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This is the second lecture of the condensed matter. Of course. Welcome back. Three quick things before we start.
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A bunch of people, after the first lecture asked me what's the title of the book so they can find it in the bookstore?
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It's the exercise basics. Now you can find it. The second thing I want to encourage you to use the message board.
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No one's used it yet. Someone be brave. Post the first message.
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Once you get going, I promise you'll find it useful. The third thing I didn't emphasise this enough last time.
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If there's any problems in the course, any sort of error, typos in the homework sets, typos in the book, anything I say that seems wrong in lecture,
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please come to me immediately and I can issue corrections and therefore not get everyone is confused as much as we can.
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So just, you know, either post a message on the message boards and the email,
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catch me off to lecture or you know, I know it's crazy, but you can call me maybe.
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Yeah. Anyway, so last time we left off, we were talking about high capacity.
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Of solids. And we started with the Boltzmann picture of a solid and Boltzmann picture.
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Boltzmann picture was that an atom in a solid should be thought of as sitting in the bottom of a harmonic oscillator oscillating back and forth.
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And from that picture, he was able to derive the law of do long petite.
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He capacity per atom is three KB. It's a pretty good result.
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Except it didn't agree at low temperature where the heat capacity of solids drops.
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Einstein figured out what was going on. He added Quantum mechanics.
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And due to quantum mechanics, the energy levels of the harmonic oscillator get quantised.
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And when the temperature drops below the quantisation energy, the oscillator gets frozen.
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His ground state and the heat capacity drops c drops at low temperature drops at low T and his prediction was it should drop exponentially.
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Exponentially. His theory of heat capacity fit the data pretty well.
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And indeed, it predicted this drop of heat capacity at low temperature.
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But unfortunately, experiment said experiment.
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Said that at low temperature the heat capacity should be proportional to t cubed, not dropping exponentially.
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So that brings us to the subject of today, which is the Devi theory.
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If I was 1912 and that in tuition was that you cannot think of a solid as a bunch of atoms in the body of a quantum harmonic.
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Well, because when one atom moves, it does get pushed back to its original position by its neighbour.
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But in the process it pushes its neighbour and then its neighbours versus its neighbour and so forth and so on.
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So in fact what we should be thinking about is the vibration in a solid is actually a wave, and in particular vibrational waves are sound.
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Now something was already known about how waves behave under the influence of quantum mechanics.
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All the way back in 1899, Max Planck had done this blackbody radiation calculation,
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and when when Max Planck did this calculation, he actually had no idea what he was doing.
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He didn't even like what he was doing. He said it was an act of complete despair, never really liked the calculation very much.
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But 13 years later, people started to understand little bits and pieces about quantum mechanics,
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and Planck's original calculations seem to make a lot more sense.
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So Debye thought that what we should do is we should quantised the sound waves.
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Just like light. Like light. The general idea is, you know, this is in modern language, you know, plankton understand this.
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But but we understand this now is from Einstein's work.
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We know that the energy of a given oscillator would be h bar omega times, the bonus factor beta h bar omega plus 0.2 energy plus one half.
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And so if we have that,
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the picture that we interpret Planck as now and with Debye understood is that each wave mode in a box intimately how you put your material in the box,
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each wave more in that box should be treated as an oscillator.
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So the total energy total inside your box is the sum overall wave modes of H bar omega
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for that mode times lowest factor beta h bar omega for the mode mode plus one half.
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And all you have to do is sum over all the modes in the box.
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Now, the reason this is going to give you something different from what Einstein had was because whenever you have a bunch of modes in the box,
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some of those modes are going to be low temperature, low energy modes or low frequency modes.
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So in Einstein's calculation, when the temperature dropped below the frequency of the oscillator, then the heat capacity dropped.
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But the idea here is we have a whole distribution of frequencies in the modes in a box.
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And so as we lower that the temperature, there will always be some modes of low enough frequency that will still have some heat capacity left.
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So the heat capacity won't drop exponentially as we go to low temperature.
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So that's the intuition.
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Now, there's going to be some differences between what plonk did and what Dubai did, because light is obviously different from sound in many ways.
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One way is that light is, of course, a lot faster than sound, but that's only sort of a quantitative difference.
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There's some more qualitative differences that we're going to have to deal with, and one in particular, light versus sound.
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Is that light has to polarisations polarisations and sound has three polarisations.
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Now you'll remember from your ENM that if a light wave is,
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or an electromagnetic wave of any sort is going in the direction the electric field has to be in the Y direction or the Z direction it can't be in.
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The extraction sound is not like that. If a sound wave is going the x direction, the polarisation of the sound wave can be in any direction.
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So it's probably easiest to explain this with a movie. So let me show you the movie of what I mean by this.
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This is what's known as a longitudinal wave,
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which means that the atoms are moving back and forth in the same direction that the wave is actually going.
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So the wave is going the X direction and the atoms are moving back and forth in the x direction.
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So this doesn't occur for light. For a light.
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If the wave is going, the extraction, the electromagnetic field is always pointing in the Y direction of the Z direction.
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More similar to light is what's known as a transverse wave, where the light is, where the wave is going in the x direction.
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But the atoms are actually moving back and forth in the Y direction or the Z direction.
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Okay. So with sound you can actually have three polarisations.
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The atoms can be moving in any direction, whereas in light the electric field has a point in only two possible directions.
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Is that clear? Yeah. Okay, good.
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So we're going to have to keep track of the fact that we have three polarisations for light and only two for the other way around.
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Three polarisation for sound and only two for light.
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There's a couple of things we're going to also do, which are going to be approximations, which are going to make our life simpler.
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One is that we're going to assume the velocity of sound.
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The sound is independent of polarisation and depth of Paul.
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This is not actually true in reality. Almost always the transverse mode is slower.
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It has a lower velocity than the longitudinal mode, but it's actually not so much more complicated to treat this properly.
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But you don't learn a whole lot more from doing it more properly. I think there's an exercise in the book that asks you to do it more properly,
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but we're just going to assume all sound waves have the same same velocity independent of whether they're longitudinal or transverse.
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As long as we're assuming things which are not really correct,
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we might as well assume the sound is independent of direction, of direction, which is often not true.
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Also, frequently when you have a real solid, the speed of sound depends on which direction the solid you're going.
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We'll discuss that a little bit more later on in the term.
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But again, it's something that if you treat it more properly, it's not that much harder and you don't learn so much more from doing it.
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So we're not going to to do it more properly, but just keep in mind that we are making these approximations.
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Now, if you remember back to Planck's calculation, which you did last year,
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the first thing you had to worry about was counting all of the modes in a box.
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Does that sound familiar? A little bit familiar. We're going to go through that again because it's important and we're going to
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use a lot of the a lot of the mechanism is to do many other things this year.
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So it's worth going through. So this is a little aside on counting waves in a box, counting waves or modes, I guess modes in box.
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So it's time to start with a one dimensional box, one D box as draw the one dimensional box.
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Here it is. It has some length l.
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We can write down a wave in that box with hard wall boundary conditions and pi x
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over l has comes to zero at both ends because of the hardware boundary conditions.
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And this is a perfectly good way to write down the waves in the box for every different positive integer.
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And you have a different wave mode. That's okay. But that's not not what we're going to do.
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What we're going to do is instead we're going to use what's known as periodic boundary conditions.
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You may have discussed this last year. Periodic boundary conditions.
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Boundaries also known as Bourn von Karman.
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Boundary conditions. Bourne von Karman. Karman Karman.
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Bourne was one of the Max Bourne is one of the creators of Quantum Mechanics.
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And Taylor von Karman was a very important mathematician.
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Max Bourne is also very significant because he was Olivia Newton-John grandfather.
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She's an important pop star. If you don't know her, she's important for my generation, at any rate.
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So anyway, people don't seem sufficiently impressed by that statement.
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It's actually it's anyway. Okay.
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So the idea of a periodic boundary condition is we're going to take our box of length L and we're going to wrap it up into a circle of circumference.
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L And we're going to measure X going around the box and this way like that.
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And the reason we're doing this is because once we do this, the waves can take the form of the i k x.
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We don't have to work with signs and cosines. If we have a periodic box in a circle, we can use exponentials.
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And if you haven't learned already by this time in your career, exponentials are just a lot easier to work with than signs and cosines.
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Now when we write a wave,
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it expects we have to be a little bit careful because the coordinate X and the coordinate x plus l are actually the same coordinate.
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If you go a distance l around the the box, you get back to exactly the same point.
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So this wave form must be exactly the same, whether we plug in X or x plus l.
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So we better have, you know, if this is going to make sense, we better have either the x equal to either the I k x plus l or otherwise.
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Either the excel has to equal one or k has to be two pi over l times an integer n and that integer can be positive or negative or whatever we like.
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Okay, everyone happy with that? So far so good.
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Okay, so this means that the spacing between allowed between between allowed allowed ks ks is two pi over l l being the circumference of our loop.
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And in particular, if we ever have to sum over all the K's, we can replace that sum by an integral decay times, a factor of l over two pi.
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Did you go through this argument last year? A little bit, yes.
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Well, okay, we're going to use it an awful lot this year. So that's why I'm going through it again, because it's important.
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Now, we, of course, don't live in one dimension, so we have to think as multidimensional people.
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We have to think in multiple dimensions. So in three D, we're going to take an L by L, by L periodic box.
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Periodic box. And at this point, you might start to be a little bit upset because there's no such thing as an l l by l periodic box.
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In the real world, you would have to have a box for which if you went a distance l in any direction,
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you came back to exactly where you started in three dimensions. You can't build such a thing.
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They just don't exist. So why is it we're going to do this? The point is that doesn't actually matter what you do with the boundaries of your system.
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For almost any quantity you're actually interested in calculating, such as the heat capacity.
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It's something that you can measure locally, just in a small region of the actual physical system.
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So it doesn't matter what you would do with the boundaries, which could be very, very far away.
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So you can use hard wall boundary conditions, you can use periodic boundary conditions,
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you can use any other type of boundary condition which is convenient.
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And for us it's convenient to use these periodic boundary conditions and we're going to get away with it.
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The reason we want to use these periodic boundary conditions is because then the waves will look like the i k vector dot x vector.
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It's an exponential exponential, so easy to work with and that's why we're doing it.
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Everyone has still happy with that. Okay, good. So in three dimensions k has to be two pi over at all times integers and x and y and z.
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And so if we ever have a sum of. Over K vectors.
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We can replace that sum with l over two pi cubed times the integral dk x integral
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d.k. y integral d kc or another way that we will most usually write this.
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We'll write this as the volume of the system r times the integral d3k over
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two pi cubed and we'll see this factor volume integral d3k over two pi cubed,
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probably C at 100 times this year before we're done. Okay.
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So what we're going to do is we're going to use what we just learned about counting waves in a box to apply to this equation here.
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So the sum over modes here and right sum over modes is going to become the sum over all possible k vectors for our modes.
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But then times a factor of three and the factor of three is from the three polarisations we can have one longitudinal
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or two transverse polarisations and then this sum and we're going to use this law here to convert it into an integral.
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So now we have three factor of the volume integral d3k over two pi cubed.
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Now, since we've assumed over here that everything is isotropic, in other words, velocities are independent of direction.
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We can then take this three dimensional Cartesian integral and turn it into a spherical polar integral.
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So we have three times the volume. Let's pull out the two pi cubed here and then we have integral for pi k squared decay from zero to infinity.
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As this familiar, the four pi is the integral over the circle directions.
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Yes, throughout complete corners. Everyone's happy with that.
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Finally, since we're thinking about sound for sound, you probably learn in your fluids course last year or last last term,
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frequency is proportional to the wave vector and the proportionality constant is the velocity velocity of the sound.
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So let's just change that integral into an integral over frequency instead of an integral of a wave vector.
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So that becomes a plot. Some factors here as plotted for pi upstairs we had a two pi cubed downstairs.
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I guess we have a velocity cubed downstairs and then we have an integral zero to infinity of omega squared the omega.
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And I'm going to write that as just defining a quantity g of omega deal mega zero to infinity.
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And this g of omega is just a bunch of those constants.
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I'm going to stick together. So I guess I have 1212 pi upstairs and I have two pi cubed downstairs.
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I guess I have a velocity cubed downstairs and then I have a volume upstairs and I'm going to write the volume upstairs in sort of a tricky way.
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I'm going to write it as the number of atoms divided by the volume downstairs.
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That's the density, and then the number of atoms upstairs that's going to be convenient.
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And then I need also that omega squared here.
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Omega squared. So this quantity here is known as swish.
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I put it over here. Do you have a mega is known as deep omega is known as the density of states.
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And I think you ran into this last year when you did blackbody radiation.
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And the idea of a density of states is that it is basically how many of these modes you have at a given frequency,
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in particular g omega d omega equals number of modes with frequency.
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Frequency between between omega and omega plus de omega.
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So if you want to sum over all the modes in the system, instead of just summing all all the modes,
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what you do is you integrate overall frequencies times the number of modes at each frequency.
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Okay. Sort of a convenient way to do things.
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I'm actually also going to simplify this a little bit further by defining another convenient quantity.
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I'm going to rewrite this g of Omega gave Omega as I'm going to write it as pull out the n the number of atoms,
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the nine omega squared over omega D cubed.
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So I'm defining omega d cubed here to be a bunch of those constants stuck together.
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Omega d cubed is six pi squared times the density number over volume times velocity of sound cubed.
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This is known as the devi frequency. After Mr. Devi, my frequency, at least for now,
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we're just going to think of it as this bunch of constants that we've conveniently defined, so it's easier to write G.
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In a moment we'll give it a more important meaning, but for now, it's just those bunch of constants.
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Okay, so.
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Having done all of that, we can then way up there at the very top of the board, we have the expression for the energy as a sum over all of the modes.
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So I can maybe I'll put it over here so I don't have to scroll out off the top of the board.
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So we'll now write the energy, the total energy at a given temperature.
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Instead of writing it as a sum over all the modes,
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I'm going to write it as an integral over frequencies equals integral the omega from zero to infinity g of omega.
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That's the sum over all modes by integrating of frequency in the thing that we want to integrate is h bar omega bose factor theta.
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H Bar Omega plus one half. Make people happy with that what we're doing?
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Yes. Yes, maybe. Yeah. Okay, good.
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All right. So this is the expression that we're going to try to evaluate.
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And if you're paying really close attention, you'll be upset with me already because it's actually infinite.
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This total energy in the box is infinite. And the reason it's infinite is because of this plus one half look at the plus one half multiplies.
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It multiplies a factor. Omega g of omega is proportional to omega squared.
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So it's the integral of omega cubed from zero to infinity that's infinite.
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So that's kind of it. You might think that that's kind of a problem.
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Actually, it didn't bother to buy a plant because they didn't know about zero point energy, so they never wrote down this plus one half.
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It's not going to bother us either for a number of reasons. The main reason it's not going to bother us is because it gives us an infinite quantity,
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but it's an infinite quantity that's independent of temperature. The only place in this expression where temperature occurs is right here.
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At the end of the day, we want heat capacity, which is the derivative of the energy with respect to temperature.
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So we're going to have to differentiate this thing with respect to temperature.
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And the one half is going to go away because it's temperature independent. So it's giving us an infinite but temperature independent contribution.
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So we don't care about it. Actually, in a few moments, we're going to see another reason why we're not going to care about that one half.
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But for now we'll just realise it's not going to change our expression for the heat capacity.
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All right. So now we can plug in the density of states into this equation here.
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So let's do that. So we get nine and I'll pull out the bar from over there.
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There's omega two by cubed here, cubed downstairs.
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And then the thing we have left to integrate integral the omega from zero to infinity,
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there's omega cubed upstairs and then the bose factor into the data h bar, omega minus one is left.
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That looks a little bit ugly, but we can take this integral.
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We can simplify a little bit by writing X equals beta h bar omega,
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and then the integral turns into one over beta h bar to the fourth times this
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integral integral d x from zero to infinity of x cubed in the x minus one.
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And I promise you, no one is ever going to ask you to evaluate this integral.
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It's just some number. The number happens to be PI to the fourth over 15.
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And if you really want to know where that number comes from you, it's in the appendix of one of the chapters of the book.
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And don't you hate it when someone says you can go read the book? But in fact, it's not so important.
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It just gives us a number.
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The important thing is because it gives us some known number so we can put that number in and get the end result that the energy,
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the total energy in the box is nine and kb t to the fourth from here divided by h bar omega Dubai cubed times this factor pi to the fourth over 15.
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Okay. Good. And then, of course, we can differentiate this to get the heat capacity.
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Heat capacity is 80. And okay, so it's an KB cavity or H bar, omega Dubai cubed and then I guess 12 pi to the fourth over five watts.
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So as the heat capacity. And so a couple of things to comment about this.
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It makes this exciting. First of all, it's proportional to t cube just like Dubai wanted.
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You get the t cubed dependence of the heat capacity as expected.
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It shouldn't be surprising that you get this t cubed heat capacity because you remember from
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the blackbody radiation calculation you did last year that the energy of of radiation,
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the energy of waves in a box that you calculated is proportional to t to the fourth.
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You should have expected the energy to be t to the fourth, and then you differentiate it once and you get t cubed.
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So this is why by expected that by treating the waves the same way we treated radiation, we would get the t cubed law.
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Furthermore, there's something really exciting about this formula here, and that is that there's no free parameters.
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Remember when Einstein did his calculation, he had this frequency, this Einstein frequency,
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the oscillator frequency, which he didn't know how to come up with this frequency.
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He just fit it to the experiment to try to make it all look nice here.
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There's no free parameter. Omega debye here is fixed by the density and the sound velocity.
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So everything is fixed here. And if you calculate this quantity and you compare it to the low temperature heat capacity of most materials,
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it actually actually agrees extremely well. But unfortunately, like, you know, you never get anything for free.
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So we still have a problem. Problem.
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At high tea. We want the law of do along petite silver and is three KB and we didn't get that.
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We got T cubed at all temperatures here and this is where the buy had to actually scratches had a little bit.
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And think okay what did I do wrong. Incidentally, if you do this calculation for electromagnetic radiation,
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it really is t to the fourth energy all the way up to arbitrarily high temperature.
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Whereas in a in a solid we know that the heat capacity, the solid at some high temperature is going to give us just 3kb, it's not going to be T cubed.
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So here we have to deviate from Planck's calculation.
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We have to do something different from what it was that plotted and Planck and Debye understood that where he was going wrong,
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where he was going wrong. He has to think about where this three comes from.
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Where is this three from? The three comes from the fact that the atom can move in three directions.
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Each atom moves in three possible directions. You can think of it as three degrees of freedom that the atom has.
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And how many degrees of freedom did he count in his box?
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Well, the total number of modes, number of modes that he counted is integral from zero infinity du omega.
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The omega and g of omega goes as omega squared. You integrate that into infinity and that gives you infinity.
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So he counted an infinite number of degrees of freedom and he said, Well, wait,
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I figured I counted an infinite number of modes in the box, but there's a finite number of degrees of freedom.
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Each atom can move in three directions, and there shouldn't be more than that many degrees of freedom in my box.
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So somehow I have to fix that problem and make the box have only a fixed number of degrees of freedom, not an infinite number of degrees of freedom.
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So how did he do that? Impose a cut-off?
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So when he's going to do is he is going to declare some frequency cut off omega,
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cut off the omega, the omega, such that there are exactly three and modes in the box.
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Now, this is an ad hoc solution and it's kind of a little weird.
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So what he's doing is he's saying, we have all these wave modes in a box.
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And when you get to some frequency known as the cut-off frequency above that, there are no more wave modes left.
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You have no sound moves above this above this frequency anymore. Seems a little strange.
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Later in the term we're going to see it's maybe not as strange as we thought,
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but at that at that time, maybe it seemed a little odd, but he's going to do it anyway.
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So we're going to do this anyway. We're going to follow him and see what happens.
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First thing we need to do is we have to figure out what this cut-off frequency is going to be.
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So it's going to give us exactly three modes. So let's calculate this thing.
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We'll plug in G of Omega. So plug in G of Omega here.
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So there's a nine and there's omega two by cubed integral zero to omega cut off omega squared D omega.
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So I just plugged in g of omega there and then let's do that integral and we get three and omega cut-off
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cubed over omega devi cubed by cubed and we want this to equal three n and in order for that to be true,
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we better choose omega cut-off equal to omega two by.
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Okay. That's why I happened to choose those particular constants as being omega two by omega two by is the cut-off frequency.
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If you cut off your modes at the by frequency only count modes at lower frequencies than the by frequency, you have exactly three and modes.
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Okay. All right.
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So now we're going to go back to this equation over here and we're going to rewrite the total energy in the box now as the integral counting up modes,
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not to infinity now only up to omega by G omega and then the Bose factor, beta age for omega and then plus one half zero point energy.
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And you'll notice now, because we've cut off our number of modes, this term no longer diverges.
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It's a finite 0.2 energy term, and that makes us a little bit happier.
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But again, we can actually ignore it. Drop this because it's temperature independent.
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It's a temperature independent, zero point energy. We're going to differentiate the thing anyway.
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So we don't really care about it. It's just going to give us some overall constant shift in the energy, which isn't very interesting.
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So we're going to drop that anyway. Okay. And then this expression should give us the energy in the box or the heat capacity in the box.
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Once you differentiate it at any temperature we choose just by plugging the temperature in here.
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But it's useful to look at various limits. So the first limit is low temperature limit.
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And by that I mean cavity much less than H bar omega devi.
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And in this limit, we get the same result, same result as we had before.
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This result here. Exactly. This nothing changes.
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Why is that? Well, the reason nothing changes is because at low temperature, this Bose factor vanishes very quickly with frequency.
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By the time you're up near the Dubai frequency, the Bose factor is essentially zero anyway.
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So it doesn't matter if you cut it off at the Devi frequency, you're twice the device frequency or half the DEVI frequency.
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The integration is zero anyway by that time. So you just.
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You don't have to worry about the cut-off at all. At low temperature, the cut-off isn't doing anything.
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So you still get the t cubed heat capacity, which is what we want.
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But at high temperature, at high temperature, we have something different.
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Well, okay, so let's look at the Bose factor at high temperature, at high temperature,
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beta h bar omega is a small number, so we can expand the exponential.
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So we get one plus beta h bar omega plus start minus one.
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So the ones cancel, we get one over beta h for omega or we get the bose factor is replaced and bose becomes k, b, t over H for Omega.
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So if we then take this Bose factor and plug it into that energy expression, again, dropping the zero point energy,
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the energy is now integral zero to sorry to omega two by up to the cut-off the omega due omega than we have H for omega.
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And then we have the Bose factor, which is k, b, t over h bar omega,
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the bar omegas cancel and we get the energy being given by pull out the K and integral zero to omega by the omega g omega.
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And this integral here has been designed to give us exactly three n.
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So we get the energy in the box is three and kb t or the heat capacity is three and kb the of two long petite.
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So by implementing this cut off the by managed to get the low temperatures.
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Heat capacity is t cubed. The high temperature heat capacity is still the law of Duong Petite, so that's pretty good.
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Let's look at some actual data. So this is the heat capacity of silver over a broad range of temperature and up at high temperature.
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You see, it's converging to the law of do long, petite and at low temperature it's roughly t cubed.
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And the debye theory agrees with the experiment extremely, extremely well.
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You can see on the same on the same plot, there's the Einstein theory,
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which fits pretty good, but not quite as well as to buy, particularly at low temperature.
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And but an additional really important improvement from Einstein to Dubai is that Dubai has no free parameters,
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that everything is fixed in the Dubai theory by just the velocity of sound, of the density.
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So there's no no, you can't muck around, you can't adjust things. It just fits by itself.
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So it's a really good result and seems to agree extremely well with the experiment.
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But we still have problems, still wrong or still well, things that are still wrong.
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One is the cut off is really ad hoc, cut off, ad hoc.
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We just sort of made this up. I mean, it was a motivated it was a motivated thing to do.
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It was an intelligent thing to do, but it wasn't really justified.
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Another thing that's kind of wrong is that we used omega proportional to wave vector the sound law here.
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That frequency will be proportional to wave vector, but we use this at high K and that's not true.
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Sound is A is a small K, a long wavelength phenomenon when you go to very small wavelengths or very high wave vector.
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This is no longer true. So this was sort of a problem that we brushed under the rug.
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And incidentally, the the wavelengths we're talking about,
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when you get up near the Dubai frequency, the wavelengths are close to the entire atomic spacing.
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So we're talking about really, really small wavelengths or really high wave vectors.
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And we can't really think about sound in that regime anymore.
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So this is sort of a problem we brushed under the rug. Another thing is that Dubai is not exact.
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It's not exact for any material, although it's pretty good, as is obvious from that plot.
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And third, our fourth metals are different.
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Metals are different. Well, for metals we have this.
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For metals.
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As I mentioned last time, we have low temperature specific heat C proportional to alpha t cubed plus gamma t where alpha is predicted by Dubai.
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So if you did the Dubai theory, you would get the coefficient alpha correct, but you wouldn't get gamma at all.
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So this is just a big question right now. We don't know where that's where that's coming from.
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So we have to figure that out. And in a few lectures time, we'll have a good idea where that's coming from.
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But for now, it's it's a bit of mystery. Now, you might say to me at this point, well, wait a second, isn't silver a metal?
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Shouldn't I see a linear heat capacity at low temperature?
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Indeed, silver is a metal, and you should see a linear heat capacity at low temperature, but you have to look at it pretty hard to see it.
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So this is blown up the very, very low temperature regime, sort of 1 to 4 or five Kelvin or something like that, a 1 to 4 Kelvin, I guess.
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And what's plotted here is the heat capacity divided by the temperature as a function of temperature squared.
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If it was a purity cube law, then this line would be a straight line and it would intersect zero.
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Obviously doesn't intersect zero. It intersects a finite intercept, which is, in fact, gamma.
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So there's a very, very small term gamma. This term gamma is pretty small, but it's clearly there.
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You have to measure pretty carefully to see it. But it's there. Okay.
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All right. So at this point, we're sort of done talking about vibrations in solids for a little while.
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We'll come back to them later in the term when we do a little bit of a better job trying to understand these things.
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But for now, we're putting this aside and we're going to switch gears and start talking about metals,
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because metals are obviously different in several ways. This is one way that they're different, but they're different in many other ways as well.
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And this is something that was this was known to be known to the ancients, even probably in caveman days.
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They they knew that there were some materials they found in the ground that just looked different from other materials.
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And by 4000 B.C., people were able to work with certain metals, things like like copper.
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And then a couple thousand years later, they were able to work with iron. And each time they were able to command a particular metal and work with it,
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they were able to make new things, new devices, new technologies, and really changed the history of humankind.
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So, you know, starting with metal ploughs, metal swords, metal armour, metal warfare of all sorts.
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Then later on metal machines, you have metal skyscrapers.
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You know, the nuclear age was brought in by, you know, heavy metals and heavy metal music, very important also.
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But metals, you know, the history of metals in some way traces the history of of mankind.
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And, you know,
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it wasn't until really the late 1800s that anyone had the remotest idea what causes metals to be different and really well into the 1900s.
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Before we really understood the properties of metals, you have to remember, you know,
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for us the the defining property of a metal is going to be that it conducts electricity and non metals don't.
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But we didn't even know it what electricity was until the late 1800s.
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It was 1897 before J.J. Thompson discovered the electron or what he called the small corpuscle of charge that
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can move around freely in the metals and could be ejected out of the metal by a sufficiently high voltage.
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And with this picture of the metal really being sort of a container for all these electrons running around,
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there became a natural thing for people to do, which was to consider these electrons running around as a gas, a gas of electrons.
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And this is what Paul Drew to did. It's known as Judah Theory of Metals.
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Of Metals. Well, our junior theory of transport applies to metals, and it's actually particularly good.
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It works. This drew the theory. It's a very crude classical kinetic theory, kinetic theory of electrons of electrons,
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very much like the kinetic theory of gases that you study last year in your thermal physics course.
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It works extremely well, despite the fact that it's very crude.
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It works extremely well for a lot of things, particularly well for semiconductors.
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And so we're going to attack, you know, electron transport in in metals, using this due to theory first and then we'll improve on it later.
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So as with your last year's kinetic theory, we have a couple of assumptions.
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Assume in order to get kinetic theory going, one, there is a scattering time.
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Scattering time. Tao should look familiar from last year, by which we mean that the probability of scattering,
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probability of Scott in time t or in time d t is equal to d t over tao.
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Now this probably looks familiar from last year when you did Kinetic Theory of Gases last year.
381
00:39:41,070 --> 00:39:45,030
When did you connect their gases? You can even predict what the scattering time Tao is.
382
00:39:45,410 --> 00:39:48,530
Based on the size of the atoms and how fast they're moving and things like that.
383
00:39:49,100 --> 00:39:52,500
This year we're not going to be so lucky for a couple of reasons.
384
00:39:52,520 --> 00:39:56,450
First of all, it isn't clear what the scattering cross-section of the electrons should be
385
00:39:56,690 --> 00:39:59,870
because the electrons interact with things via long range Coulomb interaction.
386
00:39:59,870 --> 00:40:02,990
So they can they could scatter from things very far away, potentially.
387
00:40:03,260 --> 00:40:06,680
Another thing that's going to make it difficult to figure out with towers is that
388
00:40:06,680 --> 00:40:10,280
the electron can scatter off of lots of other things besides just other electrons.
389
00:40:10,460 --> 00:40:14,960
They can scatter off of protons, it can scatter off of impurities, it can scatter off of anything.
390
00:40:14,960 --> 00:40:21,170
And that happens to be in the metal. So for us, the scattering time Tao is just going to be a phenomenological parameter.
391
00:40:22,190 --> 00:40:34,370
The second thing we're going to assume is after scattering, after a scattering event, after scatter, we will set the final momentum equal to zero.
392
00:40:35,030 --> 00:40:40,339
So imagine something, something moving along. It scatters, and then its final momentum is zero.
393
00:40:40,340 --> 00:40:48,980
Now that's not right. Generally, when something scatters, its final momentum goes off randomly in some random direction, but on average as a vector,
394
00:40:49,280 --> 00:40:54,530
the average of the vector after the scattering is pretty close to zero because it can go off in any possible direction.
395
00:40:54,740 --> 00:40:58,160
And that's going to be good enough for us to be able to make progress.
396
00:40:58,640 --> 00:41:04,190
The third point, which you probably didn't have last year, is that between scattering events,
397
00:41:04,190 --> 00:41:18,020
between scatters, the electron should see C's E and B field if they happen to be there.
398
00:41:18,470 --> 00:41:22,910
So if you're applying an electric field to your metal, the electron will accelerate due to the electric field,
399
00:41:22,910 --> 00:41:28,430
or it will curve due to the magnetic field, which seems rather natural, just like the electron were living in a vacuum.
400
00:41:28,880 --> 00:41:37,550
Okay, so given these three assumptions, we can imagine that we start with an electron that has momentum p at time.
401
00:41:37,550 --> 00:41:44,420
T So t is momentum while well, that's obvious at time t time t.
402
00:41:46,110 --> 00:41:52,009
And then we'd like to calculate what is the momentum that time T plus d t I mean,
403
00:41:52,010 --> 00:41:56,090
in some ways we're asking what's the expectation of the momentum at time t plus.
404
00:41:56,850 --> 00:41:59,660
But we'll treat it as the actual momentum at time t plus t.
405
00:42:01,620 --> 00:42:09,179
Well, there's two things that can happen in between time T and time T plus de t with probably one minus d t over time.
406
00:42:09,180 --> 00:42:16,290
Now, this is the probability. This is probability of not scattering prob of not scattering mass scattering.
407
00:42:17,520 --> 00:42:19,739
If it does not scatter, then what happens?
408
00:42:19,740 --> 00:42:27,970
Well, then it has the original momentum plus whatever force is applied to at times d t and that's just okay.
409
00:42:28,080 --> 00:42:33,900
This is just saying that dpd t if it doesn't scatter, is f Newton's law.
410
00:42:35,890 --> 00:42:43,920
Good. But in addition to this, there's also the probability probability d t over t that it does scatter.
411
00:42:44,130 --> 00:42:49,140
And if it does scatter, we give it momentum zero. Okay.
412
00:42:50,520 --> 00:42:55,200
So this is the probability of not scattering. It accelerates as usual due to the force applied to it.
413
00:42:55,500 --> 00:42:59,340
And if it does scatter, we give it momentum zero at after the scattering.
414
00:42:59,930 --> 00:43:04,260
Okay, then we can do a little bit of rearrangement here. Well, actually, we multiply this out first.
415
00:43:04,560 --> 00:43:13,750
So this is then PV T plus f, d, t minus P over tau.
416
00:43:14,880 --> 00:43:19,570
And then there's de t, and then there's plus order D squared.
417
00:43:21,480 --> 00:43:34,170
And with a little bit of rearrangement, we can write DPD T, which should be p at t plus d t minus t over d t.
418
00:43:37,280 --> 00:43:41,810
Yell at me if I start writing incomprehensibly, you know, if it really gets too painful.
419
00:43:42,290 --> 00:43:51,950
And just doing a little bit of rearrangement on that equation up there and putting together this combination, we get F minus P over Tao.
420
00:43:51,980 --> 00:44:01,740
So let me rewrite this because this is an important equation. DP d t equals force minus t over tau.
421
00:44:02,120 --> 00:44:09,320
This is known as the due to transport equation, you know. And what force are you supposed to use in it?
422
00:44:09,590 --> 00:44:22,220
Well, the force is the usual Lorentz force. Force is a minus e e plus the cross b so whatever force the electron feels goes into that, that equation.
423
00:44:22,670 --> 00:44:25,670
So this looks a lot like Newton's equation.
424
00:44:26,060 --> 00:44:32,150
D equals F is Newton's equation. But we have this additional term on the right hand side, which looks like a drag force.
425
00:44:32,360 --> 00:44:36,499
It's a force going in the opposite direction from its current momentum.
426
00:44:36,500 --> 00:44:40,100
So whichever direction is going, the force is pulling in the opposite direction.
427
00:44:40,100 --> 00:44:52,669
So let's actually do a really quick calculation here. Let's consider consider the case where there's no no electric or magnetic fields in your system.
428
00:44:52,670 --> 00:45:01,070
So you're not applying any electric or magnetic field, then you just have DPI de t is minus p over tau,
429
00:45:02,030 --> 00:45:10,160
which you can solve by saying p t is some p not some initial momentum e to the minus t over tau.
430
00:45:10,790 --> 00:45:16,730
So that tells us that the moment if I have an electron moving along with some initial momentum,
431
00:45:16,970 --> 00:45:24,380
this scattering term here slows it down exponentially to zero momentum with that time scale.
432
00:45:24,530 --> 00:45:25,849
Tau So it's like a drag.
433
00:45:25,850 --> 00:45:33,980
So the idea of the true T theory is that you treat scattering as a drag force that tries to slow everything down or hinder its motion.
434
00:45:34,160 --> 00:45:38,300
And I guess we will stop there and we'll pick up with due to theory next time. I'll see you tomorrow.
435
00:45:39,770 --> 00:45:40,070
Okay?