1 00:00:00,000 --> 00:00:02,840 The following content is provided by MIT 2 00:00:02,840 --> 00:00:06,130 OpenCourseWare, under a Creative Commons license. 3 00:00:06,130 --> 00:00:08,750 Additional information about our license and MIT 4 00:00:08,750 --> 00:00:16,040 OpenCourseWare in general is available at ocw.mit.edu 5 00:00:16,040 --> 00:00:20,350 PROFESSOR: The final quiz is scheduled for a week from 6 00:00:20,350 --> 00:00:23,330 today, on December 8. 7 00:00:23,330 --> 00:00:28,230 Following that weekend, we'll have one class on the 13th. 8 00:00:28,230 --> 00:00:32,950 And the last days of classes are Monday, Tuesday, and 9 00:00:32,950 --> 00:00:36,110 Wednesday, the 12th, 13th, and 14th. 10 00:00:36,110 --> 00:00:39,030 So I know a number of you spoke up last time and said 11 00:00:39,030 --> 00:00:42,840 that you have a conflict on Thursday, the eighth, with a 12 00:00:42,840 --> 00:00:44,250 quiz in another class. 13 00:00:44,250 --> 00:00:46,910 So could I see a show of hands again of how many of you have 14 00:00:46,910 --> 00:00:49,080 this conflict? 15 00:00:49,080 --> 00:00:55,260 So three, six people, seven people, you're going to enroll 16 00:00:55,260 --> 00:00:56,475 in the other course just so you can put 17 00:00:56,475 --> 00:00:57,725 off taking the quiz. 18 00:01:02,440 --> 00:01:04,730 That's unfortunate, I don't think anybody should have to 19 00:01:04,730 --> 00:01:06,590 take two quizzes in one day. 20 00:01:06,590 --> 00:01:07,940 We can't move it up. 21 00:01:07,940 --> 00:01:09,430 We'll have to move it back. 22 00:01:09,430 --> 00:01:14,670 So I don't know if I am violating any Institute rule, 23 00:01:14,670 --> 00:01:19,200 but I know that it is strictly illegal to give assignments 24 00:01:19,200 --> 00:01:22,090 that are due after the last day of classes, let alone have 25 00:01:22,090 --> 00:01:25,180 a quiz after the last day of classes, which has not been 26 00:01:25,180 --> 00:01:28,000 previously scheduled as a final examination. 27 00:01:28,000 --> 00:01:31,560 So we can't give it after Wednesday, the 14th, the last 28 00:01:31,560 --> 00:01:32,680 day of classes. 29 00:01:32,680 --> 00:01:37,700 So, do any of the six impacted people have a strong 30 00:01:37,700 --> 00:01:41,975 preference for when we should schedule the quiz? 31 00:01:41,975 --> 00:01:44,200 AUDIENCE: Monday? 32 00:01:44,200 --> 00:01:45,790 PROFESSOR: Due it Monday? 33 00:01:45,790 --> 00:01:48,050 It doesn't have to be that soon. 34 00:01:48,050 --> 00:01:48,470 Monday? 35 00:01:48,470 --> 00:01:50,860 AUDIENCE: Why not Tuesday? 36 00:01:50,860 --> 00:01:52,190 PROFESSOR: Because Tuesday we have a class. 37 00:01:52,190 --> 00:01:53,480 AUDIENCE: Or Wednesday? 38 00:01:53,480 --> 00:01:55,352 Well we would have a Thursday class instead. 39 00:01:55,352 --> 00:01:57,220 Correct? 40 00:01:57,220 --> 00:01:59,610 PROFESSOR: No. 41 00:01:59,610 --> 00:02:01,940 If we're doing it after the quiz next 42 00:02:01,940 --> 00:02:05,100 week, there's a class. 43 00:02:05,100 --> 00:02:07,310 We have a last meeting here on the 13th. 44 00:02:07,310 --> 00:02:10,210 And I'm not going to shift it for other people, 45 00:02:10,210 --> 00:02:11,460 just for the impacted. 46 00:02:16,920 --> 00:02:20,260 So of the six who are entitled to vote, how many would prefer 47 00:02:20,260 --> 00:02:25,320 to have it Monday, the 12th? 48 00:02:25,320 --> 00:02:28,230 Three and I know how it's going to turn out. 49 00:02:28,230 --> 00:02:31,570 And for Wednesday, the 14th? 50 00:02:31,570 --> 00:02:32,820 Three. 51 00:02:36,412 --> 00:02:39,014 AUDIENCE: I can't take the test because-- not because of 52 00:02:39,014 --> 00:02:39,828 another test-- 53 00:02:39,828 --> 00:02:40,316 I'm out of town. 54 00:02:40,316 --> 00:02:41,780 And I might come back Saturday. 55 00:02:41,780 --> 00:02:45,684 And Monday is just a little quick to take it 56 00:02:45,684 --> 00:02:46,934 after we get back. 57 00:02:51,245 --> 00:02:54,020 PROFESSOR: I think I'm going to have to make an executive-- 58 00:02:54,020 --> 00:02:54,841 yes 59 00:02:54,841 --> 00:02:57,146 AUDIENCE: Professor, when you mean by conflict, do you mean 60 00:02:57,146 --> 00:02:59,090 having an exam at the exact time? 61 00:02:59,090 --> 00:03:02,690 PROFESSOR: No, it's on the exact same day. 62 00:03:02,690 --> 00:03:05,760 Considering this is a two hour examination, to have another 63 00:03:05,760 --> 00:03:09,620 examination exactly the same day, if you're not brain dead 64 00:03:09,620 --> 00:03:13,420 after an hour or two, you will be pretty close to it. 65 00:03:13,420 --> 00:03:21,840 So I think that is an unfair penalty to pay. 66 00:03:21,840 --> 00:03:27,230 I think, to keep it as close as possible to the quiz that 67 00:03:27,230 --> 00:03:29,330 the rest of the people will be taking, why don't we make it-- 68 00:03:29,330 --> 00:03:32,170 since it's a tie vote-- on Monday, the 12th. 69 00:03:32,170 --> 00:03:34,380 Do it sooner, rather than later. 70 00:03:34,380 --> 00:03:38,000 And I'll let you know next time of where we will hold it. 71 00:03:38,000 --> 00:03:40,930 I'll have to arrange a room for it. 72 00:03:40,930 --> 00:03:45,480 So that will not be a make-up quiz. 73 00:03:45,480 --> 00:03:47,220 It would be a made-up-- 74 00:03:47,220 --> 00:03:47,980 invented-- 75 00:03:47,980 --> 00:03:49,230 quiz. 76 00:03:53,530 --> 00:03:57,150 And, as I say on Monday, I should have all of the 77 00:03:57,150 --> 00:04:01,960 homework and the previous quiz to return to you. 78 00:04:01,960 --> 00:04:04,620 Reason you don't have it now is all the little crabbed 79 00:04:04,620 --> 00:04:07,450 handwriting that you see before you in the form of 80 00:04:07,450 --> 00:04:11,470 these notes, which takes forever. 81 00:04:11,470 --> 00:04:15,120 All right, so since we've satisfied the unpleasant 82 00:04:15,120 --> 00:04:18,839 aspects of the end of the term, let's get back to 83 00:04:18,839 --> 00:04:22,390 discussing some of the other piezoelectric effects that we 84 00:04:22,390 --> 00:04:23,500 have defined. 85 00:04:23,500 --> 00:04:27,850 And then also ask the question rhetorically, is there 86 00:04:27,850 --> 00:04:33,660 anything like a representation surface for a third-ranked 87 00:04:33,660 --> 00:04:35,290 tensor property? 88 00:04:35,290 --> 00:04:37,110 And it doesn't look promising. 89 00:04:37,110 --> 00:04:40,500 But there are some things that we can do to discuss variation 90 00:04:40,500 --> 00:04:42,590 of properties with direction, and we'll see 91 00:04:42,590 --> 00:04:43,980 directly what those are. 92 00:04:43,980 --> 00:04:47,230 But first let's look at the converse piezoelectric effect. 93 00:04:55,540 --> 00:04:59,860 And this again is a third-ranked tensor property, 94 00:04:59,860 --> 00:05:05,077 but what we do is to have the elements of strain, epsilon 95 00:05:05,077 --> 00:05:09,350 ij, a second-ranked tensor. 96 00:05:09,350 --> 00:05:15,190 And to take advantage of this curious relation between the 97 00:05:15,190 --> 00:05:18,540 directed converse effects, we define the converse 98 00:05:18,540 --> 00:05:23,330 piezoelectric effect as giving you nine elements of strain in 99 00:05:23,330 --> 00:05:29,210 terms of a third-ranked tensor dijk times e sub i. 100 00:05:29,210 --> 00:05:33,660 So what is not standard is our convention for the order of 101 00:05:33,660 --> 00:05:36,260 the subscripts on the moduli. 102 00:05:36,260 --> 00:05:39,420 And we make up the rules, we can do it any way we like. 103 00:05:39,420 --> 00:05:42,640 And the advantage of defining it this way, in nonstandard 104 00:05:42,640 --> 00:05:46,710 tensor notation, is that we can use the same coefficients 105 00:05:46,710 --> 00:05:50,616 for both the direct and the converse effects. 106 00:05:50,616 --> 00:05:54,030 So let me-- to illustrate what these equations look like-- 107 00:05:54,030 --> 00:05:57,270 write out a few examples. 108 00:05:57,270 --> 00:06:07,026 Epsilon 11 would be d1jk times e sub 1. 109 00:06:07,026 --> 00:06:09,980 I'm writing it in simple fashion, and not expanding 110 00:06:09,980 --> 00:06:12,830 fully, just to save space and time. 111 00:06:12,830 --> 00:06:24,820 Epsilon 22 would be d2jk times e2. 112 00:06:24,820 --> 00:06:33,760 Epsilon 33 will be d3jk times epsilon 3. 113 00:06:33,760 --> 00:06:37,210 And let me stop after these six terms, and write, at least 114 00:06:37,210 --> 00:06:38,770 for these, an expansion. 115 00:06:38,770 --> 00:06:42,040 Because this gives us some interesting information. 116 00:06:44,750 --> 00:06:48,820 I'm not doing the equal signs in here. 117 00:06:48,820 --> 00:06:52,920 So this would say that the element of strain epsilon 11 118 00:06:52,920 --> 00:07:08,990 is d111 times e1 plus d112 times e2 plus d113 times e3. 119 00:07:08,990 --> 00:07:11,270 And I want to say this is 11k. 120 00:07:11,270 --> 00:07:13,180 And I want to say that this is 22k. 121 00:07:20,280 --> 00:07:26,610 The next term would be epsilon 22, and that would be d122 122 00:07:26,610 --> 00:07:35,850 times epsilon 2 plus d212. 123 00:07:35,850 --> 00:07:37,802 AUDIENCE: Wouldn't the 2's be balanced? 124 00:07:37,802 --> 00:07:38,290 PROFESSOR: Hm? 125 00:07:38,290 --> 00:07:39,760 AUDIENCE: Wouldn't that be d2's? 126 00:07:39,760 --> 00:07:40,320 PROFESSOR: Yeah, you're right. 127 00:07:40,320 --> 00:07:52,530 This should be epsilon jk equals dijk times e sub i. 128 00:07:52,530 --> 00:07:54,270 So this is all e1. 129 00:07:54,270 --> 00:07:55,940 This is e1. 130 00:07:55,940 --> 00:07:57,760 And this one goes with this one. 131 00:07:57,760 --> 00:08:03,230 This 2 goes with this one. 132 00:08:03,230 --> 00:08:04,580 You're right. 133 00:08:04,580 --> 00:08:10,710 d2 and this is the d311 times e11. 134 00:08:10,710 --> 00:08:18,520 This would be 122 times e1 plus 222 times e2 135 00:08:18,520 --> 00:08:26,830 plus d322 times e3. 136 00:08:26,830 --> 00:08:33,590 And the fourth one would be e33 equals d133 times e1 plus 137 00:08:33,590 --> 00:08:43,679 d233 times e2 plus d333 times e3. 138 00:08:46,490 --> 00:08:52,760 OK, these elements here are the three that appear in the 139 00:08:52,760 --> 00:08:57,930 box that I had indicated for the terms djk. 140 00:09:03,530 --> 00:09:07,410 When we write the direct piezoelectric effect, this 141 00:09:07,410 --> 00:09:09,190 would be the box of coefficients. 142 00:09:09,190 --> 00:09:12,290 When we write the converse effect, this is the box of 143 00:09:12,290 --> 00:09:13,650 coefficients. 144 00:09:13,650 --> 00:09:18,550 And now what I wanted to point out is that delta v over v is 145 00:09:18,550 --> 00:09:23,610 equal to episilon 11 plus epsilon 22 plus epsilon 33, 146 00:09:23,610 --> 00:09:26,210 which is the trace of the strain tensor. 147 00:09:26,210 --> 00:09:29,400 And we're going to get one set of terms which depends on the 148 00:09:29,400 --> 00:09:33,290 x1 component of the field, another set of three terms 149 00:09:33,290 --> 00:09:37,025 that depend on the x2 component of the field, and 150 00:09:37,025 --> 00:09:38,650 another one on the x3 component. 151 00:09:43,230 --> 00:09:46,230 If these expressions sum to 0, then there would 152 00:09:46,230 --> 00:09:47,770 be no volume change. 153 00:09:47,770 --> 00:09:51,320 So this first set of 3-- 154 00:09:51,320 --> 00:09:53,070 3 of the first 6 equations-- 155 00:09:53,070 --> 00:09:56,270 give you an indication of when the application of a field 156 00:09:56,270 --> 00:09:59,270 will result in no volume change in the sample. 157 00:09:59,270 --> 00:10:03,300 And that again, I remind you, can be for two reasons. 158 00:10:03,300 --> 00:10:10,180 It could be because all of these six of the nine 159 00:10:10,180 --> 00:10:12,430 piezoelectric moduli are 0. 160 00:10:12,430 --> 00:10:17,720 Or alternatively, if you examined the form of the 161 00:10:17,720 --> 00:10:21,750 piezoelectric moduli matrices that is required by symmetry 162 00:10:21,750 --> 00:10:25,880 constraints, you'll find that there are a substantial number 163 00:10:25,880 --> 00:10:29,750 of point groups for which some terms are 0. 164 00:10:29,750 --> 00:10:32,870 But then there is, in addition, an equality between 165 00:10:32,870 --> 00:10:36,840 some of the other terms, which make the volume change zero. 166 00:10:36,840 --> 00:10:42,250 Even though, not all of the elements within this 1/2 box 167 00:10:42,250 --> 00:10:44,270 of moduli are zero. 168 00:10:44,270 --> 00:10:47,410 So that is an interesting effect. 169 00:10:47,410 --> 00:10:49,970 And it turns out that the majority of the 170 00:10:49,970 --> 00:10:52,580 non-centrosymmetric point groups do not have a volume 171 00:10:52,580 --> 00:10:56,730 change, when you apply in a field in any way you choose. 172 00:10:56,730 --> 00:11:01,720 A few do, but it's a minority. 173 00:11:01,720 --> 00:11:03,920 Alright, but this is not the main point of writing this. 174 00:11:03,920 --> 00:11:05,900 I want to-- 175 00:11:05,900 --> 00:11:07,170 at this point-- 176 00:11:07,170 --> 00:11:10,190 point out that this tells you about the volume change. 177 00:11:10,190 --> 00:11:14,330 And then we would have additional terms, we would 178 00:11:14,330 --> 00:11:33,740 have a term of the form e1 something like e123 or e132. 179 00:11:33,740 --> 00:11:37,350 And this would be e14. 180 00:11:37,350 --> 00:11:41,710 This would also be e14 since we replace both of those 181 00:11:41,710 --> 00:11:44,260 subscripts by a single subscript. 182 00:11:44,260 --> 00:11:46,220 And the tensor elements that would go in 183 00:11:46,220 --> 00:11:52,430 here would be d123. 184 00:11:52,430 --> 00:11:54,000 We just want one of the them. 185 00:11:54,000 --> 00:12:09,300 d123 times e1, and then we want d223 times e2, and then 186 00:12:09,300 --> 00:12:14,520 d323 times e3. 187 00:12:14,520 --> 00:12:21,940 We're writing one of the specific equations for the 188 00:12:21,940 --> 00:12:23,500 shear strings. 189 00:12:23,500 --> 00:12:29,600 If we write the expression for d132, this is going to be d132 190 00:12:29,600 --> 00:12:42,400 times e1 plus d232 times e2 plus d332 times e3. 191 00:12:46,060 --> 00:12:49,690 There are two interesting consequences of this. 192 00:12:49,690 --> 00:12:55,060 The strain tensor is symmetric. 193 00:12:55,060 --> 00:13:00,130 And this element of strain-- 194 00:13:00,130 --> 00:13:03,360 why have I got three subscripts in here? 195 00:13:03,360 --> 00:13:04,840 Don't want that one in there. 196 00:13:07,850 --> 00:13:09,570 These two strains are equal. 197 00:13:09,570 --> 00:13:15,710 And therefore, if we would apply just an e1 for example, 198 00:13:15,710 --> 00:13:21,980 let e be equal to just a component of field along x1. 199 00:13:21,980 --> 00:13:24,760 The strains have to be equal. 200 00:13:24,760 --> 00:13:29,690 But we have two different tensor elements here. 201 00:13:29,690 --> 00:13:32,460 And the only way that strain can be symmetric, and it's 202 00:13:32,460 --> 00:13:41,460 defined as such, is that d123 be identical to d132. 203 00:13:41,460 --> 00:13:44,220 And that resolves the issue that came up in connection 204 00:13:44,220 --> 00:13:46,080 with the direct case electric effect. 205 00:13:46,080 --> 00:13:48,480 We said the direct piezoelectric effect depends 206 00:13:48,480 --> 00:13:53,850 just on the sum of the elements dijk and dikj. 207 00:13:53,850 --> 00:13:56,610 And since they're lumped together, all we can measure 208 00:13:56,610 --> 00:13:58,740 is the sum. 209 00:13:58,740 --> 00:14:03,160 And, so, we'll just have to call that a 210 00:14:03,160 --> 00:14:05,080 single matrix element. 211 00:14:05,080 --> 00:14:08,540 The converse piezoelectric effect tells us these tensor 212 00:14:08,540 --> 00:14:12,200 elements have to be equal, if the strain 213 00:14:12,200 --> 00:14:13,450 tensor is to be symmetric. 214 00:14:16,350 --> 00:14:26,260 So that says that, since we defined d16 as the sum of d123 215 00:14:26,260 --> 00:14:39,650 plus d132, this says that d132 is equal to d123 is equal to 216 00:14:39,650 --> 00:14:47,030 1/2 of d16, just making the equality 217 00:14:47,030 --> 00:14:49,020 in the reverse direction. 218 00:14:49,020 --> 00:14:55,860 The converse effect let's us say that 123 has to be 132, 219 00:14:55,860 --> 00:15:01,210 that any ijk has to be equal to a dijk. 220 00:15:01,210 --> 00:15:05,880 So if I tried now to write this first expression in the 221 00:15:05,880 --> 00:15:20,230 reduced subscript notation, e23 is what we let e5 be. 222 00:15:20,230 --> 00:15:26,510 And now we have this equal to and in our reduced subscripts, 223 00:15:26,510 --> 00:15:31,370 we have this as 1/2 of d16. 224 00:15:31,370 --> 00:15:34,150 And the field that's multiplying this is 225 00:15:34,150 --> 00:15:36,480 piezoelectric modulus is e1. 226 00:15:40,880 --> 00:15:45,470 And that messy factor of 2 has come back to haunt us again. 227 00:15:45,470 --> 00:15:46,955 It's like trying to stuff a jack-in-the 228 00:15:46,955 --> 00:15:48,530 box back in the box. 229 00:15:48,530 --> 00:15:49,810 It keeps popping up. 230 00:15:49,810 --> 00:15:56,550 We ate the factor of 2 in defining the matrix 231 00:15:56,550 --> 00:15:59,050 representation of the 232 00:15:59,050 --> 00:16:00,280 piezoelectric electric modulus. 233 00:16:00,280 --> 00:16:03,920 And now when we try to go to a reduced subscript notation for 234 00:16:03,920 --> 00:16:06,210 the converse piezoelectric effect, we've 235 00:16:06,210 --> 00:16:07,780 got a 1/2 in there. 236 00:16:07,780 --> 00:16:12,370 And similarly, the second equation would be e5-- 237 00:16:12,370 --> 00:16:14,670 same result epsilon 5-- 238 00:16:14,670 --> 00:16:22,050 and it's 132, but 132 is 1/2 of d16 times e1. 239 00:16:22,050 --> 00:16:33,000 And we have a similar mess for the other coefficients here. 240 00:16:33,000 --> 00:16:34,230 So what do we do? 241 00:16:34,230 --> 00:16:46,800 Do we say that the relation between strain epsilon j 242 00:16:46,800 --> 00:16:54,820 equals dij e sub j has 1/2 in front of several of the 243 00:16:54,820 --> 00:16:57,330 coefficients and not in others? 244 00:16:57,330 --> 00:17:00,670 Well, we can't really absorb the factor of 2 in the 245 00:17:00,670 --> 00:17:04,079 definition of the piezoelectric moduli, because 246 00:17:04,079 --> 00:17:05,819 we've already done that. 247 00:17:05,819 --> 00:17:11,160 So the only thing we can do is to say that we will have to 248 00:17:11,160 --> 00:17:16,710 take the off diagonal strains, and define them as having a 249 00:17:16,710 --> 00:17:19,730 1/2 in front, and we add these up. 250 00:17:25,640 --> 00:17:30,720 So we will have to write matrix strain in this reduced 251 00:17:30,720 --> 00:17:32,770 subscript notation. 252 00:17:32,770 --> 00:17:37,560 We'll have to take e11, epsilon 12, epsilon 13, 253 00:17:37,560 --> 00:17:45,250 epsilon 21, epsilon 22, epsilon 23, epsilon 31, 254 00:17:45,250 --> 00:17:50,790 epsilon 32, and epsilon 33. 255 00:17:50,790 --> 00:17:54,490 And in converting this to matrix form, we'll call this 256 00:17:54,490 --> 00:17:57,110 epsilon 1, analogous to what we did 257 00:17:57,110 --> 00:17:58,710 for the tensile stresses. 258 00:17:58,710 --> 00:18:02,980 We'll call this epsilon 2 and this epsilon 3. 259 00:18:02,980 --> 00:18:06,960 And then for all of the off-diagonal elements of 260 00:18:06,960 --> 00:18:11,250 strain, in order to avoid the factor of 2 popping up in 261 00:18:11,250 --> 00:18:13,890 front of the matrix representation of the 262 00:18:13,890 --> 00:18:17,280 piezoelectric moduli, we're going to have to put in here 263 00:18:17,280 --> 00:18:23,867 1/2 of epsilon 4, 1/2 of epsilon 5, and 1/2 of epsilon 264 00:18:23,867 --> 00:18:28,050 6, and same for the off diagonal terms 1/2 of epsilon 265 00:18:28,050 --> 00:18:33,035 5, 1/2 of epsilon 4, and 1/2 of epsilon 6. 266 00:18:36,290 --> 00:18:40,340 So the moral of this story is that you can't win, but if you 267 00:18:40,340 --> 00:18:44,410 play it right, you can come out even. 268 00:18:44,410 --> 00:18:49,060 So only if we define the reduced subscript strains in 269 00:18:49,060 --> 00:18:52,510 this fashion, can we write an expression of this form. 270 00:19:05,140 --> 00:19:08,370 So this algebra is carried through for you for the other 271 00:19:08,370 --> 00:19:10,890 elements in the notes. 272 00:19:10,890 --> 00:19:15,080 But this is the way we are forced, unless we want to have 273 00:19:15,080 --> 00:19:18,620 a factor of 2 in some terms, and not in others, is the way 274 00:19:18,620 --> 00:19:20,830 we have to define matrix strain. 275 00:19:26,700 --> 00:19:30,850 All this is formalism and definition, but I'd like to 276 00:19:30,850 --> 00:19:32,110 now do two things. 277 00:19:32,110 --> 00:19:35,240 First of all, give you some examples of real numbers for 278 00:19:35,240 --> 00:19:41,390 piezoelectric moduli, and then ask the question about 279 00:19:41,390 --> 00:19:42,765 representation surfaces. 280 00:19:45,500 --> 00:19:49,170 Once again, these numbers are in the handout for you, so you 281 00:19:49,170 --> 00:19:51,010 don't have to make note of them. 282 00:19:51,010 --> 00:19:58,280 But one of the very important piezoelectric materials is the 283 00:19:58,280 --> 00:20:00,020 quartz form of SiO2. 284 00:20:07,060 --> 00:20:10,160 SiO2 has many polymorphic forms. 285 00:20:10,160 --> 00:20:13,820 Quartz is the form that's stable at room temperature, 286 00:20:13,820 --> 00:20:25,080 and it has point group 32 asymmetric. 287 00:20:25,080 --> 00:20:28,640 And there are higher temperature 288 00:20:28,640 --> 00:20:30,820 polymorphs of SiO2. 289 00:20:30,820 --> 00:20:34,090 There's a phase transition in quartz to a more symmetric 290 00:20:34,090 --> 00:20:37,550 form, and then there are cubic forms at the highest 291 00:20:37,550 --> 00:20:38,800 temperatures. 292 00:20:40,670 --> 00:20:45,970 Now, quartz is not the material that displays the 293 00:20:45,970 --> 00:20:48,750 largest piezoelectric moduli. 294 00:20:48,750 --> 00:20:50,210 But it has the following advantages. 295 00:20:50,210 --> 00:20:55,020 One is it is a naturally occurring material that is 296 00:20:55,020 --> 00:20:56,510 very inexpensive. 297 00:20:56,510 --> 00:20:59,260 So it's not an exotic expensive material. 298 00:20:59,260 --> 00:21:00,260 Very stable. 299 00:21:00,260 --> 00:21:02,360 It's not water soluble. 300 00:21:02,360 --> 00:21:03,620 Extremely tough. 301 00:21:03,620 --> 00:21:06,620 You can take a thin wafer of quartz, for example, if you 302 00:21:06,620 --> 00:21:10,550 want to make a monochromator for a diffraction experiment. 303 00:21:10,550 --> 00:21:13,550 You can take a thin wafer of quartz, and bend it like this, 304 00:21:13,550 --> 00:21:17,910 and it does not break, very elastic. 305 00:21:17,910 --> 00:21:22,240 And if you want a material that's going to earn its 306 00:21:22,240 --> 00:21:26,130 living by being squished, you want something that doesn't 307 00:21:26,130 --> 00:21:29,480 plastically deform and something that is very hard 308 00:21:29,480 --> 00:21:31,300 and resistant to stress. 309 00:21:31,300 --> 00:21:35,070 So quartz, even though the moduli are not the largest, is 310 00:21:35,070 --> 00:21:37,840 a very attractive material, and is used in 311 00:21:37,840 --> 00:21:40,730 a variety of devices. 312 00:21:40,730 --> 00:21:43,760 There was a time when CB radios 313 00:21:43,760 --> 00:21:45,260 were very, very popular. 314 00:21:45,260 --> 00:21:48,290 Everybody had to have one in their car. 315 00:21:48,290 --> 00:21:51,990 I guess so they could pretend that they were truck drivers. 316 00:21:51,990 --> 00:21:54,550 But anyway, you don't have them anymore now that 317 00:21:54,550 --> 00:21:57,690 cellphones have come in in existence. 318 00:21:57,690 --> 00:22:01,310 But for your CB radio, you needed 319 00:22:01,310 --> 00:22:04,150 something called a crystal. 320 00:22:04,150 --> 00:22:07,000 And they were fairly expensive. 321 00:22:07,000 --> 00:22:10,020 And the number of channels on which you could communicate 322 00:22:10,020 --> 00:22:12,870 dependent on the number of crystals that you could plug 323 00:22:12,870 --> 00:22:15,090 into your CB radio. 324 00:22:15,090 --> 00:22:17,820 The so-called crystal was exactly that. 325 00:22:17,820 --> 00:22:20,550 It was a little black box that looked almost like a 326 00:22:20,550 --> 00:22:21,360 transistor. 327 00:22:21,360 --> 00:22:23,180 And there were two leads coming out of it. 328 00:22:23,180 --> 00:22:26,300 If you ever got curious and broke this thing open, what 329 00:22:26,300 --> 00:22:30,120 you found was a nice wafer of quartz. 330 00:22:30,120 --> 00:22:32,500 And on the wafer of quartz-- brazed onto it-- 331 00:22:32,500 --> 00:22:33,470 were two wires. 332 00:22:33,470 --> 00:22:35,040 And that's all there was in the box. 333 00:22:35,040 --> 00:22:37,240 The crystal really was a crystal. 334 00:22:37,240 --> 00:22:40,580 And the crystals had been very precisely ground to 335 00:22:40,580 --> 00:22:45,720 thicknesses such that when a field caused these wafers to 336 00:22:45,720 --> 00:22:51,210 hit a resonance, that resonance would be at exactly 337 00:22:51,210 --> 00:22:52,430 a particular frequency. 338 00:22:52,430 --> 00:22:54,170 And that was the frequency of that channel. 339 00:22:57,350 --> 00:23:03,610 One of the crises, during the Second World War, is that the 340 00:23:03,610 --> 00:23:08,070 highest quality natural crystals of quartz come from 341 00:23:08,070 --> 00:23:15,220 Brazil, and during the conflict the sea channels were 342 00:23:15,220 --> 00:23:16,380 essentially blocked. 343 00:23:16,380 --> 00:23:19,650 And so, people-- in order to make all these 344 00:23:19,650 --> 00:23:21,240 communication devices-- 345 00:23:21,240 --> 00:23:25,370 had to learn how to synthesize crystals of quartz 346 00:23:25,370 --> 00:23:26,790 synthetically. 347 00:23:26,790 --> 00:23:29,420 And there are a number of companies, such as Sylvania up 348 00:23:29,420 --> 00:23:32,210 on the North Shore, that developed entire buildings 349 00:23:32,210 --> 00:23:34,680 devoted to growing single crystals of quartz. 350 00:23:34,680 --> 00:23:38,040 And they're big tanks like something out of the aquarium. 351 00:23:38,040 --> 00:23:41,940 And in the center of the tank is a rod, and seeds of quartz 352 00:23:41,940 --> 00:23:42,980 are placed on the rod. 353 00:23:42,980 --> 00:23:44,880 And the thing very slowly rotates 354 00:23:44,880 --> 00:23:46,510 around in this solution. 355 00:23:46,510 --> 00:23:49,770 And on the rod, eventually, are single crystals of quartz 356 00:23:49,770 --> 00:23:50,620 that are this size. 357 00:23:50,620 --> 00:23:54,290 And its a very spectacular thing to see. 358 00:23:54,290 --> 00:23:58,270 In any case, symmetry 3 2, and the moduli 359 00:23:58,270 --> 00:24:01,370 that are 0 and non-zero. 360 00:24:01,370 --> 00:24:12,000 If we refer the reference axes to a set of coordinates with 361 00:24:12,000 --> 00:24:15,150 x1 in this direction, and x2-- 362 00:24:15,150 --> 00:24:17,960 since it has to be orthogonal to x1-- in between the 363 00:24:17,960 --> 00:24:21,360 two-fold axes, and x3. 364 00:24:21,360 --> 00:24:30,130 And for all materials of commerce that are anisotropic, 365 00:24:30,130 --> 00:24:33,070 there has to be some standard for defining 366 00:24:33,070 --> 00:24:34,330 the choice of axes. 367 00:24:34,330 --> 00:24:38,820 For example, crystallographers would say the unique axis 368 00:24:38,820 --> 00:24:42,090 should be along the z direction, the x3 direction. 369 00:24:42,090 --> 00:24:46,490 But why isn't x1 and x2 in between the two-fold axis? 370 00:24:46,490 --> 00:24:50,640 There's some professional society that is responsible 371 00:24:50,640 --> 00:24:54,500 for giving standards for representing property 372 00:24:54,500 --> 00:24:56,930 measurements in some mutually agreed upon form. 373 00:24:56,930 --> 00:25:00,550 And this is the standard set of axes for the quartz and 374 00:25:00,550 --> 00:25:02,200 symmetry 3, 2. 375 00:25:02,200 --> 00:25:10,610 The moduli, dij, not dijk, but dij, these two are constrained 376 00:25:10,610 --> 00:25:12,940 to be equal. 377 00:25:12,940 --> 00:25:13,860 This one is 0. 378 00:25:13,860 --> 00:25:28,509 This is minus 0.67, 0, 0, 0, 0, 0, 0, 0.67, 4.6, 0, 0, 0, 379 00:25:28,509 --> 00:25:30,994 0, 0, 0, 0. 380 00:25:30,994 --> 00:25:35,340 One of the strange materials for which no field 381 00:25:35,340 --> 00:25:41,450 can create a strain-- 382 00:25:41,450 --> 00:25:43,810 field along x3 cannot create a strain. 383 00:25:43,810 --> 00:25:48,570 And these are all in units of 10 to the minus 12 coulombs 384 00:25:48,570 --> 00:25:49,890 per Newton. 385 00:25:53,330 --> 00:25:57,860 An example they give you here, if you apply a field of 100 386 00:25:57,860 --> 00:26:01,110 volts per centimeter, which is not terribly large, but would 387 00:26:01,110 --> 00:26:02,950 be comparable to what you have in some 388 00:26:02,950 --> 00:26:05,140 electronic device, perhaps. 389 00:26:05,140 --> 00:26:12,040 So this, since our units are MKS, this would correspond to 390 00:26:12,040 --> 00:26:14,370 10 to the 4 volts per meter. 391 00:26:17,780 --> 00:26:23,810 The strain epsilon 1, which is d11 times e1. 392 00:26:23,810 --> 00:26:28,510 It turns out to be minus 2.3, which means it contracts. 393 00:26:28,510 --> 00:26:35,580 That's the significance of the negative times 10 to the 4. 394 00:26:35,580 --> 00:26:38,310 And that turns out to be 10 to the minus 12 395 00:26:38,310 --> 00:26:39,560 times 10 to the fourth. 396 00:26:42,910 --> 00:26:49,900 That turns out to be a strain of minus 2.3 times 10 to the 397 00:26:49,900 --> 00:26:52,900 minus eighth. 398 00:26:52,900 --> 00:26:57,800 10 to the minus eighth is not exactly a large point strain. 399 00:26:57,800 --> 00:27:01,550 You're not going to see the crystal wafer twitch and jump, 400 00:27:01,550 --> 00:27:04,210 if you apply a field of 100 volts on it. 401 00:27:04,210 --> 00:27:09,450 But yet, even a strain of this sort is more than 402 00:27:09,450 --> 00:27:12,660 enough to be useful. 403 00:27:12,660 --> 00:27:19,540 But this is just to illustrate that quartz is not the most 404 00:27:19,540 --> 00:27:21,495 sensitive of piezoelectric materials. 405 00:27:24,750 --> 00:27:31,660 Another one that I give you data for is so-called ADP. 406 00:27:31,660 --> 00:27:35,360 And this is a widely used material. 407 00:27:35,360 --> 00:27:37,665 This is ammonium dihydrogen phosphate. 408 00:27:51,080 --> 00:27:54,780 And this is one of the family of salts that have very large 409 00:27:54,780 --> 00:27:57,420 piezoelectric responses. 410 00:27:57,420 --> 00:28:01,180 The nice part about it is that it's water soluble, so you can 411 00:28:01,180 --> 00:28:05,640 grow very, very large crystals easily from solution. 412 00:28:05,640 --> 00:28:10,040 The nasty part about it is that it is water soluble, so 413 00:28:10,040 --> 00:28:13,230 you have to be careful to protect this material from 414 00:28:13,230 --> 00:28:15,990 moisture if you're going to use it in any sort of device. 415 00:28:15,990 --> 00:28:21,180 But if we look at the moduli, relative to the standard axes, 416 00:28:21,180 --> 00:28:25,720 and this has point group 4-bar 2m. 417 00:28:25,720 --> 00:28:30,655 So x3 is taken along the 4-bar access. 418 00:28:30,655 --> 00:28:35,010 Then there are two-fold axes and orientations like this. 419 00:28:35,010 --> 00:28:41,280 And since they are orthogonal, you can take both x1 and x2 420 00:28:41,280 --> 00:28:43,260 along the two-fold axes. 421 00:28:43,260 --> 00:28:54,264 And the numbers here for dij, are 0, 0, 0, 1.7, 0, 0, 0, 0, 422 00:28:54,264 --> 00:28:57,350 0, 0, 1.7, 0. 423 00:28:57,350 --> 00:29:01,140 This is one of the interesting tensors where there's a 424 00:29:01,140 --> 00:29:04,930 diagonal row of non-zero terms off on the right hand side. 425 00:29:04,930 --> 00:29:07,170 And finally, the big surprise is the third 426 00:29:07,170 --> 00:29:10,540 modulus this is 51.7. 427 00:29:10,540 --> 00:29:12,690 So you can see this has a very strong effect. 428 00:29:12,690 --> 00:29:17,070 This is over 10 times the maximum 429 00:29:17,070 --> 00:29:19,270 piezoelectric modulus in quartz. 430 00:29:19,270 --> 00:29:23,040 So this is a material that's very commonly used in 431 00:29:23,040 --> 00:29:24,340 transducers. 432 00:29:24,340 --> 00:29:27,704 This is again times ten to the minus 12 coulombs per Newton. 433 00:29:32,140 --> 00:29:36,480 One of the very, very exciting developments in recent years 434 00:29:36,480 --> 00:29:43,000 is a class of materials that are perovskites. 435 00:29:43,000 --> 00:29:46,435 And they are very, very new. 436 00:29:46,435 --> 00:29:48,710 I gave you the reference to the first one that was 437 00:29:48,710 --> 00:29:53,970 reported, and that was just in the spring of 2000. 438 00:29:53,970 --> 00:29:55,540 And these are perovskites. 439 00:29:55,540 --> 00:30:03,090 Perovskites are materials that in the type form are cubic. 440 00:30:03,090 --> 00:30:06,840 But depending on composition and temperature, they can 441 00:30:06,840 --> 00:30:11,530 transform to a distorted version of this very simple 442 00:30:11,530 --> 00:30:13,890 cubic structure that is tetragonal. 443 00:30:13,890 --> 00:30:17,480 And this material can exhibit piezoelectric effects. 444 00:30:17,480 --> 00:30:21,050 And whether it distorts or not depends on the relative sizes 445 00:30:21,050 --> 00:30:22,840 of what goes into the perovskite. 446 00:30:22,840 --> 00:30:28,800 Perovskite has a composition ABO3 like barium titanate is 447 00:30:28,800 --> 00:30:30,050 one example. 448 00:30:32,270 --> 00:30:36,410 And the material has two different cations. 449 00:30:36,410 --> 00:30:39,750 And they have different valences so they have 450 00:30:39,750 --> 00:30:41,560 different sizes. 451 00:30:41,560 --> 00:30:46,720 And I won't bother to describe the structure, but it is only 452 00:30:46,720 --> 00:30:56,940 a very restricted locus in the field RA versus RB, where both 453 00:30:56,940 --> 00:31:00,670 the A and the B can remain in contact with the oxygen 454 00:31:00,670 --> 00:31:01,445 without distortion. 455 00:31:01,445 --> 00:31:04,680 And it turns out to be a line that does something like that. 456 00:31:04,680 --> 00:31:07,250 Any other perovskite-- and there are lots of them in this 457 00:31:07,250 --> 00:31:09,260 field of radii-- 458 00:31:09,260 --> 00:31:11,990 has to have one of the ion sort of flopping around. 459 00:31:11,990 --> 00:31:15,390 And if that gets too serious, the structure distorts so that 460 00:31:15,390 --> 00:31:19,330 all these ions can remain in contact with the oxygen. 461 00:31:19,330 --> 00:31:22,130 OK in order to do that, many of them distort 462 00:31:22,130 --> 00:31:23,710 to tetragonal forms. 463 00:31:23,710 --> 00:31:27,840 Others distort to super structures, which have very, 464 00:31:27,840 --> 00:31:29,940 very large unit cells. 465 00:31:29,940 --> 00:31:35,880 But in any case, when you're right at the phase boundary 466 00:31:35,880 --> 00:31:39,460 between the distorted structure and the true 467 00:31:39,460 --> 00:31:43,340 perovskite structure, these materials sometimes have very, 468 00:31:43,340 --> 00:31:47,220 very complicated x solutions of the two phases, on a very 469 00:31:47,220 --> 00:31:48,850 sort of fine scale. 470 00:31:48,850 --> 00:31:53,480 And it's not known exactly why they have this property. 471 00:31:53,480 --> 00:31:59,000 But they have absolutely enormous piezoelectric moduli, 472 00:31:59,000 --> 00:32:01,620 very close to this phase boundary. 473 00:32:01,620 --> 00:32:06,590 And the references that I give you-- here-- is a compound 474 00:32:06,590 --> 00:32:11,540 that is a lead titanium zinc niobate. 475 00:32:15,450 --> 00:32:18,610 And the complicated composition is to get you 476 00:32:18,610 --> 00:32:20,340 close to this phase boundary. 477 00:32:20,340 --> 00:32:29,380 This has a d333 that is greater than 2,000 picocuries 478 00:32:29,380 --> 00:32:30,650 per Newton. 479 00:32:34,590 --> 00:32:39,960 And pico is 10 to the minus 12. 480 00:32:39,960 --> 00:32:48,510 So this is a piezoelectric electric modulus that is 10 to 481 00:32:48,510 --> 00:32:52,435 the 3 times d11 for quartz. 482 00:32:58,490 --> 00:33:01,430 So 3 orders of magnitude stronger than this very 483 00:33:01,430 --> 00:33:03,340 commonly used piezoelectric material. 484 00:33:05,860 --> 00:33:12,390 Some of these materials have strains getting close to 1%. 485 00:33:12,390 --> 00:33:15,630 So this is something that will actually twitch on the lab 486 00:33:15,630 --> 00:33:18,900 bench, when you apply a field to it. 487 00:33:18,900 --> 00:33:20,450 So these are entirely new. 488 00:33:20,450 --> 00:33:24,040 People still don't know the origin of this behavior. 489 00:33:24,040 --> 00:33:28,720 And it's still under considerable study. 490 00:33:28,720 --> 00:33:31,120 So this is a new family of materials. 491 00:33:31,120 --> 00:33:34,850 It's very exciting, and undergoing a lot of 492 00:33:34,850 --> 00:33:38,940 investigation and development work at the moment. 493 00:33:38,940 --> 00:33:41,100 Alright, we are almost out of time. 494 00:33:41,100 --> 00:33:42,910 Time goes fast when you're having fun. 495 00:33:46,730 --> 00:33:50,740 Let me raise the question that we'll consider next time, 496 00:33:50,740 --> 00:33:52,780 which will be one of our last lectures. 497 00:33:52,780 --> 00:34:00,000 And that is, is it possible to create representation surfaces 498 00:34:00,000 --> 00:34:03,300 that tell you how the piezoelectric properties of a 499 00:34:03,300 --> 00:34:07,300 particular material will vary with direction? 500 00:34:07,300 --> 00:34:09,219 Well, vary with what? 501 00:34:09,219 --> 00:34:13,110 Well, we talk about the direct piezoelectric effect. 502 00:34:13,110 --> 00:34:18,770 This gives us components of-- 503 00:34:18,770 --> 00:34:23,489 well let's look at the simpler one, in 504 00:34:23,489 --> 00:34:25,449 terms of what we apply. 505 00:34:25,449 --> 00:34:28,929 We have the converse piezoelectric effect that says 506 00:34:28,929 --> 00:34:36,290 that epsilon ijk is going to be dijk times e sub i. 507 00:34:39,179 --> 00:34:43,710 So we've got a piece of material, and we apply a 508 00:34:43,710 --> 00:34:46,915 field, e sub i. 509 00:34:46,915 --> 00:34:50,780 So we can vary this in space relative to a coordinate 510 00:34:50,780 --> 00:34:54,646 system x1, x2, x3. 511 00:34:54,646 --> 00:34:57,575 But how in the world are we going to show what happens? 512 00:35:00,800 --> 00:35:02,720 So I always do an extra thing in here. 513 00:35:13,730 --> 00:35:18,540 There are 9 components to the strain 514 00:35:18,540 --> 00:35:22,370 tensor, which is symmetric. 515 00:35:22,370 --> 00:35:24,865 So they're really six responses that are unique. 516 00:35:27,890 --> 00:35:30,210 So yes, we can define the direction 517 00:35:30,210 --> 00:35:33,600 of the applied field. 518 00:35:33,600 --> 00:35:37,070 But they're going to be 6 different strains. 519 00:35:37,070 --> 00:35:40,910 So we're going to need six representation surfaces. 520 00:35:40,910 --> 00:35:45,320 One for each of the three tensile strains, and one for 521 00:35:45,320 --> 00:35:48,510 each of the three shear strains. 522 00:35:48,510 --> 00:35:51,060 So you can't do it with a single surface. 523 00:35:51,060 --> 00:35:58,050 So you can't do much other than say, there are certain 524 00:35:58,050 --> 00:36:05,220 responses which are intended to emphasize one particular 525 00:36:05,220 --> 00:36:12,400 sort of strain or one particular sort of 526 00:36:12,400 --> 00:36:13,040 polarization. 527 00:36:13,040 --> 00:36:18,060 So one of the things we might do is to cut a very thin plate 528 00:36:18,060 --> 00:36:20,700 of something like quartz, and subject it 529 00:36:20,700 --> 00:36:25,010 to a uniaxial stress. 530 00:36:25,010 --> 00:36:28,050 So let's say sigma along the x3 axis. 531 00:36:28,050 --> 00:36:31,860 So we're looking at a very restricted strain tensor. 532 00:36:31,860 --> 00:36:36,530 That's 0, 0, 0, 0, 0, 0, 0, 0, sigma 3. 533 00:36:39,500 --> 00:36:41,710 And in response to that strain, there are going to be 534 00:36:41,710 --> 00:36:46,320 three different components of the polarization. 535 00:36:46,320 --> 00:36:51,740 Polarization is manifested as a charge per unit area. 536 00:36:51,740 --> 00:36:54,310 So if we make a very thin plate-- 537 00:36:54,310 --> 00:36:56,610 to be sure there will be charges induced 538 00:36:56,610 --> 00:37:00,580 on these thin edges-- 539 00:37:00,580 --> 00:37:04,790 but if it's got a surface area that's a large compared to the 540 00:37:04,790 --> 00:37:08,110 area of these thin edges, we are going to be measuring 541 00:37:08,110 --> 00:37:10,200 primarily p3-- 542 00:37:10,200 --> 00:37:13,160 the component of polarization that's 543 00:37:13,160 --> 00:37:14,970 normal to this surface-- 544 00:37:14,970 --> 00:37:20,070 and that might have a charge per unit area that is 545 00:37:20,070 --> 00:37:22,880 comparable to these other two charged surfaces. 546 00:37:22,880 --> 00:37:28,120 But because the area by design of our specimen is so large, 547 00:37:28,120 --> 00:37:31,650 the response that would be most easy to detect, and which 548 00:37:31,650 --> 00:37:38,280 would be the largest response, by design, would be p3, which 549 00:37:38,280 --> 00:37:42,620 is the charge per unit area on this surface. 550 00:37:42,620 --> 00:37:45,420 So this is an effect we can define for a particular 551 00:37:45,420 --> 00:37:51,370 sample, and for a particular special form of the 552 00:37:51,370 --> 00:37:53,290 generalized force. 553 00:37:53,290 --> 00:37:59,940 And we then can ask, what is the value of the single 554 00:37:59,940 --> 00:38:04,180 modulus that relates p3 to sigma 3? 555 00:38:04,180 --> 00:38:05,970 And that's a question we can ask. 556 00:38:05,970 --> 00:38:14,070 And we can plot that response as a function of direction of 557 00:38:14,070 --> 00:38:19,410 a plate that we consider as being cut out of a single 558 00:38:19,410 --> 00:38:23,080 crystal, and different orientations, and then ask how 559 00:38:23,080 --> 00:38:24,030 this modulus-- 560 00:38:24,030 --> 00:38:25,930 which connects the two-- 561 00:38:25,930 --> 00:38:29,900 changes with the orientation of x3. 562 00:38:29,900 --> 00:38:33,050 So that is a question we can ask. 563 00:38:33,050 --> 00:38:35,940 And these surfaces are absolutely wild, highly 564 00:38:35,940 --> 00:38:39,130 anisotropic, can be identically zero in certain 565 00:38:39,130 --> 00:38:42,600 special directions, and they are very interesting, and a 566 00:38:42,600 --> 00:38:43,940 lot of fun to look at. 567 00:38:43,940 --> 00:38:46,580 So we'll take a quick look at a couple of those, which won't 568 00:38:46,580 --> 00:38:49,690 come as a surprise because they're already worked out for 569 00:38:49,690 --> 00:38:50,940 you in the notes. 570 00:38:53,160 --> 00:38:55,520 So we'll take a look at one of those. 571 00:38:55,520 --> 00:38:58,640 And in the problem set, I invite you to amuse yourself 572 00:38:58,640 --> 00:39:02,700 by looking at such representation surfaces for 573 00:39:02,700 --> 00:39:04,920 two other point groups. 574 00:39:04,920 --> 00:39:08,550 And with that, having kept you til five after 575 00:39:08,550 --> 00:39:10,140 the hour I will quit.