1 00:00:08,189 --> 00:00:12,620 PROFESSOR: Any questions before we push bravely onward? 2 00:00:16,960 --> 00:00:19,370 OK if there's no question on any of this I shall 3 00:00:19,370 --> 00:00:20,330 clean off the board. 4 00:00:20,330 --> 00:00:25,550 And the last thing I wanted to do in connection with thermal 5 00:00:25,550 --> 00:00:31,030 expansion is to examine how the macroscopic linear thermal 6 00:00:31,030 --> 00:00:35,830 expansion coefficient relates to the nature of the potential 7 00:00:35,830 --> 00:00:37,080 minimum between-- 8 00:00:40,170 --> 00:00:43,970 in the attraction between-- neighboring atoms. 9 00:00:43,970 --> 00:00:52,280 So let me suppose we have a pair of atoms separated by 10 00:00:52,280 --> 00:00:59,350 some distance, d, and there is a cohesive force between them 11 00:00:59,350 --> 00:01:03,040 or else the material would not be in a solid state. 12 00:01:03,040 --> 00:01:06,115 And the nature of the potential-- 13 00:01:08,660 --> 00:01:10,640 and I'm going to call the distance 14 00:01:10,640 --> 00:01:12,920 between the atoms, x-- 15 00:01:12,920 --> 00:01:16,090 and this is going to be the energy. 16 00:01:20,740 --> 00:01:29,990 And I'm measuring now relative to-- 17 00:01:29,990 --> 00:01:35,150 let me call this d as I did over here-- so the energy as a 18 00:01:35,150 --> 00:01:39,800 function of d is going to consist of a very short range, 19 00:01:39,800 --> 00:01:43,930 repulsive interaction, that rises very, very rapidly when 20 00:01:43,930 --> 00:01:49,370 you attempt to squeeze the atoms any closer than a fairly 21 00:01:49,370 --> 00:01:51,190 small value. 22 00:01:51,190 --> 00:01:56,560 And then there'll be some sort of cohesive energy. 23 00:01:56,560 --> 00:02:00,340 And everybody loves to model ionic compounds because in an 24 00:02:00,340 --> 00:02:09,130 ionic compound this cohesive force would be purely a 25 00:02:09,130 --> 00:02:13,840 Coulombic force that would be, in rationalized MKS units, 1 26 00:02:13,840 --> 00:02:20,590 over 4 pi epsilon zero times the product of the charges and 27 00:02:20,590 --> 00:02:24,100 divided by the distance between them. 28 00:02:27,660 --> 00:02:30,480 Taking those two potentials together, and you've all seen 29 00:02:30,480 --> 00:02:37,460 this many times before, there is a minimum in the energy. 30 00:02:37,460 --> 00:02:38,890 And then absolute zero-- 31 00:02:38,890 --> 00:02:42,340 that is the separation at which this 32 00:02:42,340 --> 00:02:44,130 atom pair would sit-- 33 00:02:44,130 --> 00:02:49,180 and this would be the binding energy of that ionic pair. 34 00:02:49,180 --> 00:02:52,070 If you carry this through to get the entire energy of the 35 00:02:52,070 --> 00:02:56,900 crystal, you find that the energy of the crystal goes as 36 00:02:56,900 --> 00:03:00,296 1 minus 1 over-- 37 00:03:00,296 --> 00:03:02,060 haven't introduced this term yet, so 38 00:03:02,060 --> 00:03:05,310 let's hold off on that. 39 00:03:05,310 --> 00:03:11,730 What I'm going to do now is to take a look at this potential 40 00:03:11,730 --> 00:03:14,870 and expand it around the equilibrium separation just 41 00:03:14,870 --> 00:03:18,350 with an empirical collection of terms. 42 00:03:18,350 --> 00:03:22,250 But from the shape of this well, we can see why the bond 43 00:03:22,250 --> 00:03:25,790 length should increase as the energy of 44 00:03:25,790 --> 00:03:28,520 the atomic pair increases. 45 00:03:28,520 --> 00:03:36,020 At absolute zero, the separation will be this 46 00:03:36,020 --> 00:03:38,680 equilibrium separation, d zero. 47 00:03:38,680 --> 00:03:41,130 But as you increase the temperature and pump 48 00:03:41,130 --> 00:03:45,070 vibrational energy into this pair of atoms, they will get a 49 00:03:45,070 --> 00:03:48,860 certain increase in energy. 50 00:03:48,860 --> 00:03:53,330 And as a result, the pair will be able to vibrate between 51 00:03:53,330 --> 00:03:54,520 these two limits. 52 00:03:54,520 --> 00:03:57,610 Put in some more thermal energy by raising the 53 00:03:57,610 --> 00:04:00,830 temperature, and the pair will be able to vibrate between 54 00:04:00,830 --> 00:04:02,740 these limits-- 55 00:04:02,740 --> 00:04:04,510 larger limits. 56 00:04:04,510 --> 00:04:09,440 And so the average position of the atom, the separation of 57 00:04:09,440 --> 00:04:12,740 the atoms, is something that is going to increase with 58 00:04:12,740 --> 00:04:14,630 increasing temperature. 59 00:04:14,630 --> 00:04:17,800 And that is purely a consequence of the asymmetry 60 00:04:17,800 --> 00:04:21,660 in this potential well. 61 00:04:21,660 --> 00:04:25,430 If this were a Coulombic bond, we know exactly how to model 62 00:04:25,430 --> 00:04:27,190 the attractive part. 63 00:04:27,190 --> 00:04:30,460 The repulsive part is a little harder to model because that 64 00:04:30,460 --> 00:04:33,060 is something that's inherently quantum mechanical. 65 00:04:33,060 --> 00:04:36,160 As the ion pair gets close together, the electrons start 66 00:04:36,160 --> 00:04:38,910 to perturb each other's orbitals. 67 00:04:38,910 --> 00:04:43,890 And a very common thing to do is to model this as 1 over d 68 00:04:43,890 --> 00:04:46,290 to the sum power n. 69 00:04:46,290 --> 00:04:50,530 And n is typically on the order of 8 to 12. 70 00:04:50,530 --> 00:04:56,210 So this is something that rises very, very steeply. 71 00:04:56,210 --> 00:05:00,170 I was once at the Gordon conference where a fist fight 72 00:05:00,170 --> 00:05:03,200 almost broke out because somebody was giving a 73 00:05:03,200 --> 00:05:06,360 presentation and started from the beginning by introducing 74 00:05:06,360 --> 00:05:07,870 this model to the potential well. 75 00:05:07,870 --> 00:05:10,950 And he says, "As usual, we'll write this as a power law, 1 76 00:05:10,950 --> 00:05:15,260 over d to the n." And somebody in a front row jumped up and 77 00:05:15,260 --> 00:05:17,550 said, "what do you mean usually? 78 00:05:17,550 --> 00:05:21,490 Sensible people use exponentials." And the speaker 79 00:05:21,490 --> 00:05:25,920 said, "No, I'm going to use a power law." That's dumb. 80 00:05:25,920 --> 00:05:27,700 Exponentials work better. 81 00:05:27,700 --> 00:05:30,770 Power laws, exponentials, power laws, exponentials. 82 00:05:30,770 --> 00:05:35,120 But in point of fact they're both grossly empirical and one 83 00:05:35,120 --> 00:05:36,460 is not better than the other. 84 00:05:36,460 --> 00:05:40,930 The algebra changes a little bit, but people get very, very 85 00:05:40,930 --> 00:05:43,510 emotionally attached to their physical models. 86 00:05:43,510 --> 00:05:46,205 AUDIENCE: [INAUDIBLE] 87 00:05:46,205 --> 00:05:48,153 law in the exponential. 88 00:05:48,153 --> 00:05:50,831 PROFESSOR: Good, you should've been there to 89 00:05:50,831 --> 00:05:52,536 break up the fight. 90 00:05:52,536 --> 00:05:53,997 If you'd gotten between them, you'd have been likely punched 91 00:05:53,997 --> 00:05:54,971 in the nose. 92 00:05:54,971 --> 00:05:56,221 [INAUDIBLE] 93 00:05:58,380 --> 00:06:01,180 OK so I'm going to use the power law because as we've 94 00:06:01,180 --> 00:06:04,010 seen they're the same thing as exponentials. 95 00:06:04,010 --> 00:06:07,390 So now what I'm going to do is expand this in a purely 96 00:06:07,390 --> 00:06:09,340 empirical way-- 97 00:06:09,340 --> 00:06:13,650 measuring distance from the minimum of the potential well. 98 00:06:13,650 --> 00:06:17,680 So what I'm going to say is that the potential as a 99 00:06:17,680 --> 00:06:20,670 function of x-- and now I won't call it d because I'm 100 00:06:20,670 --> 00:06:25,150 going to measure distance, x, from the minimum in this well. 101 00:06:25,150 --> 00:06:27,580 I'm going to say V is a function of x. 102 00:06:27,580 --> 00:06:31,970 There is a well in which the-- 103 00:06:31,970 --> 00:06:33,960 which keeps the atoms separate. 104 00:06:33,960 --> 00:06:37,270 I'll assume that is a parabolic potential well, so 105 00:06:37,270 --> 00:06:42,180 it'll go as first some constant c times x squared. 106 00:06:42,180 --> 00:06:44,070 Then the thing that causes thermal 107 00:06:44,070 --> 00:06:46,840 expansion is the asymmetry. 108 00:06:46,840 --> 00:06:50,660 And I want the asymmetry to increase the energy more 109 00:06:50,660 --> 00:06:54,590 rapidly on the side of negative x than on the side of 110 00:06:54,590 --> 00:06:55,920 positive x. 111 00:06:55,920 --> 00:06:58,335 So I'll put in a term, g x cubed. 112 00:07:00,960 --> 00:07:04,100 And I put a negative sign in here because I want it to go 113 00:07:04,100 --> 00:07:07,580 up as x becomes negative. 114 00:07:07,580 --> 00:07:12,580 That means the ion pairs are decreasing at separation. 115 00:07:12,580 --> 00:07:19,750 And finally, a very fine touch, if x becomes very 116 00:07:19,750 --> 00:07:23,030 large, the potential well should soften. 117 00:07:23,030 --> 00:07:26,700 The edges of this are starting to level out and get lower 118 00:07:26,700 --> 00:07:30,730 than the purely parabolic part for very large separations. 119 00:07:30,730 --> 00:07:32,730 So that's what I'll use as a very simple 120 00:07:32,730 --> 00:07:36,640 model for my potential. 121 00:07:36,640 --> 00:07:41,910 This is what gives the asymmetry, and this is what 122 00:07:41,910 --> 00:07:43,160 gives softening. 123 00:07:45,860 --> 00:07:51,700 And now I'm going to ask, what is the average value of x? 124 00:07:51,700 --> 00:07:58,430 The probability of finding a particular x is going to be 125 00:07:58,430 --> 00:08:09,180 proportional to e to the minus V of x over kt. 126 00:08:09,180 --> 00:08:13,840 So I can say that the average value of x is going to be the 127 00:08:13,840 --> 00:08:19,640 integral from minus infinity to plus infinity of the value 128 00:08:19,640 --> 00:08:23,260 that I'm trying to find the average of, namely x times the 129 00:08:23,260 --> 00:08:25,480 probability of finding that x. 130 00:08:25,480 --> 00:08:28,670 And that's going to e to the minus V of x over kt. 131 00:08:33,690 --> 00:08:36,929 And I will integrate that over x. 132 00:08:36,929 --> 00:08:40,570 And then I will normalize by a term which is the integral of 133 00:08:40,570 --> 00:08:44,230 the probability from minus infinity to plus infinity of e 134 00:08:44,230 --> 00:08:48,420 to the minus V of x over kt. 135 00:08:48,420 --> 00:08:50,440 So that's a general expression for an average. 136 00:08:50,440 --> 00:08:52,820 It's the thing you want to average times the probability 137 00:08:52,820 --> 00:08:57,220 of finding the particular value divided by the integral 138 00:08:57,220 --> 00:08:58,470 of the probabilities. 139 00:09:02,130 --> 00:09:05,220 OK now let me do some very straightforward things and I'm 140 00:09:05,220 --> 00:09:08,600 just going to rattle these off very quickly. 141 00:09:08,600 --> 00:09:13,350 We'll expand this in terms of a series of terms. 142 00:09:13,350 --> 00:09:22,830 And I'm going to expand both the numerator and denominator 143 00:09:22,830 --> 00:09:26,640 using the approximation that e to the u is approximately 144 00:09:26,640 --> 00:09:30,380 equal to 1 plus u. 145 00:09:30,380 --> 00:09:32,090 So the numerator-- 146 00:09:32,090 --> 00:09:39,930 x times e to the minus cx squared over kt. 147 00:09:39,930 --> 00:09:45,620 I'm going to take the higher order terms and say that e to 148 00:09:45,620 --> 00:09:54,080 the gx cubed over kt times e to the minus-- 149 00:09:54,080 --> 00:09:56,840 this is a minus sign out in front-- 150 00:09:56,840 --> 00:10:02,300 times e to the fx fourth over kt, can be approximated by, 151 00:10:02,300 --> 00:10:06,660 since these are very, very small terms, 1 plus g x cubed 152 00:10:06,660 --> 00:10:14,590 over kt times 1 plus fx fourth over kt. 153 00:10:14,590 --> 00:10:18,530 So that's going to be, if I put an x e to the minus cx 154 00:10:18,530 --> 00:10:25,000 squared, I'm going to have a term 1 plus g x cubed over kt, 155 00:10:25,000 --> 00:10:29,180 plus f x fourth over kt. 156 00:10:29,180 --> 00:10:33,210 And then a very, very minuscule term, f times g 157 00:10:33,210 --> 00:10:36,650 times x to the 7th over kt. 158 00:10:36,650 --> 00:10:37,900 And I'm going to throw that out. 159 00:10:44,000 --> 00:10:47,460 If I look at these integrals one at a time, the integral 160 00:10:47,460 --> 00:10:52,560 from minus infinity to plus infinity of x e to the minus 161 00:10:52,560 --> 00:10:56,230 cx squared over k t, that's the first term in 162 00:10:56,230 --> 00:10:58,720 expansion, times dx. 163 00:10:58,720 --> 00:11:00,860 That's going to be zero. 164 00:11:00,860 --> 00:11:03,180 Because for positive x, the thing that we're 165 00:11:03,180 --> 00:11:05,030 integrating is plus. 166 00:11:05,030 --> 00:11:08,590 For negative x, the thing that we're integrating has the same 167 00:11:08,590 --> 00:11:11,000 exponential but has a minus sign. 168 00:11:11,000 --> 00:11:13,830 So that integral is zero. 169 00:11:13,830 --> 00:11:25,250 So the next term I want to look at is x times 170 00:11:25,250 --> 00:11:28,330 gx cubed over kt. 171 00:11:28,330 --> 00:11:34,880 And for this I turned to my handy book of integrals. 172 00:11:34,880 --> 00:11:39,030 The integral from zero to infinity of x to the 2n, where 173 00:11:39,030 --> 00:11:44,400 that quantity is even, times e to the minus a x squared dx, 174 00:11:44,400 --> 00:11:49,010 is equal to, as you all know, 1 times 3 times 5 all the way 175 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 176 00:11:56,270 --> 00:11:59,660 n times the square root of pi over a. 177 00:11:59,660 --> 00:12:04,080 And proof of that the integral is left as an exercise to the 178 00:12:04,080 --> 00:12:06,290 reckless student. 179 00:12:06,290 --> 00:12:06,625 Yes? 180 00:12:06,625 --> 00:12:07,952 AUDIENCE: The x cubed [INAUDIBLE]. 181 00:12:10,928 --> 00:12:14,896 PROFESSOR: That's why this is a 2n here. 182 00:12:14,896 --> 00:12:17,624 So I can say that the x cubed term is--but we don't have an 183 00:12:17,624 --> 00:12:18,368 x cubed term. 184 00:12:18,368 --> 00:12:24,320 The next term is going to be equal to x fourth. 185 00:12:24,320 --> 00:12:27,086 And this I'm going to drop because that's 186 00:12:27,086 --> 00:12:28,610 going to be x fifth. 187 00:12:28,610 --> 00:12:43,730 So the integral from minus infinity to plus infinity of x 188 00:12:43,730 --> 00:12:52,850 to the fourth power times g over kt times e to the minus 189 00:12:52,850 --> 00:12:56,480 cx squared over kt dx. 190 00:12:56,480 --> 00:12:59,110 If I use this formula, turns out to be a 191 00:12:59,110 --> 00:13:00,480 very, very simple thing. 192 00:13:00,480 --> 00:13:05,520 It turns out to be 3/4 times the square root of pi times g 193 00:13:05,520 --> 00:13:15,980 times kt over c to the 5/2. 194 00:13:15,980 --> 00:13:20,010 And kt is to the power 3/2. 195 00:13:23,660 --> 00:13:25,880 So that is-- 196 00:13:25,880 --> 00:13:32,520 if I throw away the term in x times x fourth which is odd, 197 00:13:32,520 --> 00:13:36,800 that will vanish and I've then thrown out only a really large 198 00:13:36,800 --> 00:13:39,530 term in x to the 7th. 199 00:13:39,530 --> 00:13:56,570 For the denominator, the integral that is there before 200 00:13:56,570 --> 00:14:03,480 us is the integral from minus infinity to plus infinity of e 201 00:14:03,480 --> 00:14:11,810 to the minus cx squared over kt times 1 plus g 202 00:14:11,810 --> 00:14:16,450 x cubed over kt. 203 00:14:16,450 --> 00:14:19,580 This thing is going to vanish because x is odd. 204 00:14:19,580 --> 00:14:24,530 And then I'll have plus fx fourth over kt. 205 00:14:24,530 --> 00:14:27,320 That will not vanish but it's going to be a relatively small 206 00:14:27,320 --> 00:14:29,390 term, so I'll go like that. 207 00:14:29,390 --> 00:14:31,770 Which means I just have to integrate 208 00:14:31,770 --> 00:14:33,760 this Gaussian factor. 209 00:14:33,760 --> 00:14:39,070 And that turns out to be equal to 2 times 1/2 times square 210 00:14:39,070 --> 00:14:47,850 root of pi kT pi times T over c. 211 00:14:47,850 --> 00:14:49,100 So this is the denominator. 212 00:14:52,180 --> 00:14:57,645 And if I take the ratio of those two integrals, I have 213 00:14:57,645 --> 00:15:04,230 that the average value of x will be equal to 3/4 square 214 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, 215 00:15:17,950 --> 00:15:24,200 divided by the square root of pi times kt over c 216 00:15:24,200 --> 00:15:26,630 to the power 1/2. 217 00:15:26,630 --> 00:15:29,560 And if I carry through the algebra correctly, this turns 218 00:15:29,560 --> 00:15:35,745 out to be 3/4 of g over c squared times kt. 219 00:15:42,540 --> 00:15:45,730 So there is the average distance between this pair of 220 00:15:45,730 --> 00:15:49,480 atoms as a function of temperature in terms of 221 00:15:49,480 --> 00:15:53,420 characteristics of the potential. 222 00:15:53,420 --> 00:15:57,040 The first thing that is very gratifying is that the bond 223 00:15:57,040 --> 00:16:00,240 length, according to the simple model, should increase 224 00:16:00,240 --> 00:16:02,160 linearly with temperature. 225 00:16:02,160 --> 00:16:05,970 So it tells us we would expect things to expand linearly with 226 00:16:05,970 --> 00:16:11,260 temperature which corresponds to reality. 227 00:16:11,260 --> 00:16:15,360 The next thing we'll see is that the amount of expansion 228 00:16:15,360 --> 00:16:19,360 goes inversely as the square. 229 00:16:19,360 --> 00:16:20,810 That's a strong dependence-- 230 00:16:20,810 --> 00:16:23,060 inversely as the square of the 231 00:16:23,060 --> 00:16:25,930 parabolic part of our potential. 232 00:16:25,930 --> 00:16:29,230 That was the c-- c was the coefficient in front of cx 233 00:16:29,230 --> 00:16:31,300 squared in our potential. 234 00:16:31,300 --> 00:16:35,800 So the stronger the bond, the smaller the thermal expansion 235 00:16:35,800 --> 00:16:40,050 coefficient is in proportion to the reciprocal of the 236 00:16:40,050 --> 00:16:43,360 square of that term. 237 00:16:43,360 --> 00:16:46,840 And then finally, the last important term to survive is 238 00:16:46,840 --> 00:16:48,650 g, the asymmetry. 239 00:16:48,650 --> 00:16:51,280 And the thermal expansion coefficient should increase 240 00:16:51,280 --> 00:16:55,070 with the first power of the term that describes the 241 00:16:55,070 --> 00:16:56,560 asymmetry of the potential well. 242 00:17:00,010 --> 00:17:02,970 So your intuition would tell you that 243 00:17:02,970 --> 00:17:05,140 that's the way the thermal-- 244 00:17:05,140 --> 00:17:07,380 linear thermal expansion coefficient should vary. 245 00:17:07,380 --> 00:17:11,250 The interesting thing is that it depends more strongly on 246 00:17:11,250 --> 00:17:14,450 the parameter that gives the parabolic part of the 247 00:17:14,450 --> 00:17:15,710 potential well. 248 00:17:15,710 --> 00:17:18,960 And secondly, that it does predict a thermal expansion 249 00:17:18,960 --> 00:17:21,680 coefficient that increases-- 250 00:17:21,680 --> 00:17:24,579 the separation that increases with-- 251 00:17:24,579 --> 00:17:25,960 in proportion to temperature. 252 00:17:25,960 --> 00:17:29,250 And therefore there is a constant linear thermal 253 00:17:29,250 --> 00:17:32,300 expansion coefficient. 254 00:17:32,300 --> 00:17:34,730 Now all this is true at elevated temperatures. 255 00:17:34,730 --> 00:17:38,340 At low temperatures, thermodynamics kicks in. 256 00:17:38,340 --> 00:17:40,990 And it turns out that the linear thermal expansion 257 00:17:40,990 --> 00:17:45,820 coefficient, as does energies of vibrating atoms, must go to 258 00:17:45,820 --> 00:17:50,880 zero as temperature approaches absolute zero. 259 00:17:50,880 --> 00:17:55,920 So a typical variation of the thermal expansion coefficient 260 00:17:55,920 --> 00:17:59,050 with temperature is something that does this. 261 00:17:59,050 --> 00:18:02,330 The same way as-- the same sort of variation as things 262 00:18:02,330 --> 00:18:03,750 like heat capacity. 263 00:18:03,750 --> 00:18:07,550 And this temperature in here is on the order of the Debye 264 00:18:07,550 --> 00:18:09,780 temperature of the particular material. 265 00:18:15,930 --> 00:18:21,650 All right, so much for linear thermal expansion. 266 00:18:21,650 --> 00:18:28,730 And I would now like to turn to something much more 267 00:18:28,730 --> 00:18:30,110 interesting. 268 00:18:30,110 --> 00:18:35,920 And that is physical properties that involve 269 00:18:35,920 --> 00:18:38,595 property tensors of rank higher than two. 270 00:18:50,420 --> 00:18:55,140 One of the important and more interesting of these is a 271 00:18:55,140 --> 00:18:56,560 property called piezoelectricity. 272 00:19:04,350 --> 00:19:08,490 Piezo means pressure, so the property literally means 273 00:19:08,490 --> 00:19:12,770 pressure electricity or pressure induced charge. 274 00:19:19,470 --> 00:19:20,720 Now what is the charge? 275 00:19:24,450 --> 00:19:29,810 First of all, this does not describe a property where-- if 276 00:19:29,810 --> 00:19:33,320 I were to pick up a chunk of crystal, and I drop it because 277 00:19:33,320 --> 00:19:35,890 I just got an electric charge. 278 00:19:35,890 --> 00:19:39,510 These are not mobile charges, these are bound charges. 279 00:19:39,510 --> 00:19:46,120 And the reason for the induced charge is the fact that we are 280 00:19:46,120 --> 00:19:50,740 squeezing or stretching an ionic material, and that's 281 00:19:50,740 --> 00:19:53,390 moving charged ions around. 282 00:19:53,390 --> 00:19:56,960 So what we're going to do if that process involves motion 283 00:19:56,960 --> 00:20:01,600 of charged ions is to induce a dipole moment per unit volume. 284 00:20:04,160 --> 00:20:08,550 I have a little primer on some quantities in electromagnetic 285 00:20:08,550 --> 00:20:10,440 theory and I-- 286 00:20:10,440 --> 00:20:13,230 I brought them with me upstairs to look them over and 287 00:20:13,230 --> 00:20:14,390 I forgot to bring them down. 288 00:20:14,390 --> 00:20:16,320 But I don't think we'll need them in what's 289 00:20:16,320 --> 00:20:17,990 left of this hour. 290 00:20:17,990 --> 00:20:25,590 So let me say that for matter and bulk, the quantity that is 291 00:20:25,590 --> 00:20:28,213 involved in piezoelectricity is the polarization. 292 00:20:30,740 --> 00:20:34,260 And polarization is dipole moment per unit volume. 293 00:20:44,600 --> 00:20:47,930 And a dipole moment, to refresh your memory, is that 294 00:20:47,930 --> 00:20:55,670 if I have a positive charge plus q and negative charge 295 00:20:55,670 --> 00:21:01,080 minus q separated by a distance, d. 296 00:21:01,080 --> 00:21:05,940 The dipole moment, which I'll indicate by a lowercase p, has 297 00:21:05,940 --> 00:21:10,790 a vector quality and it's equal to the magnitude of one 298 00:21:10,790 --> 00:21:17,240 of these charges times the distance between them. 299 00:21:17,240 --> 00:21:22,140 And we'll define the distance as going from the positive 300 00:21:22,140 --> 00:21:24,875 charge to the negative charge. 301 00:21:29,300 --> 00:21:34,220 That's a crazy way of defining a pair of charges of equal but 302 00:21:34,220 --> 00:21:36,480 opposite sign. 303 00:21:36,480 --> 00:21:40,050 The reason for defining a quantity like the dipole 304 00:21:40,050 --> 00:21:44,150 moment is that this combination of charge and 305 00:21:44,150 --> 00:21:46,910 separation of the charges comes up in all sorts of 306 00:21:46,910 --> 00:21:51,240 problems in electrostatics and electrodynamics. 307 00:21:51,240 --> 00:21:52,900 And this is the delight of people who 308 00:21:52,900 --> 00:21:55,270 teach elementary physics. 309 00:21:55,270 --> 00:21:59,610 You are given, on the first quiz, a question that says-- 310 00:21:59,610 --> 00:22:04,350 calculate the electric field that is created by a set of 311 00:22:04,350 --> 00:22:08,210 dipoles on the Great Court which spell out MIT. 312 00:22:08,210 --> 00:22:12,130 And we can come up with all sorts of crazy configurations 313 00:22:12,130 --> 00:22:16,350 of dipoles and then calculate the field that they produce. 314 00:22:16,350 --> 00:22:20,420 And the field turns out to be something that always involves 315 00:22:20,420 --> 00:22:26,120 the product of charge and separation in a vector sense. 316 00:22:26,120 --> 00:22:30,450 Probably the best illustration of this would be if I have a 317 00:22:30,450 --> 00:22:34,710 dipole charge plus q, charge minus q, separated by d. 318 00:22:34,710 --> 00:22:38,330 And I put it in the electric field. 319 00:22:38,330 --> 00:22:41,780 The electric field, which is defined as the force per unit 320 00:22:41,780 --> 00:22:45,050 charge on a positive charge is going to pull the positive 321 00:22:45,050 --> 00:22:46,590 charge this way. 322 00:22:46,590 --> 00:22:48,880 And it's going to pull the negative charge this way. 323 00:22:51,380 --> 00:22:53,630 And what that's going to do is to place a 324 00:22:53,630 --> 00:22:56,040 torque on the dipole. 325 00:22:56,040 --> 00:23:04,880 And that torque is going to be equal to a vector product of 326 00:23:04,880 --> 00:23:08,550 the electric field and a vector. 327 00:23:08,550 --> 00:23:13,440 You can express it in terms of the separation of charges 328 00:23:13,440 --> 00:23:17,220 defined in a vector sense going from the positive charge 329 00:23:17,220 --> 00:23:19,580 to the negative charge. 330 00:23:19,580 --> 00:23:23,400 So there's an example of how the torque on a dipole 331 00:23:23,400 --> 00:23:29,470 directly involves the product of charge and the separation 332 00:23:29,470 --> 00:23:34,010 between them and the angle between the 333 00:23:34,010 --> 00:23:36,260 field and the dipole. 334 00:23:36,260 --> 00:23:39,440 So it's a useful quality and, without further apology, we 335 00:23:39,440 --> 00:23:40,690 will use it in our discussion. 336 00:23:43,550 --> 00:23:45,010 Now polarization-- 337 00:23:45,010 --> 00:23:47,150 dipole moment per unit volume-- is just going to be 338 00:23:47,150 --> 00:23:51,390 the dipole moment of one dipole pair times the number 339 00:23:51,390 --> 00:23:53,000 of them per unit volume. 340 00:23:53,000 --> 00:23:58,630 And that will be a macroscopic measure of polarization. 341 00:24:02,530 --> 00:24:10,930 OK, if we subject a piece of material that contains ions to 342 00:24:10,930 --> 00:24:15,970 a stress- so here is a stress tensor, sigma ij. 343 00:24:19,520 --> 00:24:23,420 Let me call it jk. 344 00:24:23,420 --> 00:24:28,870 We will create reorientation of the dipoles. 345 00:24:28,870 --> 00:24:34,110 We will have a polarization, a dipole moment per unit volume. 346 00:24:34,110 --> 00:24:41,190 This will be a vector because there will be some net dipole 347 00:24:41,190 --> 00:24:42,350 moment per unit volume. 348 00:24:42,350 --> 00:24:44,950 And that will just be the sum of all the little individual 349 00:24:44,950 --> 00:24:48,220 dipole moments on the ions-- 350 00:24:48,220 --> 00:24:49,820 ion pairs. 351 00:24:49,820 --> 00:24:57,390 And if the stress is not too large, every component of 352 00:24:57,390 --> 00:25:02,300 polarization is going to be given by a linear combination 353 00:25:02,300 --> 00:25:04,960 of every one of the components of stress. 354 00:25:04,960 --> 00:25:13,150 So we will have an array of coefficients, d ijk The i goes 355 00:25:13,150 --> 00:25:17,745 with the P and the j and k goes with 356 00:25:17,745 --> 00:25:21,930 the elements of stress. 357 00:25:21,930 --> 00:25:23,870 This is something that's called the direct 358 00:25:23,870 --> 00:25:25,120 piezoelectric effect. 359 00:25:39,930 --> 00:25:43,870 It's the basis of a lot of every day devices-- 360 00:25:43,870 --> 00:25:49,090 pick ups on the old records which nobody uses anymore, 361 00:25:49,090 --> 00:25:51,760 cigarette lighter which nobody uses anymore because 362 00:25:51,760 --> 00:25:53,560 everybody's quit smoking-- 363 00:25:53,560 --> 00:25:57,650 but worked the cigarette lighter, in some of the more 364 00:25:57,650 --> 00:26:00,570 modern incarnations of the lighter, was the 365 00:26:00,570 --> 00:26:03,980 piezoelectric effect. 366 00:26:03,980 --> 00:26:07,880 It's important to note that the charges that are involved 367 00:26:07,880 --> 00:26:10,660 in the piezoelectric effect are bound charges. 368 00:26:10,660 --> 00:26:14,040 Again you can't get a shock by picking up a piece of a 369 00:26:14,040 --> 00:26:18,200 strongly piezoelectric material like quartz. 370 00:26:18,200 --> 00:26:22,270 Might say then, well if there's no spark, how could a 371 00:26:22,270 --> 00:26:27,090 piezoelectric material in a cigarette lighter cause the 372 00:26:27,090 --> 00:26:30,875 lighter fuel to ignite. 373 00:26:30,875 --> 00:26:32,600 The charge doesn't move. 374 00:26:32,600 --> 00:26:33,960 How could there be a spark? 375 00:26:33,960 --> 00:26:36,740 How could the lighter fluid be ignited? 376 00:26:36,740 --> 00:26:37,990 Anybody got an idea? 377 00:26:41,640 --> 00:26:47,570 Charge can induce current flow in circuit elements that are 378 00:26:47,570 --> 00:26:49,200 removed from the charge. 379 00:26:49,200 --> 00:26:53,050 So if you have a little capacitor in the lighter that 380 00:26:53,050 --> 00:26:56,770 is near the piezoelectric element that will develop the 381 00:26:56,770 --> 00:27:01,740 charge, you can induce current flow in a circuit as a result 382 00:27:01,740 --> 00:27:05,680 of that capacitor experiencing the electric charge of the 383 00:27:05,680 --> 00:27:09,000 piezoelectric induced charge. 384 00:27:09,000 --> 00:27:12,050 And that's how the little piezoelectric cigarette 385 00:27:12,050 --> 00:27:13,900 lighters worked. 386 00:27:13,900 --> 00:27:17,020 OK, this stress-- 387 00:27:17,020 --> 00:27:22,960 array of stress elements, nine of them, constitutes a tensor 388 00:27:22,960 --> 00:27:24,450 of second rank. 389 00:27:24,450 --> 00:27:31,040 Polarization is a vector per unit volume. 390 00:27:31,040 --> 00:27:33,910 This is a tensor of first rank. 391 00:27:33,910 --> 00:27:35,890 This is a tensor of second rank. 392 00:27:35,890 --> 00:27:40,070 Therefore it follows that this array of coefficients, d ijk 393 00:27:40,070 --> 00:27:44,100 will be three equations involving all 394 00:27:44,100 --> 00:27:46,700 nine elements of stress. 395 00:27:46,700 --> 00:27:54,790 So it'll be a total of 27 coefficients in the array. 396 00:27:54,790 --> 00:27:58,465 And this array constitutes a tensor of the third rank. 397 00:28:06,340 --> 00:28:10,770 So this is clearly a property tensor which relates a 398 00:28:10,770 --> 00:28:12,480 generalized force-- 399 00:28:12,480 --> 00:28:18,500 the stress tensor to a generalized displacement, 400 00:28:18,500 --> 00:28:20,420 which is the dipole moment per unit volume. 401 00:28:30,900 --> 00:28:37,830 Since this is a property of a crystal, that tensor is 402 00:28:37,830 --> 00:28:41,860 subject to symmetry restrictions. 403 00:28:41,860 --> 00:28:48,550 In particular, if there is some direction, cosine scheme 404 00:28:48,550 --> 00:28:53,010 Cij, that corresponds to the operation of a symmetry 405 00:28:53,010 --> 00:29:08,470 element, then it follows that if we evaluate the new tensor 406 00:29:08,470 --> 00:29:16,760 elements, d ijk prime in terms of the old elements, that is 407 00:29:16,760 --> 00:29:27,190 going to be a transformation of the form Cil, Cjm, Ckn 408 00:29:27,190 --> 00:29:31,555 times all 27 of the original tensor elements, d lmn. 409 00:29:36,030 --> 00:29:41,950 And the l, m, and n are dummy indices, just as with our 410 00:29:41,950 --> 00:29:43,200 second rank tensors. 411 00:29:49,014 --> 00:29:55,980 And the ijk are real indices that go with the subscripts on 412 00:29:55,980 --> 00:29:57,920 the particular tensor element. 413 00:30:00,850 --> 00:30:03,365 So there are 27 coefficients. 414 00:30:10,970 --> 00:30:18,760 And each transformed element is going to be a sum therefore 415 00:30:18,760 --> 00:30:27,080 of 27 coefficients times-- 416 00:30:27,080 --> 00:30:29,790 each times three direction cosines. 417 00:30:36,910 --> 00:30:40,800 So each transform tensor element is going to be given 418 00:30:40,800 --> 00:30:46,195 by a sum of 3 times 27 quantities. 419 00:30:49,980 --> 00:30:52,700 And to do the complete transformation would involve 420 00:30:52,700 --> 00:30:55,470 doing it 27 times. 421 00:30:55,470 --> 00:31:01,710 So 3 times 27 squared is the number of terms that we're 422 00:31:01,710 --> 00:31:06,030 going to have to write down to transform the entire tensor. 423 00:31:06,030 --> 00:31:11,380 Not a terribly encouraging prospect unless you use the 424 00:31:11,380 --> 00:31:18,310 method of direct inspection, namely that the tensor 425 00:31:18,310 --> 00:31:21,920 transforms like the product of the corresponding coordinates. 426 00:31:21,920 --> 00:31:24,180 And then it works fairly quickly, particularly for 427 00:31:24,180 --> 00:31:27,540 symmetry transformations, where there are a lot of zeros 428 00:31:27,540 --> 00:31:28,980 in the direction cosine scheme. 429 00:31:32,050 --> 00:31:36,400 So we're going to have to do this for each of the crystal-- 430 00:31:36,400 --> 00:31:38,170 crystal point groups. 431 00:31:38,170 --> 00:31:41,340 So we're going to have to do this exercise of 3 times 27 432 00:31:41,340 --> 00:31:46,620 times 27 times the 32 point groups that exist, which 433 00:31:46,620 --> 00:31:49,380 becomes an even more daunting task. 434 00:31:49,380 --> 00:31:53,370 But let's do one such transformation and let me show 435 00:31:53,370 --> 00:32:00,780 you that it, in fact, it's going to go fairly easily. 436 00:32:00,780 --> 00:32:04,340 So let me follow the tracks of what we did with second rank 437 00:32:04,340 --> 00:32:06,000 tensors and examine the 438 00:32:06,000 --> 00:32:11,550 restrictions imposed by inversion. 439 00:32:23,070 --> 00:32:27,250 OK so the transformation of the axes, as we've seen 440 00:32:27,250 --> 00:32:30,190 before, that's produced by 1 bar. 441 00:32:30,190 --> 00:32:33,435 That's going to take x1 and change its 442 00:32:33,435 --> 00:32:35,240 direction to minus x1. 443 00:32:40,050 --> 00:32:44,940 It's going to take x2 and change its sense to an x2 444 00:32:44,940 --> 00:32:46,950 prime that points in this direction. 445 00:32:46,950 --> 00:32:50,020 And take x3 and change its sense to something 446 00:32:50,020 --> 00:32:52,340 that points down here. 447 00:32:52,340 --> 00:32:54,070 So that's x3 prime. 448 00:32:54,070 --> 00:32:57,650 So the relation between the axes, before and after 449 00:32:57,650 --> 00:33:00,720 inversion, is that x1 prime equals x1-- 450 00:33:00,720 --> 00:33:02,380 minus x1-- 451 00:33:02,380 --> 00:33:05,220 x2 prime is equal to minus x2. 452 00:33:05,220 --> 00:33:10,900 And x3 prime is equal to minus x3. 453 00:33:10,900 --> 00:33:12,240 And we've seen this before. 454 00:33:12,240 --> 00:33:13,900 It's deja vu all over again. 455 00:33:13,900 --> 00:33:17,610 So Cij, the direction cosine scheme, is equal 456 00:33:17,610 --> 00:33:19,810 to minus 1, 0, 0. 457 00:33:19,810 --> 00:33:21,710 0, minus 1, 0. 458 00:33:21,710 --> 00:33:23,720 0, 0, minus 1. 459 00:33:26,450 --> 00:33:32,060 So let's transform a representative tensor element 460 00:33:32,060 --> 00:33:36,280 for a change of axes that corresponds to that direction 461 00:33:36,280 --> 00:33:37,620 cosine scheme. 462 00:33:37,620 --> 00:33:39,950 And that's the same as physically inverting the 463 00:33:39,950 --> 00:33:43,830 crystal and demanding that the property be unchanged if that 464 00:33:43,830 --> 00:33:45,630 is a symmetry transformation which 465 00:33:45,630 --> 00:33:48,160 leaves the crystal invariant. 466 00:33:48,160 --> 00:33:59,670 So if we look at some d ijk prime, that's going to be Cil, 467 00:33:59,670 --> 00:34:09,900 Cjm, Ckn, d lmn, where this is a sum over the index l, the 468 00:34:09,900 --> 00:34:20,280 dummy index m, and the double index n, from 1 to 3. 469 00:34:20,280 --> 00:34:27,159 If we sum over Ci something, the only term of the form Ci 470 00:34:27,159 --> 00:34:30,900 something, regardless of what i is, is going to be the 471 00:34:30,900 --> 00:34:40,310 diagonal term Cii, the diagonal direction cosine Cii. 472 00:34:40,310 --> 00:34:44,190 In other words, if i were 1, the only term of the form C1 473 00:34:44,190 --> 00:34:47,900 something that is non-zero, is C11. 474 00:34:47,900 --> 00:35:00,010 If I sum over m, the only term that is non-zero is C, with m 475 00:35:00,010 --> 00:35:00,770 equal to j. 476 00:35:00,770 --> 00:35:02,730 So this is going to be Cjj. 477 00:35:02,730 --> 00:35:05,160 And if I sum over n, the only term that is going 478 00:35:05,160 --> 00:35:08,750 to be left is Ckk. 479 00:35:08,750 --> 00:35:14,490 And all this will be times d ijk, the 480 00:35:14,490 --> 00:35:16,530 single tensor element. 481 00:35:16,530 --> 00:35:19,900 Regardless of the values of i, j, and k, they are all equal 482 00:35:19,900 --> 00:35:21,150 to minus 1. 483 00:35:23,450 --> 00:35:29,480 So the operation then of inversion, then says that for 484 00:35:29,480 --> 00:35:34,440 the transformed crystal, d ijk prime for any i, j, and k, is 485 00:35:34,440 --> 00:35:36,957 going to be equal to minus d ijk. 486 00:35:39,840 --> 00:35:43,800 But if this is a symmetry transformation, the tensor 487 00:35:43,800 --> 00:35:46,580 element has to be the same before and after. 488 00:35:46,580 --> 00:35:49,740 And the only number which can be equal to the negative of 489 00:35:49,740 --> 00:35:51,100 itself is zero. 490 00:35:54,250 --> 00:35:58,610 So we have, with one swell foop, 491 00:35:58,610 --> 00:36:02,600 transformed 27th tensor elements-- 492 00:36:02,600 --> 00:36:06,270 each one of them a sum of 27 elements in 493 00:36:06,270 --> 00:36:07,550 the original tensor. 494 00:36:07,550 --> 00:36:11,920 So the conclusion is, if a crystal has the operation of 495 00:36:11,920 --> 00:36:17,795 inversion, all d ijk vanish. 496 00:36:22,170 --> 00:36:24,170 What does that mean? 497 00:36:24,170 --> 00:36:25,800 Means the property just doesn't exist. 498 00:36:30,560 --> 00:36:35,640 So no crystal which has an inversion center can display 499 00:36:35,640 --> 00:36:36,890 piezoelectricity. 500 00:36:43,190 --> 00:36:46,510 Which means that none of the 11 Laue groups can be 501 00:36:46,510 --> 00:36:48,310 piezoelectric. 502 00:36:48,310 --> 00:36:52,090 And the only point groups for crystals that can be 503 00:36:52,090 --> 00:37:17,950 piezoelectric are the 32 minus 11 equals 21-- 504 00:37:17,950 --> 00:37:20,800 and this isn't even one of my good days-- 505 00:37:20,800 --> 00:37:23,840 acentric point groups-- 506 00:37:23,840 --> 00:37:25,605 the noncentrosymmetric point groups. 507 00:37:38,520 --> 00:37:43,400 And, these, in principle, can be all different. 508 00:38:10,710 --> 00:38:13,210 So we can't get away with just looking at the restrictions of 509 00:38:13,210 --> 00:38:16,200 Laue groups as we did with second rank tensors. 510 00:38:16,200 --> 00:38:19,910 We're going to have to look at every one of the point groups 511 00:38:19,910 --> 00:38:21,840 that have no inversion center. 512 00:38:21,840 --> 00:38:25,130 And these can, in principle, be different, and 513 00:38:25,130 --> 00:38:26,380 they usually are. 514 00:38:45,250 --> 00:38:49,370 Let me give you a physically-- 515 00:38:49,370 --> 00:38:53,980 physical feeling for why the piezoelectric effect cannot 516 00:38:53,980 --> 00:38:59,130 exist for a crystal that has an inversion center in it. 517 00:38:59,130 --> 00:39:04,050 Let's suppose we cut a wafer from our single crystal, 518 00:39:04,050 --> 00:39:05,300 regardless of orientation. 519 00:39:13,380 --> 00:39:15,870 And we subject it to a compressive stress. 520 00:39:23,490 --> 00:39:24,690 We will see-- 521 00:39:24,690 --> 00:39:29,080 and I've got a primer on some simple electrostatics that may 522 00:39:29,080 --> 00:39:29,990 be useful-- 523 00:39:29,990 --> 00:39:42,590 that a polarization in the x1 direction manifests itself as 524 00:39:42,590 --> 00:39:48,430 a charge per unit area on the surface and an opposite charge 525 00:39:48,430 --> 00:39:49,990 on the other surface. 526 00:39:49,990 --> 00:39:58,090 So P1 corresponds, numerically and physically, to a charge 527 00:39:58,090 --> 00:40:06,200 per unit area on a surface that's normal to x1. 528 00:40:16,450 --> 00:40:20,040 OK now let's suppose that that crystal has an inversion 529 00:40:20,040 --> 00:40:21,290 center in it. 530 00:40:23,920 --> 00:40:29,050 And that means we should be able to invert the crystal and 531 00:40:29,050 --> 00:40:33,810 measure exactly the same charge per unit area. 532 00:40:33,810 --> 00:40:38,550 Well, if this is surface a, and this is surface b, we can 533 00:40:38,550 --> 00:40:44,500 imagine ourselves doing the thought experiment in which we 534 00:40:44,500 --> 00:40:45,880 invert the crystal. 535 00:40:45,880 --> 00:40:51,270 So this surface becomes b and this surface becomes a. 536 00:40:51,270 --> 00:40:55,040 And we apply the same sigma along x1-- 537 00:40:55,040 --> 00:40:56,310 a compressive stress. 538 00:41:00,320 --> 00:41:03,300 It's really doing exactly the same thing that we were doing 539 00:41:03,300 --> 00:41:05,550 before-- inverting the crystal. 540 00:41:05,550 --> 00:41:09,000 And now we say that the property has to stay the same. 541 00:41:09,000 --> 00:41:15,470 So now we are demanding that surface b have a positive 542 00:41:15,470 --> 00:41:19,180 charge per unit area, and surface a develop a negative 543 00:41:19,180 --> 00:41:20,960 charge per unit area-- 544 00:41:20,960 --> 00:41:23,740 completely the reverse of what happened here. 545 00:41:23,740 --> 00:41:25,820 But yet what we're doing is just squeezing the plate. 546 00:41:25,820 --> 00:41:27,630 The crystal can't tell which side is up, 547 00:41:27,630 --> 00:41:28,460 which side is down. 548 00:41:28,460 --> 00:41:31,330 We're applying the same compressive effect. 549 00:41:31,330 --> 00:41:35,110 So these two distributions of charge are not the same and 550 00:41:35,110 --> 00:41:38,810 can be the same only if that charge is zero. 551 00:41:42,480 --> 00:41:45,135 So physically we can see why the crystal can't be 552 00:41:45,135 --> 00:41:45,850 piezoelectric. 553 00:41:45,850 --> 00:41:47,540 If you don't like to invert the crystal-- 554 00:41:47,540 --> 00:41:51,640 if you can't do that physically-- 555 00:41:51,640 --> 00:41:54,980 one thing we can do is leave the crystal alone and invert 556 00:41:54,980 --> 00:41:57,680 the compressive stress. 557 00:41:57,680 --> 00:42:01,070 So this is sigma top and this is sigma bottom which is 558 00:42:01,070 --> 00:42:04,930 exactly equal to it but opposite in direction. 559 00:42:04,930 --> 00:42:06,830 If we invert the stress relative-- 560 00:42:12,640 --> 00:42:13,915 relative to the crystal-- 561 00:42:16,940 --> 00:42:21,448 then what we've done is to take sigma top-- 562 00:42:21,448 --> 00:42:23,080 leave the crystal alone-- 563 00:42:23,080 --> 00:42:26,535 so this is now the sigma that was squishing the top. 564 00:42:26,535 --> 00:42:34,460 Here's the sigma that was squishing the bottom and well, 565 00:42:34,460 --> 00:42:37,090 that's clearly doing the same thing. 566 00:42:37,090 --> 00:42:40,980 So inverting the stress, relative to the crystal, is 567 00:42:40,980 --> 00:42:43,130 the same as inverting the crystal 568 00:42:43,130 --> 00:42:44,960 relative to the stress. 569 00:42:44,960 --> 00:42:48,980 And nothing is supposed to happen but clearly the sign of 570 00:42:48,980 --> 00:42:52,430 the charge is going to be reversed. 571 00:42:52,430 --> 00:42:53,410 So that just won't work. 572 00:42:53,410 --> 00:42:54,660 The charge has to disappear. 573 00:43:02,370 --> 00:43:04,770 OK this is the direct piezoelectric effect. 574 00:43:04,770 --> 00:43:08,950 It can exist only for crystals that lack an inversion center. 575 00:43:08,950 --> 00:43:11,983 Let me mention some other piezoelectric effects. 576 00:43:17,970 --> 00:43:28,760 So we mentioned that a polarization, given by a third 577 00:43:28,760 --> 00:43:35,620 rank tensor times an applied stress, sigma ij, is the 578 00:43:35,620 --> 00:43:37,470 direct piezoelectric photoelectric effect. 579 00:43:43,160 --> 00:43:55,020 However, if we have a stress, we also have a strain. 580 00:43:55,020 --> 00:43:58,750 So another piezoelectric effect, which is not dignified 581 00:43:58,750 --> 00:44:04,200 with a name, is that we can apply a stress that creates a 582 00:44:04,200 --> 00:44:06,910 given state of strain. 583 00:44:06,910 --> 00:44:12,820 And we again, we'll get a polarization per unit-- 584 00:44:12,820 --> 00:44:15,440 a charge per unit area of polarization. 585 00:44:15,440 --> 00:44:18,690 And this is a set of coefficients which are 586 00:44:18,690 --> 00:44:20,900 represented by e. 587 00:44:20,900 --> 00:44:25,870 And we can write a tensor relation of this sort-- that 588 00:44:25,870 --> 00:44:30,040 components of polarization, charge per unit area, normal 589 00:44:30,040 --> 00:44:35,910 to axis xi, is given by a linear combination of every 590 00:44:35,910 --> 00:44:38,700 one of the components of strain. 591 00:44:38,700 --> 00:44:43,090 This is something that is not dignified by any name. 592 00:44:51,300 --> 00:44:57,010 And clearly, the piezoelectric coefficients, e, have to be 593 00:44:57,010 --> 00:45:03,570 related to the piezoelectric coefficients, d, by the 594 00:45:03,570 --> 00:45:05,690 elastic constant somehow-- 595 00:45:05,690 --> 00:45:07,360 unless we are doing the same thing. 596 00:45:07,360 --> 00:45:11,310 We've got to create a stress, operationally, in order to 597 00:45:11,310 --> 00:45:12,910 create the strain. 598 00:45:12,910 --> 00:45:15,600 So a stress creates a strain. 599 00:45:15,600 --> 00:45:19,870 And that has to describe the same polarization if we've 600 00:45:19,870 --> 00:45:21,100 done the same thing. 601 00:45:21,100 --> 00:45:24,740 So the e's and the d's have to be related. 602 00:45:24,740 --> 00:45:30,630 Another piezoelectric effect involves a applying an 603 00:45:30,630 --> 00:45:33,950 electric field. 604 00:45:33,950 --> 00:45:38,530 And if we apply an electric field, we will in general 605 00:45:38,530 --> 00:45:39,780 create a strain. 606 00:45:46,070 --> 00:45:53,870 This direct piezoelectric effect is three equations that 607 00:45:53,870 --> 00:46:01,960 involve a linear combination of nine elements of stress. 608 00:46:01,960 --> 00:46:06,900 This piezoelectric relation here is going to involve 609 00:46:06,900 --> 00:46:12,130 writing an equation for each of nine elements of strain. 610 00:46:12,130 --> 00:46:14,850 And there would be three components of the electric 611 00:46:14,850 --> 00:46:15,750 field vector. 612 00:46:15,750 --> 00:46:17,070 So it'll be three this way. 613 00:46:17,070 --> 00:46:20,135 So both of these relations involve 27 elements. 614 00:46:22,910 --> 00:46:25,960 So again we're going to have a third rank tensor because the 615 00:46:25,960 --> 00:46:27,360 electric field is a vector-- 616 00:46:27,360 --> 00:46:28,930 strain is a tensor. 617 00:46:28,930 --> 00:46:33,510 And now the mind boggling thing that I will leave you 618 00:46:33,510 --> 00:46:42,280 with is exactly the same set of coefficients, d ij describe 619 00:46:42,280 --> 00:46:44,670 both effects. 620 00:46:44,670 --> 00:46:47,120 And this effect is given a special name. 621 00:46:47,120 --> 00:46:49,740 This is called the converse piezoelectric effect. 622 00:46:54,880 --> 00:47:01,750 And it's not at all clear how strain, in terms of electric 623 00:47:01,750 --> 00:47:06,930 field, should be described in terms of the same array of 27 624 00:47:06,930 --> 00:47:10,320 numbers as polarization per unit 625 00:47:10,320 --> 00:47:12,780 volume in terms of stress. 626 00:47:12,780 --> 00:47:18,070 And that is an argument that is based on thermodynamics. 627 00:47:18,070 --> 00:47:21,790 So that's a good place to quit. 628 00:47:21,790 --> 00:47:26,680 We'll have some fun in looking at a few symmetry restrictions 629 00:47:26,680 --> 00:47:32,670 imposed on the acentric point groups, 21 of them. 630 00:47:32,670 --> 00:47:34,210 We're not going to do all 21. 631 00:47:34,210 --> 00:47:37,520 If you know how to do one or two, and marvel over the 632 00:47:37,520 --> 00:47:41,470 results, you can do them all. 633 00:47:41,470 --> 00:47:45,630 But one thing that we'll be asking is how can we represent 634 00:47:45,630 --> 00:47:50,430 the variation of the piezoelectric effect with 635 00:47:50,430 --> 00:47:51,820 orientation of the crystal. 636 00:47:54,430 --> 00:47:58,155 And the answer is we can't. 637 00:47:58,155 --> 00:47:58,810 We can't. 638 00:47:58,810 --> 00:48:04,770 Because what is the direction of what we're doing? 639 00:48:04,770 --> 00:48:08,790 There are nine elements of stress, not just three 640 00:48:08,790 --> 00:48:11,930 components of a vector which defines an orientation. 641 00:48:11,930 --> 00:48:14,000 There are nine parameters here. 642 00:48:14,000 --> 00:48:17,620 And there are three components to the charge per unit area-- 643 00:48:17,620 --> 00:48:19,230 that's a vector. 644 00:48:19,230 --> 00:48:22,600 So all that one can do for these higher ranked tensors is 645 00:48:22,600 --> 00:48:27,870 to look for a special, generalized force. 646 00:48:27,870 --> 00:48:33,250 And then perhaps for a very special form of that 647 00:48:33,250 --> 00:48:36,220 generalized force, that tensor, look at how the 648 00:48:36,220 --> 00:48:37,870 polarization changes. 649 00:48:37,870 --> 00:48:40,610 But you're going to have to look at six different 650 00:48:40,610 --> 00:48:42,760 representation surfaces-- 651 00:48:42,760 --> 00:48:45,440 each one which will describe a different 652 00:48:45,440 --> 00:48:47,530 piezoelectric effect. 653 00:48:47,530 --> 00:48:51,530 So this will become less simple in many respects. 654 00:48:51,530 --> 00:48:55,370 But correspondingly, more interesting because we'll have 655 00:48:55,370 --> 00:49:01,360 some really wild, mind boggling representation 656 00:49:01,360 --> 00:49:05,740 surfaces for piezoelectric effects. 657 00:49:05,740 --> 00:49:12,120 OK so we'll stop there and resume on Thursday. 658 00:49:12,120 --> 00:49:15,350 And I'll let you get back to the MRS if that's where you 659 00:49:15,350 --> 00:49:16,600 earlier today.