This is the second lecture of the condensed matter. Of course. Welcome back. Three quick things before we start.
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?
It's the exercise basics. Now you can find it. The second thing I want to encourage you to use the message board.
No one's used it yet. Someone be brave. Post the first message.
Once you get going, I promise you'll find it useful. The third thing I didn't emphasise this enough last time.
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,
please come to me immediately and I can issue corrections and therefore not get everyone is confused as much as we can.
So just, you know, either post a message on the message boards and the email,
catch me off to lecture or you know, I know it's crazy, but you can call me maybe.
Yeah. Anyway, so last time we left off, we were talking about high capacity.
Of solids. And we started with the Boltzmann picture of a solid and Boltzmann picture.
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.
And from that picture, he was able to derive the law of do long petite.
He capacity per atom is three KB. It's a pretty good result.
Except it didn't agree at low temperature where the heat capacity of solids drops.
Einstein figured out what was going on. He added Quantum mechanics.
And due to quantum mechanics, the energy levels of the harmonic oscillator get quantised.
And when the temperature drops below the quantisation energy, the oscillator gets frozen.
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.
Exponentially. His theory of heat capacity fit the data pretty well.
And indeed, it predicted this drop of heat capacity at low temperature.
But unfortunately, experiment said experiment.
Said that at low temperature the heat capacity should be proportional to t cubed, not dropping exponentially.
So that brings us to the subject of today, which is the Devi theory.
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.
Well, because when one atom moves, it does get pushed back to its original position by its neighbour.
But in the process it pushes its neighbour and then its neighbours versus its neighbour and so forth and so on.
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.
Now something was already known about how waves behave under the influence of quantum mechanics.
All the way back in 1899, Max Planck had done this blackbody radiation calculation,
and when when Max Planck did this calculation, he actually had no idea what he was doing.
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.
But 13 years later, people started to understand little bits and pieces about quantum mechanics,
and Planck's original calculations seem to make a lot more sense.
So Debye thought that what we should do is we should quantised the sound waves.
Just like light. Like light. The general idea is, you know, this is in modern language, you know, plankton understand this.
But but we understand this now is from Einstein's work.
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.
And so if we have that,
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,
each wave more in that box should be treated as an oscillator.
So the total energy total inside your box is the sum overall wave modes of H bar omega
for that mode times lowest factor beta h bar omega for the mode mode plus one half.
And all you have to do is sum over all the modes in the box.
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,
some of those modes are going to be low temperature, low energy modes or low frequency modes.
So in Einstein's calculation, when the temperature dropped below the frequency of the oscillator, then the heat capacity dropped.
But the idea here is we have a whole distribution of frequencies in the modes in a box.
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.
So the heat capacity won't drop exponentially as we go to low temperature.
So that's the intuition.
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.
One way is that light is, of course, a lot faster than sound, but that's only sort of a quantitative difference.
There's some more qualitative differences that we're going to have to deal with, and one in particular, light versus sound.
Is that light has to polarisations polarisations and sound has three polarisations.
Now you'll remember from your ENM that if a light wave is,
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.
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.
So it's probably easiest to explain this with a movie. So let me show you the movie of what I mean by this.
This is what's known as a longitudinal wave,
which means that the atoms are moving back and forth in the same direction that the wave is actually going.
So the wave is going the X direction and the atoms are moving back and forth in the x direction.
So this doesn't occur for light. For a light.
If the wave is going, the extraction, the electromagnetic field is always pointing in the Y direction of the Z direction.
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.
But the atoms are actually moving back and forth in the Y direction or the Z direction.
Okay. So with sound you can actually have three polarisations.
The atoms can be moving in any direction, whereas in light the electric field has a point in only two possible directions.
Is that clear? Yeah. Okay, good.
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.
Three polarisation for sound and only two for light.
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.
One is that we're going to assume the velocity of sound.
The sound is independent of polarisation and depth of Paul.
This is not actually true in reality. Almost always the transverse mode is slower.
It has a lower velocity than the longitudinal mode, but it's actually not so much more complicated to treat this properly.
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,
but we're just going to assume all sound waves have the same same velocity independent of whether they're longitudinal or transverse.
As long as we're assuming things which are not really correct,
we might as well assume the sound is independent of direction, of direction, which is often not true.
Also, frequently when you have a real solid, the speed of sound depends on which direction the solid you're going.
We'll discuss that a little bit more later on in the term.
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.
So we're not going to to do it more properly, but just keep in mind that we are making these approximations.
Now, if you remember back to Planck's calculation, which you did last year,
the first thing you had to worry about was counting all of the modes in a box.
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
use a lot of the a lot of the mechanism is to do many other things this year.
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.
So it's time to start with a one dimensional box, one D box as draw the one dimensional box.
Here it is. It has some length l.
We can write down a wave in that box with hard wall boundary conditions and pi x
over l has comes to zero at both ends because of the hardware boundary conditions.
And this is a perfectly good way to write down the waves in the box for every different positive integer.
And you have a different wave mode. That's okay. But that's not not what we're going to do.
What we're going to do is instead we're going to use what's known as periodic boundary conditions.
You may have discussed this last year. Periodic boundary conditions.
Boundaries also known as Bourn von Karman.
Boundary conditions. Bourne von Karman. Karman Karman.
Bourne was one of the Max Bourne is one of the creators of Quantum Mechanics.
And Taylor von Karman was a very important mathematician.
Max Bourne is also very significant because he was Olivia Newton-John grandfather.
She's an important pop star. If you don't know her, she's important for my generation, at any rate.
So anyway, people don't seem sufficiently impressed by that statement.
It's actually it's anyway. Okay.
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.
L And we're going to measure X going around the box and this way like that.
And the reason we're doing this is because once we do this, the waves can take the form of the i k x.
We don't have to work with signs and cosines. If we have a periodic box in a circle, we can use exponentials.
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.
Now when we write a wave,
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.
If you go a distance l around the the box, you get back to exactly the same point.
So this wave form must be exactly the same, whether we plug in X or x plus l.
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.
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.
Okay, everyone happy with that? So far so good.
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.
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.
Did you go through this argument last year? A little bit, yes.
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.
Now, we, of course, don't live in one dimension, so we have to think as multidimensional people.
We have to think in multiple dimensions. So in three D, we're going to take an L by L, by L periodic box.
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.
In the real world, you would have to have a box for which if you went a distance l in any direction,
you came back to exactly where you started in three dimensions. You can't build such a thing.
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.
For almost any quantity you're actually interested in calculating, such as the heat capacity.
It's something that you can measure locally, just in a small region of the actual physical system.
So it doesn't matter what you would do with the boundaries, which could be very, very far away.
So you can use hard wall boundary conditions, you can use periodic boundary conditions,
you can use any other type of boundary condition which is convenient.
And for us it's convenient to use these periodic boundary conditions and we're going to get away with it.
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.
It's an exponential exponential, so easy to work with and that's why we're doing it.
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.
And so if we ever have a sum of. Over K vectors.
We can replace that sum with l over two pi cubed times the integral dk x integral
d.k. y integral d kc or another way that we will most usually write this.
We'll write this as the volume of the system r times the integral d3k over
two pi cubed and we'll see this factor volume integral d3k over two pi cubed,
probably C at 100 times this year before we're done. Okay.
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.
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.
But then times a factor of three and the factor of three is from the three polarisations we can have one longitudinal
or two transverse polarisations and then this sum and we're going to use this law here to convert it into an integral.
So now we have three factor of the volume integral d3k over two pi cubed.
Now, since we've assumed over here that everything is isotropic, in other words, velocities are independent of direction.
We can then take this three dimensional Cartesian integral and turn it into a spherical polar integral.
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.
As this familiar, the four pi is the integral over the circle directions.
Yes, throughout complete corners. Everyone's happy with that.
Finally, since we're thinking about sound for sound, you probably learn in your fluids course last year or last last term,
frequency is proportional to the wave vector and the proportionality constant is the velocity velocity of the sound.
So let's just change that integral into an integral over frequency instead of an integral of a wave vector.
So that becomes a plot. Some factors here as plotted for pi upstairs we had a two pi cubed downstairs.
I guess we have a velocity cubed downstairs and then we have an integral zero to infinity of omega squared the omega.
And I'm going to write that as just defining a quantity g of omega deal mega zero to infinity.
And this g of omega is just a bunch of those constants.
I'm going to stick together. So I guess I have 1212 pi upstairs and I have two pi cubed downstairs.
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.
I'm going to write it as the number of atoms divided by the volume downstairs.
That's the density, and then the number of atoms upstairs that's going to be convenient.
And then I need also that omega squared here.
Omega squared. So this quantity here is known as swish.
I put it over here. Do you have a mega is known as deep omega is known as the density of states.
And I think you ran into this last year when you did blackbody radiation.
And the idea of a density of states is that it is basically how many of these modes you have at a given frequency,
in particular g omega d omega equals number of modes with frequency.
Frequency between between omega and omega plus de omega.
So if you want to sum over all the modes in the system, instead of just summing all all the modes,
what you do is you integrate overall frequencies times the number of modes at each frequency.
Okay. Sort of a convenient way to do things.
I'm actually also going to simplify this a little bit further by defining another convenient quantity.
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,
the nine omega squared over omega D cubed.
So I'm defining omega d cubed here to be a bunch of those constants stuck together.
Omega d cubed is six pi squared times the density number over volume times velocity of sound cubed.
This is known as the devi frequency. After Mr. Devi, my frequency, at least for now,
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.
In a moment we'll give it a more important meaning, but for now, it's just those bunch of constants.
Okay, so.
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.
So I can maybe I'll put it over here so I don't have to scroll out off the top of the board.
So we'll now write the energy, the total energy at a given temperature.
Instead of writing it as a sum over all the modes,
I'm going to write it as an integral over frequencies equals integral the omega from zero to infinity g of omega.
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.
H Bar Omega plus one half. Make people happy with that what we're doing?
Yes. Yes, maybe. Yeah. Okay, good.
All right. So this is the expression that we're going to try to evaluate.
And if you're paying really close attention, you'll be upset with me already because it's actually infinite.
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.
It multiplies a factor. Omega g of omega is proportional to omega squared.
So it's the integral of omega cubed from zero to infinity that's infinite.
So that's kind of it. You might think that that's kind of a problem.
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.
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,
but it's an infinite quantity that's independent of temperature. The only place in this expression where temperature occurs is right here.
At the end of the day, we want heat capacity, which is the derivative of the energy with respect to temperature.
So we're going to have to differentiate this thing with respect to temperature.
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.
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.
But for now we'll just realise it's not going to change our expression for the heat capacity.
All right. So now we can plug in the density of states into this equation here.
So let's do that. So we get nine and I'll pull out the bar from over there.
There's omega two by cubed here, cubed downstairs.
And then the thing we have left to integrate integral the omega from zero to infinity,
there's omega cubed upstairs and then the bose factor into the data h bar, omega minus one is left.
That looks a little bit ugly, but we can take this integral.
We can simplify a little bit by writing X equals beta h bar omega,
and then the integral turns into one over beta h bar to the fourth times this
integral integral d x from zero to infinity of x cubed in the x minus one.
And I promise you, no one is ever going to ask you to evaluate this integral.
It's just some number. The number happens to be PI to the fourth over 15.
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.
And don't you hate it when someone says you can go read the book? But in fact, it's not so important.
It just gives us a number.
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,
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.
Okay. Good. And then, of course, we can differentiate this to get the heat capacity.
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.
So as the heat capacity. And so a couple of things to comment about this.
It makes this exciting. First of all, it's proportional to t cube just like Dubai wanted.
You get the t cubed dependence of the heat capacity as expected.
It shouldn't be surprising that you get this t cubed heat capacity because you remember from
the blackbody radiation calculation you did last year that the energy of of radiation,
the energy of waves in a box that you calculated is proportional to t to the fourth.
You should have expected the energy to be t to the fourth, and then you differentiate it once and you get t cubed.
So this is why by expected that by treating the waves the same way we treated radiation, we would get the t cubed law.
Furthermore, there's something really exciting about this formula here, and that is that there's no free parameters.
Remember when Einstein did his calculation, he had this frequency, this Einstein frequency,
the oscillator frequency, which he didn't know how to come up with this frequency.
He just fit it to the experiment to try to make it all look nice here.
There's no free parameter. Omega debye here is fixed by the density and the sound velocity.
So everything is fixed here. And if you calculate this quantity and you compare it to the low temperature heat capacity of most materials,
it actually actually agrees extremely well. But unfortunately, like, you know, you never get anything for free.
So we still have a problem. Problem.
At high tea. We want the law of do along petite silver and is three KB and we didn't get that.
We got T cubed at all temperatures here and this is where the buy had to actually scratches had a little bit.
And think okay what did I do wrong. Incidentally, if you do this calculation for electromagnetic radiation,
it really is t to the fourth energy all the way up to arbitrarily high temperature.
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.
So here we have to deviate from Planck's calculation.
We have to do something different from what it was that plotted and Planck and Debye understood that where he was going wrong,
where he was going wrong. He has to think about where this three comes from.
Where is this three from? The three comes from the fact that the atom can move in three directions.
Each atom moves in three possible directions. You can think of it as three degrees of freedom that the atom has.
And how many degrees of freedom did he count in his box?
Well, the total number of modes, number of modes that he counted is integral from zero infinity du omega.
The omega and g of omega goes as omega squared. You integrate that into infinity and that gives you infinity.
So he counted an infinite number of degrees of freedom and he said, Well, wait,
I figured I counted an infinite number of modes in the box, but there's a finite number of degrees of freedom.
Each atom can move in three directions, and there shouldn't be more than that many degrees of freedom in my box.
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.
So how did he do that? Impose a cut-off?
So when he's going to do is he is going to declare some frequency cut off omega,
cut off the omega, the omega, such that there are exactly three and modes in the box.
Now, this is an ad hoc solution and it's kind of a little weird.
So what he's doing is he's saying, we have all these wave modes in a box.
And when you get to some frequency known as the cut-off frequency above that, there are no more wave modes left.
You have no sound moves above this above this frequency anymore. Seems a little strange.
Later in the term we're going to see it's maybe not as strange as we thought,
but at that at that time, maybe it seemed a little odd, but he's going to do it anyway.
So we're going to do this anyway. We're going to follow him and see what happens.
First thing we need to do is we have to figure out what this cut-off frequency is going to be.
So it's going to give us exactly three modes. So let's calculate this thing.
We'll plug in G of Omega. So plug in G of Omega here.
So there's a nine and there's omega two by cubed integral zero to omega cut off omega squared D omega.
So I just plugged in g of omega there and then let's do that integral and we get three and omega cut-off
cubed over omega devi cubed by cubed and we want this to equal three n and in order for that to be true,
we better choose omega cut-off equal to omega two by.
Okay. That's why I happened to choose those particular constants as being omega two by omega two by is the cut-off frequency.
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.
Okay. All right.
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,
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.
And you'll notice now, because we've cut off our number of modes, this term no longer diverges.
It's a finite 0.2 energy term, and that makes us a little bit happier.
But again, we can actually ignore it. Drop this because it's temperature independent.
It's a temperature independent, zero point energy. We're going to differentiate the thing anyway.
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.
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.
Once you differentiate it at any temperature we choose just by plugging the temperature in here.
But it's useful to look at various limits. So the first limit is low temperature limit.
And by that I mean cavity much less than H bar omega devi.
And in this limit, we get the same result, same result as we had before.
This result here. Exactly. This nothing changes.
Why is that? Well, the reason nothing changes is because at low temperature, this Bose factor vanishes very quickly with frequency.
By the time you're up near the Dubai frequency, the Bose factor is essentially zero anyway.
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.
The integration is zero anyway by that time. So you just.
You don't have to worry about the cut-off at all. At low temperature, the cut-off isn't doing anything.
So you still get the t cubed heat capacity, which is what we want.
But at high temperature, at high temperature, we have something different.
Well, okay, so let's look at the Bose factor at high temperature, at high temperature,
beta h bar omega is a small number, so we can expand the exponential.
So we get one plus beta h bar omega plus start minus one.
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.
So if we then take this Bose factor and plug it into that energy expression, again, dropping the zero point energy,
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.
And then we have the Bose factor, which is k, b, t over h bar omega,
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.
And this integral here has been designed to give us exactly three n.
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.
So by implementing this cut off the by managed to get the low temperatures.
Heat capacity is t cubed. The high temperature heat capacity is still the law of Duong Petite, so that's pretty good.
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.
You see, it's converging to the law of do long, petite and at low temperature it's roughly t cubed.
And the debye theory agrees with the experiment extremely, extremely well.
You can see on the same on the same plot, there's the Einstein theory,
which fits pretty good, but not quite as well as to buy, particularly at low temperature.
And but an additional really important improvement from Einstein to Dubai is that Dubai has no free parameters,
that everything is fixed in the Dubai theory by just the velocity of sound, of the density.
So there's no no, you can't muck around, you can't adjust things. It just fits by itself.
So it's a really good result and seems to agree extremely well with the experiment.
But we still have problems, still wrong or still well, things that are still wrong.
One is the cut off is really ad hoc, cut off, ad hoc.
We just sort of made this up. I mean, it was a motivated it was a motivated thing to do.
It was an intelligent thing to do, but it wasn't really justified.
Another thing that's kind of wrong is that we used omega proportional to wave vector the sound law here.
That frequency will be proportional to wave vector, but we use this at high K and that's not true.
Sound is A is a small K, a long wavelength phenomenon when you go to very small wavelengths or very high wave vector.
This is no longer true. So this was sort of a problem that we brushed under the rug.
And incidentally, the the wavelengths we're talking about,
when you get up near the Dubai frequency, the wavelengths are close to the entire atomic spacing.
So we're talking about really, really small wavelengths or really high wave vectors.
And we can't really think about sound in that regime anymore.
So this is sort of a problem we brushed under the rug. Another thing is that Dubai is not exact.
It's not exact for any material, although it's pretty good, as is obvious from that plot.
And third, our fourth metals are different.
Metals are different. Well, for metals we have this.
For metals.
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.
So if you did the Dubai theory, you would get the coefficient alpha correct, but you wouldn't get gamma at all.
So this is just a big question right now. We don't know where that's where that's coming from.
So we have to figure that out. And in a few lectures time, we'll have a good idea where that's coming from.
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?
Shouldn't I see a linear heat capacity at low temperature?
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.
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.
And what's plotted here is the heat capacity divided by the temperature as a function of temperature squared.
If it was a purity cube law, then this line would be a straight line and it would intersect zero.
Obviously doesn't intersect zero. It intersects a finite intercept, which is, in fact, gamma.
So there's a very, very small term gamma. This term gamma is pretty small, but it's clearly there.
You have to measure pretty carefully to see it. But it's there. Okay.
All right. So at this point, we're sort of done talking about vibrations in solids for a little while.
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.
But for now, we're putting this aside and we're going to switch gears and start talking about metals,
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.
And this is something that was this was known to be known to the ancients, even probably in caveman days.
They they knew that there were some materials they found in the ground that just looked different from other materials.
And by 4000 B.C., people were able to work with certain metals, things like like copper.
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,
they were able to make new things, new devices, new technologies, and really changed the history of humankind.
So, you know, starting with metal ploughs, metal swords, metal armour, metal warfare of all sorts.
Then later on metal machines, you have metal skyscrapers.
You know, the nuclear age was brought in by, you know, heavy metals and heavy metal music, very important also.
But metals, you know, the history of metals in some way traces the history of of mankind.
And, you know,
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.
Before we really understood the properties of metals, you have to remember, you know,
for us the the defining property of a metal is going to be that it conducts electricity and non metals don't.
But we didn't even know it what electricity was until the late 1800s.
It was 1897 before J.J. Thompson discovered the electron or what he called the small corpuscle of charge that
can move around freely in the metals and could be ejected out of the metal by a sufficiently high voltage.
And with this picture of the metal really being sort of a container for all these electrons running around,
there became a natural thing for people to do, which was to consider these electrons running around as a gas, a gas of electrons.
And this is what Paul Drew to did. It's known as Judah Theory of Metals.
Of Metals. Well, our junior theory of transport applies to metals, and it's actually particularly good.
It works. This drew the theory. It's a very crude classical kinetic theory, kinetic theory of electrons of electrons,
very much like the kinetic theory of gases that you study last year in your thermal physics course.
It works extremely well, despite the fact that it's very crude.
It works extremely well for a lot of things, particularly well for semiconductors.
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.
So as with your last year's kinetic theory, we have a couple of assumptions.
Assume in order to get kinetic theory going, one, there is a scattering time.
Scattering time. Tao should look familiar from last year, by which we mean that the probability of scattering,
probability of Scott in time t or in time d t is equal to d t over tao.
Now this probably looks familiar from last year when you did Kinetic Theory of Gases last year.
When did you connect their gases? You can even predict what the scattering time Tao is.
Based on the size of the atoms and how fast they're moving and things like that.
This year we're not going to be so lucky for a couple of reasons.
First of all, it isn't clear what the scattering cross-section of the electrons should be
because the electrons interact with things via long range Coulomb interaction.
So they can they could scatter from things very far away, potentially.
Another thing that's going to make it difficult to figure out with towers is that
the electron can scatter off of lots of other things besides just other electrons.
They can scatter off of protons, it can scatter off of impurities, it can scatter off of anything.
And that happens to be in the metal. So for us, the scattering time Tao is just going to be a phenomenological parameter.
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.
So imagine something, something moving along. It scatters, and then its final momentum is zero.
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,
the average of the vector after the scattering is pretty close to zero because it can go off in any possible direction.
And that's going to be good enough for us to be able to make progress.
The third point, which you probably didn't have last year, is that between scattering events,
between scatters, the electron should see C's E and B field if they happen to be there.
So if you're applying an electric field to your metal, the electron will accelerate due to the electric field,
or it will curve due to the magnetic field, which seems rather natural, just like the electron were living in a vacuum.
Okay, so given these three assumptions, we can imagine that we start with an electron that has momentum p at time.
T So t is momentum while well, that's obvious at time t time t.
And then we'd like to calculate what is the momentum that time T plus d t I mean,
in some ways we're asking what's the expectation of the momentum at time t plus.
But we'll treat it as the actual momentum at time t plus t.
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.
Now, this is the probability. This is probability of not scattering prob of not scattering mass scattering.
If it does not scatter, then what happens?
Well, then it has the original momentum plus whatever force is applied to at times d t and that's just okay.
This is just saying that dpd t if it doesn't scatter, is f Newton's law.
Good. But in addition to this, there's also the probability probability d t over t that it does scatter.
And if it does scatter, we give it momentum zero. Okay.
So this is the probability of not scattering. It accelerates as usual due to the force applied to it.
And if it does scatter, we give it momentum zero at after the scattering.
Okay, then we can do a little bit of rearrangement here. Well, actually, we multiply this out first.
So this is then PV T plus f, d, t minus P over tau.
And then there's de t, and then there's plus order D squared.
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.
Yell at me if I start writing incomprehensibly, you know, if it really gets too painful.
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.
So let me rewrite this because this is an important equation. DP d t equals force minus t over tau.
This is known as the due to transport equation, you know. And what force are you supposed to use in it?
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.
So this looks a lot like Newton's equation.
D equals F is Newton's equation. But we have this additional term on the right hand side, which looks like a drag force.
It's a force going in the opposite direction from its current momentum.
So whichever direction is going, the force is pulling in the opposite direction.
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.
So you're not applying any electric or magnetic field, then you just have DPI de t is minus p over tau,
which you can solve by saying p t is some p not some initial momentum e to the minus t over tau.
So that tells us that the moment if I have an electron moving along with some initial momentum,
this scattering term here slows it down exponentially to zero momentum with that time scale.
Tau So it's like a drag.
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.
And I guess we will stop there and we'll pick up with due to theory next time. I'll see you tomorrow.
Okay?