Welcome back, everyone. It's the 19th lecture of the condensed matter course nearing the end of sixth week.
I know it's hard to keep keep going on.
At the end of six week, everyone gets pretty tired, but we're really getting pretty close to the end of this course now.
So it's in our power on through.
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.
So we're really thinking about atomic magnetism.
And we had written down the Hamiltonian, actually the Hamiltonian for one electron in one atom in a magnetic field.
And we decided we would write it as the Hamiltonian for that electron in zero magnetic field.
That is the kinetic energy and the attraction to the nucleus plus two terms.
The first term is born magnets on magnetic field dotted inter orbital angular momentum
plus g spin angular momentum and the second term e squared over 2ma squared.
Where is the vector potential del cross a equals v?
Okay. Now these two terms have names. This first term over here is known as the paramagnetic term.
And this term here is known as the Magnetic Town.
And as you might guess, it is this term that is typically responsible repairing magnetism in this term,
which is typically responsible for die magnetism.
And this term is actually fairly obvious why it is that this term is going to be responsible for power magnetism.
Generally, this is just coupling of the magnetic field to the spin or overlying the momentum of of the electron and the.
Okay, now it's a little complicated because of the minus signs,
but the spin one or the overlying momentum both at one point opposite B such that the energy of this becomes negative.
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.
So the moment points opposite of the angular momentum. So any way to or in order to lower the energy,
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,
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,
whereas this term is quadratic in B, in B, and so generally for a small B,
the linear inter in the the paramagnetic term is going to dominate as long as there is not a
particularly good reason why the paramagnetic term is going to be suppressed or somehow zero,
even zero, then the magnetic term is irrelevant and we just have to worry about the paramagnetic term.
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.
But we're going to focus on this one first,
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.
Now, one thing we can do is we can add together the angular momentum of the orbital
angle momentum and the spin angular momentum to get the total angular momentum.
J is L plus sigma, but you'll notice that's not what we actually have in the Hamiltonian.
We have L plus G Sigma.
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,
the magnitude of L, the magnitude of sigma, you can rewrite that entire term in terms of some effective g factor g effective
known as the land g factor is written with a superscript j new b b that j and
there's some formula that allows you to get the g the effect of g factor from
G and the magnitude of J and the magnitude of L and the magnitude of sigma.
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.
If you're taking the atomic physics course, you won't be asked to derive it in this course.
You may be asked to derive in the other course. Any rate, the moral is that you can just some.
You can put together the orbital and spin angular momentum and under the conditions that, you know, a j,
l and Sigma R, you can just think of this as a single angular momentum vector that reorients in a magnetic field.
And the entire term can be it can be summarised as as this is a fairly simple term in the Hamiltonian under those
conditions where we can just think of it as a single total angular momentum being reoriented by the magnetic field.
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,
many electrons in atom, not just a single electron.
And that means we have to keep track of the entire angular momentum capital.
L That's the sum over all the electrons are the orbital angular momentum of each electron, and then the total spin,
angular momentum, which is the sum of all the spin, angular momentum of each of the electrons.
And of course the total angular momentum all together is the total orbital plus, the total spin.
And as you probably learned from your adventures in Eclipse,
Gordon coefficients that you can have very complicated ways of adding together angular momentum.
For example, two two spin. One halves can add together to make a singlet or triplet, and then things get really messy.
And if you have 20 electrons in your atom, they can add together and also two crazy ways.
And so we need something to guide us to figure out how it is that these electron angle momenta are going to add up.
The first thing that we have to guide us is the shell structure of the atom.
So let's remember back to the beginning of the term shell structure.
We had two laws that helped us out.
The first was the alpha principle, which basically alpha which in German I believe means building up or construction.
It means you fill up shells one at a time. And the second principle we had was mat alongs.
Mat alongs were all which told us in which order we should start filling, filling shells.
So that is going to help us out a lot.
And the reason that is going to help us out a lot is because there's is an important principle that a filled shell.
Has L equals X equals j equals zero.
No angular momentum at all in a filled shell. Why is that?
Well, we filled every orbital with a spin up and every orbital with a spin down, so the total spin is zero.
The number of ups is equal to number of downs. How else will we spin?
We filled all the positive states and we equally filled all the negative states in in a filled shell.
So there are these all cancel each other and the total you end up getting out is zero.
So a filled shell is has no anger mentum at all.
And that means we only only look at part filled shells, look at part filled shells.
Okay. So that's going to help us out a lot.
But still, you can have a part filled shell with three, four, five, six, seven electrons in it, in a partially filled out shell.
If it's a big shell, you can have, you know, 13, 14 electrons in a partially filled shell.
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.
And these are known as ones rules, ones rules after one.
And he wrote them down actually first in 1925, a very long time ago,
back in the early days of quantum mechanics, just when people were learning about shells of atoms.
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.
So where that word is, the dogs rules. But so rule one.
Rule one electron spins. Spins a line.
A line, if they can, if they can.
In other words, you maximise s right.
So this is actually a very nice rule because it's kind of like being a power magnet in a fair magnet.
You get all your spins to align and you maximise the total spin.
It's not really a ferromagnetic because we're just talking about a single atom here.
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.
So can you understand why it is that the electron spins would want to align?
Well, okay, so why? Why rule one?
Well, the first thing, it is not because of dipole and dipole interactions.
Dipole, dipole interactions between the spins. Why not?
Yes. The spins do have dipole moments and dipole moments will interact.
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.
It is insanely small. It's so small you can just completely throw it out, totally irrelevant.
So it is not because of this that is important to keep in mind.
What it is do do is due to column interaction due to Coulomb.
Let's see if we can figure out why Coulomb interaction would care if the electron spins were aligned or not aligned well.
So if you open up most solid state physics books or most physics books of any sort,
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.
And the story is sort of qualitatively right, but it has a lot of things wrong with it.
So this is a warning. I think I gave a lot of caveats in the in the book.
So you can read the details. We'll go through this a little bit.
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.
So you have a better idea of what actually is going on.
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,
function should be decomposed into the orbital part, which depends on the position of the two atoms.
Sorry, two electrons and the spin part, which depends on the spins of the two electrons.
Now the overall wave function must be anti symmetric because electrons are fermions.
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.
If the spins are aligned then this is this is then symmetric, symmetric then the orbital path sy orb is anti symmetric.
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.
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.
Anything anti symmetric has to go to zero at zero. So what does that mean?
What that means is that electrons cannot get close to each other.
Cannot get close, close to each other.
And you might think that the Coulomb interaction would care about that.
And indeed, that is the story that's usually told. So the story goes kind of like this If the spins are aligned,
then because of the symmetry of the wave function, the electrons can't get close to each other.
And that makes the Coulomb interaction happy because the interaction doesn't want the electrons to get close to each other either.
So the energy is naturally lower because the electrons are staying further apart from each other.
It's almost right, but is not really right. And the reason it's not really right is because really more important.
More important is the electron nucleus, nucleus nucleus, not the electron electron.
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.
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.
So because the spins are anti-life, they can get close to each other.
And in particular one electron can get in between the other electron in the nucleus.
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,
it can screen the nucleus, it can screen the electron from the nucleus and make the effective nuclear charge look smaller.
As a result, the electron on the outside is weakly bound.
Because it sees a smaller, effective nuclear charge.
On the other hand, if you align the spins of the electrons, then the electrons can't get close to each other.
And in particular, this electron cannot get inside this electron's orbit and it can't screen this electron from the nucleus.
So as a result, the electrons are strongly bound.
And this is more of the reason of why it is that the that it is lower energy to have the two electrons spins
aligned because then the two electrons are strongly bound to the nucleus and that lowers their energy.
So that is more of the honest reason why our first rule applies.
Okay, so let us. We can even generalisations rule up from atoms and molecules if we want.
So suppose we have a molecule molecules. Suppose we have a molecule with two plasma nuclei and say two electrons up here?
Is exactly the same physics. If the two electron spins align,
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.
Then they can get inside of each other and they can screen the nucleus.
And in the end the binding is weaker.
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.
And the other thing that it has to compete with is a covalent bonding.
Bond once anti aligned. Anti align.
We call what we learned about covalent bonding when we had orbitals on two atoms, both with energy inward.
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.
We have two anti align their spins to make a singlet and put them both in low energy state.
So there's a competition between the molecules, there's a competition between covalent bonding physics and physics.
One of them wants the spins to align. One of them wants to spin to anti-life.
So it's a lot more complicated in a molecule to figure out if spins align or if they don't.
So we need to just be aware that those two are competing with each other.
Okay. All right. So that is the first rule.
It tells us that the spins will try to align if they can.
So if you have a shell that is less than half filled, you can put all of the electrons pointing the same direction.
When you get to half filled,
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.
But to the extent that they can, the electrons will always try to align.
Now that said, we still have to figure out what happens to the orbital degree of freedom.
So rule two is very similar, which is basically l is maximised to subject to rule one, to rule one, which is more important.
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.
So let's consider a preceding atom. This is atomic number.
Atomic number is 59. Number equals 59, which is 56 equals filled shells.
I think it's filled shells up to six s actually. So that means we have filled shells plus three electrons.
Okay. So 59 equals filled shells.
Plus three electrons and f official three electrons in four f in well just f shell f is l equals three remember
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.
There is two l plus one orbitals going from minus three lc equals minus three plus three.
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.
And we have to put three electrons in those orbitals.
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.
And then what we want to do is we want to maximise l subject to the condition that all the spins are aligned.
So we can think, well let's try to maximise LC and we will do that by putting the electrons as far
to the right as possible and we will get LC equals six three plus two plus one.
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,
which would be minus six, minus three, minus two, minus one. Okay.
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.
Good. Now we can also write down as immediately is three halfs, three spins, all aligned as equals, three has.
So that is how one rule works. And one rule.
Second rule here is also driven by Coulomb interaction,
but there aren't very good cartoon pictures to explain why it is that all is maximised in one second rule.
So I apologise about that.
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.
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?
So we have rule three where all three on the third rule, which is either.
L is parallel to S or L minus L is parallel to s, and the way it goes is the j.
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,
meaning that the spins anti align with their angular momentum and with their overlying momentum.
And as you choose. Plus, if more than half is more than half.
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.
Now, so, for example, in this case, here we have less than one half filled shell.
So J equals six minus three gas, which is the minus sign because it's less than half filled.
And that gives us j equals nine halves. So we know everything there is to know about Praesidium.
We know what it is. We know it as we know James from her roles.
Okay, so why is third rule true?
Well, one third rule is actually a little bit unlike the other two, because it's not driven by interactions.
It's actually driven by something else. It's driven by our rule three is from rule three.
From spin orbit, coupling. Spin orbit, which is actually a relativistic effect.
Spin orbit coupling. And in fact, it is this is basically that when we wrote down the Hamiltonian,
we left out a small term, the spin orbit term, which takes the following terms.
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,
orbital angular momentum of the electron dotted with the spin.
Angular momentum of the electron. Okay. Now, in this, Hamiltonian alpha is greater than zero.
And so you can sort of see that this is going to make the orbital angular momentum
and the spin angular momentum counter aligned to make this as negative as possible.
You're going to have if the spin is up, the orbital momentum wants to be down and so forth.
But you might wonder, well, what about this business? About half filled shells and less than half a thousand more than half filled shells?
That seems a little bit strange. It turns out that that little strange list is actually just from bookkeeping.
You just have to be careful about the bookkeeping. Every electronic goes in is always following the same rule one.
Rule one, follow rule two. And then try to make this as small as possible.
And if you if you follow that rule, you will always get one rule three.
Let's see how that works for a second. Let's consider a case where we have, I don't know, detail.
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.
Those are the possibilities. And let's put one electron in.
If you put one electron in it, while it's pretty simple, it wants to maximise its spin angular momentum.
So the spin while only has one choice in the spin one half.
So that is easy and it wants to maximise its orbital angular momentum as well.
So we can put this the, the electron way over here or way over here.
Now if you want to minimise the all that sigma, if we put it, if we make the spin spin down,
then you will put it over here so that sigma sigma z equals minus one half, but LC equals plus one half.
So the product of Sigma L is negative, which is what we want.
We want to make that term as negative as possible. And then, sure enough, the sum of these two G is LC minus sigma.
Sorry, LC plus sigma is three halves and as claimed with the less than half filled shell.
So j is the difference of two and one half.
So that works. Basically just telling you that when you add.
We have half less than half filled shell. The spin and the orbit want to be pointing in the opposite direction.
So spin orbit is negative. That is easy.
Okay, now what happens when we get to the shell? When we get to fill the shell?
L.Z., we have equals minus two, minus one, zero one, two.
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.
So here we have s equals five has five spins all pointing in the same direction,
but l here equals zero because we spit, we filled all the negative states and all the plus states as well.
Okay, so here it is. No problem. I mean, third roll doesn't even apply because.
Because our zero spin is just. Uh.
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.
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.
We put all these guys spin down, we add one more electron.
Where is that one electron go?
Well, Kate still wants to satisfy this condition, that it wants its own spin, angular momentum to point opposite its orbital angular momentum.
So it has to go in spin up. So we have to put it over here.
So this last electron we put in has Sigma Z equals plus one half one LC equals minus two.
And that still makes it happy because Sigma Z is pointing opposite LC.
But now notice what's going on here. The total Z, the total Z is actually still negative.
Is actually negative what is negative two? Right.
So the last electron that I put in was pointing up the total spin is actually still pointing down.
So even though the L LC is here is minus two the total because I have to have also had LC total equals zero.
We added one electron that electron is pointing up, but the total spin angular momentum is still pointing down.
So here the total spin, angular momentum and the total overlying momentum and now pointing in the same direction.
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.
So in other words, the reason why we came to this bookkeeping problem is because when we have a half filled shell,
we have s, which is nonzero, but L is zero.
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.
So that is why John's third rule has this complication. That half filled shell.
Is that clear? Yeah, hopefully. Okay.
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.
Okay. So given that we know LS and J, we can then write down our Hamiltonian.
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.
And then we can write down our effective Hamiltonian for these are angular momenta,
which is g effective j mubayi the j and we can do statistical mechanics with this kind of Hamiltonian.
So to remind you how this works, this is something that you did last year.
So example. Take a really simple example.
Let's take j equals one half in a g factor of two.
So I spend can point up with this overall and momentum can point up,
I can point down and the two possible energies are then plus and minus four magnetron times magnetic field.
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
beta you b b plus either the plus beta movie B from the partition function.
I hope this all looks familiar. We can write the free energy b t log z we can write the magnetic moment.
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.
We can then write the magnetic moment the F, the B, which I believe you did this last year.
And if you didn't, you'll do it this year again. Beta UVB, it has this cash form for small.
B are small B this becomes just beta mu b squared times magnetic field, then the magnetisation capital.
M which is moment per volume. Moment for volume is the moment per spin m small m times row the density of spins.
Then Z spins. And then we can get the susceptibility chi, which is by definition not deem to be taken at big or zero.
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.
This is known as Curies Le Curie Law.
In fact, more generally, Curie's law is any time when the susceptibility is some constant over t which in the disease.
So the important thing to notice about this Curie law susceptibility is that it diverges at low temperature.
You have a divergence susceptibility as temperature goes to zero.
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.
The system is always in the lowest energy state.
So if you have a bunch of spins and you have a tiny magnetic field, no matter how tiny that magnetic field,
there is a lowest energy state where the moment is aligned with the magnetic field.
So even with a arbitrarily small magnetic field, the moments all align and point that direction,
which means you then have an infinite susceptibility because it takes only a an
infinitesimal magnetic field to orient all the spins in the same direction.
Is that clear? So this is actually a meaningful statement.
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.
I mean, they mean point vaguely in the direction of the magnetic field, but not all exactly the direction of the magnetic field.
So the susceptibility drops as you go to a higher temperature.
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,
except we start with a partition function which is sum over GC equals minus j two plus j e to the minus beta,
the energy of the spin and state j which is j g effective for j then you b the z.
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.
Otherwise we would use a different axis and it would be if we put V in the X direction,
we would just use a different basis where we count from minus de to J.
Sorry about that. Okay. Then you go through exactly the same manipulations.
The algebra is a little bit more different. Difficult.
You get out some more complicated function and attach it to a function known as a broken function.
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.
No more messy, but the same story comes out at the end of the day. At the end of the day,
you still get a carry law in the sense that it's a constant divided by t for exactly the
same reason in infinitesimal field will orient all the moments in the same direction.
Okay, now one thing actually to note along the way, which is quite important,
is a little bit of a side, is to note that this partition function is a function of B over key only.
It is not a function of being T separately. Okay?
And that is actually important because it means that the the entropy will also be a function of B over T only.
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.
And he noticed that that would mean that you could do the following experiment.
You take a system in a magnetic field at a given temperature.
If you robotically reduce the magnetic field area, Bartok means that the entropy has to stay fixed.
With antibiotics, they reduce the magnetic field. The temperature must change proportional to the magnetic field.
This is what is known as aromatic magnetisation d magnetisation and it is actually a really good way to make a refrigerator,
very powerful refrigerator. You take a system at some temperature in some magnetic field, you reduce the magnetic field.
And in order to keep the entropy fixed, changing things at the temperature must reduce proportionately to the magnetic field.
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.
So it is a good thing to know about. And eventually the technique breaks down and basically breaks down.
When our simple approximation of just independent spins and this partition function being right eventually breaks down at very low temperatures,
will come to that in the next lecture or the one after, I think.
All right. So. This is basically the story of power, magnetism, one atom at a time.
But when you start putting together many atoms at a time, things can be different.
There is this phrase that goes around the condensed matter community.
More is different. Whenever you put lots of things together, you can get different physics.
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.
But when we start putting it in the environment of a bunch of other atoms, things can start changing.
So what can be different? What goes wrong and what is different in a solid.
What different in a solid difference in a solid.
Okay. A couple of things we can write down immediately. One, we can have electron hopping, electron hopping.
And if electrons hop from one atom to the next, we expect to get our band physics.
We expect to get electron band physics bands.
And if you have a partially filled band, you have a metal to band physics.
You can get a metal. And if you are a metal, you expect to have Pauli power magnetism and therefore Pauli power magnetism.
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.
And the reason for this is that you still try to flip over spins when you apply a magnetic field,
but you have to obey the poly exclusion principle and that prevents you from flipping over
a lot of them and it's only a small number of them will flip over in a magnetic field.
So that is one thing that can be different.
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.
So you might ask, well, when? When do we have no hopping?
When is there no hopping? Is there no hopping? Because you know why?
Why would they ever not hop? Why would the electrons ever not hop from one Adam to the next?
Well, basically, this comes from interactions. Ask where they're strong interactions.
And we discussed this before when we talked about Mott Insulators.
Remind you what a Mott Insulators is. Mott INSULATOR is when you have one electron in each site, and due to the Coulomb interaction,
it prevents you from ever having to electrons on the side.
And then you have this idea of a traffic jam of electrons where there is one electron on each
atom and no one can jump to his neighbours because there is already someone sitting there.
So everyone's frozen, but it's frozen with only one electron on each side.
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.
So these are insulators at high well, at least a high temperature.
We'll talk about them at low temperature in the next lecture on after make a fairly good paramagnetic.
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.
Environment can matter.
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.
And you might imagine that in tracking your cell like this, due to the environment,
the orbital angular momentum might want to be pointing up or pointing down directly,
might want want El pointing up or El pointing down and not in the middle.
So even if it was an electron which could point an LC equals all the way from minus three to plus three,
it is still because of the environment it might prefer to point directly up, directly down, and not anywhere in the middle.
And that can change the story quite a bit. This is what's known as crystal field splitting, field splitting,
where the environment of an atom tends to fix which angular momentum are allowed in which ones are not allowed.
And. One thing that happens in this case, which is particularly important,
is sometimes you can get a situation where Crystal Field is actually quite common forces l to basically be zero,
make some sort of superposition of owls that force them to be zero,
in which case which implies that the total j is just then the spin s and l doesn't contribute.
So even if you had f electrons or a partially filled shell where there's lots of electrons and you
might imagine that it would have a non-zero L by one role because of the environment of the atom,
l can be forced to be zero. And then you would only get the spin angular momentum and not the orbital angular momentum.
This is typical of transition metals, which should tell you is called Transition Metals, things like chromium.
This is known as give it a name.
It's known as quenching. Of L when L is basically removed from the problem due to the environment of the atom.
It is typical of transition metals but does not occur.
Does not occur. Occur for rare earths, for rare earths, at rare earth, lantern, eyes and actinide earth atoms.
And the reason for this is a little subtle, but actually not not that complicated.
The reason for this is that in a transition metal, the partially filled shell is, say, the 3D shell.
So here. We have an atom and it is maybe the furthest out shell it's filled is for us.
The partially filled shell is a 3D shell like this, the initial like this,
and the three D shell is very close to the outer edge of the atom in the transition metal.
Transition metal.
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.
Whereas in the rare earth where earth.
You can have something where the outermost shell is something like six s and the partially filled shell is four F,
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.
So basically the the partially filled shell does not even see the environment
because it's so shielded from the environment by the larger orbitals in the atom,
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.
All right, so this is everything we have to say more or less about power magnetism.
Back up to the first equation we wrote on the board here today and worry a little bit about die magnetism.
So when can we get dear magnetism? When is there die of magnetism?
Well, there are a couple ways that you can get dear magnetism.
The first sort of classic case of when you get down magnetism is when j equals zero,
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.
And for example, example, if you have a field shell like a noble gas that would have all equals as equals Jake or zero.
And you can't re-orient any magnetic moment because there is no magnetic moment to reorient.
So in that case, you can basically just throw out this term in the Hamiltonian.
The only thing left is the magnetic term of the Hamiltonian, another case where you can get this magnetism.
I should also mention that not only you can have fill shell, you can also have filled molecular orbitals.
Molecular orbitals. Which, you know, similar.
Like I mentioned this one, we talked about Van der Waals bonding.
You have an inert atom, like a nitrogen to sorry, an inert molecule, like a nitrogen two molecule,
which you can think of as being just a bunch of a filled molecular orbitals.
So it has a net j equals zero at at the molecular level.
And so there's really nothing to re-orient in the case of a,
of a nitrogen two molecule because all the molecular orbitals are filled sort of analogous to to a, a larger atom with orbitals.
There are real shells that are filled. There is another situation where you can get di magnetism,
which is when you have when when you have poly power mag are well, poly power magnetism is weak.
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.
So if you have a metal like copper or something like that,
you would expect it to be a power magnet because it has spins to reorient, but because of the exclusion principle,
they do not re orient very well and you only get a very small susceptibility,
in which case the magnetic term has a chance to compete with the paramagnetic term.
And so here's a a challenge you should ask your tutors is copper die a magnet or a power magnet?
And unless they were during this course last year, in which case students asked them the same question last year.
So for new tutors, most of them will get it wrong.
Most people will say that, well, it's a metal, so it's a poly power magnet, and indeed it has a poly power,
magnetism, physics, but that is so weak that the DI magnetic term here actually has a chance to compete with it.
And in fact, for copper it turns out to be magnetic, surprisingly enough.
So you can with with metals which have only poly power magnetism, you can get die magnetism winning overall.
So challenge your tutors with that. See how many get it right. All right.
So very quickly, I think we can do this in 5 minutes. We can calculate the effect of this down magnetic time.
So the part of the Hamiltonian we're interested in is e squared over 2ma squared.
And we'll put V in the Z direction.
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.
I guess we'll write y hat x minus x, hat y convenient enough.
So this term in the Hamiltonian we're interested in is then e squared or 8mb squared, x squared plus y squared.
Then the expectation energy of this term in the Hamiltonian, which if we're doing this at zero temperature, which is convenient to do,
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.
And so the magnetic moment is then minus the expectation of energy, the B.
Which is then minus b e squared over four m expectation of x squared plus y squared.
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.
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.
And indeed, this expectation of X squared plus Y squared is the area enclosed by the orbital of the electron.
So it is quite analogous to lenses lies really measuring the currents going around the the atom in the same
way that lenses law gives you a magnetisation associated with the current running around the loop of wire.
Calculating X squared plus y squared is not so difficult in our spherical atoms.
So if we have a spherical atom, most atoms are spherical these days.
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.
And so we can then rewrite the magnetic moment per atom here is minus b e squared over six.
M That six comes from putting together the two thirds with the one quarter their times expectation of our squared
and then the magnetisation is m times rho where rho is the density of electrons of electrons in the entire system.
And that gives us well we differentiate with that with respect to the magnetic field we'll get here chi equals nu,
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.
And this is known as the lama result lama, result lama or sometimes lama larger than larger man.
Dear Magnetic. This is more a launch of Iron Dome magnetism, and that is the expectation of this.
Now, in the remaining 2 minutes. I am going to tell you something is not examined about this.
This calculation is very standard for exams calculated by magnetism of an atom are some system of atoms.
There is nothing is non-existent of all, which is that we were a little bit too fast in throwing away the paramagnetic term.
The reason we were too fast, you know, we were we, we initially said, well,
the paramagnetic term is linear and in B the dimming determines quadratic and B so will
treat this term first and then only treat this one if this term happens to be small.
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,
we will have to also write some term that looked like this second order of
perturbation result and not equal to zero of an you b b dot l plus gs zero squared.
It is also quadratic and magnetic field e minus one is n.
This term is manifestly negative. The sign of a second order perturbation theory is manifestly negative.
And so it would actually give us a paramagnetic contribution, per mag contribution.
Known as Van Black Power Magnetism Van Black after Nobel laureate who discovered it.
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.
So we have j r l equals s, but j equals l minus as equals zero.
And the reason it can occur is because if you remember when we.
We went back here and we said that we could replace this term in the Hamiltonian with j the sum of these two terms.
We were actually presupposing that the magnitude of L, the magnitude of S and the magnitude of J were fixed.
But in fact you can have excitations where you unfixed those,
where you rotate Allen away from each other and change the value of j in the excited states.
And you can get a nonzero contribution of this.
So that is it does not tend to happen for noble gases because in noble gases you have Alan as both being zero.
But there are some elements on the periodic table which are I think it's one less than
half a filled shell where l have an equal s and the combination by rule is zero.
And this term can actually come out and give you a paramagnetic contribution as well that is not examined.
Don't worry about it. It's just for your general edification. We'll talk more about magnetism tomorrow.
Sudan.