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Before we get going, I want to do a little bit of philosophising about how it is we learn things in in physics.
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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.
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So we had to improve it with the Einstein model of the solid, and that was pretty good,
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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.
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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
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beginning and not have to go through all of these models which are all wrong?
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And it's not just because I like telling you history stories about how all of this was developed,
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but the reason we do this is because this is always how we learn things.
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In physics, you always learn the simple model, even though it's incorrect first, because it's a lot easier to think about.
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So for example, you learn classical mechanics first and it's not really right because, you know,
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there's special relativity and quantum mechanics and you layer those on top later,
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but your intuition always falls back to the simple model that's a lot easier to think about.
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And that's why we do this. And this is exactly what we're going to be doing today.
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When we talk about metals we started last time metals and a simple crude model we use for many,
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many things is the Judah picture of electrons in metals, which is basically just kinetic theory, kinetic theory for electrons for electrons.
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And at the end of last time we derived the due to transport equation.
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The momentum de t equals the force on the electron minus p over some phenomenological scattering
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time tao where the force is Lorentz force the general force that the electron feels.
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So it's basically just Newton's equation with an added drag force P over Tao,
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which sort of represents scattering, sort of slows the electron down in some way.
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Now, for many experiments we're going to be interested in, we are actually doing some sort of steady state experiment.
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You might apply a steady electric field, you might have a steady current, and you are interested in a steady state result.
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So it simplifies things a lot to look for a steady state, so the p t equals zero.
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So let's try to solve that so equal zero and put in the force on the right hand side here.
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So the force and we replace that by the Lorenz force plus V Crosby and then we need P over Tao over here.
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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.
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Now velocity is a perfectly good quantity, but it's not actually what you are likely to measure in the experiment,
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which you're much more likely to measure. An experiment is the current density.
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Current density very closely related to velocity j is the number of a lack of density of electrons.
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Density of electrons times the charge on the electron, which is minus e times his velocity.
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So the current density is just how many electrons you have with the charge they have and how fast they are moving.
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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.
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I'm also going to move over to the other side and get this equation e equals.
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One over an e j crosby and then plus m over and e squared Tao times j.
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That could. Happy with that. Too many steps at once.
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Yeah. So I just I just plugged in J for V and then moved to the other side and divide it through by any.
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Right. Okay, good. So there's two terms in this equation, and we'll call them two different things.
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Let's call this E parallel and we'll call this E hall.
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Let me draw a diagram of this. So what we have is we have a block of metal like this.
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We're going to run a current through the metal like this contessa in the out like that we might apply a magnetic field,
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maybe perpendicular to the metal like this. V And then we will have E parallel in this direction, electric field in that direction,
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and then e hall perpendicular to the current and perpendicular to the magnetic field as well.
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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.
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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.
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You may have run into this before. It is a pretty clear result of the Lorenz force.
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What's happening is you're running electrons through the material and they're trying to curve because of the magnetic field.
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And that builds up an electric field. Good.
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Everyone happy with that? More or less. Okay. So let us try to think about this equation a little more closely.
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Let's take a simple case. Case one, magnetic field equals zero.
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So in that case, we just have electric field. Let me turn it around the other way.
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Move the sides of the current is then an E squared now or m times the electric field.
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This quantity here is a conductivity sigma.
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Of course, conductivity and the conductivity because it relates the electric field to the current.
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Now the expression for the conductivity that we derived and e squared tao over m is
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the due to conductivity is the conductivity we would calculate in the jury theory.
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We just calculated it and we should have pretty easy intuition for what what is going on here.
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The conductivity has a factor of density up top because the more electrons you have,
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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,
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bigger Tao means longer scattering time and the longer scattering time you have are the less things you have to run into,
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the better your conductivity is going to be.
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The Factor Mass downstairs looks a little bit more complicated, but actually it just comes from F equals may.
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If your math is small for a fixed force, your acceleration has to be larger.
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So you apply some fixed force, the electrons move faster if their mass were smaller.
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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?
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And right now we actually don't know because Tao is some unknown number, it's some phenomenon.
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He doesn't know how to calculate it. We can just we have to put it in phenomena logically.
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So really it would fit pretty much anything at this point. So you might think of it sort of turning this on its head.
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The measurement of conductivity is actually a measurement of the parameter.
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Tao Okay. And that is frequently how it's viewed.
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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.
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And there are various ways we might think about B, not equal to zero.
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One possibility of doing this experiment, as we might imagine.
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Suppose we know. So let's call this two. Suppose we know a couple of things.
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Suppose we know the density of our electrons. Of course we know the charge on the electron.
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Suppose we know the current and we measure.
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We measure ihall. Then, according to Judith Theory, we then know.
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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.
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And, you know, it's still used frequently in modern technology now to be a little bit more detailed about this over here.
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Generally, if you have a current in a magnetic field, you will get something of the form.
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We define this quantity our age because Jay noticed that it's turned the order of J and B over there to over here.
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This is just a definition of a commonly used quantity known as the hall coefficient.
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So, you know, if you know what Jay is, you know it is measure E, get the whole coefficient, for example.
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So. Right. So intruder theory due to theory.
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Comparing to that equation. So this is the hall. Sorry, the hall.
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In June. A theory comparing to that equation over there. The whole coefficient r h is one over density times minus e.
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The minus sign comes from the fact that I flipped the order of these two compared to over there.
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So the whole electric field that you would measure would be proportional to the hall coefficient here.
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And if you're trying to build a whole sensor in order to find accurate ways of
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measuring magnetic fields are using electronics or using voltage measurements.
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You really want the whole voltage to be large, as large as possible so that you have a large electric field to measure.
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So to do that, what you do is you usually choose a material with a small density.
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So you use you small density to get big to get big, big E.
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And typical hall sensors are built with semiconductors and other materials that have
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a small electron density and will come to semiconductors later on in the term.
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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.
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And so in his case, he probably knew the magnetic field, the electron charge in the current measured measure E hall.
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And what you get, what you get is the density of electrons in your sample,
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because, of course, he didn't know the density of electrons in his materials.
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Well, so let's try this. Suppose we do this for a bunch of different materials.
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So we'll take a bunch of metals like lithium, sodium, potassium, copper.
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So typical good metals. And the first thing the jury would have noticed is that for all of these medals,
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our age is less than zero, which is what he predicted by his formula over here.
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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.
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And what he would have gotten was about 0.8 electrons per atom here,
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about 1.2 electrons per atom here about 1.1 electrons per atom 1.5 electrons per atom seems like reasonable numbers.
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A couple of electrons wanting one electron per atom more or less.
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Is this a good answer or is this a bad answer?
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Well, if you think about it for a second, you'll remember your periodic table copper is the 29th element on the periodic table.
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So it has 29 protons and 29 electrons and we measured 1.5 electrons per atom where the other 27.5 go.
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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,
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we know about the shell structure of atoms, we know a little bit about chemical bonding.
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We'll even talk a little bit about chemical bonding later on in the year.
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And we know that, in fact, many of the electrons in an atom are actually in orbitals very close to the to the nucleus.
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These core orbital electrons are basically stuck.
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So core orbitals. But all electrons or orbital electrons don't move.
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The only ones that do move out of Shell.
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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.
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And, you know, from chemistry, we actually know how many valence electrons these materials have.
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And in fact, all these four materials have valence one meaning only one electron in the outermost shell.
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So if we assume only the outermost electrons move the ones in the outermost shell, we should predict one electron per atom.
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And that is actually in fairly good agreement with what is measured experimentally.
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So that is pretty good for a four Judith theory. We get roughly one electron per atom moving around.
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So we might, you know, emboldened by our success, we might try some other materials.
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So let's try some with valence to valence equals two.
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We have materials like beryllium, we have magnesium.
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Both of them have our valence too. And all of a sudden we have a big problem.
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The big problem is that our h the whole coefficient is now greater than zero.
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And this is completely puzzling from this picture, Andrew,
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because predecessors that it should be just the density times the charge of the electron which is negative
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and so there is no way in Judith theory we're going to get our hall coefficient which is positive.
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And this must have puzzled Paul due to terribly and later on in the year when we study van structure of materials,
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we'll understand why this is, but it kind of looks like one of two things is going on possibility.
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One is that the density of electrons has gone negative for some reason, if that makes any sense.
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Possibility two is that the charge carrier,
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the thing that is moving around carrying the charge is not the electron but something else with a positive charge.
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Neither of those seem to make much sense right now, but they'll make a little bit more sense later on, hopefully,
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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,
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two that have high coefficient positive.
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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.
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Now, you studied kinetic theory last year, and one of the things that you were able to calculate in kinetic theory was thermal conductivity.
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I'm not going to go through the whole calculation because it's something that
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you probably learned very well last year by not worth going through again.
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But I'll write down the answer that you probably well should probably look familiar with thermal conductivity for a moment.
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Atomic gas is one third the density of particles heat capacity per particle.
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This is KV or an and then there's a velocity and a scattering length, which is a velocity times the scattering time.
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And in kinetic theory we can take the velocity to b square root of eight k beat over pi times the mass.
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Does this all look familiar. Vaguely familiar from last year?
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Hopefully. Okay. So what we do is we actually just plug this in.
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It's going to be V squared because V occurs in two places.
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We'll plug in the expression from kinetic theory over and we have KB from the classical gas physics.
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And what we get is the thermal conductivity is four over pi.
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So the let's see. So the pi comes from down here having gotten squared.
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I guess we got a two over there which counts on the eight to make a four.
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Then we have n tao k.v. squared t over m.
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So that's the prediction for the thermal conductivity in due theory.
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Now, again, we have this question. Is this a good answer or is this a bad answer?
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And we don't really know because we have this Tao, this unknown quantity Tao in this equation.
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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.
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So let's do that. We'll take what is known as L.
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The Lawrence number. Lawrence number. Number.
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This is not actually the same guy who is Lawrence Transformations and Lawrence Force.
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This is Lawrence without the T in his name, Lawrence number.
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Everyone seems to have the same name in Germany in the 1800s.
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So we can take the ratio of the thermal conductivity to t times.
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Sigma here the Taos Council and we're left with just pore over pi and kb over e squared.
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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.
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And this result is kind of interesting because it's, it's what we call universal.
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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.
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It doesn't pan on the density of electrons.
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It doesn't depend on anything that these fundamental quantities, Boltzmann constant and the charge of the electron.
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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.
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For all materials was known and all t0t this is known as the vitamin Frans Lau Vitamin Frans
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and it was known since the mid 1800s and no one had the foggiest idea why it was true.
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They just measure that vitamin in France where Germans who like to measure things like thermal conductivity,
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electrical conductivity in the middle of 1800s,
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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.
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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.
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For iron, it's about 2.6. So it's pretty close to Julian's prediction.
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Okay, it's off by a factor of two, but at least it's in the right ballpark.
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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.
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So this was really a great step forward. And the intuition that we should have in our heads is actually not that complicated.
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The intuition should be that whether we're thinking about heat transport or we're really thinking about electrical transport,
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we're really just thinking about electrons moving.
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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.
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Whereas if we're thinking about thermal conductivity, we're still counting how many electrons are moving,
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but each electron is moving a certain amount of heat, which is proportional to temperature.
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So that ratio of thermal conductivity divided by temperature, then divided by regular conductivity should come out roughly constant because in both,
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in all cases you're just counting the number of electrons that move. Okay.
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So this was a really good result from from from Juda explaining the Vitamin Franz law.
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But there's a really big puzzle. And the puzzle is that we used we used rather glibly the statement.
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The CV over DN is three half KB, which is a perfectly good result for a minor atomic gas.
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But it's just not true for electrons in the metal. Not true for electrons in metal.
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Why not? Well, we measured it and we saw it's not true.
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Remember, in metals, the heat capacity is alpha t cubed.
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And we identify this alpha as being vibrations or debye by theory.
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Plus Gamma Times t. And in fact, if you will, in this game, at times, tea is special for metal.
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So this is the heat capacity of the electrons running around in the metal.
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And for any reasonable temperature gamma times t is much, much less than three KB as long as T is not ginormous.
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Ginormous. That means huge, like, you know, 10,000 kelvin as long as T is not 10,000 kelvin.
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Gamma T is much less than three KB.
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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.
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Whereas, in fact, if you measure the heat capacity, the metal, that capacity is just not there.
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The only capacity the electrons have is is gamma times t, which is much, much less than three KB.
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So how did we get this right? What's going on here?
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Well, the problem here, the problem here actually becomes much, much more obvious if we look at some other quantities.
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Let's look at thermal electric properties. Thermal electric properties.
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Electric properties. Bertie, is this the kind of experiment we want to look at?
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Here's a block of metal. We want to run a current. So here's a current source.
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We run the current through the metal. So the current drags electrons through the metal this way.
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And then because the current is moving through the metal, it means the electrons are moving through the metal and therefore heat is out.
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I should have the electrons moving this way. Current was in the opposite direction, but the electrons are moving this way.
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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.
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So one can define JQ and JQ is now the heat current density.
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Heat current. Density.
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JQ is equal to some coefficient pi times the regular current and c j and pi is known as the Peltier coefficient.
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Peltier coefficient. And this is known as the Peltier Effect after Mr. Peltier, who discovered this effect in 1834.
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Another one of these people from the 1800s who like to measure thermal and electrical transport.
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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.
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You can actually build a very nice refrigerator with it, which has no moving parts.
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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,
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you just run current and it drags heat with this side cold, making that side hot.
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In fact, you can buy refrigerators that are built like this that don't have pumps and things like that.
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They're not loud. They're very quiet. Frequently, they're actually used as wine refrigerators.
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I don't know why, but but you can buy a healthier refrigerator, you know, on the open market these days,
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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
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and more current and getting it colder and colder because at some point there's going to be an ice squared are heat dissipation.
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So you'll be dissipating power and if I squared are gets too big, then you start heating things up.
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So for small currents, this side will get cold, this side will get hot.
229
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But for large currents, everything gets hot.
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So that the art form in building a good, healthier refrigerator is that you have to find the material with a large Peltier coefficient,
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but a small resistivity, and there's many, many people whose job it is to find these materials.
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All right. Anyway, we're going to try to calculate this Peltier coefficient.
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How do we do that? Well, the heat current and the JQ and we have the electrical current.
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And so when we know what the electrical current density is, it's the velocity the electrons are moving,
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the density of the electrons and the charge on the electrons, the heat, current density, very similar.
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It's a lot. While it is a factor of one third, if you're honest about it, that's a usual geometric factor.
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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.
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Which is CVT, we take the ratio of these two things.
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We'll get a Peltier coefficient pi is CBT over three times minus E.
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If we plug in the usual kinetic theory result, our CV is three halves.
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KB, which are a warning you is not a good idea.
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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,
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which is pi over t s is known as the C back coefficient.
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Quebec was another person from the 1800s who liked to measure electrical and thermal properties.
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He I believe he was from Estonia, actually. For anyone who happens to be Estonian anyway.
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So if you take that ratio, we just get the universal constant KBE over two times minus E,
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which has a numerical value of -0.4 times ten to the minus ten times for volts for Kelvin.
248
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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.
249
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So this is an obvious much more obvious error in in kinetic theory.
250
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Kinetic theory is going way wrong. And the source of this problem is that we were using this heat capacity for the electrons.
251
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That is just wrong. And the problem comes from y y the problem.
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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.
253
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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?
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When we calculate the thermal conductivity, we got the ratio of the thermal conductivity to the regular conductivity.
255
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We got the Lorenz number almost exactly right. Whereas the seabed coefficient is completely wrong, so something's inconsistent.
256
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Somehow or other, the victim in France law is coming out right. The seabed coefficient is coming out completely wrong.
257
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And the reason that this happens, the reason we got the VMA in France.
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All right. The reason why. Why copper is right.
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Is because we made two cancelling errors, two cancelling errors.
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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.
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But there is another one. We also use V squared expectation of V.
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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.
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And these errors in the thermal conductivity, they almost exactly cancel.
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So we get the thermal conductivity almost exactly right. Whereas this quantity B square does not enter in the seabed coefficient.
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So we get it completely wrong because we use this, but not this.
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Okay. Now, the first chance of the year to win fabulous prizes.
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One very high quality chocolate bar. Actually, my favourite chocolate bar was sold out.
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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.
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What do we do wrong? What do we leave out? Come on, come on, come on, come on, come on, come on.
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When we forget someone. Someone? Yes. So I'm not going to be the reporter for that, but it's a good idea.
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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.
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So it's a small error. So there's various debates as to which one is better to use in this case.
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I agree that that's a potential problem. Anyone else? Yes. Yes.
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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.
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So the thing we left out was probably exclusion principle.
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Electrons are fermions. You can't put all the electrons in the same state.
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They're not a classical gas in any sense. They are Fermi Gas.
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And so we're going to have to treat that properly. Maybe I'll read that, write that down because it's really important.
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Fermi Statistics. Fermi statistics.
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So we have to deal with that, and that's what we're going to do next.
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But before doing that, you can do a little bit of a summary of to theory due to summary.
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So due to summary. Many things. Right, many things.
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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,
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it's still used very heavily due to theory is used very heavily for understanding metals and semiconductors,
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particularly good for semiconductors, but it has problems.
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And some of the problems are, well, our age can have wrong sign have wrong sign.
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Fine. You can have CV or PN is actually much, much less than three KB where we use three house kbps in the calculations.
288
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And you can get the seatback wrong by wrong by 100, wrong by 100 and so forth.
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In fact, you can get the sign of the seabed coefficient wrong as well.
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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.
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Okay.
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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.
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Before Garuda, people didn't understand medals at all and still do the theory theories.
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Extremely good way to roughly understand medals,
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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.
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But round about 1925, there are a lot of sort of simultaneous enormous advances in physics.
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1925 was the Polish exclusion principle, 1926 with the SchrÃ¶dinger equation, also 1926 with the Fermi Dirac statistics.
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And in 1927, someone came along.
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Guy named Arnold Sommerfeld. 1927 Sommerfeld.
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Sommerfeld was nominated for a Nobel Prize 81 times and never got it.
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Poor guy. But his idea was he was going to treat medals with promise statistics.
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Treat medals. Metals with Fermi statistics, with Fermi statistics.
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In other words, we're going to account for the fact that the electrons are actually fermions.
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And so now we have to do a little bit of a review of what we know about Fermi statistics.
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And the most important thing we have to remember is the Fermi occupation function
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and F of Beta Epsilon minus MU is one over either beta epsilon minus mu plus one.
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Here is the chemical potential, chem potential and this NF thing.
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And F is the probability.
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The problem. Then I can state that and I and I at energy pi at energy and Energy Epsilon is occupied.
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Good question. Now. Yeah. Good. All right, good.
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So just plot this thing. So we have energy on this axis.
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We have an F on this axis. And somewhere over here, we have the chemical potential new at zero temperature.
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The Fermi function goes from 1 to 0 as a step function that's supposed to be flat up there.
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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.
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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.
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This with here is roughly cubed. Okay.
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Does this look vaguely familiar from last year? I hope, yeah.
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Okay. So now is it useful to have a couple of definitions that we're going to use?
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Definition The chemical potential new at equals zero is known as is called the Fermi Energy.
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F the Fermi Energy. For me energy.
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You'll discover that a lot of things are named after Mr. Fermi.
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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.
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Some people think that the chemical potential and the Fermi energy are actually synonymous mean the same thing.
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Most people, I believe, think that the chemical potential at zero temperature is the Fermi energy,
325
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and that is the definition we're going to work with.
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It's actually it makes sense to do that because why would you just have two terms that mean exactly the same thing?
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You just call it the chemical potential of you mean chemical potential, potential zero temperature.
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We call the Fermi Energy for electrons, which are free electron waves.
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We generally have H bar squared K squared over two m is the energy is an energy.
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If you have a a wave with wave vector K, its energy will be quite k squared over two m.
331
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So if we fix this thing to be the Fermi energy, then we define k f to satisfy this equation.
332
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So k f is defined by this equation.
333
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This defines k f is the Fermi momentum.
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Uh, Fermi. Momentum.
335
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So if we know that from the energy, we just calculate from this equation with the Fermi momentum, is everyone happy with that?
336
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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.
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We write in the total number of electrons is the sum over all eigen states ags of the Fermi occupation factor of data,
338
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epsilon of the eigen state minus mu. So why do I write it like this?
339
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So this is the sum of all possible states in the system, the probability that that state will be occupied.
340
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And if you sum up over all eigen states, you get the total number of particles in the system.
341
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Good. Yeah. You can also turn this on its head.
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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.
343
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Now, as we did in the Dubai theory, we had to frequently sum over overall possible plane waves.
344
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So one thing we should be familiar with at this time,
345
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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.
346
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So every eigen state is a particular plane wave in our box.
347
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So the sum becomes an integral D-3 cubed over two pi times a factor of the volume.
348
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The only thing that's different is I'm going to put a factor of two out front.
349
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And the factor of two is four spins that electrons have two possible spins in one particular plane wave.
350
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It can be either spin up or spin down. So two possible states.
351
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And then we're integrating the Fermi factor, beta epsilon K minus new.
352
00:38:33,270 --> 00:38:36,700
Okay. So far, so good. All right.
353
00:38:37,330 --> 00:38:44,470
So at low temperature, at low t, low t, this thing here is a step function.
354
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This is step which the step function would tell us we should integrate up until integrated number one.
355
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Up until the energy is the Fermi energy.
356
00:38:56,410 --> 00:39:03,340
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.
357
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So we can rewrite an is to the let's pull out the two pi cubed integral d3k up to k less than kf.
358
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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.
359
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Good. Yeah. Okay. Now, this is what this is telling us, is that the filled states here are filled states.
360
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States form a ball. A ball of radius.
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Chaos and filled states at zero temperature are usually known as the Fermi C again something
362
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named after Fermi and the surface of the ball surface is known as the Fermi surface.
363
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So the sphere and everywhere around the sphere, the surface of this ball, the energies are all f.
364
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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.
365
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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.
366
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So we have to be over two pi cubed.
367
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And the value of that integral is just going to give us the volume, which is 4/3 pi k f cubed.
368
00:40:44,720 --> 00:40:52,010
And then with a little bit of rearrangement, we can move the V over to this side, cancel some factors,
369
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we'll get an over V, which is the density and electron density density which is F cubed over three pi squared.
370
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Then solve this for k f or get k f equals three hi squared times the density to the one third.
371
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So if we know the density of electrons, we know k f and then we can plug K back in to our equation,
372
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which I'm just about to scroll off the top to get f so f is h bar squared kf squared over
373
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two m equals h bar squared over two m times three pi squared and to the two thirds.
374
00:41:47,560 --> 00:41:52,270
And this is a rather important equation. We're going to use it later on.
375
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So it tells us where is the Fermi Energy in terms of the density.
376
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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.
377
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Well, let's try try density of about one electron per atom,
378
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which seemed to work pretty well for through the theory, at least for some of these metals.
379
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If we do that, say for copper, x copper we get our F is approximately seven electron volts.
380
00:42:28,700 --> 00:42:35,899
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.
381
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So all the fine Fermi temperature again is something named after Fermi temp, which is t f equals F over k b.
382
00:42:47,250 --> 00:42:53,340
And for copper. For copper, t f is about 80,000 kelvin.
383
00:42:56,340 --> 00:43:03,750
Huge, huge. Huge, huge. So typical Fermi energies and Fermi temperatures in metals are absolutely enormous.
384
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And the reason for this is because there's a lot a lot of electrons.
385
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If one electron per atom, there's a lot of atoms.
386
00:43:09,990 --> 00:43:15,990
When you start putting electrons into your system, you fill up the small, the low energy states first, the ones with the smallest K,
387
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but then those are filled and you have to start filling up higher and higher and higher energy states.
388
00:43:19,950 --> 00:43:27,120
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.
389
00:43:27,540 --> 00:43:30,780
So let's actually plot the Fermi function again.
390
00:43:31,770 --> 00:43:39,900
So here is an F, here's one. Here is F one chemical potential, approximately 80,000 kelvin.
391
00:43:41,610 --> 00:43:45,450
If I take out a, B and the Fermi function will look like.
392
00:43:47,180 --> 00:43:51,620
This. It drops in a very, very small range. This is cavity tea room.
393
00:43:53,360 --> 00:43:59,479
It's even more exaggerated than that. I've drawn it's even narrower, a little drop than I've then I've drawn,
394
00:43:59,480 --> 00:44:07,940
because this distance here as a temperature is 80,000 kelvin versus our room temperature, which is 300 kelvin.
395
00:44:07,940 --> 00:44:18,620
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.
396
00:44:18,890 --> 00:44:26,470
And in fact. This picture here gives us a hint as to why the heat capacity of the metal is so low,
397
00:44:26,490 --> 00:44:29,690
so much lower than we would have guessed in order to have heat capacity.
398
00:44:29,870 --> 00:44:31,910
You have to be able to absorb some energy.
399
00:44:32,210 --> 00:44:39,680
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.
400
00:44:39,920 --> 00:44:48,410
Well, all of these electrons down here, they can't absorb any energy because all the iron states near them are already filled.
401
00:44:48,470 --> 00:44:52,430
There's nowhere for them to go. They're completely frozen. They can't absorb any energy at all.
402
00:44:52,670 --> 00:44:56,870
The only things that can absorb energy are the things close to the Fermi surface,
403
00:44:56,870 --> 00:45:02,870
where they can jump above the Fermi surface and absorb energy because there's an empty state there for them to go into.
404
00:45:02,870 --> 00:45:08,180
If there's no empty state to go into. There's no way they can absorb any energy at all.
405
00:45:08,420 --> 00:45:13,670
So that gives us a hint as to why the heat capacity is so low.
406
00:45:13,970 --> 00:45:22,880
One final thing is we can look at the typical velocity, typical velocity, which is known as the Fermi velocity.
407
00:45:24,050 --> 00:45:28,490
Fermi velocity, apparently.
408
00:45:29,270 --> 00:45:41,360
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.
409
00:45:41,630 --> 00:45:49,010
Huge equals huge approximately 1% of the speed of light, which might surprise you.
410
00:45:49,250 --> 00:45:57,290
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,
411
00:45:57,290 --> 00:46:02,000
running around at speeds of 1%, the speed of light, or even greater.
412
00:46:02,240 --> 00:46:09,350
Now, that might sound surprising to you. And in fact, relativity starts to become important if you're doing things carefully at those speeds.
413
00:46:09,620 --> 00:46:15,080
But in fact, it's not surprising once you think about how many electrons there are,
414
00:46:15,260 --> 00:46:19,430
there are tons and tons and tons of electrons, one for every atom or several for every atom in some cases.
415
00:46:19,700 --> 00:46:24,620
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.
416
00:46:24,800 --> 00:46:27,890
And the electrons on the top of the Fermi surface are on the Fermi.
417
00:46:28,220 --> 00:46:33,230
On the Fermi surface in the outer edge of the Fermi ball are extraordinarily high,
418
00:46:33,260 --> 00:46:38,569
have extraordinarily high energy, high kinetic energy, therefore extremely high velocity.
419
00:46:38,570 --> 00:46:40,730
And I guess we stop there and I guess I see you Monday.