WEBVTT 00:00:00.260 --> 00:00:03.720 The following content is provided by MIT OpenCourseWare 00:00:03.720 --> 00:00:06.120 under a Creative Commons license. 00:00:06.120 --> 00:00:08.740 Additional information about our license, and MIT 00:00:08.740 --> 00:00:13.590 OpenCourseWare in general, is available at ocw.mit.edu. 00:00:16.379 --> 00:00:16.830 PROFESSOR: OK. 00:00:16.830 --> 00:00:18.910 Let's resume. 00:00:18.910 --> 00:00:22.600 I cut things off at a time when we had the final answer. 00:00:22.600 --> 00:00:25.180 And I left you hanging because we don't know what the final 00:00:25.180 --> 00:00:27.050 answer is telling us. 00:00:27.050 --> 00:00:31.920 This says that as we change the orientation of the normal 00:00:31.920 --> 00:00:39.710 to the plate, relative to x3 and, secondarily, as we change 00:00:39.710 --> 00:00:46.340 the angle between the threefold axis and x2, we get 00:00:46.340 --> 00:00:52.270 this strange third-rank trigonometric function. 00:00:52.270 --> 00:00:58.020 Let's convert these cosines into the appropriate angles. 00:00:58.020 --> 00:01:00.180 This was the twofold axis. 00:01:00.180 --> 00:01:02.320 And this was x1. 00:01:02.320 --> 00:01:04.610 This was the direction of the threefold axis. 00:01:04.610 --> 00:01:06.290 This was x3. 00:01:06.290 --> 00:01:10.290 And this is the direction x2. 00:01:10.290 --> 00:01:13.630 And that comes out in between a pair of twofold axes. 00:01:13.630 --> 00:01:19.380 And we may want to look at this from up above, relative 00:01:19.380 --> 00:01:21.450 to these twofold axes that occur. 00:01:26.680 --> 00:01:31.270 This thing here looks like a trigonometric identity, 00:01:31.270 --> 00:01:33.840 doesn't it? 00:01:33.840 --> 00:01:39.420 Let's let this angle here be theta. 00:01:39.420 --> 00:01:44.600 And l1, then, is cosine of theta. 00:01:44.600 --> 00:01:53.790 And so our geometry is that this is l1 cosine of theta. 00:01:53.790 --> 00:01:59.730 This is l2, which is the cosine of the angle between 00:01:59.730 --> 00:02:01.640 our direction and x2. 00:02:01.640 --> 00:02:10.070 And that is the cosine of pi over 2, minus theta. 00:02:12.780 --> 00:02:16.500 And that, then, is equal to sine of theta. 00:02:20.540 --> 00:02:22.790 So this identity, as you all know-- 00:02:22.790 --> 00:02:25.210 I'm not telling you anything that you don't already know-- 00:02:25.210 --> 00:02:28.770 this is d is equal to d1,1 1, times the 00:02:28.770 --> 00:02:31.180 cosine of 3 theta, right? 00:02:31.180 --> 00:02:33.380 You knew that. 00:02:33.380 --> 00:02:34.220 Believe it or not, it is. 00:02:34.220 --> 00:02:37.090 It's one of these obscure trigonometric identities, 00:02:37.090 --> 00:02:40.400 which wouldn't occur to you in a million years unless you go 00:02:40.400 --> 00:02:42.110 digging through some handbooks. 00:02:42.110 --> 00:02:45.460 So this is rather astonishingly simple. 00:02:45.460 --> 00:02:47.420 It's simply a cosine function. 00:02:47.420 --> 00:02:51.330 But the interesting thing is it goes as a 00:02:51.330 --> 00:02:55.040 function of 3 theta. 00:02:55.040 --> 00:02:59.640 So this goes through one cycle between a pair of adjacent 00:02:59.640 --> 00:03:00.660 twofold axes. 00:03:00.660 --> 00:03:05.260 So it starts out as d1,1. 00:03:05.260 --> 00:03:08.090 And it finishes up at d1,1. 00:03:08.090 --> 00:03:12.420 And in between it is minus d1,1. 00:03:12.420 --> 00:03:15.150 And so it goes down. 00:03:15.150 --> 00:03:18.220 And now we come to something that I do differently than 00:03:18.220 --> 00:03:19.200 most people do. 00:03:19.200 --> 00:03:26.710 Most people will say it goes as minus d1,1. 00:03:26.710 --> 00:03:34.760 But minus d1,1 I like to show as a lobe going off in this 00:03:34.760 --> 00:03:36.705 direction, with a minus sign. 00:03:39.968 --> 00:03:42.896 AUDIENCE: How do you get that cosine of 3 theta? 00:03:42.896 --> 00:03:43.384 PROFESSOR: Hmm? 00:03:43.384 --> 00:03:46.800 AUDIENCE: How do you get d equals d1 cosine 3 theta? 00:03:46.800 --> 00:03:47.776 PROFESSOR: How do I get that? 00:03:47.776 --> 00:03:48.264 AUDIENCE: Yeah. 00:03:48.264 --> 00:03:51.192 PROFESSOR: That is just a trigonometric identify, 00:03:51.192 --> 00:03:55.430 believe it or not, a well-known 00:03:55.430 --> 00:03:57.970 trigonometric identify. 00:03:57.970 --> 00:04:00.200 Actually, it's an exceedingly obscure 00:04:00.200 --> 00:04:02.590 trigonometric identity. 00:04:02.590 --> 00:04:05.350 So the way this is going to go from x1 is it's going to be a 00:04:05.350 --> 00:04:07.830 positive lobe around x1. 00:04:07.830 --> 00:04:13.060 There's going to be a negative lobe along x2 and then a 00:04:13.060 --> 00:04:19.130 positive lobe again about the twofold axis that's 120 00:04:19.130 --> 00:04:20.300 degrees away. 00:04:20.300 --> 00:04:24.770 And then a negative lobe, and then a positive lobe opposite 00:04:24.770 --> 00:04:28.550 this negative lobe, and a negative lobe opposite this 00:04:28.550 --> 00:04:29.630 positive lobe. 00:04:29.630 --> 00:04:32.950 So it's a six-membered-- 00:04:32.950 --> 00:04:35.000 six lobes. 00:04:35.000 --> 00:04:42.120 There is always a positive lobe opposite a negative lobe. 00:04:42.120 --> 00:04:47.820 And what this means is that the charge is of opposite sign 00:04:47.820 --> 00:04:51.620 on opposite ends of the twofold axes. 00:04:51.620 --> 00:04:54.980 So the response peaks up on twofold axes. 00:04:54.980 --> 00:04:58.700 Now the thing that I don't like is that what Nye does is 00:04:58.700 --> 00:05:02.540 to say, OK, this is a negative value. 00:05:02.540 --> 00:05:06.410 So you should plot the radius in a negative direction. 00:05:06.410 --> 00:05:11.110 And that puts it over here, right on top of 00:05:11.110 --> 00:05:12.560 this positive lobe. 00:05:12.560 --> 00:05:18.890 So what Nye shows is the polar plot of this result, in the 00:05:18.890 --> 00:05:23.680 plane of the twofold axes, is simply this. 00:05:23.680 --> 00:05:25.550 This is x1. 00:05:25.550 --> 00:05:27.860 And he shows a lobe here. 00:05:27.860 --> 00:05:29.580 And he shows a lobe here. 00:05:29.580 --> 00:05:32.950 And he shows a lobe here. 00:05:32.950 --> 00:05:35.530 And if you interpret that as a polar plot you say, well, I 00:05:35.530 --> 00:05:37.210 know what the value is in here. 00:05:37.210 --> 00:05:39.320 It's decreasing. 00:05:39.320 --> 00:05:41.550 And there's nothing going on in here. 00:05:41.550 --> 00:05:43.220 So the response must be 0. 00:05:43.220 --> 00:05:45.180 And then it starts coming up again. 00:05:45.180 --> 00:05:47.860 And there's a response in different directions here. 00:05:47.860 --> 00:05:49.770 And then it goes back down to 0. 00:05:49.770 --> 00:05:52.115 And in this range, there's nothing going on. 00:05:52.115 --> 00:05:53.500 And that's not true. 00:05:53.500 --> 00:06:00.090 What's going on is a negative value of the modulus. 00:06:00.090 --> 00:06:03.860 And to me, that becomes abundantly clear if you just 00:06:03.860 --> 00:06:07.590 put a sign that labels the sign of the modulus within 00:06:07.590 --> 00:06:08.840 those lobes. 00:06:12.310 --> 00:06:17.070 Now I submit, that's pretty anisotropic, isn't it? 00:06:17.070 --> 00:06:17.620 Yes, Steve? 00:06:17.620 --> 00:06:20.020 AUDIENCE: Did you just arbitrarily choose where your 00:06:20.020 --> 00:06:23.586 positive and negatives go, is it [INAUDIBLE] 00:06:23.586 --> 00:06:24.062 out here? 00:06:24.062 --> 00:06:24.540 PROFESSOR: No. 00:06:24.540 --> 00:06:25.530 This comes out here. 00:06:25.530 --> 00:06:30.340 When l1 is 0, that's-- 00:06:30.340 --> 00:06:31.590 excuse me. 00:06:33.950 --> 00:06:37.830 When the angle is 0, l1 is plus 1. 00:06:41.420 --> 00:06:44.340 Remember that l1 is the cosine of this angle. 00:06:44.340 --> 00:06:47.480 When that angle is 0, then the value is plus 1. 00:06:50.530 --> 00:06:54.500 So I was careful to put the label x1 on this. 00:06:54.500 --> 00:06:57.850 Now the other thing that we should examine is how these 00:06:57.850 --> 00:07:02.840 lobes vary in a direction perpendicular to x, in a 00:07:02.840 --> 00:07:04.830 direction that includes x3. 00:07:04.830 --> 00:07:13.790 So let us look at how the function varies in a direction 00:07:13.790 --> 00:07:15.105 perpendicular to-- 00:07:18.440 --> 00:07:20.930 that includes x1 and x3. 00:07:20.930 --> 00:07:22.180 OK. 00:07:25.280 --> 00:07:28.190 Call this angle phi. 00:07:28.190 --> 00:07:40.070 And in the x1, x3 plane, d1,1,1 prime turns out to be 00:07:40.070 --> 00:07:44.800 l1 cubed d1,1. 00:07:44.800 --> 00:07:48.960 Before you go to our general expression. 00:07:48.960 --> 00:07:54.020 And l2 is cosine of 90. 00:07:54.020 --> 00:07:55.160 This thing drops out. 00:07:55.160 --> 00:07:57.830 We're left with simply d1,1 prime equals 00:07:57.830 --> 00:08:00.300 l cubed times d1,1. 00:08:00.300 --> 00:08:08.090 So this then goes as cosine cubed of phi, which is a lobe 00:08:08.090 --> 00:08:10.580 that starts out at plus 1. 00:08:10.580 --> 00:08:16.740 And then because it's cosine cubed, this dies out very, 00:08:16.740 --> 00:08:18.920 very rapidly. 00:08:18.920 --> 00:08:25.090 So these lobes are very flat in the x1, x2 plane, die out 00:08:25.090 --> 00:08:29.610 very rapidly as the cube of phi, where phi is the angle 00:08:29.610 --> 00:08:31.380 between the twofold axis and x3. 00:08:35.630 --> 00:08:39.000 So the interesting thing about this surface is that if you 00:08:39.000 --> 00:08:42.610 decided to pick up the random fragment of crystal and 00:08:42.610 --> 00:08:46.050 determine what its Piezoelectric Modulus is. 00:08:49.110 --> 00:08:52.540 One way of doing this, a poor man's test for 00:08:52.540 --> 00:08:58.330 piezoelectricity is to just clamp a fragment of crystal 00:08:58.330 --> 00:09:04.570 between two electrodes and then hook this up to a 00:09:04.570 --> 00:09:08.700 variable frequency generator that sweeps through a range of 00:09:08.700 --> 00:09:11.920 frequencies, changes the frequency of a voltage across 00:09:11.920 --> 00:09:14.730 these plates, and then comes back and sweeps again. 00:09:14.730 --> 00:09:17.160 Or alternatively, have a knob one on that 00:09:17.160 --> 00:09:18.780 lets you change frequency. 00:09:18.780 --> 00:09:24.290 Then put a pair of earphones across the crystal. 00:09:24.290 --> 00:09:28.640 And if we would do that, by having this variable frequency 00:09:28.640 --> 00:09:33.960 generator, as we turned it, when we hit a resonant 00:09:33.960 --> 00:09:38.800 frequency that set up either a full wavelength of a 00:09:38.800 --> 00:09:40.710 vibrational wave in the crystal-- 00:09:40.710 --> 00:09:44.690 or half wavelength, or one wavelength, or 3/2 00:09:44.690 --> 00:09:45.680 wavelength-- 00:09:45.680 --> 00:09:47.060 there'd be a resonance. 00:09:47.060 --> 00:09:49.430 And the capacitance does something crazy. 00:09:49.430 --> 00:09:51.560 It does like this as you go through 00:09:51.560 --> 00:09:53.390 the resonant frequency. 00:09:53.390 --> 00:09:54.860 And then your simple detector-- 00:09:54.860 --> 00:09:55.820 a pair of earphones-- 00:09:55.820 --> 00:09:58.870 as you tuned the frequency, you would hear static as you 00:09:58.870 --> 00:10:03.190 went through this discontinuity in capacitance. 00:10:03.190 --> 00:10:07.750 Or alternatively, you could put this on a cathode ray tube 00:10:07.750 --> 00:10:13.230 and just have a sweep frequency and put the voltage 00:10:13.230 --> 00:10:18.530 across the electrodes on the oscilloscope screen. 00:10:18.530 --> 00:10:20.660 And you would see something like this, and then this 00:10:20.660 --> 00:10:24.560 discontinuity, and then maybe second harmonic. 00:10:24.560 --> 00:10:25.640 And it would work just fine. 00:10:25.640 --> 00:10:28.520 This is a poor man's way of detecting piezoelectricity. 00:10:28.520 --> 00:10:30.440 But what if-- 00:10:30.440 --> 00:10:32.260 what if-- 00:10:32.260 --> 00:10:35.040 you happen to have your piece of quartz with the electrodes 00:10:35.040 --> 00:10:37.470 directly on the c-axis? 00:10:37.470 --> 00:10:41.080 The modulus would be 0, and you wouldn't detect anything. 00:10:41.080 --> 00:10:44.830 Or if you put on the probes such that they were in a 00:10:44.830 --> 00:10:49.070 direction that was in between these lobes where, again, the 00:10:49.070 --> 00:10:51.580 Piezoelectric Modulus has gone to 0. 00:10:51.580 --> 00:10:53.390 You wouldn't find anything. 00:10:53.390 --> 00:10:56.790 So measuring the Piezoelectric Modulus for a random chunk of 00:10:56.790 --> 00:11:00.160 material is dangerous, if you just look at one direction and 00:11:00.160 --> 00:11:03.780 say nothing's going on; it's not piezoelectric. 00:11:03.780 --> 00:11:05.940 The other thing is the material could have a 00:11:05.940 --> 00:11:07.910 piezoelectric response that's so weak you just 00:11:07.910 --> 00:11:10.430 can't measure it. 00:11:10.430 --> 00:11:11.260 Yeah, OK? 00:11:11.260 --> 00:11:12.670 AUDIENCE: This stuff's only for single crystals, right? 00:11:12.670 --> 00:11:14.315 Because if you [INAUDIBLE] polycrystal material, you're 00:11:14.315 --> 00:11:15.490 getting [INAUDIBLE] averaging. 00:11:15.490 --> 00:11:17.380 PROFESSOR: You're averaging over all directions. 00:11:17.380 --> 00:11:18.910 So this is for a single crystal. 00:11:18.910 --> 00:11:23.770 And that's the only time you get these exotic, very 00:11:23.770 --> 00:11:25.020 anisotropic surfaces. 00:11:31.360 --> 00:11:34.190 Another method that I've seen in a rather old 00:11:34.190 --> 00:11:37.320 book is really nice. 00:11:37.320 --> 00:11:42.840 What you do is you cut your crystal into a little plate. 00:11:42.840 --> 00:11:50.110 And then you drive the plate by means of putting electrodes 00:11:50.110 --> 00:11:52.500 on it and hook this up to a variable frequency. 00:11:52.500 --> 00:11:56.540 And if the material is transparent, you will set up-- 00:11:56.540 --> 00:12:00.740 depending on the velocity of sound in the material-- 00:12:00.740 --> 00:12:05.470 you will set up standing waves when you 00:12:05.470 --> 00:12:07.220 hit the right frequency. 00:12:07.220 --> 00:12:12.110 And these lines that I've drawn are places where the 00:12:12.110 --> 00:12:14.420 displacement is always 0. 00:12:14.420 --> 00:12:17.750 And these things that I've indicated at little squares is 00:12:17.750 --> 00:12:20.780 a region where the displacement goes up and down 00:12:20.780 --> 00:12:23.630 between its maximum and minimum extreme. 00:12:23.630 --> 00:12:27.460 Now if this crystal is transparent and you shine a 00:12:27.460 --> 00:12:35.230 light through it, these little regions bounded by lines of 00:12:35.230 --> 00:12:37.470 zero displacement act like little lenses. 00:12:37.470 --> 00:12:43.870 So if you pass a beam of light through it, you get a bunch of 00:12:43.870 --> 00:12:47.950 right maxima, little focused spots on the sheet. 00:12:47.950 --> 00:12:51.120 And you can determine for a particular frequency if this 00:12:51.120 --> 00:12:54.745 is the 1 and 1/2 wavelengths along a dimension that you 00:12:54.745 --> 00:13:01.560 know, you can again find the Piezoelectric Moduli. 00:13:01.560 --> 00:13:06.010 So there are lots of ways of detecting this effect. 00:13:06.010 --> 00:13:08.050 But when you have a single crystal, you've got to really 00:13:08.050 --> 00:13:09.940 look at this as a function of direction, to 00:13:09.940 --> 00:13:11.190 be absolutely sure. 00:13:15.160 --> 00:13:24.530 One other Piezoelectric Modulus is in the problem set. 00:13:24.530 --> 00:13:29.590 4 bar 3m, this is the structure of sphalerite and a 00:13:29.590 --> 00:13:34.800 lot of the compound semiconductors. 00:13:34.800 --> 00:13:36.000 This is not in the notes. 00:13:36.000 --> 00:13:38.620 But if you try the problem, what you find is, again, a 00:13:38.620 --> 00:13:42.210 highly, highly anisotropic behavior. 00:13:42.210 --> 00:13:47.340 And this is the direction of the unit cell of the crystal. 00:13:47.340 --> 00:13:50.290 You find that the piezoelectric response is a 00:13:50.290 --> 00:13:55.070 series of very sharp lobes, positive and negative, going 00:13:55.070 --> 00:13:58.260 along the directions of the body diagonal, namely the 1, 00:13:58.260 --> 00:14:01.410 1, 1 directions in the crystal. 00:14:01.410 --> 00:14:03.420 And then there's another lobe that goes like 00:14:03.420 --> 00:14:06.040 this, negative, positive. 00:14:06.040 --> 00:14:09.850 Another lobe that goes up here, positive and negative. 00:14:09.850 --> 00:14:13.630 So very sharp lobes along the body diagonal, alternately 00:14:13.630 --> 00:14:18.500 positive, negative, positive, negative as you go around 0, 00:14:18.500 --> 00:14:19.750 0, 1 plane. 00:14:25.030 --> 00:14:25.330 OK. 00:14:25.330 --> 00:14:30.890 That's what the longitudinal piezoelectric effect is like. 00:14:30.890 --> 00:14:36.900 You can invent single-crystal devices for weighing fish. 00:14:36.900 --> 00:14:38.720 And then you have an interesting question. 00:14:38.720 --> 00:14:42.930 If you hang the weight on the crystal, and it's a flat 00:14:42.930 --> 00:14:48.010 plate, how would you orient the crystal to get the optimum 00:14:48.010 --> 00:14:49.990 sensitivity? 00:14:49.990 --> 00:14:53.460 So you define a scalar modulus for the crystal in the 00:14:53.460 --> 00:14:56.810 particular direction you're exerting a uniaxial stress. 00:14:56.810 --> 00:15:01.480 And then you express this module in terms of theta, the 00:15:01.480 --> 00:15:07.020 angle of rotation, within the crystographic plane. 00:15:07.020 --> 00:15:10.290 In the case of the sup cell if you looked at that geometry, 00:15:10.290 --> 00:15:14.660 that is a flat plate subject to compression, where you 00:15:14.660 --> 00:15:20.920 measure the surface charge perpendicular to the direction 00:15:20.920 --> 00:15:23.230 of the uniaxial stress, not parallel to it 00:15:23.230 --> 00:15:24.540 as we've done here. 00:15:24.540 --> 00:15:26.850 And then there's a question if you rotate the plane of the 00:15:26.850 --> 00:15:31.240 crystal about the imposed stress, what orientation gives 00:15:31.240 --> 00:15:33.890 you optimum, maximum response? 00:15:33.890 --> 00:15:35.970 So there are fun things that you could do with that. 00:15:40.100 --> 00:15:42.610 We have a quiz next time. 00:15:42.610 --> 00:15:48.000 There's a lame-duck period after the quiz will be over. 00:15:48.000 --> 00:15:55.550 And I want to say a little bit about the tensor aspects of 00:15:55.550 --> 00:15:58.440 elastic moduli. 00:15:58.440 --> 00:16:03.640 Mechanical behavior is something that you have or 00:16:03.640 --> 00:16:09.020 will cover in great detail in a graduate-level class on 00:16:09.020 --> 00:16:12.770 mechanics, but I think very few people who deal with 00:16:12.770 --> 00:16:16.960 things other than cubic materials or with 00:16:16.960 --> 00:16:20.850 single-crystal materials where the elastic constants are 00:16:20.850 --> 00:16:25.210 functions of crystal symmetry and direction. 00:16:25.210 --> 00:16:29.670 So let me, at least at the time available, set up a 00:16:29.670 --> 00:16:37.270 definition of stress in terms of strain. 00:16:37.270 --> 00:16:39.980 Stress is a second-rank tensor. 00:16:39.980 --> 00:16:46.080 There are six unique components. 00:16:46.080 --> 00:16:50.230 So we could write an expression for sigma 1,1; 00:16:50.230 --> 00:16:58.780 sigma 2,2; sigma 3,3; sigma 4,4; and sigma 5,5. 00:16:58.780 --> 00:17:04.092 And sigma-- 00:17:04.092 --> 00:17:05.750 AUDIENCE: --1,2. 00:17:05.750 --> 00:17:13.300 PROFESSOR: No, I want to go down like this. 00:17:13.300 --> 00:17:23.940 So this would be sigma 2,3; sigma 1,3; and sigma 1,2; 00:17:23.940 --> 00:17:28.160 eventually to be known as sigma 1, sigma 2, sigma 3, 00:17:28.160 --> 00:17:33.420 sigma 4, sigma 5, and sigma six. 00:17:33.420 --> 00:17:35.430 But actually, if we're writing out a full tensor relation, we 00:17:35.430 --> 00:17:37.450 should write down all six elements here. 00:17:37.450 --> 00:17:44.080 So let me do that and put in a 3,2. 00:17:44.080 --> 00:17:46.820 Put in all nine of them, because before we condense to 00:17:46.820 --> 00:17:50.080 matrix form, that is what we're going to have. 00:17:50.080 --> 00:17:52.910 So we'll have a sigma 2,3; we'll have a sigma 3,2; we'll 00:17:52.910 --> 00:18:00.650 have a sigma 1,3; a sigma 3,1; a sigma 1,2; and a sigma 2,1. 00:18:00.650 --> 00:18:04.460 And we can express each of the stresses in terms of the 00:18:04.460 --> 00:18:05.720 elements of strain. 00:18:05.720 --> 00:18:10.110 So I'll have an epsilon 1,1; an epsilon 1,2; an epsilon 00:18:10.110 --> 00:18:18.132 2,2; an epsilon 3,3; an epsilon 2,3; an epsilon 3,2; 00:18:18.132 --> 00:18:26.190 an epsilon 1,3; an epsilon 3,1; an epsilon 1,2; and an 00:18:26.190 --> 00:18:27.740 epsilon 2,1. 00:18:27.740 --> 00:18:29.290 Nine terms, nine terms. 00:18:29.290 --> 00:18:33.810 And in between is going to be a tensor 00:18:33.810 --> 00:18:38.550 consisting of 9 by 9 elements. 00:18:38.550 --> 00:18:41.030 There will be 81 different coefficients in here. 00:18:43.720 --> 00:18:46.900 If we want to derive symmetry constraints, we're going to 00:18:46.900 --> 00:18:50.210 have to, for every one of those 81 elements, do a 00:18:50.210 --> 00:18:51.700 transformation. 00:18:51.700 --> 00:18:56.050 And each transformed element is going to be a sum of four 00:18:56.050 --> 00:19:00.540 direction cosines times one tensor element, repeated 81 00:19:00.540 --> 00:19:02.690 different times. 00:19:02.690 --> 00:19:05.990 Not something to be undertaken on a short afternoon. 00:19:08.750 --> 00:19:09.090 OK. 00:19:09.090 --> 00:19:13.640 What are the tensor elements in here? 00:19:13.640 --> 00:19:18.170 They are represented by the symbol c. 00:19:18.170 --> 00:19:21.840 This would be c1,1,1,1. 00:19:21.840 --> 00:19:30.880 This would be c1,1,2,2 times epsilon 2,2; a c1,1,3,3; a 00:19:30.880 --> 00:19:33.830 c1,1,2,3; and so on. 00:19:33.830 --> 00:19:39.030 And these c's, in one of the great perversions of 00:19:39.030 --> 00:19:41.960 scientific notation, are called stiffnesses. 00:19:51.270 --> 00:19:54.010 The other thing we could do is to write 00:19:54.010 --> 00:19:57.870 strain in terms of stress. 00:19:57.870 --> 00:19:59.420 And really, stress is something you 00:19:59.420 --> 00:20:00.570 can do to the crystal. 00:20:00.570 --> 00:20:04.380 So stress is an independent variable. 00:20:04.380 --> 00:20:08.010 And this is something that I feel more at home with. 00:20:08.010 --> 00:20:14.190 So I'll have an epsilon 1,1; 2,2; 3,3; epsilon 3,2; 2,3; 00:20:14.190 --> 00:20:21.010 epsilon 1,3; epsilon 3,1; epsilon 2,1; 00:20:21.010 --> 00:20:23.160 and an epsilon 1,2. 00:20:23.160 --> 00:20:26.260 Again, nine elements of strain. 00:20:26.260 --> 00:20:30.120 And these will be given by coefficients times each of the 00:20:30.120 --> 00:20:37.440 nine elements of stress, sigma 1,1; sigma 2,2; sigma 3,3; 00:20:37.440 --> 00:20:42.940 sigma 2,3; sigma 3,2; and so on for nine different 00:20:42.940 --> 00:20:44.790 components. 00:20:44.790 --> 00:20:51.020 Andy the coefficients here are designated with the symbol s. 00:20:51.020 --> 00:20:52.800 These are the tensor elements. 00:20:52.800 --> 00:20:55.020 So this is s1,1,1. 00:20:55.020 --> 00:20:57.890 This is s1,1,2,2. 00:20:57.890 --> 00:21:00.900 And the s's stand for compliances. 00:21:06.890 --> 00:21:10.410 Now English is a very strange language. 00:21:10.410 --> 00:21:19.350 But to call compliance s and to call stiffness c is surely 00:21:19.350 --> 00:21:24.510 a perversion that can be designed for no other purpose 00:21:24.510 --> 00:21:28.830 than to confuse the introductory student and give 00:21:28.830 --> 00:21:32.210 the instructor some feeling of superiority. 00:21:32.210 --> 00:21:34.800 I don't feel superior, I feel embarrassed that I have to 00:21:34.800 --> 00:21:36.510 explain this. 00:21:36.510 --> 00:21:42.110 Stiffness, c; compliance, s. 00:21:42.110 --> 00:21:44.300 It's got be one of the atrocities 00:21:44.300 --> 00:21:46.630 of scientific notation. 00:21:46.630 --> 00:21:49.920 To remember which goes with which, I'll tell you what 00:21:49.920 --> 00:21:50.630 works for me. 00:21:50.630 --> 00:21:52.620 If it doesn't work for you, forget it. 00:21:52.620 --> 00:21:55.380 The s goes with sigma. 00:22:00.940 --> 00:22:05.530 And the little c goes with the epsilon. 00:22:05.530 --> 00:22:07.920 That's a c with a bar in it. 00:22:07.920 --> 00:22:08.850 That works for me. 00:22:08.850 --> 00:22:12.160 If it doesn't work for you, use your own mnemonic device. 00:22:18.050 --> 00:22:22.010 English does a lot of things strangely, but nothing is as 00:22:22.010 --> 00:22:23.260 perverse as this. 00:22:26.620 --> 00:22:27.750 Now what's some examples? 00:22:27.750 --> 00:22:29.090 Consider this. 00:22:29.090 --> 00:22:34.120 You park your car in your driveway, but you drive your 00:22:34.120 --> 00:22:35.370 car on a parkway. 00:22:38.010 --> 00:22:40.970 Why don't you drive your car on a driveway and park your 00:22:40.970 --> 00:22:42.110 car in a parkway? 00:22:42.110 --> 00:22:43.160 No, it's the other way around. 00:22:43.160 --> 00:22:46.560 It's almost as bad as this. 00:22:46.560 --> 00:22:47.810 What are some other ones? 00:22:50.000 --> 00:22:53.490 In my basement, I have something that's called a hot 00:22:53.490 --> 00:22:55.260 water heater. 00:22:55.260 --> 00:22:57.890 You don't heat hot water, you heat cold water. 00:22:57.890 --> 00:23:02.210 Why isn't it called a cold water heater? 00:23:02.210 --> 00:23:06.900 Even better, I once worked for a year and a half in 00:23:06.900 --> 00:23:07.600 Switzerland. 00:23:07.600 --> 00:23:12.490 And in the kitchen of my home I had, in German, an 00:23:12.490 --> 00:23:13.740 elektrowarmwasse rsheissungsapparat. 00:23:16.070 --> 00:23:20.440 That's one word that's 11 syllables. 00:23:20.440 --> 00:23:23.380 The Germans have a knack with the language that we have 00:23:23.380 --> 00:23:26.090 never approached in English. 00:23:26.090 --> 00:23:29.666 But there's another one you've probably heard. 00:23:29.666 --> 00:23:31.950 You know what the prefix "pro" means. 00:23:31.950 --> 00:23:34.905 That means for, and "con" means against. 00:23:34.905 --> 00:23:38.760 And so progress is moving forward, and I leave it to you 00:23:38.760 --> 00:23:40.360 to decide what Congress means. 00:23:45.550 --> 00:23:46.080 OK. 00:23:46.080 --> 00:23:47.520 So I could go on and on. 00:23:47.520 --> 00:23:53.130 I probably shouldn't, unless you egg me on. 00:23:53.130 --> 00:23:53.730 No, I won't. 00:23:53.730 --> 00:23:55.440 But anyway, there are-- 00:23:55.440 --> 00:23:56.740 OK, I'll give another one. 00:23:56.740 --> 00:24:01.025 Why is what a doctor does called "practice"? 00:24:03.640 --> 00:24:07.590 This is not reassuring at all. 00:24:07.590 --> 00:24:11.540 Let me invite you over to my practice. 00:24:11.540 --> 00:24:17.170 And why is it when you take an airplane your flight ends with 00:24:17.170 --> 00:24:19.880 a terminal? 00:24:19.880 --> 00:24:22.600 That's doesn't sound very encouraging either. 00:24:22.600 --> 00:24:23.845 So anyway, language is silly. 00:24:23.845 --> 00:24:27.400 And I guess if you want to call c stiffnesses and s 00:24:27.400 --> 00:24:29.640 compliances, it's OK. 00:24:29.640 --> 00:24:32.360 The verbal description of these terms means something. 00:24:32.360 --> 00:24:39.180 Because if the stiffness is very high, that says that a 00:24:39.180 --> 00:24:42.850 little bit of strain requires that you haven't posed a very 00:24:42.850 --> 00:24:44.970 large stress. 00:24:44.970 --> 00:24:48.080 So stiff is when the material is resistant to stress. 00:24:48.080 --> 00:24:51.630 So you have to have a very, very large strain to get 00:24:51.630 --> 00:24:53.910 yourself a given level of stress. 00:24:53.910 --> 00:24:58.290 Conversely, if the material is very compliant, a very small 00:24:58.290 --> 00:25:00.750 stress for a compliant material should 00:25:00.750 --> 00:25:01.960 give you a big strain. 00:25:01.960 --> 00:25:04.890 And that's exactly what a large value of s will do. 00:25:04.890 --> 00:25:07.080 It will give you a large strain for a 00:25:07.080 --> 00:25:08.850 relatively small stress. 00:25:08.850 --> 00:25:11.730 So the words describe what's going on. 00:25:15.990 --> 00:25:16.550 OK. 00:25:16.550 --> 00:25:23.620 There is great utility in reducing these equations to a 00:25:23.620 --> 00:25:34.160 matrix form, which is going to cut us down from a 9 by 9, or 00:25:34.160 --> 00:25:41.590 a tensor with 81 elements, to a smaller number of subscripts 00:25:41.590 --> 00:25:48.160 and a smaller number of matrix elements, so if we let sigma 1 00:25:48.160 --> 00:25:55.460 be sigma 1,1; if we let sigma 2 be equal to sigma 2,2 and 00:25:55.460 --> 00:26:00.330 sigma 3 be equal to sigma 3. 00:26:00.330 --> 00:26:05.080 And then if we write the inequalities we've had before, 00:26:05.080 --> 00:26:12.420 and we let epsilon 1,1 to be replaced by epsilon 1, then we 00:26:12.420 --> 00:26:18.550 would have out in front here a term simply c1,1, a 1 for the 00:26:18.550 --> 00:26:21.130 sigma and a 1 for the epsilon. 00:26:21.130 --> 00:26:26.620 The next term would be c1,2 times epsilon 2, where 00:26:26.620 --> 00:26:30.930 obviously epsilon 2 has replaced epsilon 2,2, and 1 00:26:30.930 --> 00:26:33.370 stands for the two indices, 1,1. 00:26:33.370 --> 00:26:37.300 So this will go c1,3; epsilon 3. 00:26:37.300 --> 00:26:41.140 And then you hit these messy factors of two. 00:26:41.140 --> 00:26:42.750 You'll have a pair of terms. 00:26:42.750 --> 00:26:46.950 You'll have c1,4. 00:26:46.950 --> 00:26:58.300 And that will stand for a term c1,1,2,3. 00:26:58.300 --> 00:27:04.300 And then there'll be another term c1,1,3,2. 00:27:04.300 --> 00:27:08.760 And if you want to write this as a matrix, c1,4 has to be 00:27:08.760 --> 00:27:14.140 equal to the sum of these two things, times epsilon 4. 00:27:14.140 --> 00:27:16.570 That's the only way you can write it as a matrix. 00:27:16.570 --> 00:27:20.950 And if you think it was bad worrying about a factor of 2 00:27:20.950 --> 00:27:23.490 in front of half of the terms on the right-hand side of the 00:27:23.490 --> 00:27:29.520 equation, we're going to be dealing with factors of 2 for 00:27:29.520 --> 00:27:34.000 terms in the upper 3 by 3 array. 00:27:34.000 --> 00:27:38.730 We're going to be worrying about a factor of 4, somehow 00:27:38.730 --> 00:27:43.010 or other, down in the lower 3 by 3 array. 00:27:43.010 --> 00:27:47.360 So the way you handle the factor of 2 really is a 00:27:47.360 --> 00:27:51.260 nightmare for relations in elasticity. 00:27:51.260 --> 00:28:02.260 But continuing on with the first line, c1,5 times epsilon 00:28:02.260 --> 00:28:11.710 5 would be a combination of c1,1,3,1 plus 1,1,1,3. 00:28:11.710 --> 00:28:17.860 And then finally, when you get to c1,6 times epsilon 6, that 00:28:17.860 --> 00:28:28.520 would represent a combination of c1,1,2,1 plus c1,1,1,2. 00:28:28.520 --> 00:28:31.400 You have to make a decision where to swallow the 2. 00:28:31.400 --> 00:28:35.790 And probably the best thing I can do is to give you 00:28:35.790 --> 00:28:39.270 conventions for relabeling stress, strain, stiffness, and 00:28:39.270 --> 00:28:43.870 compliance, which I'll pass around to you. 00:28:43.870 --> 00:28:49.700 No big deal, except that this is a case where you can eat 00:28:49.700 --> 00:28:52.690 the 2's early on, but you can't even break even. 00:29:00.220 --> 00:29:05.150 So very, very briefly, what you do is you convert the 00:29:05.150 --> 00:29:09.510 tensor elements of stress to matrix elements in exactly the 00:29:09.510 --> 00:29:10.930 same way we did with piezoelectricity. 00:29:13.630 --> 00:29:21.230 For strains, we do exactly the same thing that we did with 00:29:21.230 --> 00:29:23.950 the converse piezoelectric effect. 00:29:23.950 --> 00:29:27.810 The factor of 2 that you swallowed in definition of 00:29:27.810 --> 00:29:32.350 stress pops up to haunt you when you 00:29:32.350 --> 00:29:33.560 deal with the strains. 00:29:33.560 --> 00:29:39.160 And you cannot write a nice matrix relation unless you 00:29:39.160 --> 00:29:43.500 divide the off-diagonal elements of strain by 1/2. 00:29:43.500 --> 00:29:45.770 And that was exactly the same thing we encountered with the 00:29:45.770 --> 00:29:50.110 converse piezoelectric effect, which related strain to an 00:29:50.110 --> 00:29:53.640 applied electric field. 00:29:53.640 --> 00:29:59.790 In defining the matrix stiffnesses, the c's, you 00:29:59.790 --> 00:30:04.460 forge straight ahead, and you let ci,j,k,l, which is 00:30:04.460 --> 00:30:05.780 identical to ci,j,l,k-- 00:30:09.830 --> 00:30:13.180 that's the term that relates the two equal shear stresses-- 00:30:13.180 --> 00:30:16.760 and you define that as a matrix term with two 00:30:16.760 --> 00:30:17.810 subscripts. 00:30:17.810 --> 00:30:21.580 No factors of 2 or 4 are involved. 00:30:21.580 --> 00:30:28.150 Then when you hit the s's, there's a nightmare. 00:30:28.150 --> 00:30:35.810 si,j,k,l is sm,n if m and n are 1, 2, or 3; that is to 00:30:35.810 --> 00:30:38.720 say, not 4, 5, or 6. 00:30:38.720 --> 00:30:44.640 si,j,k,l, which is the same as s,i,j,l,k, has to be defined 00:30:44.640 --> 00:30:52.560 as one half of sm,n, where m or n is 4, 5, or 6. 00:30:52.560 --> 00:30:56.250 And then finally, you have to throw in a factor of 4 when 00:30:56.250 --> 00:31:00.900 both m and n are 4, 5, or six or, in other words, m and n 00:31:00.900 --> 00:31:05.640 are not 1, 2, or 3. 00:31:05.640 --> 00:31:06.680 So it is a mess. 00:31:06.680 --> 00:31:09.750 And these are rules that you have to bear in mind if you're 00:31:09.750 --> 00:31:15.030 ever going to go from matrix form, which works absolutely 00:31:15.030 --> 00:31:18.620 lovely in a fixed coordinate system. 00:31:18.620 --> 00:31:20.775 But if you have a single crystal and you want to refer 00:31:20.775 --> 00:31:25.520 it to different axes, you have to be prepared to resurrect 00:31:25.520 --> 00:31:27.320 the full tensor form. 00:31:27.320 --> 00:31:33.490 And then and only then can you work symmetry transformations. 00:31:33.490 --> 00:31:37.380 I don't want to go through the simple algebra of expanding 00:31:37.380 --> 00:31:39.150 and contracting these terms. 00:31:39.150 --> 00:31:43.330 So the next two sheets show you how you condense from 00:31:43.330 --> 00:31:48.050 tensor to matrix notation, and then go back from matrix 00:31:48.050 --> 00:31:49.940 notation to tensor notation. 00:31:49.940 --> 00:31:53.700 This is for the compliances si,j,k. 00:31:53.700 --> 00:31:57.280 And the next page does the same thing, if you care to 00:31:57.280 --> 00:32:00.190 write stress in terms of strain. 00:32:00.190 --> 00:32:02.570 And there are a lot more factors of 00:32:02.570 --> 00:32:04.130 2 that appear there. 00:32:04.130 --> 00:32:07.490 But again, it shows you how you go from tensor notation, 00:32:07.490 --> 00:32:11.160 for subscripts on the c's, down to matrix notation with 2 00:32:11.160 --> 00:32:15.080 subscripts, and then go back up again to tensor notation. 00:32:15.080 --> 00:32:16.860 So it's a tedious business. 00:32:16.860 --> 00:32:21.010 And you've got to keep careful tracks of your factors of 2 00:32:21.010 --> 00:32:22.260 and your factors of 4. 00:32:26.530 --> 00:32:29.470 How about symmetry restrictions? 00:32:29.470 --> 00:32:30.550 Holy mackerel. 00:32:30.550 --> 00:32:33.570 Transformation of 81 different elements. 00:32:33.570 --> 00:32:36.480 Well, in order to do the tensor transformations, you 00:32:36.480 --> 00:32:40.940 have to go to the full-force subscript notation. 00:32:40.940 --> 00:32:44.450 Only then is a law of transformation defined. 00:32:44.450 --> 00:32:48.230 Fourth-rank properties are even tensors. 00:32:48.230 --> 00:32:50.370 So mercifully, there are not as many different 00:32:50.370 --> 00:32:52.370 possibilities. 00:32:52.370 --> 00:32:56.070 All point groups that differ by presence or absence of 00:32:56.070 --> 00:32:59.020 inversion have exactly the same form of the tensor. 00:32:59.020 --> 00:33:02.430 So symmetry 2, m, and 2/m look alike. 00:33:02.430 --> 00:33:07.250 Symmetry 2, 2mm, and 2/m, 2/m, 2/m look alike. 00:33:07.250 --> 00:33:11.550 So there's only one orthorhombic tensor, only one 00:33:11.550 --> 00:33:15.160 kind of monoclinic tensor, only one kind of triclinic 00:33:15.160 --> 00:33:16.500 tensor, and so on. 00:33:16.500 --> 00:33:20.540 And the only place that you have more than one form of the 00:33:20.540 --> 00:33:26.710 tensor for a particular set of point groups is looking at the 00:33:26.710 --> 00:33:31.780 point groups that are based on single-rotation axes, 00:33:31.780 --> 00:33:35.870 something like 4, 4 bar 4m, and those that are based on 00:33:35.870 --> 00:33:37.940 the axial arrangements 4, 2, 2. 00:33:37.940 --> 00:33:40.850 So they're different, just as they were for second-rank 00:33:40.850 --> 00:33:42.790 tensor properties. 00:33:42.790 --> 00:33:46.980 For cubic, a rather remarkable result. 00:33:46.980 --> 00:33:53.540 There are three independent compliances. 00:33:53.540 --> 00:33:58.920 But cubic crystals are not elastically isotropic. 00:33:58.920 --> 00:34:01.190 They are anisotropic. 00:34:01.190 --> 00:34:03.830 And it's just the nature of the tensor that requires that. 00:34:03.830 --> 00:34:06.060 So rather than letting you hang by your thumbs wondering 00:34:06.060 --> 00:34:09.900 what's on these pages, let me pass around the summary of 00:34:09.900 --> 00:34:11.270 symmetry constraints. 00:34:11.270 --> 00:34:15.060 If you understood what we did for third-rank tensors or even 00:34:15.060 --> 00:34:17.900 second-rank tensors, you how to do this. 00:34:17.900 --> 00:34:20.310 And fortunately, the terms almost over. 00:34:20.310 --> 00:34:22.920 So I can't make you do it, which is probably an enormous 00:34:22.920 --> 00:34:24.170 relief to you. 00:34:33.020 --> 00:34:37.350 Some additional bits of information. 00:34:37.350 --> 00:34:41.850 The transformations for hexagonal crystals are 00:34:41.850 --> 00:34:46.630 complicated because threefold and sixfold axes do not change 00:34:46.630 --> 00:34:50.980 the reference axes x1, x2, x3 into one another. 00:34:50.980 --> 00:34:59.330 And the equations that relate the individual tensor elements 00:34:59.330 --> 00:35:01.230 are more complex. 00:35:01.230 --> 00:35:08.400 For the threefold axis, for example, 2 s1,1 minus 2 s1,2 00:35:08.400 --> 00:35:11.420 is equal to s4,4. 00:35:11.420 --> 00:35:15.890 So they're not simple relations between them. 00:35:15.890 --> 00:35:18.160 And that's simply because you're not changing one 00:35:18.160 --> 00:35:19.920 reference access into another. 00:35:19.920 --> 00:35:23.120 You're changing x1, for example, into a linear 00:35:23.120 --> 00:35:26.980 combination of x1 and x2. 00:35:26.980 --> 00:35:31.860 Two things down here of interest. 00:35:31.860 --> 00:35:35.300 I said that cubic crystals are not isotropic. 00:35:35.300 --> 00:35:37.780 What would have to be the case if the 00:35:37.780 --> 00:35:42.630 material were to be isotropic? 00:35:42.630 --> 00:35:46.120 Well, it turns out that if you do this for the compliances, 00:35:46.120 --> 00:35:53.100 s, if 1/2 of s4,4 is equal to s1,1 minus s1,2, then the 00:35:53.100 --> 00:35:56.680 material is elastically isotropic. 00:35:56.680 --> 00:36:01.325 If you do this in terms stiffnesses, the c's, then 2 00:36:01.325 --> 00:36:06.780 c4,4 is equal to c1,1 minus c1,2 if the material is 00:36:06.780 --> 00:36:09.065 supposed to be isotropic or going to be isotropic. 00:36:12.990 --> 00:36:18.330 Then there is one other inequality 00:36:18.330 --> 00:36:20.620 that depends on structure. 00:36:20.620 --> 00:36:25.200 And this is something known as the Cauchy relation. 00:36:25.200 --> 00:36:30.610 And it depends on the interatomic forces. 00:36:30.610 --> 00:36:34.500 The first condition that has to be met is that the forces 00:36:34.500 --> 00:36:37.970 between the atoms should be central forces. 00:36:37.970 --> 00:36:39.150 What is a central force? 00:36:39.150 --> 00:36:43.890 Well, this is a case where the attractive force between the 00:36:43.890 --> 00:36:48.610 atoms is directly along the line joining their centers. 00:36:48.610 --> 00:36:51.170 Isn't that always the case? 00:36:51.170 --> 00:36:54.100 Don't crystals hold together because there's an attractive 00:36:54.100 --> 00:36:57.722 force between atoms? 00:36:57.722 --> 00:37:03.660 Well actually, for metallic crystals and ionic crystals, 00:37:03.660 --> 00:37:06.270 maybe that's not a bad assumption, particularly for 00:37:06.270 --> 00:37:07.460 ionic crystals. 00:37:07.460 --> 00:37:11.670 But if you had a covalent material, where the bonding 00:37:11.670 --> 00:37:16.730 was due to overlap of orbitals like this, then the thing that 00:37:16.730 --> 00:37:20.580 holds the crystal together is overlap 00:37:20.580 --> 00:37:22.240 between these orbitals. 00:37:22.240 --> 00:37:26.080 And the force holding the atoms together is a force that 00:37:26.080 --> 00:37:28.580 goes through this shared electron pair. 00:37:28.580 --> 00:37:30.530 And that's not a central force. 00:37:30.530 --> 00:37:34.780 And so the Cauchy relation generally fails rather badly 00:37:34.780 --> 00:37:38.230 for covalently bonded materials. 00:37:38.230 --> 00:37:42.080 Second assumption is that each atom is 00:37:42.080 --> 00:37:44.970 at a center of symmetry. 00:37:44.970 --> 00:37:47.580 And the reason for that is so the force in the plus-x 00:37:47.580 --> 00:37:49.120 direction is the same as the force 00:37:49.120 --> 00:37:51.260 in the minus-x direction. 00:37:51.260 --> 00:37:54.780 And that is not true for any material that has tetrahedral 00:37:54.780 --> 00:37:56.580 coordination. 00:37:56.580 --> 00:37:59.830 So it's not true for any of the forms of SiO2. 00:37:59.830 --> 00:38:04.480 It's not true for any of the compound semiconductors that 00:38:04.480 --> 00:38:07.520 are based on tetrahedral units. 00:38:07.520 --> 00:38:12.010 So there, the atom inside of these tetrahedra is decidedly 00:38:12.010 --> 00:38:15.740 not at an inversion center. 00:38:15.740 --> 00:38:18.760 And finally, you assume that the crystal is under no state 00:38:18.760 --> 00:38:20.250 of initial stress. 00:38:20.250 --> 00:38:23.950 Because if it is under initial stress, you've squished it, 00:38:23.950 --> 00:38:25.700 you've stretched the bonds. 00:38:25.700 --> 00:38:28.330 And the forces-- 00:38:28.330 --> 00:38:30.720 not to a major degree, to be sure, but-- the forces will 00:38:30.720 --> 00:38:33.460 not truly be perfect central forces. 00:38:33.460 --> 00:38:38.100 So if all three of these assumptions are satisfied, 00:38:38.100 --> 00:38:40.120 then you have an additional equality 00:38:40.120 --> 00:38:44.980 c1,2 is equal to c4,4. 00:38:44.980 --> 00:38:48.650 What I'll do next time is I'll bring in some examples of data 00:38:48.650 --> 00:38:52.720 for stiffnesses and compliances and, in 00:38:52.720 --> 00:38:58.620 particular, show you how some of these elastic tensor 00:38:58.620 --> 00:39:00.430 elements change with temperature. 00:39:00.430 --> 00:39:04.040 And we can examine them to see how well the Cauchy equality 00:39:04.040 --> 00:39:06.050 or the isotropy condition is satisfied. 00:39:09.850 --> 00:39:10.070 All right. 00:39:10.070 --> 00:39:13.300 That's about all that we'll do about the definition of 00:39:13.300 --> 00:39:15.070 fourth-rank tensor properties. 00:39:15.070 --> 00:39:18.680 On next Tuesday to wrap the term up, we'll look at the 00:39:18.680 --> 00:39:21.720 variation of some elastic moduli, such as Young's 00:39:21.720 --> 00:39:25.400 Modulus or the shear modulus, for different symmetries as a 00:39:25.400 --> 00:39:28.260 function of direction in the crystal, OK?