1 00:00:00,030 --> 00:00:06,150 Before we get going, I want to do a little bit of philosophising about how it is we learn things in in physics. 2 00:00:06,360 --> 00:00:12,719 At the beginning of the week, we started with the Boltzmann model of vibrations in solids, and then we decided that that wasn't quite right. 3 00:00:12,720 --> 00:00:16,890 So we had to improve it with the Einstein model of the solid, and that was pretty good, 4 00:00:16,890 --> 00:00:22,650 but it didn't get the low temperature behaviour right, so we had to improve it with the device theory of solids and that was even better. 5 00:00:22,650 --> 00:00:29,129 But we're going to have to improve it later on in the year as well. And a perfectly valid question is why didn't we learn the right thing at the 6 00:00:29,130 --> 00:00:32,820 beginning and not have to go through all of these models which are all wrong? 7 00:00:33,210 --> 00:00:38,250 And it's not just because I like telling you history stories about how all of this was developed, 8 00:00:38,550 --> 00:00:41,580 but the reason we do this is because this is always how we learn things. 9 00:00:41,580 --> 00:00:46,650 In physics, you always learn the simple model, even though it's incorrect first, because it's a lot easier to think about. 10 00:00:46,920 --> 00:00:51,120 So for example, you learn classical mechanics first and it's not really right because, you know, 11 00:00:51,120 --> 00:00:54,720 there's special relativity and quantum mechanics and you layer those on top later, 12 00:00:54,930 --> 00:00:59,489 but your intuition always falls back to the simple model that's a lot easier to think about. 13 00:00:59,490 --> 00:01:02,610 And that's why we do this. And this is exactly what we're going to be doing today. 14 00:01:03,360 --> 00:01:11,399 When we talk about metals we started last time metals and a simple crude model we use for many, 15 00:01:11,400 --> 00:01:24,330 many things is the Judah picture of electrons in metals, which is basically just kinetic theory, kinetic theory for electrons for electrons. 16 00:01:24,930 --> 00:01:30,330 And at the end of last time we derived the due to transport equation. 17 00:01:30,750 --> 00:01:39,809 The momentum de t equals the force on the electron minus p over some phenomenological scattering 18 00:01:39,810 --> 00:01:49,590 time tao where the force is Lorentz force the general force that the electron feels. 19 00:01:49,770 --> 00:01:54,780 So it's basically just Newton's equation with an added drag force P over Tao, 20 00:01:54,780 --> 00:01:59,280 which sort of represents scattering, sort of slows the electron down in some way. 21 00:01:59,580 --> 00:02:05,370 Now, for many experiments we're going to be interested in, we are actually doing some sort of steady state experiment. 22 00:02:05,370 --> 00:02:11,520 You might apply a steady electric field, you might have a steady current, and you are interested in a steady state result. 23 00:02:11,820 --> 00:02:22,320 So it simplifies things a lot to look for a steady state, so the p t equals zero. 24 00:02:22,980 --> 00:02:27,360 So let's try to solve that so equal zero and put in the force on the right hand side here. 25 00:02:27,660 --> 00:02:37,110 So the force and we replace that by the Lorenz force plus V Crosby and then we need P over Tao over here. 26 00:02:37,110 --> 00:02:46,560 But I'm going replace P by V over Tao just because we have a velocity here, there's the same velocity here, we'll put another velocity there. 27 00:02:46,780 --> 00:02:53,339 Now velocity is a perfectly good quantity, but it's not actually what you are likely to measure in the experiment, 28 00:02:53,340 --> 00:02:56,790 which you're much more likely to measure. An experiment is the current density. 29 00:02:58,780 --> 00:03:07,720 Current density very closely related to velocity j is the number of a lack of density of electrons. 30 00:03:08,680 --> 00:03:16,990 Density of electrons times the charge on the electron, which is minus e times his velocity. 31 00:03:18,100 --> 00:03:23,740 So the current density is just how many electrons you have with the charge they have and how fast they are moving. 32 00:03:24,100 --> 00:03:32,469 So every time I have velocity in that equation, I'm going to plug in J instead and divide by an and in a minus e and one step. 33 00:03:32,470 --> 00:03:40,300 I'm also going to move over to the other side and get this equation e equals. 34 00:03:41,820 --> 00:03:54,750 One over an e j crosby and then plus m over and e squared Tao times j. 35 00:03:57,670 --> 00:04:01,590 That could. Happy with that. Too many steps at once. 36 00:04:03,360 --> 00:04:08,910 Yeah. So I just I just plugged in J for V and then moved to the other side and divide it through by any. 37 00:04:10,510 --> 00:04:15,710 Right. Okay, good. So there's two terms in this equation, and we'll call them two different things. 38 00:04:15,740 --> 00:04:21,830 Let's call this E parallel and we'll call this E hall. 39 00:04:24,020 --> 00:04:30,020 Let me draw a diagram of this. So what we have is we have a block of metal like this. 40 00:04:32,400 --> 00:04:40,290 We're going to run a current through the metal like this contessa in the out like that we might apply a magnetic field, 41 00:04:40,290 --> 00:04:48,600 maybe perpendicular to the metal like this. V And then we will have E parallel in this direction, electric field in that direction, 42 00:04:48,840 --> 00:04:57,329 and then e hall perpendicular to the current and perpendicular to the magnetic field as well. 43 00:04:57,330 --> 00:05:07,650 E Hall And it is called the Hall Electric Field because it was discovered by Edwin Hall, who is doing exactly these kind of experiments in 1879. 44 00:05:08,010 --> 00:05:14,879 And he discovered that when he ran a current through a metal in a magnetic field, he ended up with an electric field perpendicular, both of them. 45 00:05:14,880 --> 00:05:19,320 You may have run into this before. It is a pretty clear result of the Lorenz force. 46 00:05:19,320 --> 00:05:24,780 What's happening is you're running electrons through the material and they're trying to curve because of the magnetic field. 47 00:05:24,930 --> 00:05:28,400 And that builds up an electric field. Good. 48 00:05:28,850 --> 00:05:37,160 Everyone happy with that? More or less. Okay. So let us try to think about this equation a little more closely. 49 00:05:37,170 --> 00:05:42,620 Let's take a simple case. Case one, magnetic field equals zero. 50 00:05:43,560 --> 00:05:48,930 So in that case, we just have electric field. Let me turn it around the other way. 51 00:05:49,230 --> 00:05:57,120 Move the sides of the current is then an E squared now or m times the electric field. 52 00:05:57,720 --> 00:06:01,440 This quantity here is a conductivity sigma. 53 00:06:01,800 --> 00:06:10,260 Of course, conductivity and the conductivity because it relates the electric field to the current. 54 00:06:10,590 --> 00:06:16,290 Now the expression for the conductivity that we derived and e squared tao over m is 55 00:06:16,290 --> 00:06:20,219 the due to conductivity is the conductivity we would calculate in the jury theory. 56 00:06:20,220 --> 00:06:25,560 We just calculated it and we should have pretty easy intuition for what what is going on here. 57 00:06:25,950 --> 00:06:32,429 The conductivity has a factor of density up top because the more electrons you have, 58 00:06:32,430 --> 00:06:38,880 the more current you're going to get and the more conductivity you're going to get. It has a factor of tao up top because longer time, 59 00:06:38,930 --> 00:06:44,700 bigger Tao means longer scattering time and the longer scattering time you have are the less things you have to run into, 60 00:06:44,700 --> 00:06:46,290 the better your conductivity is going to be. 61 00:06:46,800 --> 00:06:52,500 The Factor Mass downstairs looks a little bit more complicated, but actually it just comes from F equals may. 62 00:06:52,800 --> 00:06:57,180 If your math is small for a fixed force, your acceleration has to be larger. 63 00:06:57,510 --> 00:07:02,819 So you apply some fixed force, the electrons move faster if their mass were smaller. 64 00:07:02,820 --> 00:07:11,070 That's why the mass comes out downstairs. Now, we might ask or I might ask, is this a good answer or is this a bad answer? 65 00:07:11,340 --> 00:07:17,969 And right now we actually don't know because Tao is some unknown number, it's some phenomenon. 66 00:07:17,970 --> 00:07:22,640 He doesn't know how to calculate it. We can just we have to put it in phenomena logically. 67 00:07:22,680 --> 00:07:28,680 So really it would fit pretty much anything at this point. So you might think of it sort of turning this on its head. 68 00:07:29,040 --> 00:07:33,149 The measurement of conductivity is actually a measurement of the parameter. 69 00:07:33,150 --> 00:07:36,870 Tao Okay. And that is frequently how it's viewed. 70 00:07:37,970 --> 00:07:46,100 All right, now let's do a little something a little bit more complicated case to case two, which is B, not equal to zero. 71 00:07:46,700 --> 00:07:50,300 And there are various ways we might think about B, not equal to zero. 72 00:07:50,540 --> 00:07:54,290 One possibility of doing this experiment, as we might imagine. 73 00:07:54,590 --> 00:08:00,400 Suppose we know. So let's call this two. Suppose we know a couple of things. 74 00:08:00,420 --> 00:08:05,690 Suppose we know the density of our electrons. Of course we know the charge on the electron. 75 00:08:05,930 --> 00:08:08,960 Suppose we know the current and we measure. 76 00:08:11,450 --> 00:08:17,209 We measure ihall. Then, according to Judith Theory, we then know. 77 00:08:17,210 --> 00:08:27,470 Then we get a magnetic field. B. And in fact, this is a very frequently used method for measuring magnetic fields is known as a hall sensor. 78 00:08:31,230 --> 00:08:39,480 And, you know, it's still used frequently in modern technology now to be a little bit more detailed about this over here. 79 00:08:40,830 --> 00:08:47,580 Generally, if you have a current in a magnetic field, you will get something of the form. 80 00:08:48,180 --> 00:08:57,370 We define this quantity our age because Jay noticed that it's turned the order of J and B over there to over here. 81 00:08:57,390 --> 00:09:01,500 This is just a definition of a commonly used quantity known as the hall coefficient. 82 00:09:06,540 --> 00:09:15,150 So, you know, if you know what Jay is, you know it is measure E, get the whole coefficient, for example. 83 00:09:17,600 --> 00:09:22,550 So. Right. So intruder theory due to theory. 84 00:09:24,580 --> 00:09:28,690 Comparing to that equation. So this is the hall. Sorry, the hall. 85 00:09:29,920 --> 00:09:40,210 In June. A theory comparing to that equation over there. The whole coefficient r h is one over density times minus e. 86 00:09:40,600 --> 00:09:45,310 The minus sign comes from the fact that I flipped the order of these two compared to over there. 87 00:09:46,960 --> 00:09:53,140 So the whole electric field that you would measure would be proportional to the hall coefficient here. 88 00:09:53,440 --> 00:09:58,419 And if you're trying to build a whole sensor in order to find accurate ways of 89 00:09:58,420 --> 00:10:03,040 measuring magnetic fields are using electronics or using voltage measurements. 90 00:10:03,430 --> 00:10:10,570 You really want the whole voltage to be large, as large as possible so that you have a large electric field to measure. 91 00:10:10,840 --> 00:10:15,490 So to do that, what you do is you usually choose a material with a small density. 92 00:10:16,870 --> 00:10:25,090 So you use you small density to get big to get big, big E. 93 00:10:27,250 --> 00:10:31,329 And typical hall sensors are built with semiconductors and other materials that have 94 00:10:31,330 --> 00:10:36,880 a small electron density and will come to semiconductors later on in the term. 95 00:10:37,660 --> 00:10:46,810 Now, let's turn this experiment now on its head, probably the way that Paul Duda thought about it in his case, he probably knew no. 96 00:10:49,090 --> 00:10:58,690 And so in his case, he probably knew the magnetic field, the electron charge in the current measured measure E hall. 97 00:10:59,770 --> 00:11:07,350 And what you get, what you get is the density of electrons in your sample, 98 00:11:07,360 --> 00:11:11,050 because, of course, he didn't know the density of electrons in his materials. 99 00:11:11,500 --> 00:11:16,030 Well, so let's try this. Suppose we do this for a bunch of different materials. 100 00:11:16,030 --> 00:11:22,419 So we'll take a bunch of metals like lithium, sodium, potassium, copper. 101 00:11:22,420 --> 00:11:30,410 So typical good metals. And the first thing the jury would have noticed is that for all of these medals, 102 00:11:30,750 --> 00:11:36,410 our age is less than zero, which is what he predicted by his formula over here. 103 00:11:36,420 --> 00:11:45,690 So that much is good. And then he can actually put in measure the magnitude of Ihal and try to extract the density of electrons. 104 00:11:45,690 --> 00:11:49,680 And what he would have gotten was about 0.8 electrons per atom here, 105 00:11:50,850 --> 00:11:58,559 about 1.2 electrons per atom here about 1.1 electrons per atom 1.5 electrons per atom seems like reasonable numbers. 106 00:11:58,560 --> 00:12:02,220 A couple of electrons wanting one electron per atom more or less. 107 00:12:03,420 --> 00:12:05,340 Is this a good answer or is this a bad answer? 108 00:12:05,730 --> 00:12:12,240 Well, if you think about it for a second, you'll remember your periodic table copper is the 29th element on the periodic table. 109 00:12:12,540 --> 00:12:21,060 So it has 29 protons and 29 electrons and we measured 1.5 electrons per atom where the other 27.5 go. 110 00:12:22,170 --> 00:12:28,860 So this might have been a little bit puzzling to Judge Ruda, but actually, now that we know a little bit about the atomic structure, 111 00:12:28,860 --> 00:12:32,040 we know about the shell structure of atoms, we know a little bit about chemical bonding. 112 00:12:32,040 --> 00:12:34,380 We'll even talk a little bit about chemical bonding later on in the year. 113 00:12:35,550 --> 00:12:42,450 And we know that, in fact, many of the electrons in an atom are actually in orbitals very close to the to the nucleus. 114 00:12:42,450 --> 00:12:45,900 These core orbital electrons are basically stuck. 115 00:12:46,230 --> 00:12:54,750 So core orbitals. But all electrons or orbital electrons don't move. 116 00:12:58,340 --> 00:13:00,830 The only ones that do move out of Shell. 117 00:13:02,980 --> 00:13:18,190 Electrons, also known as valence electrons move so that electrons that are running around in the in the solid are the so called valence electrons. 118 00:13:18,490 --> 00:13:25,420 And, you know, from chemistry, we actually know how many valence electrons these materials have. 119 00:13:25,660 --> 00:13:32,170 And in fact, all these four materials have valence one meaning only one electron in the outermost shell. 120 00:13:33,010 --> 00:13:41,200 So if we assume only the outermost electrons move the ones in the outermost shell, we should predict one electron per atom. 121 00:13:41,440 --> 00:13:46,089 And that is actually in fairly good agreement with what is measured experimentally. 122 00:13:46,090 --> 00:13:52,750 So that is pretty good for a four Judith theory. We get roughly one electron per atom moving around. 123 00:13:53,870 --> 00:13:59,509 So we might, you know, emboldened by our success, we might try some other materials. 124 00:13:59,510 --> 00:14:04,280 So let's try some with valence to valence equals two. 125 00:14:04,580 --> 00:14:08,000 We have materials like beryllium, we have magnesium. 126 00:14:08,450 --> 00:14:12,830 Both of them have our valence too. And all of a sudden we have a big problem. 127 00:14:13,130 --> 00:14:18,020 The big problem is that our h the whole coefficient is now greater than zero. 128 00:14:19,240 --> 00:14:22,690 And this is completely puzzling from this picture, Andrew, 129 00:14:22,870 --> 00:14:27,490 because predecessors that it should be just the density times the charge of the electron which is negative 130 00:14:27,700 --> 00:14:32,679 and so there is no way in Judith theory we're going to get our hall coefficient which is positive. 131 00:14:32,680 --> 00:14:39,190 And this must have puzzled Paul due to terribly and later on in the year when we study van structure of materials, 132 00:14:39,190 --> 00:14:44,469 we'll understand why this is, but it kind of looks like one of two things is going on possibility. 133 00:14:44,470 --> 00:14:49,270 One is that the density of electrons has gone negative for some reason, if that makes any sense. 134 00:14:49,480 --> 00:14:53,170 Possibility two is that the charge carrier, 135 00:14:53,170 --> 00:14:58,420 the thing that is moving around carrying the charge is not the electron but something else with a positive charge. 136 00:14:58,870 --> 00:15:04,480 Neither of those seem to make much sense right now, but they'll make a little bit more sense later on, hopefully, 137 00:15:05,500 --> 00:15:14,500 but in general is a very brave person and he decided that we're going to ignore these two materials that don't seem to fit the materials with valence, 138 00:15:14,500 --> 00:15:17,500 two that have high coefficient positive. 139 00:15:18,070 --> 00:15:24,820 And we're going to go ahead and try to calculate some other quantities that kinetic theory or due to theory should be able to predict. 140 00:15:25,060 --> 00:15:33,220 Now, you studied kinetic theory last year, and one of the things that you were able to calculate in kinetic theory was thermal conductivity. 141 00:15:37,120 --> 00:15:40,989 I'm not going to go through the whole calculation because it's something that 142 00:15:40,990 --> 00:15:44,590 you probably learned very well last year by not worth going through again. 143 00:15:44,920 --> 00:15:50,290 But I'll write down the answer that you probably well should probably look familiar with thermal conductivity for a moment. 144 00:15:50,290 --> 00:15:57,429 Atomic gas is one third the density of particles heat capacity per particle. 145 00:15:57,430 --> 00:16:07,510 This is KV or an and then there's a velocity and a scattering length, which is a velocity times the scattering time. 146 00:16:07,870 --> 00:16:16,899 And in kinetic theory we can take the velocity to b square root of eight k beat over pi times the mass. 147 00:16:16,900 --> 00:16:19,900 Does this all look familiar. Vaguely familiar from last year? 148 00:16:20,380 --> 00:16:25,600 Hopefully. Okay. So what we do is we actually just plug this in. 149 00:16:25,690 --> 00:16:28,570 It's going to be V squared because V occurs in two places. 150 00:16:28,810 --> 00:16:38,920 We'll plug in the expression from kinetic theory over and we have KB from the classical gas physics. 151 00:16:39,370 --> 00:16:45,580 And what we get is the thermal conductivity is four over pi. 152 00:16:45,970 --> 00:16:50,110 So the let's see. So the pi comes from down here having gotten squared. 153 00:16:50,440 --> 00:16:53,590 I guess we got a two over there which counts on the eight to make a four. 154 00:16:53,860 --> 00:16:59,560 Then we have n tao k.v. squared t over m. 155 00:17:01,670 --> 00:17:06,050 So that's the prediction for the thermal conductivity in due theory. 156 00:17:06,350 --> 00:17:11,030 Now, again, we have this question. Is this a good answer or is this a bad answer? 157 00:17:11,210 --> 00:17:16,610 And we don't really know because we have this Tao, this unknown quantity Tao in this equation. 158 00:17:17,360 --> 00:17:26,660 But it's not so bad because we also had sigma is an E squared tao over m and we can look at the ratio of these two quantities and get rid of Tao. 159 00:17:26,960 --> 00:17:30,140 So let's do that. We'll take what is known as L. 160 00:17:30,140 --> 00:17:35,390 The Lawrence number. Lawrence number. Number. 161 00:17:36,050 --> 00:17:40,580 This is not actually the same guy who is Lawrence Transformations and Lawrence Force. 162 00:17:40,820 --> 00:17:43,970 This is Lawrence without the T in his name, Lawrence number. 163 00:17:45,080 --> 00:17:48,230 Everyone seems to have the same name in Germany in the 1800s. 164 00:17:51,470 --> 00:17:56,400 So we can take the ratio of the thermal conductivity to t times. 165 00:17:56,450 --> 00:18:05,420 Sigma here the Taos Council and we're left with just pore over pi and kb over e squared. 166 00:18:05,790 --> 00:18:15,970 That's it. And this result, while if you put in numbers, you discover it's one times ten to the minus eight watt arms per kelvin squared. 167 00:18:16,600 --> 00:18:21,070 And this result is kind of interesting because it's, it's what we call universal. 168 00:18:21,280 --> 00:18:27,129 It does not depend on temperature, it doesn't depend on the mass of electronics, doesn't depend on the scattering time of the electron. 169 00:18:27,130 --> 00:18:28,750 It doesn't pan on the density of electrons. 170 00:18:28,930 --> 00:18:36,340 It doesn't depend on anything that these fundamental quantities, Boltzmann constant and the charge of the electron. 171 00:18:36,460 --> 00:18:48,730 It's kind of an interesting prediction and in fact the fact that the Lorenz number is fixed l the same the same for all materials. 172 00:18:48,910 --> 00:19:03,430 For all materials was known and all t0t this is known as the vitamin Frans Lau Vitamin Frans 173 00:19:04,360 --> 00:19:09,729 and it was known since the mid 1800s and no one had the foggiest idea why it was true. 174 00:19:09,730 --> 00:19:15,790 They just measure that vitamin in France where Germans who like to measure things like thermal conductivity, 175 00:19:15,790 --> 00:19:17,650 electrical conductivity in the middle of 1800s, 176 00:19:18,430 --> 00:19:26,110 and they noticed that if you take this ratio is comes out pretty much the same for all materials I guess if we want to be precise about it. 177 00:19:26,110 --> 00:19:38,110 In fact, for lithium, it comes out about 2.2 what times ten to the minus eight watt ohms per kelvin squared for copper is about 2.0. 178 00:19:38,620 --> 00:19:45,219 For iron, it's about 2.6. So it's pretty close to Julian's prediction. 179 00:19:45,220 --> 00:19:49,390 Okay, it's off by a factor of two, but at least it's in the right ballpark. 180 00:19:49,390 --> 00:19:59,379 It's a very crude theory. And before Drew, no one had had any idea why this ratio should be fixed for all materials, all at all. 181 00:19:59,380 --> 00:20:06,190 So this was really a great step forward. And the intuition that we should have in our heads is actually not that complicated. 182 00:20:06,490 --> 00:20:12,640 The intuition should be that whether we're thinking about heat transport or we're really thinking about electrical transport, 183 00:20:12,820 --> 00:20:15,280 we're really just thinking about electrons moving. 184 00:20:15,550 --> 00:20:22,150 If we're thinking about regular conductivity, we're sort of counting how many electrons move and each electron has a certain amount of charge on it. 185 00:20:22,330 --> 00:20:27,940 Whereas if we're thinking about thermal conductivity, we're still counting how many electrons are moving, 186 00:20:28,090 --> 00:20:32,350 but each electron is moving a certain amount of heat, which is proportional to temperature. 187 00:20:32,590 --> 00:20:42,010 So that ratio of thermal conductivity divided by temperature, then divided by regular conductivity should come out roughly constant because in both, 188 00:20:42,010 --> 00:20:45,820 in all cases you're just counting the number of electrons that move. Okay. 189 00:20:46,660 --> 00:20:51,580 So this was a really good result from from from Juda explaining the Vitamin Franz law. 190 00:20:51,790 --> 00:21:04,309 But there's a really big puzzle. And the puzzle is that we used we used rather glibly the statement. 191 00:21:04,310 --> 00:21:11,240 The CV over DN is three half KB, which is a perfectly good result for a minor atomic gas. 192 00:21:11,480 --> 00:21:19,250 But it's just not true for electrons in the metal. Not true for electrons in metal. 193 00:21:19,970 --> 00:21:23,180 Why not? Well, we measured it and we saw it's not true. 194 00:21:23,420 --> 00:21:27,709 Remember, in metals, the heat capacity is alpha t cubed. 195 00:21:27,710 --> 00:21:31,370 And we identify this alpha as being vibrations or debye by theory. 196 00:21:32,490 --> 00:21:39,610 Plus Gamma Times t. And in fact, if you will, in this game, at times, tea is special for metal. 197 00:21:39,610 --> 00:21:42,940 So this is the heat capacity of the electrons running around in the metal. 198 00:21:43,330 --> 00:21:53,740 And for any reasonable temperature gamma times t is much, much less than three KB as long as T is not ginormous. 199 00:21:54,220 --> 00:22:01,790 Ginormous. That means huge, like, you know, 10,000 kelvin as long as T is not 10,000 kelvin. 200 00:22:02,540 --> 00:22:05,210 Gamma T is much less than three KB. 201 00:22:05,570 --> 00:22:14,220 So somehow or other, even though we use this thing that was clearly incorrect, we used we gave the electrons a heat capacity of three KB each. 202 00:22:14,660 --> 00:22:19,760 Whereas, in fact, if you measure the heat capacity, the metal, that capacity is just not there. 203 00:22:20,090 --> 00:22:25,340 The only capacity the electrons have is is gamma times t, which is much, much less than three KB. 204 00:22:25,610 --> 00:22:29,240 So how did we get this right? What's going on here? 205 00:22:29,630 --> 00:22:37,250 Well, the problem here, the problem here actually becomes much, much more obvious if we look at some other quantities. 206 00:22:37,520 --> 00:22:42,050 Let's look at thermal electric properties. Thermal electric properties. 207 00:22:43,160 --> 00:22:53,380 Electric properties. Bertie, is this the kind of experiment we want to look at? 208 00:22:53,830 --> 00:22:57,909 Here's a block of metal. We want to run a current. So here's a current source. 209 00:22:57,910 --> 00:23:03,130 We run the current through the metal. So the current drags electrons through the metal this way. 210 00:23:03,790 --> 00:23:11,380 And then because the current is moving through the metal, it means the electrons are moving through the metal and therefore heat is out. 211 00:23:11,860 --> 00:23:17,140 I should have the electrons moving this way. Current was in the opposite direction, but the electrons are moving this way. 212 00:23:17,440 --> 00:23:26,140 So then the heat. The heat current gets dragged in the same direction because each electron is dragging a certain amount of heat with it. 213 00:23:26,530 --> 00:23:32,980 So one can define JQ and JQ is now the heat current density. 214 00:23:33,790 --> 00:23:37,440 Heat current. Density. 215 00:23:40,740 --> 00:23:50,010 JQ is equal to some coefficient pi times the regular current and c j and pi is known as the Peltier coefficient. 216 00:23:50,820 --> 00:24:03,980 Peltier coefficient. And this is known as the Peltier Effect after Mr. Peltier, who discovered this effect in 1834. 217 00:24:04,160 --> 00:24:09,080 Another one of these people from the 1800s who like to measure thermal and electrical transport. 218 00:24:10,950 --> 00:24:21,480 Anyway we can actually I should comment here as a bit of an aside, that penalty effect is actually a very technologically useful effect to know about. 219 00:24:21,660 --> 00:24:25,230 You can actually build a very nice refrigerator with it, which has no moving parts. 220 00:24:25,260 --> 00:24:33,030 The idea is that if you want to move heat from one side, the inside of your refrigerator to the outside of the refrigerator to move heat out, 221 00:24:33,330 --> 00:24:38,790 you just run current and it drags heat with this side cold, making that side hot. 222 00:24:39,000 --> 00:24:43,130 In fact, you can buy refrigerators that are built like this that don't have pumps and things like that. 223 00:24:43,140 --> 00:24:47,790 They're not loud. They're very quiet. Frequently, they're actually used as wine refrigerators. 224 00:24:47,790 --> 00:24:53,790 I don't know why, but but you can buy a healthier refrigerator, you know, on the open market these days, 225 00:24:54,090 --> 00:25:00,510 that one thing that one has to keep in mind and probably a refrigerator is that you can't make you can't just keep running more 226 00:25:00,510 --> 00:25:07,320 and more current and getting it colder and colder because at some point there's going to be an ice squared are heat dissipation. 227 00:25:07,470 --> 00:25:12,560 So you'll be dissipating power and if I squared are gets too big, then you start heating things up. 228 00:25:12,570 --> 00:25:16,110 So for small currents, this side will get cold, this side will get hot. 229 00:25:16,260 --> 00:25:18,420 But for large currents, everything gets hot. 230 00:25:19,600 --> 00:25:26,440 So that the art form in building a good, healthier refrigerator is that you have to find the material with a large Peltier coefficient, 231 00:25:26,740 --> 00:25:32,710 but a small resistivity, and there's many, many people whose job it is to find these materials. 232 00:25:33,190 --> 00:25:36,190 All right. Anyway, we're going to try to calculate this Peltier coefficient. 233 00:25:37,360 --> 00:25:43,749 How do we do that? Well, the heat current and the JQ and we have the electrical current. 234 00:25:43,750 --> 00:25:47,950 And so when we know what the electrical current density is, it's the velocity the electrons are moving, 235 00:25:47,950 --> 00:25:53,110 the density of the electrons and the charge on the electrons, the heat, current density, very similar. 236 00:25:53,320 --> 00:25:58,210 It's a lot. While it is a factor of one third, if you're honest about it, that's a usual geometric factor. 237 00:25:58,540 --> 00:26:04,900 Don't worry about it too much. The velocity electrons are moving, the density of the electrons in the heat that each electron is carrying. 238 00:26:05,740 --> 00:26:09,069 Which is CVT, we take the ratio of these two things. 239 00:26:09,070 --> 00:26:16,360 We'll get a Peltier coefficient pi is CBT over three times minus E. 240 00:26:16,600 --> 00:26:21,000 If we plug in the usual kinetic theory result, our CV is three halves. 241 00:26:21,070 --> 00:26:24,730 KB, which are a warning you is not a good idea. 242 00:26:25,600 --> 00:26:37,749 We end up getting KB over two times minus e t and frequently what people do is they actually consider the ratio known as S, 243 00:26:37,750 --> 00:26:41,410 which is pi over t s is known as the C back coefficient. 244 00:26:43,560 --> 00:26:48,600 Quebec was another person from the 1800s who liked to measure electrical and thermal properties. 245 00:26:48,950 --> 00:26:55,649 He I believe he was from Estonia, actually. For anyone who happens to be Estonian anyway. 246 00:26:55,650 --> 00:27:01,290 So if you take that ratio, we just get the universal constant KBE over two times minus E, 247 00:27:01,590 --> 00:27:10,020 which has a numerical value of -0.4 times ten to the minus ten times for volts for Kelvin. 248 00:27:11,410 --> 00:27:25,060 And the problem is here that if you actually measure the seabed coefficient for any metal you discover actually actually as is 100 times smaller. 249 00:27:29,360 --> 00:27:35,360 So this is an obvious much more obvious error in in kinetic theory. 250 00:27:35,360 --> 00:27:42,560 Kinetic theory is going way wrong. And the source of this problem is that we were using this heat capacity for the electrons. 251 00:27:42,800 --> 00:27:47,660 That is just wrong. And the problem comes from y y the problem. 252 00:27:50,650 --> 00:27:59,920 Why the problem? Because in fact, silver end is much, much less than three KB K and it's much, much less than three has KB. 253 00:28:00,520 --> 00:28:10,480 Well, if this is true, if we're using this capacity that's so wrong, why is it that we did okay when we calculated way up there? 254 00:28:11,020 --> 00:28:17,559 When we calculate the thermal conductivity, we got the ratio of the thermal conductivity to the regular conductivity. 255 00:28:17,560 --> 00:28:24,460 We got the Lorenz number almost exactly right. Whereas the seabed coefficient is completely wrong, so something's inconsistent. 256 00:28:24,640 --> 00:28:29,230 Somehow or other, the victim in France law is coming out right. The seabed coefficient is coming out completely wrong. 257 00:28:29,500 --> 00:28:33,570 And the reason that this happens, the reason we got the VMA in France. 258 00:28:33,580 --> 00:28:39,490 All right. The reason why. Why copper is right. 259 00:28:42,170 --> 00:28:47,270 Is because we made two cancelling errors, two cancelling errors. 260 00:28:50,060 --> 00:28:58,910 I was the first one I mentioned already with Silver and CV and is much much less than three have CCB and we use CCB. 261 00:28:59,210 --> 00:29:04,370 But there is another one. We also use V squared expectation of V. 262 00:29:04,580 --> 00:29:16,280 We plugged in the kinetic theory result A.C.T. over pi m and in in actuality these squared is actually much much greater than A.B overpay m. 263 00:29:17,310 --> 00:29:22,600 And these errors in the thermal conductivity, they almost exactly cancel. 264 00:29:22,620 --> 00:29:29,070 So we get the thermal conductivity almost exactly right. Whereas this quantity B square does not enter in the seabed coefficient. 265 00:29:29,340 --> 00:29:32,820 So we get it completely wrong because we use this, but not this. 266 00:29:33,200 --> 00:29:38,759 Okay. Now, the first chance of the year to win fabulous prizes. 267 00:29:38,760 --> 00:29:42,540 One very high quality chocolate bar. Actually, my favourite chocolate bar was sold out. 268 00:29:42,540 --> 00:29:50,130 So this is a slightly less than very high quality chocolate bar. So if anyone who can tell me why we made both of these mistakes. 269 00:29:51,170 --> 00:29:56,410 What do we do wrong? What do we leave out? Come on, come on, come on, come on, come on, come on. 270 00:29:56,920 --> 00:30:08,770 When we forget someone. Someone? Yes. So I'm not going to be the reporter for that, but it's a good idea. 271 00:30:09,130 --> 00:30:16,030 Generally, generally, using expectation of security in the square will make an error, but it's usually only a factor of two kind of error. 272 00:30:16,330 --> 00:30:21,070 So it's a small error. So there's various debates as to which one is better to use in this case. 273 00:30:21,220 --> 00:30:29,710 I agree that that's a potential problem. Anyone else? Yes. Yes. 274 00:30:29,720 --> 00:30:34,370 Yes, that's it for me. And yes, so I would throw this to you, but it would probably miss so you can come down and get it afterwards. 275 00:30:34,730 --> 00:30:38,360 So the thing we left out was probably exclusion principle. 276 00:30:38,600 --> 00:30:41,959 Electrons are fermions. You can't put all the electrons in the same state. 277 00:30:41,960 --> 00:30:45,620 They're not a classical gas in any sense. They are Fermi Gas. 278 00:30:45,920 --> 00:30:51,080 And so we're going to have to treat that properly. Maybe I'll read that, write that down because it's really important. 279 00:30:51,320 --> 00:30:54,800 Fermi Statistics. Fermi statistics. 280 00:30:58,380 --> 00:31:01,480 So we have to deal with that, and that's what we're going to do next. 281 00:31:01,500 --> 00:31:06,060 But before doing that, you can do a little bit of a summary of to theory due to summary. 282 00:31:10,200 --> 00:31:14,579 So due to summary. Many things. Right, many things. 283 00:31:14,580 --> 00:31:22,860 Right. Many transport properties you get right. And I think for homework, your study, a couple of other things that you get right in judiciary, 284 00:31:22,860 --> 00:31:27,870 it's still used very heavily due to theory is used very heavily for understanding metals and semiconductors, 285 00:31:27,870 --> 00:31:31,950 particularly good for semiconductors, but it has problems. 286 00:31:33,370 --> 00:31:38,980 And some of the problems are, well, our age can have wrong sign have wrong sign. 287 00:31:42,600 --> 00:31:53,460 Fine. You can have CV or PN is actually much, much less than three KB where we use three house kbps in the calculations. 288 00:31:53,670 --> 00:32:01,350 And you can get the seatback wrong by wrong by 100, wrong by 100 and so forth. 289 00:32:01,360 --> 00:32:04,170 In fact, you can get the sign of the seabed coefficient wrong as well. 290 00:32:04,410 --> 00:32:11,880 But is being that we're off by a matter in magnitude by a factor of 100, it seems a little bit superfluous to worry about the sign all of a sudden. 291 00:32:12,210 --> 00:32:12,510 Okay. 292 00:32:12,510 --> 00:32:22,950 Anyway, so these are some of the summaries of duty theory, but nonetheless, this was how people understood metals for the first time in the 1900s. 293 00:32:22,950 --> 00:32:27,059 Before Garuda, people didn't understand medals at all and still do the theory theories. 294 00:32:27,060 --> 00:32:29,910 Extremely good way to roughly understand medals, 295 00:32:30,240 --> 00:32:37,920 but nothing really progressed for about 25 years and the theory was all there was and people didn't understand why these things didn't come out right. 296 00:32:38,280 --> 00:32:44,550 But round about 1925, there are a lot of sort of simultaneous enormous advances in physics. 297 00:32:44,700 --> 00:32:52,950 1925 was the Polish exclusion principle, 1926 with the Schrödinger equation, also 1926 with the Fermi Dirac statistics. 298 00:32:53,250 --> 00:32:56,690 And in 1927, someone came along. 299 00:32:57,060 --> 00:33:00,840 Guy named Arnold Sommerfeld. 1927 Sommerfeld. 300 00:33:03,130 --> 00:33:07,300 Sommerfeld was nominated for a Nobel Prize 81 times and never got it. 301 00:33:08,650 --> 00:33:14,020 Poor guy. But his idea was he was going to treat medals with promise statistics. 302 00:33:14,530 --> 00:33:24,060 Treat medals. Metals with Fermi statistics, with Fermi statistics. 303 00:33:24,660 --> 00:33:30,240 In other words, we're going to account for the fact that the electrons are actually fermions. 304 00:33:30,720 --> 00:33:36,360 And so now we have to do a little bit of a review of what we know about Fermi statistics. 305 00:33:36,360 --> 00:33:40,829 And the most important thing we have to remember is the Fermi occupation function 306 00:33:40,830 --> 00:33:50,580 and F of Beta Epsilon minus MU is one over either beta epsilon minus mu plus one. 307 00:33:51,150 --> 00:33:57,870 Here is the chemical potential, chem potential and this NF thing. 308 00:33:58,260 --> 00:34:01,380 And F is the probability. 309 00:34:01,640 --> 00:34:15,270 The problem. Then I can state that and I and I at energy pi at energy and Energy Epsilon is occupied. 310 00:34:19,590 --> 00:34:22,890 Good question. Now. Yeah. Good. All right, good. 311 00:34:24,210 --> 00:34:30,140 So just plot this thing. So we have energy on this axis. 312 00:34:30,150 --> 00:34:37,740 We have an F on this axis. And somewhere over here, we have the chemical potential new at zero temperature. 313 00:34:37,770 --> 00:34:43,590 The Fermi function goes from 1 to 0 as a step function that's supposed to be flat up there. 314 00:34:43,860 --> 00:34:51,630 So this is at T equals zero. It's a perfect step from one down to zero at the chemical potential a t not equal to zero. 315 00:34:51,870 --> 00:35:00,660 It's somewhat smoother. Looks like this. So this is t greater than zero and the width over which it drops from 1 to 0. 316 00:35:01,080 --> 00:35:05,030 This with here is roughly cubed. Okay. 317 00:35:05,130 --> 00:35:08,430 Does this look vaguely familiar from last year? I hope, yeah. 318 00:35:08,850 --> 00:35:13,500 Okay. So now is it useful to have a couple of definitions that we're going to use? 319 00:35:14,100 --> 00:35:24,000 Definition The chemical potential new at equals zero is known as is called the Fermi Energy. 320 00:35:24,210 --> 00:35:31,020 F the Fermi Energy. For me energy. 321 00:35:31,020 --> 00:35:34,230 You'll discover that a lot of things are named after Mr. Fermi. 322 00:35:37,650 --> 00:35:45,000 Now, I should warn you that there is some disagreement in the literature, in the books, as to what you're supposed to call the Fermi Energy. 323 00:35:45,270 --> 00:35:51,120 Some people think that the chemical potential and the Fermi energy are actually synonymous mean the same thing. 324 00:35:51,390 --> 00:35:56,850 Most people, I believe, think that the chemical potential at zero temperature is the Fermi energy, 325 00:35:57,150 --> 00:35:59,160 and that is the definition we're going to work with. 326 00:35:59,370 --> 00:36:04,660 It's actually it makes sense to do that because why would you just have two terms that mean exactly the same thing? 327 00:36:04,720 --> 00:36:08,670 You just call it the chemical potential of you mean chemical potential, potential zero temperature. 328 00:36:08,850 --> 00:36:14,430 We call the Fermi Energy for electrons, which are free electron waves. 329 00:36:14,610 --> 00:36:21,140 We generally have H bar squared K squared over two m is the energy is an energy. 330 00:36:21,150 --> 00:36:27,420 If you have a a wave with wave vector K, its energy will be quite k squared over two m. 331 00:36:28,380 --> 00:36:35,280 So if we fix this thing to be the Fermi energy, then we define k f to satisfy this equation. 332 00:36:35,280 --> 00:36:38,910 So k f is defined by this equation. 333 00:36:39,660 --> 00:36:43,920 This defines k f is the Fermi momentum. 334 00:36:44,400 --> 00:36:48,240 Uh, Fermi. Momentum. 335 00:36:50,250 --> 00:36:57,480 So if we know that from the energy, we just calculate from this equation with the Fermi momentum, is everyone happy with that? 336 00:36:58,110 --> 00:37:04,800 All right. So if we want to if we have some physical system, we want to know how many electrons are in that physical system. 337 00:37:05,100 --> 00:37:14,070 We write in the total number of electrons is the sum over all eigen states ags of the Fermi occupation factor of data, 338 00:37:14,550 --> 00:37:19,730 epsilon of the eigen state minus mu. So why do I write it like this? 339 00:37:19,740 --> 00:37:25,410 So this is the sum of all possible states in the system, the probability that that state will be occupied. 340 00:37:25,410 --> 00:37:29,880 And if you sum up over all eigen states, you get the total number of particles in the system. 341 00:37:30,300 --> 00:37:34,980 Good. Yeah. You can also turn this on its head. 342 00:37:35,160 --> 00:37:41,700 If you know the number of particles in the system, you can use this equation to figure out what the chemical potential is if you have to do that. 343 00:37:42,670 --> 00:37:50,050 Now, as we did in the Dubai theory, we had to frequently sum over overall possible plane waves. 344 00:37:50,290 --> 00:37:52,990 So one thing we should be familiar with at this time, 345 00:37:53,140 --> 00:38:01,480 I promised you we would use this a lot is a sum over all possible plane waves gets replaced by an integral ad3k over two pi cubed. 346 00:38:03,250 --> 00:38:06,460 So every eigen state is a particular plane wave in our box. 347 00:38:06,670 --> 00:38:10,990 So the sum becomes an integral D-3 cubed over two pi times a factor of the volume. 348 00:38:11,260 --> 00:38:14,740 The only thing that's different is I'm going to put a factor of two out front. 349 00:38:15,100 --> 00:38:22,530 And the factor of two is four spins that electrons have two possible spins in one particular plane wave. 350 00:38:22,540 --> 00:38:25,540 It can be either spin up or spin down. So two possible states. 351 00:38:25,780 --> 00:38:31,930 And then we're integrating the Fermi factor, beta epsilon K minus new. 352 00:38:33,270 --> 00:38:36,700 Okay. So far, so good. All right. 353 00:38:37,330 --> 00:38:44,470 So at low temperature, at low t, low t, this thing here is a step function. 354 00:38:44,830 --> 00:38:53,650 This is step which the step function would tell us we should integrate up until integrated number one. 355 00:38:53,800 --> 00:38:56,260 Up until the energy is the Fermi energy. 356 00:38:56,410 --> 00:39:03,340 In other words, all the electrons are filled, all the states are filled below the Fermi energy and everything above the Fermi energy is empty. 357 00:39:04,600 --> 00:39:18,670 So we can rewrite an is to the let's pull out the two pi cubed integral d3k up to k less than kf. 358 00:39:19,420 --> 00:39:28,270 So saying that the absolute value of K is less than if the Fermi momentum is equivalent to saying the energy must be less than the Fermi energy. 359 00:39:28,660 --> 00:39:37,720 Good. Yeah. Okay. Now, this is what this is telling us, is that the filled states here are filled states. 360 00:39:39,370 --> 00:39:45,640 States form a ball. A ball of radius. 361 00:39:48,090 --> 00:39:55,650 Chaos and filled states at zero temperature are usually known as the Fermi C again something 362 00:39:56,100 --> 00:40:03,870 named after Fermi and the surface of the ball surface is known as the Fermi surface. 363 00:40:07,430 --> 00:40:14,230 So the sphere and everywhere around the sphere, the surface of this ball, the energies are all f. 364 00:40:14,540 --> 00:40:23,330 So if you have a a sphere of radius chaos, everything on the surface of that sphere has the same energy and the energy is the Fermi energy can. 365 00:40:24,690 --> 00:40:33,120 All right. So last thing we have to do is we have to actually calculate the volume of that sphere, the volume of that ball. 366 00:40:33,540 --> 00:40:36,540 So we have to be over two pi cubed. 367 00:40:36,780 --> 00:40:43,380 And the value of that integral is just going to give us the volume, which is 4/3 pi k f cubed. 368 00:40:44,720 --> 00:40:52,010 And then with a little bit of rearrangement, we can move the V over to this side, cancel some factors, 369 00:40:52,400 --> 00:41:07,010 we'll get an over V, which is the density and electron density density which is F cubed over three pi squared. 370 00:41:08,430 --> 00:41:20,230 Then solve this for k f or get k f equals three hi squared times the density to the one third. 371 00:41:20,250 --> 00:41:26,490 So if we know the density of electrons, we know k f and then we can plug K back in to our equation, 372 00:41:27,330 --> 00:41:35,520 which I'm just about to scroll off the top to get f so f is h bar squared kf squared over 373 00:41:35,520 --> 00:41:46,170 two m equals h bar squared over two m times three pi squared and to the two thirds. 374 00:41:47,560 --> 00:41:52,270 And this is a rather important equation. We're going to use it later on. 375 00:41:54,680 --> 00:41:58,970 So it tells us where is the Fermi Energy in terms of the density. 376 00:41:59,540 --> 00:42:06,229 So, you know, how big is the Fermi energy? Well, let us first we have to figure out what the density is, what densities we use. 377 00:42:06,230 --> 00:42:11,570 Well, let's try try density of about one electron per atom, 378 00:42:11,900 --> 00:42:17,240 which seemed to work pretty well for through the theory, at least for some of these metals. 379 00:42:18,470 --> 00:42:28,160 If we do that, say for copper, x copper we get our F is approximately seven electron volts. 380 00:42:28,700 --> 00:42:35,899 And is that a big number or a small number? Well, it is useful to convert it to a temperature which is known as the Fermi temperature. 381 00:42:35,900 --> 00:42:45,950 So all the fine Fermi temperature again is something named after Fermi temp, which is t f equals F over k b. 382 00:42:47,250 --> 00:42:53,340 And for copper. For copper, t f is about 80,000 kelvin. 383 00:42:56,340 --> 00:43:03,750 Huge, huge. Huge, huge. So typical Fermi energies and Fermi temperatures in metals are absolutely enormous. 384 00:43:03,960 --> 00:43:07,380 And the reason for this is because there's a lot a lot of electrons. 385 00:43:07,530 --> 00:43:09,780 If one electron per atom, there's a lot of atoms. 386 00:43:09,990 --> 00:43:15,990 When you start putting electrons into your system, you fill up the small, the low energy states first, the ones with the smallest K, 387 00:43:16,230 --> 00:43:19,740 but then those are filled and you have to start filling up higher and higher and higher energy states. 388 00:43:19,950 --> 00:43:27,120 And by the time you put that last electron in, you've gotten up to an enormous energy because the density of these electrons is really, really big. 389 00:43:27,540 --> 00:43:30,780 So let's actually plot the Fermi function again. 390 00:43:31,770 --> 00:43:39,900 So here is an F, here's one. Here is F one chemical potential, approximately 80,000 kelvin. 391 00:43:41,610 --> 00:43:45,450 If I take out a, B and the Fermi function will look like. 392 00:43:47,180 --> 00:43:51,620 This. It drops in a very, very small range. This is cavity tea room. 393 00:43:53,360 --> 00:43:59,479 It's even more exaggerated than that. I've drawn it's even narrower, a little drop than I've then I've drawn, 394 00:43:59,480 --> 00:44:07,940 because this distance here as a temperature is 80,000 kelvin versus our room temperature, which is 300 kelvin. 395 00:44:07,940 --> 00:44:18,620 So this distance over which the, the Fermi function drops from from 1 to 0 is a very, very narrow sliver at the top of the Fermi surface. 396 00:44:18,890 --> 00:44:26,470 And in fact. This picture here gives us a hint as to why the heat capacity of the metal is so low, 397 00:44:26,490 --> 00:44:29,690 so much lower than we would have guessed in order to have heat capacity. 398 00:44:29,870 --> 00:44:31,910 You have to be able to absorb some energy. 399 00:44:32,210 --> 00:44:39,680 So you imagine an electron in some iron state, it absorbs some energy and it jumps up to a nearby eigen state with a little more energy. 400 00:44:39,920 --> 00:44:48,410 Well, all of these electrons down here, they can't absorb any energy because all the iron states near them are already filled. 401 00:44:48,470 --> 00:44:52,430 There's nowhere for them to go. They're completely frozen. They can't absorb any energy at all. 402 00:44:52,670 --> 00:44:56,870 The only things that can absorb energy are the things close to the Fermi surface, 403 00:44:56,870 --> 00:45:02,870 where they can jump above the Fermi surface and absorb energy because there's an empty state there for them to go into. 404 00:45:02,870 --> 00:45:08,180 If there's no empty state to go into. There's no way they can absorb any energy at all. 405 00:45:08,420 --> 00:45:13,670 So that gives us a hint as to why the heat capacity is so low. 406 00:45:13,970 --> 00:45:22,880 One final thing is we can look at the typical velocity, typical velocity, which is known as the Fermi velocity. 407 00:45:24,050 --> 00:45:28,490 Fermi velocity, apparently. 408 00:45:29,270 --> 00:45:41,360 Right. Okay. The F which would be h berkoff divided by the mass is k, k, f, and if you put that in, you get a huge number. 409 00:45:41,630 --> 00:45:49,010 Huge equals huge approximately 1% of the speed of light, which might surprise you. 410 00:45:49,250 --> 00:45:57,290 This is every metal that you've ever run into contact with. Things like copper, lead, silver, tungsten, whatever it is, it has electrons in it, 411 00:45:57,290 --> 00:46:02,000 running around at speeds of 1%, the speed of light, or even greater. 412 00:46:02,240 --> 00:46:09,350 Now, that might sound surprising to you. And in fact, relativity starts to become important if you're doing things carefully at those speeds. 413 00:46:09,620 --> 00:46:15,080 But in fact, it's not surprising once you think about how many electrons there are, 414 00:46:15,260 --> 00:46:19,430 there are tons and tons and tons of electrons, one for every atom or several for every atom in some cases. 415 00:46:19,700 --> 00:46:24,620 And so all the low energy states are completely full. And you just have to keep building up to higher and higher and higher energy states. 416 00:46:24,800 --> 00:46:27,890 And the electrons on the top of the Fermi surface are on the Fermi. 417 00:46:28,220 --> 00:46:33,230 On the Fermi surface in the outer edge of the Fermi ball are extraordinarily high, 418 00:46:33,260 --> 00:46:38,569 have extraordinarily high energy, high kinetic energy, therefore extremely high velocity. 419 00:46:38,570 --> 00:46:40,730 And I guess we stop there and I guess I see you Monday.