WEBVTT 00:00:00.000 --> 00:00:02.415 align:middle line:90% [SQUEAKING] 00:00:02.415 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:06.762 align:middle line:90% [CLICKING] 00:00:06.762 --> 00:00:10.630 align:middle line:90% 00:00:10.630 --> 00:00:13.720 align:middle line:84% JACK HARE: So today, and for the remainder of this semester, 00:00:13.720 --> 00:00:19.120 align:middle line:84% we will be discussing Thomson scattering. 00:00:19.120 --> 00:00:20.215 align:middle line:90% Please hold your applause. 00:00:20.215 --> 00:00:22.860 align:middle line:90% 00:00:22.860 --> 00:00:24.000 align:middle line:90% You won't like it. 00:00:24.000 --> 00:00:26.856 align:middle line:90% [LAUGHS] 00:00:26.856 --> 00:00:33.520 align:middle line:90% 00:00:33.520 --> 00:00:37.090 align:middle line:84% So Thomson scattering is an extremely powerful and subtle 00:00:37.090 --> 00:00:40.120 align:middle line:84% diagnostic, and it can be used over a huge range 00:00:40.120 --> 00:00:43.990 align:middle line:84% of different plasma parameters to provide local information 00:00:43.990 --> 00:00:45.830 align:middle line:90% about our plasma. 00:00:45.830 --> 00:00:48.550 align:middle line:84% And so that makes it extremely powerful in comparison 00:00:48.550 --> 00:00:50.860 align:middle line:84% to a large number of line integration techniques 00:00:50.860 --> 00:00:54.460 align:middle line:90% that we've seen so far. 00:00:54.460 --> 00:00:57.950 align:middle line:84% The basic idea of Thomson scattering is very simple. 00:00:57.950 --> 00:01:03.180 align:middle line:84% We have our vacuum chamber with some plasma inside that we 00:01:03.180 --> 00:01:05.590 align:middle line:90% would like to diagnose. 00:01:05.590 --> 00:01:11.290 align:middle line:84% And we take a laser beam, and we focus it inside the plasma. 00:01:11.290 --> 00:01:16.240 align:middle line:90% 00:01:16.240 --> 00:01:19.960 align:middle line:84% And the light from the laser scatters off 00:01:19.960 --> 00:01:21.850 align:middle line:84% the particles in the plasma, and it 00:01:21.850 --> 00:01:24.760 align:middle line:84% will scatter off in lots of different directions. 00:01:24.760 --> 00:01:28.760 align:middle line:90% 00:01:28.760 --> 00:01:34.380 align:middle line:84% Our job, then, is simply to place some sort of detector 00:01:34.380 --> 00:01:37.710 align:middle line:84% to look at this scattered light here. 00:01:37.710 --> 00:01:40.290 align:middle line:84% That detector is most likely to be 00:01:40.290 --> 00:01:41.730 align:middle line:90% something like a spectrometer. 00:01:41.730 --> 00:01:45.530 align:middle line:90% 00:01:45.530 --> 00:01:48.100 align:middle line:84% If we use a spectrometer, we can see 00:01:48.100 --> 00:01:50.170 align:middle line:84% that the initial laser that we put in, 00:01:50.170 --> 00:01:53.170 align:middle line:84% which we will choose to have a very narrow line width-- so it 00:01:53.170 --> 00:01:58.570 align:middle line:84% will be at just one wavelength or just one frequency-- 00:01:58.570 --> 00:02:06.050 align:middle line:84% that narrow line width will undergo Doppler 00:02:06.050 --> 00:02:08.330 align:middle line:84% drift due to the velocity of the plasma. 00:02:08.330 --> 00:02:17.210 align:middle line:84% And it will undergo Doppler broadening 00:02:17.210 --> 00:02:20.120 align:middle line:84% due to the thermal motion of the plasma. 00:02:20.120 --> 00:02:23.390 align:middle line:84% So this will give us out two key quantities. 00:02:23.390 --> 00:02:26.290 align:middle line:90% 00:02:26.290 --> 00:02:28.000 align:middle line:90% Cool. 00:02:28.000 --> 00:02:29.490 align:middle line:90% Could you mute, please, Yang? 00:02:29.490 --> 00:02:32.130 align:middle line:90% 00:02:32.130 --> 00:02:33.510 align:middle line:90% AUDIENCE: Yes, I can hear you. 00:02:33.510 --> 00:02:34.920 align:middle line:84% JACK HARE: Yes, but could you mute yourself? 00:02:34.920 --> 00:02:36.995 align:middle line:84% Because we can hear something in the background. 00:02:36.995 --> 00:02:38.220 align:middle line:90% AUDIENCE: Oh, sorry. 00:02:38.220 --> 00:02:39.512 align:middle line:90% JACK HARE: Thank you very much. 00:02:39.512 --> 00:02:44.770 align:middle line:90% 00:02:44.770 --> 00:02:46.045 align:middle line:90% Ah, tediousness of the Zoom. 00:02:46.045 --> 00:02:54.470 align:middle line:90% 00:02:54.470 --> 00:02:57.200 align:middle line:90% OK, good. 00:02:57.200 --> 00:02:59.630 align:middle line:84% And so this is going to give us things 00:02:59.630 --> 00:03:02.000 align:middle line:84% like the flow velocity within the plasma 00:03:02.000 --> 00:03:05.930 align:middle line:84% and the temperature of the plasma-- so the ion temperature 00:03:05.930 --> 00:03:07.650 align:middle line:90% and the electron temperature. 00:03:07.650 --> 00:03:09.560 align:middle line:90% And so this seems simple enough. 00:03:09.560 --> 00:03:11.700 align:middle line:84% And yet we're going to spend four lectures on it. 00:03:11.700 --> 00:03:13.542 align:middle line:84% And so it is not particularly simple at all. 00:03:13.542 --> 00:03:15.500 align:middle line:84% And, in fact, if you keep analyzing the Thomson 00:03:15.500 --> 00:03:18.020 align:middle line:84% scattering, you can also get out quantities 00:03:18.020 --> 00:03:22.670 align:middle line:84% like the average ionization state of the ions. 00:03:22.670 --> 00:03:25.610 align:middle line:84% We can get out the electron density. 00:03:25.610 --> 00:03:31.010 align:middle line:84% We can even start measuring the local current within the plasma. 00:03:31.010 --> 00:03:35.880 align:middle line:84% We can measure the magnetic field locally within the plasma. 00:03:35.880 --> 00:03:40.580 align:middle line:84% And we may be able to get out the full electron and ion 00:03:40.580 --> 00:03:44.270 align:middle line:84% distribution function, which is particularly important when 00:03:44.270 --> 00:03:47.642 align:middle line:84% these distribution functions are not Maxwellian. 00:03:47.642 --> 00:03:51.086 align:middle line:90% 00:03:51.086 --> 00:03:52.625 align:middle line:84% If they were Maxwellian, of course, 00:03:52.625 --> 00:03:54.250 align:middle line:84% they would fully characterized, once we 00:03:54.250 --> 00:03:58.730 align:middle line:84% knew the temperature and the flow velocity and the density. 00:03:58.730 --> 00:03:59.450 align:middle line:90% Right. 00:03:59.450 --> 00:04:01.910 align:middle line:84% So the nice thing about Thomson scattering, 00:04:01.910 --> 00:04:05.090 align:middle line:84% from my point of view, is it unites a lot of plasma physics. 00:04:05.090 --> 00:04:08.520 align:middle line:90% 00:04:08.520 --> 00:04:15.260 align:middle line:84% We have electromagnetism that we need to understand to do this. 00:04:15.260 --> 00:04:26.620 align:middle line:84% We need this to get the particle orbits and the radiation 00:04:26.620 --> 00:04:30.810 align:middle line:90% from these particles. 00:04:30.810 --> 00:04:34.320 align:middle line:84% We need to understand our statistical mechanics. 00:04:34.320 --> 00:04:37.250 align:middle line:90% 00:04:37.250 --> 00:04:45.970 align:middle line:84% In particular, we need kinetic theory 00:04:45.970 --> 00:04:48.490 align:middle line:84% because, although we're going to end up 00:04:48.490 --> 00:04:51.100 align:middle line:84% a lot of the time with gross fluid quantities 00:04:51.100 --> 00:04:54.190 align:middle line:84% like temperature and flow velocity, in order to get them, 00:04:54.190 --> 00:04:58.390 align:middle line:84% we still need to first do the theory using distribution 00:04:58.390 --> 00:05:00.820 align:middle line:84% functions, even if those distribution functions turn out 00:05:00.820 --> 00:05:02.880 align:middle line:90% to be Maxwellian. 00:05:02.880 --> 00:05:05.880 align:middle line:84% And, of course, this involves lasers. 00:05:05.880 --> 00:05:10.300 align:middle line:84% Lasers are just simply very cool. 00:05:10.300 --> 00:05:13.005 align:middle line:90% It also unites a lot of plasmas. 00:05:13.005 --> 00:05:16.290 align:middle line:90% 00:05:16.290 --> 00:05:19.002 align:middle line:84% So I'll give you some ideas of the sorts of plasmas 00:05:19.002 --> 00:05:20.210 align:middle line:90% that we can measure stuff in. 00:05:20.210 --> 00:05:23.390 align:middle line:84% If I have a little table here with the electron temperature 00:05:23.390 --> 00:05:25.730 align:middle line:84% and the electron density, we can start out 00:05:25.730 --> 00:05:28.730 align:middle line:84% with a pretty low-temperature plasma, 1 electron-volt, 00:05:28.730 --> 00:05:33.720 align:middle line:84% and also a pretty low density, 10 to 17 per meter cubed. 00:05:33.720 --> 00:05:38.370 align:middle line:84% This sort of density could be a low-temperature plasma used, 00:05:38.370 --> 00:05:42.260 align:middle line:84% for example, for converting carbon dioxide into carbon 00:05:42.260 --> 00:05:46.130 align:middle line:84% monoxide and oxygen. This is an interesting plasma-catalyzed 00:05:46.130 --> 00:05:47.980 align:middle line:90% process. 00:05:47.980 --> 00:05:50.020 align:middle line:84% It could be used for doing carbon capture. 00:05:50.020 --> 00:05:54.290 align:middle line:90% 00:05:54.290 --> 00:05:57.560 align:middle line:84% The plasma catalysis here is because the electron 00:05:57.560 --> 00:06:00.322 align:middle line:84% distribution function is non-Maxwellian. 00:06:00.322 --> 00:06:03.700 align:middle line:90% 00:06:03.700 --> 00:06:05.410 align:middle line:84% And it's actually the fast electrons 00:06:05.410 --> 00:06:08.470 align:middle line:84% in the tail of this distribution which do the plasma catalysis 00:06:08.470 --> 00:06:10.360 align:middle line:84% process and make it thermodynamically 00:06:10.360 --> 00:06:11.890 align:middle line:90% and economically feasible. 00:06:11.890 --> 00:06:13.607 align:middle line:84% But in these sorts of plasmas, people 00:06:13.607 --> 00:06:15.940 align:middle line:84% have been able to characterize the non-Maxwellian nature 00:06:15.940 --> 00:06:18.460 align:middle line:84% of this distribution function extremely well 00:06:18.460 --> 00:06:20.400 align:middle line:84% by repetitively pulsing this plasma 00:06:20.400 --> 00:06:22.150 align:middle line:84% and doing Thomson scattering and acquiring 00:06:22.150 --> 00:06:24.430 align:middle line:90% signals over hours or days. 00:06:24.430 --> 00:06:26.470 align:middle line:84% And so even in this very low-density, 00:06:26.470 --> 00:06:28.360 align:middle line:84% low-temperature plasma, Thomson scattering 00:06:28.360 --> 00:06:30.080 align:middle line:84% has given us a lot of information. 00:06:30.080 --> 00:06:31.990 align:middle line:84% We could go up to a plasma which is 00:06:31.990 --> 00:06:35.830 align:middle line:84% much hotter, 6 KeV, and a density of around about 10 00:06:35.830 --> 00:06:37.790 align:middle line:90% to the 20. 00:06:37.790 --> 00:06:40.200 align:middle line:90% This will be a plasma-- 00:06:40.200 --> 00:06:45.050 align:middle line:84% the specific example here is TFTR, which was a fusion reactor 00:06:45.050 --> 00:06:46.790 align:middle line:90% prototype at Princeton. 00:06:46.790 --> 00:06:49.400 align:middle line:90% And this was a D-T plasma. 00:06:49.400 --> 00:06:51.917 align:middle line:90% 00:06:51.917 --> 00:06:54.250 align:middle line:84% And it produced a significant number of alpha particles. 00:06:54.250 --> 00:06:55.958 align:middle line:84% There was quite a lot of fusion going on. 00:06:55.958 --> 00:06:59.830 align:middle line:84% It got to very high temperatures and moderately high densities. 00:06:59.830 --> 00:07:08.810 align:middle line:84% Here they did time-resolved Thomson scattering. 00:07:08.810 --> 00:07:10.970 align:middle line:84% And they were able to image an event called 00:07:10.970 --> 00:07:16.190 align:middle line:84% a sawtooth crash, which is a catastrophic loss of confinement 00:07:16.190 --> 00:07:17.390 align:middle line:90% due to reconnection. 00:07:17.390 --> 00:07:19.872 align:middle line:84% And they made a movie of this, effectively 00:07:19.872 --> 00:07:21.080 align:middle line:90% using the Thomson scattering. 00:07:21.080 --> 00:07:24.230 align:middle line:84% So they were able to better understand this process. 00:07:24.230 --> 00:07:27.830 align:middle line:84% And then we can go to similar temperatures here 00:07:27.830 --> 00:07:29.990 align:middle line:90% but much higher densities-- 00:07:29.990 --> 00:07:32.730 align:middle line:90% 10 to the 27 per meter cubed. 00:07:32.730 --> 00:07:35.810 align:middle line:84% And this is a measurement using Thomson scattering 00:07:35.810 --> 00:07:40.430 align:middle line:84% inside an inertial confinement fusion hohlraum, which 00:07:40.430 --> 00:07:44.120 align:middle line:84% is that little gold cylinder that has 192 laser 00:07:44.120 --> 00:07:47.030 align:middle line:84% beams hitting the inside of it to produce an X-ray bath. 00:07:47.030 --> 00:07:53.930 align:middle line:84% And it was in this hohlraum with all the lasers coming in. 00:07:53.930 --> 00:07:56.983 align:middle line:84% There's a large amount of plasma on the walls. 00:07:56.983 --> 00:07:58.900 align:middle line:84% Remember, the eventual goal is that the X-rays 00:07:58.900 --> 00:07:59.960 align:middle line:90% compress this capsule. 00:07:59.960 --> 00:08:02.500 align:middle line:84% But the plasma on the walls is pretty hard to diagnose. 00:08:02.500 --> 00:08:05.125 align:middle line:84% But folks were able to use Thomson scattering 00:08:05.125 --> 00:08:07.190 align:middle line:84% to diagnose the temperature and density here, 00:08:07.190 --> 00:08:08.650 align:middle line:84% which is very important if you want 00:08:08.650 --> 00:08:12.020 align:middle line:84% to know what your X-ray flux is going to be onto the capsule. 00:08:12.020 --> 00:08:16.000 align:middle line:84% So this is 10 orders of magnitude in terms of density, 00:08:16.000 --> 00:08:19.100 align:middle line:84% but it is fundamentally the same technique. 00:08:19.100 --> 00:08:21.710 align:middle line:84% So how are we going to tackle this? 00:08:21.710 --> 00:08:23.680 align:middle line:84% Oh, no, I've got another bit of reasons 00:08:23.680 --> 00:08:25.555 align:middle line:84% why you should care about Thomson scattering. 00:08:25.555 --> 00:08:27.040 align:middle line:90% Let's keep going. 00:08:27.040 --> 00:08:29.770 align:middle line:84% OK, so Thomson scattering also changed the course 00:08:29.770 --> 00:08:31.840 align:middle line:90% of fusion research forever. 00:08:31.840 --> 00:08:39.520 align:middle line:90% 00:08:39.520 --> 00:08:43.570 align:middle line:84% Now, I may have hopped on about this a little bit already. 00:08:43.570 --> 00:08:45.970 align:middle line:84% Those of you who took 2262 with me 00:08:45.970 --> 00:08:49.870 align:middle line:84% will remember the famous or infamous ZETA 00:08:49.870 --> 00:08:52.280 align:middle line:90% Reactor which was in the UK. 00:08:52.280 --> 00:08:56.050 align:middle line:84% So this was back in the '50s or so. 00:08:56.050 --> 00:08:59.790 align:middle line:84% 1958, they say they've got fusion. 00:08:59.790 --> 00:09:02.787 align:middle line:90% 00:09:02.787 --> 00:09:04.370 align:middle line:84% It turns out that the reason that they 00:09:04.370 --> 00:09:06.020 align:middle line:84% think they saw fusion is they didn't 00:09:06.020 --> 00:09:07.340 align:middle line:90% understand their diagnostics. 00:09:07.340 --> 00:09:12.300 align:middle line:84% So this is poor diagnostic interpretation. 00:09:12.300 --> 00:09:14.190 align:middle line:84% They interpreted their neutron spectra 00:09:14.190 --> 00:09:17.700 align:middle line:84% as being due to isotropic neutrons when, in fact, it 00:09:17.700 --> 00:09:18.960 align:middle line:90% was beam-target fusion. 00:09:18.960 --> 00:09:27.420 align:middle line:90% 00:09:27.420 --> 00:09:29.520 align:middle line:84% This, of course, was rather humiliating. 00:09:29.520 --> 00:09:33.580 align:middle line:90% 00:09:33.580 --> 00:09:35.620 align:middle line:84% But it did lead to a rather good result, 00:09:35.620 --> 00:09:41.280 align:middle line:84% which was the development of Thomson scattering, 00:09:41.280 --> 00:09:48.360 align:middle line:84% because it was clear that we needed much better diagnostics 00:09:48.360 --> 00:09:50.550 align:middle line:84% in order to measure our plasma conditions 00:09:50.550 --> 00:09:52.557 align:middle line:84% and actually show that fusion is happening. 00:09:52.557 --> 00:09:55.140 align:middle line:84% We need to know if the plasma is even hot enough to do fusion. 00:09:55.140 --> 00:09:57.120 align:middle line:84% The problem with the ZETA machine 00:09:57.120 --> 00:09:58.890 align:middle line:84% is it was far too cold anyway, wherever 00:09:58.890 --> 00:09:59.790 align:middle line:90% we've got fusion happening. 00:09:59.790 --> 00:10:00.990 align:middle line:84% And the British should have realized this 00:10:00.990 --> 00:10:03.540 align:middle line:84% before they claimed that fusion was occurring here. 00:10:03.540 --> 00:10:06.180 align:middle line:84% But they did develop this Thomson scattering diagnostic 00:10:06.180 --> 00:10:07.560 align:middle line:90% with some of the first lasers. 00:10:07.560 --> 00:10:10.225 align:middle line:84% I'll point out that the laser was just invented in 1961. 00:10:10.225 --> 00:10:12.600 align:middle line:84% So they didn't really have a chance to do it before then. 00:10:12.600 --> 00:10:15.520 align:middle line:84% But after then, there was really no excuse. 00:10:15.520 --> 00:10:18.490 align:middle line:84% And this was very handy because only a few years later, 00:10:18.490 --> 00:10:22.890 align:middle line:90% there was a new device, T3. 00:10:22.890 --> 00:10:25.170 align:middle line:90% This was in the USSR. 00:10:25.170 --> 00:10:29.550 align:middle line:84% And it was something that they were calling a tokamak. 00:10:29.550 --> 00:10:32.310 align:middle line:84% Some of you may have heard of this. 00:10:32.310 --> 00:10:37.470 align:middle line:84% And in 1968, these guys claimed that they got temperatures 00:10:37.470 --> 00:10:41.010 align:middle line:84% in excess of 1 kiloelectron volt, which 00:10:41.010 --> 00:10:44.770 align:middle line:84% was significantly better than any other machine at the time. 00:10:44.770 --> 00:10:46.050 align:middle line:90% So this was really remarkable. 00:10:46.050 --> 00:10:48.300 align:middle line:84% And, of course, there was a huge amount of skepticism, 00:10:48.300 --> 00:10:51.630 align:middle line:84% both from the UK, who felt a little bit burnt after the ZETA 00:10:51.630 --> 00:10:53.460 align:middle line:90% incident, and also from the US. 00:10:53.460 --> 00:10:55.920 align:middle line:84% And so there were needed to be a way 00:10:55.920 --> 00:10:59.930 align:middle line:84% to verify the claim of the Soviets. 00:10:59.930 --> 00:11:02.540 align:middle line:84% And what happened was kind of a remarkable piece of Cold War 00:11:02.540 --> 00:11:08.360 align:middle line:84% history, which was a team from the UK went to the USSR 00:11:08.360 --> 00:11:11.810 align:middle line:84% with their Thomson scattering diagnostic in a load of crates, 00:11:11.810 --> 00:11:15.680 align:middle line:84% consisting of a laser and a load of spectrometers here. 00:11:15.680 --> 00:11:19.820 align:middle line:84% And they used it to confirm these temperatures. 00:11:19.820 --> 00:11:23.380 align:middle line:90% 00:11:23.380 --> 00:11:25.000 align:middle line:84% And there's a beautiful Nature paper 00:11:25.000 --> 00:11:28.420 align:middle line:84% by Peacock, the name of the lead scientist on this group, 00:11:28.420 --> 00:11:31.690 align:middle line:84% showing the Thomson scattering spectra from the T3 tokamak. 00:11:31.690 --> 00:11:33.550 align:middle line:84% And this caused an absolute firestorm 00:11:33.550 --> 00:11:35.500 align:middle line:90% across the world of fusion. 00:11:35.500 --> 00:11:38.600 align:middle line:84% Instantly, everyone wanted to build tokamaks instead. 00:11:38.600 --> 00:11:41.410 align:middle line:84% So at the Princeton Plasma Physics Lab, 00:11:41.410 --> 00:11:52.640 align:middle line:84% they had the model C stellarator, 00:11:52.640 --> 00:11:57.520 align:middle line:84% which sort of looks like this kind of racetrack-shaped thing, 00:11:57.520 --> 00:11:59.020 align:middle line:90% like this. 00:11:59.020 --> 00:12:03.040 align:middle line:84% And within one year, they'd gone, all right, 00:12:03.040 --> 00:12:04.940 align:middle line:90% get rid of that. 00:12:04.940 --> 00:12:06.010 align:middle line:90% Now look. 00:12:06.010 --> 00:12:06.730 align:middle line:90% It's a tokamak. 00:12:06.730 --> 00:12:08.500 align:middle line:84% They converted the machine into a tokamak 00:12:08.500 --> 00:12:11.270 align:middle line:84% and immediately started to get very good performance out of it. 00:12:11.270 --> 00:12:14.050 align:middle line:84% So if you work on tokamaks, which many of you do, 00:12:14.050 --> 00:12:15.790 align:middle line:84% you should thank Thomson scattering. 00:12:15.790 --> 00:12:18.610 align:middle line:84% And it's the reason why everyone was so convinced straight away 00:12:18.610 --> 00:12:19.900 align:middle line:90% that they were the way to go. 00:12:19.900 --> 00:12:20.733 align:middle line:90% All right, question? 00:12:20.733 --> 00:12:23.300 align:middle line:90% 00:12:23.300 --> 00:12:25.630 align:middle line:84% AUDIENCE: What diagnostics did they [AUDIO OUT]? 00:12:25.630 --> 00:12:27.787 align:middle line:84% JACK HARE: Mostly neutron diagnostics. 00:12:27.787 --> 00:12:30.370 align:middle line:84% They had neutron detectors, and they were looking for neutrons 00:12:30.370 --> 00:12:31.870 align:middle line:84% because they were like, if we have neutrons, 00:12:31.870 --> 00:12:32.830 align:middle line:90% there must be fusion. 00:12:32.830 --> 00:12:34.580 align:middle line:84% And there were fusion reactions happening, 00:12:34.580 --> 00:12:37.330 align:middle line:84% but they were beam-target fusion reactions caused by [INAUDIBLE]. 00:12:37.330 --> 00:12:39.280 align:middle line:90% They were not isotropic. 00:12:39.280 --> 00:12:41.380 align:middle line:84% If they had more neutron detectors, 00:12:41.380 --> 00:12:43.090 align:middle line:84% they could have detected the anisotropy, 00:12:43.090 --> 00:12:45.465 align:middle line:84% and that would have been a key signature that they didn't 00:12:45.465 --> 00:12:46.550 align:middle line:90% have thermonuclear fusion. 00:12:46.550 --> 00:12:47.440 align:middle line:90% Yeah? 00:12:47.440 --> 00:12:49.114 align:middle line:90% AUDIENCE: Did the [INAUDIBLE]? 00:12:49.114 --> 00:12:53.320 align:middle line:90% 00:12:53.320 --> 00:12:55.550 align:middle line:84% JACK HARE: Well, I mean, I don't know. 00:12:55.550 --> 00:12:58.240 align:middle line:84% But I do know that Peacock wrote a book about it 00:12:58.240 --> 00:13:01.370 align:middle line:84% called Lasers over the Cherry Orchard or something like that, 00:13:01.370 --> 00:13:03.650 align:middle line:84% which I've not read but I'm told is quite good. 00:13:03.650 --> 00:13:04.730 align:middle line:84% And I should read it at some point. 00:13:04.730 --> 00:13:06.188 align:middle line:84% So maybe he would have more insight 00:13:06.188 --> 00:13:10.430 align:middle line:84% than I do into geopolitics in 1968. 00:13:10.430 --> 00:13:13.980 align:middle line:90% OK, other questions? 00:13:13.980 --> 00:13:14.480 align:middle line:90% Good. 00:13:14.480 --> 00:13:15.938 align:middle line:84% I haven't really said anything yet. 00:13:15.938 --> 00:13:17.260 align:middle line:90% So let's do some math. 00:13:17.260 --> 00:13:19.740 align:middle line:90% Is it math next? 00:13:19.740 --> 00:13:20.690 align:middle line:90% Yes. 00:13:20.690 --> 00:13:21.470 align:middle line:90% OK, good. 00:13:21.470 --> 00:13:25.060 align:middle line:90% 00:13:25.060 --> 00:13:29.700 align:middle line:84% OK, so I gave you a very handwavy explanation of it 00:13:29.700 --> 00:13:31.350 align:middle line:90% through plasmas and lasers. 00:13:31.350 --> 00:13:34.080 align:middle line:84% Let's break this down to just a single charge. 00:13:34.080 --> 00:13:34.800 align:middle line:90% Here it is. 00:13:34.800 --> 00:13:37.930 align:middle line:90% 00:13:37.930 --> 00:13:40.460 align:middle line:84% And this charge is initially at rest. 00:13:40.460 --> 00:13:42.650 align:middle line:90% And there's some wave packet. 00:13:42.650 --> 00:13:45.650 align:middle line:84% So think of this as roughly a plane wave, 00:13:45.650 --> 00:13:48.935 align:middle line:84% but I'm just going to draw it as a wave packet. 00:13:48.935 --> 00:13:52.520 align:middle line:84% It increases in amplitude and decreases in amplitude again. 00:13:52.520 --> 00:13:54.980 align:middle line:84% And this wave packet is traveling along 00:13:54.980 --> 00:13:56.130 align:middle line:90% in this direction. 00:13:56.130 --> 00:13:57.920 align:middle line:84% So this is our laser beam going through. 00:13:57.920 --> 00:14:00.740 align:middle line:84% It's got some finite extent because the pulse of the laser 00:14:00.740 --> 00:14:02.000 align:middle line:90% beam has some finite width. 00:14:02.000 --> 00:14:04.208 align:middle line:84% There would obviously be many, many more oscillations 00:14:04.208 --> 00:14:05.557 align:middle line:90% within the wave packet. 00:14:05.557 --> 00:14:07.140 align:middle line:84% But then you won't be able to see them 00:14:07.140 --> 00:14:09.730 align:middle line:90% when I drew them on the board. 00:14:09.730 --> 00:14:24.990 align:middle line:84% So just to be clear, particle, and this is an EM wave. 00:14:24.990 --> 00:14:28.250 align:middle line:84% So, for example, this could be electric field here. 00:14:28.250 --> 00:14:32.090 align:middle line:90% 00:14:32.090 --> 00:14:34.670 align:middle line:84% What does this particle do in this electric field? 00:14:34.670 --> 00:14:37.510 align:middle line:90% 00:14:37.510 --> 00:14:38.450 align:middle line:90% Yeah? 00:14:38.450 --> 00:14:39.610 align:middle line:90% It will oscillate. 00:14:39.610 --> 00:14:41.740 align:middle line:84% This particle will see the electric field going up 00:14:41.740 --> 00:14:43.690 align:middle line:90% and down, and it will join it. 00:14:43.690 --> 00:14:46.330 align:middle line:84% It will continue going up and down. 00:14:46.330 --> 00:14:50.980 align:middle line:84% OK, now let's imagine some time a little bit later, 00:14:50.980 --> 00:14:55.840 align:middle line:84% we now have our particle oscillating still. 00:14:55.840 --> 00:14:58.240 align:middle line:84% And the wave packet has passed on. 00:14:58.240 --> 00:15:01.425 align:middle line:90% 00:15:01.425 --> 00:15:02.800 align:middle line:84% What does this particle do there? 00:15:02.800 --> 00:15:08.572 align:middle line:90% 00:15:08.572 --> 00:15:10.510 align:middle line:90% Hmm? 00:15:10.510 --> 00:15:11.830 align:middle line:90% It's an accelerating charge. 00:15:11.830 --> 00:15:13.695 align:middle line:84% So it will continue to radiate away. 00:15:13.695 --> 00:15:17.892 align:middle line:90% 00:15:17.892 --> 00:15:19.350 align:middle line:84% Presumably, it'll be radiating away 00:15:19.350 --> 00:15:23.300 align:middle line:90% little wave packets of its own. 00:15:23.300 --> 00:15:24.830 align:middle line:90% Yeah? 00:15:24.830 --> 00:15:26.925 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:15:26.925 --> 00:15:29.300 align:middle line:84% JACK HARE: Yeah, I guess we're still dealing with a state 00:15:29.300 --> 00:15:31.290 align:middle line:84% where there is still some electric field trailing off. 00:15:31.290 --> 00:15:33.790 align:middle line:84% So this is radiating while the electric field is going past. 00:15:33.790 --> 00:15:39.180 align:middle line:90% 00:15:39.180 --> 00:15:40.770 align:middle line:84% But because it's being accelerated 00:15:40.770 --> 00:15:42.630 align:middle line:84% by the electric field, it will be emitted. 00:15:42.630 --> 00:15:45.390 align:middle line:84% So this is the geometry of the problem 00:15:45.390 --> 00:15:46.830 align:middle line:84% that we're going to consider here. 00:15:46.830 --> 00:15:48.900 align:middle line:84% We're going to consider just plane waves here 00:15:48.900 --> 00:15:51.367 align:middle line:84% because it's much easier than doing the Fourier-- 00:15:51.367 --> 00:15:53.700 align:middle line:84% well, it's much easier when we go into doing the Fourier 00:15:53.700 --> 00:15:54.200 align:middle line:90% analysis. 00:15:54.200 --> 00:15:56.320 align:middle line:84% So we're just going to consider some plane waves. 00:15:56.320 --> 00:15:59.820 align:middle line:84% We're going to consider a plane wave coming in. 00:15:59.820 --> 00:16:05.490 align:middle line:84% And this plane wave has some wave vector Ki, which 00:16:05.490 --> 00:16:08.400 align:middle line:90% is in the i hat direction. 00:16:08.400 --> 00:16:12.580 align:middle line:84% That is just simply Ki divided by the size of Ki. 00:16:12.580 --> 00:16:17.990 align:middle line:84% And there's a load of parallel wavefronts like this. 00:16:17.990 --> 00:16:21.540 align:middle line:84% So these are just constant phase. 00:16:21.540 --> 00:16:23.960 align:middle line:90% 00:16:23.960 --> 00:16:25.710 align:middle line:84% You can't get away with a geometric optics 00:16:25.710 --> 00:16:27.920 align:middle line:90% picture of Thomson scattering. 00:16:27.920 --> 00:16:29.030 align:middle line:90% We've got some particle. 00:16:29.030 --> 00:16:33.470 align:middle line:90% 00:16:33.470 --> 00:16:36.920 align:middle line:84% And this particle is moving in some direction 00:16:36.920 --> 00:16:39.515 align:middle line:84% with a velocity v, which is a vector. 00:16:39.515 --> 00:16:42.120 align:middle line:90% 00:16:42.120 --> 00:16:43.770 align:middle line:84% And it's doing it at a time t prime. 00:16:43.770 --> 00:16:46.710 align:middle line:84% We'll get back to this t prime in a moment. 00:16:46.710 --> 00:16:50.040 align:middle line:84% There's some subtleties here that we have to deal with. 00:16:50.040 --> 00:16:52.190 align:middle line:84% We also need to have an origin in our system. 00:16:52.190 --> 00:16:54.680 align:middle line:84% This is the point where we measure 00:16:54.680 --> 00:16:57.420 align:middle line:84% all of our displacement vectors from. 00:16:57.420 --> 00:17:01.780 align:middle line:84% And so we can say that this particle is at some distance. 00:17:01.780 --> 00:17:04.060 align:middle line:84% I'm going to draw it somewhere slightly different so I 00:17:04.060 --> 00:17:07.090 align:middle line:84% don't have an accidental coincidence between my vectors. 00:17:07.090 --> 00:17:10.910 align:middle line:90% 00:17:10.910 --> 00:17:16.560 align:middle line:84% This particle is at some position r t prime, like this. 00:17:16.560 --> 00:17:19.250 align:middle line:84% So this specifies this single particle very well, 00:17:19.250 --> 00:17:23.520 align:middle line:84% and this specifies our incoming electromagnetic wave. 00:17:23.520 --> 00:17:24.930 align:middle line:84% Now, we assume that there's going 00:17:24.930 --> 00:17:27.220 align:middle line:84% to be scattering in lots of different directions. 00:17:27.220 --> 00:17:30.060 align:middle line:84% We haven't exactly solved where all the scattered light is 00:17:30.060 --> 00:17:31.942 align:middle line:84% going to go to, but maybe it goes lots 00:17:31.942 --> 00:17:33.150 align:middle line:90% of different directions here. 00:17:33.150 --> 00:17:34.680 align:middle line:84% So we're just going to consider what 00:17:34.680 --> 00:17:40.810 align:middle line:84% happens if we have an observer down here looking 00:17:40.810 --> 00:17:42.640 align:middle line:84% at the scattered light from this particle. 00:17:42.640 --> 00:17:45.190 align:middle line:84% Well, if we have an observer, that 00:17:45.190 --> 00:17:49.000 align:middle line:84% means that the light that we are seeing 00:17:49.000 --> 00:17:53.500 align:middle line:84% must be heading towards it with a wave vector Ks that 00:17:53.500 --> 00:17:56.270 align:middle line:90% is in the s hat direction. 00:17:56.270 --> 00:17:59.920 align:middle line:84% And these are more plane waves coming towards our observer. 00:17:59.920 --> 00:18:05.040 align:middle line:84% And we also need a position vector for our observer. 00:18:05.040 --> 00:18:08.570 align:middle line:84% And that is a position capital R. 00:18:08.570 --> 00:18:11.107 align:middle line:84% We can assume that the observer is stationary in time. 00:18:11.107 --> 00:18:13.190 align:middle line:84% And so I don't need to put a time coordinate here. 00:18:13.190 --> 00:18:14.960 align:middle line:84% We're not going to move our spectrometer 00:18:14.960 --> 00:18:18.080 align:middle line:90% during this experiment. 00:18:18.080 --> 00:18:22.890 align:middle line:84% And there also has to be a vector which joins the particle 00:18:22.890 --> 00:18:24.060 align:middle line:90% through the observer here. 00:18:24.060 --> 00:18:27.040 align:middle line:84% And that is just capital R prime, like that. 00:18:27.040 --> 00:18:30.660 align:middle line:84% And so just from standard vector algebra, 00:18:30.660 --> 00:18:38.390 align:middle line:84% we have R equals R prime plus r of t prime 00:18:38.390 --> 00:18:42.323 align:middle line:84% is just forming this little triangle here. 00:18:42.323 --> 00:18:44.490 align:middle line:84% And now we need to think a little bit about the time 00:18:44.490 --> 00:18:46.860 align:middle line:84% because when the observer sees things, 00:18:46.860 --> 00:18:49.110 align:middle line:90% it's happening at a time t. 00:18:49.110 --> 00:18:51.420 align:middle line:84% But because it's taken the light some time 00:18:51.420 --> 00:18:53.550 align:middle line:84% to get from the particle to the observer, 00:18:53.550 --> 00:18:56.820 align:middle line:84% the observer is observing the state of the particle at a time 00:18:56.820 --> 00:18:57.870 align:middle line:90% t prime. 00:18:57.870 --> 00:19:03.420 align:middle line:84% And that time t prime is t, but it's earlier by a factor of R 00:19:03.420 --> 00:19:05.820 align:middle line:90% prime upon c, like this. 00:19:05.820 --> 00:19:07.320 align:middle line:84% So this is just simply the distance. 00:19:07.320 --> 00:19:09.240 align:middle line:84% Whenever I drop the vector notation, 00:19:09.240 --> 00:19:11.650 align:middle line:84% I'm just taking the size of the vector here. 00:19:11.650 --> 00:19:14.170 align:middle line:90% So this is no longer a vector. 00:19:14.170 --> 00:19:18.260 align:middle line:84% OK, and this is often called the retarded time. 00:19:18.260 --> 00:19:26.700 align:middle line:90% 00:19:26.700 --> 00:19:28.685 align:middle line:84% We have sort of seen this setup before when 00:19:28.685 --> 00:19:30.810 align:middle line:84% we're talking about radiation from a moving charge. 00:19:30.810 --> 00:19:32.700 align:middle line:84% I just want to run through it again 00:19:32.700 --> 00:19:34.575 align:middle line:84% because it's very important for understanding 00:19:34.575 --> 00:19:35.710 align:middle line:90% Thomson scattering. 00:19:35.710 --> 00:19:40.320 align:middle line:84% So what we do at this point is we write down two 00:19:40.320 --> 00:19:42.300 align:middle line:90% of Maxwell's equations again-- 00:19:42.300 --> 00:19:51.280 align:middle line:84% curl of curl of E plus 1 upon c squared, 00:19:51.280 --> 00:19:54.940 align:middle line:84% second partial derivative of the electric field with respect 00:19:54.940 --> 00:19:56.060 align:middle line:90% to time. 00:19:56.060 --> 00:20:02.330 align:middle line:84% And this is equal to minus mu 0, partial J, 00:20:02.330 --> 00:20:06.200 align:middle line:90% the electric current, partial t. 00:20:06.200 --> 00:20:10.340 align:middle line:84% Normally, when we're solving Maxwell's equations for light 00:20:10.340 --> 00:20:13.490 align:middle line:84% in a vacuum, we just get rid of this left-hand-- right-hand side 00:20:13.490 --> 00:20:16.190 align:middle line:84% and set it to 0 because there's no current in a vacuum. 00:20:16.190 --> 00:20:18.710 align:middle line:84% But now we're dealing with a plasma here. 00:20:18.710 --> 00:20:21.120 align:middle line:84% And the current in this system here-- 00:20:21.120 --> 00:20:22.920 align:middle line:84% well, there's only a single particle. 00:20:22.920 --> 00:20:29.925 align:middle line:84% So this J is simply equal to qv t. 00:20:29.925 --> 00:20:33.190 align:middle line:84% And I guess we should really have a delta function saying 00:20:33.190 --> 00:20:37.445 align:middle line:84% that the current is localized at that point there. 00:20:37.445 --> 00:20:40.170 align:middle line:84% I haven't specified whether this is an electron or an ion, 00:20:40.170 --> 00:20:42.510 align:middle line:84% but we'll get on to that later on. 00:20:42.510 --> 00:20:43.620 align:middle line:90% OK, questions? 00:20:43.620 --> 00:20:49.795 align:middle line:90% 00:20:49.795 --> 00:20:50.670 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:20:50.670 --> 00:20:58.980 align:middle line:84% JACK HARE: [CHUCKLES] We will be more precise later on. 00:20:58.980 --> 00:21:00.340 align:middle line:90% Does it matter? 00:21:00.340 --> 00:21:01.140 align:middle line:90% Yes. 00:21:01.140 --> 00:21:02.460 align:middle line:90% [LAUGHS] 00:21:02.460 --> 00:21:03.627 align:middle line:90% But this is on the subtlety. 00:21:03.627 --> 00:21:05.127 align:middle line:84% So this is one of the things where-- 00:21:05.127 --> 00:21:06.720 align:middle line:84% if you go and look at Jackson, he's 00:21:06.720 --> 00:21:08.180 align:middle line:90% very precise about all of these. 00:21:08.180 --> 00:21:09.930 align:middle line:84% Here I'm going to be a little bit handwavy 00:21:09.930 --> 00:21:10.990 align:middle line:90% until it really matters. 00:21:10.990 --> 00:21:11.760 align:middle line:90% But yeah. 00:21:11.760 --> 00:21:12.540 align:middle line:90% Oh, yeah? 00:21:12.540 --> 00:21:22.490 align:middle line:90% 00:21:22.490 --> 00:21:24.410 align:middle line:84% Well, that's true for this single particle. 00:21:24.410 --> 00:21:26.600 align:middle line:84% What if I have a second particle and I 00:21:26.600 --> 00:21:29.340 align:middle line:84% want to look at the scattering from two particles? 00:21:29.340 --> 00:21:32.010 align:middle line:84% I can't put the origin in both the same place, both of them. 00:21:32.010 --> 00:21:33.190 align:middle line:90% They're in different places. 00:21:33.190 --> 00:21:36.282 align:middle line:84% So I'm better off just starting with a more generalized way 00:21:36.282 --> 00:21:36.990 align:middle line:90% of looking at it. 00:21:36.990 --> 00:21:42.300 align:middle line:90% 00:21:42.300 --> 00:21:42.960 align:middle line:90% OK, yeah. 00:21:42.960 --> 00:21:43.710 align:middle line:90% We could do this. 00:21:43.710 --> 00:21:45.470 align:middle line:90% This is slightly more general. 00:21:45.470 --> 00:21:46.910 align:middle line:90% What if have multiple observers? 00:21:46.910 --> 00:21:48.952 align:middle line:84% So, for example, if I'm doing Thomson scattering, 00:21:48.952 --> 00:21:51.585 align:middle line:84% I might observe from multiple different angles. 00:21:51.585 --> 00:21:53.460 align:middle line:84% But then I couldn't get away with that trick. 00:21:53.460 --> 00:21:56.590 align:middle line:84% So this more general formula will probably be useful. 00:21:56.590 --> 00:21:58.250 align:middle line:90% Yeah? 00:21:58.250 --> 00:22:00.650 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:22:00.650 --> 00:22:02.957 align:middle line:90% 00:22:02.957 --> 00:22:03.540 align:middle line:90% JACK HARE: No. 00:22:03.540 --> 00:22:06.165 align:middle line:84% So, really, at the moment, we're just being like, say we've got 00:22:06.165 --> 00:22:08.655 align:middle line:84% some v of t, which, of course, has to be changing. 00:22:08.655 --> 00:22:10.940 align:middle line:90% 00:22:10.940 --> 00:22:13.440 align:middle line:84% It has to be changed in time for us to get radiation at all. 00:22:13.440 --> 00:22:14.982 align:middle line:84% And we're going to go back, and we'll 00:22:14.982 --> 00:22:17.730 align:middle line:84% solve it for the oscillations due to the electric field. 00:22:17.730 --> 00:22:20.340 align:middle line:84% So this is really the very general "imagine you have 00:22:20.340 --> 00:22:23.770 align:middle line:84% a moving accelerating charge, what's the radiation from it," 00:22:23.770 --> 00:22:25.410 align:middle line:84% which you have probably seen before. 00:22:25.410 --> 00:22:27.660 align:middle line:84% But we're just going through it to set up 00:22:27.660 --> 00:22:30.020 align:middle line:84% the mathematics and the diagram and all the vectors 00:22:30.020 --> 00:22:32.520 align:middle line:84% so that I can then use them for doing the Thomson scattering 00:22:32.520 --> 00:22:33.150 align:middle line:90% derivation. 00:22:33.150 --> 00:22:34.820 align:middle line:90% But yeah. 00:22:34.820 --> 00:22:37.605 align:middle line:90% Yeah? 00:22:37.605 --> 00:22:40.854 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:22:40.854 --> 00:22:44.233 align:middle line:90% 00:22:44.233 --> 00:22:46.650 align:middle line:84% JACK HARE: It doesn't matter for this because as we found, 00:22:46.650 --> 00:22:48.930 align:middle line:84% the refractive index only varies by a very, very 00:22:48.930 --> 00:22:50.840 align:middle line:90% small amount from 1. 00:22:50.840 --> 00:22:53.360 align:middle line:84% So, in fact, you're still going to-- 00:22:53.360 --> 00:22:55.490 align:middle line:84% for most reasonable cases, if we're operating 00:22:55.490 --> 00:22:56.900 align:middle line:90% below the critical density-- 00:22:56.900 --> 00:22:58.400 align:middle line:84% if you're operating near the critical density, 00:22:58.400 --> 00:22:59.870 align:middle line:84% then your laser beam will probably 00:22:59.870 --> 00:23:02.090 align:middle line:84% refract before it gets to the center of the plasma 00:23:02.090 --> 00:23:02.930 align:middle line:90% and can scatter. 00:23:02.930 --> 00:23:04.060 align:middle line:90% So this is-- 00:23:04.060 --> 00:23:05.730 align:middle line:84% I guess we're making an assumption here. 00:23:05.730 --> 00:23:06.230 align:middle line:90% Yeah. 00:23:06.230 --> 00:23:09.260 align:middle line:90% 00:23:09.260 --> 00:23:11.520 align:middle line:90% No. 00:23:11.520 --> 00:23:12.030 align:middle line:90% Yeah. 00:23:12.030 --> 00:23:14.363 align:middle line:84% And I will say, actually, in this example I'm giving you 00:23:14.363 --> 00:23:16.417 align:middle line:90% right now, there is no plasma. 00:23:16.417 --> 00:23:17.250 align:middle line:90% There's just vacuum. 00:23:17.250 --> 00:23:18.480 align:middle line:90% So it is just it. 00:23:18.480 --> 00:23:19.620 align:middle line:90% Yeah. 00:23:19.620 --> 00:23:20.915 align:middle line:90% But that's a good point. 00:23:20.915 --> 00:23:22.290 align:middle line:84% Maybe when we get into plasma, we 00:23:22.290 --> 00:23:24.748 align:middle line:84% should be a little bit more careful about what we're doing. 00:23:24.748 --> 00:23:27.992 align:middle line:84% And, in fact, we will substitute out a dispersion relationship 00:23:27.992 --> 00:23:29.700 align:middle line:84% in the plasma that takes into account how 00:23:29.700 --> 00:23:31.110 align:middle line:90% the wave propagates through it. 00:23:31.110 --> 00:23:33.120 align:middle line:90% Yeah, OK. 00:23:33.120 --> 00:23:34.353 align:middle line:90% Other questions? 00:23:34.353 --> 00:23:41.610 align:middle line:90% 00:23:41.610 --> 00:23:43.860 align:middle line:90% OK. 00:23:43.860 --> 00:23:47.010 align:middle line:84% So, again, we're not going to solve these equations. 00:23:47.010 --> 00:23:49.020 align:middle line:84% We're just going to remind ourselves 00:23:49.020 --> 00:23:55.740 align:middle line:84% that we have a near-field term where the electric field drops 00:23:55.740 --> 00:23:58.320 align:middle line:90% off as 1 upon R cubed. 00:23:58.320 --> 00:24:05.020 align:middle line:84% And we're going to have a far-field term where 00:24:05.020 --> 00:24:08.125 align:middle line:84% the electric field drops off as 1 upon R. 00:24:08.125 --> 00:24:10.000 align:middle line:84% And we're going to be far enough away that we 00:24:10.000 --> 00:24:11.920 align:middle line:90% don't see the near-field term. 00:24:11.920 --> 00:24:14.500 align:middle line:90% We only see the far-field term. 00:24:14.500 --> 00:24:18.190 align:middle line:84% We're also going to make some approximations here. 00:24:18.190 --> 00:24:21.890 align:middle line:84% We're going to say that the length of R prime 00:24:21.890 --> 00:24:25.718 align:middle line:84% is approximately equal to R. I have a terrifying feeling 00:24:25.718 --> 00:24:27.260 align:middle line:84% that this means that I've effectively 00:24:27.260 --> 00:24:29.343 align:middle line:90% put my origin on the particle. 00:24:29.343 --> 00:24:31.010 align:middle line:84% So maybe Brian was right all along that. 00:24:31.010 --> 00:24:33.240 align:middle line:84% We could have just saved some time here. 00:24:33.240 --> 00:24:36.080 align:middle line:84% But this has the effect of us being 00:24:36.080 --> 00:24:38.660 align:middle line:84% able to simplify our expression for the retarded time 00:24:38.660 --> 00:24:48.974 align:middle line:84% to be t prime is equal to or approximately equal to minus 00:24:48.974 --> 00:24:56.190 align:middle line:90% R s hat dot R over c. 00:24:56.190 --> 00:24:59.230 align:middle line:90% 00:24:59.230 --> 00:25:01.270 align:middle line:84% This, again, is just looking at this 00:25:01.270 --> 00:25:03.130 align:middle line:84% and using some of the vectors we've got. 00:25:03.130 --> 00:25:06.037 align:middle line:84% You see there's an R here, s here, this sort of thing. 00:25:06.037 --> 00:25:07.870 align:middle line:84% I think this is actually saying that this is 00:25:07.870 --> 00:25:09.280 align:middle line:90% an isosceles triangle, right? 00:25:09.280 --> 00:25:12.920 align:middle line:84% So we're just saying these two distances are similar. 00:25:12.920 --> 00:25:16.015 align:middle line:84% We're trying to calculate this small term here. 00:25:16.015 --> 00:25:17.640 align:middle line:84% This is similar to putting our observer 00:25:17.640 --> 00:25:19.860 align:middle line:84% far away from the particle and the origin 00:25:19.860 --> 00:25:25.080 align:middle line:90% of our coordinate system here. 00:25:25.080 --> 00:25:29.550 align:middle line:84% OK, and if we do all of that, we can go back and take a formula 00:25:29.550 --> 00:25:36.110 align:middle line:84% from something like Jackson for the far-field radiation. 00:25:36.110 --> 00:25:57.990 align:middle line:90% 00:25:57.990 --> 00:26:00.240 align:middle line:84% So the scattered radiation-- so this 00:26:00.240 --> 00:26:05.170 align:middle line:84% is the one coming towards our observer. 00:26:05.170 --> 00:26:07.990 align:middle line:84% And I'm going to use subscript s to show that it's the scattered 00:26:07.990 --> 00:26:14.540 align:middle line:84% radiation observed at the observer R prime at some time 00:26:14.540 --> 00:26:17.750 align:middle line:84% t-- so at the observer's time, not at the particle's time-- 00:26:17.750 --> 00:26:25.600 align:middle line:84% is going to be equal to q over 4 pi epsilon 0 capital 00:26:25.600 --> 00:26:29.740 align:middle line:84% R, which, again, is approximately 00:26:29.740 --> 00:26:33.410 align:middle line:90% equal to half of R prime. 00:26:33.410 --> 00:26:35.215 align:middle line:84% And then we get this interesting term, 00:26:35.215 --> 00:26:44.250 align:middle line:84% s hat cross s hat cross s hat minus theta. 00:26:44.250 --> 00:26:46.740 align:middle line:84% I'll define beta again in a moment, but some of you 00:26:46.740 --> 00:26:48.510 align:middle line:90% will remember it from before. 00:26:48.510 --> 00:26:50.820 align:middle line:84% And all of that inside these brackets 00:26:50.820 --> 00:26:53.070 align:middle line:90% is crossed with theta dot. 00:26:53.070 --> 00:26:56.020 align:middle line:90% 00:26:56.020 --> 00:26:58.270 align:middle line:84% And I'm going to put a little subscript right on here. 00:26:58.270 --> 00:26:59.960 align:middle line:84% We'll get back to it in a moment. 00:26:59.960 --> 00:27:08.330 align:middle line:84% And this is divided by 1 minus s hat dot theta cubed, like this. 00:27:08.330 --> 00:27:12.080 align:middle line:84% OK, so two things in here that I didn't define straight away-- 00:27:12.080 --> 00:27:15.020 align:middle line:84% beta is simply defined as the velocity 00:27:15.020 --> 00:27:17.840 align:middle line:84% of the particle normalized to the speed of light. 00:27:17.840 --> 00:27:20.090 align:middle line:84% This is particularly useful because a lot of the time, 00:27:20.090 --> 00:27:22.310 align:middle line:84% we can just take a nonrelativistic approximation, 00:27:22.310 --> 00:27:24.360 align:middle line:84% and this makes the betas quite small. 00:27:24.360 --> 00:27:28.610 align:middle line:84% And we can drop pesky terms in the denominator here. 00:27:28.610 --> 00:27:37.440 align:middle line:84% And this is evaluated at the retarded time 00:27:37.440 --> 00:27:40.410 align:middle line:84% because we're seeing the light some time later at our observer, 00:27:40.410 --> 00:27:43.960 align:middle line:84% which is emitted some time earlier at our particle. 00:27:43.960 --> 00:27:47.490 align:middle line:84% And if we do take this nonrelativistic approximation, 00:27:47.490 --> 00:27:49.470 align:middle line:84% where v upon c is much less than 1, 00:27:49.470 --> 00:27:53.290 align:middle line:84% then we end up with a formula for the scattered radiation, 00:27:53.290 --> 00:28:01.350 align:middle line:84% which is Es at R prime t is equal to q upon 4 pi epsilon 00:28:01.350 --> 00:28:14.390 align:middle line:84% 0 c s hat cross s hat cross theta dot over R, 00:28:14.390 --> 00:28:21.480 align:middle line:84% where the numerator is still evaluated at the retarded time. 00:28:21.480 --> 00:28:23.820 align:middle line:84% The key thing here is that in order 00:28:23.820 --> 00:28:26.550 align:middle line:84% to get the scattered electric field, 00:28:26.550 --> 00:28:32.190 align:middle line:84% we now need to know this beta dot, which is the particle's 00:28:32.190 --> 00:28:34.270 align:middle line:90% trajectory or orbit. 00:28:34.270 --> 00:28:37.510 align:middle line:84% And this is where, as Sean was asking, 00:28:37.510 --> 00:28:39.793 align:middle line:84% what exactly v are we using, we'll then go back 00:28:39.793 --> 00:28:41.710 align:middle line:84% and we'll work out how the particle oscillates 00:28:41.710 --> 00:28:44.230 align:middle line:84% in the electric field of our probe. 00:28:44.230 --> 00:28:45.860 align:middle line:84% But you might already be thinking, 00:28:45.860 --> 00:28:47.800 align:middle line:84% what if my particle is also gyrating 00:28:47.800 --> 00:28:50.140 align:middle line:90% due to some magnetic field? 00:28:50.140 --> 00:28:51.790 align:middle line:84% We'll not get on to that straight away, 00:28:51.790 --> 00:28:54.110 align:middle line:84% but it is a good question to ask yourself. 00:28:54.110 --> 00:28:55.900 align:middle line:90% OK, how are we doing so far? 00:28:55.900 --> 00:29:12.060 align:middle line:90% 00:29:12.060 --> 00:29:13.740 align:middle line:90% OK, let's keep going. 00:29:13.740 --> 00:29:16.435 align:middle line:90% 00:29:16.435 --> 00:29:18.310 align:middle line:84% I'm just going to move this word "trajectory" 00:29:18.310 --> 00:29:20.470 align:middle line:90% because it's bit of board. 00:29:20.470 --> 00:29:34.110 align:middle line:90% 00:29:34.110 --> 00:29:42.000 align:middle line:84% So now we want to find the trajectory of the particle beta 00:29:42.000 --> 00:29:45.850 align:middle line:90% for our incident electric field. 00:29:45.850 --> 00:29:50.640 align:middle line:84% So this is the electric field that is coming in and setting 00:29:50.640 --> 00:29:51.810 align:middle line:90% the whole thing in motion. 00:29:51.810 --> 00:29:56.090 align:middle line:84% It's got a subscript i for incident. 00:29:56.090 --> 00:30:05.810 align:middle line:84% So Ei at some position R at some time t prime-- 00:30:05.810 --> 00:30:07.870 align:middle line:84% so this is looking at the retarded time here, 00:30:07.870 --> 00:30:09.700 align:middle line:84% this is where things will get complicated-- 00:30:09.700 --> 00:30:13.420 align:middle line:84% is equal to whatever the strength and polarization 00:30:13.420 --> 00:30:15.160 align:middle line:90% of that electric field is. 00:30:15.160 --> 00:30:17.000 align:middle line:84% And we'll just have it be a plane wave. 00:30:17.000 --> 00:30:19.750 align:middle line:84% So it's oscillating sinusoidally, 00:30:19.750 --> 00:30:27.600 align:middle line:84% and it's got a phase factor Ki dot r minus omega i t prime. 00:30:27.600 --> 00:30:30.490 align:middle line:90% 00:30:30.490 --> 00:30:33.010 align:middle line:84% We know that the equation of motion for this particle 00:30:33.010 --> 00:30:41.580 align:middle line:84% is m dv dt is equal to minus q Ei. 00:30:41.580 --> 00:30:44.720 align:middle line:84% I don't see the need to write all the rest of it. 00:30:44.720 --> 00:30:46.160 align:middle line:90% Good? 00:30:46.160 --> 00:30:50.240 align:middle line:84% And we will assume that the particle is initially 00:30:50.240 --> 00:30:50.810 align:middle line:90% stationary. 00:30:50.810 --> 00:30:55.500 align:middle line:84% So v at time 0 is equal to 0 here. 00:30:55.500 --> 00:30:57.710 align:middle line:84% So this is equivalent to saying the thermal motion is 00:30:57.710 --> 00:30:59.570 align:middle line:84% pretty negligible compared to the motion 00:30:59.570 --> 00:31:00.800 align:middle line:90% that we get from this. 00:31:00.800 --> 00:31:02.720 align:middle line:84% And we will relax that assumption later on. 00:31:02.720 --> 00:31:04.100 align:middle line:84% If we want to measure the temperature, 00:31:04.100 --> 00:31:06.308 align:middle line:84% we're going to need to care about the thermal motion. 00:31:06.308 --> 00:31:09.740 align:middle line:84% But for now, we're just going to deal with this [INAUDIBLE]. 00:31:09.740 --> 00:31:13.830 align:middle line:84% And so we solve this to get out the beta dot, which 00:31:13.830 --> 00:31:16.560 align:middle line:84% is effectively the acceleration of the particle 00:31:16.560 --> 00:31:23.020 align:middle line:84% is equal to minus q upon mc times the i here. 00:31:23.020 --> 00:31:24.970 align:middle line:84% The particle is being accelerated 00:31:24.970 --> 00:31:27.820 align:middle line:84% with the electric field, parallel or antiparallel, 00:31:27.820 --> 00:31:29.170 align:middle line:90% depending on the charge here. 00:31:29.170 --> 00:31:32.500 align:middle line:84% And we can put all of that back into our previous equation, 00:31:32.500 --> 00:31:37.210 align:middle line:84% and we can get out that the scattered light now 00:31:37.210 --> 00:31:42.590 align:middle line:84% that's observed at a time at a position prime at time t 00:31:42.590 --> 00:31:52.400 align:middle line:84% is equal to q squared upon 4 pi epsilon 0 mc squared 00:31:52.400 --> 00:31:59.480 align:middle line:84% times s hat cross s hat cross Ei 0, which 00:31:59.480 --> 00:32:01.070 align:middle line:84% was the strength and polarization 00:32:01.070 --> 00:32:04.490 align:middle line:90% of the initial electric field. 00:32:04.490 --> 00:32:07.400 align:middle line:84% This is all evaluated at the retarded time. 00:32:07.400 --> 00:32:09.760 align:middle line:84% And then there's also a factor out here, 00:32:09.760 --> 00:32:18.310 align:middle line:84% cosine Ki dot r minus omega i t prime. 00:32:18.310 --> 00:32:21.410 align:middle line:90% 00:32:21.410 --> 00:32:31.000 align:middle line:84% This term here we're going to say is roughly constant. 00:32:31.000 --> 00:32:33.810 align:middle line:84% And the reason is that r is not going to change very much. 00:32:33.810 --> 00:32:38.660 align:middle line:84% Remember, our particle is just oscillating up and down. 00:32:38.660 --> 00:32:40.520 align:middle line:84% We assume it doesn't oscillate very far. 00:32:40.520 --> 00:32:42.960 align:middle line:84% Then its mean position doesn't change very much. 00:32:42.960 --> 00:32:45.860 align:middle line:84% And so, therefore, we can just take this as a constant term 00:32:45.860 --> 00:32:46.380 align:middle line:90% here. 00:32:46.380 --> 00:32:48.755 align:middle line:84% So the only term we have to worry about actually changing 00:32:48.755 --> 00:32:51.590 align:middle line:90% here is omega i t prime. 00:32:51.590 --> 00:32:55.290 align:middle line:84% OK, someone tell me something interesting about this formula. 00:32:55.290 --> 00:32:55.800 align:middle line:90% Yeah? 00:32:55.800 --> 00:32:59.195 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:32:59.195 --> 00:33:04.540 align:middle line:90% 00:33:04.540 --> 00:33:07.030 align:middle line:84% JACK HARE: Yeah, I'm trying to work out if I've dropped it 00:33:07.030 --> 00:33:08.740 align:middle line:84% or whether it disappeared for some other reason. 00:33:08.740 --> 00:33:10.370 align:middle line:84% I can't see a good reason to drop it. 00:33:10.370 --> 00:33:12.730 align:middle line:84% So I'm going to put it back in for now. 00:33:12.730 --> 00:33:13.240 align:middle line:90% Thank you. 00:33:13.240 --> 00:33:16.060 align:middle line:90% 00:33:16.060 --> 00:33:18.130 align:middle line:84% We would have trouble conserving energy 00:33:18.130 --> 00:33:21.940 align:middle line:84% if we didn't put it there-- so probably for the best. 00:33:21.940 --> 00:33:24.047 align:middle line:90% OK. 00:33:24.047 --> 00:33:26.130 align:middle line:84% Tell me something interesting about this equation. 00:33:26.130 --> 00:33:30.140 align:middle line:90% 00:33:30.140 --> 00:33:32.568 align:middle line:84% Do I care more about scattering in a plasma 00:33:32.568 --> 00:33:33.860 align:middle line:90% from the electrons or the ions? 00:33:33.860 --> 00:33:37.115 align:middle line:90% 00:33:37.115 --> 00:33:37.615 align:middle line:90% Why? 00:33:37.615 --> 00:33:40.510 align:middle line:90% 00:33:40.510 --> 00:33:42.590 align:middle line:90% OK. 00:33:42.590 --> 00:33:45.770 align:middle line:84% So electron scattering dominates. 00:33:45.770 --> 00:33:47.270 align:middle line:90% And we will no longer-- 00:33:47.270 --> 00:33:50.840 align:middle line:84% from this point onwards, we will not consider the ion scattering 00:33:50.840 --> 00:33:53.570 align:middle line:84% at all, because it'll be smaller by a factor of at least 2,000 00:33:53.570 --> 00:33:54.340 align:middle line:90% in the electrons. 00:33:54.340 --> 00:33:55.410 align:middle line:90% It's irrelevant. 00:33:55.410 --> 00:33:57.800 align:middle line:84% It may be there, but it won't be easy to measure over 00:33:57.800 --> 00:33:59.360 align:middle line:90% the electrons. 00:33:59.360 --> 00:34:01.610 align:middle line:84% Later on, we will now be talking about scattering 00:34:01.610 --> 00:34:03.980 align:middle line:84% from the ion feature and the electron feature. 00:34:03.980 --> 00:34:05.060 align:middle line:90% I want to be very clear. 00:34:05.060 --> 00:34:07.160 align:middle line:84% Whenever we're talking about Thomson scattering, 00:34:07.160 --> 00:34:09.260 align:middle line:90% we are scattering off electrons. 00:34:09.260 --> 00:34:12.409 align:middle line:84% Some of those electrons know about what the ions are up to. 00:34:12.409 --> 00:34:14.090 align:middle line:84% Some of those electrons know about what 00:34:14.090 --> 00:34:15.264 align:middle line:90% the electrons are up to. 00:34:15.264 --> 00:34:16.639 align:middle line:84% It's to do with the bi-shielding. 00:34:16.639 --> 00:34:17.790 align:middle line:90% We'll talk about it more. 00:34:17.790 --> 00:34:21.020 align:middle line:84% But if anyone ever asks you, what is Thomson scattering from, 00:34:21.020 --> 00:34:22.429 align:middle line:90% it is from the electron. 00:34:22.429 --> 00:34:24.830 align:middle line:84% You cannot measure easily Thomson scattering from 00:34:24.830 --> 00:34:26.340 align:middle line:90% the ions. 00:34:26.340 --> 00:34:28.980 align:middle line:84% The second interesting thing about this formula 00:34:28.980 --> 00:34:30.570 align:middle line:90% is, of course, it's a mess. 00:34:30.570 --> 00:34:33.270 align:middle line:84% We're trying to evaluate this at some time t. 00:34:33.270 --> 00:34:35.159 align:middle line:90% But there's t primes here. 00:34:35.159 --> 00:34:36.427 align:middle line:90% There's a retarded time. 00:34:36.427 --> 00:34:38.219 align:middle line:84% So if we want to make any good use of this, 00:34:38.219 --> 00:34:39.900 align:middle line:84% we're going to have to sort all of this 00:34:39.900 --> 00:34:43.600 align:middle line:84% out and write it in terms of some consistent time here. 00:34:43.600 --> 00:34:45.540 align:middle line:84% But before we get on to that, I want 00:34:45.540 --> 00:34:48.677 align:middle line:84% to just look at how much power is being scattered from this. 00:34:48.677 --> 00:34:50.969 align:middle line:84% So this is the electric field which is being scattered. 00:34:50.969 --> 00:34:54.153 align:middle line:84% But we also want to know what the power being scattered is. 00:34:54.153 --> 00:34:55.570 align:middle line:84% Is that all I have to say on that? 00:34:55.570 --> 00:34:56.070 align:middle line:90% Yeah. 00:34:56.070 --> 00:34:58.280 align:middle line:90% 00:34:58.280 --> 00:35:03.710 align:middle line:90% So this is the scattered power. 00:35:03.710 --> 00:35:12.310 align:middle line:90% 00:35:12.310 --> 00:35:18.080 align:middle line:84% So this has units of watts per steradian, per solid angle. 00:35:18.080 --> 00:35:22.120 align:middle line:84% And we often write it as dt d omega s. 00:35:22.120 --> 00:35:28.040 align:middle line:84% This omega here is some differential solid angle. 00:35:28.040 --> 00:35:29.920 align:middle line:90% So this is ds like that. 00:35:29.920 --> 00:35:32.890 align:middle line:84% We have some scattering source here scattered 00:35:32.890 --> 00:35:35.030 align:middle line:84% through some area at some distance. 00:35:35.030 --> 00:35:36.020 align:middle line:90% That's the solid angle. 00:35:36.020 --> 00:35:37.480 align:middle line:90% And so this ds can change. 00:35:37.480 --> 00:35:39.940 align:middle line:84% We can have-- for example, if our electron is oscillating 00:35:39.940 --> 00:35:42.148 align:middle line:84% up and down like this, there's no good reason 00:35:42.148 --> 00:35:44.440 align:middle line:84% to expect that it's the same amount of energy scattered 00:35:44.440 --> 00:35:48.310 align:middle line:84% into this volume as there is scattered into this volume. 00:35:48.310 --> 00:35:50.180 align:middle line:84% We'll talk about that in just a moment. 00:35:50.180 --> 00:35:51.490 align:middle line:90% So this is what this means. 00:35:51.490 --> 00:35:55.155 align:middle line:84% You have to actually know which direction you're scattering to. 00:35:55.155 --> 00:35:56.530 align:middle line:84% And there's an equation from this 00:35:56.530 --> 00:35:58.690 align:middle line:84% that you can find, Jackson or similar, which 00:35:58.690 --> 00:36:02.470 align:middle line:84% looks like r squared c epsilon 0, 00:36:02.470 --> 00:36:08.560 align:middle line:84% and then the time average of the scattered electric field 00:36:08.560 --> 00:36:12.640 align:middle line:84% averaged with itself, dotted with itself. 00:36:12.640 --> 00:36:14.890 align:middle line:84% Yeah, it's a good thing I did put that r back in there 00:36:14.890 --> 00:36:17.182 align:middle line:84% because that's going to cancel out with this r squared. 00:36:17.182 --> 00:36:19.640 align:middle line:84% So now I don't have to think about it anymore. 00:36:19.640 --> 00:36:21.760 align:middle line:84% And the key thing here is we're going 00:36:21.760 --> 00:36:28.745 align:middle line:84% to end up with a term that looks like s hat cross s hat cross Ei 00:36:28.745 --> 00:36:32.220 align:middle line:90% 0 squared. 00:36:32.220 --> 00:36:34.880 align:middle line:84% And if we say that our electric field is 00:36:34.880 --> 00:36:38.990 align:middle line:84% in this direction, which is also the direction the particle is 00:36:38.990 --> 00:36:42.830 align:middle line:84% oscillating in, and we define some angle with respect 00:36:42.830 --> 00:36:45.427 align:middle line:84% to that electric field, phi here-- 00:36:45.427 --> 00:36:47.010 align:middle line:84% so this is like a polar angle, I don't 00:36:47.010 --> 00:36:49.950 align:middle line:84% care about the azimuthal angle, just the polar angle here-- 00:36:49.950 --> 00:36:52.830 align:middle line:84% then you can do some vector algebra 00:36:52.830 --> 00:37:01.270 align:middle line:84% and find out that this can be written as sine squared of phi 00:37:01.270 --> 00:37:05.190 align:middle line:90% Ei 0 squared, like that. 00:37:05.190 --> 00:37:10.125 align:middle line:84% So our total scattered power looks like-- 00:37:10.125 --> 00:37:12.500 align:middle line:84% did I skip the thing about the classical electron radius? 00:37:12.500 --> 00:37:17.021 align:middle line:90% 00:37:17.021 --> 00:37:18.850 align:middle line:90% Oh, I did. 00:37:18.850 --> 00:37:19.350 align:middle line:90% Excuse me. 00:37:19.350 --> 00:37:22.760 align:middle line:90% 00:37:22.760 --> 00:37:26.000 align:middle line:84% So if we say that these are electrons-- so we 00:37:26.000 --> 00:37:31.130 align:middle line:84% say that q equals minus e and m equals m e, 00:37:31.130 --> 00:37:36.850 align:middle line:84% then this whole term here looks like the classical electron 00:37:36.850 --> 00:37:37.350 align:middle line:90% radius. 00:37:37.350 --> 00:37:39.600 align:middle line:84% So this is, using classical physics, how big you would 00:37:39.600 --> 00:37:40.970 align:middle line:90% estimate an electron to be. 00:37:40.970 --> 00:37:42.400 align:middle line:84% There is nothing that is actually 00:37:42.400 --> 00:37:44.150 align:middle line:84% the size of the classical electron radius. 00:37:44.150 --> 00:37:45.262 align:middle line:90% It's just a length scale. 00:37:45.262 --> 00:37:47.720 align:middle line:84% And this is a useful length scale that we'll be using here. 00:37:47.720 --> 00:37:50.720 align:middle line:84% Notice this kind of has to be the case because dimensionally, 00:37:50.720 --> 00:37:52.070 align:middle line:90% we've got electric field here. 00:37:52.070 --> 00:37:53.630 align:middle line:90% We've got electric fields here. 00:37:53.630 --> 00:37:54.945 align:middle line:90% These are unit vectors. 00:37:54.945 --> 00:37:56.820 align:middle line:84% We've got some of these dimensions of length. 00:37:56.820 --> 00:37:58.160 align:middle line:84% So to make this balance, you knew 00:37:58.160 --> 00:37:59.600 align:middle line:84% this was going to have some length scale. 00:37:59.600 --> 00:38:01.880 align:middle line:84% And we just call it the classical electron radius-- 00:38:01.880 --> 00:38:06.037 align:middle line:84% just some length scale, not very big. 00:38:06.037 --> 00:38:07.870 align:middle line:84% And so then we can write the scattered power 00:38:07.870 --> 00:38:13.300 align:middle line:84% as classical electron radius squared 00:38:13.300 --> 00:38:18.810 align:middle line:84% sine squared phi speed of light epsilon 0, 00:38:18.810 --> 00:38:26.050 align:middle line:84% size of or strength of the incident electric field here. 00:38:26.050 --> 00:38:28.930 align:middle line:84% This amount of light we're scattering into some solid angle 00:38:28.930 --> 00:38:30.200 align:middle line:90% in some direction. 00:38:30.200 --> 00:38:34.760 align:middle line:90% And if we sketch this-- 00:38:34.760 --> 00:38:36.640 align:middle line:90% so this is the electric field-- 00:38:36.640 --> 00:38:38.608 align:middle line:84% and we do a polar plot of this, we 00:38:38.608 --> 00:38:40.900 align:middle line:84% find out that most of the light is scattered, actually, 00:38:40.900 --> 00:38:42.850 align:middle line:90% at 90 degrees electric field. 00:38:42.850 --> 00:38:45.947 align:middle line:90% 00:38:45.947 --> 00:38:47.530 align:middle line:84% You imagine this being rotated around. 00:38:47.530 --> 00:38:50.260 align:middle line:84% This is sort of like a red blood cell type shape, 00:38:50.260 --> 00:38:52.570 align:middle line:84% where you have most of your emission out here, 00:38:52.570 --> 00:38:54.990 align:middle line:84% you have a very small emission in this direction, 00:38:54.990 --> 00:38:57.850 align:middle line:84% and you have no emission along E-- 00:38:57.850 --> 00:39:05.390 align:middle line:84% so dp d omega s parallel to E is equal to 0. 00:39:05.390 --> 00:39:08.780 align:middle line:90% 00:39:08.780 --> 00:39:14.360 align:middle line:84% This is for the case with nonrelativistic. 00:39:14.360 --> 00:39:16.040 align:middle line:84% Does anyone know what happens if we 00:39:16.040 --> 00:39:19.150 align:middle line:84% look at the scattering from a relativistic particle? 00:39:19.150 --> 00:39:21.610 align:middle line:90% Yeah? 00:39:21.610 --> 00:39:25.250 align:middle line:84% AUDIENCE: You get more scattering [INAUDIBLE]. 00:39:25.250 --> 00:39:29.460 align:middle line:84% JACK HARE: Yeah, it sort of starts looking like this. 00:39:29.460 --> 00:39:31.400 align:middle line:90% So this is often called beaming. 00:39:31.400 --> 00:39:38.647 align:middle line:90% 00:39:38.647 --> 00:39:40.730 align:middle line:84% And it goes in the direction of the electric field 00:39:40.730 --> 00:39:45.495 align:middle line:84% and, therefore, in the direction of the motion of the particle. 00:39:45.495 --> 00:39:47.125 align:middle line:84% I guess when the particle reverses, 00:39:47.125 --> 00:39:48.250 align:middle line:90% this also reverses as well. 00:39:48.250 --> 00:39:49.750 align:middle line:84% So this is important if you're doing 00:39:49.750 --> 00:39:52.020 align:middle line:84% Thomson scattering in plasmas which are relativistic. 00:39:52.020 --> 00:39:54.090 align:middle line:84% You might want to think about this. 00:39:54.090 --> 00:39:55.940 align:middle line:84% If you're not in a relativistic regime, 00:39:55.940 --> 00:39:57.570 align:middle line:84% if you're a nice classical regime, 00:39:57.570 --> 00:40:00.305 align:middle line:84% then it's important to note that your choice of polarization 00:40:00.305 --> 00:40:01.680 align:middle line:84% of your laser beam dictates where 00:40:01.680 --> 00:40:02.847 align:middle line:90% the light will be scattered. 00:40:02.847 --> 00:40:04.560 align:middle line:84% If you fire a laser beam like this, 00:40:04.560 --> 00:40:07.690 align:middle line:84% and it's polarized in this direction pointing upwards, 00:40:07.690 --> 00:40:10.300 align:middle line:84% and you put a spectrometer above looking down, 00:40:10.300 --> 00:40:11.250 align:middle line:90% you will see no light. 00:40:11.250 --> 00:40:13.800 align:middle line:84% And I have done this experiment, accidentally. 00:40:13.800 --> 00:40:15.310 align:middle line:90% And it's very frustrating. 00:40:15.310 --> 00:40:16.560 align:middle line:90% So you want to make sure-- 00:40:16.560 --> 00:40:19.945 align:middle line:84% honest-- that if you want to collect upwards, 00:40:19.945 --> 00:40:21.820 align:middle line:84% you want to have your polarization like that. 00:40:21.820 --> 00:40:24.660 align:middle line:84% Otherwise, you just won't see anything at all. 00:40:24.660 --> 00:40:27.660 align:middle line:84% But it doesn't matter, with respect to the electric field, 00:40:27.660 --> 00:40:31.520 align:middle line:84% where you collect, because this has no as azimuthal dependence 00:40:31.520 --> 00:40:32.020 align:middle line:90% here. 00:40:32.020 --> 00:40:33.870 align:middle line:84% So I have my electric-- my light going 00:40:33.870 --> 00:40:36.240 align:middle line:84% through this, my electric field pointing up, 00:40:36.240 --> 00:40:40.120 align:middle line:84% I'm going to get scattering in every direction like this. 00:40:40.120 --> 00:40:42.568 align:middle line:84% And so I can put lots of spectrometers 00:40:42.568 --> 00:40:44.610 align:middle line:84% at lots of different positions around the plasma, 00:40:44.610 --> 00:40:45.930 align:middle line:90% and they will all see light. 00:40:45.930 --> 00:40:46.545 align:middle line:90% OK, question? 00:40:46.545 --> 00:40:51.240 align:middle line:90% 00:40:51.240 --> 00:40:53.066 align:middle line:84% AUDIENCE: So where is [INAUDIBLE]? 00:40:53.066 --> 00:41:06.348 align:middle line:90% 00:41:06.348 --> 00:41:07.890 align:middle line:84% JACK HARE: Well, there's this t here, 00:41:07.890 --> 00:41:09.285 align:middle line:90% and there's this t prime here. 00:41:09.285 --> 00:41:17.520 align:middle line:84% AUDIENCE: [INAUDIBLE] because if the cosine term is [INAUDIBLE]. 00:41:17.520 --> 00:41:18.270 align:middle line:90% JACK HARE: Yeah. 00:41:18.270 --> 00:41:20.310 align:middle line:90% AUDIENCE: --the cross product-- 00:41:20.310 --> 00:41:23.478 align:middle line:84% JACK HARE: This is being evaluated at t prime as well. 00:41:23.478 --> 00:41:26.550 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:41:26.550 --> 00:41:28.590 align:middle line:84% JACK HARE: Yeah, but this is the equation at t. 00:41:28.590 --> 00:41:31.457 align:middle line:84% So we still need to do a little bit of substitution 00:41:31.457 --> 00:41:32.790 align:middle line:90% to get between the two of these. 00:41:32.790 --> 00:41:34.140 align:middle line:84% And this just gets a little bit subtle 00:41:34.140 --> 00:41:36.300 align:middle line:84% because we're going to start Fourier transforming them soon. 00:41:36.300 --> 00:41:36.800 align:middle line:90% Yeah. 00:41:36.800 --> 00:41:38.710 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:41:38.710 --> 00:41:40.940 align:middle line:90% 00:41:40.940 --> 00:41:44.190 align:middle line:84% JACK HARE: Yes, and we see stuff that happened R upon c earlier, 00:41:44.190 --> 00:41:44.690 align:middle line:90% right? 00:41:44.690 --> 00:41:47.850 align:middle line:84% But you could imagine, especially-- 00:41:47.850 --> 00:41:50.560 align:middle line:84% this becomes especially important if you don't have 00:41:50.560 --> 00:41:52.810 align:middle line:84% a plane wave, which is just a continuous wave in time, 00:41:52.810 --> 00:41:54.007 align:middle line:90% but if you have a pulse. 00:41:54.007 --> 00:41:55.590 align:middle line:84% Then it starts to get very complicated 00:41:55.590 --> 00:41:57.210 align:middle line:84% to keep track of all these t primes. 00:41:57.210 --> 00:41:58.620 align:middle line:84% So you're going to see a lot of them come up, 00:41:58.620 --> 00:42:00.412 align:middle line:84% and I'm going to do my best to get it right 00:42:00.412 --> 00:42:02.370 align:middle line:84% but will probably make a mistake somewhere. 00:42:02.370 --> 00:42:03.015 align:middle line:90% Yeah? 00:42:03.015 --> 00:42:04.140 align:middle line:90% AUDIENCE: What [INAUDIBLE]? 00:42:04.140 --> 00:42:12.965 align:middle line:90% 00:42:12.965 --> 00:42:13.590 align:middle line:90% JACK HARE: Yes. 00:42:13.590 --> 00:42:15.007 align:middle line:84% But, again, it becomes complicated 00:42:15.007 --> 00:42:18.210 align:middle line:84% when you do the Fourier transform because what we need 00:42:18.210 --> 00:42:19.807 align:middle line:90% to do is find out a spectra. 00:42:19.807 --> 00:42:22.140 align:middle line:84% So at the moment, this is a time-varying electric field. 00:42:22.140 --> 00:42:24.100 align:middle line:84% But we don't tend to measure that. 00:42:24.100 --> 00:42:27.150 align:middle line:84% So imagine that your laser beam is going through the plasma. 00:42:27.150 --> 00:42:28.950 align:middle line:90% It's a visible green laser beam. 00:42:28.950 --> 00:42:31.720 align:middle line:84% And so its frequency is extremely large. 00:42:31.720 --> 00:42:33.720 align:middle line:84% We don't have a detector that can digitize that. 00:42:33.720 --> 00:42:36.150 align:middle line:84% What we have is a spectrometer that disperses light 00:42:36.150 --> 00:42:37.502 align:middle line:90% depending on its frequency. 00:42:37.502 --> 00:42:39.960 align:middle line:84% So we have to do all of this instead in the Fourier domain. 00:42:39.960 --> 00:42:41.470 align:middle line:84% I'm setting it up in the time domain, 00:42:41.470 --> 00:42:43.387 align:middle line:84% but we're going to Fourier transform later on. 00:42:43.387 --> 00:42:46.210 align:middle line:84% And that's when you have to really start thinking, like, 00:42:46.210 --> 00:42:49.680 align:middle line:84% am I doing E to the i omega t or E to the i omega t prime? 00:42:49.680 --> 00:42:53.340 align:middle line:84% And if it's t prime, all of a sudden, 00:42:53.340 --> 00:42:54.610 align:middle line:90% there's an R inside there. 00:42:54.610 --> 00:42:56.730 align:middle line:84% So now I'm doing a Fourier transform with respect to space 00:42:56.730 --> 00:42:58.022 align:middle line:90% as well as time in the same go. 00:42:58.022 --> 00:42:59.020 align:middle line:90% This gets complicated. 00:42:59.020 --> 00:43:00.480 align:middle line:84% So I'm just sort of setting you up 00:43:00.480 --> 00:43:02.670 align:middle line:84% to think this will be hard, though I agree right now 00:43:02.670 --> 00:43:03.628 align:middle line:90% it looks quite trivial. 00:43:03.628 --> 00:43:04.590 align:middle line:90% Yeah. 00:43:04.590 --> 00:43:06.816 align:middle line:90% Was there a question online? 00:43:06.816 --> 00:43:07.590 align:middle line:90% Oh, maybe not. 00:43:07.590 --> 00:43:10.280 align:middle line:90% Other questions? 00:43:10.280 --> 00:43:12.410 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:43:12.410 --> 00:43:21.655 align:middle line:90% 00:43:21.655 --> 00:43:23.030 align:middle line:84% JACK HARE: Yeah, so you're saying 00:43:23.030 --> 00:43:24.380 align:middle line:84% that as the particle starts to oscillate, 00:43:24.380 --> 00:43:26.005 align:middle line:84% it will have some velocity, and then it 00:43:26.005 --> 00:43:27.650 align:middle line:90% should feel the magnetic field. 00:43:27.650 --> 00:43:30.380 align:middle line:84% We are going to assume that the velocity is still 00:43:30.380 --> 00:43:32.760 align:middle line:84% quite small so that we don't have to worry about that. 00:43:32.760 --> 00:43:34.640 align:middle line:84% So that's kind of saying that the particle doesn't actually 00:43:34.640 --> 00:43:37.098 align:middle line:84% get accelerated very far Before the electric field reverses 00:43:37.098 --> 00:43:38.250 align:middle line:90% and sends it the other way. 00:43:38.250 --> 00:43:40.010 align:middle line:84% So its velocity never gets large enough 00:43:40.010 --> 00:43:41.060 align:middle line:84% that we have to worry about that, 00:43:41.060 --> 00:43:42.720 align:middle line:84% because that sounds like a nightmare. 00:43:42.720 --> 00:43:45.680 align:middle line:84% But, yeah, you could, in theory, extend this treatment 00:43:45.680 --> 00:43:46.430 align:middle line:90% to include that. 00:43:46.430 --> 00:43:47.980 align:middle line:90% Yeah. 00:43:47.980 --> 00:43:50.670 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:43:50.670 --> 00:43:53.862 align:middle line:90% 00:43:53.862 --> 00:43:54.820 align:middle line:90% JACK HARE: Right, yeah. 00:43:54.820 --> 00:43:57.070 align:middle line:84% So the electromagnetic wave has a magnetic field. 00:43:57.070 --> 00:43:59.320 align:middle line:84% And the force on it is v cross B. 00:43:59.320 --> 00:44:02.470 align:middle line:84% So if I don't want that force to be large, I just keep v small. 00:44:02.470 --> 00:44:05.350 align:middle line:84% I keep v small by making my frequency large enough 00:44:05.350 --> 00:44:07.223 align:middle line:84% that the electron never really has a chance 00:44:07.223 --> 00:44:09.640 align:middle line:84% to accelerate very fast or the electric field turns around 00:44:09.640 --> 00:44:11.830 align:middle line:90% and turns it the other way. 00:44:11.830 --> 00:44:13.737 align:middle line:84% And this very simple treatment to get us 00:44:13.737 --> 00:44:15.820 align:middle line:84% through just scattering of a single particle is 0. 00:44:15.820 --> 00:44:18.440 align:middle line:84% And, of course, very soon, we need it to be nonzero. 00:44:18.440 --> 00:44:20.260 align:middle line:90% So, yeah, great. 00:44:20.260 --> 00:44:21.599 align:middle line:90% Any other questions? 00:44:21.599 --> 00:44:26.090 align:middle line:90% 00:44:26.090 --> 00:44:29.240 align:middle line:90% Ooh, boy. 00:44:29.240 --> 00:44:32.000 align:middle line:84% OK, another quantity we want to calculate 00:44:32.000 --> 00:44:33.262 align:middle line:90% is the total cross-section. 00:44:33.262 --> 00:44:34.970 align:middle line:84% So this is the differential cross-section 00:44:34.970 --> 00:44:35.975 align:middle line:90% here in terms of power. 00:44:35.975 --> 00:44:38.866 align:middle line:90% 00:44:38.866 --> 00:44:39.480 align:middle line:90% Ah, sorry. 00:44:39.480 --> 00:44:44.086 align:middle line:84% This is the-- what's the right word for this? 00:44:44.086 --> 00:44:45.530 align:middle line:90% This is the scattered power. 00:44:45.530 --> 00:44:48.030 align:middle line:84% We also want to calculate a quantity called the differential 00:44:48.030 --> 00:44:48.807 align:middle line:90% cross-section. 00:44:48.807 --> 00:44:50.640 align:middle line:84% This is where I wished had a bit more space, 00:44:50.640 --> 00:44:53.160 align:middle line:84% but I think I'm just going to use this. 00:44:53.160 --> 00:45:00.350 align:middle line:90% 00:45:00.350 --> 00:45:03.320 align:middle line:84% So the differential cross-section 00:45:03.320 --> 00:45:06.440 align:middle line:84% is the probability of this scattering event happening 00:45:06.440 --> 00:45:09.320 align:middle line:84% and scattering a photon in a certain direction 00:45:09.320 --> 00:45:11.790 align:middle line:84% through a certain solid angle, d mu s. 00:45:11.790 --> 00:45:15.650 align:middle line:84% And it's basically equal to the scattered power, 00:45:15.650 --> 00:45:18.540 align:middle line:84% but we're no longer interested in the power. 00:45:18.540 --> 00:45:24.460 align:middle line:84% So we just divide through by c epsilon 0 Ei squared. 00:45:24.460 --> 00:45:25.840 align:middle line:90% So that's the incident power. 00:45:25.840 --> 00:45:30.230 align:middle line:90% 00:45:30.230 --> 00:45:31.590 align:middle line:90% And we're just dividing it out. 00:45:31.590 --> 00:45:34.910 align:middle line:84% So now we'll have something that just simply looks 00:45:34.910 --> 00:45:39.350 align:middle line:90% like r e sine squared phi. 00:45:39.350 --> 00:45:41.510 align:middle line:84% The reason to do that is to say, well, this 00:45:41.510 --> 00:45:45.890 align:middle line:84% is the probability of a photon scattering into some solid angle 00:45:45.890 --> 00:45:47.900 align:middle line:90% in some direction omega s. 00:45:47.900 --> 00:45:50.240 align:middle line:84% What's just the probability of it scattering? 00:45:50.240 --> 00:45:52.610 align:middle line:84% Well, that probability, the total probability 00:45:52.610 --> 00:45:54.050 align:middle line:84% of any scattering event happening, 00:45:54.050 --> 00:46:02.800 align:middle line:84% is then equal to our E squared sine squared phi d omega, 00:46:02.800 --> 00:46:05.860 align:middle line:84% integrating over all the solid angle, 00:46:05.860 --> 00:46:13.830 align:middle line:84% that the omega is 2 pi sine phi d phi. 00:46:13.830 --> 00:46:15.930 align:middle line:84% So this is just the infinitesimal area element 00:46:15.930 --> 00:46:18.762 align:middle line:84% for a cylindrically symmetric system. 00:46:18.762 --> 00:46:20.220 align:middle line:84% And when you perform this integral, 00:46:20.220 --> 00:46:28.750 align:middle line:84% you get out 8 upon 3 pi r e squared. 00:46:28.750 --> 00:46:33.712 align:middle line:84% OK, this is extremely important because these are all constants. 00:46:33.712 --> 00:46:36.170 align:middle line:84% It doesn't depend on your laser or your plasma or anything. 00:46:36.170 --> 00:46:37.878 align:middle line:84% This is just the chance of scattering off 00:46:37.878 --> 00:46:39.490 align:middle line:90% a particle, off an electron. 00:46:39.490 --> 00:46:47.640 align:middle line:84% And this number is 6.7 times 10 to the minus 29 meters squared. 00:46:47.640 --> 00:46:51.950 align:middle line:84% So if we have a plasma with a density of any, 00:46:51.950 --> 00:46:56.570 align:middle line:84% any particles that scatter off, and it's got some length L-- so 00:46:56.570 --> 00:47:04.170 align:middle line:84% where the laser beam is going through some distance L 00:47:04.170 --> 00:47:06.810 align:middle line:84% with some number of particles per unit volume ne, 00:47:06.810 --> 00:47:10.270 align:middle line:84% and we times this by the scattering cross-section here, 00:47:10.270 --> 00:47:12.870 align:middle line:84% and if I put this in some numbers 00:47:12.870 --> 00:47:17.428 align:middle line:84% here-- so ne is maybe 10 to 14 per meter cubed. 00:47:17.428 --> 00:47:18.720 align:middle line:90% I don't know why I picked this. 00:47:18.720 --> 00:47:20.520 align:middle line:84% It seems kind of dense for a tokamak. 00:47:20.520 --> 00:47:22.560 align:middle line:84% But-- I don't know, not very-- yeah, kind 00:47:22.560 --> 00:47:25.410 align:middle line:84% of dense we're talking about, not very dense for other things. 00:47:25.410 --> 00:47:31.280 align:middle line:90% 00:47:31.280 --> 00:47:32.240 align:middle line:90% That's not that dense. 00:47:32.240 --> 00:47:35.693 align:middle line:90% 00:47:35.693 --> 00:47:37.610 align:middle line:84% Yeah, it is meant to be sent to me centimeters 00:47:37.610 --> 00:47:39.985 align:middle line:84% because then the number I write down doesn't work either. 00:47:39.985 --> 00:47:43.020 align:middle line:84% OK, that's a pretty dense for a tokamak, right? 00:47:43.020 --> 00:47:43.520 align:middle line:90% Yeah? 00:47:43.520 --> 00:47:45.050 align:middle line:90% OK. 00:47:45.050 --> 00:47:47.400 align:middle line:84% And then let's say this length looks like 1 meter. 00:47:47.400 --> 00:47:50.150 align:middle line:84% So this is looking a lot more like a tokamak here. 00:47:50.150 --> 00:47:55.910 align:middle line:84% Then this is equal to 10 to the minus 8. 00:47:55.910 --> 00:47:58.685 align:middle line:90% So this is scattered photons-- 00:47:58.685 --> 00:48:03.810 align:middle line:90% 00:48:03.810 --> 00:48:16.215 align:middle line:84% scattered-- over incident photons. 00:48:16.215 --> 00:48:19.980 align:middle line:90% 00:48:19.980 --> 00:48:25.550 align:middle line:84% So if I put 10 to the 8 photons in, I get 1 photon out. 00:48:25.550 --> 00:48:28.180 align:middle line:84% Can someone tell me something interesting about Thomson 00:48:28.180 --> 00:48:29.152 align:middle line:90% scattering? 00:48:29.152 --> 00:48:42.436 align:middle line:90% 00:48:42.436 --> 00:48:43.413 align:middle line:90% So that's very true. 00:48:43.413 --> 00:48:44.830 align:middle line:84% So, yeah, we definitely don't want 00:48:44.830 --> 00:48:47.205 align:middle line:84% to put our detector somewhere where the beam will hit it. 00:48:47.205 --> 00:48:50.480 align:middle line:84% What if we put the detector somewhere else? 00:48:50.480 --> 00:48:51.700 align:middle line:90% Is this an easy technique? 00:48:51.700 --> 00:48:55.290 align:middle line:90% Is it a hard technique? 00:48:55.290 --> 00:48:57.450 align:middle line:84% I'm putting a lot of laser-- so if I 00:48:57.450 --> 00:49:00.420 align:middle line:84% put in a joule of laser energy, I'm 00:49:00.420 --> 00:49:04.170 align:middle line:84% going to get out 10 to the minus 8, 10 nanojoules of laser energy 00:49:04.170 --> 00:49:06.790 align:middle line:90% distributed in 4 pi. 00:49:06.790 --> 00:49:09.150 align:middle line:84% So when I put my detector there and it's 00:49:09.150 --> 00:49:11.490 align:middle line:84% got a solid angle that is not 4 pi, 00:49:11.490 --> 00:49:13.420 align:middle line:84% that number will go down even further. 00:49:13.420 --> 00:49:14.880 align:middle line:84% So this is a very hard technique. 00:49:14.880 --> 00:49:17.340 align:middle line:84% Thomson scattering is difficult. There's not very much 00:49:17.340 --> 00:49:18.060 align:middle line:90% scattered light. 00:49:18.060 --> 00:49:21.660 align:middle line:84% You need a very powerful laser source in order to do it. 00:49:21.660 --> 00:49:23.490 align:middle line:84% Can someone tell me something else 00:49:23.490 --> 00:49:25.185 align:middle line:90% interesting about this result? 00:49:25.185 --> 00:49:28.890 align:middle line:90% 00:49:28.890 --> 00:49:31.160 align:middle line:90% Yeah? 00:49:31.160 --> 00:49:33.770 align:middle line:84% So denser plasmas, it's easier, definitely. 00:49:33.770 --> 00:49:34.310 align:middle line:90% Yeah. 00:49:34.310 --> 00:49:35.000 align:middle line:90% That's true. 00:49:35.000 --> 00:49:38.025 align:middle line:84% But, of course, yeah, it depends on this length as well 00:49:38.025 --> 00:49:39.650 align:middle line:84% a little bit, and how you're collecting 00:49:39.650 --> 00:49:43.110 align:middle line:84% the light because you're not collecting it-- 00:49:43.110 --> 00:49:46.600 align:middle line:84% if you have a spectrometer looking here, 00:49:46.600 --> 00:49:49.155 align:middle line:84% you're only collecting it over a relatively short length. 00:49:49.155 --> 00:49:50.530 align:middle line:84% You're not probably collecting it 00:49:50.530 --> 00:49:51.770 align:middle line:84% over the total length of the plasma. 00:49:51.770 --> 00:49:53.350 align:middle line:84% So this is an upper bound on the total number 00:49:53.350 --> 00:49:54.340 align:middle line:90% of photons scattered. 00:49:54.340 --> 00:49:56.860 align:middle line:84% And as we said with solid angle and your small collection 00:49:56.860 --> 00:50:00.140 align:middle line:84% volume, you'll scatter many fewer photons. 00:50:00.140 --> 00:50:01.390 align:middle line:90% What about double scattering? 00:50:01.390 --> 00:50:03.850 align:middle line:84% Should I ever worry about a photon scattering off 00:50:03.850 --> 00:50:06.445 align:middle line:84% an electron and then, having done 00:50:06.445 --> 00:50:08.320 align:middle line:84% some weird shift in frequency, scattering off 00:50:08.320 --> 00:50:09.070 align:middle line:90% a second electron? 00:50:09.070 --> 00:50:11.260 align:middle line:84% Do I need to account for that in my calculation? 00:50:11.260 --> 00:50:13.230 align:middle line:90% Never, ever, ever, ever, right? 00:50:13.230 --> 00:50:16.060 align:middle line:84% I mean, maybe if you got to extremely high densities, 00:50:16.060 --> 00:50:18.740 align:middle line:84% but this is not something that we need to worry about. 00:50:18.740 --> 00:50:21.770 align:middle line:84% So we're just looking for scattering of single particles. 00:50:21.770 --> 00:50:23.422 align:middle line:90% This is very important. 00:50:23.422 --> 00:50:38.210 align:middle line:90% 00:50:38.210 --> 00:50:40.870 align:middle line:84% And this is why this is a local measurement, 00:50:40.870 --> 00:50:43.570 align:middle line:84% because if you scatter off a particle here, 00:50:43.570 --> 00:50:46.870 align:middle line:84% the photon will leave the plasma without scattering again. 00:50:46.870 --> 00:50:49.150 align:middle line:84% It's not like radiation transport, 00:50:49.150 --> 00:50:52.500 align:middle line:84% where there was a chance of the scattered photon being absorbed. 00:50:52.500 --> 00:50:54.250 align:middle line:84% Well, it could still be absorbed, I guess. 00:50:54.250 --> 00:50:56.740 align:middle line:84% But it's not going to be scattered twice. 00:50:56.740 --> 00:50:58.600 align:middle line:84% So we know if we see a scattered photon 00:50:58.600 --> 00:51:00.190 align:middle line:84% that it came from the volume we think 00:51:00.190 --> 00:51:02.200 align:middle line:84% it came from and then straight out the plasma 00:51:02.200 --> 00:51:04.860 align:middle line:90% without scattering again. 00:51:04.860 --> 00:51:07.120 align:middle line:90% OK, any questions? 00:51:07.120 --> 00:51:10.170 align:middle line:84% We're going to relax B 0 equals 0. 00:51:10.170 --> 00:51:11.110 align:middle line:90% Yes? 00:51:11.110 --> 00:51:13.550 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:51:13.550 --> 00:51:15.900 align:middle line:90% 00:51:15.900 --> 00:51:17.390 align:middle line:90% JACK HARE: Yes. 00:51:17.390 --> 00:51:19.071 align:middle line:90% AUDIENCE: Why is that? 00:51:19.071 --> 00:51:21.630 align:middle line:84% JACK HARE: I think if you take the nonclassical version, 00:51:21.630 --> 00:51:24.890 align:middle line:84% there is a dependent frequency there. 00:51:24.890 --> 00:51:27.200 align:middle line:84% But in this treatment here-- yeah, 00:51:27.200 --> 00:51:29.190 align:middle line:84% again, this is very classical treatment. 00:51:29.190 --> 00:51:30.950 align:middle line:84% This is the classical electron radius. 00:51:30.950 --> 00:51:33.320 align:middle line:84% And we dropped relativistic effects 00:51:33.320 --> 00:51:34.500 align:middle line:90% to get this formula here. 00:51:34.500 --> 00:51:35.450 align:middle line:90% So we don't know about quantum. 00:51:35.450 --> 00:51:36.710 align:middle line:90% We don't know about those. 00:51:36.710 --> 00:51:37.997 align:middle line:90% I suspect there is. 00:51:37.997 --> 00:51:40.580 align:middle line:84% Thomson scattering turns into Compton scattering and the limit 00:51:40.580 --> 00:51:41.730 align:middle line:90% of high-energy photons. 00:51:41.730 --> 00:51:44.450 align:middle line:84% So we're not going to cover Compton scattering. 00:51:44.450 --> 00:51:47.477 align:middle line:84% But this is technically a limit of it. 00:51:47.477 --> 00:51:49.310 align:middle line:84% So I guess I could derive Compton scattering 00:51:49.310 --> 00:51:50.760 align:middle line:84% and then get Thomson scattering from it. 00:51:50.760 --> 00:51:52.843 align:middle line:84% But this is hard enough already that we don't need 00:51:52.843 --> 00:51:53.990 align:middle line:90% to inflict that extra pain. 00:51:53.990 --> 00:51:55.010 align:middle line:90% Yeah. 00:51:55.010 --> 00:51:58.850 align:middle line:90% OK, any other questions? 00:51:58.850 --> 00:51:59.350 align:middle line:90% OK. 00:51:59.350 --> 00:51:59.770 align:middle line:90% Yeah? 00:51:59.770 --> 00:52:00.270 align:middle line:90% Go on. 00:52:00.270 --> 00:52:07.270 align:middle line:90% 00:52:07.270 --> 00:52:09.440 align:middle line:90% Yes. 00:52:09.440 --> 00:52:10.398 align:middle line:90% Yes. 00:52:10.398 --> 00:52:12.440 align:middle line:84% Almost all of your laser light will pass directly 00:52:12.440 --> 00:52:13.830 align:middle line:90% through the plasma. 00:52:13.830 --> 00:52:15.580 align:middle line:84% And then, on the other side of the plasma, 00:52:15.580 --> 00:52:17.538 align:middle line:84% you have to work out what happens to that beam. 00:52:17.538 --> 00:52:19.570 align:middle line:84% If it reflects off the wall of your chamber, 00:52:19.570 --> 00:52:23.708 align:middle line:84% the reflected light will be quite large, even from a Thomson 00:52:23.708 --> 00:52:26.250 align:middle line:84% scattering perspective, because the electron density of metal 00:52:26.250 --> 00:52:28.470 align:middle line:84% is extremely high, 10 to the 30 or so. 00:52:28.470 --> 00:52:31.560 align:middle line:84% And so that reflected light will bounce a few times 00:52:31.560 --> 00:52:33.715 align:middle line:84% and eventually end up in your spectrometer, where 00:52:33.715 --> 00:52:35.590 align:middle line:84% it will completely drown out scattered light. 00:52:35.590 --> 00:52:37.350 align:middle line:84% So instead, you have to do something 00:52:37.350 --> 00:52:40.312 align:middle line:84% with that unscattered light, which is most of your light. 00:52:40.312 --> 00:52:42.270 align:middle line:84% You have to have a beam dump, which is normally 00:52:42.270 --> 00:52:44.430 align:middle line:84% a very, very long pipe with a very, very 00:52:44.430 --> 00:52:47.730 align:middle line:84% dark box at the far end, which you hope absorbs all the light 00:52:47.730 --> 00:52:49.060 align:middle line:90% and stops the backscatter. 00:52:49.060 --> 00:52:50.610 align:middle line:84% So, yeah, that's a quite important 00:52:50.610 --> 00:52:52.500 align:middle line:90% practical consideration. 00:52:52.500 --> 00:52:53.760 align:middle line:90% Yeah? 00:52:53.760 --> 00:52:58.530 align:middle line:84% AUDIENCE: So if the scattering probability is so low, 00:52:58.530 --> 00:53:01.650 align:middle line:84% how could you do time-resolved Thomson scattering? 00:53:01.650 --> 00:53:05.680 align:middle line:84% How could you get sufficient [INAUDIBLE] time? 00:53:05.680 --> 00:53:06.180 align:middle line:90% Because-- 00:53:06.180 --> 00:53:07.460 align:middle line:90% JACK HARE: A big laser. 00:53:07.460 --> 00:53:10.150 align:middle line:90% AUDIENCE: OK. 00:53:10.150 --> 00:53:10.775 align:middle line:90% JACK HARE: Yes. 00:53:10.775 --> 00:53:15.420 align:middle line:90% 00:53:15.420 --> 00:53:16.980 align:middle line:90% OK. 00:53:16.980 --> 00:53:20.890 align:middle line:84% Let's make life more complicated. 00:53:20.890 --> 00:53:23.680 align:middle line:90% 00:53:23.680 --> 00:53:25.555 align:middle line:90% Yeah? 00:53:25.555 --> 00:53:28.030 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:53:28.030 --> 00:53:38.440 align:middle line:90% 00:53:38.440 --> 00:53:40.400 align:middle line:84% JACK HARE: So you could, in principle, 00:53:40.400 --> 00:53:42.590 align:middle line:90% do something clever with it. 00:53:42.590 --> 00:53:45.460 align:middle line:84% I've seen people do Thomson where they also 00:53:45.460 --> 00:53:47.350 align:middle line:84% look at the rotation of the polarization 00:53:47.350 --> 00:53:48.923 align:middle line:84% by magnetic fields in the plasma. 00:53:48.923 --> 00:53:50.590 align:middle line:84% So they've done Faraday at the same time 00:53:50.590 --> 00:53:52.180 align:middle line:90% with the same proton beam. 00:53:52.180 --> 00:53:55.620 align:middle line:84% So there are some things you can do. 00:53:55.620 --> 00:53:57.870 align:middle line:84% In general, it's better to have dedicated laser beams 00:53:57.870 --> 00:54:00.138 align:middle line:90% with different angles. 00:54:00.138 --> 00:54:01.600 align:middle line:90% Yeah. 00:54:01.600 --> 00:54:03.410 align:middle line:90% OK. 00:54:03.410 --> 00:54:10.250 align:middle line:84% So let's now say we have 0 is not 0. 00:54:10.250 --> 00:54:13.210 align:middle line:90% 00:54:13.210 --> 00:54:17.110 align:middle line:84% Our position at time t prime is now 00:54:17.110 --> 00:54:25.140 align:middle line:84% equal to wherever we started at at time 0 plus velocity t 00:54:25.140 --> 00:54:27.360 align:middle line:90% prime later. 00:54:27.360 --> 00:54:30.720 align:middle line:84% And this means that our retarded time is now 00:54:30.720 --> 00:54:41.410 align:middle line:84% equal to t minus R minus s dot r over c, like this. 00:54:41.410 --> 00:54:45.770 align:middle line:84% By the way, we've assumed at this point this particle is just 00:54:45.770 --> 00:54:48.750 align:middle line:90% going in a straight line. 00:54:48.750 --> 00:54:52.660 align:middle line:84% So this is before the electric field gets here and mucks things 00:54:52.660 --> 00:54:53.218 align:middle line:90% up-- 00:54:53.218 --> 00:54:54.760 align:middle line:84% just going to let this particle move. 00:54:54.760 --> 00:54:57.510 align:middle line:90% 00:54:57.510 --> 00:55:01.370 align:middle line:84% Now let me see what I'm saying here. 00:55:01.370 --> 00:55:05.330 align:middle line:90% No, that's not true. 00:55:05.330 --> 00:55:07.760 align:middle line:84% This particle is going roughly in a straight line, 00:55:07.760 --> 00:55:10.760 align:middle line:84% but the electric field is causing it to deviate 00:55:10.760 --> 00:55:12.600 align:middle line:90% only a very small amount here. 00:55:12.600 --> 00:55:19.000 align:middle line:84% So there's some small deviation delta v from the incident 00:55:19.000 --> 00:55:20.050 align:middle line:90% electric field. 00:55:20.050 --> 00:55:25.283 align:middle line:84% But that delta v is much, much smaller than this velocity here. 00:55:25.283 --> 00:55:27.450 align:middle line:84% So the particle is roughly going in a straight line, 00:55:27.450 --> 00:55:30.750 align:middle line:84% and it's oscillating ever so slightly. 00:55:30.750 --> 00:55:31.250 align:middle line:90% OK. 00:55:31.250 --> 00:55:34.140 align:middle line:90% 00:55:34.140 --> 00:55:36.083 align:middle line:84% Now, I have a load of algebra here, 00:55:36.083 --> 00:55:38.500 align:middle line:84% which I worked out because I couldn't find it in the book. 00:55:38.500 --> 00:55:40.080 align:middle line:84% But I'm not going to go through it. 00:55:40.080 --> 00:55:42.660 align:middle line:84% You can have the joy of doing it yourself. 00:55:42.660 --> 00:55:45.910 align:middle line:90% You can rewrite this equation. 00:55:45.910 --> 00:55:48.360 align:middle line:84% And this is actually much less trivial 00:55:48.360 --> 00:55:56.360 align:middle line:84% than it looks as t minus R minus s dot r of 0. 00:55:56.360 --> 00:56:00.080 align:middle line:84% We're taking this expression for r, substituting it into here. 00:56:00.080 --> 00:56:01.970 align:middle line:90% All of this is divided by c. 00:56:01.970 --> 00:56:04.810 align:middle line:90% 00:56:04.810 --> 00:56:12.690 align:middle line:84% And this whole thing is divided by 1 minus s hat dot beta. 00:56:12.690 --> 00:56:17.380 align:middle line:90% 00:56:17.380 --> 00:56:22.270 align:middle line:84% This term down here is a Doppler shift. 00:56:22.270 --> 00:56:25.000 align:middle line:84% We'll see another one of those occur in a moment. 00:56:25.000 --> 00:56:34.070 align:middle line:90% 00:56:34.070 --> 00:56:41.210 align:middle line:84% So now we want to calculate the phase factor inside the cosine 00:56:41.210 --> 00:56:44.720 align:middle line:84% that we had for our incident electric field. 00:56:44.720 --> 00:56:48.650 align:middle line:84% Remember, previously, this was Ki dot 00:56:48.650 --> 00:56:57.310 align:middle line:84% r of t prime minus omega i of t prime. 00:56:57.310 --> 00:56:59.115 align:middle line:84% And we said that this was roughly constant 00:56:59.115 --> 00:57:00.490 align:middle line:84% because the particle didn't move. 00:57:00.490 --> 00:57:01.823 align:middle line:90% Well, now the particle's moving. 00:57:01.823 --> 00:57:03.520 align:middle line:90% So it's not constant anymore. 00:57:03.520 --> 00:57:12.770 align:middle line:84% This factor becomes equal to Ki dot r of 0 00:57:12.770 --> 00:57:24.200 align:middle line:84% plus v prime of K dot v minus omega i. 00:57:24.200 --> 00:57:26.870 align:middle line:84% This is from substituting this into here. 00:57:26.870 --> 00:57:29.910 align:middle line:90% 00:57:29.910 --> 00:57:31.565 align:middle line:90% I think that should be a Ki. 00:57:31.565 --> 00:57:37.353 align:middle line:90% 00:57:37.353 --> 00:57:38.520 align:middle line:90% I'm going to put it as a Ki. 00:57:38.520 --> 00:57:40.320 align:middle line:84% It doesn't have a subscript for my notes-- 00:57:40.320 --> 00:57:41.195 align:middle line:90% definitely a mistake. 00:57:41.195 --> 00:57:47.940 align:middle line:90% 00:57:47.940 --> 00:57:50.080 align:middle line:90% OK. 00:57:50.080 --> 00:57:53.590 align:middle line:84% And then we can rewrite this altogether 00:57:53.590 --> 00:58:01.205 align:middle line:84% as Ks R. This is just some phase that doesn't change in time, 00:58:01.205 --> 00:58:03.580 align:middle line:84% which is just due to the fact that the scattered wave has 00:58:03.580 --> 00:58:06.620 align:middle line:84% some distance to go to get to our detector. 00:58:06.620 --> 00:58:09.205 align:middle line:84% So this is just another constant phase-- 00:58:09.205 --> 00:58:14.890 align:middle line:90% 00:58:14.890 --> 00:58:19.000 align:middle line:84% minus a term that looks like scattered frequency, 00:58:19.000 --> 00:58:22.090 align:middle line:84% omega s times t minus a term that looks 00:58:22.090 --> 00:58:33.010 align:middle line:90% like Ks minus Ki dot r of 0. 00:58:33.010 --> 00:58:35.240 align:middle line:84% And I've introduced several new terms here. 00:58:35.240 --> 00:58:37.552 align:middle line:90% One of them is this omega s. 00:58:37.552 --> 00:58:40.330 align:middle line:90% 00:58:40.330 --> 00:58:42.730 align:middle line:84% And this is the frequency of the scattered wave 00:58:42.730 --> 00:58:44.140 align:middle line:90% which is seen at the observer. 00:58:44.140 --> 00:58:46.600 align:middle line:84% So by switching from t prime to t, 00:58:46.600 --> 00:58:49.480 align:middle line:84% I've had to take into account this change in time. 00:58:49.480 --> 00:58:52.670 align:middle line:84% And the Doppler shift starts coming in here. 00:58:52.670 --> 00:58:54.730 align:middle line:84% So this scattered frequency looks 00:58:54.730 --> 00:59:04.046 align:middle line:84% like 1 minus i hat dot beta over 1 minus s hat 00:59:04.046 --> 00:59:08.900 align:middle line:90% dot beta times by omega i. 00:59:08.900 --> 00:59:12.310 align:middle line:84% This is a scattered frequency, and it's interesting 00:59:12.310 --> 00:59:15.490 align:middle line:84% because it's got two Doppler shifts inside it. 00:59:15.490 --> 00:59:19.270 align:middle line:84% It's got a Doppler shift due to the light 00:59:19.270 --> 00:59:24.320 align:middle line:84% coming in being Doppler shifted by this particle's motion. 00:59:24.320 --> 00:59:27.440 align:middle line:84% So the particle sees a different frequency of the incoming light, 00:59:27.440 --> 00:59:30.530 align:middle line:84% and then the particle scatters that incoming light 00:59:30.530 --> 00:59:32.310 align:middle line:90% in some other direction. 00:59:32.310 --> 00:59:36.390 align:middle line:84% And that light is Doppler shifted again to the observer. 00:59:36.390 --> 00:59:41.060 align:middle line:84% So we get two Doppler shift factors here. 00:59:41.060 --> 00:59:46.150 align:middle line:84% And the other new term I introduced is this Ks here. 00:59:46.150 --> 00:59:49.780 align:middle line:84% And this is simply from the dispersion relationship 00:59:49.780 --> 00:59:52.910 align:middle line:84% for electromagnetic radiation in a vacuum. 00:59:52.910 --> 00:59:58.420 align:middle line:84% So it's omega equals tK, and this is going in the s hat 00:59:58.420 --> 01:00:01.110 align:middle line:90% direction here. 01:00:01.110 --> 01:00:05.850 align:middle line:84% So if you put this equation into this equation, 01:00:05.850 --> 01:00:07.230 align:middle line:84% turn through all the mathematics, 01:00:07.230 --> 01:00:09.390 align:middle line:84% you find out you get a new phase term. 01:00:09.390 --> 01:00:14.160 align:middle line:84% In a moment, I'll explain what's going on here. 01:00:14.160 --> 01:00:15.870 align:middle line:90% OK, any questions on that? 01:00:15.870 --> 01:00:47.340 align:middle line:90% 01:00:47.340 --> 01:00:49.610 align:middle line:84% This is the phase factor inside the cosine 01:00:49.610 --> 01:00:53.420 align:middle line:90% for our scattered radiation. 01:00:53.420 --> 01:00:55.710 align:middle line:84% Previously, we just had this term here. 01:00:55.710 --> 01:00:58.460 align:middle line:84% So our scattered radiation just looked like a plane wave 01:00:58.460 --> 01:01:05.040 align:middle line:84% with the same wavelength and frequency as the incoming wave. 01:01:05.040 --> 01:01:08.840 align:middle line:84% But now it's oscillating at the frequency 01:01:08.840 --> 01:01:11.870 align:middle line:84% of the scattered wave, which it should do because that's 01:01:11.870 --> 01:01:12.870 align:middle line:90% how we've defined it. 01:01:12.870 --> 01:01:15.410 align:middle line:84% But interestingly, we've got this extra phase 01:01:15.410 --> 01:01:18.330 align:middle line:84% term that comes from the difference between these two. 01:01:18.330 --> 01:01:23.580 align:middle line:84% So we can now define two more variables 01:01:23.580 --> 01:01:25.030 align:middle line:90% which don't have subscripts. 01:01:25.030 --> 01:01:32.310 align:middle line:84% So this is K, and this is simply equal to Ks minus Ki, which 01:01:32.310 --> 01:01:33.330 align:middle line:90% is this term here. 01:01:33.330 --> 01:01:37.270 align:middle line:90% 01:01:37.270 --> 01:01:39.810 align:middle line:84% And so it looks like a plane wave which is actually 01:01:39.810 --> 01:01:43.338 align:middle line:84% moving with a wave number K because we've 01:01:43.338 --> 01:01:44.880 align:middle line:84% got the difference between these two. 01:01:44.880 --> 01:01:47.010 align:middle line:84% And we can also define a frequency, 01:01:47.010 --> 01:01:49.650 align:middle line:84% omega with no subscript, which is 01:01:49.650 --> 01:01:53.760 align:middle line:84% equal to the difference between omega s and omega i. 01:01:53.760 --> 01:02:00.757 align:middle line:84% And this is simply K dotted with v. So this is the Doppler shift. 01:02:00.757 --> 01:02:03.090 align:middle line:84% This is the shift in frequency between the incident wave 01:02:03.090 --> 01:02:04.740 align:middle line:84% and the scattered wave-- that's omega. 01:02:04.740 --> 01:02:06.780 align:middle line:84% And that Doppler shift, as we expect, 01:02:06.780 --> 01:02:12.090 align:middle line:84% is the wave number, this difference wave number K, 01:02:12.090 --> 01:02:13.170 align:middle line:90% dotted into the velocity. 01:02:13.170 --> 01:02:16.390 align:middle line:84% This is what we normally expect for the Doppler shift here. 01:02:16.390 --> 01:02:18.210 align:middle line:84% What's interesting about these equations 01:02:18.210 --> 01:02:20.260 align:middle line:90% is this is a vector equation. 01:02:20.260 --> 01:02:27.650 align:middle line:84% And it implies that we have some particle with velocity v, 01:02:27.650 --> 01:02:29.010 align:middle line:90% like this. 01:02:29.010 --> 01:02:34.500 align:middle line:84% We have some input wave of light, our probing laser 01:02:34.500 --> 01:02:38.070 align:middle line:84% beam, that crosses that particle in this direction. 01:02:38.070 --> 01:02:44.130 align:middle line:84% And we observe the scattered light at some other point, Ks, 01:02:44.130 --> 01:02:45.390 align:middle line:90% like this. 01:02:45.390 --> 01:02:48.270 align:middle line:84% Then it appears that what we're actually measuring 01:02:48.270 --> 01:02:53.350 align:middle line:84% are the properties of some wave that goes from-- 01:02:53.350 --> 01:02:57.010 align:middle line:84% well, has this wave number K, which 01:02:57.010 --> 01:02:59.630 align:middle line:84% is the difference between these two wave vectors here, 01:02:59.630 --> 01:03:03.080 align:middle line:84% and it has some frequency, omega, 01:03:03.080 --> 01:03:06.380 align:middle line:90% which is that K dot v here. 01:03:06.380 --> 01:03:08.230 align:middle line:84% So in reality, what we're doing when 01:03:08.230 --> 01:03:12.580 align:middle line:84% we measure with our spectrometer at omega s and at Ks is 01:03:12.580 --> 01:03:14.740 align:middle line:84% we're measuring the properties of this wave, which 01:03:14.740 --> 01:03:17.440 align:middle line:84% is transferring us from the incident radiation 01:03:17.440 --> 01:03:18.710 align:middle line:90% to the scattered radiation. 01:03:18.710 --> 01:03:22.270 align:middle line:84% So this looks an awful lot like an equation for the conservation 01:03:22.270 --> 01:03:28.140 align:middle line:90% of momentum and energy. 01:03:28.140 --> 01:03:31.325 align:middle line:90% 01:03:31.325 --> 01:03:36.810 align:middle line:84% If I just times these by h bar and think about them as photons, 01:03:36.810 --> 01:03:38.280 align:middle line:90% you can see that straight away. 01:03:38.280 --> 01:03:44.440 align:middle line:90% 01:03:44.440 --> 01:03:45.610 align:middle line:90% This is the mathematics. 01:03:45.610 --> 01:03:48.430 align:middle line:84% We'll get to an interpretation of it quite soon. 01:03:48.430 --> 01:03:50.620 align:middle line:90% Hold on if you're struggling. 01:03:50.620 --> 01:03:51.820 align:middle line:90% We will get there. 01:03:51.820 --> 01:03:53.530 align:middle line:84% Any questions on the mathematics so far? 01:03:53.530 --> 01:04:03.510 align:middle line:90% 01:04:03.510 --> 01:04:04.010 align:middle line:90% OK. 01:04:04.010 --> 01:04:36.710 align:middle line:90% 01:04:36.710 --> 01:04:39.560 align:middle line:84% So a couple of things to say about these new vectors we 01:04:39.560 --> 01:04:44.090 align:middle line:84% introduced-- the size of this new vector K, which 01:04:44.090 --> 01:04:47.420 align:middle line:84% is simply equal, again, to the size of the difference 01:04:47.420 --> 01:04:56.770 align:middle line:84% between these two vectors, is equal to Ks squared plus Ki 01:04:56.770 --> 01:05:03.970 align:middle line:84% squared minus 2 Ki Ks cosine of theta 01:05:03.970 --> 01:05:09.120 align:middle line:84% to the half, where theta is the angle between these two. 01:05:09.120 --> 01:05:12.160 align:middle line:84% So this is just vector algebra here. 01:05:12.160 --> 01:05:17.190 align:middle line:84% Now, for nonrelativistic particles, 01:05:17.190 --> 01:05:20.880 align:middle line:84% we find that the incident light is 01:05:20.880 --> 01:05:24.600 align:middle line:84% going to have a frequency very similar to the scattered light. 01:05:24.600 --> 01:05:27.420 align:middle line:84% And so these two wave numbers will also 01:05:27.420 --> 01:05:29.350 align:middle line:90% be similar in magnitude here. 01:05:29.350 --> 01:05:31.080 align:middle line:84% So let me write that more explicitly. 01:05:31.080 --> 01:05:34.080 align:middle line:84% For nonrelativistic systems, the Doppler shift is small. 01:05:34.080 --> 01:05:41.330 align:middle line:84% So omega equals K dot v is much, much less than omega s or omega 01:05:41.330 --> 01:05:42.210 align:middle line:90% i. 01:05:42.210 --> 01:05:46.020 align:middle line:84% That means omega s is roughly omega i. 01:05:46.020 --> 01:05:49.080 align:middle line:84% We haven't seen a significant Doppler shift here. 01:05:49.080 --> 01:05:52.560 align:middle line:84% And that means that the size of these two K vectors, 01:05:52.560 --> 01:05:57.760 align:middle line:84% Ks, is roughly equal to the size of Ki, like this. 01:05:57.760 --> 01:06:00.250 align:middle line:84% And if we put that into this equation, 01:06:00.250 --> 01:06:09.360 align:middle line:84% then we find that the size of K is roughly equal to 2 times Ki 01:06:09.360 --> 01:06:11.850 align:middle line:90% sine theta upon 2. 01:06:11.850 --> 01:06:16.060 align:middle line:90% 01:06:16.060 --> 01:06:23.820 align:middle line:84% This is a very useful formula to remember because it relates 01:06:23.820 --> 01:06:26.400 align:middle line:84% the scattering angle that we're observing with respect 01:06:26.400 --> 01:06:32.010 align:middle line:84% to the incident laser beam and the wavelength of the incident 01:06:32.010 --> 01:06:34.140 align:middle line:84% laser beam to the wavelength of the mode 01:06:34.140 --> 01:06:35.340 align:middle line:90% that we're scattering on. 01:06:35.340 --> 01:06:38.220 align:middle line:84% And we'll get back to what this mode with scattering off 01:06:38.220 --> 01:06:40.870 align:middle line:90% means physically in a moment. 01:06:40.870 --> 01:06:43.770 align:middle line:84% Let's make it clear that this is sine of theta 01:06:43.770 --> 01:06:46.320 align:middle line:90% upon 2 within the brackets here. 01:06:46.320 --> 01:06:49.080 align:middle line:90% 01:06:49.080 --> 01:06:52.040 align:middle line:84% And the other thing to note is that we actually choose 01:06:52.040 --> 01:06:53.390 align:middle line:90% several of these things, right? 01:06:53.390 --> 01:06:57.770 align:middle line:84% We choose our initial laser frequency. 01:06:57.770 --> 01:07:02.060 align:middle line:84% And we choose what direction our laser beam goes in. 01:07:02.060 --> 01:07:05.293 align:middle line:84% And we choose what direction the light scatterers in. 01:07:05.293 --> 01:07:06.960 align:middle line:84% You might say, no, we don't choose that. 01:07:06.960 --> 01:07:07.950 align:middle line:90% It scatters everywhere. 01:07:07.950 --> 01:07:09.680 align:middle line:90% We choose which light to see. 01:07:09.680 --> 01:07:11.430 align:middle line:84% We choose where to place our spectrometer. 01:07:11.430 --> 01:07:13.950 align:middle line:90% So we set Ks, right? 01:07:13.950 --> 01:07:19.400 align:middle line:84% So all of these three things, we choose these. 01:07:19.400 --> 01:07:23.120 align:middle line:90% 01:07:23.120 --> 01:07:29.480 align:middle line:84% Again, I've got some plasma with some laser going through it, 01:07:29.480 --> 01:07:38.570 align:middle line:84% going along Ki, and I set up my spectrometer to look at Ks. 01:07:38.570 --> 01:07:40.655 align:middle line:84% And I'll redraw these so they're the same length. 01:07:40.655 --> 01:07:46.113 align:middle line:90% 01:07:46.113 --> 01:07:48.030 align:middle line:84% I have controlled most of the parameters here. 01:07:48.030 --> 01:07:50.360 align:middle line:84% And so that means that because I've 01:07:50.360 --> 01:07:53.880 align:middle line:84% chosen these two things, when I'm measuring my spectrum, 01:07:53.880 --> 01:07:56.090 align:middle line:84% I know that I'm measuring modes, which, again, 01:07:56.090 --> 01:07:58.700 align:middle line:90% correspond to this K here. 01:07:58.700 --> 01:08:02.070 align:middle line:90% 01:08:02.070 --> 01:08:13.930 align:middle line:84% So just to write that again, if we launch a wave with omega i, 01:08:13.930 --> 01:08:21.979 align:middle line:84% Ki, and we detect a wave being scattered 01:08:21.979 --> 01:08:27.170 align:middle line:84% at a certain frequency in a certain direction here, 01:08:27.170 --> 01:08:36.120 align:middle line:84% then this implies that some wave omega 01:08:36.120 --> 01:08:47.060 align:middle line:84% K, where omega is omega s minus omega i, and K is Ks minus Ki, 01:08:47.060 --> 01:08:50.880 align:middle line:90% implies that this wave exists. 01:08:50.880 --> 01:08:53.130 align:middle line:84% But there is something within the plasma that 01:08:53.130 --> 01:08:56.220 align:middle line:84% is capable of taking away or adding the momentum 01:08:56.220 --> 01:08:58.649 align:middle line:84% and the energy we need in order to be 01:08:58.649 --> 01:09:02.229 align:middle line:90% able to see this mode here. 01:09:02.229 --> 01:09:10.340 align:middle line:84% So, for example, if we have a spectrum 01:09:10.340 --> 01:09:12.229 align:middle line:90% on our spectrometer here-- 01:09:12.229 --> 01:09:16.880 align:middle line:84% and this is in units of omega, omega s minus omega i, 01:09:16.880 --> 01:09:18.380 align:middle line:84% so I'm just showing you the shift-- 01:09:18.380 --> 01:09:22.160 align:middle line:84% then our incident laser is going to be 01:09:22.160 --> 01:09:25.399 align:middle line:90% at omega i, which is just 0. 01:09:25.399 --> 01:09:27.407 align:middle line:84% So if we accidentally reflect this laser beam 01:09:27.407 --> 01:09:29.990 align:middle line:84% off the metal inside the chamber and catch the scattered light 01:09:29.990 --> 01:09:35.149 align:middle line:84% on our spectrometer, we will see it at omega i unshifted, 01:09:35.149 --> 01:09:36.800 align:middle line:90% like that. 01:09:36.800 --> 01:09:41.260 align:middle line:84% If we now put a detector at 10 degrees, 01:09:41.260 --> 01:09:45.250 align:middle line:84% we might, for example, see scattering here. 01:09:45.250 --> 01:09:50.450 align:middle line:84% And this scattering here, omega equals 01:09:50.450 --> 01:10:01.130 align:middle line:84% K dot v, equals 2 Ki sine theta over 2 v, 01:10:01.130 --> 01:10:04.610 align:middle line:84% will be proportional not only to the velocity of the particle 01:10:04.610 --> 01:10:06.320 align:middle line:84% that we scattered off, but it will also 01:10:06.320 --> 01:10:07.710 align:middle line:90% be related to the angle. 01:10:07.710 --> 01:10:16.110 align:middle line:84% So now if we put another detector at a different angle, 01:10:16.110 --> 01:10:19.760 align:middle line:84% we will see an omega which is larger because the angle is 01:10:19.760 --> 01:10:20.630 align:middle line:90% larger as well. 01:10:20.630 --> 01:10:27.780 align:middle line:90% 01:10:27.780 --> 01:10:29.390 align:middle line:84% So what exactly are we scattering off? 01:10:29.390 --> 01:10:33.572 align:middle line:84% Well, in this picture here, what we've drawn above, 01:10:33.572 --> 01:10:36.030 align:middle line:84% we're talking about something called incoherent scattering. 01:10:36.030 --> 01:10:38.610 align:middle line:84% And we'll get back to what this means later on. 01:10:38.610 --> 01:10:41.730 align:middle line:84% But incoherent scattering, we scatter off a particle. 01:10:41.730 --> 01:10:44.202 align:middle line:90% 01:10:44.202 --> 01:10:45.410 align:middle line:90% That's what we've drawn here. 01:10:45.410 --> 01:10:46.490 align:middle line:90% There's some particle. 01:10:46.490 --> 01:10:52.460 align:middle line:84% And that particle has a velocity v. And that v causes a shift, 01:10:52.460 --> 01:10:55.400 align:middle line:90% omega equals K dot v. 01:10:55.400 --> 01:10:58.940 align:middle line:84% So the particle is responsible for balancing this momentum 01:10:58.940 --> 01:11:00.170 align:middle line:90% and energy equation. 01:11:00.170 --> 01:11:03.350 align:middle line:84% It is the K, and it is effectively the omega 01:11:03.350 --> 01:11:05.380 align:middle line:90% here as well. 01:11:05.380 --> 01:11:07.800 align:middle line:84% There's also another regime of Thomson scattering, though, 01:11:07.800 --> 01:11:11.520 align:middle line:84% and this is called coherent Thomson scattering. 01:11:11.520 --> 01:11:17.050 align:middle line:84% And in coherent scattering, we scatter off waves in the plasma. 01:11:17.050 --> 01:11:21.500 align:middle line:84% And waves also have omega and K. And they 01:11:21.500 --> 01:11:25.730 align:middle line:84% have some distribution which links these-- for example, omega 01:11:25.730 --> 01:11:29.460 align:middle line:90% equals Cs K. 01:11:29.460 --> 01:11:40.830 align:middle line:84% This Cs, the sound speed, square root of ZTe plus Ti upon Mi 01:11:40.830 --> 01:11:46.100 align:middle line:84% means that if we now measure a shift on our spectrometer, 01:11:46.100 --> 01:11:49.730 align:middle line:84% we can interpret that omega as being due to, for example, 01:11:49.730 --> 01:11:50.870 align:middle line:90% this sound wave. 01:11:50.870 --> 01:11:53.240 align:middle line:84% And from the shift that we see, we 01:11:53.240 --> 01:11:55.410 align:middle line:84% can now infer the plasma temperature. 01:11:55.410 --> 01:11:57.260 align:middle line:84% So this very roughly is giving you 01:11:57.260 --> 01:11:59.880 align:middle line:84% an idea of how we will use coherent waves. 01:11:59.880 --> 01:12:02.940 align:middle line:84% It will make this very, very mathematical. 01:12:02.940 --> 01:12:05.090 align:middle line:84% OK, there's a lot going on there. 01:12:05.090 --> 01:12:07.910 align:middle line:90% Question? 01:12:07.910 --> 01:12:09.188 align:middle line:90% Yes? 01:12:09.188 --> 01:12:15.440 align:middle line:84% AUDIENCE: How-- you said that [INAUDIBLE]. 01:12:15.440 --> 01:12:17.750 align:middle line:90% JACK HARE: Yes. 01:12:17.750 --> 01:12:19.060 align:middle line:90% AUDIENCE: How [INAUDIBLE]? 01:12:19.060 --> 01:12:23.183 align:middle line:90% 01:12:23.183 --> 01:12:24.350 align:middle line:90% JACK HARE: Yeah, absolutely. 01:12:24.350 --> 01:12:25.880 align:middle line:90% So there's your background. 01:12:25.880 --> 01:12:27.740 align:middle line:84% You've got bremsstrahlung, and you've got line emission, 01:12:27.740 --> 01:12:28.460 align:middle line:90% and you've got synchrotron. 01:12:28.460 --> 01:12:29.752 align:middle line:90% You've got all sorts of things. 01:12:29.752 --> 01:12:33.530 align:middle line:84% You pick a probe, as in you pick a laser with an incident 01:12:33.530 --> 01:12:37.010 align:middle line:84% frequency which is not near any other strong features 01:12:37.010 --> 01:12:37.910 align:middle line:90% in your plasma. 01:12:37.910 --> 01:12:39.600 align:middle line:84% Bremsstrahlung is a nice background. 01:12:39.600 --> 01:12:41.840 align:middle line:90% So if you have bremsstrahlung-- 01:12:41.840 --> 01:12:44.600 align:middle line:90% this is wavelength here. 01:12:44.600 --> 01:12:47.570 align:middle line:84% If you have bremsstrahlung, and you're doing Thomson scattering, 01:12:47.570 --> 01:12:49.700 align:middle line:84% you should see a very clear peak from it. 01:12:49.700 --> 01:12:52.993 align:middle line:84% If you've got bremsstrahlung, and you've got lots of lines, 01:12:52.993 --> 01:12:54.410 align:middle line:84% then it is a bit difficult to tell 01:12:54.410 --> 01:12:57.448 align:middle line:84% the difference between the lines and your Thomson scattering. 01:12:57.448 --> 01:12:59.240 align:middle line:84% So you want to pick your Thomson scattering 01:12:59.240 --> 01:13:01.848 align:middle line:84% probe so it's not close to any other features there. 01:13:01.848 --> 01:13:03.890 align:middle line:84% Then it's usually pretty clear because it's like, 01:13:03.890 --> 01:13:05.012 align:middle line:90% what else could it be? 01:13:05.012 --> 01:13:06.470 align:middle line:84% If we see scattered light there, we 01:13:06.470 --> 01:13:07.928 align:middle line:84% know there aren't any atomic lines. 01:13:07.928 --> 01:13:09.396 align:middle line:90% So it is likely [INAUDIBLE]. 01:13:09.396 --> 01:13:14.330 align:middle line:84% AUDIENCE: [INAUDIBLE] very common. 01:13:14.330 --> 01:13:16.300 align:middle line:90% JACK HARE: Yes. 01:13:16.300 --> 01:13:16.800 align:middle line:90% Yes. 01:13:16.800 --> 01:13:18.100 align:middle line:90% There are also advantages. 01:13:18.100 --> 01:13:19.558 align:middle line:84% For example, a lot of the radiation 01:13:19.558 --> 01:13:22.110 align:middle line:84% that's emitted from the plasma is unpolarized, 01:13:22.110 --> 01:13:24.600 align:middle line:84% whereas the Thomson scattering radiation is polarized. 01:13:24.600 --> 01:13:27.030 align:middle line:84% And so we can use a polarizer to help reject some 01:13:27.030 --> 01:13:28.410 align:middle line:90% of that unpolarized light-- 01:13:28.410 --> 01:13:29.320 align:middle line:90% half of it. 01:13:29.320 --> 01:13:31.482 align:middle line:84% So there are various techniques you can do. 01:13:31.482 --> 01:13:33.940 align:middle line:84% The Thomson scattering probe is usually very short in time. 01:13:33.940 --> 01:13:35.610 align:middle line:84% So if you have a gated camera, you only 01:13:35.610 --> 01:13:37.450 align:middle line:84% collect light while the probe is active. 01:13:37.450 --> 01:13:39.492 align:middle line:84% So you don't collect so much of the self-emission 01:13:39.492 --> 01:13:42.180 align:middle line:84% from the rest of the time that the plasma is there. 01:13:42.180 --> 01:13:44.322 align:middle line:84% But it is challenging to get above the background. 01:13:44.322 --> 01:13:46.530 align:middle line:84% And, again, this is the answer to Lansing's question. 01:13:46.530 --> 01:13:47.700 align:middle line:90% You need a big laser. 01:13:47.700 --> 01:13:49.705 align:middle line:84% You need a lot of scattered light. 01:13:49.705 --> 01:13:50.205 align:middle line:90% Yeah. 01:13:50.205 --> 01:13:54.010 align:middle line:90% 01:13:54.010 --> 01:13:57.460 align:middle line:90% Joules in nanoseconds? 01:13:57.460 --> 01:13:58.790 align:middle line:90% Yeah. 01:13:58.790 --> 01:14:00.140 align:middle line:90% That's really big. 01:14:00.140 --> 01:14:02.300 align:middle line:90% [LAUGHS] Yeah. 01:14:02.300 --> 01:14:05.500 align:middle line:90% 01:14:05.500 --> 01:14:08.215 align:middle line:84% Yeah, for example, on the [INAUDIBLE] tokamak, 01:14:08.215 --> 01:14:10.840 align:middle line:84% where they're trying to do this in a sort of time-resolved way, 01:14:10.840 --> 01:14:13.840 align:middle line:84% they have a huge bank of Nd:YAG lasers that they trigger 01:14:13.840 --> 01:14:15.950 align:middle line:84% sequentially and send down the same beams. 01:14:15.950 --> 01:14:17.590 align:middle line:84% They're like, pulse, pulse, pulse, pulse, pulse, and then 01:14:17.590 --> 01:14:19.870 align:middle line:84% they have a load of spectrometers which are timed 01:14:19.870 --> 01:14:20.953 align:middle line:90% with those pulses as well. 01:14:20.953 --> 01:14:22.953 align:middle line:84% And so then they're doing Thomson scattering at, 01:14:22.953 --> 01:14:25.540 align:middle line:84% like, I think, 100 kilohertz or something like that using 01:14:25.540 --> 01:14:26.830 align:middle line:90% the staggered bank of lasers. 01:14:26.830 --> 01:14:28.510 align:middle line:84% In my experiments, I can't get the laser 01:14:28.510 --> 01:14:31.810 align:middle line:84% to go twice during a microsecond because you can't easily 01:14:31.810 --> 01:14:34.490 align:middle line:84% get, like, a joule at a megahertz from a laser. 01:14:34.490 --> 01:14:35.830 align:middle line:90% It's terrifying. 01:14:35.830 --> 01:14:38.210 align:middle line:90% So then we just have one time. 01:14:38.210 --> 01:14:40.062 align:middle line:84% And then we have to gate around that time 01:14:40.062 --> 01:14:42.520 align:middle line:84% and make sure that we only collect Thomson scattering light 01:14:42.520 --> 01:14:43.853 align:middle line:90% during the time the laser is on. 01:14:43.853 --> 01:14:47.850 align:middle line:90% 01:14:47.850 --> 01:14:50.470 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:14:50.470 --> 01:14:57.100 align:middle line:90% 01:14:57.100 --> 01:14:59.100 align:middle line:84% JACK HARE: Yeah, so the technique I talked about 01:14:59.100 --> 01:15:01.650 align:middle line:84% was the low-temperature plasmas with the CO2. 01:15:01.650 --> 01:15:03.410 align:middle line:84% For example, they had a pulsed system. 01:15:03.410 --> 01:15:05.160 align:middle line:84% So they were constantly making the plasma. 01:15:05.160 --> 01:15:08.760 align:middle line:84% What they did is they had the laser synchronized with that-- 01:15:08.760 --> 01:15:12.720 align:middle line:84% so at whatever rep rate the plasma was being produced-- 01:15:12.720 --> 01:15:13.595 align:middle line:90% and they integrated. 01:15:13.595 --> 01:15:14.970 align:middle line:84% So they just added up the photons 01:15:14.970 --> 01:15:16.710 align:middle line:84% from every single experiment until they 01:15:16.710 --> 01:15:18.660 align:middle line:84% had enough photons to get a good spectrum 01:15:18.660 --> 01:15:21.870 align:middle line:84% because the trouble is the density is so low there 01:15:21.870 --> 01:15:24.780 align:middle line:84% that just from a single shot, you don't really 01:15:24.780 --> 01:15:26.610 align:middle line:90% get very much scattered light. 01:15:26.610 --> 01:15:29.133 align:middle line:84% But if you do it for hours at a kilohertz, 01:15:29.133 --> 01:15:31.800 align:middle line:84% you start to get enough photons that you build up your spectrum. 01:15:31.800 --> 01:15:34.290 align:middle line:84% For that to work, you have to be really sure your plasma is 01:15:34.290 --> 01:15:35.820 align:middle line:90% the same every time. 01:15:35.820 --> 01:15:38.360 align:middle line:84% And they were, or they felt that they were sure that it 01:15:38.360 --> 01:15:39.360 align:middle line:90% was the same every time. 01:15:39.360 --> 01:15:40.943 align:middle line:84% Obviously, if your plasma is changing, 01:15:40.943 --> 01:15:43.410 align:middle line:84% if we're dealing with a very unstable system where 01:15:43.410 --> 01:15:46.060 align:middle line:84% the plasma is moving around, and then the Thomson scattering 01:15:46.060 --> 01:15:48.060 align:middle line:84% probe will be sampling a different bit of plasma 01:15:48.060 --> 01:15:49.410 align:middle line:84% every time, that's not really reasonable. 01:15:49.410 --> 01:15:51.540 align:middle line:84% But if you've got a nice homogeneous plasma, 01:15:51.540 --> 01:15:52.563 align:middle line:90% you've got it. 01:15:52.563 --> 01:15:54.480 align:middle line:84% So that's a nice situation to be in, actually, 01:15:54.480 --> 01:15:57.040 align:middle line:84% because if you want better signal-to-noise, 01:15:57.040 --> 01:15:59.617 align:middle line:84% you can just run the system for, like, another day. 01:15:59.617 --> 01:16:02.200 align:middle line:84% Of course, your signal-to-noise usually goes to square root n. 01:16:02.200 --> 01:16:03.760 align:middle line:90% So you get diminishing returns. 01:16:03.760 --> 01:16:05.640 align:middle line:90% But yeah. 01:16:05.640 --> 01:16:09.000 align:middle line:90%