WEBVTT 00:00:00.000 --> 00:00:01.485 align:middle line:90% 00:00:01.485 --> 00:00:02.475 align:middle line:90% [SQUEAKING] 00:00:02.475 --> 00:00:03.465 align:middle line:90% [RUSTLING] 00:00:03.465 --> 00:00:05.940 align:middle line:90% [CLICKING] 00:00:05.940 --> 00:00:09.910 align:middle line:90% 00:00:09.910 --> 00:00:13.400 align:middle line:84% JACK HARE: So just a quick recap of what we covered on Tuesday, 00:00:13.400 --> 00:00:15.880 align:middle line:84% and then we will continue and look 00:00:15.880 --> 00:00:18.900 align:middle line:90% at electron cyclotron emission. 00:00:18.900 --> 00:00:25.660 align:middle line:84% So first of all, we had a look at the radiation transport 00:00:25.660 --> 00:00:26.160 align:middle line:90% equation. 00:00:26.160 --> 00:00:40.190 align:middle line:90% 00:00:40.190 --> 00:00:42.830 align:middle line:84% And we wrote this radiation transport equation 00:00:42.830 --> 00:00:47.330 align:middle line:84% in terms of the intensity of light-- 00:00:47.330 --> 00:00:49.400 align:middle line:84% spectral radiance, to be correct-- 00:00:49.400 --> 00:00:53.600 align:middle line:84% coming out of the plasma and how it changes along the path 00:00:53.600 --> 00:00:54.530 align:middle line:90% through the plasma. 00:00:54.530 --> 00:00:57.920 align:middle line:84% So we've got a little plasma here 00:00:57.920 --> 00:00:59.840 align:middle line:84% and we've got some path going through it 00:00:59.840 --> 00:01:01.850 align:middle line:90% that we parameterize by s. 00:01:01.850 --> 00:01:04.099 align:middle line:84% The change in this spectral radiance 00:01:04.099 --> 00:01:10.130 align:middle line:84% is equal to a quantity j, the emissivity, minus alpha times 00:01:10.130 --> 00:01:11.240 align:middle line:90% the spectral radiance. 00:01:11.240 --> 00:01:13.760 align:middle line:90% Alpha is the opacity here. 00:01:13.760 --> 00:01:16.910 align:middle line:84% And it's convenient to rewrite this equation in terms 00:01:16.910 --> 00:01:21.740 align:middle line:84% of a dimensionless parameter tau by dividing through by alpha. 00:01:21.740 --> 00:01:25.280 align:middle line:84% And then we get this compact form of equation, 00:01:25.280 --> 00:01:32.760 align:middle line:84% di d tau is equal to j from the alpha minus i. 00:01:32.760 --> 00:01:35.550 align:middle line:84% And what we've done is we've defined this parameter 00:01:35.550 --> 00:01:41.540 align:middle line:84% tau as being equal to the integral of alpha ds here. 00:01:41.540 --> 00:01:43.880 align:middle line:84% And again, this is an indefinite integral. 00:01:43.880 --> 00:01:46.430 align:middle line:84% It's only when we start putting limits on this 00:01:46.430 --> 00:01:48.920 align:middle line:84% that we know whereabouts we're integrating from 00:01:48.920 --> 00:01:50.490 align:middle line:90% and integrating to. 00:01:50.490 --> 00:01:53.240 align:middle line:84% And the other thing to say is that this equation 00:01:53.240 --> 00:01:59.160 align:middle line:90% is true for all frequencies. 00:01:59.160 --> 00:02:03.870 align:middle line:84% Maybe we'll write these in terms of cyclic frequency or rads 00:02:03.870 --> 00:02:07.860 align:middle line:84% per second or wavelength or energy, whatever it is. 00:02:07.860 --> 00:02:11.610 align:middle line:84% Just some way that we're measuring the difference 00:02:11.610 --> 00:02:12.990 align:middle line:90% in the spectrum here. 00:02:12.990 --> 00:02:15.628 align:middle line:84% And this equation then can be solved separately 00:02:15.628 --> 00:02:17.670 align:middle line:84% for all the different frequencies in your system. 00:02:17.670 --> 00:02:20.087 align:middle line:84% In our assumption here, we're not allowing the frequencies 00:02:20.087 --> 00:02:21.120 align:middle line:90% to interact in any way. 00:02:21.120 --> 00:02:23.400 align:middle line:84% There's no second harmonic generation. 00:02:23.400 --> 00:02:25.590 align:middle line:84% There's no absorption and fluorescence 00:02:25.590 --> 00:02:27.430 align:middle line:90% and things like that. 00:02:27.430 --> 00:02:30.690 align:middle line:84% And this tau parameter is a nice dimensionless parameter, 00:02:30.690 --> 00:02:32.190 align:middle line:84% so of course the first thing we want 00:02:32.190 --> 00:02:35.130 align:middle line:84% to do is work out what it means for that dimensionless parameter 00:02:35.130 --> 00:02:38.220 align:middle line:84% to be small and what it means for that dimensionless parameter 00:02:38.220 --> 00:02:40.020 align:middle line:90% to be large. 00:02:40.020 --> 00:02:42.630 align:middle line:84% In the case where the parameter is small, 00:02:42.630 --> 00:02:46.170 align:middle line:84% we find a plasma which we call optically thin, which 00:02:46.170 --> 00:02:48.750 align:middle line:84% means the radiation streams through it without being 00:02:48.750 --> 00:02:51.690 align:middle line:84% absorbed and there's minimal emission. 00:02:51.690 --> 00:02:54.420 align:middle line:84% In the opposite case where tau is greater than one, 00:02:54.420 --> 00:02:59.350 align:middle line:84% our plasma is thick and incident radiation is strongly absorbed 00:02:59.350 --> 00:03:03.220 align:middle line:84% and the plasma is also strongly emitting. 00:03:03.220 --> 00:03:06.070 align:middle line:84% And in fact, we found out that in this thick case, 00:03:06.070 --> 00:03:08.140 align:middle line:84% our plasma is going to be so strongly absorbing 00:03:08.140 --> 00:03:10.270 align:middle line:84% that it's going to approximate a black body. 00:03:10.270 --> 00:03:14.920 align:middle line:90% 00:03:14.920 --> 00:03:19.580 align:middle line:84% The black body has a spectral radiance 00:03:19.580 --> 00:03:24.570 align:middle line:84% that looks like mu squared upon c squared times the temperature. 00:03:24.570 --> 00:03:26.810 align:middle line:84% So it doesn't depend on any particular properties 00:03:26.810 --> 00:03:27.740 align:middle line:90% of your material. 00:03:27.740 --> 00:03:29.270 align:middle line:84% It just depends on its temperature. 00:03:29.270 --> 00:03:31.460 align:middle line:84% This is the ratty jeans approximation. 00:03:31.460 --> 00:03:33.290 align:middle line:84% The full black body is more complicated, 00:03:33.290 --> 00:03:36.650 align:middle line:84% but most of the time we can use this ratty jeans approximation. 00:03:36.650 --> 00:03:38.420 align:middle line:84% And the neat thing about that is we 00:03:38.420 --> 00:03:41.700 align:middle line:84% found that there was a correspondence 00:03:41.700 --> 00:03:45.060 align:middle line:84% between this equation here and the optically thick limit 00:03:45.060 --> 00:03:47.670 align:middle line:84% and the black body equation, which is Kirchhoff's law. 00:03:47.670 --> 00:03:53.310 align:middle line:90% 00:03:53.310 --> 00:03:56.370 align:middle line:84% And Kirchhoff's law lets us set the emissivity 00:03:56.370 --> 00:04:00.600 align:middle line:84% over the opacity equal to the frequency squared 00:04:00.600 --> 00:04:03.730 align:middle line:84% over the speed of light squared times the temperature. 00:04:03.730 --> 00:04:07.200 align:middle line:84% This effectively tells us that the emissivity and the opacity 00:04:07.200 --> 00:04:08.650 align:middle line:90% are linked together. 00:04:08.650 --> 00:04:11.340 align:middle line:84% And so we've calculated, one we automatically 00:04:11.340 --> 00:04:12.720 align:middle line:90% have a calculation of the other. 00:04:12.720 --> 00:04:14.970 align:middle line:84% And simultaneously, if we have a region 00:04:14.970 --> 00:04:17.250 align:middle line:84% with high emission at some frequency, 00:04:17.250 --> 00:04:20.500 align:middle line:84% that's also a region with high absorption as well. 00:04:20.500 --> 00:04:22.220 align:middle line:84% So these two quantities are linked. 00:04:22.220 --> 00:04:30.820 align:middle line:90% 00:04:30.820 --> 00:04:31.570 align:middle line:90% OK. 00:04:31.570 --> 00:04:35.050 align:middle line:84% And then we talked about how to actually calculate 00:04:35.050 --> 00:04:38.920 align:middle line:84% j, and therefore alpha, so that we can use this radiation 00:04:38.920 --> 00:04:42.040 align:middle line:84% transport equation to see how plasma 00:04:42.040 --> 00:04:45.640 align:middle line:84% allows electromagnetic radiation to propagate through it. 00:04:45.640 --> 00:04:49.070 align:middle line:84% And we did a sort of pictorial representation 00:04:49.070 --> 00:04:52.000 align:middle line:84% and Tucker sent around a very good web demo 00:04:52.000 --> 00:04:53.560 align:middle line:84% that you can play with to play with 00:04:53.560 --> 00:04:57.790 align:middle line:84% that pictorial representation of y accelerating charges radiate. 00:04:57.790 --> 00:05:00.010 align:middle line:84% And we also wrote down a very long equation 00:05:00.010 --> 00:05:02.800 align:middle line:84% and then immediately dropped the near field term. 00:05:02.800 --> 00:05:07.180 align:middle line:84% But the remaining term, the far field electromagnetic radiation 00:05:07.180 --> 00:05:10.660 align:middle line:84% from moving charge, was equal to something 00:05:10.660 --> 00:05:15.280 align:middle line:84% with a bunch of constants and a few new variables related 00:05:15.280 --> 00:05:20.110 align:middle line:84% to where we are with respect to the particle. 00:05:20.110 --> 00:05:22.120 align:middle line:84% But the key part of it was that we 00:05:22.120 --> 00:05:24.670 align:middle line:84% started getting these cross products here 00:05:24.670 --> 00:05:29.020 align:middle line:84% that have within them some properties of the particle 00:05:29.020 --> 00:05:36.450 align:middle line:84% itself, namely its velocity crossed with its acceleration. 00:05:36.450 --> 00:05:38.970 align:middle line:84% So it's clear that in order to estimate the radiation 00:05:38.970 --> 00:05:41.010 align:middle line:84% from a moving charge, we're going to have 00:05:41.010 --> 00:05:43.170 align:middle line:90% to know these two properties. 00:05:43.170 --> 00:05:48.860 align:middle line:84% And so the recipe that we're going to follow 00:05:48.860 --> 00:05:56.363 align:middle line:84% is we're going to firstly calculate v and v dot. 00:05:56.363 --> 00:05:58.655 align:middle line:84% And we're going to do that with the equation of motion. 00:05:58.655 --> 00:06:05.040 align:middle line:90% 00:06:05.040 --> 00:06:08.070 align:middle line:84% And then secondly, now we're given the radiation 00:06:08.070 --> 00:06:09.510 align:middle line:90% from just a single charge. 00:06:09.510 --> 00:06:15.360 align:middle line:84% We then need to integrate this over the distribution function 00:06:15.360 --> 00:06:19.620 align:middle line:84% f of v v3v to get out the emission from all 00:06:19.620 --> 00:06:21.780 align:middle line:84% of the charges in the plasma together. 00:06:21.780 --> 00:06:27.670 align:middle line:84% And that is going to give us the emissivity at a single point. 00:06:27.670 --> 00:06:29.550 align:middle line:84% And then of course, because they're linked, 00:06:29.550 --> 00:06:31.050 align:middle line:90% also the opacity. 00:06:31.050 --> 00:06:34.020 align:middle line:84% So this is what we're going after. 00:06:34.020 --> 00:06:37.785 align:middle line:84% Any questions on that before we move on to an example of this? 00:06:37.785 --> 00:06:40.290 align:middle line:90% 00:06:40.290 --> 00:06:41.160 align:middle line:90% Yes? 00:06:41.160 --> 00:06:43.920 align:middle line:84% STUDENT: Can you explain again how we got the near field 00:06:43.920 --> 00:06:45.593 align:middle line:90% in the electric field? 00:06:45.593 --> 00:06:46.260 align:middle line:90% JACK HARE: Yeah. 00:06:46.260 --> 00:06:48.810 align:middle line:84% So the near field term-- which I haven't written down here. 00:06:48.810 --> 00:07:02.780 align:middle line:90% 00:07:02.780 --> 00:07:07.700 align:middle line:84% The near field term goes as 1 upon r 00:07:07.700 --> 00:07:12.170 align:middle line:84% squared whereas the far field term goes as 1 upon r. 00:07:12.170 --> 00:07:17.405 align:middle line:84% So when you calculate s is equal to e cross b-- 00:07:17.405 --> 00:07:20.820 align:middle line:90% 00:07:20.820 --> 00:07:23.930 align:middle line:90% [INAUDIBLE] 00:07:23.930 --> 00:07:26.990 align:middle line:84% The magnetic field is going to have the same radial dependence, 00:07:26.990 --> 00:07:30.212 align:middle line:84% the same drop off in distance as the electric field. 00:07:30.212 --> 00:07:31.670 align:middle line:84% And so the Poynting vector is going 00:07:31.670 --> 00:07:34.280 align:middle line:90% to go as one upon r to the 4. 00:07:34.280 --> 00:07:37.130 align:middle line:84% And that means that if you go far enough away 00:07:37.130 --> 00:07:40.940 align:middle line:84% from the particle, that electric fields and therefore 00:07:40.940 --> 00:07:42.800 align:middle line:84% the radiation from it is going to drop off 00:07:42.800 --> 00:07:44.210 align:middle line:90% faster than r squared. 00:07:44.210 --> 00:07:47.160 align:middle line:84% And so you're not going to see it from any reasonable distance 00:07:47.160 --> 00:07:47.660 align:middle line:90% away. 00:07:47.660 --> 00:07:49.368 align:middle line:84% In fact, the distance you need to be away 00:07:49.368 --> 00:07:51.290 align:middle line:84% is a few wavelengths of the radiation, 00:07:51.290 --> 00:07:53.250 align:middle line:84% and so that's usually quite short. 00:07:53.250 --> 00:07:54.798 align:middle line:84% So we don't have to worry about this. 00:07:54.798 --> 00:07:56.840 align:middle line:84% Effectively what your electric field looks like-- 00:07:56.840 --> 00:07:59.420 align:middle line:84% and I had a little chat with Sean about this after the class, 00:07:59.420 --> 00:08:01.760 align:middle line:90% so hopefully I remember it. 00:08:01.760 --> 00:08:03.510 align:middle line:84% The electric field that you as an observer 00:08:03.510 --> 00:08:05.670 align:middle line:84% would see at some function of distance 00:08:05.670 --> 00:08:11.670 align:middle line:84% has 1 over r component, which is the far field, a much 00:08:11.670 --> 00:08:17.920 align:middle line:84% more steeply dropping 1 over r to the 4 component, 00:08:17.920 --> 00:08:19.110 align:middle line:90% which is near field. 00:08:19.110 --> 00:08:20.517 align:middle line:84% And so you don't, as an observer, 00:08:20.517 --> 00:08:22.350 align:middle line:84% get to see the difference between these two. 00:08:22.350 --> 00:08:23.320 align:middle line:90% They look the same. 00:08:23.320 --> 00:08:26.160 align:middle line:84% What you'll end up seeing is just some field 00:08:26.160 --> 00:08:27.930 align:middle line:90% that looks like this. 00:08:27.930 --> 00:08:29.610 align:middle line:84% The near field component won't show up 00:08:29.610 --> 00:08:31.950 align:middle line:84% at any sensible distance away from the plasma 00:08:31.950 --> 00:08:33.090 align:middle line:90% so we never need to-- 00:08:33.090 --> 00:08:36.030 align:middle line:84% we do not need to worry about it in plasma physics 00:08:36.030 --> 00:08:36.870 align:middle line:90% as far as I know. 00:08:36.870 --> 00:08:38.820 align:middle line:84% But maybe there is some special case 00:08:38.820 --> 00:08:40.330 align:middle line:90% you do need to think about it. 00:08:40.330 --> 00:08:42.820 align:middle line:84% We're going to drop it in these lectures. 00:08:42.820 --> 00:08:43.535 align:middle line:90% Yeah, Shawn? 00:08:43.535 --> 00:08:45.910 align:middle line:84% STUDENT: It makes sense that we don't need the near field 00:08:45.910 --> 00:08:48.070 align:middle line:84% for trying to observe the radiation that's emitted, 00:08:48.070 --> 00:08:50.710 align:middle line:84% but if you're trying to couple the emitted radiation back 00:08:50.710 --> 00:08:53.110 align:middle line:84% to the plasma for the absorption, 00:08:53.110 --> 00:08:55.240 align:middle line:84% shouldn't the near field contribute to that 00:08:55.240 --> 00:08:56.680 align:middle line:84% because you do have particles that 00:08:56.680 --> 00:08:59.890 align:middle line:90% are near the emitting particle? 00:08:59.890 --> 00:09:01.817 align:middle line:90% 00:09:01.817 --> 00:09:03.900 align:middle line:84% JACK HARE: OK, so to the benefit of people online, 00:09:03.900 --> 00:09:06.930 align:middle line:84% the question is, doesn't the near field perturb the plasma? 00:09:06.930 --> 00:09:07.472 align:middle line:90% I don't know. 00:09:07.472 --> 00:09:09.763 align:middle line:84% I could handwave and say something about the y lengths. 00:09:09.763 --> 00:09:12.390 align:middle line:84% The plasma is very good at shielding out electric fields. 00:09:12.390 --> 00:09:13.320 align:middle line:90% Right? 00:09:13.320 --> 00:09:16.470 align:middle line:84% And so maybe it will depend on the wavelength of your radiation 00:09:16.470 --> 00:09:19.650 align:middle line:84% compared to the Debye length of that. 00:09:19.650 --> 00:09:21.030 align:middle line:90% But I'm just spitballing here. 00:09:21.030 --> 00:09:22.590 align:middle line:90% I don't know the answer to that. 00:09:22.590 --> 00:09:23.770 align:middle line:90% Yeah. 00:09:23.770 --> 00:09:24.270 align:middle line:90% Thank you. 00:09:24.270 --> 00:09:25.590 align:middle line:90% You're going to love my next-- 00:09:25.590 --> 00:09:27.370 align:middle line:84% I went to talk to Paul Bernoulli and I'm like, 00:09:27.370 --> 00:09:28.110 align:middle line:90% hey, Paul, is this right? 00:09:28.110 --> 00:09:29.250 align:middle line:84% And he's like, yeah, that sounds right. 00:09:29.250 --> 00:09:31.875 align:middle line:84% So you're going to love my next approximation and a little bit, 00:09:31.875 --> 00:09:32.970 align:middle line:90% so. 00:09:32.970 --> 00:09:33.540 align:middle line:90% OK. 00:09:33.540 --> 00:09:34.390 align:middle line:90% Any other questions? 00:09:34.390 --> 00:09:35.057 align:middle line:90% Anything online? 00:09:35.057 --> 00:09:41.020 align:middle line:90% 00:09:41.020 --> 00:09:41.560 align:middle line:90% OK. 00:09:41.560 --> 00:09:42.280 align:middle line:90% Let's keep going. 00:09:42.280 --> 00:09:53.780 align:middle line:90% 00:09:53.780 --> 00:10:07.450 align:middle line:84% So we're going to be looking at electron cyclotron 00:10:07.450 --> 00:10:14.640 align:middle line:90% emission or ECE. 00:10:14.640 --> 00:10:18.870 align:middle line:84% This is a key diagnostic in magnetically confined flasks. 00:10:18.870 --> 00:10:21.030 align:middle line:84% This is a very, very important diagnostic, 00:10:21.030 --> 00:10:23.280 align:middle line:84% which is why we're going to spend a significant amount 00:10:23.280 --> 00:10:25.200 align:middle line:90% of time on it. 00:10:25.200 --> 00:10:27.720 align:middle line:84% This diagnostic works on the basis 00:10:27.720 --> 00:10:32.010 align:middle line:84% that we have magnetic field lines inside our plasma 00:10:32.010 --> 00:10:34.560 align:middle line:84% and we have particles, electrons, 00:10:34.560 --> 00:10:38.640 align:middle line:84% which are spiraling around those magnetic field lines. 00:10:38.640 --> 00:10:42.540 align:middle line:84% There is also ion cyclotron emission as well, which 00:10:42.540 --> 00:10:44.430 align:middle line:90% we are not going to cover here. 00:10:44.430 --> 00:10:47.100 align:middle line:84% I believe that's because it has a much harder time propagating 00:10:47.100 --> 00:10:48.030 align:middle line:90% out of the plasma. 00:10:48.030 --> 00:10:51.030 align:middle line:84% We'll discuss a little bit about that later on. 00:10:51.030 --> 00:10:54.820 align:middle line:84% At the moment, we're just focusing on the electrons here. 00:10:54.820 --> 00:10:58.000 align:middle line:84% And so what we want to do, as we said before, 00:10:58.000 --> 00:10:59.565 align:middle line:84% is write down an equation of motion. 00:10:59.565 --> 00:11:02.680 align:middle line:90% 00:11:02.680 --> 00:11:04.630 align:middle line:84% And that equation of motion is going 00:11:04.630 --> 00:11:07.810 align:middle line:84% to be simply that the time rate of change 00:11:07.810 --> 00:11:11.870 align:middle line:84% of the momentum of our electron is 00:11:11.870 --> 00:11:14.670 align:middle line:84% going to be equal to the force on our electron. 00:11:14.670 --> 00:11:19.220 align:middle line:90% So this is ev cross B here. 00:11:19.220 --> 00:11:21.650 align:middle line:84% We're assuming in this formula that there 00:11:21.650 --> 00:11:23.703 align:middle line:90% are no electric fields. 00:11:23.703 --> 00:11:25.370 align:middle line:84% So we're just dealing with the particles 00:11:25.370 --> 00:11:27.140 align:middle line:90% gyrating the magnetic fields. 00:11:27.140 --> 00:11:28.280 align:middle line:90% You've seen this before. 00:11:28.280 --> 00:11:29.510 align:middle line:90% You've solved this before. 00:11:29.510 --> 00:11:31.340 align:middle line:84% The one thing that Hutchinson puts 00:11:31.340 --> 00:11:35.460 align:middle line:84% inside here that we didn't have previously is this factor gamma. 00:11:35.460 --> 00:11:38.360 align:middle line:84% So this is the relativistic gamma here, 00:11:38.360 --> 00:11:46.310 align:middle line:84% and that is 1 minus v squared upon c squared minus 1/2. 00:11:46.310 --> 00:11:49.190 align:middle line:84% The gamma is always greater than 1. 00:11:49.190 --> 00:11:51.800 align:middle line:84% And this is a factor that effectively in this case, 00:11:51.800 --> 00:11:54.050 align:middle line:84% you want to think about it as increasing the rest 00:11:54.050 --> 00:11:57.560 align:middle line:84% mass of our particle, and this is because our particles tend 00:11:57.560 --> 00:12:01.070 align:middle line:84% to be moving in a MCF plasma where the electrons are 00:12:01.070 --> 00:12:03.260 align:middle line:90% at 10 keV or 20 keV. 00:12:03.260 --> 00:12:05.570 align:middle line:84% That's a significant fraction of their rest mass, 00:12:05.570 --> 00:12:08.570 align:middle line:84% and so there will be some relativistic effects 00:12:08.570 --> 00:12:10.560 align:middle line:90% on top of this. 00:12:10.560 --> 00:12:15.980 align:middle line:84% So in this case here, we know that this-- 00:12:15.980 --> 00:12:17.810 align:middle line:84% because we've solved this equation before, 00:12:17.810 --> 00:12:21.125 align:middle line:84% we know that this electron is going to oscillate. 00:12:21.125 --> 00:12:22.520 align:middle line:90% I guess I can be precise. 00:12:22.520 --> 00:12:31.740 align:middle line:84% I can say it's going to gyrate at a frequency omega 00:12:31.740 --> 00:12:35.670 align:middle line:90% c, the cyclotron frequency. 00:12:35.670 --> 00:12:37.600 align:middle line:84% And this is just ever so slightly different 00:12:37.600 --> 00:12:39.350 align:middle line:84% from our normal gyrofrequency because it's 00:12:39.350 --> 00:12:41.430 align:middle line:84% different by a factor of gamma here. 00:12:41.430 --> 00:12:43.650 align:middle line:84% So the gamma changes a little bit, 00:12:43.650 --> 00:12:50.000 align:middle line:84% and so this is just e0 over electron mass times my gamma. 00:12:50.000 --> 00:12:53.150 align:middle line:84% Again, if you just keep the gamma sticking with the electron 00:12:53.150 --> 00:12:56.780 align:middle line:84% mass you won't go too badly inside this. 00:12:56.780 --> 00:13:00.260 align:middle line:84% Because the electron is gyrating at this frequency, 00:13:00.260 --> 00:13:03.830 align:middle line:84% it is reasonable to believe that there is an electric field that 00:13:03.830 --> 00:13:06.900 align:middle line:84% generates which is also gyrating at that frequency. 00:13:06.900 --> 00:13:20.510 align:middle line:84% So this means we expect to have radiation at this frequency 00:13:20.510 --> 00:13:22.070 align:middle line:90% as well. 00:13:22.070 --> 00:13:22.760 align:middle line:90% And that's true. 00:13:22.760 --> 00:13:24.520 align:middle line:84% We do have radiation at frequency. 00:13:24.520 --> 00:13:26.080 align:middle line:84% But the thing I've been struggling 00:13:26.080 --> 00:13:28.310 align:middle line:84% to find an intuitive explanation for-- 00:13:28.310 --> 00:13:30.220 align:middle line:84% I went to go talk to Dr. Bernoulli about this 00:13:30.220 --> 00:13:31.540 align:middle line:90% just before this class-- 00:13:31.540 --> 00:13:34.240 align:middle line:84% is that we don't just get radiation at this frequency. 00:13:34.240 --> 00:13:37.700 align:middle line:84% When you go into the mathematics of it, 00:13:37.700 --> 00:13:39.550 align:middle line:84% you find out that you get radiation 00:13:39.550 --> 00:13:42.200 align:middle line:84% at integer multiples of that, at harmonics of that. 00:13:42.200 --> 00:13:48.380 align:middle line:84% So we also have this factor of m here where 00:13:48.380 --> 00:13:52.810 align:middle line:90% m is some natural number. 00:13:52.810 --> 00:13:56.270 align:middle line:84% Now mathematically, I can tell you where this comes from. 00:13:56.270 --> 00:14:02.350 align:middle line:84% It comes from, for example, Hutchinson's equation 5.2.7. 00:14:02.350 --> 00:14:04.810 align:middle line:84% What this is is there's an exponential 00:14:04.810 --> 00:14:07.838 align:middle line:84% of some horrific quantity and Hutchinson 00:14:07.838 --> 00:14:09.880 align:middle line:84% and everyone else says, well, fortunately there's 00:14:09.880 --> 00:14:12.910 align:middle line:84% a really great trigonometric identity that 00:14:12.910 --> 00:14:15.160 align:middle line:84% converts the exponential of this horrific quantity 00:14:15.160 --> 00:14:18.040 align:middle line:84% into a sum over an infinite number of Bessel functions. 00:14:18.040 --> 00:14:20.788 align:middle line:84% And those Bessel functions are all oscillating at harmonics 00:14:20.788 --> 00:14:23.080 align:middle line:84% here, and so therefore, you've got all these harmonics. 00:14:23.080 --> 00:14:23.500 align:middle line:90% Right? 00:14:23.500 --> 00:14:25.042 align:middle line:84% So mathematically it all makes sense. 00:14:25.042 --> 00:14:26.620 align:middle line:90% No one is upset about that. 00:14:26.620 --> 00:14:29.165 align:middle line:84% The trouble is that it doesn't make any physical sense. 00:14:29.165 --> 00:14:30.790 align:middle line:84% As far as I can tell, there's no reason 00:14:30.790 --> 00:14:34.030 align:middle line:84% why electrons should have to care about Bessel functions 00:14:34.030 --> 00:14:35.410 align:middle line:90% any more than I do. 00:14:35.410 --> 00:14:38.060 align:middle line:84% So something is going on physically, 00:14:38.060 --> 00:14:40.250 align:middle line:84% and so I went to go talk to Paul and ask about it. 00:14:40.250 --> 00:14:44.390 align:middle line:84% And the thing we came up with is whenever 00:14:44.390 --> 00:14:47.115 align:middle line:84% you have harmonics in a system, in some sort 00:14:47.115 --> 00:14:48.740 align:middle line:84% of simple physical system, it's usually 00:14:48.740 --> 00:14:50.270 align:middle line:90% due to some nonlinearity. 00:14:50.270 --> 00:14:51.920 align:middle line:84% So we need a nonlinearity in our system 00:14:51.920 --> 00:14:55.130 align:middle line:84% where the system acts back upon itself in order to do so. 00:14:55.130 --> 00:14:58.580 align:middle line:84% And what our best guess is that the electric field that you 00:14:58.580 --> 00:15:02.630 align:middle line:84% produce at the fundamental frequency here, omega C, 00:15:02.630 --> 00:15:08.730 align:middle line:84% is going to go back and couple back into the electron 00:15:08.730 --> 00:15:09.640 align:middle line:90% equation of motion. 00:15:09.640 --> 00:15:11.140 align:middle line:84% So effectively the electron is going 00:15:11.140 --> 00:15:14.100 align:middle line:84% to be forcing itself slightly out of phase, 00:15:14.100 --> 00:15:16.902 align:middle line:84% and that is going to give rise to these other harmonics here. 00:15:16.902 --> 00:15:19.110 align:middle line:84% So this is sort of a way of trying to make the system 00:15:19.110 --> 00:15:23.220 align:middle line:84% nonlinear because, of course, this electric field depends 00:15:23.220 --> 00:15:26.310 align:middle line:84% on v dot and v, and that's the thing we're 00:15:26.310 --> 00:15:28.050 align:middle line:90% solving for over here. 00:15:28.050 --> 00:15:30.670 align:middle line:84% So we think that it's a nonlinearity. 00:15:30.670 --> 00:15:35.040 align:middle line:84% So we think that these harmonics are due to nonlinearity. 00:15:35.040 --> 00:15:36.330 align:middle line:90% We looked at Sticks. 00:15:36.330 --> 00:15:37.860 align:middle line:90% We looked at Chen. 00:15:37.860 --> 00:15:40.290 align:middle line:90% We looked at [INAUDIBLE] piece. 00:15:40.290 --> 00:15:43.140 align:middle line:84% We couldn't find anyone who was willing to tell us 00:15:43.140 --> 00:15:44.100 align:middle line:90% what the answer is. 00:15:44.100 --> 00:15:48.660 align:middle line:84% So if anyone has any insight into why it's the case 00:15:48.660 --> 00:15:52.180 align:middle line:84% and if it's better than just waving my hands 00:15:52.180 --> 00:15:54.350 align:middle line:84% and saying nonlinearity, then please go for it. 00:15:54.350 --> 00:15:56.140 align:middle line:84% But the alternative is you just have 00:15:56.140 --> 00:15:57.880 align:middle line:84% to treat this as a mathematical exercise 00:15:57.880 --> 00:15:59.050 align:middle line:84% and churn through all the algebra. 00:15:59.050 --> 00:16:01.340 align:middle line:84% And I assure you if you go and have a look at this, 00:16:01.340 --> 00:16:03.548 align:middle line:84% you'll see that it really is an awful lot of algebra. 00:16:03.548 --> 00:16:06.160 align:middle line:84% So I'm trying to keep the mathematics out of this as much 00:16:06.160 --> 00:16:07.400 align:middle line:90% as possible. 00:16:07.400 --> 00:16:12.310 align:middle line:84% But we're going to do this properly, you find out-- 00:16:12.310 --> 00:16:19.700 align:middle line:90% and this is equation 5.2.16. 00:16:19.700 --> 00:16:24.410 align:middle line:84% You're going to get a sense of peaks in your emission, 00:16:24.410 --> 00:16:29.880 align:middle line:84% and those peaks are going to be at frequencies omega n, which 00:16:29.880 --> 00:16:33.780 align:middle line:84% are equal to m omega c, and they're actually 00:16:33.780 --> 00:16:37.770 align:middle line:84% going to be modified for future relativity. 00:16:37.770 --> 00:16:41.332 align:middle line:84% They're going to be 1 minus beta squared to the 1/2-- 00:16:41.332 --> 00:16:43.290 align:middle line:84% I'm going to explain what beta is in a moment-- 00:16:43.290 --> 00:16:47.100 align:middle line:90% 1 minus beta parallel theta. 00:16:47.100 --> 00:16:51.640 align:middle line:84% I'm going to explain all of these terms in a moment here. 00:16:51.640 --> 00:16:54.810 align:middle line:84% So we've still got this m inside here. 00:16:54.810 --> 00:16:56.100 align:middle line:90% It's just that there's also-- 00:16:56.100 --> 00:16:58.960 align:middle line:90% 00:16:58.960 --> 00:17:01.310 align:middle line:90% m is the control number. 00:17:01.310 --> 00:17:04.420 align:middle line:84% It's just that we also have some modification 00:17:04.420 --> 00:17:06.140 align:middle line:84% due to relativistic effects here. 00:17:06.140 --> 00:17:07.359 align:middle line:90% So what are these betas? 00:17:07.359 --> 00:17:09.109 align:middle line:84% Some of you have come across these before. 00:17:09.109 --> 00:17:11.589 align:middle line:84% This is absolutely not the plasma beta. 00:17:11.589 --> 00:17:13.540 align:middle line:84% It has nothing to do with the plasma beta. 00:17:13.540 --> 00:17:17.762 align:middle line:90% The beta is defined as b upon c. 00:17:17.762 --> 00:17:19.595 align:middle line:84% So it's the ratio of the particle's velocity 00:17:19.595 --> 00:17:21.089 align:middle line:90% to the speed of light. 00:17:21.089 --> 00:17:24.380 align:middle line:84% And we also have theta parallel, which 00:17:24.380 --> 00:17:28.069 align:middle line:90% is equal to b parallel upon c. 00:17:28.069 --> 00:17:34.750 align:middle line:84% That parallel is with respect to the magnetic field, c0 here. 00:17:34.750 --> 00:17:37.310 align:middle line:84% I don't think it comes up, but you can also 00:17:37.310 --> 00:17:41.670 align:middle line:84% define beta perpendicular if you want to as well. 00:17:41.670 --> 00:17:42.690 align:middle line:90% OK. 00:17:42.690 --> 00:17:46.510 align:middle line:84% There's something else I wanted to mention in all this. 00:17:46.510 --> 00:17:47.087 align:middle line:90% Hm? 00:17:47.087 --> 00:17:48.250 align:middle line:90% STUDENT: [INAUDIBLE]. 00:17:48.250 --> 00:17:49.210 align:middle line:90% JACK HARE: What? 00:17:49.210 --> 00:17:50.547 align:middle line:90% STUDENT: [INAUDIBLE]. 00:17:50.547 --> 00:17:51.130 align:middle line:90% JACK HARE: Oh. 00:17:51.130 --> 00:17:51.838 align:middle line:90% That was the one. 00:17:51.838 --> 00:17:53.780 align:middle line:90% Thank you. 00:17:53.780 --> 00:17:55.670 align:middle line:84% Yes, I forgot to build a diagram early on. 00:17:55.670 --> 00:17:58.490 align:middle line:84% So theta, in this case, is the same theta 00:17:58.490 --> 00:18:00.740 align:middle line:84% that we had when we're talking about x mode and o mode 00:18:00.740 --> 00:18:01.865 align:middle line:90% and all that sort of stuff. 00:18:01.865 --> 00:18:06.710 align:middle line:84% It's the angle between the observer and the magnetic field 00:18:06.710 --> 00:18:08.880 align:middle line:90% here. 00:18:08.880 --> 00:18:11.240 align:middle line:84% So this is the term that is largest 00:18:11.240 --> 00:18:15.582 align:middle line:84% when you're looking along the electron trajectory towards you 00:18:15.582 --> 00:18:17.540 align:middle line:84% and smallest when you're looking perpendicular. 00:18:17.540 --> 00:18:20.555 align:middle line:84% And so you can see straight away this is a signed quantity. 00:18:20.555 --> 00:18:23.180 align:middle line:84% It's going to end up looking a little bit like a Doppler shift. 00:18:23.180 --> 00:18:27.700 align:middle line:84% So that is the Doppler shift that you were expecting to see. 00:18:27.700 --> 00:18:30.330 align:middle line:84% But it's relativistic Doppler shift here. 00:18:30.330 --> 00:18:34.440 align:middle line:90% 00:18:34.440 --> 00:18:39.510 align:middle line:84% So for a single particle, just some electron moving 00:18:39.510 --> 00:18:47.050 align:middle line:84% on a magnetic field, we would have 00:18:47.050 --> 00:18:51.400 align:middle line:84% an emissivity that looked like a series of these peaks. 00:18:51.400 --> 00:18:51.940 align:middle line:90% OK? 00:18:51.940 --> 00:18:54.140 align:middle line:84% And these are delta functions in the theory. 00:18:54.140 --> 00:18:57.220 align:middle line:84% So you just have these very sharp emission lines 00:18:57.220 --> 00:18:58.495 align:middle line:90% and they're evenly spaced. 00:18:58.495 --> 00:19:01.270 align:middle line:90% 00:19:01.270 --> 00:19:02.650 align:middle line:90% STUDENT: Are heights the same? 00:19:02.650 --> 00:19:03.830 align:middle line:84% JACK HARE: The heights are not the same. 00:19:03.830 --> 00:19:05.320 align:middle line:84% If you want to get the heights, you 00:19:05.320 --> 00:19:07.990 align:middle line:84% have to go look at all those nasty Bessel functions, which 00:19:07.990 --> 00:19:09.080 align:middle line:90% I'm hiding from you. 00:19:09.080 --> 00:19:12.550 align:middle line:84% But if you want the height, the argument of the n-th Bessel 00:19:12.550 --> 00:19:13.240 align:middle line:90% function. 00:19:13.240 --> 00:19:15.340 align:middle line:84% And the thing that goes into the Bessel function 00:19:15.340 --> 00:19:18.110 align:middle line:84% is some complicated factor of all these other things here. 00:19:18.110 --> 00:19:19.180 align:middle line:84% So again, I'm not going to explain it, 00:19:19.180 --> 00:19:21.190 align:middle line:84% but they do not have to have the same height. 00:19:21.190 --> 00:19:23.050 align:middle line:84% I've drawn them a little bit like this. 00:19:23.050 --> 00:19:25.360 align:middle line:84% And if that particle is moving particularly fast, 00:19:25.360 --> 00:19:27.250 align:middle line:84% then all of these will be shifted in one way. 00:19:27.250 --> 00:19:28.510 align:middle line:84% If you're looking at it from a certain angle, 00:19:28.510 --> 00:19:29.510 align:middle line:90% we shift it another way. 00:19:29.510 --> 00:19:31.518 align:middle line:84% But they will all be shifted in the same way 00:19:31.518 --> 00:19:33.310 align:middle line:84% because this is just for a single particle. 00:19:33.310 --> 00:19:43.480 align:middle line:90% 00:19:43.480 --> 00:19:44.440 align:middle line:90% OK. 00:19:44.440 --> 00:19:46.210 align:middle line:84% Any questions on that before we find out 00:19:46.210 --> 00:19:48.670 align:middle line:84% what happens when we have more than one electron? 00:19:48.670 --> 00:19:49.694 align:middle line:90% [INAUDIBLE] 00:19:49.694 --> 00:19:55.390 align:middle line:90% 00:19:55.390 --> 00:19:57.350 align:middle line:90% OK. 00:19:57.350 --> 00:20:04.400 align:middle line:84% So the next thing we do is we look at many particles. 00:20:04.400 --> 00:20:08.040 align:middle line:90% Many electrons. 00:20:08.040 --> 00:20:12.990 align:middle line:84% And for this, it's useful to split our three dimensional 00:20:12.990 --> 00:20:19.440 align:middle line:84% distribution function f or b the vector v3v into a distribution 00:20:19.440 --> 00:20:25.850 align:middle line:84% function in beta parallel and beta perpendicular. 00:20:25.850 --> 00:20:27.415 align:middle line:90% It would be like this. 00:20:27.415 --> 00:20:29.540 align:middle line:84% So this is looking at particles which are streaming 00:20:29.540 --> 00:20:30.677 align:middle line:90% along the magnetic wall. 00:20:30.677 --> 00:20:32.510 align:middle line:84% This is looking at the component of velocity 00:20:32.510 --> 00:20:34.760 align:middle line:84% along the magnetic field and the component of velocity 00:20:34.760 --> 00:20:36.322 align:middle line:90% perpendicular to magnetic field. 00:20:36.322 --> 00:20:38.030 align:middle line:84% That's because this problem is symmetric. 00:20:38.030 --> 00:20:40.310 align:middle line:84% It doesn't matter how I rotate my axes 00:20:40.310 --> 00:20:41.930 align:middle line:90% around the magnetic field. 00:20:41.930 --> 00:20:44.040 align:middle line:84% The velocity overall is going to be the same. 00:20:44.040 --> 00:20:47.750 align:middle line:84% So I don't have to actually deal with three dimensions here. 00:20:47.750 --> 00:20:50.640 align:middle line:84% I've just turned it into a two dimensional thing. 00:20:50.640 --> 00:20:52.850 align:middle line:84% And if you do this integration, you'll 00:20:52.850 --> 00:20:55.145 align:middle line:84% find that your betas are different from 00:20:55.145 --> 00:20:56.270 align:middle line:90% different particles, right? 00:20:56.270 --> 00:20:59.530 align:middle line:90% Remember your Maxwellian. 00:20:59.530 --> 00:21:03.492 align:middle line:84% I just take a one dimensional slice through it, 00:21:03.492 --> 00:21:06.640 align:middle line:84% it has lots of different velocities available. 00:21:06.640 --> 00:21:08.280 align:middle line:84% And so that means some particles are 00:21:08.280 --> 00:21:10.290 align:middle line:84% going to be going with a velocity, which 00:21:10.290 --> 00:21:13.620 align:middle line:84% changes the sign of this to be bigger than one 00:21:13.620 --> 00:21:15.135 align:middle line:84% and some part of this denominator 00:21:15.135 --> 00:21:17.260 align:middle line:84% to be bigger than one, some to make it smaller one. 00:21:17.260 --> 00:21:19.302 align:middle line:84% So we can see that we're going to have broadening 00:21:19.302 --> 00:21:22.230 align:middle line:84% of these peaks, and this is the Doppler broadening 00:21:22.230 --> 00:21:23.440 align:middle line:90% that we expect. 00:21:23.440 --> 00:21:27.010 align:middle line:84% So let's just draw a single peak. 00:21:27.010 --> 00:21:34.650 align:middle line:84% It was a delta function at m omega c. 00:21:34.650 --> 00:21:38.582 align:middle line:84% It doesn't matter which harmonic it is for our purposes. 00:21:38.582 --> 00:21:40.790 align:middle line:84% This delta function is now going to be broadened out. 00:21:40.790 --> 00:21:44.660 align:middle line:84% So the delta function was always rather unscientific, unphysical, 00:21:44.660 --> 00:21:46.310 align:middle line:84% and so it's going to be broadened out 00:21:46.310 --> 00:21:48.190 align:middle line:84% by these natural broadening processes. 00:21:48.190 --> 00:21:49.190 align:middle line:90% And there's two of them. 00:21:49.190 --> 00:21:52.310 align:middle line:84% There's this top process and the bottom process here. 00:21:52.310 --> 00:21:54.950 align:middle line:84% I'll just write them out again down here. 00:21:54.950 --> 00:22:02.410 align:middle line:84% So we have a factor of 1 minus beta squared to the half over 1 00:22:02.410 --> 00:22:08.470 align:middle line:84% minus beta parallel cosine theta. 00:22:08.470 --> 00:22:10.980 align:middle line:84% So let's start with the numerator-- 00:22:10.980 --> 00:22:14.120 align:middle line:84% the denominator here, the term on the bottom. 00:22:14.120 --> 00:22:17.850 align:middle line:84% For some symmetric Maxwellian like distribution function, 00:22:17.850 --> 00:22:20.150 align:middle line:84% what does this term do to this line? 00:22:20.150 --> 00:22:23.490 align:middle line:90% How does it broaden it? 00:22:23.490 --> 00:22:26.820 align:middle line:90% Is it symmetric, asymmetric? 00:22:26.820 --> 00:22:29.070 align:middle line:84% Imagine we're looking at a fixed angle because we are. 00:22:29.070 --> 00:22:31.153 align:middle line:84% We don't have eyes all the way around the tokamak, 00:22:31.153 --> 00:22:33.660 align:middle line:84% so you can just fix theta to be some angle. 00:22:33.660 --> 00:22:35.880 align:middle line:84% Pick an angle where theta isn't zero 00:22:35.880 --> 00:22:37.470 align:middle line:84% or otherwise it gets very boring. 00:22:37.470 --> 00:22:40.170 align:middle line:84% What will this term 1 over beta-- 00:22:40.170 --> 00:22:41.640 align:middle line:90% 1 minus beta parallel do? 00:22:41.640 --> 00:22:44.480 align:middle line:90% 00:22:44.480 --> 00:22:45.260 align:middle line:90% Yeah? 00:22:45.260 --> 00:22:46.640 align:middle line:84% STUDENT: It'll broaden it symmetrically. 00:22:46.640 --> 00:22:47.750 align:middle line:84% JACK HARE: Broaden it symmetrically, right? 00:22:47.750 --> 00:22:49.280 align:middle line:90% This is going to look like-- 00:22:49.280 --> 00:22:51.110 align:middle line:90% say I look along the field line. 00:22:51.110 --> 00:22:52.700 align:middle line:90% Cos theta equals what? 00:22:52.700 --> 00:22:54.140 align:middle line:90% Theta equals zero. 00:22:54.140 --> 00:23:00.035 align:middle line:84% This is going to be 1 plus or minus b upon c. 00:23:00.035 --> 00:23:03.303 align:middle line:84% The plus or minus there is saying for every particle going 00:23:03.303 --> 00:23:04.970 align:middle line:84% along the field line towards me, there's 00:23:04.970 --> 00:23:07.920 align:middle line:84% another particle going away to the other side. 00:23:07.920 --> 00:23:10.820 align:middle line:84% And this is also going to be very small because the particles 00:23:10.820 --> 00:23:13.010 align:middle line:84% are not-- we're not dealing with actually 00:23:13.010 --> 00:23:14.940 align:middle line:84% ultrarelativistic particles here. 00:23:14.940 --> 00:23:17.932 align:middle line:84% And so this is going to look 1 plus or minus delta, 00:23:17.932 --> 00:23:19.640 align:middle line:84% and that means that it's going to broaden 00:23:19.640 --> 00:23:23.730 align:middle line:90% this peak symmetrically. 00:23:23.730 --> 00:23:25.470 align:middle line:90% What about this term at the top? 00:23:25.470 --> 00:23:27.720 align:middle line:84% So you can call this the Doppler term. 00:23:27.720 --> 00:23:32.403 align:middle line:90% 00:23:32.403 --> 00:23:33.820 align:middle line:84% What does this term on the top do? 00:23:33.820 --> 00:23:39.120 align:middle line:90% 00:23:39.120 --> 00:23:39.830 align:middle line:90% Anyone online? 00:23:39.830 --> 00:23:47.000 align:middle line:90% 00:23:47.000 --> 00:23:50.120 align:middle line:84% Is it always bigger than 1, always less than 1, 00:23:50.120 --> 00:23:54.480 align:middle line:84% sometimes bigger than 1, sometimes less than 1? 00:23:54.480 --> 00:23:55.320 align:middle line:90% Yes? 00:23:55.320 --> 00:23:57.153 align:middle line:84% STUDENT: It's always going to be less than 1 00:23:57.153 --> 00:23:58.780 align:middle line:84% since theta squared is always positive. 00:23:58.780 --> 00:24:01.815 align:middle line:84% JACK HARE: So this is always less than 1 because, as you say, 00:24:01.815 --> 00:24:03.190 align:middle line:84% theta squared is always positive. 00:24:03.190 --> 00:24:05.160 align:middle line:84% So it doesn't care about the sine of v. 00:24:05.160 --> 00:24:07.350 align:middle line:84% So it's going to be asymmetric here. 00:24:07.350 --> 00:24:09.525 align:middle line:84% So this is the relativistic mass term. 00:24:09.525 --> 00:24:20.610 align:middle line:90% 00:24:20.610 --> 00:24:22.620 align:middle line:84% And that will broaden the peak further, 00:24:22.620 --> 00:24:26.200 align:middle line:84% but it will only broaden it in one direction like that. 00:24:26.200 --> 00:24:29.475 align:middle line:84% So your peaks end up looking like this. 00:24:29.475 --> 00:24:32.420 align:middle line:90% 00:24:32.420 --> 00:24:35.750 align:middle line:84% If you want to know which of these two terms is larger, 00:24:35.750 --> 00:24:40.580 align:middle line:84% then you're asking the question, when is beta cosine-- 00:24:40.580 --> 00:24:46.640 align:middle line:84% beta parallel cosine theta greater than theta squared? 00:24:46.640 --> 00:24:53.420 align:middle line:84% That's the case when cosine theta is greater than dt 00:24:53.420 --> 00:24:57.230 align:middle line:84% upon c, the thermal velocity of the electrons. 00:24:57.230 --> 00:25:02.060 align:middle line:84% And that's the case for beta less than but 00:25:02.060 --> 00:25:04.760 align:middle line:90% almost up to pi upon 2. 00:25:04.760 --> 00:25:11.850 align:middle line:84% For almost all angles, this Doppler term dominates. 00:25:11.850 --> 00:25:13.840 align:middle line:90% STUDENT: [INAUDIBLE] 00:25:13.840 --> 00:25:22.715 align:middle line:90% 00:25:22.715 --> 00:25:23.340 align:middle line:90% JACK HARE: Yes. 00:25:23.340 --> 00:25:23.882 align:middle line:90% You're right. 00:25:23.882 --> 00:25:30.200 align:middle line:90% 00:25:30.200 --> 00:25:32.930 align:middle line:84% JACK HARE: Almost all theta apart from theta exactly equal 00:25:32.930 --> 00:25:35.687 align:middle line:84% to pi upon 2, the Doppler term dominates. 00:25:35.687 --> 00:25:38.370 align:middle line:90% 00:25:38.370 --> 00:25:40.460 align:middle line:84% So you'd see mostly symmetric lines 00:25:40.460 --> 00:25:42.920 align:middle line:84% with some small asymmetric shift. 00:25:42.920 --> 00:25:46.342 align:middle line:84% Apart from if you're exactly pi upon 2, which is a place 00:25:46.342 --> 00:25:48.550 align:middle line:84% you're often at when you're looking inside a tokamak. 00:25:48.550 --> 00:25:50.520 align:middle line:84% So this could happen quite a lot, in which case 00:25:50.520 --> 00:25:54.200 align:middle line:84% you're going to see a line that is very asymmetric indeed. 00:25:54.200 --> 00:25:56.230 align:middle line:90% OK. 00:25:56.230 --> 00:25:57.469 align:middle line:90% Questions on this? 00:25:57.469 --> 00:26:01.600 align:middle line:90% 00:26:01.600 --> 00:26:02.620 align:middle line:90% Yes? 00:26:02.620 --> 00:26:05.110 align:middle line:84% STUDENT: What is the j in the single particle graph? 00:26:05.110 --> 00:26:09.850 align:middle line:84% JACK HARE: That is the emissivity. 00:26:09.850 --> 00:26:11.822 align:middle line:84% The thing we're trying to calculate. 00:26:11.822 --> 00:26:14.660 align:middle line:90% 00:26:14.660 --> 00:26:18.320 align:middle line:84% Hutchinson writes it in a sort of differential form, which 00:26:18.320 --> 00:26:20.330 align:middle line:84% is nice for keeping track of the units. 00:26:20.330 --> 00:26:22.820 align:middle line:90% I'm just going to call it j. 00:26:22.820 --> 00:26:24.680 align:middle line:84% Sometimes there's some steradians and stuff 00:26:24.680 --> 00:26:26.347 align:middle line:84% lying around that you have to deal with. 00:26:26.347 --> 00:26:28.880 align:middle line:84% But roughly this corresponds to your-- 00:26:28.880 --> 00:26:31.440 align:middle line:84% this corresponds to your intuition of brightness. 00:26:31.440 --> 00:26:31.940 align:middle line:90% Right? 00:26:31.940 --> 00:26:36.390 align:middle line:84% This is how bright the plasma is with this magnetic field 00:26:36.390 --> 00:26:38.930 align:middle line:84% and this velocity for this single particle. 00:26:38.930 --> 00:26:40.730 align:middle line:84% And then of course, all of these lines 00:26:40.730 --> 00:26:43.350 align:middle line:84% would be broadened by these terms. 00:26:43.350 --> 00:26:43.850 align:middle line:90% OK. 00:26:43.850 --> 00:26:44.240 align:middle line:90% Yeah? 00:26:44.240 --> 00:26:46.115 align:middle line:84% STUDENT: I'm a little confused in the picture 00:26:46.115 --> 00:26:47.515 align:middle line:90% with the asymmetric peak. 00:26:47.515 --> 00:26:48.140 align:middle line:90% JACK HARE: Yes. 00:26:48.140 --> 00:26:48.950 align:middle line:90% STUDENT: Right? 00:26:48.950 --> 00:26:50.840 align:middle line:84% The orange equation on the right is 00:26:50.840 --> 00:26:55.130 align:middle line:84% saying that slower particles will have a-- 00:26:55.130 --> 00:26:58.130 align:middle line:90% like less reduced. 00:26:58.130 --> 00:27:02.334 align:middle line:84% I'm just trying to think about the way that you drew the-- 00:27:02.334 --> 00:27:04.697 align:middle line:84% JACK HARE: The relativistic mass correction, or the-- 00:27:04.697 --> 00:27:05.530 align:middle line:90% STUDENT: Yeah, yeah. 00:27:05.530 --> 00:27:07.030 align:middle line:84% JACK HARE: Well, actually this part, 00:27:07.030 --> 00:27:08.630 align:middle line:90% this is due to fast particles. 00:27:08.630 --> 00:27:09.130 align:middle line:90% STUDENT: OK. 00:27:09.130 --> 00:27:09.340 align:middle line:90% Yeah. 00:27:09.340 --> 00:27:10.660 align:middle line:84% JACK HARE: They're so fast, they're so heavy, 00:27:10.660 --> 00:27:12.077 align:middle line:84% their gyrofrequency has gone down. 00:27:12.077 --> 00:27:12.843 align:middle line:90% STUDENT: Oh, OK. 00:27:12.843 --> 00:27:13.510 align:middle line:90% JACK HARE: Yeah. 00:27:13.510 --> 00:27:14.290 align:middle line:90% Good question. 00:27:14.290 --> 00:27:14.790 align:middle line:90% Yeah. 00:27:14.790 --> 00:27:18.360 align:middle line:90% 00:27:18.360 --> 00:27:19.270 align:middle line:90% Yeah, you're right. 00:27:19.270 --> 00:27:21.895 align:middle line:84% That's really counterintuitive because actually for the Doppler 00:27:21.895 --> 00:27:24.697 align:middle line:84% shift, the particles down here are the fast ones. 00:27:24.697 --> 00:27:25.780 align:middle line:90% That's true in both cases. 00:27:25.780 --> 00:27:25.800 align:middle line:90% Yeah. 00:27:25.800 --> 00:27:26.980 align:middle line:84% It's the fast ones in both cases. 00:27:26.980 --> 00:27:27.420 align:middle line:90% Yeah. 00:27:27.420 --> 00:27:29.087 align:middle line:84% These ones up here are the fastest, but. 00:27:29.087 --> 00:27:32.640 align:middle line:90% 00:27:32.640 --> 00:27:33.540 align:middle line:90% Any questions online? 00:27:33.540 --> 00:27:38.115 align:middle line:90% 00:27:38.115 --> 00:27:38.615 align:middle line:90% OK. 00:27:38.615 --> 00:27:49.360 align:middle line:90% 00:27:49.360 --> 00:27:55.120 align:middle line:84% So we use this in practice by having 00:27:55.120 --> 00:27:59.440 align:middle line:84% a very good idea of what the magnetic field profile is inside 00:27:59.440 --> 00:28:00.220 align:middle line:90% our plasma. 00:28:00.220 --> 00:28:02.140 align:middle line:84% And a good example of a plasma where 00:28:02.140 --> 00:28:04.780 align:middle line:84% we have a very good idea of magnetic field profile 00:28:04.780 --> 00:28:08.350 align:middle line:84% is a tokamak, or maybe something like a stellarator. 00:28:08.350 --> 00:28:09.850 align:middle line:90% Something that's very low beta. 00:28:09.850 --> 00:28:14.910 align:middle line:84% So I'll show you what we do in that situation. 00:28:14.910 --> 00:28:22.780 align:middle line:84% So this tends to be used in magnetic confinement 00:28:22.780 --> 00:28:27.100 align:middle line:90% fusion with beta. 00:28:27.100 --> 00:28:31.785 align:middle line:84% That's the thermal pressure over the magnetic pressure-- 00:28:31.785 --> 00:28:33.240 align:middle line:90% I don't make the rules-- 00:28:33.240 --> 00:28:35.040 align:middle line:90% is much less than 1. 00:28:35.040 --> 00:28:37.320 align:middle line:84% Nothing to do with the other theta. 00:28:37.320 --> 00:28:38.820 align:middle line:84% And this condition effectively means 00:28:38.820 --> 00:28:40.528 align:middle line:84% that no matter what your plasma is doing, 00:28:40.528 --> 00:28:42.790 align:middle line:84% your magnetic field is pretty much the same. 00:28:42.790 --> 00:28:44.580 align:middle line:84% So you know your magnetic field profile. 00:28:44.580 --> 00:28:47.430 align:middle line:84% And a good example of this would be inside something 00:28:47.430 --> 00:28:55.390 align:middle line:84% like a tokamak where we have an axis of symmetry here 00:28:55.390 --> 00:29:00.750 align:middle line:84% and I have some cross-section of my tokamak like this, 00:29:00.750 --> 00:29:05.240 align:middle line:84% and maybe the magnetic field in this tokamak 00:29:05.240 --> 00:29:11.240 align:middle line:84% is going to be dropping off as 1 upon r. 00:29:11.240 --> 00:29:15.380 align:middle line:90% 00:29:15.380 --> 00:29:17.210 align:middle line:84% That just comes from Ampere's law, the fact 00:29:17.210 --> 00:29:19.100 align:middle line:84% that you have current flowing up through the magnets 00:29:19.100 --> 00:29:19.760 align:middle line:90% in the middle. 00:29:19.760 --> 00:29:25.425 align:middle line:84% It can hardly be anything other than 1 upon r, and that's true. 00:29:25.425 --> 00:29:26.800 align:middle line:84% And so the interesting thing here 00:29:26.800 --> 00:29:31.110 align:middle line:84% is that different regions of the plasma 00:29:31.110 --> 00:29:33.063 align:middle line:90% have different magnetic fields. 00:29:33.063 --> 00:29:34.980 align:middle line:84% Maybe some of you can see where this is going. 00:29:34.980 --> 00:29:42.370 align:middle line:90% 00:29:42.370 --> 00:29:52.530 align:middle line:84% So we can call these, for example, r3, r1, and r2. 00:29:52.530 --> 00:29:55.620 align:middle line:90% Make sense? 00:29:55.620 --> 00:29:56.120 align:middle line:90% r1. 00:29:56.120 --> 00:29:58.562 align:middle line:90% Three different radii here. 00:29:58.562 --> 00:30:00.020 align:middle line:84% And what we're going to do is we're 00:30:00.020 --> 00:30:06.570 align:middle line:84% going to observe this plasma with some microwave 00:30:06.570 --> 00:30:08.400 align:middle line:90% antenna, some sort of horn. 00:30:08.400 --> 00:30:10.350 align:middle line:84% That's going to collect the electron cyclotron 00:30:10.350 --> 00:30:13.080 align:middle line:84% emission, which is in this gigahertz range here, 00:30:13.080 --> 00:30:14.910 align:middle line:84% and that electron cyclotron emission 00:30:14.910 --> 00:30:17.550 align:middle line:84% is going to stream out in every direction. 00:30:17.550 --> 00:30:20.070 align:middle line:84% But one of the directions it's going to stream out into 00:30:20.070 --> 00:30:22.020 align:middle line:84% is our horn, so hopefully we can catch it. 00:30:22.020 --> 00:30:30.130 align:middle line:90% 00:30:30.130 --> 00:30:32.760 align:middle line:84% And then what we'll do with the signal that comes into this 00:30:32.760 --> 00:30:36.290 align:middle line:84% is we will split it a load of different times. 00:30:36.290 --> 00:30:41.080 align:middle line:84% I'm just going to draw n times here into n separate channels. 00:30:41.080 --> 00:30:45.350 align:middle line:84% We'll have a little bandpass filter 00:30:45.350 --> 00:30:50.210 align:middle line:84% on each of these which selects out for different frequencies. 00:30:50.210 --> 00:30:55.260 align:middle line:90% 00:30:55.260 --> 00:30:58.490 align:middle line:90% This is bandpass. 00:30:58.490 --> 00:31:06.580 align:middle line:90% 00:31:06.580 --> 00:31:10.700 align:middle line:84% And then each of these will go on to some detector. 00:31:10.700 --> 00:31:16.880 align:middle line:90% 00:31:16.880 --> 00:31:19.230 align:middle line:84% And that detector will measure the power 00:31:19.230 --> 00:31:23.830 align:middle line:84% as a function of time contained within this frequency band here. 00:31:23.830 --> 00:31:25.350 align:middle line:84% And so this is effectively a time 00:31:25.350 --> 00:31:28.440 align:middle line:84% resolved spectrometer kind of system. 00:31:28.440 --> 00:31:30.780 align:middle line:90% OK. 00:31:30.780 --> 00:31:32.640 align:middle line:90% Why is this helpful? 00:31:32.640 --> 00:31:39.610 align:middle line:84% It's because, as you said before, B of r is well known. 00:31:39.610 --> 00:31:50.940 align:middle line:90% 00:31:50.940 --> 00:31:54.710 align:middle line:84% We think of the intensity that's detected 00:31:54.710 --> 00:31:56.270 align:middle line:90% on our detector over here. 00:31:56.270 --> 00:31:59.680 align:middle line:90% 00:31:59.680 --> 00:32:04.170 align:middle line:84% We're going to have some emission that's coming from r1. 00:32:04.170 --> 00:32:07.860 align:middle line:84% r1 has the highest magnetic field, therefore, 00:32:07.860 --> 00:32:10.060 align:middle line:90% it has the highest frequencies. 00:32:10.060 --> 00:32:14.580 align:middle line:84% So its first emission is going to be, for example, 00:32:14.580 --> 00:32:20.280 align:middle line:84% at this frequency here, which is omega 1, like that. 00:32:20.280 --> 00:32:23.260 align:middle line:84% This is the frequency corresponding to position r1 00:32:23.260 --> 00:32:25.320 align:middle line:84% and it's the first harmonic of it. 00:32:25.320 --> 00:32:28.620 align:middle line:84% In the same way, r2 corresponds to the middle of the plasma. 00:32:28.620 --> 00:32:32.720 align:middle line:84% The magnetic field is lower, the frequency is lower. 00:32:32.720 --> 00:32:36.770 align:middle line:84% Omega 2 and r3 corresponds to the lowest magnetic field. 00:32:36.770 --> 00:32:40.030 align:middle line:90% 00:32:40.030 --> 00:32:43.240 align:middle line:84% There's no good reason for these to be the same in general, 00:32:43.240 --> 00:32:45.382 align:middle line:90% aside from [INAUDIBLE]. 00:32:45.382 --> 00:32:47.590 align:middle line:84% But zigs around a little bit, so it's clear that they 00:32:47.590 --> 00:32:48.970 align:middle line:90% don't have to be [INAUDIBLE]. 00:32:48.970 --> 00:32:50.770 align:middle line:84% But of course, as well as that, you're 00:32:50.770 --> 00:32:52.570 align:middle line:90% also going to get the harmonics. 00:32:52.570 --> 00:32:54.070 align:middle line:90% You're going to get a harmonic. 00:32:54.070 --> 00:32:56.528 align:middle line:84% This is why I should have drawn these much closer together. 00:32:56.528 --> 00:33:03.600 align:middle line:84% It would have made it much less clear at 2 omega 3 and at 2 00:33:03.600 --> 00:33:09.210 align:middle line:90% omega 2 and at 2 omega 1. 00:33:09.210 --> 00:33:11.820 align:middle line:84% And then of course at the third harmonic as well, now 00:33:11.820 --> 00:33:14.070 align:middle line:84% you can see, oh, dear, these are beginning to overlap. 00:33:14.070 --> 00:33:15.570 align:middle line:84% This is getting a little bit tricky. 00:33:15.570 --> 00:33:18.097 align:middle line:84% So maybe I shouldn't have done it like this. 00:33:18.097 --> 00:33:18.680 align:middle line:90% But they will. 00:33:18.680 --> 00:33:19.993 align:middle line:90% Eventually they will overlap. 00:33:19.993 --> 00:33:21.410 align:middle line:84% With high enough harmonics, you're 00:33:21.410 --> 00:33:24.380 align:middle line:84% going to have different harmonics all interacting 00:33:24.380 --> 00:33:25.200 align:middle line:90% with each other. 00:33:25.200 --> 00:33:30.930 align:middle line:84% So these are the third harmonics up here. 00:33:30.930 --> 00:33:37.560 align:middle line:84% But if you focus just down in this region, for example, 00:33:37.560 --> 00:33:45.490 align:middle line:84% every frequency corresponds to a specific magnetic field, 00:33:45.490 --> 00:33:48.880 align:middle line:84% which corresponds to a specific radial location. 00:33:48.880 --> 00:33:51.942 align:middle line:84% So if you're seeing emission at this frequency, 00:33:51.942 --> 00:33:53.400 align:middle line:84% you know that an emission is coming 00:33:53.400 --> 00:33:55.420 align:middle line:90% from this bit of the plasma. 00:33:55.420 --> 00:33:59.670 align:middle line:84% So this is extremely powerful because you have, as well 00:33:59.670 --> 00:34:02.160 align:middle line:84% as the spectral resolution and time resolution, 00:34:02.160 --> 00:34:04.440 align:middle line:90% you now have spatial resolution. 00:34:04.440 --> 00:34:16.460 align:middle line:90% 00:34:16.460 --> 00:34:18.389 align:middle line:90% OK. 00:34:18.389 --> 00:34:20.300 align:middle line:90% Questions on that? 00:34:20.300 --> 00:34:21.260 align:middle line:90% Yeah? 00:34:21.260 --> 00:34:22.962 align:middle line:90% Yes, question on that? 00:34:22.962 --> 00:34:24.170 align:middle line:90% STUDENT: Hi there, professor. 00:34:24.170 --> 00:34:28.489 align:middle line:84% So the spatial-- you can get spatial resolution 00:34:28.489 --> 00:34:30.409 align:middle line:84% because of the decreasing r field, 00:34:30.409 --> 00:34:35.210 align:middle line:84% but is there not going to be any overlap with the earlier 00:34:35.210 --> 00:34:36.590 align:middle line:90% harmonics in those? 00:34:36.590 --> 00:34:44.370 align:middle line:84% Couldn't you be closer to 1, then they'll overlap and-- 00:34:44.370 --> 00:34:46.949 align:middle line:84% JACK HARE: Depending on where your tokamak sits out here 00:34:46.949 --> 00:34:48.960 align:middle line:84% that gives you the gradient in magnetic field-- 00:34:48.960 --> 00:34:50.418 align:middle line:84% and that gradient in magnetic field 00:34:50.418 --> 00:34:53.070 align:middle line:84% is going to determine the spread of frequencies that you have 00:34:53.070 --> 00:34:54.330 align:middle line:90% that you're observing from. 00:34:54.330 --> 00:34:56.980 align:middle line:84% If your frequencies are closer together because, for example, 00:34:56.980 --> 00:34:59.250 align:middle line:84% your tokamak is further out in the 1 over r field, 00:34:59.250 --> 00:35:01.262 align:middle line:84% it's a large aspect ratio of tokamak, 00:35:01.262 --> 00:35:02.970 align:middle line:84% if those frequencies are closer together, 00:35:02.970 --> 00:35:05.538 align:middle line:84% you'll simply have to have a better spectrometer. 00:35:05.538 --> 00:35:07.080 align:middle line:84% There, of course, comes a point where 00:35:07.080 --> 00:35:11.383 align:middle line:84% all of the broadening that we talked about before 00:35:11.383 --> 00:35:13.050 align:middle line:84% is going to screw you over a little bit, 00:35:13.050 --> 00:35:15.050 align:middle line:84% and so that's going to fundamentally limit 00:35:15.050 --> 00:35:16.820 align:middle line:90% the spatial resolution you have. 00:35:16.820 --> 00:35:19.310 align:middle line:84% But broadly, you can take this signal 00:35:19.310 --> 00:35:21.350 align:middle line:84% and maybe deconvolve it and say, I've 00:35:21.350 --> 00:35:23.030 align:middle line:84% got some spatial resolution here. 00:35:23.030 --> 00:35:24.680 align:middle line:90% Yeah. 00:35:24.680 --> 00:35:26.330 align:middle line:84% Really, your spatial resolution is also 00:35:26.330 --> 00:35:28.130 align:middle line:84% going to be limited by the number of these detectors 00:35:28.130 --> 00:35:29.150 align:middle line:90% that you can field. 00:35:29.150 --> 00:35:31.692 align:middle line:84% So, like, people will do a 12 channel system, 00:35:31.692 --> 00:35:33.650 align:middle line:84% and then they'll effectively be able to resolve 00:35:33.650 --> 00:35:37.350 align:middle line:84% 12 different spatial locations across the plasma. 00:35:37.350 --> 00:35:40.620 align:middle line:84% STUDENT: And the radiation field-- 00:35:40.620 --> 00:35:42.375 align:middle line:84% JACK HARE: You're breaking up quite badly. 00:35:42.375 --> 00:35:44.250 align:middle line:84% I don't know if it's your microphone or what, 00:35:44.250 --> 00:35:46.780 align:middle line:84% but I can't really hear you very well. 00:35:46.780 --> 00:35:48.030 align:middle line:90% STUDENT: Can you hear me now? 00:35:48.030 --> 00:35:49.155 align:middle line:90% JACK HARE: Slightly better. 00:35:49.155 --> 00:35:50.190 align:middle line:90% Yeah, go for it. 00:35:50.190 --> 00:35:53.901 align:middle line:84% STUDENT: The detectors, the emissions, 00:35:53.901 --> 00:35:57.670 align:middle line:84% they're not happening isotropically, are they? 00:35:57.670 --> 00:35:59.940 align:middle line:84% Are they happening all over the tokamak, or? 00:35:59.940 --> 00:36:01.950 align:middle line:84% JACK HARE: They're happening in every direction. 00:36:01.950 --> 00:36:03.450 align:middle line:84% They're not happening isotropically, 00:36:03.450 --> 00:36:06.160 align:middle line:84% and we haven't derived the pattern that they make. 00:36:06.160 --> 00:36:07.618 align:middle line:90% And it's in Hutchinson as well. 00:36:07.618 --> 00:36:09.660 align:middle line:84% And you haven't asked about the polarization yet, 00:36:09.660 --> 00:36:11.340 align:middle line:84% and they've got different polarizations too. 00:36:11.340 --> 00:36:12.720 align:middle line:84% So it's definitely not isotropic, 00:36:12.720 --> 00:36:14.095 align:middle line:84% but you should definitely imagine 00:36:14.095 --> 00:36:17.150 align:middle line:84% that these are going out in lots of different directions. 00:36:17.150 --> 00:36:18.900 align:middle line:84% And if you're one of these sorts of people 00:36:18.900 --> 00:36:22.050 align:middle line:84% who likes putting things on the high field side of your tokamak, 00:36:22.050 --> 00:36:24.600 align:middle line:84% then you could also measure this just as well from that side 00:36:24.600 --> 00:36:25.278 align:middle line:90% as well. 00:36:25.278 --> 00:36:26.820 align:middle line:84% STUDENT: So you can put the detectors 00:36:26.820 --> 00:36:28.080 align:middle line:90% anywhere in the tokamak? 00:36:28.080 --> 00:36:29.997 align:middle line:84% JACK HARE: You can put the detectors anywhere. 00:36:29.997 --> 00:36:30.820 align:middle line:90% We tend to-- yes. 00:36:30.820 --> 00:36:31.320 align:middle line:90% Yes. 00:36:31.320 --> 00:36:32.778 align:middle line:84% You can put the detectors anywhere. 00:36:32.778 --> 00:36:34.950 align:middle line:84% Obviously this line of sight works particularly well 00:36:34.950 --> 00:36:38.340 align:middle line:84% because you're looking along the gradient in magnetic field. 00:36:38.340 --> 00:36:40.890 align:middle line:84% If you look from above, it's a little bit more complicated. 00:36:40.890 --> 00:36:44.730 align:middle line:84% But there was a very good paper just published on TCV, 00:36:44.730 --> 00:36:48.860 align:middle line:84% the tokamak at UTFL, where they had vertical electron cyclotron 00:36:48.860 --> 00:36:49.360 align:middle line:90% emission. 00:36:49.360 --> 00:36:51.235 align:middle line:84% Apparently it helps with some issues as well. 00:36:51.235 --> 00:36:53.620 align:middle line:84% So just saying, you can put it anywhere. 00:36:53.620 --> 00:36:56.350 align:middle line:84% It would definitely make more sense to put it in the direction 00:36:56.350 --> 00:36:58.720 align:middle line:84% or against the direction of the magnetic field gradients 00:36:58.720 --> 00:37:01.510 align:middle line:84% because you have this advantage that all 00:37:01.510 --> 00:37:03.760 align:middle line:84% the different frequencies then very clearly correspond 00:37:03.760 --> 00:37:05.840 align:middle line:84% to different places in the plasma. 00:37:05.840 --> 00:37:09.840 align:middle line:90% STUDENT: [INAUDIBLE] plane? 00:37:09.840 --> 00:37:10.680 align:middle line:90% JACK HARE: Yes. 00:37:10.680 --> 00:37:11.920 align:middle line:90% I think so. 00:37:11.920 --> 00:37:12.420 align:middle line:90% OK. 00:37:12.420 --> 00:37:13.290 align:middle line:90% Do you have a question? 00:37:13.290 --> 00:37:13.740 align:middle line:90% STUDENT: Yeah. 00:37:13.740 --> 00:37:15.570 align:middle line:84% I was wondering for the higher harmonics, 00:37:15.570 --> 00:37:17.880 align:middle line:84% you had the intensity increasing. 00:37:17.880 --> 00:37:20.670 align:middle line:84% Would the intensity of the higher harmonics be larger? 00:37:20.670 --> 00:37:24.803 align:middle line:90% 00:37:24.803 --> 00:37:26.720 align:middle line:84% JACK HARE: So I haven't really talked too much 00:37:26.720 --> 00:37:28.715 align:middle line:84% about what the intensity of the harmonics is. 00:37:28.715 --> 00:37:30.440 align:middle line:84% There are formulas to work it out. 00:37:30.440 --> 00:37:31.850 align:middle line:84% I think in general they drop off, 00:37:31.850 --> 00:37:35.160 align:middle line:84% but we're about to talk about opacity effects on these lines. 00:37:35.160 --> 00:37:37.940 align:middle line:84% And so what actually happens is the lower harmonics are absorbed 00:37:37.940 --> 00:37:39.810 align:middle line:84% more and the higher harmonics aren't. 00:37:39.810 --> 00:37:41.690 align:middle line:84% And so at least, I think, Hutchinson 00:37:41.690 --> 00:37:44.193 align:middle line:84% has a figure in his textbook where the second harmonics are 00:37:44.193 --> 00:37:45.860 align:middle line:84% the strongest and the third are actually 00:37:45.860 --> 00:37:48.650 align:middle line:84% stronger than the first and things like that. 00:37:48.650 --> 00:37:50.180 align:middle line:84% So it's complicated because there's 00:37:50.180 --> 00:37:51.555 align:middle line:84% a lot of factors at play in terms 00:37:51.555 --> 00:37:52.847 align:middle line:90% of what the final intensity is. 00:37:52.847 --> 00:37:53.347 align:middle line:90% Yeah. 00:37:53.347 --> 00:37:55.460 align:middle line:84% So here I'm just drawing them as different heights 00:37:55.460 --> 00:37:57.460 align:middle line:84% to show that they don't have to all be the same. 00:37:57.460 --> 00:37:59.660 align:middle line:84% But I don't mean anything by the specific heights 00:37:59.660 --> 00:38:01.280 align:middle line:90% that I chose for anything. 00:38:01.280 --> 00:38:02.530 align:middle line:90% Yeah. 00:38:02.530 --> 00:38:03.193 align:middle line:90% Shawn? 00:38:03.193 --> 00:38:05.960 align:middle line:84% STUDENT: On tokamaks people tend to neglect the poloidal field 00:38:05.960 --> 00:38:08.503 align:middle line:84% effect on the total magnetic field. 00:38:08.503 --> 00:38:09.170 align:middle line:90% JACK HARE: Yeah. 00:38:09.170 --> 00:38:11.966 align:middle line:84% So the question is, do you neglect the poloidal field? 00:38:11.966 --> 00:38:12.765 align:middle line:90% You do neglect it. 00:38:12.765 --> 00:38:14.390 align:middle line:84% Of course, you don't really neglect it. 00:38:14.390 --> 00:38:15.230 align:middle line:90% You would take it into account. 00:38:15.230 --> 00:38:16.790 align:middle line:84% But in this simple sketch here, I 00:38:16.790 --> 00:38:18.860 align:middle line:84% don't need to put the poloidal magnetic field in. 00:38:18.860 --> 00:38:20.402 align:middle line:84% But you could imagine that if you did 00:38:20.402 --> 00:38:22.360 align:middle line:84% know what the poloidal field was because you've 00:38:22.360 --> 00:38:24.320 align:middle line:84% got all your wonderful magnetic diagnostics 00:38:24.320 --> 00:38:26.430 align:middle line:84% and you can do a magnetic field reconstruction, 00:38:26.430 --> 00:38:28.590 align:middle line:84% then you know what the magnetic field is 00:38:28.590 --> 00:38:30.920 align:middle line:84% at every point in your tokamak and then you 00:38:30.920 --> 00:38:32.880 align:middle line:84% know what the frequency is and so then you 00:38:32.880 --> 00:38:34.540 align:middle line:90% can back this back out again. 00:38:34.540 --> 00:38:36.540 align:middle line:84% So it is possible to have poloidal field 00:38:36.540 --> 00:38:37.910 align:middle line:90% and still do this technique. 00:38:37.910 --> 00:38:40.410 align:middle line:84% It's just the sketch I'm giving you here doesn't include it. 00:38:40.410 --> 00:38:42.937 align:middle line:84% STUDENT: Then is it more complicated to get the theta, 00:38:42.937 --> 00:38:45.270 align:middle line:84% because it's now the magnetic field is not just straight 00:38:45.270 --> 00:38:46.110 align:middle line:90% around the torus? 00:38:46.110 --> 00:38:46.658 align:middle line:90% It has some-- 00:38:46.658 --> 00:38:48.450 align:middle line:84% JACK HARE: Well, you're still going to be-- 00:38:48.450 --> 00:38:50.730 align:middle line:84% even if you've got a twisted magnetic field 00:38:50.730 --> 00:38:53.640 align:middle line:84% line, if you're looking from the outside in it's 00:38:53.640 --> 00:38:55.110 align:middle line:90% still going to be at 90 degrees. 00:38:55.110 --> 00:38:56.730 align:middle line:84% It's just that it's going to be 90 degrees like that, 00:38:56.730 --> 00:38:59.025 align:middle line:84% not 90 degrees like this, because your magnetic field 00:38:59.025 --> 00:39:00.150 align:middle line:90% might be tilted up or down. 00:39:00.150 --> 00:39:01.330 align:middle line:90% It's still 90 degrees. 00:39:01.330 --> 00:39:02.913 align:middle line:84% If you start looking at another angle, 00:39:02.913 --> 00:39:04.630 align:middle line:84% not at 90 degrees, then, of course, 00:39:04.630 --> 00:39:06.420 align:middle line:84% you have to think about the geometry a lot more carefully. 00:39:06.420 --> 00:39:06.630 align:middle line:90% STUDENT: OK. 00:39:06.630 --> 00:39:08.213 align:middle line:84% JACK HARE: So if you're looking from-- 00:39:08.213 --> 00:39:09.307 align:middle line:90% yeah. 00:39:09.307 --> 00:39:10.890 align:middle line:84% If you're looking tangentially, right? 00:39:10.890 --> 00:39:13.050 align:middle line:84% If you're doing tangential viewing of ECE, electron 00:39:13.050 --> 00:39:15.270 align:middle line:84% cyclotron, like trying to look along the field lines, 00:39:15.270 --> 00:39:17.520 align:middle line:84% then the poloidal field will make a bigger difference. 00:39:17.520 --> 00:39:19.818 align:middle line:84% Here it doesn't make a huge difference. 00:39:19.818 --> 00:39:20.610 align:middle line:90% STUDENT: Thank you. 00:39:20.610 --> 00:39:20.820 align:middle line:90% JACK HARE: Cool. 00:39:20.820 --> 00:39:21.570 align:middle line:90% Lots of questions. 00:39:21.570 --> 00:39:22.570 align:middle line:90% OK, we'll go with Grant. 00:39:22.570 --> 00:39:24.028 align:middle line:84% STUDENT: I'm just a little confused 00:39:24.028 --> 00:39:25.480 align:middle line:90% on the spatial resolution. 00:39:25.480 --> 00:39:28.030 align:middle line:84% So is the idea we know the magnetic field 00:39:28.030 --> 00:39:30.690 align:middle line:84% so we can back out where the emission came from? 00:39:30.690 --> 00:39:31.230 align:middle line:90% Is that? 00:39:31.230 --> 00:39:32.130 align:middle line:90% JACK HARE: Yes. 00:39:32.130 --> 00:39:34.110 align:middle line:84% STUDENT: But that also relies on us 00:39:34.110 --> 00:39:35.812 align:middle line:84% knowing the velocity of the electrons 00:39:35.812 --> 00:39:37.770 align:middle line:84% to know their rest mass-- or not the rest mass. 00:39:37.770 --> 00:39:39.687 align:middle line:90% Their relativistic mass, right? 00:39:39.687 --> 00:39:41.520 align:middle line:84% JACK HARE: What you're saying is these peaks 00:39:41.520 --> 00:39:44.490 align:middle line:84% could be sufficiently broadened by these mechanisms 00:39:44.490 --> 00:39:45.400 align:middle line:90% that they overlap. 00:39:45.400 --> 00:39:45.900 align:middle line:90% Right? 00:39:45.900 --> 00:39:48.060 align:middle line:84% And of course, effectively, they will always 00:39:48.060 --> 00:39:51.413 align:middle line:84% overlap because I've just gone three spatial locations, 00:39:51.413 --> 00:39:53.580 align:middle line:84% but there's an infinite number of spatial locations. 00:39:53.580 --> 00:39:54.872 align:middle line:90% And so you're absolutely right. 00:39:54.872 --> 00:39:58.440 align:middle line:84% That broadening sets a limit on your frequency resolution. 00:39:58.440 --> 00:40:00.960 align:middle line:84% You're effectively convolved your signal 00:40:00.960 --> 00:40:03.512 align:middle line:90% with that asymmetric peak. 00:40:03.512 --> 00:40:05.970 align:middle line:84% And because you've set a limit on the frequency resolution, 00:40:05.970 --> 00:40:07.650 align:middle line:84% it sets a limit on how well you can 00:40:07.650 --> 00:40:09.660 align:middle line:84% resolve different magnetic field regions. 00:40:09.660 --> 00:40:11.190 align:middle line:84% And because of that, it sets a limit 00:40:11.190 --> 00:40:14.087 align:middle line:84% on how well you can resolve different spatial regions. 00:40:14.087 --> 00:40:14.670 align:middle line:90% STUDENT: Sure. 00:40:14.670 --> 00:40:16.810 align:middle line:84% But even just for a single particle, 00:40:16.810 --> 00:40:18.690 align:middle line:84% so there wouldn't be the broadening effect. 00:40:18.690 --> 00:40:23.260 align:middle line:84% If I gave you a set of peaks and a magnetic field profile, 00:40:23.260 --> 00:40:25.760 align:middle line:84% would you be able to figure out where those peaks came from? 00:40:25.760 --> 00:40:28.910 align:middle line:84% Would you also need to know what energy electron was-- 00:40:28.910 --> 00:40:32.540 align:middle line:84% like the mass was different to figure out? 00:40:32.540 --> 00:40:35.030 align:middle line:84% JACK HARE: I think this broadening is very small 00:40:35.030 --> 00:40:39.170 align:middle line:84% And so this is not a huge change to your signal. 00:40:39.170 --> 00:40:41.870 align:middle line:84% And then to me it seems like a deconvolution problem, 00:40:41.870 --> 00:40:44.510 align:middle line:84% which, they're hard, but not impossible to deconvolve 00:40:44.510 --> 00:40:45.230 align:middle line:90% signals. 00:40:45.230 --> 00:40:45.730 align:middle line:90% STUDENT: OK. 00:40:45.730 --> 00:40:47.780 align:middle line:90% Maybe I'm overcomplicating then. 00:40:47.780 --> 00:40:48.380 align:middle line:84% JACK HARE: I think you're thinking 00:40:48.380 --> 00:40:50.240 align:middle line:84% about reasonable complications, but not ones 00:40:50.240 --> 00:40:52.845 align:middle line:84% that are the biggest problem with this diagnostic. 00:40:52.845 --> 00:40:55.220 align:middle line:84% We're about to go on to the fact that most of these lines 00:40:55.220 --> 00:40:57.428 align:middle line:84% are optically thick and so you don't see them anyway. 00:40:57.428 --> 00:40:58.040 align:middle line:90% So yeah. 00:40:58.040 --> 00:40:58.730 align:middle line:90% But OK. 00:40:58.730 --> 00:40:59.450 align:middle line:90% Other questions. 00:40:59.450 --> 00:41:00.650 align:middle line:90% Yeah? 00:41:00.650 --> 00:41:04.250 align:middle line:84% STUDENT: What's the wavelength and frequency of these? 00:41:04.250 --> 00:41:06.130 align:middle line:84% JACK HARE: What it would be a typical frame-- 00:41:06.130 --> 00:41:06.800 align:middle line:90% yeah, like is there any-- 00:41:06.800 --> 00:41:08.240 align:middle line:90% STUDENT: Picking up other-- 00:41:08.240 --> 00:41:09.930 align:middle line:84% JACK HARE: Like 100 gigahertz or so. 00:41:09.930 --> 00:41:10.490 align:middle line:90% STUDENT: OK. 00:41:10.490 --> 00:41:12.210 align:middle line:84% JACK HARE: It depends strongly on the magnetic field. 00:41:12.210 --> 00:41:13.877 align:middle line:84% So in something like [INAUDIBLE] c model 00:41:13.877 --> 00:41:16.470 align:middle line:84% spark, it will be much higher than it is on other devices. 00:41:16.470 --> 00:41:18.470 align:middle line:84% STUDENT: If you get some other kind of radiation 00:41:18.470 --> 00:41:21.358 align:middle line:84% that kind of falls in the same spectrum, then-- 00:41:21.358 --> 00:41:23.400 align:middle line:84% JACK HARE: We will actually talk about that, yes, 00:41:23.400 --> 00:41:25.358 align:middle line:84% when we talk about the accessibility condition. 00:41:25.358 --> 00:41:28.230 align:middle line:84% This frequency may be close to other natural frequencies 00:41:28.230 --> 00:41:31.050 align:middle line:84% in the plasma, and that may modify what we can observe. 00:41:31.050 --> 00:41:31.980 align:middle line:90% So good question. 00:41:31.980 --> 00:41:32.890 align:middle line:90% Yeah. 00:41:32.890 --> 00:41:33.390 align:middle line:90% OK. 00:41:33.390 --> 00:41:35.655 align:middle line:84% Any other questions online or in the room? 00:41:35.655 --> 00:41:42.630 align:middle line:90% 00:41:42.630 --> 00:41:43.785 align:middle line:90% OK, let's keep going. 00:41:43.785 --> 00:42:04.690 align:middle line:90% 00:42:04.690 --> 00:42:06.310 align:middle line:90% OK. 00:42:06.310 --> 00:42:11.090 align:middle line:84% So because you know the emissivity 00:42:11.090 --> 00:42:12.980 align:middle line:84% or you can know the emissivity-- hopefully 00:42:12.980 --> 00:42:15.355 align:middle line:84% you believe me that it is possible to calculate with lots 00:42:15.355 --> 00:42:16.620 align:middle line:90% of algebra, the emissivity. 00:42:16.620 --> 00:42:18.260 align:middle line:90% We can also calculate alpha. 00:42:18.260 --> 00:42:25.990 align:middle line:90% 00:42:25.990 --> 00:42:28.520 align:middle line:84% That means that you can ask yourself 00:42:28.520 --> 00:42:33.200 align:middle line:84% what is the optical depth of one of these waves propagating 00:42:33.200 --> 00:42:34.190 align:middle line:90% through this plasma. 00:42:34.190 --> 00:42:39.900 align:middle line:84% You can solve tau is equal to alpha dx. 00:42:39.900 --> 00:42:43.330 align:middle line:84% And what you will find if you do that is that for the lowest 00:42:43.330 --> 00:42:49.930 align:middle line:84% harmonics m equals 1, m equals 2, often but not always, 00:42:49.930 --> 00:42:53.620 align:middle line:90% that tau is much greater than 1. 00:42:53.620 --> 00:42:58.330 align:middle line:84% So for these lower frequency harmonics, 00:42:58.330 --> 00:42:59.560 align:middle line:90% you are optically thin. 00:42:59.560 --> 00:43:11.840 align:middle line:90% 00:43:11.840 --> 00:43:15.860 align:middle line:84% Now that's bad because you can no longer use detailed structure 00:43:15.860 --> 00:43:17.630 align:middle line:84% inside these in order to work out 00:43:17.630 --> 00:43:19.170 align:middle line:90% clever things about your plasma. 00:43:19.170 --> 00:43:22.070 align:middle line:84% But it's actually incredibly useful 00:43:22.070 --> 00:43:24.680 align:middle line:84% because what we find in practice is 00:43:24.680 --> 00:43:27.830 align:middle line:84% that the intensity at these different frequencies 00:43:27.830 --> 00:43:30.570 align:middle line:84% is now just going to be the black body intensity. 00:43:30.570 --> 00:43:31.070 align:middle line:90% Right? 00:43:31.070 --> 00:43:36.170 align:middle line:84% So the intensity i of mu is just equal to nu 00:43:36.170 --> 00:43:39.660 align:middle line:90% squared upon c squared times t. 00:43:39.660 --> 00:43:42.720 align:middle line:84% You'll notice I continuously switch between omega and nu, 00:43:42.720 --> 00:43:47.250 align:middle line:84% and hope you realize that omega is just 2 pi nu 00:43:47.250 --> 00:43:49.140 align:middle line:84% and are not too confused by this. 00:43:49.140 --> 00:43:52.670 align:middle line:84% Hutchinson uses nu because most diagnostics work in Hertz, 00:43:52.670 --> 00:43:54.420 align:middle line:84% or at least we think about things in Hertz 00:43:54.420 --> 00:43:57.823 align:middle line:84% when we're digitizing them rather than omega. 00:43:57.823 --> 00:44:00.240 align:middle line:84% But to be honest, I think you should be comfortable enough 00:44:00.240 --> 00:44:02.310 align:middle line:84% to switch between these unit systems. 00:44:02.310 --> 00:44:03.300 align:middle line:90% OK. 00:44:03.300 --> 00:44:08.100 align:middle line:84% So the neat thing about this is that you will then have, 00:44:08.100 --> 00:44:17.120 align:middle line:84% for this region of the spectra, a signal that 00:44:17.120 --> 00:44:22.410 align:middle line:90% maybe looks like this. 00:44:22.410 --> 00:44:27.110 align:middle line:84% And again, this is frequency here and intensity here. 00:44:27.110 --> 00:44:30.880 align:middle line:84% And that means that at each frequency-- 00:44:30.880 --> 00:44:31.900 align:middle line:90% what colors did I use? 00:44:31.900 --> 00:44:38.710 align:middle line:90% 00:44:38.710 --> 00:44:40.330 align:middle line:90% Actually, I will go like that. 00:44:40.330 --> 00:44:51.270 align:middle line:90% 00:44:51.270 --> 00:44:54.778 align:middle line:84% This is just the black body curve. 00:44:54.778 --> 00:44:56.570 align:middle line:84% I've actually gone and done the full thing, 00:44:56.570 --> 00:44:58.850 align:middle line:90% not just the nu squared. 00:44:58.850 --> 00:44:59.970 align:middle line:90% Do we want to say that? 00:44:59.970 --> 00:45:00.780 align:middle line:90% No, don't say that. 00:45:00.780 --> 00:45:01.100 align:middle line:90% Ignore that. 00:45:01.100 --> 00:45:01.975 align:middle line:90% That's not the scope. 00:45:01.975 --> 00:45:07.930 align:middle line:90% 00:45:07.930 --> 00:45:08.430 align:middle line:90% Right. 00:45:08.430 --> 00:45:10.420 align:middle line:84% You'll get a spectrum that looks like this. 00:45:10.420 --> 00:45:13.890 align:middle line:84% And that means for every frequency, 00:45:13.890 --> 00:45:15.810 align:middle line:90% you can measure the intensity. 00:45:15.810 --> 00:45:18.600 align:middle line:84% And so therefore for every frequency, 00:45:18.600 --> 00:45:20.280 align:middle line:84% you know the black body temperature 00:45:20.280 --> 00:45:21.810 align:middle line:90% that corresponds to it. 00:45:21.810 --> 00:45:26.670 align:middle line:90% So we have omega. 00:45:26.670 --> 00:45:30.690 align:middle line:84% This frequency codes for a very specific magnetic field, which 00:45:30.690 --> 00:45:31.890 align:middle line:90% codes for a very specific. 00:45:31.890 --> 00:45:34.410 align:middle line:90% 00:45:34.410 --> 00:45:35.590 align:middle line:90% Also the plasma. 00:45:35.590 --> 00:45:40.880 align:middle line:84% And this intensity divided by nu squared times 00:45:40.880 --> 00:45:45.880 align:middle line:84% by c squared codes for a very specific temperature. 00:45:45.880 --> 00:45:50.760 align:middle line:84% And so from this, we get out temperature 00:45:50.760 --> 00:45:51.840 align:middle line:90% as a function of radius. 00:45:51.840 --> 00:45:55.150 align:middle line:90% 00:45:55.150 --> 00:46:05.470 align:middle line:84% So we can convert this spectrum into a plot of temperature 00:46:05.470 --> 00:46:06.580 align:middle line:90% inside our tokamak. 00:46:06.580 --> 00:46:10.190 align:middle line:90% 00:46:10.190 --> 00:46:14.880 align:middle line:84% Again, we are measuring in this toy example 00:46:14.880 --> 00:46:16.610 align:middle line:84% that these three frequencies are this. 00:46:16.610 --> 00:46:20.810 align:middle line:90% 00:46:20.810 --> 00:46:24.540 align:middle line:84% But we can, for example, get out of the peak temperature profile. 00:46:24.540 --> 00:46:26.120 align:middle line:90% So this is extremely powerful. 00:46:26.120 --> 00:46:29.510 align:middle line:84% If we look at just the optically thick lines 00:46:29.510 --> 00:46:32.030 align:middle line:84% down at low frequencies, the intensity 00:46:32.030 --> 00:46:33.710 align:middle line:84% of those optically thick lines depends 00:46:33.710 --> 00:46:37.040 align:middle line:84% only on the plasma temperature and not on any other complicated 00:46:37.040 --> 00:46:38.000 align:middle line:90% physics. 00:46:38.000 --> 00:46:40.970 align:middle line:84% Because of that, because we have this very strong mapping 00:46:40.970 --> 00:46:43.430 align:middle line:84% in frequency of magnetic fields position, 00:46:43.430 --> 00:46:46.213 align:middle line:84% we can also take the intensity map at the temperature 00:46:46.213 --> 00:46:47.630 align:middle line:84% and we can get out the temperature 00:46:47.630 --> 00:46:48.713 align:middle line:90% as a function of position. 00:46:48.713 --> 00:46:52.130 align:middle line:84% So this is why ECE is so incredibly important. 00:46:52.130 --> 00:46:54.860 align:middle line:84% Because of course, as I said before, it's also time resolved. 00:46:54.860 --> 00:46:59.890 align:middle line:84% You get temperature as a function of r and time. 00:46:59.890 --> 00:47:02.740 align:middle line:84% And then tokamaks, because of the existence of flux surfaces, 00:47:02.740 --> 00:47:05.350 align:middle line:84% once you measure the temperature along the line, 00:47:05.350 --> 00:47:07.930 align:middle line:84% you also know the temperature along all the flux surfaces. 00:47:07.930 --> 00:47:09.518 align:middle line:84% So from your magnetic reconstruction, 00:47:09.518 --> 00:47:11.560 align:middle line:84% you've effectively got the temperature everywhere 00:47:11.560 --> 00:47:13.810 align:middle line:90% inside your plasma. 00:47:13.810 --> 00:47:14.830 align:middle line:90% OK. 00:47:14.830 --> 00:47:17.750 align:middle line:84% This is a little bit hard to get your head around, 00:47:17.750 --> 00:47:20.470 align:middle line:84% so I'll pause there, let you think, and ask any questions. 00:47:20.470 --> 00:47:24.640 align:middle line:90% 00:47:24.640 --> 00:47:26.390 align:middle line:84% STUDENT: If this is such a great property, 00:47:26.390 --> 00:47:29.070 align:middle line:84% why does Thomson scattering exist? 00:47:29.070 --> 00:47:31.293 align:middle line:90% What's the issue here? 00:47:31.293 --> 00:47:33.710 align:middle line:84% JACK HARE: Thomson scattering can use as many more plasmas 00:47:33.710 --> 00:47:34.710 align:middle line:90% than electron cyclotron. 00:47:34.710 --> 00:47:38.240 align:middle line:84% So even in the nonexistence of tokamaks, 00:47:38.240 --> 00:47:41.390 align:middle line:84% we would still have Thomson scattering. 00:47:41.390 --> 00:47:43.535 align:middle line:84% Why does Thomson scattering exist? 00:47:43.535 --> 00:47:46.200 align:middle line:90% 00:47:46.200 --> 00:47:48.140 align:middle line:84% Why is Thomson scattering used in tokamaks? 00:47:48.140 --> 00:47:51.980 align:middle line:84% Well, this requires plasmas of sufficient temperature 00:47:51.980 --> 00:47:55.440 align:middle line:84% to make high enough frequencies to be detected. 00:47:55.440 --> 00:47:58.010 align:middle line:84% So it will not work in the edge, and edge Thomson scattering 00:47:58.010 --> 00:48:01.580 align:middle line:90% is a big, important tokamak. 00:48:01.580 --> 00:48:04.640 align:middle line:84% This Thomson scattering is defined along a chord 00:48:04.640 --> 00:48:05.750 align:middle line:90% by your laser beam. 00:48:05.750 --> 00:48:09.020 align:middle line:84% And so you can get truly local measurements of temperature, 00:48:09.020 --> 00:48:12.920 align:middle line:84% whereas this diagnostic is kind of a little bit more global. 00:48:12.920 --> 00:48:15.230 align:middle line:84% We'll talk in a moment about fluctuations and the fact 00:48:15.230 --> 00:48:17.270 align:middle line:84% that this default diagnostic is not 00:48:17.270 --> 00:48:20.210 align:middle line:84% good at seeing small temperature fluctuations. 00:48:20.210 --> 00:48:23.060 align:middle line:90% 00:48:23.060 --> 00:48:24.537 align:middle line:90% Feel like that's enough. 00:48:24.537 --> 00:48:26.120 align:middle line:84% There are probably other good reasons. 00:48:26.120 --> 00:48:28.078 align:middle line:84% Oh, we'll talk about accessibility in a moment. 00:48:28.078 --> 00:48:30.020 align:middle line:84% So the fact that, in fact, sometimes 00:48:30.020 --> 00:48:31.940 align:middle line:84% it's very hard to do electron cyclotron 00:48:31.940 --> 00:48:35.030 align:middle line:84% emission because your waves as opposed to exiting the plasma 00:48:35.030 --> 00:48:38.360 align:middle line:84% and hitting your detector here get cut off and get 00:48:38.360 --> 00:48:41.330 align:middle line:84% reflected by density, regions above the critical density 00:48:41.330 --> 00:48:42.170 align:middle line:90% inside the plasma. 00:48:42.170 --> 00:48:44.550 align:middle line:84% So we'll talk about accessibility. 00:48:44.550 --> 00:48:45.050 align:middle line:90% Yeah. 00:48:45.050 --> 00:48:47.820 align:middle line:90% I think those are good reasons. 00:48:47.820 --> 00:48:49.020 align:middle line:90% Yeah? 00:48:49.020 --> 00:48:52.800 align:middle line:84% STUDENT: Do we do anything with the higher harmonics, or? 00:48:52.800 --> 00:48:54.480 align:middle line:84% JACK HARE: In this course, I'm no longer 00:48:54.480 --> 00:48:55.897 align:middle line:84% talking about the higher harmonics 00:48:55.897 --> 00:48:57.314 align:middle line:84% because it gets quite complicated. 00:48:57.314 --> 00:48:59.064 align:middle line:84% If you go to Hutchinson you can apparently 00:48:59.064 --> 00:49:00.450 align:middle line:84% find out things about high energy 00:49:00.450 --> 00:49:02.760 align:middle line:84% electrons in the tail of your distribution 00:49:02.760 --> 00:49:04.620 align:middle line:90% from the high harmonics. 00:49:04.620 --> 00:49:09.305 align:middle line:84% And in general, your emissivity contains 00:49:09.305 --> 00:49:11.430 align:middle line:84% lots of information about the distribution function 00:49:11.430 --> 00:49:14.162 align:middle line:84% unless you end up in this optically thick regime. 00:49:14.162 --> 00:49:16.620 align:middle line:84% And so if you look at the shape of the high harmonic lines, 00:49:16.620 --> 00:49:18.330 align:middle line:84% you might be able to say something clever 00:49:18.330 --> 00:49:20.038 align:middle line:84% about the electron distribution function. 00:49:20.038 --> 00:49:22.320 align:middle line:84% So if you want to study runaway electrons or something 00:49:22.320 --> 00:49:23.445 align:middle line:90% like that, you can do that. 00:49:23.445 --> 00:49:25.318 align:middle line:84% But I had to cut something out of course 00:49:25.318 --> 00:49:26.610 align:middle line:90% and that's what I cut out here. 00:49:26.610 --> 00:49:29.190 align:middle line:84% But yeah, go to Hutchinson's book if you want to know more. 00:49:29.190 --> 00:49:30.270 align:middle line:90% Yeah. 00:49:30.270 --> 00:49:31.270 align:middle line:90% Other questions? 00:49:31.270 --> 00:49:31.980 align:middle line:90% Anything online? 00:49:31.980 --> 00:49:39.430 align:middle line:90% 00:49:39.430 --> 00:49:42.052 align:middle line:84% Gone so quiet I can hear the cryopump compressor. 00:49:42.052 --> 00:49:45.750 align:middle line:90% 00:49:45.750 --> 00:49:46.830 align:middle line:90% OK. 00:49:46.830 --> 00:49:48.340 align:middle line:90% Good. 00:49:48.340 --> 00:49:52.690 align:middle line:84% Well, actually, yeah, the only thing I will say here 00:49:52.690 --> 00:49:56.320 align:middle line:84% is that this signal will not, of course, be nice and smooth. 00:49:56.320 --> 00:49:59.270 align:middle line:90% It'll be very noisy. 00:49:59.270 --> 00:50:05.520 align:middle line:84% And the point is that in general noise limits 00:50:05.520 --> 00:50:09.930 align:middle line:84% your measurements of delta te over te. 00:50:09.930 --> 00:50:11.930 align:middle line:84% These are the temperature fluctuations. 00:50:11.930 --> 00:50:14.430 align:middle line:84% And the reason why you might be very interested in measuring 00:50:14.430 --> 00:50:17.495 align:middle line:84% temperature fluctuations is because, as we discussed 00:50:17.495 --> 00:50:19.770 align:middle line:84% I think a little bit ago with reflectometry, 00:50:19.770 --> 00:50:23.190 align:middle line:84% if you have fluctuations in density and temperature driven 00:50:23.190 --> 00:50:26.580 align:middle line:84% by turbulence, that will cause anomalous transport of particles 00:50:26.580 --> 00:50:29.670 align:middle line:84% and heat out of your plasma and limits its efficiency 00:50:29.670 --> 00:50:30.780 align:middle line:90% as a fusion reactor. 00:50:30.780 --> 00:50:34.770 align:middle line:84% So a ongoing problem is how to measure these density 00:50:34.770 --> 00:50:36.460 align:middle line:84% fluctuations, temperature fluctuations 00:50:36.460 --> 00:50:38.540 align:middle line:84% so that we can understand turbulence better. 00:50:38.540 --> 00:50:40.920 align:middle line:84% And so one thing we'll now talk about 00:50:40.920 --> 00:50:43.740 align:middle line:84% is correlation electron cyclotron emission, 00:50:43.740 --> 00:50:45.960 align:middle line:84% and that is a technique which is specifically 00:50:45.960 --> 00:50:49.310 align:middle line:84% designed to measure very small temperature fluctuations inside. 00:50:49.310 --> 00:50:51.243 align:middle line:90% 00:50:51.243 --> 00:50:53.410 align:middle line:84% Is that what I'm going to talk or am I going to talk 00:50:53.410 --> 00:50:56.400 align:middle line:90% about accessibility? 00:50:56.400 --> 00:50:58.027 align:middle line:84% I'm going to talk about accessibility. 00:50:58.027 --> 00:50:58.860 align:middle line:90% And all that run up. 00:50:58.860 --> 00:51:11.590 align:middle line:90% 00:51:11.590 --> 00:51:14.110 align:middle line:84% STUDENT: What is the noise in that [INAUDIBLE]? 00:51:14.110 --> 00:51:17.610 align:middle line:90% 00:51:17.610 --> 00:51:20.910 align:middle line:84% I'm used to thinking of thermal noise [INAUDIBLE] 00:51:20.910 --> 00:51:25.972 align:middle line:84% like you're looking at a big [INAUDIBLE]. 00:51:25.972 --> 00:51:27.680 align:middle line:84% JACK HARE: Obviously there's fluctuations 00:51:27.680 --> 00:51:29.870 align:middle line:84% in the temperature, which will produce a different intensity, 00:51:29.870 --> 00:51:31.670 align:middle line:84% but there are also thermal fluctuations 00:51:31.670 --> 00:51:33.950 align:middle line:84% in the number of photons being produced, 00:51:33.950 --> 00:51:35.880 align:middle line:84% this sort of shot noise type thing. 00:51:35.880 --> 00:51:38.720 align:middle line:84% There's also noise from the fact that your detectors are 00:51:38.720 --> 00:51:40.730 align:middle line:84% in a noisy environment and they're not perfect 00:51:40.730 --> 00:51:41.640 align:middle line:90% and things like that. 00:51:41.640 --> 00:51:44.450 align:middle line:84% So yeah, we'll talk a little bit more about the sources of noise 00:51:44.450 --> 00:51:47.573 align:middle line:84% when we talk about correlation electron cyclotron emission, 00:51:47.573 --> 00:51:49.490 align:middle line:84% which, again, I thought I was going to do now, 00:51:49.490 --> 00:51:51.198 align:middle line:84% but it turns out I'm doing accessibility. 00:51:51.198 --> 00:51:53.428 align:middle line:84% So we'll get back to that, but yeah, think about just 00:51:53.428 --> 00:51:55.220 align:middle line:84% in general, whenever we make a measurement, 00:51:55.220 --> 00:51:56.762 align:middle line:84% we're going to have some noise of it. 00:51:56.762 --> 00:52:00.220 align:middle line:84% In this case, it just is very difficult to make-- 00:52:00.220 --> 00:52:01.970 align:middle line:84% these detectors are very difficult to work 00:52:01.970 --> 00:52:04.678 align:middle line:84% with, and at 100 gigahertz or so you tend to have a lot of noise. 00:52:04.678 --> 00:52:06.200 align:middle line:90% Yeah. 00:52:06.200 --> 00:52:07.330 align:middle line:90% Any other questions? 00:52:07.330 --> 00:52:11.240 align:middle line:90% 00:52:11.240 --> 00:52:11.740 align:middle line:90% OK. 00:52:11.740 --> 00:52:24.960 align:middle line:90% 00:52:24.960 --> 00:52:26.070 align:middle line:90% OK. 00:52:26.070 --> 00:52:28.500 align:middle line:84% So the question of accessibility can 00:52:28.500 --> 00:52:34.980 align:middle line:84% be phrased as a question of can the electron cyclotron emission 00:52:34.980 --> 00:52:40.910 align:middle line:90% reach the detector? 00:52:40.910 --> 00:52:43.890 align:middle line:84% If it can't, we can't detect it, so all 00:52:43.890 --> 00:52:46.710 align:middle line:84% of this beautiful technique is useless. 00:52:46.710 --> 00:52:50.980 align:middle line:84% Now in general, when we're doing something with an MCS device, 00:52:50.980 --> 00:52:52.720 align:middle line:84% we're going to be measuring perpendicular 00:52:52.720 --> 00:52:53.740 align:middle line:90% to the magnetic field. 00:52:53.740 --> 00:52:55.960 align:middle line:84% That's just a simple geometric constraint 00:52:55.960 --> 00:53:00.660 align:middle line:84% that our device, whether it's a stellarator or a tokamak, 00:53:00.660 --> 00:53:05.310 align:middle line:84% looks like this, and therefore, you'll 00:53:05.310 --> 00:53:08.800 align:middle line:84% tend to have magnets that look like this. 00:53:08.800 --> 00:53:13.430 align:middle line:84% And so the only gaps are to look in this direction here. 00:53:13.430 --> 00:53:15.650 align:middle line:84% You, of course, can design other geometries. 00:53:15.650 --> 00:53:18.290 align:middle line:84% I'm just using this as an example here. 00:53:18.290 --> 00:53:20.720 align:middle line:84% Because the magnetic field is predominantly 00:53:20.720 --> 00:53:24.860 align:middle line:84% the toroidal magnetic field, our angle theta 00:53:24.860 --> 00:53:27.620 align:middle line:90% is equal to pi upon 2. 00:53:27.620 --> 00:53:31.100 align:middle line:84% So what modes are propagating-- what 00:53:31.100 --> 00:53:33.320 align:middle line:84% modes is the electromagnetic radiation, the ECE 00:53:33.320 --> 00:53:35.930 align:middle line:84% radiation propagating as inside our plasma? 00:53:35.930 --> 00:53:42.000 align:middle line:90% 00:53:42.000 --> 00:53:45.830 align:middle line:90% We discussed four modes. 00:53:45.830 --> 00:53:49.740 align:middle line:84% X mode, and we'll start with the O mode. 00:53:49.740 --> 00:53:52.930 align:middle line:90% It's much easier. 00:53:52.930 --> 00:53:53.935 align:middle line:90% The other mode is easy. 00:53:53.935 --> 00:53:59.160 align:middle line:90% 00:53:59.160 --> 00:54:08.130 align:middle line:84% So once again, we think about our plasma 00:54:08.130 --> 00:54:12.420 align:middle line:84% as a circular cross-section and a magnetic field, 00:54:12.420 --> 00:54:16.959 align:middle line:84% which folds up as roughly 1 upon r. 00:54:16.959 --> 00:54:22.710 align:middle line:90% 00:54:22.710 --> 00:54:30.130 align:middle line:84% And so what I'm going to plot is a drawing with the r-axis 00:54:30.130 --> 00:54:34.390 align:middle line:84% and the frequency of different modes within the plasma here. 00:54:34.390 --> 00:54:35.680 align:middle line:90% And I'll plot this r axis. 00:54:35.680 --> 00:54:36.910 align:middle line:90% I'll have zero. 00:54:36.910 --> 00:54:39.460 align:middle line:90% This is going to be r over r0. 00:54:39.460 --> 00:54:44.230 align:middle line:84% r0 is the coordinate to the middle 00:54:44.230 --> 00:54:45.970 align:middle line:90% of our circular cross-section. 00:54:45.970 --> 00:54:49.090 align:middle line:84% And so this is the central plasma one here. 00:54:49.090 --> 00:54:50.800 align:middle line:90% That's zero. 00:54:50.800 --> 00:54:51.700 align:middle line:90% Call that 2. 00:54:51.700 --> 00:54:56.800 align:middle line:84% The plasma is going to have some boundaries here and here. 00:54:56.800 --> 00:54:58.840 align:middle line:84% Very tight aspect ratio you're talking about. 00:54:58.840 --> 00:55:00.000 align:middle line:90% OK. 00:55:00.000 --> 00:55:02.500 align:middle line:84% And so for the O mode, we know that the spatial relationship 00:55:02.500 --> 00:55:06.790 align:middle line:84% is just n squared equals 1 over omega p squared 00:55:06.790 --> 00:55:11.230 align:middle line:90% upon omega squared, like that. 00:55:11.230 --> 00:55:13.690 align:middle line:84% If we think about a typical tokamak, 00:55:13.690 --> 00:55:16.720 align:middle line:84% we know that the density is going to be peaked. 00:55:16.720 --> 00:55:18.280 align:middle line:84% So we're going to assume that we have 00:55:18.280 --> 00:55:20.260 align:middle line:84% a tokamak where we've achieved good particle 00:55:20.260 --> 00:55:21.880 align:middle line:90% confinement in the core. 00:55:21.880 --> 00:55:24.637 align:middle line:84% We know the density has to go to zero at the edge anyway, 00:55:24.637 --> 00:55:26.720 align:middle line:84% otherwise there wouldn't be the end of the plasma. 00:55:26.720 --> 00:55:28.345 align:middle line:84% And so we're going to assume that we've 00:55:28.345 --> 00:55:34.030 align:middle line:84% got some density profile that's a little bit like this. 00:55:34.030 --> 00:55:36.300 align:middle line:84% And that means that at the edges of the plasma, 00:55:36.300 --> 00:55:43.670 align:middle line:84% the plasma frequency, omega p, is zero. 00:55:43.670 --> 00:55:47.480 align:middle line:84% Remember the plasma frequency goes as any squared 00:55:47.480 --> 00:55:49.740 align:middle line:90% plus a few other constants. 00:55:49.740 --> 00:55:53.450 align:middle line:84% And that means in the middle it's going to peak. 00:55:53.450 --> 00:55:55.450 align:middle line:84% It's going to be the highest possible frequency. 00:55:55.450 --> 00:55:57.100 align:middle line:84% That plasma frequency is important 00:55:57.100 --> 00:56:00.790 align:middle line:84% because waves below the plasma frequency are evanescent. 00:56:00.790 --> 00:56:03.760 align:middle line:84% So if you have a wave that's propagating 00:56:03.760 --> 00:56:07.030 align:middle line:84% in this region of frequency position space, 00:56:07.030 --> 00:56:08.260 align:middle line:90% it's going to be evanescent. 00:56:08.260 --> 00:56:09.280 align:middle line:90% It won't propagate. 00:56:09.280 --> 00:56:11.150 align:middle line:90% It won't reach us. 00:56:11.150 --> 00:56:11.870 align:middle line:90% OK. 00:56:11.870 --> 00:56:14.450 align:middle line:84% What about the cyclotron harmonics? 00:56:14.450 --> 00:56:19.250 align:middle line:84% Well, they go as magnetic fields. 00:56:19.250 --> 00:56:21.090 align:middle line:90% They also go as 1 upon r. 00:56:21.090 --> 00:56:25.570 align:middle line:84% So this is maybe the first harmonic, the second harmonic, 00:56:25.570 --> 00:56:27.210 align:middle line:90% the third harmonic. 00:56:27.210 --> 00:56:28.860 align:middle line:84% Let me just label these a bit better. 00:56:28.860 --> 00:56:34.360 align:middle line:84% Omega, 2 omega, 3 omega, like that. 00:56:34.360 --> 00:56:36.570 align:middle line:84% So in the picture I've drawn right now, 00:56:36.570 --> 00:56:40.110 align:middle line:84% if I have a cyclotron frequency, they're 00:56:40.110 --> 00:56:43.620 align:middle line:84% here born at this point near the far edge of the plasma, 00:56:43.620 --> 00:56:50.870 align:middle line:84% and I follow the trajectory of that wave outwards, 00:56:50.870 --> 00:56:52.850 align:middle line:84% I don't pass through the evanescent region. 00:56:52.850 --> 00:56:55.250 align:middle line:84% This wave is going to make it to my detector. 00:56:55.250 --> 00:56:58.250 align:middle line:84% Of course, it may also be absorbed as we talked about 00:56:58.250 --> 00:57:01.280 align:middle line:84% and become black body, but the general idea 00:57:01.280 --> 00:57:03.320 align:middle line:84% is that this wave could propagate out 00:57:03.320 --> 00:57:04.160 align:middle line:90% without absorption. 00:57:04.160 --> 00:57:06.650 align:middle line:84% And certainly the second and third harmonics 00:57:06.650 --> 00:57:08.720 align:middle line:90% could all propagate out. 00:57:08.720 --> 00:57:11.870 align:middle line:84% But what if my plasma was a little bit denser than this? 00:57:11.870 --> 00:57:16.290 align:middle line:84% What if I increase the density in the core a little bit more? 00:57:16.290 --> 00:57:20.030 align:middle line:84% Then I could end up in a scenario 00:57:20.030 --> 00:57:22.340 align:middle line:84% where there is now an evanescent region. 00:57:22.340 --> 00:57:25.190 align:middle line:84% And in fact, almost all of this wave 00:57:25.190 --> 00:57:30.080 align:middle line:84% will get reflected back to the high field side 00:57:30.080 --> 00:57:34.470 align:middle line:84% to the r towards 0 side, and only this region will propagate. 00:57:34.470 --> 00:57:44.370 align:middle line:84% So this stuff will be reflected and only this. 00:57:44.370 --> 00:57:47.360 align:middle line:84% And the stuff that's born inside here can't even be born. 00:57:47.360 --> 00:57:50.960 align:middle line:84% There's no mode in which for that electromagnetic radiation 00:57:50.960 --> 00:57:52.400 align:middle line:90% to be emitted. 00:57:52.400 --> 00:57:52.970 align:middle line:90% OK. 00:57:52.970 --> 00:57:56.260 align:middle line:90% So this can still propagate. 00:57:56.260 --> 00:58:02.370 align:middle line:90% 00:58:02.370 --> 00:58:05.760 align:middle line:84% So that means that for a sufficiently high density, 00:58:05.760 --> 00:58:08.910 align:middle line:84% if I draw my tokamak cross-section again, 00:58:08.910 --> 00:58:18.300 align:middle line:84% there is a region where omega capital omega is less than omega 00:58:18.300 --> 00:58:21.630 align:middle line:90% p-- that's this region here-- 00:58:21.630 --> 00:58:24.570 align:middle line:84% where I will not get electron cyclotron emission from. 00:58:24.570 --> 00:58:36.397 align:middle line:84% So this effectively says at high density what's ECE. 00:58:36.397 --> 00:58:38.230 align:middle line:84% And you could imagine the density could even 00:58:38.230 --> 00:58:41.840 align:middle line:84% get so high it blocks the second harmonic or something like that. 00:58:41.840 --> 00:58:44.080 align:middle line:84% So this is a challenge if you're designing a tokamak 00:58:44.080 --> 00:58:46.507 align:middle line:84% and you want to have some magnetic field and some density. 00:58:46.507 --> 00:58:48.340 align:middle line:84% And obviously, those are not free parameters 00:58:48.340 --> 00:58:51.465 align:middle line:84% because we have data limits and things like that. 00:58:51.465 --> 00:58:52.840 align:middle line:84% If you're designing some tokamak, 00:58:52.840 --> 00:58:55.570 align:middle line:84% you might end up in a situation where 00:58:55.570 --> 00:58:59.020 align:middle line:84% you don't have accessibility to this first harmonic 00:58:59.020 --> 00:59:02.560 align:middle line:84% and so you can't use it as a diagnostic like you wanted to. 00:59:02.560 --> 00:59:04.720 align:middle line:84% STUDENT: Just to ask again, what happens 00:59:04.720 --> 00:59:07.908 align:middle line:84% when it's emitted under the plasma frequency? 00:59:07.908 --> 00:59:09.700 align:middle line:84% JACK HARE: I don't think it can be emitted. 00:59:09.700 --> 00:59:11.230 align:middle line:84% There's no mode for it to go into. 00:59:11.230 --> 00:59:14.080 align:middle line:84% STUDENT: [INAUDIBLE] so you have you still 00:59:14.080 --> 00:59:16.810 align:middle line:90% have orbiting electrons. 00:59:16.810 --> 00:59:20.340 align:middle line:84% For example, the far field would get destroyed by the plasma 00:59:20.340 --> 00:59:25.210 align:middle line:84% effects or [INAUDIBLE] get reflected back [INAUDIBLE]. 00:59:25.210 --> 00:59:32.430 align:middle line:90% 00:59:32.430 --> 00:59:33.850 align:middle line:90% JACK HARE: It's a good question. 00:59:33.850 --> 00:59:36.480 align:middle line:84% For those online, the question is, what actually happens 00:59:36.480 --> 00:59:38.580 align:middle line:84% to the electric fields generated by gyrating 00:59:38.580 --> 00:59:41.520 align:middle line:84% particles in a single particle picture in this region? 00:59:41.520 --> 00:59:47.370 align:middle line:90% 00:59:47.370 --> 00:59:49.710 align:middle line:84% What happens if you try and launch a wave evanescently? 00:59:49.710 --> 00:59:51.690 align:middle line:84% Well, you were telling me the other day 00:59:51.690 --> 00:59:54.090 align:middle line:84% that if you're launching your waves from too far away 00:59:54.090 --> 00:59:55.620 align:middle line:84% from the plasma, they evanescently 00:59:55.620 --> 00:59:59.430 align:middle line:84% decay until they couple into the plasma, where there is 00:59:59.430 --> 01:00:00.970 align:middle line:90% a mode that they can fulfill. 01:00:00.970 --> 01:00:05.193 align:middle line:84% So what happens to the energy that is lost? 01:00:05.193 --> 01:00:07.110 align:middle line:84% Some of the energy is coupled into the plasma. 01:00:07.110 --> 01:00:08.693 align:middle line:84% What happens to the rest of the energy 01:00:08.693 --> 01:00:11.820 align:middle line:84% from when you do low field side launch? 01:00:11.820 --> 01:00:14.313 align:middle line:84% Is it reflected back into the antenna? 01:00:14.313 --> 01:00:16.230 align:middle line:84% STUDENT: Well, I think it's lost in the scrape 01:00:16.230 --> 01:00:18.402 align:middle line:84% off layer to some extent, but reflections also. 01:00:18.402 --> 01:00:20.610 align:middle line:84% JACK HARE: Well, so you're saying it's collisionless, 01:00:20.610 --> 01:00:22.402 align:middle line:84% but maybe this is one of those places where 01:00:22.402 --> 01:00:24.900 align:middle line:84% a small collisionality comes to the rescue and saves us. 01:00:24.900 --> 01:00:25.410 align:middle line:90% Right? 01:00:25.410 --> 01:00:27.420 align:middle line:84% So there's lots of cases where you're 01:00:27.420 --> 01:00:30.030 align:middle line:84% like, if it's collisionless, physics breaks. 01:00:30.030 --> 01:00:32.070 align:middle line:84% That probably means that something happens such 01:00:32.070 --> 01:00:33.862 align:middle line:84% that the collisionality becomes high enough 01:00:33.862 --> 01:00:36.057 align:middle line:90% to save the day again. 01:00:36.057 --> 01:00:37.515 align:middle line:84% What am I thinking of specifically? 01:00:37.515 --> 01:00:39.860 align:middle line:84% Back to the sheaths, actually, where you have-- 01:00:39.860 --> 01:00:41.400 align:middle line:84% you say my sheath is collisionless 01:00:41.400 --> 01:00:44.050 align:middle line:84% and then you violate all sorts of fundamental laws of physics. 01:00:44.050 --> 01:00:44.850 align:middle line:84% And then you're like, well, perhaps 01:00:44.850 --> 01:00:46.350 align:middle line:84% it's slightly collisional, but not enough 01:00:46.350 --> 01:00:48.750 align:middle line:84% that I care about it, but just enough to save the day. 01:00:48.750 --> 01:00:49.510 align:middle line:90% So I don't know. 01:00:49.510 --> 01:00:51.745 align:middle line:90% That's a great question. 01:00:51.745 --> 01:00:52.870 align:middle line:90% I'll have a think about it. 01:00:52.870 --> 01:00:54.480 align:middle line:84% I'm probably not going to find the solution to it, 01:00:54.480 --> 01:00:56.910 align:middle line:84% but if anyone else does, feel free to put along the answer 01:00:56.910 --> 01:00:58.410 align:middle line:90% or something for that. 01:00:58.410 --> 01:01:00.900 align:middle line:84% STUDENT: But radiation does need to be admitted or emitted 01:01:00.900 --> 01:01:01.567 align:middle line:90% at least, right? 01:01:01.567 --> 01:01:04.342 align:middle line:84% Because it's still a moving charge? 01:01:04.342 --> 01:01:05.050 align:middle line:90% JACK HARE: Right. 01:01:05.050 --> 01:01:08.740 align:middle line:84% So the question is like, does the particle lose energy? 01:01:08.740 --> 01:01:11.470 align:middle line:84% Does it just decide not to emit because it's moving? 01:01:11.470 --> 01:01:11.980 align:middle line:90% Yeah. 01:01:11.980 --> 01:01:12.820 align:middle line:90% I don't know. 01:01:12.820 --> 01:01:16.780 align:middle line:84% I suspect there are also some limitations to the WKB picture 01:01:16.780 --> 01:01:18.590 align:middle line:84% that we're using here and things like that. 01:01:18.590 --> 01:01:21.110 align:middle line:90% So I don't know the answer yet. 01:01:21.110 --> 01:01:22.960 align:middle line:84% STUDENT: I have an even more evil question. 01:01:22.960 --> 01:01:24.002 align:middle line:90% JACK HARE: OK, go for it. 01:01:24.002 --> 01:01:24.502 align:middle line:90% I see. 01:01:24.502 --> 01:01:26.800 align:middle line:84% STUDENT: [INAUDIBLE] towards the process that generates 01:01:26.800 --> 01:01:28.307 align:middle line:90% the harmonic components? 01:01:28.307 --> 01:01:29.140 align:middle line:90% JACK HARE: Oh, yeah. 01:01:29.140 --> 01:01:30.473 align:middle line:90% That would be good, wouldn't it? 01:01:30.473 --> 01:01:31.120 align:middle line:90% Yeah. 01:01:31.120 --> 01:01:33.250 align:middle line:84% No, so I guess don't know the answer to that one either, 01:01:33.250 --> 01:01:34.417 align:middle line:90% but that's a great question. 01:01:34.417 --> 01:01:35.120 align:middle line:90% Yeah. 01:01:35.120 --> 01:01:35.750 align:middle line:90% Cool. 01:01:35.750 --> 01:01:36.250 align:middle line:90% Nikola? 01:01:36.250 --> 01:01:38.500 align:middle line:84% STUDENT: So we can look at it from the way 01:01:38.500 --> 01:01:42.070 align:middle line:84% we looked last class and when you entered 01:01:42.070 --> 01:01:43.450 align:middle line:90% the evanescent [INAUDIBLE]? 01:01:43.450 --> 01:01:50.030 align:middle line:90% 01:01:50.030 --> 01:01:52.520 align:middle line:84% Anyway, the point is you never get oscillations. 01:01:52.520 --> 01:01:56.300 align:middle line:84% You still get an energy release via the wave, 01:01:56.300 --> 01:01:57.350 align:middle line:90% but it's not a wave. 01:01:57.350 --> 01:02:00.200 align:middle line:90% It's just an exponential decay. 01:02:00.200 --> 01:02:03.260 align:middle line:84% JACK HARE: I don't think you do get energy released by the wave. 01:02:03.260 --> 01:02:04.760 align:middle line:84% The point of the evanescent wave is 01:02:04.760 --> 01:02:08.300 align:middle line:84% that that energy that you've lost gets reflected back. 01:02:08.300 --> 01:02:09.980 align:middle line:84% It's not absorbed inside the plasma. 01:02:09.980 --> 01:02:12.140 align:middle line:90% That's for sure. 01:02:12.140 --> 01:02:14.600 align:middle line:84% And that's true whether it's a plasma or a block of wax 01:02:14.600 --> 01:02:15.480 align:middle line:90% with some microwaves. 01:02:15.480 --> 01:02:15.980 align:middle line:90% Right? 01:02:15.980 --> 01:02:19.250 align:middle line:84% So that's just the property of whether the waves can propagate 01:02:19.250 --> 01:02:19.820 align:middle line:90% or not. 01:02:19.820 --> 01:02:23.618 align:middle line:84% But I don't think the energy gets dissipated in the medium. 01:02:23.618 --> 01:02:25.910 align:middle line:84% STUDENT: And that's what's spooky about when you have-- 01:02:25.910 --> 01:02:26.835 align:middle line:90% JACK HARE: Yes. 01:02:26.835 --> 01:02:28.460 align:middle line:84% That's what's spooky about having a gap 01:02:28.460 --> 01:02:30.500 align:middle line:84% that the wave just sort of goes through and then 01:02:30.500 --> 01:02:31.625 align:middle line:90% appears out the other side. 01:02:31.625 --> 01:02:33.600 align:middle line:90% Yeah. 01:02:33.600 --> 01:02:35.250 align:middle line:84% I mean, so in this case here, you 01:02:35.250 --> 01:02:39.345 align:middle line:84% could imagine if you had a small enough evanescent region-- 01:02:39.345 --> 01:02:41.290 align:middle line:90% oh, that's interesting. 01:02:41.290 --> 01:02:41.790 align:middle line:90% OK. 01:02:41.790 --> 01:02:43.415 align:middle line:84% If you have a lot of evanescent region, 01:02:43.415 --> 01:02:46.260 align:middle line:84% you could have some radiation that couples to the other side 01:02:46.260 --> 01:02:48.270 align:middle line:84% where there's a mode you can propagate into. 01:02:48.270 --> 01:02:52.620 align:middle line:84% Similarly, if you have an evanescent length scale which 01:02:52.620 --> 01:02:55.350 align:middle line:84% is long enough, then the plasma inside this region 01:02:55.350 --> 01:02:57.630 align:middle line:84% could couple to some modes outside, right? 01:02:57.630 --> 01:03:00.108 align:middle line:84% And evanescentally produce radiation. 01:03:00.108 --> 01:03:02.650 align:middle line:84% I will say this is much more complicated than what Hutchinson 01:03:02.650 --> 01:03:03.370 align:middle line:90% has in his notes. 01:03:03.370 --> 01:03:04.990 align:middle line:90% I am not a ECE person. 01:03:04.990 --> 01:03:07.468 align:middle line:90% These are great questions. 01:03:07.468 --> 01:03:10.010 align:middle line:84% I'm going to keep going before someone asks another question. 01:03:10.010 --> 01:03:11.480 align:middle line:84% I've almost run out of tea as well. 01:03:11.480 --> 01:03:12.190 align:middle line:90% I'm so stressed. 01:03:12.190 --> 01:03:15.270 align:middle line:90% 01:03:15.270 --> 01:03:15.770 align:middle line:90% Good. 01:03:15.770 --> 01:03:16.340 align:middle line:90% Right. 01:03:16.340 --> 01:03:17.720 align:middle line:90% Let's do the X mode. 01:03:17.720 --> 01:03:24.490 align:middle line:90% 01:03:24.490 --> 01:03:27.378 align:middle line:84% So the X mode has n squared equals 1 minus-- 01:03:27.378 --> 01:03:28.920 align:middle line:84% you really shouldn't write this down. 01:03:28.920 --> 01:03:30.150 align:middle line:90% We've talked about modes before. 01:03:30.150 --> 01:03:31.525 align:middle line:84% I just want to write on the board 01:03:31.525 --> 01:03:34.140 align:middle line:84% so that we are staring at it a little bit. 01:03:34.140 --> 01:03:45.110 align:middle line:90% 01:03:45.110 --> 01:03:47.520 align:middle line:90% OK. 01:03:47.520 --> 01:03:50.930 align:middle line:84% And just to remind you with the symbol mode example, 01:03:50.930 --> 01:03:53.960 align:middle line:90% this surface is when n equals 0. 01:03:53.960 --> 01:03:55.310 align:middle line:90% And so this is the cut off. 01:03:55.310 --> 01:03:58.170 align:middle line:90% 01:03:58.170 --> 01:04:00.000 align:middle line:84% So the question-- and it's pretty obvious 01:04:00.000 --> 01:04:01.680 align:middle line:84% that the cutoff, in this case, is when 01:04:01.680 --> 01:04:03.240 align:middle line:90% omega equals omega p squared. 01:04:03.240 --> 01:04:06.690 align:middle line:84% What's much less obvious for the X mode is where the cutoffs are. 01:04:06.690 --> 01:04:07.290 align:middle line:90% Right? 01:04:07.290 --> 01:04:09.480 align:middle line:84% So this is a complicated formula. 01:04:09.480 --> 01:04:13.660 align:middle line:84% We can go and rearrange it and find out what we get out here. 01:04:13.660 --> 01:04:16.710 align:middle line:84% And what we find out is that we have cut 01:04:16.710 --> 01:04:27.350 align:middle line:84% offs for a wave frequency omega less than omega l. 01:04:27.350 --> 01:04:30.020 align:middle line:84% And I'll explain that to the left resonance. 01:04:30.020 --> 01:04:32.660 align:middle line:84% And also for a wave frequency that 01:04:32.660 --> 01:04:38.050 align:middle line:84% is between what's called the upper hybrid 01:04:38.050 --> 01:04:39.770 align:middle line:90% and the right resonance as well. 01:04:39.770 --> 01:04:41.290 align:middle line:84% And I haven't defined any of those. 01:04:41.290 --> 01:04:42.537 align:middle line:90% I will do that in a moment. 01:04:42.537 --> 01:04:44.620 align:middle line:84% But just to say there are now actually two regions 01:04:44.620 --> 01:04:45.910 align:middle line:90% in which we have cutoffs. 01:04:45.910 --> 01:04:49.060 align:middle line:84% Previously we just had this region with a cutoff for omega 01:04:49.060 --> 01:04:51.580 align:middle line:90% less than omega p. 01:04:51.580 --> 01:04:52.795 align:middle line:90% OK. 01:04:52.795 --> 01:04:56.440 align:middle line:84% So you can go to Chen and you can go look up all of these 01:04:56.440 --> 01:05:00.840 align:middle line:84% and you find that the upper hybrid is 01:05:00.840 --> 01:05:08.200 align:middle line:84% the combination of, well, upper hybrid squared is 01:05:08.200 --> 01:05:12.310 align:middle line:84% equal to the electron gyrotron frequency squared 01:05:12.310 --> 01:05:14.560 align:middle line:84% plus the plasma frequency squared. 01:05:14.560 --> 01:05:16.360 align:middle line:84% That's why it's called a hybrid frequency. 01:05:16.360 --> 01:05:20.410 align:middle line:84% It's like a sum of two other important plasma frequencies. 01:05:20.410 --> 01:05:26.090 align:middle line:84% And then these left and right cutoffs, which I'll write in one 01:05:26.090 --> 01:05:29.780 align:middle line:84% go, with left over right meaning there's going to be 01:05:29.780 --> 01:05:31.160 align:middle line:90% plus and minus signs. 01:05:31.160 --> 01:05:33.180 align:middle line:84% And for the left way, if you take the top sign, 01:05:33.180 --> 01:05:35.305 align:middle line:84% and for the right way, if you take the bottom sign, 01:05:35.305 --> 01:05:44.330 align:middle line:84% this is equal to a 1/2 minus a plus omega plus omega squared 01:05:44.330 --> 01:05:51.912 align:middle line:90% plus for omega p squared to 1/2. 01:05:51.912 --> 01:05:53.620 align:middle line:84% So looking at that you think, oh, someone 01:05:53.620 --> 01:05:55.680 align:middle line:90% had to solve a quadratic. 01:05:55.680 --> 01:05:57.640 align:middle line:90% Indeed, that's what happened. 01:05:57.640 --> 01:05:59.540 align:middle line:90% OK. 01:05:59.540 --> 01:06:01.610 align:middle line:84% Doesn't necessarily mean very much to 01:06:01.610 --> 01:06:03.320 align:middle line:84% you at the moment apart from these-- 01:06:03.320 --> 01:06:04.790 align:middle line:84% I'm just going to move this 2 down 01:06:04.790 --> 01:06:06.770 align:middle line:84% a little bit so it's closer to the symbol it's actually 01:06:06.770 --> 01:06:08.060 align:middle line:90% meant to be referring to. 01:06:08.060 --> 01:06:10.550 align:middle line:84% It's clear that we have a complicated set of different cut 01:06:10.550 --> 01:06:11.050 align:middle line:90% offs here. 01:06:11.050 --> 01:06:17.550 align:middle line:84% So what I'm going to do is draw again this diagram upon 01:06:17.550 --> 01:06:26.650 align:middle line:84% on r0 frequency of the mode inside plasma 1, 0, 2. 01:06:26.650 --> 01:06:30.490 align:middle line:84% And we'll do the edges of the plasma here and here. 01:06:30.490 --> 01:06:33.670 align:middle line:84% Now it turns out that the lower cutoff here 01:06:33.670 --> 01:06:37.990 align:middle line:84% is going to look very similar to this O mode cutoff. 01:06:37.990 --> 01:06:46.150 align:middle line:84% And the reason is that if we go to a region where ne at the edge 01:06:46.150 --> 01:06:50.740 align:middle line:84% is equal to zero, then omega p is equal to zero 01:06:50.740 --> 01:06:54.160 align:middle line:84% and we have minus gyrotron frequency plus gyrotron 01:06:54.160 --> 01:06:54.940 align:middle line:90% frequency. 01:06:54.940 --> 01:06:55.900 align:middle line:90% Oh, that's zero. 01:06:55.900 --> 01:06:56.587 align:middle line:90% OK. 01:06:56.587 --> 01:06:58.420 align:middle line:84% And then we can also start this a little bit 01:06:58.420 --> 01:07:01.440 align:middle line:84% and go, OK, it's going to peak in the center. 01:07:01.440 --> 01:07:08.580 align:middle line:84% So this is the lower evanescent region down here. 01:07:08.580 --> 01:07:09.880 align:middle line:90% So far so good. 01:07:09.880 --> 01:07:17.380 align:middle line:84% And again, we can draw on the same 1 01:07:17.380 --> 01:07:29.400 align:middle line:84% over r falling modes for their cyclotron emission. 01:07:29.400 --> 01:07:33.040 align:middle line:84% Now, what's very interesting is when we go and ask ourselves, 01:07:33.040 --> 01:07:34.870 align:middle line:84% well, where is this second region here? 01:07:34.870 --> 01:07:38.830 align:middle line:84% This is the region bounded by the upper hybrid and omega r. 01:07:38.830 --> 01:07:40.300 align:middle line:90% Well, let's do the same thing. 01:07:40.300 --> 01:07:44.250 align:middle line:84% Let's ask ourselves at the edge of the plasma where 01:07:44.250 --> 01:07:46.560 align:middle line:84% the density is zero and the plasma frequency is zero, 01:07:46.560 --> 01:07:48.480 align:middle line:84% the frequency of the upper hybrid 01:07:48.480 --> 01:07:51.780 align:middle line:84% is just going to be the frequency of the first harmonic 01:07:51.780 --> 01:07:53.948 align:middle line:90% here. 01:07:53.948 --> 01:07:56.490 align:middle line:84% And actually, conveniently, the frequency of the lower hybrid 01:07:56.490 --> 01:08:00.210 align:middle line:84% is also going to be the frequency of the lower harmonic. 01:08:00.210 --> 01:08:05.145 align:middle line:84% We can play the same game at the other side here and here. 01:08:05.145 --> 01:08:07.020 align:middle line:84% And if you stare at these for a little while, 01:08:07.020 --> 01:08:10.500 align:middle line:84% you'll realize that the upper harmonic is always 01:08:10.500 --> 01:08:12.760 align:middle line:90% slightly higher than-- 01:08:12.760 --> 01:08:13.260 align:middle line:90% sorry. 01:08:13.260 --> 01:08:15.010 align:middle line:84% The upper hybrid is always slightly higher 01:08:15.010 --> 01:08:16.950 align:middle line:90% than the first harmonic. 01:08:16.950 --> 01:08:21.700 align:middle line:84% And the right hand resonance is even slightly higher than that. 01:08:21.700 --> 01:08:26.840 align:middle line:84% So this is the right hand frequency 01:08:26.840 --> 01:08:29.830 align:middle line:90% and this is the upper hybrid. 01:08:29.830 --> 01:08:32.737 align:middle line:90% 01:08:32.737 --> 01:08:34.279 align:middle line:84% Of course, let me write it like that. 01:08:34.279 --> 01:08:44.870 align:middle line:90% 01:08:44.870 --> 01:08:48.370 align:middle line:84% It's all based on some excellent notes that Hutchinson made 01:08:48.370 --> 01:08:51.520 align:middle line:84% for 22-611, which is still floating around amongst the grad 01:08:51.520 --> 01:08:53.170 align:middle line:84% students if you can get a hold of them. 01:08:53.170 --> 01:08:54.760 align:middle line:90% These are very nice diagrams. 01:08:54.760 --> 01:08:57.393 align:middle line:84% They are in some form in the textbook, 01:08:57.393 --> 01:08:58.935 align:middle line:84% but the ones in his notes are better. 01:08:58.935 --> 01:09:03.130 align:middle line:90% 01:09:03.130 --> 01:09:08.198 align:middle line:84% What does this mean for accessibility? 01:09:08.198 --> 01:09:12.029 align:middle line:90% 01:09:12.029 --> 01:09:15.120 align:middle line:84% STUDENT: You need a minimum of the second harmonic. 01:09:15.120 --> 01:09:15.930 align:middle line:90% JACK HARE: Yeah. 01:09:15.930 --> 01:09:18.390 align:middle line:84% If we're going to observe from the high field side-- 01:09:18.390 --> 01:09:20.760 align:middle line:84% sorry, the low field side out at large R, 01:09:20.760 --> 01:09:23.520 align:middle line:84% which is where we tend to put our detectors, 01:09:23.520 --> 01:09:25.830 align:middle line:84% there's no way this wave is going 01:09:25.830 --> 01:09:29.189 align:middle line:84% to get through this evanescent region, evanescent coupling 01:09:29.189 --> 01:09:30.270 align:middle line:90% notwithstanding. 01:09:30.270 --> 01:09:30.870 align:middle line:90% Right? 01:09:30.870 --> 01:09:32.700 align:middle line:84% Because if we emit here and it's got 01:09:32.700 --> 01:09:35.252 align:middle line:84% to travel in this direction up, it's hit the region. 01:09:35.252 --> 01:09:36.960 align:middle line:84% If we emit here, it's going to travel up. 01:09:36.960 --> 01:09:37.990 align:middle line:90% It's hit the region. 01:09:37.990 --> 01:09:41.590 align:middle line:84% So we have got a complete cut off in assembly here. 01:09:41.590 --> 01:09:44.250 align:middle line:84% And if I'd left myself slightly more space, I would draw it. 01:09:44.250 --> 01:09:54.120 align:middle line:90% 01:09:54.120 --> 01:09:55.870 align:middle line:84% I'll draw it directly underneath this one. 01:09:55.870 --> 01:09:58.440 align:middle line:90% 01:09:58.440 --> 01:10:02.970 align:middle line:84% So now, I, again, have my tokamak cross-section like this, 01:10:02.970 --> 01:10:06.362 align:middle line:84% and I effectively have an evanescent region 01:10:06.362 --> 01:10:12.440 align:middle line:84% that looks a little bit like a banana right in the middle here. 01:10:12.440 --> 01:10:15.860 align:middle line:90% This is the upper hybrid. 01:10:15.860 --> 01:10:18.200 align:middle line:84% This is the right hand resonance. 01:10:18.200 --> 01:10:21.110 align:middle line:84% So if I have any emission anywhere 01:10:21.110 --> 01:10:25.710 align:middle line:84% at the fundamental capital omega, 01:10:25.710 --> 01:10:27.480 align:middle line:90% that's going to reflect back. 01:10:27.480 --> 01:10:31.640 align:middle line:90% 01:10:31.640 --> 01:10:33.600 align:middle line:90% Say it. 01:10:33.600 --> 01:10:34.350 align:middle line:90% STUDENT: Say what? 01:10:34.350 --> 01:10:34.820 align:middle line:90% JACK HARE: I don't know. 01:10:34.820 --> 01:10:35.660 align:middle line:84% I thought you're going to tell me 01:10:35.660 --> 01:10:37.430 align:middle line:84% I should put my detector on the high field side. 01:10:37.430 --> 01:10:39.138 align:middle line:84% STUDENT: I was going to, but now I'm not. 01:10:39.138 --> 01:10:41.750 align:middle line:90% 01:10:41.750 --> 01:10:44.030 align:middle line:84% Is this helpful in the sense of because you now 01:10:44.030 --> 01:10:47.390 align:middle line:84% have no first harmonics electron emission you 01:10:47.390 --> 01:10:50.480 align:middle line:84% can very confidently do your black body thing? 01:10:50.480 --> 01:10:52.520 align:middle line:84% JACK HARE: Well, no, you can't because that's 01:10:52.520 --> 01:10:54.560 align:middle line:84% a mission, which becomes black body. 01:10:54.560 --> 01:10:55.250 align:middle line:90% STUDENT: OK. 01:10:55.250 --> 01:10:57.230 align:middle line:84% JACK HARE: If that's blocked out you can't see it at all. 01:10:57.230 --> 01:10:57.500 align:middle line:90% STUDENT: OK. 01:10:57.500 --> 01:10:59.042 align:middle line:84% JACK HARE: Basically what it's saying 01:10:59.042 --> 01:11:03.170 align:middle line:84% is, so there is a black body spectrum at some frequency 01:11:03.170 --> 01:11:04.830 align:middle line:84% that the plasma should be emitting here 01:11:04.830 --> 01:11:06.830 align:middle line:84% at some frequency with that black body spectrum, 01:11:06.830 --> 01:11:08.810 align:middle line:90% but that frequency gets blocked. 01:11:08.810 --> 01:11:10.650 align:middle line:84% It cannot get through the evanescent region. 01:11:10.650 --> 01:11:12.440 align:middle line:84% So we can't see any of the radiation 01:11:12.440 --> 01:11:14.070 align:middle line:90% from this side on the far side. 01:11:14.070 --> 01:11:15.572 align:middle line:84% We can only see radiation from here. 01:11:15.572 --> 01:11:16.280 align:middle line:90% STUDENT: So yeah. 01:11:16.280 --> 01:11:17.390 align:middle line:84% High field side sounds like a great idea. 01:11:17.390 --> 01:11:19.280 align:middle line:84% JACK HARE: High field site is a good idea for this. 01:11:19.280 --> 01:11:19.820 align:middle line:90% Yeah. 01:11:19.820 --> 01:11:20.660 align:middle line:90% Exactly. 01:11:20.660 --> 01:11:23.300 align:middle line:84% Now didn't go into this at all, and it 01:11:23.300 --> 01:11:25.190 align:middle line:84% seems slightly obscure the exact place where 01:11:25.190 --> 01:11:26.720 align:middle line:90% it crops up in Hutchinson. 01:11:26.720 --> 01:11:31.290 align:middle line:84% But it turns out we had this conversation before about 01:11:31.290 --> 01:11:33.960 align:middle line:84% whether you have O mode or X mode. 01:11:33.960 --> 01:11:35.730 align:middle line:84% And of course, it's not at all clear to us 01:11:35.730 --> 01:11:39.850 align:middle line:84% which of these two modes the gyrating particle couples into, 01:11:39.850 --> 01:11:40.350 align:middle line:90% right? 01:11:40.350 --> 01:11:42.892 align:middle line:84% There's no reason-- we haven't done nearly enough mathematics 01:11:42.892 --> 01:11:44.010 align:middle line:90% to know that. 01:11:44.010 --> 01:11:52.300 align:middle line:84% It turns out that predominantly ECE is in the X mode 01:11:52.300 --> 01:11:55.507 align:middle line:90% by, like, a factor of 10. 01:11:55.507 --> 01:11:57.090 align:middle line:84% STUDENT: That's extremely unfortunate. 01:11:57.090 --> 01:11:58.757 align:middle line:84% JACK HARE: Extremely unfortunate for us. 01:11:58.757 --> 01:12:00.850 align:middle line:90% And yet, we soldier on. 01:12:00.850 --> 01:12:04.210 align:middle line:90% So yeah. 01:12:04.210 --> 01:12:09.540 align:middle line:84% If you have high enough density, this region, of course, 01:12:09.540 --> 01:12:10.980 align:middle line:90% could go like that. 01:12:10.980 --> 01:12:13.870 align:middle line:84% Block out the second mode as well. 01:12:13.870 --> 01:12:16.700 align:middle line:90% Completely possible, so. 01:12:16.700 --> 01:12:19.250 align:middle line:84% It depends an awful lot on the density of your tokamak 01:12:19.250 --> 01:12:20.750 align:middle line:90% or whatever system you do. 01:12:20.750 --> 01:12:24.300 align:middle line:90% 01:12:24.300 --> 01:12:26.350 align:middle line:90% All right. 01:12:26.350 --> 01:12:28.600 align:middle line:84% I want to do the correlation ECE, but I don't think 01:12:28.600 --> 01:12:30.730 align:middle line:90% we're going to-- 01:12:30.730 --> 01:12:32.815 align:middle line:84% I really want to start and not finish. 01:12:32.815 --> 01:12:38.290 align:middle line:90% 01:12:38.290 --> 01:12:39.321 align:middle line:90% Questions? 01:12:39.321 --> 01:12:51.600 align:middle line:90% 01:12:51.600 --> 01:12:53.010 align:middle line:90% Yeah? 01:12:53.010 --> 01:12:55.140 align:middle line:84% STUDENT: Do we have a general rule or intuition 01:12:55.140 --> 01:12:57.990 align:middle line:84% for the second harmonic intensity in terms 01:12:57.990 --> 01:13:03.900 align:middle line:84% of if we can completely block the fundamental frequency, if we 01:13:03.900 --> 01:13:06.780 align:middle line:84% go up to try to detect the second harmonic frequency, 01:13:06.780 --> 01:13:08.550 align:middle line:84% would that be possible practically, 01:13:08.550 --> 01:13:11.353 align:middle line:90% or is it just too low? 01:13:11.353 --> 01:13:12.270 align:middle line:90% JACK HARE: No, no, no. 01:13:12.270 --> 01:13:13.312 align:middle line:90% It's definitely possible. 01:13:13.312 --> 01:13:15.958 align:middle line:84% If you look in Hutchinson's book with his example spectra, 01:13:15.958 --> 01:13:18.000 align:middle line:84% you will see a spectrum with second harmonics in, 01:13:18.000 --> 01:13:20.100 align:middle line:90% and they're very clear. 01:13:20.100 --> 01:13:22.590 align:middle line:84% The trouble is they may be optically thick, 01:13:22.590 --> 01:13:23.340 align:middle line:90% which may be good. 01:13:23.340 --> 01:13:24.690 align:middle line:90% So that may be fine. 01:13:24.690 --> 01:13:27.107 align:middle line:84% You'd be happy if they're optically thick because you will 01:13:27.107 --> 01:13:29.550 align:middle line:90% see some emission from them. 01:13:29.550 --> 01:13:32.100 align:middle line:84% So it just is complicated is the short answer. 01:13:32.100 --> 01:13:33.810 align:middle line:84% So they should be detectable, but they 01:13:33.810 --> 01:13:37.740 align:middle line:84% may have been attenuated by the optical thickness. 01:13:37.740 --> 01:13:40.750 align:middle line:84% They may have been blocked off because our density is too high. 01:13:40.750 --> 01:13:42.250 align:middle line:84% And so there's an evanescent region. 01:13:42.250 --> 01:13:42.450 align:middle line:90% So yeah. 01:13:42.450 --> 01:13:43.408 align:middle line:90% There's lots of things. 01:13:43.408 --> 01:13:45.480 align:middle line:84% You need to do the calculations for your plasma 01:13:45.480 --> 01:13:47.190 align:middle line:84% to work out whether you're going to see them or not. 01:13:47.190 --> 01:13:47.690 align:middle line:90% Yeah. 01:13:47.690 --> 01:13:51.335 align:middle line:90% 01:13:51.335 --> 01:13:52.210 align:middle line:90% Any questions online? 01:13:52.210 --> 01:13:59.600 align:middle line:90% 01:13:59.600 --> 01:14:00.100 align:middle line:90% OK. 01:14:00.100 --> 01:14:01.600 align:middle line:84% I think we will leave it-- oh, yeah. 01:14:01.600 --> 01:14:04.070 align:middle line:90% Go on. 01:14:04.070 --> 01:14:10.302 align:middle line:84% STUDENT: Why is it that the upper hybrid [INAUDIBLE] and-- 01:14:10.302 --> 01:14:12.010 align:middle line:84% JACK HARE: The left and right resonances. 01:14:12.010 --> 01:14:12.440 align:middle line:90% Yeah. 01:14:12.440 --> 01:14:13.023 align:middle line:90% STUDENT: Yeah. 01:14:13.023 --> 01:14:16.450 align:middle line:84% Why do they have to meet the cyclotron frequency? 01:14:16.450 --> 01:14:17.650 align:middle line:90% JACK HARE: Yeah, absolutely. 01:14:17.650 --> 01:14:19.090 align:middle line:90% No, it's a great question. 01:14:19.090 --> 01:14:20.320 align:middle line:90% It's not completely obvious. 01:14:20.320 --> 01:14:24.420 align:middle line:84% The upper hybrid one, imagine we're at the edge of the plasma 01:14:24.420 --> 01:14:24.920 align:middle line:90% here. 01:14:24.920 --> 01:14:31.880 align:middle line:84% So this is the plasma edge, and at this point, ne at the edge 01:14:31.880 --> 01:14:36.090 align:middle line:84% is equal to zero, which means that our plasma frequency 01:14:36.090 --> 01:14:38.650 align:middle line:90% at the edge is equal to zero. 01:14:38.650 --> 01:14:39.150 align:middle line:90% STUDENT: OK. 01:14:39.150 --> 01:14:39.390 align:middle line:90% Yeah. 01:14:39.390 --> 01:14:41.535 align:middle line:84% JACK HARE: And so then if you stare at this for long enough, 01:14:41.535 --> 01:14:42.060 align:middle line:90% you get-- 01:14:42.060 --> 01:14:42.870 align:middle line:90% you get the idea? 01:14:42.870 --> 01:14:43.740 align:middle line:90% Cool. 01:14:43.740 --> 01:14:44.240 align:middle line:90% But yeah. 01:14:44.240 --> 01:14:45.657 align:middle line:84% When I first saw this picture, I'm 01:14:45.657 --> 01:14:47.580 align:middle line:84% like, wait, that's too much of a coincidence. 01:14:47.580 --> 01:14:48.460 align:middle line:90% Is it always like that? 01:14:48.460 --> 01:14:50.040 align:middle line:84% And the answer is, yes, it is always like that. 01:14:50.040 --> 01:14:52.338 align:middle line:84% There's no combination of density and magnetic field 01:14:52.338 --> 01:14:54.130 align:middle line:84% you can come up with where that's not true, 01:14:54.130 --> 01:14:55.047 align:middle line:90% which is kind of cool. 01:14:55.047 --> 01:15:03.640 align:middle line:90% 01:15:03.640 --> 01:15:04.140 align:middle line:90% OK. 01:15:04.140 --> 01:15:05.643 align:middle line:84% I think we're good for today then. 01:15:05.643 --> 01:15:07.060 align:middle line:84% If anyone does have any questions, 01:15:07.060 --> 01:15:08.685 align:middle line:90% they can come and ask them. 01:15:08.685 --> 01:15:10.880 align:middle line:90% See you on Tuesday. 01:15:10.880 --> 01:15:14.000 align:middle line:90%