Transcription
Okay, welcome everyone. Hello, welcome. Getting started. Hello, welcome. This is the first lecture of the third-year B6 Condensed Matter Physics course. Uh, before we get actually started in the material for the course, uh, I want to mention a couple of key resources we have that are meant to make your life easier.
Uh, the first key resource is the book. I'm going to write down book. So, I wrote this book for this course. Um, 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 will discover that the book and the lectures are extremely similar. This is not a coincidence. I wrote them both. The um, but the book is actually more detailed in many places than the lectures will be. The lectures will emphasize those things which are most important. So, that I view them sort of as being a little bit complimentary to each other. Hopefully, you'll find them both useful. You can pick up books at, uh, Blackwells. They're in many of the libraries. They're on Amazon. I do apologize, it's not quite as cheap as I would have liked it to be. I think it costs about 25 pounds a copy, so apologies for that.
The second web page, the second resource is the web page. Uh, there's a lot of interesting and useful things on the course web page. Link to my homepage. Um, one thing that you might find useful on the web page is the book. Now, the, 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, if I gave it to you for free. But the draft that we used in previous years is posted on the web page, and it's a pretty good, uh, you know, 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 and, and, uh, see what's in there. Maybe you might not even find that you need to spend the 25 pounds on it. Um, although the book is better, a lot of the errors have been corrected, the pictures are better, some of the explanations have been improved, and so forth. Um, so it's, as you choose.
Uh, other resources on the web page that are useful is 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, uh, 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 slide, 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. Um, to the largest extent possible, I will always use chalk. But there are sometimes when I just have to show a picture that I can't draw. So, those slides or videos or anything else I show will be on the web page.
You may notice there's someone in the back, Greg, filming us. That will presumably be linked to the web page also. So, if you want to re-watch 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 from the books, anything I might say in lecture that turns out to be incorrect somehow, that will be fixed on the web page.
Uh, one more thing I really want to point out is the message board that's linked to the web page as well. Uh, I'm not sure if you've 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, 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 as 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? What are we 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 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 sort of an example of how big and how broad and how diverse this field is, I'm going to use, uh, the March meeting of the American Physical Society. Every March, the American Physical Society holds a meeting of all of its condensed matter physicists, or as many as can possibly show up. Last year, it was in Baltimore, Maryland. I went there myself. There were 8,000 physicists there. It was very exciting for one week. Huge nerd fest. Very enjoyable. I recommend going if you ever get a chance. At 8:00 AM 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, fractionalized charges, quantum computation, high-pressure physics, magnetism, heavy fermion, multiferroics, and liquid crystals. And that's just at 8:00 AM on the first day. By 11:00 AM on the first day, there were just as many talks on different topics. So, the list goes on and on and on of all 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 with 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, 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 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 and you 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 yolk crucial for making a good mayonnaise?" Um, all of these things are the subject of, uh, 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 to. So, pretty much everything you see in the world around you, the world is condensed matter physics.
Reason two: 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 150 years, 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. 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 and 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 you 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 at CERN 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, condensed matter physicists were studying metals at low temperature, and they discovered that metals superconduct 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 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.
Four: reason for why we study it. Anti-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 is if you ask 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 the protons are made up of quarks, and those are made up 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 stat mech. 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 liked 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, 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, we're actually thinking 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 crystalline solids, and we'll define what we mean by crystalline solids later on in the course. We know more about solids than we know about any other state of matter. The, 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 electronics of the solid state. Um, 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 solid 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 the subject proper.
And a good place to start studying solid state or condensed matter is actually, uh, over a hundred years ago, around the turn of the century, 1900. And the reason I picked 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, uh, 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, you could measure and they could see what was going on, but they couldn't understand, understand why they were getting the results 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. 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 is heat capacity? Heat capacity is DQ DT. 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, why should I say something's wrong with this? Well, if you remember from your, uh, courses in thermodynamics last year, whenever you write a heat capacity, you should write a subscript, whether you mean heat capacity at constant pressure, CP, or heat capacity at 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 CP or CV is because in a solid, CP and CV are very close to the same number. So, we just call them a C. I mean, if you measure them really, really closely, you'll discover they're 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 learned 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. CP minus CV is volume times temperature times alpha squared over beta, where alpha is a thermal expansion coefficient, thermal expansion, and beta is the isothermal compressibility, isothermal compressibility. Um, 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, CP and CV are almost exactly the same. And at 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 were thinking about a gas, thermal expansion is very large. CP and CV, as you learned last year, can be quite different.
So, let's remind ourselves of a couple of things that we learned about heat capacity last year. Um, let's go back to gases now, since I mentioned one. For a monatomic, monatomic, monatomic atomic gas, you learned last year that CV over N, and here, since I'm talking about a gas, I have to specify CV because it's different from CP, CV over N, the heat capacity per atom, was 3/2 KB. Does that look familiar? Yeah? Okay, good. I'm glad. Um, well, it turns out that there's a very similar law for solids. Solids, C over N, the heat capacity per atom, is 3 KB. And I didn't specify CV or CP. As I explained, this law that C over N is 3 KB is known as the law of Dulong-Petit. Dulong-Petit. Dulong and Petit were 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. Until along comes Boltzmann.
So, Boltzmann was a very smart guy. Um, 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 of Dulong-Petit using his Boltzmann, man, Boltzmann, man. Um, using his statistical mechanics. Now, if you remember, what is his statistical mechanical picture of a gas? He already knew how to derive this 3/2 KB for a gas. This 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, DQ, Q is the energy, and so T is the temperature, and Q goes up as 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, in the solid, it moves off to the left, and then it gets pushed back by its neighbor. It doesn't go off, you know, completely off to the right or the left. It, 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."
So, this was the Boltzmann model of a solid, and that was in 1896. Um, 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 it's 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. It's to use the equipartition theorem, which you probably learned last year, which basically says for each degree of freedom where you can store energy, you get 1/2 KB worth of heat capacity.
So, let's remind ourselves how this works. So, for example, with a monatomic gas, monatomic gas, what are the degrees of freedom that store energy? Well, the gas molecule can be have momentum in the X direction, it can have momentum in the Y direction, it can have momentum in the Z direction. So, have PX, PY, PZ. That's 3 degrees of freedom per atom. So, CV over N is 3/2 KB. 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 X direction, momentum in the Y direction, momentum in 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 well, or to stretch the spring, if you want to think of it that way. So, C over N, six degrees of freedom is 3/2 KB. Law of Dulong-Petit. This was Boltzmann's reasoning.
Now, for your first homework assignment, you'll derive this more rigorously. And, you know, if you remember from your stat mech 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 3/2 KB.
Now, this result that Mr. Boltzmann, uh, derived in 1896 was an extremely important result for a number of reasons. The first reason is it explained this result, this law of Dulong-Petit, that had been known for almost a hundred years and no one knew why that it 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, you know, a couple of his friends believed in statistical 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 this statistical mechanics stuff more seriously. And this was one of the important results for him. So, so this was a, a great advance.
Um, but there was a problem. Something that bothered Boltzmann and something that made people think that potentially maybe Boltzmann's reasoning was wrong. And the problem was that the law of Dulong-Petit, of DP, is not always true. Not always true. Fails sometimes. Um, so if you take a material, you measure its heat capacity, and you come up with 3/2 KB per atom, um, 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 N always becomes much less than 3/2 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, um, that C over N is much less than 3/2 KB, even at room temperature. At T room, T room. So, for most materials at room temperature, you get 3/2 KB, uh, heat capacity per atom. Diamond is an exception that its heat capacity is smaller. You take any material and you cool it down, and the heat capacity will drop below, uh, 3/2 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, uh, the answer to this puzzle. He, unfortunately, committed suicide in 1905. Rather sad. And for a great scientist. And it was two years after that that, um, this was finally sorted out by a very, very bright young man by the name of Albert Einstein.
Um, 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. Of a solid, which is 1907. Um, and Einstein's model of a solid is actually exactly the same as Boltzmann's 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 of a potential well. Can oscillate back and forth. There's an oscillator frequency, omega, is square root of some spring constant divided by some mass. And then we have to treat this, this, uh, 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 Schrödinger 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 Schrödinger equation. We know about eigenstates. We know how to treat, uh, quantized eigenstates 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. You know, I'd still bet on him, but, uh, um, all right, anyway. So, what we learned about the harmonic oscillator in quantum mechanics last year is that the, the energy states, the eigenstates of the atom in the potential well should be given by the following expression: E sub n is h-bar omega n plus 1/2. 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. Um, 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 to, we know how to handle this from our stat mech course. What we do is we write down a partition function Z equals sum over the eigenstate n greater than or equal to zero of e to the minus beta E sub n, where beta is 1 over KBT in the usual way. And, um, then we're going to take this partition function and we'll differentiate it to get, uh, let's see. So, we need 1 over Z DZ D beta to give us the expectation of the energy. Um, just a comment that this expectation is both a quantum mechanical and a statistical mechanical expectation. Um, and I won't go through the algebra to do this. That's actually on the homework assignment, and actually probably did it last year as well. But if you do that algebra, you get the final, final expression: h-bar omega, N sub B of beta h-bar omega plus 1/2, where N sub B is known as the Bose factor. Bose factor. N B equals 1 over e to the beta h-bar omega minus 1. And actually, if you compare this expression for the expectation of the energy to the expression up there, N equals h-bar omega n plus 1/2, you know, they look very similar. It's just n got replaced by N sub B. And what that tells you is that the Bose factor tells you the expected level of excitation at a given temperature. So, if at a particular temperature, N sub B takes the value three, it means on average at that temperature, you're excited up to the N equals three, uh, excitation level. Okay. All right.
So, once we have an expression for the energy, we can write an expression for the heat capacity, dE dT. And if you do that algebra again, won't do it here, um, it's KB times beta h-bar omega squared, e to the beta h-bar omega over e to the beta h-bar omega minus 1 squared. Now, this is almost our final result, except we have to deal with two things. The first thing is that we only calculated the heat capacity of a single atom. So, if you have a lot of atoms, you would have to multiply this 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 an oscillator in only one dimension, and in fact, it can oscillate, you know, an 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 N is three times KB, beta h-bar omega squared, e to the beta h-bar omega over e to the beta h-bar omega minus 1 squared. And this is Einstein's final, final result.
Um, so it's worth thinking about this expression for a second and seeing what its properties should be. The first thing we might do, um, is take the high temperature limit, KBT much greater than h-bar omega, or in other words, beta h-bar omega much less than one. And if we do that, we then have e to the beta h-bar omega is approximately equal to 1 plus beta h-bar omega plus dot dot dot. So, then if we take this e to the beta h-bar omega and 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-bar omega downstairs. So, then we have beta h-bar omega downstairs, we have beta h-bar omega upstairs. Those two cancel, and we get C over N is 3 KB. Everyone good with that? Yeah? Okay. So, this recovers at high temperature the law of Dulong-Petit. And, and maybe that's not surprising because, you know, at high temperature, the limit of a quantum system, you sort of like a classical limit. It is going to give you back what Boltzmann calculated, classical physics, just like Boltzmann expected.
At low temperature, though, is quite different. So, let's try KBT much less than h-bar omega. In this case, what we have is e to the beta h-bar omega is big, very big, exponentially big. Uh, 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 e to the beta h-bar omega squared downstairs, and we'll cancel one of the things upstairs, and we'll get C over N is 3 KB times beta h-bar omega squared, e to the minus beta h-bar omega. And the important thing about this is this exponent, which is exponentially small, exponentially small, tiny. So, when you go to temperatures way below, uh, h-bar omega, the heat capacity drops like crazy.
So, let's, let's actually plot this. So, we'll plot this function, this function here, C over N on the vertical axis, and, uh, KBT over h-bar omega on the horizontal axis. And we know that at high temperature, we're going to asymptote to 3 KB. So, look something like this up at high temperature. At low temperature, we're going to be exponentially small. X small. And it connects up kind of like this. And then maybe here is something, KB over h-bar 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, a finite gap of h-bar omega before it can get into the next eigenstate. So, if its temperature is much less than the spacing between these eigenstates, it's just stuck in the bottom eigenstate 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 realized must have been going on in, uh, these physical systems.
So, what's going on with, uh, where was it? Over here, over somewhere, I lost it. Oh, up there. Yeah. What's going on with, uh, 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. So, what is this h-bar omega coming from? Um, it's the oscillator frequency up there. It's some spring constant divided by mass. You know, 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. Omega is known as the Einstein frequency. Um, and so the Einstein frequency of different materials is different. For some materials, the Einstein frequency is very low, in which case room temperature puts KBT over h-bar 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 h-bar omega is less than room temperature. But if you cool down the system to higher, to lower temperature, you'll, 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 low than h-bar omega, and so the heat capacity even at room temperature should drop.
Now, why is it that 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. It's very high up on the periodic table. It's the only, six elements 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, kappa, you might think that the toughness of the material is somehow related to its spring constant, how, how hard the material is. So, kappa is really big. So, for diamond, the frequency omega is anomalously large. It has one of the largest, uh, Einstein frequencies. Here, so the oscillator frequency is extremely big. So, room temperature is much higher than the Einstein frequency. So, on this plot, um, sorry, backwards. Room temperature is much lower than the Einstein frequency, which is very big. So, on 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, uh, 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 free fit parameter, h-bar omega. He was not able to predict what h-bar omega should be. Although he had a good argument why h-bar omega should be big for diamond, he didn't know exactly what the number should be. So, he just chose the one that fit the data the best. And it does seem, given that one parameter, it does seem to fit fairly well. At high temperature, you get 3/2, 3 KB, the law of Dulong-Petit. And at low temperature, it seems to drop.
And I just want to emphasize 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 quantization of the harmonic oscillator. An extremely important and pivotal moment in the development of quan, of quantum mechanics. And it came because he was studying condensed matter physics.
So, um, 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, uh, 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 the theory plot, the the dash 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, um, for most materials, most materials, including diamond, the heat capacity at low temperature is proportional to T cubed. At low T. The exception to this, an exception we'll come to, um, later on in maybe two or three lectures, is metals. Things like lead, copper, whatever. Um, 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, yeah, here. Specific heat of diamond. Heat capacity of diamond is a function of T. Plot is a function of T cubed. This is at very low temperature. This temperature here is about 4 Kelvin, and this temperature here is about 40 Kelvin. It fits a perfect T cubed, uh, line there, plotted against T cubed. Um, so even way back then, uh, Einstein, I think, was aware that there was a problem with his theory because it did not predict, uh, T cubed. It, uh, predicted an exponentially small heat capacity at low temperature.
And this is where the field stood, uh, in 1907. I had to wait about 5 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 improved, improved upon Einstein's result and explained where this T cubed, uh, result came from. Um, now, Debye's, incident, I should say that Einstein thought that Debye was really brilliant, and he was very, very happy that Debye managed to figure out this T cubed and overturn the Einstein model of the solid in what we will call the Debye model of the solid, which is somewhat better. He was, Einstein was a good man. Um, anyway, so is Debye. Um, although I guess recently there's just been some, some stories about how Einstein was, how Debye was, was secretly a Nazi and he hated Einstein. But it turns out they were all wrong. There was a whole book about it. Anyway, you can find out on the web. So, it's got to be true, right?
Um, so anyway, what Debye'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 a potential well, it does get pushed back by its neighbor, but in the same time, it pushes on its neighbor, and so its neighbor moves, and then it pushes on its neighbor, 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'll have to study wave motion of vibrations in the solid. And in fact, we even know what the wave motion in a solid, a vibrational wave motion in a solid, is called. It's usually called sound, a sound wave. So, what Debye realized is there's a connection between the oscillation of atoms in a solid and, in fact, sound.
Now, one further thing that Debye understood at the time, which was really important, is that waves can be quantized too, in the sense, um, the same way that the vibrational atom, the vibrational motion of an atom in the bottom of a potential well can be quantized. And the example that he looked to as to how waves should be quantized was Max Planck's, uh, quantization of light. So, what Debye wanted to do was quantize the motion, the vibrational motion of atoms, sound waves in a solid, the same way that Max Planck quantized, uh, the, uh, light, uh, 10 years earlier. And I guess we'll stop there, and I'll see you, I guess, Thursday. [Applause]