1 00:00:00,870 --> 00:00:04,200 Okay. Welcome, everyone. Hello. 2 00:00:04,620 --> 00:00:07,710 Welcome. Getting started. Hello. 3 00:00:08,520 --> 00:00:14,100 Welcome. This is the first lecture of the third year basics condensed matter physics course. 4 00:00:14,910 --> 00:00:17,460 Before we get actually started in the material for the course, 5 00:00:18,120 --> 00:00:22,200 I want to mention a couple of key resources we have that are meant to make your life easier. 6 00:00:22,800 --> 00:00:27,390 The first key source is the book. I'm going to write down book. 7 00:00:30,540 --> 00:00:33,120 So I wrote this book for this course. 8 00:00:34,260 --> 00:00:41,760 The the idea of the book was to cover everything that could possibly be asked to you on a condensed matter physics exam. 9 00:00:41,760 --> 00:00:44,250 So if you learn everything in this book, you'll be in very good shape. 10 00:00:44,730 --> 00:00:50,340 As we go through the course, you'll discover that the book and the lectures are extremely similar. 11 00:00:50,370 --> 00:01:00,310 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. 12 00:01:00,520 --> 00:01:07,860 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. 13 00:01:07,870 --> 00:01:15,010 Hopefully you'll find them both useful. You can pick up books at Blackwell's there in many of the libraries there on Amazon. 14 00:01:15,280 --> 00:01:18,960 I do apologise. It's not quite as cheap as I would have liked it to be. 15 00:01:18,970 --> 00:01:24,310 I think it costs about £25 a copy. So apologies for that. 16 00:01:24,490 --> 00:01:27,640 The second web page. The second resource is the Web page. 17 00:01:29,050 --> 00:01:33,970 There's a lot of interesting and useful things on the course web page link to my homepage. 18 00:01:34,690 --> 00:01:39,430 One thing that you might find useful on the web page is the book. 19 00:01:39,730 --> 00:01:43,290 Now, the book, the actual book is not on the web page. 20 00:01:43,300 --> 00:01:48,040 The publishers wouldn't let me do that. Presumably they figured they would never sell one if I gave it to you for free. 21 00:01:48,340 --> 00:01:54,850 But the draft that we used in previous years is posted on the web page and is a pretty good replacement for this. 22 00:01:54,880 --> 00:01:57,670 It's about 85% similar to what's actually in the book. 23 00:01:57,910 --> 00:02:05,290 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. 24 00:02:05,440 --> 00:02:09,310 Maybe you might not even find that you need to spend the £25 on it. 25 00:02:10,420 --> 00:02:12,940 Although the book is better, a lot of the areas have been corrected. 26 00:02:12,940 --> 00:02:15,790 The pictures are better, some of the explanations have been improved and so forth. 27 00:02:16,750 --> 00:02:21,430 So as you choose other resources on the web page that are useful, the suggested homeworks. 28 00:02:21,610 --> 00:02:25,270 Please check with your tutors to make sure they agree with which homework assignments you should do. 29 00:02:25,510 --> 00:02:28,840 Sample Exams. Sample solutions are also there. 30 00:02:29,350 --> 00:02:35,250 PowerPoint slides. First of all, I will never leave any sort of handouts here, so you don't have to look, 31 00:02:35,260 --> 00:02:43,720 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. 32 00:02:43,900 --> 00:02:47,340 Now, I know that people prefer chalk to PowerPoint slides. 33 00:02:47,350 --> 00:02:52,690 I feel the same way. To the largest extent possible, I will always use chalk. 34 00:02:52,870 --> 00:02:56,440 But there are some times when I just have to show a picture that I can't draw. 35 00:02:56,650 --> 00:03:01,750 So those slides or videos or anything else they show will be on the web page. 36 00:03:02,830 --> 00:03:09,069 You may notice there's someone in the back, Greg, filming us that will presumably be link to the web page also. 37 00:03:09,070 --> 00:03:15,160 So if you want to see something, some explanation or something like that, it will be on the web page. 38 00:03:15,370 --> 00:03:20,019 Any correction, I have to issue corrections to the homework, corrections from the books, 39 00:03:20,020 --> 00:03:23,919 any typos in the books, anything I might say in lecture that turns out to be incorrect. 40 00:03:23,920 --> 00:03:30,880 Somehow that will be fixed on the Web page. One more thing I really want to point out is the message board. 41 00:03:32,840 --> 00:03:37,520 That's linked to the Web page as well. I'm not sure if you had this in your other courses before. 42 00:03:37,520 --> 00:03:40,820 We tried it for the first time last year. It worked extremely well. 43 00:03:41,000 --> 00:03:42,710 Other courses are now using it as well. 44 00:03:42,920 --> 00:03:47,840 The idea is to have an online forum where people can discuss what's going on and ask questions and give answers. 45 00:03:48,170 --> 00:03:53,389 When you're outside of lecture. So the way it worked last year is someone would type the type in you said X, 46 00:03:53,390 --> 00:03:57,140 Y or Z in lecture, but that doesn't make sense because of this, that or the other. 47 00:03:57,170 --> 00:04:00,680 Can you please explain? I would write some answer, but just as often is not. 48 00:04:00,830 --> 00:04:05,330 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 49 00:04:05,330 --> 00:04:08,540 tutor would ask a question and some student would answer it or something like that. So. 50 00:04:08,790 --> 00:04:13,250 So it was just some place where people could go and just leave simple messages and get them answered. 51 00:04:13,550 --> 00:04:19,260 And I think as the course goes on, people will find that very useful. Don't be shy in trying to use that. 52 00:04:20,030 --> 00:04:25,250 So that's all I have for resources, and we can actually get started with the introduction to this course. 53 00:04:26,000 --> 00:04:31,250 The first thing you're probably asking yourself is What is this subject we're supposed to be learning? 54 00:04:31,250 --> 00:04:37,610 What is condensed matter physics? Well, to begin with, condensed matter physics is one third of physics. 55 00:04:39,310 --> 00:04:44,370 Approximately one third of physics. And what I mean by that is the following experiment. 56 00:04:44,380 --> 00:04:49,300 If you go around the world and you ask every single physicist, Are you a condensed matter physicist? 57 00:04:49,450 --> 00:04:52,600 About one third of them will tell you, Yes, I'm a condensed matter physicist. 58 00:04:52,960 --> 00:04:58,240 Condensed matter physics is the largest subfield of physics. Actually, it's the largest subfield by far. 59 00:04:58,240 --> 00:05:03,850 It's extremely broad, extremely diverse, and it includes many, many different topics within it. 60 00:05:04,720 --> 00:05:09,730 To give you a sort of an example of how big and how broad and how diverse this field is. 61 00:05:09,730 --> 00:05:14,139 I'm going to use the March meeting of the American Physical Society. 62 00:05:14,140 --> 00:05:17,140 Every march, the American Physical Society holds a meeting. 63 00:05:17,350 --> 00:05:20,620 Of all of this condensed matter, physicists are as many as can possibly show up. 64 00:05:21,010 --> 00:05:25,030 Last year was in Baltimore, Maryland. I went there myself. There were 8000 physicists there. 65 00:05:25,210 --> 00:05:28,900 It was very exciting for one week. Huge Nerd Fest, very enjoyable. 66 00:05:28,900 --> 00:05:33,100 I recommend going if you ever get a chance at 8 a.m. 67 00:05:33,100 --> 00:05:41,440 On the first day of the meeting, there were talks on the following subjects superconductors, superfluids, glasses, polymers, quantum dots, 68 00:05:41,440 --> 00:05:48,519 microfluidics crystal growth spintronics phase transitions quantum criticality Bose Condensates Fractionalised Charges Quantum Computation, 69 00:05:48,520 --> 00:05:52,330 high pressure physics magnetism, heavy fermions, multiple rocks and liquid crystals. 70 00:05:52,480 --> 00:05:58,510 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. 71 00:05:58,870 --> 00:06:05,719 So the list goes on and on and on. Of all of the interesting things that are contained within condensed matter physics, 72 00:06:05,720 --> 00:06:10,600 there tends to be a large overlap in condensed matter physics with fields such as chemistry, 73 00:06:10,810 --> 00:06:17,440 material science, biology, atomic physics, sometimes high energy physics, nanoscience, quantum sciences. 74 00:06:17,440 --> 00:06:24,790 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. 75 00:06:25,510 --> 00:06:33,010 So it includes many, many, many things in it. And you might be asking yourself at this point, why study it? 76 00:06:33,880 --> 00:06:39,940 Why study it, besides the fact that it happens to be on your syllabus? 77 00:06:40,150 --> 00:06:44,020 But that just raises the question of why did someone put it on your syllabus? 78 00:06:44,290 --> 00:06:47,620 Well, there's a lot of good reasons why we should be studying this subject. 79 00:06:48,550 --> 00:06:58,870 The first good reason is one, it is the world condensed matter. 80 00:06:58,870 --> 00:07:06,880 Physics is the world around you. We think of condensed matter as being the study of the stuff in the world around you. 81 00:07:07,210 --> 00:07:11,650 So anything you can point at, any material you can, you can look at. 82 00:07:13,110 --> 00:07:18,570 Any thing you can pick up solids, liquids, glasses, polymers, you know, metals. 83 00:07:18,690 --> 00:07:23,669 These are all condensed matter in some way. And if you point at these things, you know, ask questions. 84 00:07:23,670 --> 00:07:29,190 That's what we as scientists and we as physicists should be doing. We should be pointing at things and asking questions about them. 85 00:07:29,460 --> 00:07:32,530 If you ask questions like, Why is glass transparent? 86 00:07:32,550 --> 00:07:35,610 Why is metal shiny? Why is rubber soft and squishy? 87 00:07:35,640 --> 00:07:43,380 Why is water wet? Why is oil slippery? And even more subtle questions like Why is egg yolks crucial for making good mayonnaise? 88 00:07:44,100 --> 00:07:48,900 All of these things are the subject of condensed matter physics. 89 00:07:49,140 --> 00:07:52,950 Condensed matter physicists are in the business of explaining all of these things. 90 00:07:52,950 --> 00:07:58,230 And in that long list of questions, at least some of them, by the end of the term, we will have answers, too. 91 00:07:58,380 --> 00:08:03,060 So pretty much everything you see in the world around you. The world is condensed matter. 92 00:08:03,240 --> 00:08:07,480 Physics. Reason to. It is useful. 93 00:08:11,870 --> 00:08:19,160 Condensed matter, physics is by far the most technologically and industrially important field of physics. 94 00:08:19,470 --> 00:08:25,520 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 95 00:08:25,520 --> 00:08:31,220 an incredible control over the materials and the stuff in the world around us. 96 00:08:31,430 --> 00:08:41,030 You know, examples of this include, you know, 150 years ago, people first started being able to think about electrical currents moving in materials, 97 00:08:41,030 --> 00:08:44,570 and then they figured some materials had different electrical properties than others. 98 00:08:44,660 --> 00:08:48,860 And pretty soon they were able to make electronics and they could build up very fancy electronics. 99 00:08:48,860 --> 00:08:55,370 And pretty soon they're building things like iPads, iPhones, computers and things that we take for granted in the world around us today. 100 00:08:55,460 --> 00:08:58,910 And all of this comes from the study of condensed matter physics. 101 00:08:58,910 --> 00:09:07,190 Condensed matter physics really changes our lives as people that these things that we take for granted now come out of condensed matter physics. 102 00:09:07,430 --> 00:09:12,380 By far, the most useful field of physics is condensed matter. 103 00:09:13,190 --> 00:09:25,190 Reason three Why we study it. Three It is deep, fundamental and deep new ideas in condensed matter physics. 104 00:09:25,190 --> 00:09:30,140 It is. Now, some people have this mistaken idea that two in three are somehow contradictory. 105 00:09:30,150 --> 00:09:34,940 If it's useful, it can't be deep. If it's deep, it can't be useful. This is completely untrue. 106 00:09:35,090 --> 00:09:41,810 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. 107 00:09:42,110 --> 00:09:43,940 How do I know this? How can I prove this to you? 108 00:09:44,120 --> 00:09:48,800 The reason I know that is just as deep as any other field of physics is because they're exactly the same ideas. 109 00:09:49,100 --> 00:09:52,790 Good ideas come from one field and go into the other field and vice versa. 110 00:09:53,090 --> 00:09:58,970 And as often as not, the best ideas have come out of condensed matter and gone into other fields. 111 00:09:59,210 --> 00:10:02,810 Really good example of this. Everyone has probably heard of the Higgs boson. 112 00:10:02,960 --> 00:10:08,000 Professor Higgs won his Nobel Prize just a few months ago for prediction of this subatomic particle, 113 00:10:08,000 --> 00:10:12,320 which was then discovered and certain many years later. Where did he get the idea for the Higgs boson? 114 00:10:12,530 --> 00:10:17,780 Well, if you read paragraph one of his Nobel Prize winning paper, it says very explicitly, 115 00:10:18,020 --> 00:10:25,129 I got this idea from condensed matter physicists back before Higgs P Condensed Matter physicists were 116 00:10:25,130 --> 00:10:29,600 studying metals at low temperature and they discovered that metals superconductor at low temperature, 117 00:10:29,810 --> 00:10:36,139 they have no electrical resistance at all. And after coming to a really good understanding of what causes this condensed matter, 118 00:10:36,140 --> 00:10:41,420 physicists were suggesting to high energy physicists that something similar might be going on in a high energy, 119 00:10:41,540 --> 00:10:48,170 and that is the origin of the Higgs boson. So the deep ideas have come out of condensed matter and gone into other fields. 120 00:10:49,310 --> 00:10:52,840 For. Reason for why we study it. 121 00:10:53,680 --> 00:11:00,820 A.I. Reductionism. This is a word that I made up. 122 00:11:02,950 --> 00:11:05,640 But but reductionism is actually is a real word. 123 00:11:05,670 --> 00:11:11,229 The reductionism is the idea that you're going to learn more about something by asking what is it made of? 124 00:11:11,230 --> 00:11:13,300 What is a smaller and smaller and smaller piece? 125 00:11:13,660 --> 00:11:21,040 Anti reductionism is the idea that that is the wrong way to go, that that is completely the wrong way to understand something. 126 00:11:21,040 --> 00:11:26,469 An example that people like to use. If you asked to describe a glass of water and if you start along, 127 00:11:26,470 --> 00:11:30,340 this rabbit hole of the water is made up of molecules and molecules are made up of atoms. 128 00:11:30,340 --> 00:11:31,450 The atoms are made of protons, 129 00:11:31,450 --> 00:11:36,970 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. 130 00:11:37,510 --> 00:11:40,930 You get nowhere. You didn't learn anything about why the water is wet. 131 00:11:40,930 --> 00:11:44,590 You didn't learn why it's transparent. You didn't learn anything really useful at all. 132 00:11:44,830 --> 00:11:51,670 Reductionism completely misses the forest for the trees. And more often than not, if you want to understand the real physical properties of something, 133 00:11:51,850 --> 00:11:56,800 what you need to ask about is the big picture how the pieces act together to give you the whole. 134 00:11:57,010 --> 00:12:01,120 So anti reductionism is a good reason to study condensed matter physics. 135 00:12:01,930 --> 00:12:05,950 Five. It is a laboratory. 136 00:12:06,790 --> 00:12:21,999 Laboratory. For Quantum and step mak to a large extent condensed matter. 137 00:12:22,000 --> 00:12:27,159 Physics is the best laboratory we have for exploring the amazing things that quantum 138 00:12:27,160 --> 00:12:32,170 mechanics and statistical mechanics can do and the things that they can do together. 139 00:12:32,470 --> 00:12:39,070 So I would like you to view this entire course as an extension of what you learned in those two courses last year. 140 00:12:39,250 --> 00:12:45,100 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. 141 00:12:45,310 --> 00:12:50,050 And if you didn't like those two courses from last year, I will bet that you will like this course better. 142 00:12:51,610 --> 00:12:54,190 But there's actually a good reason to think that you'll like this course better, 143 00:12:54,400 --> 00:12:58,540 because instead of thinking about these subjects in the more abstract work, 144 00:12:58,540 --> 00:13:04,900 actually think about them in some very real and important applications and how they apply to some real, real stuff. 145 00:13:05,830 --> 00:13:13,210 Okay. So as I mentioned early on, condensed matter physics is a very broad and diverse field and we can't possibly 146 00:13:13,540 --> 00:13:18,370 study all the subfields within condensed matter all within these eight weeks. 147 00:13:18,580 --> 00:13:22,159 So we're going to focus in on just one subfield of condensed matter. 148 00:13:22,160 --> 00:13:27,970 And the particular subfield we're going to spend all of our time on or most of our time on is solid state. 149 00:13:30,340 --> 00:13:31,540 And by solid state. 150 00:13:31,540 --> 00:13:40,180 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. 151 00:13:40,390 --> 00:13:43,660 And the reason we there's several reasons why we pick solid state here. 152 00:13:43,930 --> 00:13:50,350 It's the largest, the most useful and the most successful of the subfields in condensed matter physics. 153 00:13:50,350 --> 00:13:57,250 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. 154 00:13:57,460 --> 00:14:01,360 We know more about solids than we know about any other state of matter than what we 155 00:14:01,360 --> 00:14:04,719 have learned about solids is incredible what we've been able to do with solids. 156 00:14:04,720 --> 00:14:08,740 You know, you think about it, electronics, the electronics industry, that's all made of solids. 157 00:14:08,950 --> 00:14:13,390 So all of that, that's why it's called solid state electronics, because the electronics of the solid state, 158 00:14:14,170 --> 00:14:18,520 all of that is incredibly well understood and it's a great place to start our study of this field. 159 00:14:18,670 --> 00:14:26,319 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, 160 00:14:26,320 --> 00:14:33,010 a really good jumping off point for studying other fields within condensed matter and even outside of condensed matter. 161 00:14:33,010 --> 00:14:38,140 So if one wants to go on and study something more complicated later on in life, like a liquid or a superfluid, 162 00:14:38,320 --> 00:14:43,810 what we learn in solid state is going to be a fundamental starting point for learning those things later on. 163 00:14:44,620 --> 00:14:52,719 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. 164 00:14:52,720 --> 00:14:59,530 Studying solid state or condensed matter is actually over 100 years ago, around the turn of the century 1900. 165 00:14:59,890 --> 00:15:07,270 And the reason I pick that time is because that was about the time when people first started applying things 166 00:15:07,270 --> 00:15:13,240 like statistical mechanics and later on quantum mechanics to the study of the things in the world around them. 167 00:15:14,110 --> 00:15:20,860 Over the course of the 1800s, scientists were able to make a lot of very increasingly precise measurements, 168 00:15:20,860 --> 00:15:23,919 and they had figured out a lot about the world around them. 169 00:15:23,920 --> 00:15:26,799 But there were a lot of the measurements that they could give, the measurement. 170 00:15:26,800 --> 00:15:32,170 They could see what was going on, but they couldn't understand why they were getting the results that they were getting. 171 00:15:32,440 --> 00:15:39,340 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. 172 00:15:39,520 --> 00:15:51,820 And those are experiments on heat capacity. But you probably remember from your thermodynamics course and in particular heat capacity of solids. 173 00:15:54,250 --> 00:16:00,310 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, 174 00:16:00,640 --> 00:16:08,010 but they didn't know what the results meant. So just to remind you what heat capacity is, heat capacity is d d t, 175 00:16:09,130 --> 00:16:13,420 you know how much heat you have to put into a material to raise it a certain amount in temperature. 176 00:16:13,630 --> 00:16:18,940 And this is the first equation of the course. And already someone should be objecting and saying, wait a second, that's wrong. 177 00:16:19,300 --> 00:16:22,300 So why should I say something? Something's wrong with this. 178 00:16:22,540 --> 00:16:30,520 Well, if you remember from your courses in and AMOC last year, whenever you write a heat capacity, you should write a subscript. 179 00:16:30,910 --> 00:16:39,120 Whether you mean heat capacity, a constant pressure CP or heat capacity, a constant volume CV and generally these two things are different. 180 00:16:39,130 --> 00:16:39,940 I didn't do that. 181 00:16:40,060 --> 00:16:49,840 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. 182 00:16:49,930 --> 00:16:51,190 So we just call it a C. 183 00:16:52,030 --> 00:16:57,129 I mean, if you measure really, really closely, you'll discover that slightly different, but they're very, very close to the same. 184 00:16:57,130 --> 00:17:01,870 Now, why are they very close to the same? I'll remind you of something that you learn to derive. 185 00:17:01,870 --> 00:17:07,600 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. 186 00:17:08,590 --> 00:17:18,610 The C p minus CV is volume times, temperature times, alpha squared over beta, where alpha is a thermal expansion coefficient the thermal expansion. 187 00:17:21,440 --> 00:17:25,190 And data is the isothermal compress ability as a thermal. 188 00:17:27,180 --> 00:17:36,870 Possibility the press. And the point here is that solids have very, very small thermal expansions. 189 00:17:37,230 --> 00:17:41,880 Generally, a solid will have a thermal expansion coefficient of a few parts per million per degree Kelvin. 190 00:17:42,120 --> 00:17:46,050 Very, very small number. And then that number gets squared over here on the right hand side. 191 00:17:46,260 --> 00:17:51,630 So the right hand side is really, really, really small. So they are almost exactly the same. 192 00:17:51,840 --> 00:17:57,630 And our level of understanding or our level of analysis, we can just treat them as as being exactly the same. 193 00:17:57,840 --> 00:18:02,250 Not true for a gas if we're thinking about a gas. Thermal expansion is very large. 194 00:18:02,280 --> 00:18:06,030 Cpcb As you learned last year, it can be quite different. 195 00:18:06,300 --> 00:18:10,410 So let's remind ourselves of a couple of things that we learned about heat capacity last year. 196 00:18:12,000 --> 00:18:20,220 Let's go back to gases now since I mentioned one for our modern atomic monit mana atomic, atomic, atomic gas. 197 00:18:21,030 --> 00:18:28,320 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. 198 00:18:28,710 --> 00:18:33,150 CV over and the heat capacity per atom was three KB. 199 00:18:35,550 --> 00:18:44,850 Does that look familiar? Yeah. Okay, good. I'm glad. Well, it turns out that there's a very similar law for solids. 200 00:18:45,570 --> 00:18:49,750 Solids? See over. 201 00:18:51,160 --> 00:18:59,139 The capacity per atom is 3kv and I didn't specify c, v or c p as I explained this law, 202 00:18:59,140 --> 00:19:09,640 the silver n is 3kb is known as the law of do long, petite, do, long, petite, do long and petite. 203 00:19:09,640 --> 00:19:17,770 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. 204 00:19:17,980 --> 00:19:23,920 People knew about this law, but they didn't know what caused it. And so along comes Boltzmann. 205 00:19:24,310 --> 00:19:31,600 So Boltzmann was a very smart guy, and he very much liked this statistical mechanics stuff. 206 00:19:32,140 --> 00:19:37,780 He sort of invented the field and he thought that maybe he could understand this law. 207 00:19:37,780 --> 00:19:47,350 And you long, petite using his Boltzmann machine, Boltzmann on using his statistical mechanics. 208 00:19:48,460 --> 00:19:52,270 Now, if you remember, what is this statistical mechanical picture of a gas? 209 00:19:52,270 --> 00:19:56,020 He already knew how to derive this three half KB for a gas. 210 00:19:56,020 --> 00:19:59,440 And its picture of a gas is that you have these atoms and they're flying back and forth in space. 211 00:19:59,620 --> 00:20:03,040 As you raise the temperature of the gas, they fly back and forth faster. 212 00:20:03,220 --> 00:20:09,850 And so they have more kinetic energy. So D, Q Q is the energy and two T is the temperature and Q goes up. 213 00:20:09,850 --> 00:20:17,139 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. 214 00:20:17,140 --> 00:20:21,520 You know, you have some atoms in the solids and as you raise the temperature, they move around faster. 215 00:20:21,820 --> 00:20:25,270 But it's not exactly like a gas because they're not flying around completely free. 216 00:20:25,480 --> 00:20:31,629 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. 217 00:20:31,630 --> 00:20:36,730 It doesn't go off completely off to the right or the left, and it sort of gets pushed back to its original position, 218 00:20:36,730 --> 00:20:39,730 then goes off in the other direction and then comes back to the original position. 219 00:20:40,090 --> 00:20:46,690 So he thought, well, maybe. I should make a model of the solid as the Boltzmann model of solid. 220 00:20:48,820 --> 00:20:55,330 And that was in 1896. And his model of a solid was really simple. 221 00:20:55,330 --> 00:21:01,930 It was just a harmonic. Well, with an atom in the bottom of the harmonic well, and the atom can oscillate back and forth. 222 00:21:01,930 --> 00:21:06,290 And as you raise the temperature, it oscillates more back and forth and so is storing more energy. 223 00:21:06,290 --> 00:21:12,699 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. 224 00:21:12,700 --> 00:21:24,610 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, 225 00:21:25,450 --> 00:21:27,670 which you probably learned last year, 226 00:21:27,970 --> 00:21:35,680 which basically says for each degree of freedom where you can store energy, you get one half KB worth of heat capacity. 227 00:21:36,250 --> 00:21:39,550 So let's remind ourselves how this works. So, for example. 228 00:21:40,720 --> 00:21:44,980 With a minor atomic gas, mine atomic gas. 229 00:21:46,120 --> 00:21:53,010 What are the degrees of freedom that store energy? Well, the gas molecule can be have momentum in the X direction. 230 00:21:53,020 --> 00:21:57,290 It can a momentum in the Y direction. It can momentum in Z direction. So. P. 231 00:21:57,740 --> 00:22:00,910 P. That's three degrees of freedom per atom. 232 00:22:01,150 --> 00:22:04,450 So see the over pn is three halves. 233 00:22:04,450 --> 00:22:08,180 KB k b. Okay. 234 00:22:09,440 --> 00:22:13,400 Now. What about the solid? Well, solid. Pretty similar. 235 00:22:14,570 --> 00:22:16,620 What are the degrees of freedom that can store energy? 236 00:22:16,640 --> 00:22:21,020 Well, you have momentum in the direction, momentum in the Y direction, momentum the Z direction. 237 00:22:21,590 --> 00:22:30,889 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. 238 00:22:30,890 --> 00:22:33,980 It costs you energy, even if it doesn't have any momentum, 239 00:22:33,980 --> 00:22:39,680 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. 240 00:22:40,130 --> 00:22:44,720 So C over n six degrees of freedom is three. 241 00:22:44,720 --> 00:22:49,640 KB lived long, petite. This was boatman's reasoning. 242 00:22:49,940 --> 00:22:53,839 Now for your first homework assignment, you'll derive this more rigorously. 243 00:22:53,840 --> 00:23:00,200 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. 244 00:23:00,440 --> 00:23:05,450 Differentiate the partition function to get the energy. Differentiate the energy to get the heat capacity. 245 00:23:05,600 --> 00:23:09,890 And if you do that, you'll get exactly this. Number of three has to be. 246 00:23:11,230 --> 00:23:18,220 Now, this result that Mr. Boltzmann derived in 1896 was an extremely important result for a number of reasons. 247 00:23:18,520 --> 00:23:27,940 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. 248 00:23:27,940 --> 00:23:31,600 And all of a sudden he he had this good reason why it holds. 249 00:23:32,070 --> 00:23:38,500 And but moreover, it was extremely important because in 1896, not a lot of people believed in statistical mechanics. 250 00:23:38,710 --> 00:23:43,930 Boltzmann was kind of off on his own. He and a couple of his friends believed in system mechanics, and no one else did. 251 00:23:44,290 --> 00:23:49,690 But each time Boltzmann came up with a result that he could explain and no one else in the world could explain. 252 00:23:49,900 --> 00:23:54,430 People had to take the statistical mechanics stuff more seriously, and this was one of the important results for him. 253 00:23:55,030 --> 00:23:57,910 So so this was it was a great advance. 254 00:23:58,630 --> 00:24:06,010 But there was a problem, something that bothered Boltzmann and something that made people think that potentially maybe both man's reasoning was wrong. 255 00:24:06,340 --> 00:24:13,270 And the problem was that the law of do long, petite of deep is not always true. 256 00:24:15,190 --> 00:24:20,470 Not always true. Fails sometimes. 257 00:24:23,350 --> 00:24:27,820 So if you take a material, you measure its heat capacity and you come up with three KB per atom. 258 00:24:28,810 --> 00:24:36,880 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, 259 00:24:37,840 --> 00:24:43,180 you would discover that C over an always becomes much less than three kb. 260 00:24:44,680 --> 00:24:48,700 So the heat capacity per atom drops at low temperature. 261 00:24:49,150 --> 00:25:01,690 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. 262 00:25:02,230 --> 00:25:05,230 Even at room temperature. At tea room. 263 00:25:05,830 --> 00:25:13,060 Tea room. So for most materials at room temperature, you get three kbps heat capacity per atom. 264 00:25:13,330 --> 00:25:21,190 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. 265 00:25:21,190 --> 00:25:27,920 KB So this was a puzzle to Boltzmann. And it puzzled everyone for for a number of years. 266 00:25:28,250 --> 00:25:32,710 And in fact, Boltzmann never lived to see the answer to this puzzle. 267 00:25:32,720 --> 00:25:38,270 He unfortunately committed suicide in 1905. Rather sad end for a great scientist. 268 00:25:38,270 --> 00:25:45,470 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. 269 00:25:47,320 --> 00:25:54,010 And this you know, this achievement by Einstein is actually one of the most important things that Einstein ever did. 270 00:25:54,100 --> 00:25:58,510 Way up there with relativity or photoelectric effect that people know a lot better. 271 00:25:58,660 --> 00:26:04,390 But this result by Einstein, I will argue, is just as important as as those results. 272 00:26:04,630 --> 00:26:08,340 So this is Einstein's model, Einstein model. 273 00:26:09,700 --> 00:26:16,020 Of a solid. Oh, solid. Which is 1987. 274 00:26:19,690 --> 00:26:24,580 And Einsteins model of a solid is actually exactly the same as Boltzmann model. 275 00:26:25,660 --> 00:26:29,080 Boltzmann equals Boltzmann solid. 276 00:26:31,250 --> 00:26:38,180 Plus one ingredient. And the one ingredient is quantum mechanics plus quantum. 277 00:26:40,010 --> 00:26:45,450 So just to be a little bit more specific about what we mean here about this model. 278 00:26:45,470 --> 00:26:50,270 So again, we have an atom in the bottom. A potential well can oscillate back and forth. 279 00:26:50,630 --> 00:26:56,120 There's an oscillator frequency omega is a square is some ring spring constant divided by some mass. 280 00:26:56,390 --> 00:27:00,379 And then we have to treat this this potential. 281 00:27:00,380 --> 00:27:04,040 Well, using quantum mechanics, not using classical mechanics. 282 00:27:04,280 --> 00:27:13,300 Now this. Result by Einstein was completely outrageous in 1907, and you have to really think about what was going on in 1907. 283 00:27:13,450 --> 00:27:17,499 This was 19 years before the Schrodinger equation. No one knew about quantum mechanics. 284 00:27:17,500 --> 00:27:19,510 There wasn't even a word for quantum mechanics then. 285 00:27:19,780 --> 00:27:26,900 So he was really working way out on a limb, and he was deducing from the experiment what must be going on in nature. 286 00:27:26,920 --> 00:27:31,150 He was figuring out quantum mechanics based on these experiments. 287 00:27:31,420 --> 00:27:34,450 Now we have some huge advantages over Einstein. 288 00:27:34,630 --> 00:27:41,440 We know about the shortening equation. We know about iGen states. We know how to treat Quantised Eigen states using statistical mechanics. 289 00:27:41,740 --> 00:27:46,390 So we're going to be able to do this much, much more easily and much more directly than Einstein did. 290 00:27:46,420 --> 00:27:52,060 Einstein had to take some really roundabout routes to get to the final answer, which turns out to be a pretty good answer. 291 00:27:52,300 --> 00:27:56,380 We have another huge advantage over Einstein, which is that we're alive and he's dead, 292 00:27:56,680 --> 00:28:00,220 and that that will put us on a much more equal intellectual footing. 293 00:28:03,070 --> 00:28:06,790 I still bet on him. But all right. 294 00:28:06,790 --> 00:28:13,180 Anyway, so what we learned about the harmonic oscillator in quantum mechanics last year is that the energy states, 295 00:28:13,510 --> 00:28:19,060 the eigen states of the atom in the potential well should be given by the following expression. 296 00:28:19,090 --> 00:28:24,010 S of n is h bar omega and plus one half. 297 00:28:24,280 --> 00:28:31,390 Now I should be a little bit more careful here. This is the result for a one dimensional oscillator, one dimension. 298 00:28:34,690 --> 00:28:40,000 And we do live in three dimensions. And so we're going to have to fix that up a little bit later. 299 00:28:40,240 --> 00:28:47,080 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, 300 00:28:47,350 --> 00:28:51,670 in one dimension. So now we know how we know how to handle this from our step course. 301 00:28:51,940 --> 00:28:59,650 What we do is we write down partition function, the equal sum over the eigen states and greater than equal to zero, 302 00:28:59,950 --> 00:29:05,720 either the minus beta and where beta is one over in the usual way. 303 00:29:07,210 --> 00:29:14,790 And then we're going to take this partition function and we'll differentiate it to get let's see. 304 00:29:14,800 --> 00:29:21,940 So we need one over the DB debater to give us the expectation of the energy. 305 00:29:22,780 --> 00:29:28,660 Just a comment that this expectation is both a quantum mechanical and statistical mechanical expectation. 306 00:29:29,740 --> 00:29:33,280 And I won't go through the algebra to do this that's actually on the homework assignment. 307 00:29:33,280 --> 00:29:35,230 And actually you probably did it last year as well. 308 00:29:35,770 --> 00:29:48,700 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. 309 00:29:49,420 --> 00:29:58,510 Both factor. And B equals one over eight of the beta omega minus one. 310 00:30:00,590 --> 00:30:07,100 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, 311 00:30:07,430 --> 00:30:08,509 you know, they look very similar. 312 00:30:08,510 --> 00:30:17,420 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. 313 00:30:17,630 --> 00:30:21,740 So I've had a particular temperature and Bose takes the value three. 314 00:30:21,890 --> 00:30:27,860 It means on average at that temperature you're excited up to the end equals three excitation level. 315 00:30:28,060 --> 00:30:38,320 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. 316 00:30:39,380 --> 00:30:43,100 And if you do that algebra again won't do it here. 317 00:30:44,930 --> 00:30:49,340 It's KBE times beta h bar omega squared. 318 00:30:49,820 --> 00:30:56,570 Even the beta omega over the beta h for omega minus one squared. 319 00:30:57,870 --> 00:31:01,560 Now this is almost our final result, except we have to deal with two things. 320 00:31:01,890 --> 00:31:05,400 The first thing is that we only calculate the heat capacity of a single atom. 321 00:31:05,700 --> 00:31:08,819 So if you have a lot of atoms, you would have to multiply that by the number of atoms. 322 00:31:08,820 --> 00:31:11,490 So this would be the right result for the heat capacity per atom. 323 00:31:11,850 --> 00:31:19,739 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. 324 00:31:19,740 --> 00:31:25,830 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. 325 00:31:26,100 --> 00:31:39,870 C over add is three times kbe beta omega squared either beta h for omega same expression for omega minus one squared. 326 00:31:40,760 --> 00:31:51,480 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. 327 00:31:51,490 --> 00:32:04,150 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. 328 00:32:04,720 --> 00:32:13,090 And if we do that, we then have either the beta h for omega is approximately equal to one plus beta omega. 329 00:32:13,270 --> 00:32:19,900 Plus da da da da. So then if we take this into the beta h omega, we plug it in upstairs. 330 00:32:20,140 --> 00:32:23,350 We only need to keep the one. We plug it in downstairs. 331 00:32:23,590 --> 00:32:28,330 The one cancels this minus one and we have to keep the beta h for Omega downstairs. 332 00:32:28,630 --> 00:32:31,959 So then we have beta promega downstairs, we have beta H-bomb Omega upstairs. 333 00:32:31,960 --> 00:32:36,610 Those two cancel and we get C over n is three KB. 334 00:32:38,360 --> 00:32:45,049 Everyone good with that? Yeah. Okay. So this recovers at high temperature, the law of duelling fatigue. 335 00:32:45,050 --> 00:32:51,770 And maybe that's not surprising because, you know, a high temperature limit of a quantum system is sort of like a classical limit. 336 00:32:51,950 --> 00:32:57,590 It is going to give you back what Boltzmann calculated classical physics, just like Boltzmann expected. 337 00:32:58,630 --> 00:33:05,680 Low temperature, though, is quite different. So it's tri cavity, much less than H Bar Omega. 338 00:33:06,130 --> 00:33:14,170 In this case, what we have is either the beta H Bar Omega is big, very big, exponentially big. 339 00:33:14,980 --> 00:33:20,950 In which case we can come over to this equation here. The minus one doesn't matter compared to this huge number. 340 00:33:21,130 --> 00:33:28,600 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 341 00:33:28,600 --> 00:33:36,760 we'll get C over N is three KB times theta omega squared E to the minus beta h for omega. 342 00:33:38,020 --> 00:33:42,610 The important thing about this is this exponent, which is exponentially small. 343 00:33:42,610 --> 00:33:46,780 Exponentially. Small. 344 00:33:47,740 --> 00:33:52,810 Tiny. So when you go to temperatures, way below is promega. 345 00:33:53,260 --> 00:33:58,090 The heat capacity drops like crazy. So let's let's actually plot this. 346 00:34:00,310 --> 00:34:12,760 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. 347 00:34:13,360 --> 00:34:19,390 And we know that in high temperature we're going to asymptotes to three kb. 348 00:34:20,270 --> 00:34:27,770 So look, something like this up and high temperature and low temperature, we're going to be exponentially small, X small. 349 00:34:29,480 --> 00:34:36,480 And that connects up kind of like this. And then maybe here is something KB over omega is about one. 350 00:34:37,350 --> 00:34:40,800 So this is what the function looks like. This is what Einstein derived. 351 00:34:41,100 --> 00:34:46,110 So the physics of what's going on here is that, again, high temperature is classical. 352 00:34:46,260 --> 00:34:53,280 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. 353 00:34:53,880 --> 00:34:56,580 So if the harmonic oscillator is sitting in its ground state, 354 00:34:56,820 --> 00:35:03,000 it can't absorb any energy unless it has enough thermal energy to jump all the way up to the first excited state. 355 00:35:03,000 --> 00:35:06,900 And there's a finite gap of age for Omega before it can get into the next thing and state. 356 00:35:07,170 --> 00:35:09,360 So if its temperature is much less, 357 00:35:09,660 --> 00:35:17,310 then that spacing between these eigen stays is just stuck in the bottom eigen state and it can't absorb any energy at all. 358 00:35:17,460 --> 00:35:21,050 So the heat capacity drops like crazy. That's what's going on in this picture. 359 00:35:21,210 --> 00:35:27,090 And that's what Einstein realised must have been going on in these physics physical systems. 360 00:35:27,960 --> 00:35:31,860 So what's going on with where is it? 361 00:35:31,860 --> 00:35:35,519 Over here or over somewhere. Lost it. Oh, up there. 362 00:35:35,520 --> 00:35:44,070 Yeah. What's going on with. With Diamond. Well, one thing that Einstein didn't manage to derive is this this H Bar Omega. 363 00:35:45,390 --> 00:35:50,520 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. 364 00:35:50,520 --> 00:35:55,319 So where's this H-bomb coming from? It's the oscillator frequency up there. 365 00:35:55,320 --> 00:36:00,210 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. 366 00:36:00,220 --> 00:36:03,360 You don't know how springy you know, these harmonic wells are. 367 00:36:03,570 --> 00:36:09,150 So you just have to guess at them. And so each material will have a different spring constant, a different so-called Einstein frequency. 368 00:36:09,540 --> 00:36:18,580 The Omega is known as the Einstein frequency. And so the same frequency of different materials is different for some materials. 369 00:36:18,820 --> 00:36:26,950 The Einstein frequency is very low, in which case room temperature puts KB h over H for omega to be a large number. 370 00:36:27,190 --> 00:36:29,559 So you'd be up here on the graph. 371 00:36:29,560 --> 00:36:36,700 As a matter of fact, for most materials, you're somewhere up here on the graph because Omega is less than room temperature. 372 00:36:36,970 --> 00:36:45,730 But if you cool down the system to higher to lower temperature, your the heat capacity will drop as predicted by Einstein here. 373 00:36:46,060 --> 00:36:51,010 However, some materials are different that they have a large h bar omega. 374 00:36:51,970 --> 00:36:55,750 In which case room temperature would put you here on the plot. 375 00:36:55,930 --> 00:37:01,960 Your temperature is lower than for omega, and so the heat capacity, even at room temperature, should drop. 376 00:37:02,200 --> 00:37:06,510 Now, why is it the diamond is is one of these funny materials? 377 00:37:06,520 --> 00:37:11,410 Well, diamond. Let's write again. 378 00:37:11,530 --> 00:37:16,480 Omega is square root of Kappa over. M Diamond is special for a couple of reasons. 379 00:37:16,810 --> 00:37:22,060 First of all, Diamond is carbon and carbon is a very small element, is very high. 380 00:37:22,060 --> 00:37:25,420 Up on the periodic table is the only sixth element on the periodic table. 381 00:37:25,420 --> 00:37:28,630 Only five elements are smaller, lighter than carbon. 382 00:37:28,900 --> 00:37:35,379 So M is small for diamond, but also Diamond is a really hard material. 383 00:37:35,380 --> 00:37:37,540 It's very, very tough. So. 384 00:37:37,540 --> 00:37:43,210 K you might think that the toughness of the material is somehow related to its spring constant how, how hard the material is. 385 00:37:43,450 --> 00:37:49,689 So K is really big. So for Diamond, the frequency omega is anomalously large. 386 00:37:49,690 --> 00:37:52,809 It has one of the largest Einstein frequencies here. 387 00:37:52,810 --> 00:37:58,540 So the oscillator frequency is extremely big. So room temperature is much higher than the Einstein frequency. 388 00:37:58,540 --> 00:38:02,920 So on this plot, sorry factors, room temperature is much lower. 389 00:38:03,250 --> 00:38:07,030 So that wrong room temperature much lower than the Einstein frequency, which is very big. 390 00:38:07,210 --> 00:38:12,340 So in this plot you're way down on this side. And so the heat capacity per atom is really small. 391 00:38:12,610 --> 00:38:15,760 So let me actually show you some real experimental data. 392 00:38:18,080 --> 00:38:22,490 Here we go. So this is a picture right out of Einstein's original paper. 393 00:38:23,800 --> 00:38:32,260 And what you have is the experimental data, which is these these circles, and you have the theoretical curve, which is the dashed line. 394 00:38:32,260 --> 00:38:38,660 It seems to fit fairly well. Now, one should keep in mind that there is a three fit parameter H for Omega. 395 00:38:38,680 --> 00:38:44,889 He was not able to predict what H bombmaker should be, although he had a good argument why bombmaker should be big for Diamond. 396 00:38:44,890 --> 00:38:49,330 He didn't know exactly what the numbers should be, so he just chose the one that fit the data the best. 397 00:38:50,440 --> 00:38:54,399 And it does seem, given that one parameter does seem to fit fairly well at high temperature, 398 00:38:54,400 --> 00:38:59,830 you get three three kbps the law of daylong fatigue and at low temperature it seems to drop. 399 00:39:00,040 --> 00:39:05,140 And I just want to emphasise one more time that this huge success for Einstein was important, 400 00:39:05,200 --> 00:39:10,900 not only because he was able to explain what was going on in these materials, the heat capacity at low temperature. 401 00:39:11,140 --> 00:39:15,910 But it was much more important because in order to do this, he needed to invent quantum mechanics. 402 00:39:16,120 --> 00:39:23,980 He needed to invent the quantisation of the harmonic oscillator, an extremely important and pivotal moment in the development of quantum mechanics. 403 00:39:24,160 --> 00:39:27,460 And it came because he was studying condensed matter physics. 404 00:39:28,640 --> 00:39:38,060 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. 405 00:39:38,060 --> 00:39:40,790 There are some shortcomings in this picture as well. 406 00:39:41,150 --> 00:39:47,180 And the shortcoming always seems to be because something doesn't fit perfectly now, as you probably know from your practicals. 407 00:39:47,420 --> 00:39:53,270 There's plenty of reasons why data may not fit theory. One reason might be because the data is wrong. 408 00:39:53,510 --> 00:40:01,670 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. 409 00:40:01,700 --> 00:40:05,239 Whoever measured it messed up. It's not even close to the right answer. 410 00:40:05,240 --> 00:40:10,640 It's supposed to be down here. So that was one that probably caused Einstein some nightmares. 411 00:40:10,820 --> 00:40:20,299 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, 412 00:40:20,300 --> 00:40:25,670 there's the dashed curve is below, systematically below the measured experimental data. 413 00:40:25,910 --> 00:40:31,580 And that is real. Actually, that is a real problem, is a shortcoming of Einstein's theory. 414 00:40:32,840 --> 00:40:38,930 In truth, for most materials, most materials, including diamond. 415 00:40:41,230 --> 00:40:49,240 The heat capacity at a low temperature is proportional to t cubed at low t the exception to this, 416 00:40:49,240 --> 00:40:58,360 an exception will come to later on in maybe two or three lectures is metals, things like lead, copper, whatever. 417 00:40:59,200 --> 00:41:05,380 It's slightly different alpha t, cubed plus gamma t, but in no case, 418 00:41:05,410 --> 00:41:12,430 no material at all is the heat capacity at low temperature, exponentially small, which is Einstein's prediction. 419 00:41:12,850 --> 00:41:16,840 The heat capacity never drops as fast as Einstein predicted. 420 00:41:17,900 --> 00:41:24,880 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. 421 00:41:24,890 --> 00:41:30,290 This is a very low temperature. This temperature here is about four Kelvin and this temperature here is about 40 kelvin. 422 00:41:30,560 --> 00:41:34,850 If it's a perfect t cubed line, their plot against t cubed. 423 00:41:36,740 --> 00:41:40,040 So even way back then, Einstein, 424 00:41:40,040 --> 00:41:43,279 I think was aware that there was a problem with this theory because it did not 425 00:41:43,280 --> 00:41:48,920 predict t cubed predicted exponentially small heat capacity at low temperature. 426 00:41:49,220 --> 00:41:56,750 And this is where the field stood in 1907 and had to wait about five years or so before someone came along, 427 00:41:57,320 --> 00:42:01,670 who's also very smart to figure out what was wrong with Einstein's theory. 428 00:42:02,030 --> 00:42:06,470 The person who came along was a guy by the name of Peter Debye. 429 00:42:08,900 --> 00:42:11,900 And he approved improved upon Einstein's result. 430 00:42:13,240 --> 00:42:17,110 And explained where this T cubed r results came from. 431 00:42:17,710 --> 00:42:22,720 Now Dubai's into incidentally, I should say that Einstein thought that Dubai was really brilliant and he was very, 432 00:42:22,720 --> 00:42:27,910 very happy that Dubai managed to figure out this t cubed and overturn the 433 00:42:27,910 --> 00:42:31,840 Einstein model of the solid in what we will call the Dubai model of the solid. 434 00:42:32,020 --> 00:42:35,920 Which is somewhat better. He was a Einstein was a good man. 435 00:42:36,850 --> 00:42:37,810 Anyway, so is the bike. 436 00:42:38,740 --> 00:42:46,569 Although I guess recently there's just been some stories about how Einstein was how Debye was secretly a Nazi and he hated Einstein. 437 00:42:46,570 --> 00:42:51,790 But it turns out they were all wrong. There's a whole book about it you can find out on the web. 438 00:42:51,800 --> 00:42:54,760 So it's got to be true, right? So anyway. 439 00:42:55,760 --> 00:43:04,790 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. 440 00:43:04,940 --> 00:43:10,700 Why not? Because if an atom moves to the side of potential, well, it does get pushed back by its neighbour. 441 00:43:10,970 --> 00:43:13,310 But at the same time, it pushes on its neighbour. 442 00:43:13,460 --> 00:43:18,950 And so its neighbour moves and then it pushes on its neighbour and so forth and so on and so forth and so on. 443 00:43:19,190 --> 00:43:24,260 So what you need to think about is not just the oscillation of single atoms in the bottom of potential wells, 444 00:43:24,470 --> 00:43:28,970 but you have to think about the collective motion of all the atoms together in the solid, 445 00:43:29,210 --> 00:43:33,560 and the collective motion of all the atoms in the solid makes a wave. 446 00:43:33,740 --> 00:43:37,340 So we have to study wave motion of vibrations in the solid. 447 00:43:37,350 --> 00:43:43,219 And in fact, we even know what the wave motion in a solid vibrational wave motion is. 448 00:43:43,220 --> 00:43:46,700 Solid is called. It's usually called sound a sound wave. 449 00:43:47,800 --> 00:43:54,580 So what to buy realises there's a connection between the oscillation of atoms in a solid and in fact sound. 450 00:43:54,790 --> 00:44:01,960 Now, one of the thing that Debye understood at the time, which was really important, is that waves can be quantised too, 451 00:44:02,260 --> 00:44:09,850 in the sense the same way that the vibrational atom, the vibrational motion of an atom in the bottle potential well can be quantised. 452 00:44:10,090 --> 00:44:17,110 And the example that he look to as to how ways to be quantised was max Planck's quantisation of light. 453 00:44:17,380 --> 00:44:25,060 So what Debye wanted to do was quantised the motion, the vibrational motion of atoms and sound waves in a solid, 454 00:44:25,060 --> 00:44:31,450 the same way that Max Planck quantised the light ten years earlier. 455 00:44:31,690 --> 00:44:34,090 And I guess we'll stop there and I'll see you, I guess, Thursday.