1 00:00:00,470 --> 00:00:03,720 The following content is provided by MIT OpenCourseWare 2 00:00:03,720 --> 00:00:06,130 under a Creative Commons license. 3 00:00:06,130 --> 00:00:08,750 Additional information about our license and MIT 4 00:00:08,750 --> 00:00:12,394 OpenCourseWare in general is available at ocw.mit.edu. 5 00:00:15,640 --> 00:00:17,840 PROFESSOR: --not of the vindictive sort. 6 00:00:17,840 --> 00:00:20,790 You skip class, you've skipped a lot of important stuff. 7 00:00:20,790 --> 00:00:25,220 But I'll get you on the quiz, that's all. 8 00:00:25,220 --> 00:00:27,300 Kidding aside, there were a number of things 9 00:00:27,300 --> 00:00:30,530 that I passed out. 10 00:00:30,530 --> 00:00:34,530 I think nobody needs a set of problem set number 13. 11 00:00:34,530 --> 00:00:37,440 That was the one that had something on symmetry 12 00:00:37,440 --> 00:00:39,645 constraints and working with second-rank tensors. 13 00:00:39,645 --> 00:00:45,020 If anybody missed that, you can see me during break. 14 00:00:45,020 --> 00:00:48,430 Some people, I think, missed problem set 14. 15 00:00:48,430 --> 00:00:52,200 And that's the one where you are invited to diagonalize 16 00:00:52,200 --> 00:00:57,260 some tensors, either using the method of successive 17 00:00:57,260 --> 00:00:59,705 approximations or the direct diagonalization- 18 00:00:59,705 --> 00:01:02,710 by-an-eigenvalue procedure. 19 00:01:02,710 --> 00:01:05,540 Anybody need one of that? 20 00:01:05,540 --> 00:01:10,570 And I'm sure that nobody has a copy of problem set number 15, 21 00:01:10,570 --> 00:01:12,310 which deals with piezoelectricity. 22 00:01:12,310 --> 00:01:13,880 And I know you don't have it, because 23 00:01:13,880 --> 00:01:15,450 I just put it together. 24 00:01:15,450 --> 00:01:19,930 So I'd like to hand it out and invite you to explore things 25 00:01:19,930 --> 00:01:23,090 that deal with third-rank tensors. 26 00:01:23,090 --> 00:01:29,510 And I hope that, even though doing the problem sets is 27 00:01:29,510 --> 00:01:34,120 optional, particularly at this juncture in the semester when 28 00:01:34,120 --> 00:01:38,580 things have come to a set of successive crunches. 29 00:01:38,580 --> 00:01:41,250 But if you don't know how to do it, for goodness' sakes, 30 00:01:41,250 --> 00:01:41,920 come see me. 31 00:01:41,920 --> 00:01:44,546 Or raise it in our next class. 32 00:01:44,546 --> 00:01:47,130 You know, some question like, I haven't the foggiest idea 33 00:01:47,130 --> 00:01:48,430 how to do problem number two. 34 00:01:48,430 --> 00:01:50,220 Could you say a little bit about that, please? 35 00:01:50,220 --> 00:01:51,470 And I'd be happy to oblige. 36 00:01:54,440 --> 00:01:54,800 All right. 37 00:01:54,800 --> 00:01:59,760 I will have for you next time the quizzes and also all the 38 00:01:59,760 --> 00:02:01,800 problem sets which will have been turned 39 00:02:01,800 --> 00:02:03,480 in up to that point. 40 00:02:03,480 --> 00:02:08,280 And what I spent my time doing instead is writing out notes 41 00:02:08,280 --> 00:02:13,750 for those people who missed the last lecture, and also 42 00:02:13,750 --> 00:02:15,480 notes covering what we're going to do today. 43 00:02:15,480 --> 00:02:20,370 Because it is very exquisitely intensive, algebraically. 44 00:02:20,370 --> 00:02:23,810 It's not hard, but there are a lot of variables with a lot of 45 00:02:23,810 --> 00:02:24,960 subscripts. 46 00:02:24,960 --> 00:02:27,696 So let me pass this around. 47 00:02:27,696 --> 00:02:30,740 I'll split it up into packs. 48 00:02:30,740 --> 00:02:37,240 These are notes on some basic relations in electromagnetism 49 00:02:37,240 --> 00:02:39,095 which you may or may not have forgotten. 50 00:02:41,620 --> 00:02:43,480 Take one off the top. 51 00:02:43,480 --> 00:02:45,850 And it's coming at you from either side, so you're going 52 00:02:45,850 --> 00:02:46,770 to pass it back. 53 00:02:46,770 --> 00:02:49,590 And the notes also cover everything that we're going to 54 00:02:49,590 --> 00:02:52,170 do on piezoelectricity. 55 00:02:52,170 --> 00:02:54,450 Most of it will take place today. 56 00:02:54,450 --> 00:02:56,840 And I can zip along a little more rapidly if you 57 00:02:56,840 --> 00:02:59,351 have notes to follow. 58 00:02:59,351 --> 00:03:04,510 I would like to ask you when you get a set of the notes-- 59 00:03:04,510 --> 00:03:08,730 I could really kick myself-- 60 00:03:08,730 --> 00:03:15,630 the introductory discussion reminds you of the definition 61 00:03:15,630 --> 00:03:17,130 of a dipole. 62 00:03:17,130 --> 00:03:22,750 And down in the middle of the page, on the cover sheet, two 63 00:03:22,750 --> 00:03:27,040 different types of polarizability are defined. 64 00:03:27,040 --> 00:03:30,410 And one of them involves the separation of charge on an 65 00:03:30,410 --> 00:03:32,050 individual atom. 66 00:03:32,050 --> 00:03:34,680 And that is called, quite appropriately, the electronic 67 00:03:34,680 --> 00:03:39,880 polarizability because it involves polarization of the 68 00:03:39,880 --> 00:03:46,140 electrons and protons on the individual atoms. 69 00:03:46,140 --> 00:03:50,600 And then there's another type of induced dipole moment that 70 00:03:50,600 --> 00:03:53,190 comes when the structure is ionic. 71 00:03:53,190 --> 00:03:57,160 And then an electric field will pull positive ions in one 72 00:03:57,160 --> 00:04:00,180 direction and negative ions in the opposite direction. 73 00:04:00,180 --> 00:04:03,510 And that is very often referred to as the ionic 74 00:04:03,510 --> 00:04:03,920 polarizability. 75 00:04:03,920 --> 00:04:05,870 And it's easy to keep them straight. 76 00:04:05,870 --> 00:04:08,890 One involves electrons, which all atoms have. 77 00:04:08,890 --> 00:04:10,510 The other involves ions. 78 00:04:10,510 --> 00:04:13,720 And not all structures and materials have ions. 79 00:04:13,720 --> 00:04:17,329 So the second one is unique to ionic structures. 80 00:04:17,329 --> 00:04:20,170 And then, these are sometimes also referred to as the 81 00:04:20,170 --> 00:04:21,910 dielectric polarizability. 82 00:04:21,910 --> 00:04:24,890 And I meant to purge that from the notes. 83 00:04:24,890 --> 00:04:27,690 And as I put this together to xerox it, I grabbed the 84 00:04:27,690 --> 00:04:28,770 uncorrected sheet. 85 00:04:28,770 --> 00:04:33,090 So please, just below you see the displaced positive and 86 00:04:33,090 --> 00:04:36,710 negative ion on the middle of the page, cross out 87 00:04:36,710 --> 00:04:38,730 "dielectric" polarizability. 88 00:04:38,730 --> 00:04:43,140 And change that to "ionic" polarizability. 89 00:04:43,140 --> 00:04:45,680 And I didn't catch that. 90 00:04:45,680 --> 00:04:49,070 There's one other little typo as we go partway through. 91 00:04:49,070 --> 00:04:54,010 In any case, we'll talk about third-rank tensor properties. 92 00:04:54,010 --> 00:04:56,430 We'll introduce some other ones other than 93 00:04:56,430 --> 00:04:58,460 piezoelectricity later on. 94 00:04:58,460 --> 00:05:03,050 But piezoelectricity is one of the primary examples of a 95 00:05:03,050 --> 00:05:04,960 third-rank tensor property. 96 00:05:04,960 --> 00:05:09,450 And it's one that has a lot of applications in devices-- 97 00:05:09,450 --> 00:05:16,000 pressure sensors, audio equipment, electronic devices. 98 00:05:16,000 --> 00:05:22,340 It's a very important property in terms of devices and 99 00:05:22,340 --> 00:05:24,710 present-day technology. 100 00:05:24,710 --> 00:05:25,530 But there are others. 101 00:05:25,530 --> 00:05:27,670 And we'll cover those in due course. 102 00:05:27,670 --> 00:05:32,480 Some of them are rather exotic. 103 00:05:32,480 --> 00:05:32,710 OK. 104 00:05:32,710 --> 00:05:37,170 But to review these basic concepts in electromagnetism, 105 00:05:37,170 --> 00:05:39,650 we remind you again of the definition of a dipole. 106 00:05:39,650 --> 00:05:41,350 We mentioned this last time. 107 00:05:41,350 --> 00:05:45,770 But to quickly review, a dipole is a pair of charges of 108 00:05:45,770 --> 00:05:49,510 opposite sign but equal magnitude, separated by a 109 00:05:49,510 --> 00:05:51,580 separation, d. 110 00:05:51,580 --> 00:05:55,750 And then one defines a dipole moment, which has a vector 111 00:05:55,750 --> 00:06:00,700 character, as the product of one of the charges of 112 00:06:00,700 --> 00:06:02,240 magnitude, q. 113 00:06:02,240 --> 00:06:06,940 And the vector that separates the two charges in the sense 114 00:06:06,940 --> 00:06:11,200 of the vector is defined as going from the negative charge 115 00:06:11,200 --> 00:06:14,160 and pointing towards the positive charge. 116 00:06:14,160 --> 00:06:15,490 It's purely a definition. 117 00:06:15,490 --> 00:06:19,360 But the reason it's convenient is that the vector sense and 118 00:06:19,360 --> 00:06:23,490 the product of charge and separation comes up again and 119 00:06:23,490 --> 00:06:25,830 again in all sorts of problems, among them 120 00:06:25,830 --> 00:06:30,550 definition of the piezoelectric effects. 121 00:06:30,550 --> 00:06:35,480 Also a dipole in an electric field is going to experience a 122 00:06:35,480 --> 00:06:38,760 torque because the electric field will pull on the 123 00:06:38,760 --> 00:06:40,790 positive charge in the same direction. 124 00:06:40,790 --> 00:06:43,190 It will pull on the negative charge in the opposite 125 00:06:43,190 --> 00:06:44,230 direction from the field. 126 00:06:44,230 --> 00:06:49,760 And that's going to create a torque on this little gizmo. 127 00:06:49,760 --> 00:06:49,970 OK. 128 00:06:49,970 --> 00:06:51,660 Now it's important, too-- 129 00:06:51,660 --> 00:06:53,830 and interesting to note-- that there are three different 130 00:06:53,830 --> 00:06:56,040 kinds of dipole moments. 131 00:06:56,040 --> 00:06:57,120 There are some molecules-- 132 00:06:57,120 --> 00:07:00,030 and water is the primary example. 133 00:07:00,030 --> 00:07:04,070 Water has seen an asymmetrical arrangement of hydrogen, 134 00:07:04,070 --> 00:07:07,780 relative to the oxygen ion to which they are connected. 135 00:07:07,780 --> 00:07:10,890 And that gives water a permanent dipole moment, which 136 00:07:10,890 --> 00:07:14,290 is what makes water such a darn good solvent. 137 00:07:14,290 --> 00:07:19,990 And its ability to dissolve primordial juices probably 138 00:07:19,990 --> 00:07:23,280 accounts for our fact, intelligent design 139 00:07:23,280 --> 00:07:27,080 notwithstanding, of why we are here today. 140 00:07:27,080 --> 00:07:29,520 On the other hand, dipoles, as I was just saying, can be 141 00:07:29,520 --> 00:07:33,750 induced when you impose an electric field on matter. 142 00:07:33,750 --> 00:07:36,190 And these are of two kinds. 143 00:07:36,190 --> 00:07:40,580 One is the dipole moment that is induced on 144 00:07:40,580 --> 00:07:42,180 an individual atom. 145 00:07:42,180 --> 00:07:46,270 And that results in displacement of the positive 146 00:07:46,270 --> 00:07:50,250 nucleus relative to the negative electron shell. 147 00:07:50,250 --> 00:07:54,450 It's found that the dipole moment is proportional to the 148 00:07:54,450 --> 00:07:56,590 magnitude of the electric field. 149 00:07:56,590 --> 00:07:59,350 And the proportionality constant, alpha, is called the 150 00:07:59,350 --> 00:08:00,930 polarizability. 151 00:08:00,930 --> 00:08:03,900 And for an individual atom, as I said a moment ago, it's 152 00:08:03,900 --> 00:08:07,660 defined as the electronic polarizability. 153 00:08:07,660 --> 00:08:13,620 We write it as a scalar quantity, but actually, by now 154 00:08:13,620 --> 00:08:17,390 you're probably sensitized to being a little bit skeptical 155 00:08:17,390 --> 00:08:19,530 when things that relate to vectors are 156 00:08:19,530 --> 00:08:21,440 described as a scalar. 157 00:08:21,440 --> 00:08:24,710 And in fact, the electronic polarizability is not a 158 00:08:24,710 --> 00:08:25,760 scalar, it's a tensor. 159 00:08:25,760 --> 00:08:30,510 And one should really write that the i-th component of the 160 00:08:30,510 --> 00:08:34,240 dipole moment is given by alpha i,j times the j-th 161 00:08:34,240 --> 00:08:37,500 component of the electric field. 162 00:08:37,500 --> 00:08:42,370 Second type of induced dipole moment involves, again as we 163 00:08:42,370 --> 00:08:46,290 said a moment ago, the effect of imposing an electric field 164 00:08:46,290 --> 00:08:49,420 on an ionic structure. 165 00:08:49,420 --> 00:08:52,880 And again, the field will pull the positive ions in one 166 00:08:52,880 --> 00:08:55,590 direction and negative ions in the other. 167 00:08:55,590 --> 00:08:59,292 And here quite clearly, if this pair of ions is in a 168 00:08:59,292 --> 00:09:03,050 structure, and that structure has some symmetry, we really 169 00:09:03,050 --> 00:09:07,510 have to consider this second origin to induce dipole 170 00:09:07,510 --> 00:09:09,120 moments as a tensor. 171 00:09:09,120 --> 00:09:13,400 And this is referred to as the ionic polarizability. 172 00:09:13,400 --> 00:09:17,750 So both of these types of dipole moments will be present 173 00:09:17,750 --> 00:09:21,240 in matter, in general. 174 00:09:21,240 --> 00:09:24,440 The relative importance of each depends on whether the 175 00:09:24,440 --> 00:09:29,480 electric field is a static field or an oscillatory 176 00:09:29,480 --> 00:09:30,870 electric field. 177 00:09:30,870 --> 00:09:33,680 And then the frequency dependence of these two 178 00:09:33,680 --> 00:09:38,780 polarizabilities has a consequence on the magnitude 179 00:09:38,780 --> 00:09:41,470 for fields of different frequencies. 180 00:09:41,470 --> 00:09:46,130 And not surprisingly, the ability of the ions in the 181 00:09:46,130 --> 00:09:49,910 structure to polarize is going to poop out with 182 00:09:49,910 --> 00:09:53,580 high-frequency electric fields a lot quicker than just the 183 00:09:53,580 --> 00:09:56,280 displacement of the light electrons 184 00:09:56,280 --> 00:09:58,240 about a positive nucleus. 185 00:09:58,240 --> 00:10:01,950 So there is a frequency dependence of the net 186 00:10:01,950 --> 00:10:03,230 polarizability. 187 00:10:03,230 --> 00:10:04,350 We won't go into that. 188 00:10:04,350 --> 00:10:08,720 But just keep in mind that at very high-frequency electric 189 00:10:08,720 --> 00:10:12,210 fields, the ionic polarizability will damp out. 190 00:10:14,970 --> 00:10:15,060 OK. 191 00:10:15,060 --> 00:10:20,810 Then we went through a rather simplified but amusing model 192 00:10:20,810 --> 00:10:23,970 for the electronic polarizability. 193 00:10:23,970 --> 00:10:26,870 And there's some rather severe assumptions that are made. 194 00:10:26,870 --> 00:10:29,730 But making those assumptions let's you get a rigorous 195 00:10:29,730 --> 00:10:34,040 result, which tells you something about the electronic 196 00:10:34,040 --> 00:10:34,920 polarizability. 197 00:10:34,920 --> 00:10:39,590 So we model the electron distribution on the atom as a 198 00:10:39,590 --> 00:10:44,820 uniform charge density in a distribution that goes up to 199 00:10:44,820 --> 00:10:47,280 some radius, r, and then quits. 200 00:10:47,280 --> 00:10:50,710 So there's a sharp cutoff to the distribution of electrons, 201 00:10:50,710 --> 00:10:52,350 which is obviously ridiculous. 202 00:10:52,350 --> 00:10:54,510 You know there are a collection of wave functions 203 00:10:54,510 --> 00:10:58,450 that give you charge probabilities that tail off 204 00:10:58,450 --> 00:11:01,830 slowly to large distances, getting progressively smaller 205 00:11:01,830 --> 00:11:02,680 and smaller. 206 00:11:02,680 --> 00:11:04,950 So this is not terribly realistic. 207 00:11:04,950 --> 00:11:07,820 And then the other thing we assume to make this model, 208 00:11:07,820 --> 00:11:11,580 something that we can solve exactly, is that the nucleus 209 00:11:11,580 --> 00:11:16,160 and the electron distribution displace as units. 210 00:11:16,160 --> 00:11:18,270 In other words, we start with a sphere of electrons, the 211 00:11:18,270 --> 00:11:22,240 center displaces, but it stays a sphere of uniformly 212 00:11:22,240 --> 00:11:24,150 distributed electrons. 213 00:11:24,150 --> 00:11:28,710 And then having made those assumptions and having lost 214 00:11:28,710 --> 00:11:31,910 any credibility for the model, if we carry through to see 215 00:11:31,910 --> 00:11:35,290 what the model predicts, it's rather interesting. 216 00:11:35,290 --> 00:11:40,560 We use a fact, again, known to freshman and sophomore 217 00:11:40,560 --> 00:11:45,200 students of electromagnetism, that a charge inside of a 218 00:11:45,200 --> 00:11:51,400 uniform distribution of charge experiences no force. 219 00:11:51,400 --> 00:11:53,970 And that's surprising, but it's something that you are 220 00:11:53,970 --> 00:11:56,820 very often invited to do on problem sets. 221 00:11:56,820 --> 00:12:00,670 So therefore, if the center of the electron distribution is 222 00:12:00,670 --> 00:12:04,880 displaced from the nucleus, then the restoring force 223 00:12:04,880 --> 00:12:10,350 between the electron sphere and the nucleus is simply the 224 00:12:10,350 --> 00:12:16,180 coulombic force between a nucleus of charge plus ze, and 225 00:12:16,180 --> 00:12:20,050 a fraction of the total number of electrons, namely that 226 00:12:20,050 --> 00:12:23,020 fraction of the electrons which are contained within a 227 00:12:23,020 --> 00:12:26,740 sphere that has a radius equal to the displacement. 228 00:12:26,740 --> 00:12:29,940 And that geometry and that algebra's carried out for you 229 00:12:29,940 --> 00:12:32,080 on the bottom of the first page. 230 00:12:32,080 --> 00:12:37,540 If you set that up and ask what the dipole moment will 231 00:12:37,540 --> 00:12:40,020 be, it comes out beautifully simple. 232 00:12:40,020 --> 00:12:45,230 It comes out to be equal to whatever proportionality 233 00:12:45,230 --> 00:12:48,380 constant you use in Coulomb's law. 234 00:12:48,380 --> 00:12:51,540 I use rationalized MKS units from force of habit. 235 00:12:51,540 --> 00:12:56,160 So there's a 4 pi epsilon 0 in there, then times the cube of 236 00:12:56,160 --> 00:12:59,310 the radius of the electron distribution, times the 237 00:12:59,310 --> 00:13:00,540 electric field. 238 00:13:00,540 --> 00:13:03,900 So the two items of note that come out of this simplified 239 00:13:03,900 --> 00:13:08,820 treatment is first of all, the induced dipole moment is 240 00:13:08,820 --> 00:13:11,790 proportional to the applied electric field, which is what 241 00:13:11,790 --> 00:13:13,290 we assumed. 242 00:13:13,290 --> 00:13:18,340 And so therefore, the polarizability, which is the 243 00:13:18,340 --> 00:13:21,410 quantity that relates the dipole moment to the magnitude 244 00:13:21,410 --> 00:13:24,900 of the field, is a constant. 245 00:13:24,900 --> 00:13:27,530 And it turns out to be equal to 4 pi epsilon 246 00:13:27,530 --> 00:13:30,590 0, times the radius. 247 00:13:30,590 --> 00:13:36,250 So not only does this tell us that the electronic 248 00:13:36,250 --> 00:13:42,525 polarizability is something that relates dipole moment in 249 00:13:42,525 --> 00:13:45,520 direct proportion to the magnitude of the field. 250 00:13:45,520 --> 00:13:50,240 And secondly, the electronic polarizability involves the 251 00:13:50,240 --> 00:13:53,340 radius of the charge distribution, cubed. 252 00:13:53,340 --> 00:13:57,410 And even on a qualitative basis, this is interesting. 253 00:13:57,410 --> 00:14:01,550 It tells you that high-atomic-number, big, fat 254 00:14:01,550 --> 00:14:06,930 ions are going to have a very, very large polarizability. 255 00:14:06,930 --> 00:14:10,570 And things way down in the periodic table, like beryllium 256 00:14:10,570 --> 00:14:18,440 and lithium, and other low-z atoms, are going to be tough 257 00:14:18,440 --> 00:14:22,620 little nuts that don't display much polarization at all. 258 00:14:22,620 --> 00:14:28,360 And in point of fact, several individuals have tabulated 259 00:14:28,360 --> 00:14:31,580 empirical sets of electronic polarizabilities. 260 00:14:31,580 --> 00:14:36,260 One of the earliest ones are the so-called TKS values 261 00:14:36,260 --> 00:14:39,340 published a long time ago by Tessman, Kahn, 262 00:14:39,340 --> 00:14:43,170 and "Wild Bill" Shockley. 263 00:14:43,170 --> 00:14:45,740 And I give you the reference to those. 264 00:14:45,740 --> 00:14:49,950 There is another set of values that were assembled by a 265 00:14:49,950 --> 00:14:53,020 fellow at DuPont named Bob Shannon. 266 00:14:53,020 --> 00:14:55,880 And I'll give you a citation to those values. 267 00:14:55,880 --> 00:15:01,080 But in any case, if you look at these values, you find that 268 00:15:01,080 --> 00:15:06,900 the cation that has highest electronic polarizability is 269 00:15:06,900 --> 00:15:10,680 thallium, way down on the bottom of the periodic table, 270 00:15:10,680 --> 00:15:11,750 next to lead. 271 00:15:11,750 --> 00:15:14,780 And that's just a big, fat, flabby atom that can be 272 00:15:14,780 --> 00:15:18,120 deformed very, very easily. 273 00:15:18,120 --> 00:15:20,490 How do you get these polarizabilities? 274 00:15:20,490 --> 00:15:24,420 Well, the equations at the bottom of page two-- which we 275 00:15:24,420 --> 00:15:26,620 won't make any use of but, nevertheless, will tell you 276 00:15:26,620 --> 00:15:28,010 where they come from-- 277 00:15:28,010 --> 00:15:31,500 there's a relation between the square of the index of 278 00:15:31,500 --> 00:15:37,730 refraction and the sum of the polarizabilities of the 279 00:15:37,730 --> 00:15:41,330 individual species, times the number of those species per 280 00:15:41,330 --> 00:15:42,760 unit volume. 281 00:15:42,760 --> 00:15:45,890 And that is something that's called the Lorentz-Lorenz 282 00:15:45,890 --> 00:15:48,780 equation, equation. 283 00:15:48,780 --> 00:15:50,040 I can't resist saying everything 284 00:15:50,040 --> 00:15:53,870 twice, Lorenz and Lorentz. 285 00:15:53,870 --> 00:15:56,890 You can put this in another form that involves the 286 00:15:56,890 --> 00:15:59,170 molecular weight of a molecular structure and the 287 00:15:59,170 --> 00:16:00,685 polarizability per molecule. 288 00:16:03,340 --> 00:16:08,450 It's the same equation, but for organic compounds. 289 00:16:08,450 --> 00:16:10,190 It's a useful form. 290 00:16:10,190 --> 00:16:15,100 And it turns out that the dielectric constant of the 291 00:16:15,100 --> 00:16:19,540 material is directly related to the square of the index of 292 00:16:19,540 --> 00:16:20,740 refraction. 293 00:16:20,740 --> 00:16:25,540 So you can write those two equations in terms of either 294 00:16:25,540 --> 00:16:29,590 the dielectric constant or the index of refraction. 295 00:16:29,590 --> 00:16:32,780 And if you substitute dielectric constant in place 296 00:16:32,780 --> 00:16:35,070 of n squared in those equations, the 297 00:16:35,070 --> 00:16:36,390 equations get new names. 298 00:16:36,390 --> 00:16:37,060 And they're not called the 299 00:16:37,060 --> 00:16:38,610 Lorentz-Lorenz equation, equations. 300 00:16:38,610 --> 00:16:42,290 They're called the Clausius-Mossotti equations, 301 00:16:42,290 --> 00:16:48,880 which shows you that sometimes fame and immortality can be 302 00:16:48,880 --> 00:16:53,850 gained simply by a trivial substitution of variables. 303 00:16:53,850 --> 00:16:57,590 Nice to keep in mind if you can find something like that. 304 00:16:57,590 --> 00:16:59,990 Finally, we don't see dipole moments on 305 00:16:59,990 --> 00:17:03,050 individual atoms or molecules. 306 00:17:03,050 --> 00:17:06,630 We see evidence of polarization in bulk. 307 00:17:06,630 --> 00:17:12,906 And on page three is a little model that reminds you of what 308 00:17:12,906 --> 00:17:17,390 the polarization of a material is. 309 00:17:17,390 --> 00:17:20,119 And that's defined as simply the dipole 310 00:17:20,119 --> 00:17:22,109 moment per unit volume. 311 00:17:22,109 --> 00:17:25,050 And that's represented by a capital P rather 312 00:17:25,050 --> 00:17:27,099 than a small p. 313 00:17:27,099 --> 00:17:30,800 And this is also a vector quantity. 314 00:17:30,800 --> 00:17:33,660 And we can see how this is related to the individual 315 00:17:33,660 --> 00:17:39,850 dipole moments in the solid by dividing the solid up into 316 00:17:39,850 --> 00:17:41,460 individual cells. 317 00:17:41,460 --> 00:17:44,130 And I use the term of "cell" loosely. 318 00:17:44,130 --> 00:17:45,320 Is this a unit cell? 319 00:17:45,320 --> 00:17:47,720 Is this a box around each of the atoms? 320 00:17:47,720 --> 00:17:49,370 It really doesn't matter. 321 00:17:49,370 --> 00:17:52,700 Because all of these little dipoles on the individual unit 322 00:17:52,700 --> 00:17:58,860 cells are packed together back to front in the solid. 323 00:17:58,860 --> 00:18:03,750 So internally, the negative end of one induced dipole 324 00:18:03,750 --> 00:18:06,990 moment is always adjacent to the positive end of the 325 00:18:06,990 --> 00:18:08,470 neighboring dipole moment. 326 00:18:08,470 --> 00:18:12,060 And everything cancels out internally, except for the two 327 00:18:12,060 --> 00:18:13,500 surfaces of the solid. 328 00:18:13,500 --> 00:18:17,230 So macroscopically, if you impose an electric field and 329 00:18:17,230 --> 00:18:20,900 it induces dipole moments, what happens is you see that 330 00:18:20,900 --> 00:18:23,060 there is a charge on the surface of 331 00:18:23,060 --> 00:18:25,020 the piece of material. 332 00:18:25,020 --> 00:18:28,230 And that is a real charge. 333 00:18:28,230 --> 00:18:29,570 It's a bound charge. 334 00:18:29,570 --> 00:18:32,410 You can't draw that charge off as a current. 335 00:18:32,410 --> 00:18:36,620 Because the minute you reduce and eliminate the electric 336 00:18:36,620 --> 00:18:38,390 field, those dipoles disappear. 337 00:18:38,390 --> 00:18:41,710 And the charges all go back to where they came from. 338 00:18:41,710 --> 00:18:44,760 So this is the thing that keeps the exact model that we 339 00:18:44,760 --> 00:18:48,270 use for one of the cells in the solid [INAUDIBLE] of no 340 00:18:48,270 --> 00:18:49,950 importance whatsoever. 341 00:18:49,950 --> 00:18:53,050 You could say that the dipole moment on each atom is the 342 00:18:53,050 --> 00:18:56,570 charge times the separation of the positive 343 00:18:56,570 --> 00:18:58,160 and negative charge. 344 00:18:58,160 --> 00:19:02,760 And qi is the induced charge. 345 00:19:02,760 --> 00:19:05,480 The number of cells per unit volume, if we say that they're 346 00:19:05,480 --> 00:19:10,620 little cubes of the same edge length, delta, is unit volume 347 00:19:10,620 --> 00:19:13,050 one divided by the volume per cell, which 348 00:19:13,050 --> 00:19:14,910 will be delta cubed. 349 00:19:14,910 --> 00:19:19,670 And the dipole moment per unit volume is the number of cells 350 00:19:19,670 --> 00:19:22,300 per unit volume times the polarization on each. 351 00:19:22,300 --> 00:19:24,410 And you find that everything drops out. 352 00:19:24,410 --> 00:19:29,840 And the polarization indeed turns out to be equal to 353 00:19:29,840 --> 00:19:33,690 induced charge per unit area, whether that's induced charge 354 00:19:33,690 --> 00:19:36,570 per unit area of one of our cells or on a square 355 00:19:36,570 --> 00:19:40,930 centimeter of material. 356 00:19:40,930 --> 00:19:45,560 So polarization, dipole moment per unit volume is numerically 357 00:19:45,560 --> 00:19:49,470 equal to, and physically equivalent to, an induced 358 00:19:49,470 --> 00:19:52,480 charge per unit area. 359 00:19:55,170 --> 00:19:55,480 OK. 360 00:19:55,480 --> 00:19:59,960 So much for freshman 361 00:19:59,960 --> 00:20:02,810 electromagnetism in fast forward. 362 00:20:02,810 --> 00:20:05,100 And now I'd like to turn to something that involves 363 00:20:05,100 --> 00:20:08,530 tensors and materials. 364 00:20:08,530 --> 00:20:11,380 As I mentioned a moment ago, piezoelectricity, literally 365 00:20:11,380 --> 00:20:16,990 "pressure electricity," is a very technologically important 366 00:20:16,990 --> 00:20:22,320 property, useful property, but also provides a nice example 367 00:20:22,320 --> 00:20:27,400 of a tensor property that has to be defined in terms of a 368 00:20:27,400 --> 00:20:29,880 sensor of third rank. 369 00:20:29,880 --> 00:20:31,310 There are a number of different 370 00:20:31,310 --> 00:20:33,410 piezoelectric effects. 371 00:20:33,410 --> 00:20:37,165 The first one is the so-called direct piezoelectric effect. 372 00:20:47,160 --> 00:20:53,020 And it refers to the fact that if you subject a material to 373 00:20:53,020 --> 00:20:57,160 an applied stress, it will develop a surface charge. 374 00:20:57,160 --> 00:21:01,460 Remove the applied stress, the surface charge goes away. 375 00:21:01,460 --> 00:21:04,600 The surface charge can be described in terms of a 376 00:21:04,600 --> 00:21:11,390 polarization, a dipole moment per unit volume. 377 00:21:11,390 --> 00:21:15,800 And it will be in direct proportion to a stress 378 00:21:15,800 --> 00:21:17,400 tensor sigma j,k. 379 00:21:20,820 --> 00:21:22,700 And so what we'll assume-- 380 00:21:22,700 --> 00:21:24,210 and this is an assumption. 381 00:21:24,210 --> 00:21:30,210 And it's followed, except for very extreme conditions, that 382 00:21:30,210 --> 00:21:36,070 each component of the polarization, p sub i, is 383 00:21:36,070 --> 00:21:40,960 given by a linear combination of every one of the nine 384 00:21:40,960 --> 00:21:42,900 elements of stress, sigma i,j. 385 00:21:46,090 --> 00:21:47,000 Oops, not i,j. 386 00:21:47,000 --> 00:21:48,080 I have to use a different index. 387 00:21:48,080 --> 00:21:50,880 I'm used to writing i,j. 388 00:21:50,880 --> 00:21:55,250 So each component of the polarization, where i ranges 389 00:21:55,250 --> 00:21:59,660 from 1 to 3, is given by a linear combination of all nine 390 00:21:59,660 --> 00:22:02,950 of the elements of stress, sigma j,k. 391 00:22:02,950 --> 00:22:08,520 And the proportionality constants, di,j,k, are known 392 00:22:08,520 --> 00:22:11,630 as the piezoelectric moduli. 393 00:22:11,630 --> 00:22:14,230 And this is called the direct piezoelectric effect. 394 00:22:19,160 --> 00:22:19,440 OK. 395 00:22:19,440 --> 00:22:24,290 That's simply an assumption that the polarization is 396 00:22:24,290 --> 00:22:26,620 proportional to the applied stress. 397 00:22:26,620 --> 00:22:30,920 We know that stress, however, is a field 398 00:22:30,920 --> 00:22:32,870 tensor of second rank. 399 00:22:32,870 --> 00:22:38,550 We know that the polarization has the character of a vector, 400 00:22:38,550 --> 00:22:40,460 charge times the length. 401 00:22:40,460 --> 00:22:43,980 We know how second-rank tensors transform. 402 00:22:43,980 --> 00:22:49,230 We know how first-rank tensors transform. 403 00:22:49,230 --> 00:22:52,540 And therefore, knowing that, we can say that the 404 00:22:52,540 --> 00:22:56,980 coefficients, the array of coefficients, di,j,k, will 405 00:22:56,980 --> 00:22:59,090 transform like a third-rank tensor. 406 00:22:59,090 --> 00:23:01,530 And therefore they are a tensor. 407 00:23:01,530 --> 00:23:04,840 So we will have three components of polarization. 408 00:23:04,840 --> 00:23:09,380 And we will have nine components of stress. 409 00:23:09,380 --> 00:23:14,776 And so there will be 27 piezoelectric moduli, di,j,k. 410 00:23:20,300 --> 00:23:24,940 So things go up rapidly in terms of number of 411 00:23:24,940 --> 00:23:30,560 coefficients as the rank of a tensor increases. 412 00:23:30,560 --> 00:23:37,320 Now let's take a look at the nature of these equations. 413 00:23:37,320 --> 00:23:43,550 We'll have, for example, the x1 component of P being given 414 00:23:43,550 --> 00:23:52,030 by d1,1,1 times the element of stress sigma 1,1. 415 00:23:52,030 --> 00:24:00,450 The next term will be d1,2,2 times sigma 2,2. 416 00:24:00,450 --> 00:24:02,640 I'm putting down the tensor components first. 417 00:24:02,640 --> 00:24:04,060 But this is just an equation. 418 00:24:04,060 --> 00:24:06,580 I can write the terms in any order. 419 00:24:06,580 --> 00:24:13,370 Next will be a d1,3,3 times sigma 3,3. 420 00:24:13,370 --> 00:24:26,130 And the next one would be a d1,2,3 times a shear component 421 00:24:26,130 --> 00:24:31,436 of stress sigma 2,3 and other terms. 422 00:24:31,436 --> 00:24:33,540 I don't want to write out all nine of them. 423 00:24:33,540 --> 00:24:37,870 Because I think I have written enough to make my point. 424 00:24:37,870 --> 00:24:42,850 The point is that the value of i is always tied to the 425 00:24:42,850 --> 00:24:48,680 component of the resulting vector that we're defining. 426 00:24:48,680 --> 00:24:52,940 But the second pair of subscripts, j and k, always go 427 00:24:52,940 --> 00:25:01,320 together as an unseparable pair with the component of 428 00:25:01,320 --> 00:25:04,710 stress that they modify. 429 00:25:04,710 --> 00:25:07,070 So the 1,1 here goes with the 1,1. 430 00:25:07,070 --> 00:25:08,720 The 2,2 goes with the 2,2. 431 00:25:08,720 --> 00:25:11,720 The 3,3 goes with the 3,3. 432 00:25:11,720 --> 00:25:13,680 So why in the world, if they're always going to go 433 00:25:13,680 --> 00:25:19,930 together, do we have to use two indices to define the 434 00:25:19,930 --> 00:25:22,250 piezoelectric moduli? 435 00:25:22,250 --> 00:25:24,210 Why don't we use a single symbol? 436 00:25:24,210 --> 00:25:25,960 Well, there is a good reason for not doing it. 437 00:25:25,960 --> 00:25:29,260 But we'll issue that caveat later on. 438 00:25:29,260 --> 00:25:34,610 We could use an a, a b, and a c, or an alpha or a beta or a 439 00:25:34,610 --> 00:25:37,910 gamma, or some other esoteric symbol. 440 00:25:37,910 --> 00:25:42,620 But what makes sense since we're using Arabic numerals to 441 00:25:42,620 --> 00:25:49,300 represent subscripts, let's write the pair of indices in 442 00:25:49,300 --> 00:25:55,410 terms of an index that's related to the stress tensor, 443 00:25:55,410 --> 00:26:00,280 which is sigma 1,1; sigma 2,2; sigma-- 444 00:26:00,280 --> 00:26:00,500 whoops. 445 00:26:00,500 --> 00:26:02,940 Sigma 1,2; sigma 1,3. 446 00:26:02,940 --> 00:26:08,420 And then comes sigma 2,1; sigma 2,2; sigma 2,3; sigma 447 00:26:08,420 --> 00:26:12,280 3,1; sigma 3,2; sigma 3,3. 448 00:26:12,280 --> 00:26:19,410 We know that this is a symmetric tensor. 449 00:26:19,410 --> 00:26:22,420 So these off-diagonal terms are always 450 00:26:22,420 --> 00:26:23,710 equal to one another. 451 00:26:23,710 --> 00:26:25,890 So we're numerically going to have to enter 452 00:26:25,890 --> 00:26:29,050 each of those twice. 453 00:26:29,050 --> 00:26:35,310 So to define a single index that represents the 454 00:26:35,310 --> 00:26:37,130 components of stress. 455 00:26:37,130 --> 00:26:40,820 And I'll describe it in this fashion so you can remember, 456 00:26:40,820 --> 00:26:43,290 as a mnemonic device, what makes this work. 457 00:26:43,290 --> 00:26:47,740 We'll go down the main diagonal of the stress tensor, 458 00:26:47,740 --> 00:26:48,960 this fashion. 459 00:26:48,960 --> 00:26:52,440 And then having reached the bottom, we'll go up along the 460 00:26:52,440 --> 00:26:55,940 right-hand side and then jog over to the left to pick up 461 00:26:55,940 --> 00:26:58,010 the last term. 462 00:26:58,010 --> 00:27:01,310 And we'll define a number associated with these indices 463 00:27:01,310 --> 00:27:10,060 that goes in the form 1 to 2, to 3, to 4, to 5, to 6. 464 00:27:13,260 --> 00:27:19,150 So we'll just use these three integers, ranging from 1 to 6, 465 00:27:19,150 --> 00:27:24,480 to represent pairs of integers in the stress tensor. 466 00:27:24,480 --> 00:27:26,970 So things tidy up quite nicely then. 467 00:27:26,970 --> 00:27:38,120 This would be P1 is d1,1 times sigma 1; plus d1,2 times sigma 468 00:27:38,120 --> 00:27:48,540 2; plus d1,3 times sigma 3; plus d1,4 times sigma 4. 469 00:27:52,150 --> 00:27:54,070 And now, uh-oh, Houston. 470 00:27:54,070 --> 00:27:56,040 We've got a problem. 471 00:27:56,040 --> 00:27:58,530 Because there's another term in here that is 472 00:27:58,530 --> 00:28:03,750 d1,3,2 times sigma 3,2. 473 00:28:03,750 --> 00:28:09,280 And sigma 3,2 is required, because the stress tensor is 474 00:28:09,280 --> 00:28:15,400 symmetric, to be numerically equal to sigma 2,3. 475 00:28:15,400 --> 00:28:23,510 So this is really d1,2,3 times sigma 2,3. 476 00:28:23,510 --> 00:28:28,820 And then we've got a d1,3,2 times a sigma 3,2. 477 00:28:28,820 --> 00:28:32,260 I know that this is equal to this. 478 00:28:32,260 --> 00:28:41,780 But is there any reason d1,2,3 has to be equal to d1,3,2? 479 00:28:41,780 --> 00:28:45,420 I can't see any reason why. 480 00:28:45,420 --> 00:28:45,610 OK. 481 00:28:45,610 --> 00:28:49,230 Let me confide in you that, yes, they are equal. 482 00:28:49,230 --> 00:28:53,150 But we can't show it or claim it on the basis of what we've 483 00:28:53,150 --> 00:28:55,000 got before us right now. 484 00:28:55,000 --> 00:28:56,000 It's going to come later. 485 00:28:56,000 --> 00:28:58,200 But we can show that they are equal. 486 00:28:58,200 --> 00:28:58,420 OK. 487 00:28:58,420 --> 00:29:04,210 But equal or not, this means we'll have a term d1,4 sigma 488 00:29:04,210 --> 00:29:07,900 4, and we're going to get it in there twice. 489 00:29:07,900 --> 00:29:12,580 So when the subscripts go up to 4, we're going to get a 2 490 00:29:12,580 --> 00:29:13,280 out in front. 491 00:29:13,280 --> 00:29:21,770 And then we'll get for the term sigma 1,3 and 3,1 we'll 492 00:29:21,770 --> 00:29:25,850 have a d1,5 times a sigma 5. 493 00:29:25,850 --> 00:29:29,160 This would be sigma 1,3. 494 00:29:29,160 --> 00:29:34,320 And then we'd have another d1,5 for sigma 5 again. 495 00:29:34,320 --> 00:29:37,870 And this would be sigma 3,1. 496 00:29:37,870 --> 00:29:38,760 So what are we going to do? 497 00:29:38,760 --> 00:29:41,280 We're going to say, well, it's nice we're getting rid of an 498 00:29:41,280 --> 00:29:43,350 unneeded subscript. 499 00:29:43,350 --> 00:29:49,100 P1 is the d1,1 times sigma 1; plus d1,2 times sigma 2; plus 500 00:29:49,100 --> 00:29:52,260 d1,3 times sigma 3. 501 00:29:52,260 --> 00:29:58,580 Are we then going to say 2 d1,4 times sigma four, and say 502 00:29:58,580 --> 00:30:03,900 that the coefficient here is di,j when i is 503 00:30:03,900 --> 00:30:05,180 equal to 1, 2, 3. 504 00:30:05,180 --> 00:30:12,030 But the coefficient is 2 di,j when j is 4, 5, or 6? 505 00:30:12,030 --> 00:30:14,060 Hell of a matrix that would be. 506 00:30:14,060 --> 00:30:16,900 That's going to be a bother. 507 00:30:16,900 --> 00:30:20,140 Since it doesn't seem that we can measure these two 508 00:30:20,140 --> 00:30:28,070 coefficients, d1,2,3 and d1,3,2, independently anyhow, 509 00:30:28,070 --> 00:30:32,390 let's just write our relation in reduced subscripts with the 510 00:30:32,390 --> 00:30:38,370 two added together as the element d1,4. 511 00:30:38,370 --> 00:30:50,280 So we are going to define d1,4 equals d1,3,2 plus d1,2,3. 512 00:30:50,280 --> 00:30:51,780 So that's a definition. 513 00:30:51,780 --> 00:31:01,380 And d1,5 will be defined as the sum of d1,1,3 plus d1,3,1. 514 00:31:01,380 --> 00:31:04,480 And we're going to define d1,6, finally, 515 00:31:04,480 --> 00:31:11,820 as d1,1,2 plus d1,2,1. 516 00:31:11,820 --> 00:31:12,200 OK. 517 00:31:12,200 --> 00:31:17,930 So if we do that, then and only then are we entitled to 518 00:31:17,930 --> 00:31:24,390 write a nice, simple, compact little nugget that says that 519 00:31:24,390 --> 00:31:32,940 in our redefined form, Pi is equal to di,j 520 00:31:32,940 --> 00:31:34,350 times sigma of j. 521 00:31:36,960 --> 00:31:45,095 And i goes from 1 to 3, and j goes from 1 to 6. 522 00:31:48,580 --> 00:31:50,890 And this is kind of a neat compact, 523 00:31:50,890 --> 00:31:53,830 easily managed outcome. 524 00:31:53,830 --> 00:31:57,510 So the moral of this story is, I like to think, is you can 525 00:31:57,510 --> 00:31:59,890 have your cake if you eat its 2. 526 00:32:03,570 --> 00:32:06,170 Oh come on, this is a tough, tough crowd. 527 00:32:12,570 --> 00:32:14,950 You're just all worn out from the MRS meeting. 528 00:32:14,950 --> 00:32:16,240 That'll do it to anybody. 529 00:32:18,990 --> 00:32:22,970 So why do we worry about the fact that to the direct 530 00:32:22,970 --> 00:32:28,150 piezoelectric effect should be a third-rank tensor? 531 00:32:28,150 --> 00:32:33,130 The answer, my friends, is that this is no longer a 532 00:32:33,130 --> 00:32:35,450 tensor relationship. 533 00:32:35,450 --> 00:32:38,170 It's a matrix relationship. 534 00:32:38,170 --> 00:32:41,800 A tensor is a matrix, but it's a matrix with a difference. 535 00:32:41,800 --> 00:32:46,830 It's a matrix for which a law of transformation is defined, 536 00:32:46,830 --> 00:32:50,660 if you change axes. 537 00:32:50,660 --> 00:32:53,200 There's no such law defined for the di,j's. 538 00:32:53,200 --> 00:32:57,910 So they qualify as a matrix, but they are not a tensor. 539 00:32:57,910 --> 00:33:02,950 So to attempt to say-- as you might be inclined to do when 540 00:33:02,950 --> 00:33:06,490 we raise the issue of what the symmetry restrictions are on 541 00:33:06,490 --> 00:33:07,580 these tensors-- 542 00:33:07,580 --> 00:33:11,980 if you attempt to say, well, I'm going to find d2,1 prime. 543 00:33:11,980 --> 00:33:16,785 And that's going to be c2,i c1,j times di,j. 544 00:33:19,820 --> 00:33:20,650 Wrong. 545 00:33:20,650 --> 00:33:24,850 You go down in flames because that's just nonsense. 546 00:33:24,850 --> 00:33:26,940 That's just nonsense. 547 00:33:26,940 --> 00:33:29,040 So this is a matrix relation. 548 00:33:29,040 --> 00:33:34,320 And if you want to do exercises such as slice a 549 00:33:34,320 --> 00:33:37,930 piezoelectric wafer out of a piece of quartz, and then 550 00:33:37,930 --> 00:33:43,930 having done so, refer the properties to axes taken along 551 00:33:43,930 --> 00:33:48,770 those edges of the plate that you've cut, you've got to-- in 552 00:33:48,770 --> 00:33:52,540 order to find the piezoelectric moduli for that 553 00:33:52,540 --> 00:33:54,500 plate in that new coordinate system-- 554 00:33:54,500 --> 00:33:57,700 you've got to be prepared always to go back to the full 555 00:33:57,700 --> 00:33:59,500 tensor notation. 556 00:33:59,500 --> 00:34:04,370 If you want to derive symmetry restrictions, which we're 557 00:34:04,370 --> 00:34:06,790 going to do whether you want to or not-- 558 00:34:06,790 --> 00:34:08,389 but we won't do them exhaustively-- 559 00:34:08,389 --> 00:34:13,250 you've got to go from this matrix notation back to the 560 00:34:13,250 --> 00:34:16,789 full three-subscript tensor notation. 561 00:34:16,789 --> 00:34:17,650 OK? 562 00:34:17,650 --> 00:34:17,940 Yes, sir? 563 00:34:17,940 --> 00:34:20,814 AUDIENCE: Couldn't you extend c sub i,k's to be [? 3,6's ?] 564 00:34:20,814 --> 00:34:23,690 and extend most others [INAUDIBLE]? 565 00:34:23,690 --> 00:34:24,540 PROFESSOR: Oh, yeah. 566 00:34:24,540 --> 00:34:27,310 No, if we would write all these down, we'd have-- 567 00:34:27,310 --> 00:34:29,330 I don't know if that's what you're asking-- but we'd have 568 00:34:29,330 --> 00:34:35,739 P1 is equal to d1,j times sigma sub j; P2 is equal to 569 00:34:35,739 --> 00:34:41,320 d2,j sigma sub j; and P3 is equal to d3,j 570 00:34:41,320 --> 00:34:42,620 times sigma sub j. 571 00:34:42,620 --> 00:34:46,980 So I just did that for the line with i equal to 1, 572 00:34:46,980 --> 00:34:48,350 because I was too lazy to write 573 00:34:48,350 --> 00:34:51,280 down all three relations. 574 00:34:51,280 --> 00:34:52,250 Is that what you're asking? 575 00:34:52,250 --> 00:34:54,640 AUDIENCE: No, I'm saying you could have a transformation 576 00:34:54,640 --> 00:35:00,870 model if you were to extend a c sub i,j matrix to be 3,6. 577 00:35:00,870 --> 00:35:01,340 PROFESSOR: No. 578 00:35:01,340 --> 00:35:04,250 It just won't do it. 579 00:35:04,250 --> 00:35:07,120 I mean, think of what these indices are. 580 00:35:07,120 --> 00:35:11,570 They're 1, 2, and 3 standing for the x1 direction, the x2 581 00:35:11,570 --> 00:35:14,140 direction, the x3 direction. 582 00:35:14,140 --> 00:35:16,510 What's the x5 direction? 583 00:35:16,510 --> 00:35:19,860 It's just not defined. 584 00:35:19,860 --> 00:35:21,808 Good try, but you just can't do it. 585 00:35:21,808 --> 00:35:24,248 AUDIENCE: I'll think of something next time. 586 00:35:24,248 --> 00:35:25,750 PROFESSOR: Oh, I'm sure you will. 587 00:35:25,750 --> 00:35:27,410 Whether it will work or not is an 588 00:35:27,410 --> 00:35:29,370 absolutely different question. 589 00:35:32,850 --> 00:35:34,100 OK. 590 00:35:36,710 --> 00:35:37,840 So beware. 591 00:35:37,840 --> 00:35:43,910 Do not try to transform tensors of third rank 592 00:35:43,910 --> 00:35:49,428 expressed with two subscripts to other coordinate systems. 593 00:35:49,428 --> 00:35:49,850 All right. 594 00:35:49,850 --> 00:35:54,440 What I would like to do now, then, is to examine the 595 00:35:54,440 --> 00:35:58,290 symmetry restrictions that are imposed on a third-rank tensor 596 00:35:58,290 --> 00:36:00,410 by crystallographic symmetry. 597 00:36:00,410 --> 00:36:02,690 And these are embedded in the notes that you have. 598 00:36:02,690 --> 00:36:04,690 But it's going to be bothersome to 599 00:36:04,690 --> 00:36:05,380 leaf through those. 600 00:36:05,380 --> 00:36:08,920 So I separated out the pages separately. 601 00:36:08,920 --> 00:36:12,870 And I'm going to look at a few transformations so that you 602 00:36:12,870 --> 00:36:15,940 can see how they work, and then point out some very, very 603 00:36:15,940 --> 00:36:22,490 curious consequences of this, which are non-intuitive. 604 00:36:22,490 --> 00:36:25,680 So first of all, how would we do it? 605 00:36:25,680 --> 00:36:33,260 If we want to get the form of di,j, the matrix for-- 606 00:36:33,260 --> 00:36:37,370 let's do a very, very simple one. 607 00:36:37,370 --> 00:36:40,250 Well, let's do one that we did last time, for those who were 608 00:36:40,250 --> 00:36:42,620 here, for inversion. 609 00:36:42,620 --> 00:36:45,400 For inversion, the direction cosine scheme 610 00:36:45,400 --> 00:36:49,220 is c1,1, 0; 0, 0-- 611 00:36:49,220 --> 00:36:50,810 I'm sorry. 612 00:36:50,810 --> 00:36:55,158 For 1 bar, the transformation of axes is c1,1, 0, 0; 0, 613 00:36:55,158 --> 00:36:59,200 c2,2, 0; 0, 0, c3,3. 614 00:36:59,200 --> 00:37:04,910 So we'll take di,j, change it to the full three-subscript 615 00:37:04,910 --> 00:37:08,310 notation, di,j,k. 616 00:37:08,310 --> 00:37:12,360 And the law for transformation of the third-rank tensor is 617 00:37:12,360 --> 00:37:21,830 di,j,k prime is equal to ci, capital I; cj, capital J; ck, 618 00:37:21,830 --> 00:37:30,310 capital K; times d, capital I, capital J, capital K. We've 619 00:37:30,310 --> 00:37:30,970 not done this much. 620 00:37:30,970 --> 00:37:35,200 But we've mentioned quite some time ago that this is the way 621 00:37:35,200 --> 00:37:39,630 a third-rank tensor would transform element by element. 622 00:37:39,630 --> 00:37:45,610 The only terms that are non-zero in this array, when 623 00:37:45,610 --> 00:37:48,970 the symmetry transformation is inversion-- 624 00:37:48,970 --> 00:38:00,420 where x1, x2, x3 goes to minus x1, minus x2, minus x3-- 625 00:38:00,420 --> 00:38:05,530 is whatever i is, minus 1; whatever j is-- 626 00:38:05,530 --> 00:38:07,235 only the diagonal terms are there, and 627 00:38:07,235 --> 00:38:08,650 they're all minus 1-- 628 00:38:08,650 --> 00:38:14,000 whatever k is, it's going to be minus 1, times di,j,k. 629 00:38:14,000 --> 00:38:18,240 So we find that for inversion, di,j,k prime is always, 630 00:38:18,240 --> 00:38:21,200 regardless of what the three indices are, is going to be 631 00:38:21,200 --> 00:38:22,450 minus di,j,k. 632 00:38:25,300 --> 00:38:28,500 And if inversion is a symmetry operation which the crystal 633 00:38:28,500 --> 00:38:32,690 possesses, we're demanding that the transform index be 634 00:38:32,690 --> 00:38:34,560 identical to the original index. 635 00:38:34,560 --> 00:38:36,320 But there's a minus sign in there. 636 00:38:36,320 --> 00:38:39,010 So we can say that every single element 637 00:38:39,010 --> 00:38:41,510 vanishes, has to vanish. 638 00:38:41,510 --> 00:38:46,400 So any crystal that has inversion in it is not going 639 00:38:46,400 --> 00:38:51,410 to be able to display the piezoelectric effect or any 640 00:38:51,410 --> 00:38:55,466 other third-rank tensor property. 641 00:38:55,466 --> 00:38:59,446 The electro-optic effect, piezoresistance, and there are 642 00:38:59,446 --> 00:39:02,560 a whole slew of them, which are examined for you in part 643 00:39:02,560 --> 00:39:04,480 on this sheet. 644 00:39:04,480 --> 00:39:07,990 One third-rank tensor property is the direct piezoelectric 645 00:39:07,990 --> 00:39:13,050 effect, which we've been discussing as our example. 646 00:39:13,050 --> 00:39:20,680 There is something called the converse piezoelectric effect, 647 00:39:20,680 --> 00:39:25,550 which describes the phenomenon where an applied electric 648 00:39:25,550 --> 00:39:29,440 field creates a strain. 649 00:39:29,440 --> 00:39:32,370 So the direct effect is you [? scush ?] your material, you 650 00:39:32,370 --> 00:39:34,020 develop charge. 651 00:39:34,020 --> 00:39:37,430 The converse piezoelectric effect says if you apply an 652 00:39:37,430 --> 00:39:41,680 electric field that's going to move the atoms around and 653 00:39:41,680 --> 00:39:46,480 induce polarizations, you are going to create a strain. 654 00:39:46,480 --> 00:39:54,000 And the absolutely mind-boggling thing is that 655 00:39:54,000 --> 00:40:00,600 the same array of 3 by 9 coefficients describe both the 656 00:40:00,600 --> 00:40:04,120 direct piezoelectric effect and the converse 657 00:40:04,120 --> 00:40:05,570 piezoelectric effect. 658 00:40:05,570 --> 00:40:08,470 So in one relation, we have that each component of 659 00:40:08,470 --> 00:40:15,110 polarization is di,j,k times the nine elements of stress. 660 00:40:18,490 --> 00:40:23,390 And that means we have to write three equations in nine 661 00:40:23,390 --> 00:40:27,452 variables, the elements of stress. 662 00:40:27,452 --> 00:40:30,170 And the converse piezoelectric effect, we're 663 00:40:30,170 --> 00:40:32,740 developing a strain. 664 00:40:32,740 --> 00:40:38,010 And there are, therefore, nine elements of strain. 665 00:40:38,010 --> 00:40:45,240 And we're applying a vector, e sub i, a field. 666 00:40:45,240 --> 00:40:48,080 So there are three components of that vector. 667 00:40:48,080 --> 00:40:51,651 27 elements, 3 times 9. 668 00:40:51,651 --> 00:40:55,380 27 elements, 9 times 3. 669 00:40:55,380 --> 00:40:59,960 So both of these are third-rank tensors. 670 00:40:59,960 --> 00:41:04,430 Strain transforms like a second-rank tensor. 671 00:41:04,430 --> 00:41:07,080 Field vectors transform like a first-rank tensor. 672 00:41:07,080 --> 00:41:10,760 Therefore, the coefficients are elements of 673 00:41:10,760 --> 00:41:12,270 a third-rank tensor. 674 00:41:12,270 --> 00:41:18,870 But what boggles the mind is that the same 27 numbers 675 00:41:18,870 --> 00:41:25,040 describe these two seemingly disparate phenomena. 676 00:41:25,040 --> 00:41:26,860 You don't prove this by symmetry. 677 00:41:26,860 --> 00:41:30,500 You don't prove this by the nature of stress and strain. 678 00:41:30,500 --> 00:41:33,810 This hinges on the thermodynamic argument, which 679 00:41:33,810 --> 00:41:35,440 I'm not going to go into. 680 00:41:35,440 --> 00:41:38,780 But what you do if you're willing to stretch tensor 681 00:41:38,780 --> 00:41:41,920 notation a little bit-- 682 00:41:41,920 --> 00:41:47,430 you can't have a 3 by 9 array when you've got a tensor of 683 00:41:47,430 --> 00:41:50,100 second rank on the left and a vector on the right. 684 00:41:50,100 --> 00:41:58,080 You have to write this in this fashion, that di,j,k times ei 685 00:41:58,080 --> 00:42:02,880 gives you the element of strain, epsilon j,k. 686 00:42:02,880 --> 00:42:05,910 And that's not proper tensor notation. 687 00:42:05,910 --> 00:42:08,420 But we're not going to quibble. 688 00:42:08,420 --> 00:42:13,080 Because if you allow this little departure from 689 00:42:13,080 --> 00:42:16,350 convention, then you can write both the converse 690 00:42:16,350 --> 00:42:20,050 piezoelectric effect and the direct piezoelectric effect in 691 00:42:20,050 --> 00:42:21,305 terms of the same matrix. 692 00:42:24,060 --> 00:42:26,320 But again, that is not intuitively obvious. 693 00:42:26,320 --> 00:42:28,350 It does not have to be the case. 694 00:42:28,350 --> 00:42:30,686 And it hinges on an argument in thermodynamics. 695 00:42:37,640 --> 00:42:40,790 Some other examples of piezoelectric relations that 696 00:42:40,790 --> 00:42:42,670 I've mentioned in the notes. 697 00:42:42,670 --> 00:42:47,700 If you apply a stress and you get a polarization, that 698 00:42:47,700 --> 00:42:51,140 stress produces a strain. 699 00:42:51,140 --> 00:42:54,260 So you also have to get a polarization if you're 700 00:42:54,260 --> 00:42:57,240 applying a stress, and write it in terms of the strain 701 00:42:57,240 --> 00:42:59,030 that's produced. 702 00:42:59,030 --> 00:43:03,490 Similarly, if you apply an electric field and it produces 703 00:43:03,490 --> 00:43:07,060 a strain, the crystal must be in a state of stress. 704 00:43:07,060 --> 00:43:13,650 So there must be relation between stress and applied 705 00:43:13,650 --> 00:43:16,570 electric field, which will also be a third-rank tensor. 706 00:43:16,570 --> 00:43:19,810 Those two relations are represented by coefficients 707 00:43:19,810 --> 00:43:21,250 given the symbol e. 708 00:43:21,250 --> 00:43:24,370 And again, the same array of 27 elements 709 00:43:24,370 --> 00:43:25,830 describes both effects. 710 00:43:25,830 --> 00:43:28,670 And these are not dignified with any special names. 711 00:43:28,670 --> 00:43:33,210 They're just tensor relations that have to be true because 712 00:43:33,210 --> 00:43:35,860 of the fact that stress and strain are 713 00:43:35,860 --> 00:43:39,360 coupled by elastic relations. 714 00:43:39,360 --> 00:43:42,810 Not surprisingly, then, the coefficients e must somehow 715 00:43:42,810 --> 00:43:47,630 involve the piezoelectric moduli di,j and the elastic 716 00:43:47,630 --> 00:43:48,940 properties of the material. 717 00:43:52,070 --> 00:43:56,050 Another effect which is a very interesting one is the 718 00:43:56,050 --> 00:43:58,410 electro-optic effect. 719 00:43:58,410 --> 00:44:03,900 If you apply an electric field to a material, you change the 720 00:44:03,900 --> 00:44:05,260 birefringence of the field. 721 00:44:05,260 --> 00:44:09,380 The birefringence is defined as the difference in index of 722 00:44:09,380 --> 00:44:13,435 refraction for light polarized in two orthogonal directions. 723 00:44:16,300 --> 00:44:20,980 Another tensor effect-- and I'll give you a note defining 724 00:44:20,980 --> 00:44:23,010 the terms next time we meet-- 725 00:44:23,010 --> 00:44:32,980 there is a piezoresistive effect, which you don't see 726 00:44:32,980 --> 00:44:34,250 talked about very much. 727 00:44:34,250 --> 00:44:38,340 But that's a property that Texas Instruments was 728 00:44:38,340 --> 00:44:40,070 interested in at one time. 729 00:44:40,070 --> 00:44:43,290 And I have a set of sheets defining those relations that 730 00:44:43,290 --> 00:44:47,130 the presenter of a paper at a meeting, one time, 731 00:44:47,130 --> 00:44:48,440 kindly gave to me. 732 00:44:52,590 --> 00:44:52,960 OK. 733 00:44:52,960 --> 00:44:57,870 So what other symmetry restrictions are there? 734 00:44:57,870 --> 00:45:02,040 Having shown that inversion will not permit any third-rank 735 00:45:02,040 --> 00:45:09,320 tensor property, we have gone from 32 point groups, lost 736 00:45:09,320 --> 00:45:14,300 interest in the 11 centrosymmetric point groups. 737 00:45:14,300 --> 00:45:19,000 And so there are only 21 piezoelectric point groups. 738 00:45:27,730 --> 00:45:35,260 And the way we would plod through all 21 of them would 739 00:45:35,260 --> 00:45:39,590 be to simply define, starting with a twofold axis. 740 00:45:39,590 --> 00:45:43,480 Let's say a twofold axis parallel to x3 would 741 00:45:43,480 --> 00:45:49,920 correspond to a change of axes ci,j that describes the new 742 00:45:49,920 --> 00:46:00,040 axes in terms of the original ones that consisted of minus 743 00:46:00,040 --> 00:46:05,710 1, 0, 0; 0, minus 1, 0; 0, 0, 1. 744 00:46:05,710 --> 00:46:09,220 So if we look at an element in the piezoelectric matrix-- 745 00:46:09,220 --> 00:46:15,700 something like d1,6-- 746 00:46:15,700 --> 00:46:19,890 d1,6 actually corresponds in tensor notation 747 00:46:19,890 --> 00:46:27,030 to d1,3,2 plus d1,2,3. 748 00:46:27,030 --> 00:46:42,060 And d1,3,2 prime is going to be c1,i c3,j c2,k times all of 749 00:46:42,060 --> 00:46:43,660 the original tensor elements di,j,k. 750 00:46:47,090 --> 00:46:50,050 The only term of the form ci, something that 751 00:46:50,050 --> 00:46:53,520 is non-zero is c,1,1. 752 00:46:53,520 --> 00:46:57,180 And that has a value, minus 1. 753 00:46:57,180 --> 00:46:59,960 The only term of the form c3, something which 754 00:46:59,960 --> 00:47:02,880 is non-zero is c3,3. 755 00:47:02,880 --> 00:47:05,860 And that turns out to be plus 1. 756 00:47:05,860 --> 00:47:10,380 c2, something, the only form that's non-zero is c2,2. 757 00:47:10,380 --> 00:47:13,100 And that has value, minus 1. 758 00:47:15,630 --> 00:47:21,120 And this should be times d. 759 00:47:21,120 --> 00:47:25,250 And the only value of i that stayed was 1. 760 00:47:25,250 --> 00:47:28,725 The only value of j that stayed was 3. 761 00:47:31,560 --> 00:47:34,750 And then only value of k that stayed was 2. 762 00:47:34,750 --> 00:47:41,200 So this says that d1,3,2 should be equal to d1,3,2, 763 00:47:41,200 --> 00:47:51,775 which I don't like, unless it's supposed to be negative. 764 00:48:02,296 --> 00:48:06,345 I did something wrong here. 765 00:48:06,345 --> 00:48:08,472 Well, you see how easy it is, even if it 766 00:48:08,472 --> 00:48:09,830 didn't turn out right. 767 00:48:09,830 --> 00:48:14,140 And the form of the tensor for monoclinic crystal of symmetry 768 00:48:14,140 --> 00:48:19,970 2, with a twofold access parallel to x3, has as shown 769 00:48:19,970 --> 00:48:24,190 in the lower left-hand corner of the handout on symmetry 770 00:48:24,190 --> 00:48:28,010 restrictions, it has eight non-zero terms. 771 00:48:28,010 --> 00:48:31,200 If you do the same thing for a mirror plane perpendicular to 772 00:48:31,200 --> 00:48:36,280 x3, you find that there are 10 non-zero terms. 773 00:48:36,280 --> 00:48:41,865 So we don't have the situation where all of the point groups 774 00:48:41,865 --> 00:48:44,830 that are able to show the property within a given 775 00:48:44,830 --> 00:48:47,900 crystal systems like monoclinic have exactly the 776 00:48:47,900 --> 00:48:51,650 same form of the property tensor. 777 00:48:51,650 --> 00:48:56,510 In fact, you'll notice a curious correspondence between 778 00:48:56,510 --> 00:49:00,120 the restrictions for symmetry 2 and symmetry m. 779 00:49:00,120 --> 00:49:06,210 All the terms that are 0 in symmetry 2 are non-zero in 780 00:49:06,210 --> 00:49:08,490 symmetry m, and vice versa. 781 00:49:08,490 --> 00:49:12,380 All the non-zero terms in symmetry 2 are 782 00:49:12,380 --> 00:49:14,960 0 in symmetry m. 783 00:49:14,960 --> 00:49:19,100 And the reason for that is simply that this is the form 784 00:49:19,100 --> 00:49:22,010 of the direction cosine scheme for symmetry 2. 785 00:49:22,010 --> 00:49:26,160 The form of the direction cosine scheme for symmetry m, 786 00:49:26,160 --> 00:49:30,768 where the m is perpendicular to x3, would have the form 1, 787 00:49:30,768 --> 00:49:36,660 0, 0; 0, 1, 0; 0, 0, minus 1. 788 00:49:36,660 --> 00:49:41,840 So ci,j, for a mirror plane perpendicular to x3, is 789 00:49:41,840 --> 00:49:46,290 exactly the negative of the direction cosine scheme for a 790 00:49:46,290 --> 00:49:48,360 twofold axis parallel to x3. 791 00:49:48,360 --> 00:49:52,130 And since the number of direction cosines is odd, this 792 00:49:52,130 --> 00:49:56,230 means that everything that has an equality between the 793 00:49:56,230 --> 00:50:01,170 di,j,k's for m would have the transformed element be the 794 00:50:01,170 --> 00:50:04,250 negative of the original one for 2, and vice versa. 795 00:50:04,250 --> 00:50:11,100 So that's why there's this complementary form of the 796 00:50:11,100 --> 00:50:14,250 tensors for the two monoclinic symmetries. 797 00:50:18,120 --> 00:50:21,295 Point out a couple of curious things in the tables. 798 00:50:24,880 --> 00:50:29,780 You really have to go through 21 of the symmetries 799 00:50:29,780 --> 00:50:31,650 independently. 800 00:50:31,650 --> 00:50:38,480 And you find that some of them do come out the same. 801 00:50:38,480 --> 00:50:42,810 Symmetry 4 bar 3m and symmetry 2:3 have 802 00:50:42,810 --> 00:50:44,940 restrictions of the same form. 803 00:50:44,940 --> 00:50:46,040 But for the most part-- 804 00:50:46,040 --> 00:50:46,774 Yes? 805 00:50:46,774 --> 00:50:49,380 AUDIENCE: I know why this is wrong. 806 00:50:49,380 --> 00:50:50,010 PROFESSOR: Why is that wrong? 807 00:50:50,010 --> 00:50:55,010 AUDIENCE: Because your notation of d1,6 is in fact 808 00:50:55,010 --> 00:51:00,510 not d1,3,2 but d1,2,1. 809 00:51:00,510 --> 00:51:03,510 Since you have two same indices, [? d3 ?] 810 00:51:03,510 --> 00:51:05,510 and minus [? d1. ?] 811 00:51:05,510 --> 00:51:06,746 PROFESSOR: OK. 812 00:51:06,746 --> 00:51:08,560 d1,6; d1,6. 813 00:51:08,560 --> 00:51:10,570 Ah, of course. 814 00:51:10,570 --> 00:51:14,470 Of course. d1,2,6 is the one up here. 815 00:51:14,470 --> 00:51:19,440 And that's 1, 2 and 2, 1 for the strains. 816 00:51:19,440 --> 00:51:21,400 OK. 817 00:51:21,400 --> 00:51:23,192 Thank you. 818 00:51:23,192 --> 00:51:33,520 So this is d1,1,2 plus d1,2,1. 819 00:51:33,520 --> 00:51:44,710 And so we would have 1 by 1,k and 2,k. 820 00:51:44,710 --> 00:51:46,550 1, 1, and 2. 821 00:51:50,190 --> 00:51:53,120 And 1, 1, 1. 822 00:51:59,550 --> 00:52:05,060 1,i; 1,j; and 2,j for this one. 823 00:52:05,060 --> 00:52:11,510 So we'd have c1,1 c1,1 c2,2. c c1,1 is minus 1; c1,1 is minus 824 00:52:11,510 --> 00:52:13,480 1; c2,2 is minus 1. 825 00:52:13,480 --> 00:52:21,460 So d1,1,2 is minus d1,2,1, which means they have to be 826 00:52:21,460 --> 00:52:22,280 identically 0. 827 00:52:22,280 --> 00:52:23,270 Thank you. 828 00:52:23,270 --> 00:52:26,400 I'm sure nobody cares at this point. 829 00:52:26,400 --> 00:52:28,510 Very good. 830 00:52:28,510 --> 00:52:28,770 OK. 831 00:52:28,770 --> 00:52:34,050 Another curious thing that happens is that for cubic 832 00:52:34,050 --> 00:52:37,900 symmetry 4:3:2, it's acentric. 833 00:52:37,900 --> 00:52:41,470 But every single modulus is 0. 834 00:52:41,470 --> 00:52:44,170 And the reason, the explanation, is there's so 835 00:52:44,170 --> 00:52:48,460 many different transformations which have to leave the tensor 836 00:52:48,460 --> 00:52:52,990 invariant that the poor tensor just can't do it. 837 00:52:52,990 --> 00:52:57,940 It gives up, packs up, and goes home, leaving all the 838 00:52:57,940 --> 00:52:59,480 elements zero. 839 00:52:59,480 --> 00:53:02,125 Just no way you can get all the qualities to be satisfied. 840 00:53:04,670 --> 00:53:08,080 Another curious result that I point out for some of the 841 00:53:08,080 --> 00:53:14,390 hexagonal symmetries for symmetry 3:2, for symmetry 842 00:53:14,390 --> 00:53:21,110 6:2:2, and for 3 over m and 6 bar 2m, all the elements of 843 00:53:21,110 --> 00:53:26,190 the form d3, something are identically 0. 844 00:53:26,190 --> 00:53:31,290 Which says you simply cannot create a polarization that has 845 00:53:31,290 --> 00:53:37,290 a component P3. 846 00:53:37,290 --> 00:53:40,700 You just cannot create a polarization perpendicular to 847 00:53:40,700 --> 00:53:44,000 the axis of high symmetry. 848 00:53:44,000 --> 00:53:44,390 Yeah. 849 00:53:44,390 --> 00:53:44,675 Got a question? 850 00:53:44,675 --> 00:53:47,865 AUDIENCE: So all of the cubic that has 4:3:2 symmetry or 851 00:53:47,865 --> 00:53:49,730 [? 4:1:0 ?], that means you can't get-- 852 00:53:49,730 --> 00:53:53,290 PROFESSOR: You just don't have any piezoelectric effect, even 853 00:53:53,290 --> 00:53:58,730 though the crystal is not centrosymmetric, requiring 854 00:53:58,730 --> 00:54:01,880 that all those transformations leave the tensor invariant; 855 00:54:01,880 --> 00:54:05,360 simultaneously, require that everything has to be 0. 856 00:54:08,530 --> 00:54:10,900 One final thing, and then I'm running a little bit over. 857 00:54:10,900 --> 00:54:17,550 But I'd like to go on to other aspects of piezoelectricity 858 00:54:17,550 --> 00:54:18,800 during the next hour. 859 00:54:21,350 --> 00:54:23,790 Remember something that we've shown and which I've asked you 860 00:54:23,790 --> 00:54:27,160 to look into again on one of the problem sets, and that is 861 00:54:27,160 --> 00:54:33,620 that the trace of the strain tensor gives you 862 00:54:33,620 --> 00:54:34,940 the change in volume. 863 00:54:37,760 --> 00:54:49,870 The first 3 by 3 blocks of this tensor that we've been 864 00:54:49,870 --> 00:54:58,970 looking at, were we to write the elements of strain, 865 00:54:58,970 --> 00:55:04,740 epsilon i,j, in terms of d-- 866 00:55:04,740 --> 00:55:06,020 epsilon j,k-- 867 00:55:06,020 --> 00:55:17,310 in terms of di,j,k times e sub k, it is this block in here 868 00:55:17,310 --> 00:55:26,300 which enters into the terms epsilon 1,1; epsilon 2,2; and 869 00:55:26,300 --> 00:55:30,350 epsilon 3,3. 870 00:55:30,350 --> 00:55:36,100 These terms give you the fractional change in volume. 871 00:55:36,100 --> 00:55:41,420 And the elements that are involved in the piezoelectric 872 00:55:41,420 --> 00:55:44,220 matrix are these terms in here. 873 00:55:44,220 --> 00:55:52,830 So if all of those terms are 0, it turns out that when you 874 00:55:52,830 --> 00:55:56,260 apply a stress and look at the electric field, or apply an 875 00:55:56,260 --> 00:55:59,490 electric field and look at the strain, if you apply a field 876 00:55:59,490 --> 00:56:02,200 and all of those nine terms are 0, you 877 00:56:02,200 --> 00:56:03,750 cannot create a strain. 878 00:56:03,750 --> 00:56:04,820 So there's no volume change. 879 00:56:04,820 --> 00:56:07,070 There can be a strain, but no volume change. 880 00:56:07,070 --> 00:56:10,720 The only deformation you can create is pure shear. 881 00:56:10,720 --> 00:56:14,550 So if you drive a piezoelectric oscillator with 882 00:56:14,550 --> 00:56:19,080 an electric field, for those property tensors for which 883 00:56:19,080 --> 00:56:21,810 that first 3 by 3 block are all zero, 884 00:56:21,810 --> 00:56:23,760 the thing can shear-- 885 00:56:23,760 --> 00:56:26,060 in the water, for example-- but it can't pulse. 886 00:56:26,060 --> 00:56:28,550 It can't have a volume change. 887 00:56:28,550 --> 00:56:31,900 And if you were to create a transducer for sonar 888 00:56:31,900 --> 00:56:36,930 applications, what you would want to do is to have 889 00:56:36,930 --> 00:56:41,030 piezoelectric device which expanded this way, to create a 890 00:56:41,030 --> 00:56:42,560 sound wave going through the water. 891 00:56:42,560 --> 00:56:44,840 If it just goes back and forth, it's going to slosh 892 00:56:44,840 --> 00:56:48,040 back and forth in the water and not create any sonic wave 893 00:56:48,040 --> 00:56:50,670 that could be used in sonar. 894 00:56:50,670 --> 00:56:55,610 Now that is true for elements being identically 0. 895 00:56:55,610 --> 00:57:00,340 It turns out it's also true for elements such as 4 bar, 896 00:57:00,340 --> 00:57:06,160 where two of those terms in the 3 by 3 block are the 897 00:57:06,160 --> 00:57:07,430 negative of one of the other. 898 00:57:07,430 --> 00:57:10,110 These will also have no volume change. 899 00:57:10,110 --> 00:57:13,330 So that's another very curious property of piezoelectric 900 00:57:13,330 --> 00:57:18,350 response that follows from these symmetry restrictions. 901 00:57:18,350 --> 00:57:18,595 OK. 902 00:57:18,595 --> 00:57:21,600 That's enough for our first session. 903 00:57:21,600 --> 00:57:25,300 When we come back, we'll look at the converse 904 00:57:25,300 --> 00:57:27,090 piezoelectric effect. 905 00:57:27,090 --> 00:57:32,700 And we'll also look at representation surfaces for 906 00:57:32,700 --> 00:57:40,890 specific piezoelectric devices and ask if it's possible for a 907 00:57:40,890 --> 00:57:45,550 third-rank tensor to have a representation surface that's 908 00:57:45,550 --> 00:57:48,150 analogous to the representation quadric for 909 00:57:48,150 --> 00:57:50,170 second-rank tensors. 910 00:57:50,170 --> 00:57:53,910 So I'm sure you'll all want to come back and hear the answer 911 00:57:53,910 --> 00:57:55,160 to that question.