1 00:00:09,340 --> 00:00:11,190 PROFESSOR: All right, now for 2 00:00:11,190 --> 00:00:13,600 something completely different. 3 00:00:13,600 --> 00:00:17,580 Before beginning though, I would like to raise a 4 00:00:17,580 --> 00:00:19,550 procedural question. 5 00:00:19,550 --> 00:00:24,580 It was my intent that quiz number two, which is like a 6 00:00:24,580 --> 00:00:28,790 little more than a week away, was going to be part symmetry 7 00:00:28,790 --> 00:00:31,410 and part tensors. 8 00:00:31,410 --> 00:00:34,870 I don't want to have your fortunes based by weight of 9 00:00:34,870 --> 00:00:40,700 2:1 on symmetry theory as opposed to properties. 10 00:00:40,700 --> 00:00:44,010 What I would suggest, and you can express your opinion 11 00:00:44,010 --> 00:00:49,960 either way, is that we postpone quiz number two by 12 00:00:49,960 --> 00:00:55,640 maybe as much as 10 days, so that it can be half symmetry 13 00:00:55,640 --> 00:00:59,740 and half the introductory discussion of tensors. 14 00:00:59,740 --> 00:01:04,300 And if that does not create a conflict with some of your 15 00:01:04,300 --> 00:01:08,320 other classes, I would propose that as a suggestion. 16 00:01:08,320 --> 00:01:11,030 If not, we can just have on the quiz what we're going to 17 00:01:11,030 --> 00:01:14,880 cover of tensors in the next couple days and 18 00:01:14,880 --> 00:01:18,150 have the rest symmetry. 19 00:01:18,150 --> 00:01:24,180 So what I am suggesting is the quiz was scheduled for a week 20 00:01:24,180 --> 00:01:32,090 from November 8, which is a Tuesday. 21 00:01:32,090 --> 00:01:35,920 What I would suggest is putting that off to the 18th, 22 00:01:35,920 --> 00:01:40,950 and that will give us four extra lectures-- 23 00:01:40,950 --> 00:01:43,720 six extra lectures-- on tensors and we can make the 24 00:01:43,720 --> 00:01:44,330 quiz 50-50. 25 00:01:44,330 --> 00:01:46,925 AUDIENCE: What does that do to quiz number three? 26 00:01:46,925 --> 00:01:49,240 PROFESSOR: Quiz number three is going to stay in place 27 00:01:49,240 --> 00:01:50,750 because it's right at the end of the term. 28 00:01:50,750 --> 00:01:51,970 AUDIENCE: What about the subject in three? 29 00:01:51,970 --> 00:01:52,620 PROFESSOR: It will be all tensors. 30 00:01:52,620 --> 00:01:55,550 It will be all tensors and properties. 31 00:01:55,550 --> 00:01:59,830 And the third quiz is set up for longer than you think. 32 00:01:59,830 --> 00:02:03,470 It's December 8, so it'll be about three weeks. 33 00:02:03,470 --> 00:02:04,275 AUDIENCE: So everything from here on? 34 00:02:04,275 --> 00:02:06,650 PROFESSOR: Yeah. 35 00:02:06,650 --> 00:02:07,320 Does that make sense? 36 00:02:07,320 --> 00:02:12,770 Or is somebody going to have a big, traumatic responsibility 37 00:02:12,770 --> 00:02:14,020 at that point? 38 00:02:16,634 --> 00:02:19,793 AUDIENCE: I don't know, but I think it would be probably be 39 00:02:19,793 --> 00:02:23,080 easiest for us not to have it three days 40 00:02:23,080 --> 00:02:24,980 before close of exams. 41 00:02:24,980 --> 00:02:28,800 PROFESSOR: OK, OK, that's what I wanted to hear. 42 00:02:28,800 --> 00:02:31,630 How about if we went just a little bit further then and 43 00:02:31,630 --> 00:02:33,025 did it not on the-- 44 00:02:37,470 --> 00:02:38,570 when was the suggestion? 45 00:02:38,570 --> 00:02:39,650 The 18th. 46 00:02:39,650 --> 00:02:46,836 Let's say we did it on Tuesday the 15th? 47 00:02:46,836 --> 00:02:48,260 Does that conflict with anything? 48 00:02:48,260 --> 00:02:49,088 AUDIENCE: That's a little bit. 49 00:02:49,088 --> 00:02:51,837 AUDIENCE: We were doing it on the 18th. 50 00:02:51,837 --> 00:02:52,751 AUDIENCE: Is that Friday? 51 00:02:52,751 --> 00:02:54,122 [INTERPOSING VOICES] 52 00:02:54,122 --> 00:02:55,736 PROFESSOR: I'm sorry, 18th. 53 00:02:55,736 --> 00:02:57,742 I said it would be 17th. 54 00:02:57,742 --> 00:02:58,726 Excuse me. 55 00:02:58,726 --> 00:03:00,202 AUDIENCE: That definitely would be a little better. 56 00:03:03,646 --> 00:03:07,670 PROFESSOR: OK, let me allow you to think about that, and 57 00:03:07,670 --> 00:03:12,600 that would be pushing it back one week and maybe it'd be-- 58 00:03:12,600 --> 00:03:14,650 we'll see how much material on tensors we cover. 59 00:03:14,650 --> 00:03:18,700 So let's table it for now but let's say, tentatively, let's 60 00:03:18,700 --> 00:03:21,200 consider seriously moving it to the 15th. 61 00:03:21,200 --> 00:03:22,450 OK? 62 00:03:25,190 --> 00:03:29,140 But you guys have final say if there's a 63 00:03:29,140 --> 00:03:31,850 preference that you have. 64 00:03:31,850 --> 00:03:39,220 OK let's talk about properties in first a general and rather 65 00:03:39,220 --> 00:03:41,610 philosophic way. 66 00:03:41,610 --> 00:03:48,580 When we discuss properties we very commonly lump together a 67 00:03:48,580 --> 00:03:53,730 set of behaviors which have some common phenomenology. 68 00:03:53,730 --> 00:03:56,810 So for example, we will talk about mechanical properties. 69 00:04:04,130 --> 00:04:07,820 And that's a basket in which we place many different sorts 70 00:04:07,820 --> 00:04:09,920 of behavior. 71 00:04:09,920 --> 00:04:15,630 We place things like fracture toughness, yield strength, 72 00:04:15,630 --> 00:04:20,829 elastic constants, a whole variety of things involving 73 00:04:20,829 --> 00:04:23,970 strength deformation and so on. 74 00:04:23,970 --> 00:04:26,780 Or another thing we very often lump together in a basket is 75 00:04:26,780 --> 00:04:28,833 something called electrical properties. 76 00:04:35,320 --> 00:04:38,640 And we talk about electrical conductivity, ionic 77 00:04:38,640 --> 00:04:43,500 conductivity, electronic conductivity. 78 00:04:43,500 --> 00:04:47,120 We talk about dielectric constants, we talk about 79 00:04:47,120 --> 00:04:50,750 permittivity and permeability, magnetic properties-- 80 00:04:50,750 --> 00:04:53,700 a whole bunch of other classes of behavior, which have some 81 00:04:53,700 --> 00:04:56,853 sort of root or description in terms of electromagnetism. 82 00:05:03,850 --> 00:05:09,030 There are though a lot of strange aspects to properties. 83 00:05:09,030 --> 00:05:14,030 There are some properties that are determined for material 84 00:05:14,030 --> 00:05:16,830 which no longer exists when you're through 85 00:05:16,830 --> 00:05:19,200 determining the property. 86 00:05:19,200 --> 00:05:20,780 You have a sample you want to know what the 87 00:05:20,780 --> 00:05:21,630 yield strength is. 88 00:05:21,630 --> 00:05:23,850 So you put the gradually increasing load on it until 89 00:05:23,850 --> 00:05:25,470 finally, POP, it breaks. 90 00:05:25,470 --> 00:05:27,600 And now you know what the yield strength is for a 91 00:05:27,600 --> 00:05:29,140 material that no longer exists. 92 00:05:33,270 --> 00:05:36,980 Another example is an old one. 93 00:05:36,980 --> 00:05:40,070 Back before the days of X-ray diffraction when people would 94 00:05:40,070 --> 00:05:45,220 try to characterize material, anything that is very 95 00:05:45,220 --> 00:05:49,210 intensely colored usually looks black, and that's not a 96 00:05:49,210 --> 00:05:50,880 very definitive property. 97 00:05:50,880 --> 00:05:55,410 The color of something is a very prominent characteristic. 98 00:05:55,410 --> 00:05:58,750 So for mineralogists, in particular, anybody going out 99 00:05:58,750 --> 00:06:03,460 into the hills and wanting to be prepared to determine what 100 00:06:03,460 --> 00:06:06,900 a particular rock that they tripped over was had something 101 00:06:06,900 --> 00:06:10,210 on a rope around their neck called a streak plate. 102 00:06:10,210 --> 00:06:12,190 And what the streak plate was was a little 103 00:06:12,190 --> 00:06:14,270 rectangle of porcelain. 104 00:06:14,270 --> 00:06:17,630 And they would take this black mineral and rub it on this 105 00:06:17,630 --> 00:06:20,610 piece of porcelain and it would leave a mark-- 106 00:06:20,610 --> 00:06:21,760 that was called the streak. 107 00:06:21,760 --> 00:06:24,540 And what this was was actually little bunch of fragments that 108 00:06:24,540 --> 00:06:27,150 rubbed off on to the porcelain because the porcelain was 109 00:06:27,150 --> 00:06:29,230 white and the fragments were very small. 110 00:06:29,230 --> 00:06:32,680 A black piece of rock could give you a streak that was 111 00:06:32,680 --> 00:06:35,620 brown or green or deep orange. 112 00:06:35,620 --> 00:06:38,520 And so you really were determining the color of the 113 00:06:38,520 --> 00:06:42,450 material when it was in a form finely divided enough that 114 00:06:42,450 --> 00:06:43,350 light could pass through it. 115 00:06:43,350 --> 00:06:47,240 So the streak test was a very important diagnostic for 116 00:06:47,240 --> 00:06:48,490 determining minerals. 117 00:06:54,780 --> 00:06:58,410 There are some properties that we refer to 118 00:06:58,410 --> 00:07:01,770 as structure sensitive. 119 00:07:01,770 --> 00:07:03,800 In the sense that their value-- 120 00:07:03,800 --> 00:07:05,390 conductivity is a good example-- 121 00:07:05,390 --> 00:07:08,320 the value of ionic conductivity depends on the 122 00:07:08,320 --> 00:07:10,280 impurities and point defects that are 123 00:07:10,280 --> 00:07:11,730 present in the material. 124 00:07:11,730 --> 00:07:14,550 So to say that the ionic conductivity of a certain 125 00:07:14,550 --> 00:07:18,160 material is such and such, you have to specify the purity and 126 00:07:18,160 --> 00:07:20,660 perfection of the material or the property's meaningless. 127 00:07:24,150 --> 00:07:32,110 There are some properties that are not even single-valued. 128 00:07:32,110 --> 00:07:37,400 And the best example there are dielectric or magnetic 129 00:07:37,400 --> 00:07:39,010 properties. 130 00:07:39,010 --> 00:07:44,210 If you plot the magnetization of a material, the magnetic 131 00:07:44,210 --> 00:07:48,380 moment per unit volume as a function of the applied 132 00:07:48,380 --> 00:07:53,760 magnetic field B, the material is initially non-magnetized. 133 00:07:53,760 --> 00:07:56,930 When you apply B, you get a magnetic moment per unit 134 00:07:56,930 --> 00:07:59,850 volume that eventually saturates. 135 00:07:59,850 --> 00:08:02,650 And then if you remove the magnetic field, the material 136 00:08:02,650 --> 00:08:04,290 keeps some of its magnetization. 137 00:08:04,290 --> 00:08:08,390 It doesn't go to zero when you reduce the field to zero. 138 00:08:08,390 --> 00:08:12,060 You have to reverse the field to get the property to go to 139 00:08:12,060 --> 00:08:15,340 zero and then magnetize it in another direction. 140 00:08:15,340 --> 00:08:21,130 And if you continue to cycle the magnetic field, you get a 141 00:08:21,130 --> 00:08:22,050 behavior like this. 142 00:08:22,050 --> 00:08:28,000 This is hysteresis, and this type of behavior is called 143 00:08:28,000 --> 00:08:29,840 generally hysteretic. 144 00:08:29,840 --> 00:08:33,760 Same thing would be a relation between the displacement 145 00:08:33,760 --> 00:08:35,980 vector and an applied electric field. 146 00:08:35,980 --> 00:08:39,580 But clearly you can have any value of the magnetization in 147 00:08:39,580 --> 00:08:40,780 this range. 148 00:08:40,780 --> 00:08:44,890 And if you relate the magnetization as a function of 149 00:08:44,890 --> 00:08:48,950 the applied field, you can get any value of the 150 00:08:48,950 --> 00:08:51,630 proportionality constant you like between a negative 151 00:08:51,630 --> 00:08:55,470 maximum and a positive maximum including zero. 152 00:08:55,470 --> 00:08:58,370 So there's a property that's not even single-valued. 153 00:08:58,370 --> 00:09:02,840 Depends on the past history of the sample. 154 00:09:02,840 --> 00:09:08,190 And then there are even more peculiar properties. 155 00:09:08,190 --> 00:09:12,180 There are properties that are-- 156 00:09:12,180 --> 00:09:19,330 we call them composite properties and very often 157 00:09:19,330 --> 00:09:20,580 qualitative. 158 00:09:26,320 --> 00:09:27,880 What do they mean by a composite property. 159 00:09:27,880 --> 00:09:30,930 Let me give you one example-- 160 00:09:30,930 --> 00:09:33,446 The property fuzzy. 161 00:09:33,446 --> 00:09:36,680 If I say fuzzy, you know exactly what I mean. 162 00:09:36,680 --> 00:09:40,880 It means something that has a diffuse reflectivity, 163 00:09:40,880 --> 00:09:43,970 something that has a surface texture that's yielding. 164 00:09:43,970 --> 00:09:47,710 It's not like rubbing a wire brush. 165 00:09:47,710 --> 00:09:51,640 It's soft and giving, a whole collection of different 166 00:09:51,640 --> 00:09:52,200 properties. 167 00:09:52,200 --> 00:09:56,590 But yet when I say fuzzy, you know exactly what I mean. 168 00:09:56,590 --> 00:10:01,280 Let me not be so facetious as talking 169 00:10:01,280 --> 00:10:02,530 about a fuzzy property. 170 00:10:04,870 --> 00:10:08,770 People such as ceramists or powder metallurgists very 171 00:10:08,770 --> 00:10:13,660 often will take a powder of a material and consolidate it by 172 00:10:13,660 --> 00:10:19,080 compacting it and heating it, very often subject to a 173 00:10:19,080 --> 00:10:21,840 compacting stress. 174 00:10:21,840 --> 00:10:25,990 And when you do that little necks grow between the 175 00:10:25,990 --> 00:10:28,870 particles and they hold together and 176 00:10:28,870 --> 00:10:31,120 the material is centric. 177 00:10:31,120 --> 00:10:37,090 So you refer to a property of a powder as being centerable 178 00:10:37,090 --> 00:10:41,090 or the centerability of a powder. 179 00:10:41,090 --> 00:10:43,190 You know exactly what somebody means by that. 180 00:10:43,190 --> 00:10:44,995 But what does it depend on? 181 00:10:44,995 --> 00:10:49,380 It depends on the surface energy, it depends on vapor 182 00:10:49,380 --> 00:10:52,620 pressure, depends on bulk diffusion coefficients. 183 00:10:52,620 --> 00:10:55,850 It depends on surface diffusion coefficients. 184 00:10:55,850 --> 00:10:58,230 And all of these things have to be just right to make the 185 00:10:58,230 --> 00:11:02,890 powder something that easily densifies upon heating in the 186 00:11:02,890 --> 00:11:06,170 application or not of pressure. 187 00:11:06,170 --> 00:11:08,880 So you know exactly what I mean by centerability, but it 188 00:11:08,880 --> 00:11:11,810 is a very complex property. 189 00:11:11,810 --> 00:11:17,230 And was one that really was not understood until probably 190 00:11:17,230 --> 00:11:20,390 the late 1950s. 191 00:11:20,390 --> 00:11:23,835 Until then, it was an art that was entirely empirical. 192 00:11:23,835 --> 00:11:27,790 And it was somebody here at MIT, a fellow named Robert 193 00:11:27,790 --> 00:11:31,080 Coble, who developed a theory of centering that was the 194 00:11:31,080 --> 00:11:35,480 first really workable theory that described densification 195 00:11:35,480 --> 00:11:38,040 by heating and compaction. 196 00:11:38,040 --> 00:11:42,060 OK, composite property that depends on many different 197 00:11:42,060 --> 00:11:44,535 individual properties of the material. 198 00:11:49,160 --> 00:11:55,520 OK, what we are going to examine here are equilibrium 199 00:11:55,520 --> 00:11:59,350 properties that can be rigorously defined and 200 00:11:59,350 --> 00:12:00,630 measurable. 201 00:12:00,630 --> 00:12:03,480 So we're going to leave out of the picture things like fuzzy 202 00:12:03,480 --> 00:12:06,260 and centerability. 203 00:12:06,260 --> 00:12:09,520 And let me give a very nice, obscure definition of what I 204 00:12:09,520 --> 00:12:11,210 mean by a property. 205 00:12:11,210 --> 00:12:15,300 And what I mean by a property, in terms of a formal 206 00:12:15,300 --> 00:12:19,480 statement, is the response of a material to a specific 207 00:12:19,480 --> 00:12:23,390 change in a given set of conditions. 208 00:12:23,390 --> 00:12:26,380 So the response of the material to a specific change 209 00:12:26,380 --> 00:12:33,380 in a set of conditions that relates independent properties 210 00:12:33,380 --> 00:12:39,390 and dependent properties for a particular process. 211 00:12:39,390 --> 00:12:41,660 So I'll state that again because I think 212 00:12:41,660 --> 00:12:43,140 it's terribly elegant. 213 00:12:43,140 --> 00:12:46,830 So a property is the response of the material to a specific 214 00:12:46,830 --> 00:12:50,310 change in a given set of conditions that relates 215 00:12:50,310 --> 00:12:53,180 independent and dependent quantities in 216 00:12:53,180 --> 00:12:54,430 a particular process. 217 00:12:56,950 --> 00:13:02,260 Now there are a lot of properties that have their 218 00:13:02,260 --> 00:13:06,920 roots solidly embedded in thermodynamics. 219 00:13:06,920 --> 00:13:08,980 So let me give you a few of those. 220 00:13:13,050 --> 00:13:18,860 What we'll talk about when we specify a property is 221 00:13:18,860 --> 00:13:23,270 something that we will refer to as a displacement. 222 00:13:23,270 --> 00:13:37,390 We'll talk about a generalized displacement in response to a 223 00:13:37,390 --> 00:13:38,915 generalized force. 224 00:13:54,210 --> 00:13:57,590 So some of the thermodynamic quantities that are related in 225 00:13:57,590 --> 00:13:58,645 this fashion-- 226 00:13:58,645 --> 00:14:04,210 if we list some forces and some displacements, the thing 227 00:14:04,210 --> 00:14:07,880 that happens as a result of that stimulus that's applied 228 00:14:07,880 --> 00:14:09,130 to the material. 229 00:14:11,340 --> 00:14:15,030 Temperature can be regarded as a force that results in all 230 00:14:15,030 --> 00:14:18,340 sorts of processes as a result-- 231 00:14:18,340 --> 00:14:22,360 thermal energy flow, thermal expansion, 232 00:14:22,360 --> 00:14:23,300 all sorts of things. 233 00:14:23,300 --> 00:14:27,210 But one of the things that will happen in response to a 234 00:14:27,210 --> 00:14:28,530 temperature change 235 00:14:28,530 --> 00:14:30,890 thermodynamically is an entropy. 236 00:14:34,580 --> 00:14:40,540 So you can view entropy as a generalized displacement 237 00:14:40,540 --> 00:14:43,390 resulting from the application of temperature as a 238 00:14:43,390 --> 00:14:44,640 generalized force. 239 00:14:47,470 --> 00:14:58,650 Another example is electric field and the thing that 240 00:14:58,650 --> 00:15:01,160 happens there, and electromagnetism, is the 241 00:15:01,160 --> 00:15:03,095 quantity, D, which is displacement. 242 00:15:08,690 --> 00:15:18,230 Still another example, stress, and the result of applying a 243 00:15:18,230 --> 00:15:20,480 stress, among other things, is a strain. 244 00:15:25,330 --> 00:15:29,100 What is special about these forces in this displacement is 245 00:15:29,100 --> 00:15:37,240 that their product is, in each of these cases, energy, the 246 00:15:37,240 --> 00:15:41,920 change in internal energy of the particular body. 247 00:15:41,920 --> 00:15:46,540 And when that is the case, these are said to be 248 00:15:46,540 --> 00:15:47,790 conjugate-- 249 00:15:50,136 --> 00:15:53,430 a conjugate force and displacement. 250 00:15:53,430 --> 00:15:55,570 And there are other examples that one can come up .with. 251 00:16:03,530 --> 00:16:07,685 Now to talk about a specific set of properties. 252 00:16:11,460 --> 00:16:15,680 Very often, and this crept into the earlier discussion, 253 00:16:15,680 --> 00:16:20,800 very often the thing that we do to a material, as a 254 00:16:20,800 --> 00:16:22,540 generalized force, is a vector. 255 00:16:25,800 --> 00:16:32,290 So very often the thing that we do to the material, apply 256 00:16:32,290 --> 00:16:35,140 an electric field, apply a magnetic field, apply a 257 00:16:35,140 --> 00:16:39,900 temperature gradient, apply a tensile stress, it has the 258 00:16:39,900 --> 00:16:42,813 characteristics of a vector, magnitude and direction. 259 00:16:46,440 --> 00:16:50,090 And in many cases, the thing that happens as a generalized 260 00:16:50,090 --> 00:16:52,710 displacement, we may call this in general, 261 00:16:52,710 --> 00:16:55,200 q, is also a vector. 262 00:17:00,620 --> 00:17:03,710 An example is if we apply an electric field as a 263 00:17:03,710 --> 00:17:06,670 generalized force, one thing that might happen is a current 264 00:17:06,670 --> 00:17:09,170 flow, which is also a vector. 265 00:17:09,170 --> 00:17:14,910 And we are accustomed to writing, q, the generalized 266 00:17:14,910 --> 00:17:19,829 displacement, as a proportionality constant, 267 00:17:19,829 --> 00:17:24,180 sigma, which is in the case of electric field and current 268 00:17:24,180 --> 00:17:26,789 flow, the electrical conductivity. 269 00:17:26,789 --> 00:17:29,980 Let me call it, in general terms, proportionality 270 00:17:29,980 --> 00:17:34,330 constant, a, times the applied vector, p. 271 00:17:39,040 --> 00:17:41,890 Is this something we would like to stick with as a 272 00:17:41,890 --> 00:17:44,780 general relation? 273 00:17:44,780 --> 00:17:47,970 Well let me submit that writing an expression of this 274 00:17:47,970 --> 00:17:52,820 form makes an inherent assumption. 275 00:17:52,820 --> 00:17:59,630 Namely that this will be true for small, and we'll have to 276 00:17:59,630 --> 00:18:03,690 define for each property what we mean by a 277 00:18:03,690 --> 00:18:06,000 small, applied vector. 278 00:18:06,000 --> 00:18:07,340 Let me give you an example. 279 00:18:07,340 --> 00:18:08,980 If p were-- 280 00:18:08,980 --> 00:18:13,500 the applied vector was electric field, and the 281 00:18:13,500 --> 00:18:17,110 resulting vector was current flow, and the relation between 282 00:18:17,110 --> 00:18:20,140 those is the conductivity-- 283 00:18:20,140 --> 00:18:24,650 a relation of this sort says that if you double the field, 284 00:18:24,650 --> 00:18:26,380 you double the conductivity. 285 00:18:26,380 --> 00:18:30,760 You triple the field, you triple the conductivity. 286 00:18:30,760 --> 00:18:33,350 Obviously this can't go on indefinitely because all a 287 00:18:33,350 --> 00:18:36,110 sudden, POW, dielectric breakdown. 288 00:18:36,110 --> 00:18:38,540 The sample evaporates at the smoke. 289 00:18:38,540 --> 00:18:41,570 And again, you have the property of a material which 290 00:18:41,570 --> 00:18:42,680 no longer exists. 291 00:18:42,680 --> 00:18:48,310 So a lot of properties, we inherently assume that the 292 00:18:48,310 --> 00:18:51,940 applied vector is small in order to write something in an 293 00:18:51,940 --> 00:18:53,190 expression of this form. 294 00:18:57,330 --> 00:19:01,430 So for conductivity, dielectric breakdown is going 295 00:19:01,430 --> 00:19:05,470 to destroy the nice linear relation between current flow 296 00:19:05,470 --> 00:19:06,720 and electric field. 297 00:19:09,290 --> 00:19:10,630 That's not always the case. 298 00:19:10,630 --> 00:19:13,590 Let me give you an example of another property. 299 00:19:13,590 --> 00:19:19,340 And this is a property, magnetic susceptibility, which 300 00:19:19,340 --> 00:19:28,940 relates the magnetization, which is magnetic moment per 301 00:19:28,940 --> 00:19:42,820 unit volume, and relates that to an applied electric field-- 302 00:19:42,820 --> 00:19:51,530 an applied magnetic field, H. And the proportionality 303 00:19:51,530 --> 00:19:56,440 constant, represented by a Greek chi, is the magnetic 304 00:19:56,440 --> 00:19:57,690 susceptibility. 305 00:20:08,050 --> 00:20:09,790 So let me give you a specific example. 306 00:20:09,790 --> 00:20:15,560 Suppose we have a chunk of glass and the glass contains a 307 00:20:15,560 --> 00:20:17,850 dilute concentration of iron. 308 00:20:21,140 --> 00:20:24,120 And iron carries a permanent magnetic moment. 309 00:20:24,120 --> 00:20:27,460 And the reason I want to make it a dilute concentration of 310 00:20:27,460 --> 00:20:30,620 iron is I don't want these magnetic moments close enough 311 00:20:30,620 --> 00:20:32,760 that they can interact with one another. 312 00:20:32,760 --> 00:20:36,090 I want this to, therefore, be a dilute system. 313 00:20:36,090 --> 00:20:37,800 So we have different iron atoms in this. 314 00:20:37,800 --> 00:20:40,500 Each iron atom has a magnetic moment. 315 00:20:40,500 --> 00:20:44,350 And then we put this in a magnetic field, H. And the 316 00:20:44,350 --> 00:20:48,240 magnetic field acts on each of these little magnetic moments 317 00:20:48,240 --> 00:20:50,780 just as though it were a compass needle. 318 00:20:50,780 --> 00:20:55,930 And so it will try to take each of these moments and drag 319 00:20:55,930 --> 00:21:00,260 it into coincidence with the magnetic field. 320 00:21:00,260 --> 00:21:05,180 But at a finite temperature, temperatures making these 321 00:21:05,180 --> 00:21:08,000 magnetic moments jiggle around, so at a finite 322 00:21:08,000 --> 00:21:12,190 temperature, the magnetic moments will just not simply 323 00:21:12,190 --> 00:21:15,400 zing into coincidence with the magnetic field. 324 00:21:15,400 --> 00:21:17,240 You'll have to increase the magnetic 325 00:21:17,240 --> 00:21:18,860 field to make it larger. 326 00:21:18,860 --> 00:21:24,180 When that happens, more and more of the magnetic moments 327 00:21:24,180 --> 00:21:26,170 will come into alignment. 328 00:21:26,170 --> 00:21:31,850 And if you put on a really strong magnetic field, then 329 00:21:31,850 --> 00:21:36,600 every single magnetic moment will be dragged into exact 330 00:21:36,600 --> 00:21:38,470 alignment with the magnetic field and 331 00:21:38,470 --> 00:21:40,730 the system has saturated. 332 00:21:40,730 --> 00:21:44,340 There's no way you could squeeze further magnetic 333 00:21:44,340 --> 00:21:46,580 moment per unit volume out of it. 334 00:21:46,580 --> 00:21:50,330 So again, you would expect, for this particular property, 335 00:21:50,330 --> 00:21:53,640 magnetic susceptibility of a material, if you plot it as a 336 00:21:53,640 --> 00:21:57,740 function of the applied magnetic field-- 337 00:21:57,740 --> 00:22:01,770 linear maybe be at low applied fields-- 338 00:22:01,770 --> 00:22:04,480 but eventually if you make the field strong enough, the 339 00:22:04,480 --> 00:22:06,460 system is going to saturate. 340 00:22:06,460 --> 00:22:09,210 And then again, no longer will the direction of the 341 00:22:09,210 --> 00:22:12,180 magnetization, and that magnetic moment, be parallel 342 00:22:12,180 --> 00:22:17,900 to H. But the property becomes nonlinear if the applied 343 00:22:17,900 --> 00:22:20,950 vector is strong enough. 344 00:22:20,950 --> 00:22:25,020 This is an example of a property where you would not 345 00:22:25,020 --> 00:22:31,590 go wrong at all by stating direct proportionality. 346 00:22:31,590 --> 00:22:37,670 For ordering to occur, temperature and applied field 347 00:22:37,670 --> 00:22:38,790 go hand in hand. 348 00:22:38,790 --> 00:22:41,610 Increased temperature tends to create more disorder. 349 00:22:41,610 --> 00:22:46,350 Increased magnetic field tends to align the moments. 350 00:22:46,350 --> 00:22:51,610 In the days when MIT had a national magnet laboratory, 351 00:22:51,610 --> 00:22:55,640 the magnet laboratory held the record for the strongest 352 00:22:55,640 --> 00:22:59,400 magnetic field ever produced artificially by man. 353 00:22:59,400 --> 00:23:02,720 And if you took that magnetic field and applied it to this 354 00:23:02,720 --> 00:23:09,300 system of dilute iron in a glass, you would have to lower 355 00:23:09,300 --> 00:23:14,320 the temperature of the sample to about 3 Kelvin before you 356 00:23:14,320 --> 00:23:17,020 would begin to see saturation. 357 00:23:17,020 --> 00:23:23,120 So even the strongest field that you could produce in a 358 00:23:23,120 --> 00:23:26,380 laboratory environment would not succeed in producing 359 00:23:26,380 --> 00:23:29,460 non-linearity until you cooled the sample down almost to 360 00:23:29,460 --> 00:23:30,630 absolute zero. 361 00:23:30,630 --> 00:23:33,660 So here would be a case where under any practical 362 00:23:33,660 --> 00:23:36,570 consideration whatsoever, assumption of strict 363 00:23:36,570 --> 00:23:39,580 proportionality would be right on the money. 364 00:23:39,580 --> 00:23:40,830 You'd be absolutely correct. 365 00:23:47,660 --> 00:23:52,530 But there's another assumption built into this statement. 366 00:23:52,530 --> 00:23:57,050 P, the generalized force, is a vector in the cases we're 367 00:23:57,050 --> 00:24:01,990 discussing now, and q is also a vector. 368 00:24:01,990 --> 00:24:07,290 So when we write an expression of this form you're making 369 00:24:07,290 --> 00:24:08,990 another assumption. 370 00:24:08,990 --> 00:24:14,630 And that is that the vector displacement that results is 371 00:24:14,630 --> 00:24:18,170 always exactly parallel to the vector that you apply. 372 00:24:21,710 --> 00:24:23,230 Do I make a big deal out of this? 373 00:24:23,230 --> 00:24:24,740 Isn't that always going to be the case? 374 00:24:24,740 --> 00:24:28,050 I mean whoever heard of taking a piece of metal and putting 375 00:24:28,050 --> 00:24:30,680 an electric field on it in this direction and having the 376 00:24:30,680 --> 00:24:32,330 current run off in this direction. 377 00:24:32,330 --> 00:24:33,580 It's absurd. 378 00:24:35,650 --> 00:24:38,970 Or maybe it isn't so absurd. 379 00:24:38,970 --> 00:24:44,360 So let's think of some of the atomistics of this process. 380 00:24:44,360 --> 00:24:49,080 Now, since I know I'm among friends, I will not hesitate 381 00:24:49,080 --> 00:24:51,830 to display my ignorance, total 382 00:24:51,830 --> 00:24:53,960 ignorance, of polymer chemistry. 383 00:24:53,960 --> 00:24:57,380 So suppose we had a piece of polymer. 384 00:24:57,380 --> 00:24:59,410 That's what a polymer molecule looks like. 385 00:24:59,410 --> 00:25:05,340 It's a more or less linear molecule, and so these might, 386 00:25:05,340 --> 00:25:09,050 in a very highly ordered polymer, be chains that lined 387 00:25:09,050 --> 00:25:11,580 up like this. 388 00:25:11,580 --> 00:25:16,980 And suppose we now put an electric field on this polymer 389 00:25:16,980 --> 00:25:20,320 and asked how the current will flow. 390 00:25:20,320 --> 00:25:23,190 Again, it would not be absurd to say that an electron 391 00:25:23,190 --> 00:25:27,040 sitting on this polymer chain in response to this field 392 00:25:27,040 --> 00:25:31,580 would be constrained only to flow in the direction of the 393 00:25:31,580 --> 00:25:35,180 chain and would find it rather difficult to hop from one 394 00:25:35,180 --> 00:25:37,260 chain to another. 395 00:25:37,260 --> 00:25:41,840 So maybe, just maybe, we could apply an electric field to a 396 00:25:41,840 --> 00:25:44,890 polymer and it would have a flow in this direction, that 397 00:25:44,890 --> 00:25:46,980 would be pointing in this direction, and the current 398 00:25:46,980 --> 00:25:52,100 flow, J, would be in that direction. 399 00:25:52,100 --> 00:26:00,440 So should we maybe rethink this idea of the direction of 400 00:26:00,440 --> 00:26:04,740 the generalized displacement being not parallel to the 401 00:26:04,740 --> 00:26:06,520 direction of the generalized force. 402 00:26:09,150 --> 00:26:11,740 OK, these are hypothetical examples. 403 00:26:11,740 --> 00:26:16,950 Let me now give you an example of a real property for a real 404 00:26:16,950 --> 00:26:22,170 material, for which the thing that happens, the vector that 405 00:26:22,170 --> 00:26:25,340 happens, is decidedly not parallel to the direction of 406 00:26:25,340 --> 00:26:27,120 the applied vector. 407 00:26:27,120 --> 00:26:32,250 OK, thermal conductivity is something that relates a heat 408 00:26:32,250 --> 00:26:40,160 flux, usually represented by the symbol K, and relates that 409 00:26:40,160 --> 00:26:45,345 to a temperature gradient, dt dx, which is a vector. 410 00:26:51,410 --> 00:26:58,450 The thing that gives rise to thermal conductivity can be 411 00:26:58,450 --> 00:27:03,300 either propagation of radiation as in propagation of 412 00:27:03,300 --> 00:27:06,460 light traveling through a transparent material. 413 00:27:06,460 --> 00:27:12,525 But the other mechanism for thermal conductivity is modes 414 00:27:12,525 --> 00:27:14,340 of lattice vibration. 415 00:27:14,340 --> 00:27:18,030 When a material is hot, the atoms are jiggling around and 416 00:27:18,030 --> 00:27:22,610 you can make the displacement of an individual atom be 417 00:27:22,610 --> 00:27:27,160 represented by a sum of waves moving in all different 418 00:27:27,160 --> 00:27:29,790 directions with a variety of wavelengths in a variety of 419 00:27:29,790 --> 00:27:30,670 amplitudes. 420 00:27:30,670 --> 00:27:33,750 Take all those waves, add them together at a particular time, 421 00:27:33,750 --> 00:27:35,460 and you get the displacement of the atom. 422 00:27:38,140 --> 00:27:44,400 Propagation of heat by this mechanism, by modes of thermal 423 00:27:44,400 --> 00:27:49,460 vibration, can be very, very anisotropic and probably the 424 00:27:49,460 --> 00:27:52,450 best example of this is-- 425 00:27:52,450 --> 00:27:54,130 get a single crystal of graphite. 426 00:27:59,380 --> 00:28:03,950 And have the layers, the graphite sheets, which are 427 00:28:03,950 --> 00:28:12,850 hexagonal rings in which each carbon has three neighbors, 428 00:28:12,850 --> 00:28:19,410 and have the sheets be parallel to the surface of an 429 00:28:19,410 --> 00:28:22,080 extended two-dimensional slab. 430 00:28:22,080 --> 00:28:23,140 Now we can't do that. 431 00:28:23,140 --> 00:28:25,120 Single crystals of graphite don't occur 432 00:28:25,120 --> 00:28:26,110 in sizes like that. 433 00:28:26,110 --> 00:28:29,660 But what you can do is make a material called pyrolytic 434 00:28:29,660 --> 00:28:33,530 graphite that you make by having a reaction in the vapor 435 00:28:33,530 --> 00:28:37,360 phase and having the soot that's formed settle down onto 436 00:28:37,360 --> 00:28:38,760 a substrate. 437 00:28:38,760 --> 00:28:42,610 And what happens is you nucleate a graphite crystal in 438 00:28:42,610 --> 00:28:46,160 one orientation, and that grows preferentially with its 439 00:28:46,160 --> 00:28:49,600 layers parallel to the surface on which it's nucleated. 440 00:28:49,600 --> 00:28:51,940 And these nuclei occur at random. 441 00:28:51,940 --> 00:28:57,200 So you get a bunch of single crystals of graphite, which 442 00:28:57,200 --> 00:29:02,990 all have their layers, their three coordinated layers at 443 00:29:02,990 --> 00:29:05,680 parallel, but they are oriented at 444 00:29:05,680 --> 00:29:09,150 random about their c-axis. 445 00:29:09,150 --> 00:29:11,950 And that's a material that's very easy to prepare. 446 00:29:11,950 --> 00:29:14,290 And it's called pyrolytic graphite, and it has 447 00:29:14,290 --> 00:29:17,410 interesting applications and properties. 448 00:29:17,410 --> 00:29:20,900 OK, now I'm going to suggest an experiment to you and I'll 449 00:29:20,900 --> 00:29:24,090 caution you, please do not try this at home. 450 00:29:24,090 --> 00:29:28,290 Get yourself a piece of pyrolytic graphite. 451 00:29:28,290 --> 00:29:30,374 Put a Bunsen burner underneath it. 452 00:29:34,330 --> 00:29:38,280 Play the Bunsen burner on the bottom of the sheet of 453 00:29:38,280 --> 00:29:39,790 pyrolytic graphite. 454 00:29:39,790 --> 00:29:43,640 And to determine temperature, we'll take a ball of cotton 455 00:29:43,640 --> 00:29:46,410 and put it directly above the flame. 456 00:29:46,410 --> 00:29:48,910 And then take another ball of cotton and put it down at the 457 00:29:48,910 --> 00:29:51,550 end of the sheet. 458 00:29:51,550 --> 00:29:54,210 If you do that and bring up the Bunsen burner, this piece 459 00:29:54,210 --> 00:29:56,140 of cotton will sit there and sit there and sit 460 00:29:56,140 --> 00:29:57,180 there and sit there. 461 00:29:57,180 --> 00:30:01,790 And this piece of cotton will instantly burst into flame. 462 00:30:01,790 --> 00:30:07,140 So here's a case where dt dx, the thermal gradient, points 463 00:30:07,140 --> 00:30:09,580 in this direction. 464 00:30:09,580 --> 00:30:13,980 And the heat flow goes off in a direction at right angles to 465 00:30:13,980 --> 00:30:17,490 the temperature gradient, almost 466 00:30:17,490 --> 00:30:19,520 exactly at right angles. 467 00:30:19,520 --> 00:30:22,270 It turns out that the thermal conductivity in this 468 00:30:22,270 --> 00:30:26,600 direction, the value of K in the-- 469 00:30:31,130 --> 00:30:33,810 I want to put crystallographic directions on this. 470 00:30:33,810 --> 00:30:44,170 But K parallel to the layers is equal to 10 to the third 471 00:30:44,170 --> 00:30:46,970 times K, perpendicular to the layers. 472 00:30:50,100 --> 00:30:54,710 So there's an anisotropy of the property that amounts to a 473 00:30:54,710 --> 00:30:56,890 factor of 1,000. 474 00:30:56,890 --> 00:30:58,650 Very dramatic. 475 00:30:58,650 --> 00:31:07,140 So what I'm leading up is that in a general relation between 476 00:31:07,140 --> 00:31:13,030 a generalized displacement, p, and a generalized force-- 477 00:31:13,030 --> 00:31:14,920 I'm sorry, I'm using q-- p and q-- 478 00:31:18,090 --> 00:31:22,520 we just can't simply put some constant in front. 479 00:31:22,520 --> 00:31:25,710 Because that assumes inherently that the direction 480 00:31:25,710 --> 00:31:28,920 of what happens, the vector that happens, is exactly 481 00:31:28,920 --> 00:31:34,560 parallel to the direction of the applied vector and that 482 00:31:34,560 --> 00:31:36,860 generally is not true. 483 00:31:36,860 --> 00:31:39,990 The difference may be small but it has to be taken into 484 00:31:39,990 --> 00:31:41,240 consideration. 485 00:31:44,670 --> 00:31:49,680 All right, let me now suggest a way of patching this up. 486 00:31:53,940 --> 00:31:55,600 What I'm going to do is assume-- 487 00:31:55,600 --> 00:31:56,850 and it is an assumption-- 488 00:32:00,020 --> 00:32:10,110 we're going to assume that each component of the vector 489 00:32:10,110 --> 00:32:20,690 that happens is given by-- 490 00:32:25,410 --> 00:32:28,800 in fancy terms, the vector that results as the 491 00:32:28,800 --> 00:32:43,200 generalized displacement is given by a linear combination 492 00:32:43,200 --> 00:32:54,895 of every component of the vector that's applied. 493 00:33:03,110 --> 00:33:06,485 And that's what we've defined as the generalized force. 494 00:33:15,170 --> 00:33:17,120 So how would we express this analytically? 495 00:33:22,560 --> 00:33:29,240 I am going to use as my coordinate system, not x, y, 496 00:33:29,240 --> 00:33:33,630 and z as we usually do, I'm going to call 497 00:33:33,630 --> 00:33:37,380 it x1, x2, and x3. 498 00:33:37,380 --> 00:33:42,560 The reason is we'll see some unique properties of these 499 00:33:42,560 --> 00:33:45,120 indices that are very useful algebraically. 500 00:33:49,630 --> 00:33:58,490 So I'm going to now take my applied vector, p, and I'm 501 00:33:58,490 --> 00:34:07,840 going to assume that it has three components, p1, p2, and 502 00:34:07,840 --> 00:34:15,219 p3 along x1, x2, x3, respectively. 503 00:34:15,219 --> 00:34:19,139 My coordinate system, although I did not state it explicitly, 504 00:34:19,139 --> 00:34:22,860 is going to be a Cartesian coordinate system, not a 505 00:34:22,860 --> 00:34:26,929 crystallographic coordinate system. 506 00:34:26,929 --> 00:34:31,600 So what I'm proposing here is that we take each component of 507 00:34:31,600 --> 00:34:33,489 the resulting vector q-- 508 00:34:33,489 --> 00:34:35,860 let's say q1-- 509 00:34:35,860 --> 00:34:39,179 and we'll assume that that's given by a linear combination 510 00:34:39,179 --> 00:34:42,179 of all three components of the vector p. 511 00:34:42,179 --> 00:34:47,250 So it'll be some number times p1 plus some number times p2 512 00:34:47,250 --> 00:34:49,580 plus some number times p3. 513 00:34:52,580 --> 00:34:55,969 And those numbers in general will be different. 514 00:34:55,969 --> 00:35:00,920 So let's say I call the coefficient a, and now I'm 515 00:35:00,920 --> 00:35:05,280 going to define a convention that will stay with us. 516 00:35:05,280 --> 00:35:10,250 I'm going to define each of these coefficients, a, in 517 00:35:10,250 --> 00:35:14,430 terms of the index of the component of the generalized 518 00:35:14,430 --> 00:35:21,240 displacement which is being computed, and the coefficient 519 00:35:21,240 --> 00:35:26,280 modifies the component of the generalized force for that 520 00:35:26,280 --> 00:35:27,790 particular term. 521 00:35:27,790 --> 00:35:32,900 So I'm going to call this a11, where the 1 goes with this and 522 00:35:32,900 --> 00:35:34,430 the 1 goes with this. 523 00:35:34,430 --> 00:35:39,310 I'm going to call this a12, so that the 1 again says it's a 524 00:35:39,310 --> 00:35:41,050 contribution to q1. 525 00:35:41,050 --> 00:35:45,450 The 2 says that this term modifies p2. . 526 00:35:45,450 --> 00:35:47,690 And this similarly would be a13. 527 00:35:50,510 --> 00:35:57,690 For the term component of the generalized displacement, q2, 528 00:35:57,690 --> 00:36:06,890 I'll use the coefficients a21 times p1 plus a22 times p2, 529 00:36:06,890 --> 00:36:11,540 plus 23 times p3. 530 00:36:11,540 --> 00:36:19,370 And q3 similarly will be a31 times p1, plus a32 times p2 531 00:36:19,370 --> 00:36:24,200 plus a33 times p3. 532 00:36:24,200 --> 00:36:28,550 So I've got three simultaneous equations then, one for each 533 00:36:28,550 --> 00:36:32,370 component of the vector that results, the generalized 534 00:36:32,370 --> 00:36:33,620 displacement. 535 00:36:35,590 --> 00:36:39,410 So I can sum up this set of three equations by saying that 536 00:36:39,410 --> 00:36:45,280 the i-th component of q, where this is some particular 537 00:36:45,280 --> 00:36:54,410 component, q1, q2, or q3, is given by the sum over j from 1 538 00:36:54,410 --> 00:37:05,960 to 3 of aij times p sub j. 539 00:37:05,960 --> 00:37:14,410 So I've got ai1 times p1, ai2 times p2 plus ai3 times p3. 540 00:37:16,930 --> 00:37:21,830 So this is in a nice, compact little nugget the expression 541 00:37:21,830 --> 00:37:25,910 that we are assuming will apply for all sorts of 542 00:37:25,910 --> 00:37:29,040 physical properties in which this is a vector 543 00:37:29,040 --> 00:37:30,380 and this is a vector-- 544 00:37:30,380 --> 00:37:33,420 electrical conductivity, magnetic susceptibility, 545 00:37:33,420 --> 00:37:35,910 thermal conductivity, and so on. 546 00:37:39,080 --> 00:37:44,000 OK, this is a bad point to introduce something as hairy 547 00:37:44,000 --> 00:37:45,940 as what comes next. 548 00:37:45,940 --> 00:37:50,140 But I've got one minute left, and I think I can do it. 549 00:37:50,140 --> 00:37:53,410 I'm going to introduce something called the Einstein 550 00:37:53,410 --> 00:37:58,220 convention after old Albert Einstein himself. 551 00:37:58,220 --> 00:38:00,170 I'm sorry, but nobody said everything this 552 00:38:00,170 --> 00:38:02,590 term had to be easy. 553 00:38:02,590 --> 00:38:05,410 So let me introduce, if you're ready for it, the Einstein 554 00:38:05,410 --> 00:38:06,660 convention. 555 00:38:08,330 --> 00:38:12,910 Old Albert, I think, was just as lazy as anybody else. 556 00:38:12,910 --> 00:38:18,620 And he said it is going to be a bloody pain in the butt-- 557 00:38:18,620 --> 00:38:20,305 I don't know if he put it exactly this way. 558 00:38:20,305 --> 00:38:23,100 It's going to be a pain in the butt to write this summation 559 00:38:23,100 --> 00:38:26,840 every time we want to combine three terms in a linear 560 00:38:26,840 --> 00:38:28,500 combination. 561 00:38:28,500 --> 00:38:36,840 So the Einstein convention is let us throw out the sigma and 562 00:38:36,840 --> 00:38:44,070 write this expression just as qi times aij p sub j. 563 00:38:44,070 --> 00:38:49,000 And whenever we see a subscript repeated, summation 564 00:38:49,000 --> 00:38:51,755 over repeated subscript is implied. 565 00:39:08,610 --> 00:39:11,770 OK so that's the Einstein convention. 566 00:39:11,770 --> 00:39:13,980 It will save us the trouble of writing in a lot 567 00:39:13,980 --> 00:39:15,680 of summation signs. 568 00:39:15,680 --> 00:39:21,710 But I would caution you that this convention, compact and 569 00:39:21,710 --> 00:39:28,200 convenient as it is, can define some polynomials that 570 00:39:28,200 --> 00:39:30,250 are nightmares. 571 00:39:30,250 --> 00:39:33,200 So let me give you one example of this, and this is actually 572 00:39:33,200 --> 00:39:34,450 a physical property. 573 00:39:39,200 --> 00:39:56,260 Let me say that Cijkl is given by ai capital I aj capital J 574 00:39:56,260 --> 00:40:10,450 ak capital K, al capital L times let's say D, capital I, 575 00:40:10,450 --> 00:40:16,270 capital J, capital K, capital L. That actually, believe it 576 00:40:16,270 --> 00:40:18,340 or not, means something physically and we're going to 577 00:40:18,340 --> 00:40:20,230 get to that in due course. 578 00:40:20,230 --> 00:40:24,040 But what this is, taking the Einstein convention into 579 00:40:24,040 --> 00:40:29,640 account, this is a quadruple summation over capital I, 580 00:40:29,640 --> 00:40:35,580 capital J, capital K, and capital L. These are very 581 00:40:35,580 --> 00:40:42,510 often referred to in this business as dummy indices, 582 00:40:42,510 --> 00:40:45,490 meaning that they don't really mean anything physically. 583 00:40:45,490 --> 00:40:48,570 They're just indices of summation. 584 00:40:48,570 --> 00:40:51,250 I would caution you that this is a term 585 00:40:51,250 --> 00:40:52,390 that does not permute. 586 00:40:52,390 --> 00:40:55,990 If I say those are dummy indices, it means one thing. 587 00:40:55,990 --> 00:40:58,920 If I say those are indices, dummy, it means something 588 00:40:58,920 --> 00:41:00,510 completely different, and you're apt to get 589 00:41:00,510 --> 00:41:01,760 a poke in the nose. 590 00:41:03,470 --> 00:41:04,980 OK so these are dummy indices. 591 00:41:04,980 --> 00:41:06,600 Now, what does this represent? 592 00:41:06,600 --> 00:41:10,090 I won't say what it represents physically, but this consists 593 00:41:10,090 --> 00:41:25,733 of terms in four variables times a coefficient. 594 00:41:30,090 --> 00:41:32,090 So there are five quantities in each term. 595 00:41:41,248 --> 00:41:49,730 If I sum over capital I, J, and K from 1 to 3, there are 596 00:41:49,730 --> 00:41:59,930 going to be 81 such terms, each with five 597 00:41:59,930 --> 00:42:01,250 elements in each term. 598 00:42:01,250 --> 00:42:06,850 So it's going to be on the order of 405 terms in this 599 00:42:06,850 --> 00:42:10,730 summation, and we've collapsed the whole thing down, 405 600 00:42:10,730 --> 00:42:13,260 terms in this nice summation. 601 00:42:13,260 --> 00:42:17,910 So it is a great facility for writing down expressions 602 00:42:17,910 --> 00:42:23,420 explicitly, but these terms can hide a terribly, terribly 603 00:42:23,420 --> 00:42:24,670 complex polynomial. 604 00:42:32,140 --> 00:42:35,290 All right, I think that's a good place to quit. 605 00:42:35,290 --> 00:42:38,710 It continues to amaze me, though, how some absolutely 606 00:42:38,710 --> 00:42:43,150 trivial convention, when first proposed by a great man, will 607 00:42:43,150 --> 00:42:46,850 carry that man's name no matter how stupid it is. 608 00:42:46,850 --> 00:42:51,530 So this is called the Einstein convention by everybody who 609 00:42:51,530 --> 00:42:52,820 works in this field. 610 00:42:52,820 --> 00:42:56,940 Another one, just briefly to finish up, when you discuss 611 00:42:56,940 --> 00:43:02,120 dislocations, you talk about a circuit of steps around the 612 00:43:02,120 --> 00:43:02,980 dislocation. 613 00:43:02,980 --> 00:43:04,905 That's called a Burgers circuit. 614 00:43:07,600 --> 00:43:10,430 And if you do the same circuit in a corresponding piece of 615 00:43:10,430 --> 00:43:14,150 material that doesn't have a dislocation, if this circuit 616 00:43:14,150 --> 00:43:18,130 closes, this circuit fails to close by something that's 617 00:43:18,130 --> 00:43:21,420 called the Burgers vector. 618 00:43:21,420 --> 00:43:26,350 Every book on dislocation theory says, Shockley called 619 00:43:26,350 --> 00:43:28,820 this good material. 620 00:43:28,820 --> 00:43:32,550 I'm sorry, Shockley called this good material, and 621 00:43:32,550 --> 00:43:35,432 Shockley called this bad material. 622 00:43:35,432 --> 00:43:37,120 Big deal-- 623 00:43:37,120 --> 00:43:39,320 good material, bad material because it's got a 624 00:43:39,320 --> 00:43:39,980 dislocation in it. 625 00:43:39,980 --> 00:43:43,580 But that is identified with Shockley's name in every 626 00:43:43,580 --> 00:43:44,830 discussion of dislocation. 627 00:43:48,470 --> 00:43:51,210 I'm just jealous because nobody has ever said according 628 00:43:51,210 --> 00:43:55,430 to me, such and such is-- 629 00:43:55,430 --> 00:44:00,890 all right, see you later in the week 630 00:44:00,890 --> 00:44:02,140 for more great things.