WEBVTT 00:00:00.470 --> 00:00:03.720 The following content is provided by MIT OpenCourseWare 00:00:03.720 --> 00:00:06.140 under a Creative Commons license. 00:00:06.140 --> 00:00:08.750 Additional information about our license and MIT 00:00:08.750 --> 00:00:11.852 OpenCourseWare in general is available ocw.mit.edu. 00:00:17.190 --> 00:00:19.580 PROFESSOR: All right. 00:00:19.580 --> 00:00:24.700 The quiz on Thursday will cover up through 00:00:24.700 --> 00:00:26.470 piezoelectricity. 00:00:26.470 --> 00:00:30.800 You've had a set of notes in your hands that cover just 00:00:30.800 --> 00:00:34.400 about everything that I wanted to say about piezoelectricity. 00:00:34.400 --> 00:00:38.140 There are other modulae that one could talk about. 00:00:38.140 --> 00:00:43.700 But these follow quite directly from the one or two 00:00:43.700 --> 00:00:45.810 that we will do. 00:00:45.810 --> 00:00:50.520 So we will not have anything to say about elasticity until 00:00:50.520 --> 00:00:55.470 the last lecture of the term, which is a nice outcome 00:00:55.470 --> 00:00:59.260 because you certainly don't want to take a quiz on 00:00:59.260 --> 00:01:00.790 forthright tensors. 00:01:00.790 --> 00:01:04.069 You'll spend the entire hour just writing out all these 00:01:04.069 --> 00:01:06.290 cumbersome equations. 00:01:06.290 --> 00:01:06.470 All right. 00:01:06.470 --> 00:01:08.770 So the quiz will cover up through piezoelectricity, 00:01:08.770 --> 00:01:13.020 including a few of the representation surfaces that 00:01:13.020 --> 00:01:14.290 we'll examine today. 00:01:19.430 --> 00:01:25.980 The thing that we'll be looking at is-- 00:01:25.980 --> 00:01:26.370 I'm sorry. 00:01:26.370 --> 00:01:26.930 You have a question? 00:01:26.930 --> 00:01:27.417 AUDIENCE: Yes. 00:01:27.417 --> 00:01:27.904 A question. 00:01:27.904 --> 00:01:30.826 Well, you said [INAUDIBLE] 00:01:30.826 --> 00:01:32.287 example. 00:01:32.287 --> 00:01:33.261 PROFESSOR: I'm sorry. 00:01:33.261 --> 00:01:34.722 Third-rank tensors. 00:01:34.722 --> 00:01:35.210 AUDIENCE: OK. 00:01:35.210 --> 00:01:38.620 PROFESSOR: Also, we had just a little bit of that going into 00:01:38.620 --> 00:01:40.055 the second quiz. 00:01:40.055 --> 00:01:41.890 And we didn't really ask anything about that on the 00:01:41.890 --> 00:01:43.050 second quiz. 00:01:43.050 --> 00:01:45.040 So it'd be third-rank tensors. 00:01:45.040 --> 00:01:48.530 But you have to use second-rank tensors to define 00:01:48.530 --> 00:01:49.450 third-rank tensors. 00:01:49.450 --> 00:01:53.370 So it really will be not the emphasis, but certainly you 00:01:53.370 --> 00:01:55.860 should be familiar with the early part of what we did with 00:01:55.860 --> 00:01:59.125 second-rank tensors. 00:01:59.125 --> 00:01:59.560 All right. 00:01:59.560 --> 00:02:04.940 So today we'll look at some representation surfaces to 00:02:04.940 --> 00:02:10.490 your wonder and delight at how incredibly anisotropic the 00:02:10.490 --> 00:02:15.810 variation of these third-rank properties are with direction. 00:02:15.810 --> 00:02:20.230 One of the things that I love to do for problems when we get 00:02:20.230 --> 00:02:23.422 to these different piezoelectric effects is make 00:02:23.422 --> 00:02:27.550 up hypothetical devices just for fun. 00:02:27.550 --> 00:02:34.080 So among those that I've invented, the first one is a 00:02:34.080 --> 00:02:37.550 earthquake sensing device because it's well known that 00:02:37.550 --> 00:02:41.310 California is going to split in half, and half will fall 00:02:41.310 --> 00:02:43.360 into the sea any day now. 00:02:43.360 --> 00:02:48.700 So if this is the San Andreas fault, I have developed large, 00:02:48.700 --> 00:02:54.310 prismatic monoclinic crystals that I embed into the San 00:02:54.310 --> 00:02:58.270 Andreas fault at regular intervals. 00:02:58.270 --> 00:03:00.960 The bottom of these crystals is grounded. 00:03:00.960 --> 00:03:03.220 There's an electrode at the top. 00:03:03.220 --> 00:03:06.830 And if the San Andreas fault starts to move, and there is 00:03:06.830 --> 00:03:09.960 shear on these crystals, there will be a charge 00:03:09.960 --> 00:03:11.470 developed on the top. 00:03:11.470 --> 00:03:15.130 And I ask you to relate the charge on the top in terms of 00:03:15.130 --> 00:03:18.130 the piezoelectric modulae to the shear along the San 00:03:18.130 --> 00:03:19.990 Andreas fault. 00:03:19.990 --> 00:03:24.050 So there's a very clever little device that clearly is 00:03:24.050 --> 00:03:27.130 going to be lucrative because these crystals will have to be 00:03:27.130 --> 00:03:28.120 huge in size. 00:03:28.120 --> 00:03:31.330 And they'll cost more even than silicon. 00:03:31.330 --> 00:03:36.120 Another device that I've invented is used down at the 00:03:36.120 --> 00:03:39.710 Boston fish pier. 00:03:39.710 --> 00:03:47.130 This is a crystal on which I hang a pan, and 00:03:47.130 --> 00:03:48.710 you put fish in it. 00:03:48.710 --> 00:03:51.500 That creates a tensile stress in this direction. 00:03:51.500 --> 00:03:53.830 We measure the charge on this face. 00:03:53.830 --> 00:03:56.390 And we put a little meter that measures the charge that's 00:03:56.390 --> 00:03:57.380 accumulated. 00:03:57.380 --> 00:04:00.220 So this is how the fishermen can weigh their fish after 00:04:00.220 --> 00:04:02.810 they haul them in off the boat. 00:04:02.810 --> 00:04:05.290 So you see this is really practical material we're 00:04:05.290 --> 00:04:05.910 dealing with. 00:04:05.910 --> 00:04:10.290 This has applications in all realms of life. 00:04:10.290 --> 00:04:13.790 So today I'd like to show you, also, another 00:04:13.790 --> 00:04:15.350 problem that sets up. 00:04:15.350 --> 00:04:18.630 And I pass this out just so you can, again, have a look at 00:04:18.630 --> 00:04:21.709 some of the questions that you should be equipped to answer. 00:04:21.709 --> 00:04:22.780 So here-- 00:04:22.780 --> 00:04:26.750 not fully expecting anybody to do it, but you can see the 00:04:26.750 --> 00:04:28.380 sorts of problems one might ask. 00:04:28.380 --> 00:04:34.910 Here is problem set number 16, which asks you, should you be 00:04:34.910 --> 00:04:37.830 so inclined, to think about manipulations 00:04:37.830 --> 00:04:39.660 of third-rank tensors. 00:04:39.660 --> 00:04:47.010 But then the second question is an idea for a device which 00:04:47.010 --> 00:04:50.080 the Center for Material Science and Engineering is 00:04:50.080 --> 00:04:53.410 considering employing here in building 13. 00:04:53.410 --> 00:04:54.870 And you can read all about that.. 00:04:57.570 --> 00:04:59.940 This is the famous soup cell. 00:05:06.690 --> 00:05:11.640 I think probably you will see some sort of for fun modulus 00:05:11.640 --> 00:05:13.780 for a particular device on the quiz. 00:05:13.780 --> 00:05:15.080 I haven't made one up yet. 00:05:15.080 --> 00:05:18.750 But I think you'll have a look at something like that. 00:05:18.750 --> 00:05:22.520 And when we do the longitudinal piezoelectric 00:05:22.520 --> 00:05:27.800 modulus for quartz, which we'll do momentarily, this 00:05:27.800 --> 00:05:32.630 will give you an idea of how to set up these expressions. 00:05:32.630 --> 00:05:36.770 The problem, basically, is that the direct piezoelectric 00:05:36.770 --> 00:05:45.390 effect measures a polarization in terms of all of the 00:05:45.390 --> 00:05:47.580 elements of applied stress. 00:05:47.580 --> 00:05:53.510 So there is a modulus dijk times all of the elements of 00:05:53.510 --> 00:05:56.990 stress, sigma jk. 00:05:56.990 --> 00:06:00.400 So this, until you use the condensation of subscripts, 00:06:00.400 --> 00:06:05.120 contains nine terms going this way and three 00:06:05.120 --> 00:06:06.440 equations going this way. 00:06:10.540 --> 00:06:17.140 As we discussed earlier, since only six of the nine elements 00:06:17.140 --> 00:06:22.670 of stress are independent, you can condense this down into a 00:06:22.670 --> 00:06:29.060 3 by 6 array of terms. 00:06:29.060 --> 00:06:32.300 The problem in doing that, though, even though one has 00:06:32.300 --> 00:06:38.120 only 18 different modulae to work with, instead of 27-- 00:06:38.120 --> 00:06:41.610 that's a considerable economy in notation, and an 00:06:41.610 --> 00:06:44.740 elimination of a great deal of redundancy-- 00:06:44.740 --> 00:06:51.380 the matrix form of this relation-- and I emphasize 00:06:51.380 --> 00:06:55.830 that it is a matrix, and it's no longer a tensor-- 00:06:55.830 --> 00:06:58.110 the matrix form cannot be transformed. 00:06:58.110 --> 00:07:01.030 Again, there are only six terms in the 00:07:01.030 --> 00:07:03.720 matrix elements of stress. 00:07:03.720 --> 00:07:07.250 And there are three equations, again, one for each component 00:07:07.250 --> 00:07:08.720 of polarization. 00:07:08.720 --> 00:07:13.450 So instead of having 27, one has only 18. 00:07:13.450 --> 00:07:19.280 But if you are considering changing the reference axes-- 00:07:19.280 --> 00:07:22.130 and that is one of the things that it's interesting to do 00:07:22.130 --> 00:07:25.350 for these various modulae that one can define-- 00:07:25.350 --> 00:07:29.520 change the orientation of a particular rod-shaped specimen 00:07:29.520 --> 00:07:32.600 that you cut out of a crystal to different crystallographic 00:07:32.600 --> 00:07:37.190 orientations and then ask how the scalar modulus changes as 00:07:37.190 --> 00:07:40.280 you change the direction in which you've sliced out the 00:07:40.280 --> 00:07:43.370 wafer or the rod of material. 00:07:43.370 --> 00:07:48.070 In order to do that, you want to transform the piezoelectric 00:07:48.070 --> 00:07:51.450 tensor to a new set of reference axes. 00:07:51.450 --> 00:07:56.840 And you cannot transform the dij's because they are a 00:07:56.840 --> 00:07:58.480 matrix and not a tensor. 00:07:58.480 --> 00:08:03.580 And no law of transformation is defined. 00:08:03.580 --> 00:08:08.110 So what you have to do in any generic problem of this sort 00:08:08.110 --> 00:08:12.380 is, for a particular single crystal, look up the form of 00:08:12.380 --> 00:08:19.710 the dij matrix that will have the equalities between tensor 00:08:19.710 --> 00:08:24.060 elements written in and the 0's, those modulae which are 00:08:24.060 --> 00:08:27.770 identically 0, entered into the array. 00:08:27.770 --> 00:08:30.620 And then you have to work your way backwards to get to the 00:08:30.620 --> 00:08:32.559 full tensor notation. 00:08:32.559 --> 00:08:38.929 So we'll see this when we look at the modulae for 00:08:38.929 --> 00:08:41.850 symmetry three two. 00:08:41.850 --> 00:08:47.920 You'll have to write in the exact matrix subscripts 00:08:47.920 --> 00:08:51.360 without absorbing the equalities in the notation. 00:08:51.360 --> 00:08:55.280 And then you'll have to expand the matrix terms into full 00:08:55.280 --> 00:08:58.020 three-subscript tensor terms. 00:08:58.020 --> 00:09:01.000 Then if you want to transform the axes, which is to say you 00:09:01.000 --> 00:09:04.680 want to cut out your specimen, be it a plate or a rod, in a 00:09:04.680 --> 00:09:08.550 different orientation, you have to transform the full 00:09:08.550 --> 00:09:14.250 three subscript tensor elements to a new setting. 00:09:14.250 --> 00:09:16.110 And then if you want to continue to work in that 00:09:16.110 --> 00:09:19.980 setting in the compact matrix form, collapse it back down to 00:09:19.980 --> 00:09:22.740 matrix form, insert the equalities, and then you're 00:09:22.740 --> 00:09:24.780 back to where you started from. 00:09:24.780 --> 00:09:33.290 So this problem of seeing how modulae that describe 00:09:33.290 --> 00:09:37.910 different phenomena vary with crystal symmetry, you have to 00:09:37.910 --> 00:09:41.280 go through this problem of expanding the compact form and 00:09:41.280 --> 00:09:45.100 then collapsing back down when you've got it as a function of 00:09:45.100 --> 00:09:50.810 some orientational angle or in terms of a coordinate system 00:09:50.810 --> 00:09:52.060 that you want to work in. 00:09:55.360 --> 00:09:59.610 The problem is exactly the same for elasticity. 00:09:59.610 --> 00:10:02.210 And we'll look at some of these modulae next term. 00:10:02.210 --> 00:10:04.780 You've heard these names before, I'm sure-- 00:10:04.780 --> 00:10:07.670 Young's modulus, shear modulus, and so on. 00:10:07.670 --> 00:10:11.290 We'll take a look on next Tuesday, a week from today, at 00:10:11.290 --> 00:10:14.800 Young's modulus, which is one of the more important ones. 00:10:14.800 --> 00:10:19.800 And probably are used to seeing this in the form of 00:10:19.800 --> 00:10:22.570 information for a polycrystalline material, 00:10:22.570 --> 00:10:24.740 which is essentially isotropic. 00:10:24.740 --> 00:10:27.090 The modulus is much more interesting for 00:10:27.090 --> 00:10:28.740 single crystal materials. 00:10:28.740 --> 00:10:32.610 And then the surfaces that are defined particularly for the 00:10:32.610 --> 00:10:35.490 lower symmetries are absolutely wild things with 00:10:35.490 --> 00:10:39.070 lumps and wiggles and lobes and things of that sort, 00:10:39.070 --> 00:10:42.790 nothing like the dumb old, uninteresting ellipsoids that 00:10:42.790 --> 00:10:45.515 we encountered for second-rank properties. 00:10:49.840 --> 00:10:54.750 So let's, then, take a look at how we had set up and defined 00:10:54.750 --> 00:10:58.240 the direct piezoelectric effect. 00:10:58.240 --> 00:11:03.200 We set this up as a proper tensor relation, saying that 00:11:03.200 --> 00:11:05.720 P1 is equal to d1. 00:11:05.720 --> 00:11:12.170 V1 And then, you'll recall, we have nine elements of strain-- 00:11:12.170 --> 00:11:20.690 sigma 11, sigma 12, sigma 13, sigma 21, sigma 22, sigma 23, 00:11:20.690 --> 00:11:25.800 sigm a 31, sigma 32, and sigma 33. 00:11:25.800 --> 00:11:28.950 But the tensor is symmetric, so we really only need to 00:11:28.950 --> 00:11:32.760 enter into our relation six of these nine terms explicitly. 00:11:32.760 --> 00:11:36.630 And what we did was to replace the two subscripts on the 00:11:36.630 --> 00:11:40.350 elements of strain, which, again, you need if you want to 00:11:40.350 --> 00:11:44.400 refer to those elements of stress through a different 00:11:44.400 --> 00:11:45.580 coordinate system. 00:11:45.580 --> 00:11:49.870 You have to know the elements of stress in tensor form. 00:11:49.870 --> 00:11:58.280 But we convert it to six terms by going and replacing the 00:11:58.280 --> 00:12:02.580 pairs of subscripts with a single one, two, and three, 00:12:02.580 --> 00:12:05.820 marching down the diagonal of the tensor this way and then 00:12:05.820 --> 00:12:09.080 marching up the right-hand side, calling two, three, 00:12:09.080 --> 00:12:14.300 four, and calling one, three, five, and finally ending up in 00:12:14.300 --> 00:12:15.430 this slot here. 00:12:15.430 --> 00:12:18.860 And we call that six. 00:12:18.860 --> 00:12:22.570 So that was the notation we used to get to a matrix 00:12:22.570 --> 00:12:23.520 representation. 00:12:23.520 --> 00:12:26.560 But the place where all this started is-- 00:12:26.560 --> 00:12:30.190 and I'll write just one line of this to be merciful-- 00:12:30.190 --> 00:12:37.340 d 111 times sigma 11, so this pair of subscripts goes with 00:12:37.340 --> 00:12:41.270 this pair, plus d 122. 00:12:41.270 --> 00:12:43.870 I'm putting that one in next because we're going to number 00:12:43.870 --> 00:12:48.630 the terms for stress in this order, one through six. 00:12:48.630 --> 00:13:02.230 Times sigma 22 plus d 133 times sigma 33 plus d 123 00:13:02.230 --> 00:13:23.130 times sigma 23 plus d 132 times sigma 32 plus d113 times 00:13:23.130 --> 00:13:31.340 sigma 23 plus d 132 times sigma 32. 00:13:31.340 --> 00:13:35.040 And finally we end up in slot number six. 00:13:35.040 --> 00:13:42.470 And we have a d 112 times sigma 12 plus a d 00:13:42.470 --> 00:13:46.780 121 times sigma 21. 00:13:46.780 --> 00:13:47.110 Look at that. 00:13:47.110 --> 00:13:50.280 There's an equation that covers two whole blackboards. 00:13:53.680 --> 00:13:57.670 So if we now condense this down to matrix form we would 00:13:57.670 --> 00:14:05.840 say that P1 is d 11 times sigma 1 plus d 12 times sigma 00:14:05.840 --> 00:14:11.320 2 times d 13 times sigma 3 plus-- 00:14:11.320 --> 00:14:13.910 and now we have this messy problem with the 2's-- 00:14:13.910 --> 00:14:24.020 we have a d 14 times sigma 4 plus, again, a d 14 00:14:24.020 --> 00:14:25.560 times a sigma -- 00:14:25.560 --> 00:14:29.890 2 3 is equal to 32, so I can call this 4-- 00:14:29.890 --> 00:14:36.580 and then these terms become 15 sigma 5 and, again, a 15 times 00:14:36.580 --> 00:14:45.595 sigma 5 plus a 16 times sigma 6 plus d 16 times sigma 6. 00:14:49.850 --> 00:14:52.690 Now we have to make a choice. 00:14:52.690 --> 00:14:57.190 Either we are going to have, in a general matrix relation, 00:14:57.190 --> 00:15:06.220 that P sub i is equal to dij times sigma j, if j is equal 00:15:06.220 --> 00:15:08.740 to 1, 2, or 3. 00:15:08.740 --> 00:15:14.490 But it's equal to 2 dij times sigma j if j is 00:15:14.490 --> 00:15:17.500 equal to 4, 5, or 6. 00:15:17.500 --> 00:15:20.940 And that is something we like to avoid, if possible. 00:15:20.940 --> 00:15:23.470 That's ugly. 00:15:23.470 --> 00:15:25.070 That's ugly. 00:15:25.070 --> 00:15:28.120 And it's going to be a hell of a matrix if we have factors of 00:15:28.120 --> 00:15:31.040 two in front of some of the matrix elements but not in 00:15:31.040 --> 00:15:32.090 terms of others. 00:15:32.090 --> 00:15:34.355 So this is something we could do. 00:15:34.355 --> 00:15:35.700 Hey, it's our ballgame. 00:15:35.700 --> 00:15:36.750 We make up the rules. 00:15:36.750 --> 00:15:39.790 But that's going to be an ugly thing to have to deal with. 00:15:39.790 --> 00:15:44.740 So instead, as we mentioned last time, what we will do is 00:15:44.740 --> 00:15:53.410 to lump together these terms and define d 14 not as these 00:15:53.410 --> 00:15:57.360 individual tensor elements, but define those matrix 00:15:57.360 --> 00:16:04.700 elements as the sum of these two elements. 00:16:04.700 --> 00:16:07.450 And then we saw before that-- 00:16:07.450 --> 00:16:10.470 we saw last time in our earlier meeting-- 00:16:10.470 --> 00:16:14.140 that from the converse piezoelectric effect, which 00:16:14.140 --> 00:16:18.270 expresses strain in terms of an applied field, where the 00:16:18.270 --> 00:16:22.280 elements of strain 1, 23, and 32 00:16:22.280 --> 00:16:24.300 appear in separate equations. 00:16:24.300 --> 00:16:29.380 And knowing that the same array of piezoelectric 00:16:29.380 --> 00:16:32.700 coefficients amazingly describes the converse 00:16:32.700 --> 00:16:36.020 piezoelectric effect as well as the direct piezoelectric 00:16:36.020 --> 00:16:42.920 effect, we know that d 123 equals d 132. 00:16:42.920 --> 00:16:44.933 And that is from the converse effect. 00:16:51.700 --> 00:16:59.335 So equivalent to saying this is to define d 14 as twice d 00:16:59.335 --> 00:17:03.080 123 because these two elements are equal. 00:17:03.080 --> 00:17:06.869 So we're eating the factor of two here so that we can write 00:17:06.869 --> 00:17:10.589 a nice matrix relation that doesn't 00:17:10.589 --> 00:17:12.500 involve a factor of two. 00:17:12.500 --> 00:17:18.890 So making this combination of terms for the shear stresses, 00:17:18.890 --> 00:17:25.355 we would have simply d 14 sigma 4 plus d 15 times sigma 00:17:25.355 --> 00:17:30.580 5 plus d 16 times sigma 6. 00:17:30.580 --> 00:17:33.790 And we can say, in general, for the other two equations, 00:17:33.790 --> 00:17:41.370 by analogy to this one, that P sub i is dij times sigma j-- 00:17:41.370 --> 00:17:45.510 a nice, neat matrix relation but one for which, 00:17:45.510 --> 00:17:49.550 unfortunately, there's no law of transformation for the 00:17:49.550 --> 00:17:52.900 matrix modulae dij. 00:17:52.900 --> 00:17:55.270 If you want to change to another coordinate system, we 00:17:55.270 --> 00:17:58.990 have to be prepared to resurrect this full three 00:17:58.990 --> 00:18:02.030 subscript notation on the piezoelectric modulae. 00:18:09.320 --> 00:18:09.620 OK. 00:18:09.620 --> 00:18:11.573 Comments or questions at this point? 00:18:18.475 --> 00:18:18.970 All right. 00:18:18.970 --> 00:18:23.910 If not, let me remind you that in the notes which I 00:18:23.910 --> 00:18:30.570 distributed last time, there are summarized all of the 00:18:30.570 --> 00:18:36.020 constraints imposed on the piezoelectric modulae for 00:18:36.020 --> 00:18:37.270 single crystals. 00:18:40.120 --> 00:18:47.100 And, again, these constraints, these requirements that the 00:18:47.100 --> 00:18:54.500 tensors remain invariant for the change of axes produced by 00:18:54.500 --> 00:19:00.120 a symmetry element that the crystal possesses, these 00:19:00.120 --> 00:19:05.880 transformations show that no third-rank tensor property can 00:19:05.880 --> 00:19:08.510 exist in a crystal that has inversion. 00:19:08.510 --> 00:19:12.980 So the 11 [INAUDIBLE] group, so-called, that possess 00:19:12.980 --> 00:19:17.360 inversion have absolutely no property and can be not 00:19:17.360 --> 00:19:20.150 considered further for third-rank properties. 00:19:20.150 --> 00:19:26.730 And then one must consider all of the 32 minus 11 21 point 00:19:26.730 --> 00:19:29.400 groups that lack conversion separately. 00:19:29.400 --> 00:19:32.890 There's no reason why they should behave the same way. 00:19:32.890 --> 00:19:36.320 Remember that for second-rank tensor properties we pulled 00:19:36.320 --> 00:19:41.480 the argument that inversion imposes no restrictions or 00:19:41.480 --> 00:19:43.960 constraints whatsoever on second-rank properties so, 00:19:43.960 --> 00:19:48.250 therefore, two different symmetries that differ only by 00:19:48.250 --> 00:19:51.010 the presence or absence of an inversion center. 00:19:51.010 --> 00:19:54.460 That is to say, you change 2 to 2 over 00:19:54.460 --> 00:19:56.150 m if you add inversion. 00:19:56.150 --> 00:19:59.240 But the argument was inversion requires nothing, so the 00:19:59.240 --> 00:20:01.590 constraints or symmetry, too, look exactly 00:20:01.590 --> 00:20:02.750 like those for symmetry. 00:20:02.750 --> 00:20:06.450 And here you've got to plod through every single one of 00:20:06.450 --> 00:20:09.860 the non-centrosymmetric point groups separately. 00:20:09.860 --> 00:20:14.080 And they all have tensors that have different forms. 00:20:14.080 --> 00:20:17.840 So what I'd like to do is look at one specific one. 00:20:17.840 --> 00:20:24.750 And that is the matrix for symmetry 32. 00:20:24.750 --> 00:20:28.510 And that is a point group that you'll recall has a threefold 00:20:28.510 --> 00:20:33.665 axis and twofold axes at intervals of 60 degrees. 00:20:37.060 --> 00:20:45.760 And in your list of the qualities and absences, 32, 00:20:45.760 --> 00:20:51.570 where the 3 is parallel to the axis x3, and the twofold axis 00:20:51.570 --> 00:20:53.700 is parallel to x1. 00:20:53.700 --> 00:21:01.560 So we're defining this as x1, this as x3, and x2 comes out 00:21:01.560 --> 00:21:03.720 halfway between the two. 00:21:03.720 --> 00:21:10.130 So let me draw this looking down along the threefold axis. 00:21:10.130 --> 00:21:14.000 These are all twofold axes. 00:21:14.000 --> 00:21:16.960 And we'll take x1 in this direction. 00:21:16.960 --> 00:21:21.600 x2 pokes out in between two twofold axes. 00:21:21.600 --> 00:21:26.510 And x3 comes straight up along the threefold axis. 00:21:26.510 --> 00:21:32.050 With that coordinate system, the form of the piezoelectric 00:21:32.050 --> 00:21:44.884 modulus matrix has this form-- d 11 minus d 11 0 d 14 0 0 0 0 00:21:44.884 --> 00:21:56.680 0 d 15 minus d 14 0 and in the bottom row d 31 00:21:56.680 --> 00:22:04.780 d 31 d 33 0 0 0. 00:22:04.780 --> 00:22:11.080 So this matrix with the equalities put in is obtained 00:22:11.080 --> 00:22:15.720 by looking at the full three-subscript tensor and 00:22:15.720 --> 00:22:19.810 requiring that it look the same before and after any of 00:22:19.810 --> 00:22:26.790 the rotations involved by these three distinct axes-- 00:22:26.790 --> 00:22:28.500 the threefold and the pair of twofolds. 00:22:32.870 --> 00:22:33.090 OK. 00:22:33.090 --> 00:22:37.530 Now, there are many different scalar modulae that one could 00:22:37.530 --> 00:22:42.220 define, some of them serious and worth the consideration 00:22:42.220 --> 00:22:47.940 because of the application in devices or other practical 00:22:47.940 --> 00:22:49.740 situations. 00:22:49.740 --> 00:22:58.100 The modulus that I'd like to examine is something called 00:22:58.100 --> 00:23:01.280 the longitudinal piezoelectric effect. 00:23:12.960 --> 00:23:17.390 And let's emphasize, again, that it is impossible to come 00:23:17.390 --> 00:23:22.250 up with one representation surface that fits every need 00:23:22.250 --> 00:23:28.410 because, again, the direct piezoelectric effect relates 00:23:28.410 --> 00:23:36.170 the components of a vector to a tensor, sigma ij. 00:23:36.170 --> 00:23:40.470 There are six independent tensor elements. 00:23:40.470 --> 00:23:46.110 So how can you describe how this vector is going to change 00:23:46.110 --> 00:23:51.320 as you change orientation of a crystal whose behavior is 00:23:51.320 --> 00:23:53.840 described by all of these modulae? 00:23:53.840 --> 00:23:57.250 But I would point out, however, that there are only 00:23:57.250 --> 00:23:59.160 two distinct-- 00:23:59.160 --> 00:24:00.073 oops, I'm sorry. 00:24:00.073 --> 00:24:01.890 This is d 14. 00:24:01.890 --> 00:24:04.260 I don't know how I made that d 15. 00:24:04.260 --> 00:24:09.470 There are only, in this array-- 00:24:09.470 --> 00:24:10.730 and I slipped a notch. 00:24:10.730 --> 00:24:11.980 Excuse me. 00:24:15.220 --> 00:24:19.990 Last couple of days are such that I am not able to even 00:24:19.990 --> 00:24:21.420 read from my notes. 00:24:21.420 --> 00:24:25.190 So these bottom lines, my apologies, are all 0. 00:24:25.190 --> 00:24:30.600 And there are two modulae, d 11 and d 14. 00:24:30.600 --> 00:24:36.120 So there are two independent numbers, but they appear as 00:24:36.120 --> 00:24:39.540 different matrix elements and, therefore, different tensor 00:24:39.540 --> 00:24:42.040 elements, as well. 00:24:42.040 --> 00:24:44.610 And this should be minus 2 d 14. 00:24:47.610 --> 00:24:47.970 I'm sorry. 00:24:47.970 --> 00:24:50.910 I slipped down a notch, and I got some for 32 00:24:50.910 --> 00:24:52.630 and some for 6. 00:24:52.630 --> 00:24:55.030 And I'm glad I found it at this point, or I'd really be 00:24:55.030 --> 00:24:56.280 in deep trouble. 00:24:58.970 --> 00:24:59.270 OK. 00:24:59.270 --> 00:25:04.160 So two numbers and that is, in fact, the form of the matrix 00:25:04.160 --> 00:25:06.460 for symmetry 32. 00:25:06.460 --> 00:25:10.520 So the longitudinal piezoelectric effect is one of 00:25:10.520 --> 00:25:14.730 the representation surfaces that gives you the way in 00:25:14.730 --> 00:25:19.800 which the polarization will change for one very, very 00:25:19.800 --> 00:25:22.850 specific type of stress. 00:25:22.850 --> 00:25:27.080 In particular, what we'll do is set this up as 00:25:27.080 --> 00:25:28.330 a coordinate system. 00:25:30.940 --> 00:25:38.830 And we will look at a coordinate system where this 00:25:38.830 --> 00:25:41.790 is the reference axis, x1. 00:25:41.790 --> 00:25:45.470 We'll come down with a compressive stress, sigma 11, 00:25:45.470 --> 00:25:48.680 along that axis. 00:25:48.680 --> 00:25:53.750 And, therefore, we're applying a uni-axial stress, which has 00:25:53.750 --> 00:25:55.890 just one component of stress. 00:25:55.890 --> 00:25:57.470 So that's very specialized. 00:25:57.470 --> 00:25:59.160 In general, there would be six different 00:25:59.160 --> 00:26:00.500 components of stress. 00:26:00.500 --> 00:26:04.160 But we're looking at one specific stimulus applied to 00:26:04.160 --> 00:26:05.600 this crystal plate. 00:26:05.600 --> 00:26:09.960 In response to that sigma 11 there are charges induced on 00:26:09.960 --> 00:26:11.210 all of these surfaces. 00:26:16.170 --> 00:26:20.210 And these charges, this charge per unit area, is proportional 00:26:20.210 --> 00:26:26.840 to the component of polarization P1, the component 00:26:26.840 --> 00:26:31.590 of polarization P2 along the surface out of which x 2 00:26:31.590 --> 00:26:35.610 comes, and the charge per unit area or the 00:26:35.610 --> 00:26:37.420 polarization along x3. 00:26:37.420 --> 00:26:38.880 So this would be P3. 00:26:38.880 --> 00:26:42.560 And this would be the direction of x3. 00:26:42.560 --> 00:26:47.600 Now I deliberately tried to show this sample as a thin 00:26:47.600 --> 00:26:53.900 wafer, which has a much larger surface area here than it does 00:26:53.900 --> 00:26:57.950 on the other two surfaces-- a much smaller area there. 00:26:57.950 --> 00:27:01.350 Therefore, since polarization is charge per unit area, if 00:27:01.350 --> 00:27:06.090 this area normal to x1 is a very large area, there's a lot 00:27:06.090 --> 00:27:07.810 of charge accumulated there. 00:27:07.810 --> 00:27:09.330 It's going to be easy to measure. 00:27:09.330 --> 00:27:14.620 If we make the wafer vanishingly thin, then the 00:27:14.620 --> 00:27:18.340 charge per unit area is high, but the total area is small. 00:27:18.340 --> 00:27:22.500 So there's going to be a negligible accumulation of 00:27:22.500 --> 00:27:25.240 charge on these two side surfaces. 00:27:25.240 --> 00:27:28.150 So the longitudinal piezoelectric effect and the 00:27:28.150 --> 00:27:33.710 longitudinal piezoelectric electric modulus is an effect, 00:27:33.710 --> 00:27:38.400 where we look at the component of polarization, P1, in 00:27:38.400 --> 00:27:42.050 response to an applied stress, sigma 11. 00:27:44.900 --> 00:27:45.910 So it's that simple. 00:27:45.910 --> 00:27:48.070 Look at all the terms we've thrown out. 00:27:48.070 --> 00:27:51.920 We've thrown out a whole bunch of elements of stress, which 00:27:51.920 --> 00:27:55.800 we could impose if we wanted to. 00:27:55.800 --> 00:27:58.690 And we've thrown away two of the three components of 00:27:58.690 --> 00:28:02.510 polarization by designing a specialized sample. 00:28:02.510 --> 00:28:06.580 So all that's left then is that P1 equals sigma 11. 00:28:06.580 --> 00:28:13.430 And the relation between those two parameters is the 111. 00:28:13.430 --> 00:28:17.920 So all this is going to hinge on one single piezoelectric 00:28:17.920 --> 00:28:21.940 electric modulus, d 111, and how that 00:28:21.940 --> 00:28:23.335 changes with direction. 00:28:26.410 --> 00:28:32.830 So this is for one orientation of a plate. 00:28:32.830 --> 00:28:37.110 And I had not specified how the orientation of this plate 00:28:37.110 --> 00:28:39.600 is related to the symmetry axes. 00:28:39.600 --> 00:28:40.850 So let's do that now. 00:28:45.290 --> 00:28:50.270 What I'm going to assume is that this is a crystal. 00:28:50.270 --> 00:28:52.280 It doesn't look like it's hexagonal. 00:28:52.280 --> 00:28:55.130 But imagine that this is a crystal of quartz. 00:28:55.130 --> 00:29:00.270 And we could look at an x1 that's in this direction. 00:29:00.270 --> 00:29:07.690 And imagine that we have cut out of this crystal a wafer 00:29:07.690 --> 00:29:10.070 that has a normal along x1. 00:29:10.070 --> 00:29:13.450 And then relative to this coordinate system, if this is 00:29:13.450 --> 00:29:20.120 x1, the modulus d 111 would tell us what charges 00:29:20.120 --> 00:29:21.725 accumulated on these two surfaces. 00:29:26.300 --> 00:29:29.960 But now, what we could do if we wanted to know how this 00:29:29.960 --> 00:29:35.030 modulus changed with direction would be to cut out a plate, a 00:29:35.030 --> 00:29:37.880 thin plate, in another orientation, 00:29:37.880 --> 00:29:43.300 where this is x1 prime. 00:29:43.300 --> 00:29:51.160 And this has changed relative to the orientation of the cell 00:29:51.160 --> 00:29:52.240 edges in the crystal. 00:29:52.240 --> 00:29:53.220 The crystal is fixed. 00:29:53.220 --> 00:29:56.060 We're just cutting a wafer out in a different orientation. 00:29:56.060 --> 00:29:58.280 So this is x1 prime. 00:29:58.280 --> 00:30:03.360 We're going to, again, squeeze it with a tensile 00:30:03.360 --> 00:30:06.490 stress sigma 11 prime. 00:30:06.490 --> 00:30:12.090 And we'll ask how the polarization P1 prime is 00:30:12.090 --> 00:30:13.820 related to sigma 11 prime. 00:30:13.820 --> 00:30:19.460 And the answer is that P1 prime will be a tensor element 00:30:19.460 --> 00:30:22.440 d 11 prime times sigma 11 prime. 00:30:26.220 --> 00:30:33.460 So what we are asking, essentially, is how does d 11 00:30:33.460 --> 00:30:38.280 transform when we take the direction of x1 in a different 00:30:38.280 --> 00:30:43.225 orientation and, thus, change the value of d 11 prime? 00:30:43.225 --> 00:30:44.560 It's going to change all of the 00:30:44.560 --> 00:30:46.590 piezoelectric matrix elements. 00:30:46.590 --> 00:30:50.800 But we're looking at an effect in a sample that is 00:30:50.800 --> 00:30:56.540 deliberately prepared such that we will measure only the 00:30:56.540 --> 00:30:58.780 surface charge given by P1 prime. 00:30:58.780 --> 00:31:01.780 And, therefore, the way in which the properties of this 00:31:01.780 --> 00:31:06.040 plate change as we vary the way in which we've sliced it 00:31:06.040 --> 00:31:09.570 out of the single crystal is going to be simply the 00:31:09.570 --> 00:31:13.970 variation of d 11 prime with direction. 00:31:13.970 --> 00:31:18.170 So this is the general nature of what we will do when we 00:31:18.170 --> 00:31:26.100 define any of the scalar modulae related to the 00:31:26.100 --> 00:31:27.350 piezoelectric modulus tensor. 00:31:30.600 --> 00:31:37.500 We can change our notation a little bit in that we have a 00:31:37.500 --> 00:31:41.100 modulus which I'll define as a scalar modules d. 00:31:41.100 --> 00:31:45.330 And that d is going to be d 111 prime. 00:31:49.800 --> 00:31:53.620 And I know how to evaluate that. 00:31:53.620 --> 00:32:00.990 d 111 priime will be C 1l, C1m, C 1n, where these are 00:32:00.990 --> 00:32:06.220 direction cosines, times all of the elements in the 00:32:06.220 --> 00:32:12.330 original tensors, dlmn, in the tensor referred to the 00:32:12.330 --> 00:32:13.810 original coordinate system. 00:32:13.810 --> 00:32:15.550 So even though this looks simple-- 00:32:15.550 --> 00:32:17.500 it's just one modulus-- 00:32:17.500 --> 00:32:20.210 when we transform it, we've got a product of three 00:32:20.210 --> 00:32:25.460 direction cosines out in front at every single one of the 27 00:32:25.460 --> 00:32:29.760 tensor elements in the original tensor. 00:32:29.760 --> 00:32:32.350 So it's not as trivial as it seems. 00:32:32.350 --> 00:32:35.750 So this is how this modulus that relates compressive 00:32:35.750 --> 00:32:38.630 stress to induced surface charge will change with 00:32:38.630 --> 00:32:39.620 orientation. 00:32:39.620 --> 00:32:42.930 But what are these direction cosines? 00:32:42.930 --> 00:32:44.850 These are the direction cosines-- 00:32:44.850 --> 00:32:47.650 not the full direction cosine matrix. 00:32:47.650 --> 00:32:52.590 These are the direction cosines for x1 prime. 00:32:52.590 --> 00:32:54.530 OK? 00:32:54.530 --> 00:33:00.970 So we can get rid of this two-subscript notation if it's 00:33:00.970 --> 00:33:13.190 understood that these are the direction cosines of x1 and 00:33:13.190 --> 00:33:22.270 simply call these l l, l m, and l n, just as we did for 00:33:22.270 --> 00:33:24.910 the direction cosines of a vector because we're only 00:33:24.910 --> 00:33:27.990 concerned about the orientation of one of the 00:33:27.990 --> 00:33:30.790 axes, namely x1 prime. 00:33:30.790 --> 00:33:35.260 We don't care diddly-bop about x2 prime or x3 prime because 00:33:35.260 --> 00:33:38.400 these don't enter into the modulus that we have defined. 00:33:38.400 --> 00:33:39.940 And this will be times dlmn. 00:33:44.216 --> 00:33:44.960 OK. 00:33:44.960 --> 00:33:46.210 Is what we're doing clear? 00:33:49.660 --> 00:33:57.810 So we have defined this particular effect in terms of 00:33:57.810 --> 00:34:03.900 those of the 27 piezoelectric modulae which are necessary to 00:34:03.900 --> 00:34:05.350 describe it. 00:34:05.350 --> 00:34:09.370 And then we've established how they will change with a change 00:34:09.370 --> 00:34:12.929 of the direction of one particular direction. 00:34:12.929 --> 00:34:15.119 And we don't care anything about x2 or x3. 00:34:17.690 --> 00:34:18.040 All right. 00:34:18.040 --> 00:34:23.690 Now we go through this process of inserting for matrix 00:34:23.690 --> 00:34:27.100 notation with the equalities built in. 00:34:27.100 --> 00:34:35.679 The proper matrix notation in the first term is d 11 in 00:34:35.679 --> 00:34:37.070 matrix notation. 00:34:37.070 --> 00:34:42.030 That's this term up here in the upper left-hand corner. 00:34:42.030 --> 00:34:49.150 The second term, the term that we've written as minus d 11, 00:34:49.150 --> 00:34:51.139 that's not d 11 at all. 00:34:51.139 --> 00:34:55.560 This is, by definition, d 12. 00:34:55.560 --> 00:34:58.400 And we need the true subscripts, if we're going to 00:34:58.400 --> 00:35:00.140 transform this. 00:35:00.140 --> 00:35:04.775 So this really is not even a matrix because the subscripts 00:35:04.775 --> 00:35:08.170 have lost meaning, and we're just using them to identify 00:35:08.170 --> 00:35:09.950 equalities. 00:35:09.950 --> 00:35:11.340 Then comes a 0. 00:35:11.340 --> 00:35:14.880 And next comes something that we've labeled d 14. 00:35:14.880 --> 00:35:19.880 And the subscripts there are correct. 00:35:19.880 --> 00:35:23.580 That is, indeed, the fourth term in the first row. 00:35:23.580 --> 00:35:26.870 But we're going to want to convert d 14 into a tensor 00:35:26.870 --> 00:35:28.050 element momentarily. 00:35:28.050 --> 00:35:32.010 Now let's get the rest of the terms that are non-zero. 00:35:32.010 --> 00:35:33.325 This is really d 25. 00:35:37.040 --> 00:35:41.930 So the next term that is non-zero is d 25, which just 00:35:41.930 --> 00:35:45.940 happens, because of symmetry, to be equal to minus d 14. 00:35:49.440 --> 00:35:55.360 But this is the true matrix subscripts, and this is the 00:35:55.360 --> 00:35:57.930 true matrix subscript here. 00:35:57.930 --> 00:36:02.430 The next term over to the right is the fifth and final 00:36:02.430 --> 00:36:03.720 non-zero term. 00:36:03.720 --> 00:36:06.535 This is minus 2 d 11. 00:36:09.970 --> 00:36:15.600 And this is really d 26. 00:36:18.920 --> 00:36:22.710 Those are the true matrix elements. 00:36:22.710 --> 00:36:25.960 So we put in the proper matrix subscripts. 00:36:25.960 --> 00:36:30.440 And now the next, final, step in the expansion is to convert 00:36:30.440 --> 00:36:34.710 these terms into actual tensor elements. 00:36:34.710 --> 00:36:38.230 So this is d 111. 00:36:38.230 --> 00:36:40.540 And these are tensor subscripts, so this is 00:36:40.540 --> 00:36:42.770 something we can transform. 00:36:42.770 --> 00:36:44.460 This is d 12. 00:36:44.460 --> 00:36:48.780 In tensor notation this is d 122. 00:36:48.780 --> 00:36:51.764 And that's something that has a law of transformation. 00:36:51.764 --> 00:37:05.060 d 14 is really d 123 plus d 132. 00:37:05.060 --> 00:37:07.360 We lumped two tensor elements together 00:37:07.360 --> 00:37:11.020 to define this modulus. 00:37:11.020 --> 00:37:14.750 Minus d 14 that appears in the next to the last 00:37:14.750 --> 00:37:17.110 non-zero spot is d 25. 00:37:17.110 --> 00:37:27.370 d 25 is really d 231 plus d 213. 00:37:27.370 --> 00:37:40.415 Then, finally, d 26 is d 121 plus d 112. 00:37:40.415 --> 00:37:42.690 AUDIENCE: Shouldn't that be d221? 00:37:42.690 --> 00:37:43.070 PROFESSOR: Sorry. 00:37:43.070 --> 00:37:44.100 d 26, you're right. 00:37:44.100 --> 00:37:51.520 That's down in the second row. d 221 and d 212. 00:37:51.520 --> 00:37:52.950 Now we've got something we can transform. 00:37:56.400 --> 00:37:59.983 The law for transformation is l sub l, l 00:37:59.983 --> 00:38:02.190 sub m, l sub n, dlmn. 00:38:02.190 --> 00:38:11.370 So And these are the direction cosines of x1. 00:38:11.370 --> 00:38:19.570 So this term will transform as l 1, l 1, l1 times d 111 . 00:38:19.570 --> 00:38:35.440 1 The next term will transform as l 1, l 2, l 2 times d 122. 00:38:35.440 --> 00:38:39.020 And that will be d 122 prime for different 00:38:39.020 --> 00:38:40.560 orientation of x1. 00:38:40.560 --> 00:38:42.910 This will be two terms. 00:38:42.910 --> 00:38:45.780 This will be l 123. 00:38:45.780 --> 00:38:47.250 And I can write them in any order. 00:38:47.250 --> 00:38:54.020 So this is l 123 times d 123 plus d 132. 00:38:54.020 --> 00:38:57.720 And these terms prime, when we change axes, are going to be 00:38:57.720 --> 00:39:19.050 equal to l 2, l 1, l 3 times d 231 plus d 213. 00:39:19.050 --> 00:39:26.350 And this last term will be l 1, l 2 squared times d 221 00:39:26.350 --> 00:39:30.690 plus d 212. 00:39:30.690 --> 00:39:31.130 All right. 00:39:31.130 --> 00:39:34.530 So this now is our new tensor element, d 11 prime. 00:39:38.760 --> 00:39:42.480 And that's given by this sum of terms. 00:39:42.480 --> 00:39:49.860 So we'll have a first term l 1 cubed times d 111. 00:39:49.860 --> 00:39:52.800 And if I look through these other terms, that's the only 00:39:52.800 --> 00:39:56.300 term in l 1 cubed that I'll have. 00:39:56.300 --> 00:40:00.260 The next term will involve the product of three cosines-- 00:40:00.260 --> 00:40:05.640 l 1 and l 2 squared times d 122. 00:40:05.640 --> 00:40:10.420 And if I go down here, here's an l 1, l 2 squared again. 00:40:10.420 --> 00:40:20.610 So I have plus d 221 plus d 212. 00:40:20.610 --> 00:40:24.100 And then, finally, the other coefficient that I have is 00:40:24.100 --> 00:40:31.880 plus l 1, l 2, l 3-- 00:40:31.880 --> 00:40:33.730 which is what this should be. 00:40:33.730 --> 00:40:41.450 And that will be times the sum of terms d 123 plus d 132. 00:40:41.450 --> 00:40:43.450 Up here, you've got the same thing again-- 00:40:43.450 --> 00:40:50.170 plus d 231 plus d 213. 00:40:50.170 --> 00:40:56.145 And I have a total of 1, 2, 3, 4, 5-- 00:40:56.145 --> 00:40:58.820 1, 2, 3, 4, 5 terms. 00:41:02.150 --> 00:41:08.180 OK, that is how the longitudinal piezoelectric 00:41:08.180 --> 00:41:12.950 modulus will change as we change the direction of the 00:41:12.950 --> 00:41:16.430 normal to the plate that we have cut out of the crystal. 00:41:16.430 --> 00:41:20.170 So these are direction cosines relative to the 00:41:20.170 --> 00:41:22.960 crystallographic axes. 00:41:22.960 --> 00:41:25.860 l 3 is the angle between the normal to the plate and the 00:41:25.860 --> 00:41:27.240 threefold axis. 00:41:27.240 --> 00:41:31.200 l 1 is the angle cosine to the angle between the normal to 00:41:31.200 --> 00:41:34.270 the plate and one of the twofold axes. 00:41:34.270 --> 00:41:38.680 And l 2 is the direction cosine for the normal to the 00:41:38.680 --> 00:41:41.930 threefold axis and the twofold axis. 00:41:41.930 --> 00:41:42.240 OK. 00:41:42.240 --> 00:41:45.000 So we've got it now in terms of tensor elements. 00:41:45.000 --> 00:41:45.800 And now-- 00:41:45.800 --> 00:41:46.514 yeah? 00:41:46.514 --> 00:41:48.737 AUDIENCE: Is it at all reasonable to assume instead 00:41:48.737 --> 00:41:53.430 of taking those sums in d 123, d 122, just saying 2 d 123? 00:41:53.430 --> 00:41:57.244 Is that OK in assuming? 00:41:57.244 --> 00:41:58.048 Or not necessarily? 00:41:58.048 --> 00:42:00.500 PROFESSOR: Well, we could do that. 00:42:00.500 --> 00:42:07.430 But I did it the long way to not obscure what we're doing. 00:42:07.430 --> 00:42:08.620 OK? 00:42:08.620 --> 00:42:11.830 This is a well-defined summation over subscripts. 00:42:11.830 --> 00:42:16.140 And we're going to collapse immediately down to the sums. 00:42:16.140 --> 00:42:18.420 And we're going to replace the equalities. 00:42:18.420 --> 00:42:22.110 So let's see what comes out of this, if we now, having 00:42:22.110 --> 00:42:25.650 reached the zenith, having transformed the tensor 00:42:25.650 --> 00:42:33.100 elements, go down and replace this with a consolidation of 00:42:33.100 --> 00:42:35.580 terms and an insertion of the equalities 00:42:35.580 --> 00:42:36.830 between the matrix elements. 00:42:39.810 --> 00:42:47.820 OK The first term is d 11. 00:42:47.820 --> 00:42:53.620 So I will have l 1 cubed times d 11. 00:42:53.620 --> 00:42:57.060 Notice I'm getting third powers of direction cosines, 00:42:57.060 --> 00:43:00.530 which is going to be what causes the exotic nature of 00:43:00.530 --> 00:43:02.990 these anisotropies. 00:43:02.990 --> 00:43:08.070 And then I have a product of l 1 and l 2 squared. 00:43:10.700 --> 00:43:15.040 And this is d 12. 00:43:20.600 --> 00:43:25.800 And this second term is-- 00:43:25.800 --> 00:43:26.760 where did it go? 00:43:26.760 --> 00:43:31.440 This is d 21 plus d 212. 00:43:31.440 --> 00:43:35.140 And that is what we call d 26. 00:43:42.550 --> 00:43:46.390 And then, finally, this product of three different 00:43:46.390 --> 00:43:48.332 direction cosines-- 00:43:48.332 --> 00:43:52.720 l 1, l 2, l 3. 00:43:52.720 --> 00:43:58.360 And we have d 231 plus d 213. 00:43:58.360 --> 00:43:59.680 And this is d 25. 00:44:05.740 --> 00:44:08.818 And, again, an l 1, l 2, l 3 times d 14. 00:44:13.780 --> 00:44:22.990 And the second term here is d 25, if I've done it correctly. 00:44:22.990 --> 00:44:24.240 d 25 -- 00:44:25.940 --> 00:44:29.080 this is d 24. 00:44:29.080 --> 00:44:33.870 And this one is d 24. 00:44:33.870 --> 00:44:40.800 2/4 OK. 00:44:40.800 --> 00:44:43.830 Let's now insert the equalities-- 00:44:43.830 --> 00:44:45.840 back to where we came from. 00:44:45.840 --> 00:44:48.202 d 11 is d 11. 00:44:48.202 --> 00:44:51.820 d 12, however, for symmetry 32-- 00:44:58.551 --> 00:45:02.700 I'm going to my handy-dandy chart of symmetry 00:45:02.700 --> 00:45:03.950 restrictions. 00:45:09.000 --> 00:45:09.850 I don't want to do that. 00:45:09.850 --> 00:45:12.037 That's fourth rank. 00:45:12.037 --> 00:45:13.170 AUDIENCE: It's still on the board. 00:45:13.170 --> 00:45:16.090 PROFESSOR: It's still on the board? 00:45:16.090 --> 00:45:16.880 Yes. 00:45:16.880 --> 00:45:17.710 Thank you. 00:45:17.710 --> 00:45:21.276 When your nose is in it, it's hard to see. 00:45:21.276 --> 00:45:29.490 d 12 is minus d 11. 00:45:29.490 --> 00:45:33.750 1 And d 26 is minus 2 d 11. 00:45:38.890 --> 00:45:41.490 So these two terms can be consolidated. 00:45:41.490 --> 00:45:53.380 l 1 cubed plus l 1, l 2 squared times d 11 minus d 11. 00:45:53.380 --> 00:45:55.310 So these two terms die. 00:45:55.310 --> 00:46:00.670 And I have a minus 2 d 11 that's left. 00:46:00.670 --> 00:46:06.220 If I insert the equalities here, I'll have l 1, l 2, l 3. 00:46:06.220 --> 00:46:08.840 d 25 is minus d 14. 00:46:12.790 --> 00:46:14.970 And here's a d 14 itself. 00:46:14.970 --> 00:46:17.550 So these two terms die. 00:46:17.550 --> 00:46:18.800 And then I had d 25. 00:46:21.290 --> 00:46:22.980 And that is-- 00:46:22.980 --> 00:46:24.360 AUDIENCE: You don't have d25. 00:46:24.360 --> 00:46:25.120 PROFESSOR: I don't have d 25. 00:46:25.120 --> 00:46:26.903 Where did I get the extra one? 00:46:26.903 --> 00:46:28.153 AUDIENCE: [INAUDIBLE]. 00:46:30.607 --> 00:46:31.845 PROFESSOR: OK. 00:46:31.845 --> 00:46:34.980 I'll take your word for it. 00:46:34.980 --> 00:46:38.320 And I know how it has to turn out. 00:46:40.880 --> 00:46:41.220 OK. 00:46:41.220 --> 00:46:43.340 So these two terms kill each other. 00:46:43.340 --> 00:46:48.080 And I'm left with, then, an l 1, l 2, l 3. 00:46:51.330 --> 00:46:54.190 Or have I left something out? 00:46:54.190 --> 00:46:55.440 This is d 1-- 00:47:02.370 --> 00:47:08.870 this is d 15 and d 231 and d 23 -- 00:47:11.710 --> 00:47:16.470 uh, this is d 2 -- 00:47:19.380 --> 00:47:28.870 and the combination of 13 and 31 is d 25. 00:47:28.870 --> 00:47:30.730 Right? 00:47:30.730 --> 00:47:34.090 And if I look at my equalities, this is l 1, l 2, 00:47:34.090 --> 00:47:41.480 l 3 minus d 14 plus d 14 plus d 25. 00:47:41.480 --> 00:47:45.055 And d 25 is minus d 14. 00:47:45.055 --> 00:47:47.970 AUDIENCE: Why would you add this d25? 00:47:47.970 --> 00:47:49.210 PROFESSOR: Let me check my notes and see 00:47:49.210 --> 00:47:50.370 what I've got here. 00:47:50.370 --> 00:47:51.620 AUDIENCE: [INAUDIBLE]. 00:48:02.695 --> 00:48:03.681 PROFESSOR: OK. 00:48:03.681 --> 00:48:04.931 See what I -- 00:48:08.140 --> 00:48:11.740 d 25 is minus d 14. 00:48:11.740 --> 00:48:13.750 And then I have just a d 14. 00:48:13.750 --> 00:48:17.010 I don't know -- 00:48:17.010 --> 00:48:17.660 I see what I did. 00:48:17.660 --> 00:48:18.780 I put it in the wrong slot. 00:48:18.780 --> 00:48:20.880 This is d 14. 00:48:20.880 --> 00:48:25.150 And d 25 is the one that's minus d 14. 00:48:25.150 --> 00:48:28.870 so this term dies, which is nice because that 00:48:28.870 --> 00:48:30.500 cross term is messy. 00:48:30.500 --> 00:48:34.000 So what I'm left with, then, if I check against my notes, 00:48:34.000 --> 00:48:45.400 is d l 1 cubed plus l 1, l 2 squared times d 11. 00:48:45.400 --> 00:48:48.710 And then I have minus d 1 minus-- 00:48:48.710 --> 00:48:50.390 this doesn't belong in here. 00:48:57.630 --> 00:49:08.840 This wants to end up being l 1 cubed minus 3 l 1, l 2 squared 00:49:08.840 --> 00:49:10.290 times d 11. 00:49:16.720 --> 00:49:18.310 The first term is l 1 cubed. 00:49:18.310 --> 00:49:18.960 That's correct. 00:49:18.960 --> 00:49:21.340 The second term is l 1, l 2 squared. 00:49:21.340 --> 00:49:25.260 We've got a minus d 1 plus minus 2 d 11. 00:49:33.580 --> 00:49:38.960 So I have minus 3. 00:49:38.960 --> 00:49:40.000 It should be a minus. 00:49:40.000 --> 00:49:40.855 Yeah, that carries down to a minus. 00:49:40.855 --> 00:49:44.224 So I have minus 3 l 1, l 2 squared all time d 11. 00:49:47.170 --> 00:49:47.480 OK. 00:49:47.480 --> 00:49:52.630 So what we have ended up with is an expression for the 00:49:52.630 --> 00:49:56.700 longitudinal piezoelectric modulus as a function of 00:49:56.700 --> 00:49:58.110 orientation. 00:49:58.110 --> 00:50:02.960 The surprising thing is that l 3 does not appear here at all. 00:50:02.960 --> 00:50:07.900 It doesn't depend on the angle between the normal to the 00:50:07.900 --> 00:50:12.210 plate and the threefold axis. 00:50:12.210 --> 00:50:15.510 It depends only on one modulus, and that is a 00:50:15.510 --> 00:50:17.970 remarkable thing. 00:50:17.970 --> 00:50:23.540 This says that the shape of this surface is independent, 00:50:23.540 --> 00:50:30.810 essentially, of the property, any property that relates the 00:50:30.810 --> 00:50:40.360 one one prime to a uni-axial stimulus, sigma 11. 00:50:40.360 --> 00:50:44.240 And you measure a vectory component in the same 00:50:44.240 --> 00:50:48.310 direction is always going to have this universal surface. 00:50:48.310 --> 00:50:52.770 And it involves just a geometric term and then one 00:50:52.770 --> 00:50:57.600 modulus that changes the magnitude of the longitudinal 00:50:57.600 --> 00:51:01.500 piezoelectric modulus but does not change the asymmetry. 00:51:01.500 --> 00:51:04.550 So let's see what this function looks like as a 00:51:04.550 --> 00:51:05.480 function of direction. 00:51:05.480 --> 00:51:08.000 Maybe we better wait for that until we come back because 00:51:08.000 --> 00:51:09.850 that's going to take a few minutes. 00:51:09.850 --> 00:51:12.040 So this is what we found. 00:51:12.040 --> 00:51:13.310 That is correct. 00:51:13.310 --> 00:51:16.780 And we have to now decide what this looks like, which will 00:51:16.780 --> 00:51:17.750 take a few more minutes. 00:51:17.750 --> 00:51:20.330 But let's stop here rather than run late. 00:51:23.160 --> 00:51:23.600 All right. 00:51:23.600 --> 00:51:26.120 Let's take our 10-minute break as usual.