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Welcome back, everyone. It's the 19th lecture of the condensed matter course nearing the end of sixth week.
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I know it's hard to keep keep going on.
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At the end of six week, everyone gets pretty tired, but we're really getting pretty close to the end of this course now.
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So it's in our power on through.
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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.
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So we're really thinking about atomic magnetism.
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And we had written down the Hamiltonian, actually the Hamiltonian for one electron in one atom in a magnetic field.
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And we decided we would write it as the Hamiltonian for that electron in zero magnetic field.
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That is the kinetic energy and the attraction to the nucleus plus two terms.
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The first term is born magnets on magnetic field dotted inter orbital angular momentum
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plus g spin angular momentum and the second term e squared over 2ma squared.
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Where is the vector potential del cross a equals v?
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Okay. Now these two terms have names. This first term over here is known as the paramagnetic term.
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And this term here is known as the Magnetic Town.
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And as you might guess, it is this term that is typically responsible repairing magnetism in this term,
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which is typically responsible for die magnetism.
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And this term is actually fairly obvious why it is that this term is going to be responsible for power magnetism.
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Generally, this is just coupling of the magnetic field to the spin or overlying the momentum of of the electron and the.
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Okay, now it's a little complicated because of the minus signs,
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but the spin one or the overlying momentum both at one point opposite B such that the energy of this becomes negative.
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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.
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So the moment points opposite of the angular momentum. So any way to or in order to lower the energy,
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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,
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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,
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whereas this term is quadratic in B, in B, and so generally for a small B,
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the linear inter in the the paramagnetic term is going to dominate as long as there is not a
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particularly good reason why the paramagnetic term is going to be suppressed or somehow zero,
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even zero, then the magnetic term is irrelevant and we just have to worry about the paramagnetic term.
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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.
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But we're going to focus on this one first,
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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.
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Now, one thing we can do is we can add together the angular momentum of the orbital
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angle momentum and the spin angular momentum to get the total angular momentum.
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J is L plus sigma, but you'll notice that's not what we actually have in the Hamiltonian.
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We have L plus G Sigma.
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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,
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the magnitude of L, the magnitude of sigma, you can rewrite that entire term in terms of some effective g factor g effective
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known as the land g factor is written with a superscript j new b b that j and
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there's some formula that allows you to get the g the effect of g factor from
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G and the magnitude of J and the magnitude of L and the magnitude of sigma.
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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.
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If you're taking the atomic physics course, you won't be asked to derive it in this course.
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You may be asked to derive in the other course. Any rate, the moral is that you can just some.
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You can put together the orbital and spin angular momentum and under the conditions that, you know, a j,
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l and Sigma R, you can just think of this as a single angular momentum vector that reorients in a magnetic field.
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And the entire term can be it can be summarised as as this is a fairly simple term in the Hamiltonian under those
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conditions where we can just think of it as a single total angular momentum being reoriented by the magnetic field.
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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,
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many electrons in atom, not just a single electron.
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And that means we have to keep track of the entire angular momentum capital.
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L That's the sum over all the electrons are the orbital angular momentum of each electron, and then the total spin,
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angular momentum, which is the sum of all the spin, angular momentum of each of the electrons.
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And of course the total angular momentum all together is the total orbital plus, the total spin.
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And as you probably learned from your adventures in Eclipse,
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Gordon coefficients that you can have very complicated ways of adding together angular momentum.
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For example, two two spin. One halves can add together to make a singlet or triplet, and then things get really messy.
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And if you have 20 electrons in your atom, they can add together and also two crazy ways.
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And so we need something to guide us to figure out how it is that these electron angle momenta are going to add up.
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The first thing that we have to guide us is the shell structure of the atom.
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So let's remember back to the beginning of the term shell structure.
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We had two laws that helped us out.
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The first was the alpha principle, which basically alpha which in German I believe means building up or construction.
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It means you fill up shells one at a time. And the second principle we had was mat alongs.
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Mat alongs were all which told us in which order we should start filling, filling shells.
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So that is going to help us out a lot.
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And the reason that is going to help us out a lot is because there's is an important principle that a filled shell.
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Has L equals X equals j equals zero.
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No angular momentum at all in a filled shell. Why is that?
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Well, we filled every orbital with a spin up and every orbital with a spin down, so the total spin is zero.
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The number of ups is equal to number of downs. How else will we spin?
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We filled all the positive states and we equally filled all the negative states in in a filled shell.
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So there are these all cancel each other and the total you end up getting out is zero.
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So a filled shell is has no anger mentum at all.
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And that means we only only look at part filled shells, look at part filled shells.
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Okay. So that's going to help us out a lot.
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But still, you can have a part filled shell with three, four, five, six, seven electrons in it, in a partially filled out shell.
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If it's a big shell, you can have, you know, 13, 14 electrons in a partially filled shell.
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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.
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And these are known as ones rules, ones rules after one.
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And he wrote them down actually first in 1925, a very long time ago,
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back in the early days of quantum mechanics, just when people were learning about shells of atoms.
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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.
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So where that word is, the dogs rules. But so rule one.
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Rule one electron spins. Spins a line.
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A line, if they can, if they can.
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In other words, you maximise s right.
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So this is actually a very nice rule because it's kind of like being a power magnet in a fair magnet.
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You get all your spins to align and you maximise the total spin.
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It's not really a ferromagnetic because we're just talking about a single atom here.
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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.
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So can you understand why it is that the electron spins would want to align?
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Well, okay, so why? Why rule one?
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Well, the first thing, it is not because of dipole and dipole interactions.
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Dipole, dipole interactions between the spins. Why not?
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Yes. The spins do have dipole moments and dipole moments will interact.
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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.
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It is insanely small. It's so small you can just completely throw it out, totally irrelevant.
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So it is not because of this that is important to keep in mind.
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What it is do do is due to column interaction due to Coulomb.
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Let's see if we can figure out why Coulomb interaction would care if the electron spins were aligned or not aligned well.
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So if you open up most solid state physics books or most physics books of any sort,
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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.
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And the story is sort of qualitatively right, but it has a lot of things wrong with it.
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So this is a warning. I think I gave a lot of caveats in the in the book.
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So you can read the details. We'll go through this a little bit.
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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.
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So you have a better idea of what actually is going on.
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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,
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function should be decomposed into the orbital part, which depends on the position of the two atoms.
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Sorry, two electrons and the spin part, which depends on the spins of the two electrons.
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Now the overall wave function must be anti symmetric because electrons are fermions.
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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.
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If the spins are aligned then this is this is then symmetric, symmetric then the orbital path sy orb is anti symmetric.
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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.
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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.
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Anything anti symmetric has to go to zero at zero. So what does that mean?
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What that means is that electrons cannot get close to each other.
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Cannot get close, close to each other.
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And you might think that the Coulomb interaction would care about that.
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And indeed, that is the story that's usually told. So the story goes kind of like this If the spins are aligned,
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then because of the symmetry of the wave function, the electrons can't get close to each other.
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And that makes the Coulomb interaction happy because the interaction doesn't want the electrons to get close to each other either.
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So the energy is naturally lower because the electrons are staying further apart from each other.
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It's almost right, but is not really right. And the reason it's not really right is because really more important.
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More important is the electron nucleus, nucleus nucleus, not the electron electron.
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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.
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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.
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So because the spins are anti-life, they can get close to each other.
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And in particular one electron can get in between the other electron in the nucleus.
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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,
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it can screen the nucleus, it can screen the electron from the nucleus and make the effective nuclear charge look smaller.
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As a result, the electron on the outside is weakly bound.
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Because it sees a smaller, effective nuclear charge.
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On the other hand, if you align the spins of the electrons, then the electrons can't get close to each other.
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And in particular, this electron cannot get inside this electron's orbit and it can't screen this electron from the nucleus.
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So as a result, the electrons are strongly bound.
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And this is more of the reason of why it is that the that it is lower energy to have the two electrons spins
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aligned because then the two electrons are strongly bound to the nucleus and that lowers their energy.
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So that is more of the honest reason why our first rule applies.
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Okay, so let us. We can even generalisations rule up from atoms and molecules if we want.
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So suppose we have a molecule molecules. Suppose we have a molecule with two plasma nuclei and say two electrons up here?
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Is exactly the same physics. If the two electron spins align,
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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.
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Then they can get inside of each other and they can screen the nucleus.
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And in the end the binding is weaker.
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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.
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And the other thing that it has to compete with is a covalent bonding.
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Bond once anti aligned. Anti align.
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We call what we learned about covalent bonding when we had orbitals on two atoms, both with energy inward.
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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.
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We have two anti align their spins to make a singlet and put them both in low energy state.
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So there's a competition between the molecules, there's a competition between covalent bonding physics and physics.
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One of them wants the spins to align. One of them wants to spin to anti-life.
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So it's a lot more complicated in a molecule to figure out if spins align or if they don't.
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So we need to just be aware that those two are competing with each other.
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Okay. All right. So that is the first rule.
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It tells us that the spins will try to align if they can.
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So if you have a shell that is less than half filled, you can put all of the electrons pointing the same direction.
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When you get to half filled,
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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.
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But to the extent that they can, the electrons will always try to align.
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Now that said, we still have to figure out what happens to the orbital degree of freedom.
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So rule two is very similar, which is basically l is maximised to subject to rule one, to rule one, which is more important.
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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.
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So let's consider a preceding atom. This is atomic number.
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Atomic number is 59. Number equals 59, which is 56 equals filled shells.
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I think it's filled shells up to six s actually. So that means we have filled shells plus three electrons.
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Okay. So 59 equals filled shells.
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Plus three electrons and f official three electrons in four f in well just f shell f is l equals three remember
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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.
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There is two l plus one orbitals going from minus three lc equals minus three plus three.
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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.
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And we have to put three electrons in those orbitals.
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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.
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And then what we want to do is we want to maximise l subject to the condition that all the spins are aligned.
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So we can think, well let's try to maximise LC and we will do that by putting the electrons as far
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to the right as possible and we will get LC equals six three plus two plus one.
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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,
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which would be minus six, minus three, minus two, minus one. Okay.
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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.
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Good. Now we can also write down as immediately is three halfs, three spins, all aligned as equals, three has.
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So that is how one rule works. And one rule.
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Second rule here is also driven by Coulomb interaction,
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but there aren't very good cartoon pictures to explain why it is that all is maximised in one second rule.
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So I apologise about that.
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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.
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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?
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So we have rule three where all three on the third rule, which is either.
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L is parallel to S or L minus L is parallel to s, and the way it goes is the j.
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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,
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meaning that the spins anti align with their angular momentum and with their overlying momentum.
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And as you choose. Plus, if more than half is more than half.
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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.
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Now, so, for example, in this case, here we have less than one half filled shell.
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So J equals six minus three gas, which is the minus sign because it's less than half filled.
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And that gives us j equals nine halves. So we know everything there is to know about Praesidium.
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We know what it is. We know it as we know James from her roles.
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Okay, so why is third rule true?
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Well, one third rule is actually a little bit unlike the other two, because it's not driven by interactions.
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It's actually driven by something else. It's driven by our rule three is from rule three.
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From spin orbit, coupling. Spin orbit, which is actually a relativistic effect.
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Spin orbit coupling. And in fact, it is this is basically that when we wrote down the Hamiltonian,
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we left out a small term, the spin orbit term, which takes the following terms.
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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,
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orbital angular momentum of the electron dotted with the spin.
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Angular momentum of the electron. Okay. Now, in this, Hamiltonian alpha is greater than zero.
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And so you can sort of see that this is going to make the orbital angular momentum
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and the spin angular momentum counter aligned to make this as negative as possible.
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You're going to have if the spin is up, the orbital momentum wants to be down and so forth.
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But you might wonder, well, what about this business? About half filled shells and less than half a thousand more than half filled shells?
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That seems a little bit strange. It turns out that that little strange list is actually just from bookkeeping.
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You just have to be careful about the bookkeeping. Every electronic goes in is always following the same rule one.
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Rule one, follow rule two. And then try to make this as small as possible.
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And if you if you follow that rule, you will always get one rule three.
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Let's see how that works for a second. Let's consider a case where we have, I don't know, detail.
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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.
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Those are the possibilities. And let's put one electron in.
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If you put one electron in it, while it's pretty simple, it wants to maximise its spin angular momentum.
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So the spin while only has one choice in the spin one half.
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So that is easy and it wants to maximise its orbital angular momentum as well.
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So we can put this the, the electron way over here or way over here.
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Now if you want to minimise the all that sigma, if we put it, if we make the spin spin down,
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then you will put it over here so that sigma sigma z equals minus one half, but LC equals plus one half.
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So the product of Sigma L is negative, which is what we want.
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We want to make that term as negative as possible. And then, sure enough, the sum of these two G is LC minus sigma.
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Sorry, LC plus sigma is three halves and as claimed with the less than half filled shell.
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So j is the difference of two and one half.
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So that works. Basically just telling you that when you add.
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We have half less than half filled shell. The spin and the orbit want to be pointing in the opposite direction.
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So spin orbit is negative. That is easy.
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Okay, now what happens when we get to the shell? When we get to fill the shell?
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L.Z., we have equals minus two, minus one, zero one, two.
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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.
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So here we have s equals five has five spins all pointing in the same direction,
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but l here equals zero because we spit, we filled all the negative states and all the plus states as well.
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Okay, so here it is. No problem. I mean, third roll doesn't even apply because.
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Because our zero spin is just. Uh.
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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.
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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.
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We put all these guys spin down, we add one more electron.
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Where is that one electron go?
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Well, Kate still wants to satisfy this condition, that it wants its own spin, angular momentum to point opposite its orbital angular momentum.
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So it has to go in spin up. So we have to put it over here.
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So this last electron we put in has Sigma Z equals plus one half one LC equals minus two.
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And that still makes it happy because Sigma Z is pointing opposite LC.
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But now notice what's going on here. The total Z, the total Z is actually still negative.
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Is actually negative what is negative two? Right.
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So the last electron that I put in was pointing up the total spin is actually still pointing down.
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So even though the L LC is here is minus two the total because I have to have also had LC total equals zero.
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We added one electron that electron is pointing up, but the total spin angular momentum is still pointing down.
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So here the total spin, angular momentum and the total overlying momentum and now pointing in the same direction.
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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.
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So in other words, the reason why we came to this bookkeeping problem is because when we have a half filled shell,
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we have s, which is nonzero, but L is zero.
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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.
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So that is why John's third rule has this complication. That half filled shell.
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Is that clear? Yeah, hopefully. Okay.
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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.
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Okay. So given that we know LS and J, we can then write down our Hamiltonian.
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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.
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And then we can write down our effective Hamiltonian for these are angular momenta,
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which is g effective j mubayi the j and we can do statistical mechanics with this kind of Hamiltonian.
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So to remind you how this works, this is something that you did last year.
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So example. Take a really simple example.
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Let's take j equals one half in a g factor of two.
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So I spend can point up with this overall and momentum can point up,
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I can point down and the two possible energies are then plus and minus four magnetron times magnetic field.
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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
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beta you b b plus either the plus beta movie B from the partition function.
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I hope this all looks familiar. We can write the free energy b t log z we can write the magnetic moment.
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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.
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We can then write the magnetic moment the F, the B, which I believe you did this last year.
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And if you didn't, you'll do it this year again. Beta UVB, it has this cash form for small.
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B are small B this becomes just beta mu b squared times magnetic field, then the magnetisation capital.
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M which is moment per volume. Moment for volume is the moment per spin m small m times row the density of spins.
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Then Z spins. And then we can get the susceptibility chi, which is by definition not deem to be taken at big or zero.
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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.
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This is known as Curies Le Curie Law.
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In fact, more generally, Curie's law is any time when the susceptibility is some constant over t which in the disease.
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So the important thing to notice about this Curie law susceptibility is that it diverges at low temperature.
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You have a divergence susceptibility as temperature goes to zero.
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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.
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The system is always in the lowest energy state.
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So if you have a bunch of spins and you have a tiny magnetic field, no matter how tiny that magnetic field,
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there is a lowest energy state where the moment is aligned with the magnetic field.
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So even with a arbitrarily small magnetic field, the moments all align and point that direction,
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which means you then have an infinite susceptibility because it takes only a an
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infinitesimal magnetic field to orient all the spins in the same direction.
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Is that clear? So this is actually a meaningful statement.
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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.
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I mean, they mean point vaguely in the direction of the magnetic field, but not all exactly the direction of the magnetic field.
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So the susceptibility drops as you go to a higher temperature.
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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,
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except we start with a partition function which is sum over GC equals minus j two plus j e to the minus beta,
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the energy of the spin and state j which is j g effective for j then you b the z.
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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.
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Otherwise we would use a different axis and it would be if we put V in the X direction,
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we would just use a different basis where we count from minus de to J.
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Sorry about that. Okay. Then you go through exactly the same manipulations.
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The algebra is a little bit more different. Difficult.
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You get out some more complicated function and attach it to a function known as a broken function.
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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.
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No more messy, but the same story comes out at the end of the day. At the end of the day,
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you still get a carry law in the sense that it's a constant divided by t for exactly the
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same reason in infinitesimal field will orient all the moments in the same direction.
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Okay, now one thing actually to note along the way, which is quite important,
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is a little bit of a side, is to note that this partition function is a function of B over key only.
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It is not a function of being T separately. Okay?
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And that is actually important because it means that the the entropy will also be a function of B over T only.
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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.
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And he noticed that that would mean that you could do the following experiment.
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You take a system in a magnetic field at a given temperature.
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If you robotically reduce the magnetic field area, Bartok means that the entropy has to stay fixed.
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With antibiotics, they reduce the magnetic field. The temperature must change proportional to the magnetic field.
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This is what is known as aromatic magnetisation d magnetisation and it is actually a really good way to make a refrigerator,
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very powerful refrigerator. You take a system at some temperature in some magnetic field, you reduce the magnetic field.
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And in order to keep the entropy fixed, changing things at the temperature must reduce proportionately to the magnetic field.
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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.
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So it is a good thing to know about. And eventually the technique breaks down and basically breaks down.
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When our simple approximation of just independent spins and this partition function being right eventually breaks down at very low temperatures,
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will come to that in the next lecture or the one after, I think.
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All right. So. This is basically the story of power, magnetism, one atom at a time.
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But when you start putting together many atoms at a time, things can be different.
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There is this phrase that goes around the condensed matter community.
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More is different. Whenever you put lots of things together, you can get different physics.
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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.
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But when we start putting it in the environment of a bunch of other atoms, things can start changing.
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So what can be different? What goes wrong and what is different in a solid.
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What different in a solid difference in a solid.
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Okay. A couple of things we can write down immediately. One, we can have electron hopping, electron hopping.
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And if electrons hop from one atom to the next, we expect to get our band physics.
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We expect to get electron band physics bands.
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And if you have a partially filled band, you have a metal to band physics.
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You can get a metal. And if you are a metal, you expect to have Pauli power magnetism and therefore Pauli power magnetism.
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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.
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And the reason for this is that you still try to flip over spins when you apply a magnetic field,
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but you have to obey the poly exclusion principle and that prevents you from flipping over
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a lot of them and it's only a small number of them will flip over in a magnetic field.
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So that is one thing that can be different.
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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.
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So you might ask, well, when? When do we have no hopping?
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When is there no hopping? Is there no hopping? Because you know why?
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Why would they ever not hop? Why would the electrons ever not hop from one Adam to the next?
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Well, basically, this comes from interactions. Ask where they're strong interactions.
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And we discussed this before when we talked about Mott Insulators.
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Remind you what a Mott Insulators is. Mott INSULATOR is when you have one electron in each site, and due to the Coulomb interaction,
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it prevents you from ever having to electrons on the side.
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And then you have this idea of a traffic jam of electrons where there is one electron on each
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atom and no one can jump to his neighbours because there is already someone sitting there.
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So everyone's frozen, but it's frozen with only one electron on each side.
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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.
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So these are insulators at high well, at least a high temperature.
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We'll talk about them at low temperature in the next lecture on after make a fairly good paramagnetic.
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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.
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Environment can matter.
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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.
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And you might imagine that in tracking your cell like this, due to the environment,
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the orbital angular momentum might want to be pointing up or pointing down directly,
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might want want El pointing up or El pointing down and not in the middle.
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So even if it was an electron which could point an LC equals all the way from minus three to plus three,
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it is still because of the environment it might prefer to point directly up, directly down, and not anywhere in the middle.
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And that can change the story quite a bit. This is what's known as crystal field splitting, field splitting,
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where the environment of an atom tends to fix which angular momentum are allowed in which ones are not allowed.
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And. One thing that happens in this case, which is particularly important,
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is sometimes you can get a situation where Crystal Field is actually quite common forces l to basically be zero,
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make some sort of superposition of owls that force them to be zero,
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in which case which implies that the total j is just then the spin s and l doesn't contribute.
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So even if you had f electrons or a partially filled shell where there's lots of electrons and you
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might imagine that it would have a non-zero L by one role because of the environment of the atom,
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l can be forced to be zero. And then you would only get the spin angular momentum and not the orbital angular momentum.
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This is typical of transition metals, which should tell you is called Transition Metals, things like chromium.
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This is known as give it a name.
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It's known as quenching. Of L when L is basically removed from the problem due to the environment of the atom.
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It is typical of transition metals but does not occur.
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Does not occur. Occur for rare earths, for rare earths, at rare earth, lantern, eyes and actinide earth atoms.
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And the reason for this is a little subtle, but actually not not that complicated.
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The reason for this is that in a transition metal, the partially filled shell is, say, the 3D shell.
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So here. We have an atom and it is maybe the furthest out shell it's filled is for us.
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The partially filled shell is a 3D shell like this, the initial like this,
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and the three D shell is very close to the outer edge of the atom in the transition metal.
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Transition metal.
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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.
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Whereas in the rare earth where earth.
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You can have something where the outermost shell is something like six s and the partially filled shell is four F,
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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.
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So basically the the partially filled shell does not even see the environment
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because it's so shielded from the environment by the larger orbitals in the atom,
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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.
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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?
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00:42:25,690 --> 00:42:29,590
Well, there are a couple ways that you can get dear magnetism.
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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,
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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.
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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.
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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.
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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,
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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.
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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.