1 00:00:00,690 --> 00:00:02,610 This is the fourth lecture of the cancer matter, of course, 2 00:00:03,180 --> 00:00:12,120 where we left off last time we were talking about the free electron or the Sommerfeld Theory of Metal's free electron. 3 00:00:14,570 --> 00:00:21,200 Sommerfeld. Theory of metals. 4 00:00:24,990 --> 00:00:29,490 And when Sommerfeld was doing is more or less following through this idea that 5 00:00:29,490 --> 00:00:34,140 a metal is just a gas of electrons and he's trying to apply kinetic theory. 6 00:00:34,150 --> 00:00:37,290 The only thing he was doing differently is he was respecting Fermi statistics. 7 00:00:37,560 --> 00:00:42,180 He was keeping track of the fact that you can't put two electrons in one eigen state. 8 00:00:42,690 --> 00:00:49,200 And last lecture we derived that the Fermi wave vector is related to the density of 9 00:00:49,200 --> 00:00:54,720 electrons by three pi squared and to the one third where n is the electron density. 10 00:00:56,100 --> 00:01:01,980 And unfortunately, in the last lecture I made an error. 11 00:01:02,880 --> 00:01:06,209 I mistakenly call this thing the Fermi momentum. 12 00:01:06,210 --> 00:01:11,610 It's actually the Fermi wave vector and I even wrote it on the board incorrectly. 13 00:01:13,230 --> 00:01:17,340 H five times the way vector k is the Fermi momentum. 14 00:01:20,310 --> 00:01:26,340 Sorry about that. From the Fermi wave vector. 15 00:01:26,350 --> 00:01:29,680 We can get the Fermi energy in the usual way. 16 00:01:29,860 --> 00:01:35,769 You have a squared KF squared over two m which we can substitute in our expression for 17 00:01:35,770 --> 00:01:43,330 the Fermi wave vector h bar squared over two and three pi squared and to the two thirds. 18 00:01:44,420 --> 00:01:47,900 This is a rather important relationship that we'll use again. 19 00:01:48,380 --> 00:01:53,570 But what's important to realise here is the bigger the density of electrons you have, the bigger the Fermi energy. 20 00:01:53,570 --> 00:01:57,770 And in a typical metal like iron ore lead, the density of electrons is really big, 21 00:01:57,770 --> 00:02:03,860 a couple of electrons per atom and you have a whole lot of atoms, the very high density of atoms, an atom, every couple of angstroms. 22 00:02:03,860 --> 00:02:09,110 So the Fermi energy gets to be enormous on the order of 80,000 Kelvin, or even bigger sometimes. 23 00:02:10,300 --> 00:02:17,290 Now in in this lecture, what we're going to aim to calculate is something that we discussed earlier, the heat capacity. 24 00:02:17,770 --> 00:02:26,800 And from experiments, we know the heat capacity from metals at low temperature takes this form of tube plus plus t, 25 00:02:27,160 --> 00:02:31,450 whereas this tube term comes from vibrations or debye theory. 26 00:02:31,750 --> 00:02:39,700 We discussed that already, and this gamma t term is special to metals and in fact is the heat capacity of the electrons. 27 00:02:40,330 --> 00:02:43,690 So it's this gamma t term that we're going to be interested in today. 28 00:02:44,620 --> 00:02:50,170 Now, before we actually do this, we need to do a little bit of preparatory algebra in particular. 29 00:02:50,440 --> 00:02:59,070 We're going to need to take some over eigen states and put it into a more workable form. 30 00:02:59,080 --> 00:03:06,940 And the eigen states in this case are going to be, you know, plane waves, even the I, k r and again, since we're writing this is exponentials, 31 00:03:06,940 --> 00:03:12,009 what we're implicitly doing is we're putting the thing in a periodic box for von Karman, boundary conditions, 32 00:03:12,010 --> 00:03:17,140 periodic in all directions, so we can work with exponential plane waves instead of signs and cosines. 33 00:03:18,690 --> 00:03:26,070 What we'd like to do is we'd like to take some of our iron seats and we would like to convert it into an integral over energy, 34 00:03:27,030 --> 00:03:30,830 g of energy where G is now a density of states. 35 00:03:30,840 --> 00:03:34,060 This is very similar to what we did when we did the buy theory. 36 00:03:36,420 --> 00:03:40,490 However, we can do something slightly different here, slightly different. 37 00:03:40,500 --> 00:03:44,909 We're going to remove a factor of the volume from the density of states. 38 00:03:44,910 --> 00:03:48,000 So this is now density of states per unit volume. 39 00:03:49,230 --> 00:03:53,970 And we do this because it's conventional to do so and it's convenient to do so. 40 00:03:54,270 --> 00:03:58,050 And if you didn't notice it before, conventional and convenient come from the same word. 41 00:03:58,380 --> 00:04:02,040 So it is both of those things. It's just happens to be handy to do so. 42 00:04:02,040 --> 00:04:20,579 So we're going to do it. At any rate, the definition of this density of states is that G of D is the number of states per unit volume. 43 00:04:20,580 --> 00:04:23,760 In this case the volume. 44 00:04:25,680 --> 00:04:30,680 With energies. Energies? 45 00:04:31,560 --> 00:04:39,450 Uh, between Epsilon and epsilon. 46 00:04:39,450 --> 00:04:49,060 Plus the epsilon. Very similar to what we had for Dubai Theory before, when we were thinking about 20 states per frequency. 47 00:04:50,050 --> 00:04:55,930 Now, the general idea again, is that we're going to take the sum of all the eigen states and we're actually 48 00:04:55,930 --> 00:05:00,130 going to convert that some of our individual states to an integral over energies, 49 00:05:00,370 --> 00:05:05,590 times the number of states at each energy, just a different way of writing it that makes your life a lot easier. 50 00:05:05,980 --> 00:05:10,000 All right. So what are we going to do here to get from the sum into the integral? 51 00:05:10,360 --> 00:05:13,930 Well, first thing we're going to do, some of our eigen states is really a sum of a K, 52 00:05:15,970 --> 00:05:22,150 but it actually has a factor of two out front because there are two spins per k spins. 53 00:05:23,610 --> 00:05:27,790 Electron can be spin up or it can be spin down with the same way vector. 54 00:05:28,420 --> 00:05:31,240 Then we're going to do the same manipulation we did with the by theory. 55 00:05:31,270 --> 00:05:38,720 Leave the factor of two out front is going to replace the sum over K with an integral d3k over two pi cubed. 56 00:05:38,740 --> 00:05:44,110 This is the way sums get converted into integrals and we'll make that replacement many times this year. 57 00:05:44,110 --> 00:05:50,260 A sum over K becomes the volume times integral d3 k over two pi cubed and then. 58 00:05:51,610 --> 00:05:59,020 Since we're thinking about isotropic system, we can convert the integral over three Cartesian directions into spherical polar coordinates. 59 00:05:59,410 --> 00:06:06,130 So we have to be over two pi cubed and then we have an integral zero to infinity. 60 00:06:06,580 --> 00:06:16,420 For pi k squared decay where the four pi k squared is the usual for pi as the directions on the sphere. 61 00:06:16,750 --> 00:06:21,879 So the usual spherical polar coordinates. And this is a pretty good result. 62 00:06:21,880 --> 00:06:27,010 But really we'd like to write this in terms of energy is not in terms of wave vectors. 63 00:06:27,340 --> 00:06:35,649 So we'll use epsilon as h bar squared K squared over to M or I guess we can write 64 00:06:35,650 --> 00:06:42,820 that as k is square root of two m over H for our times epsilon to the one half. 65 00:06:43,090 --> 00:06:46,239 And in particular that would give us k is the same fact. 66 00:06:46,240 --> 00:06:55,270 Well, it's a one half times the same factor. A square root of two M over h four times epsilon to the minus one half the epsilon. 67 00:06:56,160 --> 00:07:01,550 And then if we plug these things into here, what we then get is. 68 00:07:03,050 --> 00:07:06,410 Okay, so we now have to volume. 69 00:07:06,410 --> 00:07:11,510 I'll pull out the four pi and we have the two pi cubed downstairs. 70 00:07:11,980 --> 00:07:16,670 Then we have an integral zero to infinity, the epsilon. 71 00:07:16,670 --> 00:07:21,110 And then putting in those factors for the case, square decay, we get one half. 72 00:07:21,530 --> 00:07:28,940 There's three of these factors, two M square over, two M over par cubed and then epsilon to the one half. 73 00:07:30,320 --> 00:07:33,649 So this looks almost like what we want. 74 00:07:33,650 --> 00:07:37,010 It's almost the integral of g of e d. 75 00:07:37,670 --> 00:07:41,000 So we then identify, well, okay, so I'll just write it out again. 76 00:07:41,300 --> 00:07:52,520 So this thing is integral zero to infinity g of e d e where we define g of e to then the. 77 00:07:55,030 --> 00:08:06,160 Two m to three half hour h four cubed times one over two pi squared times epsilon two one half. 78 00:08:06,700 --> 00:08:09,759 So density stays proportional to epsilon two one half. 79 00:08:09,760 --> 00:08:16,270 And this is a perfectly good answer, but it's actually convenient and therefore conventional to. 80 00:08:17,780 --> 00:08:23,239 To convert this factor of two over four cubed into something that looks a little nicer. 81 00:08:23,240 --> 00:08:29,270 And the way we do that is by using this equation here. So that equation there and you should maybe write it over here. 82 00:08:29,630 --> 00:08:35,990 If I take that equation to the three halves power I get f to the three halves 83 00:08:36,410 --> 00:08:45,680 equals h bar cubed over two m to the three halves times three pi squared. 84 00:08:46,220 --> 00:08:49,360 And did I do that right? I think I did that right. Okay. 85 00:08:49,730 --> 00:08:56,580 And then you'll notice that I can turn this around or make it upside down to m to 86 00:08:56,600 --> 00:09:04,250 three halves over a cubed is then three pi squared and over the f to the three halves. 87 00:09:04,850 --> 00:09:08,060 And this factor here is this factor here. 88 00:09:08,840 --> 00:09:19,250 So plugging that in we get g of of e is then what is three pi squared then? 89 00:09:19,280 --> 00:09:20,780 See, I know this is a lot of algebra. 90 00:09:21,530 --> 00:09:34,160 It's a big algebra day because it's a monday three has one over two pi squared are epsilon two one half and then cancelling a few things. 91 00:09:34,490 --> 00:09:47,060 We get our final result. Do you have a epsilon is three half's density over e f times energy over e f to the one half. 92 00:09:48,400 --> 00:09:52,480 Don't make any mistakes. Does that look right? Anyone object? Look good. 93 00:09:53,320 --> 00:09:56,800 All right. This is going to be something fairly useful. 94 00:09:57,250 --> 00:10:02,320 And in particular, it's useful to look at the density of states at the Fermi Energy, 95 00:10:02,740 --> 00:10:10,370 which is just three halfs density over F, which I think is something that you're asked to derive in their homework as well. 96 00:10:10,370 --> 00:10:17,080 The first homework set, and I'll give you a quick hint that I think there's a easier way to get there than what I just did. 97 00:10:17,440 --> 00:10:20,530 But, you know, if you can't figure it out, you can just follow this. 98 00:10:20,890 --> 00:10:25,150 But there's there's there's a cheaper way. But this one, this is sort of the more you know. 99 00:10:26,400 --> 00:10:30,450 It is the more direct route the other way. So sneakier. Anyway, see if you can figure it out. 100 00:10:31,670 --> 00:10:35,640 Okay. At any rate, now we're going to try to use this result figure. 101 00:10:35,670 --> 00:10:38,920 We know the density of states here in a volume. 102 00:10:38,930 --> 00:10:45,660 We're going to try to use this to figure out the heat capacity of the of the electrons at low temperature. 103 00:10:46,080 --> 00:10:49,560 Now, there is always more than one way to do something. 104 00:10:49,830 --> 00:10:53,700 There is the right way. And there's the cheating way. 105 00:10:54,820 --> 00:11:00,610 And what I am going to do is I'm actually going to explain how the right way is done and then we're going to cheat. 106 00:11:01,330 --> 00:11:04,989 And the reason we're going to cheat is because the right way is algebraically 107 00:11:04,990 --> 00:11:09,790 really horrible and it's really hard to get any intuition just by doing algebra. 108 00:11:10,030 --> 00:11:15,340 We just did enough algebra and I promise you, doing it the right way is is three times more so. 109 00:11:16,150 --> 00:11:22,480 Furthermore, the actual calculation is so algebraically complicated that you'll never be asked it on any exam an Oxford. 110 00:11:22,490 --> 00:11:26,530 And that is. I mean, I can't 100% guarantee it, but I can 99% guarantee it. 111 00:11:26,860 --> 00:11:29,000 So. So. In fact. 112 00:11:29,050 --> 00:11:35,140 So because of that, we're going to just do it the cheating way, which gives you the intuition for what's going on and avoids a lot of algebra. 113 00:11:35,140 --> 00:11:39,430 But it's worth knowing at least how you would go about it if you really want to be honest. 114 00:11:39,910 --> 00:11:46,930 So if you really want to be honest, what you would do first is you would write an equation for the number of electrons in the system, 115 00:11:46,930 --> 00:11:54,159 and we wrote this equation before last time. It's the sum over all eigen states of the probability that each eigen state is filled, 116 00:11:54,160 --> 00:11:59,260 and that probability of an alien state being filled is the Fermi function of beta, 117 00:11:59,260 --> 00:12:04,600 the inverse temperature times, energy of the eigen state, minus metre, the chemical potential. 118 00:12:04,930 --> 00:12:08,649 And being that we just derived the density of states, 119 00:12:08,650 --> 00:12:14,049 we can rewrite that as an integral while volume times the integral from zero to 120 00:12:14,050 --> 00:12:21,130 infinity of the density to states per unit volume times the Fermi function. 121 00:12:21,490 --> 00:12:30,460 So it's exactly the same, the same expression, the epsilon, exactly the same expression, except instead of writing some of our eigen states, 122 00:12:30,640 --> 00:12:37,930 you integrate of overall energies the number of states of each energy, and of course you're always integrating the probability that a state is filled. 123 00:12:38,110 --> 00:12:41,530 Is everyone good with this? Yes. Yeah. Okay, good. 124 00:12:43,600 --> 00:12:46,839 Now, this equation here, you can think of it in two different ways. 125 00:12:46,840 --> 00:12:52,420 One way is, if you could fix the chemical potential and you knew the temperature, it would tell you how many electrons you have. 126 00:12:52,810 --> 00:12:55,540 But more often than not, it goes the other way around. 127 00:12:55,750 --> 00:13:00,160 You know, the number of electrons you have in your system because you know how many atoms you have or something like that, 128 00:13:00,700 --> 00:13:04,210 and you know the temperature and it enables you to figure out the chemical potential. 129 00:13:04,480 --> 00:13:09,820 So it's sort of an inverse relationship, you know, this, this. 130 00:13:09,820 --> 00:13:13,960 And so you can figure out this in principle, although it's algebraically messy to do so, 131 00:13:14,230 --> 00:13:17,650 but in principle it would allow you to figure out the chemical potential given that, you know, 132 00:13:17,650 --> 00:13:21,580 the number of particles and you know the temperature, once you had the chemical potential, 133 00:13:21,910 --> 00:13:28,840 you could write an expression for the energy in the system, integral again integrating over all states. 134 00:13:29,110 --> 00:13:34,990 But now you integrate the energy times the probability that a state is filled. 135 00:13:38,930 --> 00:13:44,870 Okay. So instead of just counting the particles, you count the particles times their energy to get the total energy in the system. 136 00:13:45,260 --> 00:13:48,490 So you find the chemical potential first. You then find the energy. 137 00:13:48,500 --> 00:13:53,090 Once you know the chemical potential. Then therefore, you would know the energy as a function of temperature. 138 00:13:53,090 --> 00:13:59,600 You can differentiate that to get the heat capacity. So in principle, from this kind of argument, you could get the heat capacity. 139 00:14:02,130 --> 00:14:07,320 If you could do these integrals. The problem is that these integrals are really nasty, and that's why we're not going to do it this way. 140 00:14:08,230 --> 00:14:14,180 Instead, we're going to make some assumptions which aren't quite right, but to see what the assumptions are. 141 00:14:14,200 --> 00:14:21,400 Let me first draw a diagram. This is the Fermi function again, which we drew last time and app. 142 00:14:22,790 --> 00:14:27,620 Zero temperature and if goes from one to right here. 143 00:14:27,670 --> 00:14:31,460 F 1 to 0. So this is this is t equals zero. 144 00:14:32,360 --> 00:14:37,760 And then finite. The Fermi function smears out a little bit like this is greater than zero. 145 00:14:39,080 --> 00:14:43,940 Okay, this all looks familiar, I hope. Okay. Incidentally, I believe this full calculation. 146 00:14:44,150 --> 00:14:47,870 I think you actually did it last year in your stat neck course, or at least the lecturer did it. 147 00:14:48,200 --> 00:14:50,990 And you probably remember that it was pretty awful. 148 00:14:51,230 --> 00:14:55,969 And it's hard to actually remember anything about the intuition of what's going on, if the algebra is really awful. 149 00:14:55,970 --> 00:15:00,140 So we're going to try to do this in a way that is going to give you the intuition a lot better. 150 00:15:00,680 --> 00:15:04,940 So one thing we're going to assume, which is not quite right, but it's pretty close to right, 151 00:15:05,330 --> 00:15:08,540 is that the chemical potential doesn't actually change as a function of temperature. 152 00:15:09,020 --> 00:15:13,490 It does change the function of temperature because of this equation, but it only changes a little bit. 153 00:15:13,880 --> 00:15:17,210 Why is it only changes a little bit? Well, if you look at the Fermi function here. 154 00:15:18,450 --> 00:15:25,880 You can imagine as you raise the temperature, what's happening here is some of the states here that we're filled now moved to here. 155 00:15:25,890 --> 00:15:29,850 So there were some electrons here. They empty out and they fill these states up here. 156 00:15:30,990 --> 00:15:35,370 Now, if the number of states that empty out and the number of states that are filled are equal to each other, 157 00:15:35,640 --> 00:15:41,580 the chemical potential doesn't have to move at all. You would keep in constant keeping the chemical potential constant. 158 00:15:41,850 --> 00:15:48,450 Now, in truth, you have to adjust the chemical potential a little bit as a function of temperature to keep the total number of particles fixed. 159 00:15:48,690 --> 00:15:53,460 But you do not have to adjust it a lot. Basically, the chemical potential stays almost exactly the same. 160 00:15:53,880 --> 00:16:07,900 So we're going to make this assumption. We're going to write it. Many here assume the View is in-depth, independent of tea. 161 00:16:08,550 --> 00:16:14,340 It's not quite right, but it's not too bad an approximation once we have that. 162 00:16:15,600 --> 00:16:22,920 Well, actually, once we had that, we could take this equation here and then say, okay, let's plug in new fix as a function of temperature. 163 00:16:23,160 --> 00:16:26,310 And then we could calculate the energy as a function of temperature here. 164 00:16:26,550 --> 00:16:29,940 Assuming new is fixed, differentiate it and get the heat capacity. 165 00:16:30,060 --> 00:16:33,480 But even that's too complicated because this integral is just really nasty. 166 00:16:33,660 --> 00:16:36,180 So we're not even going to do that. We're going to do something even simpler. 167 00:16:37,770 --> 00:16:40,950 So what we're going to do is we're going to write the energy at some temperature. 168 00:16:40,950 --> 00:16:47,550 T is the energy at zero plus two very approximate things. 169 00:16:47,760 --> 00:16:59,130 Approximate thing one is number of electrons that can get excited, that can get get excited. 170 00:17:01,980 --> 00:17:05,430 And thing two is times amount of energy. 171 00:17:07,500 --> 00:17:12,030 Of energy each absorbs. 172 00:17:16,530 --> 00:17:23,040 So you imagine starting at zero temperature and then you turn on the temperature and then there's some number of electrons can get excited. 173 00:17:23,340 --> 00:17:26,550 Each one absorbs a total amount of energy, some amount of energy. 174 00:17:26,730 --> 00:17:31,620 And the product of these two roughly gives you the total amount of energy that you've increased your system by. 175 00:17:31,890 --> 00:17:35,550 Sound reasonable? Yeah. Yes. Yes. Is it reasonable? 176 00:17:36,810 --> 00:17:40,650 It's roughly true. You know, it's roughly. Right. Okay. Bear with me. 177 00:17:41,820 --> 00:17:47,140 Okay. So now all we have to do is we have to figure out. What are these two factors? 178 00:17:48,400 --> 00:17:53,020 So the number of electrons that can get excited. So this is this is sort of the interesting piece here. 179 00:17:53,560 --> 00:18:07,330 The number of electrons that are excited is roughly number of electrons within within KB t of f y. 180 00:18:07,330 --> 00:18:17,440 So you have to be so this range here is about T and you have to be within this range of F in order to get excited. 181 00:18:17,440 --> 00:18:21,579 Because if you are further down here somewhere, you can't get excite. 182 00:18:21,580 --> 00:18:25,720 It's hard to get excited, right? So it's you can't get it. They're supposed to be funny of. 183 00:18:28,100 --> 00:18:34,579 An electron in a state down here. I can't get excited at all because all the states that we get excited into are already filled, so it can't move. 184 00:18:34,580 --> 00:18:39,290 It's just stuck in that state and it's going to be frozen there unless you turn the temperature huge. 185 00:18:39,290 --> 00:18:46,820 So it could jump all the way up to here. Okay. So it's the number of electrons within cavity of F and how many is that? 186 00:18:47,090 --> 00:18:55,250 Well, it's basically the density of states per unit volume at the Fermi Energy Times, the volume. 187 00:18:55,610 --> 00:19:05,269 So this gives you the total number of states per unit energy at the Fermi Surface Times can t so we're taking the number of states 188 00:19:05,270 --> 00:19:12,660 per unit volume at this energy and we're multiplying it by this range to give you the total number of states within KB t of F. 189 00:19:13,490 --> 00:19:18,890 And we take that factor and we multiply it by the amount of energy each electron absorbs, 190 00:19:19,100 --> 00:19:27,160 which has got to be roughly cavity, cavity, times cavity here. 191 00:19:27,470 --> 00:19:36,110 So some electron from down here within cavity of the Fermi surface got excited up to here by absorbing about cavity of energy. 192 00:19:36,620 --> 00:19:43,670 And then, you know, to be to try to be a little bit more honest about the fact that we're cheating, completely cheating here. 193 00:19:44,000 --> 00:19:47,070 Well, I am going to add a fudge factor. 194 00:19:48,140 --> 00:19:56,420 So E total is a T equals zero plus those factors there which we just derive. 195 00:19:56,430 --> 00:20:00,980 So it's V G E F times CPT squared. 196 00:20:02,270 --> 00:20:12,260 And then the fudge factor is we'll call gamma will over two and gamma twiddle is just our admission of guilt that we didn't do a real, 197 00:20:13,280 --> 00:20:18,590 real calculation. We and we know we're going to get the answer wrong by some factor of water one. 198 00:20:18,800 --> 00:20:21,890 So Gamma Tweedle could be, you know, two. It could be a half. 199 00:20:21,890 --> 00:20:25,700 It could be pi, it could be two pi. But it's not going to be 100. It's not going to be a thousand. 200 00:20:25,700 --> 00:20:32,660 It's not going to be 0.01. Okay. So it is some number of order one, which is our admission of guilt that we didn't actually do the real calculation. 201 00:20:33,110 --> 00:20:38,240 Okay, once we have this expression, we can of course differentiate it. 202 00:20:38,510 --> 00:20:51,410 C.V. The ETI, which is this is why I put the factor of two in so it goes away when you differentiate that clever the K.B. and then G, 203 00:20:51,440 --> 00:20:55,470 add the F and you're left with CPT. 204 00:20:56,270 --> 00:20:59,210 And in fact, we already derive g e f over there. 205 00:20:59,270 --> 00:21:06,860 So it's plug it in three halves and over e f and we're also going to use maybe I'll put it here convenient, 206 00:21:07,220 --> 00:21:12,350 conventional with the density times, the volume is the number of particles. 207 00:21:12,950 --> 00:21:18,259 So when I multiply this factor by write it out here. 208 00:21:18,260 --> 00:21:25,820 So I multiply three halves and over e f here by the the v and the small n give you a big n 209 00:21:26,780 --> 00:21:40,340 and we'll get this gamma twiddle factor times three halves big n kb times cavity over e f. 210 00:21:41,540 --> 00:21:43,820 Okay. So that's our final result. 211 00:21:45,170 --> 00:21:54,770 And actually, if you did the calculation really honestly, you would discover that gamma tweedle is is pi squared over three. 212 00:21:55,280 --> 00:21:58,249 And I'm sure you're not going to be held responsible for knowing that. 213 00:21:58,250 --> 00:22:04,130 But just for our general edification, that's what the number actually happens to be. 214 00:22:04,340 --> 00:22:07,700 If you really want to see how to do this calculation, you can go back to your stat, 215 00:22:07,700 --> 00:22:13,069 make notes from last year or you can, you know, it's a lot of a lot of books and so forth. 216 00:22:13,070 --> 00:22:17,570 And I hate it when people say, you know, it's in a lot of books, but trust me, you know, the algebra doesn't really teach you much. 217 00:22:17,840 --> 00:22:21,350 If you really want to see how it's done, you can you can work through it. 218 00:22:21,350 --> 00:22:27,830 Sommerfeld did work through it. He got the right answer. But we we're not going to be held responsible for it. 219 00:22:28,100 --> 00:22:33,530 Anyway, a couple of important things about this result. First of all, it is linear in temperature. 220 00:22:33,800 --> 00:22:40,400 And the reason it's linear in temperature comes fundamentally from the fact that only electrons near the Fermi surface can absorb energy. 221 00:22:41,360 --> 00:22:42,800 When temperature goes to zero, 222 00:22:43,070 --> 00:22:48,980 the range of electrons that can get excited near the Fermi surface drops to zero and you lose your heat capacity altogether. 223 00:22:49,970 --> 00:22:56,090 And in fact this is what we wanted experimentally. We wanted to get a heat capacity that is linear in temperature. 224 00:22:56,090 --> 00:23:02,210 So that's the first good thing about this. And another thing that's really nice about this expression is you'll recognise this piece here, 225 00:23:02,510 --> 00:23:11,839 this piece here as the classical result classical results for a monotonic gas three has and KB sounds familiar right and then 226 00:23:11,840 --> 00:23:22,190 is multiplied by this factor here K over F which is what it's tiny it's room temperature 300 kelvin over 80,000 Kelvin. 227 00:23:22,430 --> 00:23:26,000 Tiny, tiny tiny amount. So you have only again coming from the fact that. 228 00:23:26,100 --> 00:23:29,970 Only a few electrons are participating in the heat capacity. 229 00:23:30,480 --> 00:23:38,490 So this result in Sommerfeld went and he said, okay, this is my new prediction for what the heat capacity of a firm gas should be. 230 00:23:38,850 --> 00:23:42,450 And so let's compare it to experiment so well. 231 00:23:42,450 --> 00:23:48,750 Okay, so let's write it as gamma t all of those constants get absorbed into the overall gamma. 232 00:23:49,910 --> 00:23:58,310 And he's going to well, if he if you if you want a real theory of of how much heat capacity a particular metal should have, 233 00:23:58,460 --> 00:24:02,720 you need to know what the density of the electrons in the metal is so you can calculate EAF. 234 00:24:02,960 --> 00:24:08,570 But he did the usual thing and said, okay, let's assume one electron per atom the same way we had done before. 235 00:24:08,900 --> 00:24:19,370 And if you do that, then you write out gamma assay gamma for the experiment divided by gamma from the theory and with the theory, 236 00:24:19,370 --> 00:24:33,770 assuming one electron per atom for lithium, we get 2.3 for sodium, we get 1.3 for potassium, 1.2 copper, 1.5. 237 00:24:34,180 --> 00:24:38,959 This is a pretty good agreement. It's extremely good agreement considering that the classical theory, 238 00:24:38,960 --> 00:24:44,450 what you really had done, the three has NCBI result is too big by a factor of 100. 239 00:24:44,660 --> 00:24:48,950 And all of a sudden, we're getting results, which are really pretty close to the right answer. 240 00:24:49,100 --> 00:24:54,980 And the reason we're getting results that are close to the right answer is because we're treating Fermi statistics more honestly now. 241 00:24:55,280 --> 00:25:00,800 Okay, well, we didn't treat them at all before. And now Sommerfeld said, you put it in the Fermi statistics, you get the heat capacity, right. 242 00:25:01,130 --> 00:25:11,330 Okay. Now he went on and said, in fact, with this newfound understanding of what's going on and Fermi statistics being important, 243 00:25:11,510 --> 00:25:18,370 we can fix some of the problems with due to theory. You know, one of the things that you recall from Judith Theory we call Duda, 244 00:25:20,120 --> 00:25:27,770 one of the things we calculated which we actually got pretty close to, right, was the thermal conductivity thermal con. 245 00:25:29,360 --> 00:25:35,990 We had an expression, this kinetic theory expression capture the thermal conductivity one third density 246 00:25:36,320 --> 00:25:43,010 KV Heat capacity per per electron that it was b squared times tao scattering time. 247 00:25:44,430 --> 00:25:52,919 And drew the in this expression. Drew, they used used these things that we don't like anymore. 248 00:25:52,920 --> 00:26:07,920 The TV is three halves KB the classical result and he also used the classical result for v squared eight kb t kb t over pi m. 249 00:26:09,910 --> 00:26:23,049 So the combination CV times v squared which actually enters in the thermal conductivity, it has the value 12 over a pi and c k. 250 00:26:23,050 --> 00:26:26,570 B. T over m. Right. 251 00:26:27,260 --> 00:26:33,510 Okay. Now, Sommerfeld said both of those results are wrong, so let's use the right results. 252 00:26:33,540 --> 00:26:37,070 So his result was CV is well, 253 00:26:37,110 --> 00:26:49,639 he has the pi squared over three and then three halves can be the classical result and then kb t over e f and then he said, 254 00:26:49,640 --> 00:26:57,740 okay, and this squared is not not given by the classical kinetic theory result, 255 00:26:57,980 --> 00:27:01,879 but should instead be given by v f squared, the Fermi velocity squared. 256 00:27:01,880 --> 00:27:07,130 And this makes things a little bit simpler. We can write f is one half mass times v f squared. 257 00:27:09,270 --> 00:27:20,730 But those together we get some cancellation. Again, if we look at the factor CV times v squared, we get pi squared times kb t over m. 258 00:27:21,690 --> 00:27:27,050 So in fact, what we discover, it's not too far from the to projection. 259 00:27:27,390 --> 00:27:32,430 In fact, we're over Sommerfeld. Sommerfeld. 260 00:27:35,530 --> 00:27:40,870 Is is what? It's 12 over five cubed or something. It's about a half. 261 00:27:41,560 --> 00:27:45,010 So it gets pretty close to the same result. And I get the right pulverised cubed. 262 00:27:45,040 --> 00:27:45,760 Yeah, I think so. 263 00:27:47,650 --> 00:27:54,370 So it's pretty you get pretty close to the same result, which is good because we like the answer for the thermal conductivity that we got in. 264 00:27:54,370 --> 00:27:58,840 Due to theory, it satisfies this vitamin France law that was known to exist experimentally. 265 00:27:59,170 --> 00:28:02,920 And so we didn't want to ruin that. And in fact, Sommerfeld theory doesn't ruin it. 266 00:28:03,280 --> 00:28:06,850 Two mistakes, the heat capacity and the velocity. 267 00:28:07,150 --> 00:28:10,820 Cancel each other out. Exactly. Well, not exactly, but pretty close to. 268 00:28:10,840 --> 00:28:16,810 Exactly. But. There were other things that Drew Taggart completely wrong. 269 00:28:17,050 --> 00:28:24,310 One of them was the Peltier coefficient. Peltier, which let's see, what was Peltier coefficient. 270 00:28:24,610 --> 00:28:27,850 I think. Think I left it over here. Yeah, okay. 271 00:28:28,060 --> 00:28:41,110 It was a coefficient. Pi is kd times t over three times minus E and in due to theory, this came out 100 times too big, more or less. 272 00:28:41,350 --> 00:28:47,290 But now if we plug in the new value of V, which is 100 times smaller, where all of a sudden getting things, they start to look right. 273 00:28:47,460 --> 00:28:54,870 Okay, so this is this is good. So Sommerfeld was pretty happy with this result. 274 00:28:55,260 --> 00:28:59,610 But unfortunately, as with all as with all things, he introduced a new problem. 275 00:29:02,300 --> 00:29:09,410 What's the new problem? Well, okay, let's go back and remember the conductivity expression any squared tao over m. 276 00:29:10,600 --> 00:29:11,950 That's still the same result. 277 00:29:12,250 --> 00:29:20,049 In in Sommerfeld theory give you the same prediction and better be the same prediction because we have the ratio of, you know, 278 00:29:20,050 --> 00:29:28,710 we wanted to get the right ratio of thermal conductivity to electrical conductivity and we didn't change the thermal conductivity much. 279 00:29:28,900 --> 00:29:34,690 So we better not change the electrical conductivity much either so more or less up to maybe a factor of 12 or pi cubed. 280 00:29:34,900 --> 00:29:38,680 We expect that the conductivity should be given by this expression. 281 00:29:39,310 --> 00:29:43,940 Probably this expression is a good one to stick with and we don't we don't know how, 282 00:29:43,990 --> 00:29:52,750 but we can measure sigma measure this measure sigma to get to our get this get Tao. 283 00:29:53,260 --> 00:29:56,829 And then from Tao we can calculate the mean free path. 284 00:29:56,830 --> 00:30:04,680 Lambda mean pass. Scattering lines, which would be the Times Tao. 285 00:30:05,460 --> 00:30:11,400 Now in Sommerfeld theory we would probably replace this with VRF Times Tao because the the electrons 286 00:30:11,400 --> 00:30:17,310 are moving around at speeds close to VF and the problem is that the F is close to the speed of light. 287 00:30:17,970 --> 00:30:24,030 Well, not closely, but it's 1%. The speed of light is extremely fast, which means the mean path is extremely long. 288 00:30:24,150 --> 00:30:28,560 So lambda is huge, unreasonably big. 289 00:30:29,340 --> 00:30:41,400 How big? Well, at room temperature at tea room at tea room room lambda can be say 100 angstroms may not sound enormous, 290 00:30:41,790 --> 00:30:46,350 but at low t lambda can be a millimetre. 291 00:30:47,190 --> 00:30:53,790 And again, that may not sound huge to you, but you have to think about how many things the electron has to go past before it scatters. 292 00:30:54,060 --> 00:31:00,390 Well, every angstrom there is another atom, or every two angstrom there is another atom that the electron could bump into. 293 00:31:00,750 --> 00:31:05,760 So here it goes by 100 of them. Here it goes by a million of them before it bumps into something. 294 00:31:06,030 --> 00:31:10,620 So what can it have bumped into? It could have into the nucleus. It could have bumped into the core electrons. 295 00:31:10,740 --> 00:31:14,550 It could have bumped into the other free electrons in the free electron gas that are running around. 296 00:31:14,700 --> 00:31:18,810 And for some reason it does not bump into any of them. Really, really strange. 297 00:31:19,020 --> 00:31:25,920 And this is something that we're not going to answer until much later in the term when we study band theory of solids. 298 00:31:25,980 --> 00:31:31,410 So for now, the unreasonably long memory path is just a puzzle that we're going to have to deal with. 299 00:31:32,310 --> 00:31:36,720 So this was something that troubled Sommerfeld back then, troubles a lot of people back then. 300 00:31:38,880 --> 00:31:42,300 But, you know, Sommerfeld was brave and he decided what else? 301 00:31:42,300 --> 00:31:47,820 You know, we did pretty well over here, understanding the heat capacity, the thermal conductivity and Peltier coefficient. 302 00:31:48,090 --> 00:31:51,690 What else can we calculate that we might be able to get right? 303 00:31:51,870 --> 00:31:54,150 Not worrying about this particular little problem. 304 00:31:55,170 --> 00:32:00,930 And so now we're going to take a little bit of a out of order detour and discuss a little bit of magnetism. 305 00:32:01,230 --> 00:32:06,389 Now, the last couple of lectures of the year, week seven, are entirely about magnetism. 306 00:32:06,390 --> 00:32:08,100 So this is a little bit out of order, 307 00:32:08,430 --> 00:32:15,780 but I think it fits in here well because it really is based on exactly the same same business that we just we just went through. 308 00:32:16,870 --> 00:32:24,730 And the particular type of magnetism we're going to study is what's known as Pauli power magnetism, 309 00:32:25,360 --> 00:32:32,260 apparent magnetism of free electrons, of free electrons. 310 00:32:35,500 --> 00:32:40,120 Now, Howley, of course, we all know him. He was the exclusion principle guy. 311 00:32:40,360 --> 00:32:44,530 His also people said he was the most arrogant man who ever lived. 312 00:32:44,540 --> 00:32:49,809 You know, he used to tell his students that it was okay for them to make a mistake because he never made a mistake himself. 313 00:32:49,810 --> 00:32:52,810 But it was okay for his students to make a mistake. So he was sorry. 314 00:32:52,810 --> 00:32:59,830 That was typical of him. But he was a very great scientist and he actually did this calculation before Sommerfeld did. 315 00:32:59,830 --> 00:33:07,809 He did his work. So a lot of this maybe should be called Pauli theory, not Sommerfeld theory, at any rate, per magnetism. 316 00:33:07,810 --> 00:33:13,210 What is that? So what one does is one applies a magnetic field to your system and you measure 317 00:33:13,420 --> 00:33:18,700 the magnetisation that comes out the proportionality constant sky over mu not you, 318 00:33:18,700 --> 00:33:26,260 not here is just the usual constant, the permeability permeability and this wrong permeability. 319 00:33:26,440 --> 00:33:30,009 Is that right? Maybe that's right. Anyway, that's a constant. 320 00:33:30,010 --> 00:33:33,520 It shows up on your data sheet, you know, some fundamental constant. 321 00:33:33,850 --> 00:33:37,030 The sky here is the susceptibility is known as susceptibility. 322 00:33:40,450 --> 00:33:45,520 Susceptibility. So this equation defines chi power magnetism. 323 00:33:46,120 --> 00:33:51,010 Power magnetism. Means chi is greater than zero. 324 00:33:51,880 --> 00:33:52,570 So in other words, 325 00:33:52,570 --> 00:34:00,100 you apply a magnetic field to the physical system and it develops a magnetisation in the same direction as the field that you applied. 326 00:34:01,360 --> 00:34:02,590 So how are we going to address this? 327 00:34:02,620 --> 00:34:11,470 Well, first thing we have to do is we have to write a Hamiltonian for our electrons or I p squared over to m the usual kinetic term, 328 00:34:11,890 --> 00:34:17,920 plus a coupling of the electron spins to a magnetic field. 329 00:34:18,280 --> 00:34:21,700 You may have seen this in the atomic physics course last term. 330 00:34:22,510 --> 00:34:34,720 So Sigma here is the power of spin operators, spin up operators, and it has importantly, it has eigenvalues, x plus or minus one half. 331 00:34:36,420 --> 00:34:51,310 B is the magnetic field new here? Is the border magnets on board magnets on a fundamental constant is each bar over two m and numerically, 332 00:34:51,640 --> 00:34:58,990 it is useful to keep in mind that that number is somewhere around a Tesla, a kelvin per Tesla of Energy. 333 00:35:00,790 --> 00:35:09,639 G Here is the g factor. And for electrons, the g factor is typically, well, a free electron out in space. 334 00:35:09,640 --> 00:35:15,250 The Z factor is two, so we're just going to use two. And that's going to prevent us from having to write G and g, 335 00:35:15,250 --> 00:35:21,130 twiddle and get confused because we're already using G for density of states because it's conventional, therefore convenient. 336 00:35:21,580 --> 00:35:30,780 Okay. Anyway. If you're at this point, you might be wondering if you took the relativity course last term. 337 00:35:32,070 --> 00:35:35,070 Why is it that I wrote P-square over to him? 338 00:35:35,310 --> 00:35:43,260 If we have magnetic fields where you might have expected that instead you should have p plus e a squared over to m. 339 00:35:43,800 --> 00:35:48,020 Where A is the vector potential? Does this look familiar from? 340 00:35:48,030 --> 00:35:52,110 Yeah. Okay. So the EAA that shows up in the kinetic energy. 341 00:35:52,350 --> 00:35:57,420 This is the piece that makes the electrons curve. If you leave it out, the electrons don't curve. 342 00:35:57,790 --> 00:36:02,580 Okay, we're going to leave it out. The reason we're leaving out is twofold. 343 00:36:02,760 --> 00:36:05,250 First of all, because it's hard to treat. 344 00:36:05,520 --> 00:36:13,590 It is actually a rather complicated calculation to deal with how much the electrons curve and what that does to the magnetisation. 345 00:36:13,920 --> 00:36:22,170 But the better reason to leave it out is because the influence of this term as a term is actually less than the influence of the coupling to the spin. 346 00:36:22,200 --> 00:36:28,110 The to the spin is actually more important than the coupling to the actual physical motion of the electron. 347 00:36:28,380 --> 00:36:31,470 So we're going to treat this term. We're going to throw out this term. Okay. 348 00:36:31,960 --> 00:36:33,930 If you really want to know the answer, in fact, 349 00:36:34,320 --> 00:36:39,710 what this term will do is it will change the answer by a third in the it's all we'll get if you put this in. 350 00:36:39,720 --> 00:36:41,220 It actually has the opposite sign. 351 00:36:41,220 --> 00:36:47,280 As the result we'll get here and it's only a third is big so will make the final result of two thirds the final result. 352 00:36:47,550 --> 00:36:52,170 It's not important. We're going to ignore this for now. We're going to keep this. All right. 353 00:36:52,500 --> 00:37:00,480 So ignoring the term, just keeping the the spin term here, what we have is that the energy. 354 00:37:02,140 --> 00:37:10,690 For an electron with wave vector k and spin up is r e k not plus movie b, 355 00:37:12,130 --> 00:37:25,720 whereas energy spin down is e cannot minus UVB where e cannot is h bar squared k squared over two m the usual free electron energy. 356 00:37:26,740 --> 00:37:34,420 So the idea is that when you apply a magnetic field, the up spin electrons become more expensive, the down spin electrons become less expensive. 357 00:37:34,630 --> 00:37:37,000 And so what's going to happen is some of the other spin electrons are going to flip 358 00:37:37,000 --> 00:37:39,850 over to try to become down spin electrons because that would lower their energy. 359 00:37:40,330 --> 00:37:46,569 However, they can't all flip over because a lot of the states are already filled and they can't flip over into states that are already filled. 360 00:37:46,570 --> 00:37:49,630 So you're going to get some of them flipping over, but not a lot of them. 361 00:37:49,840 --> 00:37:54,460 Okay. So I'm going to run out of room here pretty quickly. 362 00:37:55,000 --> 00:38:00,580 So let's start with B equals zero. And here the whole calculation will do it A zero. 363 00:38:00,820 --> 00:38:05,590 That's okay because T is much, much less than T, so it's pretty close to two equals zero. 364 00:38:07,000 --> 00:38:12,219 And if we calculated the number of spin up electrons or the density of spin up 365 00:38:12,220 --> 00:38:16,570 electrons as the number of spin up electrons divided by the volume at B equals zero, 366 00:38:16,570 --> 00:38:25,330 it should be the same as the number of spin down electrons up and down to symmetric in that case v and we can write that is integral zero to infinity. 367 00:38:25,960 --> 00:38:32,740 The E actually will cut off the integral at F. 368 00:38:32,740 --> 00:38:40,660 We're going to count only the electrons that are the states that are filled over two and it's put in divided by two. 369 00:38:40,660 --> 00:38:46,300 Because here we're writing expression for only the spin ups or the spin downs, not both of them, of g of E. 370 00:38:46,400 --> 00:38:52,150 The density of states that we calculated was the total density of states of both spin ups and spin downs. 371 00:38:52,420 --> 00:39:02,020 Okay, so let's let's draw here density of states g of E for spin offs here. 372 00:39:02,350 --> 00:39:07,990 This is E, and you'll recall somewhere on the board, I think I scrolled it off the top. 373 00:39:08,260 --> 00:39:14,080 Oh, well, maybe it's over here. Yeah. 374 00:39:14,230 --> 00:39:19,990 Here it is. The GOP is proportional to each of the one half, so this thing looks like a parabola that way. 375 00:39:21,080 --> 00:39:27,470 And then we can also plot gravity for the spin down. 376 00:39:27,510 --> 00:39:34,750 It's going to look exactly the same for the spin downs like this, and they both get filled up. 377 00:39:35,240 --> 00:39:40,820 So the Fermi Energy, if this is in zero magnetic field. 378 00:39:41,660 --> 00:39:51,500 Yes. Now, when we add the magnetic field way up there, the spin up electrons are going to become more expensive. 379 00:39:51,770 --> 00:39:58,730 These guys are going to get shifted up in energy by UVB and the spin down electrons are going to become less expensive. 380 00:39:59,150 --> 00:40:04,550 So you're going to get shifted down in energy by maybe actually maybe I should raise some things here. 381 00:40:07,100 --> 00:40:12,830 So some of the spin ups are going to get going to want to turn over, has to become spin downs to lower the energy. 382 00:40:12,840 --> 00:40:14,870 So let's let's draw that. 383 00:40:15,260 --> 00:40:24,950 What happens we do this so once we add so this is for B equals zero in this picture B equals zero over here let's try to draw a B, 384 00:40:24,950 --> 00:40:30,379 not equal to zero over here. So here we have energy. 385 00:40:30,380 --> 00:40:37,430 Here we have G for the spin ups of E, and here we have G for the spin down of E. 386 00:40:40,490 --> 00:40:50,330 And what happens is that these guys got shifted up in energy this way, this distance here is movie times B and these guys got shifted down. 387 00:40:51,810 --> 00:40:55,700 Like this. Buy this much movie. Okay. 388 00:40:56,340 --> 00:41:00,610 And then we fill them both up to F. Here. 389 00:41:00,620 --> 00:41:11,120 This is the F, this is F. So you see that some of the ones that were spin up, these guys here emptied out and filled these states here. 390 00:41:13,870 --> 00:41:18,820 Is that clear how that happened? So these dates here got pushed up in energy. 391 00:41:20,790 --> 00:41:24,300 Above the Fermi surface. So they emptied out whereas these states here. 392 00:41:25,420 --> 00:41:28,850 Got pulled down in energy, so they filled up. Okay. 393 00:41:29,590 --> 00:41:33,220 Is that clear? I know this. This this gets a little bit confusing. 394 00:41:35,380 --> 00:41:46,780 Bear with me. So if we want to write an expression for the spin up density, we can write integral from zero to F. 395 00:41:48,550 --> 00:41:54,930 Minus UVB. Because actually we only want to integrate up to here. 396 00:41:56,260 --> 00:42:02,200 Because in financing only this region, not this region here, because once we had magnetic field, it's only this region here. 397 00:42:02,200 --> 00:42:10,270 It's only the smaller region here that's filled up. These guys have now emptied D of g of E, I guess over to. 398 00:42:10,900 --> 00:42:20,740 Whereas the number of spin downs is integral from zero to F plus maybe the g of e over to. 399 00:42:22,120 --> 00:42:26,950 Because we want to integrate all the we have to here since those have been pulled down in energy that clear. 400 00:42:27,950 --> 00:42:39,350 All right, so we're almost there. So now what I want is I want to calculate and down, minus and up, which is integral D from F, 401 00:42:39,980 --> 00:42:50,570 minus B, B to F plus me b, b of G of E over two, and then we can G, 402 00:42:50,570 --> 00:42:53,300 since this is all over only a little small sliver of energy, 403 00:42:53,510 --> 00:43:01,850 we can use the rectangular rule to calculate that integral and we get one half g of f times two. 404 00:43:02,210 --> 00:43:18,410 Newby cancel the two is we want to write out g of F times movie B and then finally the magnetisation, which is what we're looking for. 405 00:43:18,440 --> 00:43:23,690 The Magnetisation is new B, each electron has a magnetisation of one. 406 00:43:24,320 --> 00:43:30,290 It has a magnetic moment of one more magnets on. And then we have the density of spin downs minus the density of spin offs. 407 00:43:30,590 --> 00:43:34,250 We'll give you the magnetisation and if you want to know where the sine comes out this way, 408 00:43:34,400 --> 00:43:39,020 why it's down minus it's not up to minus downs is because the charge in the electron is negative, 409 00:43:39,290 --> 00:43:45,230 meaning the spin of the electron actually points opposite its magnetisation, which is pretty confusing, 410 00:43:46,040 --> 00:43:50,420 but that's what we have to deal with because the sign of the electron is is negative. 411 00:43:50,900 --> 00:44:01,220 Okay, so we plug in and up minus n down. So we'll get a G, E, F, B squared times the magnetic field. 412 00:44:02,150 --> 00:44:08,180 And you recall the definition of susceptibility chi over mu, not magnetic field. 413 00:44:08,180 --> 00:44:17,480 So we identify the susceptibility, the pauli paramagnetic susceptibility is mu b squared mu not G at F. 414 00:44:17,690 --> 00:44:31,009 And as a final result, again, we can take this result, compare it to the theory theory our chi experiment and do it for various different things. 415 00:44:31,010 --> 00:44:36,200 For lithium, that ratio is about 2.5. For sodium, it's 1.8. 416 00:44:36,620 --> 00:44:40,580 For potassium is 1.6, which is pretty good agreement. 417 00:44:40,790 --> 00:44:45,620 Now, if we were thinking about classical particles, classical atomic gas. 418 00:44:47,860 --> 00:44:53,320 Then in fact, when you apply the magnetic field, nothing would stop them all from flipping over. 419 00:44:53,410 --> 00:44:58,180 They would just go right into their, you know, the the flipped down state because that would be lower energy. 420 00:44:58,510 --> 00:45:02,319 And the only reason they don't all flip over is because of Fermi statistics. 421 00:45:02,320 --> 00:45:06,220 Fermi statistics prevents them all from flipping over because some of the states are already filled. 422 00:45:06,340 --> 00:45:12,850 And so you get a finite susceptibility, whereas the classical calculation would would predict an infinite susceptibility. 423 00:45:12,860 --> 00:45:18,220 So this is pretty good. We're getting results that are within a factor of two or three of the actual experiments. 424 00:45:18,820 --> 00:45:23,050 So this is a more or less all we have to say about Sommerfeld theory. 425 00:45:23,350 --> 00:45:36,510 So summarise a couple of things. Our successes, successes of free electron free electron theory theory, which means drew up by Summerfield. 426 00:45:37,150 --> 00:45:46,640 Duda plus Sommerfeld. Well, okay. 427 00:45:46,660 --> 00:45:51,460 A couple of successes. We got the heat capacity, right? We got the conductivity good. 428 00:45:51,760 --> 00:45:55,060 The thermal conductivity good. The ratio of those to were good. 429 00:45:55,390 --> 00:45:58,300 The Peltier coefficient is in the right ballpark, the susceptibility. 430 00:45:58,450 --> 00:46:02,140 And there's actually many, many other things that you can calculate that you'll get right. 431 00:46:03,340 --> 00:46:06,730 So the free electron picture is actually pretty good. 432 00:46:07,300 --> 00:46:12,940 However, there are still some problems that we're going to have to deal with. 433 00:46:13,990 --> 00:46:18,640 One is that, as I mentioned earlier in this lecture, Lambda seems too big. 434 00:46:20,230 --> 00:46:27,160 Too big. That's a big problem. Having a mean free path of a millimetre just seems completely unreasonable. 435 00:46:27,700 --> 00:46:31,540 We used that the density of electrons should be one electron per atom. 436 00:46:32,620 --> 00:46:34,570 Then we had some intuition why we should do that? 437 00:46:34,900 --> 00:46:40,230 Because, well, you know, there's a bunch of the electrons are bound in core orbitals and maybe they didn't count. 438 00:46:40,240 --> 00:46:47,200 They don't run around, they just stay fixed. But there are some other atoms, you know, that work perfectly well for sodium and potassium. 439 00:46:47,410 --> 00:46:54,670 But what about for carbon of carbon? Mixed diamond. It has four electrons in its valence, orbitals for electrons in its outermost shell. 440 00:46:54,760 --> 00:46:58,600 And in fact, it's an insulator. There are no electrons running around free, so why not? 441 00:46:58,840 --> 00:47:03,610 What's going on there? Why do we count one electron per atom? Why do you not count some electrons at all? 442 00:47:03,640 --> 00:47:07,660 So what happened to those other electrons? What about the sign of the hall effect? 443 00:47:08,680 --> 00:47:14,499 Sign of our h our hall? The hall coefficient is always supposed to have the same sign and due to theory. 444 00:47:14,500 --> 00:47:18,670 Also in Sommerfeld there is always the same sign. And we measured experimentally. 445 00:47:18,970 --> 00:47:24,130 Well, we didn't measure, but it was measured experimentally that in fact the sign comes out wrong sometimes. 446 00:47:24,880 --> 00:47:28,750 Another thing that's kind of interesting I didn't mention before optical properties. 447 00:47:32,160 --> 00:47:35,430 Are different, maybe y different. 448 00:47:36,630 --> 00:47:41,550 Y different. What I mean by this is that different metals look different. 449 00:47:41,910 --> 00:47:46,570 So, you know, gold looks kind of gold is silver. This kind of silver is copper the kind of copper it is. 450 00:47:46,710 --> 00:47:50,910 Hence the name of a tin looks kind of, you know, lighter and then looks kind of darker. 451 00:47:51,060 --> 00:47:54,630 They look differently. They have different optical properties. They reflect different colours. 452 00:47:55,020 --> 00:47:58,470 Why is that? In the free electron theory, they should all look exactly the same. 453 00:47:59,340 --> 00:48:02,550 Another thing we didn't address is magnetism. 454 00:48:05,370 --> 00:48:17,090 Magnetism in particular things like iron are ferromagnetic coming from the name for iron that you can have magnetisation not equal to zero. 455 00:48:17,370 --> 00:48:23,099 Even when even when b equals zero. You probably study this in and you're like your magnetism. 456 00:48:23,100 --> 00:48:26,910 Quite. Why is that? Real picture would never get that. 457 00:48:27,490 --> 00:48:30,630 And finally, what about Coulomb interactions? 458 00:48:31,110 --> 00:48:44,530 Coulomb interactions? When we think about the electrons running around in the solid, you know, they have a huge Fermi energy, you know, 80,000 Kelvin. 459 00:48:44,680 --> 00:48:47,629 But if you think about the the energy scale of the Coulomb interactions, 460 00:48:47,630 --> 00:48:54,160 it's just as big the interaction of the electron with the nucleus nuclei close by, also 80,000 kelvin, also a huge number. 461 00:48:54,370 --> 00:49:00,009 The interaction of the electron with other electrons that are running by it, also 80,000 Kelvin, we threw it out completely. 462 00:49:00,010 --> 00:49:04,780 We just treated these electrons as if they are free gas, no interactions whatsoever. 463 00:49:04,780 --> 00:49:08,979 We only treated the fact they have Fermi statistics to a large extent. 464 00:49:08,980 --> 00:49:14,680 All of these problems come from the same thing. They were neglecting the same thing over and over and over again. 465 00:49:14,950 --> 00:49:19,139 And we are going to have to take more seriously one particular item. 466 00:49:19,140 --> 00:49:25,540 And that one particular item is that materials have microscopic structures, detailed microscopic structures. 467 00:49:25,540 --> 00:49:31,749 Atoms are stuck together in a particular way, frequently a particular periodic way, and that completely changes their properties. 468 00:49:31,750 --> 00:49:34,870 And that's what we're going to have to deal with for much of the remainder of the term. 469 00:49:35,140 --> 00:49:44,800 All of these things to a large extent will be sorted out once we deal honestly with the fact that the atoms are arranged in some particular way. 470 00:49:45,010 --> 00:49:50,530 The first thing we're going to have to do is we're going to have to understand why it is that the atoms actually stick together to begin with. 471 00:49:50,710 --> 00:49:55,900 And so starting the next lecture, we're going to discuss a bit of chemistry and chemical bonding. 472 00:49:56,080 --> 00:50:00,010 Just a really quick thing. How many people had A-level chemistry? 473 00:50:01,590 --> 00:50:03,120 Oh, that's pretty good. How many did not? 474 00:50:04,000 --> 00:50:09,809 Okay, you're the lucky ones, because you're going to have to unlearn a lot of the things that, you know, maybe not like. 475 00:50:09,810 --> 00:50:12,000 It's always good to learn. Learn things, even chemistry. 476 00:50:13,050 --> 00:50:20,010 But but you may have to unlearn some of the things that you learned previously, because we're going to look at it from a more physics perspective. 477 00:50:20,340 --> 00:50:24,720 And I will see you Thursday, Thursday, Thursday, Thursday.