1 00:00:01,600 --> 00:00:04,780 Okay. Let's get started. Welcome back. 2 00:00:04,990 --> 00:00:08,440 This is the seventh lecture of the condensed matter, of course. 3 00:00:09,160 --> 00:00:11,200 In the last lecture, we introduced a rather important, 4 00:00:11,470 --> 00:00:18,020 simple model of vibrations in one dimension the anatomic harmonic chain, chain of masses and springs. 5 00:00:18,040 --> 00:00:23,980 Every spring is the same. Every mass is the same extremely simple model, but introduce a lot of very important concepts to us. 6 00:00:24,130 --> 00:00:26,350 We solve for all of the normal modes of the chain. 7 00:00:26,530 --> 00:00:36,459 When we quantised the system, we discovered the meaning of phonons with the quantisation of modes where we introduce the concept of crystal momentum. 8 00:00:36,460 --> 00:00:42,730 The idea that if you shift away vector by two pi over a, you get back the same exact physical wave that you started with. 9 00:00:43,000 --> 00:00:46,329 We introduced the concept of a bronze zone arranged in space, 10 00:00:46,330 --> 00:00:51,670 usually taken from minus pi over a two pi over a where all of the wave vectors are physically different waves. 11 00:00:51,670 --> 00:00:55,240 And if you go outside the broken zone, you just start repeating the waves that you have already described. 12 00:00:55,570 --> 00:00:59,590 And we also manage to calculate the heat capacity of this chain. Exactly. 13 00:01:00,340 --> 00:01:04,900 And we compared it to the Devi theory and the Einstein theory that we had looked at previously. 14 00:01:05,140 --> 00:01:12,820 Now, one of the simplifying assumptions we used going into this model was that every single mass and every single spring in this chain were the same. 15 00:01:12,820 --> 00:01:16,720 And in real materials you typically had different types of atoms. 16 00:01:16,900 --> 00:01:21,190 For example, in sodium chloride you'll have a sodium atom and a chlorine atom, and these will be different. 17 00:01:21,460 --> 00:01:24,940 So maybe a better model of a typical material would look something like this. 18 00:01:24,940 --> 00:01:29,709 You'd have a light coloured atom and a dark coloured atom and drawing everything in one dimension again. 19 00:01:29,710 --> 00:01:36,010 And then a light coloured atom and a dark coloured atom and a light coloured atom, dark coloured and so forth. 20 00:01:36,590 --> 00:01:40,420 Now let me introduce some nomenclature that's going to be useful. 21 00:01:40,660 --> 00:01:48,850 The idea of a unit cell is the repeated, repeated motif. 22 00:01:51,250 --> 00:01:58,150 So, for example, we can take our unit cell to be this box. 23 00:01:58,780 --> 00:02:06,340 And if you repeat this box over and over and you stack the boxes together with no space in between, you will reconstruct the entire chain. 24 00:02:07,030 --> 00:02:16,689 Another important word that we introduced last time is lattice constant, which we defined as well. 25 00:02:16,690 --> 00:02:29,110 We usually call it a and we define it as the distance between distance between we in equivalent atoms. 26 00:02:35,420 --> 00:02:44,290 So for example, Gladys Constantine. If you like this, we'll call it a dislike for Adam to dislike called Adam. 27 00:02:44,740 --> 00:02:50,560 You may also notice that that lattice constant also happens to be the size of the nasal. 28 00:02:56,450 --> 00:03:02,520 Right like this this distance here is also a or if I drove properly it was. 29 00:03:02,730 --> 00:03:07,950 Now one side comment is that the the unit cell is actually not unique. 30 00:03:07,950 --> 00:03:18,360 Let me draw another picture of this chain. I could have drawn the initial sell equally well like this. 31 00:03:21,030 --> 00:03:28,290 So that's something we should probably even write down because it's important in itself, not unique. 32 00:03:31,510 --> 00:03:42,070 And if I had chosen this as a unit cell, I could stack duplications of that together and build up the entire chain just as well. 33 00:03:42,340 --> 00:03:42,790 Now, 34 00:03:42,790 --> 00:03:51,880 what we're going to do today in a real calculation is we're going to try to calculate the normal modes of a train of a chain that looks like this. 35 00:03:53,680 --> 00:03:58,390 And the reason we're doing this is not just to add complexity to a problem we already solve last time. 36 00:03:58,690 --> 00:04:04,810 In fact, by having different types of atoms in our chain, we're going to see that some fundamentally new and different things occur. 37 00:04:05,680 --> 00:04:09,910 So that's why it's why we're doing it. So let me draw the chain. 38 00:04:10,270 --> 00:04:19,750 This is known as the alternating chain, frequently known as the altering chain, sometimes known as the diatomic chain. 39 00:04:20,410 --> 00:04:24,310 And it looks kind of like this. There's a dark atom, then a light atom. 40 00:04:24,640 --> 00:04:29,440 Dark atom and a light atom and dark atom and light atom. 41 00:04:29,530 --> 00:04:38,910 And they're connected together with springs. And so forth. 42 00:04:39,750 --> 00:04:45,690 And we have a lattice constant A which is the distance between identical. 43 00:04:45,960 --> 00:04:49,500 We see the difference between light and dark. And to make clear enough, you see that. 44 00:04:49,620 --> 00:04:53,999 Okay, so a is a distance between equivalent atoms. 45 00:04:54,000 --> 00:04:59,549 So between light atom and light atom. And now we have a choice as to what model we want to write down. 46 00:04:59,550 --> 00:05:04,770 And there's two common choices that people make in solving these two simple problems, but they are similar. 47 00:05:05,070 --> 00:05:10,350 One choice is is frequently made is to make the mass of the light atom, the mass of the dark atom different. 48 00:05:10,860 --> 00:05:14,579 We're not going to solve that. You can actually solve that for homework. That one's a typical choice. 49 00:05:14,580 --> 00:05:22,350 But we're going to solve instead is we're going to make the two different springs this spring Kappa one and this Spring Kappa two different. 50 00:05:22,350 --> 00:05:28,020 And we're going to let them alternate back and forth. Kappa one, capita Kappa one and two and so forth. 51 00:05:28,470 --> 00:05:37,020 And hence the name altering chain, because the springs alternate between Kappa One and Kappa to the physics of these two chains. 52 00:05:37,020 --> 00:05:41,730 Whether we're letting the masses alternate or the springs alternate is extremely, extremely similar. 53 00:05:41,910 --> 00:05:48,270 The reason we're doing this one is because it's algebraically just slightly easier to keep track of than the one you going to do for homework, 54 00:05:48,270 --> 00:05:54,770 which is slightly algebraically more complicated. So let's write down some coordinates. 55 00:05:54,780 --> 00:06:04,770 So for example, let's let the light coloured atoms have positions x, n and the dark coloured atoms have position y n. 56 00:06:04,860 --> 00:06:08,400 So for example, that's like this one, the x one. 57 00:06:08,970 --> 00:06:12,750 This one will be y one. It's going to be x2. 58 00:06:13,590 --> 00:06:16,620 This will be y two and so forth. 59 00:06:19,410 --> 00:06:26,100 And now the way we solve problems like this, these mass and spring problems, is we start by writing down Newton's equations of motion. 60 00:06:26,580 --> 00:06:28,690 So, well, okay, 61 00:06:29,020 --> 00:06:41,700 one more definition is the Delta X and delta Y are the deviations from equilibrium emissions from equilibrium position from from equilibrium. 62 00:06:46,760 --> 00:06:51,530 Okay. And we're going to write down Newton's equations in terms of these Delta X and delta Y. 63 00:06:52,130 --> 00:06:57,320 So maybe actually, maybe I'll put it on another board over here. 64 00:06:59,960 --> 00:07:04,280 So mass delta x double dot. 65 00:07:05,000 --> 00:07:08,420 Mass times acceleration equals. We have to be a little bit careful here. 66 00:07:09,170 --> 00:07:15,100 XA atom, a light atom on the right, it has a Kappa two capital. 67 00:07:16,070 --> 00:07:21,890 And then on the right of an x, Adam is a y atom with the same number. 68 00:07:21,980 --> 00:07:25,340 So Delta Y and minus delta x n. 69 00:07:26,330 --> 00:07:37,940 And then on his left is a Kappa one, plus Kappa one, and on its left is a delta y atom with the index one lower right is Delta X. 70 00:07:37,940 --> 00:07:43,219 And does everyone agree with that? That on the left of x two is y one. 71 00:07:43,220 --> 00:07:46,640 So I have two. The capital one spring has the lower index. 72 00:07:46,850 --> 00:07:54,790 Yes. Good, happy, good. And then delta Y and double that. 73 00:07:55,400 --> 00:08:03,500 Okay. So on the right of a Y is an X with the index one higher and that's a Kappa one spring. 74 00:08:04,040 --> 00:08:13,849 So Delta X and plus one minus Delta Y n and on the left, to the left of y one is x one. 75 00:08:13,850 --> 00:08:17,839 So it's an x with the same index, delta, x and minus Delta. 76 00:08:17,840 --> 00:08:20,920 Why? And I think I got that right. Okay. 77 00:08:20,990 --> 00:08:30,740 So those are our equations of motion. And the way we're going to solve this is the same way we did last time. 78 00:08:30,740 --> 00:08:34,060 We will write down a wave on that wave onwards. 79 00:08:35,650 --> 00:08:39,290 So there is some disagreement on what the word Anzacs actually meant in German. 80 00:08:39,620 --> 00:08:44,989 And I actually saw someone stop me after a lecture, a German speaker said, No, no, no, you were right about what that means. 81 00:08:44,990 --> 00:08:49,490 And then someone else said, No, no, no, you're wrong. And then I asked a couple of German speakers and they all disagreed. 82 00:08:49,760 --> 00:08:52,999 And then actually someone pointed out that that is actually an English word. 83 00:08:53,000 --> 00:08:56,809 It is in the OED. You can look it up. It means it means mathematical. 84 00:08:56,810 --> 00:09:01,129 Yes. So so I'm going to say it's in there. 85 00:09:01,130 --> 00:09:06,140 It really is. Okay. So anyway, we are going to write down our wave on that. 86 00:09:06,650 --> 00:09:24,440 So Delta X and is sum and is the I omega t minus i k and a and delta y n is oops I called x rather and a y in the eye omega t minus high k and a. 87 00:09:24,860 --> 00:09:29,990 And what we really mean as last time is we mean to actually take the real part of this whole expression. 88 00:09:30,440 --> 00:09:34,460 And because of that, we're allowed to choose omega a greater than or equal to zero. 89 00:09:34,670 --> 00:09:44,000 But K can be either sign, either sign to represent either left going or right going waves. 90 00:09:44,300 --> 00:09:54,920 Now, as with last time, we realise that if you take K and you shifted to K plans to pi over A, you get back exactly the same wave. 91 00:09:54,920 --> 00:10:04,370 And the reason for that is because if you look at the exponential factor, even minus i k plus two pi over eight times n is. 92 00:10:05,520 --> 00:10:13,890 Times a times, eight times and is equal to exactly the same thing as in the IK and a and a. 93 00:10:14,040 --> 00:10:17,699 So it matches that those two are exactly the same. 94 00:10:17,700 --> 00:10:25,470 So if you shift k by the two pi over a, you get back exactly the same way form rather important fact we found out last time. 95 00:10:26,910 --> 00:10:30,600 Just a couple other things we're going to use periodic boundary conditions, 96 00:10:30,600 --> 00:10:44,580 periodic boundaries because we use a system of length L equals N times A, so this means we have n unit cells in its cells. 97 00:10:45,870 --> 00:10:58,140 And if we have a system of size l that means the case must be on a periodic boundary condition, must be two pi over l times p where p is an integer. 98 00:10:59,730 --> 00:11:06,180 So the spacing between different l different possible values of k are two pi over l. 99 00:11:06,420 --> 00:11:09,780 So now if we want to count the number of different k's. 100 00:11:13,250 --> 00:11:16,570 We did exactly the same calculation last time, different case. 101 00:11:17,330 --> 00:11:22,430 That equals the range of possible case. That's two pi over a before we start repeating, 102 00:11:22,730 --> 00:11:32,060 since you can shift by two pi over a and get the same way back and the spacing between k's is two pi over l that gives us l over a or n. 103 00:11:32,780 --> 00:11:37,580 So the number of different k's is equal to the number of unit cells. 104 00:11:40,660 --> 00:11:44,800 In the system. A rather general rule that we will use. 105 00:11:45,100 --> 00:11:53,860 Many times. Okay. So given our wave on that up there, we can solve this set of Newton's equations. 106 00:11:53,860 --> 00:11:59,110 It's by plugging in and doing a little bit of algebra. 107 00:11:59,110 --> 00:12:03,340 And then I can see that the algebra I'm about to do is a little bit messy, but bear with me. 108 00:12:05,290 --> 00:12:09,130 So here goes. First of all, plug in for Delta x n. 109 00:12:09,550 --> 00:12:22,510 So two derivatives gives me minus omega squared m then the wave is a x in the I omega t minus i k and a and that equals 110 00:12:23,650 --> 00:12:36,700 well k equals kappa two times delta y n which is a y in the i omega t minus i k and a and then let's do the other capital. 111 00:12:36,700 --> 00:12:38,950 That's the other term over here, the other Y term. 112 00:12:39,280 --> 00:12:52,270 So that's plus kappa one e to the I omega t minus i k and then it's and minus one a is an index and minus one and then must be the two x terms. 113 00:12:52,690 --> 00:13:09,820 So we have minus kappa one plus Kappa 2ay here is in die why a x in the i omega t minus i k and a everyone happy with that. 114 00:13:09,830 --> 00:13:13,930 Did I do that correctly? No one reacts, seems to work. 115 00:13:14,410 --> 00:13:25,390 Okay. And then the other equation is minus omega squared and a y in the eye, omega t minus i, k and a equals. 116 00:13:26,170 --> 00:13:29,140 And let's do the two x terms first. 117 00:13:30,070 --> 00:13:44,110 So we have kappa 1ax in the I omega t minus i k and then the coefficient and in the second equation of the index of the x with kappa one is one. 118 00:13:44,110 --> 00:13:57,220 So it's end plus one a here and then we have plus kappa to a x the omega t minus I, k and A and then the two, 119 00:13:57,590 --> 00:14:08,469 then the two the two y terms minus kappa one has kept to a y in the eye, omega t minus K. 120 00:14:08,470 --> 00:14:16,240 And a notable thing about this equation is this the longest equation I'm going to write on the chalkboard all year long, and now it's done. 121 00:14:16,240 --> 00:14:17,860 So we never have to do that again. 122 00:14:18,340 --> 00:14:26,410 But it's not actually kind of complicated as it might look because there's a bunch of exponential factors which drop out the counts on both sides. 123 00:14:26,800 --> 00:14:41,470 So we have two equations and we can simplify it into one matrix equation and write it this way minus omega squared times the vector x, 124 00:14:41,470 --> 00:14:52,150 a y equals some big matrix times the vector x, a y, and the big matrix is what is it? 125 00:14:52,150 --> 00:15:01,210 It's a minus kappa one plus Kappa to and then minus Kappa one plus Kappa two on the diagonal, Kappa one plus two. 126 00:15:01,780 --> 00:15:06,610 And then the off diagonal has Tabata plus Kappa one. 127 00:15:07,300 --> 00:15:18,430 Either the I k it's that part of the exponential didn't cancel it comes from the plus one and then the Kappa two plus Kappa one, the minus a.k.a. 128 00:15:20,280 --> 00:15:25,620 Come coming from the minus one and the exponential. Does everyone good with that agree? 129 00:15:25,650 --> 00:15:34,500 Okay. All right. So what is this? This is an eigenvalue equation or the eigenvalue is this guy over here minus seven mega squared. 130 00:15:34,710 --> 00:15:40,860 And the way you solve an eigenvalue equation is by moving the omega squared to the diagonals over in this side, 131 00:15:41,160 --> 00:15:44,030 then you set the determinant equal to zero. 132 00:15:44,040 --> 00:15:48,900 You get the characteristic determinant or characteristic equation, or sometimes called the secular equation. 133 00:15:49,210 --> 00:15:56,700 We do that the one step and omega squared minus capital one plus capita all squared 134 00:15:57,300 --> 00:16:05,670 minus capital plus capital one in the i k a absolute value squared equals zero. 135 00:16:07,510 --> 00:16:10,570 And we can solve that rather rapidly. 136 00:16:11,470 --> 00:16:20,350 Omega squared equals capital one plus capital plus or minus absolute value capital plus capital one. 137 00:16:21,940 --> 00:16:26,320 Okay. And that's our solution. 138 00:16:26,590 --> 00:16:34,630 Now, one thing you may notice is that for each K, there are two possible solutions of Omega, the plus solution in the minus solution. 139 00:16:35,020 --> 00:16:38,080 Right. So does that mean we'll write that down over here? 140 00:16:38,620 --> 00:16:44,350 Each K has two normal modes. 141 00:16:46,780 --> 00:16:50,530 Modes? We can call them omega plus and minus. 142 00:16:51,010 --> 00:16:59,410 Okay. And how many cases do we have? The number of cases are equal to the number of unit cells, and each K has two times two normal modes. 143 00:16:59,680 --> 00:17:08,230 So number of normal modes. Number of normal modes equals two times number of human cells. 144 00:17:08,980 --> 00:17:13,660 Unit cells equals the number of masses. 145 00:17:14,530 --> 00:17:20,230 And this we should have expected because we had that number of masses as the number of degrees of freedom we have. 146 00:17:20,620 --> 00:17:25,029 So when we solve this problem, we should add exactly that many normal modes. 147 00:17:25,030 --> 00:17:28,080 And indeed we do. So that's that's good. Good. 148 00:17:28,630 --> 00:17:31,790 We're happy with that. All right. Good. 149 00:17:31,810 --> 00:17:38,530 Now, so it is useful to actually plot this thing, and I'm going to sketch it out first and then we'll sort of justify why it looks the way it does. 150 00:17:40,270 --> 00:17:46,150 So herring and fat. Here's K, here's omega. 151 00:17:46,480 --> 00:17:53,770 Okay. Pi over A is minus pi over a, minus pi over a. 152 00:17:53,800 --> 00:17:57,940 So that's the the brown zone outside of the brown zone. It's everything just repeats. 153 00:17:58,360 --> 00:18:05,650 So we should have a mode that looks like this and then a higher frequency mode maybe looks like this. 154 00:18:07,110 --> 00:18:12,360 Okay. And then you could actually reproduce this periodically outside. 155 00:18:12,360 --> 00:18:19,740 I mean, you could you can make it go onwards as you like, because as you shift everything by two pi over a, you just get back the same wave. 156 00:18:19,980 --> 00:18:26,100 But we're going to really focus on the bronze zone because within the bronze zone here, every wave you find is different from every other wave. 157 00:18:27,870 --> 00:18:35,400 Now, it's useful here to probably look a little bit more carefully at this plot and see why it looks the way I've drawn it. 158 00:18:36,120 --> 00:18:41,580 So let us look at some convenient points. Maybe at K equals zero. 159 00:18:42,750 --> 00:18:46,950 That's a good point to look at. Well, what happens with cake zero? 160 00:18:46,950 --> 00:18:58,530 So over here we have this absolute K to plus k one either the a.k.a that goes to just a K to plus k one. 161 00:19:00,570 --> 00:19:08,550 So M omega squared is either two times k one plus k to or zero. 162 00:19:08,970 --> 00:19:13,410 And so I've drawn that here, here zero at cake zero. 163 00:19:13,800 --> 00:19:18,120 And here is the higher end. You want me to draw the higher one also. 164 00:19:18,120 --> 00:19:26,730 So omega plus k equals zero is square root of two Kappa one plus Kappa two over m. 165 00:19:27,450 --> 00:19:32,640 I get that right? I think so. So that point here, we'll give it a square root of two. 166 00:19:32,850 --> 00:19:40,640 Kappa one plus Kappa two. Over. And so we have a long human with the high energy mode. 167 00:19:40,850 --> 00:19:45,130 Now the low energy mode will come back to the high energy a moment, but the low energy, 168 00:19:45,150 --> 00:19:47,750 low frequency mode is one that we should expect should have been there, 169 00:19:47,990 --> 00:19:51,050 because we should expect that there should be sound waves somewhere in the system. 170 00:19:51,290 --> 00:19:52,339 And when there are sound waves, 171 00:19:52,340 --> 00:19:59,749 we know the sound waves should have a spectrum which is has frequency linear in wave vector and comes down to zero at zero wave vector. 172 00:19:59,750 --> 00:20:02,930 Now to convince ourself that that's what's going on with this equation. 173 00:20:03,110 --> 00:20:10,130 We have to actually do a little bit of more math and expand the dispersion curve near K equals zero. 174 00:20:10,430 --> 00:20:18,460 So let's see how we do that. So let's take ak2 plus k one of the i. 175 00:20:18,960 --> 00:20:22,680 K. That quantity there is. 176 00:20:23,430 --> 00:20:38,070 Well, yeah. That thing there can be written as square root of capital squared plus capital one squared plus two capital one kappa two cosine of k. 177 00:20:38,740 --> 00:20:49,620 And just by multiplying that out and if k is small, we're going to replace this cosine by one minus k squared over two. 178 00:20:50,220 --> 00:20:55,650 So then what we have is I guess we can write it like this square root of this 179 00:20:55,650 --> 00:21:03,330 whole thing of two plus kappa one squared that takes care of the one term. 180 00:21:03,330 --> 00:21:07,230 When I put together this, this and this, I get capital one plus capital squared. 181 00:21:07,440 --> 00:21:14,970 And then what's left over is twos council, capital one, capital K squared. 182 00:21:15,570 --> 00:21:19,460 I get that. I want to leave that. Yeah. 183 00:21:20,230 --> 00:21:34,639 Okay, then this conveniently factors out to give her a capital plus capital one times the square root of one plus capital one. 184 00:21:34,640 --> 00:21:39,950 Capital squared over capital one's capital square. 185 00:21:41,900 --> 00:21:46,520 And if you remember square root of one plus x, I get a sign wrong. 186 00:21:46,520 --> 00:21:51,080 I got a sign wrong. That's minus. That's minus. 187 00:21:53,060 --> 00:21:58,460 No one gets chocolate for that is one minus X over two. 188 00:22:00,040 --> 00:22:09,410 So so then we have we can rewrite this as capital plus capital one minus capital one. 189 00:22:09,530 --> 00:22:14,010 Capita K squared over two. 190 00:22:14,660 --> 00:22:21,080 Capital one plus capital. Does that look right? 191 00:22:21,110 --> 00:22:28,250 Did I miss anything? Yeah. 192 00:22:28,500 --> 00:22:33,370 Capital kept to thank you. Cavity. 193 00:22:33,600 --> 00:22:39,090 Yeah. Yeah. Good. Thank you. Whoever said that, I owe them chocolate, so. 194 00:22:39,220 --> 00:22:43,770 Yeah. Okay. Okay, good. So I don't have one today. I was going to bring one, but I ate it, so. 195 00:22:45,330 --> 00:22:48,450 All right, so if I plug that into into here, 196 00:22:49,620 --> 00:22:55,919 we see that the Kappa one plus Kappa two with the with the minus solution cancels this this 197 00:22:55,920 --> 00:23:02,340 capital one plus capital also will cancel this and we'll end up just getting m omega squared. 198 00:23:03,480 --> 00:23:10,320 Omega squared equals capital one kappa two K squared over two, 199 00:23:10,890 --> 00:23:24,630 Kappa one plus Kappa two or equivalently omega square root Capital One, Kappa to a squared over, uh, to Kappa one plus Kappa two. 200 00:23:24,990 --> 00:23:30,600 I guess there's an m down there and then absolute k, all right. 201 00:23:30,600 --> 00:23:37,590 And I realise that's a lot of math. We're not going to do too much math that heavy this year, but I have to do it once in a while. 202 00:23:39,660 --> 00:23:46,260 Okay, so this we recognise as the sound velocity, so we just derived the sound velocity of this wave. 203 00:23:46,530 --> 00:23:51,240 The sound velocity is the slope of this curve here. 204 00:23:51,420 --> 00:23:54,990 Now usually actually I should use the proper nomenclature. 205 00:23:55,380 --> 00:24:06,570 Usually people refer to this as the acoustic mode and they tend to call it the acoustic mode, 206 00:24:06,570 --> 00:24:11,130 even all the way out to the zone boundary out here at very high K, 207 00:24:11,370 --> 00:24:19,140 even though it's really only sound when it's when it's long wavelength or small K, they call the whole branch here the acoustic mode. 208 00:24:20,240 --> 00:24:23,450 Now, you remember last time when we derived the sound velocity. 209 00:24:24,290 --> 00:24:28,460 We also were able to derive the same sound velocity by using a hydrogen panic argument. 210 00:24:28,700 --> 00:24:30,470 So let's see if we can do that again this time. 211 00:24:30,920 --> 00:24:39,950 Remember from hydrodynamics we have the v sound should equal square root of one over the mass density times the compress ability. 212 00:24:41,060 --> 00:24:49,040 The mass density is where there are two masses per lattice constant a and the compress ability while we derive last time, 213 00:24:49,040 --> 00:24:54,140 the compress ability should be one over the spring, constant times a and the spring constant. 214 00:24:54,380 --> 00:25:01,700 Well, here the the unit cell has two springs in it and the spring constant for two springs in series is Capital One, 215 00:25:01,700 --> 00:25:05,780 capital over capital one plus capital. Does that sound familiar? 216 00:25:06,500 --> 00:25:15,200 You remember that from first year. Okay. So if we plug this and this and this into here, in fact, we get exactly the same result. 217 00:25:15,680 --> 00:25:24,770 V sound equals square root capital one kappa to a squared or to kappa one for step two. 218 00:25:25,970 --> 00:25:33,530 So the hydrogen AMOC calculation gives you exactly the same result as we got by actually solving the system completely. 219 00:25:34,560 --> 00:25:37,800 Now let's talk about this mode up here. 220 00:25:37,830 --> 00:25:41,670 This mode is known in comparison to the acoustic mode. 221 00:25:41,940 --> 00:25:50,910 This mode is known as the optical mode, optical mode, high frequency mode, occasional zero. 222 00:25:51,300 --> 00:25:57,060 And we should probably I mean, I should probably tell you why the name optical mode. 223 00:25:57,360 --> 00:26:01,620 So the word optical mode comes from experiments on unreal. 224 00:26:01,770 --> 00:26:08,520 Real materials. When you shine light on the material and you induce vibrations by by shining light on it. 225 00:26:08,760 --> 00:26:10,200 Now, if you think for a second, 226 00:26:10,410 --> 00:26:21,080 what's the property of light that we know similar to sound light has frequency which is proportional to its wave vector. 227 00:26:21,090 --> 00:26:26,640 The only big difference is that C is huge compared to sound velocity. 228 00:26:26,910 --> 00:26:36,480 So if I drew the light dispersion curve on the same plot, it would be an extremely steep slope line like this. 229 00:26:36,930 --> 00:26:47,069 Probably so steep I couldn't even. Right now, if you imagine shining light on a material and creating vibrations in quantum mechanics, 230 00:26:47,070 --> 00:26:52,080 we should think about absorbing a photon and creating a phonon. 231 00:26:52,440 --> 00:26:56,669 So imagine a process where you absorb a photon and create a phone on. 232 00:26:56,670 --> 00:27:01,110 In order for that to happen, you have to conserve both energy and momentum. 233 00:27:01,380 --> 00:27:07,710 So you have to match up both the frequency and the wave vector of the light with the frequency in the wave vector of a phone on. 234 00:27:07,920 --> 00:27:15,740 The only place that can happen is right here. And the reason you're never going to match the acoustic mode because the velocity mismatch is so badly. 235 00:27:15,970 --> 00:27:20,350 But with the optical modes, since the object optical has finite frequency, even a cable zero, 236 00:27:20,500 --> 00:27:24,819 there is a point where the optical mode frequency matches life frequency and the optical 237 00:27:24,820 --> 00:27:30,880 mode momentum matches the the light momentum or even crystal momentum in this case. 238 00:27:32,440 --> 00:27:36,819 Now, so that is an area of where why people call this the optical mode. 239 00:27:36,820 --> 00:27:40,990 Whenever you have interactions with light and vibrations, it's inevitably the optical mode. 240 00:27:41,530 --> 00:27:50,470 But I should be a little bit honest that that process I described to you by which you absorb a photon and you emit a phonon actually does not occur. 241 00:27:50,710 --> 00:27:53,570 And the reason it doesn't occur is because there is a conserved quantum number, 242 00:27:53,590 --> 00:27:59,140 the spin of the photon, which I mean, photons have spin phonons do not. 243 00:27:59,170 --> 00:28:03,250 So if that process were to occur, you would violate angular momentum conservation. 244 00:28:03,430 --> 00:28:07,570 So that is, you know, definitely bad. You don't want to violate angular momentum conservation. 245 00:28:07,750 --> 00:28:13,510 So, in fact, no such process exists by which a single photon is absorbed and a single phonon is created. 246 00:28:13,780 --> 00:28:18,760 However, there are more complicated processes that can occur involving photons and phonons. 247 00:28:18,880 --> 00:28:23,140 For example, absorb two photons and emit some photons. 248 00:28:23,350 --> 00:28:27,790 That's okay, because the angular momentum of the two photons can cancel and you can still have 249 00:28:27,790 --> 00:28:33,279 conservation and is still the same principle that is very hard to conserve, 250 00:28:33,280 --> 00:28:37,810 both energy and momentum in any sort of interaction between phonons and photons, 251 00:28:37,960 --> 00:28:43,690 unless you're getting optical modes into the game because you need to get high frequencies into the game somehow with phonons. 252 00:28:43,690 --> 00:28:48,460 And the only way to get high frequencies with small wave vectors in Phonons is to use the optical mode. 253 00:28:49,930 --> 00:28:57,850 So that's where the name comes from, actually. More generally, let me give more general nomenclature here. 254 00:28:59,830 --> 00:29:13,750 Acoustic mode. Generally, acoustic mode is any mode where Omega is proportional to K at small K and optical mode. 255 00:29:15,640 --> 00:29:23,890 Optical mode is any mode where omega goes to a constant, not equal to zero at small K. 256 00:29:27,160 --> 00:29:31,690 Okay, so let's think about this, this small K regime a little bit more seriously. 257 00:29:34,240 --> 00:29:39,399 So we found the eigen value is the frequencies, but let's look at the eigenvectors. 258 00:29:39,400 --> 00:29:43,209 So the matrix were actually diagonals and is that big matrix there? 259 00:29:43,210 --> 00:29:56,620 And let's look at it at take zero. So at zero the matrix we're interested in is of the form, I guess it's minus Kappa one plus Kappa two times. 260 00:29:57,580 --> 00:30:00,790 What is it. One minus one, minus one one. 261 00:30:01,240 --> 00:30:06,660 Something like that. I get the minuses in the wrong place and I think it's right. 262 00:30:06,680 --> 00:30:15,010 Okay, so it looks something like this because when when the wave vector goes to zero, the exponential, the okay both go to one. 263 00:30:15,280 --> 00:30:18,670 And so you have a matrix of the form one, one minus one, minus one one. 264 00:30:19,330 --> 00:30:27,970 If I pull out minus one plus kappa two. So the eigenvectors we have two possible eigenvectors one is for the omega equals zero solution. 265 00:30:29,710 --> 00:30:34,870 We have an eigenvector x, a y equals one one. 266 00:30:35,590 --> 00:30:49,480 Whereas for the the optical mode solution, the other eigenvector the high energy high frequency eigenvector we have x a y equals one minus one. 267 00:30:50,710 --> 00:30:51,790 So what is this telling us? 268 00:30:52,090 --> 00:31:00,249 What this is telling us is that for the acoustic mode near k equals zero, the atoms are actually moving with their neighbours. 269 00:31:00,250 --> 00:31:06,580 They both are. The atoms in the unit cell are moving in the same direction at the same time they're moving in concert they are. 270 00:31:07,450 --> 00:31:13,510 Whereas for the optical mode, the higher frequency mode, the two atoms in the unit cell are moving opposite each other. 271 00:31:13,660 --> 00:31:20,020 And it is actually quite natural to explain then why it is that the optical mode is so much higher in frequency, 272 00:31:20,020 --> 00:31:22,960 because an optical mode you're compressing the springs maximally, 273 00:31:23,140 --> 00:31:28,720 whereas with the acoustic mode you are hardly compressing the springs at all because everyone is moving in the same direction. 274 00:31:29,110 --> 00:31:32,469 These sort of trends occur more generally. 275 00:31:32,470 --> 00:31:40,300 So General, if if there are m if there are m atoms in the unit cell. 276 00:31:40,990 --> 00:31:55,740 Atoms in units cell. We will have emojis that each k one is acoustic. 277 00:31:58,240 --> 00:32:05,490 That's the mode where everyone moves in the same direction at k equals zero, and then the remaining and minus one of them are optical, 278 00:32:07,590 --> 00:32:12,510 meaning they don't all move in the same direction and k zero in d dimensions. 279 00:32:13,290 --> 00:32:24,930 The dimensions there are d times m modes at each k at each k. 280 00:32:28,480 --> 00:32:37,660 These are optic are acoustic and the remaining D times and minus one are optical. 281 00:32:41,710 --> 00:32:48,520 Okay. Why is this? Well, we're in the total count of the modes is a total count of the number of degrees of freedom. 282 00:32:48,700 --> 00:32:55,150 If there are m, there is always the same number of ks per as as the number of unit cells in the system. 283 00:32:55,450 --> 00:33:02,910 Each each value of each unit cell has m masses in it and they can move in d dimension. 284 00:33:02,920 --> 00:33:07,800 So it is total of D.M. degrees of freedom per unit cell. 285 00:33:07,810 --> 00:33:10,960 So we expect the M modes for each K. 286 00:33:12,070 --> 00:33:20,980 Now of them there can be d of them there acoustic and that corresponds to all of the masses moving in any of the three possible directions. 287 00:33:21,730 --> 00:33:27,370 So we know what these things are. We have discussed them before. One of them is longitudinal and two of them are transverse. 288 00:33:27,700 --> 00:33:35,410 So for each K, one of them is moving in the direction of K and then two of them are moving perpendicular to K in three dimensions. 289 00:33:35,860 --> 00:33:42,490 Okay. There's one more point on this picture that we should probably look at more carefully, 290 00:33:42,700 --> 00:33:47,860 which is these interesting points here near which are at the so bronze zone boundary. 291 00:33:48,520 --> 00:33:57,190 Pi over a when we put it over here. So consider consider these boundary. 292 00:33:57,460 --> 00:34:04,000 These boundary. K equals pi over a. 293 00:34:05,620 --> 00:34:17,230 So what do we have then? Well, then this factor capital plus in the I k a kappa one is actually absolute k two minus k one. 294 00:34:17,800 --> 00:34:26,350 And so the frequency is omega squared is one over mk1 plus kappa one plus capital plus 295 00:34:26,350 --> 00:34:32,050 or minus absolute kappa one minus Kappa to a cap of two minus capital one does matter. 296 00:34:32,380 --> 00:34:36,160 So absolute value. And that means that we have two possible solutions. 297 00:34:37,390 --> 00:34:44,580 Omega equals either square root of two kappa one over M or square root of two Kappa two over M. 298 00:34:44,620 --> 00:34:55,959 So we'll mark them here. So this one we'll call Square Root of two Kappa one over M, and this one will call the square root of two Kappa two over M. 299 00:34:55,960 --> 00:34:59,620 And here I've assumed we have Kappa one is greater than Kappa two otherwise. 300 00:35:00,710 --> 00:35:05,000 We are around the higher end. You want to squeeze on top and these modes actually never cross each other. 301 00:35:05,000 --> 00:35:13,400 You can convince yourself by looking at the form of of the of the frequency these two modes can never actually cross. 302 00:35:14,810 --> 00:35:25,550 So what is going on at this at this brand's own boundary, at the brand's own boundary, the wave form delta x, delta y equals x, 303 00:35:26,150 --> 00:35:35,660 a, y, you know, the i omega t either minus i k and a this e to the minus i k and a becomes minus one to the n. 304 00:35:36,950 --> 00:35:43,819 And that tells us that what's going on is that alternate unit cells are moving out of phase with each other. 305 00:35:43,820 --> 00:35:48,560 They're moving in the opposite direction as each other. So now is see if we can get this to work. 306 00:35:51,290 --> 00:35:54,429 Uh oh. No, no, no. 307 00:35:54,430 --> 00:35:58,110 Stop, stop, stop, stop, stop. Stop. 308 00:36:03,700 --> 00:36:11,190 Know, uh. Oh, dear. 309 00:36:12,620 --> 00:36:17,280 Does this happen? Only on Mondays. Here we go. Screen up. Okay. 310 00:36:17,680 --> 00:36:20,970 Oh, but that's. Oh, wow. Amazing. Okay, so we can still do this while it's going on. 311 00:36:21,240 --> 00:36:27,960 Okay, so this program with is the thing written by by Mike Glaser and you can click alternating 312 00:36:28,260 --> 00:36:33,090 chain and longitudinal longitude means that everything oscillates in the single line. 313 00:36:33,510 --> 00:36:43,049 You can change the K here by changing the slider and you can see that the the dispersion is plotted down here is plotted from 314 00:36:43,050 --> 00:36:52,140 zero to pi instead of zero instead of minus pi over zero to pi instead of minus pi over a two pi over a like I've drawn here. 315 00:36:52,360 --> 00:36:56,760 So it is the same picture. So first let's start by taking K very, very small. 316 00:36:57,840 --> 00:37:03,479 O And also you have these little clickers up here which will change the masses on the springs that's making the masses the same, 317 00:37:03,480 --> 00:37:07,740 and C one and C two, or we have what we call Kappa one and Kappa to the two spring constants. 318 00:37:08,520 --> 00:37:12,270 So we have two different spring constants, Kappa one, a cap of two. 319 00:37:12,570 --> 00:37:17,700 And here's the dispersion, the acoustic mode down here, the optical mode up here, here we've clicked acoustic. 320 00:37:17,700 --> 00:37:22,230 So we're seeing actually the acoustic mode here. And you see that in acoustic mode is very low frequency. 321 00:37:22,410 --> 00:37:26,670 And basically all of the let's move this up a little bit so it happens a little faster. 322 00:37:26,850 --> 00:37:34,259 You can see that basically all the masses are moving in concert with each other, sort of slow sloshing back and forth. 323 00:37:34,260 --> 00:37:39,839 Hydrodynamic oscillations you can turn viewed as one big fluid. 324 00:37:39,840 --> 00:37:44,909 Everything moves together. Now if we go to K equals zero, it becomes zero frequency. 325 00:37:44,910 --> 00:37:53,879 But we can click on Optic here and we'll get the optical mode. You see, the optical mode is each mass moving opposite the other mass in the unit cell. 326 00:37:53,880 --> 00:37:59,370 And you can see that this is something that should be high frequency because it's compressing the springs maximally. 327 00:38:00,390 --> 00:38:09,120 Now if we then change the K vector to the brand zone boundary, what we have here is, 328 00:38:09,120 --> 00:38:16,409 is as predicted, that the alternate units cells are moving opposite each other. 329 00:38:16,410 --> 00:38:21,600 So this unit cell, these two masses are moving opposite of these two masses. 330 00:38:21,900 --> 00:38:27,180 And you can see actually why it is that only one of the two spring constants enters in the 331 00:38:27,180 --> 00:38:32,280 frequency because the spring between these two masses actually isn't getting compressed at all. 332 00:38:32,280 --> 00:38:37,589 It's only the spring between the unit cells that are getting compressed and not the spring within the unit cell. 333 00:38:37,590 --> 00:38:44,490 It's getting compressed now if you switch than from looking at the optical mode at the zone boundary to the acoustic mode of the zone boundary, 334 00:38:44,730 --> 00:38:47,730 it looks extremely similar, but it's a lower frequency here. 335 00:38:47,850 --> 00:38:51,590 But you'll notice this the other spring, this now getting compressed is still you know, 336 00:38:51,630 --> 00:38:56,850 you have one unit cell moving in one direction or another unit cell moving in the opposite direction. 337 00:38:57,030 --> 00:39:05,100 But now it's the opposite spring, the key to spring, which is being compressed rather than the Kappa one spring being compressed. 338 00:39:05,340 --> 00:39:12,090 Okay, so I highly recommend that people download this program and mess around with it. 339 00:39:12,360 --> 00:39:17,190 It's got a lot of other fun features. This is what we can do on a boring Saturday night or something. 340 00:39:17,550 --> 00:39:23,200 Okay, so let me redraw this picture here. 341 00:39:23,230 --> 00:39:26,740 Maybe maybe stop this for a second. Okay. 342 00:39:27,570 --> 00:39:30,870 Redraw this picture here. All right. 343 00:39:30,870 --> 00:39:36,330 I don't know why it's still still doing that, but. Okay. All right, good. 344 00:39:39,510 --> 00:39:42,870 Me redraw this picture, except I'm going to draw slightly differently. 345 00:39:44,160 --> 00:39:59,490 Here's K again, here's Omega and I'm going to put Pi over here and minus pi over here and then two pi over here and then minus two pi over here. 346 00:40:01,140 --> 00:40:08,820 I'm going to draw the acoustic mode just like I had it before. And then the optical mode, what we had before was something that looked like this. 347 00:40:11,010 --> 00:40:19,680 But I am perfectly allowed to take this piece here and shift it by two pi over a and plot it instead here. 348 00:40:24,270 --> 00:40:28,650 Okay. So I did research at this piece and I shifted it by two pi rate to move it over here. 349 00:40:29,040 --> 00:40:33,120 And then I can take this piece here and shift at two pi over a this way. 350 00:40:34,390 --> 00:40:37,600 And put it here. Okay. So now I'm going to erase. 351 00:40:37,610 --> 00:40:40,680 I know people hate when you do a race of transforms, going to raise the peace in the middle. 352 00:40:41,320 --> 00:40:45,880 And I leave these pieces here. So this piece and I have this piece. 353 00:40:47,410 --> 00:40:53,890 Okay. So I just took this piece and moved it by two pi over this way, and this piece moved by two pi over three that way. 354 00:40:54,100 --> 00:40:59,679 And I got this picture here. Now, this way of drawing things is known as the extended zone scheme. 355 00:40:59,680 --> 00:41:03,280 Extended Zone Scheme. 356 00:41:05,290 --> 00:41:11,859 And it has a nice advantage that there is only one K, there's only one mode at each frequency at each wave vector. 357 00:41:11,860 --> 00:41:16,569 K So whereas over here we have two modes at each wave vector. 358 00:41:16,570 --> 00:41:26,650 K Here we only have one, but we have used twice as much range of K and that's sometimes convenient to do just some nomenclature which is useful. 359 00:41:27,010 --> 00:41:30,040 This range here is known as the first boron zone. 360 00:41:30,490 --> 00:41:44,240 Boron Zone. And this region here is known as the second grand zone, along with this region here, second brand zone. 361 00:41:44,610 --> 00:41:52,170 So we took the optical mode and we moved it out of the first boron zone and put it into the second run zone for the extended zone scheme. 362 00:41:52,180 --> 00:41:57,059 This way of drawing things over here in comparison to the extended zone scheme is known as 363 00:41:57,060 --> 00:42:04,830 the reduced zone scheme because everything has been reduced into a single bronze zone. 364 00:42:06,050 --> 00:42:16,130 Now, why is it that it's so convenient to to spread everything out so that there is only one mode at each wave vector? 365 00:42:16,400 --> 00:42:21,229 Well, a case where this becomes extremely useful to do is the following case. 366 00:42:21,230 --> 00:42:28,010 Suppose consider Kappa one very close to Kappa two. 367 00:42:28,460 --> 00:42:31,810 So in that case, what's going to happen is that these. 368 00:42:32,060 --> 00:42:37,850 This little gap here is going to get extremely small. So let's see what that happens, what that looks like. 369 00:42:40,930 --> 00:42:51,010 So here's K, here's pi over a year to pi over a year minus pi over A and here is minus two pi over eight. 370 00:42:51,700 --> 00:42:54,940 So it's going to look like it's kind of this. 371 00:42:56,680 --> 00:43:04,110 But like this. Like that is a very small gap at the bronze zone boundary here. 372 00:43:05,010 --> 00:43:11,820 Because Capital One and Capital are very close to each other. Now, what happens when Capital One actually equals capital? 373 00:43:12,360 --> 00:43:17,610 Well, then what we have is the gap closes. It becomes a single, single connected line. 374 00:43:17,910 --> 00:43:23,570 But more importantly, we recover the monetary McCain Right. 375 00:43:23,630 --> 00:43:28,980 If Capital One equals capital, then every mass and every every spring is the same as every other spring. 376 00:43:28,980 --> 00:43:32,520 Every mass is the same as every other spring. So we get back the modern atomic chain. 377 00:43:32,760 --> 00:43:36,330 But the lattice constant has changed. 378 00:43:36,570 --> 00:43:40,290 The new lattice constant is only half as big as the old lattice constant, 379 00:43:40,470 --> 00:43:46,440 because the white and black masses now actually are the same, the same object now. 380 00:43:46,680 --> 00:43:50,440 So we don't have to make a unit. 381 00:43:50,530 --> 00:43:53,910 So with two things, we now make a unit cell with only one thing in it. Right. 382 00:43:54,450 --> 00:43:58,890 So the bronze on the bronze on the back. 383 00:43:59,850 --> 00:44:07,380 Now ranges from K is goes from minus pi over a prime to pi over a prime. 384 00:44:08,250 --> 00:44:15,210 Okay. And that is the same as two pi over a minus two pi over a two to pi over a. 385 00:44:16,630 --> 00:44:25,670 Okay. So this entire range here, one, if the two masses become the same, this entire range here becomes the first branch zone. 386 00:44:26,100 --> 00:44:32,660 First because of monotonic chain. My atomic chain. 387 00:44:33,140 --> 00:44:42,110 And then if you make Capital One capital slightly unequal, you open a small gap at the new own boundaries at PI over a. 388 00:44:42,770 --> 00:44:51,860 But really what's underlying is a single mode for the atomic chain from here to here and here to here, just a single connected node. 389 00:44:51,860 --> 00:44:58,060 And you open up make when you make capital one in capital slightly different, you open up a small gap at the bronze zone, 390 00:44:58,070 --> 00:45:04,390 at a bronze on boundary of the new smaller branch zone associated with the new larger unit self. 391 00:45:05,660 --> 00:45:11,270 Happy. Okay. In the last few minutes, I want to discuss a completely different topic, 392 00:45:11,270 --> 00:45:16,970 something we didn't we didn't get to last time, which was or the time before, which is van der Waals bonding. 393 00:45:17,510 --> 00:45:25,190 Van der Waals. You may remember this guy from the van der Waals equation of state that you probably studied in stap mac van der Waals bonds, 394 00:45:25,940 --> 00:45:31,460 also known as molecular bonds. Molecular or fluctuating dipole. 395 00:45:32,390 --> 00:45:36,290 Fluctuating dipole bonds. 396 00:45:38,270 --> 00:45:47,110 Bonds. And these are Vancouver's bonds are actually quite different from either ionic or covalent or 397 00:45:47,110 --> 00:45:53,589 hydrogen or metallic bonds in the sense that these bonds occur for inactive chemical species, 398 00:45:53,590 --> 00:46:06,550 inactive species. And what I mean by inactive species is that no electron is transferred between atoms, no electron is shared between atoms. 399 00:46:07,210 --> 00:46:10,420 Things like noble gases are classic cases of Andreev bonding. 400 00:46:10,630 --> 00:46:13,959 The noble gases are filled shell. It doesn't share its electrons. 401 00:46:13,960 --> 00:46:17,590 It doesn't donate electrons. It does not covalently bond. 402 00:46:17,590 --> 00:46:20,530 It does not, ironically bond. But it can still van der Waals bond. 403 00:46:21,670 --> 00:46:30,100 Now, another example of an inactive species are molecules such as nitrogen to nitrogen by itself as an atom is very active. 404 00:46:30,280 --> 00:46:35,370 But when you put two nitrogen atoms together, they form a nitrogen molecule, which is extremely inactive. 405 00:46:35,380 --> 00:46:41,590 You can think of it as being filled shell of a filled molecular shell, and it also does not share. 406 00:46:41,770 --> 00:46:43,899 Once you have the molecule, the molecule is very stable. 407 00:46:43,900 --> 00:46:49,300 It doesn't share its electrons with anything else and it doesn't donate its electrons to anything else. 408 00:46:49,660 --> 00:46:58,600 So nonetheless, even though you're not donating electrons and moving electrons around, you can still running out of space. 409 00:46:58,600 --> 00:47:00,220 You can still form a bond. 410 00:47:00,280 --> 00:47:10,780 The way the bond occurs is as follows Imagine that you have, say, a nice, noble gas atom over here and maybe another nice noble gas atom over here. 411 00:47:11,890 --> 00:47:18,760 And I suppose at some moment this atom has some dipole vector p want a polarisation? 412 00:47:19,390 --> 00:47:27,070 Now if you remember from your in m a distance away, there will be an electric field due to that dipole moment. 413 00:47:27,670 --> 00:47:37,210 In this case pointing in that direction and e will be equal to minus p one over four pi epsilon, not our cubed. 414 00:47:39,450 --> 00:47:46,070 And if it was, if I had drawn the angles differently, I conveniently chose R to be perpendicular to the moment there. 415 00:47:46,080 --> 00:47:52,340 But if I didn't make them perpendicular, there'd be some cosines and signs and stuff like that, which I'm leaving out now. 416 00:47:52,350 --> 00:47:59,669 When this atom experiences this electric field, it develops a polarisation, it gets an induced polarisation. 417 00:47:59,670 --> 00:48:03,930 P two which is chi some constant times. 418 00:48:03,930 --> 00:48:15,050 E This is this constant here is known as the polarised ability, polarised ability or electric susceptibility, electric susceptible. 419 00:48:18,060 --> 00:48:25,350 And you can calculate this for simple atoms like hydrogen, you can calculate the susceptibility in perturbation theory. 420 00:48:25,710 --> 00:48:31,860 And maybe if you had done that as an exercise, as an exercise in the book, they asked you to do that using quantum mechanics anyway. 421 00:48:31,860 --> 00:48:39,030 Now what we have is we have two dipole moments and you remember that the energy of two dipole moments is minus P, 422 00:48:39,030 --> 00:48:43,920 one P, two over four pi epsilon, not cubed. 423 00:48:44,250 --> 00:48:48,210 So the total energy is then just plugging these things all together. 424 00:48:48,420 --> 00:48:56,520 We get minus p one squared chi over four pi epsilon not r cubed all squared. 425 00:48:56,880 --> 00:49:00,360 And in particular this is negative and goes out to the sixth. 426 00:49:00,780 --> 00:49:04,920 So the force goes as one over to the seventh. 427 00:49:06,180 --> 00:49:10,350 Now that is more or less how the calculation goes that you have a polarisation, 428 00:49:10,350 --> 00:49:14,790 one atom induces a polarisation in the other atom and the polarisation is a track. 429 00:49:14,790 --> 00:49:24,719 But you might say, well, wait a second, this whole argument was predicated on the statement that the polarisation is not equal to zero to begin with, 430 00:49:24,720 --> 00:49:32,100 but in a spirit, the symmetric atom like like helium, the expectation of the polarisation is zero. 431 00:49:32,310 --> 00:49:37,320 Why is that? Well, you remember that the the polarisation of an atom, here's the nucleus, 432 00:49:37,590 --> 00:49:43,740 here's the electron electrons running around the atom in some sort of spherical orbit. 433 00:49:44,550 --> 00:49:52,680 So the, the position are of the electron is related to the polarisation just by minus eight times R or something like that. 434 00:49:53,820 --> 00:49:59,160 So it's really, we're just saying that the average position of the electron is actually at zero, 435 00:49:59,370 --> 00:50:06,330 but you'll notice that what actually comes into this equation is not the polarisation or the dipole moment P, but it's P squared. 436 00:50:07,140 --> 00:50:17,520 So P squared is not equal to zero so that you can get a nonzero dipole or attractive force, even though the average dipole moment is a zero. 437 00:50:17,670 --> 00:50:23,730 If you like to think about fluctuations in quantum mechanics as being some sort of dynamical fluctuation or fluctuation in time, 438 00:50:23,970 --> 00:50:27,629 what we really mean is that it's a sort of a course picture, 439 00:50:27,630 --> 00:50:32,490 a crude picture of what quantum mechanics is, but you can sort of think of it as at some moment in time, 440 00:50:32,700 --> 00:50:36,060 the electron is on one side of the nucleus, so there's a dipole moment. 441 00:50:36,210 --> 00:50:41,250 The other atom responds to that and oriented dipole moment in the opposite direction than the attract. 442 00:50:41,490 --> 00:50:52,760 And then at some later time, there's a question. Yeah. 443 00:50:52,760 --> 00:50:59,360 If there's an angle between. You're saying that if there's an angle between you two, there would be cosine theta and so forth and things like that. 444 00:50:59,600 --> 00:51:03,110 Yeah, but the force will still be attractive. You can. 445 00:51:03,110 --> 00:51:06,229 And actually there's, there's an exercise in the book that works you through it. 446 00:51:06,230 --> 00:51:09,680 And you can and you can get all the angles right. And it will always come out attractive. 447 00:51:10,100 --> 00:51:13,940 Good question, but not worth the chocolate bar. So but it's still a good question. 448 00:51:13,970 --> 00:51:18,380 Thank you. Anyway, the. 449 00:51:19,280 --> 00:51:22,819 So it's basically you get a little bit of fluctuation on one atom. 450 00:51:22,820 --> 00:51:25,280 It causes a fluctuation in the other atom and they attract. 451 00:51:25,550 --> 00:51:30,200 And no matter which direction the fluctuation is, the responding fluctuations in the opposite direction and they always attract. 452 00:51:30,530 --> 00:51:37,250 So you can still get a nonzero van of force. Now the van de Vos force is much weaker than covalent ionic or hydrogen bond, 453 00:51:37,520 --> 00:51:46,429 but it's still strong enough to cause important effects such as, you know, it holds together things like argon at low temperature. 454 00:51:46,430 --> 00:51:53,840 And when argon becomes a solid a more interesting case of where Vandervoort forces show up is with this guy. 455 00:51:54,530 --> 00:51:58,070 If you've ever been to a tropical climate, this is a gecko, a little lizard. 456 00:51:58,310 --> 00:52:02,240 They also they sell car insurance in the United States. I don't I don't know why that's true. 457 00:52:02,240 --> 00:52:05,630 This is the mascot of the Gecko Geico Gecko Company. 458 00:52:06,680 --> 00:52:16,650 Anyway. Geckos are amazing little creatures because they can crawl up almost completely flat glass walls with no trouble at all. 459 00:52:16,890 --> 00:52:21,360 And the thing that actually sticks them to the wall turns out to be Van der Waals forces. 460 00:52:21,360 --> 00:52:22,250 They have you know, 461 00:52:22,260 --> 00:52:29,190 they have a very flat foot and they stick their foot onto the glass and there's enough van der Waals force between their foot and the glass. 462 00:52:29,310 --> 00:52:33,270 And they're fairly like little creatures that they can actually walk up the wall because of that. 463 00:52:33,510 --> 00:52:36,390 All right. I apologise. I went over. I will see you on Wednesday.