1 00:00:00,050 --> 00:00:01,508 ANNOUNCER: The following content is 2 00:00:01,508 --> 00:00:04,010 provided under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,205 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,205 --> 00:00:17,830 at ocw.mit.edu. 8 00:00:20,910 --> 00:00:23,531 PROFESSOR: Then let's talk about exciting physics, the Lamb 9 00:00:23,531 --> 00:00:24,030 shift. 10 00:00:27,130 --> 00:00:32,170 So we discussed the Lamb shift last Friday, 11 00:00:32,170 --> 00:00:35,570 and the Lamb shift is really due to the fact 12 00:00:35,570 --> 00:00:38,780 that, if you have a atom consisting 13 00:00:38,780 --> 00:00:41,600 of an electron, a Coulomb field, and the proton, 14 00:00:41,600 --> 00:00:43,820 this is not the complete description. 15 00:00:43,820 --> 00:00:48,040 The atom lives in a vacuum, and the vacuum 16 00:00:48,040 --> 00:00:51,010 is filled with electromagnetic waves. 17 00:00:51,010 --> 00:00:54,070 So what we have to include, for an accurate description 18 00:00:54,070 --> 00:00:58,050 of atomic level structure, is the coupling 19 00:00:58,050 --> 00:01:02,390 of the atomic system, of the electron, to all of the modes 20 00:01:02,390 --> 00:01:07,350 the electromagnetic field, and this is radiation. 21 00:01:07,350 --> 00:01:11,480 We will talk about the quantized electromagnetic field later, 22 00:01:11,480 --> 00:01:14,230 at this point I could introduce you 23 00:01:14,230 --> 00:01:17,810 to a simple model, fairly accurate model of the Lamb 24 00:01:17,810 --> 00:01:21,190 shift, by simply assuming, and that's 25 00:01:21,190 --> 00:01:26,280 what we did on Friday, that we have fluctuating 26 00:01:26,280 --> 00:01:30,200 electric fields, those fluctuating electric fields 27 00:01:30,200 --> 00:01:34,310 shake, accelerate, the electron and the electron 28 00:01:34,310 --> 00:01:39,590 performs some oscillatory motion and this oscillatory motion 29 00:01:39,590 --> 00:01:44,890 leads to an ever reaching of the Coulomb potential. 30 00:01:44,890 --> 00:01:56,820 And similarly what we saw for the Darwin term, 31 00:01:56,820 --> 00:01:59,580 this ever reaching of the Coulomb potential 32 00:01:59,580 --> 00:02:03,750 takes away this singularity of the Coulomb potential 33 00:02:03,750 --> 00:02:10,850 and therefore lowers the binding energy of the electron. 34 00:02:10,850 --> 00:02:14,190 So today I want to just say a few more words about the result 35 00:02:14,190 --> 00:02:20,960 we derived, and then we have done 36 00:02:20,960 --> 00:02:24,900 what happens to an electron in a Coulomb field plus radiation. 37 00:02:24,900 --> 00:02:29,770 The next thing is, then, to discuss hyperfine structure. 38 00:02:29,770 --> 00:02:32,890 So let me first make one comment, when 39 00:02:32,890 --> 00:02:36,550 we integrated over all modes of the electromagnetic spectrum, 40 00:02:36,550 --> 00:02:38,580 we needed an upper cutoff, and a lower 41 00:02:38,580 --> 00:02:42,040 cutoff due to logarithmic singularities. 42 00:02:42,040 --> 00:02:46,410 Eventually, an upper cutoff is relativistic rest mass 43 00:02:46,410 --> 00:02:48,870 of the electron we have to cut off things. 44 00:02:48,870 --> 00:02:53,150 I just want to say one more word about the lower cutoff. 45 00:02:53,150 --> 00:02:56,220 I suggested, as a cut off, the orbital frequency 46 00:02:56,220 --> 00:02:57,940 of the electron. 47 00:02:57,940 --> 00:03:04,690 So the justification for that is the following, the electron, 48 00:03:04,690 --> 00:03:15,610 the free electron-- the free electron, when it's driven, 49 00:03:15,610 --> 00:03:20,420 has an amplitude which is 1 over the frequency squared. 50 00:03:20,420 --> 00:03:22,990 So if you drive it slower and slower, 51 00:03:22,990 --> 00:03:24,890 it's amplitude increases because it 52 00:03:24,890 --> 00:03:27,430 has more time to go in one direction. 53 00:03:27,430 --> 00:03:30,640 So this divergence at low frequency, of course, 54 00:03:30,640 --> 00:03:32,900 happens only for the free system. 55 00:03:32,900 --> 00:03:36,000 When you have a bound system, like a mnemonic oscillate, 56 00:03:36,000 --> 00:03:39,140 and you drive it at lower and lower frequency, 57 00:03:39,140 --> 00:03:43,270 the response converges to a constant 58 00:03:43,270 --> 00:03:47,550 and not to 1 over high frequency squared singularity. 59 00:03:47,550 --> 00:03:52,160 So therefore, when we reach the drive and the high frequencies 60 00:03:52,160 --> 00:03:58,770 on the order of-- oscillatory frequency of the bound system, 61 00:03:58,770 --> 00:04:00,800 the behavior changes. 62 00:04:00,800 --> 00:04:04,960 So when we mimic this with an effect cutoff because what 63 00:04:04,960 --> 00:04:07,970 you want to get rid of is a singularity, but in reality, 64 00:04:07,970 --> 00:04:11,870 of course, it should approach a constant at low frequency, 65 00:04:11,870 --> 00:04:14,890 and that is essentially the physics of the AC and then 66 00:04:14,890 --> 00:04:16,110 the DC Stark effect. 67 00:04:19,390 --> 00:04:27,853 So that was a rational for the cutoff 68 00:04:27,853 --> 00:04:32,610 and let me just annotate it here. 69 00:04:32,610 --> 00:04:37,690 And let's say, for a free particle 70 00:04:37,690 --> 00:04:41,490 we had the situation that the amplitude was proportional to 1 71 00:04:41,490 --> 00:04:44,390 over the high frequency. 72 00:04:44,390 --> 00:04:48,750 Whereas four bound particle is, you 73 00:04:48,750 --> 00:04:54,679 should approach a constant for low frequencies. 74 00:04:54,679 --> 00:04:56,720 And that's what we have introduced with a cutoff. 75 00:05:02,451 --> 00:05:02,950 OK. 76 00:05:05,550 --> 00:05:09,040 So we have discussed the Lamb shift, 77 00:05:09,040 --> 00:05:13,670 but you've already discussed one contribution to the Lamb shift. 78 00:05:13,670 --> 00:05:19,740 So this was a contribution that the Coulomb potential 79 00:05:19,740 --> 00:05:27,130 is effectively smeared out, and the result of this 80 00:05:27,130 --> 00:05:32,035 is that there is a weaker binding energy. 81 00:05:37,830 --> 00:05:44,555 However, there is a second contribution to the Lamb shift. 82 00:05:54,790 --> 00:05:56,700 I've sometimes seen sources where 83 00:05:56,700 --> 00:06:00,170 this is discussed as the main contribution to the Lamb shift, 84 00:06:00,170 --> 00:06:02,710 but this is not correct. 85 00:06:02,710 --> 00:06:08,800 This is only 3% off the total and contributes 27 megahertz. 86 00:06:08,800 --> 00:06:11,460 This is what is called the vacuum polarization. 87 00:06:18,452 --> 00:06:28,570 So if you have a-- the proton positive charge 88 00:06:28,570 --> 00:06:33,460 and you have the electron, and they scatter off each other, 89 00:06:33,460 --> 00:06:40,230 and we use this kind of diagram to indicate that, 90 00:06:40,230 --> 00:06:45,060 now that there is an additional diagram which 91 00:06:45,060 --> 00:06:49,120 has this bubble which is this production of e minus 92 00:06:49,120 --> 00:06:52,100 and e plus pairs. 93 00:06:52,100 --> 00:06:56,420 And you can say that if an electron and proton attract 94 00:06:56,420 --> 00:07:03,180 each other and you create, by virtual pair production-- 95 00:07:03,180 --> 00:07:05,940 because the vacuum is alive, things 96 00:07:05,940 --> 00:07:07,850 can happen in the vacuum-- it will virutally 97 00:07:07,850 --> 00:07:10,810 [INAUDIBLE] electron positron pair that now you 98 00:07:10,810 --> 00:07:16,970 create electron positron pairs, which shield the Coulomb field. 99 00:07:16,970 --> 00:07:18,960 And this is a second contribution 100 00:07:18,960 --> 00:07:22,300 in addition to the shaking motion of the electron. 101 00:07:22,300 --> 00:07:24,880 So I want you to just think about it for 10 seconds 102 00:07:24,880 --> 00:07:26,700 before I tell you the answer. 103 00:07:26,700 --> 00:07:33,870 Does this vacuum polarization strengthen or weaken 104 00:07:33,870 --> 00:07:34,670 the binding energy? 105 00:07:43,946 --> 00:07:45,320 Anybody want to offer an opinion? 106 00:07:49,670 --> 00:07:51,465 Whatever you say, you can't be wrong 107 00:07:51,465 --> 00:07:53,423 because there are to aspects to the answers so. 108 00:07:58,060 --> 00:08:00,240 Well then the actual answer is you would say, 109 00:08:00,240 --> 00:08:03,290 if you have charged-- if you create charges 110 00:08:03,290 --> 00:08:06,140 between the electron and the proton, 111 00:08:06,140 --> 00:08:09,100 you have a shielding effect and the shielding effect 112 00:08:09,100 --> 00:08:11,270 should weaken the Coulomb field. 113 00:08:11,270 --> 00:08:14,110 But the question is, what do we regard 114 00:08:14,110 --> 00:08:18,180 as the elementary charge, e, in our Schrodinger equation. 115 00:08:18,180 --> 00:08:20,645 And what we regard as the charge, what 116 00:08:20,645 --> 00:08:23,530 is measured in a Millikan Droplet experiment, 117 00:08:23,530 --> 00:08:26,250 is already the shielded charge. 118 00:08:26,250 --> 00:08:29,330 So therefore, the fact that vacuum polarization 119 00:08:29,330 --> 00:08:33,000 exists means we always measure the shielded charge, 120 00:08:33,000 --> 00:08:36,970 but because vacuum polarization happens at a finite distance, 121 00:08:36,970 --> 00:08:41,419 the electron, in an s state, can sort of penetrate the shield 122 00:08:41,419 --> 00:08:44,980 and feel a somewhat stronger Coulomb potential. 123 00:08:44,980 --> 00:08:47,930 So therefore, accounting for the fact 124 00:08:47,930 --> 00:08:50,720 that we have these virtual electron 125 00:08:50,720 --> 00:08:53,906 positron pairs actually means that the binding energy 126 00:08:53,906 --> 00:08:57,350 is increased and the vacuum polarization 127 00:08:57,350 --> 00:09:00,480 has the opposite sign as the dominant effect 128 00:09:00,480 --> 00:09:01,480 we've mentioned earlier. 129 00:09:04,160 --> 00:09:20,910 So the sum of e is, we observe the shielded charge and vacuum 130 00:09:20,910 --> 00:09:29,620 polarization implies that the s electron can penetrate 131 00:09:29,620 --> 00:09:34,375 the shield and sees a higher charge. 132 00:09:40,250 --> 00:09:43,910 So we normally observe the shielded charge, 133 00:09:43,910 --> 00:09:47,000 but the electron can see the higher charge. 134 00:09:47,000 --> 00:09:51,740 So therefore, that means, now, that we have, not an upshift 135 00:09:51,740 --> 00:09:58,060 in energy, but a downshift in energy for-- which for the two 136 00:09:58,060 --> 00:10:03,860 is one half state, is 27 megahertz, 137 00:10:03,860 --> 00:10:07,060 as I mentioned earlier. 138 00:10:07,060 --> 00:10:09,290 Anyway, I'm not really deriving it, 139 00:10:09,290 --> 00:10:16,500 but I want to sort of uncover certain myths, so the vacuum 140 00:10:16,500 --> 00:10:20,240 polarization, number one, is not dominant and number two, 141 00:10:20,240 --> 00:10:23,420 has the opposite sign as everybody would naively assume. 142 00:10:25,845 --> 00:10:26,345 Questions? 143 00:10:29,140 --> 00:10:33,430 I hope not because I don't know anything more about that. 144 00:10:33,430 --> 00:10:34,600 All right. 145 00:10:34,600 --> 00:10:40,160 So we have dealt with Lamb shift. 146 00:10:40,160 --> 00:10:43,310 So is next now, in revealing the atomic structure, 147 00:10:43,310 --> 00:10:48,160 is we want to go beyond the Coulomb field created 148 00:10:48,160 --> 00:10:49,970 by a point charge. 149 00:10:49,970 --> 00:10:54,380 And that means we want to address the fact that we don't 150 00:10:54,380 --> 00:10:57,900 have a point charge, but we have a nucleolus. 151 00:10:57,900 --> 00:11:08,200 And we are discussing, now, effects of the nucleus, which 152 00:11:08,200 --> 00:11:12,410 also go by the name hyperfine structure. 153 00:11:12,410 --> 00:11:16,610 So, just to summarize, so far we have 154 00:11:16,610 --> 00:11:23,580 treated, in pretty much complete detail, what 155 00:11:23,580 --> 00:11:30,040 happens for an atom which consists of a point charge 156 00:11:30,040 --> 00:11:30,680 and electrons. 157 00:11:33,900 --> 00:11:36,870 But now we want to bring in that, 158 00:11:36,870 --> 00:11:40,285 what creates a Coulomb field, the nucleolus, has structure. 159 00:11:47,340 --> 00:11:51,090 And there are actually four different ways 160 00:11:51,090 --> 00:11:55,730 how the nucleolus has structure and contributions 161 00:11:55,730 --> 00:11:59,580 to observable effects on the atomic structure 162 00:11:59,580 --> 00:12:01,700 and atomic energy levels. 163 00:12:01,700 --> 00:12:05,560 The most important one is that the nucleus 164 00:12:05,560 --> 00:12:09,010 has a magnetic moment associated with the angular 165 00:12:09,010 --> 00:12:13,560 momentum of the nucleus I. 166 00:12:13,560 --> 00:12:18,180 The second contribution is, in addition to magnetic moment, 167 00:12:18,180 --> 00:12:19,950 there may be a quadrupole moment. 168 00:12:22,700 --> 00:12:27,080 And since those effects can lead to a splitting-- this 169 00:12:27,080 --> 00:12:31,190 is actually, usually, called hyperfine structure, 170 00:12:31,190 --> 00:12:33,030 but then there are two more effects. 171 00:12:33,030 --> 00:12:39,180 One is the nucleus has finite mass, 172 00:12:39,180 --> 00:12:41,295 and the nucleus has a finite volume. 173 00:12:44,200 --> 00:12:51,390 Both the mass and volume effect lead to energy shifts. 174 00:12:51,390 --> 00:12:54,540 But tiny energy shifts are hard to measure 175 00:12:54,540 --> 00:12:56,680 unless you have two different shifts, 176 00:12:56,680 --> 00:13:01,860 and therefore those effects go as isotope shifts 177 00:13:01,860 --> 00:13:03,750 because when you have an atom which 178 00:13:03,750 --> 00:13:05,850 comes in two different isotopes you 179 00:13:05,850 --> 00:13:08,660 find that, due to those two effects, 180 00:13:08,660 --> 00:13:12,580 the energy levels are not the same. 181 00:13:12,580 --> 00:13:14,950 So this goes by the name of isotope shifts. 182 00:13:17,730 --> 00:13:23,780 By far the most important phenomenon is the first one. 183 00:13:23,780 --> 00:13:27,040 The fact that if a nucleus has angular momentum, 184 00:13:27,040 --> 00:13:35,170 we have hyperfine structure, and for the hydrogen atom 185 00:13:35,170 --> 00:13:41,770 that means that the ground state, the singlet S 1/2 state, 186 00:13:41,770 --> 00:13:46,760 actually splits into two states with total angular momentum 187 00:13:46,760 --> 00:13:58,180 quantum number F. 188 00:13:58,180 --> 00:14:04,610 So the relevance of hyperfine splitting 189 00:14:04,610 --> 00:14:08,870 is, it's actually a huge relevance, 190 00:14:08,870 --> 00:14:13,390 one is, you don't have a single ground state of many atoms, 191 00:14:13,390 --> 00:14:15,435 you have several ground states. 192 00:14:21,350 --> 00:14:24,470 So the lowest electronic state has several ground states, 193 00:14:24,470 --> 00:14:29,360 has several hyperfine states, due to angular momentum 194 00:14:29,360 --> 00:14:30,850 selection, where you can actually 195 00:14:30,850 --> 00:14:35,010 talk to them individually, you can prepare them individually, 196 00:14:35,010 --> 00:14:37,840 and many of you who do magnetic trapping know when 197 00:14:37,840 --> 00:14:39,990 need magnetic trapping you better prepare 198 00:14:39,990 --> 00:14:43,110 the atom in a single hyperfine state, 199 00:14:43,110 --> 00:14:46,700 otherwise you are in trouble. 200 00:14:46,700 --> 00:14:50,790 So you can prepare individual states. 201 00:14:50,790 --> 00:14:57,570 In the old days this was done by optical pumping, 202 00:14:57,570 --> 00:15:00,500 and you can use several hyperfine states 203 00:15:00,500 --> 00:15:07,360 to great advantage for the manipulation of atoms. 204 00:15:07,360 --> 00:15:10,200 For instance, if you want to, you 205 00:15:10,200 --> 00:15:13,380 can put atoms into a hyperfine state 206 00:15:13,380 --> 00:15:15,610 where they don't absorb light, and then you 207 00:15:15,610 --> 00:15:20,329 can have resonant light for the other ones blast those away. 208 00:15:20,329 --> 00:15:22,370 So you can play, sort of, your tricks because you 209 00:15:22,370 --> 00:15:28,270 have two states between which you can juggle at the atoms. 210 00:15:28,270 --> 00:15:36,360 Well, what else is relevance of hyperfine structure? 211 00:15:36,360 --> 00:15:42,000 OK if you have two levels, F equals 1, F equals 0, 212 00:15:42,000 --> 00:15:44,140 you can observe a transition. 213 00:15:44,140 --> 00:15:48,410 And the famous 21 centimeter line 214 00:15:48,410 --> 00:15:51,335 is used for astronomical observations. 215 00:15:58,250 --> 00:16:01,430 Hydrogen is the most abundant element in the universe, 216 00:16:01,430 --> 00:16:04,360 and how do you see hydrogen out there. 217 00:16:04,360 --> 00:16:08,560 Well it is due to hyperfine transition, the 21 centimeter 218 00:16:08,560 --> 00:16:10,970 line. 219 00:16:10,970 --> 00:16:16,780 And finally, another aspect why hyperfine structure 220 00:16:16,780 --> 00:16:18,930 is relevant, where people use it, 221 00:16:18,930 --> 00:16:21,970 is for the determination of nuclear properties. 222 00:16:21,970 --> 00:16:24,890 How do you know what the properties of nuclei are? 223 00:16:24,890 --> 00:16:27,540 Well there are techniques in nuclear physics, 224 00:16:27,540 --> 00:16:32,060 but a lot, a lot about the knowledge of nuclei 225 00:16:32,060 --> 00:16:35,440 comes from atomic spectroscopy. 226 00:16:35,440 --> 00:16:39,480 If you measure atomic energy levels with high accuracy, 227 00:16:39,480 --> 00:16:43,320 you figure out what the properties of the nucleus is, 228 00:16:43,320 --> 00:16:46,730 and one of the most outstanding examples we will discuss later 229 00:16:46,730 --> 00:16:49,800 on, and some of which is also on your homework assignment, 230 00:16:49,800 --> 00:16:53,390 is you can use atomic spectroscopy of hydrogen 231 00:16:53,390 --> 00:16:56,350 to contain the most accurate measurement, how 232 00:16:56,350 --> 00:16:58,820 big is the proton. 233 00:16:58,820 --> 00:17:02,260 And the big surprise is that there is, that there was, 234 00:17:02,260 --> 00:17:04,060 a surprise that people figured out 235 00:17:04,060 --> 00:17:09,050 that, until now, even in 2014, we do not fully 236 00:17:09,050 --> 00:17:11,869 understand how big the proton is but we'll 237 00:17:11,869 --> 00:17:12,930 talk about that later. 238 00:17:15,480 --> 00:17:19,119 So for the level of this-- the level of this introduction, 239 00:17:19,119 --> 00:17:27,930 we can use it for determination of nuclear properties 240 00:17:27,930 --> 00:17:31,910 and actually, you cannot only determine properties of stable 241 00:17:31,910 --> 00:17:36,370 nuclei, you can also determine properties of unstable nuclei. 242 00:17:36,370 --> 00:17:40,340 At various accelerators, they have a facility 243 00:17:40,340 --> 00:17:43,330 when, by, you know, energy collisions 244 00:17:43,330 --> 00:17:47,050 they create unstable nuclei. 245 00:17:47,050 --> 00:17:52,490 Maybe helium six, helium with four newtons, it exists. 246 00:17:52,490 --> 00:17:57,129 And you can take helium six, extract it, utilize it 247 00:17:57,129 --> 00:17:58,670 and then you've have a neutral helium 248 00:17:58,670 --> 00:18:01,960 atom which looks like your every days helium atom 249 00:18:01,960 --> 00:18:04,270 but it has two more neutrons in the nucleus. 250 00:18:04,270 --> 00:18:06,860 And by performing atomic spectroscopy, 251 00:18:06,860 --> 00:18:09,320 you can figure out what is the deformation, what 252 00:18:09,320 --> 00:18:14,780 is the structure, of this-- I want to say alpha particle, 253 00:18:14,780 --> 00:18:17,420 but it's an alpha particle plus two neutrons. 254 00:18:17,420 --> 00:18:20,440 So people have really learned to do those atomic physics 255 00:18:20,440 --> 00:18:22,910 measurements within a few seconds 256 00:18:22,910 --> 00:18:25,030 after the element has been produced, 257 00:18:25,030 --> 00:18:28,600 and such determined nuclear properties even 258 00:18:28,600 --> 00:18:29,530 of unstable nuclei. 259 00:18:44,100 --> 00:18:44,610 OK. 260 00:18:44,610 --> 00:18:45,980 So that's my introduction. 261 00:18:49,550 --> 00:18:54,780 So we are now discussing the most important effect 262 00:18:54,780 --> 00:18:57,050 due to the hyperfine structure. 263 00:18:57,050 --> 00:19:02,190 And this is the fact that the nucleolus has 264 00:19:02,190 --> 00:19:07,040 a magnetic moment, and this magnetic moment 265 00:19:07,040 --> 00:19:12,370 couples to the magnetic field even 266 00:19:12,370 --> 00:19:15,610 if you don't apply an external magnetic field, 267 00:19:15,610 --> 00:19:17,660 we talked about that on Wednesday, there 268 00:19:17,660 --> 00:19:21,090 is an internal magnetic field created 269 00:19:21,090 --> 00:19:26,250 by the electron with total angular momentum change. 270 00:19:26,250 --> 00:19:29,450 So this is, so to speak, the Zeeman Hamiltonian 271 00:19:29,450 --> 00:19:35,280 of the nucleus in the magnetic field created by the electron. 272 00:19:35,280 --> 00:19:41,280 And I will show you quickly to, is a simple derivation, 273 00:19:41,280 --> 00:19:43,240 what the result of that is. 274 00:19:43,240 --> 00:19:46,740 But before I do that, I also want to mention out-- 275 00:19:46,740 --> 00:19:49,600 mention that there is-- I want to point out 276 00:19:49,600 --> 00:19:51,830 that there is an alternative. 277 00:19:51,830 --> 00:19:56,290 Right now, I said we say the nucleus experiences 278 00:19:56,290 --> 00:20:00,620 the magnetic field created by the electron. 279 00:20:00,620 --> 00:20:02,870 But we can also take the other approach, 280 00:20:02,870 --> 00:20:05,750 the nucleus creates a vector potential because 281 00:20:05,750 --> 00:20:08,690 of its magnetic moment, and the electron, 282 00:20:08,690 --> 00:20:11,280 which goes around the nucleus, is not only 283 00:20:11,280 --> 00:20:13,760 feeling the Coulomb potential but also 284 00:20:13,760 --> 00:20:15,920 feeling a vector potential. 285 00:20:15,920 --> 00:20:18,910 And of course, both different perspectives, 286 00:20:18,910 --> 00:20:21,580 whether the electron moves in the magnetic field the nucleus, 287 00:20:21,580 --> 00:20:24,520 or the nucleus experiences the magnetic field of the electron, 288 00:20:24,520 --> 00:20:27,130 both treatments have to agree. 289 00:20:27,130 --> 00:20:29,530 I follow the more standard treatment, 290 00:20:29,530 --> 00:20:38,230 but the alternative treatment, where the electron moves 291 00:20:38,230 --> 00:20:40,940 in this electric and magnetic potential of the nucleus, 292 00:20:40,940 --> 00:20:45,770 is fully elaborated on the atomic physics wiki. 293 00:20:45,770 --> 00:20:51,870 So alternatively, electron moves in the potential 294 00:20:51,870 --> 00:21:02,720 of the nucleus, which is the Coulomb potential, 295 00:21:02,720 --> 00:21:05,750 we've discussed that, but then there is also 296 00:21:05,750 --> 00:21:10,775 vector potential created by the magnetic moment of the nucleus. 297 00:21:16,840 --> 00:21:19,090 So you simply assume this is a potential created 298 00:21:19,090 --> 00:21:21,000 with a nucleus, and then you just 299 00:21:21,000 --> 00:21:26,120 sort of Schrodinger's equation and this approach 300 00:21:26,120 --> 00:21:27,960 is carried out on the wiki. 301 00:21:31,320 --> 00:21:34,580 However, since it's a little bit more standard, 302 00:21:34,580 --> 00:21:37,410 and there's an easy semi-classical derivation, 303 00:21:37,410 --> 00:21:40,330 let me now discuss this one. 304 00:21:40,330 --> 00:21:42,590 Because what I like about it is it 305 00:21:42,590 --> 00:21:47,090 addresses one intuitive quantity, namely the fact 306 00:21:47,090 --> 00:21:49,635 that there is an internal magnetic field. 307 00:21:49,635 --> 00:21:51,510 We're not just using the Schrodinger equation 308 00:21:51,510 --> 00:21:52,968 for the whole system, we are really 309 00:21:52,968 --> 00:21:55,720 estimating what is the magnetic field, which 310 00:21:55,720 --> 00:21:59,228 the electron creates, at the position of the nucleus. 311 00:22:04,010 --> 00:22:13,090 So let's now do a semi-classical derivation 312 00:22:13,090 --> 00:22:17,040 of this internal magnetic field. 313 00:22:19,580 --> 00:22:21,810 And I have to-- I will immediately tell you, 314 00:22:21,810 --> 00:22:24,370 this derivation agrees quantitatively 315 00:22:24,370 --> 00:22:28,760 with the fully thermomechanical treatment. 316 00:22:28,760 --> 00:22:35,710 So, as often as these semi-classical derivations, 317 00:22:35,710 --> 00:22:38,240 we have to separate two parts. 318 00:22:40,960 --> 00:22:44,600 There are two ways how the electron creates 319 00:22:44,600 --> 00:22:46,630 a magnetic field at the nucleus. 320 00:22:46,630 --> 00:22:49,810 One is due to it's orbital motion, 321 00:22:49,810 --> 00:22:53,510 the electron is a ring current and creates a magnetic field, 322 00:22:53,510 --> 00:22:57,800 but then the electron has magnetic moment for-- due 323 00:22:57,800 --> 00:22:58,560 to it's spin. 324 00:23:03,170 --> 00:23:18,110 The spin part is simply the potential of a magnetic dipole, 325 00:23:18,110 --> 00:23:24,440 you will need to vector-- you will need to vector where 326 00:23:24,440 --> 00:23:30,490 the magnetic dipole moment of the electron 327 00:23:30,490 --> 00:23:33,630 is proportional to it's spin with a g factor. 328 00:23:36,490 --> 00:23:41,920 However, and you can find that in all textbooks 329 00:23:41,920 --> 00:23:44,820 on classical electrodynamics but, there 330 00:23:44,820 --> 00:23:48,910 is one important term which we have to add here, 331 00:23:48,910 --> 00:23:51,500 which is also part of classical E and M, 332 00:23:51,500 --> 00:23:57,560 and this is the delta function contribution. 333 00:24:00,400 --> 00:24:03,830 You've probably seen it, you find it in Jackson, 334 00:24:03,830 --> 00:24:06,190 if you haven't the model is that you 335 00:24:06,190 --> 00:24:09,440 can assume that a magnetic moment is created 336 00:24:09,440 --> 00:24:11,880 by ring current, and the ring current 337 00:24:11,880 --> 00:24:14,490 has-- creates a magnetic moment and you 338 00:24:14,490 --> 00:24:17,580 have the dipole potentially due to the magnetic moment. 339 00:24:17,580 --> 00:24:19,760 However, if you have a ring current, 340 00:24:19,760 --> 00:24:22,570 there is-- you can also ask, what 341 00:24:22,570 --> 00:24:27,160 is the magnetic field inside the current loop 342 00:24:27,160 --> 00:24:29,350 and then eventually do the transition where 343 00:24:29,350 --> 00:24:32,320 you allow the current loop to go to 0. 344 00:24:32,320 --> 00:24:36,090 That's how you make a point model of a magnetic dipole, 345 00:24:36,090 --> 00:24:39,770 but what remains is, sorts of-- as a delta function, 346 00:24:39,770 --> 00:24:42,480 the location inside the loop. 347 00:24:42,480 --> 00:24:45,330 I'm emphasizing it because it will be, 348 00:24:45,330 --> 00:24:48,460 eventually, the delta function contribution, 349 00:24:48,460 --> 00:24:51,090 which is important for s electrons, 350 00:24:51,090 --> 00:24:54,900 and therefore it is this contribution 351 00:24:54,900 --> 00:24:57,555 which is the dominant effect in many situations. 352 00:25:03,071 --> 00:25:03,570 OK. 353 00:25:03,570 --> 00:25:07,339 So this is the magnetic field created-- 354 00:25:07,339 --> 00:25:09,880 it's a classical expression, but it's buried in [? quantum ?] 355 00:25:09,880 --> 00:25:13,830 mechanics, the expression for the magnetic field created 356 00:25:13,830 --> 00:25:17,150 by the spin. 357 00:25:17,150 --> 00:25:22,890 The second contribution is the orbital contribution, 358 00:25:22,890 --> 00:25:28,500 and for that semi-classical, we just use Biot-Savat. 359 00:25:34,360 --> 00:25:40,060 So Biot-Savat is usually the 3-Dimensional integral 360 00:25:40,060 --> 00:25:43,010 over the current density. 361 00:25:48,660 --> 00:25:50,590 The volume integral, or you can rewrite it 362 00:25:50,590 --> 00:25:59,380 as the current I, d, r cross r, over r cubed. 363 00:25:59,380 --> 00:26:06,470 And eventually, if you now put in the electron charge 364 00:26:06,470 --> 00:26:13,670 distribution, velocity course r, well, 365 00:26:13,670 --> 00:26:18,860 velocity cross r means we get, and that's what we want, 366 00:26:18,860 --> 00:26:21,610 the orbital angular momentum. 367 00:26:21,610 --> 00:26:24,680 The 1 over r cubed term means we have 368 00:26:24,680 --> 00:26:27,960 to calculate an expectation value over the wave 369 00:26:27,960 --> 00:26:31,030 function which, is 1 over r cubed. 370 00:26:31,030 --> 00:26:37,040 And the prefactor leads us-- it's 371 00:26:37,040 --> 00:26:40,130 nothing else than two times the Bohr magenton. 372 00:26:46,720 --> 00:26:59,950 So, with those two terms, we can now obtain our final expression 373 00:26:59,950 --> 00:27:03,450 for the total magnetic field generated 374 00:27:03,450 --> 00:27:05,400 by the electron at the origin. 375 00:27:07,930 --> 00:27:11,460 And for that I use the g factor of two 376 00:27:11,460 --> 00:27:15,560 as it comes out of Dirac theory. 377 00:27:15,560 --> 00:27:18,120 So now we have the total magnetic field. 378 00:27:22,790 --> 00:27:28,240 We had this contribution L over r cubed. 379 00:27:28,240 --> 00:27:32,340 If you inspect the dipole potential of the spin 380 00:27:32,340 --> 00:27:39,310 it has a contribution S over r cubed, 381 00:27:39,310 --> 00:27:47,920 then it is the second contribution 382 00:27:47,920 --> 00:27:50,220 to the dipole potential. 383 00:27:50,220 --> 00:27:57,070 And finally, most importantly for the following discussion, 384 00:27:57,070 --> 00:28:01,740 the delta function contribution which I discussed earlier. 385 00:28:10,790 --> 00:28:20,250 If you have an s state, these first terms 386 00:28:20,250 --> 00:28:27,410 are 0 for L equals 0 because these are, sort of, terms which 387 00:28:27,410 --> 00:28:32,420 have dipole potential where positive and negative 388 00:28:32,420 --> 00:28:36,100 contributions cancel out when you do a spherical average, 389 00:28:36,100 --> 00:28:38,540 and the s electron performs a spherical average. 390 00:28:43,400 --> 00:28:49,840 So it is 0 for L equals 0 due to the spherical average. 391 00:28:53,410 --> 00:29:01,940 Whereas the second part, it would be 0 for L non-equals 392 00:29:01,940 --> 00:29:06,870 to 0 because the probability for a non-s electron 393 00:29:06,870 --> 00:29:08,802 to be at the nucleus is 0. 394 00:29:11,700 --> 00:29:16,014 So pretty much this describes that. 395 00:29:16,014 --> 00:29:17,680 So this describes a hyperfine structure. 396 00:29:21,940 --> 00:29:29,520 Well, it describes the magnetic field created by the electron, 397 00:29:29,520 --> 00:29:36,360 and now we have to do the usual projection 398 00:29:36,360 --> 00:29:38,770 in the following way. 399 00:29:38,770 --> 00:29:45,790 That the hyperfine structure is that Zeeman Hamiltonian 400 00:29:45,790 --> 00:29:48,240 of the internal magnetic field with a magnetic moment 401 00:29:48,240 --> 00:29:59,050 of the nucleus, and the magnetic moment of the nucleus 402 00:29:59,050 --> 00:30:04,120 is proportional to the angular momentum of the nucleus. 403 00:30:04,120 --> 00:30:07,510 Sort of this argument that even if it were not proportional, 404 00:30:07,510 --> 00:30:10,460 it would rapidly precess and eventually project it, 405 00:30:10,460 --> 00:30:12,090 and the only direction which survives 406 00:30:12,090 --> 00:30:14,550 is the direction of the angular momentum. 407 00:30:14,550 --> 00:30:18,240 And similarly, you can-- the magnetic field, 408 00:30:18,240 --> 00:30:20,640 you have a contribution of S and L, 409 00:30:20,640 --> 00:30:24,610 but S and L rapidly precess around the result and angular 410 00:30:24,610 --> 00:30:27,480 momentum, J, and therefore, as a result, 411 00:30:27,480 --> 00:30:31,720 the internal magnetic field must be, 412 00:30:31,720 --> 00:30:35,550 can only be, parallel to the angular momentum chain. 413 00:30:39,260 --> 00:30:41,310 If you do a fully [? quantum ?] treatment, 414 00:30:41,310 --> 00:30:42,550 it comes out immediately. 415 00:30:42,550 --> 00:30:44,750 But if you do it semi-classically, you calculate 416 00:30:44,750 --> 00:30:48,560 a magnetic field, you sort of have to fall in this argument 417 00:30:48,560 --> 00:30:52,730 that you always project on the axis of angular momentum 418 00:30:52,730 --> 00:30:56,706 and that means that the hyperfine interaction will 419 00:30:56,706 --> 00:31:00,050 be the Hamiltonian for it, or the operator, 420 00:31:00,050 --> 00:31:06,450 will be the dot product of I dot J. For fine structure, 421 00:31:06,450 --> 00:31:11,140 we had L dot S, for hyperfine structure, we had L dot J, 422 00:31:11,140 --> 00:31:13,630 this is always how we couple angular momentum with a dot 423 00:31:13,630 --> 00:31:15,080 product. 424 00:31:15,080 --> 00:31:24,320 The hyperfine constant cause by the letter 425 00:31:24,320 --> 00:31:29,830 a, and since historically a is measured in frequency units, 426 00:31:29,830 --> 00:31:34,530 in Hertz, I have to put in h, Planck's quantum. 427 00:31:34,530 --> 00:31:36,150 No it's not h bar. 428 00:31:36,150 --> 00:31:37,760 For historical reasons, it's h. 429 00:31:43,896 --> 00:31:44,860 AUDIENCE: Question. 430 00:31:44,860 --> 00:31:45,725 PROFESSOR: Yes? 431 00:31:45,725 --> 00:31:47,224 AUDIENCE: Are I and J dimensionless, 432 00:31:47,224 --> 00:31:49,679 or will they carry units in h bar? 433 00:31:55,080 --> 00:31:57,130 PROFESSOR: Here they are dimensionless, 434 00:31:57,130 --> 00:32:00,320 thanks for the question, because each is in frequency units, 435 00:32:00,320 --> 00:32:03,140 it's in Hertz, and if you multiple with h 436 00:32:03,140 --> 00:32:04,860 you have an energy. 437 00:32:04,860 --> 00:32:08,330 So therefore, I and J measure the angular momentum 438 00:32:08,330 --> 00:32:10,150 in unit of h bar. 439 00:32:10,150 --> 00:32:13,390 So it's not in that sense, it's a normalized angular momentum 440 00:32:13,390 --> 00:32:14,470 operator. 441 00:32:14,470 --> 00:32:18,580 The quantum numbers of I and J are not 1/2, or 1h bar, 442 00:32:18,580 --> 00:32:22,279 it's just 1/2 or 1. 443 00:32:22,279 --> 00:32:22,945 Other questions? 444 00:32:28,431 --> 00:32:28,930 OK. 445 00:32:28,930 --> 00:32:34,820 I can now take this expression with, you know, L and S 446 00:32:34,820 --> 00:32:37,650 and S.r and evaluate further, but I 447 00:32:37,650 --> 00:32:40,350 feel I'm not providing any insight 448 00:32:40,350 --> 00:32:43,600 and you can read about it on the wiki. 449 00:32:43,600 --> 00:32:51,350 So for a non-s state, how to simplify this expression 450 00:32:51,350 --> 00:32:55,650 and get the final textbook result, I defer to the wiki. 451 00:32:55,650 --> 00:33:00,730 I want to discuss the most important part, namely for s v 452 00:33:00,730 --> 00:33:04,210 electrons because hydrogen, all the alkaloids, 453 00:33:04,210 --> 00:33:05,660 have an s ground state. 454 00:33:08,690 --> 00:33:15,990 So, in that case, all we have to consider is the delta function 455 00:33:15,990 --> 00:33:21,750 part and if we project the magnetic field 456 00:33:21,750 --> 00:33:31,570 onto the angular momentum axis, we 457 00:33:31,570 --> 00:33:37,925 get the probability of the s electron to be at the origin. 458 00:33:43,120 --> 00:33:54,060 And therefore, for s states, the hyperfine constant 459 00:33:54,060 --> 00:34:06,130 is-- oh, I forgot one thing. 460 00:34:06,130 --> 00:34:11,889 We have to parametrize the magnetic moment of the nucleus, 461 00:34:11,889 --> 00:34:17,420 and that is done by using a nuclear magnetron. 462 00:34:17,420 --> 00:34:19,620 It's the same as a Bhor magneton, where 463 00:34:19,620 --> 00:34:26,520 you have replaced the electron mass by the proton mass, 464 00:34:26,520 --> 00:34:29,250 and you have the nuclear g factor. 465 00:34:29,250 --> 00:34:33,620 Just as a reminder, the g factor of the proton is 5.6, 466 00:34:33,620 --> 00:34:36,600 the g factor of the neutron is minus 3.8. 467 00:34:36,600 --> 00:34:40,290 So the g factor has nothing to do, not even close, 468 00:34:40,290 --> 00:34:43,920 to the factor of 2, which we obtained in the Dirac equation 469 00:34:43,920 --> 00:34:45,030 for the electron. 470 00:34:45,030 --> 00:34:49,389 That just shows that the nucleons, protons and neutrons, 471 00:34:49,389 --> 00:34:50,889 are more complicated. 472 00:34:50,889 --> 00:34:53,080 Well, they have quarks inside, they 473 00:34:53,080 --> 00:34:54,780 have a complicated internal structure. 474 00:34:58,090 --> 00:34:58,590 OK. 475 00:34:58,590 --> 00:35:03,930 So therefore the hyperfine constant involves, 476 00:35:03,930 --> 00:35:08,890 now, the g factor of the nucleus, 477 00:35:08,890 --> 00:35:12,580 the product of the nuclear magnetron, 478 00:35:12,580 --> 00:35:22,530 with a Bohr magneton, and for hydrogen. 479 00:35:29,630 --> 00:35:43,680 This gives the famous result of 1420 megahertz. 480 00:35:43,680 --> 00:35:47,670 So this is hydrogen, and this h is now the Hamiltonian. 481 00:35:47,670 --> 00:35:50,900 So the hyperfine coupling Hamiltonian, which has I 482 00:35:50,900 --> 00:36:02,730 dot J, by using the expression for the total angular momentum 483 00:36:02,730 --> 00:36:08,990 I plus J and then we square it. 484 00:36:08,990 --> 00:36:11,850 When you evaluate this square you get, 485 00:36:11,850 --> 00:36:19,010 on the right hand side, an expression for I dot J. 486 00:36:19,010 --> 00:36:23,920 So I dot J is nothing else than 1/2. 487 00:36:23,920 --> 00:36:32,050 F squared, minus I squared, minus J squared. 488 00:36:35,090 --> 00:36:43,950 And therefore, for hydrogen, where I, J and S are all 1/2, 489 00:36:43,950 --> 00:36:45,970 the proton has been 1/2, the electron 490 00:36:45,970 --> 00:36:48,730 has been 1/2, that's it. 491 00:36:48,730 --> 00:36:53,790 You have only two values of the result and total angular 492 00:36:53,790 --> 00:36:59,400 momentum, 1/2 and 1/2 can add up to 1 or 0. 493 00:36:59,400 --> 00:37:13,130 And now the hyperfine splitting is into an F equals 1 494 00:37:13,130 --> 00:37:17,070 and F equals 0 state. 495 00:37:17,070 --> 00:37:21,630 And one thing to remember is, if you inspect the above formula 496 00:37:21,630 --> 00:37:25,110 with the [? quantum ?] numbers you find immediately 497 00:37:25,110 --> 00:37:34,530 that, compared to the degenerate line, 498 00:37:34,530 --> 00:37:37,220 without hyperfine splitting, this-- 499 00:37:37,220 --> 00:37:44,030 so what comes out of the Dirac equation, 500 00:37:44,030 --> 00:37:53,890 the splitting is the fight-- is 1/4 and 3/4 501 00:37:53,890 --> 00:37:56,770 of the hyperfine constant. 502 00:37:56,770 --> 00:38:01,040 Since F equals 1 has a multiplicity of 3, 2 and F 503 00:38:01,040 --> 00:38:03,860 quantum numbers, plus minus 1 and 0, 504 00:38:03,860 --> 00:38:09,640 the rule is that the center of mass of this level 505 00:38:09,640 --> 00:38:11,595 does not change due to hyperfine splitting. 506 00:38:16,300 --> 00:38:24,160 So the center of mass of hyperfine states 507 00:38:24,160 --> 00:38:25,990 is not changing. 508 00:38:29,800 --> 00:38:32,500 So we introduce a level splitting, 509 00:38:32,500 --> 00:38:33,860 but no overall shift. 510 00:38:40,450 --> 00:38:44,950 Any questions about magnetic hyperfine structure? 511 00:38:44,950 --> 00:38:45,450 Yes? 512 00:38:45,450 --> 00:38:48,200 AUDIENCE: Would you explain again the center of mass 513 00:38:48,200 --> 00:38:48,940 is not changing. 514 00:38:48,940 --> 00:38:50,740 Is this just for hydrogen? 515 00:38:50,740 --> 00:38:52,090 Or is this a general rule? 516 00:38:52,090 --> 00:38:53,920 PROFESSOR: No, this is a general property, 517 00:38:53,920 --> 00:38:57,330 and you could actually show that when you evaluate 518 00:38:57,330 --> 00:39:00,130 the product of I dot J. So, the product of I 519 00:39:00,130 --> 00:39:05,270 dot J means that for arbitrary I and arbitrary J, if you 520 00:39:05,270 --> 00:39:07,980 calculate the hyperfine structure, the center of mass 521 00:39:07,980 --> 00:39:08,560 is the same. 522 00:39:14,480 --> 00:39:14,980 OK. 523 00:39:22,110 --> 00:39:26,400 I've not done any experiment in my life 524 00:39:26,400 --> 00:39:33,260 where higher order moments became important, 525 00:39:33,260 --> 00:39:37,240 but I want to teach you about it because the discussion 526 00:39:37,240 --> 00:39:38,970 about whether those higher moments exist 527 00:39:38,970 --> 00:39:42,360 or not is really an interesting discussion about what 528 00:39:42,360 --> 00:39:44,580 is allowed by symmetry and what's not. 529 00:39:44,580 --> 00:39:48,490 So I'm bringing in a higher moments, 530 00:39:48,490 --> 00:39:51,990 not so much because you need it to understand the level 531 00:39:51,990 --> 00:39:53,910 structure of your favourite atom, 532 00:39:53,910 --> 00:39:58,510 but because it teaches us a really nice piece of physics. 533 00:39:58,510 --> 00:40:03,310 So let me now discuss higher order moments and the leading 534 00:40:03,310 --> 00:40:06,960 one is the electric quadripole moment. 535 00:40:06,960 --> 00:40:09,810 So I want to raise, in general, the question, 536 00:40:09,810 --> 00:40:14,190 what further moments can a nucleus have. 537 00:40:14,190 --> 00:40:19,120 What we have discussed so far is the magnetic moment, mu. 538 00:40:19,120 --> 00:40:39,290 So beyond mu, what further electric or magnetic moments 539 00:40:39,290 --> 00:40:42,250 can a nucleus have? 540 00:40:42,250 --> 00:40:46,780 Well, it's a question about symmetry 541 00:40:46,780 --> 00:40:58,410 and if you look at the parity of multiples, 542 00:40:58,410 --> 00:41:05,790 if you have an electric multiple-- well, 543 00:41:05,790 --> 00:41:09,300 you know if you have a dipole, plus minus, you invert it, 544 00:41:09,300 --> 00:41:12,460 the dipole becomes minus the dipole. 545 00:41:12,460 --> 00:41:17,070 So for L equals 1, it is minus 1, L is 1, 546 00:41:17,070 --> 00:41:19,200 but in general the it is minus L. 547 00:41:19,200 --> 00:41:22,030 If you have a quadripole, you invert your coordinate system. 548 00:41:22,030 --> 00:41:25,780 It's L equals 2, plus plus, minus minus, 549 00:41:25,780 --> 00:41:27,735 you invert it nothing changes. 550 00:41:27,735 --> 00:41:29,860 So these I've shown you, for dipole and quadripole, 551 00:41:29,860 --> 00:41:32,040 that this formula is correct. 552 00:41:32,040 --> 00:41:36,630 So an electric multiple has this parity, 553 00:41:36,630 --> 00:41:41,990 and for, magnetic keys-- now, you 554 00:41:41,990 --> 00:41:46,310 know, magnetic we have axial vectors versus polar vectors, 555 00:41:46,310 --> 00:41:49,150 there's always an extra factor of minus 1. 556 00:41:49,150 --> 00:41:57,390 So therefore if you go magnetic, magnetic multiples with L 557 00:41:57,390 --> 00:41:59,410 have a parity of minus L plus 1. 558 00:42:02,710 --> 00:42:08,120 So this really restricts what multiples are possible. 559 00:42:08,120 --> 00:42:11,020 Instead of giving you a general discussion, 560 00:42:11,020 --> 00:42:16,160 let me just look at the very important keys. 561 00:42:16,160 --> 00:42:20,210 Whether a magnet with a nucleus can have a electric dipole 562 00:42:20,210 --> 00:42:23,430 moment, and you will immediately see what it needs to-- and then 563 00:42:23,430 --> 00:42:26,610 I give you a general result. 564 00:42:26,610 --> 00:42:39,710 So let's assume a nucleus has angular momentum I, 565 00:42:39,710 --> 00:42:48,080 and there is a magnetic moment, mu associated with it. 566 00:42:51,540 --> 00:43:02,100 So the general result is that odd electric and even 567 00:43:02,100 --> 00:43:18,010 magnetic multiple moments would violate not just one, 568 00:43:18,010 --> 00:43:28,410 but two symmetries, would violate parity and time 569 00:43:28,410 --> 00:43:32,055 reversal symmetry. 570 00:43:34,720 --> 00:43:37,920 So the argument goes as follows. 571 00:43:37,920 --> 00:43:41,450 Let us assume this is the magnetic moment, mu 572 00:43:41,450 --> 00:43:43,930 or the angular momentum I, it's a vector. 573 00:43:46,810 --> 00:43:52,590 And now we are asking, is it possible to have 574 00:43:52,590 --> 00:43:56,560 a vector of the dipole moment. 575 00:43:56,560 --> 00:43:59,820 And dipole moment, you should just think about it 576 00:43:59,820 --> 00:44:02,795 as a plus minus charge, separated. 577 00:44:05,680 --> 00:44:09,280 We can now do the parity operation and the time reversal 578 00:44:09,280 --> 00:44:10,590 operation. 579 00:44:10,590 --> 00:44:12,900 If you do the time reversal operation, 580 00:44:12,900 --> 00:44:16,120 the current, which generates a magnetic moment 581 00:44:16,120 --> 00:44:18,420 if you want to think about it in this picture, 582 00:44:18,420 --> 00:44:19,680 goes the other way. 583 00:44:19,680 --> 00:44:22,920 So mu flips but nothing moves, of course, 584 00:44:22,920 --> 00:44:25,880 for an electric dipole moment, so reversing time 585 00:44:25,880 --> 00:44:29,120 is not changing anything. 586 00:44:29,120 --> 00:44:31,410 So in other words, time reversal symmetry 587 00:44:31,410 --> 00:44:36,670 transforms parallel mu and d into anti-parallel mu and d. 588 00:44:36,670 --> 00:44:41,830 And similar, parity is not changing mu 589 00:44:41,830 --> 00:44:44,250 but it is changing d. 590 00:44:44,250 --> 00:44:50,420 So in both cases, would parity, or time reversal symmetry, 591 00:44:50,420 --> 00:44:55,640 if you had mu-d, a scalar product of mu 592 00:44:55,640 --> 00:45:10,970 and d which would be known 0, then both P or P 593 00:45:10,970 --> 00:45:14,330 would change the sign. 594 00:45:14,330 --> 00:45:18,150 But that would mean that two kinds of particles would exist, 595 00:45:18,150 --> 00:45:21,330 one where the sign is positive, one where the sign is negative. 596 00:45:21,330 --> 00:45:24,290 But we have assumed that we have 1 nucleus and only 1 nucleus 597 00:45:24,290 --> 00:45:27,800 of this kind, so we cannot have one nucleus which has 598 00:45:27,800 --> 00:45:29,960 the properties of having, simultaneously, 599 00:45:29,960 --> 00:45:33,180 a magnetic and electric dipole moment. 600 00:45:33,180 --> 00:45:42,010 So therefore, we conclude that mu times d has to be 0. 601 00:45:42,010 --> 00:45:44,440 And if you generalize this argument, 602 00:45:44,440 --> 00:45:48,820 we have ruled out that there is an electric dipole 603 00:45:48,820 --> 00:45:51,820 moment, an odd electric moment. 604 00:45:51,820 --> 00:45:55,080 The first, the lowest, possible electric moment 605 00:45:55,080 --> 00:46:04,570 is the quadripole moment, so the leading electric moment is not 606 00:46:04,570 --> 00:46:08,180 L equals 2, L equals 2, it's not L equals 1, 607 00:46:08,180 --> 00:46:12,560 the dipole, it's L equals 2 the quadripole. 608 00:46:12,560 --> 00:46:15,080 And of course if you generalize the argument, 609 00:46:15,080 --> 00:46:17,340 L equals 4, L equals 6, would be possible 610 00:46:17,340 --> 00:46:19,940 but those effects would be very small. 611 00:46:28,530 --> 00:46:29,450 Questions so far? 612 00:46:31,891 --> 00:46:32,390 OK. 613 00:46:32,390 --> 00:46:36,470 So we've talked about parity and time reversal symmetry, which 614 00:46:36,470 --> 00:46:41,190 restricts what kind of magnetic and electric dipole moments 615 00:46:41,190 --> 00:46:42,940 particles may have. 616 00:46:42,940 --> 00:46:44,650 And maybe in this context, I should just 617 00:46:44,650 --> 00:46:47,820 mention that John Doyle, Jerry Gabrielse, and Dave Demille 618 00:46:47,820 --> 00:46:51,420 at Harvard and Yale, they just published the most accurate 619 00:46:51,420 --> 00:46:56,800 result for the electric dipole moment of the electron. 620 00:46:56,800 --> 00:46:59,920 They found a bound, which was more than an order of magnitude 621 00:46:59,920 --> 00:47:04,550 lower than the best upper bound before 622 00:47:04,550 --> 00:47:06,380 and this has really made headlines. 623 00:47:06,380 --> 00:47:11,210 So to measure, accurately, that the electric dipole 624 00:47:11,210 --> 00:47:14,230 moment vanishes, in this case of the electron, 625 00:47:14,230 --> 00:47:17,020 but other people do it also for neutrons, 626 00:47:17,020 --> 00:47:19,530 is testing fundamental symmetries. 627 00:47:19,530 --> 00:47:22,810 In particular, it tests whether nature 628 00:47:22,810 --> 00:47:24,740 is time reversal invariant. 629 00:47:28,240 --> 00:47:32,070 And the reason why, until now, everybody 630 00:47:32,070 --> 00:47:35,380 has found that the results are compatible with 0 631 00:47:35,380 --> 00:47:40,440 is pretty much based on the argument I just gave you. 632 00:47:40,440 --> 00:47:45,140 OK so we have discussed those fundamental symmetries, 633 00:47:45,140 --> 00:47:52,420 but now I want to discuss something else related to it. 634 00:47:52,420 --> 00:47:57,350 So let's assume you have a nucleus, 635 00:47:57,350 --> 00:48:01,150 and I want to discuss with you, is a minimum requirement 636 00:48:01,150 --> 00:48:04,830 for the nucleus for the angular momentum of the nucleus 637 00:48:04,830 --> 00:48:13,410 in order to have a magnetic dipole, 638 00:48:13,410 --> 00:48:26,220 or to have an electric quadripole. 639 00:48:26,220 --> 00:48:29,700 So let's formulate it as a quicker question. 640 00:48:29,700 --> 00:48:36,160 So here let's assume a is-- it's possible for any nucleus 641 00:48:36,160 --> 00:48:40,230 no matter what the angular momentum is. 642 00:48:40,230 --> 00:48:46,700 Here, we put in 1/2, or larger than 1, 643 00:48:46,700 --> 00:48:51,800 and for the electric quadripole we have the same choices. 644 00:48:51,800 --> 00:48:54,820 So you should decide if I tell you, 645 00:48:54,820 --> 00:48:57,010 a nucleolus has a magnetic dipole. 646 00:48:57,010 --> 00:48:59,750 Does that imply that there is a minimum amount 647 00:48:59,750 --> 00:49:03,590 of angular momentum of the nucleus there? 648 00:49:03,590 --> 00:49:05,120 And then repeat the same question 649 00:49:05,120 --> 00:49:06,286 for the electric quadripole. 650 00:49:09,860 --> 00:49:12,240 So please tell me your opinion. 651 00:49:21,610 --> 00:49:24,600 So right now, what is a minimum requirement 652 00:49:24,600 --> 00:49:28,677 for I, for magnetic dipole, and if yes, what is it? 653 00:49:35,140 --> 00:49:37,630 Three, two, one. 654 00:49:37,630 --> 00:49:38,130 Stop. 655 00:49:42,440 --> 00:49:53,950 OK, so-- and let's immediately consider 656 00:49:53,950 --> 00:49:56,650 the electric quadripole moment. 657 00:49:56,650 --> 00:49:58,170 So this was a majority opinion. 658 00:49:58,170 --> 00:50:01,246 I gave this answer for both of them together. 659 00:50:07,240 --> 00:50:10,870 So the question is, what is the minimum requirement 660 00:50:10,870 --> 00:50:14,881 for a nucleus to have an electric quadripole moment. 661 00:50:29,980 --> 00:50:31,680 OK, stop. 662 00:50:31,680 --> 00:50:32,180 Display. 663 00:50:35,360 --> 00:50:37,410 No requirement. 664 00:50:37,410 --> 00:50:39,570 OK. 665 00:50:39,570 --> 00:50:43,650 So the majority onset was a here. 666 00:50:43,650 --> 00:50:47,360 So let me to discuss it, and I know some of you 667 00:50:47,360 --> 00:50:50,900 will have-- I always get into heated discussions with it. 668 00:50:50,900 --> 00:50:54,410 I actually just had a few minute discussion with one 669 00:50:54,410 --> 00:50:59,310 of my colleagues about it, who at least wanted 670 00:50:59,310 --> 00:51:01,720 to look at it from a different perspective than I did. 671 00:51:01,720 --> 00:51:03,700 So let me give you the short answer 672 00:51:03,700 --> 00:51:05,650 and we can go from there. 673 00:51:05,650 --> 00:51:10,310 In order to the magnetic dipole you have to take an object, 674 00:51:10,310 --> 00:51:13,350 turn it around, and figure out that there's 675 00:51:13,350 --> 00:51:15,380 a different energy. 676 00:51:15,380 --> 00:51:18,270 So therefore, unless you have an object, 677 00:51:18,270 --> 00:51:21,950 which is has two orientations, you cannot figure out if it has 678 00:51:21,950 --> 00:51:23,800 a magnetic dipole or not. 679 00:51:23,800 --> 00:51:28,560 If you're in a state without angular momentum, I equals 0, 680 00:51:28,560 --> 00:51:30,650 there is no distinguishable states. 681 00:51:30,650 --> 00:51:35,000 You cannot orient a 0 angular momentum object in space. 682 00:51:35,000 --> 00:51:36,990 So therefore you can never figure out 683 00:51:36,990 --> 00:51:39,490 that there's a magnetic moment, and now I'll 684 00:51:39,490 --> 00:51:41,350 make a bold statement, and this means 685 00:51:41,350 --> 00:51:44,300 there is no magnetic moment. 686 00:51:44,300 --> 00:51:53,120 So therefore, you need this because you 687 00:51:53,120 --> 00:51:59,766 need a minimum number of true orientations. 688 00:52:02,310 --> 00:52:03,120 OK. 689 00:52:03,120 --> 00:52:08,490 Now an electric quadripole means that the charge density 690 00:52:08,490 --> 00:52:10,050 is like ellipse. 691 00:52:10,050 --> 00:52:11,178 It has-- 692 00:52:11,178 --> 00:52:12,053 AUDIENCE: [INAUDIBLE] 693 00:52:14,940 --> 00:52:17,250 PROFESSOR: Oh, 1/2, sorry. 694 00:52:17,250 --> 00:52:17,750 Thank you. 695 00:52:27,160 --> 00:52:30,440 So I should get larger than 1/2 because we 696 00:52:30,440 --> 00:52:33,670 need a minimum of two possible orientations. 697 00:52:33,670 --> 00:52:38,160 Now my question for you is, you think something is elliptical 698 00:52:38,160 --> 00:52:41,550 but how many different orientations do 699 00:52:41,550 --> 00:52:45,783 you need to figure out that it is elliptical and not round. 700 00:52:54,960 --> 00:52:58,830 If I equal 0, you can only look at it, you can't rotate it, 701 00:52:58,830 --> 00:53:01,520 so will you never find any energies [? breathing. ?] 702 00:53:01,520 --> 00:53:04,960 It's just there and you cannot say where it's round or whether 703 00:53:04,960 --> 00:53:06,950 it has a quadripole or deformation. 704 00:53:06,950 --> 00:53:11,680 If I is equal to 1/2, you can take it and flip it around 705 00:53:11,680 --> 00:53:15,700 but can you tell from that that it's an ellipse. 706 00:53:15,700 --> 00:53:19,130 No because if you turn an ellipse around nothing changes. 707 00:53:19,130 --> 00:53:21,660 So it could be, as well, a sphere. 708 00:53:21,660 --> 00:53:23,330 You can only figure out that it's 709 00:53:23,330 --> 00:53:26,610 an ellipse if you have an intermediate rotation, 710 00:53:26,610 --> 00:53:28,840 let's say, by 90 degrees. 711 00:53:28,840 --> 00:53:32,090 So in order to assess that something is elliptical, 712 00:53:32,090 --> 00:53:36,390 you have to at least resolve three positions, three angles, 713 00:53:36,390 --> 00:53:40,290 and three angles require that you have to three sublevels 714 00:53:40,290 --> 00:53:43,700 and this requires that I is larger, or equal, than one. 715 00:53:47,330 --> 00:53:48,710 Sorry. 716 00:53:48,710 --> 00:53:55,260 Then one goes-- OK. 717 00:54:00,350 --> 00:54:03,280 I'm want to give you a formal argument using 718 00:54:03,280 --> 00:54:08,560 this spherical tensor, but I guess 719 00:54:08,560 --> 00:54:11,490 someone you are waiting for something simpler. 720 00:54:11,490 --> 00:54:12,420 So. 721 00:54:12,420 --> 00:54:14,770 I mean who wants to know the answer of the question, 722 00:54:14,770 --> 00:54:17,670 but what happens if it has a deformation. 723 00:54:17,670 --> 00:54:19,290 Does it just mean we can't measure it, 724 00:54:19,290 --> 00:54:20,331 but it has a deformation? 725 00:54:23,220 --> 00:54:24,620 Let me explain that. 726 00:54:24,620 --> 00:54:29,380 So if I have a pencil and the pencil has zero angular 727 00:54:29,380 --> 00:54:31,900 momentum, I can't really figure out 728 00:54:31,900 --> 00:54:34,440 that it's an elongated object, because all I measure is 729 00:54:34,440 --> 00:54:35,960 a symmetric wave function. 730 00:54:35,960 --> 00:54:38,150 It's completely spherically symmetric. 731 00:54:38,150 --> 00:54:40,840 The only way to figure out that it's a pencil 732 00:54:40,840 --> 00:54:43,540 is I have to localize it, that I can 733 00:54:43,540 --> 00:54:45,550 see that it's pointing somewhere. 734 00:54:45,550 --> 00:54:48,680 But to localize an object in space 735 00:54:48,680 --> 00:54:51,070 is actually an angular wave packet. 736 00:54:51,070 --> 00:54:52,280 It's not isotropic. 737 00:54:52,280 --> 00:54:53,700 It points somewhere. 738 00:54:53,700 --> 00:54:57,030 And an angular wave packet is a superposition 739 00:54:57,030 --> 00:55:00,870 of states of different angular momenta. 740 00:55:00,870 --> 00:55:04,270 So therefore, without assuming that there is a state with 741 00:55:04,270 --> 00:55:06,730 angular momentum, I cannot orient this pencil. 742 00:55:09,280 --> 00:55:10,020 OK. 743 00:55:10,020 --> 00:55:14,940 I know you would all agree that even if this pencil is cooled 744 00:55:14,940 --> 00:55:16,920 to the ground state with zero angular momentum, 745 00:55:16,920 --> 00:55:19,040 it is a pencil. 746 00:55:19,040 --> 00:55:21,600 But what you're using here is now your knowledge 747 00:55:21,600 --> 00:55:25,540 that this object has higher angular momentum states. 748 00:55:25,540 --> 00:55:27,220 And those higher angular momentum 749 00:55:27,220 --> 00:55:30,730 states have nothing to do with the structure or the appearance 750 00:55:30,730 --> 00:55:33,020 of this object of being a pencil. 751 00:55:33,020 --> 00:55:35,670 So you sort of know that in addition to the i 752 00:55:35,670 --> 00:55:39,560 equals 0 state, there are i equals 1, 2, 3, 4, 5 states, 753 00:55:39,560 --> 00:55:42,040 and the pencil looks the same. 754 00:55:42,040 --> 00:55:44,230 But if you have a nucleus, an i equals 755 00:55:44,230 --> 00:55:48,270 0 state requires a certain configuration of quarks. 756 00:55:48,270 --> 00:55:51,390 And you cannot create an i equals 2, and i equals 4, 757 00:55:51,390 --> 00:55:55,000 higher states without messing around with the internal 758 00:55:55,000 --> 00:55:55,740 structure. 759 00:55:55,740 --> 00:55:59,540 So with a nucleus, all you have is an i equals 0 state. 760 00:55:59,540 --> 00:56:03,560 And to say that this i equals 0 state has a quadrupolar 761 00:56:03,560 --> 00:56:06,180 deformation doesn't make any sense. 762 00:56:06,180 --> 00:56:11,310 If you would know that this nuclear state could be rotated 763 00:56:11,310 --> 00:56:14,640 without changing its internal structure, then you would say, 764 00:56:14,640 --> 00:56:18,550 yes, it has a quadrupole moment, I just can't see it. 765 00:56:18,550 --> 00:56:20,745 But usually, you cannot make this assumption. 766 00:56:20,745 --> 00:56:23,870 If all you have is an i equals 0 ground state, 767 00:56:23,870 --> 00:56:27,030 and the i equals 2 state is very, very different, 768 00:56:27,030 --> 00:56:29,310 because the quarks are spinning around each other 769 00:56:29,310 --> 00:56:31,820 in a different way, you have to see, 770 00:56:31,820 --> 00:56:34,950 i equals 0 is completely spherical. 771 00:56:34,950 --> 00:56:37,690 It doesn't couple to anything externally. 772 00:56:37,690 --> 00:56:40,500 And therefore, it has no moments whatsoever. 773 00:56:44,350 --> 00:56:46,160 So that's the story. 774 00:56:46,160 --> 00:56:48,130 A lot of people get confused, because they 775 00:56:48,130 --> 00:56:52,610 think an object can have a deformation without rotating. 776 00:56:52,610 --> 00:56:55,080 But you need the rotation to resolve it. 777 00:56:55,080 --> 00:56:58,550 If you cannot create an angular wave packet, 778 00:56:58,550 --> 00:57:01,070 which is a superposition state of angular momenta, 779 00:57:01,070 --> 00:57:05,080 you can never figure out that there is a deformation. 780 00:57:05,080 --> 00:57:08,980 And quantum mechanically, if you cannot figure out that there is 781 00:57:08,980 --> 00:57:12,140 a deformation, there is no deformation, 782 00:57:12,140 --> 00:57:16,680 because you can only use, in the language of quantum mechanics, 783 00:57:16,680 --> 00:57:21,605 where you have at least the possibility to measure it. 784 00:57:21,605 --> 00:57:23,060 Questions about that? 785 00:57:25,881 --> 00:57:26,380 OK. 786 00:57:26,380 --> 00:57:31,916 I think that makes it now-- let me now kind of just 787 00:57:31,916 --> 00:57:32,665 give you a formal. 788 00:57:40,070 --> 00:57:41,150 A formal derivation. 789 00:57:41,150 --> 00:57:41,650 But I agree. 790 00:57:41,650 --> 00:57:43,066 I mean, the formal derivation, I'm 791 00:57:43,066 --> 00:57:45,460 just throwing a few equations at you, and say, that's it 792 00:57:45,460 --> 00:57:46,870 and everything follows from that. 793 00:57:46,870 --> 00:57:49,160 But I provided the insight for you. 794 00:57:49,160 --> 00:57:55,390 Formally, you can define the quadrupole moment 795 00:57:55,390 --> 00:58:01,110 by the expectation operator. 796 00:58:01,110 --> 00:58:05,290 You take the nucleus with maximum MI. 797 00:58:05,290 --> 00:58:12,875 And now you calculate the expectation value 798 00:58:12,875 --> 00:58:14,125 of this operator. 799 00:58:18,170 --> 00:58:24,200 This is, of course, motivated by just electrostatics. 800 00:58:24,200 --> 00:58:34,550 If you take an expansion of the classical electrostatic energy 801 00:58:34,550 --> 00:58:39,580 into multi-poles, you find the quadrupole configuration 802 00:58:39,580 --> 00:58:44,600 to be related to a quadripole moment. 803 00:58:44,600 --> 00:58:50,200 Quadrupole moments couple to the derivative of electric fields. 804 00:58:50,200 --> 00:58:53,910 And then in this purely classical description, 805 00:58:53,910 --> 00:59:02,290 you have this term, where beta is 806 00:59:02,290 --> 00:59:08,400 the angle between two symmetry axis, namely, 807 00:59:08,400 --> 00:59:23,440 between the symmetry axis of the electric field gradient 808 00:59:23,440 --> 00:59:30,010 and the quadrupole tensor-- just the classic quadrupole tensor 809 00:59:30,010 --> 00:59:31,230 as it comes out of Jackson. 810 00:59:41,100 --> 00:59:42,130 Yes. 811 00:59:42,130 --> 00:59:44,390 You can see the quantum mechanical definition 812 00:59:44,390 --> 00:59:48,240 of the quadrupole moment, or more generally. 813 00:59:48,240 --> 00:59:49,940 So this is quadrupole moment. 814 00:59:49,940 --> 00:59:53,800 If you have a moment with l, the operator, 815 00:59:53,800 --> 00:59:56,760 which tells you whether you have a non-vanishing moment, 816 00:59:56,760 --> 01:00:00,460 a non-vanishing deformation, is actually 817 01:00:00,460 --> 01:00:02,726 a spherical tensor, Tlm. 818 01:00:08,360 --> 01:00:10,610 And what you see above is a spherical tensor, 819 01:00:10,610 --> 01:00:15,830 T20-- l equals 2, m equals 0. 820 01:00:15,830 --> 01:00:18,970 And those spherical tensors are defined by the fact 821 01:00:18,970 --> 01:00:22,190 that they transform as spherical harmonics. 822 01:00:22,190 --> 01:00:25,300 And now you sort of realize what it means. 823 01:00:25,300 --> 01:00:32,030 If you want a magnetic or electric moment with l, 824 01:00:32,030 --> 01:00:35,920 the operator for the moment transforms 825 01:00:35,920 --> 01:00:38,530 like angular momentum l. 826 01:00:38,530 --> 01:00:41,370 And now you realize that you have the triangle rule. 827 01:00:41,370 --> 01:00:47,460 If you want a matrix element where I and l overlap with I, 828 01:00:47,460 --> 01:00:50,600 you want to make sure that I, l, and I couple. 829 01:00:50,600 --> 01:00:52,870 And you have a triangle rule. 830 01:00:52,870 --> 01:00:57,530 So therefore, if you want a magnetic moment, 831 01:00:57,530 --> 01:01:02,800 or electric moment, of l, and you evaluate this expectation 832 01:01:02,800 --> 01:01:08,050 value, well, at least the triangle rules 833 01:01:08,050 --> 01:01:10,600 can only be justified like this. 834 01:01:10,600 --> 01:01:14,740 Or in other words, you can only get a non-vanishing moment 835 01:01:14,740 --> 01:01:18,020 if l is smaller than 2I. 836 01:01:18,020 --> 01:01:20,940 And this is what we discussed in the clicker question 837 01:01:20,940 --> 01:01:25,520 for the two cases of l equals 1 and l equals 2. 838 01:01:25,520 --> 01:01:28,490 So ultimately, it's a selection rule 839 01:01:28,490 --> 01:01:31,460 which is related to the triangle rule 840 01:01:31,460 --> 01:01:35,470 for the addition of angular momenta. 841 01:01:35,470 --> 01:01:38,190 But I like much better the argument, 842 01:01:38,190 --> 01:01:39,870 how many orientations do you need 843 01:01:39,870 --> 01:01:43,970 to find out that something is elliptical? 844 01:01:43,970 --> 01:01:46,960 It's formalized here. 845 01:01:46,960 --> 01:01:47,760 All right. 846 01:01:51,780 --> 01:01:57,930 Let's just spend one more minute on the quadrupolar structure. 847 01:01:57,930 --> 01:02:04,330 So based on the expansion of the electrostatic energy, what 848 01:02:04,330 --> 01:02:07,410 determines the quadrupolar structure 849 01:02:07,410 --> 01:02:13,660 is this cosine angle, which is the angle 850 01:02:13,660 --> 01:02:20,100 between the axis of the nucleus and the axis 851 01:02:20,100 --> 01:02:21,966 of an electric field gradient. 852 01:02:24,560 --> 01:02:26,920 And that means it is the angle. 853 01:02:30,490 --> 01:02:35,100 It's a cosine of the angle between J, 854 01:02:35,100 --> 01:02:41,120 the outer environment, and I, the axis of the nucleus. 855 01:02:41,120 --> 01:02:46,630 So therefore, when we would derive-- I'm not deriving it, 856 01:02:46,630 --> 01:02:49,480 but if we would derive an expression for quadrupolar 857 01:02:49,480 --> 01:02:52,480 structure, the quadrupolar structure 858 01:02:52,480 --> 01:02:56,290 would be proportional to a quantity C, which 859 01:02:56,290 --> 01:03:03,020 is nothing else than the dot product of I and J. 860 01:03:03,020 --> 01:03:06,460 And at least in my notes now, I and J have units of h bar. 861 01:03:06,460 --> 01:03:09,150 So I'm dividing it out here. 862 01:03:09,150 --> 01:03:14,720 And as you know, I, dot, J can be expressed 863 01:03:14,720 --> 01:03:19,730 by quantum numbers F, F plus 1, minus I, I plus 1, 864 01:03:19,730 --> 01:03:24,670 minus J, J plus 1. 865 01:03:24,670 --> 01:03:32,810 So therefore, the quadrupolar energies-- E2, 866 01:03:32,810 --> 01:03:39,650 El equals 2-- involve the classical expression 867 01:03:39,650 --> 01:03:42,060 at cosine square. 868 01:03:42,060 --> 01:03:49,260 So therefore, you would expect there is a quadrupole constant. 869 01:03:49,260 --> 01:03:52,640 And then it is cosine square. 870 01:03:52,640 --> 01:03:55,040 But well, usually, quantum mechanic 871 01:03:55,040 --> 01:03:57,460 even, we have the square of a quantity, 872 01:03:57,460 --> 01:04:03,480 we have to write it as quantity times quantity plus 1. 873 01:04:03,480 --> 01:04:06,780 So this is the quadrupolar structure. 874 01:04:06,780 --> 01:04:09,010 And to remind you, we just discussed 875 01:04:09,010 --> 01:04:13,670 that for the hydrogen atom, the magnetic hyperfine structure 876 01:04:13,670 --> 01:04:21,480 had invoiced the same product of I, dot, J, but in a linear way. 877 01:04:25,100 --> 01:04:28,570 So the reason why I'm not discussing quadrupolar 878 01:04:28,570 --> 01:04:33,390 structure in more detail is usually 879 01:04:33,390 --> 01:04:36,340 the hyperfine structure associated 880 01:04:36,340 --> 01:04:41,040 with quadrupole moments is much, much smaller 881 01:04:41,040 --> 01:04:43,450 than the hyperfine structure associated 882 01:04:43,450 --> 01:04:53,050 with magnetic moment, typically by a factor of 100. 883 01:04:53,050 --> 01:04:55,840 The only exceptions are molecules, 884 01:04:55,840 --> 01:04:58,960 because molecules can have-- because 885 01:04:58,960 --> 01:05:01,590 of molecular binding mechanisms-- a much, much 886 01:05:01,590 --> 01:05:04,830 larger electric field gradient. 887 01:05:04,830 --> 01:05:07,800 So therefore, in molecules, quadrupolar structure 888 01:05:07,800 --> 01:05:10,830 is more important than in atoms. 889 01:05:16,910 --> 01:05:19,848 Questions about that? 890 01:05:26,930 --> 01:05:32,620 With that, we have discussed the two effects 891 01:05:32,620 --> 01:05:37,050 of hyperfine structure due to magnetic moment of the nucleus. 892 01:05:37,050 --> 01:05:42,120 And we also discussed further deformations of the nucleus, 893 01:05:42,120 --> 01:05:45,050 in particular, the quadrupolar deformation. 894 01:05:45,050 --> 01:05:48,330 Let me now use the last 10 minutes 895 01:05:48,330 --> 01:05:50,525 to quickly discuss with you isotope effects. 896 01:06:09,600 --> 01:06:12,900 And I know that many people here know about isotope effects, 897 01:06:12,900 --> 01:06:15,450 because if you lock your laser to a lithium cell 898 01:06:15,450 --> 01:06:20,000 or to a rubidium cell, you find lithium-6 and lithium-7 peaks. 899 01:06:20,000 --> 01:06:24,590 And in rubidium, rubidium-85 and rubidium-87. 900 01:06:24,590 --> 01:06:28,440 So there are two peaks which are spectrally very, very 901 01:06:28,440 --> 01:06:30,150 well resolved. 902 01:06:30,150 --> 01:06:34,870 And now I tell you what causes the splitting between the lines 903 01:06:34,870 --> 01:06:36,590 of rubidium-85 and rubidium-87. 904 01:06:39,520 --> 01:06:42,650 Well, the first effect is really trivial. 905 01:06:42,650 --> 01:06:45,200 It's the mass effect. 906 01:06:45,200 --> 01:06:48,270 And I have to remind you that the Rydberg 907 01:06:48,270 --> 01:06:56,640 formula for a single electron energy level 908 01:06:56,640 --> 01:06:59,300 contains the reduced mass. 909 01:06:59,300 --> 01:07:01,900 In other words, the energy levels 910 01:07:01,900 --> 01:07:04,600 are the energy levels-- if you assume 911 01:07:04,600 --> 01:07:08,670 that the mass of the nucleus is infinite, that means you just 912 01:07:08,670 --> 01:07:12,460 take for the electron mass in the Rydberg constant 913 01:07:12,460 --> 01:07:14,050 the electron mass. 914 01:07:14,050 --> 01:07:23,450 But in general, you have to take the reduced mass-- 915 01:07:23,450 --> 01:07:28,030 the big M is the nucleus. 916 01:07:28,030 --> 01:07:29,825 And small m is the electron. 917 01:07:32,340 --> 01:07:38,020 The simplest case is if you set the nuclear mass to infinity. 918 01:07:38,020 --> 01:07:41,015 Then you simply have the Rydberg constant with the electron 919 01:07:41,015 --> 01:07:43,530 mass. 920 01:07:43,530 --> 01:07:47,080 So in the limit that the nuclear mass is 921 01:07:47,080 --> 01:07:49,020 much larger than the electron mass, 922 01:07:49,020 --> 01:07:54,220 this correction factor is 1 minus little m over big M. 923 01:07:54,220 --> 01:07:57,840 So therefore, the correction factor is on the order of 10 924 01:07:57,840 --> 01:08:02,290 to the 4 or 10 to the 5. 925 01:08:02,290 --> 01:08:06,750 Visible frequencies are on the order of 10 to the 14. 926 01:08:06,750 --> 01:08:09,910 So 10 to the minus 4 or 10 to the minus 5 of it 927 01:08:09,910 --> 01:08:12,850 is between 1 and 10 gigahertz. 928 01:08:12,850 --> 01:08:16,446 So this is the scale for mass corrections 929 01:08:16,446 --> 01:08:17,529 due to the isotope effect. 930 01:08:24,140 --> 01:08:27,950 What is the sine of the isotope effect? 931 01:08:27,950 --> 01:08:33,890 Does the fact that the nucleus has a finite mass-- 932 01:08:33,890 --> 01:08:36,970 does that mean that the binding energy of the electron 933 01:08:36,970 --> 01:08:38,290 is smaller or larger? 934 01:08:52,540 --> 01:08:56,824 AUDIENCE: [INAUDIBLE] it would be larger. 935 01:08:56,824 --> 01:09:02,620 PROFESSOR: The absolute value of the binding energy-- 936 01:09:02,620 --> 01:09:03,990 look at your signs. 937 01:09:03,990 --> 01:09:06,090 I would say it in the following. 938 01:09:06,090 --> 01:09:08,010 Let's start out with an infinite nucleus, 939 01:09:08,010 --> 01:09:10,290 and only the electron is moving. 940 01:09:10,290 --> 01:09:12,580 But now we make the nucleus lighter. 941 01:09:12,580 --> 01:09:15,290 And that means the nucleus also has to move, 942 01:09:15,290 --> 01:09:16,990 because it's a two-body problem. 943 01:09:16,990 --> 01:09:19,260 And there is additional kinetic energy, 944 01:09:19,260 --> 01:09:21,550 additional kinetic energy of the nucleus. 945 01:09:21,550 --> 01:09:25,500 And kinetic energy is positive and weakens the binding energy 946 01:09:25,500 --> 01:09:27,260 of the total system. 947 01:09:27,260 --> 01:09:31,770 So therefore, the fact that the effective mass correction means 948 01:09:31,770 --> 01:09:36,040 the lighter the nucleus is, the more kinetic energy 949 01:09:36,040 --> 01:09:38,810 has to be added to the system for the nuclear motion, 950 01:09:38,810 --> 01:09:41,284 and the more the binding energy is weakened. 951 01:09:43,790 --> 01:09:50,920 The most dramatic example is not the hydrogen atom. 952 01:09:50,920 --> 01:09:59,510 It is positronium, where your nucleus is not a nucleus, 953 01:09:59,510 --> 01:10:02,260 it's a positively charged electron to the positronium. 954 01:10:02,260 --> 01:10:06,610 And in this situation, you have an effective mass 955 01:10:06,610 --> 01:10:11,350 which is only 1/2 of the electron mass. 956 01:10:11,350 --> 01:10:30,090 And the 1s-2s transition, which is Lyman alpha for hydrogen, 957 01:10:30,090 --> 01:10:32,260 is now not even in the vacuum UV, 958 01:10:32,260 --> 01:10:35,710 it just happens at ordinary UV transitions. 959 01:10:35,710 --> 01:10:39,610 And that means now the 1s-2s energy 960 01:10:39,610 --> 01:10:41,570 is smaller by a factor of 2. 961 01:10:41,570 --> 01:10:44,770 That really means the binding energy between an electron 962 01:10:44,770 --> 01:10:49,050 and a positron is only 50% of the binding energy 963 01:10:49,050 --> 01:10:50,710 of an electron in the hydrogen atom. 964 01:10:54,040 --> 01:10:54,540 OK. 965 01:10:54,540 --> 01:10:56,630 That's all I want to say about the mass effect. 966 01:11:02,470 --> 01:11:04,235 Let's now talk about the volume effect. 967 01:11:11,690 --> 01:11:16,420 So if you would look at the charge distribution 968 01:11:16,420 --> 01:11:27,410 in a nucleus as a function of r, if we go from one isotope 969 01:11:27,410 --> 01:11:30,810 to a heavier isotope with more neutrons, 970 01:11:30,810 --> 01:11:34,450 the nuclear radius becomes larger 971 01:11:34,450 --> 01:11:36,521 and the charge becomes more spread out. 972 01:11:40,850 --> 01:11:50,040 So if I plot now the electrostatic potential, 973 01:11:50,040 --> 01:11:54,780 the electrostatic potential is, of course, the Coulomb 974 01:11:54,780 --> 01:12:01,770 potential, 1 over r, until we enter the charge distribution. 975 01:12:01,770 --> 01:12:10,130 And then, as you know from electrostatics, it continues. 976 01:12:14,240 --> 01:12:15,770 This is 1 over r. 977 01:12:15,770 --> 01:12:16,995 And then it's flattened off. 978 01:12:16,995 --> 01:12:21,230 It continues quadratically for the heavier-- oops, 979 01:12:21,230 --> 01:12:22,940 I wanted to change color. 980 01:12:22,940 --> 01:12:25,800 For the heavier nucleus, it is like this, 981 01:12:25,800 --> 01:12:29,630 and for the lighter nucleus, it is like this. 982 01:12:29,630 --> 01:12:32,160 So in other words, the finite size of the nucleus 983 01:12:32,160 --> 01:12:34,960 is cutting off the Coulomb potential 984 01:12:34,960 --> 01:12:36,250 where it is strongest. 985 01:12:36,250 --> 01:12:38,870 And this happens the earlier, the larger, 986 01:12:38,870 --> 01:12:40,900 or the heavier the nucleus is. 987 01:12:43,530 --> 01:12:47,195 So therefore, what you obtain is you obtain, 988 01:12:47,195 --> 01:12:51,250 in perturbation theory, a level shift. 989 01:12:51,250 --> 01:12:53,880 Since it only affects the electron when 990 01:12:53,880 --> 01:12:56,990 it's very close to the origin, this level shift 991 01:12:56,990 --> 01:13:00,750 is, as other effects we have discussed today, 992 01:13:00,750 --> 01:13:03,130 proportional to the probability of the electron 993 01:13:03,130 --> 01:13:04,900 to be at the center. 994 01:13:04,900 --> 01:13:09,320 This is only s electrons are effected. 995 01:13:20,180 --> 01:13:23,040 What is the effect in terms of energy? 996 01:13:23,040 --> 01:13:26,690 Well, it's clear the Coulomb potential is weakened. 997 01:13:26,690 --> 01:13:32,740 Therefore, this effect, the volume effect weakens, 998 01:13:32,740 --> 01:13:42,380 decreases the binding energy of the electron. 999 01:13:42,380 --> 01:13:48,090 So we have two effects now-- we have the volume effect, which 1000 01:13:48,090 --> 01:13:52,180 is the stronger the bigger and the heavier the nucleus is. 1001 01:13:59,710 --> 01:14:06,760 So it's largest for heavy nuclei. 1002 01:14:06,760 --> 01:14:12,020 And the mass effect, or the effective mass effect, 1003 01:14:12,020 --> 01:14:14,715 is of course largest for the lightest nuclei, 1004 01:14:14,715 --> 01:14:17,210 with the extreme example of positronium. 1005 01:14:33,560 --> 01:14:34,485 Any questions? 1006 01:14:42,410 --> 01:14:43,880 Well, we have five minutes left. 1007 01:14:43,880 --> 01:14:44,496 But Cody. 1008 01:14:44,496 --> 01:14:45,870 AUDIENCE: What sort of scales are 1009 01:14:45,870 --> 01:14:46,690 associated with the volume effect? 1010 01:14:46,690 --> 01:14:47,582 PROFESSOR: Pardon? 1011 01:14:47,582 --> 01:14:48,920 AUDIENCE: What sort of energy scales 1012 01:14:48,920 --> 01:14:51,503 are associated with the volume effect? [INAUDIBLE] comparable? 1013 01:14:51,503 --> 01:14:55,130 PROFESSOR: What energy is-- no, it's actually-- wait. 1014 01:14:55,130 --> 01:14:56,902 Let's estimate it. 1015 01:14:56,902 --> 01:14:58,110 You're working with rubidium. 1016 01:14:58,110 --> 01:15:01,270 What is your isotope shift in rubidium between 85 and 87? 1017 01:15:01,270 --> 01:15:04,336 AUDIENCE: I actually don't know. 1018 01:15:04,336 --> 01:15:08,171 PROFESSOR: Isn't it 170, 90-- no, 1019 01:15:08,171 --> 01:15:09,920 I'm getting confused now with [INAUDIBLE]. 1020 01:15:13,694 --> 01:15:15,110 So many people work with rubidium. 1021 01:15:15,110 --> 01:15:17,810 What is the difference in transition frequency 1022 01:15:17,810 --> 01:15:20,114 between rubidium-85 and -87? 1023 01:15:20,114 --> 01:15:21,922 AUDIENCE: [INAUDIBLE]. 1024 01:15:21,922 --> 01:15:24,558 PROFESSOR: It's much more than gigahertz. 1025 01:15:24,558 --> 01:15:26,480 You really have to tune your laser. 1026 01:15:26,480 --> 01:15:28,370 I think if you look at the wave per cell, 1027 01:15:28,370 --> 01:15:32,400 you will never accidentally see a rubidium-85 line. 1028 01:15:32,400 --> 01:15:35,206 I just don't recall the number. 1029 01:15:35,206 --> 01:15:36,190 AUDIENCE: [INAUDIBLE]. 1030 01:15:42,094 --> 01:15:46,030 AUDIENCE: [INAUDIBLE] together, you can see both of them. 1031 01:15:46,030 --> 01:15:49,490 PROFESSOR: OK. 1032 01:15:49,490 --> 01:15:51,840 And for the mass effect, we said it's 1033 01:15:51,840 --> 01:15:55,980 sort of-- then they would be comparable. 1034 01:15:55,980 --> 01:15:57,980 The mass effect is easy to estimate, 1035 01:15:57,980 --> 01:16:01,730 because the mass correction is one part in a few thousand. 1036 01:16:01,730 --> 01:16:06,530 So that would mean on the order of 10 to the 10 gigahertz. 1037 01:16:06,530 --> 01:16:08,880 So the volume effect-- it really depends. 1038 01:16:08,880 --> 01:16:11,520 It's tiny for light elements. 1039 01:16:11,520 --> 01:16:14,510 Rubidium is already heavy. 1040 01:16:14,510 --> 01:16:17,170 So right now I would say they are comparable. 1041 01:16:17,170 --> 01:16:20,170 But since it's of interest, I will give you 1042 01:16:20,170 --> 01:16:23,070 more accurate numbers on Wednesday. 1043 01:16:23,070 --> 01:16:24,745 Any other questions? 1044 01:16:24,745 --> 01:16:25,245 Yes? 1045 01:16:25,245 --> 01:16:25,911 AUDIENCE: Sorry. 1046 01:16:25,911 --> 01:16:28,040 Could you explain the graph of the previous page, 1047 01:16:28,040 --> 01:16:29,992 how the lines are joined off? 1048 01:16:32,920 --> 01:16:34,090 PROFESSOR: The potential? 1049 01:16:34,090 --> 01:16:35,374 AUDIENCE: Yes. 1050 01:16:35,374 --> 01:16:36,040 PROFESSOR: Yeah. 1051 01:16:36,040 --> 01:16:41,900 So what happens is-- so if we're being cryptic, 1052 01:16:41,900 --> 01:16:44,540 this is the charge distribution. 1053 01:16:44,540 --> 01:16:47,710 And what I'm doing now is I'm solving Laplace equation. 1054 01:16:47,710 --> 01:16:50,000 I'm solving Laplace equation and integrating 1055 01:16:50,000 --> 01:16:51,720 from r equals infinity. 1056 01:16:51,720 --> 01:16:54,980 And as long as I'm outside the charge radius, 1057 01:16:54,980 --> 01:16:57,540 I get the 1 over r Coulomb potential. 1058 01:16:57,540 --> 01:17:01,370 But the moment I heat the surface of the nucleus, 1059 01:17:01,370 --> 01:17:04,600 I continue to indicate Laplace equation. 1060 01:17:04,600 --> 01:17:12,040 But what is now inside is a smaller and smaller charge. 1061 01:17:12,040 --> 01:17:14,280 I can also say I use a form of Gauss's law. 1062 01:17:14,280 --> 01:17:18,670 So I'm therefore not continuing on the 1 over r. 1063 01:17:18,670 --> 01:17:20,750 And if you look at Jackson, or if you 1064 01:17:20,750 --> 01:17:26,460 look at maybe a PSET you have solved, the 1 over r potential 1065 01:17:26,460 --> 01:17:29,260 becomes now a parabola. 1066 01:17:29,260 --> 01:17:31,270 And so what I wanted to sort of indicate here 1067 01:17:31,270 --> 01:17:34,360 is once you heat the surface of the nucleus, 1068 01:17:34,360 --> 01:17:38,030 you're not continuing on the 1 over r trajectory. 1069 01:17:38,030 --> 01:17:41,640 You have in a wave which is continuous and has 1070 01:17:41,640 --> 01:17:45,900 a continuous deriviative-- you have to fit in a parabola. 1071 01:17:45,900 --> 01:17:47,640 And for the heavier nucleus, this 1072 01:17:47,640 --> 01:17:51,460 leads us to this potential. 1073 01:17:51,460 --> 01:17:54,490 As for the lighter nucleus, the potential is deeper. 1074 01:17:54,490 --> 01:17:58,040 And that explains why the binding energy for heavier 1075 01:17:58,040 --> 01:18:00,140 nuclei is smaller than for lighter nuclei. 1076 01:18:02,930 --> 01:18:03,430 Yes? 1077 01:18:03,430 --> 01:18:05,406 AUDIENCE: So for the other orbitals, is it 1078 01:18:05,406 --> 01:18:07,382 exactly zero or [? near to ?] zero? 1079 01:18:10,346 --> 01:18:11,420 PROFESSOR: OK. 1080 01:18:11,420 --> 01:18:14,970 For the other orbitals, what we have to do 1081 01:18:14,970 --> 01:18:18,370 is-- and actually, you have a homework assignment 1082 01:18:18,370 --> 01:18:20,720 to do it for hydrogen and the proton. 1083 01:18:20,720 --> 01:18:23,530 You take the difference between the actual potential 1084 01:18:23,530 --> 01:18:26,630 and the Coulomb potential. 1085 01:18:26,630 --> 01:18:28,710 So you take the difference. 1086 01:18:28,710 --> 01:18:31,950 And the difference is your perturbation operator 1087 01:18:31,950 --> 01:18:34,230 for the finite size of the nucleus. 1088 01:18:34,230 --> 01:18:36,730 And now you take this perturbation operator 1089 01:18:36,730 --> 01:18:39,385 between your wave function and calculate the lowest order 1090 01:18:39,385 --> 01:18:42,230 of the expectation value. 1091 01:18:42,230 --> 01:18:44,900 For the s electron, you can immediately 1092 01:18:44,900 --> 01:18:49,280 factor out the probability for the electronic s equals 0. 1093 01:18:49,280 --> 01:18:53,430 But s-- we discussed, if you have orbital angular momentum, 1094 01:18:53,430 --> 01:18:56,510 let me just scribble it down here, 1095 01:18:56,510 --> 01:19:00,961 the wave function is proportional to r to the l. 1096 01:19:00,961 --> 01:19:04,730 So therefore, you have actually an r 1097 01:19:04,730 --> 01:19:13,320 to the l-- the wave function is exactly 0 only at r equals 0. 1098 01:19:13,320 --> 01:19:15,830 And then it slowly grows. 1099 01:19:15,830 --> 01:19:19,370 So therefore, given the finite size of the nucleus, 1100 01:19:19,370 --> 01:19:23,590 you will get a tiny, an absolutely tiny effect 1101 01:19:23,590 --> 01:19:26,180 if you integrate a wave function, r to the l, 1102 01:19:26,180 --> 01:19:29,290 over your perturbation operator. 1103 01:19:29,290 --> 01:19:31,520 So it's not mathematically zero. 1104 01:19:31,520 --> 01:19:35,300 But for all practical purposes, it vanishes. 1105 01:19:35,300 --> 01:19:37,328 Nancy. 1106 01:19:37,328 --> 01:19:40,590 AUDIENCE: For the different nuclei-- for example, 1107 01:19:40,590 --> 01:19:43,789 in rubidium-- is mass and volume affecting anything? 1108 01:19:43,789 --> 01:19:48,719 Or is [INAUDIBLE] structure also important in these isotopes? 1109 01:19:48,719 --> 01:19:52,170 Because the nuclear structure would be different [INAUDIBLE]. 1110 01:19:55,128 --> 01:19:56,390 PROFESSOR: Oh, yeah. 1111 01:19:56,390 --> 01:19:58,850 When we talk isotope effects, I was 1112 01:19:58,850 --> 01:20:01,360 talking about the isotope effects of mass and volume. 1113 01:20:01,360 --> 01:20:04,530 But different isotopes will, in general, 1114 01:20:04,530 --> 01:20:10,470 have different magnetic moments or different quadrupolar 1115 01:20:10,470 --> 01:20:11,370 deformations. 1116 01:20:11,370 --> 01:20:14,630 So what I discussed about hyperfine structure also 1117 01:20:14,630 --> 01:20:16,420 applies to isotopes. 1118 01:20:16,420 --> 01:20:19,140 I only separated it, because usually when 1119 01:20:19,140 --> 01:20:21,790 you don't have isotopes, you don't talk about, like, sodium. 1120 01:20:21,790 --> 01:20:24,300 Who has ever talked about the mass shift or volume 1121 01:20:24,300 --> 01:20:25,820 effect in sodium? 1122 01:20:25,820 --> 01:20:27,780 Usually you don't, because sodium 1123 01:20:27,780 --> 01:20:31,590 has 100% natural abundance in sodium-23. 1124 01:20:31,590 --> 01:20:33,890 So therefore, the hyperfine effects, 1125 01:20:33,890 --> 01:20:36,200 they lead to observable splittings 1126 01:20:36,200 --> 01:20:39,710 even if you have only one isotope. 1127 01:20:39,710 --> 01:20:42,490 But in general, yes, different isotopes 1128 01:20:42,490 --> 01:20:45,516 differ in all four effects-- the mass effect, the volume effect, 1129 01:20:45,516 --> 01:20:47,807 the deformation effect, and the magnetic moment effect. 1130 01:20:47,807 --> 01:20:52,101 AUDIENCE: [INAUDIBLE] mass and volume effects are used more 1131 01:20:52,101 --> 01:20:54,586 than [INAUDIBLE] splitting? 1132 01:20:54,586 --> 01:20:57,071 Because I find there is a different rate 1133 01:20:57,071 --> 01:21:00,053 for different isotopes. 1134 01:21:00,053 --> 01:21:02,041 Like, in the starting of the lecture, 1135 01:21:02,041 --> 01:21:04,526 you were talking about adding two positrons 1136 01:21:04,526 --> 01:21:06,017 to the alpha particles. 1137 01:21:06,017 --> 01:21:09,496 That would change [INAUDIBLE]. 1138 01:21:09,496 --> 01:21:11,510 PROFESSOR: Oh, yeah, this would change. 1139 01:21:11,510 --> 01:21:14,080 But you can separate those effects, 1140 01:21:14,080 --> 01:21:17,070 because I mentioned that in hyperfine structure, 1141 01:21:17,070 --> 01:21:20,560 the center of mass of the energy levels is the same. 1142 01:21:20,560 --> 01:21:24,180 So if you see a splitting but take the center of mass, 1143 01:21:24,180 --> 01:21:26,310 then the center of mass will only 1144 01:21:26,310 --> 01:21:28,800 depend on volume and mass effects. 1145 01:21:31,790 --> 01:21:32,290 OK. 1146 01:21:32,290 --> 01:21:35,560 Let's continue on Wednesday.