WEBVTT 00:00:00.000 --> 00:00:02.460 align:middle line:90% [SQUEAKING] 00:00:02.460 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:11.290 align:middle line:90% [CLICKING] 00:00:11.290 --> 00:00:13.690 align:middle line:84% JACK HARE: So once again, we are going 00:00:13.690 --> 00:00:15.190 align:middle line:84% to be discussing current scattering. 00:00:15.190 --> 00:00:16.565 align:middle line:84% And we're going to derive Thomson 00:00:16.565 --> 00:00:18.160 align:middle line:90% scattering for the third time. 00:00:18.160 --> 00:00:19.740 align:middle line:90% Huzzah! 00:00:19.740 --> 00:00:20.240 align:middle line:90% Huzzah. 00:00:20.240 --> 00:00:20.720 align:middle line:90% Thank you. 00:00:20.720 --> 00:00:21.220 align:middle line:90% Good. 00:00:21.220 --> 00:00:24.470 align:middle line:90% 00:00:24.470 --> 00:00:25.460 align:middle line:90% Well done. 00:00:25.460 --> 00:00:27.810 align:middle line:90% A grade. 00:00:27.810 --> 00:00:33.660 align:middle line:84% Coherent scattering-- so just as a quick review 00:00:33.660 --> 00:00:39.220 align:middle line:84% of what we discussed at the end of the last lecture, 00:00:39.220 --> 00:00:42.750 align:middle line:84% we are looking at scattering when 00:00:42.750 --> 00:00:47.400 align:middle line:84% the wavelength of whatever the mode is that we're scattering of 00:00:47.400 --> 00:00:50.980 align:middle line:84% is greater than the Debye length here. 00:00:50.980 --> 00:00:57.450 align:middle line:84% So this wavelength is 2 pi upon the size of k, where 00:00:57.450 --> 00:01:02.110 align:middle line:90% k is ks minus ki, like this. 00:01:02.110 --> 00:01:05.437 align:middle line:84% So we're not talking about the wavelength of our laser. 00:01:05.437 --> 00:01:07.770 align:middle line:84% Though, this wavelength may be similar to the wavelength 00:01:07.770 --> 00:01:10.140 align:middle line:84% of our laser, but we're talking specifically 00:01:10.140 --> 00:01:13.950 align:middle line:84% about the wavelength that we get from drawing this diagram where 00:01:13.950 --> 00:01:18.930 align:middle line:84% we have some laser beam coming in with a wave vector ki. 00:01:18.930 --> 00:01:23.490 align:middle line:84% We have some scatterer at a vector ks. 00:01:23.490 --> 00:01:27.870 align:middle line:90% And then we go, OK, ks minus ki. 00:01:27.870 --> 00:01:30.768 align:middle line:84% Aha, this is the k that we're talking about. 00:01:30.768 --> 00:01:32.560 align:middle line:84% We're talking about the wavelength of this. 00:01:32.560 --> 00:01:35.185 align:middle line:84% And the reason we're interested in things larger than the Debye 00:01:35.185 --> 00:01:37.860 align:middle line:84% length is on scales larger than the Debye length, the wavelength 00:01:37.860 --> 00:01:39.930 align:middle line:84% of this scattering vector is going 00:01:39.930 --> 00:01:42.870 align:middle line:84% to be sampling the collective motion of the plasma, 00:01:42.870 --> 00:01:44.640 align:middle line:84% or the coherent motion of plasma. 00:01:44.640 --> 00:01:47.730 align:middle line:84% So you might see this called collective Thomson scattering, 00:01:47.730 --> 00:01:50.380 align:middle line:84% as well as coherent Thomson scattering. 00:01:50.380 --> 00:01:53.530 align:middle line:84% So modes which exist in the plasma that 00:01:53.530 --> 00:01:56.270 align:middle line:84% fulfill this condition, they have a frequency, 00:01:56.270 --> 00:01:59.608 align:middle line:84% which is much, much less than our laser frequency in general. 00:01:59.608 --> 00:02:01.150 align:middle line:84% And this is a consequence of the fact 00:02:01.150 --> 00:02:04.240 align:middle line:84% that we're doing relatively low energy scattering. 00:02:04.240 --> 00:02:08.590 align:middle line:84% So our photon energy is much, much less 00:02:08.590 --> 00:02:13.580 align:middle line:84% than the kinetic energy of the particles in our plasma. 00:02:13.580 --> 00:02:20.680 align:middle line:84% But the size of the k vector may well be on the same order 00:02:20.680 --> 00:02:22.420 align:middle line:84% as we see in this sort of vector diagram 00:02:22.420 --> 00:02:24.460 align:middle line:84% as the k vector for our free space mode. 00:02:24.460 --> 00:02:26.950 align:middle line:84% And so we still have modes which carry a lot of momentum, 00:02:26.950 --> 00:02:29.350 align:middle line:84% even if they carry relatively little energy. 00:02:29.350 --> 00:02:31.480 align:middle line:84% And these two facts together mean 00:02:31.480 --> 00:02:35.530 align:middle line:84% that our phase velocity, which is defined as omega upon k, 00:02:35.530 --> 00:02:38.020 align:middle line:84% is much, much less than the speed of light. 00:02:38.020 --> 00:02:40.750 align:middle line:84% And the modes that we identified that fulfill this 00:02:40.750 --> 00:02:42.100 align:middle line:90% were the ion acoustic waves. 00:02:42.100 --> 00:02:47.450 align:middle line:90% 00:02:47.450 --> 00:02:50.240 align:middle line:84% And these have a dispersion relationship, 00:02:50.240 --> 00:02:57.890 align:middle line:84% omega equals square root of z te plus ti over mi-- 00:02:57.890 --> 00:03:00.320 align:middle line:84% Could you mute your microphone, please? 00:03:00.320 --> 00:03:01.460 align:middle line:90% Times k. 00:03:01.460 --> 00:03:04.130 align:middle line:84% And we also have the electron plasma waves. 00:03:04.130 --> 00:03:15.180 align:middle line:90% 00:03:15.180 --> 00:03:17.030 align:middle line:84% And these have a dispersion relationship 00:03:17.030 --> 00:03:27.908 align:middle line:84% of omega squared equals omega p squared plus 3 bte k squared. 00:03:27.908 --> 00:03:29.450 align:middle line:84% So these are the sorts of modes we're 00:03:29.450 --> 00:03:31.200 align:middle line:90% going to be scattering off. 00:03:31.200 --> 00:03:35.110 align:middle line:84% This is a very low frequency mode. 00:03:35.110 --> 00:03:37.582 align:middle line:84% And this is merely a low frequency mode. 00:03:37.582 --> 00:03:40.180 align:middle line:90% 00:03:40.180 --> 00:03:42.060 align:middle line:84% So when I say scatter off, it would 00:03:42.060 --> 00:03:45.690 align:middle line:84% be reasonable to ask why am I so sure that these modes exist 00:03:45.690 --> 00:03:47.370 align:middle line:90% in my plasma. 00:03:47.370 --> 00:03:50.670 align:middle line:84% Is there something in my plasma that is launching ion acoustic 00:03:50.670 --> 00:03:52.530 align:middle line:90% waves or electron plasma waves? 00:03:52.530 --> 00:03:57.510 align:middle line:84% And the answer is my plasma, just by virtue of sitting here, 00:03:57.510 --> 00:04:01.110 align:middle line:84% is a bath of these different modes, these fluctuations going 00:04:01.110 --> 00:04:02.260 align:middle line:90% on all the time. 00:04:02.260 --> 00:04:04.440 align:middle line:84% We have in every single direction ion 00:04:04.440 --> 00:04:07.620 align:middle line:84% acoustic waves flying off from every part of the plasma 00:04:07.620 --> 00:04:09.250 align:middle line:90% to every part of the plasma. 00:04:09.250 --> 00:04:14.300 align:middle line:84% So if you try and fire your laser beam through here, 00:04:14.300 --> 00:04:15.770 align:middle line:84% you are certain within the region 00:04:15.770 --> 00:04:19.010 align:middle line:84% of plasma you're interested in to find a wave going 00:04:19.010 --> 00:04:20.839 align:middle line:90% in the direction you want here. 00:04:20.839 --> 00:04:24.580 align:middle line:90% And you will get scattering. 00:04:24.580 --> 00:04:27.300 align:middle line:84% And we will be scattering off, as I've 00:04:27.300 --> 00:04:28.365 align:middle line:90% drawn several times now-- 00:04:28.365 --> 00:04:33.260 align:middle line:90% 00:04:33.260 --> 00:04:35.960 align:middle line:84% we'll be scattering off ion acoustic waves going 00:04:35.960 --> 00:04:39.350 align:middle line:84% in this direction at the sound speed here, 00:04:39.350 --> 00:04:42.560 align:middle line:84% but we will be observing them over here with our spectrometer, 00:04:42.560 --> 00:04:43.556 align:middle line:90% looking along ks. 00:04:43.556 --> 00:04:50.517 align:middle line:90% 00:04:50.517 --> 00:04:53.100 align:middle line:84% So I think this is one of the things that confuses people when 00:04:53.100 --> 00:04:55.410 align:middle line:84% they first start learning about collective Thomson scattering. 00:04:55.410 --> 00:04:57.872 align:middle line:84% They're wondering, why are these waves here to start with? 00:04:57.872 --> 00:04:59.580 align:middle line:84% The fact is there is always a possibility 00:04:59.580 --> 00:05:00.790 align:middle line:90% of there being a wave. 00:05:00.790 --> 00:05:03.270 align:middle line:84% And as you scatter off it, you transfer more energy 00:05:03.270 --> 00:05:04.930 align:middle line:90% and momentum to that wave. 00:05:04.930 --> 00:05:07.860 align:middle line:84% So you actually transfer energy from your probe laser 00:05:07.860 --> 00:05:08.613 align:middle line:90% to the wave. 00:05:08.613 --> 00:05:10.530 align:middle line:84% This is sort of like a Landau damping process. 00:05:10.530 --> 00:05:11.850 align:middle line:90% And you will grow that mode. 00:05:11.850 --> 00:05:15.150 align:middle line:84% And so that mode will definitely be there for you to scatter off. 00:05:15.150 --> 00:05:19.320 align:middle line:84% And the intensity of the scattering-- 00:05:19.320 --> 00:05:22.800 align:middle line:84% so again, we can write this intensity 00:05:22.800 --> 00:05:25.590 align:middle line:84% as the amount of power scattered into a solid angle scattered 00:05:25.590 --> 00:05:28.860 align:middle line:84% into a certain frequency will be proportional 00:05:28.860 --> 00:05:35.970 align:middle line:84% to the fluctuation strength for the number of modes-- 00:05:35.970 --> 00:05:38.720 align:middle line:90% fluctuation strength. 00:05:38.720 --> 00:05:42.250 align:middle line:90% 00:05:42.250 --> 00:05:45.040 align:middle line:84% Or if you want to think of this in a quasiparticle picture, 00:05:45.040 --> 00:05:47.680 align:middle line:84% you can think of it as like the number of phonons, 00:05:47.680 --> 00:05:49.810 align:middle line:84% or something like that, which are available. 00:05:49.810 --> 00:05:54.420 align:middle line:90% 00:05:54.420 --> 00:05:56.540 align:middle line:84% So the spectrum that we expect to get if we do 00:05:56.540 --> 00:05:57.980 align:middle line:90% all the mathematics correctly-- 00:05:57.980 --> 00:06:00.590 align:middle line:90% 00:06:00.590 --> 00:06:01.970 align:middle line:90% this is omega. 00:06:01.970 --> 00:06:05.105 align:middle line:84% This is what our spectrometer measures. 00:06:05.105 --> 00:06:09.330 align:middle line:90% 00:06:09.330 --> 00:06:13.365 align:middle line:84% If we expect to see some peaks at very low frequencies, 00:06:13.365 --> 00:06:16.300 align:middle line:84% there's 0 in the middle here, corresponding 00:06:16.300 --> 00:06:17.680 align:middle line:90% to the ion acoustic waves. 00:06:17.680 --> 00:06:22.040 align:middle line:90% 00:06:22.040 --> 00:06:23.975 align:middle line:84% And we'd also expect to see some peaks 00:06:23.975 --> 00:06:28.670 align:middle line:84% at slightly higher frequencies, corresponding 00:06:28.670 --> 00:06:31.592 align:middle line:84% to the electron plasma wave, the EPWs. 00:06:31.592 --> 00:06:37.010 align:middle line:90% 00:06:37.010 --> 00:06:39.890 align:middle line:84% And just remember I'm using this omega 00:06:39.890 --> 00:06:44.360 align:middle line:84% here, which is the difference between the scattered light 00:06:44.360 --> 00:06:46.657 align:middle line:90% and the incident light. 00:06:46.657 --> 00:06:48.740 align:middle line:84% So if I didn't have any plasma and I scattered off 00:06:48.740 --> 00:06:51.710 align:middle line:84% a lump of metal, I would just have a very big spectra 00:06:51.710 --> 00:06:54.170 align:middle line:84% right here in the center, omega equals 0. 00:06:54.170 --> 00:06:56.000 align:middle line:84% When there's plasma, I scatter off 00:06:56.000 --> 00:06:58.013 align:middle line:90% these modes inside the plasma. 00:06:58.013 --> 00:06:59.430 align:middle line:84% And there'll be two possibilities. 00:06:59.430 --> 00:07:02.030 align:middle line:84% I can scatter off a mode, which is going this way, 00:07:02.030 --> 00:07:04.130 align:middle line:84% or I can scatter off a mode that's going this way. 00:07:04.130 --> 00:07:06.950 align:middle line:84% And they both fulfill that condition 00:07:06.950 --> 00:07:14.975 align:middle line:84% that omega equals omega s minus omega i and k equals 00:07:14.975 --> 00:07:17.610 align:middle line:90% ks minus ki. 00:07:17.610 --> 00:07:20.490 align:middle line:84% So I'll see both of these peaks here. 00:07:20.490 --> 00:07:21.930 align:middle line:84% Now, in reality, they're not going 00:07:21.930 --> 00:07:23.790 align:middle line:84% to show up as little delta functions, 00:07:23.790 --> 00:07:26.735 align:middle line:84% even though if you look at the dispersion relationship, 00:07:26.735 --> 00:07:28.860 align:middle line:84% it looks like there's only one solution for a given 00:07:28.860 --> 00:07:30.210 align:middle line:90% omega and a given k. 00:07:30.210 --> 00:07:33.510 align:middle line:84% In reality, they'll be thermal effects, which broaden these. 00:07:33.510 --> 00:07:37.650 align:middle line:84% So what we will see is some sort of peak like this. 00:07:37.650 --> 00:07:39.900 align:middle line:84% It's good, because we don't like singularities. 00:07:39.900 --> 00:07:41.317 align:middle line:84% They don't really occur in nature. 00:07:41.317 --> 00:07:43.990 align:middle line:90% 00:07:43.990 --> 00:07:46.330 align:middle line:84% This heuristically is what we expect to get. 00:07:46.330 --> 00:07:48.760 align:middle line:84% But now we have to go out and do the mathematics 00:07:48.760 --> 00:07:50.840 align:middle line:90% and actually go and get it all. 00:07:50.840 --> 00:07:53.440 align:middle line:84% So any questions before we launch into that? 00:07:53.440 --> 00:07:55.000 align:middle line:90% Yes. 00:07:55.000 --> 00:07:55.990 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:07:55.990 --> 00:08:16.895 align:middle line:90% 00:08:16.895 --> 00:08:18.270 align:middle line:84% JACK HARE: Scattering gets chosen 00:08:18.270 --> 00:08:20.772 align:middle line:84% by where we put our detector, our spectrometer. 00:08:20.772 --> 00:08:22.230 align:middle line:84% So if you would like to think of it 00:08:22.230 --> 00:08:24.480 align:middle line:84% as a fiber-coupled spectrometer, I 00:08:24.480 --> 00:08:29.540 align:middle line:84% would have a little fiber optic bundle here. 00:08:29.540 --> 00:08:31.850 align:middle line:84% And that would go to my spectrometer, 00:08:31.850 --> 00:08:36.700 align:middle line:84% which disperses the light in the frequency space like this. 00:08:36.700 --> 00:08:38.210 align:middle line:84% There'll be some dispersive element. 00:08:38.210 --> 00:08:40.780 align:middle line:84% And so for example, if I have a lens here, 00:08:40.780 --> 00:08:43.990 align:middle line:84% I would collect light from this region here. 00:08:43.990 --> 00:08:45.490 align:middle line:84% And this spectrometer, therefore, 00:08:45.490 --> 00:08:49.315 align:middle line:90% defines the k vector, ks. 00:08:49.315 --> 00:08:50.940 align:middle line:84% Now, you bring up a good point, because 00:08:50.940 --> 00:08:55.050 align:middle line:84% for any realistic system, you actually collect a range of ks. 00:08:55.050 --> 00:08:57.540 align:middle line:84% You can't just collect along a single vector, 00:08:57.540 --> 00:09:00.328 align:middle line:84% because different angles of light 00:09:00.328 --> 00:09:01.620 align:middle line:90% will be collected by your lens. 00:09:01.620 --> 00:09:04.600 align:middle line:84% And that will also contribute to this broadening as well. 00:09:04.600 --> 00:09:07.110 align:middle line:84% But fundamentally, you can make that very, very small. 00:09:07.110 --> 00:09:08.610 align:middle line:84% And you still wouldn't get these delta functions. 00:09:08.610 --> 00:09:10.318 align:middle line:84% You're still going to get some broadening 00:09:10.318 --> 00:09:12.560 align:middle line:90% due to thermal effects. 00:09:12.560 --> 00:09:16.238 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:09:16.238 --> 00:09:18.030 align:middle line:84% JACK HARE: Well, actually, you don't really 00:09:18.030 --> 00:09:20.190 align:middle line:90% choose the magnitude of k. 00:09:20.190 --> 00:09:21.750 align:middle line:84% You only choose the direction of k. 00:09:21.750 --> 00:09:24.460 align:middle line:84% The magnitude is set by the dispersion relationship 00:09:24.460 --> 00:09:26.860 align:middle line:84% and the frequency relationship here. 00:09:26.860 --> 00:09:30.180 align:middle line:84% So when I see a wave with a certain frequency, 00:09:30.180 --> 00:09:31.860 align:middle line:84% I can use the dispersion relationship 00:09:31.860 --> 00:09:33.600 align:middle line:84% to work out what k was going to be at. 00:09:33.600 --> 00:09:35.880 align:middle line:84% So really, this is just setting the angle. 00:09:35.880 --> 00:09:39.100 align:middle line:84% So it's setting the vector relationship here. 00:09:39.100 --> 00:09:40.698 align:middle line:90% It's the size of k. 00:09:40.698 --> 00:09:42.240 align:middle line:84% Remember, we had this nice expression 00:09:42.240 --> 00:09:45.870 align:middle line:84% for the size of k, which was the size of k 00:09:45.870 --> 00:09:53.180 align:middle line:84% is equal to 2 ki size of ki sine theta upon 2. 00:09:53.180 --> 00:09:56.240 align:middle line:84% And we got that from saying that we're doing elastic scattering. 00:09:56.240 --> 00:10:01.740 align:middle line:84% So the size of ki is roughly the size of ks. 00:10:01.740 --> 00:10:05.100 align:middle line:84% So what you do is once you've chosen your angle 00:10:05.100 --> 00:10:08.310 align:middle line:84% and you know what your incident wavelength is, 00:10:08.310 --> 00:10:11.310 align:middle line:84% you have effectively chosen the size of this k vector 00:10:11.310 --> 00:10:12.180 align:middle line:90% straight away. 00:10:12.180 --> 00:10:14.070 align:middle line:84% And then when you choose the angle, 00:10:14.070 --> 00:10:16.720 align:middle line:84% you've chosen the other part of the vector, which is its angle. 00:10:16.720 --> 00:10:18.330 align:middle line:90% So this is the magnitude. 00:10:18.330 --> 00:10:23.320 align:middle line:84% And this equation determines theta and this is the magnitude. 00:10:23.320 --> 00:10:25.660 align:middle line:84% And then you've set everything about k. 00:10:25.660 --> 00:10:27.670 align:middle line:84% And so if you look at a certain frequency 00:10:27.670 --> 00:10:29.840 align:middle line:84% and you see a wave show up at that frequency, 00:10:29.840 --> 00:10:31.630 align:middle line:90% you know what your k is. 00:10:31.630 --> 00:10:35.200 align:middle line:84% And therefore, from the frequency you measure, 00:10:35.200 --> 00:10:37.270 align:middle line:84% you can work out what this term has to be. 00:10:37.270 --> 00:10:39.640 align:middle line:84% So for these peaks here, I can work out 00:10:39.640 --> 00:10:41.890 align:middle line:84% that this frequency shift is going 00:10:41.890 --> 00:10:47.620 align:middle line:84% to be proportional to square root of z te plus ti-- 00:10:47.620 --> 00:10:49.810 align:middle line:84% I don't know why I made a temperature vector there-- 00:10:49.810 --> 00:10:52.010 align:middle line:90% ti upon mi. 00:10:52.010 --> 00:10:56.740 align:middle line:90% 00:10:56.740 --> 00:10:57.730 align:middle line:90% Other questions? 00:10:57.730 --> 00:10:58.859 align:middle line:90% Yeah. 00:10:58.859 --> 00:11:02.352 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:11:02.352 --> 00:11:12.877 align:middle line:90% 00:11:12.877 --> 00:11:14.710 align:middle line:84% JACK HARE: At this point, this is like, yes, 00:11:14.710 --> 00:11:18.280 align:middle line:84% I guess this is for the k in the center of our lens here. 00:11:18.280 --> 00:11:20.290 align:middle line:84% And then one side, we're going to have 00:11:20.290 --> 00:11:22.210 align:middle line:84% k's, which are a slightly larger angle, and the other side, 00:11:22.210 --> 00:11:24.002 align:middle line:84% we'll have k's at a slightly smaller angle. 00:11:24.002 --> 00:11:25.810 align:middle line:84% And you can see from this that corresponds 00:11:25.810 --> 00:11:29.470 align:middle line:84% to slightly bigger and smaller magnitudes of k, which 00:11:29.470 --> 00:11:32.020 align:middle line:84% in turn will, for the same temperature in the plasma, 00:11:32.020 --> 00:11:34.610 align:middle line:84% correspond to slightly bigger and smaller frequencies. 00:11:34.610 --> 00:11:38.590 align:middle line:84% So this corresponds, for example, to theta 0. 00:11:38.590 --> 00:11:43.060 align:middle line:84% And on this side, this is theta 0 minus delta theta. 00:11:43.060 --> 00:11:47.060 align:middle line:84% And this is theta 0 plus delta theta, 00:11:47.060 --> 00:11:50.540 align:middle line:84% where we're talking about this being the angle theta. 00:11:50.540 --> 00:11:53.350 align:middle line:84% And then this being the angle delta 00:11:53.350 --> 00:11:56.380 align:middle line:84% theta between the sort of main angle of your thing, 00:11:56.380 --> 00:11:59.683 align:middle line:90% and the spread of angles there. 00:11:59.683 --> 00:12:02.392 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:12:02.392 --> 00:12:02.975 align:middle line:90% JACK HARE: No. 00:12:02.975 --> 00:12:04.990 align:middle line:84% It's the multiple effects that are going on. 00:12:04.990 --> 00:12:06.060 align:middle line:84% But the angle is one of them that 00:12:06.060 --> 00:12:08.160 align:middle line:84% would also stop you getting singularities as well. 00:12:08.160 --> 00:12:10.050 align:middle line:84% And you do need to think about it, 00:12:10.050 --> 00:12:13.320 align:middle line:84% because a good trick-- we talked about how few photons we have. 00:12:13.320 --> 00:12:15.810 align:middle line:84% So it makes sense to have a bigger solid angle to collect 00:12:15.810 --> 00:12:16.710 align:middle line:90% more scattered light. 00:12:16.710 --> 00:12:17.985 align:middle line:84% The trouble is that each of the photons 00:12:17.985 --> 00:12:18.960 align:middle line:90% are coming at a different angle. 00:12:18.960 --> 00:12:20.590 align:middle line:84% They actually have a different spectrum. 00:12:20.590 --> 00:12:22.090 align:middle line:84% So it'll also broaden your spectrum, 00:12:22.090 --> 00:12:24.660 align:middle line:84% which will make it hard to measure things. 00:12:24.660 --> 00:12:25.590 align:middle line:90% Other questions? 00:12:25.590 --> 00:12:30.140 align:middle line:90% 00:12:30.140 --> 00:12:32.045 align:middle line:84% Anyone online who has a question? 00:12:32.045 --> 00:12:34.940 align:middle line:90% 00:12:34.940 --> 00:12:35.915 align:middle line:90% Let's do some maths. 00:12:35.915 --> 00:12:41.620 align:middle line:90% 00:12:41.620 --> 00:12:45.740 align:middle line:90% This is a two-step process. 00:12:45.740 --> 00:12:53.630 align:middle line:84% Our first step is calculate the scattered power spectrum, 00:12:53.630 --> 00:12:58.370 align:middle line:84% which, again, we're writing as d 2 p d omega d nu. 00:12:58.370 --> 00:13:01.140 align:middle line:90% 00:13:01.140 --> 00:13:07.290 align:middle line:84% And we want to know how this relates to density fluctuations. 00:13:07.290 --> 00:13:16.125 align:middle line:84% So fluctuations in the density, delta ne of r and t. 00:13:16.125 --> 00:13:18.000 align:middle line:84% Then we're going to quickly Fourier transform 00:13:18.000 --> 00:13:20.370 align:middle line:84% these and actually be looking at them in terms 00:13:20.370 --> 00:13:23.760 align:middle line:84% of a spectrum of density fluctuations with certain k 00:13:23.760 --> 00:13:25.710 align:middle line:84% vector and omega, because everything is going 00:13:25.710 --> 00:13:27.430 align:middle line:90% to be Fourier transformed here. 00:13:27.430 --> 00:13:32.640 align:middle line:84% So we relate this to this quantity, a relatively simple 00:13:32.640 --> 00:13:33.480 align:middle line:90% formula. 00:13:33.480 --> 00:13:35.400 align:middle line:84% And then the more complicated bit 00:13:35.400 --> 00:13:42.970 align:middle line:84% is calculate what these density fluctuations are, delta-- 00:13:42.970 --> 00:13:45.260 align:middle line:84% well, we're just going to go call them 00:13:45.260 --> 00:13:48.320 align:middle line:90% any k and omega in a plasma. 00:13:48.320 --> 00:13:52.657 align:middle line:90% 00:13:52.657 --> 00:13:54.240 align:middle line:84% So this goes back to my previous point 00:13:54.240 --> 00:13:56.770 align:middle line:84% that there are always these fluctuations in a plasma. 00:13:56.770 --> 00:13:59.653 align:middle line:84% So we just have to work out the size of these fluctuations. 00:13:59.653 --> 00:14:02.070 align:middle line:84% And we're going to do that using a test particle treatment 00:14:02.070 --> 00:14:05.170 align:middle line:84% and a little bit of Landau damping along the way. 00:14:05.170 --> 00:14:09.030 align:middle line:84% And one thing I want to say is that this treatment here 00:14:09.030 --> 00:14:18.240 align:middle line:84% is going to reproduce the incoherent case as a limit. 00:14:18.240 --> 00:14:21.350 align:middle line:84% So our treatment now-- sort of derivation 00:14:21.350 --> 00:14:23.390 align:middle line:84% includes what we've done previously. 00:14:23.390 --> 00:14:25.610 align:middle line:90% It just includes it as a limit. 00:14:25.610 --> 00:14:28.880 align:middle line:84% And I'll try and point out where in our derivation 00:14:28.880 --> 00:14:30.707 align:middle line:84% we've exceeded what we did previously. 00:14:30.707 --> 00:14:33.290 align:middle line:84% But you will be able to get back all of the Thomson scattering 00:14:33.290 --> 00:14:35.210 align:middle line:84% spectrum, not just the collective part, 00:14:35.210 --> 00:14:38.880 align:middle line:84% from the equations at the end here. 00:14:38.880 --> 00:14:44.120 align:middle line:84% So we want to recall a couple of things. 00:14:44.120 --> 00:14:50.550 align:middle line:84% We had our Fourier transformed scattered electric field. 00:14:50.550 --> 00:14:52.670 align:middle line:84% So it's in terms of frequency, which 00:14:52.670 --> 00:14:54.440 align:middle line:84% was equal to the classical electron 00:14:54.440 --> 00:14:58.160 align:middle line:84% radius over the distance to the scattering volume. 00:14:58.160 --> 00:15:01.460 align:middle line:84% And we were doing a Fourier transform on this. 00:15:01.460 --> 00:15:06.350 align:middle line:84% And we had this pi dot Ei0, which 00:15:06.350 --> 00:15:10.280 align:middle line:84% was just the thing that gave us the shape of our scattering. 00:15:10.280 --> 00:15:12.560 align:middle line:84% So it's got some angular dependence. 00:15:12.560 --> 00:15:17.150 align:middle line:84% Actually, I can just write this as Ei in terms of r and t prime. 00:15:17.150 --> 00:15:20.570 align:middle line:84% And we were Fourier transforming it. 00:15:20.570 --> 00:15:33.830 align:middle line:84% So we had an exponential of minus i omega s and e. 00:15:33.830 --> 00:15:35.990 align:middle line:84% And we replaced our t with the t prime, 00:15:35.990 --> 00:15:37.860 align:middle line:84% because that's what the electric field was. 00:15:37.860 --> 00:15:42.590 align:middle line:84% And when we replaced this t with a t prime, this becomes t prime. 00:15:42.590 --> 00:15:49.250 align:middle line:84% And we get a second term that was minus ks dot r at here. 00:15:49.250 --> 00:15:55.120 align:middle line:84% And now our integration is respect to d prime instead of t. 00:15:55.120 --> 00:15:57.880 align:middle line:84% So this was from a single particle 00:15:57.880 --> 00:16:03.360 align:middle line:84% what our scattered electric field was going to be. 00:16:03.360 --> 00:16:06.480 align:middle line:84% So now we want to know the scattering from a distribution 00:16:06.480 --> 00:16:07.530 align:middle line:90% of particles. 00:16:07.530 --> 00:16:09.960 align:middle line:84% And we're going to take as our distribution function 00:16:09.960 --> 00:16:11.620 align:middle line:90% the Klimontovich distribution. 00:16:11.620 --> 00:16:13.710 align:middle line:84% So I'm going to use a capital F here. 00:16:13.710 --> 00:16:16.900 align:middle line:84% And this is a distribution function 00:16:16.900 --> 00:16:19.270 align:middle line:90% in space and velocity and time. 00:16:19.270 --> 00:16:22.090 align:middle line:84% And in some sense, it's a very dumb distribution function, 00:16:22.090 --> 00:16:25.300 align:middle line:84% because we just say it's the sum over the position 00:16:25.300 --> 00:16:28.270 align:middle line:84% of every particle, which we write as a delta function, 00:16:28.270 --> 00:16:33.850 align:middle line:84% r minus r sub j of t, and another delta function, which is 00:16:33.850 --> 00:16:35.560 align:middle line:90% the velocity of every particle. 00:16:35.560 --> 00:16:39.140 align:middle line:90% 00:16:39.140 --> 00:16:41.810 align:middle line:84% So sort of like an obviously true statement, 00:16:41.810 --> 00:16:43.910 align:middle line:84% but the Klimontovich distribution is not usually 00:16:43.910 --> 00:16:45.200 align:middle line:90% very useful to work with. 00:16:45.200 --> 00:16:47.947 align:middle line:84% But we'll get back to why we do it later on. 00:16:47.947 --> 00:16:50.030 align:middle line:84% You may have seen this in particle kinetic theory. 00:16:50.030 --> 00:16:57.050 align:middle line:90% 00:16:57.050 --> 00:16:59.180 align:middle line:84% And we're going to say what we want now is 00:16:59.180 --> 00:17:00.530 align:middle line:90% the scattered electric field. 00:17:00.530 --> 00:17:02.030 align:middle line:84% We want the scattered electric field 00:17:02.030 --> 00:17:05.579 align:middle line:84% from every particle in this Klimontovich distribution. 00:17:05.579 --> 00:17:08.954 align:middle line:84% So we're going to call this scattered electric field Es 00:17:08.954 --> 00:17:09.454 align:middle line:90% total. 00:17:09.454 --> 00:17:12.970 align:middle line:90% 00:17:12.970 --> 00:17:19.160 align:middle line:84% And we're going to do an integral over all of the plasma, 00:17:19.160 --> 00:17:23.390 align:middle line:84% all of the particle velocities, our Klimontovich distribution, 00:17:23.390 --> 00:17:28.980 align:middle line:84% and our electric field from a single particle here. 00:17:28.980 --> 00:17:30.926 align:middle line:90% So this is the total field. 00:17:30.926 --> 00:17:37.250 align:middle line:90% 00:17:37.250 --> 00:17:39.800 align:middle line:84% The key difference in what we did in the incoherent case, 00:17:39.800 --> 00:17:42.680 align:middle line:84% I believe, is that now in the incoherent case, 00:17:42.680 --> 00:17:47.510 align:middle line:84% we kind of handwaved our way over this integral over space. 00:17:47.510 --> 00:17:50.950 align:middle line:84% And I think we effectively assume that all particles are 00:17:50.950 --> 00:17:51.985 align:middle line:90% in the incoherent case. 00:17:51.985 --> 00:17:56.370 align:middle line:90% 00:17:56.370 --> 00:18:01.450 align:middle line:84% All the particles j were at r equals 0. 00:18:01.450 --> 00:18:05.290 align:middle line:84% And so now we're going to do the coherent case, 00:18:05.290 --> 00:18:07.457 align:middle line:84% and we're not going to make that assumption. 00:18:07.457 --> 00:18:10.380 align:middle line:90% 00:18:10.380 --> 00:18:13.550 align:middle line:90% So r equals rj. 00:18:13.550 --> 00:18:15.600 align:middle line:84% That's what the delta function up here does. 00:18:15.600 --> 00:18:19.100 align:middle line:84% And that means that if we have our spectrometer, our observer 00:18:19.100 --> 00:18:21.770 align:middle line:84% over here, and our origin here, and we 00:18:21.770 --> 00:18:28.240 align:middle line:84% have two different particles at positions R1 and R2, 00:18:28.240 --> 00:18:31.930 align:middle line:84% the electric fields coming from these, 00:18:31.930 --> 00:18:36.940 align:middle line:84% because they're scattering waves in this direction, 00:18:36.940 --> 00:18:38.330 align:middle line:90% they may be a phase difference. 00:18:38.330 --> 00:18:41.640 align:middle line:84% And so we may get interference from them. 00:18:41.640 --> 00:18:45.210 align:middle line:84% Previously, because we assumed that all the particles were 00:18:45.210 --> 00:18:47.580 align:middle line:84% in the same place, we couldn't get a path difference. 00:18:47.580 --> 00:18:49.440 align:middle line:84% We couldn't get any interference. 00:18:49.440 --> 00:18:51.630 align:middle line:84% And so we just summed up their contribution. 00:18:51.630 --> 00:18:53.460 align:middle line:84% Now, when we do this integral, we're 00:18:53.460 --> 00:18:56.280 align:middle line:84% going to very carefully take into account 00:18:56.280 --> 00:18:59.200 align:middle line:84% the difference in position of each of the particles inside Fe. 00:18:59.200 --> 00:19:01.632 align:middle line:84% And so when all the waves gather up together, 00:19:01.632 --> 00:19:03.090 align:middle line:84% we're going to be interested in how 00:19:03.090 --> 00:19:04.382 align:middle line:90% they interfere with each other. 00:19:04.382 --> 00:19:06.718 align:middle line:84% So this is the key difference in this derivation. 00:19:06.718 --> 00:19:10.070 align:middle line:90% 00:19:10.070 --> 00:19:11.880 align:middle line:84% A couple of things to note, of course, 00:19:11.880 --> 00:19:15.500 align:middle line:84% is that our electron density, ne is simply 00:19:15.500 --> 00:19:19.700 align:middle line:84% the integral of our distribution function over all velocities. 00:19:19.700 --> 00:19:20.990 align:middle line:90% That's still true. 00:19:20.990 --> 00:19:26.910 align:middle line:84% Just to be clear, this is ne as a position of space and time. 00:19:26.910 --> 00:19:28.790 align:middle line:84% So we've lost the velocity component. 00:19:28.790 --> 00:19:29.960 align:middle line:90% We've integrated over it. 00:19:29.960 --> 00:19:33.780 align:middle line:90% 00:19:33.780 --> 00:19:41.190 align:middle line:84% And so-- do we have space to write this? 00:19:41.190 --> 00:19:41.690 align:middle line:90% No. 00:19:41.690 --> 00:19:46.520 align:middle line:90% 00:19:46.520 --> 00:19:49.370 align:middle line:84% This total scattered electric field, where has it gone? 00:19:49.370 --> 00:19:50.300 align:middle line:90% That one up there. 00:19:50.300 --> 00:19:58.640 align:middle line:90% 00:19:58.640 --> 00:20:02.450 align:middle line:84% It's going to be equal to re on R. 00:20:02.450 --> 00:20:05.690 align:middle line:84% There's going to be this boring phase factor out front, 00:20:05.690 --> 00:20:09.050 align:middle line:84% iks dot R, which will vanish very soon. 00:20:09.050 --> 00:20:13.275 align:middle line:84% This is the phase that all of the waves pick up by virtue-- 00:20:13.275 --> 00:20:14.900 align:middle line:84% this is kind of like the average phase. 00:20:14.900 --> 00:20:17.660 align:middle line:84% It's the phase that they pick up from going from this point 00:20:17.660 --> 00:20:18.585 align:middle line:90% to here. 00:20:18.585 --> 00:20:20.210 align:middle line:84% So it's going to be a very boring phase 00:20:20.210 --> 00:20:21.085 align:middle line:90% that will cancel out. 00:20:21.085 --> 00:20:23.360 align:middle line:84% But we're still interested in the additional phase 00:20:23.360 --> 00:20:26.000 align:middle line:84% from the exact locations of each particle. 00:20:26.000 --> 00:20:29.530 align:middle line:84% There's going to be a factor of this pi dotted 00:20:29.530 --> 00:20:32.720 align:middle line:90% into the polarization. 00:20:32.720 --> 00:20:34.280 align:middle line:84% Again, it's just a shape function. 00:20:34.280 --> 00:20:37.580 align:middle line:84% But now, we're going to be an integral over pi 00:20:37.580 --> 00:20:44.810 align:middle line:84% and an integral over volume of the density of R and T prime. 00:20:44.810 --> 00:20:46.670 align:middle line:84% So I've carried out the velocity integral 00:20:46.670 --> 00:20:51.630 align:middle line:84% and replaced this with the density. 00:20:51.630 --> 00:20:56.850 align:middle line:84% And now I have the factor from the electric field. 00:20:56.850 --> 00:20:59.350 align:middle line:84% I didn't write it down again, did I? 00:20:59.350 --> 00:20:59.850 align:middle line:90% OK. 00:20:59.850 --> 00:21:03.310 align:middle line:90% I shall put it here. 00:21:03.310 --> 00:21:14.080 align:middle line:84% Our incident electric field is equal to Ei 0. 00:21:14.080 --> 00:21:15.302 align:middle line:90% I'm going to do it this way. 00:21:15.302 --> 00:21:26.910 align:middle line:84% Cosine exponential of i ki dot r minus omega i t prime. 00:21:26.910 --> 00:21:28.470 align:middle line:84% This is the incident electric field. 00:21:28.470 --> 00:21:35.300 align:middle line:90% 00:21:35.300 --> 00:21:38.540 align:middle line:84% So if I now substitute in the electric field, 00:21:38.540 --> 00:21:40.920 align:middle line:84% I'm going to get that exponential term here. 00:21:40.920 --> 00:21:47.430 align:middle line:84% But that's going to be exponential of i ai dot 00:21:47.430 --> 00:21:52.110 align:middle line:90% r minus omega t prime. 00:21:52.110 --> 00:21:54.810 align:middle line:84% And then I'm also going to have this exponential here. 00:21:54.810 --> 00:21:56.910 align:middle line:84% I'm just going to write directly below, 00:21:56.910 --> 00:22:08.323 align:middle line:84% exponential of minus i omega s t prime minus ks dot r. 00:22:08.323 --> 00:22:14.590 align:middle line:90% 00:22:14.590 --> 00:22:18.860 align:middle line:84% So far this looks a great deal like what we did before. 00:22:18.860 --> 00:22:23.170 align:middle line:84% But if you want to evaluate this electric field carefully, 00:22:23.170 --> 00:22:25.960 align:middle line:84% and then work out the scattered power very carefully, 00:22:25.960 --> 00:22:29.050 align:middle line:84% you have to pay an awful lot of attention to exactly-- 00:22:29.050 --> 00:22:32.440 align:middle line:84% sorry, I've forgotten to put in the limits here of this. 00:22:32.440 --> 00:22:34.990 align:middle line:90% d t prime d3 r. 00:22:34.990 --> 00:22:38.353 align:middle line:84% So integrating over time and space. 00:22:38.353 --> 00:22:40.520 align:middle line:84% To do this properly so that we can take into account 00:22:40.520 --> 00:22:44.240 align:middle line:84% the interference requires you to very carefully go through this 00:22:44.240 --> 00:22:46.940 align:middle line:84% and pick out some important terms. 00:22:46.940 --> 00:22:49.760 align:middle line:84% And I had it in my notes from last year. 00:22:49.760 --> 00:22:53.240 align:middle line:84% And it was not a very educational exercise 00:22:53.240 --> 00:22:54.450 align:middle line:90% to do on the board. 00:22:54.450 --> 00:22:57.830 align:middle line:84% So what I'm going to say is if you are interested in seeing 00:22:57.830 --> 00:23:01.850 align:middle line:84% the mathematics behind this-- this is Hutchinson's 7.3.8 00:23:01.850 --> 00:23:05.420 align:middle line:90% to 7.3.12. 00:23:05.420 --> 00:23:08.640 align:middle line:84% And I'm simply going to quote the result now instead, 00:23:08.640 --> 00:23:12.640 align:middle line:84% which is that the total scattered power 00:23:12.640 --> 00:23:17.480 align:middle line:84% into some solid angle, into some frequency here 00:23:17.480 --> 00:23:22.460 align:middle line:84% is equal to the classical electron radius squared, 00:23:22.460 --> 00:23:27.440 align:middle line:84% the power of the laser over the cross-sectional area 00:23:27.440 --> 00:23:32.276 align:middle line:84% of the laser, this shape function pi 00:23:32.276 --> 00:23:42.510 align:middle line:84% dot i hat squared times ne times the volume 00:23:42.510 --> 00:23:46.110 align:middle line:84% that we're scattering from trying to find all of this times 00:23:46.110 --> 00:23:52.700 align:middle line:90% something called s of k omega. 00:23:52.700 --> 00:23:56.760 align:middle line:84% This is the important quantity called the spectral density 00:23:56.760 --> 00:23:57.260 align:middle line:90% function. 00:23:57.260 --> 00:24:06.998 align:middle line:90% 00:24:06.998 --> 00:24:09.040 align:middle line:84% And it's important because all of the information 00:24:09.040 --> 00:24:11.830 align:middle line:84% about this spectrum is held inside there. 00:24:11.830 --> 00:24:14.252 align:middle line:84% This is where the k and omega dependence are that tell us 00:24:14.252 --> 00:24:15.460 align:middle line:90% what the spectrum looks like. 00:24:15.460 --> 00:24:17.020 align:middle line:84% These are all just scaling constants 00:24:17.020 --> 00:24:18.580 align:middle line:90% that tell us the intensity. 00:24:18.580 --> 00:24:20.830 align:middle line:84% And often, because we don't have absolutely calibrated 00:24:20.830 --> 00:24:23.390 align:middle line:84% spectrometers, we can't really measure this. 00:24:23.390 --> 00:24:24.430 align:middle line:90% So we don't really care. 00:24:24.430 --> 00:24:27.040 align:middle line:84% But we can measure the spectrum, the relative intensity. 00:24:27.040 --> 00:24:29.170 align:middle line:84% And that's all hidden within this function. 00:24:29.170 --> 00:24:35.070 align:middle line:84% And this function is equal to 1 upon the time 00:24:35.070 --> 00:24:38.290 align:middle line:84% of our integration, the volume of our integration, 00:24:38.290 --> 00:24:42.420 align:middle line:84% and then the average of the Fourier 00:24:42.420 --> 00:24:49.510 align:middle line:84% transformed electron density squared 00:24:49.510 --> 00:24:54.010 align:middle line:84% over the average electron density. 00:24:54.010 --> 00:24:56.640 align:middle line:84% So these are the fluctuating term, 00:24:56.640 --> 00:25:00.557 align:middle line:84% and this is the sort of steady state ne 0 electron. 00:25:00.557 --> 00:25:02.390 align:middle line:84% I should really make that square very clear. 00:25:02.390 --> 00:25:03.605 align:middle line:90% It's very important. 00:25:03.605 --> 00:25:11.160 align:middle line:90% 00:25:11.160 --> 00:25:13.710 align:middle line:90% Question? 00:25:13.710 --> 00:25:15.420 align:middle line:90% Yeah. 00:25:15.420 --> 00:25:24.375 align:middle line:84% AUDIENCE: So why are we the r instead of [INAUDIBLE]? 00:25:24.375 --> 00:25:27.630 align:middle line:90% 00:25:27.630 --> 00:25:28.452 align:middle line:90% JACK HARE: Yeah. 00:25:28.452 --> 00:25:30.660 align:middle line:84% This is a generic Klimontovich distribution function. 00:25:30.660 --> 00:25:32.620 align:middle line:90% We'll be using it. 00:25:32.620 --> 00:25:34.450 align:middle line:90% Yeah, you're right. 00:25:34.450 --> 00:25:37.170 align:middle line:84% When it goes into here, I did put t prime 00:25:37.170 --> 00:25:38.943 align:middle line:90% in a Klimontovich distribution. 00:25:38.943 --> 00:25:39.610 align:middle line:90% So you're right. 00:25:39.610 --> 00:25:41.752 align:middle line:84% That's just the Klimontovich distribution. 00:25:41.752 --> 00:25:42.460 align:middle line:90% But you're right. 00:25:42.460 --> 00:25:45.570 align:middle line:84% We're evaluating a Klimontovich distribution at time t prime. 00:25:45.570 --> 00:25:48.310 align:middle line:90% So this should be t prime here. 00:25:48.310 --> 00:25:53.740 align:middle line:84% And this should be function of r d and t prime here. 00:25:53.740 --> 00:25:56.380 align:middle line:84% I haven't generally been using primes on the r's. 00:25:56.380 --> 00:25:58.030 align:middle line:84% Though, Hutchinson does to remind us 00:25:58.030 --> 00:26:00.970 align:middle line:84% that it's r evaluated at t prime. 00:26:00.970 --> 00:26:05.540 align:middle line:84% So he writes r prime is equal to r at t prime. 00:26:05.540 --> 00:26:08.720 align:middle line:84% I tend to drop the prime because I remember. 00:26:08.720 --> 00:26:12.110 align:middle line:90% But maybe I shouldn't drop it. 00:26:12.110 --> 00:26:14.070 align:middle line:84% Other people don't use that notation. 00:26:14.070 --> 00:26:16.940 align:middle line:84% So just if you note, if you go into Hutchinson. 00:26:16.940 --> 00:26:21.500 align:middle line:84% In the syllabus, I recommended a book by Froula and Sheffield 00:26:21.500 --> 00:26:24.110 align:middle line:84% called Plasma Scattering of Electromagnetic Radiation, which 00:26:24.110 --> 00:26:26.690 align:middle line:90% is very good and very thorough. 00:26:26.690 --> 00:26:29.120 align:middle line:84% And I've kind of blended their treatment with Hutchinson's 00:26:29.120 --> 00:26:30.590 align:middle line:90% treatments to get some of this. 00:26:30.590 --> 00:26:33.140 align:middle line:84% So not all of this looks exactly like how Hutchinson does it, 00:26:33.140 --> 00:26:35.930 align:middle line:84% but I do think that Hutchinson's treatment is pretty 00:26:35.930 --> 00:26:39.310 align:middle line:90% good for most of this stuff. 00:26:39.310 --> 00:26:40.538 align:middle line:90% Yeah. 00:26:40.538 --> 00:26:45.980 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:26:45.980 --> 00:26:46.820 align:middle line:90% JACK HARE: This one? 00:26:46.820 --> 00:26:50.400 align:middle line:90% 00:26:50.400 --> 00:26:52.960 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:26:52.960 --> 00:26:56.080 align:middle line:84% JACK HARE: So it comes in as part of this term. 00:26:56.080 --> 00:26:57.120 align:middle line:90% So let's be clear. 00:26:57.120 --> 00:27:02.970 align:middle line:84% So this, it actually should be back here as well. 00:27:02.970 --> 00:27:06.170 align:middle line:90% I have missed it out. 00:27:06.170 --> 00:27:12.370 align:middle line:84% So there's a times e to the i as dot r factor in there as well. 00:27:12.370 --> 00:27:14.620 align:middle line:84% And where it comes from is originally 00:27:14.620 --> 00:27:22.630 align:middle line:84% this is exponential of minus i omega s t dt. 00:27:22.630 --> 00:27:27.280 align:middle line:84% And when we substitute out t equals 00:27:27.280 --> 00:27:30.940 align:middle line:84% t prime with a load of other bits, one of the other bits 00:27:30.940 --> 00:27:34.900 align:middle line:84% becomes ks dot r, and the other bit comes ks dot capital R. 00:27:34.900 --> 00:27:36.340 align:middle line:90% And we can take that outside. 00:27:36.340 --> 00:27:38.060 align:middle line:84% But yeah, I forgot to write it down here. 00:27:38.060 --> 00:27:39.560 align:middle line:84% It should have been in here as well. 00:27:39.560 --> 00:27:42.430 align:middle line:84% And what that represents, that relatively boring term, 00:27:42.430 --> 00:27:46.250 align:middle line:84% is just the fact that all of the electromagnetic fields from all 00:27:46.250 --> 00:27:50.990 align:middle line:84% of the plasma have to travel some distance r to our detector. 00:27:50.990 --> 00:27:53.600 align:middle line:90% But some of them travel r. 00:27:53.600 --> 00:27:58.220 align:middle line:84% So if this is the r vector here, some of them 00:27:58.220 --> 00:28:03.320 align:middle line:84% travel r plus delta R or minus delta R, 00:28:03.320 --> 00:28:04.968 align:middle line:90% or whatever else like that. 00:28:04.968 --> 00:28:07.010 align:middle line:84% So we're kind of accounting for the overall phase 00:28:07.010 --> 00:28:08.630 align:middle line:84% that they all pick up as the waves travel. 00:28:08.630 --> 00:28:10.047 align:middle line:84% And then we're trying to calculate 00:28:10.047 --> 00:28:11.600 align:middle line:90% these very small differences. 00:28:11.600 --> 00:28:13.683 align:middle line:84% And those are the ones which interfere and give us 00:28:13.683 --> 00:28:15.105 align:middle line:84% the interesting coherent effects. 00:28:15.105 --> 00:28:17.923 align:middle line:90% 00:28:17.923 --> 00:28:18.590 align:middle line:90% Other questions? 00:28:18.590 --> 00:28:19.585 align:middle line:90% Yes. 00:28:19.585 --> 00:28:28.810 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:28:28.810 --> 00:28:31.630 align:middle line:84% JACK HARE: That's the relativistic treatment, 00:28:31.630 --> 00:28:35.050 align:middle line:84% the relativistic treatment where we keep terms in b upon c. 00:28:35.050 --> 00:28:36.050 align:middle line:90% And we've dropped those. 00:28:36.050 --> 00:28:40.040 align:middle line:84% So this is the non-relativistic treatment. 00:28:40.040 --> 00:28:40.700 align:middle line:90% Good point. 00:28:40.700 --> 00:28:45.610 align:middle line:90% 00:28:45.610 --> 00:28:46.808 align:middle line:90% Any other questions? 00:28:46.808 --> 00:28:51.410 align:middle line:90% 00:28:51.410 --> 00:28:55.380 align:middle line:90% So we've now done step one. 00:28:55.380 --> 00:28:58.260 align:middle line:84% We have calculated, to some extent, 00:28:58.260 --> 00:29:01.030 align:middle line:84% the power spectrum in terms of density fluctuations. 00:29:01.030 --> 00:29:02.670 align:middle line:84% So now the question is, step two, 00:29:02.670 --> 00:29:04.830 align:middle line:84% what are these density fluctuations? 00:29:04.830 --> 00:29:14.808 align:middle line:90% 00:29:14.808 --> 00:29:16.350 align:middle line:84% What we're going to do is we're going 00:29:16.350 --> 00:29:20.220 align:middle line:84% to consider the response of the plasma to a test particle. 00:29:20.220 --> 00:29:22.050 align:middle line:84% So we imagine we have some plasma. 00:29:22.050 --> 00:29:27.020 align:middle line:84% And we put some test particle through it. 00:29:27.020 --> 00:29:29.720 align:middle line:84% And then we asked ourselves, what does the rest of the plasma 00:29:29.720 --> 00:29:32.468 align:middle line:84% do in response to this test particle? 00:29:32.468 --> 00:29:34.010 align:middle line:84% The test particle, in this case, what 00:29:34.010 --> 00:29:35.677 align:middle line:84% we're going to do is we're going to pick 00:29:35.677 --> 00:29:39.440 align:middle line:84% each electron in the plasma and each ion in the plasma, 00:29:39.440 --> 00:29:40.940 align:middle line:84% work out what the rest of the plasma 00:29:40.940 --> 00:29:43.670 align:middle line:84% does in response to each electron and each ion moving. 00:29:43.670 --> 00:29:47.250 align:middle line:84% And then we'll sum all of those responses together. 00:29:47.250 --> 00:29:52.350 align:middle line:84% So at the moment, this could be an electron or an ion. 00:29:52.350 --> 00:29:54.960 align:middle line:84% Remember, we're not interested in the scattering of the ions. 00:29:54.960 --> 00:29:58.933 align:middle line:84% But if the ion motion perturbs the cloud of electrons, 00:29:58.933 --> 00:30:01.100 align:middle line:84% and then we can scatter off that cloud of electrons, 00:30:01.100 --> 00:30:03.450 align:middle line:90% we will see that light here. 00:30:03.450 --> 00:30:05.870 align:middle line:84% So this cloud here is a sort of Debye cloud here. 00:30:05.870 --> 00:30:10.560 align:middle line:84% These are the particles which are shielding our test particle. 00:30:10.560 --> 00:30:21.240 align:middle line:90% 00:30:21.240 --> 00:30:27.370 align:middle line:84% If our test particle is an electron, 00:30:27.370 --> 00:30:34.790 align:middle line:84% then the shielding particles are an absence of electrons. 00:30:34.790 --> 00:30:41.230 align:middle line:90% 00:30:41.230 --> 00:30:43.120 align:middle line:84% The ions can't move out the way fast enough 00:30:43.120 --> 00:30:46.120 align:middle line:84% as an electron goes through, but the electrons 00:30:46.120 --> 00:30:48.558 align:middle line:90% will move out of the way. 00:30:48.558 --> 00:30:50.850 align:middle line:84% And you can imagine that as this particle goes through, 00:30:50.850 --> 00:30:52.230 align:middle line:84% and the electrons move out of the way, 00:30:52.230 --> 00:30:54.150 align:middle line:84% it will cause a perturbation to the density. 00:30:54.150 --> 00:30:55.618 align:middle line:84% It'll be a negative perturbation. 00:30:55.618 --> 00:30:56.910 align:middle line:90% So it'll actually scatter less. 00:30:56.910 --> 00:30:58.170 align:middle line:84% But that means that these electrons 00:30:58.170 --> 00:30:59.250 align:middle line:90% have to end up somewhere else. 00:30:59.250 --> 00:31:00.420 align:middle line:90% The density will be higher. 00:31:00.420 --> 00:31:02.010 align:middle line:90% They'll be scattering off those. 00:31:02.010 --> 00:31:08.630 align:middle line:84% If the test particle is an ion, then it 00:31:08.630 --> 00:31:13.650 align:middle line:84% turns out that half of the particles, or half 00:31:13.650 --> 00:31:16.896 align:middle line:84% of the shielding that's made up that shields this ion charge 00:31:16.896 --> 00:31:22.930 align:middle line:84% are due to electrons being attracted. 00:31:22.930 --> 00:31:26.510 align:middle line:90% 00:31:26.510 --> 00:31:34.130 align:middle line:84% And the other half of the shield is by ions being propelled. 00:31:34.130 --> 00:31:36.050 align:middle line:84% So it's slightly different from the argument 00:31:36.050 --> 00:31:39.020 align:middle line:84% for the electrons, where only electrons can shield electrons 00:31:39.020 --> 00:31:39.770 align:middle line:90% from ions. 00:31:39.770 --> 00:31:42.020 align:middle line:84% Some of the shielding comes from drawing electrons in. 00:31:42.020 --> 00:31:44.450 align:middle line:84% The rest of the shielding comes from pushing ions out. 00:31:44.450 --> 00:31:48.450 align:middle line:84% And that can happen because the ions can move as fast as an ion. 00:31:48.450 --> 00:31:51.640 align:middle line:84% And so we can see that shielding here. 00:31:51.640 --> 00:31:56.650 align:middle line:84% So what we get is a perturbed charge density, rho 0, 00:31:56.650 --> 00:32:03.540 align:middle line:84% at some position r and t, due to our test particle, which 00:32:03.540 --> 00:32:09.900 align:middle line:90% is q delta r minus vt. 00:32:09.900 --> 00:32:15.120 align:middle line:84% We're just saying our test particle perturbed density looks 00:32:15.120 --> 00:32:16.080 align:middle line:90% like this. 00:32:16.080 --> 00:32:18.240 align:middle line:84% As it moves through, it's at some position r, 00:32:18.240 --> 00:32:21.160 align:middle line:84% moving at some velocity v in a straight line. 00:32:21.160 --> 00:32:27.580 align:middle line:84% And that means that its Fourier transformed contribution 00:32:27.580 --> 00:32:32.710 align:middle line:84% to the charge density is equal to q2 pi 00:32:32.710 --> 00:32:38.445 align:middle line:90% delta a dot b minus omega. 00:32:38.445 --> 00:32:39.820 align:middle line:84% So this is the Fourier transform. 00:32:39.820 --> 00:32:43.520 align:middle line:90% 00:32:43.520 --> 00:32:48.740 align:middle line:84% Now, we're going to take some equations from electromagnetism 00:32:48.740 --> 00:32:51.020 align:middle line:84% and talk about this in terms of a polarization 00:32:51.020 --> 00:32:53.020 align:middle line:84% and a polarization charge density. 00:32:53.020 --> 00:32:54.920 align:middle line:84% So it'll be free and bound charges. 00:32:54.920 --> 00:32:58.880 align:middle line:84% So we have a polarization vector, 00:32:58.880 --> 00:33:02.510 align:middle line:84% which is equal to the susceptibility 00:33:02.510 --> 00:33:08.070 align:middle line:84% times the permittivity of free space times the electric field. 00:33:08.070 --> 00:33:12.660 align:middle line:84% And this is also equal to pi upon epsilon, 00:33:12.660 --> 00:33:15.980 align:middle line:84% which is just the permittivity, and the d field. 00:33:15.980 --> 00:33:17.900 align:middle line:84% And the e and d fields are defined 00:33:17.900 --> 00:33:27.470 align:middle line:84% as divergence of the electric field is equal to rho 0. 00:33:27.470 --> 00:33:34.970 align:middle line:84% And divergence of the field is equal to rho e. 00:33:34.970 --> 00:33:38.180 align:middle line:90% So this is the free charge. 00:33:38.180 --> 00:33:41.787 align:middle line:90% 00:33:41.787 --> 00:33:43.120 align:middle line:90% And these are the bound charges. 00:33:43.120 --> 00:33:45.903 align:middle line:90% 00:33:45.903 --> 00:33:47.320 align:middle line:84% And in our case, our bound charges 00:33:47.320 --> 00:33:48.362 align:middle line:90% are our shielded charges. 00:33:48.362 --> 00:33:51.490 align:middle line:90% 00:33:51.490 --> 00:33:53.230 align:middle line:84% I ran out of space to write shielding. 00:33:53.230 --> 00:33:54.412 align:middle line:90% There we go. 00:33:54.412 --> 00:33:55.120 align:middle line:90% You get the idea. 00:33:55.120 --> 00:33:57.640 align:middle line:90% 00:33:57.640 --> 00:34:01.200 align:middle line:84% And we can solve these equations to work out 00:34:01.200 --> 00:34:03.600 align:middle line:84% what the size of our shielding charge is. 00:34:03.600 --> 00:34:06.420 align:middle line:84% And our shielding charge, rho e, is 00:34:06.420 --> 00:34:14.909 align:middle line:84% equal to minus chi e upon epsilon times by this Fourier 00:34:14.909 --> 00:34:24.989 align:middle line:84% transformed test particle charge q2 pi delta a dot b minus omega. 00:34:24.989 --> 00:34:28.380 align:middle line:84% What this is effectively saying is as this test charge moves 00:34:28.380 --> 00:34:31.739 align:middle line:84% through or as it exists as a mode in Fourier space, 00:34:31.739 --> 00:34:34.139 align:middle line:84% it induces some bound charge to shield it. 00:34:34.139 --> 00:34:36.280 align:middle line:84% And that bound charge has a charge density, 00:34:36.280 --> 00:34:38.940 align:middle line:84% which is to do with the susceptibility 00:34:38.940 --> 00:34:44.060 align:middle line:90% of the permittivity here. 00:34:44.060 --> 00:34:50.179 align:middle line:90% Then we can-- this board-- 00:34:50.179 --> 00:35:25.410 align:middle line:90% 00:35:25.410 --> 00:35:32.310 align:middle line:84% say that our electron density, ne of k and omega, 00:35:32.310 --> 00:35:36.210 align:middle line:84% is going to be equal to a density, which 00:35:36.210 --> 00:35:39.330 align:middle line:84% is due to the electrons, and the density, which 00:35:39.330 --> 00:35:42.310 align:middle line:84% is due to the ions, which have been perturbed in this system. 00:35:42.310 --> 00:35:50.110 align:middle line:84% So for the electrons, we sum over j electrons. 00:35:50.110 --> 00:35:51.870 align:middle line:90% And we have a 1. 00:35:51.870 --> 00:35:53.493 align:middle line:84% This is the test particle charge. 00:35:53.493 --> 00:35:55.410 align:middle line:84% So whenever we have an electron test particle, 00:35:55.410 --> 00:35:59.420 align:middle line:84% it's going to contribute 1 to the electron density here. 00:35:59.420 --> 00:36:02.120 align:middle line:84% And we're going to subtract off it the number of electrons 00:36:02.120 --> 00:36:05.450 align:middle line:84% which were repelled by that test particle, pi upon epsilon. 00:36:05.450 --> 00:36:12.330 align:middle line:84% And both of these are 2 pi delta k dot bj 00:36:12.330 --> 00:36:16.320 align:middle line:90% for the electrons minus omega. 00:36:16.320 --> 00:36:20.080 align:middle line:90% So this is the electrons. 00:36:20.080 --> 00:36:23.510 align:middle line:84% This is our test particle electron. 00:36:23.510 --> 00:36:26.510 align:middle line:84% And these are our shielding electrons. 00:36:26.510 --> 00:36:30.970 align:middle line:90% 00:36:30.970 --> 00:36:33.490 align:middle line:84% And then we also have a term that's for the ions. 00:36:33.490 --> 00:36:38.420 align:middle line:90% We sum over l ions. 00:36:38.420 --> 00:36:44.990 align:middle line:84% And here, we simply have a term pi e upon epsilon here. 00:36:44.990 --> 00:36:53.390 align:middle line:84% The ions have a charge of z 2 pi delta k dot velocity 00:36:53.390 --> 00:36:56.340 align:middle line:90% of the ions, vl, minus omega. 00:36:56.340 --> 00:36:58.275 align:middle line:84% This is the ion contribution here. 00:36:58.275 --> 00:37:01.600 align:middle line:90% 00:37:01.600 --> 00:37:04.600 align:middle line:84% Just to be very clear, this is the contribution 00:37:04.600 --> 00:37:06.340 align:middle line:84% of the electrons shielding the ions. 00:37:06.340 --> 00:37:07.840 align:middle line:84% We're only talking about the electron density 00:37:07.840 --> 00:37:10.190 align:middle line:84% here because we're only interested in electron density. 00:37:10.190 --> 00:37:13.480 align:middle line:84% So this is the contribution of the electron test particle 00:37:13.480 --> 00:37:15.490 align:middle line:84% and the electron shielding the electron. 00:37:15.490 --> 00:37:18.070 align:middle line:84% And this is the contribution of the ion test particle. 00:37:18.070 --> 00:37:21.460 align:middle line:84% So we don't get a 1, because we don't add. 00:37:21.460 --> 00:37:24.040 align:middle line:84% The ion density of the test particle 00:37:24.040 --> 00:37:25.970 align:middle line:84% doesn't go into the electron density overall. 00:37:25.970 --> 00:37:28.780 align:middle line:84% So we merely have the response of the shielding electrons, 00:37:28.780 --> 00:37:33.670 align:middle line:84% these ones, which are following the ions around here. 00:37:33.670 --> 00:37:36.530 align:middle line:84% Again, we're only interested in ion. 00:37:36.530 --> 00:37:39.080 align:middle line:84% So this, in case you're wondering where we got 00:37:39.080 --> 00:37:47.360 align:middle line:84% to in Hutchinson's book, is his equation 7.3.19. 00:37:47.360 --> 00:37:54.580 align:middle line:84% And we said that our spectral density function, s of k omega, 00:37:54.580 --> 00:37:58.270 align:middle line:84% was equal to 1 upon the average electron density, the time 00:37:58.270 --> 00:38:00.190 align:middle line:84% and the volume that we're integrating over, 00:38:00.190 --> 00:38:09.290 align:middle line:84% and then the ensemble average of k omega squared like that. 00:38:09.290 --> 00:38:13.340 align:middle line:90% We can plug this in. 00:38:13.340 --> 00:38:15.390 align:middle line:84% We note that we've got two terms. 00:38:15.390 --> 00:38:17.600 align:middle line:84% So we're going to have a cross product, 00:38:17.600 --> 00:38:19.430 align:middle line:84% but the cross between these two terms 00:38:19.430 --> 00:38:22.190 align:middle line:84% is going to be a correlation between the electron 00:38:22.190 --> 00:38:23.180 align:middle line:90% and the ions. 00:38:23.180 --> 00:38:24.360 align:middle line:90% And there is no correlation. 00:38:24.360 --> 00:38:25.730 align:middle line:90% So those cross terms vanish. 00:38:25.730 --> 00:38:29.370 align:middle line:84% So we simply get this term squared and this term squared. 00:38:29.370 --> 00:38:30.890 align:middle line:90% Let me write some of this down. 00:38:30.890 --> 00:38:36.500 align:middle line:84% So we would get the electron contribution 00:38:36.500 --> 00:38:43.790 align:middle line:84% squared plus 2 ion times electron plus ion contribution 00:38:43.790 --> 00:38:45.750 align:middle line:90% squared. 00:38:45.750 --> 00:38:53.510 align:middle line:84% This cross term goes to 0, because we have uncorrelated 00:38:53.510 --> 00:38:54.470 align:middle line:90% electrons and ions. 00:38:54.470 --> 00:39:02.860 align:middle line:90% 00:39:02.860 --> 00:39:06.940 align:middle line:84% And we also use our dodgy identity from last week 00:39:06.940 --> 00:39:11.850 align:middle line:84% that the square of a delta function in frequency space 00:39:11.850 --> 00:39:17.320 align:middle line:84% is equal to p upon 2 chi of that delta function. 00:39:17.320 --> 00:39:19.872 align:middle line:84% So when we take the square of this term and this term, 00:39:19.872 --> 00:39:21.580 align:middle line:84% and we have a square of a delta function, 00:39:21.580 --> 00:39:23.372 align:middle line:84% we're just going to replace it with itself, 00:39:23.372 --> 00:39:25.900 align:middle line:84% with a t over 2 pi in front here. 00:39:25.900 --> 00:39:35.390 align:middle line:84% And so all of this gives us 2 pi upon the electron density times 00:39:35.390 --> 00:39:44.860 align:middle line:84% the volume summed upon j electrons 1 minus pi e 00:39:44.860 --> 00:39:54.840 align:middle line:84% upon epsilon squared delta of k dot bj minus omega. 00:39:54.840 --> 00:39:57.200 align:middle line:84% So this is the electron term, summing over 00:39:57.200 --> 00:40:04.060 align:middle line:84% j electrons, plus a term that looks like sum upon the ions, 00:40:04.060 --> 00:40:16.440 align:middle line:84% pi e upon epsilon squared delta k dot vl, the ions minus omega. 00:40:16.440 --> 00:40:20.040 align:middle line:84% And I need to put some big ensemble average 00:40:20.040 --> 00:40:21.600 align:middle line:90% brackets around this. 00:40:21.600 --> 00:40:25.650 align:middle line:84% And I actually want to go back and put in some absolute value 00:40:25.650 --> 00:40:28.230 align:middle line:84% signs inside here, which is why we end up 00:40:28.230 --> 00:40:29.617 align:middle line:90% with these absolute value signs. 00:40:29.617 --> 00:40:33.552 align:middle line:90% 00:40:33.552 --> 00:40:36.260 align:middle line:84% We're about to convert this into something that makes more sense, 00:40:36.260 --> 00:40:38.360 align:middle line:90% but any questions at this point? 00:40:38.360 --> 00:40:46.210 align:middle line:90% 00:40:46.210 --> 00:40:48.640 align:middle line:84% The size of the density fluctuations in Fourier space, 00:40:48.640 --> 00:40:50.640 align:middle line:84% these can be complex quantities, because they've 00:40:50.640 --> 00:40:52.060 align:middle line:90% got a phase attached to them. 00:40:52.060 --> 00:40:54.183 align:middle line:90% So this is any times any star. 00:40:54.183 --> 00:40:55.100 align:middle line:90% It's the same as that. 00:40:55.100 --> 00:40:59.290 align:middle line:84% So we're taking the-- the power that we measure depends directly 00:40:59.290 --> 00:41:00.730 align:middle line:90% on s of k and omega. 00:41:00.730 --> 00:41:02.710 align:middle line:84% The power we measure has to be a real quantity. 00:41:02.710 --> 00:41:04.390 align:middle line:84% We can't measure an imaginary power. 00:41:04.390 --> 00:41:07.230 align:middle line:84% And so therefore, this needs to be a real quantity as well. 00:41:07.230 --> 00:41:08.730 align:middle line:84% So we're just making sure we've only 00:41:08.730 --> 00:41:11.140 align:middle line:84% got the magnitude of it, not its phase. 00:41:11.140 --> 00:41:12.486 align:middle line:90% Yeah. 00:41:12.486 --> 00:41:19.890 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:41:19.890 --> 00:41:21.600 align:middle line:84% JACK HARE: What we're really doing 00:41:21.600 --> 00:41:25.380 align:middle line:84% is the cross terms would involve delta functions of ions 00:41:25.380 --> 00:41:27.580 align:middle line:84% and electrons times by each other. 00:41:27.580 --> 00:41:30.240 align:middle line:84% So if you can imagine, it's very unlikely 00:41:30.240 --> 00:41:32.700 align:middle line:84% that an ion electron is in the same place at the same time. 00:41:32.700 --> 00:41:34.620 align:middle line:84% These are delta functions that are very, very sharp. 00:41:34.620 --> 00:41:36.210 align:middle line:84% So really, what we're saying is not 00:41:36.210 --> 00:41:38.280 align:middle line:84% so much they're uncorrelated, but they 00:41:38.280 --> 00:41:40.210 align:middle line:90% don't occupy the same space. 00:41:40.210 --> 00:41:41.790 align:middle line:90% So they're different locations. 00:41:41.790 --> 00:41:48.480 align:middle line:90% 00:41:48.480 --> 00:41:53.030 align:middle line:84% So if I did a little diagram of some x-coordinate here, 00:41:53.030 --> 00:41:58.010 align:middle line:84% I might have an ion here and I might have an electron here. 00:41:58.010 --> 00:42:00.380 align:middle line:84% But those delta functions will not overlap in general. 00:42:00.380 --> 00:42:03.960 align:middle line:90% 00:42:03.960 --> 00:42:05.529 align:middle line:90% Any other questions? 00:42:05.529 --> 00:42:17.210 align:middle line:90% 00:42:17.210 --> 00:42:18.460 align:middle line:90% Yes, we'll do this. 00:42:18.460 --> 00:42:40.744 align:middle line:90% 00:42:40.744 --> 00:42:42.530 align:middle line:84% This still does not look very useful 00:42:42.530 --> 00:42:45.170 align:middle line:84% because we're summing over all like 10 to the 23 00:42:45.170 --> 00:42:46.860 align:middle line:84% or whatever electrons in our system. 00:42:46.860 --> 00:42:48.908 align:middle line:84% So this is a difficult sum to do. 00:42:48.908 --> 00:42:50.700 align:middle line:84% But you must have spotted this and thought, 00:42:50.700 --> 00:42:54.830 align:middle line:84% OK, if I'm doing a sum over a load of particles 00:42:54.830 --> 00:43:00.450 align:middle line:84% delta k dot d minus omega, this is 00:43:00.450 --> 00:43:03.540 align:middle line:84% looking an awful lot like my Klimontovich distribution again. 00:43:03.540 --> 00:43:09.870 align:middle line:84% And in fact, this looks like a sort of integral over space 00:43:09.870 --> 00:43:15.240 align:middle line:84% and over velocity of my Klimontovich distribution, r 00:43:15.240 --> 00:43:23.960 align:middle line:84% and v and t, times by some delta function k dot v minus omega. 00:43:23.960 --> 00:43:28.070 align:middle line:84% Remember, this consists of a load of delta functions-- 00:43:28.070 --> 00:43:36.230 align:middle line:84% delta r minus ri delta v minus vi. 00:43:36.230 --> 00:43:39.117 align:middle line:84% And so if I integrate this up over this delta function, 00:43:39.117 --> 00:43:40.950 align:middle line:84% this is going to pick out all the particles. 00:43:40.950 --> 00:43:44.390 align:middle line:84% So I can replace this by my Klimontovich distribution. 00:43:44.390 --> 00:43:47.810 align:middle line:90% 00:43:47.810 --> 00:43:49.630 align:middle line:84% And I'm going to end up with a term that 00:43:49.630 --> 00:43:54.530 align:middle line:84% looks like the volume from our d3v integral. 00:43:54.530 --> 00:44:00.770 align:middle line:84% There's a 1/k that I'm going to swap out omega equals kv. 00:44:00.770 --> 00:44:03.395 align:middle line:84% I'm going to swap out my v integral, the v for-- 00:44:03.395 --> 00:44:08.180 align:middle line:90% 00:44:08.180 --> 00:44:11.000 align:middle line:84% when I do that, I'm going to pick up a 1/k factor. 00:44:11.000 --> 00:44:14.420 align:middle line:84% I'm going to have an average over my Klimontovich 00:44:14.420 --> 00:44:18.550 align:middle line:84% distribution function, omega upon k. 00:44:18.550 --> 00:44:20.980 align:middle line:84% And here, this is like an ensemble average 00:44:20.980 --> 00:44:22.000 align:middle line:90% of lots of particles. 00:44:22.000 --> 00:44:24.652 align:middle line:84% The Klimontovich distribution function is very spiky. 00:44:24.652 --> 00:44:26.360 align:middle line:84% And we don't really want to work with it. 00:44:26.360 --> 00:44:31.140 align:middle line:84% So I wave my hands and say, this is V 1/k, some nice, 00:44:31.140 --> 00:44:36.030 align:middle line:84% smooth distribution function that you can actually work with. 00:44:36.030 --> 00:44:38.420 align:middle line:84% So I'm not going to claim that's a particularly 00:44:38.420 --> 00:44:39.930 align:middle line:90% rigorous argument. 00:44:39.930 --> 00:44:41.840 align:middle line:84% But you can have a look in Froula's book 00:44:41.840 --> 00:44:44.210 align:middle line:84% and in Hutchinson's book if you want 00:44:44.210 --> 00:44:47.282 align:middle line:84% to see maybe a slightly better argument for this. 00:44:47.282 --> 00:44:48.740 align:middle line:84% So we're going to basically replace 00:44:48.740 --> 00:44:51.920 align:middle line:84% these sums over a large number of particles-- 00:44:51.920 --> 00:44:55.290 align:middle line:90% this should have been vi here-- 00:44:55.290 --> 00:44:57.900 align:middle line:84% with the distribution function, which is kind of, obviously, 00:44:57.900 --> 00:44:58.620 align:middle line:90% what it is. 00:44:58.620 --> 00:45:01.000 align:middle line:84% If you have a sum over a load of particles, 00:45:01.000 --> 00:45:03.138 align:middle line:84% it's going to be some distribution function. 00:45:03.138 --> 00:45:04.680 align:middle line:84% But this distribution function is now 00:45:04.680 --> 00:45:08.360 align:middle line:84% being evaluated in terms of omega and k. 00:45:08.360 --> 00:45:09.860 align:middle line:84% So I'll write that down, and then we 00:45:09.860 --> 00:45:12.580 align:middle line:90% can talk about what it is. 00:45:12.580 --> 00:45:13.180 align:middle line:90% Yeah. 00:45:13.180 --> 00:45:19.727 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:45:19.727 --> 00:45:21.560 align:middle line:84% JACK HARE: Well, our test particle treatment 00:45:21.560 --> 00:45:23.510 align:middle line:84% we talk about a single particle moving through. 00:45:23.510 --> 00:45:24.927 align:middle line:84% And that's what starts introducing 00:45:24.927 --> 00:45:26.460 align:middle line:90% all of these delta functions. 00:45:26.460 --> 00:45:30.133 align:middle line:84% And so I think that that is why we do it, 00:45:30.133 --> 00:45:32.300 align:middle line:84% because when we've done the test particle treatment, 00:45:32.300 --> 00:45:34.580 align:middle line:84% we end up with all of these, and it's easier to get rid of them 00:45:34.580 --> 00:45:35.910 align:middle line:90% by going through a Klimontovich. 00:45:35.910 --> 00:45:37.430 align:middle line:84% So I think we started with a Klimontovich, 00:45:37.430 --> 00:45:38.990 align:middle line:84% knowing that this would happen later. 00:45:38.990 --> 00:45:41.930 align:middle line:84% Then we take out the Klimontovich later on. 00:45:41.930 --> 00:45:43.535 align:middle line:84% There may be other ways to do it. 00:45:43.535 --> 00:45:46.160 align:middle line:84% And it's not like I'm claiming that what I've done on the board 00:45:46.160 --> 00:45:48.110 align:middle line:90% is particularly rigorous. 00:45:48.110 --> 00:45:49.973 align:middle line:84% So there may be an equally rigorous way 00:45:49.973 --> 00:45:51.890 align:middle line:84% to do it without the Klimontovich distribution 00:45:51.890 --> 00:45:52.530 align:middle line:90% function. 00:45:52.530 --> 00:45:54.363 align:middle line:84% But I think if you want to do it rigorously, 00:45:54.363 --> 00:45:56.540 align:middle line:90% you have to put this. 00:45:56.540 --> 00:45:59.150 align:middle line:84% So if we do this substitution, we then 00:45:59.150 --> 00:46:02.690 align:middle line:84% end up with the very, very key formula for the spectral density 00:46:02.690 --> 00:46:05.600 align:middle line:84% function in terms of things that we actually 00:46:05.600 --> 00:46:07.080 align:middle line:90% stand a chance of evaluating. 00:46:07.080 --> 00:46:14.005 align:middle line:84% So s, k, and omega, 2 pi upon k times the average electron 00:46:14.005 --> 00:46:14.505 align:middle line:90% density. 00:46:14.505 --> 00:46:17.470 align:middle line:90% 00:46:17.470 --> 00:46:23.930 align:middle line:84% Then we have the brackets, the first term, 00:46:23.930 --> 00:46:27.440 align:middle line:84% 1 minus the susceptibility of the electrons 00:46:27.440 --> 00:46:31.860 align:middle line:84% over the permittivity of the entire system squared. 00:46:31.860 --> 00:46:34.910 align:middle line:84% Now, the distribution function of the electrons 00:46:34.910 --> 00:46:38.270 align:middle line:84% in the direction of k, so this is our incoherent scattering 00:46:38.270 --> 00:46:40.130 align:middle line:84% thing again, where we're only taking 00:46:40.130 --> 00:46:42.780 align:middle line:84% a slice of the distribution function in one direction. 00:46:42.780 --> 00:46:45.910 align:middle line:84% So a one-dimensional distribution function here. 00:46:45.910 --> 00:46:48.790 align:middle line:84% Plus a term that looks like susceptibility 00:46:48.790 --> 00:46:51.880 align:middle line:84% of electrons over the permittivity 00:46:51.880 --> 00:46:53.890 align:middle line:90% of the entire system squared. 00:46:53.890 --> 00:46:59.400 align:middle line:84% And then, formally here, we have to sum over every single ion 00:46:59.400 --> 00:47:00.370 align:middle line:90% species. 00:47:00.370 --> 00:47:04.390 align:middle line:84% So now we're doing a sum over different ion species. 00:47:04.390 --> 00:47:06.780 align:middle line:84% So if you've got carbon and hydrogen, things like that, 00:47:06.780 --> 00:47:08.670 align:middle line:84% in a quasi neutral plasma, you need 00:47:08.670 --> 00:47:12.540 align:middle line:84% to sum over the charge of each of these ion species 00:47:12.540 --> 00:47:17.565 align:middle line:84% and the distribution function for each of these ion species. 00:47:17.565 --> 00:47:19.190 align:middle line:84% But again, if everything is Maxwellian, 00:47:19.190 --> 00:47:20.520 align:middle line:90% this becomes much simpler. 00:47:20.520 --> 00:47:24.290 align:middle line:84% But this is the fully general formula here. 00:47:24.290 --> 00:47:26.040 align:middle line:84% And if you didn't want to write this down, 00:47:26.040 --> 00:47:30.920 align:middle line:84% this is Hutchinson's 7.3.22, or you can check in case I've made 00:47:30.920 --> 00:47:32.420 align:middle line:90% a typo here. 00:47:32.420 --> 00:47:34.000 align:middle line:84% So I'm going to talk through what 00:47:34.000 --> 00:47:37.150 align:middle line:84% each of these terms physically means, physically motivated. 00:47:37.150 --> 00:47:42.840 align:middle line:84% So this here, the distribution function at omega k 00:47:42.840 --> 00:47:49.560 align:middle line:84% is the number of electrons with a velocity v equals omega 00:47:49.560 --> 00:47:50.640 align:middle line:90% upon k. 00:47:50.640 --> 00:47:53.400 align:middle line:84% So the number of electrons in this distribution 00:47:53.400 --> 00:47:56.865 align:middle line:84% whose velocity is equal to the phase velocity of the mode 00:47:56.865 --> 00:47:57.990 align:middle line:90% that we're scattering from. 00:47:57.990 --> 00:48:00.760 align:middle line:84% Remember, we've been setting omega and k all this time. 00:48:00.760 --> 00:48:02.760 align:middle line:84% So this tells you how many scatterers there 00:48:02.760 --> 00:48:04.150 align:middle line:90% are which are electrons. 00:48:04.150 --> 00:48:14.430 align:middle line:84% And this tells you the response of the plasma 00:48:14.430 --> 00:48:17.843 align:middle line:90% to each of these electrons. 00:48:17.843 --> 00:48:19.760 align:middle line:84% So each of these-- there may be more than one. 00:48:19.760 --> 00:48:21.560 align:middle line:84% Each of these electrons are flying around 00:48:21.560 --> 00:48:22.820 align:middle line:90% as a test particle. 00:48:22.820 --> 00:48:25.520 align:middle line:84% And the plasma is going whoa and moving out of the way 00:48:25.520 --> 00:48:26.850 align:middle line:90% in order to accommodate them. 00:48:26.850 --> 00:48:28.670 align:middle line:84% So this is the response of the plasma. 00:48:28.670 --> 00:48:30.740 align:middle line:84% We've got a contribution from those electrons. 00:48:30.740 --> 00:48:31.580 align:middle line:90% That's the one. 00:48:31.580 --> 00:48:34.040 align:middle line:84% And then the minus is all the electrons getting out 00:48:34.040 --> 00:48:35.760 align:middle line:90% the way of those electrons. 00:48:35.760 --> 00:48:37.610 align:middle line:84% So that's the total amount of electrons 00:48:37.610 --> 00:48:39.260 align:middle line:90% you have to scatter off. 00:48:39.260 --> 00:48:41.000 align:middle line:84% We have the density of those electrons, 00:48:41.000 --> 00:48:43.940 align:middle line:84% but we also have a depletion of density of other electrons 00:48:43.940 --> 00:48:47.390 align:middle line:90% getting out of their way. 00:48:47.390 --> 00:48:53.780 align:middle line:84% This over here is the number of ions 00:48:53.780 --> 00:48:58.040 align:middle line:90% with a velocity omega upon k. 00:48:58.040 --> 00:48:59.930 align:middle line:90% And this is that response again. 00:48:59.930 --> 00:49:02.910 align:middle line:90% 00:49:02.910 --> 00:49:05.620 align:middle line:84% But note that we don't have a 1, because we don't scatter off 00:49:05.620 --> 00:49:06.120 align:middle line:90% ions. 00:49:06.120 --> 00:49:07.380 align:middle line:90% Their scattering is tiny. 00:49:07.380 --> 00:49:09.997 align:middle line:84% So all we get to see when the ion is moving-- 00:49:09.997 --> 00:49:12.580 align:middle line:84% the scattering we get is just the scattering of the electrons, 00:49:12.580 --> 00:49:14.370 align:middle line:90% which are pulled into the ion. 00:49:14.370 --> 00:49:16.420 align:middle line:84% Note there's a difference in sign here. 00:49:16.420 --> 00:49:17.490 align:middle line:84% Well, I guess-- yeah, it doesn't matter. 00:49:17.490 --> 00:49:19.282 align:middle line:84% When we square it, we get rid of it anyway. 00:49:19.282 --> 00:49:23.430 align:middle line:84% But these are the ions which are drawn into the region where 00:49:23.430 --> 00:49:24.108 align:middle line:90% the-- 00:49:24.108 --> 00:49:25.650 align:middle line:84% sorry-- these are the electrons which 00:49:25.650 --> 00:49:27.903 align:middle line:84% are drawn into the region where the ion is. 00:49:27.903 --> 00:49:29.320 align:middle line:84% And so we're scattering off those. 00:49:29.320 --> 00:49:37.920 align:middle line:84% So we only look at the response of electrons to these ions here. 00:49:37.920 --> 00:49:41.310 align:middle line:84% This is what your total scattering spectral density 00:49:41.310 --> 00:49:44.770 align:middle line:90% function looks like. 00:49:44.770 --> 00:49:47.950 align:middle line:84% Where does this function have peaks? 00:49:47.950 --> 00:49:51.700 align:middle line:84% Where am I looking in this for that spectrum what 00:49:51.700 --> 00:49:56.425 align:middle line:90% I drew of r here? 00:49:56.425 --> 00:50:24.170 align:middle line:90% 00:50:24.170 --> 00:50:27.800 align:middle line:84% Where do we get peaks in s of k and omega? 00:50:27.800 --> 00:50:29.977 align:middle line:84% There will be a singularity, shall I say? 00:50:29.977 --> 00:50:31.310 align:middle line:90% I'll try and make it very clear. 00:50:31.310 --> 00:50:33.540 align:middle line:84% Where will I get singularities in s of k and omega, 00:50:33.540 --> 00:50:35.990 align:middle line:84% which will look like nice, sharp, defined resonances 00:50:35.990 --> 00:50:37.850 align:middle line:90% scattering off specific waves? 00:50:37.850 --> 00:50:49.830 align:middle line:90% 00:50:49.830 --> 00:51:09.270 align:middle line:84% We will get peaks, singularities for epsilon 00:51:09.270 --> 00:51:12.850 align:middle line:90% equals 0 in the denominator. 00:51:12.850 --> 00:51:15.870 align:middle line:84% So whenever epsilon tends to 0, this function 00:51:15.870 --> 00:51:17.310 align:middle line:90% will be extremely large. 00:51:17.310 --> 00:51:22.320 align:middle line:84% Epsilon is also-- by the way, this is chi of k and omega. 00:51:22.320 --> 00:51:25.312 align:middle line:84% This is also epsilon of k and omega. 00:51:25.312 --> 00:51:28.020 align:middle line:84% So there's k's and omegas hiding everywhere inside here, not just 00:51:28.020 --> 00:51:29.550 align:middle line:90% within these fusion functions. 00:51:29.550 --> 00:51:33.460 align:middle line:84% These depend on the speed of the test particle as well. 00:51:33.460 --> 00:51:36.600 align:middle line:84% And so if the test particles are going at a certain speed, 00:51:36.600 --> 00:51:38.910 align:middle line:84% they will induce a certain response to plasma. 00:51:38.910 --> 00:51:40.620 align:middle line:90% That's a wave. 00:51:40.620 --> 00:51:43.830 align:middle line:84% So this is equivalent to saying, epsilon equals 00:51:43.830 --> 00:51:47.010 align:middle line:84% 0 is equivalent to finding the normal modes of the plasma, 00:51:47.010 --> 00:51:51.260 align:middle line:84% just like you do with determinants of matrices 00:51:51.260 --> 00:51:52.300 align:middle line:90% and stuff like that. 00:51:52.300 --> 00:51:56.320 align:middle line:90% 00:51:56.320 --> 00:51:56.945 align:middle line:90% So are we done? 00:51:56.945 --> 00:51:59.518 align:middle line:90% 00:51:59.518 --> 00:52:01.810 align:middle line:84% Can you plot me a Thomson scattering spectra from this? 00:52:01.810 --> 00:52:10.600 align:middle line:90% 00:52:10.600 --> 00:52:12.150 align:middle line:90% You can? 00:52:12.150 --> 00:52:14.910 align:middle line:90% You were nodding. 00:52:14.910 --> 00:52:17.810 align:middle line:84% We don't know anything about chi and epsilon. 00:52:17.810 --> 00:52:20.610 align:middle line:84% So now we're going to have to go do some [INAUDIBLE]. 00:52:20.610 --> 00:52:21.230 align:middle line:90% Don't worry. 00:52:21.230 --> 00:52:22.730 align:middle line:90% We're almost there. 00:52:22.730 --> 00:52:25.685 align:middle line:84% But in principle, if you did know these for some reason 00:52:25.685 --> 00:52:27.560 align:middle line:84% and you did know your distribution functions, 00:52:27.560 --> 00:52:29.685 align:middle line:84% you could work out what the scattering spectra was. 00:52:29.685 --> 00:52:32.360 align:middle line:84% But we need to know what these chis are. 00:52:32.360 --> 00:52:34.175 align:middle line:90% And just-- yeah. 00:52:34.175 --> 00:52:43.013 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:52:43.013 --> 00:52:44.680 align:middle line:84% JACK HARE: So again, still, we are never 00:52:44.680 --> 00:52:46.090 align:middle line:90% scattering off the ions. 00:52:46.090 --> 00:52:50.380 align:middle line:84% But people will talk about scattering off the ion feature. 00:52:50.380 --> 00:52:52.300 align:middle line:84% So these are actually-- these two terms 00:52:52.300 --> 00:52:54.440 align:middle line:84% tend to be large at different frequencies. 00:52:54.440 --> 00:52:57.520 align:middle line:84% This term is large, around the ion acoustic waves. 00:52:57.520 --> 00:53:00.100 align:middle line:84% This term is large around the electron plasma waves. 00:53:00.100 --> 00:53:02.110 align:middle line:84% And so people call this the ion feature. 00:53:02.110 --> 00:53:06.200 align:middle line:90% 00:53:06.200 --> 00:53:07.970 align:middle line:84% But just, it's important to remember 00:53:07.970 --> 00:53:10.715 align:middle line:84% that no ions were scattered off in the making of this spectrum. 00:53:10.715 --> 00:53:20.120 align:middle line:90% 00:53:20.120 --> 00:53:23.100 align:middle line:84% Any other questions on this before we keep going? 00:53:23.100 --> 00:53:27.720 align:middle line:84% And I don't think I said it explicitly anywhere before. 00:53:27.720 --> 00:53:32.620 align:middle line:84% But your permittivity is always equal to 1 00:53:32.620 --> 00:53:37.990 align:middle line:84% plus all of the susceptibilities in your system. 00:53:37.990 --> 00:53:40.660 align:middle line:84% This is just true for some electromagnetic system. 00:53:40.660 --> 00:53:42.710 align:middle line:84% It could be dielectrics and stuff like that. 00:53:42.710 --> 00:53:45.760 align:middle line:84% So in this case, our permittivity 00:53:45.760 --> 00:53:50.890 align:middle line:84% is equal to 1 plus the electron susceptibility plus the ion 00:53:50.890 --> 00:53:52.048 align:middle line:90% susceptibility. 00:53:52.048 --> 00:53:53.590 align:middle line:84% And technically, because I've written 00:53:53.590 --> 00:53:55.660 align:middle line:84% this in terms of a load of ion species, 00:53:55.660 --> 00:53:59.860 align:middle line:84% it's the sum over all of the ion susceptibilities. 00:53:59.860 --> 00:54:02.050 align:middle line:84% So what we're heading out to find 00:54:02.050 --> 00:54:07.270 align:middle line:84% is the electron susceptibility and the ion susceptibility. 00:54:07.270 --> 00:54:09.370 align:middle line:84% And that will give us epsilon straight away. 00:54:09.370 --> 00:54:13.390 align:middle line:84% And we'll also find that when epsilon equals 0. 00:54:13.390 --> 00:54:17.050 align:middle line:84% Remember, epsilon is some function of k and omega. 00:54:17.050 --> 00:54:19.720 align:middle line:84% So this will be some function of k and omega equals 0. 00:54:19.720 --> 00:54:25.070 align:middle line:84% And all of a sudden, out will pop omega equals csk like that. 00:54:25.070 --> 00:54:27.143 align:middle line:84% So that will be a solution to epsilon equals 0. 00:54:27.143 --> 00:54:28.810 align:middle line:84% And it'll be the dispersion relationship 00:54:28.810 --> 00:54:31.300 align:middle line:84% for the ion acoustic waves, or the electron plasma waves, 00:54:31.300 --> 00:54:34.090 align:middle line:84% or whatever wave you've got in your system. 00:54:34.090 --> 00:54:34.930 align:middle line:90% Questions? 00:54:34.930 --> 00:54:43.720 align:middle line:90% 00:54:43.720 --> 00:54:46.266 align:middle line:90% Run out of tea. 00:54:46.266 --> 00:54:46.850 align:middle line:90% Oh, no. 00:54:46.850 --> 00:54:47.642 align:middle line:90% There's a bit left. 00:54:47.642 --> 00:54:56.450 align:middle line:90% 00:54:56.450 --> 00:54:57.750 align:middle line:90% Well-- yeah. 00:54:57.750 --> 00:54:58.730 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:54:58.730 --> 00:55:00.064 align:middle line:90% JACK HARE: Please. 00:55:00.064 --> 00:55:02.350 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:55:02.350 --> 00:55:03.520 align:middle line:90% JACK HARE: Yes. 00:55:03.520 --> 00:55:05.620 align:middle line:84% AUDIENCE: So when we get a measurement, 00:55:05.620 --> 00:55:08.343 align:middle line:90% we get a direct [INAUDIBLE]. 00:55:08.343 --> 00:55:09.010 align:middle line:90% JACK HARE: Yeah. 00:55:09.010 --> 00:55:12.190 align:middle line:90% We effectively do. 00:55:12.190 --> 00:55:13.706 align:middle line:90% Do I have it anywhere? 00:55:13.706 --> 00:55:18.570 align:middle line:90% 00:55:18.570 --> 00:55:20.610 align:middle line:84% What we measure is this quantity. 00:55:20.610 --> 00:55:24.930 align:middle line:84% We measure watts into some solid angle into-- 00:55:24.930 --> 00:55:27.150 align:middle line:90% at some frequency range here. 00:55:27.150 --> 00:55:30.000 align:middle line:84% That depends on the cross-section, 00:55:30.000 --> 00:55:32.340 align:middle line:84% the intensity of our laser, how well we focused it, 00:55:32.340 --> 00:55:33.960 align:middle line:84% if we put our spectrometer in the right place 00:55:33.960 --> 00:55:36.293 align:middle line:84% to actually see the scattered light, how many scatterers 00:55:36.293 --> 00:55:40.911 align:middle line:84% there are, how big the volume is I'm scattering off, and s. 00:55:40.911 --> 00:55:42.890 align:middle line:84% This is the only thing that has a shape. 00:55:42.890 --> 00:55:43.810 align:middle line:90% This is just a number. 00:55:43.810 --> 00:55:47.710 align:middle line:84% All of these together for a spectrometer in a certain place, 00:55:47.710 --> 00:55:50.180 align:middle line:90% this is like two. 00:55:50.180 --> 00:55:53.870 align:middle line:84% Whereas this s is actually this function 00:55:53.870 --> 00:56:00.090 align:middle line:84% that looks like it has peaks at different frequencies 00:56:00.090 --> 00:56:01.030 align:middle line:90% and stuff like that. 00:56:01.030 --> 00:56:02.350 align:middle line:90% So that's omega. 00:56:02.350 --> 00:56:04.000 align:middle line:90% This is s, like that. 00:56:04.000 --> 00:56:06.240 align:middle line:84% So this is the only interesting thing. 00:56:06.240 --> 00:56:09.240 align:middle line:84% If we had absolutely calibrated our spectrometer 00:56:09.240 --> 00:56:11.820 align:middle line:84% so that we could calculate this value, 00:56:11.820 --> 00:56:14.838 align:middle line:84% we can, in principle, get the density of our plasma as well. 00:56:14.838 --> 00:56:17.130 align:middle line:84% It turns out the absolute calibration of a spectrometer 00:56:17.130 --> 00:56:18.610 align:middle line:90% is very hard. 00:56:18.610 --> 00:56:21.075 align:middle line:84% So very few people use it to get this density. 00:56:21.075 --> 00:56:24.710 align:middle line:84% It also turns out that the electron plasma waves carry 00:56:24.710 --> 00:56:25.960 align:middle line:90% information about the density. 00:56:25.960 --> 00:56:27.752 align:middle line:84% So you're usually better off just measuring 00:56:27.752 --> 00:56:28.980 align:middle line:90% the frequency shift of these. 00:56:28.980 --> 00:56:32.490 align:middle line:84% So this is like the spectrum, what 00:56:32.490 --> 00:56:35.400 align:middle line:84% we used to think [INAUDIBLE] all of our lines and our k-shell, 00:56:35.400 --> 00:56:36.550 align:middle line:90% and all that sort of stuff. 00:56:36.550 --> 00:56:38.175 align:middle line:84% And this is just some constant factor 00:56:38.175 --> 00:56:40.300 align:middle line:84% that we probably won't be able to do anything with. 00:56:40.300 --> 00:56:41.717 align:middle line:84% So it does tell us that if we want 00:56:41.717 --> 00:56:43.770 align:middle line:84% to get more light onto our spectrometer, 00:56:43.770 --> 00:56:45.840 align:middle line:84% we should use a more powerful laser. 00:56:45.840 --> 00:56:47.267 align:middle line:90% We should focus it more closely. 00:56:47.267 --> 00:56:49.350 align:middle line:84% We should put our spectrometer in the right place. 00:56:49.350 --> 00:56:50.970 align:middle line:84% We should make our plasma denser. 00:56:50.970 --> 00:56:53.410 align:middle line:84% And we should collect the light from a larger region. 00:56:53.410 --> 00:56:54.930 align:middle line:84% So it's sort of useful information 00:56:54.930 --> 00:56:57.050 align:middle line:84% to know how to do Thomson scattering better. 00:56:57.050 --> 00:56:59.008 align:middle line:84% But this is the sort of thing that you actually 00:56:59.008 --> 00:57:00.970 align:middle line:90% spend your time fitting to. 00:57:00.970 --> 00:57:04.090 align:middle line:84% So you alter the temperature and density of your model plasma 00:57:04.090 --> 00:57:11.110 align:middle line:84% until your model sk omega matches the data 00:57:11.110 --> 00:57:12.260 align:middle line:90% that you've recorded. 00:57:12.260 --> 00:57:13.040 align:middle line:90% Yeah. 00:57:13.040 --> 00:57:13.540 align:middle line:90% Cool. 00:57:13.540 --> 00:57:14.590 align:middle line:90% But good question, yeah. 00:57:14.590 --> 00:57:15.573 align:middle line:90% Thank you. 00:57:15.573 --> 00:57:18.680 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:57:18.680 --> 00:57:20.806 align:middle line:84% JACK HARE: We have not assumed it's Maxwellian yet. 00:57:20.806 --> 00:57:22.565 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:57:22.565 --> 00:57:23.190 align:middle line:90% JACK HARE: Yes. 00:57:23.190 --> 00:57:24.600 align:middle line:84% If we did assume it was Maxwellian, then 00:57:24.600 --> 00:57:25.740 align:middle line:90% we could get a temperature. 00:57:25.740 --> 00:57:26.640 align:middle line:90% We'll go through it. 00:57:26.640 --> 00:57:29.070 align:middle line:90% We will derive-- 00:57:29.070 --> 00:57:32.247 align:middle line:84% I can't even remember which board we're on now. 00:57:32.247 --> 00:57:33.830 align:middle line:84% We would derive these susceptibilities 00:57:33.830 --> 00:57:35.300 align:middle line:84% from an arbitrary distribution function. 00:57:35.300 --> 00:57:36.710 align:middle line:84% And then we would discuss it in the case 00:57:36.710 --> 00:57:38.543 align:middle line:84% of a Maxwellian distribution function, which 00:57:38.543 --> 00:57:42.420 align:middle line:90% is a pretty common case. 00:57:42.420 --> 00:57:45.797 align:middle line:90% Any other questions? 00:57:45.797 --> 00:57:46.630 align:middle line:90% We're getting there. 00:57:46.630 --> 00:57:49.937 align:middle line:90% 00:57:49.937 --> 00:57:50.770 align:middle line:90% What shall we erase? 00:57:50.770 --> 00:58:25.290 align:middle line:90% 00:58:25.290 --> 00:58:28.200 align:middle line:84% The refractive index of a medium is simply 00:58:28.200 --> 00:58:32.470 align:middle line:84% equal to its permittivity to the half. 00:58:32.470 --> 00:58:34.210 align:middle line:90% So if we were dealing-- 00:58:34.210 --> 00:58:39.100 align:middle line:84% and for example, for a plasma, a cold plasma, 00:58:39.100 --> 00:58:44.290 align:middle line:84% where the thermal velocity was much less than the phase 00:58:44.290 --> 00:58:48.190 align:middle line:84% velocity, we derived, for example, [INAUDIBLE] mode, 00:58:48.190 --> 00:58:52.180 align:middle line:84% this was something like omega p squared on omega. 00:58:52.180 --> 00:58:55.980 align:middle line:90% So I'll write that as n squared. 00:58:55.980 --> 00:58:58.750 align:middle line:84% So if we were dealing with a plasma like this, 00:58:58.750 --> 00:59:02.110 align:middle line:84% it would be very easy to find the permittivity. 00:59:02.110 --> 00:59:02.905 align:middle line:90% But we're not. 00:59:02.905 --> 00:59:05.650 align:middle line:90% 00:59:05.650 --> 00:59:08.440 align:middle line:84% Remember, we're dealing with plasmas where the dispersion 00:59:08.440 --> 00:59:11.110 align:middle line:84% relationships we're interested in look like omega 00:59:11.110 --> 00:59:18.290 align:middle line:84% squared omega pe squared plus bk bte squared. 00:59:18.290 --> 00:59:21.130 align:middle line:84% So that means that our phase velocity now 00:59:21.130 --> 00:59:24.050 align:middle line:84% is on the order of the thermal velocity. 00:59:24.050 --> 00:59:26.770 align:middle line:84% So we can no longer do a cold plasma treatment. 00:59:26.770 --> 00:59:29.230 align:middle line:90% We need to do a warm plasma. 00:59:29.230 --> 00:59:32.990 align:middle line:90% 00:59:32.990 --> 00:59:34.790 align:middle line:84% And because we're doing a warm plasma, 00:59:34.790 --> 00:59:36.415 align:middle line:84% that means we're going to have to think 00:59:36.415 --> 00:59:37.946 align:middle line:90% about Landau [INAUDIBLE]. 00:59:37.946 --> 00:59:43.750 align:middle line:90% 00:59:43.750 --> 00:59:50.910 align:middle line:84% This is because our distribution function, something 00:59:50.910 --> 00:59:53.520 align:middle line:84% like this, previously, for this case, 00:59:53.520 --> 00:59:57.400 align:middle line:84% for the cold plasma, our wave was all the way up here-- 00:59:57.400 --> 01:00:00.750 align:middle line:84% v phase much, much larger than vte. 01:00:00.750 --> 01:00:03.870 align:middle line:84% There were no particles which had a velocity 01:00:03.870 --> 01:00:05.220 align:middle line:90% close to the phase velocity. 01:00:05.220 --> 01:00:07.470 align:middle line:84% And so therefore, from the point of view of this wave, 01:00:07.470 --> 01:00:10.000 align:middle line:84% the plasma was just cold and stationary. 01:00:10.000 --> 01:00:13.850 align:middle line:84% What we're doing now is we're looking 01:00:13.850 --> 01:00:17.180 align:middle line:84% at the phase on the order of vte. 01:00:17.180 --> 01:00:20.120 align:middle line:84% And so we have to consider how the wave interacts with 01:00:20.120 --> 01:00:21.900 align:middle line:90% and damps on these particles. 01:00:21.900 --> 01:00:28.170 align:middle line:84% So you will recall that we start with the Vlasov equation, 01:00:28.170 --> 01:00:32.180 align:middle line:84% where we're looking at the time derivative of some distribution 01:00:32.180 --> 01:00:34.820 align:middle line:84% function, plus the velocity dotted 01:00:34.820 --> 01:00:38.210 align:middle line:84% into the spatial derivative of a distribution 01:00:38.210 --> 01:00:42.140 align:middle line:90% function, plus the force-- 01:00:42.140 --> 01:00:45.620 align:middle line:84% the acceleration from the electric field, q of fe 01:00:45.620 --> 01:00:49.070 align:middle line:84% dot the change in the distribution 01:00:49.070 --> 01:00:50.750 align:middle line:90% function in velocity space. 01:00:50.750 --> 01:00:54.190 align:middle line:84% And we're going to set this equal to 0. 01:00:54.190 --> 01:00:58.220 align:middle line:84% So this is explicitly a collisionless system. 01:00:58.220 --> 01:01:01.790 align:middle line:84% Now, it turns out you can derive Thomson scattering 01:01:01.790 --> 01:01:04.160 align:middle line:84% for collisional systems, but it also 01:01:04.160 --> 01:01:07.460 align:middle line:84% turns out that most Thomson scattering 01:01:07.460 --> 01:01:09.320 align:middle line:90% systems are collisionless. 01:01:09.320 --> 01:01:11.013 align:middle line:84% I'm going to spend 30 seconds on this, 01:01:11.013 --> 01:01:12.680 align:middle line:84% and I hope I don't confuse you too much. 01:01:12.680 --> 01:01:14.990 align:middle line:84% You may be doing Thomson scattering in a system, which 01:01:14.990 --> 01:01:16.250 align:middle line:90% is collisional. 01:01:16.250 --> 01:01:20.090 align:middle line:84% A collisional system is one in which the length scale is 01:01:20.090 --> 01:01:23.090 align:middle line:90% larger than the mean free path. 01:01:23.090 --> 01:01:27.200 align:middle line:84% But when you're doing Thomson scattering, 01:01:27.200 --> 01:01:29.810 align:middle line:84% the length scale we're interested in is lambda. 01:01:29.810 --> 01:01:34.220 align:middle line:84% Remember, this lambda is 2 pi upon k. 01:01:34.220 --> 01:01:38.780 align:middle line:84% And that lambda may be on the order of the Debye length. 01:01:38.780 --> 01:01:41.210 align:middle line:84% If we're down towards the incoherent limit, 01:01:41.210 --> 01:01:42.860 align:middle line:84% this is just the Debye length here. 01:01:42.860 --> 01:01:46.580 align:middle line:84% And on these length scales here, the particles never collide. 01:01:46.580 --> 01:01:48.360 align:middle line:90% The mean free path is huge. 01:01:48.360 --> 01:01:51.290 align:middle line:84% And so these waves never see a particle collision. 01:01:51.290 --> 01:01:53.720 align:middle line:84% And so it's still legitimate to use 01:01:53.720 --> 01:01:55.550 align:middle line:84% this collisionless treatment, where 01:01:55.550 --> 01:01:58.340 align:middle line:84% lambda mean free path is much, much larger 01:01:58.340 --> 01:01:59.730 align:middle line:90% than our scattering wavelength. 01:01:59.730 --> 01:02:01.670 align:middle line:84% So even in a very collisional plasma, 01:02:01.670 --> 01:02:04.520 align:middle line:84% I can still do collisionless Thomson scattering. 01:02:04.520 --> 01:02:05.730 align:middle line:90% And that's good. 01:02:05.730 --> 01:02:07.520 align:middle line:84% If I wanted to do collisional, I'd 01:02:07.520 --> 01:02:11.257 align:middle line:84% have to put some collision operator in here. 01:02:11.257 --> 01:02:13.340 align:middle line:84% And if any of you have done plasma kinetic theory, 01:02:13.340 --> 01:02:13.970 align:middle line:90% it's a pain. 01:02:13.970 --> 01:02:16.730 align:middle line:90% 01:02:16.730 --> 01:02:17.893 align:middle line:90% But people do it. 01:02:17.893 --> 01:02:20.060 align:middle line:84% There's some really nice theory out there on Thomson 01:02:20.060 --> 01:02:22.055 align:middle line:84% scattering in collisional plasmas. 01:02:22.055 --> 01:02:25.380 align:middle line:90% 01:02:25.380 --> 01:02:29.472 align:middle line:84% We are going to assume that we have some distribution 01:02:29.472 --> 01:02:31.680 align:middle line:84% function that looks like some background distribution 01:02:31.680 --> 01:02:35.550 align:middle line:84% function plus a perturbed distribution function, which 01:02:35.550 --> 01:02:39.900 align:middle line:84% is oscillating, so some Fourier decomposition here, 01:02:39.900 --> 01:02:45.570 align:middle line:84% so e to the i k dot x minus omega t. 01:02:45.570 --> 01:02:48.910 align:middle line:84% When we substitute that back into this equation, 01:02:48.910 --> 01:02:56.460 align:middle line:84% we get minus i omega f prime plus k dot v 01:02:56.460 --> 01:03:06.995 align:middle line:84% f prime plus q on m e dot df dv all equal to 0. 01:03:06.995 --> 01:03:09.580 align:middle line:90% 01:03:09.580 --> 01:03:16.530 align:middle line:84% And we can define a current, which 01:03:16.530 --> 01:03:20.370 align:middle line:84% is equal to this perturbed distribution 01:03:20.370 --> 01:03:23.220 align:middle line:84% function times the charge of the particles 01:03:23.220 --> 01:03:27.900 align:middle line:84% we're talking about times the velocity integrated d3v. 01:03:27.900 --> 01:03:31.590 align:middle line:84% And we can rearrange this to be an equation for f prime, 01:03:31.590 --> 01:03:33.360 align:middle line:90% substituting into this. 01:03:33.360 --> 01:03:38.040 align:middle line:84% And we get q squared upon i times the mass, 01:03:38.040 --> 01:03:42.650 align:middle line:84% integral of velocity dotted into-- 01:03:42.650 --> 01:03:45.800 align:middle line:84% sorry-- velocity times the electric field 01:03:45.800 --> 01:03:57.410 align:middle line:84% dotted into the df dv over omega minus k dot v dv like this. 01:03:57.410 --> 01:04:00.560 align:middle line:90% 01:04:00.560 --> 01:04:03.880 align:middle line:84% We then say this current here, j, 01:04:03.880 --> 01:04:10.880 align:middle line:84% is equal to some conductivity times the electric field. 01:04:10.880 --> 01:04:14.160 align:middle line:84% And if we have a conductivity, we 01:04:14.160 --> 01:04:17.730 align:middle line:84% know that the susceptibility is equal to the conductivity 01:04:17.730 --> 01:04:21.300 align:middle line:90% over minus i omega epsilon 0. 01:04:21.300 --> 01:04:23.880 align:middle line:84% So this is just from electromagnetism here. 01:04:23.880 --> 01:04:26.250 align:middle line:84% We have taken all of the complicated plasma physics. 01:04:26.250 --> 01:04:28.750 align:middle line:84% And we're going to chuck it inside this sigma, calculate it, 01:04:28.750 --> 01:04:30.833 align:middle line:84% and then we're going to say from electromagnetism, 01:04:30.833 --> 01:04:32.640 align:middle line:84% we know the link between the conductivity 01:04:32.640 --> 01:04:35.660 align:middle line:90% and the susceptibility here. 01:04:35.660 --> 01:04:39.400 align:middle line:84% So this chi is equal to q squared 01:04:39.400 --> 01:04:52.040 align:middle line:84% over epsilon 0 mk the integral of partial f partial v 01:04:52.040 --> 01:04:57.750 align:middle line:90% over omega minus a0v dv. 01:04:57.750 --> 01:05:01.350 align:middle line:84% So now for some distribution function f, 01:05:01.350 --> 01:05:05.460 align:middle line:90% we can calculate this chi here. 01:05:05.460 --> 01:05:15.070 align:middle line:84% And I need to make this f in the direction of our k vector 01:05:15.070 --> 01:05:18.130 align:middle line:84% here, because along the way, I've stopped this 01:05:18.130 --> 01:05:19.270 align:middle line:90% from being a tensor. 01:05:19.270 --> 01:05:20.830 align:middle line:84% And I've just made it into a scalar. 01:05:20.830 --> 01:05:23.020 align:middle line:84% And we're taking the longitudinal component 01:05:23.020 --> 01:05:31.130 align:middle line:84% because these waves are all longitudinal waves. 01:05:31.130 --> 01:05:36.010 align:middle line:84% So this is the zz component of that tensor, which would 01:05:36.010 --> 01:05:37.525 align:middle line:90% otherwise be a 3 by 3 tensor. 01:05:37.525 --> 01:05:44.182 align:middle line:90% 01:05:44.182 --> 01:05:45.890 align:middle line:84% So we're just interested in the component 01:05:45.890 --> 01:05:48.800 align:middle line:84% of this along the electric field, which is causing 01:05:48.800 --> 01:05:51.250 align:middle line:90% all of this in the first place. 01:05:51.250 --> 01:05:56.020 align:middle line:84% When you do this, you remember that there is a singularity 01:05:56.020 --> 01:05:57.610 align:middle line:90% in the denominator here. 01:05:57.610 --> 01:05:59.000 align:middle line:90% And you have to be careful. 01:05:59.000 --> 01:06:06.460 align:middle line:84% There's a pole at v equals omega upon k. 01:06:06.460 --> 01:06:16.050 align:middle line:84% And then you do your integral in the complex plane. 01:06:16.050 --> 01:06:18.840 align:middle line:84% And you close it so that you don't go around this pole. 01:06:18.840 --> 01:06:21.870 align:middle line:84% This is called the Landau [INAUDIBLE]. 01:06:21.870 --> 01:06:24.730 align:middle line:84% I assume that you have all seen this at some point. 01:06:24.730 --> 01:06:29.450 align:middle line:84% And I'm just reminding you exactly what's going on here. 01:06:29.450 --> 01:06:31.366 align:middle line:90% So now we have-- 01:06:31.366 --> 01:06:34.500 align:middle line:84% we do this integral properly using the Landau contour, 01:06:34.500 --> 01:06:35.210 align:middle line:90% we have chi. 01:06:35.210 --> 01:06:40.030 align:middle line:90% 01:06:40.030 --> 01:06:45.625 align:middle line:84% And then we also have epsilon is equal to 1 plus chi e plus chi 01:06:45.625 --> 01:06:47.500 align:middle line:84% i, where these subscripts here are going 01:06:47.500 --> 01:06:50.710 align:middle line:84% to come from us putting in the distribution function for these. 01:06:50.710 --> 01:06:55.000 align:middle line:90% So this is chi j. 01:06:55.000 --> 01:06:58.225 align:middle line:84% And this is f in the k direction for the j-th species. 01:06:58.225 --> 01:07:01.930 align:middle line:90% 01:07:01.930 --> 01:07:02.980 align:middle line:90% Any questions on that? 01:07:02.980 --> 01:07:12.200 align:middle line:90% 01:07:12.200 --> 01:07:13.950 align:middle line:84% Hopefully you've seen this in more detail. 01:07:13.950 --> 01:07:30.620 align:middle line:90% 01:07:30.620 --> 01:07:33.350 align:middle line:84% So now let's consider a Maxwellian distribution. 01:07:33.350 --> 01:08:08.803 align:middle line:90% 01:08:08.803 --> 01:08:10.720 align:middle line:84% So there's an arbitrary distribution function. 01:08:10.720 --> 01:08:12.250 align:middle line:84% If your plasma is not Maxwellian, 01:08:12.250 --> 01:08:14.950 align:middle line:84% you'll get some complicated looking spectrum you 01:08:14.950 --> 01:08:16.490 align:middle line:90% can, in principle, calculate. 01:08:16.490 --> 01:08:26.310 align:middle line:84% But if we specify a Maxwellian such that our f for the j-th 01:08:26.310 --> 01:08:30.130 align:middle line:84% species in the k direction-- so it's a one-dimensional 01:08:30.130 --> 01:08:31.680 align:middle line:90% distribution function-- 01:08:31.680 --> 01:08:35.000 align:middle line:84% is a function of velocity is equal to whatever 01:08:35.000 --> 01:08:39.547 align:middle line:84% density of particles, some constants to make sure 01:08:39.547 --> 01:08:41.630 align:middle line:84% that this is normalized so we get density out when 01:08:41.630 --> 01:08:44.710 align:middle line:90% we integrate over velocity. 01:08:44.710 --> 01:08:46.779 align:middle line:84% Thermal velocity of the j-th particle 01:08:46.779 --> 01:08:51.740 align:middle line:84% electrons or ions exponential of minus v 01:08:51.740 --> 01:08:55.410 align:middle line:84% squared upon that thermal velocity squared. 01:08:55.410 --> 01:08:57.979 align:middle line:90% So this is just the Maxwellian. 01:08:57.979 --> 01:09:01.109 align:middle line:84% Then we find that chi, in this case, 01:09:01.109 --> 01:09:02.910 align:middle line:84% susceptibility for the j-th particle 01:09:02.910 --> 01:09:08.430 align:middle line:84% looks like omega p the j-th particle upon k 01:09:08.430 --> 01:09:12.520 align:middle line:84% squared thermal velocity the j-th particle squared. 01:09:12.520 --> 01:09:16.819 align:middle line:84% I've still got this 1 upon square root 2 pi hanging around. 01:09:16.819 --> 01:09:20.270 align:middle line:84% Then we have this integral of minus 01:09:20.270 --> 01:09:29.569 align:middle line:84% b upon btj exponential of minus v upon btj word 01:09:29.569 --> 01:09:33.819 align:middle line:90% over omega minus kv-- 01:09:33.819 --> 01:09:37.569 align:middle line:90% sorry-- omega upon k minus vbv. 01:09:37.569 --> 01:09:41.950 align:middle line:90% 01:09:41.950 --> 01:09:46.779 align:middle line:84% Now, finally, we get back to alpha because we recognize 01:09:46.779 --> 01:09:50.109 align:middle line:84% that this term here is just the Debye length 01:09:50.109 --> 01:09:51.979 align:middle line:90% for the j-th particle squared. 01:09:51.979 --> 01:09:57.520 align:middle line:84% So we can say that chi j is equal to 1 01:09:57.520 --> 01:10:02.350 align:middle line:90% upon k lambda Debye squared. 01:10:02.350 --> 01:10:06.820 align:middle line:84% That is effectively 1 upon alpha squared, our coupling parameter. 01:10:06.820 --> 01:10:08.320 align:middle line:84% So this is why this has come back in 01:10:08.320 --> 01:10:09.695 align:middle line:84% and why I didn't find [INAUDIBLE] 01:10:09.695 --> 01:10:12.910 align:middle line:84% 2 pis, because at the moment, it's [INAUDIBLE]. 01:10:12.910 --> 01:10:15.280 align:middle line:84% And then out in front here, we're 01:10:15.280 --> 01:10:17.440 align:middle line:84% going to have a term that looks like the charge 01:10:17.440 --> 01:10:18.400 align:middle line:90% of the j-th species. 01:10:18.400 --> 01:10:20.200 align:middle line:84% For electrons, this would just be minus 1. 01:10:20.200 --> 01:10:22.480 align:middle line:84% But for ions, it could be some bigger number-- 01:10:22.480 --> 01:10:27.440 align:middle line:84% the density of these species times the electron temperature 01:10:27.440 --> 01:10:31.100 align:middle line:84% over the electron temperature times the temperature 01:10:31.100 --> 01:10:32.930 align:middle line:90% of the j-th species. 01:10:32.930 --> 01:10:34.970 align:middle line:90% And we've got a squared there. 01:10:34.970 --> 01:10:37.820 align:middle line:84% We've pulled a load of terms out because what 01:10:37.820 --> 01:10:39.950 align:middle line:84% I'm about to write next is a different function, 01:10:39.950 --> 01:10:42.530 align:middle line:84% but I just want to point out if you stare at this a little 01:10:42.530 --> 01:10:46.710 align:middle line:84% while, this is equal to 1 for electrons. 01:10:46.710 --> 01:10:49.815 align:middle line:84% For electrons, this term just vanishes here. 01:10:49.815 --> 01:10:52.500 align:middle line:84% It's kind of obvious for one electron 01:10:52.500 --> 01:10:54.780 align:middle line:84% density, electron temperature, electron density, 01:10:54.780 --> 01:10:55.770 align:middle line:90% electron temperature. 01:10:55.770 --> 01:10:56.710 align:middle line:90% This all cancels out. 01:10:56.710 --> 01:10:59.070 align:middle line:84% But for ions, it will be a different number. 01:10:59.070 --> 01:11:00.753 align:middle line:84% And that will be mostly due to the fact 01:11:00.753 --> 01:11:03.420 align:middle line:84% that the ions may have different temperatures than the electrons 01:11:03.420 --> 01:11:05.670 align:middle line:84% and due to the fact that the ions will 01:11:05.670 --> 01:11:06.690 align:middle line:90% have different charges. 01:11:06.690 --> 01:11:08.850 align:middle line:84% And then the thing that is left is 01:11:08.850 --> 01:11:12.170 align:middle line:90% this function, which we call w. 01:11:12.170 --> 01:11:14.630 align:middle line:84% And it's a function of some dimensionless parameter 01:11:14.630 --> 01:11:15.845 align:middle line:90% squiggle of j. 01:11:15.845 --> 01:11:18.530 align:middle line:90% 01:11:18.530 --> 01:11:21.348 align:middle line:90% This squiggle parameter-- 01:11:21.348 --> 01:11:22.890 align:middle line:84% I don't remember how to pronounce it, 01:11:22.890 --> 01:11:24.800 align:middle line:90% so I may as well be honest-- 01:11:24.800 --> 01:11:30.320 align:middle line:84% is equal to the phase velocity over the thermal velocity. 01:11:30.320 --> 01:11:34.340 align:middle line:84% Remember, the phase velocity is omega upon k. 01:11:34.340 --> 01:11:36.890 align:middle line:84% So this is one when we're talking 01:11:36.890 --> 01:11:39.418 align:middle line:84% about particles or thermal velocities that 01:11:39.418 --> 01:11:40.710 align:middle line:90% are same as the phase velocity. 01:11:40.710 --> 01:11:42.710 align:middle line:90% And it's less than 1 or greater. 01:11:42.710 --> 01:11:44.828 align:middle line:90% This function here is the-- 01:11:44.828 --> 01:11:47.816 align:middle line:90% 01:11:47.816 --> 01:11:49.265 align:middle line:84% I think it's the Wolf-Z function. 01:11:49.265 --> 01:11:54.750 align:middle line:90% 01:11:54.750 --> 01:11:56.580 align:middle line:84% And it's simply a function, which 01:11:56.580 --> 01:12:01.270 align:middle line:84% is defined to be-- to involve the solution to this integral 01:12:01.270 --> 01:12:02.650 align:middle line:90% in a dimensionless way. 01:12:02.650 --> 01:12:11.870 align:middle line:84% And this function w of squiggle is equal to 1 minus 2 squiggle e 01:12:11.870 --> 01:12:17.370 align:middle line:84% to the minus squiggle squared integral from 0 to squiggle e 01:12:17.370 --> 01:12:23.370 align:middle line:84% to the x squared ex plus a term, which is the residue due 01:12:23.370 --> 01:12:31.530 align:middle line:84% to the Landau contour, i pi to the 1/2 squiggle e to the minus 01:12:31.530 --> 01:12:34.060 align:middle line:90% squiggle squared. 01:12:34.060 --> 01:12:38.580 align:middle line:84% Obvious-- take it home to meet your parents. 01:12:38.580 --> 01:12:42.000 align:middle line:84% This is a very ugly function, but because we use Maxwellian 01:12:42.000 --> 01:12:45.540 align:middle line:84% so often, this is a function which can be tabulated. 01:12:45.540 --> 01:12:48.810 align:middle line:84% So we can just calculate it for various values of epsilon. 01:12:48.810 --> 01:12:53.790 align:middle line:84% We can make a table of-- sorry, not epsilons-- of squiggles 01:12:53.790 --> 01:12:56.190 align:middle line:84% from very small numbers to very large numbers. 01:12:56.190 --> 01:12:57.750 align:middle line:84% And we can just calculate this up. 01:12:57.750 --> 01:13:00.730 align:middle line:84% Remember, it's a complex number, a complex value here. 01:13:00.730 --> 01:13:02.700 align:middle line:84% But we can have a lookup table here. 01:13:02.700 --> 01:13:08.510 align:middle line:84% And there is something called the plasma dispersion function. 01:13:08.510 --> 01:13:17.160 align:middle line:90% 01:13:17.160 --> 01:13:21.410 align:middle line:84% I think it's normally written as z. 01:13:21.410 --> 01:13:23.990 align:middle line:84% That is related to this function here. 01:13:23.990 --> 01:13:25.010 align:middle line:90% And it's related. 01:13:25.010 --> 01:13:26.030 align:middle line:90% It's sort of like-- 01:13:26.030 --> 01:13:29.420 align:middle line:84% I can't remember what it is, but the z is equal to squiggle 01:13:29.420 --> 01:13:33.060 align:middle line:84% to the 1/2 w of squiggle, or something like that. 01:13:33.060 --> 01:13:35.090 align:middle line:84% I can't remember exactly what it is right now. 01:13:35.090 --> 01:13:36.830 align:middle line:84% I prefer to work with this thing, 01:13:36.830 --> 01:13:38.540 align:middle line:84% because this is a thing that you'll 01:13:38.540 --> 01:13:46.800 align:middle line:84% find in Python, and Julia, and Matlab, 01:13:46.800 --> 01:13:50.270 align:middle line:84% because it shows up in other contexts as well. 01:13:50.270 --> 01:13:52.630 align:middle line:84% And so, people, there are very nice fast implementations 01:13:52.630 --> 01:13:54.888 align:middle line:84% of this function in all of these languages 01:13:54.888 --> 01:13:55.930 align:middle line:90% and many other languages. 01:13:55.930 --> 01:13:57.740 align:middle line:90% Fortran as well, no doubt. 01:13:57.740 --> 01:14:01.060 align:middle line:84% So it's almost always better if you're writing your own code 01:14:01.060 --> 01:14:02.748 align:middle line:84% to write it in terms of this function, 01:14:02.748 --> 01:14:04.540 align:middle line:84% rather than the plasma dispersion function. 01:14:04.540 --> 01:14:06.623 align:middle line:84% But you may have come across the plasma dispersion 01:14:06.623 --> 01:14:08.620 align:middle line:84% function in other Landau damping contexts. 01:14:08.620 --> 01:14:10.490 align:middle line:84% And there they're very, very similar. 01:14:10.490 --> 01:14:14.050 align:middle line:84% This is also related to the error function, 01:14:14.050 --> 01:14:20.170 align:middle line:84% the complex error function, and a related family of functions 01:14:20.170 --> 01:14:23.080 align:middle line:84% that crop up in all sorts of other interesting contexts 01:14:23.080 --> 01:14:24.400 align:middle line:90% as well. 01:14:24.400 --> 01:14:26.710 align:middle line:84% The link between them all is that they have these fun 01:14:26.710 --> 01:14:32.483 align:middle line:84% integrals and e to the something squared inside them all. 01:14:32.483 --> 01:14:38.000 align:middle line:90%