WEBVTT 00:00:00.000 --> 00:00:01.000 align:middle line:90% 00:00:01.000 --> 00:00:02.000 align:middle line:90% [SQUEAKING] 00:00:02.000 --> 00:00:03.000 align:middle line:90% [RUSTLING] 00:00:03.000 --> 00:00:10.690 align:middle line:90% [CLICKING] 00:00:10.690 --> 00:00:14.620 align:middle line:84% JACK HARE: Welcome, everyone, to the final lecture. 00:00:14.620 --> 00:00:17.100 align:middle line:84% We will be discussing some of the ramifications 00:00:17.100 --> 00:00:20.070 align:middle line:84% of the collective Thomson scattering spectrum, which 00:00:20.070 --> 00:00:24.180 align:middle line:90% we derived last week. 00:00:24.180 --> 00:00:26.850 align:middle line:84% So just to remind you, we derive first of all 00:00:26.850 --> 00:00:31.410 align:middle line:84% that the scattered power into a given solid angle into a given 00:00:31.410 --> 00:00:36.540 align:middle line:84% frequency seen by our spectrometer 00:00:36.540 --> 00:00:40.920 align:middle line:84% was equal to the classical electron radius squared, 00:00:40.920 --> 00:00:44.730 align:middle line:84% the intensity of our laser and its power over the area. 00:00:44.730 --> 00:00:51.350 align:middle line:84% There was some shape function like this 00:00:51.350 --> 00:00:53.990 align:middle line:84% that told us where we should look to see the scattered light. 00:00:53.990 --> 00:00:57.133 align:middle line:84% It was the polarization of the electric field. 00:00:57.133 --> 00:00:59.550 align:middle line:84% It depends on the number of electrons we're scattering off 00:00:59.550 --> 00:01:02.220 align:middle line:84% and how many electrons are within our scattering volume, 00:01:02.220 --> 00:01:05.129 align:middle line:84% and then it depends on this interesting object, 00:01:05.129 --> 00:01:07.230 align:middle line:84% the spectral density function, which 00:01:07.230 --> 00:01:10.860 align:middle line:84% is what contains all of the actual frequency resolved 00:01:10.860 --> 00:01:12.390 align:middle line:90% information here. 00:01:12.390 --> 00:01:17.070 align:middle line:84% And this spectral density function, we found, 00:01:17.070 --> 00:01:18.750 align:middle line:84% depended on the time and the volume 00:01:18.750 --> 00:01:20.340 align:middle line:90% over which we were integrating. 00:01:20.340 --> 00:01:26.160 align:middle line:84% And it was an average over the absolute value of the Fourier 00:01:26.160 --> 00:01:31.280 align:middle line:84% transform of the electron density, squared-- 00:01:31.280 --> 00:01:34.320 align:middle line:84% this normalized by the electron density. 00:01:34.320 --> 00:01:36.350 align:middle line:84% So after we derive that, we said, OK. 00:01:36.350 --> 00:01:40.460 align:middle line:84% Now we need to find out what these electron density 00:01:40.460 --> 00:01:41.480 align:middle line:90% fluctuations look like. 00:01:41.480 --> 00:01:44.000 align:middle line:84% And we did that using a test particle formalism, where 00:01:44.000 --> 00:01:47.180 align:middle line:84% we let some electrons or ions flow through our plasma. 00:01:47.180 --> 00:01:49.190 align:middle line:84% And we looked at the response of the plasma 00:01:49.190 --> 00:01:52.777 align:middle line:84% to these electrons' ions, with the electrons 00:01:52.777 --> 00:01:54.860 align:middle line:84% being repelled from other electrons and ions being 00:01:54.860 --> 00:01:57.300 align:middle line:84% attracted and all sorts of things like that. 00:01:57.300 --> 00:01:59.600 align:middle line:84% And from that, we were able to write down 00:01:59.600 --> 00:02:02.000 align:middle line:84% a formula for the spectral density 00:02:02.000 --> 00:02:05.300 align:middle line:84% function for some arbitrary distribution of electrons 00:02:05.300 --> 00:02:06.200 align:middle line:90% and ions. 00:02:06.200 --> 00:02:11.360 align:middle line:84% And this was 2 pi upon k and E. And then 00:02:11.360 --> 00:02:15.560 align:middle line:84% there was a factor that looked like 1 minus pi E 00:02:15.560 --> 00:02:19.030 align:middle line:90% upon epsilon squared. 00:02:19.030 --> 00:02:21.490 align:middle line:84% The electron distribution function 00:02:21.490 --> 00:02:25.840 align:middle line:84% in the k direction, evaluated at the phase velocity of the wave, 00:02:25.840 --> 00:02:30.370 align:middle line:84% omega upon k, and the component that looked like pi E 00:02:30.370 --> 00:02:35.290 align:middle line:90% upon epsilon squared times pi v. 00:02:35.290 --> 00:02:39.370 align:middle line:84% Charge on the ions and the ion distribution function 00:02:39.370 --> 00:02:47.500 align:middle line:84% in the k direction, evaluated at omega upon k, like that. 00:02:47.500 --> 00:02:55.100 align:middle line:84% And we said that this was the number of electrons at omega 00:02:55.100 --> 00:02:58.130 align:middle line:84% on k, moving with that velocity, and that this 00:02:58.130 --> 00:03:08.970 align:middle line:84% was the response of the plasma to this number of electrons. 00:03:08.970 --> 00:03:10.020 align:middle line:90% And the same here-- 00:03:10.020 --> 00:03:18.940 align:middle line:84% we have the number of ions and the plasma response here. 00:03:18.940 --> 00:03:21.180 align:middle line:84% And so these we call, respectively, 00:03:21.180 --> 00:03:26.160 align:middle line:84% the electron component and the ion component. 00:03:26.160 --> 00:03:28.590 align:middle line:84% And as I've stressed several times before, 00:03:28.590 --> 00:03:31.675 align:middle line:84% we are only ever doing scattering from electrons. 00:03:31.675 --> 00:03:33.300 align:middle line:84% This turns out to be Thomson scattering 00:03:33.300 --> 00:03:36.390 align:middle line:84% from electrons looking at other electrons being repelled 00:03:36.390 --> 00:03:36.960 align:middle line:90% by them. 00:03:36.960 --> 00:03:38.835 align:middle line:84% And this is Thomson scattering from electrons 00:03:38.835 --> 00:03:41.460 align:middle line:84% which were following the ion particles around. 00:03:41.460 --> 00:03:43.890 align:middle line:84% So we're scattering off electrons in both cases, 00:03:43.890 --> 00:03:46.620 align:middle line:84% but some of the electrons tell about other electrons, 00:03:46.620 --> 00:03:51.660 align:middle line:84% and some of the electrons tell us about ions. 00:03:51.660 --> 00:03:53.850 align:middle line:84% And we said that if we specify this now 00:03:53.850 --> 00:03:57.570 align:middle line:84% so our distribution function is a Maxwellian so we have 00:03:57.570 --> 00:04:00.690 align:middle line:84% for the j species, the distribution function in the k 00:04:00.690 --> 00:04:02.940 align:middle line:84% direction is a function of velocity. 00:04:02.940 --> 00:04:07.020 align:middle line:84% That's going to be the number density of the j 00:04:07.020 --> 00:04:10.590 align:middle line:84% species from factors to do with normalization-- 00:04:10.590 --> 00:04:14.280 align:middle line:84% 1 over square root 2 pi, 1 over the thermal velocity 00:04:14.280 --> 00:04:19.260 align:middle line:84% of the j species, exponential of minus v squared 00:04:19.260 --> 00:04:22.770 align:middle line:84% upon the thermal velocity of the j species 00:04:22.770 --> 00:04:26.050 align:middle line:84% squared, where this thermal velocity 00:04:26.050 --> 00:04:31.000 align:middle line:84% is equal to the square root of 2 Tj upon mj. 00:04:31.000 --> 00:04:32.710 align:middle line:84% So if we assume that this is Maxwellian 00:04:32.710 --> 00:04:35.020 align:middle line:84% for the electrons and the ions, which 00:04:35.020 --> 00:04:37.790 align:middle line:90% may have different temperatures. 00:04:37.790 --> 00:04:41.890 align:middle line:84% So we're still allowing Te to not equal the I here. 00:04:41.890 --> 00:04:48.100 align:middle line:84% We found that these susceptibilities, chi, 00:04:48.100 --> 00:04:51.670 align:middle line:84% for the electrons was equal to alpha squared 00:04:51.670 --> 00:04:55.240 align:middle line:84% times this mysterious w function, squiggle e. 00:04:55.240 --> 00:04:57.160 align:middle line:84% I'm going to define all of these in a moment. 00:04:57.160 --> 00:05:05.000 align:middle line:84% And for the ions, was equal to alpha squared zT 00:05:05.000 --> 00:05:12.820 align:middle line:84% e upon TI w squiggle of I, where if alpha 00:05:12.820 --> 00:05:15.670 align:middle line:84% is our collective scattering parameter and it's 00:05:15.670 --> 00:05:19.582 align:middle line:84% defined as 1 upon k lambda Debye. 00:05:19.582 --> 00:05:21.790 align:middle line:84% And if you're curious about which Debye length it is, 00:05:21.790 --> 00:05:24.790 align:middle line:84% it's the Debye length of the electrons. 00:05:24.790 --> 00:05:35.600 align:middle line:84% And this function, w of squiggle is omega upon k times vTj. 00:05:35.600 --> 00:05:38.250 align:middle line:84% So squiggle for the j species looks like that. 00:05:38.250 --> 00:05:40.700 align:middle line:84% So it's the ratio of the phase velocity 00:05:40.700 --> 00:05:45.260 align:middle line:84% of our wave or mode over the thermal velocity 00:05:45.260 --> 00:05:46.730 align:middle line:90% of our distribution function. 00:05:46.730 --> 00:05:59.250 align:middle line:84% And this w here of squiggle as a form that looks like 1 minus 2 00:05:59.250 --> 00:06:03.570 align:middle line:84% squiggle e to the minus squiggle squared, integral from 0 00:06:03.570 --> 00:06:06.990 align:middle line:90% to squiggle, e to the x squared. 00:06:06.990 --> 00:06:12.080 align:middle line:90% The x, that's the real part. 00:06:12.080 --> 00:06:14.680 align:middle line:84% And then there's an imaginary part, 00:06:14.680 --> 00:06:21.840 align:middle line:84% I square root of pi squiggle aetla minus squiggle squared. 00:06:21.840 --> 00:06:25.590 align:middle line:84% And this, of course, is the imaginary part of the function. 00:06:25.590 --> 00:06:33.360 align:middle line:84% And we often write this as equal to the real part 00:06:33.360 --> 00:06:40.740 align:middle line:84% of w of squiggle plus I times the imaginary part, w squiggle. 00:06:40.740 --> 00:06:44.100 align:middle line:84% And we're breaking it down into the imaginary part 00:06:44.100 --> 00:06:47.700 align:middle line:90% several times in what's to come. 00:06:47.700 --> 00:06:48.300 align:middle line:90% OK. 00:06:48.300 --> 00:06:51.420 align:middle line:84% The physical meaning of this squiggle parameter 00:06:51.420 --> 00:06:54.870 align:middle line:84% here is to do with what part of the distribution 00:06:54.870 --> 00:06:57.220 align:middle line:84% function you're interacting with. 00:06:57.220 --> 00:07:03.190 align:middle line:84% So if I draw a little plot here of velocity versus log 00:07:03.190 --> 00:07:07.480 align:middle line:84% of the distribution function here, and I say for Maxwellian, 00:07:07.480 --> 00:07:11.440 align:middle line:84% this is just going to look like minus v squared so it'll 00:07:11.440 --> 00:07:14.150 align:middle line:90% be something like that. 00:07:14.150 --> 00:07:17.440 align:middle line:84% So this is the distribution function for the ions. 00:07:17.440 --> 00:07:23.938 align:middle line:84% Then if I'm looking at how modes with low phase velocity compared 00:07:23.938 --> 00:07:26.230 align:middle line:84% to the thermal velocity interact with this distribution 00:07:26.230 --> 00:07:27.710 align:middle line:90% function, maybe down here. 00:07:27.710 --> 00:07:31.690 align:middle line:84% So this is squiggle ion much less than 1. 00:07:31.690 --> 00:07:34.390 align:middle line:84% Or I could look at high frequency modes 00:07:34.390 --> 00:07:36.985 align:middle line:84% up where there are very few particles with a velocity that 00:07:36.985 --> 00:07:38.110 align:middle line:90% matches the phase velocity. 00:07:38.110 --> 00:07:40.360 align:middle line:84% This would be squiggle greater than 1. 00:07:40.360 --> 00:07:42.580 align:middle line:84% And of course, if I'm somewhere in the middle, 00:07:42.580 --> 00:07:46.150 align:middle line:84% this is squiggle about 1 here, where 00:07:46.150 --> 00:07:49.555 align:middle line:84% we're sampling in the middle of this distribution function. 00:07:49.555 --> 00:07:50.930 align:middle line:84% And it's worth noting, of course, 00:07:50.930 --> 00:07:52.930 align:middle line:84% that for roughly equal temperatures, 00:07:52.930 --> 00:07:56.380 align:middle line:84% the electron distribution function at e 00:07:56.380 --> 00:07:58.430 align:middle line:90% is going to be much broader. 00:07:58.430 --> 00:08:01.810 align:middle line:84% And so that means that, in general, squiggle for e 00:08:01.810 --> 00:08:04.030 align:middle line:84% is going to be less than squiggle for I 00:08:04.030 --> 00:08:08.180 align:middle line:90% when the temperatures are equal. 00:08:08.180 --> 00:08:10.450 align:middle line:84% So if you're asking, how do the ions 00:08:10.450 --> 00:08:12.863 align:middle line:84% interact with the mode of a certain frequency? 00:08:12.863 --> 00:08:15.280 align:middle line:84% Then the electrons will be interacting as if that mode had 00:08:15.280 --> 00:08:16.270 align:middle line:90% even lower frequency. 00:08:16.270 --> 00:08:18.640 align:middle line:84% Because the electrons-- this line, for example, 00:08:18.640 --> 00:08:21.760 align:middle line:84% here, which was high frequency for the ions 00:08:21.760 --> 00:08:23.770 align:middle line:84% is going to be low frequency for the electrons. 00:08:23.770 --> 00:08:27.130 align:middle line:84% And you'd have to go all the way out here 00:08:27.130 --> 00:08:31.360 align:middle line:84% to find a mode where squiggle e is much, much greater than 1 00:08:31.360 --> 00:08:31.940 align:middle line:90% here. 00:08:31.940 --> 00:08:34.065 align:middle line:84% So in general, the fact that the ions and electrons 00:08:34.065 --> 00:08:38.200 align:middle line:84% have very different masses will divide the spectra 00:08:38.200 --> 00:08:39.892 align:middle line:84% that we see into modes which interact 00:08:39.892 --> 00:08:42.309 align:middle line:84% with the ions, the modes that interact with the electrons. 00:08:42.309 --> 00:08:44.860 align:middle line:84% And those are the anacoustic waves and the electron plasma 00:08:44.860 --> 00:08:48.560 align:middle line:84% waves that we heuristically discussed already. 00:08:48.560 --> 00:08:52.420 align:middle line:90% So that is my summary. 00:08:52.420 --> 00:08:55.990 align:middle line:84% What we're going to do next is take this rather complicated 00:08:55.990 --> 00:08:59.320 align:middle line:84% looking formula and do some limits on it in order 00:08:59.320 --> 00:09:01.768 align:middle line:84% to recover the modes that we know are hidden within all 00:09:01.768 --> 00:09:02.935 align:middle line:90% of this hideous mathematics. 00:09:02.935 --> 00:09:07.402 align:middle line:84% But before we keep going, does anyone have any questions? 00:09:07.402 --> 00:09:08.860 align:middle line:90% Yeah. 00:09:08.860 --> 00:09:10.037 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:09:10.037 --> 00:09:10.620 align:middle line:90% JACK HARE: No. 00:09:10.620 --> 00:09:15.200 align:middle line:90% 00:09:15.200 --> 00:09:16.790 align:middle line:90% Well, yeah. 00:09:16.790 --> 00:09:18.690 align:middle line:90% You can solve it for a bit. 00:09:18.690 --> 00:09:20.499 align:middle line:84% I'm pretty certain it's positive. 00:09:20.499 --> 00:09:28.490 align:middle line:90% 00:09:28.490 --> 00:09:29.270 align:middle line:90% No. 00:09:29.270 --> 00:09:30.230 align:middle line:90% It's chi e. 00:09:30.230 --> 00:09:33.530 align:middle line:84% So if you remember, again, we're just doing the electrons. 00:09:33.530 --> 00:09:40.890 align:middle line:84% The term here, the 1 here, is just the test particle itself. 00:09:40.890 --> 00:09:43.490 align:middle line:84% So each electron in the distribution function. 00:09:43.490 --> 00:09:51.510 align:middle line:84% This term here is e minus repelled from the test particle. 00:09:51.510 --> 00:09:55.020 align:middle line:84% Over here, there would be a 1 if we had Thomson scattering off 00:09:55.020 --> 00:09:56.730 align:middle line:90% the ions, but we don't. 00:09:56.730 --> 00:10:00.940 align:middle line:84% So we don't see any scattered light from the ions themselves. 00:10:00.940 --> 00:10:02.370 align:middle line:90% So there's no 1 here. 00:10:02.370 --> 00:10:06.780 align:middle line:84% All we have is electrons attracted to the ions. 00:10:06.780 --> 00:10:16.390 align:middle line:90% 00:10:16.390 --> 00:10:19.390 align:middle line:84% One sort of formula that maybe you remember, but I will write 00:10:19.390 --> 00:10:21.280 align:middle line:84% down just to be very, very clear, 00:10:21.280 --> 00:10:29.590 align:middle line:84% is that this epsilon is equal to 1 plus pi e plus pi I, 00:10:29.590 --> 00:10:30.230 align:middle line:90% like that. 00:10:30.230 --> 00:10:32.650 align:middle line:84% So that's the permittivity, and it's 00:10:32.650 --> 00:10:36.010 align:middle line:84% related to the two susceptibilities like that. 00:10:36.010 --> 00:10:36.730 align:middle line:90% OK. 00:10:36.730 --> 00:10:37.540 align:middle line:90% Another question? 00:10:37.540 --> 00:10:43.290 align:middle line:90% 00:10:43.290 --> 00:10:53.223 align:middle line:84% AUDIENCE: Wouldn't it be 1 over [INAUDIBLE] 00:10:53.223 --> 00:10:53.890 align:middle line:90% JACK HARE: Yeah. 00:10:53.890 --> 00:10:57.740 align:middle line:90% 00:10:57.740 --> 00:11:02.270 align:middle line:84% The volumes can cancel very nicely here. 00:11:02.270 --> 00:11:05.310 align:middle line:84% I think the time comes from the integration of the incident 00:11:05.310 --> 00:11:05.810 align:middle line:90% power. 00:11:05.810 --> 00:11:08.355 align:middle line:90% 00:11:08.355 --> 00:11:09.623 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:11:09.623 --> 00:11:10.290 align:middle line:90% JACK HARE: Yeah. 00:11:10.290 --> 00:11:14.720 align:middle line:84% Once we rewrite it in terms of this, they've disappeared. 00:11:14.720 --> 00:11:16.280 align:middle line:90% I'd have to check my notes. 00:11:16.280 --> 00:11:22.180 align:middle line:84% I think it might be related to one of the conversions between f 00:11:22.180 --> 00:11:23.440 align:middle line:90% and density, yeah. 00:11:23.440 --> 00:11:29.650 align:middle line:90% 00:11:29.650 --> 00:11:34.477 align:middle line:84% No, this is like an average over an ensemble of articles. 00:11:34.477 --> 00:11:36.310 align:middle line:84% AUDIENCE: I don't think it's a time average. 00:11:36.310 --> 00:11:36.977 align:middle line:90% JACK HARE: Yeah. 00:11:36.977 --> 00:11:39.323 align:middle line:90% 00:11:39.323 --> 00:11:40.990 align:middle line:84% I can't remember off the top of my head. 00:11:40.990 --> 00:11:42.530 align:middle line:84% I can certainly check it out later. 00:11:42.530 --> 00:11:43.030 align:middle line:90% OK. 00:11:43.030 --> 00:11:46.010 align:middle line:90% Any other questions? 00:11:46.010 --> 00:11:46.510 align:middle line:90% All right. 00:11:46.510 --> 00:11:49.540 align:middle line:84% So now we are going to approximate this w here. 00:11:49.540 --> 00:11:54.580 align:middle line:90% 00:11:54.580 --> 00:11:56.838 align:middle line:84% And just to rewrite it here, we have w of x. 00:11:56.838 --> 00:11:58.630 align:middle line:84% I'm just going to use x instead of squiggle 00:11:58.630 --> 00:12:01.600 align:middle line:90% because it's easier to write. 00:12:01.600 --> 00:12:07.660 align:middle line:84% It's equal to the real part of w plus the imaginary part of w. 00:12:07.660 --> 00:12:15.780 align:middle line:84% And for x less than 1, it turns out that the real part of w 00:12:15.780 --> 00:12:19.800 align:middle line:84% has the following expansion as an approximate solution-- 00:12:19.800 --> 00:12:30.960 align:middle line:84% 1 minus 2x squared times 1 minus 2x cubed upon 3 plus a load 00:12:30.960 --> 00:12:34.380 align:middle line:84% of other terms that you could keep going to, but we 00:12:34.380 --> 00:12:36.490 align:middle line:90% won't in this class. 00:12:36.490 --> 00:12:41.970 align:middle line:84% And then for x much, much greater than 1, 00:12:41.970 --> 00:12:49.160 align:middle line:84% we have this real part of x being approximately minus 1 upon 00:12:49.160 --> 00:12:58.950 align:middle line:84% 2x squared, 1 plus 3 over 2 x squared, plus some other terms. 00:12:58.950 --> 00:13:03.320 align:middle line:84% And the way you get this is by carefully 00:13:03.320 --> 00:13:06.170 align:middle line:84% thinking about what's going on with this real part here. 00:13:06.170 --> 00:13:09.020 align:middle line:84% It's not completely trivial to get. 00:13:09.020 --> 00:13:13.910 align:middle line:84% And just to write down again that the imaginary part 00:13:13.910 --> 00:13:20.050 align:middle line:84% for all x is just equal to square root of pi x 00:13:20.050 --> 00:13:24.590 align:middle line:90% e to the minus x squared. 00:13:24.590 --> 00:13:27.200 align:middle line:84% In general, we're very often going 00:13:27.200 --> 00:13:29.900 align:middle line:84% to try and take this imaginary part to 0 for small x. 00:13:29.900 --> 00:13:33.140 align:middle line:84% But when it goes to x, it goes to 0 for small x 00:13:33.140 --> 00:13:35.240 align:middle line:84% and for large x, which is the two regimes we're 00:13:35.240 --> 00:13:37.190 align:middle line:90% going to be working in. 00:13:37.190 --> 00:13:39.140 align:middle line:84% There is one place later on where 00:13:39.140 --> 00:13:41.300 align:middle line:84% I'm going to talk about the actual value for this. 00:13:41.300 --> 00:13:43.175 align:middle line:84% But we tend to be working in these two limits 00:13:43.175 --> 00:13:46.250 align:middle line:84% because that's what makes these results analytically tractable. 00:13:46.250 --> 00:13:48.440 align:middle line:84% Of course, you can always just go and calculate 00:13:48.440 --> 00:13:53.180 align:middle line:84% the full s of k omega using this full tabulated function 00:13:53.180 --> 00:13:55.543 align:middle line:84% that you can find in Python and Julia and Matlab 00:13:55.543 --> 00:13:56.960 align:middle line:84% and all sorts of things like that. 00:13:56.960 --> 00:14:01.980 align:middle line:90% 00:14:01.980 --> 00:14:10.640 align:middle line:84% So the first limit, we're going to pick the limit 00:14:10.640 --> 00:14:15.610 align:middle line:90% where alpha squared goes to 0. 00:14:15.610 --> 00:14:21.323 align:middle line:90% 00:14:21.323 --> 00:14:22.615 align:middle line:90% Yeah, I wish I had more boards. 00:14:22.615 --> 00:14:27.492 align:middle line:90% 00:14:27.492 --> 00:14:29.200 align:middle line:84% You can see the principal thing that this 00:14:29.200 --> 00:14:32.320 align:middle line:84% is going to do is that these susceptibilities both scale 00:14:32.320 --> 00:14:33.440 align:middle line:90% as alpha squared. 00:14:33.440 --> 00:14:36.070 align:middle line:84% So for both of them, the susceptibilities 00:14:36.070 --> 00:14:36.970 align:middle line:90% are going to go to 0. 00:14:36.970 --> 00:14:41.290 align:middle line:84% So chi e and chi I are going to go to 0. 00:14:41.290 --> 00:14:43.780 align:middle line:84% Remember the limit where alpha squared goes to 0 00:14:43.780 --> 00:14:46.480 align:middle line:84% is the same as saying that the wavelength of the fluctuations 00:14:46.480 --> 00:14:50.560 align:middle line:84% we are scattering off is less than the electron Debye length. 00:14:50.560 --> 00:14:53.140 align:middle line:84% So this is scattering within the Debye cloud, 00:14:53.140 --> 00:14:55.300 align:middle line:84% and we expect that this scattering will therefore 00:14:55.300 --> 00:14:57.760 align:middle line:84% give us our incoherent spectra, which 00:14:57.760 --> 00:15:01.090 align:middle line:84% we derived a long time ago under much simpler circumstances. 00:15:01.090 --> 00:15:04.810 align:middle line:84% But I did promise that this new formulation would recover that. 00:15:04.810 --> 00:15:09.650 align:middle line:84% So as these two go to 0, our connectivity, which is just 1 00:15:09.650 --> 00:15:17.280 align:middle line:84% plus e plus pi I just goes to 1 like that-- quite boring. 00:15:17.280 --> 00:15:21.520 align:middle line:84% Because it goes to 1 for all frequencies, 00:15:21.520 --> 00:15:26.380 align:middle line:90% we don't have any waves here. 00:15:26.380 --> 00:15:29.485 align:middle line:84% The permittivity is never 0, so there are no waves. 00:15:29.485 --> 00:15:32.380 align:middle line:90% 00:15:32.380 --> 00:15:39.660 align:middle line:84% And indeed, what we see is that s of k omega, which 00:15:39.660 --> 00:15:48.580 align:middle line:84% is proportional to 1 minus chi e squared upon chi e upon epsilon 00:15:48.580 --> 00:16:00.210 align:middle line:84% squared f of e omega upon k plus Ie upon epsilon squared, f of I 00:16:00.210 --> 00:16:02.080 align:middle line:90% omega upon k. 00:16:02.080 --> 00:16:07.560 align:middle line:84% When we send these chi e's to 0, we just 00:16:07.560 --> 00:16:12.440 align:middle line:84% get that this is proportional to the electron distribution 00:16:12.440 --> 00:16:16.860 align:middle line:84% function, exactly as we found for the incoherent case. 00:16:16.860 --> 00:16:19.610 align:middle line:84% So in this case, we've dropped the response of the plasma. 00:16:19.610 --> 00:16:21.890 align:middle line:84% We've lost all information about the plasma response. 00:16:21.890 --> 00:16:24.170 align:middle line:84% We've just kept the scattering off each 00:16:24.170 --> 00:16:26.520 align:middle line:84% of the individual electrons here. 00:16:26.520 --> 00:16:28.880 align:middle line:90% So no plasma response. 00:16:28.880 --> 00:16:36.950 align:middle line:90% 00:16:36.950 --> 00:16:39.810 align:middle line:90% Just individual electrons. 00:16:39.810 --> 00:16:48.774 align:middle line:90% 00:16:48.774 --> 00:16:50.140 align:middle line:90% Does that make sense? 00:16:50.140 --> 00:16:51.000 align:middle line:90% Any questions? 00:16:51.000 --> 00:17:03.460 align:middle line:90% 00:17:03.460 --> 00:17:06.880 align:middle line:84% I just want to clarify when I say let's take alpha to 0. 00:17:06.880 --> 00:17:09.790 align:middle line:84% This is the thing that you have complete control over. 00:17:09.790 --> 00:17:17.079 align:middle line:84% So alpha is defined as being 1 over k lambda divide e. 00:17:17.079 --> 00:17:19.030 align:middle line:90% This is the size of k. 00:17:19.030 --> 00:17:29.440 align:middle line:84% So this is this, which is equal to 2kI sine theta upon 2. 00:17:29.440 --> 00:17:31.810 align:middle line:84% You've chosen the wavelength of your probe 00:17:31.810 --> 00:17:34.725 align:middle line:84% if you make that wavelength larger. 00:17:34.725 --> 00:17:36.100 align:middle line:84% So you go to a longer wavelength, 00:17:36.100 --> 00:17:37.225 align:middle line:90% you'll get a smaller alpha. 00:17:37.225 --> 00:17:41.600 align:middle line:90% 00:17:41.600 --> 00:17:46.290 align:middle line:84% So larger-- actually, the other way around. 00:17:46.290 --> 00:17:47.655 align:middle line:90% Sorry. 00:17:47.655 --> 00:17:48.155 align:middle line:90% Yes. 00:17:48.155 --> 00:17:53.340 align:middle line:90% 00:17:53.340 --> 00:17:53.850 align:middle line:90% Larger. 00:17:53.850 --> 00:17:55.433 align:middle line:84% I'll write it in terms of wavelengths. 00:17:55.433 --> 00:18:03.270 align:middle line:84% So we tend to think of lasers like they're smaller, lower, 00:18:03.270 --> 00:18:04.230 align:middle line:90% alpha. 00:18:04.230 --> 00:18:07.380 align:middle line:84% Or we can also do the same thing with this. 00:18:07.380 --> 00:18:10.515 align:middle line:90% We can have a larger theta. 00:18:10.515 --> 00:18:24.080 align:middle line:90% 00:18:24.080 --> 00:18:26.580 align:middle line:84% Taking this limit here is not some mathematical abstraction. 00:18:26.580 --> 00:18:28.650 align:middle line:84% This is a choice that you've made as an experimentalist. 00:18:28.650 --> 00:18:30.608 align:middle line:84% Now, of course you might be constrained in what 00:18:30.608 --> 00:18:31.743 align:middle line:90% your laser wavelength is. 00:18:31.743 --> 00:18:34.160 align:middle line:84% You might be constrained where you can place the detector. 00:18:34.160 --> 00:18:37.260 align:middle line:84% And then you'll go, OK, well, it turns out my alpha is very low. 00:18:37.260 --> 00:18:39.748 align:middle line:84% So I guess my Thomson scattering spectrum is incoherent. 00:18:39.748 --> 00:18:42.290 align:middle line:84% But perhaps there's a chance to place your detector somewhere 00:18:42.290 --> 00:18:43.748 align:middle line:84% else or use a different wavelength. 00:18:43.748 --> 00:18:47.090 align:middle line:84% And you could start looking at the coherent particle spectrum, 00:18:47.090 --> 00:18:48.240 align:middle line:90% which we'll discuss now. 00:18:48.240 --> 00:18:48.740 align:middle line:90% OK. 00:18:48.740 --> 00:18:50.345 align:middle line:84% So any questions on that before we keep going? 00:18:50.345 --> 00:18:50.960 align:middle line:90% Yes. 00:18:50.960 --> 00:18:51.765 align:middle line:84% AUDIENCE: Is measuring the electron distribution 00:18:51.765 --> 00:18:52.130 align:middle line:90% function cytoplasm useful? 00:18:52.130 --> 00:18:53.120 align:middle line:90% JACK HARE: Very. 00:18:53.120 --> 00:18:55.550 align:middle line:84% Is measuring the electron distribution function cytoplasm 00:18:55.550 --> 00:18:56.050 align:middle line:90% useful? 00:18:56.050 --> 00:18:57.440 align:middle line:90% Yes, extremely useful. 00:18:57.440 --> 00:19:00.050 align:middle line:84% Because this could include information 00:19:00.050 --> 00:19:02.750 align:middle line:84% about fast particles, like runaway electrons, 00:19:02.750 --> 00:19:05.408 align:middle line:84% particles which have been heated by IDRF. 00:19:05.408 --> 00:19:07.700 align:middle line:84% Non-thermal particles doing chemical reactions in a low 00:19:07.700 --> 00:19:08.540 align:middle line:90% temperature plasma-- 00:19:08.540 --> 00:19:10.040 align:middle line:90% this is incredibly useful, yeah. 00:19:10.040 --> 00:19:11.260 align:middle line:90% 00:19:11.260 --> 00:19:12.970 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:19:12.970 --> 00:19:13.990 align:middle line:84% JACK HARE: It depends what you're trying to do. 00:19:13.990 --> 00:19:16.310 align:middle line:84% In general, it's very hard to get both in the same experiment, 00:19:16.310 --> 00:19:18.435 align:middle line:84% but if you can, you can do some really cool things. 00:19:18.435 --> 00:19:20.120 align:middle line:90% Yeah. 00:19:20.120 --> 00:19:20.620 align:middle line:90% OK. 00:19:20.620 --> 00:19:22.335 align:middle line:90% Other questions? 00:19:22.335 --> 00:19:26.914 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:19:26.914 --> 00:19:27.890 align:middle line:90% JACK HARE: Oh. 00:19:27.890 --> 00:19:29.850 align:middle line:90% What I mean-- sorry. 00:19:29.850 --> 00:19:32.220 align:middle line:84% Yeah, there's no mode to the plasma. 00:19:32.220 --> 00:19:35.920 align:middle line:84% What we found when we looked at s of k. 00:19:35.920 --> 00:19:37.420 align:middle line:84% What we found when we looked at this 00:19:37.420 --> 00:19:40.660 align:middle line:84% before is we said, OK, if we do get epsilon going to 0, 00:19:40.660 --> 00:19:43.720 align:middle line:84% then there will be singularities in f of k omega. 00:19:43.720 --> 00:19:44.920 align:middle line:90% There'll be resonances. 00:19:44.920 --> 00:19:47.050 align:middle line:84% And so our spectrum will contain resonances 00:19:47.050 --> 00:19:50.740 align:middle line:84% which are related to modes in the plasma, waves in the plasma. 00:19:50.740 --> 00:19:52.180 align:middle line:84% But I'm talking about plasma waves 00:19:52.180 --> 00:19:53.800 align:middle line:90% here, not electromagnetic waves. 00:19:53.800 --> 00:19:57.320 align:middle line:84% I'm just being a little bit brief here. 00:19:57.320 --> 00:19:58.190 align:middle line:90% You're quite right. 00:19:58.190 --> 00:20:05.176 align:middle line:90% 00:20:05.176 --> 00:20:10.505 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:20:10.505 --> 00:20:11.130 align:middle line:90% JACK HARE: Yes. 00:20:11.130 --> 00:20:13.230 align:middle line:84% You just scatter off the individual particles. 00:20:13.230 --> 00:20:15.450 align:middle line:84% The wavelength of the scattering is 00:20:15.450 --> 00:20:18.668 align:middle line:84% smaller than the Debye length, and so we only 00:20:18.668 --> 00:20:19.710 align:middle line:90% see individual particles. 00:20:19.710 --> 00:20:21.210 align:middle line:84% We don't see these collective plasma 00:20:21.210 --> 00:20:22.543 align:middle line:90% effects do Debye shielding. 00:20:22.543 --> 00:20:24.210 align:middle line:84% And it's those collective plasma effects 00:20:24.210 --> 00:20:27.030 align:middle line:84% which give rise to the modes and the waves within the plasma. 00:20:27.030 --> 00:20:28.530 align:middle line:84% So that's a reasonable way of thinking about it, 00:20:28.530 --> 00:20:29.260 align:middle line:90% just too fast. 00:20:29.260 --> 00:20:29.760 align:middle line:90% Yeah. 00:20:29.760 --> 00:20:38.545 align:middle line:90% 00:20:38.545 --> 00:20:39.045 align:middle line:90% Yes. 00:20:39.045 --> 00:20:45.040 align:middle line:90% 00:20:45.040 --> 00:20:48.255 align:middle line:84% What we're going to see very soon is that yes, of course, 00:20:48.255 --> 00:20:49.630 align:middle line:84% you get the electron distribution 00:20:49.630 --> 00:20:50.960 align:middle line:90% and the distribution function. 00:20:50.960 --> 00:20:52.630 align:middle line:84% But they're multiplied by this thing, 00:20:52.630 --> 00:20:55.310 align:middle line:84% which is a very complicated function of frequency. 00:20:55.310 --> 00:20:57.672 align:middle line:84% And so it's like your electron distribution function 00:20:57.672 --> 00:20:58.630 align:middle line:90% put through the mangle. 00:20:58.630 --> 00:21:00.250 align:middle line:84% And so you may not be able to work out 00:21:00.250 --> 00:21:03.040 align:middle line:84% what your original electron distribution function was 00:21:03.040 --> 00:21:07.280 align:middle line:84% from that-- again, because as we're about to show, 00:21:07.280 --> 00:21:10.790 align:middle line:84% these are present when epsilon goes to 0. 00:21:10.790 --> 00:21:12.950 align:middle line:90% And so at those particular-- 00:21:12.950 --> 00:21:13.700 align:middle line:90% I should be clear. 00:21:13.700 --> 00:21:19.970 align:middle line:84% All of these here-- this is pi e of omega upon k and epsilon 00:21:19.970 --> 00:21:21.980 align:middle line:90% of omega upon k. 00:21:21.980 --> 00:21:27.530 align:middle line:84% So what this means is that for a specific omega upon k 00:21:27.530 --> 00:21:30.260 align:middle line:84% which makes epsilon go to 0, this function 00:21:30.260 --> 00:21:31.220 align:middle line:90% will really blow up. 00:21:31.220 --> 00:21:34.220 align:middle line:84% So maybe before, you had scattering just 00:21:34.220 --> 00:21:36.290 align:middle line:84% off the electron distribution function, 00:21:36.290 --> 00:21:37.640 align:middle line:90% and it looked like this. 00:21:37.640 --> 00:21:42.115 align:middle line:84% But now as you raise alpha, the presence of these waves 00:21:42.115 --> 00:21:42.990 align:middle line:90% become more apparent. 00:21:42.990 --> 00:21:44.090 align:middle line:84% And these waves are a singularity, 00:21:44.090 --> 00:21:45.757 align:middle line:84% so the scattering is absolutely massive. 00:21:45.757 --> 00:21:49.070 align:middle line:84% So now your spectrum looks like this. 00:21:49.070 --> 00:21:52.070 align:middle line:84% And so now you can't really see the electron distribution 00:21:52.070 --> 00:21:52.830 align:middle line:90% function at all. 00:21:52.830 --> 00:21:54.655 align:middle line:84% It's just some busy background noise. 00:21:54.655 --> 00:21:56.030 align:middle line:84% And all you're actually seeing is 00:21:56.030 --> 00:21:58.490 align:middle line:84% the scattering of the electron plasma waves 00:21:58.490 --> 00:22:01.280 align:middle line:90% on anacoustic waves. 00:22:01.280 --> 00:22:02.180 align:middle line:90% Any other questions? 00:22:02.180 --> 00:22:04.097 align:middle line:84% But it sounds like we should go on to the next 00:22:04.097 --> 00:22:06.320 align:middle line:84% because we're getting ahead of ourselves. 00:22:06.320 --> 00:22:07.190 align:middle line:90% OK, cool. 00:22:07.190 --> 00:22:16.510 align:middle line:84% So now let's talk about arbitrary alpha squared. 00:22:16.510 --> 00:22:18.750 align:middle line:84% So alpha, not necessarily going to 0. 00:22:18.750 --> 00:22:22.550 align:middle line:84% And I'm going to cover two cases here. 00:22:22.550 --> 00:22:27.210 align:middle line:84% The first case is the high frequency waves. 00:22:27.210 --> 00:22:31.050 align:middle line:90% 00:22:31.050 --> 00:22:34.160 align:middle line:84% So for the high frequency waves, we're 00:22:34.160 --> 00:22:37.790 align:middle line:84% going to say that squiggle I is much, much greater than 1. 00:22:37.790 --> 00:22:39.950 align:middle line:84% These waves of frequencies way out, 00:22:39.950 --> 00:22:42.860 align:middle line:84% they have phase velocities much higher 00:22:42.860 --> 00:22:44.440 align:middle line:90% than the thermal velocity. 00:22:44.440 --> 00:22:46.610 align:middle line:84% So there's very few ions around here. 00:22:46.610 --> 00:22:53.290 align:middle line:84% And this effectively means that w of squiggle 00:22:53.290 --> 00:23:00.345 align:middle line:84% goes to 0, which is the same as saying that there's no ion 00:23:00.345 --> 00:23:00.845 align:middle line:90% response. 00:23:00.845 --> 00:23:05.250 align:middle line:90% 00:23:05.250 --> 00:23:08.570 align:middle line:84% So just to be clear here, what we've done 00:23:08.570 --> 00:23:12.590 align:middle line:84% is we've taken this limit here as x gets very, very large, 00:23:12.590 --> 00:23:15.363 align:middle line:84% and we've just gone, well, the first order effect 00:23:15.363 --> 00:23:16.280 align:middle line:90% gets very, very large. 00:23:16.280 --> 00:23:18.130 align:middle line:90% Even this term vanishes. 00:23:18.130 --> 00:23:20.173 align:middle line:84% There's no just one term up front here, 00:23:20.173 --> 00:23:21.590 align:middle line:84% so we're just going to approximate 00:23:21.590 --> 00:23:23.810 align:middle line:84% the real part of this as 0, and the imaginary part 00:23:23.810 --> 00:23:26.180 align:middle line:84% will also go to 0, because for very large x, 00:23:26.180 --> 00:23:30.840 align:middle line:84% this exponential for minus x squared kills off this as well. 00:23:30.840 --> 00:23:33.710 align:middle line:84% So we don't have to worry about what the ions are doing anymore. 00:23:33.710 --> 00:23:35.010 align:middle line:90% They're irrelevant. 00:23:35.010 --> 00:23:37.140 align:middle line:84% What you do find with the electrons 00:23:37.140 --> 00:23:47.560 align:middle line:84% is that you get s of k omega is equal to 2 square root pi. 00:23:47.560 --> 00:23:49.926 align:middle line:84% And then you get the term which is quite interesting. 00:23:49.926 --> 00:23:55.580 align:middle line:84% And I'll put the kve e squared here. 00:23:55.580 --> 00:23:59.110 align:middle line:84% You've got a term that looks like exponential 00:23:59.110 --> 00:24:05.990 align:middle line:84% of minus squiggle e squared upon something 00:24:05.990 --> 00:24:07.220 align:middle line:90% a bit more complicated. 00:24:07.220 --> 00:24:13.280 align:middle line:84% This is 1 plus alpha squared real part of w, 00:24:13.280 --> 00:24:15.370 align:middle line:90% evaluated for squiggle e. 00:24:15.370 --> 00:24:18.520 align:middle line:90% 00:24:18.520 --> 00:24:25.220 align:middle line:84% All of that squared plus alpha squared 00:24:25.220 --> 00:24:30.715 align:middle line:84% imaginary part of w, the squiggly e, all squared. 00:24:30.715 --> 00:24:33.710 align:middle line:90% 00:24:33.710 --> 00:24:36.650 align:middle line:84% This is just by taking the previous expression I gave you 00:24:36.650 --> 00:24:40.490 align:middle line:84% and reducing the susceptibility down towards 0. 00:24:40.490 --> 00:24:41.990 align:middle line:84% The reason this is interesting is 00:24:41.990 --> 00:24:45.950 align:middle line:84% it goes directly back to Grant's question. 00:24:45.950 --> 00:24:53.660 align:middle line:84% This here, this looks exactly like e to the minus 00:24:53.660 --> 00:24:56.740 align:middle line:90% v squared upon eTe squared. 00:24:56.740 --> 00:25:01.230 align:middle line:84% So this is equal to your Maxwellian. 00:25:01.230 --> 00:25:03.690 align:middle line:84% You've got a spectrum that knows about the Maxwellian, 00:25:03.690 --> 00:25:05.460 align:middle line:84% but the Maxwellian is being modified 00:25:05.460 --> 00:25:08.370 align:middle line:84% by this horrific denominator down here. 00:25:08.370 --> 00:25:11.720 align:middle line:84% I think there should be another closing square bracket there. 00:25:11.720 --> 00:25:13.940 align:middle line:84% Maybe I didn't need those, but there. 00:25:13.940 --> 00:25:16.770 align:middle line:84% So you're not just getting back the electron distribution 00:25:16.770 --> 00:25:17.270 align:middle line:90% function. 00:25:17.270 --> 00:25:19.790 align:middle line:84% You're getting back the electron distribution function, 00:25:19.790 --> 00:25:21.920 align:middle line:90% and it is strongly modified. 00:25:21.920 --> 00:25:25.000 align:middle line:90% 00:25:25.000 --> 00:25:28.540 align:middle line:84% So now we want to go see what modifications 00:25:28.540 --> 00:25:32.310 align:middle line:84% this interesting denominator gives us. 00:25:32.310 --> 00:25:38.270 align:middle line:84% So we're going to be looking for resonances, as always, where 00:25:38.270 --> 00:25:45.080 align:middle line:84% the real part of the permittivity goes to 0. 00:25:45.080 --> 00:25:51.140 align:middle line:84% We're going to be using, again, high frequency. 00:25:51.140 --> 00:25:57.620 align:middle line:84% So we're going to say that the electron squiggle factor is 00:25:57.620 --> 00:25:59.540 align:middle line:90% much, much greater than 1. 00:25:59.540 --> 00:26:01.490 align:middle line:84% Note we said the same thing for the ions. 00:26:01.490 --> 00:26:04.260 align:middle line:90% It's not as big as the ions. 00:26:04.260 --> 00:26:06.380 align:middle line:84% So squiggle for the ions is still 00:26:06.380 --> 00:26:08.570 align:middle line:84% much bigger than squiggle for the electrons, which 00:26:08.570 --> 00:26:09.680 align:middle line:90% is much bigger than 1. 00:26:09.680 --> 00:26:11.900 align:middle line:84% So we're not going to be as brutal in cutting off 00:26:11.900 --> 00:26:14.720 align:middle line:84% all of the terms for electrons as we did for the ions. 00:26:14.720 --> 00:26:19.610 align:middle line:84% We're still going to take the imaginary part of w for squiggle 00:26:19.610 --> 00:26:21.020 align:middle line:90% e to be 0. 00:26:21.020 --> 00:26:23.060 align:middle line:84% We're going to ignore the damping here. 00:26:23.060 --> 00:26:26.867 align:middle line:84% But we're going to take the real part of w to your e, 00:26:26.867 --> 00:26:28.700 align:middle line:84% and we're going to keep the first two terms. 00:26:28.700 --> 00:26:33.290 align:middle line:84% And those first two terms are minus 1 upon 2 squiggly squared 00:26:33.290 --> 00:26:39.710 align:middle line:84% minus upon 3 upon 4 squiggly to the 4th. 00:26:39.710 --> 00:26:42.110 align:middle line:84% And so if we're seeking solutions 00:26:42.110 --> 00:26:45.380 align:middle line:84% where this is equal to 0, we'll end up 00:26:45.380 --> 00:26:50.060 align:middle line:84% with something that looks like 1 plus alpha squared, 00:26:50.060 --> 00:26:55.880 align:middle line:84% real part of w, squiggle e is equal to 0. 00:26:55.880 --> 00:26:58.710 align:middle line:84% Because again, just looking at this denominator here, 00:26:58.710 --> 00:27:03.030 align:middle line:90% this is permitivity. 00:27:03.030 --> 00:27:05.680 align:middle line:84% And that gives us out a dispersion relationship, 00:27:05.680 --> 00:27:14.776 align:middle line:84% which is omega squared epw is equal to omega pe squared plus 3 00:27:14.776 --> 00:27:19.450 align:middle line:90% Te on me k squared. 00:27:19.450 --> 00:27:23.770 align:middle line:84% So we have recovered from all of this complicated mathematics 00:27:23.770 --> 00:27:26.090 align:middle line:90% the electron plasma waves here. 00:27:26.090 --> 00:27:28.635 align:middle line:90% And so our s of k omega-- 00:27:28.635 --> 00:27:32.450 align:middle line:90% 00:27:32.450 --> 00:27:33.700 align:middle line:90% I'm going to put it like this. 00:27:33.700 --> 00:27:36.460 align:middle line:84% I'm going to put some little crosses here 00:27:36.460 --> 00:27:39.070 align:middle line:84% to suggest that these are pretty high frequency compared 00:27:39.070 --> 00:27:42.050 align:middle line:84% to other ways we'll be talking about. 00:27:42.050 --> 00:27:45.966 align:middle line:84% It's going to have some little peaks in it. 00:27:45.966 --> 00:27:49.160 align:middle line:90% 00:27:49.160 --> 00:27:51.910 align:middle line:84% Remember, all of our analysis so far has been for high frequency, 00:27:51.910 --> 00:27:53.952 align:middle line:84% so we don't actually know what the spectrum looks 00:27:53.952 --> 00:27:56.408 align:middle line:84% like at these low frequencies here. 00:27:56.408 --> 00:27:58.700 align:middle line:84% But we do know what it looks like at large frequencies. 00:27:58.700 --> 00:28:00.740 align:middle line:84% It's got these very sharp resonances. 00:28:00.740 --> 00:28:05.560 align:middle line:84% The reason that they are not actually singularities 00:28:05.560 --> 00:28:07.810 align:middle line:84% is because there will be a small amount of damping. 00:28:07.810 --> 00:28:09.610 align:middle line:84% We've neglected it in our treatments here just so 00:28:09.610 --> 00:28:11.120 align:middle line:84% that we can get the dispersion relationship. 00:28:11.120 --> 00:28:13.360 align:middle line:84% But if you allow a small amount of damping back in, 00:28:13.360 --> 00:28:17.830 align:middle line:84% that will prevent the denominator from going to 0. 00:28:17.830 --> 00:28:19.330 align:middle line:84% And what you find with these-- these 00:28:19.330 --> 00:28:25.280 align:middle line:84% are occurring at this frequency, the electron plasma wave, 00:28:25.280 --> 00:28:27.020 align:middle line:90% and at minus the frequency. 00:28:27.020 --> 00:28:35.440 align:middle line:90% 00:28:35.440 --> 00:28:36.950 align:middle line:84% I think I'll cover that later on. 00:28:36.950 --> 00:28:38.860 align:middle line:84% And so the position of these depends 00:28:38.860 --> 00:28:40.030 align:middle line:90% on the plasma parameters. 00:28:40.030 --> 00:28:44.420 align:middle line:84% This depends on the density, and this of course, very, 00:28:44.420 --> 00:28:46.510 align:middle line:84% very clearly depends on the temperature. 00:28:46.510 --> 00:28:48.010 align:middle line:84% So the exact location of these peaks 00:28:48.010 --> 00:28:50.510 align:middle line:84% gives you information about the temperature and the density. 00:28:50.510 --> 00:28:53.800 align:middle line:84% And it turns out the broadening of the peaks-- 00:28:53.800 --> 00:28:57.860 align:middle line:84% so this sort of thickness in frequency space 00:28:57.860 --> 00:29:02.250 align:middle line:84% is related to the temperature of the electrons as well. 00:29:02.250 --> 00:29:04.820 align:middle line:84% And you can get that out if you do this more thoroughly 00:29:04.820 --> 00:29:07.823 align:middle line:84% with the Landau damping term, this imaginary component. 00:29:07.823 --> 00:29:09.740 align:middle line:84% That imaginary component broadens these peaks, 00:29:09.740 --> 00:29:11.282 align:middle line:84% and the amount of broadening is to do 00:29:11.282 --> 00:29:13.750 align:middle line:90% with how hot the electrons are. 00:29:13.750 --> 00:29:14.250 align:middle line:90% OK. 00:29:14.250 --> 00:29:17.130 align:middle line:84% So that is the first mode we wanted to get. 00:29:17.130 --> 00:29:22.410 align:middle line:84% And this is the mode related to the electron plasma waves, 00:29:22.410 --> 00:29:24.280 align:middle line:90% or Langmuir waves, if you like. 00:29:24.280 --> 00:29:28.357 align:middle line:84% And these are what we call high frequency waves. 00:29:28.357 --> 00:29:29.940 align:middle line:84% So of course, their frequency is still 00:29:29.940 --> 00:29:33.910 align:middle line:84% much less than the frequencies of the radiation you've heard. 00:29:33.910 --> 00:29:35.080 align:middle line:90% OK. 00:29:35.080 --> 00:29:35.807 align:middle line:90% Questions? 00:29:35.807 --> 00:29:51.740 align:middle line:90% 00:29:51.740 --> 00:29:52.700 align:middle line:90% Yeah. 00:29:52.700 --> 00:29:54.680 align:middle line:84% You can rewrite this so it looks like something 00:29:54.680 --> 00:29:56.730 align:middle line:90% plus alpha squared. 00:29:56.730 --> 00:29:59.810 align:middle line:84% And then in the limit where alpha is really big so you're 00:29:59.810 --> 00:30:02.420 align:middle line:84% very collective, it turns out that it really 00:30:02.420 --> 00:30:03.590 align:middle line:90% just depends on the density. 00:30:03.590 --> 00:30:05.810 align:middle line:84% And this is the limit that lots of people work in. 00:30:05.810 --> 00:30:08.990 align:middle line:84% So you'll find papers where they're like, 00:30:08.990 --> 00:30:12.130 align:middle line:84% this frequency omega is proportional to density, 00:30:12.130 --> 00:30:14.140 align:middle line:84% and the width is proportional to temperature. 00:30:14.140 --> 00:30:16.890 align:middle line:90% 00:30:16.890 --> 00:30:20.370 align:middle line:84% But that's only true for alpha much greater than 1. 00:30:20.370 --> 00:30:22.892 align:middle line:90% If your alpha is like 3-ish-- 00:30:22.892 --> 00:30:24.600 align:middle line:84% and I've done experiments in this regime. 00:30:24.600 --> 00:30:27.000 align:middle line:84% Then the position depends on both of these parameters. 00:30:27.000 --> 00:30:29.730 align:middle line:84% And so you have to go find the width. 00:30:29.730 --> 00:30:32.040 align:middle line:84% That gives you the temperature, and then 00:30:32.040 --> 00:30:36.067 align:middle line:84% you sub the temperature out from the displacement 00:30:36.067 --> 00:30:37.650 align:middle line:84% of the location of the peaks, and that 00:30:37.650 --> 00:30:38.370 align:middle line:90% will give you the density. 00:30:38.370 --> 00:30:39.690 align:middle line:84% In reality, you don't do any of those things. 00:30:39.690 --> 00:30:40.600 align:middle line:90% You fit. 00:30:40.600 --> 00:30:43.110 align:middle line:84% You do a least squares fit to the full spectral density 00:30:43.110 --> 00:30:43.620 align:middle line:90% function. 00:30:43.620 --> 00:30:45.620 align:middle line:84% But if you don't want to do a least squares fit, 00:30:45.620 --> 00:30:47.970 align:middle line:84% you can still actually estimate all these parameters. 00:30:47.970 --> 00:30:49.987 align:middle line:84% And that's what my old Russian professor did. 00:30:49.987 --> 00:30:51.570 align:middle line:84% He would just go and point and go, ah, 00:30:51.570 --> 00:30:53.140 align:middle line:90% this is the frequency shift. 00:30:53.140 --> 00:30:55.265 align:middle line:84% And he would calculate temperature and density just 00:30:55.265 --> 00:30:57.600 align:middle line:90% from looking at the raw data. 00:30:57.600 --> 00:30:59.970 align:middle line:84% You can also do it using computers as well. 00:30:59.970 --> 00:31:04.190 align:middle line:90% 00:31:04.190 --> 00:31:05.457 align:middle line:90% Any other questions? 00:31:05.457 --> 00:31:09.610 align:middle line:90% 00:31:09.610 --> 00:31:10.110 align:middle line:90% OK. 00:31:10.110 --> 00:31:12.669 align:middle line:84% We're going to go to the low frequency. 00:31:12.669 --> 00:31:49.630 align:middle line:90% 00:31:49.630 --> 00:31:55.520 align:middle line:84% So just to remind you, we did arbitrary alpha, alpha squared, 00:31:55.520 --> 00:31:57.430 align:middle line:90% and high frequency. 00:31:57.430 --> 00:31:59.980 align:middle line:84% Now we're going to go looking for low frequency waves here. 00:31:59.980 --> 00:32:09.110 align:middle line:90% 00:32:09.110 --> 00:32:10.880 align:middle line:84% So for these low frequency waves, 00:32:10.880 --> 00:32:13.030 align:middle line:84% we know that squiggle for the electrons 00:32:13.030 --> 00:32:15.940 align:middle line:84% is going to be much, much less than 1. 00:32:15.940 --> 00:32:19.270 align:middle line:84% And so we're just going to say that the real part of squiggle 00:32:19.270 --> 00:32:22.870 align:middle line:84% for the electrons is going to be about 1. 00:32:22.870 --> 00:32:27.280 align:middle line:90% Just to remind you, the small x. 00:32:27.280 --> 00:32:31.060 align:middle line:84% This is 1 minus 2x squared plus a load of other terms. 00:32:31.060 --> 00:32:34.360 align:middle line:84% So for very small x, I can just drop this term 00:32:34.360 --> 00:32:36.190 align:middle line:90% and just get the 1 here. 00:32:36.190 --> 00:32:38.700 align:middle line:90% OK. 00:32:38.700 --> 00:32:43.920 align:middle line:84% So now we find that our permittivity, which is equal, 00:32:43.920 --> 00:32:50.890 align:middle line:84% as always, to 1 plus pi e plus pi I, is equal to 1 00:32:50.890 --> 00:32:56.830 align:middle line:84% plus alpha squared I, the imaginary part of w 00:32:56.830 --> 00:32:58.450 align:middle line:90% for the electrons. 00:32:58.450 --> 00:33:00.040 align:middle line:84% I haven't dropped the imaginary part 00:33:00.040 --> 00:33:01.915 align:middle line:84% because they play an extremely important role 00:33:01.915 --> 00:33:03.230 align:middle line:90% in damping these waves. 00:33:03.230 --> 00:33:05.390 align:middle line:90% So I've got to keep this. 00:33:05.390 --> 00:33:12.250 align:middle line:84% Plus zte upon TI times alpha squared, 00:33:12.250 --> 00:33:18.040 align:middle line:84% the real part of squiggle I plus I 00:33:18.040 --> 00:33:24.710 align:middle line:84% times the imaginary part of squiggle I. 00:33:24.710 --> 00:33:27.530 align:middle line:84% And again, we seek resonances where 00:33:27.530 --> 00:33:34.930 align:middle line:84% the real part of the permittivity epsilon equal 0. 00:33:34.930 --> 00:33:38.330 align:middle line:84% And if you play around with this formula for long enough, 00:33:38.330 --> 00:33:46.950 align:middle line:84% you'll find that these resonances are damped. 00:33:46.950 --> 00:33:48.180 align:middle line:90% That means they don't appear. 00:33:48.180 --> 00:33:51.900 align:middle line:90% 00:33:51.900 --> 00:33:57.720 align:middle line:84% And they're damped for the condition that zTe over TI 00:33:57.720 --> 00:34:02.310 align:middle line:90% is less than 3. 00:34:02.310 --> 00:34:03.600 align:middle line:90% I haven't proved this at all. 00:34:03.600 --> 00:34:06.270 align:middle line:84% You can go and prove it to yourself 00:34:06.270 --> 00:34:08.310 align:middle line:84% by trying to solve this condition here. 00:34:08.310 --> 00:34:11.199 align:middle line:84% And it's in Froula's book, and it takes a few lines of algebra. 00:34:11.199 --> 00:34:12.900 align:middle line:84% So I'm not doing it on the board. 00:34:12.900 --> 00:34:16.409 align:middle line:84% So we don't get any resonances if we have this condition. 00:34:16.409 --> 00:34:19.520 align:middle line:84% So damped in this case means don't appear. 00:34:19.520 --> 00:34:26.820 align:middle line:90% 00:34:26.820 --> 00:34:29.850 align:middle line:84% So let's just quickly take a look at this. 00:34:29.850 --> 00:34:32.790 align:middle line:84% We're very used to having plasmas where the ion electron 00:34:32.790 --> 00:34:34.014 align:middle line:90% temperatures are equal. 00:34:34.014 --> 00:34:35.639 align:middle line:84% But also, many of us are used to having 00:34:35.639 --> 00:34:38.610 align:middle line:90% plasmas where z is equal to 1. 00:34:38.610 --> 00:34:43.830 align:middle line:84% So in this case, this would be Te is less than 3 TI 00:34:43.830 --> 00:34:45.090 align:middle line:90% for a hydrogenic plasma. 00:34:45.090 --> 00:34:48.010 align:middle line:84% So this would be ions hotter than electrons. 00:34:48.010 --> 00:34:51.080 align:middle line:84% So that's not a plasma you necessarily find very often. 00:34:51.080 --> 00:34:53.460 align:middle line:84% So this is a slightly uncommon thing 00:34:53.460 --> 00:34:55.739 align:middle line:84% to not have these resonances appear. 00:34:55.739 --> 00:34:58.770 align:middle line:84% In fact, most of the time, we're going to be in the regime 00:34:58.770 --> 00:35:04.860 align:middle line:84% where zTe over TI is greater than 3. 00:35:04.860 --> 00:35:07.920 align:middle line:84% And this will be possible even with electron and ion 00:35:07.920 --> 00:35:11.160 align:middle line:84% temperatures being equal, if the charge on the ions 00:35:11.160 --> 00:35:12.840 align:middle line:90% was 3 or more. 00:35:12.840 --> 00:35:16.050 align:middle line:84% So this could definitely happen if you have a carbon plasma. 00:35:16.050 --> 00:35:18.330 align:middle line:84% The electrons and ions are in thermal equilibrium, 00:35:18.330 --> 00:35:21.413 align:middle line:84% but because of the charge on the ions, 00:35:21.413 --> 00:35:22.830 align:middle line:84% this condition might be fulfilled. 00:35:22.830 --> 00:35:25.840 align:middle line:84% So this is relatively easy to find. 00:35:25.840 --> 00:35:28.450 align:middle line:90% And we're going to often-- 00:35:28.450 --> 00:35:30.640 align:middle line:84% this greater than 3, this is really the same 00:35:30.640 --> 00:35:31.960 align:middle line:90% as being much bigger than 1. 00:35:31.960 --> 00:35:34.990 align:middle line:84% Because as we all know, 3 is much larger than 1. 00:35:34.990 --> 00:35:36.730 align:middle line:84% So if you fulfill this condition, 00:35:36.730 --> 00:35:41.510 align:middle line:84% then you find that the velocity of your anacoustic waves here, 00:35:41.510 --> 00:35:49.130 align:middle line:84% which is equal to zTe upon mI at 1/2 must be much, 00:35:49.130 --> 00:35:53.240 align:middle line:84% much larger than the thermal velocity of your ions, 00:35:53.240 --> 00:36:00.090 align:middle line:84% which is equal to TI upon mI at 1/2. 00:36:00.090 --> 00:36:04.020 align:middle line:84% Just by plonking this condition that it's much greater than 1 00:36:04.020 --> 00:36:06.450 align:middle line:84% into here, we can see that the modes that we're looking 00:36:06.450 --> 00:36:08.565 align:middle line:84% for-- and these will be the anacoustic waves-- 00:36:08.565 --> 00:36:12.980 align:middle line:90% 00:36:12.980 --> 00:36:15.920 align:middle line:84% are going to be moving much faster than the thermal velocity 00:36:15.920 --> 00:36:18.360 align:middle line:90% of the ions themselves. 00:36:18.360 --> 00:36:21.080 align:middle line:84% And so that means that we can now write down 00:36:21.080 --> 00:36:26.410 align:middle line:84% that squiggle for the ions is going to be much greater than 1. 00:36:26.410 --> 00:36:35.550 align:middle line:84% That means that the real part of w or the ions is roughly minus 1 00:36:35.550 --> 00:36:42.870 align:middle line:84% upon 2 squiggle ion squared minus 3 upon four squiggle ion 00:36:42.870 --> 00:36:43.710 align:middle line:90% to the 4th. 00:36:43.710 --> 00:36:45.850 align:middle line:84% Remember, for the electron plasma waves, 00:36:45.850 --> 00:36:48.970 align:middle line:84% this is the limiting factor that we use for the electrons. 00:36:48.970 --> 00:36:51.810 align:middle line:84% But now we're doing it for the ions here. 00:36:51.810 --> 00:36:54.700 align:middle line:84% And we're again going to assume that there 00:36:54.700 --> 00:36:56.790 align:middle line:90% are no damping from the ions. 00:36:56.790 --> 00:36:59.190 align:middle line:84% And that's because these waves are 00:36:59.190 --> 00:37:02.230 align:middle line:84% velocities much faster than most of the ions in the system. 00:37:02.230 --> 00:37:04.500 align:middle line:84% And so there's no ions to damp up. 00:37:04.500 --> 00:37:07.200 align:middle line:84% So we're going to set the ion damping to 0. 00:37:07.200 --> 00:37:09.960 align:middle line:84% Just remember, we have not set the electron damping to 0. 00:37:09.960 --> 00:37:12.330 align:middle line:84% So the reason that these resonances don't blow up 00:37:12.330 --> 00:37:15.830 align:middle line:84% is because they damp on the electrons. 00:37:15.830 --> 00:37:17.830 align:middle line:84% And then we can do the same thing we did before. 00:37:17.830 --> 00:37:22.210 align:middle line:84% We can substitute all of this in to the permittivity. 00:37:22.210 --> 00:37:24.820 align:middle line:84% We can find the resonances when this is equal to 0. 00:37:24.820 --> 00:37:27.100 align:middle line:84% And we find that we have resonances now 00:37:27.100 --> 00:37:29.990 align:middle line:84% at the anacoustic wave frequency. 00:37:29.990 --> 00:37:33.100 align:middle line:90% And this is equal to k squared. 00:37:33.100 --> 00:37:35.080 align:middle line:90% What follows may surprise you-- 00:37:35.080 --> 00:37:47.220 align:middle line:84% alpha squared 1 plus alpha squared zTe upon mI plus 3 TI 00:37:47.220 --> 00:37:49.570 align:middle line:90% upon mI. 00:37:49.570 --> 00:37:55.004 align:middle line:84% Why do I write it like that, not just like this? 00:37:55.004 --> 00:37:57.004 align:middle line:84% Oh, because actually nobody asked that question. 00:37:57.004 --> 00:38:04.920 align:middle line:90% 00:38:04.920 --> 00:38:07.290 align:middle line:84% So this looks a little bit like what you're expecting, 00:38:07.290 --> 00:38:08.993 align:middle line:90% the anacoustic waves. 00:38:08.993 --> 00:38:10.410 align:middle line:84% But it looks most like what you're 00:38:10.410 --> 00:38:14.610 align:middle line:84% expecting if alpha squared is much, much greater than 1. 00:38:14.610 --> 00:38:24.960 align:middle line:84% This goes to zTe plus 3zI upon m hat, because this factor, 00:38:24.960 --> 00:38:26.730 align:middle line:90% this goes to 1 at that point. 00:38:26.730 --> 00:38:29.937 align:middle line:84% And that looks a fair bit more like the anacoustic velocity 00:38:29.937 --> 00:38:31.020 align:middle line:90% that you're familiar with. 00:38:31.020 --> 00:38:36.070 align:middle line:90% 00:38:36.070 --> 00:38:37.680 align:middle line:84% So now we know that this spectrum 00:38:37.680 --> 00:38:41.010 align:middle line:84% is going to have resonances which are appearing 00:38:41.010 --> 00:38:42.968 align:middle line:90% at this anacoustic frequency. 00:38:42.968 --> 00:38:45.510 align:middle line:84% So questions on that before I sketch what this spectrum looks 00:38:45.510 --> 00:38:46.010 align:middle line:90% like? 00:38:46.010 --> 00:38:54.247 align:middle line:90% 00:38:54.247 --> 00:38:55.730 align:middle line:90% Yeah. 00:38:55.730 --> 00:39:02.115 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:39:02.115 --> 00:39:04.240 align:middle line:84% JACK HARE: There's no [INAUDIBLE] or radii in here. 00:39:04.240 --> 00:39:08.350 align:middle line:90% No point to-- sorry. 00:39:08.350 --> 00:39:10.420 align:middle line:90% What does it mean physically? 00:39:10.420 --> 00:39:14.450 align:middle line:84% If alpha is small now, this term just goes away. 00:39:14.450 --> 00:39:14.950 align:middle line:90% OK. 00:39:14.950 --> 00:39:18.010 align:middle line:84% I'm going to show you what it means physically in a moment. 00:39:18.010 --> 00:39:19.030 align:middle line:90% Actually, no. 00:39:19.030 --> 00:39:20.230 align:middle line:90% That's not in these notes. 00:39:20.230 --> 00:39:22.360 align:middle line:90% That's something else. 00:39:22.360 --> 00:39:23.770 align:middle line:90% Alpha is small. 00:39:23.770 --> 00:39:26.950 align:middle line:90% 00:39:26.950 --> 00:39:31.030 align:middle line:84% If alpha is small, although these resonances will appear, 00:39:31.030 --> 00:39:36.080 align:middle line:84% they will disappear because a small alpha-- the ions 00:39:36.080 --> 00:39:38.240 align:middle line:84% and electrons don't respond anyway. 00:39:38.240 --> 00:39:41.220 align:middle line:84% And we get back to our incoherent scattering spectrum. 00:39:41.220 --> 00:39:44.990 align:middle line:84% So if I made alpha smaller here, although it 00:39:44.990 --> 00:39:47.150 align:middle line:84% would look like this term would drop out, 00:39:47.150 --> 00:39:50.030 align:middle line:84% and I'd get anacoustic resonances happening at 3 TI 00:39:50.030 --> 00:39:53.360 align:middle line:84% upon mI, in reality those anacoustic resonances wouldn't 00:39:53.360 --> 00:39:58.320 align:middle line:84% occur because we'd just be doing incoherent scattering. 00:39:58.320 --> 00:40:00.177 align:middle line:84% Can I explain exactly why it's alpha square 00:40:00.177 --> 00:40:01.260 align:middle line:90% upon 1 plus alpha squared? 00:40:01.260 --> 00:40:02.490 align:middle line:90% No, not at the moment. 00:40:02.490 --> 00:40:17.120 align:middle line:90% 00:40:17.120 --> 00:40:20.990 align:middle line:84% This comes about because we have modes. 00:40:20.990 --> 00:40:28.520 align:middle line:90% 00:40:28.520 --> 00:40:29.540 align:middle line:90% But alpha squared. 00:40:29.540 --> 00:40:36.808 align:middle line:90% 00:40:36.808 --> 00:40:37.350 align:middle line:90% I don't know. 00:40:37.350 --> 00:40:39.630 align:middle line:84% We have derived this assuming that alpha is-- 00:40:39.630 --> 00:40:42.297 align:middle line:84% no, we haven't derived assuming alpha is not small because we've 00:40:42.297 --> 00:40:44.030 align:middle line:90% done it for arbitrary alpha. 00:40:44.030 --> 00:40:46.130 align:middle line:84% So how would we get back the incoherent result? 00:40:46.130 --> 00:40:53.470 align:middle line:90% 00:40:53.470 --> 00:40:53.970 align:middle line:90% OK. 00:40:53.970 --> 00:40:56.490 align:middle line:84% So the interesting thing is that-- 00:40:56.490 --> 00:40:59.670 align:middle line:84% I think I might have erased it, sadly. 00:40:59.670 --> 00:41:00.540 align:middle line:90% Let me just go back. 00:41:00.540 --> 00:41:12.090 align:middle line:90% 00:41:12.090 --> 00:41:17.850 align:middle line:84% Chi e is equal to alpha squared w of squiggle e. 00:41:17.850 --> 00:41:26.220 align:middle line:84% And chi I is equal to alpha squared zTe upon TI 00:41:26.220 --> 00:41:28.060 align:middle line:90% squiggle I, like that. 00:41:28.060 --> 00:41:30.330 align:middle line:84% So when we're doing the incoherent, 00:41:30.330 --> 00:41:32.580 align:middle line:84% we're just looking at this first term, 00:41:32.580 --> 00:41:35.010 align:middle line:90% and we're taking that to 0. 00:41:35.010 --> 00:41:37.860 align:middle line:84% So no matter what w is doing-- and in this case here, 00:41:37.860 --> 00:41:40.590 align:middle line:84% W has got all sorts of funky resonances in it. 00:41:40.590 --> 00:41:43.440 align:middle line:84% Mathematically, at least, that will just not 00:41:43.440 --> 00:41:46.320 align:middle line:84% be important because whatever this response of the plasma 00:41:46.320 --> 00:41:51.390 align:middle line:84% is goes away because we are scattering with modes smaller 00:41:51.390 --> 00:41:56.580 align:middle line:84% than Debye length so that we can't see this part anyway. 00:41:56.580 --> 00:41:58.590 align:middle line:84% So it's like they're two separate arguments. 00:41:58.590 --> 00:42:01.380 align:middle line:84% We use the alpha going to alpha squared going to 0 argument 00:42:01.380 --> 00:42:03.150 align:middle line:84% to get rid of this term to make it small, 00:42:03.150 --> 00:42:05.980 align:middle line:90% setting these equal to 0. 00:42:05.980 --> 00:42:08.670 align:middle line:90% If we did that again-- 00:42:08.670 --> 00:42:17.525 align:middle line:84% I'm trying to think what you would end up with. 00:42:17.525 --> 00:42:18.900 align:middle line:84% But I have a feeling you're going 00:42:18.900 --> 00:42:23.420 align:middle line:84% to end up with a term that looks like alpha 2 the 4 00:42:23.420 --> 00:42:25.700 align:middle line:84% or the strength of this resonance. 00:42:25.700 --> 00:42:28.040 align:middle line:84% And then that is still going to go to 0. 00:42:28.040 --> 00:42:29.975 align:middle line:84% So we could feed all of this back through. 00:42:29.975 --> 00:42:32.770 align:middle line:90% 00:42:32.770 --> 00:42:33.990 align:middle line:90% Yeah. 00:42:33.990 --> 00:42:36.182 align:middle line:84% I don't know the full answer to it. 00:42:36.182 --> 00:42:38.390 align:middle line:84% I think we're talking about two separate things here. 00:42:38.390 --> 00:42:39.990 align:middle line:84% I think that the limit that we took 00:42:39.990 --> 00:42:42.240 align:middle line:84% to get the incoherent scattering is separate from what 00:42:42.240 --> 00:42:43.180 align:middle line:90% we've done here. 00:42:43.180 --> 00:42:45.300 align:middle line:84% So although this term would still 00:42:45.300 --> 00:42:47.730 align:middle line:84% exist in the coherent/incoherent limit, 00:42:47.730 --> 00:42:50.370 align:middle line:84% it would be inside this bit of the equations. 00:42:50.370 --> 00:42:55.390 align:middle line:84% And so whatever these are doing is irrelevant because this. 00:42:55.390 --> 00:43:01.650 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:43:01.650 --> 00:43:03.040 align:middle line:90% JACK HARE: Yeah. 00:43:03.040 --> 00:43:04.738 align:middle line:84% We took this very brutally to start 00:43:04.738 --> 00:43:06.780 align:middle line:84% with so we didn't even have to think about these. 00:43:06.780 --> 00:43:09.810 align:middle line:84% Once we restore the alpha being some arbitrary value, 00:43:09.810 --> 00:43:12.230 align:middle line:84% then we can go on about these things. 00:43:12.230 --> 00:43:21.680 align:middle line:84% AUDIENCE: It's not like [INAUDIBLE] 00:43:21.680 --> 00:43:23.390 align:middle line:84% JACK HARE: No, I don't believe so. 00:43:23.390 --> 00:43:29.030 align:middle line:84% There's nothing related to alpha inside w, yeah. 00:43:29.030 --> 00:43:32.600 align:middle line:84% w really has to do with the distribution function, 00:43:32.600 --> 00:43:35.240 align:middle line:84% whereas alpha is more fundamental than that. 00:43:35.240 --> 00:43:36.830 align:middle line:84% Alpha can be evaluated regardless 00:43:36.830 --> 00:43:37.890 align:middle line:90% of distribution function. 00:43:37.890 --> 00:43:39.560 align:middle line:84% So probably if you're trying to work out 00:43:39.560 --> 00:43:41.960 align:middle line:84% what the Debye length is for a nonthermal plasma, 00:43:41.960 --> 00:43:43.050 align:middle line:90% you also have problems. 00:43:43.050 --> 00:43:45.240 align:middle line:84% So I'm not quite sure how this gets defined 00:43:45.240 --> 00:43:47.372 align:middle line:90% if you're very non-Maxwellian. 00:43:47.372 --> 00:43:48.330 align:middle line:90% We're going to move on. 00:43:48.330 --> 00:43:51.600 align:middle line:90% 00:43:51.600 --> 00:43:59.340 align:middle line:84% if you happen to be in the case back here where zTe upon TI 00:43:59.340 --> 00:44:01.210 align:middle line:90% is less than-- 00:44:01.210 --> 00:44:04.710 align:middle line:84% we've got 3 here, but I'm going to say less than 1. 00:44:04.710 --> 00:44:09.720 align:middle line:84% It turns out you get a spectrum s of a omega. 00:44:09.720 --> 00:44:13.220 align:middle line:90% 00:44:13.220 --> 00:44:14.717 align:middle line:90% And this is for low frequencies. 00:44:14.717 --> 00:44:17.300 align:middle line:84% So we don't care about what the electron plasma wave is doing. 00:44:17.300 --> 00:44:21.660 align:middle line:84% You get a spectrum that actually looks like the ion distribution 00:44:21.660 --> 00:44:22.160 align:middle line:90% function. 00:44:22.160 --> 00:44:27.870 align:middle line:90% 00:44:27.870 --> 00:44:31.800 align:middle line:84% Now, this is because the anacoustic waves are strongly 00:44:31.800 --> 00:44:33.370 align:middle line:90% damped in this limit here. 00:44:33.370 --> 00:44:36.970 align:middle line:84% So the anacoustic waves would be somewhere here. 00:44:36.970 --> 00:44:39.610 align:middle line:84% But they're Landau damping on the ions 00:44:39.610 --> 00:44:42.040 align:middle line:84% when we have this criteria fulfilled. 00:44:42.040 --> 00:44:44.470 align:middle line:84% And so the waves are completely damped. 00:44:44.470 --> 00:44:49.252 align:middle line:90% 00:44:49.252 --> 00:44:50.960 align:middle line:84% As far as I know, I've never seen a paper 00:44:50.960 --> 00:44:52.127 align:middle line:90% where someone has done this. 00:44:52.127 --> 00:44:54.170 align:middle line:84% This is actually a pretty strange place to be, 00:44:54.170 --> 00:44:57.740 align:middle line:84% where you have really rather hot ions and cold electrons. 00:44:57.740 --> 00:45:00.590 align:middle line:84% I assume that it is doable, but I've just never 00:45:00.590 --> 00:45:03.320 align:middle line:90% seen it done in an experiment. 00:45:03.320 --> 00:45:11.870 align:middle line:84% The next limit we could have is where zTe upon TI is roughly 1. 00:45:11.870 --> 00:45:13.388 align:middle line:90% So that's in between. 00:45:13.388 --> 00:45:14.930 align:middle line:84% The resonances are completely damped, 00:45:14.930 --> 00:45:17.030 align:middle line:84% or the resonances are not damped. 00:45:17.030 --> 00:45:19.850 align:middle line:84% And so what we end up with is a spectra that 00:45:19.850 --> 00:45:21.320 align:middle line:90% looks like a strange hybrid. 00:45:21.320 --> 00:45:24.370 align:middle line:90% 00:45:24.370 --> 00:45:30.990 align:middle line:84% It almost looks like a flat top like this, which is effectively 00:45:30.990 --> 00:45:36.255 align:middle line:84% adding some anacoustic waves onto your distribution function. 00:45:36.255 --> 00:45:40.950 align:middle line:90% 00:45:40.950 --> 00:45:46.220 align:middle line:84% These anacoustic waves are still pretty damped, 00:45:46.220 --> 00:45:48.180 align:middle line:84% but they're not as strongly damped here. 00:45:48.180 --> 00:45:51.666 align:middle line:84% So there is an anacoustic wave contribution. 00:45:51.666 --> 00:46:01.600 align:middle line:90% 00:46:01.600 --> 00:46:11.450 align:middle line:84% Finally, we have the case where zTe upon TI 00:46:11.450 --> 00:46:13.670 align:middle line:90% is much bigger than 1. 00:46:13.670 --> 00:46:16.070 align:middle line:84% This is the case we just derived, 00:46:16.070 --> 00:46:19.640 align:middle line:84% where we're going to have very strong anacoustic resonances. 00:46:19.640 --> 00:46:21.080 align:middle line:90% And in this case, your spectra-- 00:46:21.080 --> 00:46:24.260 align:middle line:90% 00:46:24.260 --> 00:46:26.810 align:middle line:84% although it does have some contribution from this f of I, 00:46:26.810 --> 00:46:29.570 align:middle line:84% as we've discussed before, the contribution from the resonances 00:46:29.570 --> 00:46:30.500 align:middle line:90% are much larger. 00:46:30.500 --> 00:46:33.280 align:middle line:84% You get something that looks a little bit like that. 00:46:33.280 --> 00:46:35.530 align:middle line:90% There we go. 00:46:35.530 --> 00:46:38.260 align:middle line:84% And these occur at the anacoustic free. 00:46:38.260 --> 00:46:50.750 align:middle line:90% 00:46:50.750 --> 00:46:55.807 align:middle line:84% So now we have a large ionic contribution. 00:46:55.807 --> 00:47:01.780 align:middle line:90% 00:47:01.780 --> 00:47:04.180 align:middle line:84% And of course, if you write a code to implement 00:47:04.180 --> 00:47:07.390 align:middle line:84% all of these equations, and you change the value of z upon TI, 00:47:07.390 --> 00:47:09.100 align:middle line:84% you will get these different shapes. 00:47:09.100 --> 00:47:11.440 align:middle line:84% I'm just describing to you hand-wavingly 00:47:11.440 --> 00:47:13.570 align:middle line:84% what shapes you get and why they end up 00:47:13.570 --> 00:47:16.560 align:middle line:90% looking a little bit like that. 00:47:16.560 --> 00:47:17.060 align:middle line:90% OK. 00:47:17.060 --> 00:47:19.715 align:middle line:84% Any questions on the anacoustic waves? 00:47:19.715 --> 00:47:37.950 align:middle line:90% 00:47:37.950 --> 00:47:39.450 align:middle line:84% Wherever you want to be-- it depends 00:47:39.450 --> 00:47:40.492 align:middle line:90% what you want to measure. 00:47:40.492 --> 00:47:44.290 align:middle line:90% 00:47:44.290 --> 00:47:45.820 align:middle line:90% No. 00:47:45.820 --> 00:47:46.780 align:middle line:90% it's pretty irritating. 00:47:46.780 --> 00:47:49.960 align:middle line:90% I ended up there once. 00:47:49.960 --> 00:47:51.580 align:middle line:90% You'd much prefer to be here. 00:47:51.580 --> 00:47:53.270 align:middle line:84% You can learn an awful lot more from it. 00:47:53.270 --> 00:47:54.982 align:middle line:90% This is hard to fit properly. 00:47:54.982 --> 00:47:57.190 align:middle line:84% And there's other things that can cause your spectrum 00:47:57.190 --> 00:47:58.273 align:middle line:90% to look like that as well. 00:47:58.273 --> 00:48:00.590 align:middle line:84% So there's probably a better place to be, 00:48:00.590 --> 00:48:03.040 align:middle line:84% but of course you don't get to choose this in your plasma. 00:48:03.040 --> 00:48:05.540 align:middle line:84% Unlike where we're choosing alpha by moving our spectrometer 00:48:05.540 --> 00:48:07.285 align:middle line:84% and changing the wavelength of light, 00:48:07.285 --> 00:48:09.910 align:middle line:84% I have no choice whatsoever on what my temperature of my plasma 00:48:09.910 --> 00:48:10.030 align:middle line:90% is. 00:48:10.030 --> 00:48:11.300 align:middle line:84% That's the thing I'm trying to measure. 00:48:11.300 --> 00:48:13.217 align:middle line:84% So sometimes you just end up here by accident. 00:48:13.217 --> 00:48:18.740 align:middle line:90% 00:48:18.740 --> 00:48:19.240 align:middle line:90% OK. 00:48:19.240 --> 00:48:22.940 align:middle line:84% So we derived the anacoustic waves and the electron plasma 00:48:22.940 --> 00:48:23.440 align:middle line:90% waves. 00:48:23.440 --> 00:48:25.630 align:middle line:84% And a reasonable question is, in my spectrum, 00:48:25.630 --> 00:48:27.458 align:middle line:84% which of these is more important? 00:48:27.458 --> 00:48:28.750 align:middle line:90% Which do I see more light from? 00:48:28.750 --> 00:48:41.365 align:middle line:90% 00:48:41.365 --> 00:48:42.740 align:middle line:84% Because that might tell you which 00:48:42.740 --> 00:48:45.450 align:middle line:84% ones are going to be easier to detect. 00:48:45.450 --> 00:48:48.800 align:middle line:84% So again, we might have a spectrum 00:48:48.800 --> 00:48:53.960 align:middle line:84% where alpha is much bigger than 1 and zTe 00:48:53.960 --> 00:48:56.493 align:middle line:90% upon TI is much bigger than 1. 00:48:56.493 --> 00:48:58.160 align:middle line:84% And we might go, OK, we're going to have 00:48:58.160 --> 00:49:00.990 align:middle line:90% some anacoustic peaks here. 00:49:00.990 --> 00:49:03.810 align:middle line:84% And then at some much higher frequency, because they really 00:49:03.810 --> 00:49:08.080 align:middle line:84% are very, very different frequencies. 00:49:08.080 --> 00:49:10.550 align:middle line:84% We're going to have some electron plasma waves, 00:49:10.550 --> 00:49:11.050 align:middle line:90% like this. 00:49:11.050 --> 00:49:20.190 align:middle line:90% 00:49:20.190 --> 00:49:22.630 align:middle line:84% And which of these contribute more to our spectrum? 00:49:22.630 --> 00:49:24.240 align:middle line:84% So we get this by calculating what's 00:49:24.240 --> 00:49:27.660 align:middle line:84% called the total cross-section, which we call s sub T, 00:49:27.660 --> 00:49:30.360 align:middle line:84% as a function of a, where our detector is. 00:49:30.360 --> 00:49:36.750 align:middle line:84% And this is just the frequency integrated cross-section of s 00:49:36.750 --> 00:49:39.270 align:middle line:90% of k omega. 00:49:39.270 --> 00:49:44.710 align:middle line:84% So we can do this analytically for the ion and electron 00:49:44.710 --> 00:49:47.107 align:middle line:90% temperatures being equal. 00:49:47.107 --> 00:49:48.940 align:middle line:84% So you have to make this assumption in order 00:49:48.940 --> 00:49:50.230 align:middle line:90% to make progress analytically. 00:49:50.230 --> 00:49:53.380 align:middle line:84% And this is called the saltpeter approximation. 00:49:53.380 --> 00:49:54.810 align:middle line:84% The saltpeter was the first person 00:49:54.810 --> 00:49:57.310 align:middle line:84% to come up with it when he was looking at Thomson scattering 00:49:57.310 --> 00:49:59.820 align:middle line:90% from the ionosphere. 00:49:59.820 --> 00:50:03.630 align:middle line:84% And he showed that our total scattered cross-section 00:50:03.630 --> 00:50:10.330 align:middle line:84% looks, therefore, like 2 pi upon 1 plus alpha squared. 00:50:10.330 --> 00:50:13.005 align:middle line:84% This is the contribution from the electron feature. 00:50:13.005 --> 00:50:17.190 align:middle line:90% 00:50:17.190 --> 00:50:20.540 align:middle line:90% So this region. 00:50:20.540 --> 00:50:23.900 align:middle line:84% And then there's another term, which looks like 2 pi 00:50:23.900 --> 00:50:31.770 align:middle line:84% upon 1 plus alpha squared, z alpha to the fourth power, 1 00:50:31.770 --> 00:50:41.410 align:middle line:84% plus alpha squared plus alpha squared, the Te upon TI. 00:50:41.410 --> 00:50:44.230 align:middle line:84% You're going to say, we assume these are equal. 00:50:44.230 --> 00:50:46.270 align:middle line:84% We assume they are approximately equal. 00:50:46.270 --> 00:50:46.870 align:middle line:90% There we go. 00:50:46.870 --> 00:50:47.370 align:middle line:90% Good. 00:50:47.370 --> 00:50:50.140 align:middle line:84% So this can still be slightly different than 1. 00:50:50.140 --> 00:50:52.880 align:middle line:84% And of course, the z makes a big difference here. 00:50:52.880 --> 00:50:56.810 align:middle line:84% And this is due to the scattering from the ion feature. 00:50:56.810 --> 00:51:01.795 align:middle line:90% 00:51:01.795 --> 00:51:03.420 align:middle line:84% But now we look at this, and we go, OK. 00:51:03.420 --> 00:51:05.040 align:middle line:84% Well, depending on my alpha parameter, 00:51:05.040 --> 00:51:07.330 align:middle line:84% this seems to depend very, very strongly on alpha. 00:51:07.330 --> 00:51:13.020 align:middle line:84% So if I look at alpha squared going to 0 here, 00:51:13.020 --> 00:51:15.300 align:middle line:84% it's very clear that this term, the aisle term, 00:51:15.300 --> 00:51:16.450 align:middle line:90% is just going to drop off. 00:51:16.450 --> 00:51:18.900 align:middle line:84% And we're just going to have electron feature only. 00:51:18.900 --> 00:51:31.812 align:middle line:90% 00:51:31.812 --> 00:51:33.770 align:middle line:84% Which is what we've seen several times already. 00:51:33.770 --> 00:51:35.290 align:middle line:84% But when we go to incoherence, we're 00:51:35.290 --> 00:51:38.540 align:middle line:84% just scattering off the individual electrons. 00:51:38.540 --> 00:51:42.910 align:middle line:84% If we have alpha equals to 1-- so some intermediate alpha 00:51:42.910 --> 00:51:44.920 align:middle line:84% parameter-- then we're going to get something 00:51:44.920 --> 00:52:00.070 align:middle line:84% that looks like pi times 1 plus z over 2 plus zTe upon TI. 00:52:00.070 --> 00:52:02.430 align:middle line:84% And so there are two limits to this. 00:52:02.430 --> 00:52:05.730 align:middle line:84% There's the limit we already saw, zTe upon TI 00:52:05.730 --> 00:52:08.920 align:middle line:90% is much, much less than 1. 00:52:08.920 --> 00:52:10.720 align:middle line:90% Then we got both these terms. 00:52:10.720 --> 00:52:12.370 align:middle line:84% So again, this is the electron term. 00:52:12.370 --> 00:52:15.600 align:middle line:90% This is the ion term. 00:52:15.600 --> 00:52:21.780 align:middle line:84% This gives both e minus and ions. 00:52:21.780 --> 00:52:28.810 align:middle line:84% And we actually end up with coherent electrons 00:52:28.810 --> 00:52:31.130 align:middle line:90% and incoherent ions-- 00:52:31.130 --> 00:52:33.586 align:middle line:84% or incoherent ions scattering from the ions. 00:52:33.586 --> 00:52:36.090 align:middle line:90% 00:52:36.090 --> 00:52:37.840 align:middle line:84% This is a strange situation, as I've said. 00:52:37.840 --> 00:52:40.007 align:middle line:84% It requires you to have hot ions and cold electrons, 00:52:40.007 --> 00:52:41.830 align:middle line:90% which we don't often encounter. 00:52:41.830 --> 00:52:47.050 align:middle line:84% The other one would be zTe over TI is much greater than 1. 00:52:47.050 --> 00:52:51.610 align:middle line:84% And then we get coherent electrons and ions. 00:52:51.610 --> 00:53:00.230 align:middle line:90% 00:53:00.230 --> 00:53:01.610 align:middle line:90% And finally, just so-- 00:53:01.610 --> 00:53:03.530 align:middle line:84% well, I'll create some space here. 00:53:03.530 --> 00:53:18.540 align:middle line:90% 00:53:18.540 --> 00:53:21.315 align:middle line:84% We could look at when our alpha parameter is very large. 00:53:21.315 --> 00:53:26.470 align:middle line:90% 00:53:26.470 --> 00:53:27.820 align:middle line:90% Alpha much greater than 1. 00:53:27.820 --> 00:53:31.930 align:middle line:84% And then we'd have this s of k omega 00:53:31.930 --> 00:53:40.930 align:middle line:84% go to sk total go towards 0 for the electrons plus z 00:53:40.930 --> 00:53:47.874 align:middle line:84% over 1 plus zTe over TI, the ions. 00:53:47.874 --> 00:53:52.980 align:middle line:90% 00:53:52.980 --> 00:53:58.010 align:middle line:84% And so we just have ions dominating this. 00:53:58.010 --> 00:54:01.340 align:middle line:90% 00:54:01.340 --> 00:54:04.690 align:middle line:84% If we have a very large alpha, your spectrum 00:54:04.690 --> 00:54:05.650 align:middle line:90% would just look like-- 00:54:05.650 --> 00:54:09.110 align:middle line:90% 00:54:09.110 --> 00:54:12.360 align:middle line:90% and then flat out to here. 00:54:12.360 --> 00:54:14.790 align:middle line:84% In some of these other cases, for example, we just 00:54:14.790 --> 00:54:16.700 align:middle line:90% have electron scattering. 00:54:16.700 --> 00:54:18.955 align:middle line:84% And we wouldn't have any contributions from the ions. 00:54:18.955 --> 00:54:20.580 align:middle line:84% And this is important because depending 00:54:20.580 --> 00:54:22.240 align:middle line:84% on what alpha you have, if you're like, 00:54:22.240 --> 00:54:26.730 align:middle line:84% I really want to go look at the electron distribution function 00:54:26.730 --> 00:54:29.133 align:middle line:84% directly, but then you calculate your alphas large. 00:54:29.133 --> 00:54:31.800 align:middle line:84% But you know you're not going to see any of the electron feature 00:54:31.800 --> 00:54:32.370 align:middle line:90% at all. 00:54:32.370 --> 00:54:33.960 align:middle line:84% The electron feature is going to disappear. 00:54:33.960 --> 00:54:36.377 align:middle line:84% So you won't learn anything directly about your electrons. 00:54:36.377 --> 00:54:39.930 align:middle line:84% You'll only learn something about your ions. 00:54:39.930 --> 00:54:42.120 align:middle line:84% And this is important because in general, 00:54:42.120 --> 00:54:49.230 align:middle line:84% the eTw frequency is much, much larger than the Iaw frequency, 00:54:49.230 --> 00:54:50.910 align:middle line:90% as I alluded to here. 00:54:50.910 --> 00:54:54.270 align:middle line:84% So it's very hard to measure both on the same spectrometer, 00:54:54.270 --> 00:54:55.095 align:middle line:90% so we can't. 00:54:55.095 --> 00:55:01.550 align:middle line:90% 00:55:01.550 --> 00:55:08.688 align:middle line:84% The eTw's and the Iaw's on one spectrometer. 00:55:08.688 --> 00:55:14.215 align:middle line:90% 00:55:14.215 --> 00:55:15.840 align:middle line:84% Because if you set up your spectrometer 00:55:15.840 --> 00:55:19.350 align:middle line:84% to have a nice, wide frequency wave to get the eTw's, then 00:55:19.350 --> 00:55:21.910 align:middle line:84% the Iaw's will just appear on a couple of pixels. 00:55:21.910 --> 00:55:25.170 align:middle line:84% And if you set it up to see the Iaw's, then the eTw's 00:55:25.170 --> 00:55:26.790 align:middle line:90% will be way off to the side. 00:55:26.790 --> 00:55:28.290 align:middle line:84% And so that means you generally have 00:55:28.290 --> 00:55:30.450 align:middle line:84% to choose which of these you're going to measure 00:55:30.450 --> 00:55:32.260 align:middle line:84% or you have to buy an alpha spectrometer. 00:55:32.260 --> 00:55:36.112 align:middle line:84% So it is actually a pretty big practical concern. 00:55:36.112 --> 00:55:38.070 align:middle line:84% And so you want to know which of these features 00:55:38.070 --> 00:55:40.980 align:middle line:84% is going to be brightest so that you can set up your spectrometer 00:55:40.980 --> 00:55:42.397 align:middle line:84% to go look at that, because that's 00:55:42.397 --> 00:55:44.520 align:middle line:90% going to make life better. 00:55:44.520 --> 00:55:45.020 align:middle line:90% OK. 00:55:45.020 --> 00:55:48.220 align:middle line:90% Questions on this? 00:55:48.220 --> 00:55:49.200 align:middle line:90% Yeah. 00:55:49.200 --> 00:55:53.120 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:55:53.120 --> 00:55:54.020 align:middle line:90% JACK HARE: Yes. 00:55:54.020 --> 00:55:56.145 align:middle line:84% We haven't talked about magnetic fields at all yet. 00:55:56.145 --> 00:55:58.676 align:middle line:90% 00:55:58.676 --> 00:55:59.390 align:middle line:90% Oh, yes. 00:55:59.390 --> 00:56:01.400 align:middle line:84% If we have time, we'll get there. 00:56:01.400 --> 00:56:06.670 align:middle line:84% But the very, very, very brief answer is in certain conditions, 00:56:06.670 --> 00:56:09.730 align:middle line:84% the anacoustic feature gets a modulation 00:56:09.730 --> 00:56:11.440 align:middle line:90% so it wobbles up and down. 00:56:11.440 --> 00:56:15.470 align:middle line:84% And that wobble is the ion cyclotron frequency. 00:56:15.470 --> 00:56:18.860 align:middle line:84% And the electron feature gets a wobble, 00:56:18.860 --> 00:56:20.960 align:middle line:90% and that wobble is the electron. 00:56:20.960 --> 00:56:24.140 align:middle line:84% But you have to be looking precisely perpendicular 00:56:24.140 --> 00:56:26.180 align:middle line:84% to the magnetic field, which is very hard. 00:56:26.180 --> 00:56:28.790 align:middle line:84% If you look even half a degree off, 00:56:28.790 --> 00:56:30.832 align:middle line:84% this effect washes out completely. 00:56:30.832 --> 00:56:32.790 align:middle line:84% So it's a very challenging measurement to make. 00:56:32.790 --> 00:56:35.440 align:middle line:84% I haven't seen anyone do it since 1972. 00:56:35.440 --> 00:56:36.740 align:middle line:90% I didn't see it myself. 00:56:36.740 --> 00:56:39.170 align:middle line:84% But that's the last time someone published a paper 00:56:39.170 --> 00:56:41.420 align:middle line:84% actually measuring it in Thomson scattering. 00:56:41.420 --> 00:56:44.350 align:middle line:90% And I've not seen anyone do it. 00:56:44.350 --> 00:56:46.260 align:middle line:90% It's a really hard measurement. 00:56:46.260 --> 00:56:48.710 align:middle line:84% So people say, oh, Thomson scattering 00:56:48.710 --> 00:56:50.760 align:middle line:90% can measure magnetic fields. 00:56:50.760 --> 00:56:52.600 align:middle line:84% Because you measure the cyclotron frequency, 00:56:52.600 --> 00:56:54.690 align:middle line:84% so you're like, OK, now I know the magnetic field. 00:56:54.690 --> 00:56:59.170 align:middle line:84% But in reality, it's really hard to do. 00:56:59.170 --> 00:57:00.100 align:middle line:90% Other questions? 00:57:00.100 --> 00:57:01.060 align:middle line:84% I will show you something in a moment 00:57:01.060 --> 00:57:03.185 align:middle line:84% that can help you infer the magnetic field, though. 00:57:03.185 --> 00:57:27.770 align:middle line:90% 00:57:27.770 --> 00:57:28.760 align:middle line:90% OK. 00:57:28.760 --> 00:57:31.490 align:middle line:84% So now I'm going to talk about electron ion drift. 00:57:31.490 --> 00:57:43.910 align:middle line:90% 00:57:43.910 --> 00:57:46.430 align:middle line:84% And as you probably realize, if your electrons and ions 00:57:46.430 --> 00:57:49.040 align:middle line:84% are drifting with respect to each other, 00:57:49.040 --> 00:57:52.900 align:middle line:84% that means you have a net current in your system. 00:57:52.900 --> 00:57:55.770 align:middle line:84% And so if we can measure how much electrons and ions are 00:57:55.770 --> 00:57:57.600 align:middle line:84% drifting with respect to each other, 00:57:57.600 --> 00:57:59.820 align:middle line:84% we can measure the current locally 00:57:59.820 --> 00:58:01.920 align:middle line:84% within the plasma, which is pretty cool. 00:58:01.920 --> 00:58:10.220 align:middle line:90% 00:58:10.220 --> 00:58:15.890 align:middle line:84% So we're going to say that there is some drift velocity which 00:58:15.890 --> 00:58:19.550 align:middle line:84% is equal to the difference between the electron 00:58:19.550 --> 00:58:23.390 align:middle line:84% velocity and the ion velocity, like this. 00:58:23.390 --> 00:58:26.840 align:middle line:84% Now, this is not the velocity inside your distribution 00:58:26.840 --> 00:58:27.410 align:middle line:90% function. 00:58:27.410 --> 00:58:31.620 align:middle line:84% This is the bulk velocity of your entire distribution. 00:58:31.620 --> 00:58:34.190 align:middle line:90% So this is a big capital V here. 00:58:34.190 --> 00:58:36.200 align:middle line:84% So your entire distribution of ions 00:58:36.200 --> 00:58:40.010 align:middle line:84% is moving with some VI, plus the thermal velocity on top of that. 00:58:40.010 --> 00:58:41.510 align:middle line:84% The entire distribution of electrons 00:58:41.510 --> 00:58:43.370 align:middle line:90% is moving with some Ve. 00:58:43.370 --> 00:58:45.110 align:middle line:84% And we define this drift velocity 00:58:45.110 --> 00:58:48.590 align:middle line:84% to be the difference between the flows of the electron ion 00:58:48.590 --> 00:58:50.470 align:middle line:90% species. 00:58:50.470 --> 00:58:52.785 align:middle line:84% And we know from all the other Thomson scattering 00:58:52.785 --> 00:58:55.410 align:middle line:84% that we've done that we're only really sensitive to a component 00:58:55.410 --> 00:58:57.880 align:middle line:90% of this in the k direction. 00:58:57.880 --> 00:59:02.700 align:middle line:84% So we can only measure Vd dotted into k 00:59:02.700 --> 00:59:06.210 align:middle line:84% because everything we've done we only measure it along the wave 00:59:06.210 --> 00:59:08.040 align:middle line:90% vector of the scattering mode. 00:59:08.040 --> 00:59:12.060 align:middle line:84% And so that means that we can define a squiggle, which 00:59:12.060 --> 00:59:14.760 align:middle line:84% we call squiggle d, which is defined 00:59:14.760 --> 00:59:22.460 align:middle line:90% as Vd dotted into k over Vde. 00:59:22.460 --> 00:59:28.680 align:middle line:84% Once again, this is analogous to the other squiggles, where we 00:59:28.680 --> 00:59:29.940 align:middle line:90% were using the phase velocity. 00:59:29.940 --> 00:59:33.270 align:middle line:84% Now we've got the component of the drift velocity along k. 00:59:33.270 --> 00:59:38.170 align:middle line:84% And that means we're going to replace the real part of the w 00:59:38.170 --> 00:59:40.570 align:middle line:84% function, which was normally in terms of squiggle. 00:59:40.570 --> 00:59:42.790 align:middle line:84% And we're going to now evaluate it 00:59:42.790 --> 00:59:49.550 align:middle line:84% in terms of squiggle e minus squiggle d. 00:59:49.550 --> 00:59:52.190 align:middle line:84% So we're effectively looking at the damping 00:59:52.190 --> 00:59:55.230 align:middle line:84% from a different part of the electron distribution function. 00:59:55.230 --> 00:59:57.812 align:middle line:84% So this is our shifted Maxwellian. 00:59:57.812 --> 01:00:12.090 align:middle line:90% 01:00:12.090 --> 01:00:14.190 align:middle line:84% The reason this is important is because it 01:00:14.190 --> 01:00:17.250 align:middle line:84% was the electrons which were damping the anacoustic waves. 01:00:17.250 --> 01:00:19.890 align:middle line:84% And both the anacoustic waves were damped by the same amount, 01:00:19.890 --> 01:00:21.520 align:middle line:90% and they had the same intensity. 01:00:21.520 --> 01:00:24.780 align:middle line:84% But if we shift the electrons, some of the anacoustic waves 01:00:24.780 --> 01:00:27.390 align:middle line:84% are going to see a different part of the electron 01:00:27.390 --> 01:00:29.250 align:middle line:84% distribution function than the ones going 01:00:29.250 --> 01:00:30.220 align:middle line:90% the opposite direction. 01:00:30.220 --> 01:00:32.160 align:middle line:90% So we've broken that symmetry. 01:00:32.160 --> 01:00:38.230 align:middle line:84% Previously, for example, our electron distribution function 01:00:38.230 --> 01:00:42.690 align:middle line:84% was symmetric and our anacoustic waves were, say, here and here. 01:00:42.690 --> 01:00:46.750 align:middle line:84% And they both saw the same, df, dV, which 01:00:46.750 --> 01:00:48.220 align:middle line:90% is what causes the damping. 01:00:48.220 --> 01:00:51.390 align:middle line:84% If now we've shifted our distribution function, 01:00:51.390 --> 01:00:54.270 align:middle line:84% we can see that this wave now sees no df 01:00:54.270 --> 01:00:56.515 align:middle line:90% dV because it's at the maxima. 01:00:56.515 --> 01:00:58.140 align:middle line:84% This wave, these are different amounts. 01:00:58.140 --> 01:01:00.960 align:middle line:84% So we can imagine that the damping will be differently 01:01:00.960 --> 01:01:03.660 align:middle line:84% effective for the plus and minus frequency waves, 01:01:03.660 --> 01:01:05.410 align:middle line:84% and so they'll have different heights. 01:01:05.410 --> 01:01:11.860 align:middle line:84% So just to make that clear here, the damping 01:01:11.860 --> 01:01:17.390 align:middle line:84% of the anacoustic waves is proportional to the gradient 01:01:17.390 --> 01:01:25.020 align:middle line:84% of the electron velocity, e, at the phase velocity of the wave, 01:01:25.020 --> 01:01:26.730 align:middle line:90% omega upon k. 01:01:26.730 --> 01:01:28.500 align:middle line:84% And that, it turns out, is proportional 01:01:28.500 --> 01:01:31.245 align:middle line:84% to this imaginary part of the w function 01:01:31.245 --> 01:01:33.120 align:middle line:84% that we've been carting around all this time. 01:01:33.120 --> 01:01:37.000 align:middle line:84% So it's proportional to the imagined part of that at x. 01:01:37.000 --> 01:01:43.953 align:middle line:84% And that is proportional to x exponential minus x squared. 01:01:43.953 --> 01:01:45.620 align:middle line:84% And you shouldn't be surprised that it's 01:01:45.620 --> 01:01:48.050 align:middle line:84% x exponential minus x squared because that looks 01:01:48.050 --> 01:01:51.000 align:middle line:84% like V exponential minus V squared, 01:01:51.000 --> 01:01:53.720 align:middle line:84% which would be the derivative of our Maxwellian distribution 01:01:53.720 --> 01:01:54.490 align:middle line:90% function. 01:01:54.490 --> 01:01:55.910 align:middle line:84% And so this is what we're saying, 01:01:55.910 --> 01:01:57.410 align:middle line:84% that the amount of damping we have 01:01:57.410 --> 01:02:00.110 align:middle line:84% is directly proportional to the derivative of the distribution 01:02:00.110 --> 01:02:02.020 align:middle line:90% function. 01:02:02.020 --> 01:02:05.190 align:middle line:84% So if I now draw this again, but now I 01:02:05.190 --> 01:02:13.650 align:middle line:84% draw it in terms of the damping, I have axes with squiggle I 01:02:13.650 --> 01:02:15.530 align:middle line:90% down here. 01:02:15.530 --> 01:02:20.065 align:middle line:90% And up here, I have log damping. 01:02:20.065 --> 01:02:24.232 align:middle line:90% 01:02:24.232 --> 01:02:25.690 align:middle line:84% Previously, for example, I may have 01:02:25.690 --> 01:02:29.320 align:middle line:84% had the anacoustic resonances, these frequencies, 01:02:29.320 --> 01:02:36.010 align:middle line:90% minus omega Iaw plus omega Iaw. 01:02:36.010 --> 01:02:38.470 align:middle line:84% And the damping coming from the ions 01:02:38.470 --> 01:02:43.632 align:middle line:90% would have looked like this. 01:02:43.632 --> 01:02:46.470 align:middle line:90% 01:02:46.470 --> 01:02:49.200 align:middle line:84% This is just the derivative of the Maxwellian here. 01:02:49.200 --> 01:02:52.060 align:middle line:90% It's just this function plotted. 01:02:52.060 --> 01:02:53.680 align:middle line:84% And the damping from the electrons 01:02:53.680 --> 01:02:58.940 align:middle line:90% would have looked like this. 01:02:58.940 --> 01:03:01.420 align:middle line:84% So the ions are feeding the ion acoustic waves 01:03:01.420 --> 01:03:02.570 align:middle line:90% because of the slope here. 01:03:02.570 --> 01:03:05.470 align:middle line:84% And the electrons are damping these waves. 01:03:05.470 --> 01:03:07.510 align:middle line:90% If we shift the electrons now-- 01:03:07.510 --> 01:03:14.140 align:middle line:90% 01:03:14.140 --> 01:03:17.210 align:middle line:84% I put the anacoustic waves in the same place. 01:03:17.210 --> 01:03:20.147 align:middle line:90% I put the ion damping-- 01:03:20.147 --> 01:03:22.480 align:middle line:84% I'm going to try to make this one a bit more symmetric-- 01:03:22.480 --> 01:03:25.520 align:middle line:90% 01:03:25.520 --> 01:03:27.020 align:middle line:90% in the same place. 01:03:27.020 --> 01:03:29.300 align:middle line:84% But I now shift the electrons over 01:03:29.300 --> 01:03:36.200 align:middle line:84% so that they've got some drift velocity like this. 01:03:36.200 --> 01:03:47.270 align:middle line:84% We can see that we have increased damping here 01:03:47.270 --> 01:03:48.800 align:middle line:90% and reduced damping here. 01:03:48.800 --> 01:03:56.820 align:middle line:90% 01:03:56.820 --> 01:04:01.170 align:middle line:84% And what that means, in terms of the spectrum, 01:04:01.170 --> 01:04:02.820 align:middle line:84% is that we go from something that 01:04:02.820 --> 01:04:11.950 align:middle line:84% looks like, in frequency space on our spectrometer, this. 01:04:11.950 --> 01:04:18.000 align:middle line:84% And we end up with something that looks like this instead. 01:04:18.000 --> 01:04:19.830 align:middle line:84% There's a very pronounced asymmetry 01:04:19.830 --> 01:04:21.390 align:middle line:90% between these two peaks here. 01:04:21.390 --> 01:04:30.367 align:middle line:90% 01:04:30.367 --> 01:04:32.200 align:middle line:84% Now, unfortunately there's no simple formula 01:04:32.200 --> 01:04:36.520 align:middle line:84% I can give you for the ratio of these two peaks 01:04:36.520 --> 01:04:40.300 align:middle line:84% that gives you the drift velocity straight away. 01:04:40.300 --> 01:04:42.795 align:middle line:90% This is pretty complex. 01:04:42.795 --> 01:04:44.170 align:middle line:84% In general, as far as I know, you 01:04:44.170 --> 01:04:46.960 align:middle line:84% have to fit the full s of k omega, 01:04:46.960 --> 01:04:50.380 align:middle line:84% allowing there to be a drift of the electrons here. 01:04:50.380 --> 01:04:53.260 align:middle line:90% And it depends on the exact-- 01:04:53.260 --> 01:04:55.150 align:middle line:84% the ratio of these two peaks depends 01:04:55.150 --> 01:04:59.620 align:middle line:84% on exactly what the drift velocity is, what zTe upon TI 01:04:59.620 --> 01:05:03.440 align:middle line:90% is, what the alpha parameter is. 01:05:03.440 --> 01:05:10.090 align:middle line:84% And so we have to fit the full s of k omega to our data. 01:05:10.090 --> 01:05:13.440 align:middle line:84% But once we've done that, we can get out this drift velocity 01:05:13.440 --> 01:05:14.020 align:middle line:90% here. 01:05:14.020 --> 01:05:19.110 align:middle line:84% And then we know that our current is equal to minus 01:05:19.110 --> 01:05:27.850 align:middle line:90% en V Ve plus ze and I VI. 01:05:27.850 --> 01:05:32.710 align:middle line:84% So this is simply equal to e and e Vd. 01:05:32.710 --> 01:05:35.890 align:middle line:84% So if we have some way of knowing the density-- 01:05:35.890 --> 01:05:38.380 align:middle line:84% for example, from interferometry or from scattering 01:05:38.380 --> 01:05:41.440 align:middle line:84% of electron plasma waves of a separate spectrometer-- 01:05:41.440 --> 01:05:44.900 align:middle line:84% by inferring the drift velocity from the spectrum, 01:05:44.900 --> 01:05:47.620 align:middle line:84% we can now get out the local current. 01:05:47.620 --> 01:05:54.000 align:middle line:84% Of course, it is the local current in the k direction. 01:05:54.000 --> 01:05:55.780 align:middle line:84% So if your current is pointing upwards, 01:05:55.780 --> 01:05:57.530 align:middle line:84% but you've chosen your scattering geometry 01:05:57.530 --> 01:06:00.450 align:middle line:84% so that k is perpendicular to it, you won't measure anything. 01:06:00.450 --> 01:06:02.390 align:middle line:84% So this doesn't tell you the direction of k. 01:06:02.390 --> 01:06:03.765 align:middle line:84% It just tells you the size of it. 01:06:03.765 --> 01:06:07.040 align:middle line:84% If you know the direction of J from symmetry, 01:06:07.040 --> 01:06:10.100 align:middle line:84% and you have measurements of J at different positions-- 01:06:10.100 --> 01:06:12.650 align:middle line:84% for example, if I do Thomson scattering 01:06:12.650 --> 01:06:14.660 align:middle line:90% at several positions along x. 01:06:14.660 --> 01:06:17.390 align:middle line:84% I've got my laser beam going through the plasma, 01:06:17.390 --> 01:06:22.480 align:middle line:84% and I collect the scattered light several places along here. 01:06:22.480 --> 01:06:27.960 align:middle line:84% And I calculate J at each of these points, 01:06:27.960 --> 01:06:30.670 align:middle line:84% then with some symmetry arguments, 01:06:30.670 --> 01:06:35.970 align:middle line:84% I may be able to solve curl of B equals mu 0 J 01:06:35.970 --> 01:06:38.130 align:middle line:84% and locally find the magnetic field. 01:06:38.130 --> 01:06:40.420 align:middle line:84% But that requires a lot of assumptions about symmetry. 01:06:40.420 --> 01:06:42.940 align:middle line:84% So it may not be completely general, 01:06:42.940 --> 01:06:45.135 align:middle line:84% but even getting to current is pretty cool. 01:06:45.135 --> 01:06:46.260 align:middle line:90% You get the magnetic field. 01:06:46.260 --> 01:06:48.650 align:middle line:90% That's really nice. 01:06:48.650 --> 01:06:49.310 align:middle line:90% OK. 01:06:49.310 --> 01:06:50.120 align:middle line:90% Questions on that? 01:06:50.120 --> 01:06:53.150 align:middle line:90% 01:06:53.150 --> 01:06:55.660 align:middle line:90% Yeah. 01:06:55.660 --> 01:06:58.747 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:06:58.747 --> 01:06:59.330 align:middle line:90% JACK HARE: No. 01:06:59.330 --> 01:07:01.220 align:middle line:84% We're just going to work in the ion frame. 01:07:01.220 --> 01:07:04.970 align:middle line:84% So this Vd here, I've basically set this equal to 0. 01:07:04.970 --> 01:07:09.540 align:middle line:84% And the ions we can assume just aren't drifting and [INAUDIBLE]. 01:07:09.540 --> 01:07:13.805 align:middle line:90% 01:07:13.805 --> 01:07:16.450 align:middle line:84% Just choose a frame, whether or not. 01:07:16.450 --> 01:07:18.937 align:middle line:84% So I can always choose a frame with the arms stationary. 01:07:18.937 --> 01:07:23.240 align:middle line:90% 01:07:23.240 --> 01:07:25.010 align:middle line:84% This only depends on the relative drift 01:07:25.010 --> 01:07:26.302 align:middle line:90% between the electrons and ions. 01:07:26.302 --> 01:07:28.320 align:middle line:84% It doesn't matter to the Thomson scattering 01:07:28.320 --> 01:07:29.612 align:middle line:90% which of them is actually true. 01:07:29.612 --> 01:07:32.740 align:middle line:90% 01:07:32.740 --> 01:07:34.360 align:middle line:90% Yes, exactly. 01:07:34.360 --> 01:07:36.422 align:middle line:84% I've never seen a case where the-- 01:07:36.422 --> 01:07:38.880 align:middle line:84% I don't think I've seen a case where the ions are drifting, 01:07:38.880 --> 01:07:39.810 align:middle line:90% and you're like-- 01:07:39.810 --> 01:07:41.310 align:middle line:84% that's unusual because the electrons 01:07:41.310 --> 01:07:42.780 align:middle line:90% tend to be the ones that move. 01:07:42.780 --> 01:07:45.660 align:middle line:90% Yeah. 01:07:45.660 --> 01:07:47.980 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:07:47.980 --> 01:07:49.330 align:middle line:90% JACK HARE: Yes. 01:07:49.330 --> 01:07:51.675 align:middle line:90% AUDIENCE: Why do we have this? 01:07:51.675 --> 01:07:53.300 align:middle line:84% JACK HARE: Oh, I'm just saying in terms 01:07:53.300 --> 01:07:54.300 align:middle line:90% of solving the equation. 01:07:54.300 --> 01:07:58.700 align:middle line:90% 01:07:58.700 --> 01:07:59.780 align:middle line:90% Yes, it definitely is. 01:07:59.780 --> 01:08:03.710 align:middle line:84% And the solution only depends on the difference between the two. 01:08:03.710 --> 01:08:04.210 align:middle line:90% Yeah. 01:08:04.210 --> 01:08:05.627 align:middle line:84% So all the Thomson scattering will 01:08:05.627 --> 01:08:07.752 align:middle line:84% tell you is the difference between the two of them. 01:08:07.752 --> 01:08:09.400 align:middle line:84% It's then up to you to say, and I 01:08:09.400 --> 01:08:11.530 align:middle line:84% believe that the ions are stationary 01:08:11.530 --> 01:08:14.260 align:middle line:84% in my lab frame or the electrons are stationary in my lab frame 01:08:14.260 --> 01:08:14.968 align:middle line:90% or whatever else. 01:08:14.968 --> 01:08:18.647 align:middle line:84% But the drift velocity will be invariant, regardless 01:08:18.647 --> 01:08:19.689 align:middle line:90% of what frame you choose. 01:08:19.689 --> 01:08:22.609 align:middle line:90% 01:08:22.609 --> 01:08:23.708 align:middle line:90% Other questions? 01:08:23.708 --> 01:08:26.814 align:middle line:90% 01:08:26.814 --> 01:08:28.189 align:middle line:84% We talked a little bit about what 01:08:28.189 --> 01:08:32.330 align:middle line:84% you can learn from the electron plasma waves before. 01:08:32.330 --> 01:08:34.609 align:middle line:84% Just in terms of what you can learn about these-- 01:08:34.609 --> 01:08:40.420 align:middle line:84% these anacoustic peaks here, their frequency 01:08:40.420 --> 01:08:45.979 align:middle line:90% depends on z times Te. 01:08:45.979 --> 01:08:48.500 align:middle line:84% So it depends on the charge of the ions times the electron 01:08:48.500 --> 01:08:49.430 align:middle line:90% temperature. 01:08:49.430 --> 01:08:55.189 align:middle line:84% The width of these turns out to depend on the ion temperature 01:08:55.189 --> 01:08:56.810 align:middle line:90% here. 01:08:56.810 --> 01:09:01.040 align:middle line:84% And to some extent, the ratio between the peaks and the center 01:09:01.040 --> 01:09:03.500 align:middle line:84% here is also sensitive to the ion temperature. 01:09:03.500 --> 01:09:05.450 align:middle line:84% So you can use both of these two things 01:09:05.450 --> 01:09:07.220 align:middle line:90% to get the ion temperature out. 01:09:07.220 --> 01:09:09.470 align:middle line:84% Because these are relatively low frequency modes, 01:09:09.470 --> 01:09:12.060 align:middle line:84% you can also see bulk shifts of the plasma. 01:09:12.060 --> 01:09:14.450 align:middle line:84% So if the ions are moving, your spectra 01:09:14.450 --> 01:09:15.529 align:middle line:90% will be shifted slightly. 01:09:15.529 --> 01:09:18.680 align:middle line:90% 01:09:18.680 --> 01:09:23.830 align:middle line:84% And that shift here will be proportional to the ion bulk 01:09:23.830 --> 01:09:24.529 align:middle line:90% flow velocity. 01:09:24.529 --> 01:09:26.649 align:middle line:84% So you can measure the whole plasma moving. 01:09:26.649 --> 01:09:29.840 align:middle line:84% This ion velocity is the V in the mhg equation 01:09:29.840 --> 01:09:31.660 align:middle line:84% because the ions have much more momentum. 01:09:31.660 --> 01:09:33.760 align:middle line:84% And then finally, as we just discussed 01:09:33.760 --> 01:09:39.520 align:middle line:84% from the ratio between these two peaks-- if they are asymmetric, 01:09:39.520 --> 01:09:46.418 align:middle line:84% we can get Vd dot k equals J dot k. 01:09:46.418 --> 01:09:48.050 align:middle line:90% That gives us J dot k. 01:09:48.050 --> 01:09:50.990 align:middle line:84% So from the anacoustic spectrum, you can measure an awful lot. 01:09:50.990 --> 01:09:53.930 align:middle line:84% You can measure the electron temperature, the ion 01:09:53.930 --> 01:09:56.028 align:middle line:84% temperature, the ion flow velocity, 01:09:56.028 --> 01:09:58.070 align:middle line:84% and the drift between the electrons and the ions. 01:09:58.070 --> 01:10:00.350 align:middle line:84% The only thing you can't get directly from the spectra 01:10:00.350 --> 01:10:01.910 align:middle line:90% is the electron density. 01:10:01.910 --> 01:10:04.040 align:middle line:84% From that, you need the electron plasma waves. 01:10:04.040 --> 01:10:06.500 align:middle line:84% But still, there's a remarkably rich amount of information 01:10:06.500 --> 01:10:10.440 align:middle line:84% we get out of this by scattering off these anacoustic waves. 01:10:10.440 --> 01:10:10.940 align:middle line:90% OK. 01:10:10.940 --> 01:10:15.590 align:middle line:90% 01:10:15.590 --> 01:10:19.080 align:middle line:84% So I have many pages left, and I knew I wasn't going to finish. 01:10:19.080 --> 01:10:23.010 align:middle line:84% So we discussed magnetic fields already, 01:10:23.010 --> 01:10:25.110 align:middle line:84% to the extent I said they were complicated. 01:10:25.110 --> 01:10:30.060 align:middle line:90% 01:10:30.060 --> 01:10:34.200 align:middle line:90% We vaguely discussed collisions. 01:10:34.200 --> 01:10:37.552 align:middle line:84% The only thing I want to say about collisions here-- 01:10:37.552 --> 01:10:44.570 align:middle line:84% if you do include collisions, and you 01:10:44.570 --> 01:10:46.640 align:middle line:90% do this in the form of some-- 01:10:46.640 --> 01:10:49.175 align:middle line:90% 01:10:49.175 --> 01:10:49.675 align:middle line:90% sorry. 01:10:49.675 --> 01:10:53.380 align:middle line:90% 01:10:53.380 --> 01:10:55.990 align:middle line:84% Some collision operator acting on the distribution 01:10:55.990 --> 01:10:58.160 align:middle line:84% function inside your Vlasov equation. 01:10:58.160 --> 01:11:00.350 align:middle line:84% So it'll be about Vlasov Fokker-Planck equation. 01:11:00.350 --> 01:11:02.320 align:middle line:90% You include collisions here. 01:11:02.320 --> 01:11:04.270 align:middle line:84% The main result I want to tell you about 01:11:04.270 --> 01:11:12.040 align:middle line:84% is that we end up usually damping coherent features. 01:11:12.040 --> 01:11:13.785 align:middle line:84% So we don't get these nice waves anymore. 01:11:13.785 --> 01:11:15.285 align:middle line:84% We just get the incoherent features. 01:11:15.285 --> 01:11:18.200 align:middle line:90% 01:11:18.200 --> 01:11:22.090 align:middle line:90% And it also further-- 01:11:22.090 --> 01:11:23.350 align:middle line:90% you might think, OK. 01:11:23.350 --> 01:11:25.690 align:middle line:84% We just get the coherent spectrum, incoherent spectrum 01:11:25.690 --> 01:11:28.390 align:middle line:84% where s of k omega is just related to the electron 01:11:28.390 --> 01:11:29.990 align:middle line:90% distribution function. 01:11:29.990 --> 01:11:33.040 align:middle line:84% But it turns out that this is times another factor 01:11:33.040 --> 01:11:34.340 align:middle line:90% to do with collisions. 01:11:34.340 --> 01:11:37.510 align:middle line:84% So if you have collisions present, 01:11:37.510 --> 01:11:40.220 align:middle line:84% they will modify what you're measuring. 01:11:40.220 --> 01:11:42.370 align:middle line:84% And so if you don't include the collisions 01:11:42.370 --> 01:11:44.233 align:middle line:84% in your theoretical treatment, you 01:11:44.233 --> 01:11:45.900 align:middle line:84% will be interpreting the spectrum wrong. 01:11:45.900 --> 01:11:47.290 align:middle line:90% So collisions can be a problem. 01:11:47.290 --> 01:11:49.570 align:middle line:84% As we discussed before, to be collision 01:11:49.570 --> 01:11:51.490 align:middle line:84% for Thomson scattering, you really 01:11:51.490 --> 01:11:53.725 align:middle line:84% need to have very short mean-free paths. 01:11:53.725 --> 01:11:56.350 align:middle line:84% And most of the time, people are not in the collisional regime, 01:11:56.350 --> 01:11:58.115 align:middle line:84% even if their plasma is collisional, 01:11:58.115 --> 01:11:59.740 align:middle line:84% in the sense of having a mean-free path 01:11:59.740 --> 01:12:01.115 align:middle line:84% less than the size of the plasma. 01:12:01.115 --> 01:12:09.130 align:middle line:90% 01:12:09.130 --> 01:12:10.345 align:middle line:90% OK. 01:12:10.345 --> 01:12:12.475 align:middle line:90% I like finite volume. 01:12:12.475 --> 01:12:37.420 align:middle line:90% 01:12:37.420 --> 01:12:40.380 align:middle line:84% So a lot of the arguments I use here also 01:12:40.380 --> 01:12:42.800 align:middle line:84% apply to finite time measurements. 01:12:42.800 --> 01:12:44.550 align:middle line:84% So all of our measurements will take place 01:12:44.550 --> 01:12:47.790 align:middle line:84% in a finite volume of plasma or over a finite time scale. 01:12:47.790 --> 01:12:50.610 align:middle line:84% We'll have some plasma like this. 01:12:50.610 --> 01:12:55.370 align:middle line:84% We'll focus our laser beam through the plasma 01:12:55.370 --> 01:13:00.660 align:middle line:84% and we will collect the scattered light 01:13:00.660 --> 01:13:03.600 align:middle line:90% along the direction ks. 01:13:03.600 --> 01:13:06.790 align:middle line:84% This is the collection volume of our spectrometer, 01:13:06.790 --> 01:13:11.310 align:middle line:84% and so we'll be looking at light scattered from this region here. 01:13:11.310 --> 01:13:13.250 align:middle line:84% So of course, one thing that could trip you up 01:13:13.250 --> 01:13:19.880 align:middle line:84% is that your f of V and r, and I guess time 01:13:19.880 --> 01:13:21.530 align:middle line:90% can have spatial variations. 01:13:21.530 --> 01:13:28.592 align:middle line:90% 01:13:28.592 --> 01:13:30.050 align:middle line:84% So if, for example, the temperature 01:13:30.050 --> 01:13:31.430 align:middle line:84% on one side of your collection volume 01:13:31.430 --> 01:13:32.810 align:middle line:84% is different from the temperature on the other side 01:13:32.810 --> 01:13:34.393 align:middle line:84% of the collection volume, you're going 01:13:34.393 --> 01:13:36.055 align:middle line:84% to be collecting two different spectra, 01:13:36.055 --> 01:13:37.430 align:middle line:84% but they're both going to show up 01:13:37.430 --> 01:13:40.460 align:middle line:84% on top of each other on your spectrometer. 01:13:40.460 --> 01:13:43.610 align:middle line:90% So that can be quite hard to do. 01:13:43.610 --> 01:13:47.700 align:middle line:84% If it's not temperature, maybe it's something like velocity. 01:13:47.700 --> 01:13:50.900 align:middle line:84% So if we have velocity shear, I have a plasma 01:13:50.900 --> 01:13:54.050 align:middle line:84% where across this collection volume, 01:13:54.050 --> 01:13:56.850 align:middle line:90% the velocity looks like this. 01:13:56.850 --> 01:14:00.960 align:middle line:84% And I look at the anacoustic waves, 01:14:00.960 --> 01:14:05.630 align:middle line:84% then the spectra coming from this point 01:14:05.630 --> 01:14:09.650 align:middle line:84% here would be Doppler shifted in one direction. 01:14:09.650 --> 01:14:12.380 align:middle line:84% The spectra coming from the middle of my collection volume 01:14:12.380 --> 01:14:13.930 align:middle line:90% would look like this. 01:14:13.930 --> 01:14:15.680 align:middle line:84% And the spectra coming from the other side 01:14:15.680 --> 01:14:18.670 align:middle line:84% would be Doppler shifted in the other way. 01:14:18.670 --> 01:14:19.920 align:middle line:90% And so what you'd have to do-- 01:14:19.920 --> 01:14:21.450 align:middle line:84% what you see in your spectrometer 01:14:21.450 --> 01:14:23.910 align:middle line:84% is all of these averaged together. 01:14:23.910 --> 01:14:26.080 align:middle line:84% And maybe you end up with something 01:14:26.080 --> 01:14:31.490 align:middle line:84% like this, where the peaks are much wider than they 01:14:31.490 --> 01:14:33.260 align:middle line:90% were in your original case. 01:14:33.260 --> 01:14:36.010 align:middle line:90% 01:14:36.010 --> 01:14:38.350 align:middle line:84% If you didn't know that you had gradients in velocity 01:14:38.350 --> 01:14:41.170 align:middle line:84% within your collection volume, you would interpret this plasma 01:14:41.170 --> 01:14:43.020 align:middle line:90% as being hotter. 01:14:43.020 --> 01:14:55.040 align:middle line:84% So TI looks larger than it is due to velocity gradient. 01:14:55.040 --> 01:15:01.330 align:middle line:90% 01:15:01.330 --> 01:15:03.730 align:middle line:84% One way around this is actually by observing 01:15:03.730 --> 01:15:05.330 align:middle line:90% from multiple angles. 01:15:05.330 --> 01:15:15.340 align:middle line:84% So if you have ks1 but you also observe ks2 or maybe even ks3 01:15:15.340 --> 01:15:19.320 align:middle line:84% as well, the spectra that you get out 01:15:19.320 --> 01:15:21.970 align:middle line:84% will not all be broadened in exactly the same way. 01:15:21.970 --> 01:15:25.290 align:middle line:84% And so you may be able to infer from these different angles what 01:15:25.290 --> 01:15:27.780 align:middle line:84% the true temperature is, but also what the true velocity 01:15:27.780 --> 01:15:28.457 align:middle line:90% gradient is. 01:15:28.457 --> 01:15:29.790 align:middle line:90% But of course, that's expensive. 01:15:29.790 --> 01:15:31.623 align:middle line:84% Now you have to have multiple spectrometers. 01:15:31.623 --> 01:15:33.750 align:middle line:90% You might not want to do that. 01:15:33.750 --> 01:15:35.970 align:middle line:84% This problem gets much worse if you have 01:15:35.970 --> 01:15:37.743 align:middle line:90% a plasma which is turbulent. 01:15:37.743 --> 01:15:39.660 align:middle line:84% So instead of having a nice velocity gradient, 01:15:39.660 --> 01:15:41.035 align:middle line:84% you have a plasma where there are 01:15:41.035 --> 01:15:45.590 align:middle line:84% lots of little hydrodynamic eddies inside here. 01:15:45.590 --> 01:15:50.170 align:middle line:84% If you collect your scattered light 01:15:50.170 --> 01:15:52.207 align:middle line:84% from over this turbulent region here, 01:15:52.207 --> 01:15:54.790 align:middle line:84% then you're going to be having all sorts of different velocity 01:15:54.790 --> 01:15:55.900 align:middle line:90% components inside. 01:15:55.900 --> 01:15:59.410 align:middle line:84% And what might start out as a nice anacoustic feature 01:15:59.410 --> 01:16:03.910 align:middle line:84% might turn out to be just some gigantic, featureless blur. 01:16:03.910 --> 01:16:06.500 align:middle line:84% And then it's very, very hard to infer anything from that. 01:16:06.500 --> 01:16:09.280 align:middle line:84% So in general, you've got to have a nice, small collection 01:16:09.280 --> 01:16:11.350 align:middle line:84% volume so that you have smaller velocity 01:16:11.350 --> 01:16:12.752 align:middle line:90% gradients over the volume. 01:16:12.752 --> 01:16:14.710 align:middle line:84% And as you make the volume smaller and smaller, 01:16:14.710 --> 01:16:16.360 align:middle line:90% you get less scattered light. 01:16:16.360 --> 01:16:18.250 align:middle line:84% And so therefore, you get less signal, 01:16:18.250 --> 01:16:19.750 align:middle line:84% and then your signal to noise drops. 01:16:19.750 --> 01:16:21.230 align:middle line:90% So you can't really win this. 01:16:21.230 --> 01:16:23.105 align:middle line:84% Then you'd have to get a more powerful laser. 01:16:23.105 --> 01:16:25.870 align:middle line:90% 01:16:25.870 --> 01:16:27.558 align:middle line:90% That's a problem. 01:16:27.558 --> 01:16:30.100 align:middle line:84% And we also discussed this, but I'll just mention it briefly, 01:16:30.100 --> 01:16:32.518 align:middle line:90% is our finite collection angle. 01:16:32.518 --> 01:16:49.980 align:middle line:90% 01:16:49.980 --> 01:16:53.180 align:middle line:84% So again, if we've got some sort of plasma 01:16:53.180 --> 01:16:58.990 align:middle line:84% and we've got our laser going through it, 01:16:58.990 --> 01:17:02.250 align:middle line:84% we're generally going to be collecting the light that's 01:17:02.250 --> 01:17:06.000 align:middle line:84% coming out of here using a lens or something 01:17:06.000 --> 01:17:08.040 align:middle line:84% like that-- something with a finite solid angle. 01:17:08.040 --> 01:17:10.630 align:middle line:84% This is the e omega we kept going on about it-- 01:17:10.630 --> 01:17:13.650 align:middle line:84% until it goes on to our spectrometer here. 01:17:13.650 --> 01:17:18.130 align:middle line:84% But that means that there's going to be several different k 01:17:18.130 --> 01:17:18.630 align:middle line:90% vectors. 01:17:18.630 --> 01:17:22.190 align:middle line:84% There's going to be ks0 here, the one that 01:17:22.190 --> 01:17:23.772 align:middle line:90% goes to the center of our lens. 01:17:23.772 --> 01:17:25.730 align:middle line:84% But we're also going to have light that's going 01:17:25.730 --> 01:17:28.580 align:middle line:90% to an angle, ks0 plus delta k. 01:17:28.580 --> 01:17:33.435 align:middle line:84% And on the other side, ks0 minus delta k. 01:17:33.435 --> 01:17:36.060 align:middle line:84% And these are all going to have slightly different vectors that 01:17:36.060 --> 01:17:37.780 align:middle line:84% will have slightly different spectra. 01:17:37.780 --> 01:17:45.730 align:middle line:84% So for example, you can imagine that one of your spectra 01:17:45.730 --> 01:17:49.000 align:middle line:84% looks like this, one of them is slightly more 01:17:49.000 --> 01:17:52.600 align:middle line:84% in the collective regime, an incoherent regime 01:17:52.600 --> 01:17:55.210 align:middle line:84% where alpha is smaller than the other spectra. 01:17:55.210 --> 01:17:58.130 align:middle line:90% It looks like this. 01:17:58.130 --> 01:18:00.950 align:middle line:84% And one of them is even more in the collective regime, 01:18:00.950 --> 01:18:04.470 align:middle line:90% and it looks like this. 01:18:04.470 --> 01:18:05.970 align:middle line:84% When you sum those three together, 01:18:05.970 --> 01:18:07.980 align:middle line:84% you're also going to get a signal that 01:18:07.980 --> 01:18:09.750 align:middle line:84% looks a little bit like something you 01:18:09.750 --> 01:18:13.300 align:middle line:84% should be able to fit with your standard fitting techniques. 01:18:13.300 --> 01:18:15.190 align:middle line:84% But you'll never be able to quite fit this. 01:18:15.190 --> 01:18:17.860 align:middle line:84% Or if you do fit it, you'll infer the wrong temperature 01:18:17.860 --> 01:18:18.860 align:middle line:90% and density. 01:18:18.860 --> 01:18:22.330 align:middle line:84% So the solution here is to make your lens as small as possible 01:18:22.330 --> 01:18:25.030 align:middle line:84% because if you make your lens nice and small, 01:18:25.030 --> 01:18:28.360 align:middle line:84% then most of the light is being scattered with the same k 01:18:28.360 --> 01:18:29.110 align:middle line:90% vector. 01:18:29.110 --> 01:18:30.400 align:middle line:84% And of course, if you make your lens small, 01:18:30.400 --> 01:18:32.200 align:middle line:84% then you collect less scattered light, 01:18:32.200 --> 01:18:34.100 align:middle line:84% and so then you have worse signal to noise, 01:18:34.100 --> 01:18:36.010 align:middle line:84% so then you need to buy a bigger laser. 01:18:36.010 --> 01:18:39.710 align:middle line:84% But there are no ways to win in Thomson scattering. 01:18:39.710 --> 01:18:41.490 align:middle line:90% OK. 01:18:41.490 --> 01:18:43.990 align:middle line:84% I don't think I have anything else I really desperately need 01:18:43.990 --> 01:18:44.600 align:middle line:90% to tell you. 01:18:44.600 --> 01:18:45.758 align:middle line:90% Any questions on that? 01:18:45.758 --> 01:18:51.178 align:middle line:90% 01:18:51.178 --> 01:18:54.905 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:18:54.905 --> 01:18:55.530 align:middle line:90% JACK HARE: Yes. 01:18:55.530 --> 01:18:56.670 align:middle line:84% If your laser gets more powerful, 01:18:56.670 --> 01:18:58.680 align:middle line:84% then the plasma starts to absorb the laser power 01:18:58.680 --> 01:18:59.820 align:middle line:90% through inverse Bremsstrahlung. 01:18:59.820 --> 01:19:01.620 align:middle line:84% And there are laser plasma instabilities, 01:19:01.620 --> 01:19:03.537 align:middle line:84% like filamentation, so there's an upper limits 01:19:03.537 --> 01:19:07.110 align:middle line:84% as well on the laser power, too, as well as upper limits 01:19:07.110 --> 01:19:10.110 align:middle line:90% on your budget for a laser. 01:19:10.110 --> 01:19:11.520 align:middle line:90% Yeah, so it's not-- 01:19:11.520 --> 01:19:14.910 align:middle line:84% so a laser to do Thomson scattering is like a $200,000 01:19:14.910 --> 01:19:17.580 align:middle line:90% to $500,000 object. 01:19:17.580 --> 01:19:20.770 align:middle line:84% And then bigger lasers are more expensive than that. 01:19:20.770 --> 01:19:22.860 align:middle line:84% So this already could end up costing more 01:19:22.860 --> 01:19:27.010 align:middle line:84% than the thing making the plasma along with [INAUDIBLE], yeah. 01:19:27.010 --> 01:19:30.230 align:middle line:90% 01:19:30.230 --> 01:19:33.005 align:middle line:90% Any other questions? 01:19:33.005 --> 01:19:33.505 align:middle line:90% Yes. 01:19:33.505 --> 01:19:36.225 align:middle line:90% 01:19:36.225 --> 01:19:40.360 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:19:40.360 --> 01:19:42.640 align:middle line:84% JACK HARE: What's new at the moment? 01:19:42.640 --> 01:19:45.860 align:middle line:84% A lot of work on non-Maxwellian distribution functions. 01:19:45.860 --> 01:19:48.110 align:middle line:84% So I've erased it from the board, 01:19:48.110 --> 01:19:50.980 align:middle line:84% but for Maxwellian, we have this nice function, w, 01:19:50.980 --> 01:19:53.300 align:middle line:84% that people have worked out and they've tabulated it. 01:19:53.300 --> 01:19:55.508 align:middle line:84% But if you have an arbitrary non-Maxwellian function, 01:19:55.508 --> 01:19:57.010 align:middle line:84% you have to do Landau integration 01:19:57.010 --> 01:19:59.710 align:middle line:84% for every single omega over k-- so 01:19:59.710 --> 01:20:01.300 align:middle line:84% for every point in the complex plane. 01:20:01.300 --> 01:20:03.430 align:middle line:84% And evaluating that function is extremely slow, 01:20:03.430 --> 01:20:05.150 align:middle line:84% so you can't really do fitting with it. 01:20:05.150 --> 01:20:06.820 align:middle line:84% So people are doing machine learning 01:20:06.820 --> 01:20:10.330 align:middle line:84% or they're doing other ways of speeding up those evaluations. 01:20:10.330 --> 01:20:12.310 align:middle line:84% People are also doing experiments 01:20:12.310 --> 01:20:16.580 align:middle line:84% when they observe the light from many, many, many angles. 01:20:16.580 --> 01:20:18.940 align:middle line:84% And so the Thomson scattering experiment there 01:20:18.940 --> 01:20:21.970 align:middle line:84% looks like you've got a plasma, you've 01:20:21.970 --> 01:20:25.390 align:middle line:84% got some laser light going through. 01:20:25.390 --> 01:20:30.270 align:middle line:84% It gets scattered off in loads of different directions, 01:20:30.270 --> 01:20:31.980 align:middle line:84% and you put a convex mirror here. 01:20:31.980 --> 01:20:35.880 align:middle line:84% And that convex mirror focuses the light onto a spectrometer. 01:20:35.880 --> 01:20:38.880 align:middle line:84% And each different bit of the spectrometer 01:20:38.880 --> 01:20:40.990 align:middle line:84% corresponds to a different angle. 01:20:40.990 --> 01:20:44.095 align:middle line:84% So then you're recovering the full three-dimensional Thomson 01:20:44.095 --> 01:20:45.970 align:middle line:84% scattering distribution function, or at least 01:20:45.970 --> 01:20:48.470 align:middle line:84% the two-dimensional Thomson scattering distribution function 01:20:48.470 --> 01:20:50.050 align:middle line:84% there using these very clever optics. 01:20:50.050 --> 01:20:52.050 align:middle line:84% So those are two things people are doing. 01:20:52.050 --> 01:20:53.550 align:middle line:84% It's a very active area of research. 01:20:53.550 --> 01:20:58.750 align:middle line:90% 01:20:58.750 --> 01:20:59.250 align:middle line:90% OK. 01:20:59.250 --> 01:21:00.840 align:middle line:84% Good luck on your final, those of you 01:21:00.840 --> 01:21:02.080 align:middle line:90% who haven't submitted it yet. 01:21:02.080 --> 01:21:05.520 align:middle line:84% And as I said on Canvas, I'd be very grateful for any feedback 01:21:05.520 --> 01:21:08.120 align:middle line:90% from this evaluation. 01:21:08.120 --> 01:21:11.000 align:middle line:90%