WEBVTT 00:00:08.189 --> 00:00:12.620 PROFESSOR: Any questions before we push bravely onward? 00:00:16.960 --> 00:00:19.370 OK if there's no question on any of this I shall 00:00:19.370 --> 00:00:20.330 clean off the board. 00:00:20.330 --> 00:00:25.550 And the last thing I wanted to do in connection with thermal 00:00:25.550 --> 00:00:31.030 expansion is to examine how the macroscopic linear thermal 00:00:31.030 --> 00:00:35.830 expansion coefficient relates to the nature of the potential 00:00:35.830 --> 00:00:37.080 minimum between-- 00:00:40.170 --> 00:00:43.970 in the attraction between-- neighboring atoms. 00:00:43.970 --> 00:00:52.280 So let me suppose we have a pair of atoms separated by 00:00:52.280 --> 00:00:59.350 some distance, d, and there is a cohesive force between them 00:00:59.350 --> 00:01:03.040 or else the material would not be in a solid state. 00:01:03.040 --> 00:01:06.115 And the nature of the potential-- 00:01:08.660 --> 00:01:10.640 and I'm going to call the distance 00:01:10.640 --> 00:01:12.920 between the atoms, x-- 00:01:12.920 --> 00:01:16.090 and this is going to be the energy. 00:01:20.740 --> 00:01:29.990 And I'm measuring now relative to-- 00:01:29.990 --> 00:01:35.150 let me call this d as I did over here-- so the energy as a 00:01:35.150 --> 00:01:39.800 function of d is going to consist of a very short range, 00:01:39.800 --> 00:01:43.930 repulsive interaction, that rises very, very rapidly when 00:01:43.930 --> 00:01:49.370 you attempt to squeeze the atoms any closer than a fairly 00:01:49.370 --> 00:01:51.190 small value. 00:01:51.190 --> 00:01:56.560 And then there'll be some sort of cohesive energy. 00:01:56.560 --> 00:02:00.340 And everybody loves to model ionic compounds because in an 00:02:00.340 --> 00:02:09.130 ionic compound this cohesive force would be purely a 00:02:09.130 --> 00:02:13.840 Coulombic force that would be, in rationalized MKS units, 1 00:02:13.840 --> 00:02:20.590 over 4 pi epsilon zero times the product of the charges and 00:02:20.590 --> 00:02:24.100 divided by the distance between them. 00:02:27.660 --> 00:02:30.480 Taking those two potentials together, and you've all seen 00:02:30.480 --> 00:02:37.460 this many times before, there is a minimum in the energy. 00:02:37.460 --> 00:02:38.890 And then absolute zero-- 00:02:38.890 --> 00:02:42.340 that is the separation at which this 00:02:42.340 --> 00:02:44.130 atom pair would sit-- 00:02:44.130 --> 00:02:49.180 and this would be the binding energy of that ionic pair. 00:02:49.180 --> 00:02:52.070 If you carry this through to get the entire energy of the 00:02:52.070 --> 00:02:56.900 crystal, you find that the energy of the crystal goes as 00:02:56.900 --> 00:03:00.296 1 minus 1 over-- 00:03:00.296 --> 00:03:02.060 haven't introduced this term yet, so 00:03:02.060 --> 00:03:05.310 let's hold off on that. 00:03:05.310 --> 00:03:11.730 What I'm going to do now is to take a look at this potential 00:03:11.730 --> 00:03:14.870 and expand it around the equilibrium separation just 00:03:14.870 --> 00:03:18.350 with an empirical collection of terms. 00:03:18.350 --> 00:03:22.250 But from the shape of this well, we can see why the bond 00:03:22.250 --> 00:03:25.790 length should increase as the energy of 00:03:25.790 --> 00:03:28.520 the atomic pair increases. 00:03:28.520 --> 00:03:36.020 At absolute zero, the separation will be this 00:03:36.020 --> 00:03:38.680 equilibrium separation, d zero. 00:03:38.680 --> 00:03:41.130 But as you increase the temperature and pump 00:03:41.130 --> 00:03:45.070 vibrational energy into this pair of atoms, they will get a 00:03:45.070 --> 00:03:48.860 certain increase in energy. 00:03:48.860 --> 00:03:53.330 And as a result, the pair will be able to vibrate between 00:03:53.330 --> 00:03:54.520 these two limits. 00:03:54.520 --> 00:03:57.610 Put in some more thermal energy by raising the 00:03:57.610 --> 00:04:00.830 temperature, and the pair will be able to vibrate between 00:04:00.830 --> 00:04:02.740 these limits-- 00:04:02.740 --> 00:04:04.510 larger limits. 00:04:04.510 --> 00:04:09.440 And so the average position of the atom, the separation of 00:04:09.440 --> 00:04:12.740 the atoms, is something that is going to increase with 00:04:12.740 --> 00:04:14.630 increasing temperature. 00:04:14.630 --> 00:04:17.800 And that is purely a consequence of the asymmetry 00:04:17.800 --> 00:04:21.660 in this potential well. 00:04:21.660 --> 00:04:25.430 If this were a Coulombic bond, we know exactly how to model 00:04:25.430 --> 00:04:27.190 the attractive part. 00:04:27.190 --> 00:04:30.460 The repulsive part is a little harder to model because that 00:04:30.460 --> 00:04:33.060 is something that's inherently quantum mechanical. 00:04:33.060 --> 00:04:36.160 As the ion pair gets close together, the electrons start 00:04:36.160 --> 00:04:38.910 to perturb each other's orbitals. 00:04:38.910 --> 00:04:43.890 And a very common thing to do is to model this as 1 over d 00:04:43.890 --> 00:04:46.290 to the sum power n. 00:04:46.290 --> 00:04:50.530 And n is typically on the order of 8 to 12. 00:04:50.530 --> 00:04:56.210 So this is something that rises very, very steeply. 00:04:56.210 --> 00:05:00.170 I was once at the Gordon conference where a fist fight 00:05:00.170 --> 00:05:03.200 almost broke out because somebody was giving a 00:05:03.200 --> 00:05:06.360 presentation and started from the beginning by introducing 00:05:06.360 --> 00:05:07.870 this model to the potential well. 00:05:07.870 --> 00:05:10.950 And he says, "As usual, we'll write this as a power law, 1 00:05:10.950 --> 00:05:15.260 over d to the n." And somebody in a front row jumped up and 00:05:15.260 --> 00:05:17.550 said, "what do you mean usually? 00:05:17.550 --> 00:05:21.490 Sensible people use exponentials." And the speaker 00:05:21.490 --> 00:05:25.920 said, "No, I'm going to use a power law." That's dumb. 00:05:25.920 --> 00:05:27.700 Exponentials work better. 00:05:27.700 --> 00:05:30.770 Power laws, exponentials, power laws, exponentials. 00:05:30.770 --> 00:05:35.120 But in point of fact they're both grossly empirical and one 00:05:35.120 --> 00:05:36.460 is not better than the other. 00:05:36.460 --> 00:05:40.930 The algebra changes a little bit, but people get very, very 00:05:40.930 --> 00:05:43.510 emotionally attached to their physical models. 00:05:43.510 --> 00:05:46.205 AUDIENCE: [INAUDIBLE] 00:05:46.205 --> 00:05:48.153 law in the exponential. 00:05:48.153 --> 00:05:50.831 PROFESSOR: Good, you should've been there to 00:05:50.831 --> 00:05:52.536 break up the fight. 00:05:52.536 --> 00:05:53.997 If you'd gotten between them, you'd have been likely punched 00:05:53.997 --> 00:05:54.971 in the nose. 00:05:54.971 --> 00:05:56.221 [INAUDIBLE] 00:05:58.380 --> 00:06:01.180 OK so I'm going to use the power law because as we've 00:06:01.180 --> 00:06:04.010 seen they're the same thing as exponentials. 00:06:04.010 --> 00:06:07.390 So now what I'm going to do is expand this in a purely 00:06:07.390 --> 00:06:09.340 empirical way-- 00:06:09.340 --> 00:06:13.650 measuring distance from the minimum of the potential well. 00:06:13.650 --> 00:06:17.680 So what I'm going to say is that the potential as a 00:06:17.680 --> 00:06:20.670 function of x-- and now I won't call it d because I'm 00:06:20.670 --> 00:06:25.150 going to measure distance, x, from the minimum in this well. 00:06:25.150 --> 00:06:27.580 I'm going to say V is a function of x. 00:06:27.580 --> 00:06:31.970 There is a well in which the-- 00:06:31.970 --> 00:06:33.960 which keeps the atoms separate. 00:06:33.960 --> 00:06:37.270 I'll assume that is a parabolic potential well, so 00:06:37.270 --> 00:06:42.180 it'll go as first some constant c times x squared. 00:06:42.180 --> 00:06:44.070 Then the thing that causes thermal 00:06:44.070 --> 00:06:46.840 expansion is the asymmetry. 00:06:46.840 --> 00:06:50.660 And I want the asymmetry to increase the energy more 00:06:50.660 --> 00:06:54.590 rapidly on the side of negative x than on the side of 00:06:54.590 --> 00:06:55.920 positive x. 00:06:55.920 --> 00:06:58.335 So I'll put in a term, g x cubed. 00:07:00.960 --> 00:07:04.100 And I put a negative sign in here because I want it to go 00:07:04.100 --> 00:07:07.580 up as x becomes negative. 00:07:07.580 --> 00:07:12.580 That means the ion pairs are decreasing at separation. 00:07:12.580 --> 00:07:19.750 And finally, a very fine touch, if x becomes very 00:07:19.750 --> 00:07:23.030 large, the potential well should soften. 00:07:23.030 --> 00:07:26.700 The edges of this are starting to level out and get lower 00:07:26.700 --> 00:07:30.730 than the purely parabolic part for very large separations. 00:07:30.730 --> 00:07:32.730 So that's what I'll use as a very simple 00:07:32.730 --> 00:07:36.640 model for my potential. 00:07:36.640 --> 00:07:41.910 This is what gives the asymmetry, and this is what 00:07:41.910 --> 00:07:43.160 gives softening. 00:07:45.860 --> 00:07:51.700 And now I'm going to ask, what is the average value of x? 00:07:51.700 --> 00:07:58.430 The probability of finding a particular x is going to be 00:07:58.430 --> 00:08:09.180 proportional to e to the minus V of x over kt. 00:08:09.180 --> 00:08:13.840 So I can say that the average value of x is going to be the 00:08:13.840 --> 00:08:19.640 integral from minus infinity to plus infinity of the value 00:08:19.640 --> 00:08:23.260 that I'm trying to find the average of, namely x times the 00:08:23.260 --> 00:08:25.480 probability of finding that x. 00:08:25.480 --> 00:08:28.670 And that's going to e to the minus V of x over kt. 00:08:33.690 --> 00:08:36.929 And I will integrate that over x. 00:08:36.929 --> 00:08:40.570 And then I will normalize by a term which is the integral of 00:08:40.570 --> 00:08:44.230 the probability from minus infinity to plus infinity of e 00:08:44.230 --> 00:08:48.420 to the minus V of x over kt. 00:08:48.420 --> 00:08:50.440 So that's a general expression for an average. 00:08:50.440 --> 00:08:52.820 It's the thing you want to average times the probability 00:08:52.820 --> 00:08:57.220 of finding the particular value divided by the integral 00:08:57.220 --> 00:08:58.470 of the probabilities. 00:09:02.130 --> 00:09:05.220 OK now let me do some very straightforward things and I'm 00:09:05.220 --> 00:09:08.600 just going to rattle these off very quickly. 00:09:08.600 --> 00:09:13.350 We'll expand this in terms of a series of terms. 00:09:13.350 --> 00:09:22.830 And I'm going to expand both the numerator and denominator 00:09:22.830 --> 00:09:26.640 using the approximation that e to the u is approximately 00:09:26.640 --> 00:09:30.380 equal to 1 plus u. 00:09:30.380 --> 00:09:32.090 So the numerator-- 00:09:32.090 --> 00:09:39.930 x times e to the minus cx squared over kt. 00:09:39.930 --> 00:09:45.620 I'm going to take the higher order terms and say that e to 00:09:45.620 --> 00:09:54.080 the gx cubed over kt times e to the minus-- 00:09:54.080 --> 00:09:56.840 this is a minus sign out in front-- 00:09:56.840 --> 00:10:02.300 times e to the fx fourth over kt, can be approximated by, 00:10:02.300 --> 00:10:06.660 since these are very, very small terms, 1 plus g x cubed 00:10:06.660 --> 00:10:14.590 over kt times 1 plus fx fourth over kt. 00:10:14.590 --> 00:10:18.530 So that's going to be, if I put an x e to the minus cx 00:10:18.530 --> 00:10:25.000 squared, I'm going to have a term 1 plus g x cubed over kt, 00:10:25.000 --> 00:10:29.180 plus f x fourth over kt. 00:10:29.180 --> 00:10:33.210 And then a very, very minuscule term, f times g 00:10:33.210 --> 00:10:36.650 times x to the 7th over kt. 00:10:36.650 --> 00:10:37.900 And I'm going to throw that out. 00:10:44.000 --> 00:10:47.460 If I look at these integrals one at a time, the integral 00:10:47.460 --> 00:10:52.560 from minus infinity to plus infinity of x e to the minus 00:10:52.560 --> 00:10:56.230 cx squared over k t, that's the first term in 00:10:56.230 --> 00:10:58.720 expansion, times dx. 00:10:58.720 --> 00:11:00.860 That's going to be zero. 00:11:00.860 --> 00:11:03.180 Because for positive x, the thing that we're 00:11:03.180 --> 00:11:05.030 integrating is plus. 00:11:05.030 --> 00:11:08.590 For negative x, the thing that we're integrating has the same 00:11:08.590 --> 00:11:11.000 exponential but has a minus sign. 00:11:11.000 --> 00:11:13.830 So that integral is zero. 00:11:13.830 --> 00:11:25.250 So the next term I want to look at is x times 00:11:25.250 --> 00:11:28.330 gx cubed over kt. 00:11:28.330 --> 00:11:34.880 And for this I turned to my handy book of integrals. 00:11:34.880 --> 00:11:39.030 The integral from zero to infinity of x to the 2n, where 00:11:39.030 --> 00:11:44.400 that quantity is even, times e to the minus a x squared dx, 00:11:44.400 --> 00:11:49.010 is equal to, as you all know, 1 times 3 times 5 all the way 00:11:49.010 --> 00:11:56.270 up to 2n minus 1 divided by 2 to the n plus 1 times a to the 00:11:56.270 --> 00:11:59.660 n times the square root of pi over a. 00:11:59.660 --> 00:12:04.080 And proof of that the integral is left as an exercise to the 00:12:04.080 --> 00:12:06.290 reckless student. 00:12:06.290 --> 00:12:06.625 Yes? 00:12:06.625 --> 00:12:07.952 AUDIENCE: The x cubed [INAUDIBLE]. 00:12:10.928 --> 00:12:14.896 PROFESSOR: That's why this is a 2n here. 00:12:14.896 --> 00:12:17.624 So I can say that the x cubed term is--but we don't have an 00:12:17.624 --> 00:12:18.368 x cubed term. 00:12:18.368 --> 00:12:24.320 The next term is going to be equal to x fourth. 00:12:24.320 --> 00:12:27.086 And this I'm going to drop because that's 00:12:27.086 --> 00:12:28.610 going to be x fifth. 00:12:28.610 --> 00:12:43.730 So the integral from minus infinity to plus infinity of x 00:12:43.730 --> 00:12:52.850 to the fourth power times g over kt times e to the minus 00:12:52.850 --> 00:12:56.480 cx squared over kt dx. 00:12:56.480 --> 00:12:59.110 If I use this formula, turns out to be a 00:12:59.110 --> 00:13:00.480 very, very simple thing. 00:13:00.480 --> 00:13:05.520 It turns out to be 3/4 times the square root of pi times g 00:13:05.520 --> 00:13:15.980 times kt over c to the 5/2. 00:13:15.980 --> 00:13:20.010 And kt is to the power 3/2. 00:13:23.660 --> 00:13:25.880 So that is-- 00:13:25.880 --> 00:13:32.520 if I throw away the term in x times x fourth which is odd, 00:13:32.520 --> 00:13:36.800 that will vanish and I've then thrown out only a really large 00:13:36.800 --> 00:13:39.530 term in x to the 7th. 00:13:39.530 --> 00:13:56.570 For the denominator, the integral that is there before 00:13:56.570 --> 00:14:03.480 us is the integral from minus infinity to plus infinity of e 00:14:03.480 --> 00:14:11.810 to the minus cx squared over kt times 1 plus g 00:14:11.810 --> 00:14:16.450 x cubed over kt. 00:14:16.450 --> 00:14:19.580 This thing is going to vanish because x is odd. 00:14:19.580 --> 00:14:24.530 And then I'll have plus fx fourth over kt. 00:14:24.530 --> 00:14:27.320 That will not vanish but it's going to be a relatively small 00:14:27.320 --> 00:14:29.390 term, so I'll go like that. 00:14:29.390 --> 00:14:31.770 Which means I just have to integrate 00:14:31.770 --> 00:14:33.760 this Gaussian factor. 00:14:33.760 --> 00:14:39.070 And that turns out to be equal to 2 times 1/2 times square 00:14:39.070 --> 00:14:47.850 root of pi kT pi times T over c. 00:14:47.850 --> 00:14:49.100 So this is the denominator. 00:14:52.180 --> 00:14:57.645 And if I take the ratio of those two integrals, I have 00:14:57.645 --> 00:15:04.230 that the average value of x will be equal to 3/4 square 00:15:04.230 --> 00:15:17.950 root of pi times g times kt to the 3/2 over c to the 5/2, 00:15:17.950 --> 00:15:24.200 divided by the square root of pi times kt over c 00:15:24.200 --> 00:15:26.630 to the power 1/2. 00:15:26.630 --> 00:15:29.560 And if I carry through the algebra correctly, this turns 00:15:29.560 --> 00:15:35.745 out to be 3/4 of g over c squared times kt. 00:15:42.540 --> 00:15:45.730 So there is the average distance between this pair of 00:15:45.730 --> 00:15:49.480 atoms as a function of temperature in terms of 00:15:49.480 --> 00:15:53.420 characteristics of the potential. 00:15:53.420 --> 00:15:57.040 The first thing that is very gratifying is that the bond 00:15:57.040 --> 00:16:00.240 length, according to the simple model, should increase 00:16:00.240 --> 00:16:02.160 linearly with temperature. 00:16:02.160 --> 00:16:05.970 So it tells us we would expect things to expand linearly with 00:16:05.970 --> 00:16:11.260 temperature which corresponds to reality. 00:16:11.260 --> 00:16:15.360 The next thing we'll see is that the amount of expansion 00:16:15.360 --> 00:16:19.360 goes inversely as the square. 00:16:19.360 --> 00:16:20.810 That's a strong dependence-- 00:16:20.810 --> 00:16:23.060 inversely as the square of the 00:16:23.060 --> 00:16:25.930 parabolic part of our potential. 00:16:25.930 --> 00:16:29.230 That was the c-- c was the coefficient in front of cx 00:16:29.230 --> 00:16:31.300 squared in our potential. 00:16:31.300 --> 00:16:35.800 So the stronger the bond, the smaller the thermal expansion 00:16:35.800 --> 00:16:40.050 coefficient is in proportion to the reciprocal of the 00:16:40.050 --> 00:16:43.360 square of that term. 00:16:43.360 --> 00:16:46.840 And then finally, the last important term to survive is 00:16:46.840 --> 00:16:48.650 g, the asymmetry. 00:16:48.650 --> 00:16:51.280 And the thermal expansion coefficient should increase 00:16:51.280 --> 00:16:55.070 with the first power of the term that describes the 00:16:55.070 --> 00:16:56.560 asymmetry of the potential well. 00:17:00.010 --> 00:17:02.970 So your intuition would tell you that 00:17:02.970 --> 00:17:05.140 that's the way the thermal-- 00:17:05.140 --> 00:17:07.380 linear thermal expansion coefficient should vary. 00:17:07.380 --> 00:17:11.250 The interesting thing is that it depends more strongly on 00:17:11.250 --> 00:17:14.450 the parameter that gives the parabolic part of the 00:17:14.450 --> 00:17:15.710 potential well. 00:17:15.710 --> 00:17:18.960 And secondly, that it does predict a thermal expansion 00:17:18.960 --> 00:17:21.680 coefficient that increases-- 00:17:21.680 --> 00:17:24.579 the separation that increases with-- 00:17:24.579 --> 00:17:25.960 in proportion to temperature. 00:17:25.960 --> 00:17:29.250 And therefore there is a constant linear thermal 00:17:29.250 --> 00:17:32.300 expansion coefficient. 00:17:32.300 --> 00:17:34.730 Now all this is true at elevated temperatures. 00:17:34.730 --> 00:17:38.340 At low temperatures, thermodynamics kicks in. 00:17:38.340 --> 00:17:40.990 And it turns out that the linear thermal expansion 00:17:40.990 --> 00:17:45.820 coefficient, as does energies of vibrating atoms, must go to 00:17:45.820 --> 00:17:50.880 zero as temperature approaches absolute zero. 00:17:50.880 --> 00:17:55.920 So a typical variation of the thermal expansion coefficient 00:17:55.920 --> 00:17:59.050 with temperature is something that does this. 00:17:59.050 --> 00:18:02.330 The same way as-- the same sort of variation as things 00:18:02.330 --> 00:18:03.750 like heat capacity. 00:18:03.750 --> 00:18:07.550 And this temperature in here is on the order of the Debye 00:18:07.550 --> 00:18:09.780 temperature of the particular material. 00:18:15.930 --> 00:18:21.650 All right, so much for linear thermal expansion. 00:18:21.650 --> 00:18:28.730 And I would now like to turn to something much more 00:18:28.730 --> 00:18:30.110 interesting. 00:18:30.110 --> 00:18:35.920 And that is physical properties that involve 00:18:35.920 --> 00:18:38.595 property tensors of rank higher than two. 00:18:50.420 --> 00:18:55.140 One of the important and more interesting of these is a 00:18:55.140 --> 00:18:56.560 property called piezoelectricity. 00:19:04.350 --> 00:19:08.490 Piezo means pressure, so the property literally means 00:19:08.490 --> 00:19:12.770 pressure electricity or pressure induced charge. 00:19:19.470 --> 00:19:20.720 Now what is the charge? 00:19:24.450 --> 00:19:29.810 First of all, this does not describe a property where-- if 00:19:29.810 --> 00:19:33.320 I were to pick up a chunk of crystal, and I drop it because 00:19:33.320 --> 00:19:35.890 I just got an electric charge. 00:19:35.890 --> 00:19:39.510 These are not mobile charges, these are bound charges. 00:19:39.510 --> 00:19:46.120 And the reason for the induced charge is the fact that we are 00:19:46.120 --> 00:19:50.740 squeezing or stretching an ionic material, and that's 00:19:50.740 --> 00:19:53.390 moving charged ions around. 00:19:53.390 --> 00:19:56.960 So what we're going to do if that process involves motion 00:19:56.960 --> 00:20:01.600 of charged ions is to induce a dipole moment per unit volume. 00:20:04.160 --> 00:20:08.550 I have a little primer on some quantities in electromagnetic 00:20:08.550 --> 00:20:10.440 theory and I-- 00:20:10.440 --> 00:20:13.230 I brought them with me upstairs to look them over and 00:20:13.230 --> 00:20:14.390 I forgot to bring them down. 00:20:14.390 --> 00:20:16.320 But I don't think we'll need them in what's 00:20:16.320 --> 00:20:17.990 left of this hour. 00:20:17.990 --> 00:20:25.590 So let me say that for matter and bulk, the quantity that is 00:20:25.590 --> 00:20:28.213 involved in piezoelectricity is the polarization. 00:20:30.740 --> 00:20:34.260 And polarization is dipole moment per unit volume. 00:20:44.600 --> 00:20:47.930 And a dipole moment, to refresh your memory, is that 00:20:47.930 --> 00:20:55.670 if I have a positive charge plus q and negative charge 00:20:55.670 --> 00:21:01.080 minus q separated by a distance, d. 00:21:01.080 --> 00:21:05.940 The dipole moment, which I'll indicate by a lowercase p, has 00:21:05.940 --> 00:21:10.790 a vector quality and it's equal to the magnitude of one 00:21:10.790 --> 00:21:17.240 of these charges times the distance between them. 00:21:17.240 --> 00:21:22.140 And we'll define the distance as going from the positive 00:21:22.140 --> 00:21:24.875 charge to the negative charge. 00:21:29.300 --> 00:21:34.220 That's a crazy way of defining a pair of charges of equal but 00:21:34.220 --> 00:21:36.480 opposite sign. 00:21:36.480 --> 00:21:40.050 The reason for defining a quantity like the dipole 00:21:40.050 --> 00:21:44.150 moment is that this combination of charge and 00:21:44.150 --> 00:21:46.910 separation of the charges comes up in all sorts of 00:21:46.910 --> 00:21:51.240 problems in electrostatics and electrodynamics. 00:21:51.240 --> 00:21:52.900 And this is the delight of people who 00:21:52.900 --> 00:21:55.270 teach elementary physics. 00:21:55.270 --> 00:21:59.610 You are given, on the first quiz, a question that says-- 00:21:59.610 --> 00:22:04.350 calculate the electric field that is created by a set of 00:22:04.350 --> 00:22:08.210 dipoles on the Great Court which spell out MIT. 00:22:08.210 --> 00:22:12.130 And we can come up with all sorts of crazy configurations 00:22:12.130 --> 00:22:16.350 of dipoles and then calculate the field that they produce. 00:22:16.350 --> 00:22:20.420 And the field turns out to be something that always involves 00:22:20.420 --> 00:22:26.120 the product of charge and separation in a vector sense. 00:22:26.120 --> 00:22:30.450 Probably the best illustration of this would be if I have a 00:22:30.450 --> 00:22:34.710 dipole charge plus q, charge minus q, separated by d. 00:22:34.710 --> 00:22:38.330 And I put it in the electric field. 00:22:38.330 --> 00:22:41.780 The electric field, which is defined as the force per unit 00:22:41.780 --> 00:22:45.050 charge on a positive charge is going to pull the positive 00:22:45.050 --> 00:22:46.590 charge this way. 00:22:46.590 --> 00:22:48.880 And it's going to pull the negative charge this way. 00:22:51.380 --> 00:22:53.630 And what that's going to do is to place a 00:22:53.630 --> 00:22:56.040 torque on the dipole. 00:22:56.040 --> 00:23:04.880 And that torque is going to be equal to a vector product of 00:23:04.880 --> 00:23:08.550 the electric field and a vector. 00:23:08.550 --> 00:23:13.440 You can express it in terms of the separation of charges 00:23:13.440 --> 00:23:17.220 defined in a vector sense going from the positive charge 00:23:17.220 --> 00:23:19.580 to the negative charge. 00:23:19.580 --> 00:23:23.400 So there's an example of how the torque on a dipole 00:23:23.400 --> 00:23:29.470 directly involves the product of charge and the separation 00:23:29.470 --> 00:23:34.010 between them and the angle between the 00:23:34.010 --> 00:23:36.260 field and the dipole. 00:23:36.260 --> 00:23:39.440 So it's a useful quality and, without further apology, we 00:23:39.440 --> 00:23:40.690 will use it in our discussion. 00:23:43.550 --> 00:23:45.010 Now polarization-- 00:23:45.010 --> 00:23:47.150 dipole moment per unit volume-- is just going to be 00:23:47.150 --> 00:23:51.390 the dipole moment of one dipole pair times the number 00:23:51.390 --> 00:23:53.000 of them per unit volume. 00:23:53.000 --> 00:23:58.630 And that will be a macroscopic measure of polarization. 00:24:02.530 --> 00:24:10.930 OK, if we subject a piece of material that contains ions to 00:24:10.930 --> 00:24:15.970 a stress- so here is a stress tensor, sigma ij. 00:24:19.520 --> 00:24:23.420 Let me call it jk. 00:24:23.420 --> 00:24:28.870 We will create reorientation of the dipoles. 00:24:28.870 --> 00:24:34.110 We will have a polarization, a dipole moment per unit volume. 00:24:34.110 --> 00:24:41.190 This will be a vector because there will be some net dipole 00:24:41.190 --> 00:24:42.350 moment per unit volume. 00:24:42.350 --> 00:24:44.950 And that will just be the sum of all the little individual 00:24:44.950 --> 00:24:48.220 dipole moments on the ions-- 00:24:48.220 --> 00:24:49.820 ion pairs. 00:24:49.820 --> 00:24:57.390 And if the stress is not too large, every component of 00:24:57.390 --> 00:25:02.300 polarization is going to be given by a linear combination 00:25:02.300 --> 00:25:04.960 of every one of the components of stress. 00:25:04.960 --> 00:25:13.150 So we will have an array of coefficients, d ijk The i goes 00:25:13.150 --> 00:25:17.745 with the P and the j and k goes with 00:25:17.745 --> 00:25:21.930 the elements of stress. 00:25:21.930 --> 00:25:23.870 This is something that's called the direct 00:25:23.870 --> 00:25:25.120 piezoelectric effect. 00:25:39.930 --> 00:25:43.870 It's the basis of a lot of every day devices-- 00:25:43.870 --> 00:25:49.090 pick ups on the old records which nobody uses anymore, 00:25:49.090 --> 00:25:51.760 cigarette lighter which nobody uses anymore because 00:25:51.760 --> 00:25:53.560 everybody's quit smoking-- 00:25:53.560 --> 00:25:57.650 but worked the cigarette lighter, in some of the more 00:25:57.650 --> 00:26:00.570 modern incarnations of the lighter, was the 00:26:00.570 --> 00:26:03.980 piezoelectric effect. 00:26:03.980 --> 00:26:07.880 It's important to note that the charges that are involved 00:26:07.880 --> 00:26:10.660 in the piezoelectric effect are bound charges. 00:26:10.660 --> 00:26:14.040 Again you can't get a shock by picking up a piece of a 00:26:14.040 --> 00:26:18.200 strongly piezoelectric material like quartz. 00:26:18.200 --> 00:26:22.270 Might say then, well if there's no spark, how could a 00:26:22.270 --> 00:26:27.090 piezoelectric material in a cigarette lighter cause the 00:26:27.090 --> 00:26:30.875 lighter fuel to ignite. 00:26:30.875 --> 00:26:32.600 The charge doesn't move. 00:26:32.600 --> 00:26:33.960 How could there be a spark? 00:26:33.960 --> 00:26:36.740 How could the lighter fluid be ignited? 00:26:36.740 --> 00:26:37.990 Anybody got an idea? 00:26:41.640 --> 00:26:47.570 Charge can induce current flow in circuit elements that are 00:26:47.570 --> 00:26:49.200 removed from the charge. 00:26:49.200 --> 00:26:53.050 So if you have a little capacitor in the lighter that 00:26:53.050 --> 00:26:56.770 is near the piezoelectric element that will develop the 00:26:56.770 --> 00:27:01.740 charge, you can induce current flow in a circuit as a result 00:27:01.740 --> 00:27:05.680 of that capacitor experiencing the electric charge of the 00:27:05.680 --> 00:27:09.000 piezoelectric induced charge. 00:27:09.000 --> 00:27:12.050 And that's how the little piezoelectric cigarette 00:27:12.050 --> 00:27:13.900 lighters worked. 00:27:13.900 --> 00:27:17.020 OK, this stress-- 00:27:17.020 --> 00:27:22.960 array of stress elements, nine of them, constitutes a tensor 00:27:22.960 --> 00:27:24.450 of second rank. 00:27:24.450 --> 00:27:31.040 Polarization is a vector per unit volume. 00:27:31.040 --> 00:27:33.910 This is a tensor of first rank. 00:27:33.910 --> 00:27:35.890 This is a tensor of second rank. 00:27:35.890 --> 00:27:40.070 Therefore it follows that this array of coefficients, d ijk 00:27:40.070 --> 00:27:44.100 will be three equations involving all 00:27:44.100 --> 00:27:46.700 nine elements of stress. 00:27:46.700 --> 00:27:54.790 So it'll be a total of 27 coefficients in the array. 00:27:54.790 --> 00:27:58.465 And this array constitutes a tensor of the third rank. 00:28:06.340 --> 00:28:10.770 So this is clearly a property tensor which relates a 00:28:10.770 --> 00:28:12.480 generalized force-- 00:28:12.480 --> 00:28:18.500 the stress tensor to a generalized displacement, 00:28:18.500 --> 00:28:20.420 which is the dipole moment per unit volume. 00:28:30.900 --> 00:28:37.830 Since this is a property of a crystal, that tensor is 00:28:37.830 --> 00:28:41.860 subject to symmetry restrictions. 00:28:41.860 --> 00:28:48.550 In particular, if there is some direction, cosine scheme 00:28:48.550 --> 00:28:53.010 Cij, that corresponds to the operation of a symmetry 00:28:53.010 --> 00:29:08.470 element, then it follows that if we evaluate the new tensor 00:29:08.470 --> 00:29:16.760 elements, d ijk prime in terms of the old elements, that is 00:29:16.760 --> 00:29:27.190 going to be a transformation of the form Cil, Cjm, Ckn 00:29:27.190 --> 00:29:31.555 times all 27 of the original tensor elements, d lmn. 00:29:36.030 --> 00:29:41.950 And the l, m, and n are dummy indices, just as with our 00:29:41.950 --> 00:29:43.200 second rank tensors. 00:29:49.014 --> 00:29:55.980 And the ijk are real indices that go with the subscripts on 00:29:55.980 --> 00:29:57.920 the particular tensor element. 00:30:00.850 --> 00:30:03.365 So there are 27 coefficients. 00:30:10.970 --> 00:30:18.760 And each transformed element is going to be a sum therefore 00:30:18.760 --> 00:30:27.080 of 27 coefficients times-- 00:30:27.080 --> 00:30:29.790 each times three direction cosines. 00:30:36.910 --> 00:30:40.800 So each transform tensor element is going to be given 00:30:40.800 --> 00:30:46.195 by a sum of 3 times 27 quantities. 00:30:49.980 --> 00:30:52.700 And to do the complete transformation would involve 00:30:52.700 --> 00:30:55.470 doing it 27 times. 00:30:55.470 --> 00:31:01.710 So 3 times 27 squared is the number of terms that we're 00:31:01.710 --> 00:31:06.030 going to have to write down to transform the entire tensor. 00:31:06.030 --> 00:31:11.380 Not a terribly encouraging prospect unless you use the 00:31:11.380 --> 00:31:18.310 method of direct inspection, namely that the tensor 00:31:18.310 --> 00:31:21.920 transforms like the product of the corresponding coordinates. 00:31:21.920 --> 00:31:24.180 And then it works fairly quickly, particularly for 00:31:24.180 --> 00:31:27.540 symmetry transformations, where there are a lot of zeros 00:31:27.540 --> 00:31:28.980 in the direction cosine scheme. 00:31:32.050 --> 00:31:36.400 So we're going to have to do this for each of the crystal-- 00:31:36.400 --> 00:31:38.170 crystal point groups. 00:31:38.170 --> 00:31:41.340 So we're going to have to do this exercise of 3 times 27 00:31:41.340 --> 00:31:46.620 times 27 times the 32 point groups that exist, which 00:31:46.620 --> 00:31:49.380 becomes an even more daunting task. 00:31:49.380 --> 00:31:53.370 But let's do one such transformation and let me show 00:31:53.370 --> 00:32:00.780 you that it, in fact, it's going to go fairly easily. 00:32:00.780 --> 00:32:04.340 So let me follow the tracks of what we did with second rank 00:32:04.340 --> 00:32:06.000 tensors and examine the 00:32:06.000 --> 00:32:11.550 restrictions imposed by inversion. 00:32:23.070 --> 00:32:27.250 OK so the transformation of the axes, as we've seen 00:32:27.250 --> 00:32:30.190 before, that's produced by 1 bar. 00:32:30.190 --> 00:32:33.435 That's going to take x1 and change its 00:32:33.435 --> 00:32:35.240 direction to minus x1. 00:32:40.050 --> 00:32:44.940 It's going to take x2 and change its sense to an x2 00:32:44.940 --> 00:32:46.950 prime that points in this direction. 00:32:46.950 --> 00:32:50.020 And take x3 and change its sense to something 00:32:50.020 --> 00:32:52.340 that points down here. 00:32:52.340 --> 00:32:54.070 So that's x3 prime. 00:32:54.070 --> 00:32:57.650 So the relation between the axes, before and after 00:32:57.650 --> 00:33:00.720 inversion, is that x1 prime equals x1-- 00:33:00.720 --> 00:33:02.380 minus x1-- 00:33:02.380 --> 00:33:05.220 x2 prime is equal to minus x2. 00:33:05.220 --> 00:33:10.900 And x3 prime is equal to minus x3. 00:33:10.900 --> 00:33:12.240 And we've seen this before. 00:33:12.240 --> 00:33:13.900 It's deja vu all over again. 00:33:13.900 --> 00:33:17.610 So Cij, the direction cosine scheme, is equal 00:33:17.610 --> 00:33:19.810 to minus 1, 0, 0. 00:33:19.810 --> 00:33:21.710 0, minus 1, 0. 00:33:21.710 --> 00:33:23.720 0, 0, minus 1. 00:33:26.450 --> 00:33:32.060 So let's transform a representative tensor element 00:33:32.060 --> 00:33:36.280 for a change of axes that corresponds to that direction 00:33:36.280 --> 00:33:37.620 cosine scheme. 00:33:37.620 --> 00:33:39.950 And that's the same as physically inverting the 00:33:39.950 --> 00:33:43.830 crystal and demanding that the property be unchanged if that 00:33:43.830 --> 00:33:45.630 is a symmetry transformation which 00:33:45.630 --> 00:33:48.160 leaves the crystal invariant. 00:33:48.160 --> 00:33:59.670 So if we look at some d ijk prime, that's going to be Cil, 00:33:59.670 --> 00:34:09.900 Cjm, Ckn, d lmn, where this is a sum over the index l, the 00:34:09.900 --> 00:34:20.280 dummy index m, and the double index n, from 1 to 3. 00:34:20.280 --> 00:34:27.159 If we sum over Ci something, the only term of the form Ci 00:34:27.159 --> 00:34:30.900 something, regardless of what i is, is going to be the 00:34:30.900 --> 00:34:40.310 diagonal term Cii, the diagonal direction cosine Cii. 00:34:40.310 --> 00:34:44.190 In other words, if i were 1, the only term of the form C1 00:34:44.190 --> 00:34:47.900 something that is non-zero, is C11. 00:34:47.900 --> 00:35:00.010 If I sum over m, the only term that is non-zero is C, with m 00:35:00.010 --> 00:35:00.770 equal to j. 00:35:00.770 --> 00:35:02.730 So this is going to be Cjj. 00:35:02.730 --> 00:35:05.160 And if I sum over n, the only term that is going 00:35:05.160 --> 00:35:08.750 to be left is Ckk. 00:35:08.750 --> 00:35:14.490 And all this will be times d ijk, the 00:35:14.490 --> 00:35:16.530 single tensor element. 00:35:16.530 --> 00:35:19.900 Regardless of the values of i, j, and k, they are all equal 00:35:19.900 --> 00:35:21.150 to minus 1. 00:35:23.450 --> 00:35:29.480 So the operation then of inversion, then says that for 00:35:29.480 --> 00:35:34.440 the transformed crystal, d ijk prime for any i, j, and k, is 00:35:34.440 --> 00:35:36.957 going to be equal to minus d ijk. 00:35:39.840 --> 00:35:43.800 But if this is a symmetry transformation, the tensor 00:35:43.800 --> 00:35:46.580 element has to be the same before and after. 00:35:46.580 --> 00:35:49.740 And the only number which can be equal to the negative of 00:35:49.740 --> 00:35:51.100 itself is zero. 00:35:54.250 --> 00:35:58.610 So we have, with one swell foop, 00:35:58.610 --> 00:36:02.600 transformed 27th tensor elements-- 00:36:02.600 --> 00:36:06.270 each one of them a sum of 27 elements in 00:36:06.270 --> 00:36:07.550 the original tensor. 00:36:07.550 --> 00:36:11.920 So the conclusion is, if a crystal has the operation of 00:36:11.920 --> 00:36:17.795 inversion, all d ijk vanish. 00:36:22.170 --> 00:36:24.170 What does that mean? 00:36:24.170 --> 00:36:25.800 Means the property just doesn't exist. 00:36:30.560 --> 00:36:35.640 So no crystal which has an inversion center can display 00:36:35.640 --> 00:36:36.890 piezoelectricity. 00:36:43.190 --> 00:36:46.510 Which means that none of the 11 Laue groups can be 00:36:46.510 --> 00:36:48.310 piezoelectric. 00:36:48.310 --> 00:36:52.090 And the only point groups for crystals that can be 00:36:52.090 --> 00:37:17.950 piezoelectric are the 32 minus 11 equals 21-- 00:37:17.950 --> 00:37:20.800 and this isn't even one of my good days-- 00:37:20.800 --> 00:37:23.840 acentric point groups-- 00:37:23.840 --> 00:37:25.605 the noncentrosymmetric point groups. 00:37:38.520 --> 00:37:43.400 And, these, in principle, can be all different. 00:38:10.710 --> 00:38:13.210 So we can't get away with just looking at the restrictions of 00:38:13.210 --> 00:38:16.200 Laue groups as we did with second rank tensors. 00:38:16.200 --> 00:38:19.910 We're going to have to look at every one of the point groups 00:38:19.910 --> 00:38:21.840 that have no inversion center. 00:38:21.840 --> 00:38:25.130 And these can, in principle, be different, and 00:38:25.130 --> 00:38:26.380 they usually are. 00:38:45.250 --> 00:38:49.370 Let me give you a physically-- 00:38:49.370 --> 00:38:53.980 physical feeling for why the piezoelectric effect cannot 00:38:53.980 --> 00:38:59.130 exist for a crystal that has an inversion center in it. 00:38:59.130 --> 00:39:04.050 Let's suppose we cut a wafer from our single crystal, 00:39:04.050 --> 00:39:05.300 regardless of orientation. 00:39:13.380 --> 00:39:15.870 And we subject it to a compressive stress. 00:39:23.490 --> 00:39:24.690 We will see-- 00:39:24.690 --> 00:39:29.080 and I've got a primer on some simple electrostatics that may 00:39:29.080 --> 00:39:29.990 be useful-- 00:39:29.990 --> 00:39:42.590 that a polarization in the x1 direction manifests itself as 00:39:42.590 --> 00:39:48.430 a charge per unit area on the surface and an opposite charge 00:39:48.430 --> 00:39:49.990 on the other surface. 00:39:49.990 --> 00:39:58.090 So P1 corresponds, numerically and physically, to a charge 00:39:58.090 --> 00:40:06.200 per unit area on a surface that's normal to x1. 00:40:16.450 --> 00:40:20.040 OK now let's suppose that that crystal has an inversion 00:40:20.040 --> 00:40:21.290 center in it. 00:40:23.920 --> 00:40:29.050 And that means we should be able to invert the crystal and 00:40:29.050 --> 00:40:33.810 measure exactly the same charge per unit area. 00:40:33.810 --> 00:40:38.550 Well, if this is surface a, and this is surface b, we can 00:40:38.550 --> 00:40:44.500 imagine ourselves doing the thought experiment in which we 00:40:44.500 --> 00:40:45.880 invert the crystal. 00:40:45.880 --> 00:40:51.270 So this surface becomes b and this surface becomes a. 00:40:51.270 --> 00:40:55.040 And we apply the same sigma along x1-- 00:40:55.040 --> 00:40:56.310 a compressive stress. 00:41:00.320 --> 00:41:03.300 It's really doing exactly the same thing that we were doing 00:41:03.300 --> 00:41:05.550 before-- inverting the crystal. 00:41:05.550 --> 00:41:09.000 And now we say that the property has to stay the same. 00:41:09.000 --> 00:41:15.470 So now we are demanding that surface b have a positive 00:41:15.470 --> 00:41:19.180 charge per unit area, and surface a develop a negative 00:41:19.180 --> 00:41:20.960 charge per unit area-- 00:41:20.960 --> 00:41:23.740 completely the reverse of what happened here. 00:41:23.740 --> 00:41:25.820 But yet what we're doing is just squeezing the plate. 00:41:25.820 --> 00:41:27.630 The crystal can't tell which side is up, 00:41:27.630 --> 00:41:28.460 which side is down. 00:41:28.460 --> 00:41:31.330 We're applying the same compressive effect. 00:41:31.330 --> 00:41:35.110 So these two distributions of charge are not the same and 00:41:35.110 --> 00:41:38.810 can be the same only if that charge is zero. 00:41:42.480 --> 00:41:45.135 So physically we can see why the crystal can't be 00:41:45.135 --> 00:41:45.850 piezoelectric. 00:41:45.850 --> 00:41:47.540 If you don't like to invert the crystal-- 00:41:47.540 --> 00:41:51.640 if you can't do that physically-- 00:41:51.640 --> 00:41:54.980 one thing we can do is leave the crystal alone and invert 00:41:54.980 --> 00:41:57.680 the compressive stress. 00:41:57.680 --> 00:42:01.070 So this is sigma top and this is sigma bottom which is 00:42:01.070 --> 00:42:04.930 exactly equal to it but opposite in direction. 00:42:04.930 --> 00:42:06.830 If we invert the stress relative-- 00:42:12.640 --> 00:42:13.915 relative to the crystal-- 00:42:16.940 --> 00:42:21.448 then what we've done is to take sigma top-- 00:42:21.448 --> 00:42:23.080 leave the crystal alone-- 00:42:23.080 --> 00:42:26.535 so this is now the sigma that was squishing the top. 00:42:26.535 --> 00:42:34.460 Here's the sigma that was squishing the bottom and well, 00:42:34.460 --> 00:42:37.090 that's clearly doing the same thing. 00:42:37.090 --> 00:42:40.980 So inverting the stress, relative to the crystal, is 00:42:40.980 --> 00:42:43.130 the same as inverting the crystal 00:42:43.130 --> 00:42:44.960 relative to the stress. 00:42:44.960 --> 00:42:48.980 And nothing is supposed to happen but clearly the sign of 00:42:48.980 --> 00:42:52.430 the charge is going to be reversed. 00:42:52.430 --> 00:42:53.410 So that just won't work. 00:42:53.410 --> 00:42:54.660 The charge has to disappear. 00:43:02.370 --> 00:43:04.770 OK this is the direct piezoelectric effect. 00:43:04.770 --> 00:43:08.950 It can exist only for crystals that lack an inversion center. 00:43:08.950 --> 00:43:11.983 Let me mention some other piezoelectric effects. 00:43:17.970 --> 00:43:28.760 So we mentioned that a polarization, given by a third 00:43:28.760 --> 00:43:35.620 rank tensor times an applied stress, sigma ij, is the 00:43:35.620 --> 00:43:37.470 direct piezoelectric photoelectric effect. 00:43:43.160 --> 00:43:55.020 However, if we have a stress, we also have a strain. 00:43:55.020 --> 00:43:58.750 So another piezoelectric effect, which is not dignified 00:43:58.750 --> 00:44:04.200 with a name, is that we can apply a stress that creates a 00:44:04.200 --> 00:44:06.910 given state of strain. 00:44:06.910 --> 00:44:12.820 And we again, we'll get a polarization per unit-- 00:44:12.820 --> 00:44:15.440 a charge per unit area of polarization. 00:44:15.440 --> 00:44:18.690 And this is a set of coefficients which are 00:44:18.690 --> 00:44:20.900 represented by e. 00:44:20.900 --> 00:44:25.870 And we can write a tensor relation of this sort-- that 00:44:25.870 --> 00:44:30.040 components of polarization, charge per unit area, normal 00:44:30.040 --> 00:44:35.910 to axis xi, is given by a linear combination of every 00:44:35.910 --> 00:44:38.700 one of the components of strain. 00:44:38.700 --> 00:44:43.090 This is something that is not dignified by any name. 00:44:51.300 --> 00:44:57.010 And clearly, the piezoelectric coefficients, e, have to be 00:44:57.010 --> 00:45:03.570 related to the piezoelectric coefficients, d, by the 00:45:03.570 --> 00:45:05.690 elastic constant somehow-- 00:45:05.690 --> 00:45:07.360 unless we are doing the same thing. 00:45:07.360 --> 00:45:11.310 We've got to create a stress, operationally, in order to 00:45:11.310 --> 00:45:12.910 create the strain. 00:45:12.910 --> 00:45:15.600 So a stress creates a strain. 00:45:15.600 --> 00:45:19.870 And that has to describe the same polarization if we've 00:45:19.870 --> 00:45:21.100 done the same thing. 00:45:21.100 --> 00:45:24.740 So the e's and the d's have to be related. 00:45:24.740 --> 00:45:30.630 Another piezoelectric effect involves a applying an 00:45:30.630 --> 00:45:33.950 electric field. 00:45:33.950 --> 00:45:38.530 And if we apply an electric field, we will in general 00:45:38.530 --> 00:45:39.780 create a strain. 00:45:46.070 --> 00:45:53.870 This direct piezoelectric effect is three equations that 00:45:53.870 --> 00:46:01.960 involve a linear combination of nine elements of stress. 00:46:01.960 --> 00:46:06.900 This piezoelectric relation here is going to involve 00:46:06.900 --> 00:46:12.130 writing an equation for each of nine elements of strain. 00:46:12.130 --> 00:46:14.850 And there would be three components of the electric 00:46:14.850 --> 00:46:15.750 field vector. 00:46:15.750 --> 00:46:17.070 So it'll be three this way. 00:46:17.070 --> 00:46:20.135 So both of these relations involve 27 elements. 00:46:22.910 --> 00:46:25.960 So again we're going to have a third rank tensor because the 00:46:25.960 --> 00:46:27.360 electric field is a vector-- 00:46:27.360 --> 00:46:28.930 strain is a tensor. 00:46:28.930 --> 00:46:33.510 And now the mind boggling thing that I will leave you 00:46:33.510 --> 00:46:42.280 with is exactly the same set of coefficients, d ij describe 00:46:42.280 --> 00:46:44.670 both effects. 00:46:44.670 --> 00:46:47.120 And this effect is given a special name. 00:46:47.120 --> 00:46:49.740 This is called the converse piezoelectric effect. 00:46:54.880 --> 00:47:01.750 And it's not at all clear how strain, in terms of electric 00:47:01.750 --> 00:47:06.930 field, should be described in terms of the same array of 27 00:47:06.930 --> 00:47:10.320 numbers as polarization per unit 00:47:10.320 --> 00:47:12.780 volume in terms of stress. 00:47:12.780 --> 00:47:18.070 And that is an argument that is based on thermodynamics. 00:47:18.070 --> 00:47:21.790 So that's a good place to quit. 00:47:21.790 --> 00:47:26.680 We'll have some fun in looking at a few symmetry restrictions 00:47:26.680 --> 00:47:32.670 imposed on the acentric point groups, 21 of them. 00:47:32.670 --> 00:47:34.210 We're not going to do all 21. 00:47:34.210 --> 00:47:37.520 If you know how to do one or two, and marvel over the 00:47:37.520 --> 00:47:41.470 results, you can do them all. 00:47:41.470 --> 00:47:45.630 But one thing that we'll be asking is how can we represent 00:47:45.630 --> 00:47:50.430 the variation of the piezoelectric effect with 00:47:50.430 --> 00:47:51.820 orientation of the crystal. 00:47:54.430 --> 00:47:58.155 And the answer is we can't. 00:47:58.155 --> 00:47:58.810 We can't. 00:47:58.810 --> 00:48:04.770 Because what is the direction of what we're doing? 00:48:04.770 --> 00:48:08.790 There are nine elements of stress, not just three 00:48:08.790 --> 00:48:11.930 components of a vector which defines an orientation. 00:48:11.930 --> 00:48:14.000 There are nine parameters here. 00:48:14.000 --> 00:48:17.620 And there are three components to the charge per unit area-- 00:48:17.620 --> 00:48:19.230 that's a vector. 00:48:19.230 --> 00:48:22.600 So all that one can do for these higher ranked tensors is 00:48:22.600 --> 00:48:27.870 to look for a special, generalized force. 00:48:27.870 --> 00:48:33.250 And then perhaps for a very special form of that 00:48:33.250 --> 00:48:36.220 generalized force, that tensor, look at how the 00:48:36.220 --> 00:48:37.870 polarization changes. 00:48:37.870 --> 00:48:40.610 But you're going to have to look at six different 00:48:40.610 --> 00:48:42.760 representation surfaces-- 00:48:42.760 --> 00:48:45.440 each one which will describe a different 00:48:45.440 --> 00:48:47.530 piezoelectric effect. 00:48:47.530 --> 00:48:51.530 So this will become less simple in many respects. 00:48:51.530 --> 00:48:55.370 But correspondingly, more interesting because we'll have 00:48:55.370 --> 00:49:01.360 some really wild, mind boggling representation 00:49:01.360 --> 00:49:05.740 surfaces for piezoelectric effects. 00:49:05.740 --> 00:49:12.120 OK so we'll stop there and resume on Thursday. 00:49:12.120 --> 00:49:15.350 And I'll let you get back to the MRS if that's where you 00:49:15.350 --> 00:49:16.600 earlier today.