,560 --> 00:15:08,140 thought experiments. 276 00:15:08,140 --> 00:15:12,460 Suppose we had decided that we want to measure the electrical 277 00:15:12,460 --> 00:15:17,560 conductivity along the 1, 0, 0 direction of an 278 00:15:17,560 --> 00:15:18,770 orthorhombic crystal. 279 00:15:18,770 --> 00:15:20,290 What would we do? 280 00:15:20,290 --> 00:15:22,640 Well, we get a hold of that crystal. 281 00:15:22,640 --> 00:15:27,620 And we would cut surfaces on it such that this was the 282 00:15:27,620 --> 00:15:31,680 direction of 1, 0, 0, if that's what we wanted to do. 283 00:15:31,680 --> 00:15:41,620 We would glom on electrodes and put a voltage across this 284 00:15:41,620 --> 00:15:45,420 crystal, this chunk of crystal that we had cut out. 285 00:15:45,420 --> 00:15:48,280 So when we say we're going to measure the property along 1, 286 00:15:48,280 --> 00:15:52,740 0, 0 or 1, 1, 1, what we really mean is that we're 287 00:15:52,740 --> 00:15:57,400 going to cut the crystal normal to that direction and 288 00:15:57,400 --> 00:16:00,330 then apply some sort of electrodes or attach 289 00:16:00,330 --> 00:16:03,390 some sort of probe. 290 00:16:03,390 --> 00:16:06,780 So when we talk about the direction of the property, the 291 00:16:06,780 --> 00:16:09,430 direction of the property will be the direction of the 292 00:16:09,430 --> 00:16:10,680 applied vector. 293 00:16:29,640 --> 00:16:34,990 And we had a very fancy name for the applied vector. 294 00:16:34,990 --> 00:16:38,430 We said we could think of that as a generalized force. 295 00:16:38,430 --> 00:16:41,510 And what happens is a generalized displacement. 296 00:16:41,510 --> 00:16:46,370 Notice that, when we apply the field, there are all sorts of 297 00:16:46,370 --> 00:16:50,730 components to J. And so the direction of the property, 298 00:16:50,730 --> 00:16:53,310 although we would get a number out of this experiment, the 299 00:16:53,310 --> 00:16:54,910 property really is a tensor. 300 00:16:54,910 --> 00:16:59,200 And there are nine different quantities involved in it. 301 00:16:59,200 --> 00:17:01,970 OK, so the direction of the property is going to be the 302 00:17:01,970 --> 00:17:05,140 direction in which we apply the generalized 303 00:17:05,140 --> 00:17:08,220 force of the stimulus. 304 00:17:08,220 --> 00:17:12,640 When we do that, however, that is going to unequivocally, at 305 00:17:12,640 --> 00:17:15,980 least in this experiment, place the electric field in a 306 00:17:15,980 --> 00:17:19,740 direction along the crystallographic orientation 307 00:17:19,740 --> 00:17:20,359 of interest. 308 00:17:20,359 --> 00:17:23,980 Because the electric field in a pair of electrodes like this 309 00:17:23,980 --> 00:17:27,109 extends normal to those electrodes. 310 00:17:27,109 --> 00:17:32,720 The current flow, the charge per unit area per unit time, 311 00:17:32,720 --> 00:17:35,490 goes off in some direction like this. 312 00:17:35,490 --> 00:17:40,310 Well, conductivity relates current flow to applied 313 00:17:40,310 --> 00:17:41,410 electric field. 314 00:17:41,410 --> 00:17:44,480 So would we say that the conductivity, when we apply 315 00:17:44,480 --> 00:17:50,510 field in that direction, is simply J/E, magnitude of J 316 00:17:50,510 --> 00:17:53,820 over magnitude of E since these are both vectors 317 00:17:53,820 --> 00:17:55,876 question mark? 318 00:17:55,876 --> 00:17:59,730 The answer to that, if we do this in terms of a thought 319 00:17:59,730 --> 00:18:01,550 experiment, is no. 320 00:18:01,550 --> 00:18:03,080 That is not what we're measuring. 321 00:18:05,980 --> 00:18:14,060 What we can imagine is that between our electrodes is some 322 00:18:14,060 --> 00:18:19,213 window of unit area between those electrodes. 323 00:18:24,500 --> 00:18:27,730 And what we're going to measure as the current flow is 324 00:18:27,730 --> 00:18:40,580 the amount of charge per unit area per unit time that passes 325 00:18:40,580 --> 00:18:44,190 between the electrodes, namely the charge per unit area per 326 00:18:44,190 --> 00:18:47,860 unit time that gets through this window and reaches the 327 00:18:47,860 --> 00:18:49,970 other electrode. 328 00:18:49,970 --> 00:18:52,130 That's what we're going to measure. 329 00:18:52,130 --> 00:18:56,330 And what we're going to want to call the conductivity is 330 00:18:56,330 --> 00:19:02,490 the ratio of this part of the current flux, let me call that 331 00:19:02,490 --> 00:19:08,220 J parallel, over the magnitude of E. And that's what we'll 332 00:19:08,220 --> 00:19:11,470 mean by the conductivity in the direction in which we've 333 00:19:11,470 --> 00:19:13,440 applied the field. 334 00:19:13,440 --> 00:19:16,060 The actual flow of current goes off 335 00:19:16,060 --> 00:19:17,410 in some other direction. 336 00:19:17,410 --> 00:19:20,640 But what we measure, at least in this experiment, is the 337 00:19:20,640 --> 00:19:24,350 component of that charge flux that goes through a window, 338 00:19:24,350 --> 00:19:28,540 that pair of unit area, that's parallel to the electrode. 339 00:19:28,540 --> 00:19:31,440 And we don't really care if the current is going off in a 340 00:19:31,440 --> 00:19:32,430 different direction. 341 00:19:32,430 --> 00:19:36,000 We don't even care if it jumps around in majestic 342 00:19:36,000 --> 00:19:37,370 loop-de-loops. 343 00:19:37,370 --> 00:19:40,640 What we're going to measure is the amount of that charge 344 00:19:40,640 --> 00:19:46,380 motion that moves normal to our window and gets through 345 00:19:46,380 --> 00:19:51,190 one unit area from one electrode to another. 346 00:19:51,190 --> 00:19:53,920 So the conductivity and the direction of E then is going 347 00:19:53,920 --> 00:19:59,190 to be measured by the parallel part of the resulting 348 00:19:59,190 --> 00:20:02,340 generalized displacement divided by the magnitude of 349 00:20:02,340 --> 00:20:03,970 the applied generalized force. 350 00:20:06,780 --> 00:20:08,920 If you're not convinced and completely overwhelmed by 351 00:20:08,920 --> 00:20:12,860 that, let me give you another experiment. 352 00:20:12,860 --> 00:20:17,280 Another tensor relation, another tensor property, is 353 00:20:17,280 --> 00:20:21,070 the diffusivity of a material. 354 00:20:21,070 --> 00:20:28,840 And if you put a material in a concentration gradient dc dx, 355 00:20:28,840 --> 00:20:33,580 you produce a flux of matter. 356 00:20:33,580 --> 00:20:36,650 And the proportionality constant is the diffusion 357 00:20:36,650 --> 00:20:38,640 coefficient. 358 00:20:38,640 --> 00:20:43,660 And this has units of length per unit time if you put in 359 00:20:43,660 --> 00:20:48,150 the units of the concentration gradient and the flux. 360 00:20:48,150 --> 00:20:52,280 Well, dc dx is a gradient. 361 00:20:52,280 --> 00:20:54,070 That's a vector really. 362 00:20:54,070 --> 00:20:57,970 So a proper tensor relation, instead of a scalar relation 363 00:20:57,970 --> 00:21:02,890 like this, would be the change of concentration with the i-th 364 00:21:02,890 --> 00:21:04,400 coordinate. 365 00:21:04,400 --> 00:21:08,990 And out in front would be a diffusion coefficient Dij. 366 00:21:08,990 --> 00:21:15,300 And this tensor relation would give us the components of the 367 00:21:15,300 --> 00:21:20,190 mass flux J1 J2 J3. 368 00:21:20,190 --> 00:21:23,040 Actually, thermodynamically we shouldn't be talking about a 369 00:21:23,040 --> 00:21:24,390 concentration gradient. 370 00:21:24,390 --> 00:21:27,640 We should be talking about a gradient in free energy. 371 00:21:27,640 --> 00:21:30,380 The concentration works well enough. 372 00:21:30,380 --> 00:21:33,170 And the concentration is what one would measure. 373 00:21:33,170 --> 00:21:36,510 So what might a typical experiment be? 374 00:21:36,510 --> 00:21:40,110 If you wanted to know what the diffusion was in the 1, 1, 1 375 00:21:40,110 --> 00:21:44,060 direction of some crystal, you would, again, cut a plate that 376 00:21:44,060 --> 00:21:46,900 is normal to the direction of interest. 377 00:21:46,900 --> 00:21:49,860 And then your boundary conditions can be different. 378 00:21:49,860 --> 00:21:53,190 But what you would typically do for a simple experiment is 379 00:21:53,190 --> 00:21:58,920 to put some sort of solute on the surface of the crystal. 380 00:21:58,920 --> 00:22:04,250 And what you would see after heating the sample up for a 381 00:22:04,250 --> 00:22:08,900 period of time is that the material would have migrated 382 00:22:08,900 --> 00:22:11,680 into the sample. 383 00:22:11,680 --> 00:22:14,820 And if you plotted concentration as a function of 384 00:22:14,820 --> 00:22:18,760 distance along the normal to the plate that you've cut, 385 00:22:18,760 --> 00:22:20,280 this would do something like that. 386 00:22:25,040 --> 00:22:27,790 What can you say about this process after you've done the 387 00:22:27,790 --> 00:22:29,040 experiment? 388 00:22:30,780 --> 00:22:35,970 All that you can say is that some front of concentration 389 00:22:35,970 --> 00:22:40,580 has advanced parallel to the surface. 390 00:22:40,580 --> 00:22:43,780 And you have no idea what the individual atoms have done. 391 00:22:43,780 --> 00:22:46,240 Then they could be doing loop-de-loops 392 00:22:46,240 --> 00:22:47,510 or little hip hops. 393 00:22:47,510 --> 00:22:51,750 And all that you measure is the rate at which material 394 00:22:51,750 --> 00:22:54,910 advances in the direction of the current gradient. 395 00:22:54,910 --> 00:22:57,610 So again, I think I've convinced you, hopefully, that 396 00:22:57,610 --> 00:23:01,360 what you're measuring is the flux of matter unit area per 397 00:23:01,360 --> 00:23:05,840 unit time that is in the direction of the applied 398 00:23:05,840 --> 00:23:08,500 concentration gradient. 399 00:23:08,500 --> 00:23:11,880 And you do not measure, and in fact, in this instance, I 400 00:23:11,880 --> 00:23:14,830 don't think I could come up with even a very clever 401 00:23:14,830 --> 00:23:18,580 experiment that would let you measure in a single experiment 402 00:23:18,580 --> 00:23:21,380 the magnitude and direction of the net flux. 403 00:23:26,410 --> 00:23:29,730 All right, so if we're measuring a property then like 404 00:23:29,730 --> 00:23:34,490 conductivity, the value of the conductivity in the direction 405 00:23:34,490 --> 00:23:38,370 of the applied field is going to be the part of the 406 00:23:38,370 --> 00:23:41,720 conductivity that is parallel to the field divided by the 407 00:23:41,720 --> 00:23:44,020 magnitude of the field. 408 00:23:44,020 --> 00:23:45,330 We all agree on that. 409 00:23:45,330 --> 00:23:47,370 Let's have a show of hands. 410 00:23:47,370 --> 00:23:49,050 Fine. 411 00:23:49,050 --> 00:23:52,440 I saw just a few tired hands saying, OK, I believe, I 412 00:23:52,440 --> 00:23:54,610 believe, let's get on with it. 413 00:23:54,610 --> 00:23:57,990 All right, let me now put this using our tensor relationship 414 00:23:57,990 --> 00:24:02,680 into a nice analytic form that we can not only do something 415 00:24:02,680 --> 00:24:04,400 with but can gain some insight. 416 00:24:09,960 --> 00:24:15,950 The components of the charge density vector are given by 417 00:24:15,950 --> 00:24:20,480 the conductivity tensor sigma 1, 1 times the component of 418 00:24:20,480 --> 00:24:25,460 the electric field E1 plus sigma 1, 2 times the component 419 00:24:25,460 --> 00:24:30,200 of field E2 plus sigma 1, 3 times E3. 420 00:24:30,200 --> 00:24:33,050 Or in general, I don't have to write out every equation. 421 00:24:33,050 --> 00:24:38,200 We can say that the i-th component of J, our old friend 422 00:24:38,200 --> 00:24:40,960 the second ranked tensor property, again, is given by 423 00:24:40,960 --> 00:24:45,430 sigma ij times each component of applied field. 424 00:24:48,060 --> 00:24:51,820 If we specify the direction relative to the same 425 00:24:51,820 --> 00:24:55,820 coordinate system in which the electric field is being 426 00:24:55,820 --> 00:25:02,330 applied, we can specify that direction by a set of three 427 00:25:02,330 --> 00:25:07,160 direction cosines, l1, l2, and l3. 428 00:25:07,160 --> 00:25:11,480 So we can write as the three components of the electric 429 00:25:11,480 --> 00:25:17,870 field E sub i as the magnitude of E times the direction 430 00:25:17,870 --> 00:25:22,390 cosines l1, l2, and l3. 431 00:25:22,390 --> 00:25:24,690 So here we now have the components of 432 00:25:24,690 --> 00:25:28,100 the electric field. 433 00:25:28,100 --> 00:25:31,080 If these are the components of the electric field, we can say 434 00:25:31,080 --> 00:25:34,500 that J sub i is going to be equal to sigma 435 00:25:34,500 --> 00:25:37,710 ij times E sub j. 436 00:25:37,710 --> 00:25:42,930 And I can write for E sub j the magnitude of E times the 437 00:25:42,930 --> 00:25:45,980 direction cosines l sub j where these are the same 438 00:25:45,980 --> 00:25:48,200 direction cosines of the applied field. 439 00:25:53,950 --> 00:25:58,820 OK, now we have two relations in reduced subscript form that 440 00:25:58,820 --> 00:26:00,070 we can do something with. 441 00:26:03,450 --> 00:26:11,760 The magnitude of the conductivity in the direction 442 00:26:11,760 --> 00:26:16,060 of E, that is in the direction that has direction cosines li, 443 00:26:16,060 --> 00:26:21,950 is going to be J parallel over the magnitude of E. We can 444 00:26:21,950 --> 00:26:33,630 write that as J.E, except that involves the magnitude of J, 445 00:26:33,630 --> 00:26:37,980 the magnitude of E times the cosine of the 446 00:26:37,980 --> 00:26:40,350 angle between them. 447 00:26:40,350 --> 00:26:43,990 And so if I just want J parallel, I'd want to divide 448 00:26:43,990 --> 00:26:47,990 this by the magnitude of E. And that will give me just the 449 00:26:47,990 --> 00:26:53,940 part of J that falls parallel to E. And then down here in 450 00:26:53,940 --> 00:26:57,280 the denominator, I'll have magnitude of E. 451 00:26:57,280 --> 00:27:01,130 So if I tidy this up a little bit to bring everything on the 452 00:27:01,130 --> 00:27:08,260 same level, we can write the dot product of J and E as Ji 453 00:27:08,260 --> 00:27:16,800 Ei, like J1 E1 plus J2 E2 plus J3 E3. 454 00:27:16,800 --> 00:27:19,530 And then I will have in the denominator 455 00:27:19,530 --> 00:27:21,170 magnitude of E squared. 456 00:27:25,190 --> 00:27:27,805 But we know what Ei is. 457 00:27:33,080 --> 00:27:38,850 It's going to be magnitude of E times li. 458 00:27:38,850 --> 00:27:43,060 I know how to find the i-th component of J. That's going 459 00:27:43,060 --> 00:27:52,790 to be sigma ij times E sub j over magnitude of E squared. 460 00:27:52,790 --> 00:28:01,150 And I can write E sub j as magnitude of E times l sub j 461 00:28:01,150 --> 00:28:05,970 times l sub i times magnitude of E divided by 462 00:28:05,970 --> 00:28:07,960 magnitude of E squared. 463 00:28:07,960 --> 00:28:10,970 Magnitude of E drops out. 464 00:28:10,970 --> 00:28:13,890 Again, the conductivity is linear the way 465 00:28:13,890 --> 00:28:14,740 we've defined it. 466 00:28:14,740 --> 00:28:17,780 It shouldn't depend on the magnitude of E. And what I'm 467 00:28:17,780 --> 00:28:20,290 left with then is a very simple and a 468 00:28:20,290 --> 00:28:21,860 rather profound relation. 469 00:28:21,860 --> 00:28:26,400 It says that in a direction specified by the set of 470 00:28:26,400 --> 00:28:32,550 direction cosines l sub i that the magnitude of conductivity 471 00:28:32,550 --> 00:28:41,090 in that direction is going to be li lj times sigma ij. 472 00:28:41,090 --> 00:28:45,230 So that is an expression that involves the three direction 473 00:28:45,230 --> 00:28:48,530 cosines in which you're applying the field and 474 00:28:48,530 --> 00:28:50,950 measuring the property. 475 00:28:50,950 --> 00:28:54,440 But this is a linear combination of every one of 476 00:28:54,440 --> 00:28:57,040 the elements in the original tensor. 477 00:28:57,040 --> 00:28:59,220 So if I write this out, because this is an important 478 00:28:59,220 --> 00:29:03,710 relation, it's going to be l1 squared sigma 1, 1 plus l2 479 00:29:03,710 --> 00:29:13,550 squared sigma 2, 2 plus l3 squared sigma 3, 3. 480 00:29:16,360 --> 00:29:21,050 And then there will be cross-terms of the form 2 l1 481 00:29:21,050 --> 00:29:34,220 l2 sigma 1, 2 plus 2 l2 l3 times sigma 2, 3 plus 2 l3 l1 482 00:29:34,220 --> 00:29:37,360 times sigma 1, 3. 483 00:29:41,420 --> 00:29:45,730 That has made an assumption. 484 00:29:45,730 --> 00:29:48,470 And probably I should not have done it at this point. 485 00:29:48,470 --> 00:29:54,430 That's assuming that the term l1 l2 sigma 1, 2 and the term 486 00:29:54,430 --> 00:29:58,260 l2 l1 sigma 2, 1 can be lumped together. 487 00:29:58,260 --> 00:29:59,860 And maybe I shouldn't write it that way. 488 00:29:59,860 --> 00:30:03,650 Because that assumes that the tensor is symmetric. 489 00:30:03,650 --> 00:30:06,440 And it doesn't have to be for some properties. 490 00:30:06,440 --> 00:30:09,910 So let me back off and write it without any assumptions. 491 00:30:09,910 --> 00:30:22,010 So this would be sigma 1, 2 l1 l2 plus sigma 1, 3 l1 l3 plus 492 00:30:22,010 --> 00:30:32,230 sigma 2, 1 times l2 l1 and then corresponding terms-- 493 00:30:32,230 --> 00:30:35,080 let me put this one down here-- 494 00:30:35,080 --> 00:30:43,010 sigma 2, 1 times l2 l1 plus sigma 3, 1 l3 l1. 495 00:30:43,010 --> 00:30:47,030 And then the one that I'm missing is sigma 2, 3 times l2 496 00:30:47,030 --> 00:30:52,900 l3 plus sigma 3, 2 times l2 l3. 497 00:30:52,900 --> 00:30:58,950 So there is an explicit statement of this more compact 498 00:30:58,950 --> 00:31:03,840 expression written in reduced subscript notation. 499 00:31:03,840 --> 00:31:09,010 OK, so this is something that is going to tell us in a very 500 00:31:09,010 --> 00:31:13,970 neat sort of way what the anisotropy of the property is. 501 00:31:13,970 --> 00:31:16,090 You pick the direction in which you're interested. 502 00:31:16,090 --> 00:31:19,860 And this will tell you the value of the property in that 503 00:31:19,860 --> 00:31:24,570 direction and, therefore, by inference how things will 504 00:31:24,570 --> 00:31:25,900 change with direction. 505 00:31:30,250 --> 00:31:32,600 Let me quickly, and then I'll pause to see if you have any 506 00:31:32,600 --> 00:31:38,090 questions, give you an interpretation of the meaning 507 00:31:38,090 --> 00:31:40,770 of some of these elements of the tensor. 508 00:31:40,770 --> 00:31:42,750 We've really not been able to do that. 509 00:31:42,750 --> 00:31:45,710 We've been able to use them to give the value of the property 510 00:31:45,710 --> 00:31:49,750 in a given direction but not their individual meaning. 511 00:31:49,750 --> 00:31:56,320 Suppose I ask what the value of the property would be along 512 00:31:56,320 --> 00:31:59,070 the x1 direction. 513 00:31:59,070 --> 00:32:02,810 So suppose we ask what value of the conductivity would we 514 00:32:02,810 --> 00:32:07,960 measure along x1 for a given tensor sigma ij. 515 00:32:07,960 --> 00:32:11,580 If that is the case, l1 is equal to 1. 516 00:32:11,580 --> 00:32:16,510 The angle between the x1 axis and x2 is 0. 517 00:32:16,510 --> 00:32:19,160 So l2 is 0. 518 00:32:19,160 --> 00:32:21,950 90 degrees cosine of 90 degrees is zero. 519 00:32:21,950 --> 00:32:24,270 l3 would be equal to 0. 520 00:32:24,270 --> 00:32:29,060 So if I put this set of direction cosines into this 521 00:32:29,060 --> 00:32:33,720 complicated polynomial, the term l1 squared 522 00:32:33,720 --> 00:32:35,530 sigma 1, 1 stays in. 523 00:32:35,530 --> 00:32:37,010 But l2 is 0. 524 00:32:37,010 --> 00:32:38,200 l3 is 0. 525 00:32:38,200 --> 00:32:42,030 And I've got either an l2 or an l3 or both in all of these 526 00:32:42,030 --> 00:32:43,090 other terms. 527 00:32:43,090 --> 00:32:49,260 So along the x1 direction, the only term that stays in is 528 00:32:49,260 --> 00:32:56,170 sigma 1, 1 squared times l1 squared. 529 00:32:56,170 --> 00:32:57,830 But l1 is unity. 530 00:32:57,830 --> 00:33:08,390 So along the x1 direction the value of sigma is sigma 1, 1. 531 00:33:11,550 --> 00:33:14,980 And by inference the value of the property that we would 532 00:33:14,980 --> 00:33:18,400 measure along x2 would be sigma 2, 2. 533 00:33:18,400 --> 00:33:20,660 And the value of the property along x3, guess 534 00:33:20,660 --> 00:33:22,780 what, sigma 3, 3. 535 00:33:22,780 --> 00:33:27,620 So in this three by three array of numbers for sigma ij, 536 00:33:27,620 --> 00:33:31,260 there's a very direct meaning to the diagonal 537 00:33:31,260 --> 00:33:32,540 values of the tensor. 538 00:33:32,540 --> 00:33:35,860 These are the values that you would measure along x1, along 539 00:33:35,860 --> 00:33:38,580 x2, and along x3 directly. 540 00:33:44,710 --> 00:33:49,190 So the off diagonal terms are going to be saying something 541 00:33:49,190 --> 00:33:54,700 about how the extreme values of the property are aligned 542 00:33:54,700 --> 00:33:56,505 relative to our reference axes. 543 00:34:14,510 --> 00:34:17,370 OK, any comment or questions at this point? 544 00:34:23,105 --> 00:34:24,000 Yeah, [INAUDIBLE]? 545 00:34:24,000 --> 00:34:25,380 AUDIENCE: [INAUDIBLE]. 546 00:34:25,380 --> 00:34:30,260 PROFESSOR: Yeah, we've defined those as the cosines of the 547 00:34:30,260 --> 00:34:33,840 angles between the reference axes. 548 00:34:33,840 --> 00:34:36,870 And this is the direction in which we're applying our 549 00:34:36,870 --> 00:34:40,590 electric field or our generalized forced temperature 550 00:34:40,590 --> 00:34:43,300 gradient, concentration gradient, magnetic field, or 551 00:34:43,300 --> 00:34:44,550 what have you. 552 00:35:01,790 --> 00:35:03,870 OK, I've got some time left. 553 00:35:03,870 --> 00:35:14,260 So let me push this further into a different form by using 554 00:35:14,260 --> 00:35:20,165 the elements of the tensor to define a locus in space. 555 00:35:25,500 --> 00:35:28,710 Again, what I'll do is something that may seem silly. 556 00:35:28,710 --> 00:35:33,080 But we'll see that there are some very useful and profound 557 00:35:33,080 --> 00:35:36,720 conclusions to be drawn from it. 558 00:35:36,720 --> 00:35:42,520 Let me take the elements of the tensor and use them as 559 00:35:42,520 --> 00:35:46,640 coefficients to define a surface. 560 00:35:46,640 --> 00:35:52,790 I'm going to take sigma 1, 1 and multiply it by the product 561 00:35:52,790 --> 00:35:56,610 of the coordinate x1 times x1. 562 00:35:56,610 --> 00:35:59,720 What I'm viewing this now is a variable. 563 00:35:59,720 --> 00:36:03,340 I'm going to take sigma 2, 2 and multiply it by the 564 00:36:03,340 --> 00:36:07,330 coordinate x2 and x2 again. 565 00:36:07,330 --> 00:36:13,620 Sigma 3, 3 and I multiply that by x3 times x3. 566 00:36:13,620 --> 00:36:18,940 And then I'll have these cross-terms, sigma 1, 2, x1, 567 00:36:18,940 --> 00:36:21,240 x2, and so on. 568 00:36:21,240 --> 00:36:24,340 Or in short, I'm going to define a 569 00:36:24,340 --> 00:36:30,590 function sigma ij xi xj. 570 00:36:30,590 --> 00:36:33,360 And xi and xj are running variables. 571 00:36:33,360 --> 00:36:38,110 x1, x2, x3, they can extend from 0 to infinity. 572 00:36:38,110 --> 00:36:45,980 But now I'm going to take this sum of nine terms and I'm 573 00:36:45,980 --> 00:36:47,455 going to set it equal to a constant. 574 00:36:50,250 --> 00:36:53,470 And what constant could be neater and cleaner and more 575 00:36:53,470 --> 00:36:55,990 abstract than unity? 576 00:36:55,990 --> 00:37:01,420 But I could say that some function of x1, x2, x3 equal 577 00:37:01,420 --> 00:37:04,880 to a constant is going to define a locus of points in 578 00:37:04,880 --> 00:37:08,470 space, which satisfy that equation. 579 00:37:08,470 --> 00:37:09,760 And we do that all the time. 580 00:37:09,760 --> 00:37:13,420 We specify some functional relationship that defines in 581 00:37:13,420 --> 00:37:23,115 the space some surface f of xi xj equals a constant. 582 00:37:23,115 --> 00:37:25,230 And that's done all time in mathematics. 583 00:37:25,230 --> 00:37:28,340 I've yet to encounter yet a department of mathematics that 584 00:37:28,340 --> 00:37:32,440 did not have in its corridors some glass case that was 585 00:37:32,440 --> 00:37:36,200 filled with yellowing plaster figures, that some of them 586 00:37:36,200 --> 00:37:41,210 cracking or already cracked, that represented exotic 587 00:37:41,210 --> 00:37:45,220 surfaces, like elliptic paraboloids and 588 00:37:45,220 --> 00:37:46,750 things of that sort. 589 00:37:46,750 --> 00:37:49,350 I think the last time I walked down the corridor of building 590 00:37:49,350 --> 00:37:52,450 two there really was one of those in our mathematics 591 00:37:52,450 --> 00:37:55,300 department in the little door that opened out onto the great 592 00:37:55,300 --> 00:37:57,410 court about halfway down that corridor. 593 00:37:57,410 --> 00:38:01,405 But you've seen these things I'm sure, elliptic cylinders, 594 00:38:01,405 --> 00:38:03,880 elliptic cones, and all sorts of glorious things. 595 00:38:07,920 --> 00:38:13,320 This function, though, is a very special one. 596 00:38:13,320 --> 00:38:16,470 This is a quadratic equation in the second order of 597 00:38:16,470 --> 00:38:17,660 coordinates. 598 00:38:17,660 --> 00:38:22,280 And this is referred to as a quadratic form. 599 00:38:27,400 --> 00:38:30,510 And sometimes for short, since that's something of a 600 00:38:30,510 --> 00:38:33,555 mouthful, one refers to this as a quadric. 601 00:38:37,420 --> 00:38:42,480 And we are going to very shortly demonstrate that this 602 00:38:42,480 --> 00:38:47,480 particular function tells you everything you could possibly 603 00:38:47,480 --> 00:38:51,650 want to know about the variation of a tensor property 604 00:38:51,650 --> 00:38:55,650 with direction, how the magnitude of the property 605 00:38:55,650 --> 00:38:59,510 changes, how the direction of the resulting current flow or 606 00:38:59,510 --> 00:39:01,660 flux is oriented. 607 00:39:01,660 --> 00:39:05,305 And this is, consequently, called, when applied to a 608 00:39:05,305 --> 00:39:07,100 tensor, the representation quadric. 609 00:39:16,800 --> 00:39:20,310 Because it really does represent anything you would 610 00:39:20,310 --> 00:39:22,350 like to know about the physical property. 611 00:39:26,270 --> 00:39:30,440 This will be demonstrated in short order. 612 00:39:30,440 --> 00:39:34,410 Let's first notice, though, that there's only a limited 613 00:39:34,410 --> 00:39:44,660 number of surfaces that can be represented by this equation. 614 00:39:44,660 --> 00:39:46,390 One of them is one that we're all 615 00:39:46,390 --> 00:39:48,500 familiar with, an ellipsoid. 616 00:39:51,020 --> 00:39:55,430 And that's most easily represented when we've 617 00:39:55,430 --> 00:39:58,400 referred it to the principal axes of the ellipsoid. 618 00:40:01,910 --> 00:40:05,070 So an ellipsoid is a quasi-sausage 619 00:40:05,070 --> 00:40:06,730 like thing like this. 620 00:40:06,730 --> 00:40:13,250 These sections perpendicular to x3, x1, and x2 are all 621 00:40:13,250 --> 00:40:15,330 ellipsoids. 622 00:40:15,330 --> 00:40:24,800 And if this semi-axis is a and this semi-axis is b and this 623 00:40:24,800 --> 00:40:31,690 semi-axis is c, the equation of the surface in that special 624 00:40:31,690 --> 00:40:38,110 orientation is x1 squared over a squared plus x2 squared over 625 00:40:38,110 --> 00:40:43,280 b squared plus x3 squared over c squared equals 1. 626 00:40:52,240 --> 00:40:55,980 OK, the surface that we have defined 627 00:40:55,980 --> 00:40:59,070 involves tensor elements. 628 00:40:59,070 --> 00:41:04,150 And if the equation is ever going to get into this form, 629 00:41:04,150 --> 00:41:07,640 we've had to get rid of these cross-terms. 630 00:41:07,640 --> 00:41:12,710 And we are going to have principal axes that are some 631 00:41:12,710 --> 00:41:16,800 function of the tensor elements that would be out 632 00:41:16,800 --> 00:41:22,820 here in the form of some aggregate sigma ij prime. 633 00:41:22,820 --> 00:41:26,730 So we'll have to see how one could go from a tensor that 634 00:41:26,730 --> 00:41:30,090 describes an ellipsoid in a general orientation to 635 00:41:30,090 --> 00:41:33,190 something that has been diagonalized and has the 636 00:41:33,190 --> 00:41:36,450 coordinate system taken along the principal axes. 637 00:41:36,450 --> 00:41:38,460 But that's only one possible surface 638 00:41:38,460 --> 00:41:39,710 that we could encounter. 639 00:41:44,540 --> 00:41:51,830 The second one is one that we might encounter that, when we 640 00:41:51,830 --> 00:41:58,520 put it in diagonal form, has this form, x1 squared over a 641 00:41:58,520 --> 00:42:07,580 squared plus x2 squared over b squared minus z squared over c 642 00:42:07,580 --> 00:42:09,430 squared equals 1. 643 00:42:14,550 --> 00:42:17,695 This is something that has a peculiar shape. 644 00:42:17,695 --> 00:42:21,948 It's sort of an hourglass like figure. 645 00:42:21,948 --> 00:42:24,870 It has an ellipsoidal cross-section. 646 00:42:27,950 --> 00:42:28,906 This is x3. 647 00:42:28,906 --> 00:42:29,860 This is x1. 648 00:42:29,860 --> 00:42:31,275 This is x2. 649 00:42:31,275 --> 00:42:36,020 It has an ellipsoidal section perpendicular to x3. 650 00:42:36,020 --> 00:42:41,950 But it has, as a cross-section in planes that are parallel to 651 00:42:41,950 --> 00:42:45,680 x3, paraboloids. 652 00:42:45,680 --> 00:42:48,180 And the shape of the paraboloids depends on which 653 00:42:48,180 --> 00:42:52,500 particular section that you take through x3. 654 00:42:52,500 --> 00:42:54,380 This is a surface that's called a hyperboloid. 655 00:43:00,910 --> 00:43:03,110 And it's all one continuous surface. 656 00:43:03,110 --> 00:43:09,560 So this is called a hyperboloid of one sheet, 657 00:43:09,560 --> 00:43:11,775 sheet in the sense of surface. 658 00:43:15,230 --> 00:43:19,520 And then down in here when x3 is 0, it would again be an 659 00:43:19,520 --> 00:43:23,400 ellipsoidal cross-section but a little bit smaller than when 660 00:43:23,400 --> 00:43:26,600 we increase x3 above 0. 661 00:43:35,390 --> 00:43:39,920 OK, and the other surface, another surface, a third kind 662 00:43:39,920 --> 00:43:48,470 of surface is something that occurs when the terms in front 663 00:43:48,470 --> 00:43:56,390 of two coordinates are negative, x2 squared over b 664 00:43:56,390 --> 00:44:01,160 squared x3 squared over c squared equals 1. 665 00:44:01,160 --> 00:44:06,380 This is something that looks like two 666 00:44:06,380 --> 00:44:10,680 surfaces nose to nose. 667 00:44:10,680 --> 00:44:16,140 The cross-section of these two different sheets are ellipsis. 668 00:44:16,140 --> 00:44:20,180 The cross-sections passing through-- 669 00:44:20,180 --> 00:44:20,580 let's see. 670 00:44:20,580 --> 00:44:23,990 This would be the shape for x1. 671 00:44:23,990 --> 00:44:26,250 That would be the special direction. 672 00:44:30,060 --> 00:44:35,320 And then this could be x2 and this x3. 673 00:44:35,320 --> 00:44:37,500 OK, this is called an hyperboloid of two sheets. 674 00:44:49,910 --> 00:44:55,850 And it has the property that the distance to the surface in 675 00:44:55,850 --> 00:45:01,530 orientations in between the asymptotes of these hyperbolic 676 00:45:01,530 --> 00:45:04,476 sections, these radii, are imaginary. 677 00:45:17,520 --> 00:45:22,140 And between the asymptotes, if we take a section, we would 678 00:45:22,140 --> 00:45:28,380 have two hyperboloids in a cross-section. 679 00:45:28,380 --> 00:45:31,320 And again, a range of directions between the 680 00:45:31,320 --> 00:45:35,700 asymptotes in which the radius would be imaginary in 681 00:45:35,700 --> 00:45:40,020 directions along the coordinate which has the 682 00:45:40,020 --> 00:45:43,370 positive sign we'd have a minimum radius. 683 00:45:43,370 --> 00:45:47,890 And this would get progressively larger and go to 684 00:45:47,890 --> 00:45:49,545 infinity along the asymptote. 685 00:45:53,960 --> 00:46:00,200 Finally, there's a third surface in which, by 686 00:46:00,200 --> 00:46:09,170 extension, all three terms in x1, x2, and x3 687 00:46:09,170 --> 00:46:10,535 have negative signs. 688 00:46:16,095 --> 00:46:17,690 And what does this look like? 689 00:46:17,690 --> 00:46:19,378 AUDIENCE: [INAUDIBLE] 690 00:46:19,378 --> 00:46:20,360 PROFESSOR: No, it does. 691 00:46:20,360 --> 00:46:23,220 It's something that's called an imaginary ellipsoid. 692 00:46:28,150 --> 00:46:34,140 So called because the radius in all direction is imaginary. 693 00:46:34,140 --> 00:46:36,290 And what does it look like? 694 00:46:36,290 --> 00:46:38,100 You just have to use your imagination. 695 00:46:38,100 --> 00:46:38,710 That's all. 696 00:46:38,710 --> 00:46:41,610 Because it's an imaginary ellipsoid. 697 00:46:41,610 --> 00:46:45,030 Is that something that we could ever get if we were 698 00:46:45,030 --> 00:46:50,960 using the elements of a property tensor? 699 00:46:50,960 --> 00:46:54,220 The answer, he says with a big wink, yes. 700 00:46:54,220 --> 00:46:56,960 It's unusual but yes. 701 00:46:56,960 --> 00:47:00,140 You can get physical properties whose magnitude as 702 00:47:00,140 --> 00:47:04,495 a function of direction do this. 703 00:47:04,495 --> 00:47:09,260 All right, I think that is a sufficiently formal and 704 00:47:09,260 --> 00:47:12,340 deadening component to our presentation that it would be 705 00:47:12,340 --> 00:47:16,280 appropriate to take a break and stretch a little bit. 706 00:47:19,690 --> 00:47:23,960 When we return, I will show you an amazing connection 707 00:47:23,960 --> 00:47:29,290 between the anisotropy of a physical property and this 708 00:47:29,290 --> 00:47:30,750 representation surface. 709 00:47:30,750 --> 00:47:32,720 So called because it tells you how the 710 00:47:32,720 --> 00:47:34,430 property is going to change.