1 00:00:00,990 --> 00:00:07,380 Welcome back, everyone. It's the 19th lecture of the condensed matter course nearing the end of sixth week. 2 00:00:08,520 --> 00:00:11,129 I know it's hard to keep keep going on. 3 00:00:11,130 --> 00:00:15,600 At the end of six week, everyone gets pretty tired, but we're really getting pretty close to the end of this course now. 4 00:00:15,600 --> 00:00:17,970 So it's in our power on through. 5 00:00:18,390 --> 00:00:25,650 So when we left off last time, we were talking about magnetism and we decided a good place to start thinking about magnetism was one atom at a time. 6 00:00:25,650 --> 00:00:27,570 So we're really thinking about atomic magnetism. 7 00:00:32,940 --> 00:00:40,410 And we had written down the Hamiltonian, actually the Hamiltonian for one electron in one atom in a magnetic field. 8 00:00:40,740 --> 00:00:46,379 And we decided we would write it as the Hamiltonian for that electron in zero magnetic field. 9 00:00:46,380 --> 00:00:49,920 That is the kinetic energy and the attraction to the nucleus plus two terms. 10 00:00:49,920 --> 00:00:56,340 The first term is born magnets on magnetic field dotted inter orbital angular momentum 11 00:00:56,340 --> 00:01:04,530 plus g spin angular momentum and the second term e squared over 2ma squared. 12 00:01:05,040 --> 00:01:11,130 Where is the vector potential del cross a equals v? 13 00:01:12,990 --> 00:01:19,860 Okay. Now these two terms have names. This first term over here is known as the paramagnetic term. 14 00:01:23,720 --> 00:01:26,750 And this term here is known as the Magnetic Town. 15 00:01:29,300 --> 00:01:34,100 And as you might guess, it is this term that is typically responsible repairing magnetism in this term, 16 00:01:34,280 --> 00:01:37,189 which is typically responsible for die magnetism. 17 00:01:37,190 --> 00:01:42,080 And this term is actually fairly obvious why it is that this term is going to be responsible for power magnetism. 18 00:01:42,320 --> 00:01:49,640 Generally, this is just coupling of the magnetic field to the spin or overlying the momentum of of the electron and the. 19 00:01:49,880 --> 00:01:52,340 Okay, now it's a little complicated because of the minus signs, 20 00:01:52,340 --> 00:01:59,000 but the spin one or the overlying momentum both at one point opposite B such that the energy of this becomes negative. 21 00:01:59,270 --> 00:02:06,350 And if the spin or momentum is opposite B, that means the magnetic moment is the same direction is B because the charge of the electron is negative. 22 00:02:06,590 --> 00:02:11,720 So the moment points opposite of the angular momentum. So any way to or in order to lower the energy, 23 00:02:11,960 --> 00:02:20,150 the spins line up opposite B and normal argument aluminium lines up opposite B and therefore the moment lines up in the same direction is B, 24 00:02:20,150 --> 00:02:28,370 and you have power of magnetism. Now, one thing you may notice about these two terms is this paramagnetic magnetic term is linear in B, 25 00:02:29,960 --> 00:02:39,050 whereas this term is quadratic in B, in B, and so generally for a small B, 26 00:02:39,230 --> 00:02:44,719 the linear inter in the the paramagnetic term is going to dominate as long as there is not a 27 00:02:44,720 --> 00:02:49,570 particularly good reason why the paramagnetic term is going to be suppressed or somehow zero, 28 00:02:49,640 --> 00:02:54,530 even zero, then the magnetic term is irrelevant and we just have to worry about the paramagnetic term. 29 00:02:54,710 --> 00:03:02,120 We will find occasions where this term is zero or is depressed, and then the magnetic term will have a chance to dominate the physics. 30 00:03:02,120 --> 00:03:04,010 But we're going to focus on this one first, 31 00:03:04,220 --> 00:03:13,070 figuring it's going to be more important that to take this term and what we're going to want to do is we're going to want to simplify it a little bit. 32 00:03:13,070 --> 00:03:17,809 Now, one thing we can do is we can add together the angular momentum of the orbital 33 00:03:17,810 --> 00:03:20,840 angle momentum and the spin angular momentum to get the total angular momentum. 34 00:03:20,840 --> 00:03:26,420 J is L plus sigma, but you'll notice that's not what we actually have in the Hamiltonian. 35 00:03:26,660 --> 00:03:28,550 We have L plus G Sigma. 36 00:03:28,820 --> 00:03:35,330 Now, you may have learned in your or you should have learned in your atomic physics course that under conditions where you know the magnitude of j, 37 00:03:35,330 --> 00:03:43,670 the magnitude of L, the magnitude of sigma, you can rewrite that entire term in terms of some effective g factor g effective 38 00:03:43,670 --> 00:03:51,350 known as the land g factor is written with a superscript j new b b that j and 39 00:03:51,350 --> 00:03:56,270 there's some formula that allows you to get the g the effect of g factor from 40 00:03:56,270 --> 00:03:59,569 G and the magnitude of J and the magnitude of L and the magnitude of sigma. 41 00:03:59,570 --> 00:04:04,910 If you know this, that formula is not part of this course, but I believe it's part of your other course, so you probably should know it. 42 00:04:05,450 --> 00:04:09,469 If you're taking the atomic physics course, you won't be asked to derive it in this course. 43 00:04:09,470 --> 00:04:13,490 You may be asked to derive in the other course. Any rate, the moral is that you can just some. 44 00:04:13,490 --> 00:04:18,049 You can put together the orbital and spin angular momentum and under the conditions that, you know, a j, 45 00:04:18,050 --> 00:04:24,380 l and Sigma R, you can just think of this as a single angular momentum vector that reorients in a magnetic field. 46 00:04:24,650 --> 00:04:31,850 And the entire term can be it can be summarised as as this is a fairly simple term in the Hamiltonian under those 47 00:04:31,850 --> 00:04:37,610 conditions where we can just think of it as a single total angular momentum being reoriented by the magnetic field. 48 00:04:38,480 --> 00:04:45,650 But in fact, the story is a little more complicated than this and is more complicated than this because we have many electrons in an atom, 49 00:04:45,650 --> 00:04:49,160 many electrons in atom, not just a single electron. 50 00:04:49,520 --> 00:04:54,469 And that means we have to keep track of the entire angular momentum capital. 51 00:04:54,470 --> 00:05:00,470 L That's the sum over all the electrons are the orbital angular momentum of each electron, and then the total spin, 52 00:05:00,470 --> 00:05:05,390 angular momentum, which is the sum of all the spin, angular momentum of each of the electrons. 53 00:05:05,390 --> 00:05:11,210 And of course the total angular momentum all together is the total orbital plus, the total spin. 54 00:05:11,540 --> 00:05:14,359 And as you probably learned from your adventures in Eclipse, 55 00:05:14,360 --> 00:05:20,149 Gordon coefficients that you can have very complicated ways of adding together angular momentum. 56 00:05:20,150 --> 00:05:25,700 For example, two two spin. One halves can add together to make a singlet or triplet, and then things get really messy. 57 00:05:25,700 --> 00:05:29,870 And if you have 20 electrons in your atom, they can add together and also two crazy ways. 58 00:05:30,050 --> 00:05:37,190 And so we need something to guide us to figure out how it is that these electron angle momenta are going to add up. 59 00:05:37,820 --> 00:05:42,530 The first thing that we have to guide us is the shell structure of the atom. 60 00:05:42,530 --> 00:05:46,760 So let's remember back to the beginning of the term shell structure. 61 00:05:48,410 --> 00:05:51,200 We had two laws that helped us out. 62 00:05:51,200 --> 00:05:58,639 The first was the alpha principle, which basically alpha which in German I believe means building up or construction. 63 00:05:58,640 --> 00:06:02,930 It means you fill up shells one at a time. And the second principle we had was mat alongs. 64 00:06:03,620 --> 00:06:10,219 Mat alongs were all which told us in which order we should start filling, filling shells. 65 00:06:10,220 --> 00:06:11,780 So that is going to help us out a lot. 66 00:06:11,990 --> 00:06:18,710 And the reason that is going to help us out a lot is because there's is an important principle that a filled shell. 67 00:06:21,770 --> 00:06:27,320 Has L equals X equals j equals zero. 68 00:06:27,530 --> 00:06:30,560 No angular momentum at all in a filled shell. Why is that? 69 00:06:30,800 --> 00:06:36,760 Well, we filled every orbital with a spin up and every orbital with a spin down, so the total spin is zero. 70 00:06:36,770 --> 00:06:40,290 The number of ups is equal to number of downs. How else will we spin? 71 00:06:40,310 --> 00:06:47,900 We filled all the positive states and we equally filled all the negative states in in a filled shell. 72 00:06:47,930 --> 00:06:52,120 So there are these all cancel each other and the total you end up getting out is zero. 73 00:06:52,130 --> 00:06:55,430 So a filled shell is has no anger mentum at all. 74 00:06:55,640 --> 00:07:01,520 And that means we only only look at part filled shells, look at part filled shells. 75 00:07:08,890 --> 00:07:10,510 Okay. So that's going to help us out a lot. 76 00:07:10,750 --> 00:07:17,829 But still, you can have a part filled shell with three, four, five, six, seven electrons in it, in a partially filled out shell. 77 00:07:17,830 --> 00:07:21,879 If it's a big shell, you can have, you know, 13, 14 electrons in a partially filled shell. 78 00:07:21,880 --> 00:07:30,670 So this can still be pretty complicated. Fortunately, we have a set of rules that helps us figure out how these annual momenta add together. 79 00:07:30,880 --> 00:07:36,670 And these are known as ones rules, ones rules after one. 80 00:07:37,030 --> 00:07:40,870 And he wrote them down actually first in 1925, a very long time ago, 81 00:07:40,870 --> 00:07:45,850 back in the early days of quantum mechanics, just when people were learning about shells of atoms. 82 00:07:46,120 --> 00:07:52,210 Now, I may not know what that means in German. I may not know what Alpha means in German, but I know that hunt means dog in German. 83 00:07:52,510 --> 00:07:58,840 So where that word is, the dogs rules. But so rule one. 84 00:08:00,370 --> 00:08:06,040 Rule one electron spins. Spins a line. 85 00:08:06,940 --> 00:08:10,150 A line, if they can, if they can. 86 00:08:13,270 --> 00:08:20,800 In other words, you maximise s right. 87 00:08:21,400 --> 00:08:26,740 So this is actually a very nice rule because it's kind of like being a power magnet in a fair magnet. 88 00:08:26,950 --> 00:08:31,090 You get all your spins to align and you maximise the total spin. 89 00:08:32,110 --> 00:08:34,980 It's not really a ferromagnetic because we're just talking about a single atom here. 90 00:08:34,990 --> 00:08:42,250 So it's just a finite number of electrons and not a whole, you know, a whole big solid of electrons, but at least has the same qualitative physics. 91 00:08:42,550 --> 00:08:46,420 So can you understand why it is that the electron spins would want to align? 92 00:08:46,750 --> 00:08:50,620 Well, okay, so why? Why rule one? 93 00:08:52,330 --> 00:08:56,710 Well, the first thing, it is not because of dipole and dipole interactions. 94 00:08:57,250 --> 00:09:02,770 Dipole, dipole interactions between the spins. Why not? 95 00:09:03,580 --> 00:09:07,870 Yes. The spins do have dipole moments and dipole moments will interact. 96 00:09:07,870 --> 00:09:16,750 But if you try to calculate the dipole dipole interaction strength between two dipoles whose dipole moment is a more magnets than in a single atom. 97 00:09:16,930 --> 00:09:21,520 It is insanely small. It's so small you can just completely throw it out, totally irrelevant. 98 00:09:21,820 --> 00:09:26,590 So it is not because of this that is important to keep in mind. 99 00:09:26,770 --> 00:09:31,720 What it is do do is due to column interaction due to Coulomb. 100 00:09:32,860 --> 00:09:42,310 Let's see if we can figure out why Coulomb interaction would care if the electron spins were aligned or not aligned well. 101 00:09:43,430 --> 00:09:47,720 So if you open up most solid state physics books or most physics books of any sort, 102 00:09:48,110 --> 00:09:56,270 they usually tell you some story about why it is that electron spins want to align and why it is that one's rule is obeyed in this respect. 103 00:09:56,420 --> 00:10:00,469 And the story is sort of qualitatively right, but it has a lot of things wrong with it. 104 00:10:00,470 --> 00:10:05,570 So this is a warning. I think I gave a lot of caveats in the in the book. 105 00:10:05,570 --> 00:10:08,719 So you can read the details. We'll go through this a little bit. 106 00:10:08,720 --> 00:10:13,610 We'll tell the story, first of all, the fairy tale first. Then I'll explain to you why the fairy tale isn't exactly right. 107 00:10:13,610 --> 00:10:16,100 So you have a better idea of what actually is going on. 108 00:10:16,550 --> 00:10:23,270 So in order to figure out why it is a Coulomb interaction care, let's recall that if you have a wave function for two electrons that way, 109 00:10:23,280 --> 00:10:29,540 function should be decomposed into the orbital part, which depends on the position of the two atoms. 110 00:10:29,960 --> 00:10:36,200 Sorry, two electrons and the spin part, which depends on the spins of the two electrons. 111 00:10:36,590 --> 00:10:42,650 Now the overall wave function must be anti symmetric because electrons are fermions. 112 00:10:43,790 --> 00:10:55,579 And so that tells us that if k spin equals having both spins pointing in the same direction, like up up, down, down or both pointing sideways. 113 00:10:55,580 --> 00:11:07,160 If the spins are aligned then this is this is then symmetric, symmetric then the orbital path sy orb is anti symmetric. 114 00:11:09,890 --> 00:11:21,830 And what that means is if we write side orbital as a function of R one minus R two, we take R one to our to the wave function. 115 00:11:21,830 --> 00:11:30,110 It has to go to zero because the if you take an anti symmetric function and you take its argument to zero, the function must go to zero. 116 00:11:30,110 --> 00:11:35,030 Anything anti symmetric has to go to zero at zero. So what does that mean? 117 00:11:35,030 --> 00:11:40,160 What that means is that electrons cannot get close to each other. 118 00:11:40,730 --> 00:11:48,860 Cannot get close, close to each other. 119 00:11:50,990 --> 00:11:54,860 And you might think that the Coulomb interaction would care about that. 120 00:11:54,860 --> 00:12:00,530 And indeed, that is the story that's usually told. So the story goes kind of like this If the spins are aligned, 121 00:12:00,800 --> 00:12:05,210 then because of the symmetry of the wave function, the electrons can't get close to each other. 122 00:12:05,420 --> 00:12:10,760 And that makes the Coulomb interaction happy because the interaction doesn't want the electrons to get close to each other either. 123 00:12:11,000 --> 00:12:15,500 So the energy is naturally lower because the electrons are staying further apart from each other. 124 00:12:17,000 --> 00:12:25,250 It's almost right, but is not really right. And the reason it's not really right is because really more important. 125 00:12:26,240 --> 00:12:40,400 More important is the electron nucleus, nucleus nucleus, not the electron electron. 126 00:12:40,790 --> 00:12:49,370 And it's sort of a very subtle difference between electron nucleus and electron electron because it's all Coulomb interaction at the end of the day. 127 00:12:49,610 --> 00:13:00,500 But let me try to explain what I mean by this. There is imagine we have a nucleus here and then we have two spins whose spins are anti aligned. 128 00:13:00,740 --> 00:13:03,709 So because the spins are anti-life, they can get close to each other. 129 00:13:03,710 --> 00:13:09,560 And in particular one electron can get in between the other electron in the nucleus. 130 00:13:10,490 --> 00:13:18,530 Now remember when we went back and we talked about electronegativity and ionisation energy, when one electron can get inside the orbit of the other, 131 00:13:18,770 --> 00:13:26,090 it can screen the nucleus, it can screen the electron from the nucleus and make the effective nuclear charge look smaller. 132 00:13:26,360 --> 00:13:30,170 As a result, the electron on the outside is weakly bound. 133 00:13:32,420 --> 00:13:36,770 Because it sees a smaller, effective nuclear charge. 134 00:13:37,070 --> 00:13:42,780 On the other hand, if you align the spins of the electrons, then the electrons can't get close to each other. 135 00:13:42,780 --> 00:13:51,450 And in particular, this electron cannot get inside this electron's orbit and it can't screen this electron from the nucleus. 136 00:13:51,470 --> 00:13:55,130 So as a result, the electrons are strongly bound. 137 00:13:59,370 --> 00:14:07,919 And this is more of the reason of why it is that the that it is lower energy to have the two electrons spins 138 00:14:07,920 --> 00:14:12,120 aligned because then the two electrons are strongly bound to the nucleus and that lowers their energy. 139 00:14:12,660 --> 00:14:17,670 So that is more of the honest reason why our first rule applies. 140 00:14:18,330 --> 00:14:25,560 Okay, so let us. We can even generalisations rule up from atoms and molecules if we want. 141 00:14:25,800 --> 00:14:36,370 So suppose we have a molecule molecules. Suppose we have a molecule with two plasma nuclei and say two electrons up here? 142 00:14:37,770 --> 00:14:41,440 Is exactly the same physics. If the two electron spins align, 143 00:14:41,650 --> 00:14:49,510 then the electrons have to stay apart from each other and they can't screen each other from the nucleus where the spins are anti aligned. 144 00:14:49,690 --> 00:14:53,110 Then they can get inside of each other and they can screen the nucleus. 145 00:14:53,110 --> 00:14:55,660 And in the end the binding is weaker. 146 00:14:55,840 --> 00:15:10,510 So the rule, once the spins to be anti one rule are still one spins a line even in a molecule, but that has to compete with something else. 147 00:15:10,900 --> 00:15:15,100 And the other thing that it has to compete with is a covalent bonding. 148 00:15:17,020 --> 00:15:21,690 Bond once anti aligned. Anti align. 149 00:15:23,640 --> 00:15:29,610 We call what we learned about covalent bonding when we had orbitals on two atoms, both with energy inward. 150 00:15:29,850 --> 00:15:37,380 When the atoms come close together, they form a bonding in the anti bonding orbital and we want to put both electrons in the anti bonding orbital. 151 00:15:37,560 --> 00:15:42,300 We have two anti align their spins to make a singlet and put them both in low energy state. 152 00:15:42,600 --> 00:15:49,590 So there's a competition between the molecules, there's a competition between covalent bonding physics and physics. 153 00:15:49,770 --> 00:15:53,580 One of them wants the spins to align. One of them wants to spin to anti-life. 154 00:15:53,620 --> 00:15:58,319 So it's a lot more complicated in a molecule to figure out if spins align or if they don't. 155 00:15:58,320 --> 00:16:01,380 So we need to just be aware that those two are competing with each other. 156 00:16:01,680 --> 00:16:05,040 Okay. All right. So that is the first rule. 157 00:16:05,160 --> 00:16:07,740 It tells us that the spins will try to align if they can. 158 00:16:08,040 --> 00:16:14,370 So if you have a shell that is less than half filled, you can put all of the electrons pointing the same direction. 159 00:16:14,610 --> 00:16:15,659 When you get to half filled, 160 00:16:15,660 --> 00:16:21,840 you have to start putting electrons in in the opposite direction because you've filled all the states with all the electrons in the same direction. 161 00:16:22,080 --> 00:16:25,740 But to the extent that they can, the electrons will always try to align. 162 00:16:26,520 --> 00:16:31,589 Now that said, we still have to figure out what happens to the orbital degree of freedom. 163 00:16:31,590 --> 00:16:47,220 So rule two is very similar, which is basically l is maximised to subject to rule one, to rule one, which is more important. 164 00:16:48,780 --> 00:16:54,900 So let's see if we can do an example of this. Probably the easiest way to explain this is to show how it works. 165 00:16:55,200 --> 00:17:00,010 So let's consider a preceding atom. This is atomic number. 166 00:17:00,760 --> 00:17:08,740 Atomic number is 59. Number equals 59, which is 56 equals filled shells. 167 00:17:10,330 --> 00:17:19,370 I think it's filled shells up to six s actually. So that means we have filled shells plus three electrons. 168 00:17:19,400 --> 00:17:22,960 Okay. So 59 equals filled shells. 169 00:17:26,530 --> 00:17:38,940 Plus three electrons and f official three electrons in four f in well just f shell f is l equals three remember 170 00:17:39,520 --> 00:17:51,520 speed f g so f0123f is l equals three so we can write out all the orbitals one the c for f l equals three. 171 00:17:51,820 --> 00:17:57,670 There is two l plus one orbitals going from minus three lc equals minus three plus three. 172 00:17:57,670 --> 00:18:09,760 So let's write them all out. So LC equals we have minus three minus two minus one zero 1 to 3 minus three minus two minus one zero 1 to 3. 173 00:18:10,030 --> 00:18:13,689 And we have to put three electrons in those orbitals. 174 00:18:13,690 --> 00:18:21,700 Now the first rule is first rule says that the electrons we put in have to have their spins aligned threads so we know we have to have a line spins. 175 00:18:21,910 --> 00:18:27,160 And then what we want to do is we want to maximise l subject to the condition that all the spins are aligned. 176 00:18:27,430 --> 00:18:33,489 So we can think, well let's try to maximise LC and we will do that by putting the electrons as far 177 00:18:33,490 --> 00:18:38,170 to the right as possible and we will get LC equals six three plus two plus one. 178 00:18:38,530 --> 00:18:44,049 Now we can also try to put them as part of the left as possible and we will get the smallest value of LC possible, 179 00:18:44,050 --> 00:18:46,980 which would be minus six, minus three, minus two, minus one. Okay. 180 00:18:47,410 --> 00:18:58,990 So if the largest LC we can get is plus six and the smallest LC we can get is minus six, it means that the L we have L equals six in this case. 181 00:19:00,620 --> 00:19:09,950 Good. Now we can also write down as immediately is three halfs, three spins, all aligned as equals, three has. 182 00:19:10,060 --> 00:19:13,060 So that is how one rule works. And one rule. 183 00:19:13,060 --> 00:19:16,090 Second rule here is also driven by Coulomb interaction, 184 00:19:16,090 --> 00:19:22,870 but there aren't very good cartoon pictures to explain why it is that all is maximised in one second rule. 185 00:19:22,870 --> 00:19:24,129 So I apologise about that. 186 00:19:24,130 --> 00:19:31,630 It's just a you know, it's an observation that John made and no one's really come up with many very simple explanations for it at this point. 187 00:19:32,810 --> 00:19:41,350 Okay. So this tells us almost everything we need to know, but we still have the question of how does L align or anti align with S? 188 00:19:41,680 --> 00:19:48,880 So we have rule three where all three on the third rule, which is either. 189 00:19:51,530 --> 00:20:02,570 L is parallel to S or L minus L is parallel to s, and the way it goes is the j. 190 00:20:02,900 --> 00:20:15,830 Absolute value of gay is absolute l plus or minus absolute s where we choose minus if we have if less than half filled shall less than half filled, 191 00:20:20,600 --> 00:20:26,180 meaning that the spins anti align with their angular momentum and with their overlying momentum. 192 00:20:26,420 --> 00:20:32,240 And as you choose. Plus, if more than half is more than half. 193 00:20:39,040 --> 00:20:47,760 Okay. There is one third rule which tells you how the orbital and spin angle meant to either align with each other or anti align with each other. 194 00:20:48,120 --> 00:20:53,790 Now, so, for example, in this case, here we have less than one half filled shell. 195 00:20:54,060 --> 00:20:59,610 So J equals six minus three gas, which is the minus sign because it's less than half filled. 196 00:20:59,850 --> 00:21:04,560 And that gives us j equals nine halves. So we know everything there is to know about Praesidium. 197 00:21:04,740 --> 00:21:08,310 We know what it is. We know it as we know James from her roles. 198 00:21:09,050 --> 00:21:12,900 Okay, so why is third rule true? 199 00:21:13,290 --> 00:21:18,990 Well, one third rule is actually a little bit unlike the other two, because it's not driven by interactions. 200 00:21:19,240 --> 00:21:25,320 It's actually driven by something else. It's driven by our rule three is from rule three. 201 00:21:25,950 --> 00:21:32,340 From spin orbit, coupling. Spin orbit, which is actually a relativistic effect. 202 00:21:32,400 --> 00:21:38,700 Spin orbit coupling. And in fact, it is this is basically that when we wrote down the Hamiltonian, 203 00:21:38,910 --> 00:21:43,350 we left out a small term, the spin orbit term, which takes the following terms. 204 00:21:43,350 --> 00:21:50,100 All right. As Delta H, a small piece of the Hamiltonian we left out, which we can write as alpha sum over all the electrons, 205 00:21:51,450 --> 00:21:54,540 orbital angular momentum of the electron dotted with the spin. 206 00:21:54,540 --> 00:22:01,860 Angular momentum of the electron. Okay. Now, in this, Hamiltonian alpha is greater than zero. 207 00:22:02,160 --> 00:22:06,899 And so you can sort of see that this is going to make the orbital angular momentum 208 00:22:06,900 --> 00:22:11,760 and the spin angular momentum counter aligned to make this as negative as possible. 209 00:22:11,790 --> 00:22:15,270 You're going to have if the spin is up, the orbital momentum wants to be down and so forth. 210 00:22:15,660 --> 00:22:20,900 But you might wonder, well, what about this business? About half filled shells and less than half a thousand more than half filled shells? 211 00:22:20,910 --> 00:22:26,520 That seems a little bit strange. It turns out that that little strange list is actually just from bookkeeping. 212 00:22:26,550 --> 00:22:33,030 You just have to be careful about the bookkeeping. Every electronic goes in is always following the same rule one. 213 00:22:33,180 --> 00:22:37,050 Rule one, follow rule two. And then try to make this as small as possible. 214 00:22:37,290 --> 00:22:41,550 And if you if you follow that rule, you will always get one rule three. 215 00:22:41,760 --> 00:22:47,800 Let's see how that works for a second. Let's consider a case where we have, I don't know, detail. 216 00:22:48,090 --> 00:22:58,410 So in addition, we have a LC equals minus two, minus one and one, two, three, else equals minus two, minus one, zero, one, two. 217 00:22:58,440 --> 00:23:00,720 Those are the possibilities. And let's put one electron in. 218 00:23:01,140 --> 00:23:06,299 If you put one electron in it, while it's pretty simple, it wants to maximise its spin angular momentum. 219 00:23:06,300 --> 00:23:10,770 So the spin while only has one choice in the spin one half. 220 00:23:10,770 --> 00:23:16,230 So that is easy and it wants to maximise its orbital angular momentum as well. 221 00:23:16,260 --> 00:23:21,060 So we can put this the, the electron way over here or way over here. 222 00:23:21,390 --> 00:23:29,340 Now if you want to minimise the all that sigma, if we put it, if we make the spin spin down, 223 00:23:29,610 --> 00:23:37,439 then you will put it over here so that sigma sigma z equals minus one half, but LC equals plus one half. 224 00:23:37,440 --> 00:23:41,170 So the product of Sigma L is negative, which is what we want. 225 00:23:41,200 --> 00:23:51,030 We want to make that term as negative as possible. And then, sure enough, the sum of these two G is LC minus sigma. 226 00:23:51,690 --> 00:23:57,540 Sorry, LC plus sigma is three halves and as claimed with the less than half filled shell. 227 00:23:57,750 --> 00:24:03,870 So j is the difference of two and one half. 228 00:24:04,410 --> 00:24:08,100 So that works. Basically just telling you that when you add. 229 00:24:10,110 --> 00:24:14,759 We have half less than half filled shell. The spin and the orbit want to be pointing in the opposite direction. 230 00:24:14,760 --> 00:24:18,020 So spin orbit is negative. That is easy. 231 00:24:18,160 --> 00:24:22,020 Okay, now what happens when we get to the shell? When we get to fill the shell? 232 00:24:22,530 --> 00:24:28,890 L.Z., we have equals minus two, minus one, zero one, two. 233 00:24:28,900 --> 00:24:38,490 We have these five states. Now, once we get a half filled shell, we have to put in all five spins align because we want to align all the spins. 234 00:24:38,490 --> 00:24:44,550 So here we have s equals five has five spins all pointing in the same direction, 235 00:24:44,940 --> 00:24:51,510 but l here equals zero because we spit, we filled all the negative states and all the plus states as well. 236 00:24:51,990 --> 00:24:57,030 Okay, so here it is. No problem. I mean, third roll doesn't even apply because. 237 00:24:57,450 --> 00:25:03,760 Because our zero spin is just. Uh. 238 00:25:04,870 --> 00:25:10,359 Yeah, absolutely. It's just it's just it's just a spinning your mentum here so we don't have to worry about about. 239 00:25:10,360 --> 00:25:22,630 Well, now let's add one more electron. So when you add one more electron, LC equals go to five minus two, minus one, zero one, two. 240 00:25:22,810 --> 00:25:26,560 We put all these guys spin down, we add one more electron. 241 00:25:26,740 --> 00:25:28,360 Where is that one electron go? 242 00:25:28,360 --> 00:25:37,089 Well, Kate still wants to satisfy this condition, that it wants its own spin, angular momentum to point opposite its orbital angular momentum. 243 00:25:37,090 --> 00:25:40,420 So it has to go in spin up. So we have to put it over here. 244 00:25:40,660 --> 00:25:47,320 So this last electron we put in has Sigma Z equals plus one half one LC equals minus two. 245 00:25:47,620 --> 00:25:51,550 And that still makes it happy because Sigma Z is pointing opposite LC. 246 00:25:51,940 --> 00:25:59,350 But now notice what's going on here. The total Z, the total Z is actually still negative. 247 00:25:59,590 --> 00:26:03,190 Is actually negative what is negative two? Right. 248 00:26:03,640 --> 00:26:09,970 So the last electron that I put in was pointing up the total spin is actually still pointing down. 249 00:26:10,750 --> 00:26:19,780 So even though the L LC is here is minus two the total because I have to have also had LC total equals zero. 250 00:26:20,020 --> 00:26:25,870 We added one electron that electron is pointing up, but the total spin angular momentum is still pointing down. 251 00:26:26,200 --> 00:26:32,260 So here the total spin, angular momentum and the total overlying momentum and now pointing in the same direction. 252 00:26:32,500 --> 00:26:39,790 And we were still just following the same rule that each electron wants to counter align his spin and his orbital angular momentum with each other. 253 00:26:39,970 --> 00:26:45,040 So in other words, the reason why we came to this bookkeeping problem is because when we have a half filled shell, 254 00:26:45,250 --> 00:26:48,820 we have s, which is nonzero, but L is zero. 255 00:26:49,000 --> 00:26:58,210 When we add one more electron, the net spin of the whole system is pointing in the opposite direction of the spin of that one electron we added. 256 00:26:58,450 --> 00:27:02,110 So that is why John's third rule has this complication. That half filled shell. 257 00:27:02,380 --> 00:27:05,710 Is that clear? Yeah, hopefully. Okay. 258 00:27:06,220 --> 00:27:12,310 All right. Sorry. I mean, I know it's confusing, but that's the way life is sometimes, so we have to deal with it. 259 00:27:12,880 --> 00:27:20,680 Okay. So given that we know LS and J, we can then write down our Hamiltonian. 260 00:27:20,700 --> 00:27:32,910 Well, first thing we can do is we can take L plus g as do the same land g factor and rewrite that as some g effective times j where j is l plus else. 261 00:27:34,060 --> 00:27:39,700 And then we can write down our effective Hamiltonian for these are angular momenta, 262 00:27:39,820 --> 00:27:51,820 which is g effective j mubayi the j and we can do statistical mechanics with this kind of Hamiltonian. 263 00:27:52,120 --> 00:27:56,080 So to remind you how this works, this is something that you did last year. 264 00:27:56,350 --> 00:28:01,060 So example. Take a really simple example. 265 00:28:01,240 --> 00:28:05,780 Let's take j equals one half in a g factor of two. 266 00:28:06,460 --> 00:28:10,360 So I spend can point up with this overall and momentum can point up, 267 00:28:10,640 --> 00:28:17,860 I can point down and the two possible energies are then plus and minus four magnetron times magnetic field. 268 00:28:18,990 --> 00:28:25,540 And we can then write the partition function for this system, which is the sum over the two states of the of the angular momentum in the minus 269 00:28:25,540 --> 00:28:32,529 beta you b b plus either the plus beta movie B from the partition function. 270 00:28:32,530 --> 00:28:40,840 I hope this all looks familiar. We can write the free energy b t log z we can write the magnetic moment. 271 00:28:41,320 --> 00:28:49,540 Magnetic moment is gas with the minus sign don't miss minus sign here somewhere here there's a minus sign missing here free energy. 272 00:28:50,350 --> 00:28:58,690 We can then write the magnetic moment the F, the B, which I believe you did this last year. 273 00:28:58,690 --> 00:29:04,839 And if you didn't, you'll do it this year again. Beta UVB, it has this cash form for small. 274 00:29:04,840 --> 00:29:17,709 B are small B this becomes just beta mu b squared times magnetic field, then the magnetisation capital. 275 00:29:17,710 --> 00:29:30,340 M which is moment per volume. Moment for volume is the moment per spin m small m times row the density of spins. 276 00:29:32,920 --> 00:29:47,390 Then Z spins. And then we can get the susceptibility chi, which is by definition not deem to be taken at big or zero. 277 00:29:47,960 --> 00:30:00,740 And if you go through that, take that limit, you get Roe may not be squared over cubed, which hopefully maybe looks familiar from last year. 278 00:30:00,980 --> 00:30:03,650 This is known as Curies Le Curie Law. 279 00:30:04,040 --> 00:30:11,990 In fact, more generally, Curie's law is any time when the susceptibility is some constant over t which in the disease. 280 00:30:12,350 --> 00:30:19,070 So the important thing to notice about this Curie law susceptibility is that it diverges at low temperature. 281 00:30:19,220 --> 00:30:22,550 You have a divergence susceptibility as temperature goes to zero. 282 00:30:22,790 --> 00:30:30,410 And this is actually correct and it is meaningful. The reason the susceptibility diverges at two at zero temperature is because it's zero temperature. 283 00:30:30,710 --> 00:30:32,930 The system is always in the lowest energy state. 284 00:30:33,470 --> 00:30:39,110 So if you have a bunch of spins and you have a tiny magnetic field, no matter how tiny that magnetic field, 285 00:30:39,320 --> 00:30:43,640 there is a lowest energy state where the moment is aligned with the magnetic field. 286 00:30:43,910 --> 00:30:48,950 So even with a arbitrarily small magnetic field, the moments all align and point that direction, 287 00:30:48,950 --> 00:30:52,840 which means you then have an infinite susceptibility because it takes only a an 288 00:30:52,880 --> 00:30:57,650 infinitesimal magnetic field to orient all the spins in the same direction. 289 00:30:57,700 --> 00:30:59,900 Is that clear? So this is actually a meaningful statement. 290 00:31:00,140 --> 00:31:06,320 When you raise the temperature, then the spin start fluctuating all over the place and they do not all point in the direction of the magnetic field. 291 00:31:06,320 --> 00:31:12,230 I mean, they mean point vaguely in the direction of the magnetic field, but not all exactly the direction of the magnetic field. 292 00:31:12,470 --> 00:31:15,620 So the susceptibility drops as you go to a higher temperature. 293 00:31:15,920 --> 00:31:27,770 Now, more generally than this, this simple spin, one half calculation for for a spin j, for spin j in general, we follow the same same story, 294 00:31:27,950 --> 00:31:37,760 except we start with a partition function which is sum over GC equals minus j two plus j e to the minus beta, 295 00:31:38,210 --> 00:31:47,140 the energy of the spin and state j which is j g effective for j then you b the z. 296 00:31:50,610 --> 00:32:00,150 I think I implied here without without stating it, that that B is pointing in the Z direction that we have to apply to be in the Z direction. 297 00:32:00,420 --> 00:32:04,319 Otherwise we would use a different axis and it would be if we put V in the X direction, 298 00:32:04,320 --> 00:32:08,400 we would just use a different basis where we count from minus de to J. 299 00:32:08,940 --> 00:32:13,620 Sorry about that. Okay. Then you go through exactly the same manipulations. 300 00:32:13,770 --> 00:32:16,200 The algebra is a little bit more different. Difficult. 301 00:32:16,440 --> 00:32:22,649 You get out some more complicated function and attach it to a function known as a broken function. 302 00:32:22,650 --> 00:32:29,340 It's a messy thing. I think you calculated last year in your step met course it has some hyperbolic signs and hyperbolic cosines in it. 303 00:32:29,550 --> 00:32:33,030 No more messy, but the same story comes out at the end of the day. At the end of the day, 304 00:32:33,030 --> 00:32:37,290 you still get a carry law in the sense that it's a constant divided by t for exactly the 305 00:32:37,290 --> 00:32:42,210 same reason in infinitesimal field will orient all the moments in the same direction. 306 00:32:42,660 --> 00:32:46,740 Okay, now one thing actually to note along the way, which is quite important, 307 00:32:47,580 --> 00:32:54,450 is a little bit of a side, is to note that this partition function is a function of B over key only. 308 00:32:54,710 --> 00:32:57,270 It is not a function of being T separately. Okay? 309 00:32:57,840 --> 00:33:07,350 And that is actually important because it means that the the entropy will also be a function of B over T only. 310 00:33:08,580 --> 00:33:18,270 And that was noticed way back in the early 1900s by one of our heroes, Peter Debye, who has shown up several times already in this course. 311 00:33:18,510 --> 00:33:21,780 And he noticed that that would mean that you could do the following experiment. 312 00:33:22,050 --> 00:33:25,390 You take a system in a magnetic field at a given temperature. 313 00:33:25,650 --> 00:33:32,010 If you robotically reduce the magnetic field area, Bartok means that the entropy has to stay fixed. 314 00:33:32,250 --> 00:33:37,230 With antibiotics, they reduce the magnetic field. The temperature must change proportional to the magnetic field. 315 00:33:37,620 --> 00:33:54,450 This is what is known as aromatic magnetisation d magnetisation and it is actually a really good way to make a refrigerator, 316 00:33:54,870 --> 00:34:01,080 very powerful refrigerator. You take a system at some temperature in some magnetic field, you reduce the magnetic field. 317 00:34:01,200 --> 00:34:08,850 And in order to keep the entropy fixed, changing things at the temperature must reduce proportionately to the magnetic field. 318 00:34:09,120 --> 00:34:15,530 This is the way people build some of the world's most powerful refrigerators that get down to Miller, Kelvin or even micro Kelvin temperature. 319 00:34:15,540 --> 00:34:22,750 So it is a good thing to know about. And eventually the technique breaks down and basically breaks down. 320 00:34:22,870 --> 00:34:30,040 When our simple approximation of just independent spins and this partition function being right eventually breaks down at very low temperatures, 321 00:34:30,280 --> 00:34:33,700 will come to that in the next lecture or the one after, I think. 322 00:34:34,000 --> 00:34:40,550 All right. So. This is basically the story of power, magnetism, one atom at a time. 323 00:34:40,820 --> 00:34:44,690 But when you start putting together many atoms at a time, things can be different. 324 00:34:44,720 --> 00:34:47,780 There is this phrase that goes around the condensed matter community. 325 00:34:47,960 --> 00:34:51,650 More is different. Whenever you put lots of things together, you can get different physics. 326 00:34:51,980 --> 00:34:59,180 So I'm still going to think one in some sense, one spin at a time or one electron at a time, one atom at a time. 327 00:34:59,360 --> 00:35:04,400 But when we start putting it in the environment of a bunch of other atoms, things can start changing. 328 00:35:04,430 --> 00:35:07,550 So what can be different? What goes wrong and what is different in a solid. 329 00:35:07,970 --> 00:35:14,360 What different in a solid difference in a solid. 330 00:35:15,960 --> 00:35:22,910 Okay. A couple of things we can write down immediately. One, we can have electron hopping, electron hopping. 331 00:35:24,530 --> 00:35:29,989 And if electrons hop from one atom to the next, we expect to get our band physics. 332 00:35:29,990 --> 00:35:33,260 We expect to get electron band physics bands. 333 00:35:34,340 --> 00:35:37,819 And if you have a partially filled band, you have a metal to band physics. 334 00:35:37,820 --> 00:35:46,490 You can get a metal. And if you are a metal, you expect to have Pauli power magnetism and therefore Pauli power magnetism. 335 00:35:49,280 --> 00:36:00,530 You get a Fermi surface and a power magnetism, and if you have power magnetism, the susceptibility is smaller by a factor of roughly 80 over F. 336 00:36:01,520 --> 00:36:06,290 And the reason for this is that you still try to flip over spins when you apply a magnetic field, 337 00:36:06,650 --> 00:36:11,780 but you have to obey the poly exclusion principle and that prevents you from flipping over 338 00:36:11,780 --> 00:36:15,410 a lot of them and it's only a small number of them will flip over in a magnetic field. 339 00:36:15,800 --> 00:36:17,600 So that is one thing that can be different. 340 00:36:17,600 --> 00:36:25,310 If the electrons are hopping from one atom to another, then you might expect to have a metal and poly paramount is a much smaller susceptibility. 341 00:36:26,390 --> 00:36:30,580 So you might ask, well, when? When do we have no hopping? 342 00:36:30,590 --> 00:36:39,630 When is there no hopping? Is there no hopping? Because you know why? 343 00:36:39,820 --> 00:36:44,640 Why would they ever not hop? Why would the electrons ever not hop from one Adam to the next? 344 00:36:44,880 --> 00:36:49,680 Well, basically, this comes from interactions. Ask where they're strong interactions. 345 00:36:51,840 --> 00:36:57,060 And we discussed this before when we talked about Mott Insulators. 346 00:36:57,390 --> 00:37:06,690 Remind you what a Mott Insulators is. Mott INSULATOR is when you have one electron in each site, and due to the Coulomb interaction, 347 00:37:07,050 --> 00:37:09,690 it prevents you from ever having to electrons on the side. 348 00:37:09,690 --> 00:37:13,950 And then you have this idea of a traffic jam of electrons where there is one electron on each 349 00:37:13,950 --> 00:37:18,150 atom and no one can jump to his neighbours because there is already someone sitting there. 350 00:37:18,390 --> 00:37:22,580 So everyone's frozen, but it's frozen with only one electron on each side. 351 00:37:22,590 --> 00:37:29,700 And so you can worry about that electron spin or angular momentum and that can be reoriented and makes a very nice power magnet that way. 352 00:37:29,700 --> 00:37:33,570 So these are insulators at high well, at least a high temperature. 353 00:37:33,840 --> 00:37:39,960 We'll talk about them at low temperature in the next lecture on after make a fairly good paramagnetic. 354 00:37:41,050 --> 00:37:50,290 So that is point one. What can be different in a solid point two of what can be different is a solid is that the environment can matter. 355 00:37:50,890 --> 00:37:52,120 Environment can matter. 356 00:37:56,880 --> 00:38:07,890 And to give you an example of why that can be, let's imagine that you have an atom in a long triangle unit cell, which is much taller than it is wide. 357 00:38:08,250 --> 00:38:13,380 And you might imagine that in tracking your cell like this, due to the environment, 358 00:38:13,680 --> 00:38:20,880 the orbital angular momentum might want to be pointing up or pointing down directly, 359 00:38:21,210 --> 00:38:31,200 might want want El pointing up or El pointing down and not in the middle. 360 00:38:31,440 --> 00:38:37,440 So even if it was an electron which could point an LC equals all the way from minus three to plus three, 361 00:38:38,280 --> 00:38:44,639 it is still because of the environment it might prefer to point directly up, directly down, and not anywhere in the middle. 362 00:38:44,640 --> 00:38:54,270 And that can change the story quite a bit. This is what's known as crystal field splitting, field splitting, 363 00:38:57,180 --> 00:39:04,680 where the environment of an atom tends to fix which angular momentum are allowed in which ones are not allowed. 364 00:39:05,700 --> 00:39:10,730 And. One thing that happens in this case, which is particularly important, 365 00:39:10,970 --> 00:39:22,850 is sometimes you can get a situation where Crystal Field is actually quite common forces l to basically be zero, 366 00:39:23,300 --> 00:39:27,020 make some sort of superposition of owls that force them to be zero, 367 00:39:27,350 --> 00:39:34,850 in which case which implies that the total j is just then the spin s and l doesn't contribute. 368 00:39:35,090 --> 00:39:39,950 So even if you had f electrons or a partially filled shell where there's lots of electrons and you 369 00:39:39,950 --> 00:39:45,800 might imagine that it would have a non-zero L by one role because of the environment of the atom, 370 00:39:46,040 --> 00:39:50,870 l can be forced to be zero. And then you would only get the spin angular momentum and not the orbital angular momentum. 371 00:39:51,230 --> 00:40:03,230 This is typical of transition metals, which should tell you is called Transition Metals, things like chromium. 372 00:40:05,150 --> 00:40:08,960 This is known as give it a name. 373 00:40:09,080 --> 00:40:19,830 It's known as quenching. Of L when L is basically removed from the problem due to the environment of the atom. 374 00:40:20,880 --> 00:40:24,720 It is typical of transition metals but does not occur. 375 00:40:25,680 --> 00:40:36,630 Does not occur. Occur for rare earths, for rare earths, at rare earth, lantern, eyes and actinide earth atoms. 376 00:40:37,400 --> 00:40:42,900 And the reason for this is a little subtle, but actually not not that complicated. 377 00:40:43,650 --> 00:40:49,740 The reason for this is that in a transition metal, the partially filled shell is, say, the 3D shell. 378 00:40:50,190 --> 00:40:56,910 So here. We have an atom and it is maybe the furthest out shell it's filled is for us. 379 00:40:57,150 --> 00:41:01,590 The partially filled shell is a 3D shell like this, the initial like this, 380 00:41:01,800 --> 00:41:06,330 and the three D shell is very close to the outer edge of the atom in the transition metal. 381 00:41:06,810 --> 00:41:07,590 Transition metal. 382 00:41:13,130 --> 00:41:21,470 And so the partially filled shell sees the environment of the atom quite strongly because the electrons are very close to the boundary of the atom. 383 00:41:21,860 --> 00:41:24,680 Whereas in the rare earth where earth. 384 00:41:26,810 --> 00:41:34,550 You can have something where the outermost shell is something like six s and the partially filled shell is four F, 385 00:41:34,910 --> 00:41:41,120 and so it is very protected from the environment because the partially filled shell has a much smaller radius than the atom as a whole. 386 00:41:41,300 --> 00:41:45,830 So basically the the partially filled shell does not even see the environment 387 00:41:45,830 --> 00:41:50,480 because it's so shielded from the environment by the larger orbitals in the atom, 388 00:41:50,720 --> 00:42:00,890 which is more or less explains why it is that the rare earths do not have quenching of angular momentum, whereas the transition metals do. 389 00:42:01,400 --> 00:42:07,190 All right, so this is everything we have to say more or less about power magnetism. 390 00:42:08,000 --> 00:42:13,549 Back up to the first equation we wrote on the board here today and worry a little bit about die magnetism. 391 00:42:13,550 --> 00:42:20,390 So when can we get dear magnetism? When is there die of magnetism? 392 00:42:25,690 --> 00:42:29,590 Well, there are a couple ways that you can get dear magnetism. 393 00:42:29,830 --> 00:42:34,840 The first sort of classic case of when you get down magnetism is when j equals zero, 394 00:42:35,290 --> 00:42:42,369 when you just so happen to have a situation where there is no angular momentum to re-orient, there is no power of magnetism because there is no no j. 395 00:42:42,370 --> 00:42:51,520 And for example, example, if you have a field shell like a noble gas that would have all equals as equals Jake or zero. 396 00:42:51,610 --> 00:42:55,750 And you can't re-orient any magnetic moment because there is no magnetic moment to reorient. 397 00:42:55,990 --> 00:42:59,050 So in that case, you can basically just throw out this term in the Hamiltonian. 398 00:42:59,260 --> 00:43:05,800 The only thing left is the magnetic term of the Hamiltonian, another case where you can get this magnetism. 399 00:43:06,610 --> 00:43:12,580 I should also mention that not only you can have fill shell, you can also have filled molecular orbitals. 400 00:43:13,240 --> 00:43:19,490 Molecular orbitals. Which, you know, similar. 401 00:43:19,550 --> 00:43:21,980 Like I mentioned this one, we talked about Van der Waals bonding. 402 00:43:21,980 --> 00:43:27,830 You have an inert atom, like a nitrogen to sorry, an inert molecule, like a nitrogen two molecule, 403 00:43:28,070 --> 00:43:31,610 which you can think of as being just a bunch of a filled molecular orbitals. 404 00:43:31,620 --> 00:43:36,200 So it has a net j equals zero at at the molecular level. 405 00:43:36,410 --> 00:43:39,139 And so there's really nothing to re-orient in the case of a, 406 00:43:39,140 --> 00:43:48,110 of a nitrogen two molecule because all the molecular orbitals are filled sort of analogous to to a, a larger atom with orbitals. 407 00:43:48,110 --> 00:43:52,610 There are real shells that are filled. There is another situation where you can get di magnetism, 408 00:43:53,540 --> 00:44:04,520 which is when you have when when you have poly power mag are well, poly power magnetism is weak. 409 00:44:04,820 --> 00:44:14,840 It is always weak because its chi is smaller by I wrote this down above by k over e f, which is a pretty big factor. 410 00:44:15,350 --> 00:44:19,670 So if you have a metal like copper or something like that, 411 00:44:20,390 --> 00:44:26,660 you would expect it to be a power magnet because it has spins to reorient, but because of the exclusion principle, 412 00:44:26,810 --> 00:44:30,380 they do not re orient very well and you only get a very small susceptibility, 413 00:44:30,560 --> 00:44:35,060 in which case the magnetic term has a chance to compete with the paramagnetic term. 414 00:44:35,300 --> 00:44:42,380 And so here's a a challenge you should ask your tutors is copper die a magnet or a power magnet? 415 00:44:42,650 --> 00:44:47,630 And unless they were during this course last year, in which case students asked them the same question last year. 416 00:44:47,930 --> 00:44:50,270 So for new tutors, most of them will get it wrong. 417 00:44:50,270 --> 00:44:56,059 Most people will say that, well, it's a metal, so it's a poly power magnet, and indeed it has a poly power, 418 00:44:56,060 --> 00:45:02,000 magnetism, physics, but that is so weak that the DI magnetic term here actually has a chance to compete with it. 419 00:45:02,180 --> 00:45:06,080 And in fact, for copper it turns out to be magnetic, surprisingly enough. 420 00:45:06,590 --> 00:45:14,030 So you can with with metals which have only poly power magnetism, you can get die magnetism winning overall. 421 00:45:14,030 --> 00:45:17,940 So challenge your tutors with that. See how many get it right. All right. 422 00:45:18,240 --> 00:45:23,790 So very quickly, I think we can do this in 5 minutes. We can calculate the effect of this down magnetic time. 423 00:45:23,790 --> 00:45:29,130 So the part of the Hamiltonian we're interested in is e squared over 2ma squared. 424 00:45:30,460 --> 00:45:32,800 And we'll put V in the Z direction. 425 00:45:33,130 --> 00:45:48,280 So B equals B Z hat, and so we can write a vector potential one half B cross r so that del across equals B and that will then be B over two. 426 00:45:48,580 --> 00:45:54,040 I guess we'll write y hat x minus x, hat y convenient enough. 427 00:45:54,340 --> 00:46:01,930 So this term in the Hamiltonian we're interested in is then e squared or 8mb squared, x squared plus y squared. 428 00:46:03,400 --> 00:46:12,129 Then the expectation energy of this term in the Hamiltonian, which if we're doing this at zero temperature, which is convenient to do, 429 00:46:12,130 --> 00:46:22,420 that's the change in the energy is the same as the change in the free energy a squared over 8mb squared expectation of x squared plus y squared. 430 00:46:23,020 --> 00:46:30,400 And so the magnetic moment is then minus the expectation of energy, the B. 431 00:46:32,040 --> 00:46:39,200 Which is then minus b e squared over four m expectation of x squared plus y squared. 432 00:46:39,230 --> 00:46:46,220 Now remember in the last lecture I mentioned to you that dear magnetism is a little bit like lenses, lies like a loop of wire. 433 00:46:46,670 --> 00:46:53,390 And if you only have a loop of wire, the amount of magnetic moment you get from a current going around that loop of wire is proportional to its area. 434 00:46:53,780 --> 00:47:01,009 And indeed, this expectation of X squared plus Y squared is the area enclosed by the orbital of the electron. 435 00:47:01,010 --> 00:47:07,760 So it is quite analogous to lenses lies really measuring the currents going around the the atom in the same 436 00:47:07,760 --> 00:47:12,800 way that lenses law gives you a magnetisation associated with the current running around the loop of wire. 437 00:47:14,890 --> 00:47:19,510 Calculating X squared plus y squared is not so difficult in our spherical atoms. 438 00:47:19,510 --> 00:47:24,070 So if we have a spherical atom, most atoms are spherical these days. 439 00:47:24,850 --> 00:47:35,380 Atom we can write that as then two thirds of expectation of x squared plus y squared plus z squared or two thirds of expectation of our square. 440 00:47:36,160 --> 00:47:43,959 And so we can then rewrite the magnetic moment per atom here is minus b e squared over six. 441 00:47:43,960 --> 00:47:50,380 M That six comes from putting together the two thirds with the one quarter their times expectation of our squared 442 00:47:51,550 --> 00:48:03,100 and then the magnetisation is m times rho where rho is the density of electrons of electrons in the entire system. 443 00:48:03,520 --> 00:48:12,849 And that gives us well we differentiate with that with respect to the magnetic field we'll get here chi equals nu, 444 00:48:12,850 --> 00:48:28,780 not the m the B which is n minus nu, not e squared rho size of the atom expectation r squared divided by six times the mass of the electron. 445 00:48:29,110 --> 00:48:36,340 And this is known as the lama result lama, result lama or sometimes lama larger than larger man. 446 00:48:38,920 --> 00:48:47,330 Dear Magnetic. This is more a launch of Iron Dome magnetism, and that is the expectation of this. 447 00:48:47,340 --> 00:48:53,030 Now, in the remaining 2 minutes. I am going to tell you something is not examined about this. 448 00:48:53,030 --> 00:49:00,500 This calculation is very standard for exams calculated by magnetism of an atom are some system of atoms. 449 00:49:02,310 --> 00:49:07,920 There is nothing is non-existent of all, which is that we were a little bit too fast in throwing away the paramagnetic term. 450 00:49:08,310 --> 00:49:12,840 The reason we were too fast, you know, we were we, we initially said, well, 451 00:49:12,840 --> 00:49:18,090 the paramagnetic term is linear and in B the dimming determines quadratic and B so will 452 00:49:18,090 --> 00:49:22,260 treat this term first and then only treat this one if this term happens to be small. 453 00:49:22,680 --> 00:49:30,540 But actually we should be a little more careful that we should worry about this term at second order as well, that if we are careful about it, 454 00:49:30,930 --> 00:49:36,120 we will have to also write some term that looked like this second order of 455 00:49:36,120 --> 00:49:48,030 perturbation result and not equal to zero of an you b b dot l plus gs zero squared. 456 00:49:48,360 --> 00:49:53,790 It is also quadratic and magnetic field e minus one is n. 457 00:49:54,150 --> 00:50:01,740 This term is manifestly negative. The sign of a second order perturbation theory is manifestly negative. 458 00:50:02,010 --> 00:50:08,820 And so it would actually give us a paramagnetic contribution, per mag contribution. 459 00:50:11,410 --> 00:50:19,690 Known as Van Black Power Magnetism Van Black after Nobel laureate who discovered it. 460 00:50:21,630 --> 00:50:33,390 It tends to be small, but it can occur in particular in cases when J equals zero, but L and S are not equal to zero. 461 00:50:33,390 --> 00:50:39,750 So we have j r l equals s, but j equals l minus as equals zero. 462 00:50:40,260 --> 00:50:44,160 And the reason it can occur is because if you remember when we. 463 00:50:45,290 --> 00:50:52,430 We went back here and we said that we could replace this term in the Hamiltonian with j the sum of these two terms. 464 00:50:52,670 --> 00:50:59,060 We were actually presupposing that the magnitude of L, the magnitude of S and the magnitude of J were fixed. 465 00:50:59,360 --> 00:51:03,799 But in fact you can have excitations where you unfixed those, 466 00:51:03,800 --> 00:51:09,590 where you rotate Allen away from each other and change the value of j in the excited states. 467 00:51:09,590 --> 00:51:11,510 And you can get a nonzero contribution of this. 468 00:51:12,000 --> 00:51:18,470 So that is it does not tend to happen for noble gases because in noble gases you have Alan as both being zero. 469 00:51:18,650 --> 00:51:22,510 But there are some elements on the periodic table which are I think it's one less than 470 00:51:22,530 --> 00:51:27,350 half a filled shell where l have an equal s and the combination by rule is zero. 471 00:51:27,530 --> 00:51:33,050 And this term can actually come out and give you a paramagnetic contribution as well that is not examined. 472 00:51:33,260 --> 00:51:38,089 Don't worry about it. It's just for your general edification. We'll talk more about magnetism tomorrow. 473 00:51:38,090 --> 00:51:38,480 Sudan.