WEBVTT 00:00:00.000 --> 00:00:01.470 align:middle line:90% [SQUEAKING] 00:00:01.470 --> 00:00:02.440 align:middle line:90% [CLICKING] 00:00:02.440 --> 00:00:02.940 align:middle line:90% [SQUEAKING] 00:00:02.940 --> 00:00:04.410 align:middle line:90% [RUSTLING] 00:00:04.410 --> 00:00:05.880 align:middle line:90% [CLICKING] 00:00:05.880 --> 00:00:10.300 align:middle line:90% 00:00:10.300 --> 00:00:11.920 align:middle line:84% PROFESSOR: So today we are going to be 00:00:11.920 --> 00:00:15.595 align:middle line:84% talking about proton radiography and proton imaging. 00:00:15.595 --> 00:00:27.490 align:middle line:90% 00:00:27.490 --> 00:00:30.180 align:middle line:84% So we're starting a series of three lectures in which we're 00:00:30.180 --> 00:00:33.060 align:middle line:84% looking at how to use particles as opposed to the plasma 00:00:33.060 --> 00:00:33.840 align:middle line:90% previously. 00:00:33.840 --> 00:00:36.030 align:middle line:84% You remember we used electromagnetic radiation 00:00:36.030 --> 00:00:37.980 align:middle line:84% and we looked at the emission from the plasma, 00:00:37.980 --> 00:00:41.400 align:middle line:84% but now we're actively pushing particles through the plasma 00:00:41.400 --> 00:00:42.640 align:middle line:90% and seeing what happens. 00:00:42.640 --> 00:00:45.420 align:middle line:84% So imagine-- and we'll talk about exactly how you do this 00:00:45.420 --> 00:00:48.990 align:middle line:84% later on-- but you have some source of protons, 00:00:48.990 --> 00:00:52.990 align:middle line:84% a beam of protons, with some spread. 00:00:52.990 --> 00:00:55.420 align:middle line:84% Maybe it's emitting into 4 pi steradians, 00:00:55.420 --> 00:00:57.505 align:middle line:84% maybe it's a tightly focused beam. 00:00:57.505 --> 00:00:59.570 align:middle line:90% It doesn't really matter. 00:00:59.570 --> 00:01:03.250 align:middle line:90% And you have some plasma here. 00:01:03.250 --> 00:01:07.250 align:middle line:84% And maybe within that plasma, you have some magnetic fields. 00:01:07.250 --> 00:01:11.660 align:middle line:84% So you have magnetic field lines that 00:01:11.660 --> 00:01:15.140 align:middle line:84% maybe look toroidal from the point of view of the protons 00:01:15.140 --> 00:01:15.640 align:middle line:90% here. 00:01:15.640 --> 00:01:21.460 align:middle line:90% So they're looping like this. 00:01:21.460 --> 00:01:24.850 align:middle line:84% Then as our protons go through here, 00:01:24.850 --> 00:01:31.340 align:middle line:84% they're going to feel a force, which is equal to that charge-- 00:01:31.340 --> 00:01:33.640 align:middle line:90% I'll just write a z-- 00:01:33.640 --> 00:01:35.930 align:middle line:90% when you cross B. 00:01:35.930 --> 00:01:38.100 align:middle line:84% But they've got velocity in this direction. 00:01:38.100 --> 00:01:41.100 align:middle line:84% For example here, the magnetic field is pointing into the page, 00:01:41.100 --> 00:01:43.490 align:middle line:84% and so the particles are going to feel the force upwards. 00:01:43.490 --> 00:01:47.200 align:middle line:90% 00:01:47.200 --> 00:01:51.200 align:middle line:84% And these particles here are going to feel a force downwards 00:01:51.200 --> 00:01:52.880 align:middle line:90% like this. 00:01:52.880 --> 00:02:00.650 align:middle line:84% And if we put some sort of detector some distance away, 00:02:00.650 --> 00:02:03.730 align:middle line:84% we should see that our source, which maybe was initially 00:02:03.730 --> 00:02:07.450 align:middle line:84% uniform, so we had a uniform number of protons per centimeter 00:02:07.450 --> 00:02:10.870 align:middle line:84% squared, millimeter squared, now that's uniform intensity 00:02:10.870 --> 00:02:13.040 align:middle line:90% is going to be changed. 00:02:13.040 --> 00:02:17.140 align:middle line:84% So for example, if we draw a little line drawing of intensity 00:02:17.140 --> 00:02:22.940 align:middle line:84% over the initial intensity in the absence of this plasma, 00:02:22.940 --> 00:02:25.080 align:middle line:84% our intensity profile would just be 1. 00:02:25.080 --> 00:02:26.820 align:middle line:84% There's nothing to change the intensity. 00:02:26.820 --> 00:02:29.790 align:middle line:84% But in this case here we can see that this magnetic field 00:02:29.790 --> 00:02:31.740 align:middle line:84% configuration has created a void. 00:02:31.740 --> 00:02:34.470 align:middle line:84% It's pushed the protons outwards, effectively 00:02:34.470 --> 00:02:35.710 align:middle line:90% defocus them. 00:02:35.710 --> 00:02:38.970 align:middle line:84% So we see that we're going to have relatively few protons 00:02:38.970 --> 00:02:39.690 align:middle line:90% in the middle. 00:02:39.690 --> 00:02:42.090 align:middle line:84% We're going to have a pile up of protons at the edge. 00:02:42.090 --> 00:02:44.430 align:middle line:84% And then far away from the plasma, 00:02:44.430 --> 00:02:47.305 align:middle line:84% we're just going to have our standard fluence of 1 00:02:47.305 --> 00:02:48.930 align:middle line:84% because there'll be some particles that 00:02:48.930 --> 00:02:50.940 align:middle line:90% just go around the edge here. 00:02:50.940 --> 00:02:53.160 align:middle line:84% And here in this little drawing, it's 00:02:53.160 --> 00:02:55.980 align:middle line:90% going to be nice and symmetric. 00:02:55.980 --> 00:03:00.820 align:middle line:84% So what this would look like if we looked front on potentially 00:03:00.820 --> 00:03:10.260 align:middle line:84% is a bright ring where we have lots of protons 00:03:10.260 --> 00:03:11.610 align:middle line:90% and a void in the center. 00:03:11.610 --> 00:03:16.660 align:middle line:90% 00:03:16.660 --> 00:03:18.190 align:middle line:90% That's the magnetic fields. 00:03:18.190 --> 00:03:24.010 align:middle line:84% We could also have a plasma that has electric fields as well. 00:03:24.010 --> 00:03:28.190 align:middle line:84% And although we think electric fields are quite small due 00:03:28.190 --> 00:03:31.820 align:middle line:84% to bi-shielding, on short timescales in HED plasmas 00:03:31.820 --> 00:03:34.225 align:middle line:84% where these are often used, the electric fields 00:03:34.225 --> 00:03:36.350 align:middle line:84% can be relatively large because the plasma does not 00:03:36.350 --> 00:03:38.210 align:middle line:84% have to be quasi neutral, so we could 00:03:38.210 --> 00:03:40.820 align:middle line:84% have some sort of electric field like this. 00:03:40.820 --> 00:03:43.550 align:middle line:84% And then our electric field would also 00:03:43.550 --> 00:03:47.840 align:middle line:90% put a force on these protons. 00:03:47.840 --> 00:03:49.800 align:middle line:84% In this case, I've drawn it in the same way 00:03:49.800 --> 00:03:53.790 align:middle line:84% so the electric field is pushing the protons out from here. 00:03:53.790 --> 00:03:57.740 align:middle line:84% So if I draw some more protons coming from a point source 00:03:57.740 --> 00:04:00.590 align:middle line:84% and I draw some trajectories going into the plasma, 00:04:00.590 --> 00:04:03.520 align:middle line:84% I'll have protons which are being bent upwards 00:04:03.520 --> 00:04:06.530 align:middle line:90% and downwards like this. 00:04:06.530 --> 00:04:09.470 align:middle line:84% And so on my detector I could, for example, 00:04:09.470 --> 00:04:12.640 align:middle line:84% see a very similar pattern, where 00:04:12.640 --> 00:04:15.040 align:middle line:84% I have a void in the middle, I have some regions 00:04:15.040 --> 00:04:18.100 align:middle line:84% of increased fluence on the limbs, 00:04:18.100 --> 00:04:20.529 align:middle line:84% and then I have a region where there's 00:04:20.529 --> 00:04:22.750 align:middle line:84% undisturbed fluence because I've got particles 00:04:22.750 --> 00:04:24.355 align:middle line:90% which are just going around. 00:04:24.355 --> 00:04:27.575 align:middle line:90% 00:04:27.575 --> 00:04:29.450 align:middle line:84% So this is a technique called proton imaging. 00:04:29.450 --> 00:04:32.520 align:middle line:84% It's a way of imaging electric and magnetic fields. 00:04:32.520 --> 00:04:36.360 align:middle line:84% But of course, it feels both the electric and the magnetic field. 00:04:36.360 --> 00:04:38.330 align:middle line:84% So that's one fundamental limitation with it, 00:04:38.330 --> 00:04:41.060 align:middle line:84% is that you never quite sure whether your deflection is 00:04:41.060 --> 00:04:43.930 align:middle line:84% due to electric or magnetic fields. 00:04:43.930 --> 00:04:47.890 align:middle line:84% There is a third form of proton imaging, which is technically 00:04:47.890 --> 00:04:49.450 align:middle line:90% called proton radiography. 00:04:49.450 --> 00:04:59.640 align:middle line:90% 00:04:59.640 --> 00:05:01.680 align:middle line:84% Proton radiography, the radiography 00:05:01.680 --> 00:05:05.460 align:middle line:84% here is to make you think about X-rays and X-ray 00:05:05.460 --> 00:05:13.040 align:middle line:84% radiography, which is where you have your beam of particles, 00:05:13.040 --> 00:05:15.640 align:middle line:84% and you have some very, very dense plasma here. 00:05:15.640 --> 00:05:18.960 align:middle line:90% 00:05:18.960 --> 00:05:21.090 align:middle line:84% And because it's so dense, these particles 00:05:21.090 --> 00:05:22.980 align:middle line:84% don't stream through it and just fuel 00:05:22.980 --> 00:05:25.410 align:middle line:84% the electric or magnetic fields, the particles 00:05:25.410 --> 00:05:28.410 align:middle line:84% start scattering, and colliding, and maybe 00:05:28.410 --> 00:05:32.180 align:middle line:84% they never escape out the other side. 00:05:32.180 --> 00:05:35.420 align:middle line:84% And then on your detector, you would 00:05:35.420 --> 00:05:41.280 align:middle line:84% have similar to an X-ray, an image which, for example, has 00:05:41.280 --> 00:05:45.760 align:middle line:84% a region which is a void where there are no protons, 00:05:45.760 --> 00:05:48.400 align:middle line:84% and that's because all the protons have been scattered. 00:05:48.400 --> 00:05:52.580 align:middle line:90% 00:05:52.580 --> 00:05:58.840 align:middle line:84% So again, this only happens at high densities. 00:05:58.840 --> 00:06:02.890 align:middle line:84% At lower densities, the particles should stream through. 00:06:02.890 --> 00:06:05.560 align:middle line:84% Unfortunately a lot of people call this proton imaging 00:06:05.560 --> 00:06:07.410 align:middle line:84% technique proton radiography because that's 00:06:07.410 --> 00:06:09.160 align:middle line:84% what it was initially called, but it's not 00:06:09.160 --> 00:06:10.780 align:middle line:90% a radiographic technique. 00:06:10.780 --> 00:06:13.630 align:middle line:84% Radiography refers to your beam being attenuated 00:06:13.630 --> 00:06:14.720 align:middle line:90% by density in the way. 00:06:14.720 --> 00:06:17.345 align:middle line:84% So it's not really a right name, but you'll see a lot of people 00:06:17.345 --> 00:06:19.240 align:middle line:90% call this proton radiography. 00:06:19.240 --> 00:06:23.320 align:middle line:84% Some people have been trying to call it proton imaging instead. 00:06:23.320 --> 00:06:27.010 align:middle line:84% Even in the case where you don't have such a high density, 00:06:27.010 --> 00:06:30.400 align:middle line:84% you may still end up with enough density in this plasma 00:06:30.400 --> 00:06:34.120 align:middle line:84% that it will knock a particle off course, 00:06:34.120 --> 00:06:36.040 align:middle line:84% and that will lead to some sort of blurring. 00:06:36.040 --> 00:06:44.850 align:middle line:84% So these weak collisions can lead to a blurring of the image. 00:06:44.850 --> 00:06:47.720 align:middle line:84% So you may still want to take into account the [INAUDIBLE] 00:06:47.720 --> 00:06:48.380 align:middle line:90% a little bit. 00:06:48.380 --> 00:06:55.150 align:middle line:90% 00:06:55.150 --> 00:06:56.590 align:middle line:90% OK. 00:06:56.590 --> 00:07:02.990 align:middle line:84% And the references for this lecture are two papers. 00:07:02.990 --> 00:07:04.485 align:middle line:90% First one is by Hoogland. 00:07:04.485 --> 00:07:08.020 align:middle line:90% 00:07:08.020 --> 00:07:11.320 align:middle line:84% You can find it in Review of Scientific Instruments, RSI, 00:07:11.320 --> 00:07:13.200 align:middle line:90% from 2012. 00:07:13.200 --> 00:07:15.890 align:middle line:84% A lot of the mathematics we'll be using is based on this paper. 00:07:15.890 --> 00:07:19.880 align:middle line:84% And very recently, there was a review article by Schaeffer. 00:07:19.880 --> 00:07:22.586 align:middle line:90% 00:07:22.586 --> 00:07:25.090 align:middle line:90% It hasn't been published yet. 00:07:25.090 --> 00:07:28.220 align:middle line:90% It's on the archive. 00:07:28.220 --> 00:07:30.833 align:middle line:84% And that was published last year. 00:07:30.833 --> 00:07:32.750 align:middle line:84% And this review article is really, really good 00:07:32.750 --> 00:07:35.390 align:middle line:84% at the nitty gritty of how proton radiography is actually 00:07:35.390 --> 00:07:38.270 align:middle line:84% done, and we'll talk an awful lot, or a little bit at least, 00:07:38.270 --> 00:07:41.215 align:middle line:84% about how these sources of protons are made later on. 00:07:41.215 --> 00:07:42.590 align:middle line:84% For now, for the first bit, we're 00:07:42.590 --> 00:07:44.940 align:middle line:84% just going to assume there is some source of protons. 00:07:44.940 --> 00:07:48.500 align:middle line:84% So before we get on to a more mathematical treatment of this, 00:07:48.500 --> 00:07:50.260 align:middle line:90% does anyone have any questions? 00:07:50.260 --> 00:07:51.618 align:middle line:90% Yes. 00:07:51.618 --> 00:07:55.803 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:07:55.803 --> 00:07:57.220 align:middle line:84% PROFESSOR: Yeah, absolutely right. 00:07:57.220 --> 00:07:59.380 align:middle line:84% So at the edges here, where they have less distance 00:07:59.380 --> 00:08:01.122 align:middle line:84% to travel through, you could have 00:08:01.122 --> 00:08:03.580 align:middle line:84% something that gets knocked around, but still gets through. 00:08:03.580 --> 00:08:06.130 align:middle line:84% But maybe another particle traveling a very similar path 00:08:06.130 --> 00:08:07.930 align:middle line:84% doesn't get through, and so you'd 00:08:07.930 --> 00:08:09.680 align:middle line:90% have grayscale you'd have here. 00:08:09.680 --> 00:08:13.113 align:middle line:90% So just like with X-rays, yeah. 00:08:13.113 --> 00:08:13.780 align:middle line:90% Other questions. 00:08:13.780 --> 00:08:14.730 align:middle line:90% Any questions online? 00:08:14.730 --> 00:08:21.230 align:middle line:90% 00:08:21.230 --> 00:08:23.180 align:middle line:84% OK, so mathematically this does look 00:08:23.180 --> 00:08:26.010 align:middle line:84% very similar to shadowgraphy, but there are some differences. 00:08:26.010 --> 00:08:28.280 align:middle line:84% So I'm going to go through the mathematics again. 00:08:28.280 --> 00:08:30.080 align:middle line:84% So you may notice some similarities, 00:08:30.080 --> 00:08:31.550 align:middle line:90% some differences in notation. 00:08:31.550 --> 00:08:33.200 align:middle line:90% That's all OK. 00:08:33.200 --> 00:08:36.409 align:middle line:84% So we're going to start by making some assumptions. 00:08:36.409 --> 00:08:41.760 align:middle line:90% 00:08:41.760 --> 00:08:44.355 align:middle line:84% Our first assumption is that we have a point source. 00:08:44.355 --> 00:08:49.660 align:middle line:90% 00:08:49.660 --> 00:08:55.960 align:middle line:84% In reality this means that if we have some source size here, 00:08:55.960 --> 00:09:01.650 align:middle line:84% which has a size, say, D source, that has to be much, much less 00:09:01.650 --> 00:09:05.640 align:middle line:84% than a length scale the size of our plasma. 00:09:05.640 --> 00:09:09.690 align:middle line:84% Because if that's true, we have some finite source size here 00:09:09.690 --> 00:09:12.870 align:middle line:84% and some very big plasma, from the perspective of the plasma, 00:09:12.870 --> 00:09:14.970 align:middle line:90% this will roughly be a point. 00:09:14.970 --> 00:09:17.100 align:middle line:84% So we can make this more rigorous 00:09:17.100 --> 00:09:19.320 align:middle line:84% by introducing a dimensionless parameter 00:09:19.320 --> 00:09:22.060 align:middle line:84% and insisting that parameter is small. 00:09:22.060 --> 00:09:25.080 align:middle line:84% This is a pretty good approximation. 00:09:25.080 --> 00:09:33.810 align:middle line:84% The next thing we need to assume is a uniform beam, 00:09:33.810 --> 00:09:38.880 align:middle line:84% which is to say that we have the same number of protons 00:09:38.880 --> 00:09:44.570 align:middle line:84% coming through every little infinitesimal solid angle unit 00:09:44.570 --> 00:09:45.070 align:middle line:90% here. 00:09:45.070 --> 00:09:47.590 align:middle line:84% So that in the absence of any plasma, 00:09:47.590 --> 00:09:50.150 align:middle line:84% when I put my detector here, I do, in fact, 00:09:50.150 --> 00:09:53.910 align:middle line:90% get a nice uniform signal on it. 00:09:53.910 --> 00:09:57.660 align:middle line:90% So I over I0 is just 1 here. 00:09:57.660 --> 00:10:01.440 align:middle line:84% If my signal is not uniform like this, 00:10:01.440 --> 00:10:03.155 align:middle line:84% it complicates the interpretation. 00:10:03.155 --> 00:10:04.530 align:middle line:84% It turns out you can do something 00:10:04.530 --> 00:10:07.260 align:middle line:84% with a non-uniform beam of protons, 00:10:07.260 --> 00:10:08.950 align:middle line:90% but mathematically it's harder. 00:10:08.950 --> 00:10:11.100 align:middle line:84% So we're going to insist on a uniform beam. 00:10:11.100 --> 00:10:15.540 align:middle line:84% So we'll say the gradient of the beam intensity is equal to 0, 00:10:15.540 --> 00:10:20.815 align:middle line:84% at least over the target, over the plasma, I guess. 00:10:20.815 --> 00:10:25.580 align:middle line:90% 00:10:25.580 --> 00:10:26.790 align:middle line:90% OK. 00:10:26.790 --> 00:10:29.930 align:middle line:84% Third approximation we'll make is the paraxial approximation 00:10:29.930 --> 00:10:30.980 align:middle line:90% from geometric optics. 00:10:30.980 --> 00:10:35.800 align:middle line:90% 00:10:35.800 --> 00:10:38.470 align:middle line:84% The paraxial approximation is basically 00:10:38.470 --> 00:10:41.500 align:middle line:84% saying that although these particles are following 00:10:41.500 --> 00:10:45.580 align:middle line:84% some path that we can call DL, our detector 00:10:45.580 --> 00:10:47.720 align:middle line:90% is relatively small. 00:10:47.720 --> 00:10:51.730 align:middle line:84% And so most of the particles are going almost parallel 00:10:51.730 --> 00:10:55.290 align:middle line:90% to dz hat here. 00:10:55.290 --> 00:10:58.960 align:middle line:90% 00:10:58.960 --> 00:11:01.890 align:middle line:84% So this means we can approximate any path integrals 00:11:01.890 --> 00:11:04.230 align:middle line:84% as just integrals in the z direction, 00:11:04.230 --> 00:11:07.230 align:middle line:84% and this also means we can approximate angles like tan 00:11:07.230 --> 00:11:12.210 align:middle line:84% alpha to be approximately sine alpha to be approximately alpha. 00:11:12.210 --> 00:11:16.120 align:middle line:84% This makes the mathematics very easy as well. 00:11:16.120 --> 00:11:18.510 align:middle line:84% And the final assumption we'll make 00:11:18.510 --> 00:11:26.210 align:middle line:84% is it all protons are test particles. 00:11:26.210 --> 00:11:29.650 align:middle line:84% So you may have seen a test particle formalism 00:11:29.650 --> 00:11:31.750 align:middle line:90% in some bits of plasma theory. 00:11:31.750 --> 00:11:36.040 align:middle line:84% The idea is that your test particle merely experiences 00:11:36.040 --> 00:11:37.360 align:middle line:90% the fields inside your plasma. 00:11:37.360 --> 00:11:39.500 align:middle line:90% It doesn't contribute to them. 00:11:39.500 --> 00:11:42.070 align:middle line:84% So the fact that we have this beam of protons streaming 00:11:42.070 --> 00:11:45.490 align:middle line:84% through here, we assume it has negligible perturbation 00:11:45.490 --> 00:11:47.630 align:middle line:90% on the plasma itself. 00:11:47.630 --> 00:11:50.410 align:middle line:84% So we have no deflection, for example, 00:11:50.410 --> 00:11:53.790 align:middle line:90% between protons within the beam. 00:11:53.790 --> 00:12:00.920 align:middle line:90% 00:12:00.920 --> 00:12:04.630 align:middle line:84% And this is a statement that the average potential 00:12:04.630 --> 00:12:08.770 align:middle line:84% that the protons are creating divided by their kinetic energy 00:12:08.770 --> 00:12:11.020 align:middle line:90% is much, much less than 1. 00:12:11.020 --> 00:12:13.725 align:middle line:84% That is to say they're going so fast that even if they do feel 00:12:13.725 --> 00:12:15.850 align:middle line:84% a small electric field that deflects them slightly, 00:12:15.850 --> 00:12:18.370 align:middle line:84% it's negligible because they're moving so fast. 00:12:18.370 --> 00:12:23.050 align:middle line:84% It doesn't change their overall proton kinetic energy here. 00:12:23.050 --> 00:12:25.030 align:middle line:84% OK, any questions on assumptions? 00:12:25.030 --> 00:12:25.998 align:middle line:90% Yes. 00:12:25.998 --> 00:12:29.484 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:12:29.484 --> 00:12:37.987 align:middle line:90% 00:12:37.987 --> 00:12:40.320 align:middle line:84% PROFESSOR: Yeah, the paraxial regime takes care of that. 00:12:40.320 --> 00:12:41.280 align:middle line:90% Yeah, exactly. 00:12:41.280 --> 00:12:43.020 align:middle line:84% But you're right, but we're really 00:12:43.020 --> 00:12:45.420 align:middle line:84% talking about less than a degree subtended. 00:12:45.420 --> 00:12:47.160 align:middle line:90% So it's very, very small angle. 00:12:47.160 --> 00:12:51.963 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:12:51.963 --> 00:12:53.880 align:middle line:84% PROFESSOR: The only place that appeared so far 00:12:53.880 --> 00:12:55.680 align:middle line:84% is we've assumed that it's very, very hot. 00:12:55.680 --> 00:12:58.740 align:middle line:84% So these protons tend to be, and we'll get on to the generation 00:12:58.740 --> 00:13:03.140 align:middle line:84% mechanisms, but these tend to be mega electron volt energy 00:13:03.140 --> 00:13:03.830 align:middle line:90% protons here. 00:13:03.830 --> 00:13:08.540 align:middle line:84% So they're much, much faster than any of the other particles 00:13:08.540 --> 00:13:09.710 align:middle line:90% in our system generally. 00:13:09.710 --> 00:13:15.222 align:middle line:84% And because the collisonality, the mean free path goes as-- 00:13:15.222 --> 00:13:16.430 align:middle line:90% I'm going to get this right-- 00:13:16.430 --> 00:13:21.290 align:middle line:84% velocity cubed, then these very fast particles here 00:13:21.290 --> 00:13:22.640 align:middle line:90% have a very long mean free path. 00:13:22.640 --> 00:13:24.432 align:middle line:84% Their collisonality is very low, so they're 00:13:24.432 --> 00:13:28.320 align:middle line:84% unlikely to collide with any particles in the path. 00:13:28.320 --> 00:13:29.440 align:middle line:90% We'll talk about that. 00:13:29.440 --> 00:13:30.148 align:middle line:90% Yeah, no, no, no. 00:13:30.148 --> 00:13:31.140 align:middle line:90% It's great. 00:13:31.140 --> 00:13:34.260 align:middle line:84% For the purposes of this, we're going to do it in such a way 00:13:34.260 --> 00:13:36.060 align:middle line:84% that we will just solve for the-- there 00:13:36.060 --> 00:13:38.790 align:middle line:84% are some protons with path energy 1, 00:13:38.790 --> 00:13:41.493 align:middle line:84% and some protons of energy 2, and our solution is separable. 00:13:41.493 --> 00:13:43.410 align:middle line:84% And I guess that comes back to here-- the fact 00:13:43.410 --> 00:13:44.850 align:middle line:84% that beam doesn't interact with itself, 00:13:44.850 --> 00:13:46.433 align:middle line:84% but we'll talk a lot about that later. 00:13:46.433 --> 00:13:48.510 align:middle line:90% Yeah, Alan? 00:13:48.510 --> 00:13:52.380 align:middle line:84% AUDIENCE: Why not use [INAUDIBLE]? 00:13:52.380 --> 00:13:54.470 align:middle line:84% PROFESSOR: They're much lighter and so they'll 00:13:54.470 --> 00:13:56.190 align:middle line:90% scatter more easily. 00:13:56.190 --> 00:14:00.240 align:middle line:84% But people have started doing electron imaging as well. 00:14:00.240 --> 00:14:03.480 align:middle line:84% But then you need GeV beams and stuff like that, 00:14:03.480 --> 00:14:05.240 align:middle line:84% so the protons are better for this. 00:14:05.240 --> 00:14:07.490 align:middle line:84% AUDIENCE: Would there be advantages to that path? 00:14:07.490 --> 00:14:09.740 align:middle line:84% PROFESSOR: I don't know enough about the electron beam 00:14:09.740 --> 00:14:10.580 align:middle line:90% techniques at the moment. 00:14:10.580 --> 00:14:12.165 align:middle line:84% So a lot of what I'm going to show you 00:14:12.165 --> 00:14:13.790 align:middle line:84% is through very specific detectors that 00:14:13.790 --> 00:14:15.307 align:middle line:84% have been created for protons, and I 00:14:15.307 --> 00:14:17.390 align:middle line:84% don't know whether that detector technology exists 00:14:17.390 --> 00:14:19.970 align:middle line:90% for electrons in the same way. 00:14:19.970 --> 00:14:21.470 align:middle line:90% There's another question. 00:14:21.470 --> 00:14:22.840 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:14:22.840 --> 00:14:31.345 align:middle line:90% 00:14:31.345 --> 00:14:32.970 align:middle line:84% PROFESSOR: Yeah, it's very hard to make 00:14:32.970 --> 00:14:34.738 align:middle line:90% a collimated beam of protons. 00:14:34.738 --> 00:14:36.780 align:middle line:84% And when we talk about the generation mechanisms, 00:14:36.780 --> 00:14:40.440 align:middle line:84% they tend to make point sources or things with very small spot 00:14:40.440 --> 00:14:41.010 align:middle line:90% sizes. 00:14:41.010 --> 00:14:43.980 align:middle line:84% So it's a technological limitation. 00:14:43.980 --> 00:14:46.740 align:middle line:84% In the very early days people used particle accelerators 00:14:46.740 --> 00:14:47.550 align:middle line:90% to do this. 00:14:47.550 --> 00:14:49.740 align:middle line:84% But although particle accelerators 00:14:49.740 --> 00:14:52.660 align:middle line:84% can accelerate a lot of particles to high energy, 00:14:52.660 --> 00:14:55.830 align:middle line:84% it's not enough particles on the timescale of these experiments. 00:14:55.830 --> 00:14:58.230 align:middle line:84% These experiments are over in a nanosecond, 00:14:58.230 --> 00:15:00.773 align:middle line:84% and so you need a very bright burst of protons. 00:15:00.773 --> 00:15:02.190 align:middle line:84% And it turns out the techniques we 00:15:02.190 --> 00:15:04.020 align:middle line:84% have for making those bursts of protons 00:15:04.020 --> 00:15:05.820 align:middle line:90% make a nice point source. 00:15:05.820 --> 00:15:06.945 align:middle line:90% Yeah, was there a question? 00:15:06.945 --> 00:15:11.290 align:middle line:90% 00:15:11.290 --> 00:15:14.530 align:middle line:84% AUDIENCE: When you say point source, 00:15:14.530 --> 00:15:17.110 align:middle line:84% you mean like just from a specific one location, not 00:15:17.110 --> 00:15:20.392 align:middle line:84% the point source hitting the plasma, correct? 00:15:20.392 --> 00:15:22.600 align:middle line:84% PROFESSOR: Sorry, could you say that a different way? 00:15:22.600 --> 00:15:25.910 align:middle line:84% AUDIENCE: Yeah, so the point source is the beams. 00:15:25.910 --> 00:15:28.750 align:middle line:84% And when you were talking about the particle acceleration, 00:15:28.750 --> 00:15:31.120 align:middle line:84% you said it was coming from a single area. 00:15:31.120 --> 00:15:33.880 align:middle line:84% Are they still doing that with however else 00:15:33.880 --> 00:15:36.497 align:middle line:84% they're generating the protons these days? 00:15:36.497 --> 00:15:38.080 align:middle line:84% PROFESSOR: Yeah, we'll skip over that, 00:15:38.080 --> 00:15:40.180 align:middle line:84% and we'll get to how the protons are being generated later, 00:15:40.180 --> 00:15:41.300 align:middle line:84% and it might make a bit more sense. 00:15:41.300 --> 00:15:43.090 align:middle line:84% So now you can imagine this as a point 00:15:43.090 --> 00:15:45.430 align:middle line:84% source that, for example, is emitting 00:15:45.430 --> 00:15:47.560 align:middle line:90% protons in every direction. 00:15:47.560 --> 00:15:50.077 align:middle line:84% It just turns out that our plasma is only in one direction. 00:15:50.077 --> 00:15:51.910 align:middle line:84% And so from the point of view of the plasma, 00:15:51.910 --> 00:15:54.368 align:middle line:84% there's just a little point that's sending protons its way. 00:15:54.368 --> 00:15:56.800 align:middle line:84% AUDIENCE: But in reality, that's not exactly happening. 00:15:56.800 --> 00:15:59.440 align:middle line:90% I see, OK. 00:15:59.440 --> 00:16:01.133 align:middle line:84% PROFESSOR: OK, any other questions? 00:16:01.133 --> 00:16:05.080 align:middle line:90% 00:16:05.080 --> 00:16:08.680 align:middle line:84% OK, so let's set up the geometry of our problem. 00:16:08.680 --> 00:16:12.930 align:middle line:90% 00:16:12.930 --> 00:16:16.300 align:middle line:84% So we've got our point source over here. 00:16:16.300 --> 00:16:21.280 align:middle line:90% We've got our plasma here. 00:16:21.280 --> 00:16:27.740 align:middle line:84% The distance between the two of these is lowercase L. 00:16:27.740 --> 00:16:33.980 align:middle line:84% Our plasma has some characteristic size, a, here. 00:16:33.980 --> 00:16:40.150 align:middle line:84% We have a detector some distance away, 00:16:40.150 --> 00:16:46.940 align:middle line:84% and we'll call that distance capital L. Within the plasma, 00:16:46.940 --> 00:16:51.550 align:middle line:84% we have a coordinate system and we label points 00:16:51.550 --> 00:16:55.960 align:middle line:84% within the plasma using the position vector, x0. 00:16:55.960 --> 00:16:59.590 align:middle line:84% Yeah if I have a position with a 0 subscript, 00:16:59.590 --> 00:17:01.600 align:middle line:84% that's somewhere within the plasma. 00:17:01.600 --> 00:17:07.040 align:middle line:84% And then on the detector, I have a position vector, x. 00:17:07.040 --> 00:17:08.790 align:middle line:84% So what we're going to be doing is looking 00:17:08.790 --> 00:17:12.300 align:middle line:84% at how the protons are deflected at position x0 00:17:12.300 --> 00:17:15.030 align:middle line:84% and they will end up somewhere on the detector at x. 00:17:15.030 --> 00:17:16.692 align:middle line:84% And what our job is, therefore, is 00:17:16.692 --> 00:17:18.150 align:middle line:84% to link what we see on our detector 00:17:18.150 --> 00:17:22.829 align:middle line:84% x back to the properties of the plasma at x0. 00:17:22.829 --> 00:17:28.280 align:middle line:84% And we can think of a proton coming out like this 00:17:28.280 --> 00:17:31.880 align:middle line:84% and within the plasma undergoing some deflection. 00:17:31.880 --> 00:17:36.350 align:middle line:90% 00:17:36.350 --> 00:17:40.370 align:middle line:84% And it's deflected from its original trajectory by an angle, 00:17:40.370 --> 00:17:41.360 align:middle line:90% alpha. 00:17:41.360 --> 00:17:45.052 align:middle line:84% And this alpha, in general, can be a vector. 00:17:45.052 --> 00:17:46.760 align:middle line:84% So we can have an angle in this direction 00:17:46.760 --> 00:17:49.730 align:middle line:84% and we can have an angle in this direction here. 00:17:49.730 --> 00:17:51.740 align:middle line:84% I'll be drawing it mostly in 1D, but remember, 00:17:51.740 --> 00:17:56.640 align:middle line:84% the particles can be deflected in two dimensions as well. 00:17:56.640 --> 00:17:58.950 align:middle line:84% So from this simple geometry, we can already 00:17:58.950 --> 00:18:01.620 align:middle line:84% write down what we're going to see on the detector x. 00:18:01.620 --> 00:18:06.990 align:middle line:84% So x is a location of a proton which 00:18:06.990 --> 00:18:10.740 align:middle line:90% streams through the point x0. 00:18:10.740 --> 00:18:18.600 align:middle line:84% And then there's a magnification factor, capital L over lowercase 00:18:18.600 --> 00:18:20.520 align:middle line:84% L. This is just similar triangles. 00:18:20.520 --> 00:18:27.040 align:middle line:84% This just says in the absence of any deflection, 00:18:27.040 --> 00:18:29.620 align:middle line:84% I would have ended up at this point here, 00:18:29.620 --> 00:18:31.780 align:middle line:84% and that point is magnified because capital L 00:18:31.780 --> 00:18:34.820 align:middle line:90% and lowercase L are different. 00:18:34.820 --> 00:18:37.190 align:middle line:84% So far this is just what you'd get without any plasma. 00:18:37.190 --> 00:18:39.350 align:middle line:84% But when you put the plasma in, of course 00:18:39.350 --> 00:18:44.060 align:middle line:84% you get some deflection, alpha, and that alpha is times 00:18:44.060 --> 00:18:47.340 align:middle line:84% by the distance between the plasma and the detector here. 00:18:47.340 --> 00:18:49.250 align:middle line:90% So again, this is just geometry. 00:18:49.250 --> 00:18:53.660 align:middle line:84% It has nothing to do with plasmas, or protons, or anything 00:18:53.660 --> 00:18:56.070 align:middle line:90% in there. 00:18:56.070 --> 00:18:59.000 align:middle line:84% So just to clarify what all these terms are, 00:18:59.000 --> 00:19:02.690 align:middle line:90% this is at the detector. 00:19:02.690 --> 00:19:04.795 align:middle line:90% This is the object. 00:19:04.795 --> 00:19:08.160 align:middle line:90% 00:19:08.160 --> 00:19:12.900 align:middle line:84% This term here is due to the divergence of the beam. 00:19:12.900 --> 00:19:16.030 align:middle line:90% 00:19:16.030 --> 00:19:18.970 align:middle line:84% We wouldn't have this term if we had a collimated beam 00:19:18.970 --> 00:19:20.170 align:middle line:90% like we had with lasers. 00:19:20.170 --> 00:19:22.690 align:middle line:84% Remember with the laser, we started out with a load of rays 00:19:22.690 --> 00:19:26.350 align:middle line:84% which parallel, and so this wouldn't have happened 00:19:26.350 --> 00:19:27.430 align:middle line:90% with a collimated beam. 00:19:27.430 --> 00:19:28.490 align:middle line:90% But we don't have that. 00:19:28.490 --> 00:19:29.560 align:middle line:90% We got a point source. 00:19:29.560 --> 00:19:31.810 align:middle line:90% We got a diverging beam. 00:19:31.810 --> 00:19:35.020 align:middle line:84% And then finally, this is to do with the deflection. 00:19:35.020 --> 00:19:39.830 align:middle line:90% 00:19:39.830 --> 00:19:51.220 align:middle line:84% So this is like shadowgraphy with a diverging beam. 00:19:51.220 --> 00:20:01.910 align:middle line:90% 00:20:01.910 --> 00:20:03.400 align:middle line:90% OK, any questions on that? 00:20:03.400 --> 00:20:18.270 align:middle line:90% 00:20:18.270 --> 00:20:29.360 align:middle line:84% Typically we set up our system so uppercase L upon lowercase L 00:20:29.360 --> 00:20:31.820 align:middle line:90% is much, much larger than 1. 00:20:31.820 --> 00:20:38.670 align:middle line:84% So that is to say we have our source, we have our plasma, 00:20:38.670 --> 00:20:44.270 align:middle line:84% and then quite a long distance away, we have our detector here. 00:20:44.270 --> 00:20:48.360 align:middle line:84% Remember, this is lowercase L, this is uppercase L. 00:20:48.360 --> 00:20:49.810 align:middle line:90% This is for two reasons. 00:20:49.810 --> 00:20:52.060 align:middle line:84% Firstly, the plasmas tend to be very small. 00:20:52.060 --> 00:20:53.590 align:middle line:84% And so by setting it up like this, 00:20:53.590 --> 00:20:55.170 align:middle line:90% we get a massive magnification. 00:20:55.170 --> 00:20:59.430 align:middle line:84% So because we've got these diverging protons here mapping 00:20:59.430 --> 00:21:02.820 align:middle line:84% out over there, we're going to have a nice big image 00:21:02.820 --> 00:21:03.640 align:middle line:90% to look at. 00:21:03.640 --> 00:21:05.520 align:middle line:84% So if this is only 100 microns across, 00:21:05.520 --> 00:21:08.628 align:middle line:84% maybe on your image over here it's 10 millimeters across, 00:21:08.628 --> 00:21:10.920 align:middle line:84% which is still pretty small, but maybe you can actually 00:21:10.920 --> 00:21:12.360 align:middle line:90% see something on your detector. 00:21:12.360 --> 00:21:14.190 align:middle line:84% The other reason is just practical. 00:21:14.190 --> 00:21:17.310 align:middle line:84% You cannot put your detector very close to your plasma. 00:21:17.310 --> 00:21:19.550 align:middle line:84% This is something like a laser-driven ICF implosion. 00:21:19.550 --> 00:21:22.050 align:middle line:84% You really don't want to have your detector close because it 00:21:22.050 --> 00:21:25.030 align:middle line:84% will get trashed, so you've got to put it a long way away. 00:21:25.030 --> 00:21:28.210 align:middle line:84% So this is typically the regime that we work in. 00:21:28.210 --> 00:21:30.630 align:middle line:84% And then, as you can see up here, 00:21:30.630 --> 00:21:36.400 align:middle line:84% that means that we have a term that looks like x0 1 00:21:36.400 --> 00:21:40.960 align:middle line:84% plus capital L over lowercase L, and we can just 00:21:40.960 --> 00:21:43.810 align:middle line:84% drop that term there as being small 00:21:43.810 --> 00:21:47.420 align:middle line:90% compared to the second term. 00:21:47.420 --> 00:21:51.350 align:middle line:84% And we can then convert that equation 00:21:51.350 --> 00:22:04.558 align:middle line:84% so that it reads x equals L x0 upon lowercase L plus alpha. 00:22:04.558 --> 00:22:08.520 align:middle line:84% So I've just taken the capital L outside the brackets here 00:22:08.520 --> 00:22:12.160 align:middle line:84% and now we have these two terms inside. 00:22:12.160 --> 00:22:15.660 align:middle line:84% So this first term, as I said, is simply magnification. 00:22:15.660 --> 00:22:24.430 align:middle line:90% 00:22:24.430 --> 00:22:25.570 align:middle line:90% Very poorly written. 00:22:25.570 --> 00:22:29.490 align:middle line:84% And this second term here is distortion. 00:22:29.490 --> 00:22:34.158 align:middle line:90% 00:22:34.158 --> 00:22:35.700 align:middle line:84% Because the rays are being deflected. 00:22:35.700 --> 00:22:37.470 align:middle line:84% If we didn't have the distortion, 00:22:37.470 --> 00:22:40.410 align:middle line:84% this would just be an imaging system with magnification. 00:22:40.410 --> 00:22:43.260 align:middle line:84% Because the rays are being deflected, we have distortion. 00:22:43.260 --> 00:22:47.370 align:middle line:84% And because it's distortion, proton imaging is not imaging. 00:22:47.370 --> 00:22:49.980 align:middle line:90% 00:22:49.980 --> 00:22:52.357 align:middle line:84% So perhaps they should call it proton deflectometry. 00:22:52.357 --> 00:22:53.940 align:middle line:84% You can argue about the name for ages. 00:22:53.940 --> 00:22:55.850 align:middle line:84% But this is not an imaging technique, 00:22:55.850 --> 00:22:57.050 align:middle line:90% like with shadowgraphy. 00:22:57.050 --> 00:22:58.820 align:middle line:84% If you see something on your detector, 00:22:58.820 --> 00:23:01.460 align:middle line:84% it doesn't have a one to one correspondence with what's 00:23:01.460 --> 00:23:02.420 align:middle line:90% going on in the plasma. 00:23:02.420 --> 00:23:05.000 align:middle line:84% Features are enhanced and enlarged. 00:23:05.000 --> 00:23:09.540 align:middle line:84% The void that we were producing here, that void, for example, 00:23:09.540 --> 00:23:11.960 align:middle line:84% is going to be larger than the real size of the plasma. 00:23:11.960 --> 00:23:13.543 align:middle line:84% The protons are being pushed outwards. 00:23:13.543 --> 00:23:16.580 align:middle line:90% 00:23:16.580 --> 00:23:19.400 align:middle line:84% So mathematically what we're doing 00:23:19.400 --> 00:23:27.300 align:middle line:84% is we're taking some area inside the plasma, the s0, which 00:23:27.300 --> 00:23:32.020 align:middle line:84% has a certain number of protons going through it, 00:23:32.020 --> 00:23:34.600 align:middle line:84% and we're mapping that out onto some area 00:23:34.600 --> 00:23:36.700 align:middle line:84% on the detector, which does not have 00:23:36.700 --> 00:23:39.280 align:middle line:84% to be anywhere close to the same shape because 00:23:39.280 --> 00:23:41.950 align:middle line:84% of all the deflections going on inside here. 00:23:41.950 --> 00:23:46.890 align:middle line:90% 00:23:46.890 --> 00:23:49.950 align:middle line:90% And that area is just ds. 00:23:49.950 --> 00:23:53.940 align:middle line:84% So we're keeping our subscript notation here. 00:23:53.940 --> 00:23:55.210 align:middle line:90% This is the object. 00:23:55.210 --> 00:23:58.690 align:middle line:84% This is the image that's forming here. 00:23:58.690 --> 00:24:01.020 align:middle line:84% And the way to link these is to say 00:24:01.020 --> 00:24:08.040 align:middle line:84% that ds is equal to the magnitude of this quantity, D 00:24:08.040 --> 00:24:11.960 align:middle line:90% at x0 times ds. 00:24:11.960 --> 00:24:14.690 align:middle line:84% And this is a quantity that some of you may know. 00:24:14.690 --> 00:24:15.560 align:middle line:90% It's the Jacobian. 00:24:15.560 --> 00:24:21.590 align:middle line:90% 00:24:21.590 --> 00:24:33.365 align:middle line:84% And it has the form partial x and y over partial x0 y0. 00:24:33.365 --> 00:24:35.740 align:middle line:84% Of course, we haven't told you how to calculate that yet, 00:24:35.740 --> 00:24:38.960 align:middle line:84% but it's going to be related to this here. 00:24:38.960 --> 00:24:40.960 align:middle line:84% So this is just a general statement. 00:24:40.960 --> 00:24:43.630 align:middle line:84% When you're mapping some area to another area, 00:24:43.630 --> 00:24:46.660 align:middle line:84% you can do it using this Jacobian transform, 00:24:46.660 --> 00:24:48.940 align:middle line:84% and the exact way the Jacobian transform works 00:24:48.940 --> 00:24:51.970 align:middle line:84% will be related to this, and we'll solve it explicitly. 00:24:51.970 --> 00:24:54.560 align:middle line:90% 00:24:54.560 --> 00:24:57.320 align:middle line:90% Sorry, the left hand side here. 00:24:57.320 --> 00:24:58.220 align:middle line:90% Thank you. 00:24:58.220 --> 00:25:01.060 align:middle line:84% Yes, didn't make much sense otherwise. 00:25:01.060 --> 00:25:04.240 align:middle line:84% And guess this is a vector again, 00:25:04.240 --> 00:25:06.250 align:middle line:84% but I've written it out explicitly in terms 00:25:06.250 --> 00:25:08.800 align:middle line:90% of the x and y components. 00:25:08.800 --> 00:25:12.170 align:middle line:84% Here I just put it in this more compact vector notation. 00:25:12.170 --> 00:25:13.100 align:middle line:90% OK, questions? 00:25:13.100 --> 00:25:22.030 align:middle line:90% 00:25:22.030 --> 00:25:25.360 align:middle line:84% OK, we're going to assume that we can serve particles. 00:25:25.360 --> 00:25:27.130 align:middle line:84% Maybe this was another assumption up here, 00:25:27.130 --> 00:25:29.390 align:middle line:84% but none of our particles are scattered and lost. 00:25:29.390 --> 00:25:40.100 align:middle line:84% But if we can serve the number of protons, 00:25:40.100 --> 00:25:43.550 align:middle line:84% then the intensity on our detector 00:25:43.550 --> 00:25:45.950 align:middle line:84% is going to be equal to the intensity going 00:25:45.950 --> 00:25:51.370 align:middle line:84% through the object divided by this Jacobian. 00:25:51.370 --> 00:25:53.260 align:middle line:84% It's equivalent to the statement that we're 00:25:53.260 --> 00:25:56.650 align:middle line:84% conserving the area of this Jacobian transformation. 00:25:56.650 --> 00:25:59.050 align:middle line:84% Here we're conserving intensity here. 00:25:59.050 --> 00:26:09.340 align:middle line:84% And this Jacobian is going to be equal to capital 00:26:09.340 --> 00:26:15.320 align:middle line:84% L squared upon lowercase L squared times 1 plus then 00:26:15.320 --> 00:26:21.230 align:middle line:84% we start doing a Taylor expansion of this equation here. 00:26:21.230 --> 00:26:30.840 align:middle line:84% L radiant of alpha plus L squared 00:26:30.840 --> 00:26:37.050 align:middle line:84% partial alpha x component partial x0 partial alpha y 00:26:37.050 --> 00:26:40.275 align:middle line:90% component partial y0. 00:26:40.275 --> 00:26:42.870 align:middle line:84% And then another term, which follows symmetrically 00:26:42.870 --> 00:26:48.390 align:middle line:84% from this, which is minus partial alpha x component 00:26:48.390 --> 00:26:57.050 align:middle line:84% partial y0 partial alpha y component partial x0. 00:26:57.050 --> 00:27:00.060 align:middle line:84% Plus some other terms that we're going to drop as being small 00:27:00.060 --> 00:27:02.060 align:middle line:84% here, but there's probably a more compact vector 00:27:02.060 --> 00:27:04.130 align:middle line:84% notation for this, but I just want to write it 00:27:04.130 --> 00:27:05.210 align:middle line:90% out more explicitly. 00:27:05.210 --> 00:27:10.760 align:middle line:84% This one is clearly just partial alpha x partial x 00:27:10.760 --> 00:27:14.660 align:middle line:90% plus partial alpha y partial y. 00:27:14.660 --> 00:27:17.480 align:middle line:84% And these are all 0 subscripts because we're 00:27:17.480 --> 00:27:20.700 align:middle line:84% doing the derivative inside the plasma. 00:27:20.700 --> 00:27:22.430 align:middle line:84% So this is looking at derivatives 00:27:22.430 --> 00:27:24.320 align:middle line:84% of how the angle changes inside the plasma. 00:27:24.320 --> 00:27:27.870 align:middle line:90% 00:27:27.870 --> 00:27:30.150 align:middle line:84% So if you can calculate your angle alpha, 00:27:30.150 --> 00:27:31.860 align:middle line:84% and you might think, I can do that if I 00:27:31.860 --> 00:27:33.985 align:middle line:84% know the electric and magnetic fields in my system, 00:27:33.985 --> 00:27:35.520 align:middle line:84% and you'd be right, then you can now 00:27:35.520 --> 00:27:39.240 align:middle line:84% calculate how alpha changes within the plasma, how 00:27:39.240 --> 00:27:41.070 align:middle line:84% the protons get distorted, and now 00:27:41.070 --> 00:27:43.470 align:middle line:84% you can work out what your intensity is compared 00:27:43.470 --> 00:27:45.600 align:middle line:90% to your initial intensity. 00:27:45.600 --> 00:27:49.110 align:middle line:84% And it turns out that the sorts of images you get out of here, 00:27:49.110 --> 00:27:51.720 align:middle line:84% there's a very useful dimensionless parameter that you 00:27:51.720 --> 00:27:54.060 align:middle line:84% can write down to characterize the different images, 00:27:54.060 --> 00:27:56.640 align:middle line:84% and this parameter is called the contrast parameter. 00:27:56.640 --> 00:28:05.970 align:middle line:90% 00:28:05.970 --> 00:28:11.970 align:middle line:84% And it's written as mu, and it is defined as lowercase L 00:28:11.970 --> 00:28:12.990 align:middle line:90% over a-- 00:28:12.990 --> 00:28:15.510 align:middle line:84% lowercase L was the distance from the point source 00:28:15.510 --> 00:28:19.480 align:middle line:84% to the plasma and a is the size of our plasma-- 00:28:19.480 --> 00:28:22.540 align:middle line:84% times the modulus of this deflection angle 00:28:22.540 --> 00:28:26.950 align:middle line:84% vector, so the size of this deflection angle. 00:28:26.950 --> 00:28:30.040 align:middle line:84% And it turns out if you're in a regime where this parameter is 00:28:30.040 --> 00:28:33.550 align:middle line:84% much, much less than 1, then you find out 00:28:33.550 --> 00:28:39.970 align:middle line:84% that your D, your Jacobian here, is approximately constant 00:28:39.970 --> 00:28:42.320 align:middle line:84% and so your intensity is approximately constant. 00:28:42.320 --> 00:28:44.070 align:middle line:84% So we talked about this with shadowgraphy. 00:28:44.070 --> 00:28:47.835 align:middle line:84% This is a regime where we only have small intensity variations. 00:28:47.835 --> 00:28:49.710 align:middle line:84% And when you have small intensity variations, 00:28:49.710 --> 00:28:53.620 align:middle line:84% you have a chance of being able to undo this transformation, 00:28:53.620 --> 00:28:55.620 align:middle line:84% and work out what D was, and therefore, work out 00:28:55.620 --> 00:28:58.300 align:middle line:90% something about your plasma. 00:28:58.300 --> 00:29:02.580 align:middle line:84% But if you have a contrast parameter where mu is about 1, 00:29:02.580 --> 00:29:09.000 align:middle line:84% this corresponds to having your Jacobian with singularities 00:29:09.000 --> 00:29:11.820 align:middle line:90% or with zeros in it. 00:29:11.820 --> 00:29:14.310 align:middle line:84% And you can see if you have zeros in D, 00:29:14.310 --> 00:29:16.380 align:middle line:84% then all of a sudden, you're going to have 00:29:16.380 --> 00:29:18.690 align:middle line:90% your initial intensity over 0. 00:29:18.690 --> 00:29:23.760 align:middle line:84% Your intensity will go to some very, very large number. 00:29:23.760 --> 00:29:26.070 align:middle line:84% And these are the caustic patterns we talked about 00:29:26.070 --> 00:29:27.180 align:middle line:90% in shadowgraphy. 00:29:27.180 --> 00:29:32.000 align:middle line:90% 00:29:32.000 --> 00:29:35.540 align:middle line:84% I wish I had some more space, but I don't. 00:29:35.540 --> 00:29:38.190 align:middle line:84% Well, actually, maybe I have just about enough space. 00:29:38.190 --> 00:29:41.600 align:middle line:84% So in this case here, you can show 00:29:41.600 --> 00:29:43.610 align:middle line:84% that your intensity on your detector 00:29:43.610 --> 00:29:51.380 align:middle line:84% is equal to your source intensity times 1 minus L 00:29:51.380 --> 00:29:55.518 align:middle line:90% divergence of alpha like this. 00:29:55.518 --> 00:29:57.560 align:middle line:84% And this is equivalent to the equation we came up 00:29:57.560 --> 00:29:59.720 align:middle line:84% with for shadowgraphy, where we've implicitly 00:29:59.720 --> 00:30:02.850 align:middle line:84% assumed that all of these terms are very small. 00:30:02.850 --> 00:30:05.660 align:middle line:84% If you look here, our contrast parameter has an L 00:30:05.660 --> 00:30:06.800 align:middle line:90% and alpha has an a. 00:30:06.800 --> 00:30:09.230 align:middle line:84% We've got an L, an alpha, and this gradient 00:30:09.230 --> 00:30:12.960 align:middle line:84% is going to be on the order of the size 00:30:12.960 --> 00:30:14.380 align:middle line:90% of the plasma like that. 00:30:14.380 --> 00:30:17.370 align:middle line:84% So we've implicitly assumed that this contrast parameter is small 00:30:17.370 --> 00:30:18.730 align:middle line:90% in order to get the solution. 00:30:18.730 --> 00:30:21.460 align:middle line:84% And so what we see is we have small variations in intensity 00:30:21.460 --> 00:30:24.800 align:middle line:84% because it's 1 minus a very small number. 00:30:24.800 --> 00:30:28.040 align:middle line:84% If we have zeros here, then we actually 00:30:28.040 --> 00:30:37.890 align:middle line:84% have to solve fully for D everywhere inside the plasma. 00:30:37.890 --> 00:30:40.280 align:middle line:84% And this is where we start getting these singularities. 00:30:40.280 --> 00:30:41.915 align:middle line:90% So I've got some space. 00:30:41.915 --> 00:30:56.120 align:middle line:90% 00:30:56.120 --> 00:30:58.900 align:middle line:90% So we have D equals 0. 00:30:58.900 --> 00:31:15.630 align:middle line:84% We get singularities in I. I is equal to I0 over the size of D. 00:31:15.630 --> 00:31:18.690 align:middle line:84% And this corresponds to proton trajectories crossing. 00:31:18.690 --> 00:31:22.580 align:middle line:84% So we have deflection within the plasma 00:31:22.580 --> 00:31:25.840 align:middle line:84% and we have two protons, which take different paths, 00:31:25.840 --> 00:31:27.405 align:middle line:90% but end up at the same place. 00:31:27.405 --> 00:31:28.780 align:middle line:84% And if you can imagine if there's 00:31:28.780 --> 00:31:31.150 align:middle line:84% multiple of these happening, you end up 00:31:31.150 --> 00:31:33.290 align:middle line:84% with a huge amount of fluence in one spot, 00:31:33.290 --> 00:31:35.200 align:middle line:84% and so you get very, very high intensity. 00:31:35.200 --> 00:31:40.780 align:middle line:90% 00:31:40.780 --> 00:31:42.640 align:middle line:84% And this is usually called a caustic. 00:31:42.640 --> 00:31:45.304 align:middle line:90% 00:31:45.304 --> 00:31:47.790 align:middle line:84% So these caustic features are very, very strong. 00:31:47.790 --> 00:31:50.430 align:middle line:84% You can see them very, very easily, 00:31:50.430 --> 00:31:54.540 align:middle line:84% but they mean that there is no longer a unique reconstruction. 00:31:54.540 --> 00:31:59.220 align:middle line:84% So we can no longer uniquely take our modulation intensity, 00:31:59.220 --> 00:32:02.940 align:middle line:84% I, on our detector and work out the properties of the plasma 00:32:02.940 --> 00:32:06.940 align:middle line:90% through D. 00:32:06.940 --> 00:32:09.790 align:middle line:84% So we want to avoid being in this regime. 00:32:09.790 --> 00:32:15.260 align:middle line:90% 00:32:15.260 --> 00:32:19.580 align:middle line:84% Avoid being in the regime where mu is approximately 1. 00:32:19.580 --> 00:32:21.350 align:middle line:84% And to avoid being in that regime, 00:32:21.350 --> 00:32:23.390 align:middle line:84% we just have to look at the definition of mu 00:32:23.390 --> 00:32:25.830 align:middle line:84% and try and make it as small as possible. 00:32:25.830 --> 00:32:32.480 align:middle line:84% So we want L over a much, much less than 1. 00:32:32.480 --> 00:32:36.908 align:middle line:84% So again, that's the distance between the source-- 00:32:36.908 --> 00:32:37.950 align:middle line:90% I've got it here already. 00:32:37.950 --> 00:32:40.670 align:middle line:90% 00:32:40.670 --> 00:32:42.740 align:middle line:84% The source distance to the plasma 00:32:42.740 --> 00:32:46.490 align:middle line:84% is L and the size of the plasma is a. 00:32:46.490 --> 00:32:48.560 align:middle line:84% So obviously in this case, I haven't succeeded. 00:32:48.560 --> 00:32:49.970 align:middle line:90% L over a is roughly 1. 00:32:49.970 --> 00:32:53.810 align:middle line:84% But I could put the source much further away, and then L over a 00:32:53.810 --> 00:32:57.650 align:middle line:84% would be much smaller, and that would help me make mu smaller. 00:32:57.650 --> 00:33:04.120 align:middle line:84% And the other thing I can do is have the size of alpha 00:33:04.120 --> 00:33:05.980 align:middle line:90% be much, much less than 1. 00:33:05.980 --> 00:33:09.100 align:middle line:84% So that's effectively insisting that my deflections are small. 00:33:09.100 --> 00:33:11.372 align:middle line:84% Now for a given plasma, there are some given 00:33:11.372 --> 00:33:12.580 align:middle line:90% electric and magnetic fields. 00:33:12.580 --> 00:33:14.870 align:middle line:84% I can't just ask the deflections to be small. 00:33:14.870 --> 00:33:17.950 align:middle line:84% What I can do, however, as we'll see in a moment, 00:33:17.950 --> 00:33:21.700 align:middle line:84% is use very fast particles or fast protons. 00:33:21.700 --> 00:33:25.410 align:middle line:90% 00:33:25.410 --> 00:33:27.330 align:middle line:84% Because if we make the protons go very fast, 00:33:27.330 --> 00:33:29.220 align:middle line:84% their deflections become smaller and smaller 00:33:29.220 --> 00:33:31.823 align:middle line:90% as we're about to show. 00:33:31.823 --> 00:33:33.240 align:middle line:84% Now none of this really has talked 00:33:33.240 --> 00:33:34.650 align:middle line:90% about protons at any point. 00:33:34.650 --> 00:33:36.900 align:middle line:84% All of this applies just the same to shadowgraphy. 00:33:36.900 --> 00:33:39.300 align:middle line:84% We're just thinking about particles or rays that 00:33:39.300 --> 00:33:41.430 align:middle line:84% get deflected going through a plasma, 00:33:41.430 --> 00:33:46.170 align:middle line:84% and we've put some geometric formalism on top of it 00:33:46.170 --> 00:33:50.300 align:middle line:84% to tell us what we expect to see on our detector. 00:33:50.300 --> 00:33:52.880 align:middle line:84% So bearing that in mind, any questions before we go on 00:33:52.880 --> 00:33:54.800 align:middle line:84% and actually talk about protons, and plasmas, 00:33:54.800 --> 00:33:55.925 align:middle line:90% and electromagnetic fields? 00:33:55.925 --> 00:34:17.420 align:middle line:90% 00:34:17.420 --> 00:34:20.534 align:middle line:84% I saw some folks still writing, but yeah, taking questions. 00:34:20.534 --> 00:34:24.166 align:middle line:84% AUDIENCE: The L being [INAUDIBLE]. 00:34:24.166 --> 00:34:26.761 align:middle line:90% 00:34:26.761 --> 00:34:29.219 align:middle line:84% PROFESSOR: Oh, it looks like I got it the wrong way around, 00:34:29.219 --> 00:34:30.030 align:middle line:90% didn't I? 00:34:30.030 --> 00:34:32.984 align:middle line:84% Yes, L needs to be much less than a. 00:34:32.984 --> 00:34:34.658 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:34:34.658 --> 00:34:47.090 align:middle line:90% 00:34:47.090 --> 00:34:49.909 align:middle line:84% PROFESSOR: No, I can still do that if the plasma is spherical, 00:34:49.909 --> 00:34:50.409 align:middle line:90% right? 00:34:50.409 --> 00:34:53.330 align:middle line:84% Because if I have, for example, a source 00:34:53.330 --> 00:34:59.120 align:middle line:84% here and a spherical plasma over here, this is a and this is L, 00:34:59.120 --> 00:35:03.140 align:middle line:84% and it's pretty clear that L over a is much greater than 1. 00:35:03.140 --> 00:35:06.600 align:middle line:90% 00:35:06.600 --> 00:35:07.680 align:middle line:90% Well, no, no, no, no. 00:35:07.680 --> 00:35:09.570 align:middle line:84% This is the condition that you want. 00:35:09.570 --> 00:35:11.710 align:middle line:84% What I said earlier is incorrect. 00:35:11.710 --> 00:35:12.960 align:middle line:90% You don't want this condition. 00:35:12.960 --> 00:35:15.790 align:middle line:84% You want to have your plasma very close to your source. 00:35:15.790 --> 00:35:18.720 align:middle line:84% So my words were wrong, but the algebra is right here. 00:35:18.720 --> 00:35:19.320 align:middle line:90% Yeah. 00:35:19.320 --> 00:35:20.766 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:35:20.766 --> 00:35:23.658 align:middle line:90% 00:35:23.658 --> 00:35:25.310 align:middle line:90% PROFESSOR: Oh, I see. 00:35:25.310 --> 00:35:26.780 align:middle line:90% Yes, I take your point. 00:35:26.780 --> 00:35:28.790 align:middle line:84% Yeah, if your plasma is spherical. 00:35:28.790 --> 00:35:30.680 align:middle line:84% A lot of the time people are imaging plasmas 00:35:30.680 --> 00:35:33.095 align:middle line:90% which are quite thin. 00:35:33.095 --> 00:35:35.650 align:middle line:90% But yeah, you're right. 00:35:35.650 --> 00:35:37.240 align:middle line:90% OK, other questions? 00:35:37.240 --> 00:35:37.930 align:middle line:90% Yes. 00:35:37.930 --> 00:35:39.160 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:35:39.160 --> 00:35:43.332 align:middle line:90% 00:35:43.332 --> 00:35:44.290 align:middle line:90% PROFESSOR: Oh, radians. 00:35:44.290 --> 00:35:45.290 align:middle line:90% Everything's in radians. 00:35:45.290 --> 00:35:46.980 align:middle line:90% AUDIENCE: Less than 1? 00:35:46.980 --> 00:35:49.830 align:middle line:84% PROFESSOR: Yes, much less than 1 radian. 00:35:49.830 --> 00:35:52.308 align:middle line:84% It's in radians because we're using this approximation. 00:35:52.308 --> 00:35:53.850 align:middle line:84% So if we're using this approximation, 00:35:53.850 --> 00:35:55.853 align:middle line:84% alpha has to be in radians for that to be true. 00:35:55.853 --> 00:35:59.670 align:middle line:84% AUDIENCE: [INAUDIBLE] there's nothing special about 1 radian? 00:35:59.670 --> 00:36:01.090 align:middle line:90% PROFESSOR: No, no, no, no. 00:36:01.090 --> 00:36:04.020 align:middle line:84% No, there's nothing special about 1 radian. 00:36:04.020 --> 00:36:06.302 align:middle line:84% We want it to be as small as possible. 00:36:06.302 --> 00:36:08.010 align:middle line:84% 1 is a number that it can be small there, 00:36:08.010 --> 00:36:11.200 align:middle line:84% but guess could have raised it 10 to 6. 00:36:11.200 --> 00:36:12.960 align:middle line:90% Yeah, sure, sure, sure. 00:36:12.960 --> 00:36:17.070 align:middle line:84% In particular, we said that we can get a less than 1. 00:36:17.070 --> 00:36:19.620 align:middle line:84% So if we want to get mu much less than 1, 00:36:19.620 --> 00:36:22.620 align:middle line:84% that means that alpha has to be much less than 1 00:36:22.620 --> 00:36:25.145 align:middle line:90% from that point of view. 00:36:25.145 --> 00:36:27.270 align:middle line:84% I think this is a reasonable [INAUDIBLE], but yeah. 00:36:27.270 --> 00:36:29.230 align:middle line:90% OK, any questions online? 00:36:29.230 --> 00:36:31.745 align:middle line:90% 00:36:31.745 --> 00:36:34.120 align:middle line:84% OK, let's have a look at what the actual deflection, what 00:36:34.120 --> 00:36:36.040 align:middle line:90% this alpha, is here. 00:36:36.040 --> 00:36:37.960 align:middle line:84% Because that's what you need to in order 00:36:37.960 --> 00:36:40.090 align:middle line:84% to work out this quantity, which is D, 00:36:40.090 --> 00:36:42.170 align:middle line:84% and work out all the rest of the stuff. 00:36:42.170 --> 00:36:44.400 align:middle line:90% So which board should I use? 00:36:44.400 --> 00:37:22.880 align:middle line:90% 00:37:22.880 --> 00:37:28.020 align:middle line:84% OK, so alpha for electric and magnetic fields 00:37:28.020 --> 00:37:32.990 align:middle line:84% together is simply the integral of that force, which 00:37:32.990 --> 00:37:41.200 align:middle line:84% is E plus v cross B along the path, which remember, 00:37:41.200 --> 00:37:48.440 align:middle line:84% we're approximating as dz like that. 00:37:48.440 --> 00:37:50.940 align:middle line:84% And the deflection angle has to do as the charge, of course, 00:37:50.940 --> 00:37:53.160 align:middle line:84% because that's what goes into the force. 00:37:53.160 --> 00:37:57.660 align:middle line:84% But there's also a factor of 2 and w out the front here. 00:37:57.660 --> 00:38:06.050 align:middle line:84% This w is, again, the kinetic energy of our proton m proton v 00:38:06.050 --> 00:38:06.785 align:middle line:90% proton squared. 00:38:06.785 --> 00:38:10.260 align:middle line:90% 00:38:10.260 --> 00:38:12.105 align:middle line:84% You can work this out if you want to. 00:38:12.105 --> 00:38:13.770 align:middle line:90% I haven't proven this exactly. 00:38:13.770 --> 00:38:16.320 align:middle line:84% So what we notice is interestingly 00:38:16.320 --> 00:38:20.980 align:middle line:84% we got a w that depends on v and here we've got v inside this. 00:38:20.980 --> 00:38:24.000 align:middle line:84% And so we can separately write out that there's going to be 00:38:24.000 --> 00:38:26.890 align:middle line:84% a deflection due to the electric field, 00:38:26.890 --> 00:38:34.073 align:middle line:84% which is e over 2w integral electric field dz. 00:38:34.073 --> 00:38:34.990 align:middle line:90% That's pretty obvious. 00:38:34.990 --> 00:38:36.550 align:middle line:90% I've just taken the first-- ah. 00:38:36.550 --> 00:38:41.370 align:middle line:84% And we have a deflection due to the magnetic field Alpha B 00:38:41.370 --> 00:38:46.395 align:middle line:90% equals E over square root 2 mw. 00:38:46.395 --> 00:38:48.942 align:middle line:90% 00:38:48.942 --> 00:38:50.900 align:middle line:84% We're taking this velocity and we're 00:38:50.900 --> 00:38:54.790 align:middle line:84% combining it with the velocity squared over there. 00:38:54.790 --> 00:39:01.640 align:middle line:84% Integral of dL crossed with B. Whereas of course 00:39:01.640 --> 00:39:06.252 align:middle line:84% this dL is really dz because of our paraxial approximation. 00:39:06.252 --> 00:39:09.090 align:middle line:90% 00:39:09.090 --> 00:39:11.090 align:middle line:84% So we're going to have a deflection angle that 00:39:11.090 --> 00:39:16.633 align:middle line:84% depends on the cross product of the path and the magnetic field. 00:39:16.633 --> 00:39:18.050 align:middle line:84% What's interesting about these two 00:39:18.050 --> 00:39:19.550 align:middle line:84% is that you have different scalings. 00:39:19.550 --> 00:39:27.670 align:middle line:90% 00:39:27.670 --> 00:39:30.190 align:middle line:84% One has a scaling with the particle energy. 00:39:30.190 --> 00:39:33.130 align:middle line:84% One has a scaling with the square root of the particle 00:39:33.130 --> 00:39:33.920 align:middle line:90% energy. 00:39:33.920 --> 00:39:36.610 align:middle line:84% So this suggests to us straight away 00:39:36.610 --> 00:39:41.770 align:middle line:84% how to distinguish between E and B. 00:39:41.770 --> 00:39:45.090 align:middle line:90% 00:39:45.090 --> 00:39:49.900 align:middle line:84% Or should I call them alpha E and alpha B. So 00:39:49.900 --> 00:39:53.380 align:middle line:84% the deflection is due to electric and magnetic fields, 00:39:53.380 --> 00:40:01.685 align:middle line:84% and we distinguish between them by using two or more proton 00:40:01.685 --> 00:40:02.185 align:middle line:90% energies. 00:40:02.185 --> 00:40:07.750 align:middle line:90% 00:40:07.750 --> 00:40:11.350 align:middle line:84% If we have a source that produces 1 MeV protons and 10 00:40:11.350 --> 00:40:14.350 align:middle line:84% MeV protons, they will go through the same plasma, 00:40:14.350 --> 00:40:19.450 align:middle line:84% but the 1 MeV protons will be much more sensitive to-- 00:40:19.450 --> 00:40:20.710 align:middle line:90% I've got to get it right now-- 00:40:20.710 --> 00:40:22.310 align:middle line:90% lower energy. 00:40:22.310 --> 00:40:25.090 align:middle line:84% So this is a bigger number to the electric field than the 10 00:40:25.090 --> 00:40:26.210 align:middle line:90% MeV protons. 00:40:26.210 --> 00:40:29.260 align:middle line:84% And so we can use a differential measurement effectively 00:40:29.260 --> 00:40:32.470 align:middle line:84% to work out the contribution of alpha E and alpha B. 00:40:32.470 --> 00:40:34.570 align:middle line:84% There'll be two simultaneous equations 00:40:34.570 --> 00:40:37.700 align:middle line:84% and we'll have two things we want to solve for. 00:40:37.700 --> 00:40:39.910 align:middle line:90% So we'll be able to get it out. 00:40:39.910 --> 00:40:44.610 align:middle line:84% So this looks an awful lot like when we talked about two color 00:40:44.610 --> 00:40:49.390 align:middle line:84% interferometry to determine the difference between vibrations 00:40:49.390 --> 00:40:51.130 align:middle line:90% or neutrals in our plasma. 00:40:51.130 --> 00:40:55.238 align:middle line:90% 00:40:55.238 --> 00:40:56.780 align:middle line:84% And people do this a lot because they 00:40:56.780 --> 00:40:59.300 align:middle line:84% feel doing an experiment you don't a priori know 00:40:59.300 --> 00:41:00.948 align:middle line:84% whether the electric fields are small 00:41:00.948 --> 00:41:02.240 align:middle line:90% compared to the magnetic field. 00:41:02.240 --> 00:41:03.830 align:middle line:84% Most people want to use this technique 00:41:03.830 --> 00:41:05.810 align:middle line:90% to measure magnetic fields. 00:41:05.810 --> 00:41:08.188 align:middle line:84% And usually it's the case of magnetic fields do dominate, 00:41:08.188 --> 00:41:10.730 align:middle line:84% but you still need to check, and you can do it with these two 00:41:10.730 --> 00:41:11.522 align:middle line:90% different energies. 00:41:11.522 --> 00:41:14.860 align:middle line:90% 00:41:14.860 --> 00:41:18.510 align:middle line:84% So we've been talking a lot about alpha here, 00:41:18.510 --> 00:41:21.650 align:middle line:84% but there's a rather neat formulation, 00:41:21.650 --> 00:41:24.850 align:middle line:84% which says, well, instead of talking about this vector alpha, 00:41:24.850 --> 00:41:27.910 align:middle line:84% and that deflection angle alpha depends 00:41:27.910 --> 00:41:30.520 align:middle line:84% on where you are in the plasma, so it depends on your position 00:41:30.520 --> 00:41:32.020 align:middle line:90% x0 within the plasma. 00:41:32.020 --> 00:41:34.270 align:middle line:84% If this is a vector, then we can just write it 00:41:34.270 --> 00:41:37.090 align:middle line:90% as the gradient of some scalar. 00:41:37.090 --> 00:41:38.950 align:middle line:90% That's just generally true. 00:41:38.950 --> 00:41:43.960 align:middle line:84% So we can say what if there is some scalar, phi, of x0 00:41:43.960 --> 00:41:45.970 align:middle line:90% and alpha is just its gradient. 00:41:45.970 --> 00:41:47.950 align:middle line:84% And the reason you want to do this is 00:41:47.950 --> 00:41:50.200 align:middle line:84% can now write down an expression for phi, 00:41:50.200 --> 00:41:53.230 align:middle line:84% which works for shadowgraphy as well as proton imaging. 00:41:53.230 --> 00:41:56.330 align:middle line:84% And then all of the mathematics that you've done previously, 00:41:56.330 --> 00:41:59.590 align:middle line:84% you just do it in terms of phi, and it's 00:41:59.590 --> 00:42:01.810 align:middle line:84% agnostic to the exact technique that we use. 00:42:01.810 --> 00:42:12.590 align:middle line:84% So for shadowgraphy you find that your deflection 00:42:12.590 --> 00:42:19.980 align:middle line:84% potential is equal to minus the integral of the logarithm 00:42:19.980 --> 00:42:23.090 align:middle line:90% of the refractive index n. 00:42:23.090 --> 00:42:24.620 align:middle line:84% And I'll explicitly write this out 00:42:24.620 --> 00:42:26.855 align:middle line:84% in terms of the position within the plasma. 00:42:26.855 --> 00:42:30.860 align:middle line:90% 00:42:30.860 --> 00:42:32.480 align:middle line:90% The z0. 00:42:32.480 --> 00:42:36.410 align:middle line:84% So the integral along through the plasma in the z direction. 00:42:36.410 --> 00:42:40.740 align:middle line:84% And this logarithm here, or this n here is, of course, 00:42:40.740 --> 00:42:45.890 align:middle line:84% the one where we have n squared equals 1 minus nE over nc 00:42:45.890 --> 00:42:47.990 align:middle line:84% that we saw in all our refractive index things. 00:42:47.990 --> 00:42:49.730 align:middle line:84% So of course it could be whatever-- 00:42:49.730 --> 00:42:51.348 align:middle line:84% O modes, X modes-- but people don't do 00:42:51.348 --> 00:42:52.640 align:middle line:90% proton radiography [INAUDIBLE]. 00:42:52.640 --> 00:42:56.150 align:middle line:84% So the plasmas we're working with are mostly unmagnetized. 00:42:56.150 --> 00:42:58.380 align:middle line:90% This is fine. 00:42:58.380 --> 00:43:15.610 align:middle line:84% But E fields, this potential here is just equal to q over 2w. 00:43:15.610 --> 00:43:16.450 align:middle line:90% I use q over there? 00:43:16.450 --> 00:43:18.117 align:middle line:84% No, I switched to E, so I'll put it back 00:43:18.117 --> 00:43:26.300 align:middle line:84% to E. Integral of the real electrostatic potential, 00:43:26.300 --> 00:43:34.350 align:middle line:90% lowercase phi, x0 y0 z0 E z0. 00:43:34.350 --> 00:43:35.730 align:middle line:90% So they are related. 00:43:35.730 --> 00:43:38.370 align:middle line:84% They don't have to be, but they are in this case related. 00:43:38.370 --> 00:43:40.590 align:middle line:84% The deflection potential is related 00:43:40.590 --> 00:43:42.270 align:middle line:90% to the electrostatic potential. 00:43:42.270 --> 00:43:44.625 align:middle line:84% And then the final one is the magnetic fields. 00:43:44.625 --> 00:43:51.870 align:middle line:90% 00:43:51.870 --> 00:44:03.010 align:middle line:84% Capital phi of x0 is equal to E upon square root of 2mw integral 00:44:03.010 --> 00:44:08.560 align:middle line:90% of A dot dz 0. 00:44:08.560 --> 00:44:15.700 align:middle line:84% And this A is the magnetic vector potential. 00:44:15.700 --> 00:44:21.440 align:middle line:90% So B is equal to the curl of A. 00:44:21.440 --> 00:44:23.810 align:middle line:84% So the idea is that we will seek techniques 00:44:23.810 --> 00:44:29.490 align:middle line:84% which will take our intensity variation, I, on our detector, 00:44:29.490 --> 00:44:32.270 align:middle line:84% and we will use them to infer some sort of deflection 00:44:32.270 --> 00:44:32.872 align:middle line:90% potential. 00:44:32.872 --> 00:44:35.330 align:middle line:84% And then the neat thing is, it doesn't matter whether we're 00:44:35.330 --> 00:44:37.288 align:middle line:84% looking at electric fields, or magnetic fields, 00:44:37.288 --> 00:44:39.170 align:middle line:90% or shadowgraphy. 00:44:39.170 --> 00:44:41.730 align:middle line:84% That's a detail that the end user has to use, 00:44:41.730 --> 00:44:43.230 align:middle line:84% but the algorithms are all the same. 00:44:43.230 --> 00:44:44.272 align:middle line:90% So this is very powerful. 00:44:44.272 --> 00:44:47.150 align:middle line:84% We can share techniques between very disparate looking 00:44:47.150 --> 00:44:49.017 align:middle line:90% diagnostics. 00:44:49.017 --> 00:44:49.850 align:middle line:90% AUDIENCE: Professor? 00:44:49.850 --> 00:44:51.080 align:middle line:90% PROFESSOR: Yes. 00:44:51.080 --> 00:44:53.810 align:middle line:84% AUDIENCE: Does the B is A, where B 00:44:53.810 --> 00:45:01.340 align:middle line:84% equals the curl of A, is that from the Maxwell's law B-- 00:45:01.340 --> 00:45:03.350 align:middle line:90% like the divergence is 0? 00:45:03.350 --> 00:45:06.290 align:middle line:90% Is where that comes from? 00:45:06.290 --> 00:45:09.270 align:middle line:90% PROFESSOR: Gosh, yeah? 00:45:09.270 --> 00:45:10.910 align:middle line:84% Can divergence free fields be written 00:45:10.910 --> 00:45:12.160 align:middle line:90% as the curl of another vector? 00:45:12.160 --> 00:45:13.710 align:middle line:90% I think so, yeah. 00:45:13.710 --> 00:45:17.500 align:middle line:84% AUDIENCE: OK, then where does the potential come from? 00:45:17.500 --> 00:45:19.540 align:middle line:84% Is it just a regular electric potential? 00:45:19.540 --> 00:45:21.540 align:middle line:84% PROFESSOR: Yeah, this is the electric potential. 00:45:21.540 --> 00:45:24.460 align:middle line:84% This is the potential that's defined by Poisson's equation. 00:45:24.460 --> 00:45:27.743 align:middle line:90% 00:45:27.743 --> 00:45:28.410 align:middle line:90% AUDIENCE: I see. 00:45:28.410 --> 00:45:29.762 align:middle line:90% OK. 00:45:29.762 --> 00:45:30.720 align:middle line:90% I just wanted to check. 00:45:30.720 --> 00:45:31.220 align:middle line:90% Thank you. 00:45:31.220 --> 00:45:34.110 align:middle line:90% PROFESSOR: Yeah, no worries. 00:45:34.110 --> 00:45:34.910 align:middle line:90% OK. 00:45:34.910 --> 00:45:35.880 align:middle line:90% Any questions on that? 00:45:35.880 --> 00:45:36.810 align:middle line:84% The next thing I'm going to cover 00:45:36.810 --> 00:45:38.490 align:middle line:84% is how do we make these beams of protons, 00:45:38.490 --> 00:45:39.730 align:middle line:84% so we're going to switch gears a little bit. 00:45:39.730 --> 00:45:42.255 align:middle line:84% So if you have any questions on what's happened so far, 00:45:42.255 --> 00:45:43.380 align:middle line:90% this is a good time to ask. 00:45:43.380 --> 00:45:47.880 align:middle line:90% 00:45:47.880 --> 00:45:49.094 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:45:49.094 --> 00:45:58.830 align:middle line:90% 00:45:58.830 --> 00:46:00.930 align:middle line:84% PROFESSOR: These are contradictory, yes. 00:46:00.930 --> 00:46:02.760 align:middle line:84% So the paraxial approximation also 00:46:02.760 --> 00:46:05.540 align:middle line:84% makes it very hard to have your object very close. 00:46:05.540 --> 00:46:07.480 align:middle line:84% But if you were able to do that, you 00:46:07.480 --> 00:46:09.100 align:middle line:84% would reduce the contrast parameter, 00:46:09.100 --> 00:46:11.870 align:middle line:84% but you would also break all of the mathematics we've done, 00:46:11.870 --> 00:46:14.240 align:middle line:84% which is use the paraxial approximation so far. 00:46:14.240 --> 00:46:18.010 align:middle line:84% So in practice, your best bet is to reduce alpha instead. 00:46:18.010 --> 00:46:20.590 align:middle line:84% And you reduce alpha by making your protons 00:46:20.590 --> 00:46:22.060 align:middle line:90% as fast as possible. 00:46:22.060 --> 00:46:24.760 align:middle line:84% Because the faster your protons are, the higher their energy, 00:46:24.760 --> 00:46:26.480 align:middle line:90% the lower their deflections. 00:46:26.480 --> 00:46:27.970 align:middle line:84% So this is what motivates us using 00:46:27.970 --> 00:46:30.210 align:middle line:90% MeV or higher beams of protons. 00:46:30.210 --> 00:46:31.670 align:middle line:90% Yeah. 00:46:31.670 --> 00:46:32.600 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:46:32.600 --> 00:46:40.050 align:middle line:90% 00:46:40.050 --> 00:46:43.600 align:middle line:84% PROFESSOR: So the particle is being deflected a little bit, 00:46:43.600 --> 00:46:46.248 align:middle line:84% but I could approximate mostly in the same direction. 00:46:46.248 --> 00:46:47.790 align:middle line:84% It's certainly not being decelerated. 00:46:47.790 --> 00:46:50.160 align:middle line:90% Its velocity is constant. 00:46:50.160 --> 00:46:52.545 align:middle line:84% So the size of its velocity, its speed, 00:46:52.545 --> 00:46:54.420 align:middle line:84% doesn't change as it goes through the plasma. 00:46:54.420 --> 00:46:56.910 align:middle line:84% That's also consistent with our test particle picture, 00:46:56.910 --> 00:46:59.580 align:middle line:84% where the test particle is experiencing the field, 00:46:59.580 --> 00:47:02.080 align:middle line:90% but it's not being slowed down. 00:47:02.080 --> 00:47:02.820 align:middle line:90% Yeah. 00:47:02.820 --> 00:47:06.078 align:middle line:84% AUDIENCE: Is that like when [INAUDIBLE]? 00:47:06.078 --> 00:47:08.120 align:middle line:84% PROFESSOR: The particles are really, really fast. 00:47:08.120 --> 00:47:10.213 align:middle line:90% They don't scatter at all. 00:47:10.213 --> 00:47:11.880 align:middle line:84% They're way faster than everything else. 00:47:11.880 --> 00:47:15.410 align:middle line:84% So this is a completely reasonable approximation. 00:47:15.410 --> 00:47:18.590 align:middle line:84% You can treat the plasma as this static, unmoving thing 00:47:18.590 --> 00:47:20.960 align:middle line:84% on the time scale of these protons going through. 00:47:20.960 --> 00:47:22.970 align:middle line:84% And the protons barely see it apart from they 00:47:22.970 --> 00:47:24.712 align:middle line:90% get deflected a tiny amount. 00:47:24.712 --> 00:47:26.420 align:middle line:84% And it's only because we put our detector 00:47:26.420 --> 00:47:28.790 align:middle line:84% so far away that those tiny deflections show up 00:47:28.790 --> 00:47:33.060 align:middle line:84% as significant changes to the fluence. 00:47:33.060 --> 00:47:34.120 align:middle line:90% OK, any other questions? 00:47:34.120 --> 00:47:34.787 align:middle line:90% Anything online? 00:47:34.787 --> 00:47:39.370 align:middle line:90% 00:47:39.370 --> 00:47:39.870 align:middle line:90% Yeah. 00:47:39.870 --> 00:47:49.590 align:middle line:90% 00:47:49.590 --> 00:47:52.230 align:middle line:84% Yeah, the deflection potentials add linearly, 00:47:52.230 --> 00:47:54.570 align:middle line:90% which is neat as well. 00:47:54.570 --> 00:47:58.050 align:middle line:84% And then when you use two different energies, 00:47:58.050 --> 00:48:00.600 align:middle line:84% you can solve separately for the two deflection potentials 00:48:00.600 --> 00:48:03.960 align:middle line:90% and then see how big they are. 00:48:03.960 --> 00:48:06.270 align:middle line:84% OK, let's talk about how to make a beam of protons. 00:48:06.270 --> 00:48:12.392 align:middle line:90% 00:48:12.392 --> 00:48:13.850 align:middle line:84% This is actually quite frustrating. 00:48:13.850 --> 00:48:15.435 align:middle line:84% When I taught this in 2021, there 00:48:15.435 --> 00:48:17.060 align:middle line:84% was no good reference on this and I had 00:48:17.060 --> 00:48:19.760 align:middle line:90% to spend ages looking it up. 00:48:19.760 --> 00:48:22.155 align:middle line:84% And then Derek Schaeffer put his paper on archives 00:48:22.155 --> 00:48:24.030 align:middle line:84% and explains it all really straightforwardly. 00:48:24.030 --> 00:48:27.480 align:middle line:84% So if you want to understand it, you can just read his paper. 00:48:27.480 --> 00:48:30.960 align:middle line:84% But yeah, it's interesting stuff because this technique 00:48:30.960 --> 00:48:32.250 align:middle line:90% is relatively modern. 00:48:32.250 --> 00:48:34.177 align:middle line:84% It's only within about the last 20 years 00:48:34.177 --> 00:48:35.760 align:middle line:84% that people have really been using it. 00:48:35.760 --> 00:48:37.677 align:middle line:84% And a lot of the development of this technique 00:48:37.677 --> 00:48:40.367 align:middle line:84% was led by the HED group here at MIT. 00:48:40.367 --> 00:48:42.450 align:middle line:84% So a lot of the things I'm going to tell you about 00:48:42.450 --> 00:48:47.250 align:middle line:90% are MIT diagnostic development. 00:48:47.250 --> 00:48:54.340 align:middle line:90% 00:48:54.340 --> 00:48:58.160 align:middle line:90% So we need to have a source. 00:48:58.160 --> 00:49:01.590 align:middle line:90% 00:49:01.590 --> 00:49:04.500 align:middle line:84% And for each source, there is a detector. 00:49:04.500 --> 00:49:09.060 align:middle line:90% 00:49:09.060 --> 00:49:11.780 align:middle line:84% So the first source that I'm going to talk about 00:49:11.780 --> 00:49:14.715 align:middle line:90% is a technique called TNSA. 00:49:14.715 --> 00:49:18.708 align:middle line:90% 00:49:18.708 --> 00:49:22.310 align:middle line:90% This stands for Target-- 00:49:22.310 --> 00:49:24.080 align:middle line:90% and it doesn't really matter-- 00:49:24.080 --> 00:49:27.630 align:middle line:90% 00:49:27.630 --> 00:49:38.480 align:middle line:90% Normal Sheath Acceleration. 00:49:38.480 --> 00:49:42.170 align:middle line:84% You can't quite sing it to the tune of Teenage Mutant Ninja 00:49:42.170 --> 00:49:44.910 align:middle line:84% Turtles, sadly, but it's getting pretty close. 00:49:44.910 --> 00:49:47.810 align:middle line:84% So the idea of TNSA, and I will never 00:49:47.810 --> 00:49:50.270 align:middle line:84% say target normal sheath acceleration again-- 00:49:50.270 --> 00:49:54.570 align:middle line:90% is you have a metal foil. 00:49:54.570 --> 00:49:59.010 align:middle line:90% 00:49:59.010 --> 00:50:05.900 align:middle line:84% And it's thin-- tens of microns thin. 00:50:05.900 --> 00:50:08.540 align:middle line:84% And any metal that you have is going 00:50:08.540 --> 00:50:13.710 align:middle line:84% to have absorbed on its surface lots of hydrocarbons, which 00:50:13.710 --> 00:50:16.890 align:middle line:90% means lots of little protons. 00:50:16.890 --> 00:50:20.310 align:middle line:84% So again, we're not accelerating copper or something like this 00:50:20.310 --> 00:50:20.810 align:middle line:90% here. 00:50:20.810 --> 00:50:22.665 align:middle line:84% We're still accelerating protons, 00:50:22.665 --> 00:50:25.040 align:middle line:84% it just turns out there's lots of little protons absorbed 00:50:25.040 --> 00:50:26.810 align:middle line:84% in the structure at the front here. 00:50:26.810 --> 00:50:33.770 align:middle line:84% And we take our laser beam, and we focus it down, and hit 00:50:33.770 --> 00:50:35.540 align:middle line:90% the back side of this target. 00:50:35.540 --> 00:50:39.125 align:middle line:84% And our laser beam is ultrashort pulse. 00:50:39.125 --> 00:50:44.470 align:middle line:90% 00:50:44.470 --> 00:50:48.870 align:middle line:84% So this tends to be something like a 50 Joule laser 00:50:48.870 --> 00:50:52.810 align:middle line:84% in 1 picosecond, 10 to -12 seconds. 00:50:52.810 --> 00:50:55.920 align:middle line:84% So this is over 100 terawatts of laser power. 00:50:55.920 --> 00:50:57.990 align:middle line:84% Or if you've got a really short pulse, 00:50:57.990 --> 00:51:02.310 align:middle line:84% you can do 1 Joule in 50 femtoseconds. 00:51:02.310 --> 00:51:04.500 align:middle line:84% A femtosecond is 10 to the -15 seconds. 00:51:04.500 --> 00:51:07.440 align:middle line:84% So these are seriously short laser pulses 00:51:07.440 --> 00:51:10.680 align:middle line:84% and this is a seriously intense focal spot 00:51:10.680 --> 00:51:17.070 align:middle line:84% here down to about 10 microns, we can focus the laser down to. 00:51:17.070 --> 00:51:18.900 align:middle line:84% So that's a lot of electrical energy 00:51:18.900 --> 00:51:23.670 align:middle line:84% being focused into a very small spot on a very short time scale. 00:51:23.670 --> 00:51:27.660 align:middle line:84% And what that does is it initially blasts 00:51:27.660 --> 00:51:29.310 align:middle line:90% electrons off the foil. 00:51:29.310 --> 00:51:33.210 align:middle line:84% So there is some heating, there are some strong electric fields. 00:51:33.210 --> 00:51:35.640 align:middle line:84% For reasons that we're not going to go into very much 00:51:35.640 --> 00:51:38.730 align:middle line:84% this drives electrons off the foil at a high speed, 00:51:38.730 --> 00:51:41.115 align:middle line:84% and they form a little cloud going outwards. 00:51:41.115 --> 00:51:43.440 align:middle line:84% And because there's been electrons ejected, 00:51:43.440 --> 00:51:45.330 align:middle line:90% we then have an electric field. 00:51:45.330 --> 00:51:49.530 align:middle line:90% 00:51:49.530 --> 00:51:52.340 align:middle line:84% And that electric field drags these protons 00:51:52.340 --> 00:51:58.820 align:middle line:90% off the surface like that. 00:51:58.820 --> 00:52:01.640 align:middle line:84% And this is how the protons are accelerated. 00:52:01.640 --> 00:52:04.460 align:middle line:84% So this technique, some good things about it, 00:52:04.460 --> 00:52:08.050 align:middle line:90% it gives us many protons. 00:52:08.050 --> 00:52:12.042 align:middle line:84% It's a very efficient way of accelerating the protons here. 00:52:12.042 --> 00:52:13.375 align:middle line:90% It gives us a range of energies. 00:52:13.375 --> 00:52:17.185 align:middle line:90% 00:52:17.185 --> 00:52:19.840 align:middle line:84% Now remember, we said we'd like several different energies 00:52:19.840 --> 00:52:23.510 align:middle line:84% for telling difference between electric and magnetic fields. 00:52:23.510 --> 00:52:30.130 align:middle line:84% The spot size was around about 10 microns. 00:52:30.130 --> 00:52:32.340 align:middle line:84% So as we said, that's a nice small spot. 00:52:32.340 --> 00:52:34.260 align:middle line:84% That's something we wanted in order 00:52:34.260 --> 00:52:36.670 align:middle line:84% for all our paraxial approximation and stuff to work. 00:52:36.670 --> 00:52:40.200 align:middle line:90% And it's a beam. 00:52:40.200 --> 00:52:41.873 align:middle line:90% It's in one direction. 00:52:41.873 --> 00:52:43.290 align:middle line:84% And we'll see why that's important 00:52:43.290 --> 00:52:46.393 align:middle line:84% when we compare it to other [INAUDIBLE] experiments. 00:52:46.393 --> 00:52:49.850 align:middle line:90% 00:52:49.850 --> 00:52:52.760 align:middle line:84% The big disadvantage of this technique 00:52:52.760 --> 00:53:01.040 align:middle line:84% is you need a petawatt class laser or higher. 00:53:01.040 --> 00:53:03.870 align:middle line:84% So you need to have one of these ultrashort pulse lasers. 00:53:03.870 --> 00:53:05.270 align:middle line:90% Not every facility has that. 00:53:05.270 --> 00:53:07.680 align:middle line:84% These are very expensive things to build. 00:53:07.680 --> 00:53:10.305 align:middle line:84% So it's not like you can just go down to the shop and get this. 00:53:10.305 --> 00:53:12.805 align:middle line:84% I've thought about trying to get one of these for the PUFFIN 00:53:12.805 --> 00:53:15.710 align:middle line:84% facility, and we got a quote for about $2 million for a laser 00:53:15.710 --> 00:53:18.200 align:middle line:90% just to do not very good TNSA. 00:53:18.200 --> 00:53:21.090 align:middle line:90% So these are very expensive. 00:53:21.090 --> 00:53:24.530 align:middle line:90% [INAUDIBLE] 00:53:24.530 --> 00:53:26.720 align:middle line:90% So that's your source. 00:53:26.720 --> 00:53:29.542 align:middle line:90% Your detector-- 00:53:29.542 --> 00:53:35.410 align:middle line:90% 00:53:35.410 --> 00:53:37.570 align:middle line:84% Is something which is called radiochromic film. 00:53:37.570 --> 00:53:45.690 align:middle line:90% 00:53:45.690 --> 00:53:47.790 align:middle line:84% Another acronym I'm not going to repeat, 00:53:47.790 --> 00:53:51.120 align:middle line:90% we'll call it RCF from now on. 00:53:51.120 --> 00:53:55.260 align:middle line:84% Now RCF changes color when it's hit by protons. 00:53:55.260 --> 00:53:56.970 align:middle line:90% That's the chromic part of it. 00:53:56.970 --> 00:53:59.890 align:middle line:90% 00:53:59.890 --> 00:54:02.920 align:middle line:84% And the more protons that hit it, the more it changes color. 00:54:02.920 --> 00:54:04.570 align:middle line:84% It sort of starts off quite clear, 00:54:04.570 --> 00:54:06.700 align:middle line:84% and it turns this beautiful ghostly blue, 00:54:06.700 --> 00:54:08.950 align:middle line:84% and then goes into a very dark, deep blue. 00:54:08.950 --> 00:54:14.000 align:middle line:84% They're very beautiful raw proton radiographic film images. 00:54:14.000 --> 00:54:17.900 align:middle line:84% Now what you end up doing is you have stacks of this film. 00:54:17.900 --> 00:54:19.595 align:middle line:90% You have multiple sheets. 00:54:19.595 --> 00:54:22.330 align:middle line:90% 00:54:22.330 --> 00:54:26.810 align:middle line:84% So imagine these are squares of film 00:54:26.810 --> 00:54:30.120 align:middle line:84% and I've just made a stack out of lots of them like this. 00:54:30.120 --> 00:54:32.450 align:middle line:84% And the reason you make lots of stacks out of these 00:54:32.450 --> 00:54:35.810 align:middle line:84% is due to a peculiarity of charged particle stopping, 00:54:35.810 --> 00:54:38.630 align:middle line:84% and this peculiarity is called the Bragg 00:54:38.630 --> 00:54:43.490 align:middle line:84% peak, which some of you may have come across before. 00:54:43.490 --> 00:54:47.120 align:middle line:84% The idea of the Bragg peak is your charged particles 00:54:47.120 --> 00:54:51.270 align:middle line:84% tend to stop in a very specific place for material, which 00:54:51.270 --> 00:54:52.990 align:middle line:90% is to do with their energy. 00:54:52.990 --> 00:54:58.170 align:middle line:84% So for example, if this is the stopping location x, 00:54:58.170 --> 00:54:59.700 align:middle line:84% we'll call this the x-coordinate. 00:54:59.700 --> 00:55:02.640 align:middle line:84% I guess I've called it z everywhere else, but whatever. 00:55:02.640 --> 00:55:05.670 align:middle line:84% You tend to have a lot of deposition of your particles 00:55:05.670 --> 00:55:09.130 align:middle line:84% all at one place, and this is energy-dependent. 00:55:09.130 --> 00:55:14.140 align:middle line:84% So this is energy-dependent deposition. 00:55:14.140 --> 00:55:23.180 align:middle line:90% 00:55:23.180 --> 00:55:27.870 align:middle line:84% The reason this is peculiar is you remember that for photons we 00:55:27.870 --> 00:55:32.690 align:middle line:84% just had this exponential decay here 00:55:32.690 --> 00:55:35.420 align:middle line:84% that was to do with the opacity of it. 00:55:35.420 --> 00:55:40.917 align:middle line:84% But that's not what we get for particles. 00:55:40.917 --> 00:55:43.000 align:middle line:84% The fact that all the particles of the same energy 00:55:43.000 --> 00:55:45.340 align:middle line:84% roughly stop in the same bit of film 00:55:45.340 --> 00:55:48.580 align:middle line:84% means that each film corresponds to a specific energy. 00:55:48.580 --> 00:55:53.290 align:middle line:84% Maybe the first film stack here corresponds to 1 MeV, 00:55:53.290 --> 00:55:58.090 align:middle line:84% and the next one to 2 3, 4, 5, 6, and so on. 00:55:58.090 --> 00:56:00.790 align:middle line:84% You can put different filters between these 00:56:00.790 --> 00:56:03.377 align:middle line:84% to select different proton energies, 00:56:03.377 --> 00:56:05.210 align:middle line:84% but it means that each of these bits of film 00:56:05.210 --> 00:56:07.060 align:middle line:84% now corresponds to a different energy, which 00:56:07.060 --> 00:56:09.970 align:middle line:84% is very useful if you want to do your multi-energy trick 00:56:09.970 --> 00:56:13.040 align:middle line:84% of unfolding the electric and magnetic fields. 00:56:13.040 --> 00:56:20.890 align:middle line:84% So this first result of this is that we have 00:56:20.890 --> 00:56:28.022 align:middle line:90% we call it 1 energy per film. 00:56:28.022 --> 00:56:29.730 align:middle line:84% It's obviously not quite true because you 00:56:29.730 --> 00:56:32.770 align:middle line:84% have some bleed through, some particles which stopped early. 00:56:32.770 --> 00:56:35.230 align:middle line:84% So there's a little bit of crosstalk between them. 00:56:35.230 --> 00:56:37.620 align:middle line:84% The other neat thing about this, if you 00:56:37.620 --> 00:56:40.343 align:middle line:84% don't want to use these in order to back out 00:56:40.343 --> 00:56:42.510 align:middle line:84% the difference between electric and magnetic fields, 00:56:42.510 --> 00:56:44.130 align:middle line:84% you can make a different argument. 00:56:44.130 --> 00:56:48.320 align:middle line:84% And you can argue, in fact, that my particles were born here, 00:56:48.320 --> 00:56:51.070 align:middle line:84% and they were born at different energies, 00:56:51.070 --> 00:56:54.100 align:middle line:84% and they streamed through this plasma here. 00:56:54.100 --> 00:56:58.840 align:middle line:84% But that means that a particle going at 10 MeV 00:56:58.840 --> 00:57:02.260 align:middle line:90% reaches the plasma at time T0. 00:57:02.260 --> 00:57:05.930 align:middle line:84% So instead of 10, I'll write 6 because I've got 6 here. 00:57:05.930 --> 00:57:11.090 align:middle line:84% So these particles all stream through at T0. 00:57:11.090 --> 00:57:15.050 align:middle line:84% But a particle at 1 MeV is going more slowly, 00:57:15.050 --> 00:57:18.840 align:middle line:84% going six times more slowly modulo, some relativistic stuff. 00:57:18.840 --> 00:57:21.560 align:middle line:84% And so that means that this particle arrived 00:57:21.560 --> 00:57:25.670 align:middle line:90% at T0 plus delta T. 00:57:25.670 --> 00:57:28.760 align:middle line:84% And so not only did the different energies maybe 00:57:28.760 --> 00:57:31.160 align:middle line:84% give you a way of unfolding the electric magnetic fields, 00:57:31.160 --> 00:57:33.630 align:middle line:84% the different energies are different slices in time. 00:57:33.630 --> 00:57:36.930 align:middle line:84% So if this is energy in this direction, 00:57:36.930 --> 00:57:39.050 align:middle line:84% it means that if you read the films back, 00:57:39.050 --> 00:57:41.190 align:middle line:84% they correspond to time in this direction. 00:57:41.190 --> 00:57:43.700 align:middle line:84% And so you can make a little movie of how 00:57:43.700 --> 00:57:46.290 align:middle line:90% your plasma evolves in time. 00:57:46.290 --> 00:57:47.540 align:middle line:90% So that's really cool. 00:57:47.540 --> 00:57:48.902 align:middle line:90% You get time resolution. 00:57:48.902 --> 00:57:50.360 align:middle line:84% And the more different bits of film 00:57:50.360 --> 00:57:52.485 align:middle line:84% you have, and the wider range of energies you have, 00:57:52.485 --> 00:57:55.690 align:middle line:84% the better time resolution you can get. 00:57:55.690 --> 00:57:56.440 align:middle line:90% Questions? 00:57:56.440 --> 00:58:05.970 align:middle line:90% 00:58:05.970 --> 00:58:06.470 align:middle line:90% Yeah. 00:58:06.470 --> 00:58:09.150 align:middle line:90% 00:58:09.150 --> 00:58:10.374 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:58:10.374 --> 00:58:17.817 align:middle line:90% 00:58:17.817 --> 00:58:19.400 align:middle line:84% PROFESSOR: Yeah, the radiographic film 00:58:19.400 --> 00:58:21.840 align:middle line:84% isn't sensitive to other particles. 00:58:21.840 --> 00:58:26.245 align:middle line:84% So not to neutrons, and gammas, and things like that. 00:58:26.245 --> 00:58:27.620 align:middle line:84% In this case, I think it's almost 00:58:27.620 --> 00:58:31.870 align:middle line:84% that the lower energy particles and photons you could filter out 00:58:31.870 --> 00:58:33.620 align:middle line:84% and the high energy ones, like gamma rays, 00:58:33.620 --> 00:58:35.703 align:middle line:84% will just go straight through without interacting. 00:58:35.703 --> 00:58:37.992 align:middle line:84% So this is pretty much just sensitive to the protons. 00:58:37.992 --> 00:58:39.200 align:middle line:90% Yeah, that's a good question. 00:58:39.200 --> 00:58:43.510 align:middle line:90% 00:58:43.510 --> 00:58:44.280 align:middle line:90% Yes. 00:58:44.280 --> 00:58:45.456 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:58:45.456 --> 00:58:48.538 align:middle line:90% 00:58:48.538 --> 00:58:49.830 align:middle line:90% PROFESSOR: It's the same thing. 00:58:49.830 --> 00:58:54.950 align:middle line:84% So the protons are all emitted in a very, very narrow burst. 00:58:54.950 --> 00:58:57.140 align:middle line:90% Yeah, that's great. 00:58:57.140 --> 00:58:57.800 align:middle line:90% Thank you. 00:58:57.800 --> 00:59:03.145 align:middle line:84% Let's have a little figure here of time and fluence. 00:59:03.145 --> 00:59:06.530 align:middle line:90% 00:59:06.530 --> 00:59:12.670 align:middle line:84% So at time 0 all of the protons are emitted. 00:59:12.670 --> 00:59:14.580 align:middle line:90% They're very narrow. 00:59:14.580 --> 00:59:17.440 align:middle line:84% Remember, this is picosecond or femtosecond here. 00:59:17.440 --> 00:59:19.560 align:middle line:90% So I can say this is picosecond. 00:59:19.560 --> 00:59:23.140 align:middle line:84% And maybe I've got two energies here the blue energy, 00:59:23.140 --> 00:59:28.312 align:middle line:84% which is 1 MeV, and the purple energy, which is 10 MeV. 00:59:28.312 --> 00:59:31.930 align:middle line:90% 00:59:31.930 --> 00:59:33.970 align:middle line:84% They have to travel some distance 00:59:33.970 --> 00:59:38.050 align:middle line:84% from the source to the plasma, and they will go through it 00:59:38.050 --> 00:59:39.160 align:middle line:90% at different times. 00:59:39.160 --> 00:59:41.860 align:middle line:84% The 10 MeV one will get there first 00:59:41.860 --> 00:59:45.440 align:middle line:84% and the 1 MeV one will get there second. 00:59:45.440 --> 00:59:49.307 align:middle line:84% And then finally, they will travel through to the detector. 00:59:49.307 --> 00:59:51.890 align:middle line:84% And they'll arrive there much later, but we don't really care. 00:59:51.890 --> 00:59:53.390 align:middle line:84% The detector is time integrating. 00:59:53.390 --> 00:59:55.430 align:middle line:84% So it will see protons, which sampled 00:59:55.430 --> 00:59:57.770 align:middle line:90% the plasma at an earlier time. 00:59:57.770 --> 01:00:00.920 align:middle line:84% Sorry, the 10 MeV film will see protons 01:00:00.920 --> 01:00:03.858 align:middle line:84% that sampled the plasma earlier than the 1 MeV film. 01:00:03.858 --> 01:00:05.900 align:middle line:84% But you do actually have to have that requirement 01:00:05.900 --> 01:00:09.110 align:middle line:84% that you have a very, very short burst of photons. 01:00:09.110 --> 01:00:10.060 align:middle line:90% Yeah. 01:00:10.060 --> 01:00:11.485 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:00:11.485 --> 01:00:15.342 align:middle line:90% 01:00:15.342 --> 01:00:17.550 align:middle line:84% PROFESSOR: Yeah, so you can only do one or the other. 01:00:17.550 --> 01:00:19.760 align:middle line:84% You can either claim that there are no electric fields, 01:00:19.760 --> 01:00:20.960 align:middle line:84% and you know it's all magnetic fields, 01:00:20.960 --> 01:00:22.340 align:middle line:90% and then you get a time series. 01:00:22.340 --> 01:00:26.583 align:middle line:84% Or you can claim that this plasma doesn't change in time, 01:00:26.583 --> 01:00:28.500 align:middle line:84% and therefore, you can use a different energy. 01:00:28.500 --> 01:00:29.600 align:middle line:84% So you've got to be a little bit careful 01:00:29.600 --> 01:00:32.240 align:middle line:84% when you see someone claiming this proton radiography. 01:00:32.240 --> 01:00:33.450 align:middle line:90% It's difficult to do both. 01:00:33.450 --> 01:00:36.080 align:middle line:84% But you might have a simulation or some other experiment that 01:00:36.080 --> 01:00:38.772 align:middle line:84% suggests the electric fields are minimal, in which case 01:00:38.772 --> 01:00:41.480 align:middle line:84% you can just say, OK, everything I see is due to magnetic fields. 01:00:41.480 --> 01:00:42.960 align:middle line:90% Now I have a time series. 01:00:42.960 --> 01:00:45.650 align:middle line:84% People do that and it may be reasonable. 01:00:45.650 --> 01:00:49.420 align:middle line:84% Was there another question do I see? 01:00:49.420 --> 01:00:52.630 align:middle line:84% OK, let's talk about this second method, which 01:00:52.630 --> 01:00:54.835 align:middle line:90% is an imploding capsule. 01:00:54.835 --> 01:01:25.630 align:middle line:90% 01:01:25.630 --> 01:01:29.530 align:middle line:84% The second method would be, for example, a D helium-3 implosion. 01:01:29.530 --> 01:01:33.050 align:middle line:90% 01:01:33.050 --> 01:01:36.230 align:middle line:84% So this is literally a little inertial confinement fusion 01:01:36.230 --> 01:01:42.320 align:middle line:84% target here filled with deuterium and helium-3. 01:01:42.320 --> 01:01:49.100 align:middle line:84% And you get a load of direct drive lasers pushing on it. 01:01:49.100 --> 01:01:54.420 align:middle line:84% I'm making a nice symmetric-ish implosion of this. 01:01:54.420 --> 01:01:56.450 align:middle line:84% And then when this implodes, it's 01:01:56.450 --> 01:01:59.005 align:middle line:84% going to send out all of the particles that 01:01:59.005 --> 01:02:01.130 align:middle line:84% come from the fusion reaction between the deuterium 01:02:01.130 --> 01:02:01.910 align:middle line:90% and the helium-3. 01:02:01.910 --> 01:02:06.140 align:middle line:84% So just to say for this, you need to have multiple, 01:02:06.140 --> 01:02:11.320 align:middle line:84% I'm going to write many, kilojoule class beams 01:02:11.320 --> 01:02:13.150 align:middle line:90% with a 1 nanosecond pulse. 01:02:13.150 --> 01:02:16.870 align:middle line:84% 1 nanosecond in the laser world is considered a long pulse. 01:02:16.870 --> 01:02:19.450 align:middle line:84% So these are much longer than picoseconds and femtosecond, 01:02:19.450 --> 01:02:22.330 align:middle line:90% but these are long pulse lasers. 01:02:22.330 --> 01:02:25.510 align:middle line:84% So the interesting thing is that this reaction 01:02:25.510 --> 01:02:29.215 align:middle line:84% produces two different energies from it, but only two. 01:02:29.215 --> 01:02:32.530 align:middle line:90% 01:02:32.530 --> 01:02:39.248 align:middle line:90% We have the 3 MeV DD protons. 01:02:39.248 --> 01:02:41.040 align:middle line:84% Remember, we're not interested in neutrons. 01:02:41.040 --> 01:02:42.415 align:middle line:84% There may be neutrons coming out, 01:02:42.415 --> 01:02:44.610 align:middle line:84% but they're not being used for this diagnostic. 01:02:44.610 --> 01:02:53.660 align:middle line:84% And we have the 14.7 MeV D helium-3 protons as well. 01:02:53.660 --> 01:02:57.250 align:middle line:84% So this could be a positive or a negative. 01:02:57.250 --> 01:02:59.980 align:middle line:84% The positive is you know exactly what your particle energies are. 01:02:59.980 --> 01:03:01.270 align:middle line:90% They're set by fusion. 01:03:01.270 --> 01:03:02.937 align:middle line:84% They're going to be shifted a little bit 01:03:02.937 --> 01:03:04.900 align:middle line:84% because there's thermal motion here, 01:03:04.900 --> 01:03:07.180 align:middle line:84% but roughly what the energies are exactly. 01:03:07.180 --> 01:03:10.735 align:middle line:84% But it's good so the positive is we know the energy. 01:03:10.735 --> 01:03:15.870 align:middle line:90% 01:03:15.870 --> 01:03:17.790 align:middle line:84% The negative is there's only two. 01:03:17.790 --> 01:03:21.800 align:middle line:90% 01:03:21.800 --> 01:03:24.530 align:middle line:84% People have recently add a little bit 01:03:24.530 --> 01:03:27.290 align:middle line:84% of spice in the form of some tritium, 01:03:27.290 --> 01:03:32.000 align:middle line:84% and then you also get the tritium helium-3 reaction, which 01:03:32.000 --> 01:03:34.550 align:middle line:90% gives you a proton at 12 MeV. 01:03:34.550 --> 01:03:36.865 align:middle line:84% So that gives you a tri particle source. 01:03:36.865 --> 01:03:37.490 align:middle line:90% So that's nice. 01:03:37.490 --> 01:03:38.910 align:middle line:90% Now you have three particles. 01:03:38.910 --> 01:03:41.550 align:middle line:90% So that's pretty good. 01:03:41.550 --> 01:03:45.925 align:middle line:84% Another negative here is the spot size. 01:03:45.925 --> 01:03:50.310 align:middle line:90% 01:03:50.310 --> 01:03:53.460 align:middle line:84% These implosions tend to get down to slightly larger sizes 01:03:53.460 --> 01:03:56.190 align:middle line:84% than you can reach with the focal spot for TNSA, 01:03:56.190 --> 01:03:59.400 align:middle line:84% so the spot size is rather about 40 microns or so. 01:03:59.400 --> 01:04:02.580 align:middle line:84% It's not too much worse, but it is significantly larger 01:04:02.580 --> 01:04:05.690 align:middle line:90% than the TNSA size. 01:04:05.690 --> 01:04:13.460 align:middle line:84% The other problem is that we have isotropic emission. 01:04:13.460 --> 01:04:16.240 align:middle line:84% So the nice thing about TNSA is all the particles 01:04:16.240 --> 01:04:18.490 align:middle line:84% are going roughly in the direction you want to. 01:04:18.490 --> 01:04:22.550 align:middle line:84% Here, if your plasma is over here, most of your particles 01:04:22.550 --> 01:04:25.190 align:middle line:84% are just flying off in completely random directions, 01:04:25.190 --> 01:04:28.770 align:middle line:84% and you don't get to capture them at all. 01:04:28.770 --> 01:04:34.000 align:middle line:84% And so this means that we have few particles. 01:04:34.000 --> 01:04:37.680 align:middle line:90% 01:04:37.680 --> 01:04:39.290 align:middle line:84% This is in contrast to TNSA, where 01:04:39.290 --> 01:04:43.327 align:middle line:84% we were able to create a very large number of particles. 01:04:43.327 --> 01:04:45.160 align:middle line:84% But the fact that we have very few particles 01:04:45.160 --> 01:04:47.305 align:middle line:84% also leads to a change in detector. 01:04:47.305 --> 01:04:51.370 align:middle line:90% 01:04:51.370 --> 01:04:55.438 align:middle line:84% And this is the big workhorse of the MIT HED group 01:04:55.438 --> 01:04:56.980 align:middle line:84% because they realized that they could 01:04:56.980 --> 01:04:58.970 align:middle line:90% use a detector called CR-39. 01:04:58.970 --> 01:05:02.170 align:middle line:90% 01:05:02.170 --> 01:05:03.820 align:middle line:90% Now CR-39 is a polymer. 01:05:03.820 --> 01:05:07.430 align:middle line:90% 01:05:07.430 --> 01:05:11.920 align:middle line:84% The interesting thing about it you have a sheet of CR-39, 01:05:11.920 --> 01:05:14.510 align:middle line:84% you have a proton going through it, 01:05:14.510 --> 01:05:19.100 align:middle line:84% is It causes a little area of damage, a little track 01:05:19.100 --> 01:05:20.570 align:middle line:90% as a proton goes through. 01:05:20.570 --> 01:05:23.060 align:middle line:84% And you can then dip your polymer 01:05:23.060 --> 01:05:25.850 align:middle line:84% after the shot in an etching solution. 01:05:25.850 --> 01:05:30.550 align:middle line:90% 01:05:30.550 --> 01:05:33.580 align:middle line:84% And it turns each of these tracks into little holes. 01:05:33.580 --> 01:05:37.390 align:middle line:84% So now if you look at your CR-39, 01:05:37.390 --> 01:05:43.490 align:middle line:84% there's all sorts of little holes on it. 01:05:43.490 --> 01:05:50.290 align:middle line:84% It turns out the size of the hole 01:05:50.290 --> 01:05:52.640 align:middle line:84% is in some way related to the energy. 01:05:52.640 --> 01:05:56.030 align:middle line:84% So you can tell the difference between the 3 MeV and the 14.7 01:05:56.030 --> 01:05:56.530 align:middle line:90% MeV. 01:05:56.530 --> 01:05:59.280 align:middle line:90% 01:05:59.280 --> 01:06:02.250 align:middle line:84% And theoretically, though again I'm not sure 01:06:02.250 --> 01:06:04.650 align:middle line:84% that anyone has done this since it was first suggested, 01:06:04.650 --> 01:06:07.117 align:middle line:84% if you have multiple pieces of CR-39 01:06:07.117 --> 01:06:08.700 align:middle line:84% and protons are just going through it, 01:06:08.700 --> 01:06:11.910 align:middle line:84% you should be able to correlate the tracks in different bits 01:06:11.910 --> 01:06:14.940 align:middle line:84% of film and go, aha, a track in this bit of film 01:06:14.940 --> 01:06:18.810 align:middle line:84% is now over here and I should be able to work out 01:06:18.810 --> 01:06:20.170 align:middle line:90% the angle of the particles. 01:06:20.170 --> 01:06:24.790 align:middle line:84% So you may be able to correlate tracks. 01:06:24.790 --> 01:06:26.920 align:middle line:84% But I'm not sure anyone's actually used 01:06:26.920 --> 01:06:28.840 align:middle line:90% that as a serious diagnostic. 01:06:28.840 --> 01:06:32.290 align:middle line:90% Just in one of the early papers. 01:06:32.290 --> 01:06:34.600 align:middle line:84% So what do you actually do with all these tracks here? 01:06:34.600 --> 01:06:37.090 align:middle line:84% Well, you set up a very good microscope, 01:06:37.090 --> 01:06:39.940 align:middle line:84% and you put one of these bits of CR-39 in it. 01:06:39.940 --> 01:06:42.580 align:middle line:84% And the microscope looks at a small square here, 01:06:42.580 --> 01:06:46.480 align:middle line:84% and it counts all of the holes in this little square. 01:06:46.480 --> 01:06:48.400 align:middle line:84% And this square represents a pixel, 01:06:48.400 --> 01:06:52.030 align:middle line:84% and it outputs, for example, a hundred holes. 01:06:52.030 --> 01:06:56.280 align:middle line:84% And now it goes to the next one and it outputs 102. 01:06:56.280 --> 01:06:58.740 align:middle line:84% So it's effectively reading off the number of protons, 01:06:58.740 --> 01:07:01.350 align:middle line:84% one by one, individually, automated, but still 01:07:01.350 --> 01:07:04.270 align:middle line:84% individually, and building up that intensity. 01:07:04.270 --> 01:07:05.935 align:middle line:84% Whereas the radiochromic film, we 01:07:05.935 --> 01:07:07.560 align:middle line:84% don't know the exact number of protons, 01:07:07.560 --> 01:07:10.680 align:middle line:84% we just have a different darkness of the film 01:07:10.680 --> 01:07:13.350 align:middle line:84% depending on the proton fluence here. 01:07:13.350 --> 01:07:19.590 align:middle line:84% So I did a scientific poll at APS DPP 2021. 01:07:19.590 --> 01:07:22.330 align:middle line:84% Three out of three physicists preferred-- 01:07:22.330 --> 01:07:25.970 align:middle line:90% 01:07:25.970 --> 01:07:27.728 align:middle line:84% I probably spelled preferred wrong-- 01:07:27.728 --> 01:07:32.260 align:middle line:90% PNFA plus radiochromic film. 01:07:32.260 --> 01:07:34.090 align:middle line:84% However, there are many facilities 01:07:34.090 --> 01:07:36.730 align:middle line:84% which do not have a sufficiently powerful short pulse laser, 01:07:36.730 --> 01:07:39.520 align:middle line:84% but they do have an awful lot of kilojoule class 01:07:39.520 --> 01:07:42.100 align:middle line:84% laser beams floating around that they don't have to use. 01:07:42.100 --> 01:07:44.560 align:middle line:84% So for example, on omega or on NIF, 01:07:44.560 --> 01:07:46.090 align:middle line:90% you have lots of these beams. 01:07:46.090 --> 01:07:48.790 align:middle line:84% You can use some of the beams to drive your target plasma, 01:07:48.790 --> 01:07:51.700 align:middle line:84% and the other beams to drive your imploding capsule, 01:07:51.700 --> 01:07:53.950 align:middle line:84% and you'll still be able to do proton radiography. 01:07:53.950 --> 01:07:56.600 align:middle line:84% So there's definitely a place for this technique. 01:07:56.600 --> 01:08:00.730 align:middle line:84% It's just that it seems like people prefer, in general, TNSA. 01:08:00.730 --> 01:08:04.290 align:middle line:90% OK, questions? 01:08:04.290 --> 01:08:05.120 align:middle line:90% Yes. 01:08:05.120 --> 01:08:06.386 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:08:06.386 --> 01:08:13.703 align:middle line:90% 01:08:13.703 --> 01:08:16.370 align:middle line:84% PROFESSOR: I don't know whether people have done that, but yeah, 01:08:16.370 --> 01:08:16.870 align:middle line:90% possibly. 01:08:16.870 --> 01:08:20.189 align:middle line:84% You need to have magnetic fields in this. 01:08:20.189 --> 01:08:23.000 align:middle line:84% And so in general, people don't think too much 01:08:23.000 --> 01:08:25.677 align:middle line:84% about the magnetic fields in ICF, but they might be there. 01:08:25.677 --> 01:08:28.010 align:middle line:84% There are mechanisms which could self-generate the field 01:08:28.010 --> 01:08:29.130 align:middle line:90% like the [INAUDIBLE] battery. 01:08:29.130 --> 01:08:31.189 align:middle line:84% I don't know if anyone's done that, but it could be done. 01:08:31.189 --> 01:08:32.522 align:middle line:90% I imagine it's then complicated. 01:08:32.522 --> 01:08:34.850 align:middle line:84% You have to work out exactly what you're measuring. 01:08:34.850 --> 01:08:37.040 align:middle line:84% But yeah, it's an interesting idea. 01:08:37.040 --> 01:08:38.346 align:middle line:90% Yeah, sure. 01:08:38.346 --> 01:08:40.520 align:middle line:84% AUDIENCE: You mentioned [INAUDIBLE]. 01:08:40.520 --> 01:08:43.570 align:middle line:90% 01:08:43.570 --> 01:08:47.450 align:middle line:90% Why not ELF particles? 01:08:47.450 --> 01:08:49.264 align:middle line:84% PROFESSOR: Why don't we do ELF particles? 01:08:49.264 --> 01:08:52.777 align:middle line:84% AUDIENCE: Because they are lower energy [INAUDIBLE]. 01:08:52.777 --> 01:08:54.319 align:middle line:84% PROFESSOR: Yeah, I would have thought 01:08:54.319 --> 01:08:58.279 align:middle line:84% that the velocity is a fair bit lower because of the higher 01:08:58.279 --> 01:09:00.080 align:middle line:90% mass. 01:09:00.080 --> 01:09:02.308 align:middle line:84% I don't know exactly, so good question. 01:09:02.308 --> 01:09:04.850 align:middle line:84% I mean, in general, the reason why people haven't put tritium 01:09:04.850 --> 01:09:06.590 align:middle line:84% in this, is you don't necessarily 01:09:06.590 --> 01:09:08.250 align:middle line:90% want to be playing with tritium. 01:09:08.250 --> 01:09:10.939 align:middle line:84% So this is relatively easy to fill this capsule, 01:09:10.939 --> 01:09:12.740 align:middle line:84% and this is just a pain in the ass. 01:09:12.740 --> 01:09:14.600 align:middle line:84% So it's not like everyone has rushed out 01:09:14.600 --> 01:09:16.189 align:middle line:84% to start doing the tri particle thing. 01:09:16.189 --> 01:09:17.120 align:middle line:90% Yeah. 01:09:17.120 --> 01:09:18.120 align:middle line:90% OK, other questions? 01:09:18.120 --> 01:09:18.787 align:middle line:90% Anything online? 01:09:18.787 --> 01:09:26.700 align:middle line:90% 01:09:26.700 --> 01:09:28.560 align:middle line:84% OK, now we'll go through some issues 01:09:28.560 --> 01:09:30.210 align:middle line:84% with doing this diagnostic practically. 01:09:30.210 --> 01:09:32.085 align:middle line:84% We've got our source, we've got our detector, 01:09:32.085 --> 01:09:34.649 align:middle line:84% we understand how to make a map of the intensity 01:09:34.649 --> 01:09:38.335 align:middle line:84% after the plasma, but are there any problems with doing that? 01:09:38.335 --> 01:09:38.835 align:middle line:90% Yes. 01:09:38.835 --> 01:09:56.220 align:middle line:90% 01:09:56.220 --> 01:10:03.700 align:middle line:90% So issue 1. 01:10:03.700 --> 01:10:08.175 align:middle line:84% This condition, but our beam was uniform. 01:10:08.175 --> 01:10:13.410 align:middle line:90% 01:10:13.410 --> 01:10:17.590 align:middle line:84% Remember, we need that because we're trying to determine from I 01:10:17.590 --> 01:10:21.070 align:middle line:84% equals I0 over the size of the Jacobian. 01:10:21.070 --> 01:10:23.020 align:middle line:84% And remember, the Jacobian is the thing 01:10:23.020 --> 01:10:24.280 align:middle line:90% that has all the physics in. 01:10:24.280 --> 01:10:31.950 align:middle line:90% 01:10:31.950 --> 01:10:34.590 align:middle line:84% If we want to work out what the size of the Jacobian is, 01:10:34.590 --> 01:10:37.170 align:middle line:84% we are measuring I, but we need to make some assumption 01:10:37.170 --> 01:10:38.190 align:middle line:90% about what I0 is. 01:10:38.190 --> 01:10:40.800 align:middle line:84% We can't measure it because we do the target normal sheaf 01:10:40.800 --> 01:10:42.720 align:middle line:90% acceleration through the plasma. 01:10:42.720 --> 01:10:44.580 align:middle line:84% We can't put a beam splitter in and measure 01:10:44.580 --> 01:10:46.110 align:middle line:84% what the protons look like because we don't have 01:10:46.110 --> 01:10:47.530 align:middle line:90% beam splitters and protons. 01:10:47.530 --> 01:10:50.130 align:middle line:84% So knowing I0 and knowing that it's 01:10:50.130 --> 01:10:53.968 align:middle line:84% uniform is very, very difficult. And so you could say, well, 01:10:53.968 --> 01:10:55.260 align:middle line:90% what if I have reproducibility? 01:10:55.260 --> 01:10:57.180 align:middle line:84% What if I do the TNSA over and over again 01:10:57.180 --> 01:10:59.040 align:middle line:84% and I measure it without any plasma 01:10:59.040 --> 01:11:00.550 align:middle line:90% and I see that it's uniform? 01:11:00.550 --> 01:11:02.820 align:middle line:84% Well, the trouble is we see that it's not uniform. 01:11:02.820 --> 01:11:05.678 align:middle line:90% So we have poor reproducibility. 01:11:05.678 --> 01:11:14.950 align:middle line:90% 01:11:14.950 --> 01:11:20.800 align:middle line:84% So a solution to this is that you 01:11:20.800 --> 01:11:22.630 align:middle line:90% include a grid in your system. 01:11:22.630 --> 01:11:26.410 align:middle line:84% You still have your point source and your protons streaming 01:11:26.410 --> 01:11:32.840 align:middle line:84% out through your plasma, but on the other side of the plasma 01:11:32.840 --> 01:11:34.770 align:middle line:90% you put a grid like this. 01:11:34.770 --> 01:11:37.850 align:middle line:84% So in the absence of the plasma, this splits up your beam 01:11:37.850 --> 01:11:40.010 align:middle line:90% into a series of beamlets. 01:11:40.010 --> 01:11:43.850 align:middle line:84% So for example, we have regions where the protons 01:11:43.850 --> 01:11:45.080 align:middle line:90% can pass through the grid-- 01:11:45.080 --> 01:11:48.020 align:middle line:84% I'm just going to draw nine of them here-- 01:11:48.020 --> 01:11:51.935 align:middle line:84% and we effectively have little points on a grid like this. 01:11:51.935 --> 01:11:53.735 align:middle line:90% This is the case without plasma. 01:11:53.735 --> 01:11:59.630 align:middle line:90% 01:11:59.630 --> 01:12:02.860 align:middle line:84% Now when the plasma is present, these protons 01:12:02.860 --> 01:12:05.110 align:middle line:84% are going to be deflected in different ways, 01:12:05.110 --> 01:12:08.030 align:middle line:90% and we'll see these points move. 01:12:08.030 --> 01:12:11.020 align:middle line:84% So for example, this point might move to here. 01:12:11.020 --> 01:12:13.335 align:middle line:90% This point might move to here. 01:12:13.335 --> 01:12:14.585 align:middle line:90% This point might move to here. 01:12:14.585 --> 01:12:24.540 align:middle line:90% 01:12:24.540 --> 01:12:27.110 align:middle line:84% And actually, we can now directly 01:12:27.110 --> 01:12:32.950 align:middle line:84% calculate the deflection angle, alpha, and therefore 01:12:32.950 --> 01:12:34.720 align:middle line:84% back out the electric and magnetic field 01:12:34.720 --> 01:12:36.670 align:middle line:90% from this technique. 01:12:36.670 --> 01:12:41.415 align:middle line:84% So this is very good because we can calculate alpha directly. 01:12:41.415 --> 01:12:45.790 align:middle line:90% 01:12:45.790 --> 01:12:50.180 align:middle line:84% But the downside of it is we have very low spatial resolution 01:12:50.180 --> 01:12:52.400 align:middle line:90% now. 01:12:52.400 --> 01:12:55.760 align:middle line:84% Because you've gone from having a continuous image 01:12:55.760 --> 01:12:58.172 align:middle line:90% into a discrete image instead. 01:12:58.172 --> 01:12:59.880 align:middle line:84% But this technique is still very powerful 01:12:59.880 --> 01:13:02.460 align:middle line:84% because it does give you a direct angle on alpha 01:13:02.460 --> 01:13:05.080 align:middle line:84% without needing to know what I0 was. 01:13:05.080 --> 01:13:05.580 align:middle line:90% OK. 01:13:05.580 --> 01:13:08.510 align:middle line:90% 01:13:08.510 --> 01:13:12.210 align:middle line:84% The second issue I want to talk about. 01:13:12.210 --> 01:13:24.500 align:middle line:90% 01:13:24.500 --> 01:13:33.070 align:middle line:84% Is if we have a uniform magnetic field as well 01:13:33.070 --> 01:13:35.455 align:middle line:90% as some varying magnetic field. 01:13:35.455 --> 01:13:39.588 align:middle line:90% 01:13:39.588 --> 01:13:42.950 align:middle line:84% If we have a uniform magnetic field all of the particles 01:13:42.950 --> 01:13:45.515 align:middle line:84% are going to be swept in the same direction as well. 01:13:45.515 --> 01:13:53.630 align:middle line:90% 01:13:53.630 --> 01:13:56.613 align:middle line:84% It's actually extremely hard to align these detectors 01:13:56.613 --> 01:13:59.030 align:middle line:84% and know exactly where the center is because we're talking 01:13:59.030 --> 01:14:01.410 align:middle line:90% about submillimeter precision. 01:14:01.410 --> 01:14:05.530 align:middle line:84% So if you see this grid initially-- 01:14:05.530 --> 01:14:15.270 align:middle line:90% 01:14:15.270 --> 01:14:17.350 align:middle line:84% If you see a grid like that, that 01:14:17.350 --> 01:14:21.010 align:middle line:84% could be a grid that's been formed without any background 01:14:21.010 --> 01:14:22.900 align:middle line:84% magnetic field or it could be a grid that's 01:14:22.900 --> 01:14:25.750 align:middle line:84% been displaced some distance by a background magnetic field. 01:14:25.750 --> 01:14:28.167 align:middle line:84% You're not going to be able to tell the difference between 01:14:28.167 --> 01:14:30.850 align:middle line:84% those two, so you cannot measure this background magnetic field. 01:14:30.850 --> 01:14:34.060 align:middle line:84% And in many experiments where this is externally 01:14:34.060 --> 01:14:36.687 align:middle line:84% applied with some magnetic coils, 01:14:36.687 --> 01:14:38.270 align:middle line:84% they really want to be able to measure 01:14:38.270 --> 01:14:41.545 align:middle line:84% that background magnetic field externally applied. 01:14:41.545 --> 01:14:45.560 align:middle line:90% 01:14:45.560 --> 01:14:50.280 align:middle line:84% So a very neat technique that's come up recently 01:14:50.280 --> 01:14:54.990 align:middle line:84% is to use a second detector, which is sensitive to X-rays 01:14:54.990 --> 01:14:55.930 align:middle line:90% here. 01:14:55.930 --> 01:15:02.290 align:middle line:84% So the idea, once again, is that we have our source of protons, 01:15:02.290 --> 01:15:04.240 align:middle line:84% but alongside the protons we're also 01:15:04.240 --> 01:15:08.010 align:middle line:84% going to have a source of X-rays because this thing is hot. 01:15:08.010 --> 01:15:11.273 align:middle line:84% And in this case, I drew the mesh afterwards. 01:15:11.273 --> 01:15:13.440 align:middle line:84% Some of you may have realized I think the mesh needs 01:15:13.440 --> 01:15:14.744 align:middle line:90% to be before this part. 01:15:14.744 --> 01:15:17.350 align:middle line:90% 01:15:17.350 --> 01:15:22.420 align:middle line:84% Now we have our mesh again here and we have our plasma. 01:15:22.420 --> 01:15:25.140 align:middle line:90% 01:15:25.140 --> 01:15:30.180 align:middle line:84% And the neat thing is we have, for example, our RCF, 01:15:30.180 --> 01:15:32.100 align:middle line:84% which is sensitive to the protons, 01:15:32.100 --> 01:15:36.225 align:middle line:84% and we have some image plate, which 01:15:36.225 --> 01:15:37.350 align:middle line:90% is sensitive to the X-rays. 01:15:37.350 --> 01:15:45.990 align:middle line:90% 01:15:45.990 --> 01:15:49.710 align:middle line:84% Then although our protons will be swept up by magnetic fields 01:15:49.710 --> 01:15:55.127 align:middle line:84% and do all sorts of strange things, 01:15:55.127 --> 01:15:57.460 align:middle line:84% the X-rays are just going to go in straight lines, which 01:15:57.460 --> 01:16:02.070 align:middle line:84% are completely unaffected by any magnetic fields here. 01:16:02.070 --> 01:16:04.910 align:middle line:84% So we're going to use high energy X-rays here, such 01:16:04.910 --> 01:16:08.540 align:middle line:84% that the refractive index at 10 kilo electron volts 01:16:08.540 --> 01:16:11.510 align:middle line:84% is roughly 1, and so we don't see 01:16:11.510 --> 01:16:13.490 align:middle line:90% any refraction of the X-rays. 01:16:13.490 --> 01:16:18.440 align:middle line:84% And so that means what we end up with is we'll have 01:16:18.440 --> 01:16:19.970 align:middle line:90% a grid of the X-ray points. 01:16:19.970 --> 01:16:24.720 align:middle line:90% 01:16:24.720 --> 01:16:28.050 align:middle line:84% In the absence of any background magnetic field 01:16:28.050 --> 01:16:29.640 align:middle line:84% and in the absence of any plasma, 01:16:29.640 --> 01:16:32.010 align:middle line:84% we should see all of the protons end up 01:16:32.010 --> 01:16:34.590 align:middle line:84% in the same place as the X-rays, if there 01:16:34.590 --> 01:16:36.750 align:middle line:84% is some uniform magnetic field that 01:16:36.750 --> 01:16:43.740 align:middle line:84% will shift all of the protons like this, this is uniform. 01:16:43.740 --> 01:16:47.257 align:middle line:90% 01:16:47.257 --> 01:16:49.090 align:middle line:84% And then if there's a uniform magnetic field 01:16:49.090 --> 01:16:50.860 align:middle line:84% plus a perturbed magnetic field, it 01:16:50.860 --> 01:16:53.155 align:middle line:84% will further shift them on top of that. 01:16:53.155 --> 01:16:56.180 align:middle line:90% 01:16:56.180 --> 01:16:59.350 align:middle line:84% So the X-ray grid acts as a really nice fiducial. 01:16:59.350 --> 01:17:01.450 align:middle line:84% It allows us to directly measure the deflections 01:17:01.450 --> 01:17:03.460 align:middle line:84% without having to worry about the uniform magnetic field 01:17:03.460 --> 01:17:04.252 align:middle line:90% screwing things up. 01:17:04.252 --> 01:17:11.180 align:middle line:84% So this is very nice bit of work by Johnson and others. 01:17:11.180 --> 01:17:15.460 align:middle line:84% And this is in Review of Scientific Instruments published 01:17:15.460 --> 01:17:17.340 align:middle line:90% last year. 01:17:17.340 --> 01:17:20.000 align:middle line:90%