Okay. Welcome, everyone. Hello.
Welcome. Getting started. Hello.
Welcome. This is the first lecture of the third year basics condensed matter physics course.
Before we get actually started in the material for the course,
I want to mention a couple of key resources we have that are meant to make your life easier.
The first key source is the book. I'm going to write down book.
So I wrote this book for this course.
The the idea of the book was to cover everything that could possibly be asked to you on a condensed matter physics exam.
So if you learn everything in this book, you'll be in very good shape.
As we go through the course, you'll discover that the book and the lectures are extremely similar.
This is not a coincidence. I wrote them both. The. But the book is actually more detailed in many places than the lectures will be.
The lectures will emphasise those things which are most important so that I view them sort of as being a little bit complementary to each other.
Hopefully you'll find them both useful. You can pick up books at Blackwell's there in many of the libraries there on Amazon.
I do apologise. It's not quite as cheap as I would have liked it to be.
I think it costs about £25 a copy. So apologies for that.
The second web page. The second resource is the Web page.
There's a lot of interesting and useful things on the course web page link to my homepage.
One thing that you might find useful on the web page is the book.
Now, the book, the actual book is not on the web page.
The publishers wouldn't let me do that. Presumably they figured they would never sell one if I gave it to you for free.
But the draft that we used in previous years is posted on the web page and is a pretty good replacement for this.
It's about 85% similar to what's actually in the book.
So if you find yourself in a library, not having your book or something like that, you can just open up the web page in and see what's in there.
Maybe you might not even find that you need to spend the £25 on it.
Although the book is better, a lot of the areas have been corrected.
The pictures are better, some of the explanations have been improved and so forth.
So as you choose other resources on the web page that are useful, the suggested homeworks.
Please check with your tutors to make sure they agree with which homework assignments you should do.
Sample Exams. Sample solutions are also there.
PowerPoint slides. First of all, I will never leave any sort of handouts here, so you don't have to look,
I promise you that anything that I need to show you in the lecture in terms of a PowerPoint slides, some sort of figure that will be on the Web page.
Now, I know that people prefer chalk to PowerPoint slides.
I feel the same way. To the largest extent possible, I will always use chalk.
But there are some times when I just have to show a picture that I can't draw.
So those slides or videos or anything else they show will be on the web page.
You may notice there's someone in the back, Greg, filming us that will presumably be link to the web page also.
So if you want to see something, some explanation or something like that, it will be on the web page.
Any correction, I have to issue corrections to the homework, corrections from the books,
any typos in the books, anything I might say in lecture that turns out to be incorrect.
Somehow that will be fixed on the Web page. One more thing I really want to point out is the message board.
That's linked to the Web page as well. I'm not sure if you had this in your other courses before.
We tried it for the first time last year. It worked extremely well.
Other courses are now using it as well.
The idea is to have an online forum where people can discuss what's going on and ask questions and give answers.
When you're outside of lecture. So the way it worked last year is someone would type the type in you said X,
Y or Z in lecture, but that doesn't make sense because of this, that or the other.
Can you please explain? I would write some answer, but just as often is not.
It wouldn't be me giving an answer, but some other student would give an answer or a tutor would give an answer or a
tutor would ask a question and some student would answer it or something like that. So.
So it was just some place where people could go and just leave simple messages and get them answered.
And I think as the course goes on, people will find that very useful. Don't be shy in trying to use that.
So that's all I have for resources, and we can actually get started with the introduction to this course.
The first thing you're probably asking yourself is What is this subject we're supposed to be learning?
What is condensed matter physics? Well, to begin with, condensed matter physics is one third of physics.
Approximately one third of physics. And what I mean by that is the following experiment.
If you go around the world and you ask every single physicist, Are you a condensed matter physicist?
About one third of them will tell you, Yes, I'm a condensed matter physicist.
Condensed matter physics is the largest subfield of physics. Actually, it's the largest subfield by far.
It's extremely broad, extremely diverse, and it includes many, many different topics within it.
To give you a sort of an example of how big and how broad and how diverse this field is.
I'm going to use the March meeting of the American Physical Society.
Every march, the American Physical Society holds a meeting.
Of all of this condensed matter, physicists are as many as can possibly show up.
Last year was in Baltimore, Maryland. I went there myself. There were 8000 physicists there.
It was very exciting for one week. Huge Nerd Fest, very enjoyable.
I recommend going if you ever get a chance at 8 a.m.
On the first day of the meeting, there were talks on the following subjects superconductors, superfluids, glasses, polymers, quantum dots,
microfluidics crystal growth spintronics phase transitions quantum criticality Bose Condensates Fractionalised Charges Quantum Computation,
high pressure physics magnetism, heavy fermions, multiple rocks and liquid crystals.
And that's just at 8 a.m. on the first day, by 11 a.m., the first day, there were just as many talks on different topics.
So the list goes on and on and on. Of all of the interesting things that are contained within condensed matter physics,
there tends to be a large overlap in condensed matter physics with fields such as chemistry,
material science, biology, atomic physics, sometimes high energy physics, nanoscience, quantum sciences.
And more and more often we're getting overlaps of black hole physics and string theory and gravity, as well as many other fields of physics as well.
So it includes many, many, many things in it. And you might be asking yourself at this point, why study it?
Why study it, besides the fact that it happens to be on your syllabus?
But that just raises the question of why did someone put it on your syllabus?
Well, there's a lot of good reasons why we should be studying this subject.
The first good reason is one, it is the world condensed matter.
Physics is the world around you. We think of condensed matter as being the study of the stuff in the world around you.
So anything you can point at, any material you can, you can look at.
Any thing you can pick up solids, liquids, glasses, polymers, you know, metals.
These are all condensed matter in some way. And if you point at these things, you know, ask questions.
That's what we as scientists and we as physicists should be doing. We should be pointing at things and asking questions about them.
If you ask questions like, Why is glass transparent?
Why is metal shiny? Why is rubber soft and squishy?
Why is water wet? Why is oil slippery? And even more subtle questions like Why is egg yolks crucial for making good mayonnaise?
All of these things are the subject of condensed matter physics.
Condensed matter physicists are in the business of explaining all of these things.
And in that long list of questions, at least some of them, by the end of the term, we will have answers, too.
So pretty much everything you see in the world around you. The world is condensed matter.
Physics. Reason to. It is useful.
Condensed matter, physics is by far the most technologically and industrially important field of physics.
You know, we as humans and we as scientists and we, as condensed matter physicists over the last hundred and 50 year have come to
an incredible control over the materials and the stuff in the world around us.
You know, examples of this include, you know, 150 years ago, people first started being able to think about electrical currents moving in materials,
and then they figured some materials had different electrical properties than others.
And pretty soon they were able to make electronics and they could build up very fancy electronics.
And pretty soon they're building things like iPads, iPhones, computers and things that we take for granted in the world around us today.
And all of this comes from the study of condensed matter physics.
Condensed matter physics really changes our lives as people that these things that we take for granted now come out of condensed matter physics.
By far, the most useful field of physics is condensed matter.
Reason three Why we study it. Three It is deep, fundamental and deep new ideas in condensed matter physics.
It is. Now, some people have this mistaken idea that two in three are somehow contradictory.
If it's useful, it can't be deep. If it's deep, it can't be useful. This is completely untrue.
The ideas that we talk about in condensed matter physics are every bit as deep as the ideas that we will run into in any other field of physics.
How do I know this? How can I prove this to you?
The reason I know that is just as deep as any other field of physics is because they're exactly the same ideas.
Good ideas come from one field and go into the other field and vice versa.
And as often as not, the best ideas have come out of condensed matter and gone into other fields.
Really good example of this. Everyone has probably heard of the Higgs boson.
Professor Higgs won his Nobel Prize just a few months ago for prediction of this subatomic particle,
which was then discovered and certain many years later. Where did he get the idea for the Higgs boson?
Well, if you read paragraph one of his Nobel Prize winning paper, it says very explicitly,
I got this idea from condensed matter physicists back before Higgs P Condensed Matter physicists were
studying metals at low temperature and they discovered that metals superconductor at low temperature,
they have no electrical resistance at all. And after coming to a really good understanding of what causes this condensed matter,
physicists were suggesting to high energy physicists that something similar might be going on in a high energy,
and that is the origin of the Higgs boson. So the deep ideas have come out of condensed matter and gone into other fields.
For. Reason for why we study it.
A.I. Reductionism. This is a word that I made up.
But but reductionism is actually is a real word.
The reductionism is the idea that you're going to learn more about something by asking what is it made of?
What is a smaller and smaller and smaller piece?
Anti reductionism is the idea that that is the wrong way to go, that that is completely the wrong way to understand something.
An example that people like to use. If you asked to describe a glass of water and if you start along,
this rabbit hole of the water is made up of molecules and molecules are made up of atoms.
The atoms are made of protons,
neutrons and electrons and neutrons and protons are made up of quarks and those are made of strings and so forth and so on and so forth and so on.
You get nowhere. You didn't learn anything about why the water is wet.
You didn't learn why it's transparent. You didn't learn anything really useful at all.
Reductionism completely misses the forest for the trees. And more often than not, if you want to understand the real physical properties of something,
what you need to ask about is the big picture how the pieces act together to give you the whole.
So anti reductionism is a good reason to study condensed matter physics.
Five. It is a laboratory.
Laboratory. For Quantum and step mak to a large extent condensed matter.
Physics is the best laboratory we have for exploring the amazing things that quantum
mechanics and statistical mechanics can do and the things that they can do together.
So I would like you to view this entire course as an extension of what you learned in those two courses last year.
And if you like those courses last year, hopefully you'll like this course just as well because it's really just the same thing.
And if you didn't like those two courses from last year, I will bet that you will like this course better.
But there's actually a good reason to think that you'll like this course better,
because instead of thinking about these subjects in the more abstract work,
actually think about them in some very real and important applications and how they apply to some real, real stuff.
Okay. So as I mentioned early on, condensed matter physics is a very broad and diverse field and we can't possibly
study all the subfields within condensed matter all within these eight weeks.
So we're going to focus in on just one subfield of condensed matter.
And the particular subfield we're going to spend all of our time on or most of our time on is solid state.
And by solid state.
What I mean is the solid state of matter as compared to the liquid state of matter or the superfluid state of matter or some other state of matter.
And the reason we there's several reasons why we pick solid state here.
It's the largest, the most useful and the most successful of the subfields in condensed matter physics.
We know more about solids and in particular about crystal in solids and will define what we mean by crystal in solids later on in the course.
We know more about solids than we know about any other state of matter than what we
have learned about solids is incredible what we've been able to do with solids.
You know, you think about it, electronics, the electronics industry, that's all made of solids.
So all of that, that's why it's called solid state electronics, because the electronics of the solid state,
all of that is incredibly well understood and it's a great place to start our study of this field.
But more than that, the reason we start with solid state is because the things we have to learn in studying Star State form a really good platform,
a really good jumping off point for studying other fields within condensed matter and even outside of condensed matter.
So if one wants to go on and study something more complicated later on in life, like a liquid or a superfluid,
what we learn in solid state is going to be a fundamental starting point for learning those things later on.
So that's my introduction to the course. And at this point we can actually get started on on the subject proper and a good place to start.
Studying solid state or condensed matter is actually over 100 years ago, around the turn of the century 1900.
And the reason I pick that time is because that was about the time when people first started applying things
like statistical mechanics and later on quantum mechanics to the study of the things in the world around them.
Over the course of the 1800s, scientists were able to make a lot of very increasingly precise measurements,
and they had figured out a lot about the world around them.
But there were a lot of the measurements that they could give, the measurement.
They could see what was going on, but they couldn't understand why they were getting the results that they were getting.
And for the next two lectures or so, we're going to focus on one particular type of experiment that they had been doing a lot of.
And those are experiments on heat capacity. But you probably remember from your thermodynamics course and in particular heat capacity of solids.
So they were able to measure all sorts of things about heat capacity and they had been doing so for close to 100 years at that point,
but they didn't know what the results meant. So just to remind you what heat capacity is, heat capacity is d d t,
you know how much heat you have to put into a material to raise it a certain amount in temperature.
And this is the first equation of the course. And already someone should be objecting and saying, wait a second, that's wrong.
So why should I say something? Something's wrong with this.
Well, if you remember from your courses in and AMOC last year, whenever you write a heat capacity, you should write a subscript.
Whether you mean heat capacity, a constant pressure CP or heat capacity, a constant volume CV and generally these two things are different.
I didn't do that.
I just wrote C and the reason I only wrote C instead of C, P or KV is because in a solid C.P and CV are very close to the same number.
So we just call it a C.
I mean, if you measure really, really closely, you'll discover that slightly different, but they're very, very close to the same.
Now, why are they very close to the same? I'll remind you of something that you learn to derive.
Last year may have even been on your exam. It's the kind of question that shows up on the exam every year or two.
The C p minus CV is volume times, temperature times, alpha squared over beta, where alpha is a thermal expansion coefficient the thermal expansion.
And data is the isothermal compress ability as a thermal.
Possibility the press. And the point here is that solids have very, very small thermal expansions.
Generally, a solid will have a thermal expansion coefficient of a few parts per million per degree Kelvin.
Very, very small number. And then that number gets squared over here on the right hand side.
So the right hand side is really, really, really small. So they are almost exactly the same.
And our level of understanding or our level of analysis, we can just treat them as as being exactly the same.
Not true for a gas if we're thinking about a gas. Thermal expansion is very large.
Cpcb As you learned last year, it can be quite different.
So let's remind ourselves of a couple of things that we learned about heat capacity last year.
Let's go back to gases now since I mentioned one for our modern atomic monit mana atomic, atomic, atomic gas.
You learned last year that CV over and here, since I'm talking about a gas, I have to specify CV because it's different from CV.
CV over and the heat capacity per atom was three KB.
Does that look familiar? Yeah. Okay, good. I'm glad. Well, it turns out that there's a very similar law for solids.
Solids? See over.
The capacity per atom is 3kv and I didn't specify c, v or c p as I explained this law,
the silver n is 3kb is known as the law of do long, petite, do, long, petite, do long and petite.
We're a French chemists who discovered this law way back in 1819 and all the way through the rest of the 1900s, almost to the very end of the 1900s.
People knew about this law, but they didn't know what caused it. And so along comes Boltzmann.
So Boltzmann was a very smart guy, and he very much liked this statistical mechanics stuff.
He sort of invented the field and he thought that maybe he could understand this law.
And you long, petite using his Boltzmann machine, Boltzmann on using his statistical mechanics.
Now, if you remember, what is this statistical mechanical picture of a gas?
He already knew how to derive this three half KB for a gas.
And its picture of a gas is that you have these atoms and they're flying back and forth in space.
As you raise the temperature of the gas, they fly back and forth faster.
And so they have more kinetic energy. So D, Q Q is the energy and two T is the temperature and Q goes up.
His T goes up. So that gives you some some heat capacity. So he thought, well, maybe, maybe a solid is really similar to a gas.
You know, you have some atoms in the solids and as you raise the temperature, they move around faster.
But it's not exactly like a gas because they're not flying around completely free.
You know, one one of the atoms in the solid, it moves off to the left and then it gets pushed back by its neighbour.
It doesn't go off completely off to the right or the left, and it sort of gets pushed back to its original position,
then goes off in the other direction and then comes back to the original position.
So he thought, well, maybe. I should make a model of the solid as the Boltzmann model of solid.
And that was in 1896. And his model of a solid was really simple.
It was just a harmonic. Well, with an atom in the bottom of the harmonic well, and the atom can oscillate back and forth.
And as you raise the temperature, it oscillates more back and forth and so is storing more energy.
And so that's some amount of heat. And so he just needs to calculate the heat capacity of this atom in the bottom of the potential.
Well, so how do you do that? Well, there's a really short and easy way to calculate the heat capacity is is to use the echo partition theorem,
which you probably learned last year,
which basically says for each degree of freedom where you can store energy, you get one half KB worth of heat capacity.
So let's remind ourselves how this works. So, for example.
With a minor atomic gas, mine atomic gas.
What are the degrees of freedom that store energy? Well, the gas molecule can be have momentum in the X direction.
It can a momentum in the Y direction. It can momentum in Z direction. So. P.
P. That's three degrees of freedom per atom.
So see the over pn is three halves.
KB k b. Okay.
Now. What about the solid? Well, solid. Pretty similar.
What are the degrees of freedom that can store energy?
Well, you have momentum in the direction, momentum in the Y direction, momentum the Z direction.
But also you can store energy in the X, Y and Z coordinates because if you take that atom, you displace it from the position zero.
It costs you energy, even if it doesn't have any momentum,
it just costs you energy to you move it up the potential world or to stretch the spring if you want to think of it that way.
So C over n six degrees of freedom is three.
KB lived long, petite. This was boatman's reasoning.
Now for your first homework assignment, you'll derive this more rigorously.
And if you remember from your stat neck course last year, the rigorous way to go about this is to write down a partition function.
Differentiate the partition function to get the energy. Differentiate the energy to get the heat capacity.
And if you do that, you'll get exactly this. Number of three has to be.
Now, this result that Mr. Boltzmann derived in 1896 was an extremely important result for a number of reasons.
The first reason is it explained this result, this law of long, petite that had been known for almost 100 years and no one knew why that had held.
And all of a sudden he he had this good reason why it holds.
And but moreover, it was extremely important because in 1896, not a lot of people believed in statistical mechanics.
Boltzmann was kind of off on his own. He and a couple of his friends believed in system mechanics, and no one else did.
But each time Boltzmann came up with a result that he could explain and no one else in the world could explain.
People had to take the statistical mechanics stuff more seriously, and this was one of the important results for him.
So so this was it was a great advance.
But there was a problem, something that bothered Boltzmann and something that made people think that potentially maybe both man's reasoning was wrong.
And the problem was that the law of do long, petite of deep is not always true.
Not always true. Fails sometimes.
So if you take a material, you measure its heat capacity and you come up with three KB per atom.
But it turns out that if you take that material and you cool it down, take it to a lower temperature at t much less than t room,
you would discover that C over an always becomes much less than three kb.
So the heat capacity per atom drops at low temperature.
For some materials, rare materials, but for some materials and one in particular, diamond for diamond that's silver and is much less than three KB.
Even at room temperature. At tea room.
Tea room. So for most materials at room temperature, you get three kbps heat capacity per atom.
Diamond is an exception. This heat capacity is smaller. You take any material and cool it down and the heat capacity will drop below three.
KB So this was a puzzle to Boltzmann. And it puzzled everyone for for a number of years.
And in fact, Boltzmann never lived to see the answer to this puzzle.
He unfortunately committed suicide in 1905. Rather sad end for a great scientist.
And it was two years after that that this was finally sorted out by a very, very bright young man by the name of Albert Einstein.
And this you know, this achievement by Einstein is actually one of the most important things that Einstein ever did.
Way up there with relativity or photoelectric effect that people know a lot better.
But this result by Einstein, I will argue, is just as important as as those results.
So this is Einstein's model, Einstein model.
Of a solid. Oh, solid. Which is 1987.
And Einsteins model of a solid is actually exactly the same as Boltzmann model.
Boltzmann equals Boltzmann solid.
Plus one ingredient. And the one ingredient is quantum mechanics plus quantum.
So just to be a little bit more specific about what we mean here about this model.
So again, we have an atom in the bottom. A potential well can oscillate back and forth.
There's an oscillator frequency omega is a square is some ring spring constant divided by some mass.
And then we have to treat this this potential.
Well, using quantum mechanics, not using classical mechanics.
Now this. Result by Einstein was completely outrageous in 1907, and you have to really think about what was going on in 1907.
This was 19 years before the Schrodinger equation. No one knew about quantum mechanics.
There wasn't even a word for quantum mechanics then.
So he was really working way out on a limb, and he was deducing from the experiment what must be going on in nature.
He was figuring out quantum mechanics based on these experiments.
Now we have some huge advantages over Einstein.
We know about the shortening equation. We know about iGen states. We know how to treat Quantised Eigen states using statistical mechanics.
So we're going to be able to do this much, much more easily and much more directly than Einstein did.
Einstein had to take some really roundabout routes to get to the final answer, which turns out to be a pretty good answer.
We have another huge advantage over Einstein, which is that we're alive and he's dead,
and that that will put us on a much more equal intellectual footing.
I still bet on him. But all right.
Anyway, so what we learned about the harmonic oscillator in quantum mechanics last year is that the energy states,
the eigen states of the atom in the potential well should be given by the following expression.
S of n is h bar omega and plus one half.
Now I should be a little bit more careful here. This is the result for a one dimensional oscillator, one dimension.
And we do live in three dimensions. And so we're going to have to fix that up a little bit later.
But let's stick with one dimension for now and see if we can figure out the heat capacity of an atom in the bottom of a potential well,
in one dimension. So now we know how we know how to handle this from our step course.
What we do is we write down partition function, the equal sum over the eigen states and greater than equal to zero,
either the minus beta and where beta is one over in the usual way.
And then we're going to take this partition function and we'll differentiate it to get let's see.
So we need one over the DB debater to give us the expectation of the energy.
Just a comment that this expectation is both a quantum mechanical and statistical mechanical expectation.
And I won't go through the algebra to do this that's actually on the homework assignment.
And actually you probably did it last year as well.
But if you do that algebra, you get the final, final expression for omega and sabi of beta omega plus one half when the B is known as the both factor.
Both factor. And B equals one over eight of the beta omega minus one.
And actually, if you compare this expression for the expectation of the energy to the expression of their E and equals H for omega and plus one half,
you know, they look very similar.
It's just and got replaced by and B and what that tells you is that both factor tells you the expected level of excitation at a given temperature.
So I've had a particular temperature and Bose takes the value three.
It means on average at that temperature you're excited up to the end equals three excitation level.
Okay. All right. So once we have an expression for the energy, we can write an expression for the heat capacity, the energy, the T.
And if you do that algebra again won't do it here.
It's KBE times beta h bar omega squared.
Even the beta omega over the beta h for omega minus one squared.
Now this is almost our final result, except we have to deal with two things.
The first thing is that we only calculate the heat capacity of a single atom.
So if you have a lot of atoms, you would have to multiply that by the number of atoms.
So this would be the right result for the heat capacity per atom.
But the second thing we have to fix is that we were considering a oscillator in only one dimension, and in fact, it can oscillate.
You know, atom can oscillate in any one of three dimensions. So we're going to multiply this result by three and get the final result.
C over add is three times kbe beta omega squared either beta h for omega same expression for omega minus one squared.
And this is my science. Final, final result. So it's worth thinking about this expression for a second and seeing what its properties should be.
The first thing we might do is take high temperature limits can be much greater than Omega or in other words, beta h, bar omega, much less than one.
And if we do that, we then have either the beta h for omega is approximately equal to one plus beta omega.
Plus da da da da. So then if we take this into the beta h omega, we plug it in upstairs.
We only need to keep the one. We plug it in downstairs.
The one cancels this minus one and we have to keep the beta h for Omega downstairs.
So then we have beta promega downstairs, we have beta H-bomb Omega upstairs.
Those two cancel and we get C over n is three KB.
Everyone good with that? Yeah. Okay. So this recovers at high temperature, the law of duelling fatigue.
And maybe that's not surprising because, you know, a high temperature limit of a quantum system is sort of like a classical limit.
It is going to give you back what Boltzmann calculated classical physics, just like Boltzmann expected.
Low temperature, though, is quite different. So it's tri cavity, much less than H Bar Omega.
In this case, what we have is either the beta H Bar Omega is big, very big, exponentially big.
In which case we can come over to this equation here. The minus one doesn't matter compared to this huge number.
And so it's just going to give us an idea. The beta age by mega squared downstairs and we'll cancel one of the things upstairs and
we'll get C over N is three KB times theta omega squared E to the minus beta h for omega.
The important thing about this is this exponent, which is exponentially small.
Exponentially. Small.
Tiny. So when you go to temperatures, way below is promega.
The heat capacity drops like crazy. So let's let's actually plot this.
So I'll plot this function. This function here, see over n on the vertical axis and kb t over bar omega on the horizontal axis.
And we know that in high temperature we're going to asymptotes to three kb.
So look, something like this up and high temperature and low temperature, we're going to be exponentially small, X small.
And that connects up kind of like this. And then maybe here is something KB over omega is about one.
So this is what the function looks like. This is what Einstein derived.
So the physics of what's going on here is that, again, high temperature is classical.
But what's going on at low temperature? What's going on at low temperature is that these harmonic oscillators are freezing into their ground state.
So if the harmonic oscillator is sitting in its ground state,
it can't absorb any energy unless it has enough thermal energy to jump all the way up to the first excited state.
And there's a finite gap of age for Omega before it can get into the next thing and state.
So if its temperature is much less,
then that spacing between these eigen stays is just stuck in the bottom eigen state and it can't absorb any energy at all.
So the heat capacity drops like crazy. That's what's going on in this picture.
And that's what Einstein realised must have been going on in these physics physical systems.
So what's going on with where is it?
Over here or over somewhere. Lost it. Oh, up there.
Yeah. What's going on with. With Diamond. Well, one thing that Einstein didn't manage to derive is this this H Bar Omega.
It was sort of a fit parameter for his theory. He knew, you know, you can figure out what the mass of the atom is with.
So where's this H-bomb coming from? It's the oscillator frequency up there.
It's some spring constant divided by mass. You know what the mass of the atom is, but you don't know what the spring constant is.
You don't know how springy you know, these harmonic wells are.
So you just have to guess at them. And so each material will have a different spring constant, a different so-called Einstein frequency.
The Omega is known as the Einstein frequency. And so the same frequency of different materials is different for some materials.
The Einstein frequency is very low, in which case room temperature puts KB h over H for omega to be a large number.
So you'd be up here on the graph.
As a matter of fact, for most materials, you're somewhere up here on the graph because Omega is less than room temperature.
But if you cool down the system to higher to lower temperature, your the heat capacity will drop as predicted by Einstein here.
However, some materials are different that they have a large h bar omega.
In which case room temperature would put you here on the plot.
Your temperature is lower than for omega, and so the heat capacity, even at room temperature, should drop.
Now, why is it the diamond is is one of these funny materials?
Well, diamond. Let's write again.
Omega is square root of Kappa over. M Diamond is special for a couple of reasons.
First of all, Diamond is carbon and carbon is a very small element, is very high.
Up on the periodic table is the only sixth element on the periodic table.
Only five elements are smaller, lighter than carbon.
So M is small for diamond, but also Diamond is a really hard material.
It's very, very tough. So.
K you might think that the toughness of the material is somehow related to its spring constant how, how hard the material is.
So K is really big. So for Diamond, the frequency omega is anomalously large.
It has one of the largest Einstein frequencies here.
So the oscillator frequency is extremely big. So room temperature is much higher than the Einstein frequency.
So on this plot, sorry factors, room temperature is much lower.
So that wrong room temperature much lower than the Einstein frequency, which is very big.
So in this plot you're way down on this side. And so the heat capacity per atom is really small.
So let me actually show you some real experimental data.
Here we go. So this is a picture right out of Einstein's original paper.
And what you have is the experimental data, which is these these circles, and you have the theoretical curve, which is the dashed line.
It seems to fit fairly well. Now, one should keep in mind that there is a three fit parameter H for Omega.
He was not able to predict what H bombmaker should be, although he had a good argument why bombmaker should be big for Diamond.
He didn't know exactly what the numbers should be, so he just chose the one that fit the data the best.
And it does seem, given that one parameter does seem to fit fairly well at high temperature,
you get three three kbps the law of daylong fatigue and at low temperature it seems to drop.
And I just want to emphasise one more time that this huge success for Einstein was important,
not only because he was able to explain what was going on in these materials, the heat capacity at low temperature.
But it was much more important because in order to do this, he needed to invent quantum mechanics.
He needed to invent the quantisation of the harmonic oscillator, an extremely important and pivotal moment in the development of quantum mechanics.
And it came because he was studying condensed matter physics.
So despite the fact that this is you know, this is a great result for for Mr. Einstein, his undoing is actually in this picture.
There are some shortcomings in this picture as well.
And the shortcoming always seems to be because something doesn't fit perfectly now, as you probably know from your practicals.
There's plenty of reasons why data may not fit theory. One reason might be because the data is wrong.
And if you look at this, this point here, which seems to be way off the curve, that picture, that point there, the data is just wrong.
Whoever measured it messed up. It's not even close to the right answer.
It's supposed to be down here. So that was one that probably caused Einstein some nightmares.
But in fact, that was not his fault. However, down here, you'll see that there's also some systematic error that this theory plot,
there's the dashed curve is below, systematically below the measured experimental data.
And that is real. Actually, that is a real problem, is a shortcoming of Einstein's theory.
In truth, for most materials, most materials, including diamond.
The heat capacity at a low temperature is proportional to t cubed at low t the exception to this,
an exception will come to later on in maybe two or three lectures is metals, things like lead, copper, whatever.
It's slightly different alpha t, cubed plus gamma t, but in no case,
no material at all is the heat capacity at low temperature, exponentially small, which is Einstein's prediction.
The heat capacity never drops as fast as Einstein predicted.
Actually, I think I have some data here specifically of diamond capacity of diamond as a function of T plot as a function of T cube.
This is a very low temperature. This temperature here is about four Kelvin and this temperature here is about 40 kelvin.
If it's a perfect t cubed line, their plot against t cubed.
So even way back then, Einstein,
I think was aware that there was a problem with this theory because it did not
predict t cubed predicted exponentially small heat capacity at low temperature.
And this is where the field stood in 1907 and had to wait about five years or so before someone came along,
who's also very smart to figure out what was wrong with Einstein's theory.
The person who came along was a guy by the name of Peter Debye.
And he approved improved upon Einstein's result.
And explained where this T cubed r results came from.
Now Dubai's into incidentally, I should say that Einstein thought that Dubai was really brilliant and he was very,
very happy that Dubai managed to figure out this t cubed and overturn the
Einstein model of the solid in what we will call the Dubai model of the solid.
Which is somewhat better. He was a Einstein was a good man.
Anyway, so is the bike.
Although I guess recently there's just been some stories about how Einstein was how Debye was secretly a Nazi and he hated Einstein.
But it turns out they were all wrong. There's a whole book about it you can find out on the web.
So it's got to be true, right? So anyway.
What Dubai's intuition was is that you can't consider an atom vibrating in a solid as just being an isolated atom in a potential well.
Why not? Because if an atom moves to the side of potential, well, it does get pushed back by its neighbour.
But at the same time, it pushes on its neighbour.
And so its neighbour moves and then it pushes on its neighbour and so forth and so on and so forth and so on.
So what you need to think about is not just the oscillation of single atoms in the bottom of potential wells,
but you have to think about the collective motion of all the atoms together in the solid,
and the collective motion of all the atoms in the solid makes a wave.
So we have to study wave motion of vibrations in the solid.
And in fact, we even know what the wave motion in a solid vibrational wave motion is.
Solid is called. It's usually called sound a sound wave.
So what to buy realises there's a connection between the oscillation of atoms in a solid and in fact sound.
Now, one of the thing that Debye understood at the time, which was really important, is that waves can be quantised too,
in the sense the same way that the vibrational atom, the vibrational motion of an atom in the bottle potential well can be quantised.
And the example that he look to as to how ways to be quantised was max Planck's quantisation of light.
So what Debye wanted to do was quantised the motion, the vibrational motion of atoms and sound waves in a solid,
the same way that Max Planck quantised the light ten years earlier.
And I guess we'll stop there and I'll see you, I guess, Thursday.