WEBVTT 00:00:00.000 --> 00:00:02.390 align:middle line:90% [SQUEAKING] 00:00:02.390 --> 00:00:04.302 align:middle line:90% [RUSTLING] 00:00:04.302 --> 00:00:08.126 align:middle line:90% [CLICKING] 00:00:08.126 --> 00:00:10.040 align:middle line:90% 00:00:10.040 --> 00:00:13.520 align:middle line:84% JACK HARE: So Thomson scattering-- as you remember, 00:00:13.520 --> 00:00:19.930 align:middle line:84% we set up the problem so that we had some scattering particle 00:00:19.930 --> 00:00:20.710 align:middle line:90% here. 00:00:20.710 --> 00:00:27.860 align:middle line:84% We had some incoming wave with a load of parallel wavefronts 00:00:27.860 --> 00:00:29.690 align:middle line:90% at some plane wave. 00:00:29.690 --> 00:00:35.930 align:middle line:84% It's got a wave vector Ki in the i hat direction here. 00:00:35.930 --> 00:00:39.650 align:middle line:84% We said that there was some origin to our system, 00:00:39.650 --> 00:00:44.600 align:middle line:84% and the particle was at a distance R from our origin. 00:00:44.600 --> 00:00:47.210 align:middle line:84% And we were going to observe the scattered light 00:00:47.210 --> 00:00:54.050 align:middle line:84% in another direction with our spectrometer here. 00:00:54.050 --> 00:00:56.030 align:middle line:84% And that scattered wave is also going 00:00:56.030 --> 00:00:59.360 align:middle line:84% to be a plane wave that's going to have a wave vector 00:00:59.360 --> 00:01:03.750 align:middle line:90% Ks in the s hat direction. 00:01:03.750 --> 00:01:06.800 align:middle line:84% There's going to be two vectors, one from the particle 00:01:06.800 --> 00:01:12.330 align:middle line:84% and one from the origin, pointing towards our observer 00:01:12.330 --> 00:01:12.830 align:middle line:90% here. 00:01:12.830 --> 00:01:19.180 align:middle line:90% And we call those R and R prime. 00:01:19.180 --> 00:01:21.550 align:middle line:84% But one of the first approximations we said 00:01:21.550 --> 00:01:25.030 align:middle line:84% is that imagine that plasma where all of our particles 00:01:25.030 --> 00:01:28.780 align:middle line:84% have some vector R is relatively small compared 00:01:28.780 --> 00:01:30.970 align:middle line:90% to the distance to our observer. 00:01:30.970 --> 00:01:34.210 align:middle line:90% And then we can say that this-- 00:01:34.210 --> 00:01:39.560 align:middle line:84% our vector is roughly the same as the R prime vector here. 00:01:39.560 --> 00:01:42.760 align:middle line:84% The position of our particle, R of t, 00:01:42.760 --> 00:01:46.060 align:middle line:84% is equal to whatever it started off 00:01:46.060 --> 00:01:49.160 align:middle line:90% plus its velocity times time. 00:01:49.160 --> 00:01:51.760 align:middle line:84% And so we're assuming that these particles are going 00:01:51.760 --> 00:01:53.710 align:middle line:90% in roughly straight lines. 00:01:53.710 --> 00:02:02.390 align:middle line:90% 00:02:02.390 --> 00:02:04.220 align:middle line:84% And we also wanted to distinguish 00:02:04.220 --> 00:02:07.310 align:middle line:84% between the time t prime at which 00:02:07.310 --> 00:02:09.350 align:middle line:84% the particle is scattering the radiation 00:02:09.350 --> 00:02:12.350 align:middle line:84% and the time t at which we're observing the radiation. 00:02:12.350 --> 00:02:14.660 align:middle line:84% And we found out through a little bit of algebra 00:02:14.660 --> 00:02:25.081 align:middle line:84% that t prime is equal to 1 minus i hat dot beta over 1 minus 00:02:25.081 --> 00:02:29.710 align:middle line:84% s hat dot beta, where beta in this case 00:02:29.710 --> 00:02:34.240 align:middle line:84% is defined as the velocity over the speed of light. 00:02:34.240 --> 00:02:36.980 align:middle line:84% And this was effectively the origin of our double Doppler 00:02:36.980 --> 00:02:37.480 align:middle line:90% shift. 00:02:37.480 --> 00:02:39.040 align:middle line:84% The particle sees a Doppler shift 00:02:39.040 --> 00:02:41.560 align:middle line:84% of the wave that's incident on it, 00:02:41.560 --> 00:02:43.390 align:middle line:84% and then the observer sees a Doppler shift 00:02:43.390 --> 00:02:47.460 align:middle line:84% of the emitted wave because the particle is moving. 00:02:47.460 --> 00:02:51.530 align:middle line:84% And we found that the electric field-- 00:02:51.530 --> 00:02:55.400 align:middle line:84% if we said that we had some incident electric fields E sub 00:02:55.400 --> 00:03:02.620 align:middle line:84% i at R and t prime, which was equal to some electric field 00:03:02.620 --> 00:03:04.678 align:middle line:90% strength and some polarization. 00:03:04.678 --> 00:03:06.220 align:middle line:84% That's why this is the little vector. 00:03:06.220 --> 00:03:08.053 align:middle line:84% And we're going to assume it's a plane wave, 00:03:08.053 --> 00:03:16.970 align:middle line:84% so it's just cosine of Ki dot r minus omega i t 00:03:16.970 --> 00:03:18.390 align:middle line:90% prime, like that. 00:03:18.390 --> 00:03:20.750 align:middle line:84% So it's this wave with the incident K 00:03:20.750 --> 00:03:23.780 align:middle line:84% vector and the incident wave frequency. 00:03:23.780 --> 00:03:31.280 align:middle line:84% Then our scattered radiation that we observe at R at time t, 00:03:31.280 --> 00:03:33.200 align:middle line:84% which is not the same as t prime, 00:03:33.200 --> 00:03:35.690 align:middle line:84% was going to be equal to the classical electron 00:03:35.690 --> 00:03:39.830 align:middle line:84% radius over the distance, because this thing is scattering 00:03:39.830 --> 00:03:41.780 align:middle line:90% into 4 pi steradians. 00:03:41.780 --> 00:03:46.070 align:middle line:84% And then there was a factor that looked like s hat cross s hat 00:03:46.070 --> 00:03:51.678 align:middle line:90% cross the incident radiation-- 00:03:51.678 --> 00:03:56.790 align:middle line:84% whoop-- at r prime and t prime like this. 00:03:56.790 --> 00:03:59.450 align:middle line:84% So we scatter some of that incident radiation. 00:03:59.450 --> 00:04:01.130 align:middle line:84% We scatter it into a shape that's 00:04:01.130 --> 00:04:03.860 align:middle line:84% given by this s hat cross s hat cross. 00:04:03.860 --> 00:04:06.230 align:middle line:84% And later on I'll start referring 00:04:06.230 --> 00:04:12.080 align:middle line:84% to this, including that second cross product, as just a tensor 00:04:12.080 --> 00:04:18.079 align:middle line:84% capital pi, which transforms the incoming electric field here. 00:04:18.079 --> 00:04:22.460 align:middle line:84% And we said that if the velocity of our particle-- this v here-- 00:04:22.460 --> 00:04:24.410 align:middle line:84% is equal to 0-- if we're just dealing 00:04:24.410 --> 00:04:27.110 align:middle line:84% with a stationary particle-- then 00:04:27.110 --> 00:04:31.460 align:middle line:84% our scattered electric field that we observe at our detector 00:04:31.460 --> 00:04:35.920 align:middle line:84% is simply going to have a very simple form, which 00:04:35.920 --> 00:04:44.583 align:middle line:84% has-- well, rE on Ri acting on the strength 00:04:44.583 --> 00:04:45.500 align:middle line:90% of the electric field. 00:04:45.500 --> 00:04:48.410 align:middle line:84% That gives us that nice doughnut-shaped scattering 00:04:48.410 --> 00:04:49.010 align:middle line:90% pattern. 00:04:49.010 --> 00:04:52.640 align:middle line:84% And then we'll have a cosine factor here that simply looks 00:04:52.640 --> 00:04:58.950 align:middle line:84% like Ki dot r of 0-- r of 0 because the particle 00:04:58.950 --> 00:05:01.830 align:middle line:84% isn't moving, so that's actually r of all time-- 00:05:01.830 --> 00:05:10.240 align:middle line:84% minus omega i t plus a factor Ki dot r. 00:05:10.240 --> 00:05:17.862 align:middle line:84% And this final Ki dot r factor here is for all our particles 00:05:17.862 --> 00:05:20.070 align:middle line:84% because we basically said they all pick up that phase 00:05:20.070 --> 00:05:20.873 align:middle line:90% as we go here. 00:05:20.873 --> 00:05:22.290 align:middle line:84% But actually all the particles are 00:05:22.290 --> 00:05:25.420 align:middle line:84% clustered in a little plasma around about here. 00:05:25.420 --> 00:05:26.700 align:middle line:90% So this was quite boring. 00:05:26.700 --> 00:05:28.710 align:middle line:84% We just get a scattered wave, which 00:05:28.710 --> 00:05:35.840 align:middle line:84% has the same frequency and K vector as the incident wave. 00:05:35.840 --> 00:05:42.500 align:middle line:84% But if we then allow our particles to move around, 00:05:42.500 --> 00:05:53.040 align:middle line:84% we found out that we get Doppler shifts, as we would expect. 00:05:53.040 --> 00:05:54.990 align:middle line:84% And we're also dealing with a regime 00:05:54.990 --> 00:06:00.720 align:middle line:84% where we're looking at elastic collisions. 00:06:00.720 --> 00:06:04.940 align:middle line:84% So this elastic collisions means that our incident K 00:06:04.940 --> 00:06:08.940 align:middle line:84% vector has roughly the same size as our scattered K vector. 00:06:08.940 --> 00:06:12.560 align:middle line:84% So we don't change the momentum of our photon very much. 00:06:12.560 --> 00:06:17.010 align:middle line:84% And this is-- bless you-- valid when our photon energy is much, 00:06:17.010 --> 00:06:18.290 align:middle line:90% much less than-- 00:06:18.290 --> 00:06:20.980 align:middle line:90% 00:06:20.980 --> 00:06:22.990 align:middle line:84% I've written it in terms of particle velocity-- 00:06:22.990 --> 00:06:28.570 align:middle line:90% interesting-- of me ve squared. 00:06:28.570 --> 00:06:32.300 align:middle line:84% But I imagine this is also true when it's non-relativistic 00:06:32.300 --> 00:06:36.470 align:middle line:84% as well, so I'll just write that down as well. 00:06:36.470 --> 00:06:38.900 align:middle line:84% If we violate this limit-- if we start 00:06:38.900 --> 00:06:41.468 align:middle line:84% scattering energy photons-- we get Compton scattering, not 00:06:41.468 --> 00:06:42.260 align:middle line:90% Thomson scattering. 00:06:42.260 --> 00:06:44.100 align:middle line:84% We have to do a full relativistic treatment. 00:06:44.100 --> 00:06:47.420 align:middle line:84% So this elastic collisions constraint 00:06:47.420 --> 00:06:50.900 align:middle line:84% comes down to us scattering off fluctuations 00:06:50.900 --> 00:06:53.060 align:middle line:84% with a K vector, which is defined 00:06:53.060 --> 00:06:55.910 align:middle line:84% as the difference between the scattered K 00:06:55.910 --> 00:07:00.230 align:middle line:84% vector and the incident K vector and a frequency, which 00:07:00.230 --> 00:07:02.990 align:middle line:84% is equal to the difference in the scattered frequency 00:07:02.990 --> 00:07:05.090 align:middle line:90% and the incident frequency. 00:07:05.090 --> 00:07:09.920 align:middle line:84% And this difference is also equal to the scattering K 00:07:09.920 --> 00:07:13.430 align:middle line:84% vector-- the difference between the scattered and the incident 00:07:13.430 --> 00:07:16.320 align:middle line:84% dotted into the velocity of the particle. 00:07:16.320 --> 00:07:18.800 align:middle line:84% So this looks like the classic Doppler shift 00:07:18.800 --> 00:07:21.240 align:middle line:90% that you're used to seeing. 00:07:21.240 --> 00:07:28.040 align:middle line:84% So the punchline of all of this is 00:07:28.040 --> 00:07:38.940 align:middle line:84% if you input a wave with Ki omega i, 00:07:38.940 --> 00:07:41.590 align:middle line:84% you put your laser beam through the plasma, 00:07:41.590 --> 00:07:51.477 align:middle line:84% and you measure some scattering in some direction Ks-- 00:07:51.477 --> 00:07:53.060 align:middle line:84% which, remember, you have control over 00:07:53.060 --> 00:07:55.460 align:middle line:84% because you get to choose where you put your spectrometer 00:07:55.460 --> 00:07:56.870 align:middle line:90% around your plasma here-- 00:07:56.870 --> 00:08:00.680 align:middle line:84% and you measure it on a spectrometer with some frequency 00:08:00.680 --> 00:08:04.700 align:middle line:84% omega s, then you can infer that, in order 00:08:04.700 --> 00:08:07.250 align:middle line:84% for you to be able to see this scattered light 00:08:07.250 --> 00:08:09.140 align:middle line:84% from your original probe signal, there 00:08:09.140 --> 00:08:13.460 align:middle line:84% has to be something within the plasma capable of carrying away 00:08:13.460 --> 00:08:15.980 align:middle line:84% a momentum K and an energy omega, 00:08:15.980 --> 00:08:19.220 align:middle line:84% or giving a momentum K and an energy omega 00:08:19.220 --> 00:08:22.060 align:middle line:90% to your photon here. 00:08:22.060 --> 00:08:27.640 align:middle line:84% And the strength of the signal, so the intensity at omega and K, 00:08:27.640 --> 00:08:31.960 align:middle line:84% is going to be proportional to the number of scattering 00:08:31.960 --> 00:08:34.220 align:middle line:90% things-- 00:08:34.220 --> 00:08:35.820 align:middle line:90% scatterers. 00:08:35.820 --> 00:08:39.850 align:middle line:84% And these scatterers could be, for example, particles, 00:08:39.850 --> 00:08:45.180 align:middle line:84% so electrons, or they could be waves, 00:08:45.180 --> 00:08:47.690 align:middle line:84% and I'll often refer to these waves as modes. 00:08:47.690 --> 00:08:52.700 align:middle line:90% 00:08:52.700 --> 00:08:57.290 align:middle line:84% And we will get on to-- and these modes in this case 00:08:57.290 --> 00:09:02.090 align:middle line:84% have an apparent velocity, which is equal to omega/K, 00:09:02.090 --> 00:09:03.762 align:middle line:84% so that's our phase velocity here. 00:09:03.762 --> 00:09:06.470 align:middle line:84% So this was a very hand-wavy way of trying to derive all of this. 00:09:06.470 --> 00:09:07.970 align:middle line:84% We're going to make it more rigorous 00:09:07.970 --> 00:09:10.280 align:middle line:84% by going through some Fourier transformations of all 00:09:10.280 --> 00:09:11.390 align:middle line:90% of these equations. 00:09:11.390 --> 00:09:14.450 align:middle line:84% But that's just a recap of what we talked about last week, so-- 00:09:14.450 --> 00:09:15.150 align:middle line:90% questions? 00:09:15.150 --> 00:09:18.910 align:middle line:90% 00:09:18.910 --> 00:09:19.410 align:middle line:90% Yes? 00:09:19.410 --> 00:09:23.785 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:09:23.785 --> 00:09:25.410 align:middle line:84% JACK HARE: So there is no vector omega. 00:09:25.410 --> 00:09:26.905 align:middle line:90% Omega is a scalar [AUDIO OUT]. 00:09:26.905 --> 00:09:27.405 align:middle line:90% That's OK. 00:09:27.405 --> 00:09:33.545 align:middle line:90% 00:09:33.545 --> 00:09:38.433 align:middle line:84% What I like to think of them as is the momentum-- if you're 00:09:38.433 --> 00:09:40.100 align:middle line:84% thinking of it as in a particle picture, 00:09:40.100 --> 00:09:43.310 align:middle line:84% then h bar K and h bar omega is the energy and momentum 00:09:43.310 --> 00:09:46.430 align:middle line:84% of the electron that scatters them. 00:09:46.430 --> 00:09:48.380 align:middle line:84% If you're thinking about it as a wave, 00:09:48.380 --> 00:09:51.140 align:middle line:84% you don't have to times it by h bar to get that. 00:09:51.140 --> 00:09:54.050 align:middle line:84% You can just say at some wave with K and omega-- 00:09:54.050 --> 00:09:58.260 align:middle line:84% so like a sound wave-- omega equals sound speed times K, 00:09:58.260 --> 00:09:58.760 align:middle line:90% for example. 00:09:58.760 --> 00:10:01.910 align:middle line:84% And we'll talk a lot more about that later on, so. 00:10:01.910 --> 00:10:04.280 align:middle line:84% But, yeah, what these represent is something 00:10:04.280 --> 00:10:06.672 align:middle line:90% that can scatter this light. 00:10:06.672 --> 00:10:08.630 align:middle line:84% If you have no plasma and you fire a laser beam 00:10:08.630 --> 00:10:10.430 align:middle line:84% through a chamber, you really shouldn't 00:10:10.430 --> 00:10:12.410 align:middle line:84% expect to see any scattered light at some random angle 00:10:12.410 --> 00:10:14.493 align:middle line:84% there because there's nothing there to scatter it. 00:10:14.493 --> 00:10:16.580 align:middle line:84% So if you see some scattering, you have to say, 00:10:16.580 --> 00:10:18.890 align:middle line:84% aha, there must be somewhere something in that chamber 00:10:18.890 --> 00:10:21.920 align:middle line:84% to scatter it, and it must be able to change 00:10:21.920 --> 00:10:25.790 align:middle line:84% the momentum of my photons by K and their energy by omega. 00:10:25.790 --> 00:10:27.740 align:middle line:84% And that limits the sorts of things that can 00:10:27.740 --> 00:10:30.470 align:middle line:90% exist inside your plasma, so. 00:10:30.470 --> 00:10:32.880 align:middle line:84% Especially for the waves and modes, 00:10:32.880 --> 00:10:35.310 align:middle line:84% they'll only be scattering at very discrete values 00:10:35.310 --> 00:10:38.960 align:middle line:84% of omega and K given by the dispersion wave. 00:10:38.960 --> 00:10:41.730 align:middle line:90% Yeah? 00:10:41.730 --> 00:10:51.620 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:10:51.620 --> 00:10:53.180 align:middle line:84% JACK HARE: Oh, you mean like these? 00:10:53.180 --> 00:10:53.763 align:middle line:90% AUDIENCE: yeah 00:10:53.763 --> 00:10:56.240 align:middle line:84% JACK HARE: We'll get on to that in a great deal of detail 00:10:56.240 --> 00:10:56.907 align:middle line:90% later on. 00:10:56.907 --> 00:11:00.950 align:middle line:84% So these four lectures are like a wave washing on the shore. 00:11:00.950 --> 00:11:03.590 align:middle line:84% So we're going to derive Thomson scattering three times. 00:11:03.590 --> 00:11:05.870 align:middle line:84% We did the first time last week, we're 00:11:05.870 --> 00:11:07.610 align:middle line:84% going to do the second time today, 00:11:07.610 --> 00:11:09.360 align:middle line:84% and we'll start on the third time as well. 00:11:09.360 --> 00:11:11.180 align:middle line:84% And each one-- the previous derivation 00:11:11.180 --> 00:11:14.750 align:middle line:84% is just a simplification of the next derivation. 00:11:14.750 --> 00:11:18.272 align:middle line:84% And previously, we got this very, very rough picture 00:11:18.272 --> 00:11:19.730 align:middle line:84% that, if we have a single particle, 00:11:19.730 --> 00:11:22.220 align:middle line:84% we might be able to say something about the particle's 00:11:22.220 --> 00:11:24.770 align:middle line:84% properties, its velocity, if we see 00:11:24.770 --> 00:11:27.958 align:middle line:84% scattering at a certain frequency and wave vector. 00:11:27.958 --> 00:11:30.125 align:middle line:84% Next, we're going to do incoherent scattering, which 00:11:30.125 --> 00:11:31.792 align:middle line:84% is where we scatter off the distribution 00:11:31.792 --> 00:11:33.290 align:middle line:90% function of electrons. 00:11:33.290 --> 00:11:35.090 align:middle line:84% But these electrons are not correlated. 00:11:35.090 --> 00:11:36.950 align:middle line:90% They're all moving incoherently. 00:11:36.950 --> 00:11:38.780 align:middle line:84% And so we basically get scattering of each 00:11:38.780 --> 00:11:40.220 align:middle line:90% of the electrons individually. 00:11:40.220 --> 00:11:42.470 align:middle line:84% And then, finally, we're going to do collective scattering, 00:11:42.470 --> 00:11:44.090 align:middle line:84% where we scatter off the collective motion 00:11:44.090 --> 00:11:46.215 align:middle line:84% of the electrons where they're all moving together, 00:11:46.215 --> 00:11:49.790 align:middle line:90% which is a wave, so yeah. 00:11:49.790 --> 00:11:50.603 align:middle line:90% Other questions? 00:11:50.603 --> 00:11:51.270 align:middle line:90% Anything online? 00:11:51.270 --> 00:11:56.950 align:middle line:90% 00:11:56.950 --> 00:12:00.640 align:middle line:84% So before we get going with Fourier transforms, 00:12:00.640 --> 00:12:03.065 align:middle line:84% I'm going to write down some assumptions. 00:12:03.065 --> 00:12:04.690 align:middle line:84% And these assumptions don't necessarily 00:12:04.690 --> 00:12:07.942 align:middle line:84% have to be true for every plasma that we're dealing with. 00:12:07.942 --> 00:12:09.400 align:middle line:84% But they are assumptions that we've 00:12:09.400 --> 00:12:11.772 align:middle line:84% made in order to do this Thomson scattering derivation. 00:12:11.772 --> 00:12:13.480 align:middle line:84% If you want to violate these assumptions, 00:12:13.480 --> 00:12:17.020 align:middle line:84% you need to go and check where this assumption comes in 00:12:17.020 --> 00:12:21.300 align:middle line:84% and re-derive your Thomson scattering. 00:12:21.300 --> 00:12:24.570 align:middle line:84% So first of all, we are scattering 00:12:24.570 --> 00:12:34.300 align:middle line:84% from some volume V, which has N electrons inside it. 00:12:34.300 --> 00:12:36.760 align:middle line:84% The electrons have a charge of e minus, 00:12:36.760 --> 00:12:43.405 align:middle line:84% and there are N/Z ions with a charge of Ze. 00:12:43.405 --> 00:12:49.880 align:middle line:90% 00:12:49.880 --> 00:12:51.150 align:middle line:90% This is pretty standard. 00:12:51.150 --> 00:12:54.960 align:middle line:84% But this is effectively a statement of quasi-neutrality. 00:12:54.960 --> 00:13:01.720 align:middle line:84% Secondly, we are mostly going to drop relativistic corrections. 00:13:01.720 --> 00:13:11.980 align:middle line:90% 00:13:11.980 --> 00:13:23.070 align:middle line:84% So if we want to look at the fully non-relativistic case, 00:13:23.070 --> 00:13:29.860 align:middle line:84% we will drop terms on the order of V/c. 00:13:29.860 --> 00:13:31.600 align:middle line:84% So if we do some sort of expansion 00:13:31.600 --> 00:13:33.980 align:middle line:84% and we see a term with V/c, we'll just drop it. 00:13:33.980 --> 00:13:36.040 align:middle line:84% So we're only keeping the zeroth-order terms. 00:13:36.040 --> 00:13:44.880 align:middle line:84% If we want to do this sort of quote marks "relativistic," 00:13:44.880 --> 00:13:51.090 align:middle line:84% then we will keep the order of V/c terms, 00:13:51.090 --> 00:13:55.770 align:middle line:84% but we'll drop the terms on the order of V squared upon c 00:13:55.770 --> 00:13:56.270 align:middle line:90% squared. 00:13:56.270 --> 00:13:59.330 align:middle line:84% So you can see this is like a Taylor expansion. 00:13:59.330 --> 00:14:01.820 align:middle line:84% We're going to claim that they're much less than 1. 00:14:01.820 --> 00:14:06.848 align:middle line:84% If you want to do the really relativistic, 00:14:06.848 --> 00:14:08.390 align:middle line:84% because obviously this still neglects 00:14:08.390 --> 00:14:13.410 align:middle line:84% some relativistic terms, then you 00:14:13.410 --> 00:14:16.433 align:middle line:84% should just go and derive Compton scattering instead, 00:14:16.433 --> 00:14:17.600 align:middle line:90% which we're not going to do. 00:14:17.600 --> 00:14:21.520 align:middle line:90% 00:14:21.520 --> 00:14:23.050 align:middle line:90% I cannot stress this enough. 00:14:23.050 --> 00:14:25.885 align:middle line:90% There will be no quantum. 00:14:25.885 --> 00:14:26.830 align:middle line:90% I don't like it. 00:14:26.830 --> 00:14:32.990 align:middle line:90% 00:14:32.990 --> 00:14:43.370 align:middle line:90% There's no scattering from ions. 00:14:43.370 --> 00:14:45.050 align:middle line:90% We discussed this already. 00:14:45.050 --> 00:14:48.020 align:middle line:84% Because of the mass difference, the ions 00:14:48.020 --> 00:14:50.060 align:middle line:90% scatter much, much less. 00:14:50.060 --> 00:14:53.062 align:middle line:84% We will no longer think about the scattering directly 00:14:53.062 --> 00:14:55.520 align:middle line:84% from the ions, but we will think a lot about the scattering 00:14:55.520 --> 00:14:58.305 align:middle line:84% from electrons which are dragged around by the ions. 00:14:58.305 --> 00:15:04.010 align:middle line:90% 00:15:04.010 --> 00:15:07.430 align:middle line:84% We're going to assume that our observer is at a distance 00:15:07.430 --> 00:15:12.680 align:middle line:84% R which is much, much larger than the volume within which 00:15:12.680 --> 00:15:15.690 align:middle line:84% the plasma is contained to the third power. 00:15:15.690 --> 00:15:17.270 align:middle line:90% So this is a length scale now-- 00:15:17.270 --> 00:15:18.710 align:middle line:90% V to 1/3. 00:15:18.710 --> 00:15:21.440 align:middle line:84% So this is effectively the max value 00:15:21.440 --> 00:15:24.670 align:middle line:90% of this little r vector here. 00:15:24.670 --> 00:15:26.910 align:middle line:84% So the plasma is small, and it's far away. 00:15:26.910 --> 00:15:29.460 align:middle line:84% And this in turn-- this length scale here-- 00:15:29.460 --> 00:15:33.420 align:middle line:84% is much, much larger than the wavelength of the incident 00:15:33.420 --> 00:15:34.830 align:middle line:90% light. 00:15:34.830 --> 00:15:37.860 align:middle line:84% So we're seeing a large number of charges within a wavelength. 00:15:37.860 --> 00:15:41.630 align:middle line:84% They're not scattering off single particles. 00:15:41.630 --> 00:15:44.480 align:middle line:84% This is equivalent to saying, of course, 00:15:44.480 --> 00:15:50.670 align:middle line:84% that there are many electrons in V-- 00:15:50.670 --> 00:15:57.560 align:middle line:84% in the volume-- and that V is small. 00:15:57.560 --> 00:16:00.070 align:middle line:84% And this is what we needed to justify that our R 00:16:00.070 --> 00:16:02.410 align:middle line:90% prime is approximately R-- 00:16:02.410 --> 00:16:03.640 align:middle line:90% our argument here. 00:16:03.640 --> 00:16:07.520 align:middle line:84% These two vectors are roughly the same. 00:16:07.520 --> 00:16:08.480 align:middle line:90% What are we up to? 00:16:08.480 --> 00:16:11.330 align:middle line:90% Five assumptions. 00:16:11.330 --> 00:16:17.980 align:middle line:84% Assumption 6-- we're going to assume that the frequency 00:16:17.980 --> 00:16:19.930 align:middle line:90% of our incident light-- 00:16:19.930 --> 00:16:23.380 align:middle line:84% the light which is being scattered-- is greater than-- 00:16:23.380 --> 00:16:27.370 align:middle line:84% significantly greater than-- the electron plasma frequency. 00:16:27.370 --> 00:16:28.420 align:middle line:90% Don't get fooled. 00:16:28.420 --> 00:16:29.980 align:middle line:84% There are going to be lots of i's showing up here. 00:16:29.980 --> 00:16:31.360 align:middle line:90% That's always for "incident." 00:16:31.360 --> 00:16:33.068 align:middle line:84% I am not thinking about the ions anymore. 00:16:33.068 --> 00:16:34.580 align:middle line:90% They're too slow and too heavy. 00:16:34.580 --> 00:16:37.030 align:middle line:84% So if we make this approximation, 00:16:37.030 --> 00:16:40.360 align:middle line:84% I can approximate the refractive index of the plasma 00:16:40.360 --> 00:16:42.970 align:middle line:84% as basically being 1 so I don't have 00:16:42.970 --> 00:16:44.920 align:middle line:90% to think about those effects. 00:16:44.920 --> 00:16:46.320 align:middle line:84% In reality, you might want to put 00:16:46.320 --> 00:16:48.820 align:middle line:84% in a small correction for this, and it's actually quite easy 00:16:48.820 --> 00:16:50.220 align:middle line:90% to do. 00:16:50.220 --> 00:16:52.100 align:middle line:90% We also have the assumption-- 00:16:52.100 --> 00:16:53.850 align:middle line:84% I think this should be number 7, frankly-- 00:16:53.850 --> 00:16:56.950 align:middle line:90% 00:16:56.950 --> 00:17:01.430 align:middle line:90% no probe attenuation. 00:17:01.430 --> 00:17:10.770 align:middle line:84% So the optical depth at tau evaluated at our probe frequency 00:17:10.770 --> 00:17:12.720 align:middle line:90% is much, much less than 1. 00:17:12.720 --> 00:17:14.250 align:middle line:84% So the beam doesn't get absorbed. 00:17:14.250 --> 00:17:16.230 align:middle line:84% This just makes it much easier to do the calculations. 00:17:16.230 --> 00:17:17.579 align:middle line:84% If your beam was getting absorbed, 00:17:17.579 --> 00:17:19.290 align:middle line:84% you'd get different amounts of scattering depending 00:17:19.290 --> 00:17:20.498 align:middle line:90% on where you are on the beam. 00:17:20.498 --> 00:17:22.960 align:middle line:84% It's a pain, so we're not going to do that. 00:17:22.960 --> 00:17:25.710 align:middle line:84% And then, finally-- we touched on this earlier-- 00:17:25.710 --> 00:17:31.470 align:middle line:84% the velocity induced from the electric field of the incident 00:17:31.470 --> 00:17:35.460 align:middle line:84% radiation we are going to assume is much, much smaller 00:17:35.460 --> 00:17:38.440 align:middle line:90% than the thermal velocity here. 00:17:38.440 --> 00:17:40.650 align:middle line:84% And so this was saying that our particles are 00:17:40.650 --> 00:17:43.350 align:middle line:84% going in straight lines at some velocity, 00:17:43.350 --> 00:17:44.610 align:middle line:90% like the thermal velocity. 00:17:44.610 --> 00:17:47.400 align:middle line:84% And the electric field is just making them wobble ever 00:17:47.400 --> 00:17:50.330 align:middle line:84% so slightly along those lines here. 00:17:50.330 --> 00:17:53.820 align:middle line:84% And this is equivalent because this equals this velocity here 00:17:53.820 --> 00:17:54.990 align:middle line:90% from the electric field-- 00:17:54.990 --> 00:17:58.710 align:middle line:84% you can work it out as being e times the strength 00:17:58.710 --> 00:18:03.460 align:middle line:84% of the electric field over m e omega i-- 00:18:03.460 --> 00:18:07.920 align:middle line:84% if you set that to be less than V thermal, 00:18:07.920 --> 00:18:10.890 align:middle line:84% then you find out that the power-- 00:18:10.890 --> 00:18:12.730 align:middle line:84% the intensity of your laser beam-- 00:18:12.730 --> 00:18:18.460 align:middle line:84% which is equal to c epsilon 0 strength of the scattering 00:18:18.460 --> 00:18:20.560 align:middle line:90% electric field squared-- 00:18:20.560 --> 00:18:22.910 align:middle line:90% it limits this power. 00:18:22.910 --> 00:18:26.590 align:middle line:84% I won't derive it directly, but-- 00:18:26.590 --> 00:18:27.910 align:middle line:90% so you can't use-- 00:18:27.910 --> 00:18:29.663 align:middle line:84% in order for this assumption to be valid, 00:18:29.663 --> 00:18:31.330 align:middle line:84% which we'll use a lot in our derivation, 00:18:31.330 --> 00:18:33.140 align:middle line:90% our laser cannot be too strong. 00:18:33.140 --> 00:18:35.020 align:middle line:84% So if you end up using a very strong laser, 00:18:35.020 --> 00:18:40.600 align:middle line:84% you have to re-derive this using the full orbit of your particle, 00:18:40.600 --> 00:18:43.300 align:middle line:84% which includes this strongly perturbing electric field, 00:18:43.300 --> 00:18:44.150 align:middle line:90% and that's a pain. 00:18:44.150 --> 00:18:45.490 align:middle line:84% So obviously, that's not a good reason 00:18:45.490 --> 00:18:46.660 align:middle line:90% not to use a very strong laser. 00:18:46.660 --> 00:18:48.460 align:middle line:84% You want to use a nice, strong laser in order 00:18:48.460 --> 00:18:50.110 align:middle line:84% to get this Thomson scattering effect. 00:18:50.110 --> 00:18:54.610 align:middle line:84% But you should be careful about whether your Thomson scattering 00:18:54.610 --> 00:18:57.550 align:middle line:84% formalism is still valid if you use a very strong laser. 00:18:57.550 --> 00:19:02.220 align:middle line:84% Any questions on any of these assumptions? 00:19:02.220 --> 00:19:02.782 align:middle line:90% Yes? 00:19:02.782 --> 00:19:06.130 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:19:06.130 --> 00:19:15.020 align:middle line:90% 00:19:15.020 --> 00:19:17.808 align:middle line:84% We are just dealing with these particles as like points. 00:19:17.808 --> 00:19:19.850 align:middle line:84% So we're going to use a Klimontovich distribution 00:19:19.850 --> 00:19:22.433 align:middle line:84% function at some point, which has delta functions representing 00:19:22.433 --> 00:19:24.990 align:middle line:84% the position of the particles in space and in velocity space. 00:19:24.990 --> 00:19:26.990 align:middle line:84% And so we don't have to worry about uncertain things 00:19:26.990 --> 00:19:27.420 align:middle line:90% like that. 00:19:27.420 --> 00:19:29.180 align:middle line:84% And so we're not thinking about particles as waves. 00:19:29.180 --> 00:19:31.430 align:middle line:84% We're not thinking about a semi-classical treatment 00:19:31.430 --> 00:19:33.775 align:middle line:84% of electromagnetic radiation scattering particles. 00:19:33.775 --> 00:19:35.150 align:middle line:84% So this is a classical treatment. 00:19:35.150 --> 00:19:41.460 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:19:41.460 --> 00:19:42.960 align:middle line:84% JACK HARE: I would have thought it's 00:19:42.960 --> 00:19:45.720 align:middle line:84% when you get to short wavelengths, 00:19:45.720 --> 00:19:50.040 align:middle line:84% so where you're getting to short wavelengths, which are getting 00:19:50.040 --> 00:19:53.100 align:middle line:84% close to the classical electron radius, 00:19:53.100 --> 00:19:55.380 align:middle line:84% so not when you're getting close to the Debye length. 00:19:55.380 --> 00:19:56.370 align:middle line:84% We'll actually talk about the fact 00:19:56.370 --> 00:19:58.453 align:middle line:84% that the Debye length is really important to this. 00:19:58.453 --> 00:20:00.040 align:middle line:84% We can definitely have wavelengths 00:20:00.040 --> 00:20:01.380 align:middle line:84% which are above or below the Debye length. 00:20:01.380 --> 00:20:03.270 align:middle line:84% But I would have thought when you get down to wavelengths 00:20:03.270 --> 00:20:04.860 align:middle line:84% on the order of the classical electron radius, 00:20:04.860 --> 00:20:06.360 align:middle line:84% then you'd need a quantum treatment. 00:20:06.360 --> 00:20:07.140 align:middle line:90% Yeah. 00:20:07.140 --> 00:20:17.750 align:middle line:90% [INAUDIBLE] 00:20:17.750 --> 00:20:18.290 align:middle line:90% Depends. 00:20:18.290 --> 00:20:20.630 align:middle line:84% If your plasma is very hot, the relativistic corrections 00:20:20.630 --> 00:20:21.500 align:middle line:90% are hard to avoid. 00:20:21.500 --> 00:20:24.540 align:middle line:84% And I'll talk a little bit about the "relativistic" corrections, 00:20:24.540 --> 00:20:27.040 align:middle line:84% which are normally used in a tokamak when you get to like 10 00:20:27.040 --> 00:20:30.770 align:middle line:84% kiloelectron volts, because then your rest mass of your electrons 00:20:30.770 --> 00:20:33.920 align:middle line:84% is 500 kiloelectron volts, and 10 kiloelectron volts is not 00:20:33.920 --> 00:20:36.260 align:middle line:84% nothing, and you need to have a small correction. 00:20:36.260 --> 00:20:40.610 align:middle line:84% I would have thought this one is often quite hard to obey, 00:20:40.610 --> 00:20:42.500 align:middle line:84% but it's also quite easy to fix up. 00:20:42.500 --> 00:20:45.050 align:middle line:84% You just need to make sure that the dispersion relationship 00:20:45.050 --> 00:20:49.160 align:middle line:84% for your incoming light and your scattered light 00:20:49.160 --> 00:20:51.200 align:middle line:84% obeys this equation as well, so that it 00:20:51.200 --> 00:20:55.960 align:middle line:84% obeys a dispersion relationship where N is not equal to 1. 00:20:55.960 --> 00:20:58.720 align:middle line:84% That's really hard to achieve in a lot of plasmas 00:20:58.720 --> 00:21:00.668 align:middle line:84% if you have inverse Bremsstrahlung. 00:21:00.668 --> 00:21:02.710 align:middle line:84% Because your plasma is relatively cold and dense, 00:21:02.710 --> 00:21:04.793 align:middle line:84% then you have very strong probe attenuation that's 00:21:04.793 --> 00:21:07.270 align:middle line:90% been seen in lots of plasmas. 00:21:07.270 --> 00:21:08.350 align:middle line:90% Yeah. 00:21:08.350 --> 00:21:10.180 align:middle line:90% Quasi-neutrality is pretty good. 00:21:10.180 --> 00:21:11.530 align:middle line:90% Quantum's pretty good. 00:21:11.530 --> 00:21:14.570 align:middle line:84% No scattering from ions is pretty good. 00:21:14.570 --> 00:21:15.570 align:middle line:90% This one is pretty good. 00:21:15.570 --> 00:21:19.283 align:middle line:84% It's actually hard to get a laser that intense, so yeah. 00:21:19.283 --> 00:21:19.950 align:middle line:90% Other questions? 00:21:19.950 --> 00:21:20.730 align:middle line:90% Yeah? 00:21:20.730 --> 00:21:22.980 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:21:22.980 --> 00:21:25.280 align:middle line:84% JACK HARE: Optical wavelengths, because they 00:21:25.280 --> 00:21:26.817 align:middle line:90% are nice to work with. 00:21:26.817 --> 00:21:29.498 align:middle line:84% [CHUCKLES] But people do X-ray Thomson scattering 00:21:29.498 --> 00:21:31.040 align:middle line:84% as well, which is really complicated, 00:21:31.040 --> 00:21:32.082 align:middle line:90% and we won't get into it. 00:21:32.082 --> 00:21:34.292 align:middle line:84% But that's where things get a bit more quantum. 00:21:34.292 --> 00:21:36.500 align:middle line:84% But, yeah, optical wavelengths are nice to work with. 00:21:36.500 --> 00:21:39.080 align:middle line:84% People have done Thomson scattering with CO2 lasers, 00:21:39.080 --> 00:21:40.860 align:middle line:90% which are about 10 microns. 00:21:40.860 --> 00:21:44.587 align:middle line:84% So that's about 10 times longer than optical wavelength. 00:21:44.587 --> 00:21:46.670 align:middle line:84% And then the first Thomson scattering measurements 00:21:46.670 --> 00:21:49.280 align:middle line:84% were actually done off the ionosphere by Salpeter, 00:21:49.280 --> 00:21:50.600 align:middle line:90% and he used radar. 00:21:50.600 --> 00:21:53.720 align:middle line:84% So he did Thomson scattering off the plasma 00:21:53.720 --> 00:21:56.552 align:middle line:84% that surrounds the Earth and measured its properties, but-- 00:21:56.552 --> 00:21:58.260 align:middle line:84% that's when the theory was first derived. 00:21:58.260 --> 00:22:01.740 align:middle line:84% But in a lab, we don't use radar because our plasmas 00:22:01.740 --> 00:22:04.178 align:middle line:90% are smaller, so-- yeah? 00:22:04.178 --> 00:22:07.850 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:22:07.850 --> 00:22:09.440 align:middle line:90% JACK HARE: No, this was just-- 00:22:09.440 --> 00:22:11.773 align:middle line:84% the previous derivation was just to give you a heuristic 00:22:11.773 --> 00:22:13.745 align:middle line:90% feel for a problem-- yeah. 00:22:13.745 --> 00:22:15.870 align:middle line:84% The previous derivation was just a single particle. 00:22:15.870 --> 00:22:18.020 align:middle line:84% So none of these really talk about single particle-- 00:22:18.020 --> 00:22:18.770 align:middle line:90% well, no, they do. 00:22:18.770 --> 00:22:20.510 align:middle line:84% Look, we've put lots of electrons 00:22:20.510 --> 00:22:21.800 align:middle line:90% inside our volume, yeah. 00:22:21.800 --> 00:22:25.090 align:middle line:90% 00:22:25.090 --> 00:22:36.270 align:middle line:90% AUDIENCE: The last [INAUDIBLE] 00:22:36.270 --> 00:22:37.211 align:middle line:90% 00:22:37.211 --> 00:22:37.836 align:middle line:90% JACK HARE: Yes. 00:22:37.836 --> 00:22:37.863 align:middle line:90% Yeah. 00:22:37.863 --> 00:22:38.363 align:middle line:90% Yeah. 00:22:38.363 --> 00:22:40.500 align:middle line:84% We're going to use this as the orbit-- 00:22:40.500 --> 00:22:43.080 align:middle line:84% the "orbit"-- of the particle as its equation of motion. 00:22:43.080 --> 00:22:46.350 align:middle line:84% And if that's not true, then you need to solve. 00:22:46.350 --> 00:22:47.410 align:middle line:90% It will be very-- 00:22:47.410 --> 00:22:48.660 align:middle line:90% it's already quite non-linear. 00:22:48.660 --> 00:22:50.610 align:middle line:84% It'll be very non-linear at that point 00:22:50.610 --> 00:22:53.197 align:middle line:84% because it will depend on how the electric field scatters. 00:22:53.197 --> 00:22:54.780 align:middle line:84% The scattered electric field will also 00:22:54.780 --> 00:22:56.520 align:middle line:84% distort the orbit of the particle. 00:22:56.520 --> 00:22:59.658 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:22:59.658 --> 00:23:00.700 align:middle line:90% JACK HARE: Yeah, exactly. 00:23:00.700 --> 00:23:03.260 align:middle line:90% 00:23:03.260 --> 00:23:04.490 align:middle line:90% Any questions on that? 00:23:04.490 --> 00:23:15.160 align:middle line:90% 00:23:15.160 --> 00:23:17.770 align:middle line:84% So let's talk about scattering for multiple electrons. 00:23:17.770 --> 00:23:29.440 align:middle line:90% 00:23:29.440 --> 00:23:36.500 align:middle line:84% So imagine I have now my incident wave coming in, Ki, 00:23:36.500 --> 00:23:40.944 align:middle line:84% and I'm still observing along Ks. 00:23:40.944 --> 00:23:43.870 align:middle line:90% 00:23:43.870 --> 00:23:48.930 align:middle line:84% And now I have a collection of particles here, 00:23:48.930 --> 00:23:55.820 align:middle line:84% and they are all oscillating, and they are all 00:23:55.820 --> 00:23:58.080 align:middle line:90% scattering light. 00:23:58.080 --> 00:24:04.230 align:middle line:84% And so I'm getting a load of e scattered from each of these. 00:24:04.230 --> 00:24:06.360 align:middle line:84% And so now the total scattered power, 00:24:06.360 --> 00:24:13.100 align:middle line:84% which is, as you remember, dP d omega s d-- 00:24:13.100 --> 00:24:15.320 align:middle line:84% oh, no, just dP d omega s OK, fine, 00:24:15.320 --> 00:24:18.880 align:middle line:84% so we're integrating over [INAUDIBLE]. 00:24:18.880 --> 00:24:21.820 align:middle line:90% Doesn't seem quite right. 00:24:21.820 --> 00:24:23.620 align:middle line:84% [VOCALIZING]-- I'll throw this back in. 00:24:23.620 --> 00:24:26.910 align:middle line:90% 00:24:26.910 --> 00:24:28.535 align:middle line:84% What did we do in the previous lecture? 00:24:28.535 --> 00:24:34.950 align:middle line:90% 00:24:34.950 --> 00:24:35.450 align:middle line:90% No. 00:24:35.450 --> 00:24:36.367 align:middle line:90% I'll leave it as that. 00:24:36.367 --> 00:24:42.510 align:middle line:90% 00:24:42.510 --> 00:24:49.200 align:middle line:84% That is now going to be equal to R squared c epsilon 0. 00:24:49.200 --> 00:24:55.200 align:middle line:84% Previously, we had a term that looked like e scattered dot e 00:24:55.200 --> 00:24:57.080 align:middle line:90% scattered star. 00:24:57.080 --> 00:24:59.550 align:middle line:90% That was for a single particle. 00:24:59.550 --> 00:25:02.540 align:middle line:84% So now we replace this with a term 00:25:02.540 --> 00:25:14.890 align:middle line:84% that looks like the sum over j and l to N of ej dot e l star. 00:25:14.890 --> 00:25:17.460 align:middle line:90% 00:25:17.460 --> 00:25:20.370 align:middle line:84% This is basically summing up all of these electric fields, 00:25:20.370 --> 00:25:27.075 align:middle line:84% and these electric fields now interfere with each other. 00:25:27.075 --> 00:25:29.230 align:middle line:84% I shouldn't have put these vertical arrows here. 00:25:29.230 --> 00:25:31.860 align:middle line:84% This is a time average, remember. 00:25:31.860 --> 00:25:33.510 align:middle line:90% There we go-- like that. 00:25:33.510 --> 00:25:36.480 align:middle line:90% 00:25:36.480 --> 00:25:38.640 align:middle line:84% And it turns out we can split this into two terms. 00:25:38.640 --> 00:25:40.725 align:middle line:90% So we split this into the term-- 00:25:40.725 --> 00:25:44.670 align:middle line:84% they're both R squared c epsilon 0. 00:25:44.670 --> 00:25:50.490 align:middle line:84% And then we have a term that looks like N times es squared. 00:25:50.490 --> 00:25:55.410 align:middle line:84% And this comes from the sum where we have j equal to l. 00:25:55.410 --> 00:25:57.870 align:middle line:84% So this is effectively the scattered light 00:25:57.870 --> 00:26:02.500 align:middle line:84% of each of the electrons summed up-- 00:26:02.500 --> 00:26:04.840 align:middle line:84% and there is N electrons scattering-- 00:26:04.840 --> 00:26:19.830 align:middle line:84% plus a term that looks like N times N minus 1 ej dot e l. 00:26:19.830 --> 00:26:22.940 align:middle line:84% And this is for j not equal to l. 00:26:22.940 --> 00:26:27.890 align:middle line:90% 00:26:27.890 --> 00:26:30.920 align:middle line:84% So we've split this sum up into the terms 00:26:30.920 --> 00:26:33.080 align:middle line:84% where we're just talking about the electric field 00:26:33.080 --> 00:26:35.430 align:middle line:84% from the same particle interfering with itself, which 00:26:35.430 --> 00:26:38.720 align:middle line:84% is obviously constructive, and the terms where these are not 00:26:38.720 --> 00:26:43.380 align:middle line:84% equal, so when the scattered electric field from one particle 00:26:43.380 --> 00:26:45.390 align:middle line:84% interferes with the scattered electric field 00:26:45.390 --> 00:26:48.380 align:middle line:90% from another particle. 00:26:48.380 --> 00:26:50.330 align:middle line:84% You might notice that there's an N minus 1 00:26:50.330 --> 00:26:53.300 align:middle line:84% here because once we've picked which particle is j, 00:26:53.300 --> 00:26:56.180 align:middle line:84% there's only N minus 1 particles to be L. 00:26:56.180 --> 00:26:59.930 align:middle line:84% But you may also realize that in a real plasma, 00:26:59.930 --> 00:27:01.820 align:middle line:84% the difference between N and N minus 1 00:27:01.820 --> 00:27:04.320 align:middle line:84% is very small because there's a lot of particles. 00:27:04.320 --> 00:27:08.980 align:middle line:84% So we can just write this as N squared most of the time. 00:27:08.980 --> 00:27:11.220 align:middle line:84% And so in this expression, this first term 00:27:11.220 --> 00:27:15.580 align:middle line:84% here is called the incoherent term 00:27:15.580 --> 00:27:20.440 align:middle line:84% because this just depends on each of the particles 00:27:20.440 --> 00:27:21.440 align:middle line:90% independently. 00:27:21.440 --> 00:27:25.660 align:middle line:84% And this second term is called the coherent term. 00:27:25.660 --> 00:27:27.403 align:middle line:84% And it's called the coherent term 00:27:27.403 --> 00:27:29.320 align:middle line:84% because, unless there is something correlating 00:27:29.320 --> 00:27:31.570 align:middle line:84% these particles so their electric fields have 00:27:31.570 --> 00:27:33.610 align:middle line:84% some correlation, you would expect them all 00:27:33.610 --> 00:27:35.930 align:middle line:84% to be random with respect to each other. 00:27:35.930 --> 00:27:39.260 align:middle line:84% And then when you dot them into each other and average them out, 00:27:39.260 --> 00:27:42.280 align:middle line:84% you would expect that randomness to average out to zero. 00:27:42.280 --> 00:27:52.280 align:middle line:84% So to be coherent, we need correlation, 00:27:52.280 --> 00:27:55.300 align:middle line:84% and we'll talk a lot about that later on. 00:27:55.300 --> 00:27:59.020 align:middle line:84% Now, what's the smallest length scale in a plasma? 00:27:59.020 --> 00:28:04.157 align:middle line:90% 00:28:04.157 --> 00:28:05.570 align:middle line:90% Yeah? 00:28:05.570 --> 00:28:07.200 align:middle line:84% The Debye length-- yeah, exactly. 00:28:07.200 --> 00:28:08.450 align:middle line:90% That's what makes it a plasma. 00:28:08.450 --> 00:28:11.200 align:middle line:90% 00:28:11.200 --> 00:28:16.680 align:middle line:84% So if you are probing this plasma on length scales 00:28:16.680 --> 00:28:18.657 align:middle line:84% smaller than the Debye length, you're 00:28:18.657 --> 00:28:20.490 align:middle line:84% effectively going to be looking at something 00:28:20.490 --> 00:28:21.630 align:middle line:90% that isn't a plasma. 00:28:21.630 --> 00:28:25.620 align:middle line:84% And by probing, I mean if our wavelength of our incoming light 00:28:25.620 --> 00:28:33.810 align:middle line:84% here, which is lambda i, which is 2 pi upon Ki-- 00:28:33.810 --> 00:28:38.310 align:middle line:84% if lambda i is smaller than the Debye length, 00:28:38.310 --> 00:28:41.665 align:middle line:84% then we end up with incoherent scattering. 00:28:41.665 --> 00:28:44.780 align:middle line:90% 00:28:44.780 --> 00:28:48.680 align:middle line:84% So this wavelength here is smaller than the length scale 00:28:48.680 --> 00:28:52.460 align:middle line:84% on which the particles can organize themselves and actually 00:28:52.460 --> 00:28:53.270 align:middle line:90% look like a plasma. 00:28:53.270 --> 00:28:56.640 align:middle line:84% So instead, we just end up looking like a random gas. 00:28:56.640 --> 00:28:59.000 align:middle line:84% But if we end up in the opposite limit, 00:28:59.000 --> 00:29:02.360 align:middle line:84% where the Debye length is less than lambda i, 00:29:02.360 --> 00:29:04.460 align:middle line:90% we get coherent scattering. 00:29:04.460 --> 00:29:07.130 align:middle line:84% Because on this length scale here, 00:29:07.130 --> 00:29:11.900 align:middle line:84% we end up being able to see the plasma as a coherent object. 00:29:11.900 --> 00:29:17.190 align:middle line:84% So this coherence is expressed in the form of waves here. 00:29:17.190 --> 00:29:20.270 align:middle line:84% And so we end up with a very important parameter 00:29:20.270 --> 00:29:24.590 align:middle line:84% that we call alpha, and we define alpha as 1 00:29:24.590 --> 00:29:29.630 align:middle line:90% upon K lambda Debye-- 00:29:29.630 --> 00:29:32.540 align:middle line:90% yeah, kind of fibbed here. 00:29:32.540 --> 00:29:34.550 align:middle line:84% We'll get back to that in a moment. 00:29:34.550 --> 00:29:37.430 align:middle line:84% And if you have alpha less than 1, 00:29:37.430 --> 00:29:39.170 align:middle line:90% you're in the incoherent regime. 00:29:39.170 --> 00:29:40.970 align:middle line:84% If you have alpha greater than 1, 00:29:40.970 --> 00:29:42.890 align:middle line:90% you're in the coherent regime. 00:29:42.890 --> 00:29:44.510 align:middle line:84% This shouldn't have been lambda i. 00:29:44.510 --> 00:29:46.520 align:middle line:84% This should have just been lambda, 00:29:46.520 --> 00:29:56.200 align:middle line:84% and this lambda here is 2 pi upon K, 00:29:56.200 --> 00:29:58.390 align:middle line:90% where K is that Ks minus Ki. 00:29:58.390 --> 00:30:01.110 align:middle line:90% 00:30:01.110 --> 00:30:04.050 align:middle line:84% So this is saying, if the mode that you're scattering off 00:30:04.050 --> 00:30:06.582 align:middle line:84% has a wavelength, which is less than the Debye length, 00:30:06.582 --> 00:30:08.790 align:middle line:84% then you're just scattering off individual particles. 00:30:08.790 --> 00:30:11.100 align:middle line:84% If the mode you're scattering off has a wavelength 00:30:11.100 --> 00:30:12.300 align:middle line:84% that's greater than the Debye length-- 00:30:12.300 --> 00:30:13.950 align:middle line:84% you're scattering off the full plasma-- 00:30:13.950 --> 00:30:16.140 align:middle line:90% you're scattering off the waves. 00:30:16.140 --> 00:30:19.770 align:middle line:84% And you get to choose which regime you're in because-- 00:30:19.770 --> 00:30:22.295 align:middle line:84% well, you don't get to choose how dense 00:30:22.295 --> 00:30:23.670 align:middle line:84% and how hot your plasma is, which 00:30:23.670 --> 00:30:24.962 align:middle line:90% is what goes into Debye length. 00:30:24.962 --> 00:30:31.090 align:middle line:84% But you do get to choose this K by where we place our scatterer. 00:30:31.090 --> 00:30:31.590 align:middle line:90% Let's see. 00:30:31.590 --> 00:30:34.440 align:middle line:84% I had that nice formula for K previously. 00:30:34.440 --> 00:30:39.930 align:middle line:90% 00:30:39.930 --> 00:30:52.160 align:middle line:84% Remember that K is equal to 2Ki sine theta upon 2, like that. 00:30:52.160 --> 00:30:57.010 align:middle line:84% So, by choosing my angle theta, I can make this Ki-- 00:30:57.010 --> 00:30:58.960 align:middle line:84% multiply by something arbitrarily small, 00:30:58.960 --> 00:31:01.480 align:middle line:84% I can make K arbitrarily small, and so I can 00:31:01.480 --> 00:31:03.200 align:middle line:90% make alpha arbitrarily large. 00:31:03.200 --> 00:31:06.040 align:middle line:84% So I get to choose by where I place my detector not 00:31:06.040 --> 00:31:08.560 align:middle line:84% just the scattering angle, but, also, I get 00:31:08.560 --> 00:31:10.810 align:middle line:90% to choose which regime I'm in. 00:31:10.810 --> 00:31:12.640 align:middle line:84% Now, it may be that, for some plasmas, 00:31:12.640 --> 00:31:15.438 align:middle line:84% this theta is so very, very, very close to the direction 00:31:15.438 --> 00:31:17.980 align:middle line:84% that the laser beam's pointing in that we can't realistically 00:31:17.980 --> 00:31:19.480 align:middle line:90% do coherent scattering. 00:31:19.480 --> 00:31:21.640 align:middle line:84% But there is still a regime where the scattering 00:31:21.640 --> 00:31:22.726 align:middle line:90% will be coherent there. 00:31:22.726 --> 00:31:27.577 align:middle line:90% 00:31:27.577 --> 00:31:29.410 align:middle line:84% And if this doesn't make sense, again, we're 00:31:29.410 --> 00:31:31.212 align:middle line:84% going to do it mathematically later on. 00:31:31.212 --> 00:31:33.670 align:middle line:84% But this is just to give you an idea of where we're headed. 00:31:33.670 --> 00:31:36.310 align:middle line:84% And the next thing that we're going to be focusing on 00:31:36.310 --> 00:31:38.830 align:middle line:90% is this incoherent regime. 00:31:38.830 --> 00:31:39.822 align:middle line:90% Yeah? 00:31:39.822 --> 00:31:46.090 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:31:46.090 --> 00:31:48.980 align:middle line:84% JACK HARE: It is-- it comes from the sum of all of these 00:31:48.980 --> 00:31:49.480 align:middle line:90% together. 00:31:49.480 --> 00:31:50.470 align:middle line:90% Yeah, exactly. 00:31:50.470 --> 00:31:55.120 align:middle line:84% Effectively, the wave correlates the electron density 00:31:55.120 --> 00:31:57.550 align:middle line:84% fluctuations such that, when you scatter off 00:31:57.550 --> 00:31:59.470 align:middle line:84% those electron density fluctuations, 00:31:59.470 --> 00:32:01.300 align:middle line:84% all of the electric fields from them 00:32:01.300 --> 00:32:03.760 align:middle line:84% end up at your spectrometer in phase, 00:32:03.760 --> 00:32:06.980 align:middle line:90% and they constructively add up. 00:32:06.980 --> 00:32:08.567 align:middle line:84% So if you've done X-ray spectroscopy 00:32:08.567 --> 00:32:10.150 align:middle line:84% and you've looked at Bragg scattering, 00:32:10.150 --> 00:32:12.460 align:middle line:84% some people like to look at this in terms of Bragg scattering. 00:32:12.460 --> 00:32:13.835 align:middle line:84% But I find most people these days 00:32:13.835 --> 00:32:16.070 align:middle line:84% haven't done X-ray spectroscopy and Bragg scattering, 00:32:16.070 --> 00:32:17.903 align:middle line:84% and so it's not a useful way to describe it. 00:32:17.903 --> 00:32:21.430 align:middle line:84% But if you've seen it like that, the wave makes a grating, 00:32:21.430 --> 00:32:23.470 align:middle line:84% and that grating, therefore, gives us 00:32:23.470 --> 00:32:25.780 align:middle line:90% constructive interference. 00:32:25.780 --> 00:32:26.847 align:middle line:90% Other question? 00:32:26.847 --> 00:32:34.310 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:32:34.310 --> 00:32:35.720 align:middle line:84% JACK HARE: Just how it's defined. 00:32:35.720 --> 00:32:38.178 align:middle line:84% You'll see later on-- if I think we didn't put it in there, 00:32:38.178 --> 00:32:41.130 align:middle line:84% there'd be lots of 2 pi's showing up everywhere. 00:32:41.130 --> 00:32:43.160 align:middle line:84% So when we derive this more formally, 00:32:43.160 --> 00:32:45.560 align:middle line:84% you'll see this factor alpha show up in the equations. 00:32:45.560 --> 00:32:48.440 align:middle line:84% And it will literally be a parameter alpha that multiplies 00:32:48.440 --> 00:32:50.840 align:middle line:84% the incoherent term and like a 1/alpha over alpha that 00:32:50.840 --> 00:32:52.490 align:middle line:84% multiplies the coherent term-- or the other way around-- 00:32:52.490 --> 00:32:53.198 align:middle line:90% other way around. 00:32:53.198 --> 00:32:56.360 align:middle line:84% And so, if alpha is largest, incoherent term gets suppressed, 00:32:56.360 --> 00:32:57.860 align:middle line:84% and the coherent term gets enhanced. 00:32:57.860 --> 00:33:00.805 align:middle line:84% And if I put it in with your lambda over lambda E, which 00:33:00.805 --> 00:33:02.180 align:middle line:84% is perfectly reasonable, I'd have 00:33:02.180 --> 00:33:04.250 align:middle line:84% like a 2 pi squared or something somewhere, 00:33:04.250 --> 00:33:05.460 align:middle line:90% which I don't want to do. 00:33:05.460 --> 00:33:06.980 align:middle line:90% So this is a better definition. 00:33:06.980 --> 00:33:08.730 align:middle line:90% Yeah. 00:33:08.730 --> 00:33:09.230 align:middle line:90% Yeah? 00:33:09.230 --> 00:33:18.430 align:middle line:90% 00:33:18.430 --> 00:33:18.940 align:middle line:90% Yeah. 00:33:18.940 --> 00:33:20.398 align:middle line:90% So, again, we will get to that. 00:33:20.398 --> 00:33:21.940 align:middle line:84% I don't think anything I've said here 00:33:21.940 --> 00:33:24.920 align:middle line:84% reaches the level of rigor where we can really say that or not. 00:33:24.920 --> 00:33:27.693 align:middle line:84% So this is just like a guide to where we're going next. 00:33:27.693 --> 00:33:29.860 align:middle line:84% But, like I said, we're going to derive the equation 00:33:29.860 --> 00:33:32.360 align:middle line:84% where alpha shows up explicitly as a parameter. 00:33:32.360 --> 00:33:35.500 align:middle line:84% And then we'll say, OK, depending on the value of alpha, 00:33:35.500 --> 00:33:37.468 align:middle line:84% you see more incoherent or coherent. 00:33:37.468 --> 00:33:39.760 align:middle line:84% And it's important to note that, when alpha is about 1, 00:33:39.760 --> 00:33:42.550 align:middle line:84% you have both of these spectra at the same time showing up-- 00:33:42.550 --> 00:33:43.640 align:middle line:90% both of these effects. 00:33:43.640 --> 00:33:46.210 align:middle line:84% So you're probably going to end up 00:33:46.210 --> 00:33:49.570 align:middle line:84% seeing some of both types of scattering. 00:33:49.570 --> 00:33:50.720 align:middle line:90% Yeah. 00:33:50.720 --> 00:33:51.670 align:middle line:90% Any questions on that? 00:33:51.670 --> 00:33:55.310 align:middle line:90% 00:33:55.310 --> 00:33:56.399 align:middle line:90% Yeah? 00:33:56.399 --> 00:34:07.646 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:34:07.646 --> 00:34:09.737 align:middle line:90% 00:34:09.737 --> 00:34:11.112 align:middle line:84% JACK HARE: Oh, yeah, that's fine. 00:34:11.112 --> 00:34:11.200 align:middle line:90% Don't worry about it. 00:34:11.200 --> 00:34:12.090 align:middle line:90% What's 2 pi between friends? 00:34:12.090 --> 00:34:13.889 align:middle line:84% So this is-- again, this is like, what does it 00:34:13.889 --> 00:34:15.120 align:middle line:90% mean to be much less than 1? 00:34:15.120 --> 00:34:16.953 align:middle line:84% What does it mean to be much greater than 1? 00:34:16.953 --> 00:34:18.620 align:middle line:84% Does a factor of 6 matter if I tell you 00:34:18.620 --> 00:34:21.120 align:middle line:84% something is much less than 1-- if it's much greater than 1? 00:34:21.120 --> 00:34:21.810 align:middle line:90% Yeah. 00:34:21.810 --> 00:34:24.010 align:middle line:84% If that was the point, yes, fine, but-- 00:34:24.010 --> 00:34:24.510 align:middle line:90% yeah. 00:34:24.510 --> 00:34:26.639 align:middle line:84% [CHUCKLES] I can rewrite these, I guess. 00:34:26.639 --> 00:34:29.277 align:middle line:84% I can rewrite like a k for Debye length, 00:34:29.277 --> 00:34:31.110 align:middle line:84% but then that's weird because you don't know 00:34:31.110 --> 00:34:34.530 align:middle line:90% the formula for that, so yeah. 00:34:34.530 --> 00:34:35.340 align:middle line:90% Let's keep going. 00:34:35.340 --> 00:34:39.265 align:middle line:90% 00:34:39.265 --> 00:34:40.890 align:middle line:84% I'm going to need a lot of board space. 00:34:40.890 --> 00:35:07.003 align:middle line:90% 00:35:07.003 --> 00:35:08.878 align:middle line:84% Now we're going to do the Fourier transforms. 00:35:08.878 --> 00:35:11.860 align:middle line:90% 00:35:11.860 --> 00:35:16.390 align:middle line:84% So we're looking at incoherent scattering. 00:35:16.390 --> 00:35:18.557 align:middle line:84% You won't even notice where I've made the assumption 00:35:18.557 --> 00:35:19.598 align:middle line:90% of incoherent scattering. 00:35:19.598 --> 00:35:21.520 align:middle line:84% You'll only spot it, probably, when we go back 00:35:21.520 --> 00:35:23.320 align:middle line:90% and do coherent scattering. 00:35:23.320 --> 00:35:25.900 align:middle line:84% But just to be clear, we're only going 00:35:25.900 --> 00:35:29.070 align:middle line:84% to get the incoherent component out of this treatment. 00:35:29.070 --> 00:35:35.150 align:middle line:84% So our scattered electric field measured at our detector at time 00:35:35.150 --> 00:35:39.770 align:middle line:84% t is equal to r e over R-- classical electron radius-- 00:35:39.770 --> 00:35:42.240 align:middle line:84% over the distance to the spectrometer-- 00:35:42.240 --> 00:35:45.020 align:middle line:84% this tensor pi, which, remember, is 00:35:45.020 --> 00:35:48.800 align:middle line:84% equivalent to s hat cross s hat cross this, 00:35:48.800 --> 00:35:52.560 align:middle line:84% is just something that tells you the shape of the emitted field 00:35:52.560 --> 00:35:53.060 align:middle line:90% here. 00:35:53.060 --> 00:35:57.460 align:middle line:84% And this tensor is acting on the scattered-- 00:35:57.460 --> 00:36:01.840 align:middle line:84% or the incident electric field at position r and at time 00:36:01.840 --> 00:36:03.320 align:middle line:90% t prime. 00:36:03.320 --> 00:36:06.860 align:middle line:84% So if I do the Fourier transformation of this and I 00:36:06.860 --> 00:36:11.000 align:middle line:84% want to know what the frequency is-- the scattered frequency-- 00:36:11.000 --> 00:36:11.750 align:middle line:90% and because-- 00:36:11.750 --> 00:36:17.180 align:middle line:84% [CHUCKLES]-- so we're going to be swapping again very freely 00:36:17.180 --> 00:36:21.500 align:middle line:84% between omega s mu s-- so, once again, 00:36:21.500 --> 00:36:23.570 align:middle line:84% factors of 2 pi flying everywhere. 00:36:23.570 --> 00:36:26.885 align:middle line:84% But it-- so I'm going to write this as a function of mu 00:36:26.885 --> 00:36:28.730 align:middle line:84% s because this is what we measure, 00:36:28.730 --> 00:36:30.887 align:middle line:84% and this is what Hutchinson uses in his book. 00:36:30.887 --> 00:36:32.720 align:middle line:84% But what I'm doing with Fourier transforms-- 00:36:32.720 --> 00:36:33.970 align:middle line:90% I'm going to be using omega s. 00:36:33.970 --> 00:36:37.550 align:middle line:84% So just feel free to keep adding 2 pi's and getting rid of them 00:36:37.550 --> 00:36:38.970 align:middle line:90% as you see fit. 00:36:38.970 --> 00:36:41.930 align:middle line:84% So this is just a standard definition 00:36:41.930 --> 00:36:43.500 align:middle line:90% of a Fourier transform. 00:36:43.500 --> 00:36:50.780 align:middle line:84% We have the electric field in time times by e to the i omega 00:36:50.780 --> 00:36:53.750 align:middle line:90% s c t dt. 00:36:53.750 --> 00:36:58.620 align:middle line:90% 00:36:58.620 --> 00:37:01.600 align:middle line:84% Now, we already mentioned we have 00:37:01.600 --> 00:37:03.790 align:middle line:84% a discrepancy between t prime, which 00:37:03.790 --> 00:37:06.760 align:middle line:84% is going to be inside this, and t out here. 00:37:06.760 --> 00:37:16.340 align:middle line:84% So t prime is equal to t minus R minus s hat 00:37:16.340 --> 00:37:19.850 align:middle line:90% dot lowercase r upon c. 00:37:19.850 --> 00:37:23.510 align:middle line:84% And we can also say that the dispersion relationship 00:37:23.510 --> 00:37:26.360 align:middle line:84% for our scattered wave is just the vacuum dispersion 00:37:26.360 --> 00:37:26.930 align:middle line:90% relationship. 00:37:26.930 --> 00:37:30.980 align:middle line:84% So K is equal to omega upon c s hat. 00:37:30.980 --> 00:37:32.510 align:middle line:84% By the way, this is one place where, 00:37:32.510 --> 00:37:35.900 align:middle line:84% if you want to modify this to take into account the O mode 00:37:35.900 --> 00:37:38.465 align:middle line:84% dispersion in a plasma, that you can modify this here. 00:37:38.465 --> 00:37:40.340 align:middle line:84% But it doesn't actually make a big difference 00:37:40.340 --> 00:37:42.120 align:middle line:90% to the derivation. 00:37:42.120 --> 00:37:48.470 align:middle line:84% This is going to allow us to replace this s hat here 00:37:48.470 --> 00:37:49.460 align:middle line:90% like that. 00:37:49.460 --> 00:37:53.850 align:middle line:84% We're also going to say that dt is 00:37:53.850 --> 00:38:02.310 align:middle line:84% equal to dt prime times a factor of 1 minus s hat dot beta, 00:38:02.310 --> 00:38:03.120 align:middle line:90% like this. 00:38:03.120 --> 00:38:08.580 align:middle line:84% And we said that anything on the first order in v/c 00:38:08.580 --> 00:38:09.810 align:middle line:90% is much less than 1-- 00:38:09.810 --> 00:38:11.680 align:middle line:90% we drop it. 00:38:11.680 --> 00:38:14.550 align:middle line:84% So, in fact, we're going to just switch between t, e t, 00:38:14.550 --> 00:38:15.685 align:middle line:90% and dt prime. 00:38:15.685 --> 00:38:17.310 align:middle line:84% We can't do the same for t prime and t. 00:38:17.310 --> 00:38:18.893 align:middle line:84% They are not the same time coordinate, 00:38:18.893 --> 00:38:21.840 align:middle line:84% but the amount they change by is the same 00:38:21.840 --> 00:38:24.450 align:middle line:84% because we're not dealing with relativistic time dilation 00:38:24.450 --> 00:38:26.340 align:middle line:90% in this treatment here. 00:38:26.340 --> 00:38:27.960 align:middle line:84% So now I can go back to this, and I 00:38:27.960 --> 00:38:32.160 align:middle line:84% can start replacing all of my t's with t primes here. 00:38:32.160 --> 00:38:38.850 align:middle line:84% And I'm going to end up with r e upon R-- 00:38:38.850 --> 00:38:44.540 align:middle line:90% 00:38:44.540 --> 00:38:46.650 align:middle line:84% am I really going to make that much of a jump? 00:38:46.650 --> 00:38:50.535 align:middle line:90% 00:38:50.535 --> 00:38:52.910 align:middle line:84% [VOCALIZING]-- I'm not going to make that much of a jump. 00:38:52.910 --> 00:38:56.450 align:middle line:84% I just need to go back up here and remind you 00:38:56.450 --> 00:39:10.392 align:middle line:84% that this Ei has a factor that looks like cosine of-- 00:39:10.392 --> 00:39:11.600 align:middle line:90% oh, this isn't going to work. 00:39:11.600 --> 00:39:13.349 align:middle line:90% I don't want to do it like that. 00:39:13.349 --> 00:39:34.560 align:middle line:90% 00:39:34.560 --> 00:39:42.450 align:middle line:84% And a factor of cosine Ki dot r minus omega i t prime. 00:39:42.450 --> 00:39:46.740 align:middle line:84% If I make the replacement of t prime with t, 00:39:46.740 --> 00:39:55.520 align:middle line:84% then this becomes cosine of Ks r, which is the shared 00:39:55.520 --> 00:39:57.040 align:middle line:90% phase for all of the particles. 00:39:57.040 --> 00:39:58.160 align:middle line:90% This is just boring-- 00:39:58.160 --> 00:40:00.290 align:middle line:84% just the phase you pick up as the wave travels 00:40:00.290 --> 00:40:09.802 align:middle line:84% to the spectrometer-- minus omega s of t minus Ks minus Ki 00:40:09.802 --> 00:40:14.190 align:middle line:84% dot r of 0 that we saw last time. 00:40:14.190 --> 00:40:16.095 align:middle line:90% So when we go back down here-- 00:40:16.095 --> 00:40:18.670 align:middle line:90% 00:40:18.670 --> 00:40:22.242 align:middle line:84% I'm going to bring this boring phase term out the front. 00:40:22.242 --> 00:40:24.200 align:middle line:84% I'm going to turn everything into exponentials. 00:40:24.200 --> 00:40:25.867 align:middle line:84% So I'm just going to replace this cosine 00:40:25.867 --> 00:40:29.180 align:middle line:84% with an exponential i times this phase factor here. 00:40:29.180 --> 00:40:38.280 align:middle line:84% I'm going to bring this phase factor exponential of iKs dot 00:40:38.280 --> 00:40:41.240 align:middle line:90% r out of the front here. 00:40:41.240 --> 00:40:43.360 align:middle line:84% But what I'm going to leave inside the integral 00:40:43.360 --> 00:40:47.220 align:middle line:84% is the integral of the vector-- the tensor 00:40:47.220 --> 00:40:55.860 align:middle line:84% pi dotted into the incident electric field at r and t prime. 00:40:55.860 --> 00:40:58.550 align:middle line:90% 00:40:58.550 --> 00:41:03.720 align:middle line:84% And all of this is multiplied by this factor-- 00:41:03.720 --> 00:41:06.100 align:middle line:90% the Fourier transform factor. 00:41:06.100 --> 00:41:08.370 align:middle line:84% But now I've taken t, and I've replaced it 00:41:08.370 --> 00:41:10.815 align:middle line:84% with t prime using this equation. 00:41:10.815 --> 00:41:14.670 align:middle line:90% 00:41:14.670 --> 00:41:16.840 align:middle line:84% And so this exponential now has a term, 00:41:16.840 --> 00:41:26.750 align:middle line:84% which looks like i omega s t prime minus Ks dot r. 00:41:26.750 --> 00:41:28.820 align:middle line:84% And this Fourier transform is now 00:41:28.820 --> 00:41:32.570 align:middle line:84% in terms of dt prime instead of dt. 00:41:32.570 --> 00:41:36.000 align:middle line:90% 00:41:36.000 --> 00:41:39.630 align:middle line:84% I then remember that this electric field here 00:41:39.630 --> 00:41:41.640 align:middle line:90% has this dependence-- 00:41:41.640 --> 00:41:45.960 align:middle line:90% Ki minus omega i t prime. 00:41:45.960 --> 00:41:48.960 align:middle line:84% If I take that as an exponential, 00:41:48.960 --> 00:41:52.380 align:middle line:84% it will add with all these exponential factors here. 00:41:52.380 --> 00:41:56.310 align:middle line:84% And we will end up with an equation that looks like r e 00:41:56.310 --> 00:42:03.000 align:middle line:84% upon R, exponential of this boring phase factor term iKs 00:42:03.000 --> 00:42:13.092 align:middle line:90% dot r, integral of i dot Ei0. 00:42:13.092 --> 00:42:15.720 align:middle line:84% So this is just the strength of the electric field 00:42:15.720 --> 00:42:17.810 align:middle line:90% and its polarization-- 00:42:17.810 --> 00:42:26.880 align:middle line:84% exponential of i omega t prime minus K 00:42:26.880 --> 00:42:35.150 align:middle line:84% dot r dt prime, where, again, I'm 00:42:35.150 --> 00:42:41.400 align:middle line:84% using those definitions of omega and K that we had previously. 00:42:41.400 --> 00:42:47.310 align:middle line:84% So this omega is omega s minus omega i, 00:42:47.310 --> 00:42:51.724 align:middle line:90% and this K is Ks minus Ki. 00:42:51.724 --> 00:42:54.390 align:middle line:90% 00:42:54.390 --> 00:42:58.170 align:middle line:84% So I've now got something where all of my t parameters 00:42:58.170 --> 00:42:59.530 align:middle line:90% are t prime parameters. 00:42:59.530 --> 00:43:00.930 align:middle line:90% So I can now do this integral. 00:43:00.930 --> 00:43:02.400 align:middle line:84% I couldn't do it up here because I 00:43:02.400 --> 00:43:05.670 align:middle line:84% had a mixture of t primes inside the scattered electric field 00:43:05.670 --> 00:43:07.805 align:middle line:90% and t's for where I really am. 00:43:07.805 --> 00:43:09.180 align:middle line:84% So you've really got to take care 00:43:09.180 --> 00:43:13.550 align:middle line:84% of doing all of these Fourier transforms carefully. 00:43:13.550 --> 00:43:17.690 align:middle line:90% This pi dot Ei0-- 00:43:17.690 --> 00:43:20.970 align:middle line:84% that can go outside the integral as well. 00:43:20.970 --> 00:43:24.470 align:middle line:90% It's just constant in time. 00:43:24.470 --> 00:43:27.320 align:middle line:84% Remember, that's just s cross s cross something like that. 00:43:27.320 --> 00:43:29.030 align:middle line:84% That's set by where our spectrometer is. 00:43:29.030 --> 00:43:30.947 align:middle line:84% And we're assuming we've got an infinite plane 00:43:30.947 --> 00:43:32.880 align:middle line:84% wave but intensive electric field change. 00:43:32.880 --> 00:43:35.384 align:middle line:84% So what is this Fourier transform? 00:43:35.384 --> 00:43:43.450 align:middle line:90% 00:43:43.450 --> 00:43:46.945 align:middle line:84% Fourier transform of an oscillating sinusoid. 00:43:46.945 --> 00:43:56.070 align:middle line:90% 00:43:56.070 --> 00:43:57.070 align:middle line:90% Just a delta function. 00:43:57.070 --> 00:43:57.570 align:middle line:90% Thank you. 00:43:57.570 --> 00:44:07.240 align:middle line:90% 00:44:07.240 --> 00:44:10.360 align:middle line:84% There's a factor of 2 pi from the Fourier transform 00:44:10.360 --> 00:44:13.760 align:middle line:84% when we get the definition of the delta function. 00:44:13.760 --> 00:44:18.920 align:middle line:84% And then we've still got this tensor dotted 00:44:18.920 --> 00:44:21.800 align:middle line:84% into the electric field vector to give us a scattering pattern. 00:44:21.800 --> 00:44:29.220 align:middle line:84% Now we have a delta function K dot v minus omega. 00:44:29.220 --> 00:44:32.190 align:middle line:84% And if you want to follow along this in Hutchinson's book, 00:44:32.190 --> 00:44:41.440 align:middle line:84% this derivation here is Hutchinson's 7.2.15-- 00:44:41.440 --> 00:44:43.280 align:middle line:90% like that. 00:44:43.280 --> 00:44:44.870 align:middle line:90% So this is actually nothing new. 00:44:44.870 --> 00:44:49.180 align:middle line:84% This is-- once again, we have just shown that our scattering 00:44:49.180 --> 00:44:50.980 align:middle line:90% is going to be off-- 00:44:50.980 --> 00:44:52.720 align:middle line:84% the scattering of the particle is 00:44:52.720 --> 00:44:56.050 align:middle line:84% going to give us a scattered frequency omega, which 00:44:56.050 --> 00:44:57.500 align:middle line:84% is K dot v. But now we've done it 00:44:57.500 --> 00:44:58.750 align:middle line:90% in terms of Fourier transform. 00:44:58.750 --> 00:45:02.850 align:middle line:84% Previously, we did it in real space and real time, 00:45:02.850 --> 00:45:04.600 align:middle line:84% and we just hand-waved our way through it. 00:45:04.600 --> 00:45:10.460 align:middle line:84% Now, we've got the spectra actually in a much more-- 00:45:10.460 --> 00:45:12.440 align:middle line:90% slightly more rigorous way. 00:45:12.440 --> 00:45:14.200 align:middle line:84% So this is still just a scattering off 00:45:14.200 --> 00:45:16.150 align:middle line:90% a single particle. 00:45:16.150 --> 00:45:20.890 align:middle line:84% If we want the scattered power off this single particle, 00:45:20.890 --> 00:45:26.232 align:middle line:84% then we again go to c epsilon 0 r squared. 00:45:26.232 --> 00:45:27.940 align:middle line:84% And now we need to do the time averaging. 00:45:27.940 --> 00:45:36.490 align:middle line:84% So to time average, we integrate between minus t/2 to t/2, 00:45:36.490 --> 00:45:38.393 align:middle line:90% and we divide by 1/t here. 00:45:38.393 --> 00:45:41.060 align:middle line:84% We're going to take the limit of t going to infinity in a moment 00:45:41.060 --> 00:45:43.352 align:middle line:84% so it will start to look more like a Fourier transform. 00:45:43.352 --> 00:45:48.956 align:middle line:84% And this average power, again, is defined in terms of the time. 00:45:48.956 --> 00:45:51.200 align:middle line:90% So Es is a function of time. 00:45:51.200 --> 00:45:55.800 align:middle line:84% Es star is a function of time, integrated d time. 00:45:55.800 --> 00:46:00.880 align:middle line:84% But if we take the limit as t goes to infinity, 00:46:00.880 --> 00:46:03.250 align:middle line:84% we can use something called Parseval's theorem, which 00:46:03.250 --> 00:46:11.850 align:middle line:84% says that this quantity here is equal to the integral Es 00:46:11.850 --> 00:46:19.050 align:middle line:84% in frequency space, dotted with Es in frequency space, 00:46:19.050 --> 00:46:23.810 align:middle line:84% integrated over frequency, still with a 1 over time out the front 00:46:23.810 --> 00:46:24.310 align:middle line:90% here. 00:46:24.310 --> 00:46:25.830 align:middle line:90% So this is a kind of-- 00:46:25.830 --> 00:46:29.100 align:middle line:84% [VOCALIZING], I'm not quite sure about the 1 over time out 00:46:29.100 --> 00:46:29.810 align:middle line:90% the front there-- 00:46:29.810 --> 00:46:32.390 align:middle line:90% Parseval's theorem. 00:46:32.390 --> 00:46:33.500 align:middle line:90% Dimensions don't work. 00:46:33.500 --> 00:46:46.538 align:middle line:90% 00:46:46.538 --> 00:46:49.080 align:middle line:84% I think I should have paused earlier and asked for questions, 00:46:49.080 --> 00:46:51.122 align:middle line:84% because this is actually an excellent derivation. 00:46:51.122 --> 00:46:52.830 align:middle line:90% So let's just go back-- 00:46:52.830 --> 00:46:54.540 align:middle line:90% draw a line under this here-- 00:46:54.540 --> 00:46:56.790 align:middle line:84% see if anyone has any questions on how we got to here. 00:46:56.790 --> 00:47:14.780 align:middle line:90% 00:47:14.780 --> 00:47:16.350 align:middle line:90% OK, then, we'll go on. 00:47:16.350 --> 00:47:19.276 align:middle line:84% So now we're trying to calculate-- yeah, all right. 00:47:19.276 --> 00:47:21.350 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:47:21.350 --> 00:47:22.490 align:middle line:90% JACK HARE: Yeah. 00:47:22.490 --> 00:47:23.530 align:middle line:90% Yeah, sure, sure, sure. 00:47:23.530 --> 00:47:26.870 align:middle line:84% I will pause in a second [INAUDIBLE]. 00:47:26.870 --> 00:47:29.408 align:middle line:84% I just want to get through this section first. 00:47:29.408 --> 00:47:33.890 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:47:33.890 --> 00:47:36.860 align:middle line:84% JACK HARE: No, this is just the incident-- i Ei0, 00:47:36.860 --> 00:47:39.530 align:middle line:84% which is just the strength and the direction of the electric 00:47:39.530 --> 00:47:41.300 align:middle line:90% field, without any of its-- 00:47:41.300 --> 00:47:49.150 align:middle line:84% so it's-- the incident electric field is Ei0 times by this 00:47:49.150 --> 00:47:49.650 align:middle line:90% cosine. 00:47:49.650 --> 00:47:52.650 align:middle line:84% So I put all of the fluctuations and interesting time and space 00:47:52.650 --> 00:47:53.970 align:middle line:90% variation in the cosine. 00:47:53.970 --> 00:47:57.670 align:middle line:84% But this is just our strength for our electric field here. 00:47:57.670 --> 00:48:00.247 align:middle line:84% So all this is saying here is that 00:48:00.247 --> 00:48:01.830 align:middle line:84% the scattered electric field is linear 00:48:01.830 --> 00:48:04.430 align:middle line:90% in the incident electric field. 00:48:04.430 --> 00:48:06.255 align:middle line:84% All of the funkiness with time and space 00:48:06.255 --> 00:48:07.630 align:middle line:84% has gone into the delta function. 00:48:07.630 --> 00:48:10.450 align:middle line:90% 00:48:10.450 --> 00:48:14.870 align:middle line:84% So here, we want to be able to evaluate this function, 00:48:14.870 --> 00:48:17.570 align:middle line:84% but we don't actually have a great expression 00:48:17.570 --> 00:48:20.390 align:middle line:90% for Es a function of time. 00:48:20.390 --> 00:48:22.730 align:middle line:84% But we do now have a great expression for Es 00:48:22.730 --> 00:48:23.970 align:middle line:90% a function of frequency. 00:48:23.970 --> 00:48:27.790 align:middle line:84% So we use Parseval's theorem to convert 00:48:27.790 --> 00:48:30.580 align:middle line:84% between the time-power spectrum and 00:48:30.580 --> 00:48:32.050 align:middle line:90% the frequency-power spectrum. 00:48:32.050 --> 00:48:40.650 align:middle line:84% And that means we can now write down our scattered power 00:48:40.650 --> 00:48:45.660 align:middle line:84% dP into some scattering volume or scattering solid angle, omega 00:48:45.660 --> 00:48:49.150 align:middle line:84% s, and some scattering frequency. 00:48:49.150 --> 00:49:01.346 align:middle line:84% And this is equal to c epsilon 0 r squared of K-- 00:49:01.346 --> 00:49:05.570 align:middle line:84% the limit of t going to infinity. 00:49:05.570 --> 00:49:08.027 align:middle line:84% So we're taking a long time slice. 00:49:08.027 --> 00:49:09.860 align:middle line:84% This allows us to use our Fourier transforms 00:49:09.860 --> 00:49:13.130 align:middle line:84% where the limits here are plus and minus infinity here. 00:49:13.130 --> 00:49:18.230 align:middle line:84% And we do the Fourier transform of this-- 00:49:18.230 --> 00:49:19.410 align:middle line:90% sorry. 00:49:19.410 --> 00:49:22.220 align:middle line:90% Don't do the Fourier transform. 00:49:22.220 --> 00:49:25.090 align:middle line:84% We've taken d nu from this side here, 00:49:25.090 --> 00:49:28.240 align:middle line:84% and we've put it down here to form our scattered power 00:49:28.240 --> 00:49:29.700 align:middle line:90% per frequency bin here. 00:49:29.700 --> 00:49:31.450 align:middle line:84% So we're getting rid of this integral now. 00:49:31.450 --> 00:49:34.300 align:middle line:84% And now we're trying to evaluate this term, which looks 00:49:34.300 --> 00:49:37.480 align:middle line:90% like Es dot Es star instead. 00:49:37.480 --> 00:49:39.590 align:middle line:84% And we have that equation up here. 00:49:39.590 --> 00:49:42.780 align:middle line:90% So we want this term-- 00:49:42.780 --> 00:49:44.100 align:middle line:90% I'm going to take it from here. 00:49:44.100 --> 00:49:48.258 align:middle line:90% 00:49:48.258 --> 00:49:49.800 align:middle line:84% So, first of all, we have a term that 00:49:49.800 --> 00:49:53.820 align:middle line:84% looks like r E squared upon R squared. 00:49:53.820 --> 00:49:58.490 align:middle line:90% 00:49:58.490 --> 00:50:02.810 align:middle line:84% Then this term will cancel out because we'll have it times 00:50:02.810 --> 00:50:05.160 align:middle line:84% by its complex conjugate, so we can get rid of it. 00:50:05.160 --> 00:50:06.050 align:middle line:90% Good riddance. 00:50:06.050 --> 00:50:11.910 align:middle line:84% There will be a term that looks like 4 pi squared here. 00:50:11.910 --> 00:50:23.510 align:middle line:84% We'll have a term that looks like pi dotted into the incident 00:50:23.510 --> 00:50:25.430 align:middle line:90% electric field squared. 00:50:25.430 --> 00:50:28.280 align:middle line:84% And then we'll have a term that looks like delta 00:50:28.280 --> 00:50:33.210 align:middle line:90% squared K dot v minus omega. 00:50:33.210 --> 00:50:37.280 align:middle line:90% 00:50:37.280 --> 00:50:41.510 align:middle line:84% The trouble with this is that we have a delta function squared. 00:50:41.510 --> 00:50:44.420 align:middle line:84% It's a pretty odd object because a delta function is already 00:50:44.420 --> 00:50:47.448 align:middle line:84% a pretty odd object, and squaring it makes it just odder. 00:50:47.448 --> 00:50:49.490 align:middle line:84% So we're going to do the sort of thing that makes 00:50:49.490 --> 00:50:51.410 align:middle line:90% the pure mathematicians wince. 00:50:51.410 --> 00:50:52.490 align:middle line:84% And we're going to say, [VOCALIZING], this 00:50:52.490 --> 00:50:54.380 align:middle line:84% is a little bit like a sinc function. 00:50:54.380 --> 00:50:58.930 align:middle line:84% So a delta function-- it's sort of like a sinc. 00:50:58.930 --> 00:51:02.280 align:middle line:90% And if our sinc function of t-- 00:51:02.280 --> 00:51:04.635 align:middle line:90% or, in this case, I guess, nu-- 00:51:04.635 --> 00:51:07.730 align:middle line:90% 00:51:07.730 --> 00:51:10.100 align:middle line:84% as this nu gets very, very small. 00:51:10.100 --> 00:51:12.680 align:middle line:84% So we have a function that looks like a sinc, 00:51:12.680 --> 00:51:14.810 align:middle line:84% and then we approximate the delta function as just 00:51:14.810 --> 00:51:16.830 align:middle line:90% a very, very, very narrow sinc. 00:51:16.830 --> 00:51:20.720 align:middle line:84% And if you do this, you can rewrite this delta squared 00:51:20.720 --> 00:51:26.690 align:middle line:84% as looking like t upon 2 pi, just times the delta 00:51:26.690 --> 00:51:27.540 align:middle line:90% function itself. 00:51:27.540 --> 00:51:30.950 align:middle line:84% So delta function squared of frequency 00:51:30.950 --> 00:51:35.510 align:middle line:84% is equal to t upon 2 pi delta function of frequency. 00:51:35.510 --> 00:51:37.430 align:middle line:84% This is by far the least satisfactory part 00:51:37.430 --> 00:51:39.170 align:middle line:90% of this derivation, by the way. 00:51:39.170 --> 00:51:41.490 align:middle line:84% But Hutchinson doesn't even do it. 00:51:41.490 --> 00:51:42.450 align:middle line:90% He just skips over it. 00:51:42.450 --> 00:51:44.060 align:middle line:84% So I'm trying to at least demystify 00:51:44.060 --> 00:51:45.890 align:middle line:90% where this might come from. 00:51:45.890 --> 00:51:48.510 align:middle line:84% But I don't know how to do this very, very rigorously. 00:51:48.510 --> 00:51:49.760 align:middle line:90% Let's just go with it for now. 00:51:49.760 --> 00:51:51.510 align:middle line:84% We're going to replace this delta function 00:51:51.510 --> 00:51:53.810 align:middle line:84% with just a delta function not squared 00:51:53.810 --> 00:51:56.560 align:middle line:84% because those are the sorts of things we can work with. 00:51:56.560 --> 00:51:59.040 align:middle line:84% Then we can run through all of this, 00:51:59.040 --> 00:52:05.940 align:middle line:84% and we can find out that we have r e squared c epsilon 0 limit 00:52:05.940 --> 00:52:15.570 align:middle line:84% of t going to infinity of 1/t times t upon 2 pi-- 00:52:15.570 --> 00:52:17.940 align:middle line:90% that's this factor here-- 00:52:17.940 --> 00:52:22.130 align:middle line:90% times by 4 pi squared. 00:52:22.130 --> 00:52:27.430 align:middle line:84% We still have this i dotted into the electric field 00:52:27.430 --> 00:52:28.990 align:middle line:90% strength squared. 00:52:28.990 --> 00:52:32.040 align:middle line:90% And now we just have-- 00:52:32.040 --> 00:52:34.190 align:middle line:90% I'm running out of space-- 00:52:34.190 --> 00:52:41.443 align:middle line:84% this delta function of K dot v minus omega not squared. 00:52:41.443 --> 00:52:43.360 align:middle line:84% The convenient thing about doing this is we've 00:52:43.360 --> 00:52:46.360 align:middle line:84% generated a factor of t which can cancel out with this t. 00:52:46.360 --> 00:52:49.270 align:middle line:84% So we've canceled out an infinity or a 0 00:52:49.270 --> 00:52:51.370 align:middle line:84% that we've accidentally introduced. 00:52:51.370 --> 00:52:54.670 align:middle line:84% And now we've also got a 2 pi, and that will cancel out some 00:52:54.670 --> 00:52:57.110 align:middle line:90% of these 2 pi's. 00:52:57.110 --> 00:53:01.190 align:middle line:84% The whole thing starts to look a little bit more tractable. 00:53:01.190 --> 00:53:03.193 align:middle line:90% And so the result is-- 00:53:03.193 --> 00:53:05.360 align:middle line:84% I don't quite have space for it on this board, which 00:53:05.360 --> 00:53:07.610 align:middle line:84% is a real shame after we've come all this way. 00:53:07.610 --> 00:53:09.761 align:middle line:90% I shall write it here-- 00:53:09.761 --> 00:53:25.240 align:middle line:84% dP d omega d nu is equal to r e squared c epsilon 0 Ei0 squared. 00:53:25.240 --> 00:53:29.570 align:middle line:90% What the hell is this? 00:53:29.570 --> 00:53:31.130 align:middle line:90% I'm going to keep it as it was-- 00:53:31.130 --> 00:53:33.800 align:middle line:90% 00:53:33.800 --> 00:53:47.970 align:middle line:84% i dotted it into Ei squared times 2 pi delta K dot v 00:53:47.970 --> 00:53:50.570 align:middle line:90% minus omega. 00:53:50.570 --> 00:53:53.300 align:middle line:84% The point is that this is what we should call our spectrum. 00:53:53.300 --> 00:53:55.790 align:middle line:84% We're looking at the power scattered in a certain direction 00:53:55.790 --> 00:53:58.620 align:middle line:84% into our spectrometer per unit frequency. 00:53:58.620 --> 00:54:00.390 align:middle line:90% And so we can now evaluate this. 00:54:00.390 --> 00:54:06.300 align:middle line:84% And once again, we can say, if we're looking at power per unit 00:54:06.300 --> 00:54:09.742 align:middle line:84% frequency-- or rather, dP d omega-- 00:54:09.742 --> 00:54:11.700 align:middle line:84% the power scattered for a certain solid angle-- 00:54:11.700 --> 00:54:14.010 align:middle line:84% this tells us that, from a single particle, 00:54:14.010 --> 00:54:17.520 align:middle line:84% our spectra will just have a little delta function then. 00:54:17.520 --> 00:54:21.790 align:middle line:84% And that delta function will be at a frequency omega, 00:54:21.790 --> 00:54:27.780 align:middle line:84% which is equal to K dot v. So if our initial laser line was 00:54:27.780 --> 00:54:32.040 align:middle line:84% at some wavelength like this-- this is omega i-- 00:54:32.040 --> 00:54:39.270 align:middle line:84% then this is at omega s equals omega i plus omega-- 00:54:39.270 --> 00:54:40.710 align:middle line:90% and we say, aha. 00:54:40.710 --> 00:54:42.090 align:middle line:90% This is blueshifted. 00:54:42.090 --> 00:54:43.570 align:middle line:90% It's of higher frequency. 00:54:43.570 --> 00:54:45.960 align:middle line:90% And so means the particle is-- 00:54:45.960 --> 00:54:48.570 align:middle line:84% got some component of velocity towards us. 00:54:48.570 --> 00:54:51.570 align:middle line:84% Again, this is all just single-particle scattering, 00:54:51.570 --> 00:54:54.060 align:middle line:84% but we're now working in the Fourier domain as opposed 00:54:54.060 --> 00:54:56.880 align:middle line:84% to in real space, which means we can actually write down 00:54:56.880 --> 00:54:58.590 align:middle line:84% equations for the spectrum, which we couldn't do when 00:54:58.590 --> 00:54:59.840 align:middle line:90% we were working in real space. 00:54:59.840 --> 00:55:03.160 align:middle line:84% We just had a hand-wavy argument for it. 00:55:03.160 --> 00:55:04.180 align:middle line:90% Now I will pause. 00:55:04.180 --> 00:55:06.370 align:middle line:90% Oh, one more thing. 00:55:06.370 --> 00:55:08.680 align:middle line:84% Just so you can have contact with the book again, 00:55:08.680 --> 00:55:15.900 align:middle line:84% this is Hutchinson's equation 7.2.19. 00:55:15.900 --> 00:55:18.815 align:middle line:84% So in five equations, he does all of this. 00:55:18.815 --> 00:55:22.890 align:middle line:84% I have tried to spend a little bit more time doing it. 00:55:22.890 --> 00:55:27.160 align:middle line:84% But I will agree with anyone who says this isn't [INAUDIBLE]. 00:55:27.160 --> 00:55:27.760 align:middle line:90% Questions? 00:55:27.760 --> 00:55:30.160 align:middle line:90% Yeah? 00:55:30.160 --> 00:55:34.820 align:middle line:84% AUDIENCE: Where did that 1/t [INAUDIBLE]? 00:55:34.820 --> 00:55:38.770 align:middle line:90% JACK HARE: [VOCALIZING] Yeah. 00:55:38.770 --> 00:55:46.980 align:middle line:84% So what I have written in my notes is 1/t here. 00:55:46.980 --> 00:55:49.740 align:middle line:84% But if you set these two equal to each other, 00:55:49.740 --> 00:55:52.890 align:middle line:84% the dimensions don't add up because of the limits 00:55:52.890 --> 00:55:55.090 align:middle line:84% here, or because of what we're integrating over. 00:55:55.090 --> 00:55:57.180 align:middle line:84% So in a panic, I changed it to be t because that 00:55:57.180 --> 00:55:58.348 align:middle line:90% made the dimensions work. 00:55:58.348 --> 00:56:00.390 align:middle line:84% But it makes the rest of the derivation not work. 00:56:00.390 --> 00:56:04.410 align:middle line:84% So probably, there's some reason why we have this 1/t in front 00:56:04.410 --> 00:56:05.923 align:middle line:84% of both of them that I've missed. 00:56:05.923 --> 00:56:08.340 align:middle line:84% And I'm very happy to go and have a look at the derivation 00:56:08.340 --> 00:56:11.160 align:middle line:90% again in Hutchinson's book-- 00:56:11.160 --> 00:56:12.610 align:middle line:84% try and work out what's going on. 00:56:12.610 --> 00:56:17.040 align:middle line:84% But what we are trying to do is replace this integral here 00:56:17.040 --> 00:56:19.830 align:middle line:84% over the electric field in space with this integral 00:56:19.830 --> 00:56:21.660 align:middle line:84% over the electric field in frequency, 00:56:21.660 --> 00:56:23.800 align:middle line:84% but the dimensions don't quite work out. 00:56:23.800 --> 00:56:26.130 align:middle line:84% So that's where the 1/t was meant to come from, 00:56:26.130 --> 00:56:29.580 align:middle line:84% and that 1/t is the one that gets canceled by the t over 2 pi 00:56:29.580 --> 00:56:32.190 align:middle line:84% that we get from our shoddy treatment of the delta function. 00:56:32.190 --> 00:56:34.870 align:middle line:90% 00:56:34.870 --> 00:56:35.370 align:middle line:90% Yeah. 00:56:35.370 --> 00:56:36.395 align:middle line:90% Other question? 00:56:36.395 --> 00:56:37.020 align:middle line:90% AUDIENCE: Yeah. 00:56:37.020 --> 00:56:45.103 align:middle line:90% So this whole [INAUDIBLE] 00:56:45.103 --> 00:56:47.270 align:middle line:84% JACK HARE: If you're doing a Fourier transformation, 00:56:47.270 --> 00:56:49.900 align:middle line:84% you have to do it over infinite time. 00:56:49.900 --> 00:56:52.680 align:middle line:90% So this is just simply-- 00:56:52.680 --> 00:56:54.990 align:middle line:84% but if you put that infinity in straight away, 00:56:54.990 --> 00:56:57.180 align:middle line:90% then you start getting 0's. 00:56:57.180 --> 00:57:00.060 align:middle line:84% So we as a t, where it's a really big t. 00:57:00.060 --> 00:57:01.720 align:middle line:84% But then eventually, we cancel out, 00:57:01.720 --> 00:57:03.845 align:middle line:84% so it's like you don't actually care how big it is. 00:57:03.845 --> 00:57:07.290 align:middle line:84% So to do this rigorously, you actually have to do Laplace 00:57:07.290 --> 00:57:10.845 align:middle line:84% transformation because the laser is not-- cannot be a plane wave. 00:57:10.845 --> 00:57:12.970 align:middle line:84% It's not traveling through the plasma for all time. 00:57:12.970 --> 00:57:14.270 align:middle line:90% It turns on at some time. 00:57:14.270 --> 00:57:16.770 align:middle line:84% And so you have to do Laplace transformations where you only 00:57:16.770 --> 00:57:18.630 align:middle line:84% take into account for t greater than 0. 00:57:18.630 --> 00:57:20.610 align:middle line:84% But the mathematics of that becomes harder, 00:57:20.610 --> 00:57:23.340 align:middle line:84% so I'm doing these Fourier transforms, and there, you 00:57:23.340 --> 00:57:25.060 align:middle line:90% need infinite time. 00:57:25.060 --> 00:57:25.560 align:middle line:90% Yeah. 00:57:25.560 --> 00:57:30.460 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:57:30.460 --> 00:57:32.560 align:middle line:90% JACK HARE: Yeah. 00:57:32.560 --> 00:57:33.060 align:middle line:90% Sorry? 00:57:33.060 --> 00:57:34.827 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:57:34.827 --> 00:57:35.660 align:middle line:90% JACK HARE: This one? 00:57:35.660 --> 00:57:36.390 align:middle line:90% AUDIENCE: Yeah. 00:57:36.390 --> 00:57:38.483 align:middle line:84% That represents the time average? 00:57:38.483 --> 00:57:39.150 align:middle line:90% JACK HARE: Yeah. 00:57:39.150 --> 00:57:42.690 align:middle line:84% This is where-- the time where we're working out 00:57:42.690 --> 00:57:45.270 align:middle line:84% the average power because we're working out 00:57:45.270 --> 00:57:47.580 align:middle line:84% the instantaneous power, we're integrating it 00:57:47.580 --> 00:57:51.410 align:middle line:84% over some interval T, and we're dividing by T. Yeah. 00:57:51.410 --> 00:57:53.100 align:middle line:84% AUDIENCE: And then we're [INAUDIBLE]. 00:57:53.100 --> 00:57:55.350 align:middle line:84% JACK HARE: And then afterwards being like, yeah, but-- 00:57:55.350 --> 00:57:56.850 align:middle line:84% how long do you need to do this for? 00:57:56.850 --> 00:57:58.650 align:middle line:84% Well, to do this for a plane wave, 00:57:58.650 --> 00:58:00.840 align:middle line:84% you should do it for a few cycles. 00:58:00.840 --> 00:58:04.710 align:middle line:84% But to be rigorous, you want to extend it out-- well, maybe 00:58:04.710 --> 00:58:05.840 align:middle line:90% not rigorous in this case. 00:58:05.840 --> 00:58:07.215 align:middle line:84% To be slightly more rigorous, you 00:58:07.215 --> 00:58:08.490 align:middle line:84% want to extend that integral to all time 00:58:08.490 --> 00:58:10.950 align:middle line:84% if you're really doing a Fourier transform type approach. 00:58:10.950 --> 00:58:12.960 align:middle line:90% Yeah. 00:58:12.960 --> 00:58:16.380 align:middle line:84% And this is the requirement of using Parseval's theorem, which 00:58:16.380 --> 00:58:18.090 align:middle line:84% I clearly don't know as well as I 00:58:18.090 --> 00:58:20.740 align:middle line:84% thought I did because of this problem with the units here. 00:58:20.740 --> 00:58:21.240 align:middle line:90% Yeah. 00:58:21.240 --> 00:58:24.940 align:middle line:90% 00:58:24.940 --> 00:58:25.573 align:middle line:90% Yeah. 00:58:25.573 --> 00:58:27.490 align:middle line:84% Yeah, I'm confused, because this is definitely 00:58:27.490 --> 00:58:30.820 align:middle line:84% something that has dimensions of electric field squared. 00:58:30.820 --> 00:58:34.810 align:middle line:84% This does not have dimension of electric field squared. 00:58:34.810 --> 00:58:36.190 align:middle line:90% Any other questions? 00:58:36.190 --> 00:58:45.013 align:middle line:90% 00:58:45.013 --> 00:58:46.430 align:middle line:84% The nice thing about our treatment 00:58:46.430 --> 00:58:50.300 align:middle line:84% is we said that, if we want to get the scattered spectrum 00:58:50.300 --> 00:58:54.670 align:middle line:84% from many particles, we are not interested in how 00:58:54.670 --> 00:58:56.670 align:middle line:84% the scattered electric fields in those particles 00:58:56.670 --> 00:58:57.610 align:middle line:90% interact with each other. 00:58:57.610 --> 00:58:58.680 align:middle line:90% That's the coherent part. 00:58:58.680 --> 00:59:00.280 align:middle line:90% This is the incoherent part. 00:59:00.280 --> 00:59:03.570 align:middle line:84% So if we now have the scattering spectrum from a single particle 00:59:03.570 --> 00:59:06.930 align:middle line:84% and it's showing up at a frequency that 00:59:06.930 --> 00:59:09.000 align:middle line:84% is to do with that single particle's velocity, 00:59:09.000 --> 00:59:11.910 align:middle line:84% if we want the scattering spectrum from many particles, 00:59:11.910 --> 00:59:15.300 align:middle line:84% we simply sum the scattering from each of the particles 00:59:15.300 --> 00:59:16.477 align:middle line:90% V together. 00:59:16.477 --> 00:59:18.060 align:middle line:84% So this now, you're probably thinking, 00:59:18.060 --> 00:59:21.370 align:middle line:84% starts to look like integration over a distribution function. 00:59:21.370 --> 00:59:22.290 align:middle line:90% So let's go do that. 00:59:22.290 --> 00:59:25.190 align:middle line:90% 00:59:25.190 --> 00:59:27.070 align:middle line:90% Do I need to write that down? 00:59:27.070 --> 00:59:28.090 align:middle line:90% No, I'm happy with that. 00:59:28.090 --> 00:59:49.250 align:middle line:90% 00:59:49.250 --> 00:59:51.620 align:middle line:84% Actually, I'll write it down again here so we've got it 00:59:51.620 --> 00:59:52.828 align:middle line:90% on the page as we look at it. 00:59:52.828 --> 00:59:57.160 align:middle line:84% So our scattered power into some solid angle 00:59:57.160 --> 01:00:08.530 align:middle line:84% into some frequency looks like r e squared times i dot-- 01:00:08.530 --> 01:00:10.480 align:middle line:90% why am I writing it -- 01:00:10.480 --> 01:00:11.753 align:middle line:90% put an i everywhere else. 01:00:11.753 --> 01:00:13.170 align:middle line:84% I'm just going to call it i again. 01:00:13.170 --> 01:00:16.790 align:middle line:90% 01:00:16.790 --> 01:00:23.720 align:middle line:84% So I've taken the electric field strength and polarization-- 01:00:23.720 --> 01:00:25.970 align:middle line:84% I've now split up even further into a term that's 01:00:25.970 --> 01:00:29.060 align:middle line:84% the electric field strength and its polarization. 01:00:29.060 --> 01:00:31.160 align:middle line:84% You get the polarization inside here. 01:00:31.160 --> 01:00:37.210 align:middle line:84% I've taken out this factor of Ei squared 01:00:37.210 --> 01:00:41.680 align:middle line:84% because, when times that by c epsilon 0, I can put all of that 01:00:41.680 --> 01:00:47.590 align:middle line:84% together and write it instead as the incident power of the laser 01:00:47.590 --> 01:00:49.880 align:middle line:90% over the area of the laser. 01:00:49.880 --> 01:00:53.800 align:middle line:84% So this is now an intensity in watts per meter squared here. 01:00:53.800 --> 01:00:57.072 align:middle line:90% 01:00:57.072 --> 01:00:58.530 align:middle line:84% And we often think about our lasers 01:00:58.530 --> 01:00:59.700 align:middle line:90% in terms of their intensity. 01:00:59.700 --> 01:01:01.170 align:middle line:90% This is nice units. 01:01:01.170 --> 01:01:03.940 align:middle line:84% And then, finally, the interesting bit, 01:01:03.940 --> 01:01:10.105 align:middle line:84% which is this 2 pi delta K dot v minus omega. 01:01:10.105 --> 01:01:12.530 align:middle line:84% So this is a scattering from one particle. 01:01:12.530 --> 01:01:15.850 align:middle line:90% 01:01:15.850 --> 01:01:19.710 align:middle line:84% And to be clear, it's one electron. 01:01:19.710 --> 01:01:23.750 align:middle line:84% So, again, nothing particularly new 01:01:23.750 --> 01:01:27.120 align:middle line:84% here but just now in frequency space. 01:01:27.120 --> 01:01:36.110 align:middle line:84% So now we want to integrate over some distribution function 01:01:36.110 --> 01:01:40.800 align:middle line:90% f of v d3v d3r. 01:01:40.800 --> 01:01:43.190 align:middle line:84% And remember, when we have a distribution function, 01:01:43.190 --> 01:01:46.760 align:middle line:84% we say that the density is the integral of f 01:01:46.760 --> 01:01:51.060 align:middle line:90% of v over velocity space here. 01:01:51.060 --> 01:01:53.150 align:middle line:84% So this is the density as a function 01:01:53.150 --> 01:01:57.942 align:middle line:84% of r-- the zeroth moment of the distribution function. 01:01:57.942 --> 01:01:59.400 align:middle line:84% What we're going to end up doing is 01:01:59.400 --> 01:02:01.950 align:middle line:84% we're going to be integrating this f 01:02:01.950 --> 01:02:12.970 align:middle line:84% of v times this delta function K dot v minus omega d3v. 01:02:12.970 --> 01:02:15.640 align:middle line:84% When we do that integral, this delta function 01:02:15.640 --> 01:02:19.660 align:middle line:84% is going to pick out certain things about this velocity 01:02:19.660 --> 01:02:20.650 align:middle line:90% distribution. 01:02:20.650 --> 01:02:23.470 align:middle line:84% It's only going to pick out particles 01:02:23.470 --> 01:02:27.280 align:middle line:84% which have a velocity which is equal to omega/K 01:02:27.280 --> 01:02:28.990 align:middle line:84% and it's only going to pick out particles 01:02:28.990 --> 01:02:32.380 align:middle line:84% which have a component of velocity parallel 01:02:32.380 --> 01:02:39.070 align:middle line:84% to K. That's the v dot K part as well. 01:02:39.070 --> 01:02:43.380 align:middle line:84% So when we do this integral, we end up with a scattered power 01:02:43.380 --> 01:02:45.580 align:middle line:90% spectrum that looks like-- 01:02:45.580 --> 01:02:48.010 align:middle line:84% scattered power, again, into some solid angle-- 01:02:48.010 --> 01:02:58.910 align:middle line:84% some frequency-- is equal to 2 pi r e squared, intensity 01:02:58.910 --> 01:03:02.730 align:middle line:84% of the laser beam, a factor which 01:03:02.730 --> 01:03:05.940 align:middle line:84% accounts for where the scattering light goes 01:03:05.940 --> 01:03:08.190 align:middle line:84% to, and then the interesting bit. 01:03:08.190 --> 01:03:11.963 align:middle line:90% And the interesting bit is-- 01:03:11.963 --> 01:03:14.130 align:middle line:84% as opposed to having the full distribution function, 01:03:14.130 --> 01:03:16.100 align:middle line:84% we now have a distribution function 01:03:16.100 --> 01:03:23.170 align:middle line:90% f sub K evaluated at omega/K. 01:03:23.170 --> 01:03:27.620 align:middle line:84% And then there's also a factor of 1/K out the front here. 01:03:27.620 --> 01:03:30.070 align:middle line:84% So let me unpack some of these bits here. 01:03:30.070 --> 01:03:32.390 align:middle line:90% This f of K-- 01:03:32.390 --> 01:03:37.440 align:middle line:84% imagine we have some full distribution 01:03:37.440 --> 01:03:41.400 align:middle line:84% function in three dimensions where 01:03:41.400 --> 01:03:46.655 align:middle line:84% we have some coordinate that is perpendicular to K 01:03:46.655 --> 01:03:51.680 align:middle line:84% and we have some coordinate, which is along K. 01:03:51.680 --> 01:03:58.090 align:middle line:84% And what we're doing is the integral of this function, 01:03:58.090 --> 01:04:01.130 align:middle line:84% integrating over all the perpendicular velocities. 01:04:01.130 --> 01:04:05.032 align:middle line:84% So our f of K is simply a one-dimensional slice 01:04:05.032 --> 01:04:06.490 align:middle line:84% through that distribution function. 01:04:06.490 --> 01:04:07.450 align:middle line:90% You can think of-- 01:04:07.450 --> 01:04:09.490 align:middle line:90% maybe, for example, in-- 01:04:09.490 --> 01:04:11.980 align:middle line:84% just in two dimensions here, we could 01:04:11.980 --> 01:04:16.480 align:middle line:84% have the K direction and the perpendicular to K direction. 01:04:16.480 --> 01:04:19.390 align:middle line:84% And our distribution function could look like-- 01:04:19.390 --> 01:04:22.970 align:middle line:84% and I'm drawing contours of f here. 01:04:22.970 --> 01:04:29.770 align:middle line:84% So, for example, this is f equals 1, 2, 3 type thing. 01:04:29.770 --> 01:04:31.272 align:middle line:84% These can be much bigger numbers, 01:04:31.272 --> 01:04:33.730 align:middle line:84% but this is just to give you an idea of some function which 01:04:33.730 --> 01:04:35.560 align:middle line:84% is peaked around about the center here, 01:04:35.560 --> 01:04:37.120 align:middle line:90% but it's not isotropic. 01:04:37.120 --> 01:04:39.970 align:middle line:84% And what our Thomson scattering is picking up 01:04:39.970 --> 01:04:45.170 align:middle line:84% is just a slice through in one direction of that distribution 01:04:45.170 --> 01:04:46.040 align:middle line:90% function here. 01:04:46.040 --> 01:04:49.700 align:middle line:84% So if I take this cut, I get out the distribution function 01:04:49.700 --> 01:04:54.880 align:middle line:84% that looks like f of K versus vK, 01:04:54.880 --> 01:04:59.421 align:middle line:90% and maybe it looks like this. 01:04:59.421 --> 01:05:02.590 align:middle line:84% So I'm only sensitive to one-- if a particle's 01:05:02.590 --> 01:05:07.350 align:middle line:84% moving in one direction-- in the K direction here. 01:05:07.350 --> 01:05:09.920 align:middle line:84% And this 1/K-- if you're wondering where that comes from, 01:05:09.920 --> 01:05:17.240 align:middle line:84% this is because dv is equal to d omega upon K. 01:05:17.240 --> 01:05:20.000 align:middle line:84% So when I change this integral from being in terms of velocity 01:05:20.000 --> 01:05:22.280 align:middle line:84% space to being in terms of frequency space, 01:05:22.280 --> 01:05:25.650 align:middle line:84% I'm going to get out this factor of 1/K here. 01:05:25.650 --> 01:05:30.020 align:middle line:90% So why did I write all of that? 01:05:30.020 --> 01:05:31.110 align:middle line:90% No, this is right. 01:05:31.110 --> 01:05:32.680 align:middle line:90% I don't need anything else. 01:05:32.680 --> 01:05:38.310 align:middle line:84% So, just to be clear, this is a 1D distribution function. 01:05:38.310 --> 01:05:42.880 align:middle line:90% 01:05:42.880 --> 01:05:45.760 align:middle line:84% Now, if your system is isotropic, 01:05:45.760 --> 01:05:47.540 align:middle line:84% this doesn't bother you at all because you 01:05:47.540 --> 01:05:49.457 align:middle line:84% can-- if you think there's lots of collisions, 01:05:49.457 --> 01:05:51.910 align:middle line:84% there's no magnetic fields, my distribution is isotropic, 01:05:51.910 --> 01:05:54.243 align:middle line:84% then I can just make one measurement of the distribution 01:05:54.243 --> 01:05:56.510 align:middle line:84% function in one direction, and I've got all I need. 01:05:56.510 --> 01:05:58.690 align:middle line:84% But if your distribution function is anisotropic 01:05:58.690 --> 01:06:01.030 align:middle line:84% because there's magnetic fields or electric fields, 01:06:01.030 --> 01:06:03.757 align:middle line:84% then you won't be able to pick up the full distribution 01:06:03.757 --> 01:06:04.840 align:middle line:90% with just one measurement. 01:06:04.840 --> 01:06:07.430 align:middle line:84% You'll need lots and lots of different measurements. 01:06:07.430 --> 01:06:12.340 align:middle line:84% For example, you'll need another K like this and another K 01:06:12.340 --> 01:06:13.900 align:middle line:90% like this. 01:06:13.900 --> 01:06:17.575 align:middle line:84% And you can effectively do a tomographic reconstruction 01:06:17.575 --> 01:06:21.220 align:middle line:84% in velocity space by taking lots of different slices 01:06:21.220 --> 01:06:22.360 align:middle line:90% at different angles here. 01:06:22.360 --> 01:06:25.690 align:middle line:90% 01:06:25.690 --> 01:06:27.797 align:middle line:84% I will just deal with a few consequences of this, 01:06:27.797 --> 01:06:29.214 align:middle line:84% and then I'll take some questions. 01:06:29.214 --> 01:06:52.970 align:middle line:90% 01:06:52.970 --> 01:06:54.490 align:middle line:90% So our scattered spectrum-- 01:06:54.490 --> 01:07:00.767 align:middle line:90% 01:07:00.767 --> 01:07:03.100 align:middle line:84% which, again, we've been writing this scattered spectrum 01:07:03.100 --> 01:07:10.287 align:middle line:84% in our differential notation as d2P d omega d nu. 01:07:10.287 --> 01:07:12.870 align:middle line:84% But every time you see this, you think, that's what I measure. 01:07:12.870 --> 01:07:16.020 align:middle line:84% I am measuring some power into some solid angle represented 01:07:16.020 --> 01:07:18.000 align:middle line:84% by my spectrometer, and I'm measuring it 01:07:18.000 --> 01:07:19.810 align:middle line:90% resolved in terms of frequency. 01:07:19.810 --> 01:07:22.100 align:middle line:90% This is what I measure here. 01:07:22.100 --> 01:07:29.570 align:middle line:84% This measures-- if I can spell "measures"-- 01:07:29.570 --> 01:07:33.710 align:middle line:90% 01:07:33.710 --> 01:07:38.540 align:middle line:84% the distribution function f of v parallel 01:07:38.540 --> 01:07:42.600 align:middle line:84% to that K scattering vector here. 01:07:42.600 --> 01:07:49.210 align:middle line:90% And so, if I have multiple Ks-- 01:07:49.210 --> 01:07:52.390 align:middle line:90% I have multiple K-- 01:07:52.390 --> 01:07:55.220 align:middle line:84% I get f of v for multiple directions. 01:07:55.220 --> 01:07:58.810 align:middle line:84% So, again, I have a plasma like this. 01:07:58.810 --> 01:08:04.480 align:middle line:84% I put my scattered light through here. 01:08:04.480 --> 01:08:07.000 align:middle line:90% This is my laser beam. 01:08:07.000 --> 01:08:10.720 align:middle line:90% It's got some Ki like that. 01:08:10.720 --> 01:08:13.480 align:middle line:84% And then I measure the scattering, for example, 01:08:13.480 --> 01:08:16.010 align:middle line:90% in this direction. 01:08:16.010 --> 01:08:16.864 align:middle line:90% We'll call that Ks1. 01:08:16.864 --> 01:08:20.160 align:middle line:90% 01:08:20.160 --> 01:08:29.069 align:middle line:84% Then my K1-- remember, K is equal to Ks minus Ki-- 01:08:29.069 --> 01:08:34.319 align:middle line:84% is equal to Ks minus Ki like that. 01:08:34.319 --> 01:08:39.600 align:middle line:84% So I measure scattering along this K1. 01:08:39.600 --> 01:08:42.689 align:middle line:84% If I measure instead at 90 degrees-- 01:08:42.689 --> 01:08:44.835 align:middle line:84% sorry, 180 degrees to that-- this is Ks2. 01:08:44.835 --> 01:08:47.520 align:middle line:90% 01:08:47.520 --> 01:08:50.759 align:middle line:90% Then I go Ks minus Ki. 01:08:50.759 --> 01:08:51.660 align:middle line:90% I end up there. 01:08:51.660 --> 01:08:56.560 align:middle line:90% 01:08:56.560 --> 01:08:58.729 align:middle line:84% This will be measuring direction K2. 01:08:58.729 --> 01:09:01.279 align:middle line:84% And these two will be measuring different cuts 01:09:01.279 --> 01:09:03.486 align:middle line:84% through our velocity distribution. 01:09:03.486 --> 01:09:05.569 align:middle line:84% So if it was anisotropic, they could be different. 01:09:05.569 --> 01:09:09.000 align:middle line:84% By the way, if these are at 180 degrees, 01:09:09.000 --> 01:09:10.560 align:middle line:84% these two should be at 90 degrees. 01:09:10.560 --> 01:09:12.768 align:middle line:84% They're not, because I'm bad at drawing that diagram, 01:09:12.768 --> 01:09:16.090 align:middle line:90% but that's where it should be. 01:09:16.090 --> 01:09:18.550 align:middle line:84% So you could measure orthogonal components of a velocity. 01:09:18.550 --> 01:09:20.620 align:middle line:84% And, of course, if it's fully three dimensional, 01:09:20.620 --> 01:09:23.109 align:middle line:84% you might want to scatter out of the page here. 01:09:23.109 --> 01:09:25.390 align:middle line:84% And then you would measure some scattering K 01:09:25.390 --> 01:09:27.229 align:middle line:90% that's also out of the page. 01:09:27.229 --> 01:09:30.069 align:middle line:84% So if you have these multiple lines of sight, 01:09:30.069 --> 01:09:41.309 align:middle line:84% you can reconstruct your three-dimensional f of v. 01:09:41.309 --> 01:09:42.684 align:middle line:84% The other cool thing about this-- 01:09:42.684 --> 01:09:45.399 align:middle line:90% 01:09:45.399 --> 01:09:55.890 align:middle line:84% is you might be able to measure non-Maxwellian distributions, 01:09:55.890 --> 01:09:57.100 align:middle line:90% even in one dimension. 01:09:57.100 --> 01:10:00.220 align:middle line:84% So if I just write it as f of v without the vector-- 01:10:00.220 --> 01:10:03.870 align:middle line:84% so we're thinking about this one-dimensional cut here-- 01:10:03.870 --> 01:10:17.420 align:middle line:84% if we look at our spectrum nu, and this is dP d omega-- 01:10:17.420 --> 01:10:22.650 align:middle line:84% our scattered light-- if we look at this on a log scale, 01:10:22.650 --> 01:10:30.210 align:middle line:84% then this dP d omega is just about log of f. 01:10:30.210 --> 01:10:33.600 align:middle line:84% On a log scale here, our Maxwellian 01:10:33.600 --> 01:10:37.320 align:middle line:90% would just look like a parabola. 01:10:37.320 --> 01:10:46.150 align:middle line:84% It would go as minus v squared, which is minus nu squared, 01:10:46.150 --> 01:10:46.820 align:middle line:90% like that. 01:10:46.820 --> 01:10:48.730 align:middle line:84% So if we had a Maxwellian distribution, 01:10:48.730 --> 01:10:51.440 align:middle line:84% we'd expect just to see a spectrum like this. 01:10:51.440 --> 01:10:53.320 align:middle line:84% But if we have a non-Maxwellian distribution, 01:10:53.320 --> 01:10:56.240 align:middle line:84% perhaps we've got scale, some tails coming out of the side 01:10:56.240 --> 01:10:56.740 align:middle line:90% here. 01:10:56.740 --> 01:11:02.890 align:middle line:90% 01:11:02.890 --> 01:11:05.550 align:middle line:84% And so, in principle, using this incoherent scattering, 01:11:05.550 --> 01:11:07.650 align:middle line:84% because we measure the full distribution function, 01:11:07.650 --> 01:11:09.420 align:middle line:84% if that distribution is not Maxwellian, 01:11:09.420 --> 01:11:12.510 align:middle line:84% we might be able to measure non-Maxwellian tails out here. 01:11:12.510 --> 01:11:16.740 align:middle line:84% Now, in general, these tails are weak scattering. 01:11:16.740 --> 01:11:21.677 align:middle line:84% There's not very many particles out in the tails. 01:11:21.677 --> 01:11:24.010 align:middle line:84% It's hard to make plasmas which are very non-Maxwellian. 01:11:24.010 --> 01:11:27.430 align:middle line:84% So it can be very, very hard to measure this. 01:11:27.430 --> 01:11:30.820 align:middle line:84% But if you are able to repeat your experiment lots of times 01:11:30.820 --> 01:11:32.890 align:middle line:84% and build up your signal like they 01:11:32.890 --> 01:11:34.955 align:middle line:84% did in these low-temperature plasma experiments, 01:11:34.955 --> 01:11:36.580 align:middle line:84% you may be able to measure the presence 01:11:36.580 --> 01:11:39.160 align:middle line:84% of fast electrons-- non-Maxwellian electrons 01:11:39.160 --> 01:11:39.730 align:middle line:90% out here. 01:11:39.730 --> 01:11:43.150 align:middle line:84% So this is two very cool uses for incoherent Thomson 01:11:43.150 --> 01:11:44.830 align:middle line:90% scattering. 01:11:44.830 --> 01:11:46.672 align:middle line:84% I will pause there and take questions. 01:11:46.672 --> 01:11:49.968 align:middle line:90% 01:11:49.968 --> 01:11:52.010 align:middle line:84% Maybe I'll put this back down so you can see what 01:11:52.010 --> 01:11:53.218 align:middle line:90% we were talking about before. 01:11:53.218 --> 01:12:26.990 align:middle line:90% 01:12:26.990 --> 01:12:27.900 align:middle line:90% Any questions online? 01:12:27.900 --> 01:12:31.765 align:middle line:90% 01:12:31.765 --> 01:12:32.265 align:middle line:90% Yeah? 01:12:32.265 --> 01:12:39.160 align:middle line:90% 01:12:39.160 --> 01:12:42.010 align:middle line:84% Yeah, so we saw that the scattering cross-section was 01:12:42.010 --> 01:12:43.225 align:middle line:90% like-- 01:12:43.225 --> 01:12:46.210 align:middle line:84% you expect 10 to the minus 8 of your particles-- photons-- 01:12:46.210 --> 01:12:46.840 align:middle line:90% to scatter. 01:12:46.840 --> 01:12:48.632 align:middle line:84% So the chance of it getting scattered twice 01:12:48.632 --> 01:12:50.930 align:middle line:84% would be 10 to the minus 16, which is very unlikely. 01:12:50.930 --> 01:12:52.105 align:middle line:84% So we don't worry about double scattering. 01:12:52.105 --> 01:12:54.438 align:middle line:84% I guess I could have put that on my list of assumptions, 01:12:54.438 --> 01:12:55.035 align:middle line:90% but yeah. 01:12:55.035 --> 01:12:56.785 align:middle line:84% So we don't worry about double scattering. 01:12:56.785 --> 01:13:01.770 align:middle line:90% 01:13:01.770 --> 01:13:04.110 align:middle line:84% Just one quick note-- a relativistic correction, 01:13:04.110 --> 01:13:05.520 align:middle line:90% because if you-- 01:13:05.520 --> 01:13:08.432 align:middle line:84% generally, in tokamaks, they do incoherent scattering 01:13:08.432 --> 01:13:10.890 align:middle line:84% because it's very hard to get into the alpha greater than 1 01:13:10.890 --> 01:13:12.750 align:middle line:90% coherent scattering regime. 01:13:12.750 --> 01:13:15.360 align:middle line:84% And in tokamaks-- if you've got a nice hot tokamak, 01:13:15.360 --> 01:13:17.190 align:middle line:84% you may well have a relativistic plasma. 01:13:17.190 --> 01:13:21.450 align:middle line:84% So just a quick note here on what the relativistic correction 01:13:21.450 --> 01:13:22.140 align:middle line:90% looks like. 01:13:22.140 --> 01:13:31.590 align:middle line:90% 01:13:31.590 --> 01:13:33.410 align:middle line:84% First of all, remember, you may need 01:13:33.410 --> 01:13:36.080 align:middle line:84% to think about relativistic beaming. 01:13:36.080 --> 01:13:42.400 align:middle line:84% So this is where the radiation pattern of your scattered light 01:13:42.400 --> 01:13:43.720 align:middle line:90% beams forward. 01:13:43.720 --> 01:13:46.390 align:middle line:84% This is v/c much, much less than 1, 01:13:46.390 --> 01:13:50.862 align:middle line:84% and this is maybe v/c less than or around about 1. 01:13:50.862 --> 01:13:52.570 align:middle line:84% So, first of all, you might want to think 01:13:52.570 --> 01:13:54.310 align:middle line:90% about this relativistic beaming. 01:13:54.310 --> 01:13:57.060 align:middle line:84% But the other thing you might want to think about 01:13:57.060 --> 01:13:58.830 align:middle line:84% is, how does this affect your spectrum? 01:13:58.830 --> 01:14:00.930 align:middle line:84% Well, there's a long derivation of this, 01:14:00.930 --> 01:14:02.470 align:middle line:84% but there's a short version of it, 01:14:02.470 --> 01:14:04.620 align:middle line:84% which is that your spectrum-- which, again, 01:14:04.620 --> 01:14:11.030 align:middle line:84% we're writing as d2P d capital omega d nu-- 01:14:11.030 --> 01:14:14.840 align:middle line:84% the relativistic version looks a lot 01:14:14.840 --> 01:14:17.880 align:middle line:84% like your non-relativistic version, 01:14:17.880 --> 01:14:21.050 align:middle line:84% which we have written several times over on the side here. 01:14:21.050 --> 01:14:24.150 align:middle line:90% 01:14:24.150 --> 01:14:26.060 align:middle line:84% But it has a correction factor to it, 01:14:26.060 --> 01:14:33.580 align:middle line:84% and that correction is 1 plus 3 omega upon 2 omega i. 01:14:33.580 --> 01:14:39.090 align:middle line:90% 01:14:39.090 --> 01:14:43.230 align:middle line:84% This means that the whole spectrum gets shifted up. 01:14:43.230 --> 01:14:46.430 align:middle line:84% Remember that omega here is equal to K dot 01:14:46.430 --> 01:14:49.140 align:middle line:84% v. If you're wondering why this is relativistic, 01:14:49.140 --> 01:14:51.160 align:middle line:90% that's where the v is coming in. 01:14:51.160 --> 01:14:54.780 align:middle line:84% So this is a correction on the order of-- 01:14:54.780 --> 01:15:01.110 align:middle line:84% and this omega i, by the way, is just cK. 01:15:01.110 --> 01:15:04.920 align:middle line:84% And so this correction here is on the order of v/c. 01:15:04.920 --> 01:15:06.420 align:middle line:84% These are the ones we said we'd keep 01:15:06.420 --> 01:15:08.960 align:middle line:90% for our relativistic correction. 01:15:08.960 --> 01:15:13.410 align:middle line:84% What this correction means is that we get a blueshift. 01:15:13.410 --> 01:15:16.230 align:middle line:84% So the spectrum is always blueshifted-- 01:15:16.230 --> 01:15:18.420 align:middle line:84% always ends up as being at a higher frequency. 01:15:18.420 --> 01:15:20.730 align:middle line:84% You can see that by the shape of this and the fact 01:15:20.730 --> 01:15:25.000 align:middle line:84% that it depends on the sign of omega here. 01:15:25.000 --> 01:15:32.620 align:middle line:84% And, in particular, this correction is for temperatures-- 01:15:32.620 --> 01:15:35.148 align:middle line:84% electron temperatures-- on the order of the keV range. 01:15:35.148 --> 01:15:36.940 align:middle line:84% So if you're working below that, you really 01:15:36.940 --> 01:15:38.350 align:middle line:90% don't need this correction. 01:15:38.350 --> 01:15:41.590 align:middle line:84% But if you're working in a 10 keV plasma, 01:15:41.590 --> 01:15:43.750 align:middle line:90% you do need this correction. 01:15:43.750 --> 01:15:50.030 align:middle line:84% Because of this blueshift, it's important to measure 01:15:50.030 --> 01:15:53.150 align:middle line:84% both the positive shifted frequencies 01:15:53.150 --> 01:15:55.130 align:middle line:84% and the negative shifted frequencies. 01:15:55.130 --> 01:15:56.660 align:middle line:84% Because if you don't measure those, 01:15:56.660 --> 01:16:00.440 align:middle line:84% then you're expecting your spectrum for a Maxwellian 01:16:00.440 --> 01:16:02.980 align:middle line:90% to be nice and symmetric. 01:16:02.980 --> 01:16:05.830 align:middle line:84% But if you are doing the relativistic one, 01:16:05.830 --> 01:16:08.530 align:middle line:84% then your spectrum might look like this. 01:16:08.530 --> 01:16:15.710 align:middle line:90% 01:16:15.710 --> 01:16:18.490 align:middle line:84% And if you only measure one half of your spectrum-- 01:16:18.490 --> 01:16:21.340 align:middle line:84% this is frequency-- we only measure the positive frequencies 01:16:21.340 --> 01:16:23.830 align:middle line:84% or negative frequencies-- we might misinterpret 01:16:23.830 --> 01:16:25.120 align:middle line:90% what's going on here. 01:16:25.120 --> 01:16:29.160 align:middle line:84% If you're starting to work with relativistic plasmas, 01:16:29.160 --> 01:16:30.910 align:middle line:84% you may be tempted with your spectrometer, 01:16:30.910 --> 01:16:33.280 align:middle line:84% because you only have a finite number of frequencies 01:16:33.280 --> 01:16:34.930 align:middle line:84% you can measure at, to only measure 01:16:34.930 --> 01:16:37.030 align:middle line:84% positive or negative frequencies because the spectrum is 01:16:37.030 --> 01:16:37.530 align:middle line:90% symmetric. 01:16:37.530 --> 01:16:40.210 align:middle line:84% But in the relativistic case, it's not. 01:16:40.210 --> 01:16:42.520 align:middle line:90% This is like d2P. 01:16:42.520 --> 01:16:43.180 align:middle line:90% OK. 01:16:43.180 --> 01:16:45.880 align:middle line:90% 01:16:45.880 --> 01:16:53.720 align:middle line:84% And this is Hutchinson's equation 7.2.28. 01:16:53.720 --> 01:16:55.610 align:middle line:84% And he does the derivation of this properly. 01:16:55.610 --> 01:17:02.290 align:middle line:84% I am just giving you the result. Any questions on that? 01:17:02.290 --> 01:17:03.280 align:middle line:90% Just like a side note. 01:17:03.280 --> 01:17:09.020 align:middle line:90% 01:17:09.020 --> 01:17:12.520 align:middle line:84% We have just enough time to introduce coherent scattering, 01:17:12.520 --> 01:17:14.110 align:middle line:84% but not enough time to actually do 01:17:14.110 --> 01:17:16.460 align:middle line:84% all the fun mathematics of it, so we 01:17:16.460 --> 01:17:17.710 align:middle line:90% will leave that for next week. 01:17:17.710 --> 01:17:44.268 align:middle line:90% 01:17:44.268 --> 01:17:46.130 align:middle line:84% So that was incoherent scattering. 01:17:46.130 --> 01:17:48.170 align:middle line:84% We derived the scattering off a single particle. 01:17:48.170 --> 01:17:50.480 align:middle line:84% Then we integrated over the distribution function 01:17:50.480 --> 01:17:52.910 align:middle line:84% and said that the scattering off each of these particles 01:17:52.910 --> 01:17:55.580 align:middle line:84% does not interfere with the scattering off any 01:17:55.580 --> 01:17:57.830 align:middle line:90% of the other particles. 01:17:57.830 --> 01:18:00.440 align:middle line:84% Or, if it does interfere, another particle's scattering 01:18:00.440 --> 01:18:04.220 align:middle line:84% will add up, and that sum over the scattering 01:18:04.220 --> 01:18:07.750 align:middle line:84% between different particles will eventually cancel out. 01:18:07.750 --> 01:18:11.293 align:middle line:84% That is not going to be the case in coherent scattering. 01:18:11.293 --> 01:18:22.410 align:middle line:90% 01:18:22.410 --> 01:18:25.463 align:middle line:84% So, again, we're just going to sketch what's going on. 01:18:25.463 --> 01:18:26.880 align:middle line:84% Before we go into the mathematics, 01:18:26.880 --> 01:18:29.910 align:middle line:84% I want you to have a heuristic understanding of what's 01:18:29.910 --> 01:18:32.880 align:middle line:84% happening first so that you have some faith that, when 01:18:32.880 --> 01:18:35.850 align:middle line:84% we wade through the math, it'll all be worth it in the end. 01:18:35.850 --> 01:18:37.950 align:middle line:84% So this is just a simple sketch for you. 01:18:37.950 --> 01:18:43.080 align:middle line:84% Again, we're talking about scattering off fluctuations 01:18:43.080 --> 01:18:45.660 align:middle line:84% which have a wavelength lambda, and that lambda 01:18:45.660 --> 01:18:49.610 align:middle line:84% is greater than the Debye length here. 01:18:49.610 --> 01:18:52.750 align:middle line:84% So these must be coherent fluctuations 01:18:52.750 --> 01:18:55.090 align:middle line:84% because we can no longer see individual particles. 01:18:55.090 --> 01:18:57.130 align:middle line:84% We can only see particles that are Debye 01:18:57.130 --> 01:18:58.720 align:middle line:90% shielding other particles. 01:18:58.720 --> 01:19:01.570 align:middle line:84% And so these particles must be moving in concert 01:19:01.570 --> 01:19:02.710 align:middle line:90% with the other particles. 01:19:02.710 --> 01:19:08.166 align:middle line:90% 01:19:08.166 --> 01:19:10.450 align:middle line:84% And just a note-- as we said before, 01:19:10.450 --> 01:19:15.440 align:middle line:84% this is the condition that alpha is greater than 1. 01:19:15.440 --> 01:19:19.580 align:middle line:84% The coherent fluctuations are modes or waves. 01:19:19.580 --> 01:19:20.735 align:middle line:90% I'll use both words. 01:19:20.735 --> 01:19:24.710 align:middle line:90% 01:19:24.710 --> 01:19:28.660 align:middle line:84% So let's consider what sort of modes and waves 01:19:28.660 --> 01:19:31.000 align:middle line:84% there are in a plasma that we can scatter off. 01:19:31.000 --> 01:19:37.430 align:middle line:84% So we consider modes with frequency omega, 01:19:37.430 --> 01:19:44.700 align:middle line:84% which is equal to, as we keep saying, omega s minus omega i. 01:19:44.700 --> 01:19:47.040 align:middle line:84% We're dealing with the difference between these two 01:19:47.040 --> 01:19:47.970 align:middle line:90% frequencies. 01:19:47.970 --> 01:19:51.840 align:middle line:84% And, again, we're looking at Doppler shifts 01:19:51.840 --> 01:19:56.400 align:middle line:84% which are on the order of K dot v. 01:19:56.400 --> 01:19:59.980 align:middle line:84% And so this is going to be a relatively small quantity. 01:19:59.980 --> 01:20:02.550 align:middle line:90% So this omega s minus omega i-- 01:20:02.550 --> 01:20:05.250 align:middle line:84% our omega is much going to be much, much less 01:20:05.250 --> 01:20:09.770 align:middle line:84% than omega i, which is basically the same as omega s. 01:20:09.770 --> 01:20:14.070 align:middle line:84% So we're looking for modes with a relatively low frequency-- 01:20:14.070 --> 01:20:16.620 align:middle line:84% much lower than the sorts of waves 01:20:16.620 --> 01:20:20.160 align:middle line:84% that we see propagating in free space, like our laser. 01:20:20.160 --> 01:20:23.890 align:middle line:90% 01:20:23.890 --> 01:20:26.740 align:middle line:84% But, despite the fact that these modes are low frequency, 01:20:26.740 --> 01:20:29.680 align:middle line:84% they may not actually have a low wave number. 01:20:29.680 --> 01:20:32.620 align:middle line:84% Because, remember, our scattering diagram-- 01:20:32.620 --> 01:20:37.140 align:middle line:84% we have some incoming wave, we measure it 01:20:37.140 --> 01:20:40.620 align:middle line:84% in some scattering direction Ks like this, 01:20:40.620 --> 01:20:42.870 align:middle line:84% and we said, because this is elastic scattering, 01:20:42.870 --> 01:20:49.707 align:middle line:84% that the size of Ks is roughly the same as the size of Ki. 01:20:49.707 --> 01:20:52.290 align:middle line:84% They didn't have to be pointing in the same direction, though. 01:20:52.290 --> 01:20:59.920 align:middle line:84% And so that means that K, which is equal to Ks minus Ki-- 01:20:59.920 --> 01:21:09.065 align:middle line:84% again, we just look at this and do this as a vector Ks minus Ki 01:21:09.065 --> 01:21:09.565 align:middle line:90% like this. 01:21:09.565 --> 01:21:12.490 align:middle line:90% 01:21:12.490 --> 01:21:18.840 align:middle line:84% And so the size of K is actually on the order of the size of Ks 01:21:18.840 --> 01:21:21.007 align:middle line:90% or the size of Ki. 01:21:21.007 --> 01:21:22.590 align:middle line:84% So, although the frequency of the mode 01:21:22.590 --> 01:21:25.500 align:middle line:84% is much lower than the frequency of a free space 01:21:25.500 --> 01:21:28.200 align:middle line:84% propagating plane wave, the momentum, 01:21:28.200 --> 01:21:31.050 align:middle line:84% or the wave number of the mode, is roughly the same here. 01:21:31.050 --> 01:21:38.345 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:21:38.345 --> 01:21:39.720 align:middle line:84% JACK HARE: Nothing I've done here 01:21:39.720 --> 01:21:41.784 align:middle line:84% actually is talking about the Debye length yet. 01:21:41.784 --> 01:21:51.188 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:21:51.188 --> 01:21:53.480 align:middle line:84% JACK HARE: Yes, though, again, none of this mathematics 01:21:53.480 --> 01:21:55.110 align:middle line:84% so far has said anything about that. 01:21:55.110 --> 01:21:58.250 align:middle line:84% We're just-- what all of this means 01:21:58.250 --> 01:22:03.290 align:middle line:84% together is-- we're looking for a mode, which has v phase which 01:22:03.290 --> 01:22:07.730 align:middle line:84% is equal to omega/K. We know that this omega 01:22:07.730 --> 01:22:13.310 align:middle line:84% is much, much less than omega i, which is equal to CKi. 01:22:13.310 --> 01:22:18.620 align:middle line:84% But we know that this K is on the order of Ki. 01:22:18.620 --> 01:22:21.170 align:middle line:84% So if we take the ratio of omega and K, 01:22:21.170 --> 01:22:23.930 align:middle line:84% we can see that this will be much, much less 01:22:23.930 --> 01:22:27.682 align:middle line:84% than the speed of light, which is-- actually, 01:22:27.682 --> 01:22:30.140 align:middle line:84% this would be the same if I replaced this with the electron 01:22:30.140 --> 01:22:32.450 align:middle line:84% velocity here, if I was doing incoherent scattering 01:22:32.450 --> 01:22:33.680 align:middle line:90% of individual electrons. 01:22:33.680 --> 01:22:36.180 align:middle line:84% But this same condition now holds here. 01:22:36.180 --> 01:22:39.410 align:middle line:84% So what we need to do is we need to identify-- 01:22:39.410 --> 01:22:42.440 align:middle line:90% 01:22:42.440 --> 01:22:52.490 align:middle line:84% find modes with-- so what modes are there 01:22:52.490 --> 01:22:54.110 align:middle line:90% in a plasma that satisfy this? 01:22:54.110 --> 01:22:57.200 align:middle line:84% They've got a low phase velocity compared to the speed of light. 01:22:57.200 --> 01:22:59.580 align:middle line:84% They have relatively low frequencies, 01:22:59.580 --> 01:23:00.815 align:middle line:90% but they have high momentum. 01:23:00.815 --> 01:23:04.476 align:middle line:90% 01:23:04.476 --> 01:23:07.130 align:middle line:84% We're not dealing with any magnetic fields here. 01:23:07.130 --> 01:23:11.480 align:middle line:84% So we're not going to scatter off alpha [INAUDIBLE]. 01:23:11.480 --> 01:23:12.410 align:middle line:90% Sound waves? 01:23:12.410 --> 01:23:15.650 align:middle line:84% What is-- and can you give me a posh name for a sound wave? 01:23:15.650 --> 01:23:18.590 align:middle line:90% 01:23:18.590 --> 01:23:19.910 align:middle line:90% An acoustic wave. 01:23:19.910 --> 01:23:20.435 align:middle line:90% There we go. 01:23:20.435 --> 01:23:24.167 align:middle line:90% 01:23:24.167 --> 01:23:25.750 align:middle line:84% So what is the dispersion relationship 01:23:25.750 --> 01:23:27.100 align:middle line:90% for an ion acoustic wave? 01:23:27.100 --> 01:23:35.080 align:middle line:90% 01:23:35.080 --> 01:23:38.690 align:middle line:84% I'm just going to take the square root there. 01:23:38.690 --> 01:23:41.030 align:middle line:90% And what is cs? 01:23:41.030 --> 01:23:47.157 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:23:47.157 --> 01:23:47.865 align:middle line:90% JACK HARE: Right. 01:23:47.865 --> 01:23:50.670 align:middle line:84% Well, we're working in a plasma that can be multiply ionized. 01:23:50.670 --> 01:23:52.900 align:middle line:84% So I'm going to put a z in there-- z in there-- 01:23:52.900 --> 01:24:00.070 align:middle line:84% and it turns out, in reality, it's z t e over t i over M i. 01:24:00.070 --> 01:24:04.695 align:middle line:84% Is this a wave which has low frequency and high momentum, 01:24:04.695 --> 01:24:06.070 align:middle line:84% which is the same as saying, does 01:24:06.070 --> 01:24:08.528 align:middle line:84% it have a phase velocity much less than the speed of light? 01:24:08.528 --> 01:24:12.240 align:middle line:90% 01:24:12.240 --> 01:24:15.880 align:middle line:90% What's its phase velocity? 01:24:15.880 --> 01:24:18.282 align:middle line:90% This isn't a trick question. 01:24:18.282 --> 01:24:19.845 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:24:19.845 --> 01:24:20.470 align:middle line:90% JACK HARE: Why? 01:24:20.470 --> 01:24:25.770 align:middle line:90% 01:24:25.770 --> 01:24:26.270 align:middle line:90% OK. 01:24:26.270 --> 01:24:30.650 align:middle line:84% [CHUCKLES] So the phase velocity of it is just the sound speed. 01:24:30.650 --> 01:24:32.570 align:middle line:84% And that sound speed is going to be much less 01:24:32.570 --> 01:24:34.130 align:middle line:90% than the speed of light. 01:24:34.130 --> 01:24:36.410 align:middle line:84% And, yes, maybe it is because ions are slow. 01:24:36.410 --> 01:24:39.032 align:middle line:84% So you'll need, effectively-- this is a relatively heavy mass. 01:24:39.032 --> 01:24:40.490 align:middle line:84% You calculate this, and this number 01:24:40.490 --> 01:24:42.247 align:middle line:84% is going to be like 10 kilometers 01:24:42.247 --> 01:24:43.580 align:middle line:90% a second or something like that. 01:24:43.580 --> 01:24:45.997 align:middle line:84% It's going to be much, much less than 3 times 10 to the 8. 01:24:45.997 --> 01:24:47.210 align:middle line:90% So these are great modes. 01:24:47.210 --> 01:24:49.730 align:middle line:84% We can definitely scatter off these modes. 01:24:49.730 --> 01:24:52.520 align:middle line:84% These are actually very low frequency modes. 01:24:52.520 --> 01:24:56.640 align:middle line:84% These are the lowest frequency modes in a plasma. 01:24:56.640 --> 01:25:00.680 align:middle line:90% What's a slightly faster wave? 01:25:00.680 --> 01:25:02.453 align:middle line:90% Yes? 01:25:02.453 --> 01:25:03.120 align:middle line:90% You're cheating. 01:25:03.120 --> 01:25:04.470 align:middle line:90% You've done this before, OK. 01:25:04.470 --> 01:25:07.110 align:middle line:90% [CHUCKLES] Go on-- 01:25:07.110 --> 01:25:08.070 align:middle line:90% Langmuir waves. 01:25:08.070 --> 01:25:11.770 align:middle line:84% I'm going to call these electron plasma waves, 01:25:11.770 --> 01:25:14.305 align:middle line:84% but they are sometimes named after Langmuir. 01:25:14.305 --> 01:25:18.920 align:middle line:90% 01:25:18.920 --> 01:25:20.900 align:middle line:84% So what is the dispersion relationship 01:25:20.900 --> 01:25:21.740 align:middle line:90% for a Langmuir wave? 01:25:21.740 --> 01:25:41.120 align:middle line:90% 01:25:41.120 --> 01:25:44.000 align:middle line:84% And we could go and work out the phase velocity for these. 01:25:44.000 --> 01:25:46.080 align:middle line:84% It's a little bit more difficult. 01:25:46.080 --> 01:25:49.680 align:middle line:84% But we'll find out that these do have a low phase velocity, 01:25:49.680 --> 01:25:52.100 align:middle line:84% but it's not quite as low as the ion acoustic waves. 01:25:52.100 --> 01:25:55.440 align:middle line:84% So these are just the low frequency waves. 01:25:55.440 --> 01:25:57.710 align:middle line:84% So if we're scattering off the ion acoustic waves, 01:25:57.710 --> 01:25:59.767 align:middle line:84% we expect to see small frequency shifts. 01:25:59.767 --> 01:26:01.850 align:middle line:84% If we're scattering off the electron plasma waves, 01:26:01.850 --> 01:26:05.890 align:middle line:84% we expect to see slightly larger frequency shifts. 01:26:05.890 --> 01:26:09.000 align:middle line:90%