1 00:00:00,260 --> 00:00:03,720 The following content is provided by MIT OpenCourseWare 2 00:00:03,720 --> 00:00:06,120 under a Creative Commons license. 3 00:00:06,120 --> 00:00:08,740 Additional information about our license, and MIT 4 00:00:08,740 --> 00:00:13,590 OpenCourseWare in general, is available at ocw.mit.edu. 5 00:00:16,379 --> 00:00:16,830 PROFESSOR: OK. 6 00:00:16,830 --> 00:00:18,910 Let's resume. 7 00:00:18,910 --> 00:00:22,600 I cut things off at a time when we had the final answer. 8 00:00:22,600 --> 00:00:25,180 And I left you hanging because we don't know what the final 9 00:00:25,180 --> 00:00:27,050 answer is telling us. 10 00:00:27,050 --> 00:00:31,920 This says that as we change the orientation of the normal 11 00:00:31,920 --> 00:00:39,710 to the plate, relative to x3 and, secondarily, as we change 12 00:00:39,710 --> 00:00:46,340 the angle between the threefold axis and x2, we get 13 00:00:46,340 --> 00:00:52,270 this strange third-rank trigonometric function. 14 00:00:52,270 --> 00:00:58,020 Let's convert these cosines into the appropriate angles. 15 00:00:58,020 --> 00:01:00,180 This was the twofold axis. 16 00:01:00,180 --> 00:01:02,320 And this was x1. 17 00:01:02,320 --> 00:01:04,610 This was the direction of the threefold axis. 18 00:01:04,610 --> 00:01:06,290 This was x3. 19 00:01:06,290 --> 00:01:10,290 And this is the direction x2. 20 00:01:10,290 --> 00:01:13,630 And that comes out in between a pair of twofold axes. 21 00:01:13,630 --> 00:01:19,380 And we may want to look at this from up above, relative 22 00:01:19,380 --> 00:01:21,450 to these twofold axes that occur. 23 00:01:26,680 --> 00:01:31,270 This thing here looks like a trigonometric identity, 24 00:01:31,270 --> 00:01:33,840 doesn't it? 25 00:01:33,840 --> 00:01:39,420 Let's let this angle here be theta. 26 00:01:39,420 --> 00:01:44,600 And l1, then, is cosine of theta. 27 00:01:44,600 --> 00:01:53,790 And so our geometry is that this is l1 cosine of theta. 28 00:01:53,790 --> 00:01:59,730 This is l2, which is the cosine of the angle between 29 00:01:59,730 --> 00:02:01,640 our direction and x2. 30 00:02:01,640 --> 00:02:10,070 And that is the cosine of pi over 2, minus theta. 31 00:02:12,780 --> 00:02:16,500 And that, then, is equal to sine of theta. 32 00:02:20,540 --> 00:02:22,790 So this identity, as you all know-- 33 00:02:22,790 --> 00:02:25,210 I'm not telling you anything that you don't already know-- 34 00:02:25,210 --> 00:02:28,770 this is d is equal to d1,1 1, times the 35 00:02:28,770 --> 00:02:31,180 cosine of 3 theta, right? 36 00:02:31,180 --> 00:02:33,380 You knew that. 37 00:02:33,380 --> 00:02:34,220 Believe it or not, it is. 38 00:02:34,220 --> 00:02:37,090 It's one of these obscure trigonometric identities, 39 00:02:37,090 --> 00:02:40,400 which wouldn't occur to you in a million years unless you go 40 00:02:40,400 --> 00:02:42,110 digging through some handbooks. 41 00:02:42,110 --> 00:02:45,460 So this is rather astonishingly simple. 42 00:02:45,460 --> 00:02:47,420 It's simply a cosine function. 43 00:02:47,420 --> 00:02:51,330 But the interesting thing is it goes as a 44 00:02:51,330 --> 00:02:55,040 function of 3 theta. 45 00:02:55,040 --> 00:02:59,640 So this goes through one cycle between a pair of adjacent 46 00:02:59,640 --> 00:03:00,660 twofold axes. 47 00:03:00,660 --> 00:03:05,260 So it starts out as d1,1. 48 00:03:05,260 --> 00:03:08,090 And it finishes up at d1,1. 49 00:03:08,090 --> 00:03:12,420 And in between it is minus d1,1. 50 00:03:12,420 --> 00:03:15,150 And so it goes down. 51 00:03:15,150 --> 00:03:18,220 And now we come to something that I do differently than 52 00:03:18,220 --> 00:03:19,200 most people do. 53 00:03:19,200 --> 00:03:26,710 Most people will say it goes as minus d1,1. 54 00:03:26,710 --> 00:03:34,760 But minus d1,1 I like to show as a lobe going off in this 55 00:03:34,760 --> 00:03:36,705 direction, with a minus sign. 56 00:03:39,968 --> 00:03:42,896 AUDIENCE: How do you get that cosine of 3 theta? 57 00:03:42,896 --> 00:03:43,384 PROFESSOR: Hmm? 58 00:03:43,384 --> 00:03:46,800 AUDIENCE: How do you get d equals d1 cosine 3 theta? 59 00:03:46,800 --> 00:03:47,776 PROFESSOR: How do I get that? 60 00:03:47,776 --> 00:03:48,264 AUDIENCE: Yeah. 61 00:03:48,264 --> 00:03:51,192 PROFESSOR: That is just a trigonometric identify, 62 00:03:51,192 --> 00:03:55,430 believe it or not, a well-known 63 00:03:55,430 --> 00:03:57,970 trigonometric identify. 64 00:03:57,970 --> 00:04:00,200 Actually, it's an exceedingly obscure 65 00:04:00,200 --> 00:04:02,590 trigonometric identity. 66 00:04:02,590 --> 00:04:05,350 So the way this is going to go from x1 is it's going to be a 67 00:04:05,350 --> 00:04:07,830 positive lobe around x1. 68 00:04:07,830 --> 00:04:13,060 There's going to be a negative lobe along x2 and then a 69 00:04:13,060 --> 00:04:19,130 positive lobe again about the twofold axis that's 120 70 00:04:19,130 --> 00:04:20,300 degrees away. 71 00:04:20,300 --> 00:04:24,770 And then a negative lobe, and then a positive lobe opposite 72 00:04:24,770 --> 00:04:28,550 this negative lobe, and a negative lobe opposite this 73 00:04:28,550 --> 00:04:29,630 positive lobe. 74 00:04:29,630 --> 00:04:32,950 So it's a six-membered-- 75 00:04:32,950 --> 00:04:35,000 six lobes. 76 00:04:35,000 --> 00:04:42,120 There is always a positive lobe opposite a negative lobe. 77 00:04:42,120 --> 00:04:47,820 And what this means is that the charge is of opposite sign 78 00:04:47,820 --> 00:04:51,620 on opposite ends of the twofold axes. 79 00:04:51,620 --> 00:04:54,980 So the response peaks up on twofold axes. 80 00:04:54,980 --> 00:04:58,700 Now the thing that I don't like is that what Nye does is 81 00:04:58,700 --> 00:05:02,540 to say, OK, this is a negative value. 82 00:05:02,540 --> 00:05:06,410 So you should plot the radius in a negative direction. 83 00:05:06,410 --> 00:05:11,110 And that puts it over here, right on top of 84 00:05:11,110 --> 00:05:12,560 this positive lobe. 85 00:05:12,560 --> 00:05:18,890 So what Nye shows is the polar plot of this result, in the 86 00:05:18,890 --> 00:05:23,680 plane of the twofold axes, is simply this. 87 00:05:23,680 --> 00:05:25,550 This is x1. 88 00:05:25,550 --> 00:05:27,860 And he shows a lobe here. 89 00:05:27,860 --> 00:05:29,580 And he shows a lobe here. 90 00:05:29,580 --> 00:05:32,950 And he shows a lobe here. 91 00:05:32,950 --> 00:05:35,530 And if you interpret that as a polar plot you say, well, I 92 00:05:35,530 --> 00:05:37,210 know what the value is in here. 93 00:05:37,210 --> 00:05:39,320 It's decreasing. 94 00:05:39,320 --> 00:05:41,550 And there's nothing going on in here. 95 00:05:41,550 --> 00:05:43,220 So the response must be 0. 96 00:05:43,220 --> 00:05:45,180 And then it starts coming up again. 97 00:05:45,180 --> 00:05:47,860 And there's a response in different directions here. 98 00:05:47,860 --> 00:05:49,770 And then it goes back down to 0. 99 00:05:49,770 --> 00:05:52,115 And in this range, there's nothing going on. 100 00:05:52,115 --> 00:05:53,500 And that's not true. 101 00:05:53,500 --> 00:06:00,090 What's going on is a negative value of the modulus. 102 00:06:00,090 --> 00:06:03,860 And to me, that becomes abundantly clear if you just 103 00:06:03,860 --> 00:06:07,590 put a sign that labels the sign of the modulus within 104 00:06:07,590 --> 00:06:08,840 those lobes. 105 00:06:12,310 --> 00:06:17,070 Now I submit, that's pretty anisotropic, isn't it? 106 00:06:17,070 --> 00:06:17,620 Yes, Steve? 107 00:06:17,620 --> 00:06:20,020 AUDIENCE: Did you just arbitrarily choose where your 108 00:06:20,020 --> 00:06:23,586 positive and negatives go, is it [INAUDIBLE] 109 00:06:23,586 --> 00:06:24,062 out here? 110 00:06:24,062 --> 00:06:24,540 PROFESSOR: No. 111 00:06:24,540 --> 00:06:25,530 This comes out here. 112 00:06:25,530 --> 00:06:30,340 When l1 is 0, that's-- 113 00:06:30,340 --> 00:06:31,590 excuse me. 114 00:06:33,950 --> 00:06:37,830 When the angle is 0, l1 is plus 1. 115 00:06:41,420 --> 00:06:44,340 Remember that l1 is the cosine of this angle. 116 00:06:44,340 --> 00:06:47,480 When that angle is 0, then the value is plus 1. 117 00:06:50,530 --> 00:06:54,500 So I was careful to put the label x1 on this. 118 00:06:54,500 --> 00:06:57,850 Now the other thing that we should examine is how these 119 00:06:57,850 --> 00:07:02,840 lobes vary in a direction perpendicular to x, in a 120 00:07:02,840 --> 00:07:04,830 direction that includes x3. 121 00:07:04,830 --> 00:07:13,790 So let us look at how the function varies in a direction 122 00:07:13,790 --> 00:07:15,105 perpendicular to-- 123 00:07:18,440 --> 00:07:20,930 that includes x1 and x3. 124 00:07:20,930 --> 00:07:22,180 OK. 125 00:07:25,280 --> 00:07:28,190 Call this angle phi. 126 00:07:28,190 --> 00:07:40,070 And in the x1, x3 plane, d1,1,1 prime turns out to be 127 00:07:40,070 --> 00:07:44,800 l1 cubed d1,1. 128 00:07:44,800 --> 00:07:48,960 Before you go to our general expression. 129 00:07:48,960 --> 00:07:54,020 And l2 is cosine of 90. 130 00:07:54,020 --> 00:07:55,160 This thing drops out. 131 00:07:55,160 --> 00:07:57,830 We're left with simply d1,1 prime equals 132 00:07:57,830 --> 00:08:00,300 l cubed times d1,1. 133 00:08:00,300 --> 00:08:08,090 So this then goes as cosine cubed of phi, which is a lobe 134 00:08:08,090 --> 00:08:10,580 that starts out at plus 1. 135 00:08:10,580 --> 00:08:16,740 And then because it's cosine cubed, this dies out very, 136 00:08:16,740 --> 00:08:18,920 very rapidly. 137 00:08:18,920 --> 00:08:25,090 So these lobes are very flat in the x1, x2 plane, die out 138 00:08:25,090 --> 00:08:29,610 very rapidly as the cube of phi, where phi is the angle 139 00:08:29,610 --> 00:08:31,380 between the twofold axis and x3. 140 00:08:35,630 --> 00:08:39,000 So the interesting thing about this surface is that if you 141 00:08:39,000 --> 00:08:42,610 decided to pick up the random fragment of crystal and 142 00:08:42,610 --> 00:08:46,050 determine what its Piezoelectric Modulus is. 143 00:08:49,110 --> 00:08:52,540 One way of doing this, a poor man's test for 144 00:08:52,540 --> 00:08:58,330 piezoelectricity is to just clamp a fragment of crystal 145 00:08:58,330 --> 00:09:04,570 between two electrodes and then hook this up to a 146 00:09:04,570 --> 00:09:08,700 variable frequency generator that sweeps through a range of 147 00:09:08,700 --> 00:09:11,920 frequencies, changes the frequency of a voltage across 148 00:09:11,920 --> 00:09:14,730 these plates, and then comes back and sweeps again. 149 00:09:14,730 --> 00:09:17,160 Or alternatively, have a knob one on that 150 00:09:17,160 --> 00:09:18,780 lets you change frequency. 151 00:09:18,780 --> 00:09:24,290 Then put a pair of earphones across the crystal. 152 00:09:24,290 --> 00:09:28,640 And if we would do that, by having this variable frequency 153 00:09:28,640 --> 00:09:33,960 generator, as we turned it, when we hit a resonant 154 00:09:33,960 --> 00:09:38,800 frequency that set up either a full wavelength of a 155 00:09:38,800 --> 00:09:40,710 vibrational wave in the crystal-- 156 00:09:40,710 --> 00:09:44,690 or half wavelength, or one wavelength, or 3/2 157 00:09:44,690 --> 00:09:45,680 wavelength-- 158 00:09:45,680 --> 00:09:47,060 there'd be a resonance. 159 00:09:47,060 --> 00:09:49,430 And the capacitance does something crazy. 160 00:09:49,430 --> 00:09:51,560 It does like this as you go through 161 00:09:51,560 --> 00:09:53,390 the resonant frequency. 162 00:09:53,390 --> 00:09:54,860 And then your simple detector-- 163 00:09:54,860 --> 00:09:55,820 a pair of earphones-- 164 00:09:55,820 --> 00:09:58,870 as you tuned the frequency, you would hear static as you 165 00:09:58,870 --> 00:10:03,190 went through this discontinuity in capacitance. 166 00:10:03,190 --> 00:10:07,750 Or alternatively, you could put this on a cathode ray tube 167 00:10:07,750 --> 00:10:13,230 and just have a sweep frequency and put the voltage 168 00:10:13,230 --> 00:10:18,530 across the electrodes on the oscilloscope screen. 169 00:10:18,530 --> 00:10:20,660 And you would see something like this, and then this 170 00:10:20,660 --> 00:10:24,560 discontinuity, and then maybe second harmonic. 171 00:10:24,560 --> 00:10:25,640 And it would work just fine. 172 00:10:25,640 --> 00:10:28,520 This is a poor man's way of detecting piezoelectricity. 173 00:10:28,520 --> 00:10:30,440 But what if-- 174 00:10:30,440 --> 00:10:32,260 what if-- 175 00:10:32,260 --> 00:10:35,040 you happen to have your piece of quartz with the electrodes 176 00:10:35,040 --> 00:10:37,470 directly on the c-axis? 177 00:10:37,470 --> 00:10:41,080 The modulus would be 0, and you wouldn't detect anything. 178 00:10:41,080 --> 00:10:44,830 Or if you put on the probes such that they were in a 179 00:10:44,830 --> 00:10:49,070 direction that was in between these lobes where, again, the 180 00:10:49,070 --> 00:10:51,580 Piezoelectric Modulus has gone to 0. 181 00:10:51,580 --> 00:10:53,390 You wouldn't find anything. 182 00:10:53,390 --> 00:10:56,790 So measuring the Piezoelectric Modulus for a random chunk of 183 00:10:56,790 --> 00:11:00,160 material is dangerous, if you just look at one direction and 184 00:11:00,160 --> 00:11:03,780 say nothing's going on; it's not piezoelectric. 185 00:11:03,780 --> 00:11:05,940 The other thing is the material could have a 186 00:11:05,940 --> 00:11:07,910 piezoelectric response that's so weak you just 187 00:11:07,910 --> 00:11:10,430 can't measure it. 188 00:11:10,430 --> 00:11:11,260 Yeah, OK? 189 00:11:11,260 --> 00:11:12,670 AUDIENCE: This stuff's only for single crystals, right? 190 00:11:12,670 --> 00:11:14,315 Because if you [INAUDIBLE] polycrystal material, you're 191 00:11:14,315 --> 00:11:15,490 getting [INAUDIBLE] averaging. 192 00:11:15,490 --> 00:11:17,380 PROFESSOR: You're averaging over all directions. 193 00:11:17,380 --> 00:11:18,910 So this is for a single crystal. 194 00:11:18,910 --> 00:11:23,770 And that's the only time you get these exotic, very 195 00:11:23,770 --> 00:11:25,020 anisotropic surfaces. 196 00:11:31,360 --> 00:11:34,190 Another method that I've seen in a rather old 197 00:11:34,190 --> 00:11:37,320 book is really nice. 198 00:11:37,320 --> 00:11:42,840 What you do is you cut your crystal into a little plate. 199 00:11:42,840 --> 00:11:50,110 And then you drive the plate by means of putting electrodes 200 00:11:50,110 --> 00:11:52,500 on it and hook this up to a variable frequency. 201 00:11:52,500 --> 00:11:56,540 And if the material is transparent, you will set up-- 202 00:11:56,540 --> 00:12:00,740 depending on the velocity of sound in the material-- 203 00:12:00,740 --> 00:12:05,470 you will set up standing waves when you 204 00:12:05,470 --> 00:12:07,220 hit the right frequency. 205 00:12:07,220 --> 00:12:12,110 And these lines that I've drawn are places where the 206 00:12:12,110 --> 00:12:14,420 displacement is always 0. 207 00:12:14,420 --> 00:12:17,750 And these things that I've indicated at little squares is 208 00:12:17,750 --> 00:12:20,780 a region where the displacement goes up and down 209 00:12:20,780 --> 00:12:23,630 between its maximum and minimum extreme. 210 00:12:23,630 --> 00:12:27,460 Now if this crystal is transparent and you shine a 211 00:12:27,460 --> 00:12:35,230 light through it, these little regions bounded by lines of 212 00:12:35,230 --> 00:12:37,470 zero displacement act like little lenses. 213 00:12:37,470 --> 00:12:43,870 So if you pass a beam of light through it, you get a bunch of 214 00:12:43,870 --> 00:12:47,950 right maxima, little focused spots on the sheet. 215 00:12:47,950 --> 00:12:51,120 And you can determine for a particular frequency if this 216 00:12:51,120 --> 00:12:54,745 is the 1 and 1/2 wavelengths along a dimension that you 217 00:12:54,745 --> 00:13:01,560 know, you can again find the Piezoelectric Moduli. 218 00:13:01,560 --> 00:13:06,010 So there are lots of ways of detecting this effect. 219 00:13:06,010 --> 00:13:08,050 But when you have a single crystal, you've got to really 220 00:13:08,050 --> 00:13:09,940 look at this as a function of direction, to 221 00:13:09,940 --> 00:13:11,190 be absolutely sure. 222 00:13:15,160 --> 00:13:24,530 One other Piezoelectric Modulus is in the problem set. 223 00:13:24,530 --> 00:13:29,590 4 bar 3m, this is the structure of sphalerite and a 224 00:13:29,590 --> 00:13:34,800 lot of the compound semiconductors. 225 00:13:34,800 --> 00:13:36,000 This is not in the notes. 226 00:13:36,000 --> 00:13:38,620 But if you try the problem, what you find is, again, a 227 00:13:38,620 --> 00:13:42,210 highly, highly anisotropic behavior. 228 00:13:42,210 --> 00:13:47,340 And this is the direction of the unit cell of the crystal. 229 00:13:47,340 --> 00:13:50,290 You find that the piezoelectric response is a 230 00:13:50,290 --> 00:13:55,070 series of very sharp lobes, positive and negative, going 231 00:13:55,070 --> 00:13:58,260 along the directions of the body diagonal, namely the 1, 232 00:13:58,260 --> 00:14:01,410 1, 1 directions in the crystal. 233 00:14:01,410 --> 00:14:03,420 And then there's another lobe that goes like 234 00:14:03,420 --> 00:14:06,040 this, negative, positive. 235 00:14:06,040 --> 00:14:09,850 Another lobe that goes up here, positive and negative. 236 00:14:09,850 --> 00:14:13,630 So very sharp lobes along the body diagonal, alternately 237 00:14:13,630 --> 00:14:18,500 positive, negative, positive, negative as you go around 0, 238 00:14:18,500 --> 00:14:19,750 0, 1 plane. 239 00:14:25,030 --> 00:14:25,330 OK. 240 00:14:25,330 --> 00:14:30,890 That's what the longitudinal piezoelectric effect is like. 241 00:14:30,890 --> 00:14:36,900 You can invent single-crystal devices for weighing fish. 242 00:14:36,900 --> 00:14:38,720 And then you have an interesting question. 243 00:14:38,720 --> 00:14:42,930 If you hang the weight on the crystal, and it's a flat 244 00:14:42,930 --> 00:14:48,010 plate, how would you orient the crystal to get the optimum 245 00:14:48,010 --> 00:14:49,990 sensitivity? 246 00:14:49,990 --> 00:14:53,460 So you define a scalar modulus for the crystal in the 247 00:14:53,460 --> 00:14:56,810 particular direction you're exerting a uniaxial stress. 248 00:14:56,810 --> 00:15:01,480 And then you express this module in terms of theta, the 249 00:15:01,480 --> 00:15:07,020 angle of rotation, within the crystographic plane. 250 00:15:07,020 --> 00:15:10,290 In the case of the sup cell if you looked at that geometry, 251 00:15:10,290 --> 00:15:14,660 that is a flat plate subject to compression, where you 252 00:15:14,660 --> 00:15:20,920 measure the surface charge perpendicular to the direction 253 00:15:20,920 --> 00:15:23,230 of the uniaxial stress, not parallel to it 254 00:15:23,230 --> 00:15:24,540 as we've done here. 255 00:15:24,540 --> 00:15:26,850 And then there's a question if you rotate the plane of the 256 00:15:26,850 --> 00:15:31,240 crystal about the imposed stress, what orientation gives 257 00:15:31,240 --> 00:15:33,890 you optimum, maximum response? 258 00:15:33,890 --> 00:15:35,970 So there are fun things that you could do with that. 259 00:15:40,100 --> 00:15:42,610 We have a quiz next time. 260 00:15:42,610 --> 00:15:48,000 There's a lame-duck period after the quiz will be over. 261 00:15:48,000 --> 00:15:55,550 And I want to say a little bit about the tensor aspects of 262 00:15:55,550 --> 00:15:58,440 elastic moduli. 263 00:15:58,440 --> 00:16:03,640 Mechanical behavior is something that you have or 264 00:16:03,640 --> 00:16:09,020 will cover in great detail in a graduate-level class on 265 00:16:09,020 --> 00:16:12,770 mechanics, but I think very few people who deal with 266 00:16:12,770 --> 00:16:16,960 things other than cubic materials or with 267 00:16:16,960 --> 00:16:20,850 single-crystal materials where the elastic constants are 268 00:16:20,850 --> 00:16:25,210 functions of crystal symmetry and direction. 269 00:16:25,210 --> 00:16:29,670 So let me, at least at the time available, set up a 270 00:16:29,670 --> 00:16:37,270 definition of stress in terms of strain. 271 00:16:37,270 --> 00:16:39,980 Stress is a second-rank tensor. 272 00:16:39,980 --> 00:16:46,080 There are six unique components. 273 00:16:46,080 --> 00:16:50,230 So we could write an expression for sigma 1,1; 274 00:16:50,230 --> 00:16:58,780 sigma 2,2; sigma 3,3; sigma 4,4; and sigma 5,5. 275 00:16:58,780 --> 00:17:04,092 And sigma-- 276 00:17:04,092 --> 00:17:05,750 AUDIENCE: --1,2. 277 00:17:05,750 --> 00:17:13,300 PROFESSOR: No, I want to go down like this. 278 00:17:13,300 --> 00:17:23,940 So this would be sigma 2,3; sigma 1,3; and sigma 1,2; 279 00:17:23,940 --> 00:17:28,160 eventually to be known as sigma 1, sigma 2, sigma 3, 280 00:17:28,160 --> 00:17:33,420 sigma 4, sigma 5, and sigma six. 281 00:17:33,420 --> 00:17:35,430 But actually, if we're writing out a full tensor relation, we 282 00:17:35,430 --> 00:17:37,450 should write down all six elements here. 283 00:17:37,450 --> 00:17:44,080 So let me do that and put in a 3,2. 284 00:17:44,080 --> 00:17:46,820 Put in all nine of them, because before we condense to 285 00:17:46,820 --> 00:17:50,080 matrix form, that is what we're going to have. 286 00:17:50,080 --> 00:17:52,910 So we'll have a sigma 2,3; we'll have a sigma 3,2; we'll 287 00:17:52,910 --> 00:18:00,650 have a sigma 1,3; a sigma 3,1; a sigma 1,2; and a sigma 2,1. 288 00:18:00,650 --> 00:18:04,460 And we can express each of the stresses in terms of the 289 00:18:04,460 --> 00:18:05,720 elements of strain. 290 00:18:05,720 --> 00:18:10,110 So I'll have an epsilon 1,1; an epsilon 1,2; an epsilon 291 00:18:10,110 --> 00:18:18,132 2,2; an epsilon 3,3; an epsilon 2,3; an epsilon 3,2; 292 00:18:18,132 --> 00:18:26,190 an epsilon 1,3; an epsilon 3,1; an epsilon 1,2; and an 293 00:18:26,190 --> 00:18:27,740 epsilon 2,1. 294 00:18:27,740 --> 00:18:29,290 Nine terms, nine terms. 295 00:18:29,290 --> 00:18:33,810 And in between is going to be a tensor 296 00:18:33,810 --> 00:18:38,550 consisting of 9 by 9 elements. 297 00:18:38,550 --> 00:18:41,030 There will be 81 different coefficients in here. 298 00:18:43,720 --> 00:18:46,900 If we want to derive symmetry constraints, we're going to 299 00:18:46,900 --> 00:18:50,210 have to, for every one of those 81 elements, do a 300 00:18:50,210 --> 00:18:51,700 transformation. 301 00:18:51,700 --> 00:18:56,050 And each transformed element is going to be a sum of four 302 00:18:56,050 --> 00:19:00,540 direction cosines times one tensor element, repeated 81 303 00:19:00,540 --> 00:19:02,690 different times. 304 00:19:02,690 --> 00:19:05,990 Not something to be undertaken on a short afternoon. 305 00:19:08,750 --> 00:19:09,090 OK. 306 00:19:09,090 --> 00:19:13,640 What are the tensor elements in here? 307 00:19:13,640 --> 00:19:18,170 They are represented by the symbol c. 308 00:19:18,170 --> 00:19:21,840 This would be c1,1,1,1. 309 00:19:21,840 --> 00:19:30,880 This would be c1,1,2,2 times epsilon 2,2; a c1,1,3,3; a 310 00:19:30,880 --> 00:19:33,830 c1,1,2,3; and so on. 311 00:19:33,830 --> 00:19:39,030 And these c's, in one of the great perversions of 312 00:19:39,030 --> 00:19:41,960 scientific notation, are called stiffnesses. 313 00:19:51,270 --> 00:19:54,010 The other thing we could do is to write 314 00:19:54,010 --> 00:19:57,870 strain in terms of stress. 315 00:19:57,870 --> 00:19:59,420 And really, stress is something you 316 00:19:59,420 --> 00:20:00,570 can do to the crystal. 317 00:20:00,570 --> 00:20:04,380 So stress is an independent variable. 318 00:20:04,380 --> 00:20:08,010 And this is something that I feel more at home with. 319 00:20:08,010 --> 00:20:14,190 So I'll have an epsilon 1,1; 2,2; 3,3; epsilon 3,2; 2,3; 320 00:20:14,190 --> 00:20:21,010 epsilon 1,3; epsilon 3,1; epsilon 2,1; 321 00:20:21,010 --> 00:20:23,160 and an epsilon 1,2. 322 00:20:23,160 --> 00:20:26,260 Again, nine elements of strain. 323 00:20:26,260 --> 00:20:30,120 And these will be given by coefficients times each of the 324 00:20:30,120 --> 00:20:37,440 nine elements of stress, sigma 1,1; sigma 2,2; sigma 3,3; 325 00:20:37,440 --> 00:20:42,940 sigma 2,3; sigma 3,2; and so on for nine different 326 00:20:42,940 --> 00:20:44,790 components. 327 00:20:44,790 --> 00:20:51,020 Andy the coefficients here are designated with the symbol s. 328 00:20:51,020 --> 00:20:52,800 These are the tensor elements. 329 00:20:52,800 --> 00:20:55,020 So this is s1,1,1. 330 00:20:55,020 --> 00:20:57,890 This is s1,1,2,2. 331 00:20:57,890 --> 00:21:00,900 And the s's stand for compliances. 332 00:21:06,890 --> 00:21:10,410 Now English is a very strange language. 333 00:21:10,410 --> 00:21:19,350 But to call compliance s and to call stiffness c is surely 334 00:21:19,350 --> 00:21:24,510 a perversion that can be designed for no other purpose 335 00:21:24,510 --> 00:21:28,830 than to confuse the introductory student and give 336 00:21:28,830 --> 00:21:32,210 the instructor some feeling of superiority. 337 00:21:32,210 --> 00:21:34,800 I don't feel superior, I feel embarrassed that I have to 338 00:21:34,800 --> 00:21:36,510 explain this. 339 00:21:36,510 --> 00:21:42,110 Stiffness, c; compliance, s. 340 00:21:42,110 --> 00:21:44,300 It's got be one of the atrocities 341 00:21:44,300 --> 00:21:46,630 of scientific notation. 342 00:21:46,630 --> 00:21:49,920 To remember which goes with which, I'll tell you what 343 00:21:49,920 --> 00:21:50,630 works for me. 344 00:21:50,630 --> 00:21:52,620 If it doesn't work for you, forget it. 345 00:21:52,620 --> 00:21:55,380 The s goes with sigma. 346 00:22:00,940 --> 00:22:05,530 And the little c goes with the epsilon. 347 00:22:05,530 --> 00:22:07,920 That's a c with a bar in it. 348 00:22:07,920 --> 00:22:08,850 That works for me. 349 00:22:08,850 --> 00:22:12,160 If it doesn't work for you, use your own mnemonic device. 350 00:22:18,050 --> 00:22:22,010 English does a lot of things strangely, but nothing is as 351 00:22:22,010 --> 00:22:23,260 perverse as this. 352 00:22:26,620 --> 00:22:27,750 Now what's some examples? 353 00:22:27,750 --> 00:22:29,090 Consider this. 354 00:22:29,090 --> 00:22:34,120 You park your car in your driveway, but you drive your 355 00:22:34,120 --> 00:22:35,370 car on a parkway. 356 00:22:38,010 --> 00:22:40,970 Why don't you drive your car on a driveway and park your 357 00:22:40,970 --> 00:22:42,110 car in a parkway? 358 00:22:42,110 --> 00:22:43,160 No, it's the other way around. 359 00:22:43,160 --> 00:22:46,560 It's almost as bad as this. 360 00:22:46,560 --> 00:22:47,810 What are some other ones? 361 00:22:50,000 --> 00:22:53,490 In my basement, I have something that's called a hot 362 00:22:53,490 --> 00:22:55,260 water heater. 363 00:22:55,260 --> 00:22:57,890 You don't heat hot water, you heat cold water. 364 00:22:57,890 --> 00:23:02,210 Why isn't it called a cold water heater? 365 00:23:02,210 --> 00:23:06,900 Even better, I once worked for a year and a half in 366 00:23:06,900 --> 00:23:07,600 Switzerland. 367 00:23:07,600 --> 00:23:12,490 And in the kitchen of my home I had, in German, an 368 00:23:12,490 --> 00:23:13,740 elektrowarmwasse rsheissungsapparat. 369 00:23:16,070 --> 00:23:20,440 That's one word that's 11 syllables. 370 00:23:20,440 --> 00:23:23,380 The Germans have a knack with the language that we have 371 00:23:23,380 --> 00:23:26,090 never approached in English. 372 00:23:26,090 --> 00:23:29,666 But there's another one you've probably heard. 373 00:23:29,666 --> 00:23:31,950 You know what the prefix "pro" means. 374 00:23:31,950 --> 00:23:34,905 That means for, and "con" means against. 375 00:23:34,905 --> 00:23:38,760 And so progress is moving forward, and I leave it to you 376 00:23:38,760 --> 00:23:40,360 to decide what Congress means. 377 00:23:45,550 --> 00:23:46,080 OK. 378 00:23:46,080 --> 00:23:47,520 So I could go on and on. 379 00:23:47,520 --> 00:23:53,130 I probably shouldn't, unless you egg me on. 380 00:23:53,130 --> 00:23:53,730 No, I won't. 381 00:23:53,730 --> 00:23:55,440 But anyway, there are-- 382 00:23:55,440 --> 00:23:56,740 OK, I'll give another one. 383 00:23:56,740 --> 00:24:01,025 Why is what a doctor does called "practice"? 384 00:24:03,640 --> 00:24:07,590 This is not reassuring at all. 385 00:24:07,590 --> 00:24:11,540 Let me invite you over to my practice. 386 00:24:11,540 --> 00:24:17,170 And why is it when you take an airplane your flight ends with 387 00:24:17,170 --> 00:24:19,880 a terminal? 388 00:24:19,880 --> 00:24:22,600 That's doesn't sound very encouraging either. 389 00:24:22,600 --> 00:24:23,845 So anyway, language is silly. 390 00:24:23,845 --> 00:24:27,400 And I guess if you want to call c stiffnesses and s 391 00:24:27,400 --> 00:24:29,640 compliances, it's OK. 392 00:24:29,640 --> 00:24:32,360 The verbal description of these terms means something. 393 00:24:32,360 --> 00:24:39,180 Because if the stiffness is very high, that says that a 394 00:24:39,180 --> 00:24:42,850 little bit of strain requires that you haven't posed a very 395 00:24:42,850 --> 00:24:44,970 large stress. 396 00:24:44,970 --> 00:24:48,080 So stiff is when the material is resistant to stress. 397 00:24:48,080 --> 00:24:51,630 So you have to have a very, very large strain to get 398 00:24:51,630 --> 00:24:53,910 yourself a given level of stress. 399 00:24:53,910 --> 00:24:58,290 Conversely, if the material is very compliant, a very small 400 00:24:58,290 --> 00:25:00,750 stress for a compliant material should 401 00:25:00,750 --> 00:25:01,960 give you a big strain. 402 00:25:01,960 --> 00:25:04,890 And that's exactly what a large value of s will do. 403 00:25:04,890 --> 00:25:07,080 It will give you a large strain for a 404 00:25:07,080 --> 00:25:08,850 relatively small stress. 405 00:25:08,850 --> 00:25:11,730 So the words describe what's going on. 406 00:25:15,990 --> 00:25:16,550 OK. 407 00:25:16,550 --> 00:25:23,620 There is great utility in reducing these equations to a 408 00:25:23,620 --> 00:25:34,160 matrix form, which is going to cut us down from a 9 by 9, or 409 00:25:34,160 --> 00:25:41,590 a tensor with 81 elements, to a smaller number of subscripts 410 00:25:41,590 --> 00:25:48,160 and a smaller number of matrix elements, so if we let sigma 1 411 00:25:48,160 --> 00:25:55,460 be sigma 1,1; if we let sigma 2 be equal to sigma 2,2 and 412 00:25:55,460 --> 00:26:00,330 sigma 3 be equal to sigma 3. 413 00:26:00,330 --> 00:26:05,080 And then if we write the inequalities we've had before, 414 00:26:05,080 --> 00:26:12,420 and we let epsilon 1,1 to be replaced by epsilon 1, then we 415 00:26:12,420 --> 00:26:18,550 would have out in front here a term simply c1,1, a 1 for the 416 00:26:18,550 --> 00:26:21,130 sigma and a 1 for the epsilon. 417 00:26:21,130 --> 00:26:26,620 The next term would be c1,2 times epsilon 2, where 418 00:26:26,620 --> 00:26:30,930 obviously epsilon 2 has replaced epsilon 2,2, and 1 419 00:26:30,930 --> 00:26:33,370 stands for the two indices, 1,1. 420 00:26:33,370 --> 00:26:37,300 So this will go c1,3; epsilon 3. 421 00:26:37,300 --> 00:26:41,140 And then you hit these messy factors of two. 422 00:26:41,140 --> 00:26:42,750 You'll have a pair of terms. 423 00:26:42,750 --> 00:26:46,950 You'll have c1,4. 424 00:26:46,950 --> 00:26:58,300 And that will stand for a term c1,1,2,3. 425 00:26:58,300 --> 00:27:04,300 And then there'll be another term c1,1,3,2. 426 00:27:04,300 --> 00:27:08,760 And if you want to write this as a matrix, c1,4 has to be 427 00:27:08,760 --> 00:27:14,140 equal to the sum of these two things, times epsilon 4. 428 00:27:14,140 --> 00:27:16,570 That's the only way you can write it as a matrix. 429 00:27:16,570 --> 00:27:20,950 And if you think it was bad worrying about a factor of 2 430 00:27:20,950 --> 00:27:23,490 in front of half of the terms on the right-hand side of the 431 00:27:23,490 --> 00:27:29,520 equation, we're going to be dealing with factors of 2 for 432 00:27:29,520 --> 00:27:34,000 terms in the upper 3 by 3 array. 433 00:27:34,000 --> 00:27:38,730 We're going to be worrying about a factor of 4, somehow 434 00:27:38,730 --> 00:27:43,010 or other, down in the lower 3 by 3 array. 435 00:27:43,010 --> 00:27:47,360 So the way you handle the factor of 2 really is a 436 00:27:47,360 --> 00:27:51,260 nightmare for relations in elasticity. 437 00:27:51,260 --> 00:28:02,260 But continuing on with the first line, c1,5 times epsilon 438 00:28:02,260 --> 00:28:11,710 5 would be a combination of c1,1,3,1 plus 1,1,1,3. 439 00:28:11,710 --> 00:28:17,860 And then finally, when you get to c1,6 times epsilon 6, that 440 00:28:17,860 --> 00:28:28,520 would represent a combination of c1,1,2,1 plus c1,1,1,2. 441 00:28:28,520 --> 00:28:31,400 You have to make a decision where to swallow the 2. 442 00:28:31,400 --> 00:28:35,790 And probably the best thing I can do is to give you 443 00:28:35,790 --> 00:28:39,270 conventions for relabeling stress, strain, stiffness, and 444 00:28:39,270 --> 00:28:43,870 compliance, which I'll pass around to you. 445 00:28:43,870 --> 00:28:49,700 No big deal, except that this is a case where you can eat 446 00:28:49,700 --> 00:28:52,690 the 2's early on, but you can't even break even. 447 00:29:00,220 --> 00:29:05,150 So very, very briefly, what you do is you convert the 448 00:29:05,150 --> 00:29:09,510 tensor elements of stress to matrix elements in exactly the 449 00:29:09,510 --> 00:29:10,930 same way we did with piezoelectricity. 450 00:29:13,630 --> 00:29:21,230 For strains, we do exactly the same thing that we did with 451 00:29:21,230 --> 00:29:23,950 the converse piezoelectric effect. 452 00:29:23,950 --> 00:29:27,810 The factor of 2 that you swallowed in definition of 453 00:29:27,810 --> 00:29:32,350 stress pops up to haunt you when you 454 00:29:32,350 --> 00:29:33,560 deal with the strains. 455 00:29:33,560 --> 00:29:39,160 And you cannot write a nice matrix relation unless you 456 00:29:39,160 --> 00:29:43,500 divide the off-diagonal elements of strain by 1/2. 457 00:29:43,500 --> 00:29:45,770 And that was exactly the same thing we encountered with the 458 00:29:45,770 --> 00:29:50,110 converse piezoelectric effect, which related strain to an 459 00:29:50,110 --> 00:29:53,640 applied electric field. 460 00:29:53,640 --> 00:29:59,790 In defining the matrix stiffnesses, the c's, you 461 00:29:59,790 --> 00:30:04,460 forge straight ahead, and you let ci,j,k,l, which is 462 00:30:04,460 --> 00:30:05,780 identical to ci,j,l,k-- 463 00:30:09,830 --> 00:30:13,180 that's the term that relates the two equal shear stresses-- 464 00:30:13,180 --> 00:30:16,760 and you define that as a matrix term with two 465 00:30:16,760 --> 00:30:17,810 subscripts. 466 00:30:17,810 --> 00:30:21,580 No factors of 2 or 4 are involved. 467 00:30:21,580 --> 00:30:28,150 Then when you hit the s's, there's a nightmare. 468 00:30:28,150 --> 00:30:35,810 si,j,k,l is sm,n if m and n are 1, 2, or 3; that is to 469 00:30:35,810 --> 00:30:38,720 say, not 4, 5, or 6. 470 00:30:38,720 --> 00:30:44,640 si,j,k,l, which is the same as s,i,j,l,k, has to be defined 471 00:30:44,640 --> 00:30:52,560 as one half of sm,n, where m or n is 4, 5, or 6. 472 00:30:52,560 --> 00:30:56,250 And then finally, you have to throw in a factor of 4 when 473 00:30:56,250 --> 00:31:00,900 both m and n are 4, 5, or six or, in other words, m and n 474 00:31:00,900 --> 00:31:05,640 are not 1, 2, or 3. 475 00:31:05,640 --> 00:31:06,680 So it is a mess. 476 00:31:06,680 --> 00:31:09,750 And these are rules that you have to bear in mind if you're 477 00:31:09,750 --> 00:31:15,030 ever going to go from matrix form, which works absolutely 478 00:31:15,030 --> 00:31:18,620 lovely in a fixed coordinate system. 479 00:31:18,620 --> 00:31:20,775 But if you have a single crystal and you want to refer 480 00:31:20,775 --> 00:31:25,520 it to different axes, you have to be prepared to resurrect 481 00:31:25,520 --> 00:31:27,320 the full tensor form. 482 00:31:27,320 --> 00:31:33,490 And then and only then can you work symmetry transformations. 483 00:31:33,490 --> 00:31:37,380 I don't want to go through the simple algebra of expanding 484 00:31:37,380 --> 00:31:39,150 and contracting these terms. 485 00:31:39,150 --> 00:31:43,330 So the next two sheets show you how you condense from 486 00:31:43,330 --> 00:31:48,050 tensor to matrix notation, and then go back from matrix 487 00:31:48,050 --> 00:31:49,940 notation to tensor notation. 488 00:31:49,940 --> 00:31:53,700 This is for the compliances si,j,k. 489 00:31:53,700 --> 00:31:57,280 And the next page does the same thing, if you care to 490 00:31:57,280 --> 00:32:00,190 write stress in terms of strain. 491 00:32:00,190 --> 00:32:02,570 And there are a lot more factors of 492 00:32:02,570 --> 00:32:04,130 2 that appear there. 493 00:32:04,130 --> 00:32:07,490 But again, it shows you how you go from tensor notation, 494 00:32:07,490 --> 00:32:11,160 for subscripts on the c's, down to matrix notation with 2 495 00:32:11,160 --> 00:32:15,080 subscripts, and then go back up again to tensor notation. 496 00:32:15,080 --> 00:32:16,860 So it's a tedious business. 497 00:32:16,860 --> 00:32:21,010 And you've got to keep careful tracks of your factors of 2 498 00:32:21,010 --> 00:32:22,260 and your factors of 4. 499 00:32:26,530 --> 00:32:29,470 How about symmetry restrictions? 500 00:32:29,470 --> 00:32:30,550 Holy mackerel. 501 00:32:30,550 --> 00:32:33,570 Transformation of 81 different elements. 502 00:32:33,570 --> 00:32:36,480 Well, in order to do the tensor transformations, you 503 00:32:36,480 --> 00:32:40,940 have to go to the full-force subscript notation. 504 00:32:40,940 --> 00:32:44,450 Only then is a law of transformation defined. 505 00:32:44,450 --> 00:32:48,230 Fourth-rank properties are even tensors. 506 00:32:48,230 --> 00:32:50,370 So mercifully, there are not as many different 507 00:32:50,370 --> 00:32:52,370 possibilities. 508 00:32:52,370 --> 00:32:56,070 All point groups that differ by presence or absence of 509 00:32:56,070 --> 00:32:59,020 inversion have exactly the same form of the tensor. 510 00:32:59,020 --> 00:33:02,430 So symmetry 2, m, and 2/m look alike. 511 00:33:02,430 --> 00:33:07,250 Symmetry 2, 2mm, and 2/m, 2/m, 2/m look alike. 512 00:33:07,250 --> 00:33:11,550 So there's only one orthorhombic tensor, only one 513 00:33:11,550 --> 00:33:15,160 kind of monoclinic tensor, only one kind of triclinic 514 00:33:15,160 --> 00:33:16,500 tensor, and so on. 515 00:33:16,500 --> 00:33:20,540 And the only place that you have more than one form of the 516 00:33:20,540 --> 00:33:26,710 tensor for a particular set of point groups is looking at the 517 00:33:26,710 --> 00:33:31,780 point groups that are based on single-rotation axes, 518 00:33:31,780 --> 00:33:35,870 something like 4, 4 bar 4m, and those that are based on 519 00:33:35,870 --> 00:33:37,940 the axial arrangements 4, 2, 2. 520 00:33:37,940 --> 00:33:40,850 So they're different, just as they were for second-rank 521 00:33:40,850 --> 00:33:42,790 tensor properties. 522 00:33:42,790 --> 00:33:46,980 For cubic, a rather remarkable result. 523 00:33:46,980 --> 00:33:53,540 There are three independent compliances. 524 00:33:53,540 --> 00:33:58,920 But cubic crystals are not elastically isotropic. 525 00:33:58,920 --> 00:34:01,190 They are anisotropic. 526 00:34:01,190 --> 00:34:03,830 And it's just the nature of the tensor that requires that. 527 00:34:03,830 --> 00:34:06,060 So rather than letting you hang by your thumbs wondering 528 00:34:06,060 --> 00:34:09,900 what's on these pages, let me pass around the summary of 529 00:34:09,900 --> 00:34:11,270 symmetry constraints. 530 00:34:11,270 --> 00:34:15,060 If you understood what we did for third-rank tensors or even 531 00:34:15,060 --> 00:34:17,900 second-rank tensors, you how to do this. 532 00:34:17,900 --> 00:34:20,310 And fortunately, the terms almost over. 533 00:34:20,310 --> 00:34:22,920 So I can't make you do it, which is probably an enormous 534 00:34:22,920 --> 00:34:24,170 relief to you. 535 00:34:33,020 --> 00:34:37,350 Some additional bits of information. 536 00:34:37,350 --> 00:34:41,850 The transformations for hexagonal crystals are 537 00:34:41,850 --> 00:34:46,630 complicated because threefold and sixfold axes do not change 538 00:34:46,630 --> 00:34:50,980 the reference axes x1, x2, x3 into one another. 539 00:34:50,980 --> 00:34:59,330 And the equations that relate the individual tensor elements 540 00:34:59,330 --> 00:35:01,230 are more complex. 541 00:35:01,230 --> 00:35:08,400 For the threefold axis, for example, 2 s1,1 minus 2 s1,2 542 00:35:08,400 --> 00:35:11,420 is equal to s4,4. 543 00:35:11,420 --> 00:35:15,890 So they're not simple relations between them. 544 00:35:15,890 --> 00:35:18,160 And that's simply because you're not changing one 545 00:35:18,160 --> 00:35:19,920 reference access into another. 546 00:35:19,920 --> 00:35:23,120 You're changing x1, for example, into a linear 547 00:35:23,120 --> 00:35:26,980 combination of x1 and x2. 548 00:35:26,980 --> 00:35:31,860 Two things down here of interest. 549 00:35:31,860 --> 00:35:35,300 I said that cubic crystals are not isotropic. 550 00:35:35,300 --> 00:35:37,780 What would have to be the case if the 551 00:35:37,780 --> 00:35:42,630 material were to be isotropic? 552 00:35:42,630 --> 00:35:46,120 Well, it turns out that if you do this for the compliances, 553 00:35:46,120 --> 00:35:53,100 s, if 1/2 of s4,4 is equal to s1,1 minus s1,2, then the 554 00:35:53,100 --> 00:35:56,680 material is elastically isotropic. 555 00:35:56,680 --> 00:36:01,325 If you do this in terms stiffnesses, the c's, then 2 556 00:36:01,325 --> 00:36:06,780 c4,4 is equal to c1,1 minus c1,2 if the material is 557 00:36:06,780 --> 00:36:09,065 supposed to be isotropic or going to be isotropic. 558 00:36:12,990 --> 00:36:18,330 Then there is one other inequality 559 00:36:18,330 --> 00:36:20,620 that depends on structure. 560 00:36:20,620 --> 00:36:25,200 And this is something known as the Cauchy relation. 561 00:36:25,200 --> 00:36:30,610 And it depends on the interatomic forces. 562 00:36:30,610 --> 00:36:34,500 The first condition that has to be met is that the forces 563 00:36:34,500 --> 00:36:37,970 between the atoms should be central forces. 564 00:36:37,970 --> 00:36:39,150 What is a central force? 565 00:36:39,150 --> 00:36:43,890 Well, this is a case where the attractive force between the 566 00:36:43,890 --> 00:36:48,610 atoms is directly along the line joining their centers. 567 00:36:48,610 --> 00:36:51,170 Isn't that always the case? 568 00:36:51,170 --> 00:36:54,100 Don't crystals hold together because there's an attractive 569 00:36:54,100 --> 00:36:57,722 force between atoms? 570 00:36:57,722 --> 00:37:03,660 Well actually, for metallic crystals and ionic crystals, 571 00:37:03,660 --> 00:37:06,270 maybe that's not a bad assumption, particularly for 572 00:37:06,270 --> 00:37:07,460 ionic crystals. 573 00:37:07,460 --> 00:37:11,670 But if you had a covalent material, where the bonding 574 00:37:11,670 --> 00:37:16,730 was due to overlap of orbitals like this, then the thing that 575 00:37:16,730 --> 00:37:20,580 holds the crystal together is overlap 576 00:37:20,580 --> 00:37:22,240 between these orbitals. 577 00:37:22,240 --> 00:37:26,080 And the force holding the atoms together is a force that 578 00:37:26,080 --> 00:37:28,580 goes through this shared electron pair. 579 00:37:28,580 --> 00:37:30,530 And that's not a central force. 580 00:37:30,530 --> 00:37:34,780 And so the Cauchy relation generally fails rather badly 581 00:37:34,780 --> 00:37:38,230 for covalently bonded materials. 582 00:37:38,230 --> 00:37:42,080 Second assumption is that each atom is 583 00:37:42,080 --> 00:37:44,970 at a center of symmetry. 584 00:37:44,970 --> 00:37:47,580 And the reason for that is so the force in the plus-x 585 00:37:47,580 --> 00:37:49,120 direction is the same as the force 586 00:37:49,120 --> 00:37:51,260 in the minus-x direction. 587 00:37:51,260 --> 00:37:54,780 And that is not true for any material that has tetrahedral 588 00:37:54,780 --> 00:37:56,580 coordination. 589 00:37:56,580 --> 00:37:59,830 So it's not true for any of the forms of SiO2. 590 00:37:59,830 --> 00:38:04,480 It's not true for any of the compound semiconductors that 591 00:38:04,480 --> 00:38:07,520 are based on tetrahedral units. 592 00:38:07,520 --> 00:38:12,010 So there, the atom inside of these tetrahedra is decidedly 593 00:38:12,010 --> 00:38:15,740 not at an inversion center. 594 00:38:15,740 --> 00:38:18,760 And finally, you assume that the crystal is under no state 595 00:38:18,760 --> 00:38:20,250 of initial stress. 596 00:38:20,250 --> 00:38:23,950 Because if it is under initial stress, you've squished it, 597 00:38:23,950 --> 00:38:25,700 you've stretched the bonds. 598 00:38:25,700 --> 00:38:28,330 And the forces-- 599 00:38:28,330 --> 00:38:30,720 not to a major degree, to be sure, but-- the forces will 600 00:38:30,720 --> 00:38:33,460 not truly be perfect central forces. 601 00:38:33,460 --> 00:38:38,100 So if all three of these assumptions are satisfied, 602 00:38:38,100 --> 00:38:40,120 then you have an additional equality 603 00:38:40,120 --> 00:38:44,980 c1,2 is equal to c4,4. 604 00:38:44,980 --> 00:38:48,650 What I'll do next time is I'll bring in some examples of data 605 00:38:48,650 --> 00:38:52,720 for stiffnesses and compliances and, in 606 00:38:52,720 --> 00:38:58,620 particular, show you how some of these elastic tensor 607 00:38:58,620 --> 00:39:00,430 elements change with temperature. 608 00:39:00,430 --> 00:39:04,040 And we can examine them to see how well the Cauchy equality 609 00:39:04,040 --> 00:39:06,050 or the isotropy condition is satisfied. 610 00:39:09,850 --> 00:39:10,070 All right. 611 00:39:10,070 --> 00:39:13,300 That's about all that we'll do about the definition of 612 00:39:13,300 --> 00:39:15,070 fourth-rank tensor properties. 613 00:39:15,070 --> 00:39:18,680 On next Tuesday to wrap the term up, we'll look at the 614 00:39:18,680 --> 00:39:21,720 variation of some elastic moduli, such as Young's 615 00:39:21,720 --> 00:39:25,400 Modulus or the shear modulus, for different symmetries as a 616 00:39:25,400 --> 00:39:28,260 function of direction in the crystal, OK?