1 00:00:00,540 --> 00:00:06,020 This is the second lecture of the condensed matter. Of course. Welcome back. Three quick things before we start. 2 00:00:06,030 --> 00:00:10,410 A bunch of people, after the first lecture asked me what's the title of the book so they can find it in the bookstore? 3 00:00:10,620 --> 00:00:17,550 It's the exercise basics. Now you can find it. The second thing I want to encourage you to use the message board. 4 00:00:17,610 --> 00:00:20,610 No one's used it yet. Someone be brave. Post the first message. 5 00:00:20,610 --> 00:00:25,260 Once you get going, I promise you'll find it useful. The third thing I didn't emphasise this enough last time. 6 00:00:25,410 --> 00:00:32,879 If there's any problems in the course, any sort of error, typos in the homework sets, typos in the book, anything I say that seems wrong in lecture, 7 00:00:32,880 --> 00:00:39,000 please come to me immediately and I can issue corrections and therefore not get everyone is confused as much as we can. 8 00:00:39,240 --> 00:00:42,600 So just, you know, either post a message on the message boards and the email, 9 00:00:42,630 --> 00:00:46,410 catch me off to lecture or you know, I know it's crazy, but you can call me maybe. 10 00:00:50,910 --> 00:00:55,330 Yeah. Anyway, so last time we left off, we were talking about high capacity. 11 00:00:59,660 --> 00:01:07,370 Of solids. And we started with the Boltzmann picture of a solid and Boltzmann picture. 12 00:01:09,290 --> 00:01:15,620 Boltzmann picture was that an atom in a solid should be thought of as sitting in the bottom of a harmonic oscillator oscillating back and forth. 13 00:01:15,830 --> 00:01:20,270 And from that picture, he was able to derive the law of do long petite. 14 00:01:20,540 --> 00:01:24,380 He capacity per atom is three KB. It's a pretty good result. 15 00:01:24,560 --> 00:01:28,400 Except it didn't agree at low temperature where the heat capacity of solids drops. 16 00:01:28,790 --> 00:01:33,350 Einstein figured out what was going on. He added Quantum mechanics. 17 00:01:33,740 --> 00:01:39,170 And due to quantum mechanics, the energy levels of the harmonic oscillator get quantised. 18 00:01:39,350 --> 00:01:44,170 And when the temperature drops below the quantisation energy, the oscillator gets frozen. 19 00:01:44,190 --> 00:01:55,910 His ground state and the heat capacity drops c drops at low temperature drops at low T and his prediction was it should drop exponentially. 20 00:01:56,720 --> 00:02:03,670 Exponentially. His theory of heat capacity fit the data pretty well. 21 00:02:03,680 --> 00:02:07,310 And indeed, it predicted this drop of heat capacity at low temperature. 22 00:02:07,550 --> 00:02:10,430 But unfortunately, experiment said experiment. 23 00:02:12,880 --> 00:02:18,970 Said that at low temperature the heat capacity should be proportional to t cubed, not dropping exponentially. 24 00:02:19,690 --> 00:02:23,500 So that brings us to the subject of today, which is the Devi theory. 25 00:02:24,130 --> 00:02:33,700 If I was 1912 and that in tuition was that you cannot think of a solid as a bunch of atoms in the body of a quantum harmonic. 26 00:02:33,700 --> 00:02:38,490 Well, because when one atom moves, it does get pushed back to its original position by its neighbour. 27 00:02:38,680 --> 00:02:42,520 But in the process it pushes its neighbour and then its neighbours versus its neighbour and so forth and so on. 28 00:02:42,880 --> 00:02:56,200 So in fact what we should be thinking about is the vibration in a solid is actually a wave, and in particular vibrational waves are sound. 29 00:02:57,880 --> 00:03:04,180 Now something was already known about how waves behave under the influence of quantum mechanics. 30 00:03:04,180 --> 00:03:08,979 All the way back in 1899, Max Planck had done this blackbody radiation calculation, 31 00:03:08,980 --> 00:03:12,460 and when when Max Planck did this calculation, he actually had no idea what he was doing. 32 00:03:12,640 --> 00:03:18,129 He didn't even like what he was doing. He said it was an act of complete despair, never really liked the calculation very much. 33 00:03:18,130 --> 00:03:22,630 But 13 years later, people started to understand little bits and pieces about quantum mechanics, 34 00:03:22,870 --> 00:03:26,860 and Planck's original calculations seem to make a lot more sense. 35 00:03:27,160 --> 00:03:30,970 So Debye thought that what we should do is we should quantised the sound waves. 36 00:03:31,270 --> 00:03:43,729 Just like light. Like light. The general idea is, you know, this is in modern language, you know, plankton understand this. 37 00:03:43,730 --> 00:03:46,940 But but we understand this now is from Einstein's work. 38 00:03:46,940 --> 00:03:58,700 We know that the energy of a given oscillator would be h bar omega times, the bonus factor beta h bar omega plus 0.2 energy plus one half. 39 00:03:59,690 --> 00:04:01,870 And so if we have that, 40 00:04:01,880 --> 00:04:11,900 the picture that we interpret Planck as now and with Debye understood is that each wave mode in a box intimately how you put your material in the box, 41 00:04:11,900 --> 00:04:15,680 each wave more in that box should be treated as an oscillator. 42 00:04:15,920 --> 00:04:25,219 So the total energy total inside your box is the sum overall wave modes of H bar omega 43 00:04:25,220 --> 00:04:33,800 for that mode times lowest factor beta h bar omega for the mode mode plus one half. 44 00:04:34,370 --> 00:04:37,190 And all you have to do is sum over all the modes in the box. 45 00:04:37,370 --> 00:04:43,699 Now, the reason this is going to give you something different from what Einstein had was because whenever you have a bunch of modes in the box, 46 00:04:43,700 --> 00:04:48,500 some of those modes are going to be low temperature, low energy modes or low frequency modes. 47 00:04:48,800 --> 00:04:56,720 So in Einstein's calculation, when the temperature dropped below the frequency of the oscillator, then the heat capacity dropped. 48 00:04:56,960 --> 00:05:00,740 But the idea here is we have a whole distribution of frequencies in the modes in a box. 49 00:05:00,950 --> 00:05:08,510 And so as we lower that the temperature, there will always be some modes of low enough frequency that will still have some heat capacity left. 50 00:05:08,720 --> 00:05:13,170 So the heat capacity won't drop exponentially as we go to low temperature. 51 00:05:13,190 --> 00:05:14,270 So that's the intuition. 52 00:05:15,020 --> 00:05:24,100 Now, there's going to be some differences between what plonk did and what Dubai did, because light is obviously different from sound in many ways. 53 00:05:24,110 --> 00:05:30,380 One way is that light is, of course, a lot faster than sound, but that's only sort of a quantitative difference. 54 00:05:30,690 --> 00:05:36,440 There's some more qualitative differences that we're going to have to deal with, and one in particular, light versus sound. 55 00:05:39,380 --> 00:05:48,700 Is that light has to polarisations polarisations and sound has three polarisations. 56 00:05:51,190 --> 00:05:55,540 Now you'll remember from your ENM that if a light wave is, 57 00:05:55,540 --> 00:06:03,249 or an electromagnetic wave of any sort is going in the direction the electric field has to be in the Y direction or the Z direction it can't be in. 58 00:06:03,250 --> 00:06:11,090 The extraction sound is not like that. If a sound wave is going the x direction, the polarisation of the sound wave can be in any direction. 59 00:06:11,290 --> 00:06:17,020 So it's probably easiest to explain this with a movie. So let me show you the movie of what I mean by this. 60 00:06:17,590 --> 00:06:20,050 This is what's known as a longitudinal wave, 61 00:06:20,920 --> 00:06:27,070 which means that the atoms are moving back and forth in the same direction that the wave is actually going. 62 00:06:27,280 --> 00:06:32,320 So the wave is going the X direction and the atoms are moving back and forth in the x direction. 63 00:06:32,770 --> 00:06:35,950 So this doesn't occur for light. For a light. 64 00:06:36,220 --> 00:06:41,380 If the wave is going, the extraction, the electromagnetic field is always pointing in the Y direction of the Z direction. 65 00:06:41,710 --> 00:06:48,910 More similar to light is what's known as a transverse wave, where the light is, where the wave is going in the x direction. 66 00:06:49,060 --> 00:06:54,170 But the atoms are actually moving back and forth in the Y direction or the Z direction. 67 00:06:54,190 --> 00:06:57,489 Okay. So with sound you can actually have three polarisations. 68 00:06:57,490 --> 00:07:03,340 The atoms can be moving in any direction, whereas in light the electric field has a point in only two possible directions. 69 00:07:03,370 --> 00:07:06,190 Is that clear? Yeah. Okay, good. 70 00:07:06,760 --> 00:07:15,450 So we're going to have to keep track of the fact that we have three polarisations for light and only two for the other way around. 71 00:07:15,460 --> 00:07:17,890 Three polarisation for sound and only two for light. 72 00:07:18,670 --> 00:07:24,670 There's a couple of things we're going to also do, which are going to be approximations, which are going to make our life simpler. 73 00:07:24,940 --> 00:07:30,520 One is that we're going to assume the velocity of sound. 74 00:07:30,610 --> 00:07:36,490 The sound is independent of polarisation and depth of Paul. 75 00:07:38,380 --> 00:07:44,410 This is not actually true in reality. Almost always the transverse mode is slower. 76 00:07:44,410 --> 00:07:52,120 It has a lower velocity than the longitudinal mode, but it's actually not so much more complicated to treat this properly. 77 00:07:52,120 --> 00:07:57,730 But you don't learn a whole lot more from doing it more properly. I think there's an exercise in the book that asks you to do it more properly, 78 00:07:58,060 --> 00:08:04,900 but we're just going to assume all sound waves have the same same velocity independent of whether they're longitudinal or transverse. 79 00:08:05,830 --> 00:08:08,920 As long as we're assuming things which are not really correct, 80 00:08:09,250 --> 00:08:16,870 we might as well assume the sound is independent of direction, of direction, which is often not true. 81 00:08:16,870 --> 00:08:23,830 Also, frequently when you have a real solid, the speed of sound depends on which direction the solid you're going. 82 00:08:24,040 --> 00:08:26,410 We'll discuss that a little bit more later on in the term. 83 00:08:26,680 --> 00:08:33,190 But again, it's something that if you treat it more properly, it's not that much harder and you don't learn so much more from doing it. 84 00:08:33,190 --> 00:08:39,250 So we're not going to to do it more properly, but just keep in mind that we are making these approximations. 85 00:08:39,490 --> 00:08:44,590 Now, if you remember back to Planck's calculation, which you did last year, 86 00:08:44,890 --> 00:08:49,030 the first thing you had to worry about was counting all of the modes in a box. 87 00:08:49,030 --> 00:08:54,129 Does that sound familiar? A little bit familiar. We're going to go through that again because it's important and we're going to 88 00:08:54,130 --> 00:08:58,750 use a lot of the a lot of the mechanism is to do many other things this year. 89 00:08:58,750 --> 00:09:10,810 So it's worth going through. So this is a little aside on counting waves in a box, counting waves or modes, I guess modes in box. 90 00:09:12,890 --> 00:09:18,470 So it's time to start with a one dimensional box, one D box as draw the one dimensional box. 91 00:09:18,650 --> 00:09:21,950 Here it is. It has some length l. 92 00:09:22,580 --> 00:09:29,240 We can write down a wave in that box with hard wall boundary conditions and pi x 93 00:09:29,750 --> 00:09:34,400 over l has comes to zero at both ends because of the hardware boundary conditions. 94 00:09:34,580 --> 00:09:39,020 And this is a perfectly good way to write down the waves in the box for every different positive integer. 95 00:09:39,020 --> 00:09:43,250 And you have a different wave mode. That's okay. But that's not not what we're going to do. 96 00:09:43,700 --> 00:09:50,620 What we're going to do is instead we're going to use what's known as periodic boundary conditions. 97 00:09:50,630 --> 00:09:54,260 You may have discussed this last year. Periodic boundary conditions. 98 00:09:54,260 --> 00:09:59,270 Boundaries also known as Bourn von Karman. 99 00:09:59,270 --> 00:10:04,580 Boundary conditions. Bourne von Karman. Karman Karman. 100 00:10:06,050 --> 00:10:11,000 Bourne was one of the Max Bourne is one of the creators of Quantum Mechanics. 101 00:10:11,000 --> 00:10:14,560 And Taylor von Karman was a very important mathematician. 102 00:10:14,570 --> 00:10:18,260 Max Bourne is also very significant because he was Olivia Newton-John grandfather. 103 00:10:18,260 --> 00:10:22,340 She's an important pop star. If you don't know her, she's important for my generation, at any rate. 104 00:10:22,790 --> 00:10:27,919 So anyway, people don't seem sufficiently impressed by that statement. 105 00:10:27,920 --> 00:10:30,080 It's actually it's anyway. Okay. 106 00:10:30,260 --> 00:10:38,360 So the idea of a periodic boundary condition is we're going to take our box of length L and we're going to wrap it up into a circle of circumference. 107 00:10:38,360 --> 00:10:43,610 L And we're going to measure X going around the box and this way like that. 108 00:10:43,910 --> 00:10:50,750 And the reason we're doing this is because once we do this, the waves can take the form of the i k x. 109 00:10:50,990 --> 00:10:57,230 We don't have to work with signs and cosines. If we have a periodic box in a circle, we can use exponentials. 110 00:10:57,500 --> 00:11:04,370 And if you haven't learned already by this time in your career, exponentials are just a lot easier to work with than signs and cosines. 111 00:11:04,700 --> 00:11:06,320 Now when we write a wave, 112 00:11:06,650 --> 00:11:16,850 it expects we have to be a little bit careful because the coordinate X and the coordinate x plus l are actually the same coordinate. 113 00:11:17,000 --> 00:11:21,800 If you go a distance l around the the box, you get back to exactly the same point. 114 00:11:22,190 --> 00:11:28,010 So this wave form must be exactly the same, whether we plug in X or x plus l. 115 00:11:28,280 --> 00:11:39,590 So we better have, you know, if this is going to make sense, we better have either the x equal to either the I k x plus l or otherwise. 116 00:11:40,100 --> 00:11:52,820 Either the excel has to equal one or k has to be two pi over l times an integer n and that integer can be positive or negative or whatever we like. 117 00:11:52,880 --> 00:11:57,140 Okay, everyone happy with that? So far so good. 118 00:11:57,200 --> 00:12:15,200 Okay, so this means that the spacing between allowed between between allowed allowed ks ks is two pi over l l being the circumference of our loop. 119 00:12:15,620 --> 00:12:25,640 And in particular, if we ever have to sum over all the K's, we can replace that sum by an integral decay times, a factor of l over two pi. 120 00:12:26,720 --> 00:12:30,140 Did you go through this argument last year? A little bit, yes. 121 00:12:30,440 --> 00:12:35,270 Well, okay, we're going to use it an awful lot this year. So that's why I'm going through it again, because it's important. 122 00:12:35,780 --> 00:12:43,609 Now, we, of course, don't live in one dimension, so we have to think as multidimensional people. 123 00:12:43,610 --> 00:12:51,470 We have to think in multiple dimensions. So in three D, we're going to take an L by L, by L periodic box. 124 00:12:52,790 --> 00:13:02,929 Periodic box. And at this point, you might start to be a little bit upset because there's no such thing as an l l by l periodic box. 125 00:13:02,930 --> 00:13:08,509 In the real world, you would have to have a box for which if you went a distance l in any direction, 126 00:13:08,510 --> 00:13:12,350 you came back to exactly where you started in three dimensions. You can't build such a thing. 127 00:13:12,350 --> 00:13:20,090 They just don't exist. So why is it we're going to do this? The point is that doesn't actually matter what you do with the boundaries of your system. 128 00:13:20,330 --> 00:13:24,920 For almost any quantity you're actually interested in calculating, such as the heat capacity. 129 00:13:25,070 --> 00:13:30,020 It's something that you can measure locally, just in a small region of the actual physical system. 130 00:13:30,260 --> 00:13:33,800 So it doesn't matter what you would do with the boundaries, which could be very, very far away. 131 00:13:34,130 --> 00:13:37,580 So you can use hard wall boundary conditions, you can use periodic boundary conditions, 132 00:13:37,580 --> 00:13:40,580 you can use any other type of boundary condition which is convenient. 133 00:13:40,790 --> 00:13:45,080 And for us it's convenient to use these periodic boundary conditions and we're going to get away with it. 134 00:13:45,620 --> 00:13:52,250 The reason we want to use these periodic boundary conditions is because then the waves will look like the i k vector dot x vector. 135 00:13:52,730 --> 00:13:56,810 It's an exponential exponential, so easy to work with and that's why we're doing it. 136 00:13:57,050 --> 00:14:08,480 Everyone has still happy with that. Okay, good. So in three dimensions k has to be two pi over at all times integers and x and y and z. 137 00:14:09,020 --> 00:14:13,170 And so if we ever have a sum of. Over K vectors. 138 00:14:13,350 --> 00:14:21,270 We can replace that sum with l over two pi cubed times the integral dk x integral 139 00:14:21,270 --> 00:14:27,509 d.k. y integral d kc or another way that we will most usually write this. 140 00:14:27,510 --> 00:14:34,559 We'll write this as the volume of the system r times the integral d3k over 141 00:14:34,560 --> 00:14:42,240 two pi cubed and we'll see this factor volume integral d3k over two pi cubed, 142 00:14:42,510 --> 00:14:45,570 probably C at 100 times this year before we're done. Okay. 143 00:14:46,320 --> 00:14:52,980 So what we're going to do is we're going to use what we just learned about counting waves in a box to apply to this equation here. 144 00:14:53,490 --> 00:15:03,420 So the sum over modes here and right sum over modes is going to become the sum over all possible k vectors for our modes. 145 00:15:03,810 --> 00:15:09,690 But then times a factor of three and the factor of three is from the three polarisations we can have one longitudinal 146 00:15:09,690 --> 00:15:17,370 or two transverse polarisations and then this sum and we're going to use this law here to convert it into an integral. 147 00:15:17,370 --> 00:15:25,770 So now we have three factor of the volume integral d3k over two pi cubed. 148 00:15:27,000 --> 00:15:35,370 Now, since we've assumed over here that everything is isotropic, in other words, velocities are independent of direction. 149 00:15:35,850 --> 00:15:41,670 We can then take this three dimensional Cartesian integral and turn it into a spherical polar integral. 150 00:15:42,360 --> 00:15:56,130 So we have three times the volume. Let's pull out the two pi cubed here and then we have integral for pi k squared decay from zero to infinity. 151 00:15:56,550 --> 00:16:00,450 As this familiar, the four pi is the integral over the circle directions. 152 00:16:00,690 --> 00:16:03,540 Yes, throughout complete corners. Everyone's happy with that. 153 00:16:04,170 --> 00:16:11,610 Finally, since we're thinking about sound for sound, you probably learn in your fluids course last year or last last term, 154 00:16:12,780 --> 00:16:22,050 frequency is proportional to the wave vector and the proportionality constant is the velocity velocity of the sound. 155 00:16:22,530 --> 00:16:28,590 So let's just change that integral into an integral over frequency instead of an integral of a wave vector. 156 00:16:28,860 --> 00:16:36,929 So that becomes a plot. Some factors here as plotted for pi upstairs we had a two pi cubed downstairs. 157 00:16:36,930 --> 00:16:44,910 I guess we have a velocity cubed downstairs and then we have an integral zero to infinity of omega squared the omega. 158 00:16:45,450 --> 00:16:54,960 And I'm going to write that as just defining a quantity g of omega deal mega zero to infinity. 159 00:16:55,380 --> 00:17:00,240 And this g of omega is just a bunch of those constants. 160 00:17:00,240 --> 00:17:07,920 I'm going to stick together. So I guess I have 1212 pi upstairs and I have two pi cubed downstairs. 161 00:17:08,400 --> 00:17:16,410 I guess I have a velocity cubed downstairs and then I have a volume upstairs and I'm going to write the volume upstairs in sort of a tricky way. 162 00:17:16,710 --> 00:17:21,150 I'm going to write it as the number of atoms divided by the volume downstairs. 163 00:17:21,360 --> 00:17:25,740 That's the density, and then the number of atoms upstairs that's going to be convenient. 164 00:17:26,250 --> 00:17:29,580 And then I need also that omega squared here. 165 00:17:30,480 --> 00:17:36,070 Omega squared. So this quantity here is known as swish. 166 00:17:36,070 --> 00:17:42,700 I put it over here. Do you have a mega is known as deep omega is known as the density of states. 167 00:17:42,910 --> 00:17:46,870 And I think you ran into this last year when you did blackbody radiation. 168 00:17:47,800 --> 00:17:56,740 And the idea of a density of states is that it is basically how many of these modes you have at a given frequency, 169 00:17:57,040 --> 00:18:06,040 in particular g omega d omega equals number of modes with frequency. 170 00:18:08,420 --> 00:18:16,910 Frequency between between omega and omega plus de omega. 171 00:18:18,410 --> 00:18:23,809 So if you want to sum over all the modes in the system, instead of just summing all all the modes, 172 00:18:23,810 --> 00:18:28,220 what you do is you integrate overall frequencies times the number of modes at each frequency. 173 00:18:28,460 --> 00:18:31,880 Okay. Sort of a convenient way to do things. 174 00:18:32,660 --> 00:18:38,540 I'm actually also going to simplify this a little bit further by defining another convenient quantity. 175 00:18:38,930 --> 00:18:47,240 I'm going to rewrite this g of Omega gave Omega as I'm going to write it as pull out the n the number of atoms, 176 00:18:47,540 --> 00:18:52,189 the nine omega squared over omega D cubed. 177 00:18:52,190 --> 00:18:56,420 So I'm defining omega d cubed here to be a bunch of those constants stuck together. 178 00:18:56,450 --> 00:19:06,770 Omega d cubed is six pi squared times the density number over volume times velocity of sound cubed. 179 00:19:07,760 --> 00:19:16,190 This is known as the devi frequency. After Mr. Devi, my frequency, at least for now, 180 00:19:16,190 --> 00:19:21,500 we're just going to think of it as this bunch of constants that we've conveniently defined, so it's easier to write G. 181 00:19:21,780 --> 00:19:26,420 In a moment we'll give it a more important meaning, but for now, it's just those bunch of constants. 182 00:19:26,960 --> 00:19:28,400 Okay, so. 183 00:19:29,900 --> 00:19:38,810 Having done all of that, we can then way up there at the very top of the board, we have the expression for the energy as a sum over all of the modes. 184 00:19:39,080 --> 00:19:44,600 So I can maybe I'll put it over here so I don't have to scroll out off the top of the board. 185 00:19:45,410 --> 00:19:50,390 So we'll now write the energy, the total energy at a given temperature. 186 00:19:50,660 --> 00:19:53,330 Instead of writing it as a sum over all the modes, 187 00:19:53,540 --> 00:20:01,489 I'm going to write it as an integral over frequencies equals integral the omega from zero to infinity g of omega. 188 00:20:01,490 --> 00:20:09,830 That's the sum over all modes by integrating of frequency in the thing that we want to integrate is h bar omega bose factor theta. 189 00:20:09,830 --> 00:20:16,740 H Bar Omega plus one half. Make people happy with that what we're doing? 190 00:20:16,770 --> 00:20:19,860 Yes. Yes, maybe. Yeah. Okay, good. 191 00:20:21,000 --> 00:20:24,330 All right. So this is the expression that we're going to try to evaluate. 192 00:20:25,260 --> 00:20:31,650 And if you're paying really close attention, you'll be upset with me already because it's actually infinite. 193 00:20:31,980 --> 00:20:41,639 This total energy in the box is infinite. And the reason it's infinite is because of this plus one half look at the plus one half multiplies. 194 00:20:41,640 --> 00:20:45,720 It multiplies a factor. Omega g of omega is proportional to omega squared. 195 00:20:45,990 --> 00:20:49,770 So it's the integral of omega cubed from zero to infinity that's infinite. 196 00:20:50,010 --> 00:20:52,800 So that's kind of it. You might think that that's kind of a problem. 197 00:20:53,520 --> 00:20:58,620 Actually, it didn't bother to buy a plant because they didn't know about zero point energy, so they never wrote down this plus one half. 198 00:20:59,040 --> 00:21:05,730 It's not going to bother us either for a number of reasons. The main reason it's not going to bother us is because it gives us an infinite quantity, 199 00:21:06,030 --> 00:21:12,450 but it's an infinite quantity that's independent of temperature. The only place in this expression where temperature occurs is right here. 200 00:21:12,780 --> 00:21:17,580 At the end of the day, we want heat capacity, which is the derivative of the energy with respect to temperature. 201 00:21:17,850 --> 00:21:20,940 So we're going to have to differentiate this thing with respect to temperature. 202 00:21:21,090 --> 00:21:27,360 And the one half is going to go away because it's temperature independent. So it's giving us an infinite but temperature independent contribution. 203 00:21:27,360 --> 00:21:32,580 So we don't care about it. Actually, in a few moments, we're going to see another reason why we're not going to care about that one half. 204 00:21:33,180 --> 00:21:38,070 But for now we'll just realise it's not going to change our expression for the heat capacity. 205 00:21:38,970 --> 00:21:44,430 All right. So now we can plug in the density of states into this equation here. 206 00:21:44,760 --> 00:21:49,110 So let's do that. So we get nine and I'll pull out the bar from over there. 207 00:21:49,440 --> 00:21:53,220 There's omega two by cubed here, cubed downstairs. 208 00:21:53,580 --> 00:21:58,710 And then the thing we have left to integrate integral the omega from zero to infinity, 209 00:21:59,190 --> 00:22:06,090 there's omega cubed upstairs and then the bose factor into the data h bar, omega minus one is left. 210 00:22:06,570 --> 00:22:10,380 That looks a little bit ugly, but we can take this integral. 211 00:22:10,530 --> 00:22:15,150 We can simplify a little bit by writing X equals beta h bar omega, 212 00:22:15,630 --> 00:22:22,830 and then the integral turns into one over beta h bar to the fourth times this 213 00:22:22,830 --> 00:22:29,310 integral integral d x from zero to infinity of x cubed in the x minus one. 214 00:22:30,420 --> 00:22:33,960 And I promise you, no one is ever going to ask you to evaluate this integral. 215 00:22:34,200 --> 00:22:37,980 It's just some number. The number happens to be PI to the fourth over 15. 216 00:22:39,600 --> 00:22:45,630 And if you really want to know where that number comes from you, it's in the appendix of one of the chapters of the book. 217 00:22:46,230 --> 00:22:49,860 And don't you hate it when someone says you can go read the book? But in fact, it's not so important. 218 00:22:49,860 --> 00:22:50,759 It just gives us a number. 219 00:22:50,760 --> 00:22:58,740 The important thing is because it gives us some known number so we can put that number in and get the end result that the energy, 220 00:22:59,070 --> 00:23:13,980 the total energy in the box is nine and kb t to the fourth from here divided by h bar omega Dubai cubed times this factor pi to the fourth over 15. 221 00:23:16,650 --> 00:23:22,110 Okay. Good. And then, of course, we can differentiate this to get the heat capacity. 222 00:23:22,440 --> 00:23:40,880 Heat capacity is 80. And okay, so it's an KB cavity or H bar, omega Dubai cubed and then I guess 12 pi to the fourth over five watts. 223 00:23:41,160 --> 00:23:44,879 So as the heat capacity. And so a couple of things to comment about this. 224 00:23:44,880 --> 00:23:49,860 It makes this exciting. First of all, it's proportional to t cube just like Dubai wanted. 225 00:23:50,070 --> 00:23:54,479 You get the t cubed dependence of the heat capacity as expected. 226 00:23:54,480 --> 00:23:58,920 It shouldn't be surprising that you get this t cubed heat capacity because you remember from 227 00:23:58,920 --> 00:24:03,780 the blackbody radiation calculation you did last year that the energy of of radiation, 228 00:24:03,780 --> 00:24:08,250 the energy of waves in a box that you calculated is proportional to t to the fourth. 229 00:24:08,490 --> 00:24:13,110 You should have expected the energy to be t to the fourth, and then you differentiate it once and you get t cubed. 230 00:24:13,410 --> 00:24:21,600 So this is why by expected that by treating the waves the same way we treated radiation, we would get the t cubed law. 231 00:24:22,110 --> 00:24:28,580 Furthermore, there's something really exciting about this formula here, and that is that there's no free parameters. 232 00:24:28,610 --> 00:24:32,610 Remember when Einstein did his calculation, he had this frequency, this Einstein frequency, 233 00:24:32,610 --> 00:24:36,870 the oscillator frequency, which he didn't know how to come up with this frequency. 234 00:24:36,870 --> 00:24:40,259 He just fit it to the experiment to try to make it all look nice here. 235 00:24:40,260 --> 00:24:47,069 There's no free parameter. Omega debye here is fixed by the density and the sound velocity. 236 00:24:47,070 --> 00:24:54,240 So everything is fixed here. And if you calculate this quantity and you compare it to the low temperature heat capacity of most materials, 237 00:24:54,240 --> 00:25:01,560 it actually actually agrees extremely well. But unfortunately, like, you know, you never get anything for free. 238 00:25:02,580 --> 00:25:05,670 So we still have a problem. Problem. 239 00:25:07,330 --> 00:25:17,240 At high tea. We want the law of do along petite silver and is three KB and we didn't get that. 240 00:25:17,440 --> 00:25:22,990 We got T cubed at all temperatures here and this is where the buy had to actually scratches had a little bit. 241 00:25:23,290 --> 00:25:29,620 And think okay what did I do wrong. Incidentally, if you do this calculation for electromagnetic radiation, 242 00:25:29,770 --> 00:25:34,600 it really is t to the fourth energy all the way up to arbitrarily high temperature. 243 00:25:34,960 --> 00:25:42,460 Whereas in a in a solid we know that the heat capacity, the solid at some high temperature is going to give us just 3kb, it's not going to be T cubed. 244 00:25:44,170 --> 00:25:47,799 So here we have to deviate from Planck's calculation. 245 00:25:47,800 --> 00:25:56,740 We have to do something different from what it was that plotted and Planck and Debye understood that where he was going wrong, 246 00:25:57,040 --> 00:26:00,729 where he was going wrong. He has to think about where this three comes from. 247 00:26:00,730 --> 00:26:06,070 Where is this three from? The three comes from the fact that the atom can move in three directions. 248 00:26:06,340 --> 00:26:12,130 Each atom moves in three possible directions. You can think of it as three degrees of freedom that the atom has. 249 00:26:12,550 --> 00:26:16,690 And how many degrees of freedom did he count in his box? 250 00:26:16,690 --> 00:26:24,460 Well, the total number of modes, number of modes that he counted is integral from zero infinity du omega. 251 00:26:24,940 --> 00:26:30,820 The omega and g of omega goes as omega squared. You integrate that into infinity and that gives you infinity. 252 00:26:31,090 --> 00:26:34,270 So he counted an infinite number of degrees of freedom and he said, Well, wait, 253 00:26:34,310 --> 00:26:39,190 I figured I counted an infinite number of modes in the box, but there's a finite number of degrees of freedom. 254 00:26:39,370 --> 00:26:44,440 Each atom can move in three directions, and there shouldn't be more than that many degrees of freedom in my box. 255 00:26:44,650 --> 00:26:52,030 So somehow I have to fix that problem and make the box have only a fixed number of degrees of freedom, not an infinite number of degrees of freedom. 256 00:26:52,330 --> 00:26:55,870 So how did he do that? Impose a cut-off? 257 00:26:58,350 --> 00:27:03,709 So when he's going to do is he is going to declare some frequency cut off omega, 258 00:27:03,710 --> 00:27:13,040 cut off the omega, the omega, such that there are exactly three and modes in the box. 259 00:27:13,760 --> 00:27:18,590 Now, this is an ad hoc solution and it's kind of a little weird. 260 00:27:18,830 --> 00:27:22,370 So what he's doing is he's saying, we have all these wave modes in a box. 261 00:27:22,370 --> 00:27:28,460 And when you get to some frequency known as the cut-off frequency above that, there are no more wave modes left. 262 00:27:28,550 --> 00:27:32,780 You have no sound moves above this above this frequency anymore. Seems a little strange. 263 00:27:32,780 --> 00:27:35,420 Later in the term we're going to see it's maybe not as strange as we thought, 264 00:27:35,750 --> 00:27:39,649 but at that at that time, maybe it seemed a little odd, but he's going to do it anyway. 265 00:27:39,650 --> 00:27:42,680 So we're going to do this anyway. We're going to follow him and see what happens. 266 00:27:43,430 --> 00:27:47,240 First thing we need to do is we have to figure out what this cut-off frequency is going to be. 267 00:27:47,450 --> 00:27:51,890 So it's going to give us exactly three modes. So let's calculate this thing. 268 00:27:52,670 --> 00:27:56,540 We'll plug in G of Omega. So plug in G of Omega here. 269 00:27:56,540 --> 00:28:06,109 So there's a nine and there's omega two by cubed integral zero to omega cut off omega squared D omega. 270 00:28:06,110 --> 00:28:13,640 So I just plugged in g of omega there and then let's do that integral and we get three and omega cut-off 271 00:28:14,930 --> 00:28:24,200 cubed over omega devi cubed by cubed and we want this to equal three n and in order for that to be true, 272 00:28:24,440 --> 00:28:29,660 we better choose omega cut-off equal to omega two by. 273 00:28:31,760 --> 00:28:39,710 Okay. That's why I happened to choose those particular constants as being omega two by omega two by is the cut-off frequency. 274 00:28:39,860 --> 00:28:47,240 If you cut off your modes at the by frequency only count modes at lower frequencies than the by frequency, you have exactly three and modes. 275 00:28:47,660 --> 00:28:49,250 Okay. All right. 276 00:28:50,150 --> 00:29:01,280 So now we're going to go back to this equation over here and we're going to rewrite the total energy in the box now as the integral counting up modes, 277 00:29:01,280 --> 00:29:12,590 not to infinity now only up to omega by G omega and then the Bose factor, beta age for omega and then plus one half zero point energy. 278 00:29:13,130 --> 00:29:18,590 And you'll notice now, because we've cut off our number of modes, this term no longer diverges. 279 00:29:18,590 --> 00:29:22,550 It's a finite 0.2 energy term, and that makes us a little bit happier. 280 00:29:23,660 --> 00:29:30,049 But again, we can actually ignore it. Drop this because it's temperature independent. 281 00:29:30,050 --> 00:29:34,130 It's a temperature independent, zero point energy. We're going to differentiate the thing anyway. 282 00:29:34,370 --> 00:29:40,099 So we don't really care about it. It's just going to give us some overall constant shift in the energy, which isn't very interesting. 283 00:29:40,100 --> 00:29:46,600 So we're going to drop that anyway. Okay. And then this expression should give us the energy in the box or the heat capacity in the box. 284 00:29:46,610 --> 00:29:51,010 Once you differentiate it at any temperature we choose just by plugging the temperature in here. 285 00:29:51,370 --> 00:29:56,800 But it's useful to look at various limits. So the first limit is low temperature limit. 286 00:29:57,220 --> 00:30:02,680 And by that I mean cavity much less than H bar omega devi. 287 00:30:03,340 --> 00:30:08,560 And in this limit, we get the same result, same result as we had before. 288 00:30:09,460 --> 00:30:12,820 This result here. Exactly. This nothing changes. 289 00:30:12,850 --> 00:30:21,820 Why is that? Well, the reason nothing changes is because at low temperature, this Bose factor vanishes very quickly with frequency. 290 00:30:22,060 --> 00:30:27,040 By the time you're up near the Dubai frequency, the Bose factor is essentially zero anyway. 291 00:30:27,040 --> 00:30:31,480 So it doesn't matter if you cut it off at the Devi frequency, you're twice the device frequency or half the DEVI frequency. 292 00:30:31,780 --> 00:30:34,990 The integration is zero anyway by that time. So you just. 293 00:30:35,020 --> 00:30:39,180 You don't have to worry about the cut-off at all. At low temperature, the cut-off isn't doing anything. 294 00:30:39,190 --> 00:30:42,520 So you still get the t cubed heat capacity, which is what we want. 295 00:30:43,030 --> 00:30:47,620 But at high temperature, at high temperature, we have something different. 296 00:30:48,600 --> 00:30:54,180 Well, okay, so let's look at the Bose factor at high temperature, at high temperature, 297 00:30:54,190 --> 00:30:58,210 beta h bar omega is a small number, so we can expand the exponential. 298 00:30:58,220 --> 00:31:03,370 So we get one plus beta h bar omega plus start minus one. 299 00:31:03,760 --> 00:31:15,280 So the ones cancel, we get one over beta h for omega or we get the bose factor is replaced and bose becomes k, b, t over H for Omega. 300 00:31:16,450 --> 00:31:24,040 So if we then take this Bose factor and plug it into that energy expression, again, dropping the zero point energy, 301 00:31:24,400 --> 00:31:33,790 the energy is now integral zero to sorry to omega two by up to the cut-off the omega due omega than we have H for omega. 302 00:31:33,790 --> 00:31:37,750 And then we have the Bose factor, which is k, b, t over h bar omega, 303 00:31:38,230 --> 00:31:50,260 the bar omegas cancel and we get the energy being given by pull out the K and integral zero to omega by the omega g omega. 304 00:31:50,650 --> 00:31:55,060 And this integral here has been designed to give us exactly three n. 305 00:31:56,660 --> 00:32:09,080 So we get the energy in the box is three and kb t or the heat capacity is three and kb the of two long petite. 306 00:32:09,950 --> 00:32:14,930 So by implementing this cut off the by managed to get the low temperatures. 307 00:32:15,140 --> 00:32:22,250 Heat capacity is t cubed. The high temperature heat capacity is still the law of Duong Petite, so that's pretty good. 308 00:32:22,280 --> 00:32:32,660 Let's look at some actual data. So this is the heat capacity of silver over a broad range of temperature and up at high temperature. 309 00:32:32,870 --> 00:32:40,309 You see, it's converging to the law of do long, petite and at low temperature it's roughly t cubed. 310 00:32:40,310 --> 00:32:44,990 And the debye theory agrees with the experiment extremely, extremely well. 311 00:32:45,170 --> 00:32:48,379 You can see on the same on the same plot, there's the Einstein theory, 312 00:32:48,380 --> 00:32:52,490 which fits pretty good, but not quite as well as to buy, particularly at low temperature. 313 00:32:52,850 --> 00:33:00,110 And but an additional really important improvement from Einstein to Dubai is that Dubai has no free parameters, 314 00:33:00,380 --> 00:33:05,060 that everything is fixed in the Dubai theory by just the velocity of sound, of the density. 315 00:33:05,300 --> 00:33:09,430 So there's no no, you can't muck around, you can't adjust things. It just fits by itself. 316 00:33:09,440 --> 00:33:13,400 So it's a really good result and seems to agree extremely well with the experiment. 317 00:33:13,820 --> 00:33:20,210 But we still have problems, still wrong or still well, things that are still wrong. 318 00:33:20,780 --> 00:33:25,710 One is the cut off is really ad hoc, cut off, ad hoc. 319 00:33:26,300 --> 00:33:31,610 We just sort of made this up. I mean, it was a motivated it was a motivated thing to do. 320 00:33:31,610 --> 00:33:34,790 It was an intelligent thing to do, but it wasn't really justified. 321 00:33:35,270 --> 00:33:45,409 Another thing that's kind of wrong is that we used omega proportional to wave vector the sound law here. 322 00:33:45,410 --> 00:33:53,810 That frequency will be proportional to wave vector, but we use this at high K and that's not true. 323 00:33:53,840 --> 00:34:00,770 Sound is A is a small K, a long wavelength phenomenon when you go to very small wavelengths or very high wave vector. 324 00:34:01,070 --> 00:34:05,780 This is no longer true. So this was sort of a problem that we brushed under the rug. 325 00:34:06,530 --> 00:34:08,960 And incidentally, the the wavelengths we're talking about, 326 00:34:09,110 --> 00:34:13,250 when you get up near the Dubai frequency, the wavelengths are close to the entire atomic spacing. 327 00:34:13,520 --> 00:34:17,299 So we're talking about really, really small wavelengths or really high wave vectors. 328 00:34:17,300 --> 00:34:20,390 And we can't really think about sound in that regime anymore. 329 00:34:20,990 --> 00:34:27,410 So this is sort of a problem we brushed under the rug. Another thing is that Dubai is not exact. 330 00:34:27,830 --> 00:34:32,960 It's not exact for any material, although it's pretty good, as is obvious from that plot. 331 00:34:33,560 --> 00:34:37,490 And third, our fourth metals are different. 332 00:34:38,390 --> 00:34:42,860 Metals are different. Well, for metals we have this. 333 00:34:46,680 --> 00:34:47,370 For metals. 334 00:34:47,370 --> 00:34:58,950 As I mentioned last time, we have low temperature specific heat C proportional to alpha t cubed plus gamma t where alpha is predicted by Dubai. 335 00:34:59,580 --> 00:35:05,010 So if you did the Dubai theory, you would get the coefficient alpha correct, but you wouldn't get gamma at all. 336 00:35:05,220 --> 00:35:09,240 So this is just a big question right now. We don't know where that's where that's coming from. 337 00:35:10,440 --> 00:35:16,530 So we have to figure that out. And in a few lectures time, we'll have a good idea where that's coming from. 338 00:35:16,530 --> 00:35:22,740 But for now, it's it's a bit of mystery. Now, you might say to me at this point, well, wait a second, isn't silver a metal? 339 00:35:23,100 --> 00:35:26,700 Shouldn't I see a linear heat capacity at low temperature? 340 00:35:27,030 --> 00:35:33,390 Indeed, silver is a metal, and you should see a linear heat capacity at low temperature, but you have to look at it pretty hard to see it. 341 00:35:33,720 --> 00:35:42,900 So this is blown up the very, very low temperature regime, sort of 1 to 4 or five Kelvin or something like that, a 1 to 4 Kelvin, I guess. 342 00:35:43,710 --> 00:35:48,570 And what's plotted here is the heat capacity divided by the temperature as a function of temperature squared. 343 00:35:48,930 --> 00:35:55,170 If it was a purity cube law, then this line would be a straight line and it would intersect zero. 344 00:35:55,830 --> 00:36:01,200 Obviously doesn't intersect zero. It intersects a finite intercept, which is, in fact, gamma. 345 00:36:01,530 --> 00:36:06,660 So there's a very, very small term gamma. This term gamma is pretty small, but it's clearly there. 346 00:36:06,840 --> 00:36:10,230 You have to measure pretty carefully to see it. But it's there. Okay. 347 00:36:11,070 --> 00:36:18,140 All right. So at this point, we're sort of done talking about vibrations in solids for a little while. 348 00:36:18,150 --> 00:36:22,590 We'll come back to them later in the term when we do a little bit of a better job trying to understand these things. 349 00:36:23,700 --> 00:36:29,280 But for now, we're putting this aside and we're going to switch gears and start talking about metals, 350 00:36:29,520 --> 00:36:36,419 because metals are obviously different in several ways. This is one way that they're different, but they're different in many other ways as well. 351 00:36:36,420 --> 00:36:41,879 And this is something that was this was known to be known to the ancients, even probably in caveman days. 352 00:36:41,880 --> 00:36:46,170 They they knew that there were some materials they found in the ground that just looked different from other materials. 353 00:36:46,170 --> 00:36:51,059 And by 4000 B.C., people were able to work with certain metals, things like like copper. 354 00:36:51,060 --> 00:36:56,910 And then a couple thousand years later, they were able to work with iron. And each time they were able to command a particular metal and work with it, 355 00:36:57,150 --> 00:37:02,900 they were able to make new things, new devices, new technologies, and really changed the history of humankind. 356 00:37:03,150 --> 00:37:09,060 So, you know, starting with metal ploughs, metal swords, metal armour, metal warfare of all sorts. 357 00:37:09,300 --> 00:37:13,200 Then later on metal machines, you have metal skyscrapers. 358 00:37:13,200 --> 00:37:18,840 You know, the nuclear age was brought in by, you know, heavy metals and heavy metal music, very important also. 359 00:37:20,370 --> 00:37:25,710 But metals, you know, the history of metals in some way traces the history of of mankind. 360 00:37:25,980 --> 00:37:27,059 And, you know, 361 00:37:27,060 --> 00:37:35,310 it wasn't until really the late 1800s that anyone had the remotest idea what causes metals to be different and really well into the 1900s. 362 00:37:35,310 --> 00:37:39,090 Before we really understood the properties of metals, you have to remember, you know, 363 00:37:39,090 --> 00:37:45,510 for us the the defining property of a metal is going to be that it conducts electricity and non metals don't. 364 00:37:45,780 --> 00:37:49,259 But we didn't even know it what electricity was until the late 1800s. 365 00:37:49,260 --> 00:37:56,370 It was 1897 before J.J. Thompson discovered the electron or what he called the small corpuscle of charge that 366 00:37:56,370 --> 00:38:01,620 can move around freely in the metals and could be ejected out of the metal by a sufficiently high voltage. 367 00:38:03,690 --> 00:38:09,060 And with this picture of the metal really being sort of a container for all these electrons running around, 368 00:38:09,330 --> 00:38:16,260 there became a natural thing for people to do, which was to consider these electrons running around as a gas, a gas of electrons. 369 00:38:16,260 --> 00:38:22,650 And this is what Paul Drew to did. It's known as Judah Theory of Metals. 370 00:38:23,100 --> 00:38:27,690 Of Metals. Well, our junior theory of transport applies to metals, and it's actually particularly good. 371 00:38:27,690 --> 00:38:38,760 It works. This drew the theory. It's a very crude classical kinetic theory, kinetic theory of electrons of electrons, 372 00:38:40,260 --> 00:38:45,360 very much like the kinetic theory of gases that you study last year in your thermal physics course. 373 00:38:46,140 --> 00:38:49,200 It works extremely well, despite the fact that it's very crude. 374 00:38:49,200 --> 00:38:52,770 It works extremely well for a lot of things, particularly well for semiconductors. 375 00:38:53,040 --> 00:39:02,280 And so we're going to attack, you know, electron transport in in metals, using this due to theory first and then we'll improve on it later. 376 00:39:02,700 --> 00:39:08,100 So as with your last year's kinetic theory, we have a couple of assumptions. 377 00:39:09,840 --> 00:39:15,360 Assume in order to get kinetic theory going, one, there is a scattering time. 378 00:39:17,370 --> 00:39:25,349 Scattering time. Tao should look familiar from last year, by which we mean that the probability of scattering, 379 00:39:25,350 --> 00:39:35,810 probability of Scott in time t or in time d t is equal to d t over tao. 380 00:39:36,270 --> 00:39:41,069 Now this probably looks familiar from last year when you did Kinetic Theory of Gases last year. 381 00:39:41,070 --> 00:39:45,030 When did you connect their gases? You can even predict what the scattering time Tao is. 382 00:39:45,410 --> 00:39:48,530 Based on the size of the atoms and how fast they're moving and things like that. 383 00:39:49,100 --> 00:39:52,500 This year we're not going to be so lucky for a couple of reasons. 384 00:39:52,520 --> 00:39:56,450 First of all, it isn't clear what the scattering cross-section of the electrons should be 385 00:39:56,690 --> 00:39:59,870 because the electrons interact with things via long range Coulomb interaction. 386 00:39:59,870 --> 00:40:02,990 So they can they could scatter from things very far away, potentially. 387 00:40:03,260 --> 00:40:06,680 Another thing that's going to make it difficult to figure out with towers is that 388 00:40:06,680 --> 00:40:10,280 the electron can scatter off of lots of other things besides just other electrons. 389 00:40:10,460 --> 00:40:14,960 They can scatter off of protons, it can scatter off of impurities, it can scatter off of anything. 390 00:40:14,960 --> 00:40:21,170 And that happens to be in the metal. So for us, the scattering time Tao is just going to be a phenomenological parameter. 391 00:40:22,190 --> 00:40:34,370 The second thing we're going to assume is after scattering, after a scattering event, after scatter, we will set the final momentum equal to zero. 392 00:40:35,030 --> 00:40:40,339 So imagine something, something moving along. It scatters, and then its final momentum is zero. 393 00:40:40,340 --> 00:40:48,980 Now that's not right. Generally, when something scatters, its final momentum goes off randomly in some random direction, but on average as a vector, 394 00:40:49,280 --> 00:40:54,530 the average of the vector after the scattering is pretty close to zero because it can go off in any possible direction. 395 00:40:54,740 --> 00:40:58,160 And that's going to be good enough for us to be able to make progress. 396 00:40:58,640 --> 00:41:04,190 The third point, which you probably didn't have last year, is that between scattering events, 397 00:41:04,190 --> 00:41:18,020 between scatters, the electron should see C's E and B field if they happen to be there. 398 00:41:18,470 --> 00:41:22,910 So if you're applying an electric field to your metal, the electron will accelerate due to the electric field, 399 00:41:22,910 --> 00:41:28,430 or it will curve due to the magnetic field, which seems rather natural, just like the electron were living in a vacuum. 400 00:41:28,880 --> 00:41:37,550 Okay, so given these three assumptions, we can imagine that we start with an electron that has momentum p at time. 401 00:41:37,550 --> 00:41:44,420 T So t is momentum while well, that's obvious at time t time t. 402 00:41:46,110 --> 00:41:52,009 And then we'd like to calculate what is the momentum that time T plus d t I mean, 403 00:41:52,010 --> 00:41:56,090 in some ways we're asking what's the expectation of the momentum at time t plus. 404 00:41:56,850 --> 00:41:59,660 But we'll treat it as the actual momentum at time t plus t. 405 00:42:01,620 --> 00:42:09,179 Well, there's two things that can happen in between time T and time T plus de t with probably one minus d t over time. 406 00:42:09,180 --> 00:42:16,290 Now, this is the probability. This is probability of not scattering prob of not scattering mass scattering. 407 00:42:17,520 --> 00:42:19,739 If it does not scatter, then what happens? 408 00:42:19,740 --> 00:42:27,970 Well, then it has the original momentum plus whatever force is applied to at times d t and that's just okay. 409 00:42:28,080 --> 00:42:33,900 This is just saying that dpd t if it doesn't scatter, is f Newton's law. 410 00:42:35,890 --> 00:42:43,920 Good. But in addition to this, there's also the probability probability d t over t that it does scatter. 411 00:42:44,130 --> 00:42:49,140 And if it does scatter, we give it momentum zero. Okay. 412 00:42:50,520 --> 00:42:55,200 So this is the probability of not scattering. It accelerates as usual due to the force applied to it. 413 00:42:55,500 --> 00:42:59,340 And if it does scatter, we give it momentum zero at after the scattering. 414 00:42:59,930 --> 00:43:04,260 Okay, then we can do a little bit of rearrangement here. Well, actually, we multiply this out first. 415 00:43:04,560 --> 00:43:13,750 So this is then PV T plus f, d, t minus P over tau. 416 00:43:14,880 --> 00:43:19,570 And then there's de t, and then there's plus order D squared. 417 00:43:21,480 --> 00:43:34,170 And with a little bit of rearrangement, we can write DPD T, which should be p at t plus d t minus t over d t. 418 00:43:37,280 --> 00:43:41,810 Yell at me if I start writing incomprehensibly, you know, if it really gets too painful. 419 00:43:42,290 --> 00:43:51,950 And just doing a little bit of rearrangement on that equation up there and putting together this combination, we get F minus P over Tao. 420 00:43:51,980 --> 00:44:01,740 So let me rewrite this because this is an important equation. DP d t equals force minus t over tau. 421 00:44:02,120 --> 00:44:09,320 This is known as the due to transport equation, you know. And what force are you supposed to use in it? 422 00:44:09,590 --> 00:44:22,220 Well, the force is the usual Lorentz force. Force is a minus e e plus the cross b so whatever force the electron feels goes into that, that equation. 423 00:44:22,670 --> 00:44:25,670 So this looks a lot like Newton's equation. 424 00:44:26,060 --> 00:44:32,150 D equals F is Newton's equation. But we have this additional term on the right hand side, which looks like a drag force. 425 00:44:32,360 --> 00:44:36,499 It's a force going in the opposite direction from its current momentum. 426 00:44:36,500 --> 00:44:40,100 So whichever direction is going, the force is pulling in the opposite direction. 427 00:44:40,100 --> 00:44:52,669 So let's actually do a really quick calculation here. Let's consider consider the case where there's no no electric or magnetic fields in your system. 428 00:44:52,670 --> 00:45:01,070 So you're not applying any electric or magnetic field, then you just have DPI de t is minus p over tau, 429 00:45:02,030 --> 00:45:10,160 which you can solve by saying p t is some p not some initial momentum e to the minus t over tau. 430 00:45:10,790 --> 00:45:16,730 So that tells us that the moment if I have an electron moving along with some initial momentum, 431 00:45:16,970 --> 00:45:24,380 this scattering term here slows it down exponentially to zero momentum with that time scale. 432 00:45:24,530 --> 00:45:25,849 Tau So it's like a drag. 433 00:45:25,850 --> 00:45:33,980 So the idea of the true T theory is that you treat scattering as a drag force that tries to slow everything down or hinder its motion. 434 00:45:34,160 --> 00:45:38,300 And I guess we will stop there and we'll pick up with due to theory next time. I'll see you tomorrow. 435 00:45:39,770 --> 00:45:40,070 Okay?