Let's a let's get started again. Welcome back.
This is now the 20th lecture of the condensed matter, of course.
When we last left off, we were talking about magnetism and we thinking about magnetism one atom at a time.
And in this lecture, we're going to upgrade and start thinking about collective magnetism.
In other words, magnetism. When we have lots of atoms collecting magnetism, using the theme that more is different.
When you put lots of things together, you can get new and different physics.
Now, we did start a little bit of collective magnetism last time when we were talking about rules.
There was the when we talked about hundreds of molecule, there was the driving force by horns roll to try to align spins between atoms in a molecule.
But there is also the covalent bonding physics which tried to anti align spin so you can make a singlet or covalent bond.
Those two things compete with each other and getting the details right to figure
out whether the spins will align or anti align is sort of a complicated process.
And we're going to, you know, brush that entirely under the rug and try to take a step back and look at a model that's going to sort
of give us much of the physics that we're interested in without getting too bogged down in the details.
So we have a couple of approximation in our model. So our toy model will have one know electron hopping.
So we don't have to worry about any sort of band physics.
We don't have to worry about tight binding model. You're not too worried about Pauli power magnetism.
All of that has been thrown out. So with no electron hopping, we can assume that on each atom there is a spin, spin.
S.I. On each atom. On each atom.
Adam I and I put spin in quotes because this spin, what I really mean is a magnetic moment.
That magnetic moment may be from an actual electron spin. It may be from many electron spins, it may be orbital angular momentum.
It may be some combination of orbital and spin, angular momentum.
But generally when you sort of make these so toI models, you just call it spin no matter what it's actually from.
And then we'll write down the Hamiltonian. So here's a Hamiltonian three H equals and then it has two pieces.
The first piece should look familiar g you b b dot s i that's just the coupling of the spin to the external magnetic field.
This is the term that gave us power magnetism. Then we're going to add to it another term, some over.
I'm going to put up minus sign and one half out front some I and j over atoms i
comma j of j i j as I thought as j so this term here is an interaction term.
So j j is the interaction between atom i and J between atoms i and J and we will set j j equal to j j i.
And this is also sometimes called also equals known as exchange, the exchange constant.
This is how the word exchange is used in condensed matter physics most of the time,
and the factor of one half is put in there to prevent over counting.
We want j12 to be the interaction between atom one and two.
But in that sum there'll be a term S1 dot S2 and another term which is S2 S1 we don't want to over count them, so we just put a one half out front.
Okay. So this model, model three, the Hamiltonian three three is known as the three meaning of the Hamiltonian three
is known as the Heisenberg model after everyone's favourite uncertain physicist.
Heisenberg model. Very frequently studied model in condensed matter physics and a very good representation of of many magnets.
Now we're going to simplify this even further. Simplify.
By setting j aj equal to j if I and J or neighbours i j neighbours, and we're going to set it equal to zero.
Otherwise, in other ways we're just going to keep track of interaction between spins that happen to be nearest neighbours with each other.
It's a fairly good approximation that spins on atoms that are close to each other,
see each other, and if they're farther from each other, they don't see each other.
And then we're also going to simplify. So this is simplification one and simplification two is four.
Now we're going to set the external magnetic field to zero, and we may put it back in later.
So our Hamiltonian now takes the form H is minus one half sum over i j and I'll write like this.
So this notation means I and J our neighbours.
Bracket notation saw a standard j.
S I thought s. J. So we're only swimming over.
I enjoy being neighbours and the constant is just a constant j here.
And there's a couple of possibilities of what you can get out of this sort of Hamiltonian.
So possibility one So let's write down case one.
Case one is that J is greater than zero, in which case in order to make the energy of that Hamiltonian as small as possible,
the two spins on adjacent sides will try to align with each other.
So spins align. And when the spins align, what you get is for magnetism, ferromagnetic magnetism, magnetisation, which is not equal to zero.
Now it doesn't matter in this Hamiltonian which direction the spins all align in, as long as the spins are all aligned with each other.
So they can't be pointing up. They can all be pointing down They can all be pointing left I'll be pointing the right same.
You'll end up with the same energy. What's important is all the spins align with each other.
Okay, so this is the easy case to study. The more complicated case to study is the case.
Two is j less than zero now with j less than zero the neighbours want to anti align.
In order to make the Hamiltonian as small as possible or the energy as small as possible.
So then what we have is if we have a let's see right here.
So if we have like a square lattice like this, each with a spin on it, if the first spin is pointing up,
then the second spin will want to point down in order to make the energy as small as possible.
And then the third spin will point up this one will point down this one, point up this one point down so forth and so on.
And we'll get these alternating spin structures.
This is what's known as an antiferromagnetic.
Or sometimes known as a nail state and e e l natural state after Louis Nile, who realised that these things actually exist in nature.
And then what's interesting about this is that they have magnetic order, magnetic order,
but they have no net magnetisation and equals zero y, they have no net magnetisation.
Well, because you have the same number of up pointing spins as you have down pointing spins.
Well, at least if I, if I drew them more aligned, it would look like there is an equal number of spin.
Pointing spins is down. Pointing spins. My drawing isn't so good.
So how is it that Louis now knew that this sort of situation could occur?
Well, on one of the homework assignments, the last homework assignment, you are meant to calculate the magnetic susceptibility of a state like this.
And there's actually some signature in the magnetic susceptibility that you can look for,
which was which was the thing that Niall is looking at and what made him realise that these Antiferromagnetic states actually exist.
But in the modern era there's actually a very direct way to see this sort of physics, which is to do neutron scattering on it.
So let me actually you write that down. You can see with neutron scattering or neutron diffraction, I guess we should call it.
So the elementary lattice constant here might be a, but the neutrons actually see a lattice constant, which is this big.
This is the unit cell now because the neutrons, if a neutron is coming in with spin up, it sees an up spin as being a different atom from a down spin.
And the reason that neutron seeing spin is being different from a down spin is the
neutrons themselves have spin and they interact differently with the neutron.
Coming in is a spin up atom. It will interact differently with this atom than with this atom because their spins are anti aligned.
So it sees a unit cell which is this big a prime. So when you do your neutron diffraction experiment,
you will see a different size of the unit cell when antiproton magnetism occurs, which will get a bigger unit cell.
I should actually comment one more thing here, which is a little subtle,
but there's a homework assignment on it as well that in this picture I've drawn or I should have drawn if I was a better artist,
outspends down spins ups, spins ups and downs, spin. And that's a very classical way of thinking.
I'm thinking of these spins as actually being, you know, vectors that you can point in any direction in space.
If you have large spins in the sense of spin, you know, five halfs or spin as equals,
two as equals, nine halves or as equals, three or four or something like that.
Then having this picture in your head of these spins is just being vectors that can point any direction in space is a fairly good classical picture,
which is fairly accurate, but if you're talking about spin one halves, then it's not so good anymore.
You really have to think in terms of quantum mechanics and you can't draw these pictures of ups and downs, spin ups and downs,
spin and spin one half Antiferromagnetic is a very complicated beast, and it's not really right to think of it this way.
This is sort of more of a classical intuition that works for large, large spins, but not so well for small spins.
And that will be presumably worked out in a homework assignment.
All right. So one more comment about this.
This case of anti from a magnetism is that you can have antiferromagnetic antiferromagnetic coupling
from ag on a triangular lattice and then you'll get something really strange triangular lattice.
So let's imagine we have a triangle here, three atoms in triangle.
And if the first one happens to have his spin up, the second one up here wants to have a spin pointing in the opposite direction from this one.
So he points spin down and then this one wants to have an opposite point in spin to each of his neighbours.
But he's in trouble. He can point up and then make the opposite direction of this guy, but the same direction as this guy.
Or he can point down which case he's opposite this guy, but same direction at this guy.
So this guy's in trouble and he can't point happily either direction.
So this is what you call a frustrated, frustrated antiferromagnetic.
Everything causes frustration these days.
Okay, so one more case of this, a.k.a. anti-terror magnetism or similar physics to anti fender magnetism that I want to mention,
is the possibility that you have a basis or let's consider a two atom basis are two, two or more or more atom basis for our lattice for our crystal.
So let's consider, for example, a big atom, a small atom, a big atom, small atom, big atom, big them small and big and small.
Big. It's not like this. So here's a two atom basis.
Okay, now imagine that we have antiferromagnetic coupling where the neighbouring atoms want to point their spins in opposite direction.
So the big atom say points up and then the small atom points down big atom points up, small and points down.
Big Adam points up. Okay, we can keep going here and you get the idea and.
Okay, up, down, up, down, up.
If the spin on the big atom is not equal to the spin on the small atom,
then what we'll have is will have magnetisation not equal to zero because you'll have more magnetisation pointing up.
Then you have magnetisation pointing down. Is that is that clear how that works?
So even though you have antiferromagnetic coupling and each neighbouring spin wants to point in the opposite direction,
you get a net magnetisation left over because some of the spins are bigger than the other spins.
And this is what is known as very magnetism, very magnetism.
And it may seem like it's kind of strange or unusual, but it's actually quite common.
In fact, the world's most common magnet occurring in nature, its ion three oxide four,
otherwise known as magnetite or lodestone that you can find in the earth sometimes.
And it's magnetised when you pull it out of the earth and it will stick. Your refrigerator directly is actually a fairy magnet.
It's made up of more than one type of atom, larger, larger moments on one type of atom,
smaller moments on the other, and they aren't aligned with each other.
Then there's an argument in the community as to whether you should call a very magnet, a subset of a magnet.
And this is where, you know, people will argue with each other till they're blue in the face.
And there's no actual good answer to this because people use it both ways.
So some people say that a very magnet is a subset of a fellow magnet because it has non-zero magnetisation.
It will stick to your refrigerator, you know, as the property that we think of as a magnet.
On the other hand, some people think that the word foul magnet should be reserved for the case where all the spins point in the same direction.
And we should call this a fairy magnet to distinguish it from the case where all the signs are pointing in the same direction.
So anyway, you should just be aware that that's there's that little conflict of nomenclature in the community.
Okay, so far so good. We're doing well still. All right. So let's move on to actually maybe write it again.
We'll write down the Heisenberg Hamiltonian H Heisenberg again,
minus one half some of our neighbours i j j as I talked SJ and as I mentioned and let's we're going to sort of focus mostly on J greater than zero.
In other words, the ferromagnetic case.
And as I mentioned this, a Hamiltonian doesn't care which direction the spins point as long as they're all pointing in the same direction.
Now, unfortunately, most materials aren't like that. Most materials actually prefer the spins to point in some particular direction.
And so this is in order to represent that, we need to add another term to the Hamiltonian.
Actually, first, let's this is something I mentioned in the last lecture,
but I'll mention again one thing that would make it obvious that the directions aren't all equivalent is if you imagine
having a very long tetrad and total unit cell where you know the crystal is very long and thin in one direction,
then you might imagine that maybe the spins want to point along, particular the long thin axes.
Maybe so you might have as wanting to be up or wanting to be down, but not any direction in between.
So this is sort of a case where you might expect this sort of anisotropy to exist.
So in order to represent that, we write down an additional term of the Hamiltonian, D, H and I,
Satrapi or H and I, Satrapi will call it D, I'm not sure which might take the form some overall atoms.
Let's put a minus a Kappa out front. That's some constant RSI in the Z direction.
Squared. So why is this going to prefer having the spin in some particular direction?
Well, it wants to maximise the magnitude of Z.
So if C is pointing in the in the direct up direction or the direct down direction, this Hamiltonian is as small as possible.
It gets as negative as you possibly can as compared to having as point in the in on the as the equals zero plane where this gets less negative.
So this kind of term, this anisotropy term is one we're going to consider a lot.
So maybe I'll even write down likes to have s up or down, not as pointing sideways.
Now for a cubic crystal. You might expect something d h cubic, which is a little more complicated.
That would allow the spin to point along any of the crystal and axes,
but not in between the crystal axes in a appropriate Hamiltonian for that might be
something like this as i x the fourth plus s i y to the fourth plus s i z to the fourth.
And the reason I wrote this to the fourth power is because if I wrote them to the squared power,
then you just get a squared percent y squared, percent z squared, which is just s squared.
And that doesn't care which direction the spin is pointing because it's just the magnitude of the spin.
But if you write it as force as the fourth, that's why the fourth is Z the fourth.
Then this prefers to have this been pointing along one of the crystal axes, but not in-between the crystal axes.
So we might consider this one a little bit as well, but mainly we're going to focus on this one.
Just one comment, comment again for small.
What we've written here is sort of based on our intuition for spins.
When we're thinking them as classical VEC, we're thinking of them as classical vectors for small spins.
Small spins are these can be trivial.
These are these terms can be trivial.
What I mean by that. Well, suppose I tried to write this down for a spin one half.
You remember this spin? One half poly operator Sigma Z.
The Pauli Spin Matrix. If you square it for spin one half, spin one half.
The Spin Matrix. Sigma Z squared is just one. It's a very uninteresting operator just gives you a constant.
So if you try to write down a term like this for spin one half spin, it would do nothing.
It wouldn't specify which direction the spin wants to point in.
However, if you did it for a spin two or spin three, it would. That would be an analogous spin matrix, which does not square to one.
It becomes non-trivial. So that's just something to be warned about. You can't write down anisotropic terms for small enough spins.
Okay. All right. So we need to think about this.
And I. Satrapi, which will prefer the spins to point in some directions over other directions.
And we need to think about two possibilities.
One possibility is that the anti Satrapi is strong and the other possibility is that the anisotropy is weak.
So maybe I'll write down the weak case over here. We can I SATRAPI By which I mean that this kappa,
this coefficient that occurs in the anisotropy term is much less than the j the exchange constant that occurs in the Heisenberg term.
Now, in this case, what one has to do is one has to first or most importantly first use spins align with neighbours, spins align with neighbours.
And when I say that first, I mean it's most important to align with your neighbours because J is the big energy scale in the system.
Once you've aligned with your neighbours, second you align along a line with anisotropy, with anisotropy axis.
So first you align with your neighbours and then everyone decides they're better off if they all align with the isotropic axis.
Now if you compare that to the case over here, strong, anisotropy,
strong and I Satrapi meaning this Kappa, this isotropic constant kappa is much bigger than J.
Then first you want to align along and I Satrapi axis align along and I Satrapi.
For example, all spins will point either up or down.
If we're using that up and down and I SATRAPI that everyone has to point either up or down
that's what you do first because that's the big energy scale in the second in the system.
But then second, you align with neighbours.
Align with neighbours. Now this may seem like it's a trivial distinction because at the end of the day the ground state is the same.
The ground state is that everyone's aligned with their neighbours and everyone is aligned along the anisotropy axis.
So why did I make a big deal about this? Well, it makes a difference. The two limits are different.
If you think about what the low energy excitations are.
So in this case, the low energy excitations, energy expectations are small, angular rotations of a whole bunch of spins.
Everyone stays more or less aligned with their neighbours, but they rotate just a little bit from the anisotropy axis.
So this is a small rotations. Small rotations staying aligned.
Aligned with neighbours. Whereas over here the low energy excitations are completely different.
Low energy excitations. Here.
The thing that's important is that you have to stay aligned with the anisotropy axis.
So the low energy excitation is a flip, some spin flips a spin, but keep it aligned with the anisotropy axis.
But stay aligned but stay on.
And I saw isotropic axis. Is it clear what the difference is between these two?
Okay. So this is an interesting case here. And it's sufficiently interesting that people study in a lot and they write
down an effective model for this one model which has the following properties.
We write some sigma I, which is plus or minus s the magnitude of the spin, and then s I is as a vector is sigma I in the Z direction.
So basically we're, we're only letting the spin have complete magnitude, pointing exactly up or full magnitude pointing exactly down.
And then we can write out a Hamiltonian very similar to the Heisenberg form in Heisenberg form h equals.
Okay, so that would be minus one half sum over i j neighbours j sigma i sigma j and then plus we'll couple it to the external field.
G might be the sigma i.
This Hamiltonian is known as the easing model.
Easing model? Because it was invented by lens.
Not easing.
Easing was the graduate student who Lenz gave this to as a project in 1925, and he solved this for his Ph.D. project in one and one dimensional case.
Now, I'm going to make a big deal about this easing model, because this easing model, it's extremely simple, and yet it's also extremely complex.
It's simple because on each side there's only one degree of freedom.
And either it points up or it points down, there's no other possibility.
So it's very simple degrees of freedom. And yet and it couples only to its neighbour.
And yet the physics of this model are extremely complex. And it's probably not an exaggeration to say the world has learned more about
statistical mechanics from this one model than from anything else we possibly study.
So using solve this model in 1925, in one dimension during World War Two,
and Sager very famously managed to solve it without magnetic field in two dimensions.
No one has solved the model, either in two dimensions with a magnetic field or in three dimensions or any higher dimension since then.
So it's been a lot of study. Thousands and thousands of papers have been written about this model since then.
And I should mention, just in the last few years, there's been a really valiant assault against solving this model in three dimensions,
and maybe within a few years there will be a solution. Maybe not.
We can be optimistic. But the reason that this is this model is so interesting is because it shows an awful lot of interesting physics,
particularly about phase transitions, even in this this model that only has one degree of freedom on each on each site.
Okay. So we'll come back to the easing model actually in the next lecture and then we will ask the question where we're going to put it here.
Now let's put it over here. Here. Okay. For any model that we write down.
So starting here, if we have a Hamiltonian, which is H of the Heisenberg form plus the anisotropy form.
And if the anisotropy is strong, we can just replace the combination with the easing Hamiltonian.
Does. Does the whole system align?
The system align align at t equals zero.
And in the model the answer is yes. In model yes, but in real magnets, no.
So in real magic magnets, not all the spins in the system actually do align with each other.
And to understand why this is, we have to realise that we actually left something important out of our Hamiltonian.
So let's do a little bit of a docking experiment. Let's imagine you have a big chunk of some magnetic material.
Another big chunk of some magnetic material. The this one has magnetisation here.
This one has a magnetisation here. Then you bring these two chunks of materials together.
Well, what happens when you've probably done this when you were a kid at some point,
and when you bring them together, you realise that they want to sit like this.
They want to sit head to tail because they have a dipolar interaction.
And so the magnets want to come together, head to tail, and they really don't want to come together head to head like this.
So this is telling us that we we left out some important piece of physics.
And what we left out we left out magnetic energy.
Magnetic energy. That's one way of thinking about it. Or dipole.
Dipole interaction.
In the last lecture, I mentioned that if you were thinking about two spins in a given atom and asking whether they want to align or not,
the dipole dipole interaction between two spins in atom is insanely small, so you could ignore it.
That's true. But if you have a big chunk of the material, you have an insanely large number of electrons there,
each with an insanely small amount of dipole moment, and you put them all together, actually becomes a reasonably big dipole moment.
And you do have to worry about this dipole dipole energy.
Another way of thinking about it, exactly the same physics is to well, okay,
if you think about the magnetic field lines in these two pictures here, magnetic field lines look kind of like this in this picture.
Whereas in this picture, the magnetic field lines kind of look like this.
And you remember with the magnetic energy, total metallic energy is integral d3r of B squared over two of you not.
And there's the total B squared is a lot smaller if you align your dipoles head to tail.
So you're reducing magnetic energy by putting your dipoles had to tail in this way.
So what actually happens in a real physical system?
So we're going to write H equals H, Heisenberg, Heisenberg plus H and I,
Satrapi and Satrapi plus we're going to add H dipolar interaction or magnetic?
Magnetic interaction, magnetic energy. Then what actually happens is you get here's your big system.
The system will break up into what are known as domains, sometimes called vice domains,
vice domains where some point up, some point down, some point up like this and break up into many little domains.
Now, what's going on here at some point in this in this material?
You'll have UPS bins right next to downs bins. And that makes the Heisenberg term in the Hamiltonian very unhappy because the
Heisenberg term of the Hamiltonian wants the UPS spins to be next to up spins.
So there's some energy cost along that, that transition along that, what's what's known as a domain wall.
Maybe I'll even write that down. This is a the transition is called a domain wall.
Between the two. So there's some Heisenberg Energy to have UPS spins next down spins on the domain wall.
But you sacrifice that Heisenberg energy on the domain wall in order to save magnetic energy or dipolar energy for the whole system as a whole.
If if you had a different type of anisotropy and I saw entropy, for example, cubic anisotropy, the domain wall structure might look different.
You might have something that looks like like this or this points up this, points right, spins left down like that.
And this is putting each dipole for a head to tail with the next dipole that improves the energy even better.
Now what happens? In this case is that overall the total magnetisation of one of these magnets is now zero.
Again, because you have just as many up domains as you have down domains and you get no magnetisation at all.
And then you might scratch your head and say, well, wait a second, I know magnets exist.
And, you know, so how is it we're going to get magnetisation out of our magnets?
And so now we have to start thinking a little bit carefully.
The first thing we can do and actually, I guess I have a slide better than drawing that pictures again.
The first thing you can do is you can imagine if you add a magnetic field externally.
To the system, the size of the domains where we re equilibrate and you know, the field that you apply is pointing up.
You'll get more up domains than down domains and you'll get a net magnetisation again.
But still, if you take away the magnetic field, the domains move back to their old equilibrium over here and you'll have no magnetisation.
So this thing that we thought was supposed to be a ferromagnetic is in fact,
a paramagnetic making can be a very good power magnet, which gives you a very high magnetisation for a very small magnetic field.
But it's still, unfortunately, a power magnet. So how are we going to get this thing to be magnetic, even in the absence of an applied magnetic field?
So the whole trick to making a real magnet, the kind of thing that sticks your refrigerator,
is to find a way to prevent the domains from going back from this configuration to this configuration.
So you train a magnet in a high magnetic field, so it gets to one of these configurations where there's more up spins,
then down spins or up moments and down moments in this picture.
But then you find a way to prevent the domains from moving back.
This is what's called pinning. Pinning of domain walls.
Pinning domain walls. And the way it's your typically painted domain wall is with disorder.
And whether one is trying to design a magnet for, you know,
for our use in electrical motors whether use trying to make a magnet that will stick to your refrigerator,
whether you're trying to make a magnet for a magnetic disk drive, the whole art to making good magnets is to figure out how to make the disorder,
prevent the magnetic domains from moving back to their original configuration, where the magnetisation goes back to zero.
Okay, so why is it that disorder is going to help you? All right, so here's here's a picture that's supposed to show us here.
So this is an easing dimension, the whole business about easing versus icing.
I didn't say that. Did. Did I say that? Okay. So let me tell you.
So easing was German. And apparently the way it's pronounced is is is easing in Germany, I believe.
But then after he finished his dissertation, he went to the United States and people called him ising for the rest of his life.
And so everyone the United States calls it the ising model. Everyone in Europe calls it the easing model.
I'll probably never get it straight, but I think it is correct to call it to call it easing.
Anyway, so this is an easing magnet. All the spins are pointing either up or down.
And the higher the interaction, the interaction between spins is unhappy along the domain wall.
And I've drawn all the unhappy spins in red here where the up spins, meet the down spins.
And if you count them from top to bottom, there are 12 of these red bonds where there's a unhappy connection.
Now in this picture, there's a little green atom there, which is supposed to be a defect someplace where there's no spin at all.
If you happen to move over the domain wall to a different position over here, so it goes through the defect.
There's only ten red bonds now, not 12.
So the energy actually drops if the domain wall goes through the defect.
It's a lower energy for domain walls to go through a defect.
So they want to stick on the defect. And that's how you pin domain wall with a with some disorder by putting defects in the system.
The once the the domain walls are in this configuration, you take away the magnetic field,
but then they can't move back because they would have to give up some energy
to move back to their original position because they're stuck on the defect.
That's the general the general physics of how it is you make a real magnet which has m not equal to zero.
So far so good. Okay.
So since the, you know, the fundamental physics is really all wrapped up in the physics of the domain wall and how the domain wall moves,
it's worth thinking about the domain while a little bit more carefully.
And it turns out that most magnets in nature are not of the easing type where the anisotropy is strong,
but there are this type and isotope is quite weak.
So we have to sort of think about what happens in four domain walls and not in the easing limit.
But in the weekend I saw a speed limit over here. There's no question of what a domain while looks like for the easing limit.
You just have, you know, up spins and then they meet down spins and nothing interesting happens in between.
But over here where we have weak and I saw entropy if you have up spins over here and you have down spins way over here,
remember that the most important thing to do for this weekend is not to the limit is to keep everyone as aligned as possible with their neighbour.
So the way you get from up into down spin is to just slightly rotate each spin just a little bit
from his neighbour and eventually make it from pointing up to pointing down with a slow rotation.
And this is what's known as, let's say this goes over and spins.
Okay, I didn't draw that very well, but you get the idea and this is what's known as a block wall.
BLOCK wall or nail wall. Now a wall now wall after Felix block the same blocks Theorem Guy and Louis.
Now the guy who realised that we should have Andy for our magnets.
So let's try to calculate the energy and the width of this on net block.
Okay. There is a slight difference between a block wall and an AL Wall.
It depends on whether the rotation of the spin is in the plane of of these atoms or it's transverse to the plane of the atoms.
It's you know, it's a detail that you probably don't have to worry about, but there's a slight difference between the two.
At any rate, I think what I've drawn here is actually a block wall.
So let's try to calculate the energy of this.
So the energy in this one dimensional picture, it's going to be some over I, I guess minus J, I see plus one.
And here you might have think you might think that I've left out a factor of one half but but I haven't
because here I since I'm summing over I only in one dimension as one as two only occurs in one term now.
Okay. So I got the factor of a half right.
And if I align all the spins aligned, I then get some over I, I guess I equals one two end of minus j as squared.
Everyone is aligned with each other and if I twist it e twist minus e aligned, I'm then going to call it Delta e equals delta e.
The amount of energy you pay to twist the spin from top to bottom that will be
sum over i equals one to end of j s squared times one minus cosine of theta i.
So for each time you twist the spin a little bit,
you have to pay this sort of cosine energy theta is is okay so if one that's theta I the rotation of the spin from one step to the next.
So let's assume that the theta is small and if we need to get from, from all the way to the left,
all the way to write in and spins, let's just declare all the theta is to be pi over n so we have to make a pi rotation and step.
So we'll say theta is pi over. And so cosine theta cosine theta is an approximately one minus one half pi over and squared.
If I plug that into here I'll get a total of and spins.
I'm summing over I have J s squared and one half pi over n squared.
Multiply that out I then get pi squared over to J.
S squared over and. And unsurprisingly, this tells me that the energy is lower.
If I rotate this, the the spins more smoothly that each spin wants to be as aligned as possible with his neighbour.
And so if we're only considering this part of the Hamiltonian, the spins would.
If they rotated they will rotate infinitely slowly and they would be most happy that way.
But they. But this has to compete with the anisotropy energy and isotropic energy.
And Isotropic, which wants the spins to be either pointing directly up or directly down.
And so we have to estimate how much energy you're going to pay to have em spins in this wall, which are neither up nor down.
And okay, this is a rather crude approximation, but we'll just say that each spin that's not pointing up and down, you pay an energy X squared.
Okay. I mean, this this number is probably off by a factor of two or pi or something like that,
but at least it would give us the order of magnitude correctly. So the total energy of the block wall e total is the sum of the Heisenberg Energy pi
squared over 2js squared over RN plus the anisotropy energy and Kappa s squared.
And then we have to minimise this veto at all.
Diane So that equal to zero and that gives us I just differentiate that we just get an equal square root of five squared over to J over Kappa.
And so not surprisingly, what this is telling us is that if the anisotropy is extremely small,
then and the, and the Heisenberg coupling is extremely strong, then the spins rotate extremely slowly.
Whereas if you're in the other limit where the and I Satrapi is large and the Heisenberg coupling small
then the spins rotate in a very small number and maybe even directly from up to down in one in one step.
Now, in real magnets, it's very often the case. Often you can have a case where N is greater than or about equal to 100 atoms.
So you can have these these block walls or these domain walls or now walls be actually quite, quite wide.
Now, this is actually quite important if you start thinking about polycrystalline material.
Polycrystalline kris. Crystal crystalline crystal in.
Magnets. So now we're talking about lots of little crystal lights separated from each other of of magnetic material.
And if the size of the crystal light in number of atoms going across is less, then this magic number the size of the domain wall.
Then the whole atom, the whole crystal light has a point, have its magnetisation pointing in just one direction, has a point and only one direction.
You know, each crystal I can point in a different direction, but each crystal light has to be pointed only in one direction.
And the reason for this is because you don't have enough space within one crystal light to rotate the spin from one direction to another.
It's too stiff. You can't. You know that the coupling between neighbours is too strong in order to get the spins to rotate within one crystal light.
And you might think that this hundred atoms is rather small, you know, because each atom is only an angstrom or two angstroms or something.
But if you have 100 arms in this direction, 100 times another direction,
100 times another direction, all of a sudden you have a million atoms you're talking about.
So these crystallised can actually have an awful lot of atoms in them.
And you can still think of them as microscopic individual magnets pointing in just one direction.
Okay, so, so, so far, so good. I will resist the urge to make jokes about one direction.
All right. Okay, so let's imagine that we take one of these crystal lights and we are going to apply.
So we have a crystal right here. And suppose it has an anisotropy axis in this ray z hat that it wants to point its magnetisation either up or down.
And let's, for the sake of argument, we'll put the magnetic field in that direction as well.
The energy and then the magnetic moment can be pointing in any direction.
We like the energy of that. Chris Light is some insight.
There's going to be minus B and then there's the anisotropy energy minus Kappa Z squared this.
That wants the the magnetisation to point either directly up or directly down.
So if we decide that, okay, so if the angle between M and B is theta, then we can rewrite this energy as E,
not minus m b cosine theta minus kappa m squared, cosine squared theta.
And in fact, we might even let x equals cosine theta.
And then what we have is a quadratic inot minus maybe x minus, kappa, m squared, x squared.
And we can plot this thing. Oops.
Money out of room, aren't I? Okay. We can plot this thing as a function of theta or as a function of x.
So this is the energy. And let's start in the case of B equals zero.
So if you have your crystal light and zero magnetic field and your energy over here is x equals minus one,
minus one, and over here is x equals plus one. What we have here is x equals minus one is theta equals pi, x equals plus one is theta equals zero.
So this over here is the magnetisation pointing down.
And over here we have the crystal light with the magnetisation pointing up and the energy is quadratic.
Looks kind of like this. So there's two stable points.
The stable point here is to have the magnetisation pointing down.
And the stable point here is to have the magnetisation pointing up.
And it's much less happy to have the magnetisation pointing in any other direction.
This is not surprising because it's the anisotropy axis wants this, the magnetisation point either up or down.
Okay. Now let's add a non-zero magnetic field.
So now will it be not equal to zero or be small, but not equal to zero zero?
So now as we plot again, we have a parabola.
It's still quadratic,
but the parabola is now shifted because of the linear term in that in the energy so that the peak of the parabola will not be in the centre anymore.
The off centre. So we'll get a parabola. It looks like this.
And again, over here, what we have is magnetisation pointing down,
even though the magnetic field is pointing up and here we have the magnetisation pointing up and the magnet and the magnetic field pointing up.
So this point here, having the magnetisation pointing in the same direction, the magnetic field, this is definitely the stable minimum.
Whereas this point over here is metastable.
In order to get from the metastable case of the magnetisation pointing in the opposite direction of the magnetic field.
In order to get down to here where it would rather be, it has to get over this activation barrier.
And this means you have to take all of your atoms, maybe a million of these atoms,
and somehow flip them all over simultaneously, either thermally or by a quantum tunnelling.
And that's a process that's so difficult that at low temperature it may never happen.
So the that these you can have these little crystals of magnets have their magnetisation stuck in the wrong direction.
Now let's turn up the magnetic field a little bit more. So now we have a large magnetic field, large B again, we have a parabola,
but the but the maximum the parabola has been shifted all the way off to the left.
The section is minus one. This is x equals plus one. And the maximum of the problem is way over here off the plot.
And so this point here with the the magnetisation pointing in the opposite direction of the magnetic field is now unstable.
And whereas this point over here is the only stable point with the magnetisation pointing the same direction as the magnetic field.
Okay, so if the magnetic field is turned up enough,
then you can flip over all of the magnetisation is to point in the direction of the magnetic field.
It's not clear how that works. Okay, so this is. A picture you've seen before.
It's a picture of a so-called hard foul magnet. Maybe even write that word down because it's used often hard feral magnets.
Hard foul magnet is when you actually have em not equal to zero in b equals zero.
And the way you get hard foul magnets and we explained that how you could get hard from magnets by pinning the domain walls.
But you can also think about getting hard foul magnets from having lots of little crystal lights.
And they you need to in order to push the crystal ice to flip over, you have to give them some coercive field.
You have to give them a finite magnetic field in the wrong direction and in the opposite direction to get them to all flip over.
If you just give out a small magnetic field in the other direction, they don't flip over.
And this is what gives these magnets, hysteresis loops. Now, if you think about it for a second,
you realise that this picture of pinning domain walls in a crystal and magnet, this picture, oops, where to go,
this picture here where the domain wall is stuck on his position because it has an energy activation barrier
to get off of that that position here where the atom gives it a lower energy if the domain wall goes to that,
that point is extremely similar to what we have here with all these little crystal.
The only difference here is that you can think with the crystal lights, the domain walls are always between the crystal lights.
If you want to think of it that way, there's some crystal lights will point in one direction, some crystals will point in the other direction.
And to move the domain wall, you have to flip over one more crystal light.
So it's extremely similar physics.
The moral of the story is that in order to get a real magnet, a hard foul magnet, somehow or other, you have to have disorder in your system,
whether it's making the system into small, little crystallise or putting disorder in, it's going to pin the position of the domain walls.
Okay. That's all for today. I will see you Thursday.
Thursday? Yeah, Thursday. Okay. Have a good weekend.