Before we get going, I want to do a little bit of philosophising about how it is we learn things in in physics.
At the beginning of the week, we started with the Boltzmann model of vibrations in solids, and then we decided that that wasn't quite right.
So we had to improve it with the Einstein model of the solid, and that was pretty good,
but it didn't get the low temperature behaviour right, so we had to improve it with the device theory of solids and that was even better.
But we're going to have to improve it later on in the year as well. And a perfectly valid question is why didn't we learn the right thing at the
beginning and not have to go through all of these models which are all wrong?
And it's not just because I like telling you history stories about how all of this was developed,
but the reason we do this is because this is always how we learn things.
In physics, you always learn the simple model, even though it's incorrect first, because it's a lot easier to think about.
So for example, you learn classical mechanics first and it's not really right because, you know,
there's special relativity and quantum mechanics and you layer those on top later,
but your intuition always falls back to the simple model that's a lot easier to think about.
And that's why we do this. And this is exactly what we're going to be doing today.
When we talk about metals we started last time metals and a simple crude model we use for many,
many things is the Judah picture of electrons in metals, which is basically just kinetic theory, kinetic theory for electrons for electrons.
And at the end of last time we derived the due to transport equation.
The momentum de t equals the force on the electron minus p over some phenomenological scattering
time tao where the force is Lorentz force the general force that the electron feels.
So it's basically just Newton's equation with an added drag force P over Tao,
which sort of represents scattering, sort of slows the electron down in some way.
Now, for many experiments we're going to be interested in, we are actually doing some sort of steady state experiment.
You might apply a steady electric field, you might have a steady current, and you are interested in a steady state result.
So it simplifies things a lot to look for a steady state, so the p t equals zero.
So let's try to solve that so equal zero and put in the force on the right hand side here.
So the force and we replace that by the Lorenz force plus V Crosby and then we need P over Tao over here.
But I'm going replace P by V over Tao just because we have a velocity here, there's the same velocity here, we'll put another velocity there.
Now velocity is a perfectly good quantity, but it's not actually what you are likely to measure in the experiment,
which you're much more likely to measure. An experiment is the current density.
Current density very closely related to velocity j is the number of a lack of density of electrons.
Density of electrons times the charge on the electron, which is minus e times his velocity.
So the current density is just how many electrons you have with the charge they have and how fast they are moving.
So every time I have velocity in that equation, I'm going to plug in J instead and divide by an and in a minus e and one step.
I'm also going to move over to the other side and get this equation e equals.
One over an e j crosby and then plus m over and e squared Tao times j.
That could. Happy with that. Too many steps at once.
Yeah. So I just I just plugged in J for V and then moved to the other side and divide it through by any.
Right. Okay, good. So there's two terms in this equation, and we'll call them two different things.
Let's call this E parallel and we'll call this E hall.
Let me draw a diagram of this. So what we have is we have a block of metal like this.
We're going to run a current through the metal like this contessa in the out like that we might apply a magnetic field,
maybe perpendicular to the metal like this. V And then we will have E parallel in this direction, electric field in that direction,
and then e hall perpendicular to the current and perpendicular to the magnetic field as well.
E Hall And it is called the Hall Electric Field because it was discovered by Edwin Hall, who is doing exactly these kind of experiments in 1879.
And he discovered that when he ran a current through a metal in a magnetic field, he ended up with an electric field perpendicular, both of them.
You may have run into this before. It is a pretty clear result of the Lorenz force.
What's happening is you're running electrons through the material and they're trying to curve because of the magnetic field.
And that builds up an electric field. Good.
Everyone happy with that? More or less. Okay. So let us try to think about this equation a little more closely.
Let's take a simple case. Case one, magnetic field equals zero.
So in that case, we just have electric field. Let me turn it around the other way.
Move the sides of the current is then an E squared now or m times the electric field.
This quantity here is a conductivity sigma.
Of course, conductivity and the conductivity because it relates the electric field to the current.
Now the expression for the conductivity that we derived and e squared tao over m is
the due to conductivity is the conductivity we would calculate in the jury theory.
We just calculated it and we should have pretty easy intuition for what what is going on here.
The conductivity has a factor of density up top because the more electrons you have,
the more current you're going to get and the more conductivity you're going to get. It has a factor of tao up top because longer time,
bigger Tao means longer scattering time and the longer scattering time you have are the less things you have to run into,
the better your conductivity is going to be.
The Factor Mass downstairs looks a little bit more complicated, but actually it just comes from F equals may.
If your math is small for a fixed force, your acceleration has to be larger.
So you apply some fixed force, the electrons move faster if their mass were smaller.
That's why the mass comes out downstairs. Now, we might ask or I might ask, is this a good answer or is this a bad answer?
And right now we actually don't know because Tao is some unknown number, it's some phenomenon.
He doesn't know how to calculate it. We can just we have to put it in phenomena logically.
So really it would fit pretty much anything at this point. So you might think of it sort of turning this on its head.
The measurement of conductivity is actually a measurement of the parameter.
Tao Okay. And that is frequently how it's viewed.
All right, now let's do a little something a little bit more complicated case to case two, which is B, not equal to zero.
And there are various ways we might think about B, not equal to zero.
One possibility of doing this experiment, as we might imagine.
Suppose we know. So let's call this two. Suppose we know a couple of things.
Suppose we know the density of our electrons. Of course we know the charge on the electron.
Suppose we know the current and we measure.
We measure ihall. Then, according to Judith Theory, we then know.
Then we get a magnetic field. B. And in fact, this is a very frequently used method for measuring magnetic fields is known as a hall sensor.
And, you know, it's still used frequently in modern technology now to be a little bit more detailed about this over here.
Generally, if you have a current in a magnetic field, you will get something of the form.
We define this quantity our age because Jay noticed that it's turned the order of J and B over there to over here.
This is just a definition of a commonly used quantity known as the hall coefficient.
So, you know, if you know what Jay is, you know it is measure E, get the whole coefficient, for example.
So. Right. So intruder theory due to theory.
Comparing to that equation. So this is the hall. Sorry, the hall.
In June. A theory comparing to that equation over there. The whole coefficient r h is one over density times minus e.
The minus sign comes from the fact that I flipped the order of these two compared to over there.
So the whole electric field that you would measure would be proportional to the hall coefficient here.
And if you're trying to build a whole sensor in order to find accurate ways of
measuring magnetic fields are using electronics or using voltage measurements.
You really want the whole voltage to be large, as large as possible so that you have a large electric field to measure.
So to do that, what you do is you usually choose a material with a small density.
So you use you small density to get big to get big, big E.
And typical hall sensors are built with semiconductors and other materials that have
a small electron density and will come to semiconductors later on in the term.
Now, let's turn this experiment now on its head, probably the way that Paul Duda thought about it in his case, he probably knew no.
And so in his case, he probably knew the magnetic field, the electron charge in the current measured measure E hall.
And what you get, what you get is the density of electrons in your sample,
because, of course, he didn't know the density of electrons in his materials.
Well, so let's try this. Suppose we do this for a bunch of different materials.
So we'll take a bunch of metals like lithium, sodium, potassium, copper.
So typical good metals. And the first thing the jury would have noticed is that for all of these medals,
our age is less than zero, which is what he predicted by his formula over here.
So that much is good. And then he can actually put in measure the magnitude of Ihal and try to extract the density of electrons.
And what he would have gotten was about 0.8 electrons per atom here,
about 1.2 electrons per atom here about 1.1 electrons per atom 1.5 electrons per atom seems like reasonable numbers.
A couple of electrons wanting one electron per atom more or less.
Is this a good answer or is this a bad answer?
Well, if you think about it for a second, you'll remember your periodic table copper is the 29th element on the periodic table.
So it has 29 protons and 29 electrons and we measured 1.5 electrons per atom where the other 27.5 go.
So this might have been a little bit puzzling to Judge Ruda, but actually, now that we know a little bit about the atomic structure,
we know about the shell structure of atoms, we know a little bit about chemical bonding.
We'll even talk a little bit about chemical bonding later on in the year.
And we know that, in fact, many of the electrons in an atom are actually in orbitals very close to the to the nucleus.
These core orbital electrons are basically stuck.
So core orbitals. But all electrons or orbital electrons don't move.
The only ones that do move out of Shell.
Electrons, also known as valence electrons move so that electrons that are running around in the in the solid are the so called valence electrons.
And, you know, from chemistry, we actually know how many valence electrons these materials have.
And in fact, all these four materials have valence one meaning only one electron in the outermost shell.
So if we assume only the outermost electrons move the ones in the outermost shell, we should predict one electron per atom.
And that is actually in fairly good agreement with what is measured experimentally.
So that is pretty good for a four Judith theory. We get roughly one electron per atom moving around.
So we might, you know, emboldened by our success, we might try some other materials.
So let's try some with valence to valence equals two.
We have materials like beryllium, we have magnesium.
Both of them have our valence too. And all of a sudden we have a big problem.
The big problem is that our h the whole coefficient is now greater than zero.
And this is completely puzzling from this picture, Andrew,
because predecessors that it should be just the density times the charge of the electron which is negative
and so there is no way in Judith theory we're going to get our hall coefficient which is positive.
And this must have puzzled Paul due to terribly and later on in the year when we study van structure of materials,
we'll understand why this is, but it kind of looks like one of two things is going on possibility.
One is that the density of electrons has gone negative for some reason, if that makes any sense.
Possibility two is that the charge carrier,
the thing that is moving around carrying the charge is not the electron but something else with a positive charge.
Neither of those seem to make much sense right now, but they'll make a little bit more sense later on, hopefully,
but in general is a very brave person and he decided that we're going to ignore these two materials that don't seem to fit the materials with valence,
two that have high coefficient positive.
And we're going to go ahead and try to calculate some other quantities that kinetic theory or due to theory should be able to predict.
Now, you studied kinetic theory last year, and one of the things that you were able to calculate in kinetic theory was thermal conductivity.
I'm not going to go through the whole calculation because it's something that
you probably learned very well last year by not worth going through again.
But I'll write down the answer that you probably well should probably look familiar with thermal conductivity for a moment.
Atomic gas is one third the density of particles heat capacity per particle.
This is KV or an and then there's a velocity and a scattering length, which is a velocity times the scattering time.
And in kinetic theory we can take the velocity to b square root of eight k beat over pi times the mass.
Does this all look familiar. Vaguely familiar from last year?
Hopefully. Okay. So what we do is we actually just plug this in.
It's going to be V squared because V occurs in two places.
We'll plug in the expression from kinetic theory over and we have KB from the classical gas physics.
And what we get is the thermal conductivity is four over pi.
So the let's see. So the pi comes from down here having gotten squared.
I guess we got a two over there which counts on the eight to make a four.
Then we have n tao k.v. squared t over m.
So that's the prediction for the thermal conductivity in due theory.
Now, again, we have this question. Is this a good answer or is this a bad answer?
And we don't really know because we have this Tao, this unknown quantity Tao in this equation.
But it's not so bad because we also had sigma is an E squared tao over m and we can look at the ratio of these two quantities and get rid of Tao.
So let's do that. We'll take what is known as L.
The Lawrence number. Lawrence number. Number.
This is not actually the same guy who is Lawrence Transformations and Lawrence Force.
This is Lawrence without the T in his name, Lawrence number.
Everyone seems to have the same name in Germany in the 1800s.
So we can take the ratio of the thermal conductivity to t times.
Sigma here the Taos Council and we're left with just pore over pi and kb over e squared.
That's it. And this result, while if you put in numbers, you discover it's one times ten to the minus eight watt arms per kelvin squared.
And this result is kind of interesting because it's, it's what we call universal.
It does not depend on temperature, it doesn't depend on the mass of electronics, doesn't depend on the scattering time of the electron.
It doesn't pan on the density of electrons.
It doesn't depend on anything that these fundamental quantities, Boltzmann constant and the charge of the electron.
It's kind of an interesting prediction and in fact the fact that the Lorenz number is fixed l the same the same for all materials.
For all materials was known and all t0t this is known as the vitamin Frans Lau Vitamin Frans
and it was known since the mid 1800s and no one had the foggiest idea why it was true.
They just measure that vitamin in France where Germans who like to measure things like thermal conductivity,
electrical conductivity in the middle of 1800s,
and they noticed that if you take this ratio is comes out pretty much the same for all materials I guess if we want to be precise about it.
In fact, for lithium, it comes out about 2.2 what times ten to the minus eight watt ohms per kelvin squared for copper is about 2.0.
For iron, it's about 2.6. So it's pretty close to Julian's prediction.
Okay, it's off by a factor of two, but at least it's in the right ballpark.
It's a very crude theory. And before Drew, no one had had any idea why this ratio should be fixed for all materials, all at all.
So this was really a great step forward. And the intuition that we should have in our heads is actually not that complicated.
The intuition should be that whether we're thinking about heat transport or we're really thinking about electrical transport,
we're really just thinking about electrons moving.
If we're thinking about regular conductivity, we're sort of counting how many electrons move and each electron has a certain amount of charge on it.
Whereas if we're thinking about thermal conductivity, we're still counting how many electrons are moving,
but each electron is moving a certain amount of heat, which is proportional to temperature.
So that ratio of thermal conductivity divided by temperature, then divided by regular conductivity should come out roughly constant because in both,
in all cases you're just counting the number of electrons that move. Okay.
So this was a really good result from from from Juda explaining the Vitamin Franz law.
But there's a really big puzzle. And the puzzle is that we used we used rather glibly the statement.
The CV over DN is three half KB, which is a perfectly good result for a minor atomic gas.
But it's just not true for electrons in the metal. Not true for electrons in metal.
Why not? Well, we measured it and we saw it's not true.
Remember, in metals, the heat capacity is alpha t cubed.
And we identify this alpha as being vibrations or debye by theory.
Plus Gamma Times t. And in fact, if you will, in this game, at times, tea is special for metal.
So this is the heat capacity of the electrons running around in the metal.
And for any reasonable temperature gamma times t is much, much less than three KB as long as T is not ginormous.
Ginormous. That means huge, like, you know, 10,000 kelvin as long as T is not 10,000 kelvin.
Gamma T is much less than three KB.
So somehow or other, even though we use this thing that was clearly incorrect, we used we gave the electrons a heat capacity of three KB each.
Whereas, in fact, if you measure the heat capacity, the metal, that capacity is just not there.
The only capacity the electrons have is is gamma times t, which is much, much less than three KB.
So how did we get this right? What's going on here?
Well, the problem here, the problem here actually becomes much, much more obvious if we look at some other quantities.
Let's look at thermal electric properties. Thermal electric properties.
Electric properties. Bertie, is this the kind of experiment we want to look at?
Here's a block of metal. We want to run a current. So here's a current source.
We run the current through the metal. So the current drags electrons through the metal this way.
And then because the current is moving through the metal, it means the electrons are moving through the metal and therefore heat is out.
I should have the electrons moving this way. Current was in the opposite direction, but the electrons are moving this way.
So then the heat. The heat current gets dragged in the same direction because each electron is dragging a certain amount of heat with it.
So one can define JQ and JQ is now the heat current density.
Heat current. Density.
JQ is equal to some coefficient pi times the regular current and c j and pi is known as the Peltier coefficient.
Peltier coefficient. And this is known as the Peltier Effect after Mr. Peltier, who discovered this effect in 1834.
Another one of these people from the 1800s who like to measure thermal and electrical transport.
Anyway we can actually I should comment here as a bit of an aside, that penalty effect is actually a very technologically useful effect to know about.
You can actually build a very nice refrigerator with it, which has no moving parts.
The idea is that if you want to move heat from one side, the inside of your refrigerator to the outside of the refrigerator to move heat out,
you just run current and it drags heat with this side cold, making that side hot.
In fact, you can buy refrigerators that are built like this that don't have pumps and things like that.
They're not loud. They're very quiet. Frequently, they're actually used as wine refrigerators.
I don't know why, but but you can buy a healthier refrigerator, you know, on the open market these days,
that one thing that one has to keep in mind and probably a refrigerator is that you can't make you can't just keep running more
and more current and getting it colder and colder because at some point there's going to be an ice squared are heat dissipation.
So you'll be dissipating power and if I squared are gets too big, then you start heating things up.
So for small currents, this side will get cold, this side will get hot.
But for large currents, everything gets hot.
So that the art form in building a good, healthier refrigerator is that you have to find the material with a large Peltier coefficient,
but a small resistivity, and there's many, many people whose job it is to find these materials.
All right. Anyway, we're going to try to calculate this Peltier coefficient.
How do we do that? Well, the heat current and the JQ and we have the electrical current.
And so when we know what the electrical current density is, it's the velocity the electrons are moving,
the density of the electrons and the charge on the electrons, the heat, current density, very similar.
It's a lot. While it is a factor of one third, if you're honest about it, that's a usual geometric factor.
Don't worry about it too much. The velocity electrons are moving, the density of the electrons in the heat that each electron is carrying.
Which is CVT, we take the ratio of these two things.
We'll get a Peltier coefficient pi is CBT over three times minus E.
If we plug in the usual kinetic theory result, our CV is three halves.
KB, which are a warning you is not a good idea.
We end up getting KB over two times minus e t and frequently what people do is they actually consider the ratio known as S,
which is pi over t s is known as the C back coefficient.
Quebec was another person from the 1800s who liked to measure electrical and thermal properties.
He I believe he was from Estonia, actually. For anyone who happens to be Estonian anyway.
So if you take that ratio, we just get the universal constant KBE over two times minus E,
which has a numerical value of -0.4 times ten to the minus ten times for volts for Kelvin.
And the problem is here that if you actually measure the seabed coefficient for any metal you discover actually actually as is 100 times smaller.
So this is an obvious much more obvious error in in kinetic theory.
Kinetic theory is going way wrong. And the source of this problem is that we were using this heat capacity for the electrons.
That is just wrong. And the problem comes from y y the problem.
Why the problem? Because in fact, silver end is much, much less than three KB K and it's much, much less than three has KB.
Well, if this is true, if we're using this capacity that's so wrong, why is it that we did okay when we calculated way up there?
When we calculate the thermal conductivity, we got the ratio of the thermal conductivity to the regular conductivity.
We got the Lorenz number almost exactly right. Whereas the seabed coefficient is completely wrong, so something's inconsistent.
Somehow or other, the victim in France law is coming out right. The seabed coefficient is coming out completely wrong.
And the reason that this happens, the reason we got the VMA in France.
All right. The reason why. Why copper is right.
Is because we made two cancelling errors, two cancelling errors.
I was the first one I mentioned already with Silver and CV and is much much less than three have CCB and we use CCB.
But there is another one. We also use V squared expectation of V.
We plugged in the kinetic theory result A.C.T. over pi m and in in actuality these squared is actually much much greater than A.B overpay m.
And these errors in the thermal conductivity, they almost exactly cancel.
So we get the thermal conductivity almost exactly right. Whereas this quantity B square does not enter in the seabed coefficient.
So we get it completely wrong because we use this, but not this.
Okay. Now, the first chance of the year to win fabulous prizes.
One very high quality chocolate bar. Actually, my favourite chocolate bar was sold out.
So this is a slightly less than very high quality chocolate bar. So if anyone who can tell me why we made both of these mistakes.
What do we do wrong? What do we leave out? Come on, come on, come on, come on, come on, come on.
When we forget someone. Someone? Yes. So I'm not going to be the reporter for that, but it's a good idea.
Generally, generally, using expectation of security in the square will make an error, but it's usually only a factor of two kind of error.
So it's a small error. So there's various debates as to which one is better to use in this case.
I agree that that's a potential problem. Anyone else? Yes. Yes.
Yes, that's it for me. And yes, so I would throw this to you, but it would probably miss so you can come down and get it afterwards.
So the thing we left out was probably exclusion principle.
Electrons are fermions. You can't put all the electrons in the same state.
They're not a classical gas in any sense. They are Fermi Gas.
And so we're going to have to treat that properly. Maybe I'll read that, write that down because it's really important.
Fermi Statistics. Fermi statistics.
So we have to deal with that, and that's what we're going to do next.
But before doing that, you can do a little bit of a summary of to theory due to summary.
So due to summary. Many things. Right, many things.
Right. Many transport properties you get right. And I think for homework, your study, a couple of other things that you get right in judiciary,
it's still used very heavily due to theory is used very heavily for understanding metals and semiconductors,
particularly good for semiconductors, but it has problems.
And some of the problems are, well, our age can have wrong sign have wrong sign.
Fine. You can have CV or PN is actually much, much less than three KB where we use three house kbps in the calculations.
And you can get the seatback wrong by wrong by 100, wrong by 100 and so forth.
In fact, you can get the sign of the seabed coefficient wrong as well.
But is being that we're off by a matter in magnitude by a factor of 100, it seems a little bit superfluous to worry about the sign all of a sudden.
Okay.
Anyway, so these are some of the summaries of duty theory, but nonetheless, this was how people understood metals for the first time in the 1900s.
Before Garuda, people didn't understand medals at all and still do the theory theories.
Extremely good way to roughly understand medals,
but nothing really progressed for about 25 years and the theory was all there was and people didn't understand why these things didn't come out right.
But round about 1925, there are a lot of sort of simultaneous enormous advances in physics.
1925 was the Polish exclusion principle, 1926 with the Schrödinger equation, also 1926 with the Fermi Dirac statistics.
And in 1927, someone came along.
Guy named Arnold Sommerfeld. 1927 Sommerfeld.
Sommerfeld was nominated for a Nobel Prize 81 times and never got it.
Poor guy. But his idea was he was going to treat medals with promise statistics.
Treat medals. Metals with Fermi statistics, with Fermi statistics.
In other words, we're going to account for the fact that the electrons are actually fermions.
And so now we have to do a little bit of a review of what we know about Fermi statistics.
And the most important thing we have to remember is the Fermi occupation function
and F of Beta Epsilon minus MU is one over either beta epsilon minus mu plus one.
Here is the chemical potential, chem potential and this NF thing.
And F is the probability.
The problem. Then I can state that and I and I at energy pi at energy and Energy Epsilon is occupied.
Good question. Now. Yeah. Good. All right, good.
So just plot this thing. So we have energy on this axis.
We have an F on this axis. And somewhere over here, we have the chemical potential new at zero temperature.
The Fermi function goes from 1 to 0 as a step function that's supposed to be flat up there.
So this is at T equals zero. It's a perfect step from one down to zero at the chemical potential a t not equal to zero.
It's somewhat smoother. Looks like this. So this is t greater than zero and the width over which it drops from 1 to 0.
This with here is roughly cubed. Okay.
Does this look vaguely familiar from last year? I hope, yeah.
Okay. So now is it useful to have a couple of definitions that we're going to use?
Definition The chemical potential new at equals zero is known as is called the Fermi Energy.
F the Fermi Energy. For me energy.
You'll discover that a lot of things are named after Mr. Fermi.
Now, I should warn you that there is some disagreement in the literature, in the books, as to what you're supposed to call the Fermi Energy.
Some people think that the chemical potential and the Fermi energy are actually synonymous mean the same thing.
Most people, I believe, think that the chemical potential at zero temperature is the Fermi energy,
and that is the definition we're going to work with.
It's actually it makes sense to do that because why would you just have two terms that mean exactly the same thing?
You just call it the chemical potential of you mean chemical potential, potential zero temperature.
We call the Fermi Energy for electrons, which are free electron waves.
We generally have H bar squared K squared over two m is the energy is an energy.
If you have a a wave with wave vector K, its energy will be quite k squared over two m.
So if we fix this thing to be the Fermi energy, then we define k f to satisfy this equation.
So k f is defined by this equation.
This defines k f is the Fermi momentum.
Uh, Fermi. Momentum.
So if we know that from the energy, we just calculate from this equation with the Fermi momentum, is everyone happy with that?
All right. So if we want to if we have some physical system, we want to know how many electrons are in that physical system.
We write in the total number of electrons is the sum over all eigen states ags of the Fermi occupation factor of data,
epsilon of the eigen state minus mu. So why do I write it like this?
So this is the sum of all possible states in the system, the probability that that state will be occupied.
And if you sum up over all eigen states, you get the total number of particles in the system.
Good. Yeah. You can also turn this on its head.
If you know the number of particles in the system, you can use this equation to figure out what the chemical potential is if you have to do that.
Now, as we did in the Dubai theory, we had to frequently sum over overall possible plane waves.
So one thing we should be familiar with at this time,
I promised you we would use this a lot is a sum over all possible plane waves gets replaced by an integral ad3k over two pi cubed.
So every eigen state is a particular plane wave in our box.
So the sum becomes an integral D-3 cubed over two pi times a factor of the volume.
The only thing that's different is I'm going to put a factor of two out front.
And the factor of two is four spins that electrons have two possible spins in one particular plane wave.
It can be either spin up or spin down. So two possible states.
And then we're integrating the Fermi factor, beta epsilon K minus new.
Okay. So far, so good. All right.
So at low temperature, at low t, low t, this thing here is a step function.
This is step which the step function would tell us we should integrate up until integrated number one.
Up until the energy is the Fermi energy.
In other words, all the electrons are filled, all the states are filled below the Fermi energy and everything above the Fermi energy is empty.
So we can rewrite an is to the let's pull out the two pi cubed integral d3k up to k less than kf.
So saying that the absolute value of K is less than if the Fermi momentum is equivalent to saying the energy must be less than the Fermi energy.
Good. Yeah. Okay. Now, this is what this is telling us, is that the filled states here are filled states.
States form a ball. A ball of radius.
Chaos and filled states at zero temperature are usually known as the Fermi C again something
named after Fermi and the surface of the ball surface is known as the Fermi surface.
So the sphere and everywhere around the sphere, the surface of this ball, the energies are all f.
So if you have a a sphere of radius chaos, everything on the surface of that sphere has the same energy and the energy is the Fermi energy can.
All right. So last thing we have to do is we have to actually calculate the volume of that sphere, the volume of that ball.
So we have to be over two pi cubed.
And the value of that integral is just going to give us the volume, which is 4/3 pi k f cubed.
And then with a little bit of rearrangement, we can move the V over to this side, cancel some factors,
we'll get an over V, which is the density and electron density density which is F cubed over three pi squared.
Then solve this for k f or get k f equals three hi squared times the density to the one third.
So if we know the density of electrons, we know k f and then we can plug K back in to our equation,
which I'm just about to scroll off the top to get f so f is h bar squared kf squared over
two m equals h bar squared over two m times three pi squared and to the two thirds.
And this is a rather important equation. We're going to use it later on.
So it tells us where is the Fermi Energy in terms of the density.
So, you know, how big is the Fermi energy? Well, let us first we have to figure out what the density is, what densities we use.
Well, let's try try density of about one electron per atom,
which seemed to work pretty well for through the theory, at least for some of these metals.
If we do that, say for copper, x copper we get our F is approximately seven electron volts.
And is that a big number or a small number? Well, it is useful to convert it to a temperature which is known as the Fermi temperature.
So all the fine Fermi temperature again is something named after Fermi temp, which is t f equals F over k b.
And for copper. For copper, t f is about 80,000 kelvin.
Huge, huge. Huge, huge. So typical Fermi energies and Fermi temperatures in metals are absolutely enormous.
And the reason for this is because there's a lot a lot of electrons.
If one electron per atom, there's a lot of atoms.
When you start putting electrons into your system, you fill up the small, the low energy states first, the ones with the smallest K,
but then those are filled and you have to start filling up higher and higher and higher energy states.
And by the time you put that last electron in, you've gotten up to an enormous energy because the density of these electrons is really, really big.
So let's actually plot the Fermi function again.
So here is an F, here's one. Here is F one chemical potential, approximately 80,000 kelvin.
If I take out a, B and the Fermi function will look like.
This. It drops in a very, very small range. This is cavity tea room.
It's even more exaggerated than that. I've drawn it's even narrower, a little drop than I've then I've drawn,
because this distance here as a temperature is 80,000 kelvin versus our room temperature, which is 300 kelvin.
So this distance over which the, the Fermi function drops from from 1 to 0 is a very, very narrow sliver at the top of the Fermi surface.
And in fact. This picture here gives us a hint as to why the heat capacity of the metal is so low,
so much lower than we would have guessed in order to have heat capacity.
You have to be able to absorb some energy.
So you imagine an electron in some iron state, it absorbs some energy and it jumps up to a nearby eigen state with a little more energy.
Well, all of these electrons down here, they can't absorb any energy because all the iron states near them are already filled.
There's nowhere for them to go. They're completely frozen. They can't absorb any energy at all.
The only things that can absorb energy are the things close to the Fermi surface,
where they can jump above the Fermi surface and absorb energy because there's an empty state there for them to go into.
If there's no empty state to go into. There's no way they can absorb any energy at all.
So that gives us a hint as to why the heat capacity is so low.
One final thing is we can look at the typical velocity, typical velocity, which is known as the Fermi velocity.
Fermi velocity, apparently.
Right. Okay. The F which would be h berkoff divided by the mass is k, k, f, and if you put that in, you get a huge number.
Huge equals huge approximately 1% of the speed of light, which might surprise you.
This is every metal that you've ever run into contact with. Things like copper, lead, silver, tungsten, whatever it is, it has electrons in it,
running around at speeds of 1%, the speed of light, or even greater.
Now, that might sound surprising to you. And in fact, relativity starts to become important if you're doing things carefully at those speeds.
But in fact, it's not surprising once you think about how many electrons there are,
there are tons and tons and tons of electrons, one for every atom or several for every atom in some cases.
And so all the low energy states are completely full. And you just have to keep building up to higher and higher and higher energy states.
And the electrons on the top of the Fermi surface are on the Fermi.
On the Fermi surface in the outer edge of the Fermi ball are extraordinarily high,
have extraordinarily high energy, high kinetic energy, therefore extremely high velocity.
And I guess we stop there and I guess I see you Monday.