WEBVTT 00:00:00.000 --> 00:00:01.446 align:middle line:90% 00:00:01.446 --> 00:00:02.892 align:middle line:90% [SQUEAKING] 00:00:02.892 --> 00:00:04.338 align:middle line:90% [RUSTLING] 00:00:04.338 --> 00:00:09.207 align:middle line:90% [CLICKING] 00:00:09.207 --> 00:00:10.790 align:middle line:84% JACK HARE: So today, we're going to be 00:00:10.790 --> 00:00:11.870 align:middle line:90% talking about Langmuir probes. 00:00:11.870 --> 00:00:14.330 align:middle line:84% Now, I'll freely admit I find Langmuir probes a little bit 00:00:14.330 --> 00:00:15.380 align:middle line:90% confusing. 00:00:15.380 --> 00:00:18.800 align:middle line:84% I would say that you need to know a lot about sheath theory 00:00:18.800 --> 00:00:21.500 align:middle line:84% to properly understand the Langmuir probe. 00:00:21.500 --> 00:00:24.840 align:middle line:84% And to do that in a single class is very, very challenging. 00:00:24.840 --> 00:00:27.380 align:middle line:84% So if you look in Hutchinson's textbook, 00:00:27.380 --> 00:00:30.560 align:middle line:84% he takes a significant amount of time on these probes. 00:00:30.560 --> 00:00:32.405 align:middle line:84% He does that because he really likes probes, 00:00:32.405 --> 00:00:33.530 align:middle line:90% and that's very reasonable. 00:00:33.530 --> 00:00:35.540 align:middle line:84% Everyone has their little biases. 00:00:35.540 --> 00:00:37.202 align:middle line:84% I'm not going to spend that much time. 00:00:37.202 --> 00:00:38.660 align:middle line:84% And what I've done in particular is 00:00:38.660 --> 00:00:43.730 align:middle line:84% I've tried to rework some of the material in Hutchinson's book 00:00:43.730 --> 00:00:45.470 align:middle line:84% in a way that makes more sense to me. 00:00:45.470 --> 00:00:47.162 align:middle line:84% Now, it may not make more sense to you, 00:00:47.162 --> 00:00:48.620 align:middle line:84% so this is why I strongly encourage 00:00:48.620 --> 00:00:50.840 align:middle line:84% you to go have a look at Hutchinson's book as well 00:00:50.840 --> 00:00:53.040 align:middle line:84% and maybe interpolate between those two. 00:00:53.040 --> 00:00:55.730 align:middle line:84% And I also included in the syllabus 00:00:55.730 --> 00:01:00.020 align:middle line:84% a reference to the book by Lieberman and Lichtenberg, 00:01:00.020 --> 00:01:02.037 align:middle line:84% which is very good on sheath theory. 00:01:02.037 --> 00:01:03.620 align:middle line:84% It's a book on low temperature plasmas 00:01:03.620 --> 00:01:05.099 align:middle line:90% where sheaths are very common. 00:01:05.099 --> 00:01:07.730 align:middle line:84% So if you are a little bit weak on sheath physics 00:01:07.730 --> 00:01:10.920 align:middle line:84% and you want to review that, I recommend that book. 00:01:10.920 --> 00:01:12.870 align:middle line:84% Now, one thing I will say is that everyone 00:01:12.870 --> 00:01:14.220 align:middle line:90% uses different notation. 00:01:14.220 --> 00:01:16.500 align:middle line:84% And in an attempt to make things clearer, 00:01:16.500 --> 00:01:19.140 align:middle line:84% I've also used a third and different notation 00:01:19.140 --> 00:01:21.750 align:middle line:84% from that used in Hutchinson and Lieberman and Lichtenberg. 00:01:21.750 --> 00:01:23.280 align:middle line:90% So good luck with that. 00:01:23.280 --> 00:01:26.560 align:middle line:84% I think mine is less confusing than Hutchinson's, but of course 00:01:26.560 --> 00:01:29.320 align:middle line:84% it's that classic thing about there are too many standards. 00:01:29.320 --> 00:01:31.200 align:middle line:84% And now I've invented a new standard. 00:01:31.200 --> 00:01:33.570 align:middle line:84% So just keep an eye on all these quantities. 00:01:33.570 --> 00:01:35.130 align:middle line:84% I'll try and explain what they are. 00:01:35.130 --> 00:01:38.040 align:middle line:84% If you're not sure about any of them, please shout out. 00:01:38.040 --> 00:01:42.270 align:middle line:84% So the Langmuir probe-- it's a very simple probe. 00:01:42.270 --> 00:01:48.190 align:middle line:84% It's simple because all we take is a conductor, some sort 00:01:48.190 --> 00:01:55.140 align:middle line:84% rod of metal, and stick it into our plasma like this. 00:01:55.140 --> 00:01:57.210 align:middle line:84% And then we measure the potential. 00:01:57.210 --> 00:02:00.340 align:middle line:90% 00:02:00.340 --> 00:02:01.860 align:middle line:84% I'm going to be calling v 0 here. 00:02:01.860 --> 00:02:05.260 align:middle line:84% This is the potential on our probe. 00:02:05.260 --> 00:02:07.420 align:middle line:84% And so if we think about what this looks like, 00:02:07.420 --> 00:02:08.770 align:middle line:90% we can draw a little probe. 00:02:08.770 --> 00:02:11.230 align:middle line:84% Maybe it's a little rod of metal like this. 00:02:11.230 --> 00:02:17.130 align:middle line:84% And it's surrounded by plasma like this. 00:02:17.130 --> 00:02:19.580 align:middle line:84% And we know from our intro to plasma physics, 00:02:19.580 --> 00:02:21.470 align:middle line:84% whenever we see a conductor in a plasma, 00:02:21.470 --> 00:02:24.300 align:middle line:84% there's going to be something forming around it. 00:02:24.300 --> 00:02:30.340 align:middle line:84% So if this is the plasma and this is the probe, 00:02:30.340 --> 00:02:34.420 align:middle line:84% then this is a region that we call the sheath. 00:02:34.420 --> 00:02:37.010 align:middle line:84% And so we're going to have to understand how sheaths work. 00:02:37.010 --> 00:02:39.040 align:middle line:84% But before we get fully going with sheaths, 00:02:39.040 --> 00:02:41.623 align:middle line:84% we're just going to give an intuitive explanation of why 00:02:41.623 --> 00:02:42.790 align:middle line:90% we're going to get a sheath. 00:02:42.790 --> 00:02:44.740 align:middle line:84% And then I'm going to give you a very brief overview 00:02:44.740 --> 00:02:46.360 align:middle line:84% of the sheath results without doing 00:02:46.360 --> 00:02:48.220 align:middle line:84% any actual formal derivation of them 00:02:48.220 --> 00:02:50.420 align:middle line:90% because it will take too long. 00:02:50.420 --> 00:02:54.500 align:middle line:84% So at the instant that we stick this probe into the plasma, 00:02:54.500 --> 00:02:58.430 align:middle line:84% it's going to be immersed in a bath of electrons and ions. 00:02:58.430 --> 00:03:00.940 align:middle line:84% And so there are going to be electrons and ions striking 00:03:00.940 --> 00:03:01.565 align:middle line:90% this conductor. 00:03:01.565 --> 00:03:04.190 align:middle line:84% And we're going to assume that when they strike this conductor, 00:03:04.190 --> 00:03:04.960 align:middle line:90% they stick to it. 00:03:04.960 --> 00:03:07.143 align:middle line:84% So effectively, it starts to gain charge. 00:03:07.143 --> 00:03:08.560 align:middle line:84% So the first thing we want to know 00:03:08.560 --> 00:03:11.090 align:middle line:84% is what is the flux of particles? 00:03:11.090 --> 00:03:13.900 align:middle line:84% So this is a flux capital gamma of particles 00:03:13.900 --> 00:03:18.492 align:middle line:84% of species, J, that is striking against this probe here. 00:03:18.492 --> 00:03:20.950 align:middle line:84% And we're going to make an approximation that this probe is 00:03:20.950 --> 00:03:22.330 align:middle line:90% a planar surface. 00:03:22.330 --> 00:03:25.670 align:middle line:84% That means we pick up a factor of 1/4, just in geometry. 00:03:25.670 --> 00:03:29.020 align:middle line:84% And that is going to be 1/4 of the number of probe particles 00:03:29.020 --> 00:03:33.610 align:middle line:84% times by the average speed that the probe particles are going 00:03:33.610 --> 00:03:34.110 align:middle line:90% at. 00:03:34.110 --> 00:03:38.550 align:middle line:84% So this is the average particle flux. 00:03:38.550 --> 00:03:40.740 align:middle line:84% I haven't said anything about charge yet. 00:03:40.740 --> 00:03:43.200 align:middle line:84% I just want to note here, I'm going 00:03:43.200 --> 00:03:47.520 align:middle line:84% to be using u for velocity, and the reason 00:03:47.520 --> 00:03:49.740 align:middle line:84% is that there's an awful lot of v's that crop up. 00:03:49.740 --> 00:03:52.860 align:middle line:84% And I think using a lowercase v for velocity 00:03:52.860 --> 00:03:54.630 align:middle line:84% and an uppercase V for potential is 00:03:54.630 --> 00:03:56.380 align:middle line:84% going to cause a huge amount of confusion. 00:03:56.380 --> 00:03:59.800 align:middle line:84% So this is the first place where I diverge from Hutchinson. 00:03:59.800 --> 00:04:02.200 align:middle line:90% So just keep that in mind. 00:04:02.200 --> 00:04:04.470 align:middle line:84% AUDIENCE: Where does the 1/4 come from? 00:04:04.470 --> 00:04:06.570 align:middle line:84% JACK HARE: The 1/4 comes from geometry. 00:04:06.570 --> 00:04:10.770 align:middle line:84% If you go look this up in some standard statistical mechanics 00:04:10.770 --> 00:04:12.930 align:middle line:84% textbook, you'll find out it's because you've 00:04:12.930 --> 00:04:15.570 align:middle line:84% got a load of particles moving in random directions, 00:04:15.570 --> 00:04:17.410 align:middle line:90% and you've got a plane here. 00:04:17.410 --> 00:04:19.110 align:middle line:84% And so it's the number of particles 00:04:19.110 --> 00:04:22.298 align:middle line:84% that have a velocity, which is directed through the plane 00:04:22.298 --> 00:04:23.590 align:middle line:90% that you're talking about here. 00:04:23.590 --> 00:04:25.020 align:middle line:90% So it's a geometric factor. 00:04:25.020 --> 00:04:26.888 align:middle line:90% I haven't derived it. 00:04:26.888 --> 00:04:27.430 align:middle line:90% AUDIENCE: Ok. 00:04:27.430 --> 00:04:28.150 align:middle line:90% Thanks. 00:04:28.150 --> 00:04:28.990 align:middle line:90% JACK HARE: Cool. 00:04:28.990 --> 00:04:31.900 align:middle line:84% And so that means that the total current that this probe is 00:04:31.900 --> 00:04:34.600 align:middle line:84% instantaneously collecting as soon as we stick it 00:04:34.600 --> 00:04:37.960 align:middle line:84% into the plasma is going to be equal to the particle 00:04:37.960 --> 00:04:40.690 align:middle line:90% charge times these two fluxes. 00:04:40.690 --> 00:04:42.670 align:middle line:90% So we're going to have-- 00:04:42.670 --> 00:04:45.257 align:middle line:84% I'm going to put a minus e at the front here. 00:04:45.257 --> 00:04:47.590 align:middle line:84% We're going to have an area which represents the surface 00:04:47.590 --> 00:04:48.640 align:middle line:90% area of the probe. 00:04:48.640 --> 00:04:51.670 align:middle line:84% We're going to have our 1/4 that we have for both species, 00:04:51.670 --> 00:04:56.560 align:middle line:84% and then we're going to have a term that looks like ni ui minus 00:04:56.560 --> 00:05:00.620 align:middle line:90% ne ue here. 00:05:00.620 --> 00:05:03.730 align:middle line:84% So that's the current that we're instantaneously collecting. 00:05:03.730 --> 00:05:07.390 align:middle line:84% Now, it turns out, of course, that the average velocity 00:05:07.390 --> 00:05:12.700 align:middle line:84% of the electrons is going to be much, much bigger 00:05:12.700 --> 00:05:15.970 align:middle line:84% than the average velocity of the ions in general, 00:05:15.970 --> 00:05:20.450 align:middle line:84% and that is just a fact because the electron mass is much, 00:05:20.450 --> 00:05:22.622 align:middle line:90% much less than the ion mass. 00:05:22.622 --> 00:05:24.580 align:middle line:84% And so if they've got the similar temperatures, 00:05:24.580 --> 00:05:26.675 align:middle line:84% the electrons are going to be moving faster, which 00:05:26.675 --> 00:05:28.300 align:middle line:84% means that at least as soon as we stick 00:05:28.300 --> 00:05:30.400 align:middle line:84% the probe into the plasma, this current is 00:05:30.400 --> 00:05:33.560 align:middle line:84% going to be completely dominated by the electrons. 00:05:33.560 --> 00:05:39.510 align:middle line:84% So it's going to be 1/4 eA ne electron density, ue, like that. 00:05:39.510 --> 00:05:41.340 align:middle line:84% And almost all of the particles which 00:05:41.340 --> 00:05:44.380 align:middle line:84% are colliding with the probe are going to be electrons. 00:05:44.380 --> 00:05:49.870 align:middle line:84% So this means that the probe charge is negative because it's 00:05:49.870 --> 00:05:51.400 align:middle line:90% collecting loads of electrons. 00:05:51.400 --> 00:06:00.524 align:middle line:84% And it keeps going until all the electrons are repelled 00:06:00.524 --> 00:06:04.110 align:middle line:84% or until a vast majority of the electrons are repelled. 00:06:04.110 --> 00:06:06.915 align:middle line:84% And it will keep going until we have no current. 00:06:06.915 --> 00:06:11.830 align:middle line:90% 00:06:11.830 --> 00:06:15.310 align:middle line:84% And by the time that the probe has reached a situation where 00:06:15.310 --> 00:06:18.100 align:middle line:84% there's no total current, we say that the probe 00:06:18.100 --> 00:06:23.717 align:middle line:84% has reached a potential, which we call the floating potential. 00:06:23.717 --> 00:06:25.300 align:middle line:84% And it's called the floating potential 00:06:25.300 --> 00:06:27.520 align:middle line:84% because if we just stick the probe in the plasma, 00:06:27.520 --> 00:06:29.240 align:middle line:90% we don't try to bias it at all. 00:06:29.240 --> 00:06:31.360 align:middle line:84% And we just let its potential adjust until it 00:06:31.360 --> 00:06:33.670 align:middle line:90% reaches this i equals 0 level. 00:06:33.670 --> 00:06:35.170 align:middle line:90% It will reach some potential. 00:06:35.170 --> 00:06:36.820 align:middle line:84% We just let it float to that potential. 00:06:36.820 --> 00:06:38.200 align:middle line:90% It's a floating potential. 00:06:38.200 --> 00:06:40.570 align:middle line:84% But the key thing here is that this is not the same 00:06:40.570 --> 00:06:42.230 align:middle line:90% as the plasma potential. 00:06:42.230 --> 00:06:45.050 align:middle line:84% So the plasma itself is going to be at some other potential, 00:06:45.050 --> 00:06:47.660 align:middle line:90% which we will derive later on. 00:06:47.660 --> 00:06:49.510 align:middle line:84% And so the floating potential, we're 00:06:49.510 --> 00:06:51.550 align:middle line:84% going to call Vf, and the plasma potential we're 00:06:51.550 --> 00:06:53.530 align:middle line:90% going to call Vp here. 00:06:53.530 --> 00:06:55.840 align:middle line:84% And this difference in potentials is due to the fact 00:06:55.840 --> 00:06:58.780 align:middle line:84% that we have perturbed the plasma. 00:06:58.780 --> 00:07:01.440 align:middle line:84% We have stuck some probe into it, 00:07:01.440 --> 00:07:03.620 align:middle line:84% which has changed the nature of the plasma. 00:07:03.620 --> 00:07:04.760 align:middle line:90% It's created this sheath. 00:07:04.760 --> 00:07:06.552 align:middle line:84% It's caused all sorts of electrons and ions 00:07:06.552 --> 00:07:08.240 align:middle line:84% to fly around, charges to fly around. 00:07:08.240 --> 00:07:10.698 align:middle line:84% So there's no good reason for it to have the same potential 00:07:10.698 --> 00:07:13.400 align:middle line:90% as the plasma at all. 00:07:13.400 --> 00:07:15.620 align:middle line:90% OK. 00:07:15.620 --> 00:07:17.055 align:middle line:90% Any questions about this? 00:07:17.055 --> 00:07:27.600 align:middle line:90% 00:07:27.600 --> 00:07:28.100 align:middle line:90% OK. 00:07:28.100 --> 00:07:28.975 align:middle line:90% We're going to go on. 00:07:28.975 --> 00:07:34.850 align:middle line:90% 00:07:34.850 --> 00:07:37.515 align:middle line:84% So what we want to do now our probe 00:07:37.515 --> 00:07:39.640 align:middle line:84% has reached a floating potential is actually change 00:07:39.640 --> 00:07:42.502 align:middle line:84% that potential and measure what current we draw. 00:07:42.502 --> 00:07:44.960 align:middle line:84% We know that, for the floating potential, we get 0 current. 00:07:44.960 --> 00:07:46.832 align:middle line:84% But if we bias the probe in some way, 00:07:46.832 --> 00:07:48.040 align:middle line:90% we'll get different currents. 00:07:48.040 --> 00:07:48.998 align:middle line:90% Nigel, I see your hand. 00:07:48.998 --> 00:07:51.675 align:middle line:90% 00:07:51.675 --> 00:07:52.300 align:middle line:90% AUDIENCE: Yeah. 00:07:52.300 --> 00:07:56.860 align:middle line:84% I wanted to ask, what ground exactly 00:07:56.860 --> 00:07:59.680 align:middle line:84% are we referencing the potential measurements to? 00:07:59.680 --> 00:08:05.290 align:middle line:84% Is it just any random ground or is it anything specific? 00:08:05.290 --> 00:08:07.450 align:middle line:84% JACK HARE: I can give you a short answer, 00:08:07.450 --> 00:08:10.240 align:middle line:84% and I probably won't give you the longer answer. 00:08:10.240 --> 00:08:12.670 align:middle line:84% The short answer is, as you know, 00:08:12.670 --> 00:08:14.845 align:middle line:90% the potential is just relative. 00:08:14.845 --> 00:08:16.720 align:middle line:84% We're only interested in relative potentials. 00:08:16.720 --> 00:08:20.770 align:middle line:84% And it makes sense in this to use the potential of the plasma 00:08:20.770 --> 00:08:25.850 align:middle line:84% very far away as our reference, v equals 0. 00:08:25.850 --> 00:08:26.720 align:middle line:90% AUDIENCE: OK. 00:08:26.720 --> 00:08:28.020 align:middle line:84% JACK HARE: And that's what we're going to do here. 00:08:28.020 --> 00:08:29.000 align:middle line:84% But of course, mathematically, we 00:08:29.000 --> 00:08:30.440 align:middle line:84% could choose any other potential, 00:08:30.440 --> 00:08:32.870 align:middle line:90% and that's absolutely fine. 00:08:32.870 --> 00:08:35.659 align:middle line:84% Another useful ground if you have a probe 00:08:35.659 --> 00:08:39.919 align:middle line:84% stuck inside, for example, a laboratory plasma, 00:08:39.919 --> 00:08:42.260 align:middle line:84% another potential-- you could use as a reference, 00:08:42.260 --> 00:08:45.020 align:middle line:84% but we're not going to-- would be the potential that the vacuum 00:08:45.020 --> 00:08:46.070 align:middle line:90% vessel takes. 00:08:46.070 --> 00:08:48.200 align:middle line:84% And another potential you could use 00:08:48.200 --> 00:08:51.500 align:middle line:84% if you are a Langmuir probe mounted to a satellite flying 00:08:51.500 --> 00:08:53.420 align:middle line:84% through space will be the potential 00:08:53.420 --> 00:08:54.875 align:middle line:90% of the satellite itself. 00:08:54.875 --> 00:08:58.730 align:middle line:90% 00:08:58.730 --> 00:09:02.090 align:middle line:84% I'm just going to mute 218 because it's 00:09:02.090 --> 00:09:03.600 align:middle line:84% very sensitive microphones in there. 00:09:03.600 --> 00:09:04.433 align:middle line:90% Thank you very much. 00:09:04.433 --> 00:09:08.790 align:middle line:84% But please feel free to unmute if you want to ask a question. 00:09:08.790 --> 00:09:09.390 align:middle line:90% OK. 00:09:09.390 --> 00:09:11.098 align:middle line:84% Any other questions before we keep going? 00:09:11.098 --> 00:09:16.070 align:middle line:90% 00:09:16.070 --> 00:09:16.910 align:middle line:90% OK, good stuff. 00:09:16.910 --> 00:09:20.610 align:middle line:90% 00:09:20.610 --> 00:09:22.470 align:middle line:84% So as I was saying, we are now going 00:09:22.470 --> 00:09:26.790 align:middle line:84% to bias our probe, which is effectively 00:09:26.790 --> 00:09:32.440 align:middle line:84% changing the probe potential, V0, 00:09:32.440 --> 00:09:35.140 align:middle line:84% and measure the current that we draw. 00:09:35.140 --> 00:09:38.510 align:middle line:90% 00:09:38.510 --> 00:09:40.580 align:middle line:84% And depending on where we bias this probe, 00:09:40.580 --> 00:09:42.372 align:middle line:84% we're going to end up in different regions. 00:09:42.372 --> 00:09:44.600 align:middle line:84% So we're going to have here as an overview, 00:09:44.600 --> 00:09:46.055 align:middle line:90% a qualitative picture. 00:09:46.055 --> 00:09:47.930 align:middle line:84% And we're going to go and do the calculations 00:09:47.930 --> 00:09:50.430 align:middle line:84% for the quantitative picture later on. 00:09:50.430 --> 00:09:54.830 align:middle line:84% So our qualitative picture, we have an IV curve, 00:09:54.830 --> 00:10:02.120 align:middle line:84% where we have I vertically like this and V along here. 00:10:02.120 --> 00:10:05.210 align:middle line:84% We're going to identify our 0 current 00:10:05.210 --> 00:10:09.140 align:middle line:84% with the floating potential, as we've already done. 00:10:09.140 --> 00:10:10.790 align:middle line:84% We're going to say that somewhere 00:10:10.790 --> 00:10:17.050 align:middle line:84% above the floating potential is the plasma potential. 00:10:17.050 --> 00:10:19.720 align:middle line:84% And it turns out that we end up with an IV 00:10:19.720 --> 00:10:22.330 align:middle line:90% curve that looks like this. 00:10:22.330 --> 00:10:25.560 align:middle line:90% 00:10:25.560 --> 00:10:27.780 align:middle line:84% And there are a few important quantities on here. 00:10:27.780 --> 00:10:31.220 align:middle line:84% There's the electron saturation current. 00:10:31.220 --> 00:10:33.260 align:middle line:84% And I'll just draw this a little bit better 00:10:33.260 --> 00:10:37.140 align:middle line:84% down here for the ions because it showed some asymptote. 00:10:37.140 --> 00:10:40.662 align:middle line:84% There is the ion saturation current down here. 00:10:40.662 --> 00:10:42.870 align:middle line:84% And I'll explain where all of these things came from. 00:10:42.870 --> 00:10:44.703 align:middle line:84% I just want you to have this drawing in mind 00:10:44.703 --> 00:10:46.650 align:middle line:90% while I start explaining it. 00:10:46.650 --> 00:10:50.800 align:middle line:84% So one place we know what the current would be, 00:10:50.800 --> 00:10:52.800 align:middle line:84% or two places we know what the current would be, 00:10:52.800 --> 00:10:55.090 align:middle line:84% are at these specific potentials here. 00:10:55.090 --> 00:10:58.170 align:middle line:84% So at the plasma potential, the current 00:10:58.170 --> 00:11:01.980 align:middle line:84% is going to be entirely due to electrons which 00:11:01.980 --> 00:11:03.270 align:middle line:90% are streaming into the probe. 00:11:03.270 --> 00:11:05.380 align:middle line:84% Because if we're up at the plasma potential, 00:11:05.380 --> 00:11:08.050 align:middle line:84% we know that there will be no ions in general. 00:11:08.050 --> 00:11:11.790 align:middle line:84% So we're just going to end up with this electron saturation 00:11:11.790 --> 00:11:16.050 align:middle line:84% current, 1/4, times the electron charge times the probe area 00:11:16.050 --> 00:11:20.010 align:middle line:84% times the density times by the random electron velocity. 00:11:20.010 --> 00:11:21.130 align:middle line:90% We showed that earlier. 00:11:21.130 --> 00:11:23.850 align:middle line:84% So this is obviously quite a large current. 00:11:23.850 --> 00:11:25.960 align:middle line:84% Another place that we know, as I said, 00:11:25.960 --> 00:11:30.150 align:middle line:84% is at the floating point here, where the current is just 00:11:30.150 --> 00:11:32.730 align:middle line:90% going to be equal to 0. 00:11:32.730 --> 00:11:33.300 align:middle line:90% OK. 00:11:33.300 --> 00:11:35.100 align:middle line:84% But then there are three other regimes 00:11:35.100 --> 00:11:36.820 align:middle line:84% that I want to talk about, and I'm 00:11:36.820 --> 00:11:41.840 align:middle line:84% going to call them A, B, and C here. 00:11:41.840 --> 00:11:46.420 align:middle line:84% So let's start with A. So A is when 00:11:46.420 --> 00:11:49.510 align:middle line:84% we have our probe potential greater than the plasma 00:11:49.510 --> 00:11:50.440 align:middle line:90% potential here. 00:11:50.440 --> 00:11:54.350 align:middle line:84% We're going to attract all the electrons. 00:11:54.350 --> 00:11:57.185 align:middle line:84% We're going to repel all the ions. 00:11:57.185 --> 00:11:59.880 align:middle line:90% 00:11:59.880 --> 00:12:02.150 align:middle line:84% And so we're just going to end up with basically 00:12:02.150 --> 00:12:02.990 align:middle line:90% the same current. 00:12:02.990 --> 00:12:07.350 align:middle line:84% So I is going to be roughly equal to the electron saturation 00:12:07.350 --> 00:12:07.850 align:middle line:90% current. 00:12:07.850 --> 00:12:09.800 align:middle line:84% Now, in general, the current actually 00:12:09.800 --> 00:12:11.340 align:middle line:90% generally starts to rise. 00:12:11.340 --> 00:12:14.217 align:middle line:84% And it still increases with voltage weekly, 00:12:14.217 --> 00:12:16.550 align:middle line:84% for reasons that we're not really going to go into here. 00:12:16.550 --> 00:12:19.640 align:middle line:84% But it's quite difficult to get the mathematics 00:12:19.640 --> 00:12:21.350 align:middle line:90% for this correct. 00:12:21.350 --> 00:12:23.180 align:middle line:84% We don't tend to work in this regime. 00:12:23.180 --> 00:12:24.770 align:middle line:84% We don't tend to bias the probe such 00:12:24.770 --> 00:12:26.360 align:middle line:84% we're drawing the electron saturation current, 00:12:26.360 --> 00:12:28.068 align:middle line:84% and we'll talk a bit about that later on. 00:12:28.068 --> 00:12:30.527 align:middle line:84% And so we're not going to go into too much detail about why 00:12:30.527 --> 00:12:31.070 align:middle line:90% that happens. 00:12:31.070 --> 00:12:33.890 align:middle line:84% But that's why it's that way on the camera. 00:12:33.890 --> 00:12:36.980 align:middle line:90% There's also the regime B here. 00:12:36.980 --> 00:12:43.010 align:middle line:84% This is where V0 is less than the plasma potential 00:12:43.010 --> 00:12:47.940 align:middle line:84% but greater than the floating potential. 00:12:47.940 --> 00:12:50.040 align:middle line:84% And here, we start getting into a region where 00:12:50.040 --> 00:12:59.300 align:middle line:84% some electrons are repelled and some ions are attracted. 00:12:59.300 --> 00:13:01.950 align:middle line:84% And that's because the probe is charging up more negative, 00:13:01.950 --> 00:13:03.980 align:middle line:84% and so it wants to repel things here. 00:13:03.980 --> 00:13:06.500 align:middle line:84% We will go into trying to find an analytical formula 00:13:06.500 --> 00:13:08.500 align:middle line:84% for exactly what this curve looks like, but just 00:13:08.500 --> 00:13:11.570 align:middle line:84% qualitatively, this is why we have a decrease in the current 00:13:11.570 --> 00:13:12.300 align:middle line:90% here. 00:13:12.300 --> 00:13:16.440 align:middle line:84% And then finally, we have this regime, C. And in this regime 00:13:16.440 --> 00:13:20.010 align:middle line:84% here, we have a potential on the probe which is less 00:13:20.010 --> 00:13:21.840 align:middle line:90% than the floating potential. 00:13:21.840 --> 00:13:24.660 align:middle line:84% And we're going to repel all the electrons 00:13:24.660 --> 00:13:30.130 align:middle line:84% and we're going to gather all the ions. 00:13:30.130 --> 00:13:33.920 align:middle line:84% And we will then be drawing the ion saturation current, 00:13:33.920 --> 00:13:39.860 align:middle line:90% which is 1/4 ea ni ui bar. 00:13:39.860 --> 00:13:42.440 align:middle line:84% So this is just the counterpart for the electron saturation 00:13:42.440 --> 00:13:42.940 align:middle line:90% current. 00:13:42.940 --> 00:13:45.020 align:middle line:84% And of course, it's smaller by the square root 00:13:45.020 --> 00:13:47.195 align:middle line:84% of the mass ratio or something like that, probably. 00:13:47.195 --> 00:13:49.820 align:middle line:84% So it's a fair bit smaller than the electron saturation current 00:13:49.820 --> 00:13:51.890 align:middle line:90% here. 00:13:51.890 --> 00:13:54.750 align:middle line:84% Now, the problem with this picture, 00:13:54.750 --> 00:13:57.080 align:middle line:84% and the reason why we can't be more quantitative is 00:13:57.080 --> 00:14:02.810 align:middle line:84% at the moment, all of our solutions-- so ise and isi-- 00:14:02.810 --> 00:14:10.528 align:middle line:84% they're functions of things like ne, ue, and ni, ui. 00:14:10.528 --> 00:14:12.695 align:middle line:84% And they're functions of these at the probe surface. 00:14:12.695 --> 00:14:16.352 align:middle line:90% 00:14:16.352 --> 00:14:18.810 align:middle line:84% That's not particularly useful because what we'd like to do 00:14:18.810 --> 00:14:24.410 align:middle line:84% is relate these quantities to the parameters 00:14:24.410 --> 00:14:28.250 align:middle line:84% inside the plasma-- things like the density and the temperature 00:14:28.250 --> 00:14:29.520 align:middle line:90% and things like that. 00:14:29.520 --> 00:14:32.180 align:middle line:84% And so these measurements of the properties 00:14:32.180 --> 00:14:33.995 align:middle line:84% of the plasma right at the probe surface 00:14:33.995 --> 00:14:35.120 align:middle line:90% aren't particularly useful. 00:14:35.120 --> 00:14:39.400 align:middle line:84% We need to have some theory that links-- 00:14:39.400 --> 00:14:40.090 align:middle line:90% excuse me. 00:14:40.090 --> 00:14:42.670 align:middle line:90% 00:14:42.670 --> 00:14:51.280 align:middle line:84% We need a theory to link the probe quantities 00:14:51.280 --> 00:14:52.990 align:middle line:90% to the plasma quantities. 00:14:52.990 --> 00:14:57.250 align:middle line:90% 00:14:57.250 --> 00:14:59.620 align:middle line:84% And that theory is unfortunately sheath theory. 00:14:59.620 --> 00:15:03.750 align:middle line:90% 00:15:03.750 --> 00:15:06.840 align:middle line:84% So we're going to have a go at deriving quantitatively 00:15:06.840 --> 00:15:08.880 align:middle line:84% some of the quantities here that I've just 00:15:08.880 --> 00:15:10.410 align:middle line:90% sketched very qualitatively. 00:15:10.410 --> 00:15:12.570 align:middle line:84% Before we go on, please ask any questions. 00:15:12.570 --> 00:15:28.490 align:middle line:90% 00:15:28.490 --> 00:15:31.387 align:middle line:84% So who can tell me what the length scale for this sheath is? 00:15:31.387 --> 00:15:33.470 align:middle line:84% What's an important length scale in sheath theory? 00:15:33.470 --> 00:15:41.090 align:middle line:90% 00:15:41.090 --> 00:15:43.682 align:middle line:90% Yeah, just please shout it out. 00:15:43.682 --> 00:15:44.640 align:middle line:90% AUDIENCE: Debye length? 00:15:44.640 --> 00:15:45.900 align:middle line:90% JACK HARE: Yeah, Debye length. 00:15:45.900 --> 00:15:48.650 align:middle line:90% 00:15:48.650 --> 00:15:50.400 align:middle line:84% I never quite remember how to spell Debye. 00:15:50.400 --> 00:15:51.858 align:middle line:84% I may have put an extra E in there. 00:15:51.858 --> 00:15:53.550 align:middle line:90% Maybe this one doesn't exist. 00:15:53.550 --> 00:15:54.180 align:middle line:90% OK. 00:15:54.180 --> 00:15:56.970 align:middle line:84% So this is often written as lambda Debye. 00:15:56.970 --> 00:16:04.590 align:middle line:84% It's equal to, in SI at least, epsilon 0 te over e squared ne 00:16:04.590 --> 00:16:06.270 align:middle line:90% to the 1/2. 00:16:06.270 --> 00:16:09.425 align:middle line:84% It's an important length scale for sheaths. 00:16:09.425 --> 00:16:12.430 align:middle line:90% 00:16:12.430 --> 00:16:15.490 align:middle line:84% But as you probably know, it's not 00:16:15.490 --> 00:16:19.570 align:middle line:84% in, for example, MHD, or other simpler, 00:16:19.570 --> 00:16:21.253 align:middle line:90% fluid pictures of the plasma. 00:16:21.253 --> 00:16:23.920 align:middle line:84% It's just important to sometimes realize the limitations of some 00:16:23.920 --> 00:16:25.045 align:middle line:90% of the other models we use. 00:16:25.045 --> 00:16:28.090 align:middle line:84% So you can't model how a Langmuir probe works using MHD. 00:16:28.090 --> 00:16:30.180 align:middle line:90% It just doesn't work. 00:16:30.180 --> 00:16:32.480 align:middle line:84% And what is important about the Debye length? 00:16:32.480 --> 00:16:34.970 align:middle line:84% What length scale does it represent? 00:16:34.970 --> 00:16:36.830 align:middle line:84% What's a physical intuition we have for it? 00:16:36.830 --> 00:16:40.740 align:middle line:90% 00:16:40.740 --> 00:16:43.260 align:middle line:84% AUDIENCE: Distance over which voltages are shielded. 00:16:43.260 --> 00:16:45.300 align:middle line:90% JACK HARE: Yeah. 00:16:45.300 --> 00:16:51.570 align:middle line:84% I guess we can say within a few lambda Debye, 00:16:51.570 --> 00:16:58.740 align:middle line:90% the probe perturbation vanishes. 00:16:58.740 --> 00:17:00.410 align:middle line:84% So although we've stuck this probe in, 00:17:00.410 --> 00:17:02.930 align:middle line:84% and it's going to be at a very different potential 00:17:02.930 --> 00:17:05.390 align:middle line:84% to the plasma, the plasma won't see it 00:17:05.390 --> 00:17:08.790 align:middle line:90% over a few lambda Debye or so. 00:17:08.790 --> 00:17:11.520 align:middle line:84% And if we take some cold plasma-- 00:17:11.520 --> 00:17:13.910 align:middle line:84% so let's say it's got a temperature of about 00:17:13.910 --> 00:17:17.690 align:middle line:84% one electron volt. It's got a density of around 10 00:17:17.690 --> 00:17:20.250 align:middle line:90% to the 17 per meter cubed. 00:17:20.250 --> 00:17:23.599 align:middle line:84% This will give us a Debye length on the order 00:17:23.599 --> 00:17:27.270 align:middle line:90% of 20 micrometers here. 00:17:27.270 --> 00:17:30.020 align:middle line:84% And so for any reasonable probe which 00:17:30.020 --> 00:17:33.530 align:middle line:84% we could construct which has some characteristic length 00:17:33.530 --> 00:17:37.850 align:middle line:84% scale here of A, this Debye length 00:17:37.850 --> 00:17:41.420 align:middle line:84% is going to be very, very, very small. 00:17:41.420 --> 00:17:45.380 align:middle line:84% So we can say that lambda Debye divided by A-- 00:17:45.380 --> 00:17:47.450 align:middle line:84% lambda Debye is much less than A. 00:17:47.450 --> 00:17:49.550 align:middle line:84% And this leads us to being able to do 00:17:49.550 --> 00:17:54.040 align:middle line:84% in most of our sheath theory what we call a quasi planar 00:17:54.040 --> 00:17:56.010 align:middle line:90% assumption. 00:17:56.010 --> 00:17:58.890 align:middle line:84% So we don't have to worry about the shape of the probe. 00:17:58.890 --> 00:18:01.870 align:middle line:84% We just worry about some infinite plane. 00:18:01.870 --> 00:18:05.460 align:middle line:84% And so we can treat this like a one-dimensional situation. 00:18:05.460 --> 00:18:07.980 align:middle line:84% You will see the sheath equations are complicated enough 00:18:07.980 --> 00:18:09.330 align:middle line:90% without doing it in 1D. 00:18:09.330 --> 00:18:13.050 align:middle line:84% So this is a nice simplification to be able to do. 00:18:13.050 --> 00:18:15.870 align:middle line:84% So the sheath equations to solve this properly-- 00:18:15.870 --> 00:18:19.650 align:middle line:90% 00:18:19.650 --> 00:18:24.420 align:middle line:84% well, you need to start doing things involving Gauss's 00:18:24.420 --> 00:18:25.800 align:middle line:90% law and the Poisson equation. 00:18:25.800 --> 00:18:29.102 align:middle line:90% 00:18:29.102 --> 00:18:31.560 align:middle line:84% And it turns out that when you start putting this properly, 00:18:31.560 --> 00:18:34.290 align:middle line:90% these equations are non-linear. 00:18:34.290 --> 00:18:38.160 align:middle line:84% You're going to need a Boltzmann factor inside here as well-- 00:18:38.160 --> 00:18:42.970 align:middle line:84% minus eV over Te and stuff like that. 00:18:42.970 --> 00:18:44.640 align:middle line:84% So these equations look very non-linear. 00:18:44.640 --> 00:18:45.807 align:middle line:90% They've got exponentials in. 00:18:45.807 --> 00:18:48.510 align:middle line:84% They've got gradients in, all sorts of stuff like that. 00:18:48.510 --> 00:18:51.700 align:middle line:84% And that means they're very complicated. 00:18:51.700 --> 00:18:55.440 align:middle line:84% They're still complicated with big assumptions. 00:18:55.440 --> 00:18:58.950 align:middle line:90% 00:18:58.950 --> 00:19:02.690 align:middle line:84% So one assumption that we almost always make in sheath theory 00:19:02.690 --> 00:19:04.970 align:middle line:90% is that the ions are cold. 00:19:04.970 --> 00:19:06.967 align:middle line:90% They've got a temperature of 0. 00:19:06.967 --> 00:19:09.050 align:middle line:84% And obviously, that may not be very well justified 00:19:09.050 --> 00:19:09.930 align:middle line:90% in your plasma. 00:19:09.930 --> 00:19:11.960 align:middle line:84% But it allows you to make a lot of progress. 00:19:11.960 --> 00:19:13.940 align:middle line:84% This actually is relatively well justified 00:19:13.940 --> 00:19:15.680 align:middle line:84% in the sorts of low temperature plasmas 00:19:15.680 --> 00:19:17.360 align:middle line:84% you tend to stick Langmuir probes in. 00:19:17.360 --> 00:19:19.152 align:middle line:84% But I'm sure that you can think of a plasma 00:19:19.152 --> 00:19:20.780 align:middle line:84% where the ion temperature is not 0 00:19:20.780 --> 00:19:23.577 align:middle line:84% or isn't much less than the electron temperature. 00:19:23.577 --> 00:19:25.160 align:middle line:84% But what I'm saying is that even if we 00:19:25.160 --> 00:19:27.077 align:middle line:84% put in these very big simplifying assumptions, 00:19:27.077 --> 00:19:29.100 align:middle line:84% the sheath equations are still very complicated. 00:19:29.100 --> 00:19:31.730 align:middle line:84% And so what I'm not going to do is give you 00:19:31.730 --> 00:19:33.920 align:middle line:84% a complete derivation of the sheath equation. 00:19:33.920 --> 00:19:36.545 align:middle line:84% This is what I did last time I taught the course two years ago, 00:19:36.545 --> 00:19:39.350 align:middle line:84% and I've never seen a class lose interest more quickly. 00:19:39.350 --> 00:19:40.703 align:middle line:90% It was a complete disaster. 00:19:40.703 --> 00:19:42.870 align:middle line:84% So I'm going to give you a very high level overview. 00:19:42.870 --> 00:19:44.537 align:middle line:84% I'm going to tell you the results of it. 00:19:44.537 --> 00:19:47.810 align:middle line:84% And I understand that might be unsatisfactory. 00:19:47.810 --> 00:19:49.580 align:middle line:84% And so I encourage you to go look 00:19:49.580 --> 00:19:52.820 align:middle line:84% at Hutchinson's book or Lieberman and Lichtenberg 00:19:52.820 --> 00:19:55.153 align:middle line:84% and learn a little bit more about it if you want to. 00:19:55.153 --> 00:19:57.320 align:middle line:84% At the end of the day, I think these Langmuir probes 00:19:57.320 --> 00:19:58.580 align:middle line:90% are a tool. 00:19:58.580 --> 00:20:01.070 align:middle line:84% We should understand how they work in some limitations, 00:20:01.070 --> 00:20:03.650 align:middle line:84% but we just want to know what the answer is. 00:20:03.650 --> 00:20:05.400 align:middle line:84% So we're going to plow on a little bit. 00:20:05.400 --> 00:20:07.730 align:middle line:84% Any questions on any of this before we keep going? 00:20:07.730 --> 00:20:16.660 align:middle line:90% 00:20:16.660 --> 00:20:19.560 align:middle line:90% OK. 00:20:19.560 --> 00:20:28.530 align:middle line:84% So this is my one-page brief overview of sheath physics. 00:20:28.530 --> 00:20:33.010 align:middle line:90% 00:20:33.010 --> 00:20:36.540 align:middle line:84% What I want you to bear in mind is we are trying to get-- 00:20:36.540 --> 00:20:41.100 align:middle line:84% what we want out of this are the electron density 00:20:41.100 --> 00:20:47.610 align:middle line:84% and the electron velocity at the probe and the ion density 00:20:47.610 --> 00:20:52.710 align:middle line:84% and the ion velocity at the probe, 00:20:52.710 --> 00:20:59.210 align:middle line:84% in terms of the density far away from the probe, 00:20:59.210 --> 00:21:03.130 align:middle line:84% which we'll call as infinity, and things like the temperature 00:21:03.130 --> 00:21:04.683 align:middle line:90% of the plasma here. 00:21:04.683 --> 00:21:06.100 align:middle line:84% So this is what we're going after. 00:21:06.100 --> 00:21:08.775 align:middle line:84% So whenever I write down an equation, think to yourself, 00:21:08.775 --> 00:21:12.410 align:middle line:84% did that help or did it not help? 00:21:12.410 --> 00:21:13.160 align:middle line:90% OK. 00:21:13.160 --> 00:21:16.790 align:middle line:84% So I am going to draw a little sketch, which 00:21:16.790 --> 00:21:19.970 align:middle line:84% is going to take me a little while to get right here. 00:21:19.970 --> 00:21:22.940 align:middle line:84% On the right-hand side of our domain, 00:21:22.940 --> 00:21:24.800 align:middle line:84% we're going to have a wall, which 00:21:24.800 --> 00:21:27.710 align:middle line:90% is our quasi-planar probe here. 00:21:27.710 --> 00:21:30.950 align:middle line:84% And we're going to have a coordinate pointing 00:21:30.950 --> 00:21:34.920 align:middle line:84% from left to right, which is the x-coordinate here. 00:21:34.920 --> 00:21:38.380 align:middle line:84% And there are a few important places on this x-coordinate. 00:21:38.380 --> 00:21:42.170 align:middle line:84% There's infinity, which is very far 00:21:42.170 --> 00:21:45.030 align:middle line:90% away in the unperturbed plasma. 00:21:45.030 --> 00:21:51.800 align:middle line:84% There is a point that I'm going to label s here, 00:21:51.800 --> 00:21:54.680 align:middle line:84% and there's a point that I'm going to label 0. 00:21:54.680 --> 00:21:58.350 align:middle line:90% So 0 is the probe. 00:21:58.350 --> 00:22:03.330 align:middle line:84% This region between s and 0 is the sheath. 00:22:03.330 --> 00:22:08.160 align:middle line:84% This is the imaginatively named presheath. 00:22:08.160 --> 00:22:10.980 align:middle line:90% and this is the plasma. 00:22:10.980 --> 00:22:16.610 align:middle line:84% So again, what we want to do is take measurements at the probe 00:22:16.610 --> 00:22:20.060 align:middle line:84% and link them back to the properties in the bulk plasma 00:22:20.060 --> 00:22:22.430 align:middle line:84% without worrying about what this sheath is doing. 00:22:22.430 --> 00:22:25.020 align:middle line:84% But what the sheath is doing is very, very important, 00:22:25.020 --> 00:22:26.990 align:middle line:84% so we need to work that out first. 00:22:26.990 --> 00:22:30.830 align:middle line:84% So first of all, I'm going to sketch the voltage. 00:22:30.830 --> 00:22:34.850 align:middle line:84% And so this voltage is going to be flat in the plasma, 00:22:34.850 --> 00:22:39.200 align:middle line:84% and it's going to have a value of 0. 00:22:39.200 --> 00:22:43.040 align:middle line:84% And we are just defining the voltage in the plasma as 0 00:22:43.040 --> 00:22:45.050 align:middle line:84% because that's our reference voltage here. 00:22:45.050 --> 00:22:49.490 align:middle line:84% The voltage is going to drop off ever so slightly 00:22:49.490 --> 00:22:50.840 align:middle line:90% in the presheath here. 00:22:50.840 --> 00:22:52.610 align:middle line:84% And I'm really exaggerating, but it's 00:22:52.610 --> 00:22:54.710 align:middle line:84% going to drop off a little bit in the presheath. 00:22:54.710 --> 00:22:59.420 align:middle line:84% And then it's going to drop down to whatever potential 00:22:59.420 --> 00:23:04.580 align:middle line:84% it is at the probe here, which is the bias point 00:23:04.580 --> 00:23:05.780 align:middle line:90% that we put our probe at. 00:23:05.780 --> 00:23:10.280 align:middle line:84% And we call that V0, and we call the potential 00:23:10.280 --> 00:23:13.940 align:middle line:84% at the edge of the sheath Vs V sheath. 00:23:13.940 --> 00:23:17.423 align:middle line:84% And I could write V infinity here, 00:23:17.423 --> 00:23:19.340 align:middle line:84% but again, we've decided that's going to be 0. 00:23:19.340 --> 00:23:21.240 align:middle line:84% So we're going to measure everything with respect to that. 00:23:21.240 --> 00:23:23.157 align:middle line:84% And you'll see in the picture I've drawn here, 00:23:23.157 --> 00:23:26.510 align:middle line:84% everything is negative with respect to that. 00:23:26.510 --> 00:23:27.080 align:middle line:90% OK. 00:23:27.080 --> 00:23:28.520 align:middle line:90% The other thing I can draw-- 00:23:28.520 --> 00:23:32.570 align:middle line:84% and you can imagine this is a separate axis down below here-- 00:23:32.570 --> 00:23:33.960 align:middle line:90% is the density. 00:23:33.960 --> 00:23:38.690 align:middle line:84% So out in the plasma, we have a density of ne, 00:23:38.690 --> 00:23:43.860 align:middle line:84% and that's equal to ni, and that's equal to n infinity. 00:23:43.860 --> 00:23:47.370 align:middle line:84% So I've invoked quasi neutrality inside my plasma here. 00:23:47.370 --> 00:23:49.080 align:middle line:84% There's no change in the potential, 00:23:49.080 --> 00:23:51.120 align:middle line:84% so there's no change in the electric field, 00:23:51.120 --> 00:23:54.410 align:middle line:84% and so there's no difference in the number of charge carriers. 00:23:54.410 --> 00:23:56.300 align:middle line:84% And I'm assuming that z equals 1 here. 00:23:56.300 --> 00:23:58.405 align:middle line:90% This is a hydrogenic plasma. 00:23:58.405 --> 00:23:59.780 align:middle line:84% Just for simplicity, of course, I 00:23:59.780 --> 00:24:03.080 align:middle line:84% can put z back in if you want to make life more complicated. 00:24:03.080 --> 00:24:06.110 align:middle line:84% Then in the presheath region, this is a little bit subtle. 00:24:06.110 --> 00:24:08.760 align:middle line:84% We are saying, yeah, the potential is changing, 00:24:08.760 --> 00:24:10.730 align:middle line:90% but it's not changing that much. 00:24:10.730 --> 00:24:12.750 align:middle line:90% It's quasi-neutral-ish. 00:24:12.750 --> 00:24:16.550 align:middle line:84% So the density is going to-- turns out it drops a little bit. 00:24:16.550 --> 00:24:20.750 align:middle line:84% But we're still going to say ne is roughly ni, 00:24:20.750 --> 00:24:25.018 align:middle line:84% but it's now no longer n infinity. 00:24:25.018 --> 00:24:26.560 align:middle line:84% Again, we can do that more rigorously 00:24:26.560 --> 00:24:28.180 align:middle line:90% if you look in the book. 00:24:28.180 --> 00:24:30.530 align:middle line:84% For now, we're just going to go with it. 00:24:30.530 --> 00:24:34.330 align:middle line:84% And then finally, this density splits 00:24:34.330 --> 00:24:39.507 align:middle line:84% so that we get the ion density and the electron density. 00:24:39.507 --> 00:24:41.590 align:middle line:84% And this is happening because in this region where 00:24:41.590 --> 00:24:43.238 align:middle line:84% the potential is changing very fast, 00:24:43.238 --> 00:24:44.530 align:middle line:90% we have strong electric fields. 00:24:44.530 --> 00:24:46.240 align:middle line:84% We no longer have quasi neutrality. 00:24:46.240 --> 00:24:48.590 align:middle line:84% We actually have electrons being repelled 00:24:48.590 --> 00:24:49.690 align:middle line:90% so there's fewer of them. 00:24:49.690 --> 00:24:53.750 align:middle line:90% And we have plenty of ions here. 00:24:53.750 --> 00:24:58.000 align:middle line:84% And so this picture is obviously applying in the region down 00:24:58.000 --> 00:25:00.730 align:middle line:84% where our probe is biased low enough 00:25:00.730 --> 00:25:03.850 align:middle line:84% that we're repelling electrons and we're gathering ions. 00:25:03.850 --> 00:25:08.120 align:middle line:84% And I'll justify that in a moment. 00:25:08.120 --> 00:25:10.240 align:middle line:90% So this sheath forms. 00:25:10.240 --> 00:25:15.650 align:middle line:90% 00:25:15.650 --> 00:25:18.670 align:middle line:90% Let me just think about this. 00:25:18.670 --> 00:25:33.710 align:middle line:84% So we only get a sheath when our V0 at the probe is less than 00:25:33.710 --> 00:25:37.490 align:middle line:90% minus Te over 2e. 00:25:37.490 --> 00:25:39.440 align:middle line:90% E So that's T subscript e. 00:25:39.440 --> 00:25:42.370 align:middle line:84% I'll try and write that a little bit more clearly. 00:25:42.370 --> 00:25:44.853 align:middle line:84% This is the first result that is completely non-obvious. 00:25:44.853 --> 00:25:46.020 align:middle line:90% I have not derived for this. 00:25:46.020 --> 00:25:47.340 align:middle line:90% This is just true. 00:25:47.340 --> 00:25:49.170 align:middle line:84% So we might think, where are we going 00:25:49.170 --> 00:25:51.300 align:middle line:84% to get some of these potentials in our system? 00:25:51.300 --> 00:25:52.800 align:middle line:90% What does our system have? 00:25:52.800 --> 00:25:55.248 align:middle line:84% Well, the only thing it's really got is a temperature. 00:25:55.248 --> 00:25:57.540 align:middle line:84% And we often measure the temperature in electron volts, 00:25:57.540 --> 00:25:59.400 align:middle line:84% so you might be thinking to yourself, a-ha. 00:25:59.400 --> 00:26:01.380 align:middle line:84% Maybe the voltages that are going 00:26:01.380 --> 00:26:03.240 align:middle line:84% to show up in this solution are going 00:26:03.240 --> 00:26:04.948 align:middle line:84% to do with the temperature of our plasma. 00:26:04.948 --> 00:26:08.070 align:middle line:84% And you're right, but the factor of 2, that's not at all obvious. 00:26:08.070 --> 00:26:11.700 align:middle line:90% It's just you can work it out. 00:26:11.700 --> 00:26:13.230 align:middle line:84% And so we only get a sheath forming 00:26:13.230 --> 00:26:16.140 align:middle line:84% when we bias the probe sufficiently negative, 00:26:16.140 --> 00:26:20.590 align:middle line:84% and that is because the voltage at the sheath-- 00:26:20.590 --> 00:26:26.640 align:middle line:84% so this Vs here is exactly minus Te over 2e. 00:26:26.640 --> 00:26:29.580 align:middle line:84% And so if you bias your probe above that potential, 00:26:29.580 --> 00:26:30.960 align:middle line:90% you don't get a sheath forming. 00:26:30.960 --> 00:26:33.990 align:middle line:84% All of this sheath physics falls apart here. 00:26:33.990 --> 00:26:38.550 align:middle line:84% And later on, we're going to show that Vf-- 00:26:38.550 --> 00:26:41.220 align:middle line:84% let me think how to write this properly. 00:26:41.220 --> 00:26:48.130 align:middle line:84% We're going to show that V0 minus Vf is about 3Te or so. 00:26:48.130 --> 00:26:52.410 align:middle line:84% So this means that our probe has to be biased somewhere 00:26:52.410 --> 00:26:57.840 align:middle line:84% around about Vf, or at least significantly lower 00:26:57.840 --> 00:26:59.530 align:middle line:90% than the plasma potential here. 00:26:59.530 --> 00:27:01.920 align:middle line:84% So we're going to be operating near the floating point. 00:27:01.920 --> 00:27:03.360 align:middle line:84% And there's other good reasons why 00:27:03.360 --> 00:27:05.700 align:middle line:84% we want to operate near that floating potential as 00:27:05.700 --> 00:27:08.190 align:middle line:90% well, but skipping ahead. 00:27:08.190 --> 00:27:09.730 align:middle line:90% That's just a preview there. 00:27:09.730 --> 00:27:13.170 align:middle line:84% So that's the first fact that we now know, 00:27:13.170 --> 00:27:19.640 align:middle line:84% is that our sheath is going to be at minus Te over 2e like 00:27:19.640 --> 00:27:20.425 align:middle line:90% that. 00:27:20.425 --> 00:27:21.800 align:middle line:84% The next thing we're going to say 00:27:21.800 --> 00:27:25.655 align:middle line:84% is that the ions are accelerated across the sheath. 00:27:25.655 --> 00:27:30.150 align:middle line:90% 00:27:30.150 --> 00:27:32.390 align:middle line:84% So although we know the potential at the sheath, 00:27:32.390 --> 00:27:34.640 align:middle line:84% the potential at the probe is still arbitrary, 00:27:34.640 --> 00:27:37.010 align:middle line:84% so this isn't particularly useful. 00:27:37.010 --> 00:27:40.100 align:middle line:84% But we can make use of the continuity equation 00:27:40.100 --> 00:27:42.990 align:middle line:84% to actually make some rather powerful arguments here. 00:27:42.990 --> 00:27:47.780 align:middle line:84% So continuity is going to say that the flux of ions which 00:27:47.780 --> 00:27:50.960 align:middle line:84% are hitting the probe, which is the quantity that we're 00:27:50.960 --> 00:27:52.200 align:middle line:90% measuring with our probe. 00:27:52.200 --> 00:27:55.970 align:middle line:90% So this is ni at 0, ui at 0. 00:27:55.970 --> 00:27:59.250 align:middle line:84% That is simply going to be equal to the flux of ions 00:27:59.250 --> 00:28:01.950 align:middle line:84% which is crossing the sheath boundary here. 00:28:01.950 --> 00:28:05.480 align:middle line:84% So ni like this and ui like this. 00:28:05.480 --> 00:28:07.527 align:middle line:84% This actually turns out to be very, very powerful 00:28:07.527 --> 00:28:09.110 align:middle line:84% because it means we don't need to know 00:28:09.110 --> 00:28:11.780 align:middle line:84% what the density and the velocity are at the probe 00:28:11.780 --> 00:28:12.440 align:middle line:90% itself. 00:28:12.440 --> 00:28:15.170 align:middle line:84% We just need to know what they are at the sheath. 00:28:15.170 --> 00:28:17.270 align:middle line:84% And the reason that's useful is because we 00:28:17.270 --> 00:28:20.810 align:middle line:84% can write the density at the sheath 00:28:20.810 --> 00:28:24.050 align:middle line:84% here as roughly equal to the electron density of the sheath. 00:28:24.050 --> 00:28:29.060 align:middle line:84% Remember, we're in this quasi-neutral regime 00:28:29.060 --> 00:28:30.060 align:middle line:90% down to here. 00:28:30.060 --> 00:28:31.640 align:middle line:84% And so at this point, we're saying, 00:28:31.640 --> 00:28:33.762 align:middle line:84% eh, the potential drop is small enough 00:28:33.762 --> 00:28:34.970 align:middle line:90% that it will cause a neutral. 00:28:34.970 --> 00:28:36.950 align:middle line:84% So we can still use quasi neutrality 00:28:36.950 --> 00:28:39.770 align:middle line:84% to write the ion density in terms of electron density. 00:28:39.770 --> 00:28:43.180 align:middle line:84% And the electrons we're going to treat with a Boltzmann factor 00:28:43.180 --> 00:28:43.680 align:middle line:90% here. 00:28:43.680 --> 00:28:46.097 align:middle line:84% So it's going to have the density that they would have out 00:28:46.097 --> 00:28:50.450 align:middle line:84% in the bulk plasma, but reduced by a factor of exponential e 00:28:50.450 --> 00:28:54.300 align:middle line:90% times the sheath over Te here. 00:28:54.300 --> 00:28:57.900 align:middle line:84% And if we plug the potential of the sheath in here, 00:28:57.900 --> 00:29:03.650 align:middle line:84% we'll get n at infinity, exponential of minus 1/2. 00:29:03.650 --> 00:29:06.380 align:middle line:84% And if you're wondering what exponential of minus 1/2 is, 00:29:06.380 --> 00:29:08.558 align:middle line:90% it's about 0.61. 00:29:08.558 --> 00:29:10.100 align:middle line:84% I'm going to leave it as exponential, 00:29:10.100 --> 00:29:12.170 align:middle line:84% as minus 1/2 in all of these calculations, 00:29:12.170 --> 00:29:15.140 align:middle line:84% just so you don't ask, where did 0.61 come from? 00:29:15.140 --> 00:29:19.558 align:middle line:84% But it is just 0.61, so it's about a half. 00:29:19.558 --> 00:29:21.350 align:middle line:84% I'm just going to keep pushing on with this 00:29:21.350 --> 00:29:23.433 align:middle line:84% and then I'll take questions, because some of them 00:29:23.433 --> 00:29:24.250 align:middle line:90% may become clear. 00:29:24.250 --> 00:29:27.070 align:middle line:84% The next thing we're going to do is conservation of energy. 00:29:27.070 --> 00:29:31.230 align:middle line:90% 00:29:31.230 --> 00:29:33.660 align:middle line:84% And we're going to assume that our ions start off 00:29:33.660 --> 00:29:35.400 align:middle line:90% with a velocity of 0. 00:29:35.400 --> 00:29:38.250 align:middle line:90% So let me rewrite that. 00:29:38.250 --> 00:29:41.290 align:middle line:90% I'll move this over here. 00:29:41.290 --> 00:29:46.750 align:middle line:84% I say that my ion velocity at infinity is equal to 0. 00:29:46.750 --> 00:29:48.640 align:middle line:84% So this is the cold ion approximation 00:29:48.640 --> 00:29:49.820 align:middle line:90% I talked about earlier. 00:29:49.820 --> 00:29:52.160 align:middle line:84% We're going to be accelerating these ions up. 00:29:52.160 --> 00:29:57.340 align:middle line:84% And so then we can say that the kinetic energy of the ions 00:29:57.340 --> 00:30:03.520 align:middle line:84% at the sheath-- so 1/2 mi ui of s squared is just going to be 00:30:03.520 --> 00:30:06.260 align:middle line:84% equal to the change in potential energy here, 00:30:06.260 --> 00:30:09.220 align:middle line:84% which is e times the sheath potential. 00:30:09.220 --> 00:30:11.470 align:middle line:84% Remember, we're referencing V at infinity 00:30:11.470 --> 00:30:14.510 align:middle line:84% is 0, so there should be a minus V at infinity here. 00:30:14.510 --> 00:30:15.520 align:middle line:90% But that disappears. 00:30:15.520 --> 00:30:16.030 align:middle line:90% Yeah, Nigel. 00:30:16.030 --> 00:30:18.520 align:middle line:90% I see your hand. 00:30:18.520 --> 00:30:21.340 align:middle line:84% AUDIENCE: Are those dashes bullet points or is that a minus 00:30:21.340 --> 00:30:21.840 align:middle line:90% 1/2? 00:30:21.840 --> 00:30:23.465 align:middle line:84% JACK HARE: Those are all bullet points. 00:30:23.465 --> 00:30:23.978 align:middle line:90% I apologize. 00:30:23.978 --> 00:30:24.520 align:middle line:90% AUDIENCE: OK. 00:30:24.520 --> 00:30:24.820 align:middle line:90% Thanks. 00:30:24.820 --> 00:30:25.970 align:middle line:90% JACK HARE: Yeah, no worries. 00:30:25.970 --> 00:30:26.845 align:middle line:90% It's a good question. 00:30:26.845 --> 00:30:29.520 align:middle line:90% 00:30:29.520 --> 00:30:30.020 align:middle line:90% Ta-da. 00:30:30.020 --> 00:30:31.910 align:middle line:84% Any remaining minuses are your problem. 00:30:31.910 --> 00:30:32.660 align:middle line:90% OK? 00:30:32.660 --> 00:30:33.705 align:middle line:90% Good. 00:30:33.705 --> 00:30:35.330 align:middle line:84% Any other questions while we're paused? 00:30:35.330 --> 00:30:41.570 align:middle line:90% 00:30:41.570 --> 00:30:42.890 align:middle line:90% OK. 00:30:42.890 --> 00:30:50.420 align:middle line:84% So from this, we can then infer the velocity at the sheath is 00:30:50.420 --> 00:30:55.970 align:middle line:84% going to be equal to the square root of Te over mI to the 1/2, 00:30:55.970 --> 00:30:59.750 align:middle line:84% where, again, I have just plugged in our sheath potential 00:30:59.750 --> 00:31:03.300 align:middle line:90% into here. 00:31:03.300 --> 00:31:04.870 align:middle line:90% OK. 00:31:04.870 --> 00:31:06.250 align:middle line:90% This is rather good. 00:31:06.250 --> 00:31:09.130 align:middle line:90% We're almost there. 00:31:09.130 --> 00:31:11.170 align:middle line:84% The final thing we might want to know 00:31:11.170 --> 00:31:16.670 align:middle line:84% is what is the velocity of the electrons at the probe surface? 00:31:16.670 --> 00:31:21.750 align:middle line:84% So this would be ne of 0, and that 00:31:21.750 --> 00:31:29.010 align:middle line:84% would be equal to n infinity exponential minus e V0 over Te. 00:31:29.010 --> 00:31:30.360 align:middle line:90% So that's going to depend. 00:31:30.360 --> 00:31:32.010 align:middle line:84% In this case, for the electrons, we're 00:31:32.010 --> 00:31:36.450 align:middle line:84% going to make it depend exactly on the probe 00:31:36.450 --> 00:31:39.090 align:middle line:84% potential for the ions we made use of this trick using 00:31:39.090 --> 00:31:41.550 align:middle line:84% the continuity equation to avoid actually knowing what 00:31:41.550 --> 00:31:44.130 align:middle line:90% their density was at the probe. 00:31:44.130 --> 00:31:47.010 align:middle line:84% Because it's clearly no longer equal to the electrons, and so 00:31:47.010 --> 00:31:49.710 align:middle line:84% we can't use the electron Boltzmann equation of state 00:31:49.710 --> 00:31:51.100 align:middle line:90% in order to get it. 00:31:51.100 --> 00:31:54.570 align:middle line:84% So we're going to put this together on the next slide, 00:31:54.570 --> 00:31:56.340 align:middle line:84% but I just want to point out right 00:31:56.340 --> 00:32:00.390 align:middle line:84% now, I think we have now written all of the quantities 00:32:00.390 --> 00:32:02.310 align:middle line:84% for the electrons and ions at the probe, 00:32:02.310 --> 00:32:05.620 align:middle line:84% in terms of quantities in the bulk of the plasma. 00:32:05.620 --> 00:32:07.950 align:middle line:84% So this is extremely powerful because we can now 00:32:07.950 --> 00:32:08.700 align:middle line:90% use that relation. 00:32:08.700 --> 00:32:10.242 align:middle line:84% And I saw a comment in the chat here, 00:32:10.242 --> 00:32:12.170 align:middle line:90% which I'm just going to read. 00:32:12.170 --> 00:32:15.350 align:middle line:84% We're assuming T is constant through the sheath. 00:32:15.350 --> 00:32:17.840 align:middle line:84% We are assuming that T is constant through the sheath. 00:32:17.840 --> 00:32:19.530 align:middle line:90% How is this apparent or valid? 00:32:19.530 --> 00:32:21.320 align:middle line:84% For example, keeping pressure constant. 00:32:21.320 --> 00:32:23.820 align:middle line:84% I believe the electric field will contribute to the pressure 00:32:23.820 --> 00:32:25.970 align:middle line:84% balance here, and so you don't have 00:32:25.970 --> 00:32:28.700 align:middle line:84% to worry about just the thermal pressure being involved. 00:32:28.700 --> 00:32:32.290 align:middle line:84% We've got electric fields involved here as well. 00:32:32.290 --> 00:32:33.790 align:middle line:90% OK. 00:32:33.790 --> 00:32:35.232 align:middle line:90% Any other questions? 00:32:35.232 --> 00:32:42.470 align:middle line:90% 00:32:42.470 --> 00:32:43.010 align:middle line:90% OK. 00:32:43.010 --> 00:32:47.340 align:middle line:90% So now if we have our probe-- 00:32:47.340 --> 00:32:54.800 align:middle line:84% so this is for V0 less than Te over 2e, 00:32:54.800 --> 00:32:57.650 align:middle line:84% which is the condition that all the sheath theory requires. 00:32:57.650 --> 00:33:03.600 align:middle line:84% And just remember, V0 here is V of the probe. 00:33:03.600 --> 00:33:06.810 align:middle line:84% Hutchinson starts using Vp here to mean 00:33:06.810 --> 00:33:09.420 align:middle line:84% the voltage of the probe, but then you might also 00:33:09.420 --> 00:33:11.070 align:middle line:84% think that's the voltage of the plasma, 00:33:11.070 --> 00:33:12.550 align:middle line:90% and I find that very confusing. 00:33:12.550 --> 00:33:14.760 align:middle line:90% So I'm writing it as V0 here. 00:33:14.760 --> 00:33:18.060 align:middle line:84% But we can now say that the current that we draw-- 00:33:18.060 --> 00:33:24.180 align:middle line:90% so I drawn is-- 00:33:24.180 --> 00:33:26.400 align:middle line:84% and we can split it up into contributions 00:33:26.400 --> 00:33:28.270 align:middle line:90% from the ion current. 00:33:28.270 --> 00:33:32.940 align:middle line:84% So this is just going to be equal to minus e times 00:33:32.940 --> 00:33:40.980 align:middle line:84% the area of the ions are passing through, ni of 0, ui of 0. 00:33:40.980 --> 00:33:46.620 align:middle line:84% And we can just replace that with ni of s, ui of s, 00:33:46.620 --> 00:33:47.355 align:middle line:90% using continuity. 00:33:47.355 --> 00:33:49.142 align:middle line:84% And we have that on the previous page. 00:33:49.142 --> 00:33:51.600 align:middle line:84% And I just want to point out, when we previously wrote down 00:33:51.600 --> 00:33:54.450 align:middle line:84% this equation, we had a factor of 1/4. 00:33:54.450 --> 00:33:57.400 align:middle line:84% That 1/4 has disappeared because, in fact, 00:33:57.400 --> 00:34:00.370 align:middle line:84% we have no random motion of our ions anymore. 00:34:00.370 --> 00:34:02.910 align:middle line:84% We set their initial temperature equal to 0. 00:34:02.910 --> 00:34:05.490 align:middle line:90% And so this is now all directed. 00:34:05.490 --> 00:34:07.770 align:middle line:84% So our geometric factor is not irrelevant 00:34:07.770 --> 00:34:10.179 align:middle line:84% because now all the ions are going in the same direction. 00:34:10.179 --> 00:34:12.750 align:middle line:84% So previously, it was 1/4 to account for random motion. 00:34:12.750 --> 00:34:18.630 align:middle line:84% Now it's just 1 because they're directed like this. 00:34:18.630 --> 00:34:21.980 align:middle line:84% And that means we can write down the ion current as minus 00:34:21.980 --> 00:34:26.360 align:middle line:84% the exponential of minus 1/2, which is just 0.61. 00:34:26.360 --> 00:34:31.670 align:middle line:84% The area times the electron charge times the electron ion 00:34:31.670 --> 00:34:35.370 align:middle line:84% density of infinity times the incoming velocity, 00:34:35.370 --> 00:34:40.412 align:middle line:84% which is Te over mi to the power of 1/2. 00:34:40.412 --> 00:34:42.820 align:middle line:84% I just want to make a quick point that I'm not 00:34:42.820 --> 00:34:44.050 align:middle line:90% going to labor too much. 00:34:44.050 --> 00:34:49.670 align:middle line:84% This area here is technically the area of the sheath. 00:34:49.670 --> 00:34:52.699 align:middle line:84% However, that is roughly the area 00:34:52.699 --> 00:34:58.580 align:middle line:84% of the probe with corrections on the order of lambda B over A. 00:34:58.580 --> 00:35:02.210 align:middle line:84% And we found that lambda tends to be very, very small. 00:35:02.210 --> 00:35:05.360 align:middle line:84% Lambda over A tends to be very, very small, 00:35:05.360 --> 00:35:07.583 align:middle line:84% so we don't really worry about those corrections. 00:35:07.583 --> 00:35:09.750 align:middle line:84% But if you end up in a regime where that's not true, 00:35:09.750 --> 00:35:12.590 align:middle line:84% you might want to worry about this a little bit more. 00:35:12.590 --> 00:35:15.140 align:middle line:90% So that's the ion current. 00:35:15.140 --> 00:35:16.370 align:middle line:90% Any questions on that? 00:35:16.370 --> 00:35:21.480 align:middle line:90% 00:35:21.480 --> 00:35:25.680 align:middle line:84% The electron current now is going to be a factor of 1/4 00:35:25.680 --> 00:35:28.320 align:middle line:84% coming back because the electrons are randomly moving. 00:35:28.320 --> 00:35:30.810 align:middle line:84% Some of them are being repelled, but in general, we're 00:35:30.810 --> 00:35:34.020 align:middle line:84% just dealing with a random Maxwellian distribution 00:35:34.020 --> 00:35:34.770 align:middle line:90% function. 00:35:34.770 --> 00:35:38.820 align:middle line:84% And then we have these factors of e and A. 00:35:38.820 --> 00:35:42.060 align:middle line:84% And then we have the density at the probe 00:35:42.060 --> 00:35:44.880 align:middle line:84% and the velocity at the probe here. 00:35:44.880 --> 00:35:47.500 align:middle line:90% 00:35:47.500 --> 00:35:53.140 align:middle line:84% This A is genuinely the area of the probe 00:35:53.140 --> 00:35:57.460 align:middle line:84% because we're working with the x-coordinate set to 0 00:35:57.460 --> 00:35:58.450 align:middle line:90% at the probe. 00:35:58.450 --> 00:36:01.390 align:middle line:84% We worked out what the density was of a Boltzmann factor. 00:36:01.390 --> 00:36:04.180 align:middle line:84% The thing that we didn't have is this. 00:36:04.180 --> 00:36:06.220 align:middle line:84% And this is just going to be Maxwellian. 00:36:06.220 --> 00:36:09.775 align:middle line:90% 00:36:09.775 --> 00:36:11.150 align:middle line:84% And so we use the standard result 00:36:11.150 --> 00:36:12.900 align:middle line:84% from a Maxwellian temperature here. 00:36:12.900 --> 00:36:19.430 align:middle line:84% So this all comes out as 1/4 times e n at infinity 00:36:19.430 --> 00:36:24.702 align:middle line:90% exponential of e V0 over Te. 00:36:24.702 --> 00:36:26.660 align:middle line:84% And then there's this factor of the Maxwellian, 00:36:26.660 --> 00:36:34.160 align:middle line:84% which is 2 times 2Te over pi me to the 1/2. 00:36:34.160 --> 00:36:36.350 align:middle line:84% So it's slightly ugly, but that's 00:36:36.350 --> 00:36:39.720 align:middle line:84% the average or the mean velocity in a Maxwellian distribution 00:36:39.720 --> 00:36:40.220 align:middle line:90% here. 00:36:40.220 --> 00:36:47.093 align:middle line:90% 00:36:47.093 --> 00:36:48.510 align:middle line:84% And then we can put this together. 00:36:48.510 --> 00:36:50.802 align:middle line:84% We can say that the total current that our probe is now 00:36:50.802 --> 00:36:54.330 align:middle line:84% drawing is equal to the ion current plus the electron 00:36:54.330 --> 00:36:54.990 align:middle line:90% current. 00:36:54.990 --> 00:36:58.470 align:middle line:84% And this is Hutchinson's equation. 00:36:58.470 --> 00:37:00.980 align:middle line:90% 00:37:00.980 --> 00:37:02.280 align:middle line:90% Gosh. 00:37:02.280 --> 00:37:04.520 align:middle line:84% I didn't update my notes, so I don't have it. 00:37:04.520 --> 00:37:06.860 align:middle line:84% But where is Hutchinson when you need him? 00:37:06.860 --> 00:37:19.050 align:middle line:90% 00:37:19.050 --> 00:37:21.480 align:middle line:84% I don't have it straight away, unfortunately here. 00:37:21.480 --> 00:37:25.560 align:middle line:84% My memory of it is it's equation 3.2.23. 00:37:25.560 --> 00:37:27.350 align:middle line:84% And perhaps someone could check that. 00:37:27.350 --> 00:37:29.234 align:middle line:90% AUDIENCE: I think it's 2.29. 00:37:29.234 --> 00:37:31.110 align:middle line:90% JACK HARE: 3.2.29? 00:37:31.110 --> 00:37:31.860 align:middle line:90% AUDIENCE: Yeah. 00:37:31.860 --> 00:37:33.152 align:middle line:90% JACK HARE: Thank you very much. 00:37:33.152 --> 00:37:34.350 align:middle line:90% I appreciate it. 00:37:34.350 --> 00:37:34.980 align:middle line:90% OK. 00:37:34.980 --> 00:37:37.000 align:middle line:90% And it's this long thing. 00:37:37.000 --> 00:37:39.152 align:middle line:84% So the reason I'm giving you the equation citation 00:37:39.152 --> 00:37:40.860 align:middle line:84% is if I make a mistake copying this down, 00:37:40.860 --> 00:37:42.730 align:middle line:84% you should go refer to Hutchinson instead. 00:37:42.730 --> 00:37:48.540 align:middle line:84% So what I've got is n infinity, electron charge, 00:37:48.540 --> 00:37:56.670 align:middle line:84% area of the probe, square root of electron temperature 00:37:56.670 --> 00:37:58.690 align:middle line:90% of the ion mass. 00:37:58.690 --> 00:38:01.630 align:middle line:84% And all of this is now times a component to do with 00:38:01.630 --> 00:38:12.250 align:middle line:84% the electrons, which is 1/2 times 2 mi over pi me to the 1/2 00:38:12.250 --> 00:38:23.170 align:middle line:84% exponential e V0 over Te minus the ion contribution, 00:38:23.170 --> 00:38:27.250 align:middle line:84% which is exponential of minus 1/2. 00:38:27.250 --> 00:38:32.590 align:middle line:84% So this is e minus, and this is the ions here. 00:38:32.590 --> 00:38:34.225 align:middle line:84% So this looks a little bit complicated, 00:38:34.225 --> 00:38:35.850 align:middle line:84% but let's have a little think about it. 00:38:35.850 --> 00:38:40.980 align:middle line:90% This ratio, mi over me is large. 00:38:40.980 --> 00:38:43.050 align:middle line:84% That just depends on fundamental quantities. 00:38:43.050 --> 00:38:47.190 align:middle line:84% But this exponential here is going to be small because we 00:38:47.190 --> 00:38:53.490 align:middle line:84% know that V0 is less than Vs. So it's less than minus Te over 2e. 00:38:53.490 --> 00:38:56.760 align:middle line:84% So this quantity here is going to be less than 0.6. 00:38:56.760 --> 00:39:00.870 align:middle line:84% And although this is large, it's large like 40 or so. 00:39:00.870 --> 00:39:02.490 align:middle line:84% And we can imagine if we decrease 00:39:02.490 --> 00:39:04.500 align:middle line:84% the voltage on the probe even more, 00:39:04.500 --> 00:39:06.090 align:middle line:84% the small thing is going to be smaller 00:39:06.090 --> 00:39:07.410 align:middle line:90% than the large thing is large. 00:39:07.410 --> 00:39:09.360 align:middle line:84% And we can see how the electron current could 00:39:09.360 --> 00:39:11.580 align:middle line:84% be completely extinguished and we're 00:39:11.580 --> 00:39:13.920 align:middle line:90% left with just the ion current. 00:39:13.920 --> 00:39:16.510 align:middle line:90% Yes, Sean. 00:39:16.510 --> 00:39:18.160 align:middle line:84% AUDIENCE: In Hutchinson's version, 00:39:18.160 --> 00:39:21.280 align:middle line:84% he has the ion term multiplied by the sheath 00:39:21.280 --> 00:39:23.950 align:middle line:90% area times the probe area. 00:39:23.950 --> 00:39:25.660 align:middle line:84% Are you just saying that that's about 1? 00:39:25.660 --> 00:39:26.930 align:middle line:90% JACK HARE: Yeah, exactly. 00:39:26.930 --> 00:39:28.240 align:middle line:90% So he puts in this ratio. 00:39:28.240 --> 00:39:29.692 align:middle line:84% And like I said, it is important. 00:39:29.692 --> 00:39:32.150 align:middle line:84% But it's one of the things I'm glossing over in this class. 00:39:32.150 --> 00:39:33.130 align:middle line:90% But you're quite right. 00:39:33.130 --> 00:39:34.540 align:middle line:90% That's there. 00:39:34.540 --> 00:39:37.300 align:middle line:84% I just said, area of the sheath or the area of the probe 00:39:37.300 --> 00:39:39.100 align:middle line:90% equal to unity. 00:39:39.100 --> 00:39:40.180 align:middle line:90% AUDIENCE: Got it, thanks. 00:39:40.180 --> 00:39:41.480 align:middle line:90% JACK HARE: Cool. 00:39:41.480 --> 00:39:42.410 align:middle line:90% Any other questions? 00:39:42.410 --> 00:39:44.510 align:middle line:84% This is our big result. But we're now 00:39:44.510 --> 00:39:46.595 align:middle line:84% going to go use this to learn about what 00:39:46.595 --> 00:39:47.720 align:middle line:90% happens to Langmuir probes. 00:39:47.720 --> 00:39:50.210 align:middle line:90% Nigel? 00:39:50.210 --> 00:39:54.940 align:middle line:84% AUDIENCE: Have we discussed how you find the area of the sheath? 00:39:54.940 --> 00:39:57.790 align:middle line:90% Or is that just a theoretical? 00:39:57.790 --> 00:39:59.320 align:middle line:84% JACK HARE: So what I've said here 00:39:59.320 --> 00:40:01.120 align:middle line:84% is that the area of the sheath is basically 00:40:01.120 --> 00:40:02.830 align:middle line:90% the area of the probe. 00:40:02.830 --> 00:40:05.308 align:middle line:84% AUDIENCE: Yeah, I mean beyond that. 00:40:05.308 --> 00:40:06.850 align:middle line:84% JACK HARE: We haven't discussed that. 00:40:06.850 --> 00:40:09.003 align:middle line:84% There's a long discussion in Hutchinson about it. 00:40:09.003 --> 00:40:10.420 align:middle line:84% Effectively, there are corrections 00:40:10.420 --> 00:40:12.410 align:middle line:84% on the order of the Debye length to it, 00:40:12.410 --> 00:40:15.270 align:middle line:84% which may or may not be important. 00:40:15.270 --> 00:40:18.320 align:middle line:84% And it turns out if you bias the probe very negative, 00:40:18.320 --> 00:40:20.660 align:middle line:84% deep into the ion saturation region, 00:40:20.660 --> 00:40:22.800 align:middle line:84% the sheath actually continues to grow. 00:40:22.800 --> 00:40:25.700 align:middle line:84% And so you continue to gather more and more ion current. 00:40:25.700 --> 00:40:27.920 align:middle line:84% And that's one of the reasons it's actually very hard 00:40:27.920 --> 00:40:32.030 align:middle line:84% to measure the ion saturation current because it's not 00:40:32.030 --> 00:40:33.140 align:middle line:90% like a flat asymptote. 00:40:33.140 --> 00:40:35.610 align:middle line:84% It's just a slowly growing function. 00:40:35.610 --> 00:40:38.480 align:middle line:84% So if you ever sit down and work with Langmuir probe data, any 00:40:38.480 --> 00:40:40.520 align:middle line:84% of you, you'll find it's incredibly frustrating 00:40:40.520 --> 00:40:43.130 align:middle line:84% because all the stuff in the textbooks doesn't really work. 00:40:43.130 --> 00:40:46.610 align:middle line:84% And last time we did this, we did have some Langmuir probe 00:40:46.610 --> 00:40:47.577 align:middle line:90% data to analyze. 00:40:47.577 --> 00:40:49.160 align:middle line:84% But you know, I thought you guys would 00:40:49.160 --> 00:40:51.868 align:middle line:84% prefer to have full problem sets in 5, and so I cut it this time. 00:40:51.868 --> 00:40:54.110 align:middle line:84% If anyone's like, give me that Langmuir probe data, 00:40:54.110 --> 00:40:55.280 align:middle line:90% I'll happily send it across. 00:40:55.280 --> 00:40:56.447 align:middle line:90% You can have a play with it. 00:40:56.447 --> 00:40:58.730 align:middle line:84% It's from Dionysus, which is Kevin Waller's 00:40:58.730 --> 00:41:01.370 align:middle line:84% machine across the street in NW14 or whatever. 00:41:01.370 --> 00:41:04.300 align:middle line:90% 00:41:04.300 --> 00:41:05.020 align:middle line:90% OK. 00:41:05.020 --> 00:41:08.202 align:middle line:84% So let's just check that this long equation matches what we 00:41:08.202 --> 00:41:09.410 align:middle line:90% thought we were going to get. 00:41:09.410 --> 00:41:15.790 align:middle line:84% So if we put v zero much, much less than minus t over 2e, 00:41:15.790 --> 00:41:19.300 align:middle line:84% we're going to get no electron currents. 00:41:19.300 --> 00:41:21.940 align:middle line:84% And that's due to this exponential 00:41:21.940 --> 00:41:24.760 align:middle line:90% here is just going to go to 0. 00:41:24.760 --> 00:41:28.385 align:middle line:84% And so I is going to be roughly equal to the ion saturation 00:41:28.385 --> 00:41:28.885 align:middle line:90% current. 00:41:28.885 --> 00:41:31.660 align:middle line:84% And the ion saturation current, we can now see, 00:41:31.660 --> 00:41:38.650 align:middle line:84% is the density at infinity e times the area of the probe Te 00:41:38.650 --> 00:41:45.820 align:middle line:84% over mi to the 1/2 exponential minus 1/2. 00:41:45.820 --> 00:41:48.400 align:middle line:84% So that's cool because we know all of those parameters, 00:41:48.400 --> 00:41:50.200 align:middle line:84% apart from the density and the temperature. 00:41:50.200 --> 00:41:52.280 align:middle line:84% And those are things we want to measure. 00:41:52.280 --> 00:41:54.623 align:middle line:84% So we can't work out from the ion saturation current 00:41:54.623 --> 00:41:56.290 align:middle line:84% exactly the density and the temperature, 00:41:56.290 --> 00:41:58.768 align:middle line:84% but we know the density times the square root 00:41:58.768 --> 00:41:59.560 align:middle line:90% of the temperature. 00:41:59.560 --> 00:42:02.740 align:middle line:84% So maybe if we can find out the temperature from somewhere else, 00:42:02.740 --> 00:42:04.520 align:middle line:84% we can crack this whole thing open. 00:42:04.520 --> 00:42:08.720 align:middle line:90% 00:42:08.720 --> 00:42:10.220 align:middle line:84% I keep clicking on the wrong monitor 00:42:10.220 --> 00:42:11.540 align:middle line:84% because I've got basically the same thing shown 00:42:11.540 --> 00:42:12.260 align:middle line:90% on three monitors. 00:42:12.260 --> 00:42:14.177 align:middle line:84% So now I've moved on to my next set of slides, 00:42:14.177 --> 00:42:16.122 align:middle line:84% but before we go on to interpreting the probes 00:42:16.122 --> 00:42:18.080 align:middle line:84% and actually measuring density and temperature, 00:42:18.080 --> 00:42:20.765 align:middle line:84% does anyone have any further questions on this key result? 00:42:20.765 --> 00:42:31.520 align:middle line:90% 00:42:31.520 --> 00:42:32.030 align:middle line:90% All right. 00:42:32.030 --> 00:42:35.370 align:middle line:90% 00:42:35.370 --> 00:42:40.695 align:middle line:84% So now we're going to go on to interpretation of probes. 00:42:40.695 --> 00:42:43.930 align:middle line:90% 00:42:43.930 --> 00:42:46.540 align:middle line:84% So let's have a little drawing of our probe. 00:42:46.540 --> 00:42:52.930 align:middle line:84% It's going to have some tip like this. 00:42:52.930 --> 00:42:57.098 align:middle line:84% That tip is going to be surrounded by some insulator, 00:42:57.098 --> 00:42:59.140 align:middle line:84% and then that, in turn, is going to be surrounded 00:42:59.140 --> 00:43:02.150 align:middle line:90% by some sort of shield. 00:43:02.150 --> 00:43:04.660 align:middle line:90% So this is a cutaway here. 00:43:04.660 --> 00:43:12.350 align:middle line:84% It's a cutaway of coaxial geometry insulator and tip. 00:43:12.350 --> 00:43:16.270 align:middle line:84% And if I draw this face on, it looks like, as I said, 00:43:16.270 --> 00:43:19.365 align:middle line:90% coaxial something like this. 00:43:19.365 --> 00:43:21.803 align:middle line:90% 00:43:21.803 --> 00:43:23.970 align:middle line:84% So you can have a lot of fun making Langmuir probes. 00:43:23.970 --> 00:43:25.950 align:middle line:84% But they are fundamentally little bits 00:43:25.950 --> 00:43:27.450 align:middle line:90% of cut-off coax cable. 00:43:27.450 --> 00:43:31.830 align:middle line:84% And there's some reasons why you might want to use them made out 00:43:31.830 --> 00:43:32.920 align:middle line:90% of certain materials. 00:43:32.920 --> 00:43:37.500 align:middle line:84% So we tend to want to work near the floating potential. 00:43:37.500 --> 00:43:40.940 align:middle line:90% So we want to have the Vf-- 00:43:40.940 --> 00:43:43.445 align:middle line:84% our pro voltage be 0 close to Vf. 00:43:43.445 --> 00:43:44.820 align:middle line:84% And there's two reasons for that. 00:43:44.820 --> 00:43:46.987 align:middle line:84% The first of all, as we've seen, is that we actually 00:43:46.987 --> 00:43:48.260 align:middle line:90% have good theory here. 00:43:48.260 --> 00:43:49.580 align:middle line:90% We have the sheath theory. 00:43:49.580 --> 00:43:50.750 align:middle line:84% We don't really know what the theory 00:43:50.750 --> 00:43:51.875 align:middle line:90% looks like in other places. 00:43:51.875 --> 00:43:54.620 align:middle line:84% Maybe we can go work it out, maybe we couldn't. 00:43:54.620 --> 00:43:57.450 align:middle line:84% So it's nice to work near the floating potential. 00:43:57.450 --> 00:44:00.710 align:middle line:84% The other reason is that it limits currents. 00:44:00.710 --> 00:44:03.950 align:middle line:90% 00:44:03.950 --> 00:44:05.420 align:middle line:84% And actually, this is a big problem 00:44:05.420 --> 00:44:09.260 align:middle line:84% because if you draw a large current, you get large heating. 00:44:09.260 --> 00:44:14.630 align:middle line:84% Or let's say this avoids heating. 00:44:14.630 --> 00:44:16.970 align:middle line:84% Because if you're sticking your probe inside a plasma, 00:44:16.970 --> 00:44:18.260 align:middle line:90% it's already going to get hot. 00:44:18.260 --> 00:44:20.280 align:middle line:84% And if you start drawing a large amount of current, 00:44:20.280 --> 00:44:21.350 align:middle line:90% it's going to get even hotter. 00:44:21.350 --> 00:44:23.660 align:middle line:84% So you don't really want to work in the electron saturation 00:44:23.660 --> 00:44:25.368 align:middle line:84% current region where the current is huge. 00:44:25.368 --> 00:44:27.620 align:middle line:84% You want to work down where the current is basically 00:44:27.620 --> 00:44:29.690 align:middle line:90% 0 to avoid melting this. 00:44:29.690 --> 00:44:31.190 align:middle line:84% Even then, you're going to be making 00:44:31.190 --> 00:44:33.680 align:middle line:84% the tip of this out of something like tungsten, which 00:44:33.680 --> 00:44:35.060 align:middle line:90% has a very high melting point. 00:44:35.060 --> 00:44:37.880 align:middle line:84% Your insulator is going to be some custom ceramic, 00:44:37.880 --> 00:44:40.230 align:middle line:84% and your shield will probably be tungsten as well. 00:44:40.230 --> 00:44:43.370 align:middle line:84% So you have to design these things very, very carefully not 00:44:43.370 --> 00:44:45.500 align:middle line:90% to melt. 00:44:45.500 --> 00:44:46.440 align:middle line:90% Cool. 00:44:46.440 --> 00:44:53.990 align:middle line:84% So what we're going to do is we're going to sweep V0 near Vf. 00:44:53.990 --> 00:44:55.730 align:middle line:84% So you can imagine that we have control 00:44:55.730 --> 00:44:59.300 align:middle line:84% over the potential of our probe with some power supply, 00:44:59.300 --> 00:45:02.040 align:middle line:84% and we measure the current through that. 00:45:02.040 --> 00:45:05.240 align:middle line:84% So this little circuit is we've got our probe 00:45:05.240 --> 00:45:08.200 align:middle line:90% and our little plasma. 00:45:08.200 --> 00:45:11.530 align:middle line:84% And we've got some sort of resistor over which we measure 00:45:11.530 --> 00:45:16.360 align:middle line:84% the current, and we have some sort of bias 00:45:16.360 --> 00:45:17.920 align:middle line:90% that we're biasing the probe to. 00:45:17.920 --> 00:45:23.140 align:middle line:84% So this is I0, and this is V0 here. 00:45:23.140 --> 00:45:25.280 align:middle line:84% And we're going to be near the floating potential. 00:45:25.280 --> 00:45:28.180 align:middle line:84% And if we go back to the equation that we had back here, 00:45:28.180 --> 00:45:30.460 align:middle line:84% and we set the current equal to 0-- 00:45:30.460 --> 00:45:34.600 align:middle line:84% so if we say I equals 0, we can rearrange this. 00:45:34.600 --> 00:45:40.960 align:middle line:84% And we can find out that the electron charge times 00:45:40.960 --> 00:45:43.960 align:middle line:84% the floating potential, which are operating here over 00:45:43.960 --> 00:45:48.760 align:middle line:84% the electron temperature, is going to be equal to 1/2 times 00:45:48.760 --> 00:45:58.530 align:middle line:84% the natural logarithm of 2 pi me over mi minus 1, 00:45:58.530 --> 00:45:59.310 align:middle line:90% all in brackets. 00:45:59.310 --> 00:45:59.580 align:middle line:90% OK. 00:45:59.580 --> 00:46:01.680 align:middle line:84% So this is a vaguely complicated looking expression, 00:46:01.680 --> 00:46:03.390 align:middle line:84% but of course the only things that matter in here 00:46:03.390 --> 00:46:05.160 align:middle line:84% are the electron mass and the ion mass. 00:46:05.160 --> 00:46:07.530 align:middle line:84% And if you put those into something like hydrogen, 00:46:07.530 --> 00:46:11.083 align:middle line:90% you get about 3 here. 00:46:11.083 --> 00:46:12.750 align:middle line:84% And of course, it doesn't matter if it's 00:46:12.750 --> 00:46:13.600 align:middle line:84% not really hydrogen because we've 00:46:13.600 --> 00:46:14.808 align:middle line:90% got a natural logarithm here. 00:46:14.808 --> 00:46:16.980 align:middle line:90% So it changes slowly. 00:46:16.980 --> 00:46:19.170 align:middle line:84% The point here is that this says that 00:46:19.170 --> 00:46:21.680 align:middle line:90% your floating potential, Vf-- 00:46:21.680 --> 00:46:25.400 align:middle line:90% 00:46:25.400 --> 00:46:30.330 align:middle line:84% e times Vf, at least is about 3 temperature units. 00:46:30.330 --> 00:46:32.810 align:middle line:84% So if your temperature is one electron volt, 00:46:32.810 --> 00:46:38.190 align:middle line:84% then your floating potential is about 3 volts. 00:46:38.190 --> 00:46:41.370 align:middle line:84% That sounds super useful, but it's not. 00:46:41.370 --> 00:46:42.360 align:middle line:90% Why is it not useful? 00:46:42.360 --> 00:46:55.980 align:middle line:90% 00:46:55.980 --> 00:46:59.387 align:middle line:84% We've measured Vf by sweeping it there, and now from this, 00:46:59.387 --> 00:47:00.470 align:middle line:90% we've got the temperature. 00:47:00.470 --> 00:47:02.140 align:middle line:90% This is incredible. 00:47:02.140 --> 00:47:05.380 align:middle line:84% AUDIENCE: Yeah, but doesn't that just cancel itself back out 00:47:05.380 --> 00:47:06.130 align:middle line:90% in that equation? 00:47:06.130 --> 00:47:09.120 align:middle line:90% 00:47:09.120 --> 00:47:11.580 align:middle line:84% JACK HARE: I'm not sure it does that. 00:47:11.580 --> 00:47:14.280 align:middle line:84% Maybe you're thinking about this in the right way. 00:47:14.280 --> 00:47:16.297 align:middle line:84% The question would be Vf relative to what? 00:47:16.297 --> 00:47:17.880 align:middle line:84% We had this whole debate at the start. 00:47:17.880 --> 00:47:20.530 align:middle line:90% 00:47:20.530 --> 00:47:27.490 align:middle line:84% So if you measure Vf in your lab as 10 volts, 00:47:27.490 --> 00:47:30.270 align:middle line:84% but you don't know what V in the plasma is 00:47:30.270 --> 00:47:33.000 align:middle line:84% so you don't know whether the 10 volts that you're measuring-- 00:47:33.000 --> 00:47:34.545 align:middle line:90% what's that reference to? 00:47:34.545 --> 00:47:36.420 align:middle line:84% And that's reference to your building ground. 00:47:36.420 --> 00:47:38.520 align:middle line:84% But you need to have Vf reference 00:47:38.520 --> 00:47:41.220 align:middle line:90% to the plasma potential. 00:47:41.220 --> 00:47:43.060 align:middle line:84% And you don't know the plasma potential. 00:47:43.060 --> 00:47:45.400 align:middle line:84% So this is actually pretty useless. 00:47:45.400 --> 00:47:54.770 align:middle line:84% So without the plasma, we have no reference for Vf. 00:47:54.770 --> 00:47:55.730 align:middle line:90% This is useless. 00:47:55.730 --> 00:48:00.010 align:middle line:90% 00:48:00.010 --> 00:48:01.410 align:middle line:90% You could try and do it. 00:48:01.410 --> 00:48:03.450 align:middle line:84% If you remember, if I go all the way 00:48:03.450 --> 00:48:07.450 align:middle line:84% back in my notes to my little sketch here-- you could say, 00:48:07.450 --> 00:48:08.020 align:middle line:90% well, look. 00:48:08.020 --> 00:48:09.600 align:middle line:90% I can measure Vf. 00:48:09.600 --> 00:48:12.520 align:middle line:84% Down here is where the current goes to 0. 00:48:12.520 --> 00:48:17.420 align:middle line:84% And then I could measure the volts between Vf and Vp here, 00:48:17.420 --> 00:48:18.580 align:middle line:90% V plasma. 00:48:18.580 --> 00:48:22.060 align:middle line:84% And I'll define the plasma as when the electron saturation 00:48:22.060 --> 00:48:25.270 align:middle line:84% current happens-- so when this curve rolls over and starts 00:48:25.270 --> 00:48:27.400 align:middle line:84% to saturate, I'll call that V plasma. 00:48:27.400 --> 00:48:30.850 align:middle line:84% And the difference here, that'll be delta Vf. 00:48:30.850 --> 00:48:33.920 align:middle line:84% And that'll be 3 times the electron temperature. 00:48:33.920 --> 00:48:35.620 align:middle line:84% So you can do it crudely like that. 00:48:35.620 --> 00:48:38.020 align:middle line:84% The trouble is that as I've sketched it here, 00:48:38.020 --> 00:48:39.340 align:middle line:90% this doesn't really roll off. 00:48:39.340 --> 00:48:41.313 align:middle line:84% It actually just keeps going up like that. 00:48:41.313 --> 00:48:42.730 align:middle line:84% And if you look at some real data, 00:48:42.730 --> 00:48:44.240 align:middle line:84% it's usually even worse than that. 00:48:44.240 --> 00:48:47.260 align:middle line:84% So it's a super inaccurate way of measuring the temperature. 00:48:47.260 --> 00:48:48.280 align:middle line:90% But you could use this. 00:48:48.280 --> 00:48:48.780 align:middle line:90% It's fine. 00:48:48.780 --> 00:48:50.363 align:middle line:84% The other problem, of course, is you'd 00:48:50.363 --> 00:48:52.720 align:middle line:84% have to bias your probe into electron saturation region 00:48:52.720 --> 00:48:53.830 align:middle line:90% and risk it melting. 00:48:53.830 --> 00:48:57.040 align:middle line:84% So good reasons why you wouldn't want to do this technique, 00:48:57.040 --> 00:48:59.920 align:middle line:84% but maybe you want to if you're desperate. 00:48:59.920 --> 00:49:02.260 align:middle line:84% But don't worry, there's still better things we can do. 00:49:02.260 --> 00:49:12.110 align:middle line:84% So again, we could use Vp at I equals the electron saturation 00:49:12.110 --> 00:49:15.500 align:middle line:90% current, but that's inaccurate. 00:49:15.500 --> 00:49:20.380 align:middle line:90% 00:49:20.380 --> 00:49:22.020 align:middle line:90% So we don't tend to do that. 00:49:22.020 --> 00:49:25.020 align:middle line:84% What we do instead is an alternative technique 00:49:25.020 --> 00:49:28.110 align:middle line:84% where we look at this equation very hard 00:49:28.110 --> 00:49:30.160 align:middle line:90% and we think to ourselves, hm. 00:49:30.160 --> 00:49:33.910 align:middle line:90% How does I change with V0 here? 00:49:33.910 --> 00:49:41.120 align:middle line:84% So we can write down analytically dI dV0. 00:49:41.120 --> 00:49:43.790 align:middle line:84% And maybe you want to do this around 00:49:43.790 --> 00:49:45.325 align:middle line:84% about the floating potential here, 00:49:45.325 --> 00:49:47.450 align:middle line:84% because that's where we want to operate where we've 00:49:47.450 --> 00:49:48.992 align:middle line:84% got all our nice sheath theory, where 00:49:48.992 --> 00:49:50.480 align:middle line:90% we've got our nice equations. 00:49:50.480 --> 00:49:52.400 align:middle line:84% And we find out that what we get out 00:49:52.400 --> 00:49:59.720 align:middle line:84% is e over Te times I minus the ion saturation current here. 00:49:59.720 --> 00:50:01.520 align:middle line:84% Now, when you do this, you also get 00:50:01.520 --> 00:50:03.300 align:middle line:90% a little term that looks like-- 00:50:03.300 --> 00:50:06.170 align:middle line:84% how does the ion saturation current change 00:50:06.170 --> 00:50:07.890 align:middle line:90% with the probe potential? 00:50:07.890 --> 00:50:11.960 align:middle line:84% And this is due to the fact that the area of the sheath 00:50:11.960 --> 00:50:15.440 align:middle line:84% is not exactly equal to the area of the probe. 00:50:15.440 --> 00:50:17.120 align:middle line:84% But for our purposes, we're going 00:50:17.120 --> 00:50:19.290 align:middle line:90% to assume that is negligible. 00:50:19.290 --> 00:50:23.070 align:middle line:84% So we're just going to set that extra term equal to 0 here. 00:50:23.070 --> 00:50:24.940 align:middle line:84% And if you look at the first term, 00:50:24.940 --> 00:50:26.690 align:middle line:84% and you think about how to rearrange this, 00:50:26.690 --> 00:50:29.720 align:middle line:84% you can see that the electron temperature then 00:50:29.720 --> 00:50:34.190 align:middle line:84% can be written as e times I minus I sI, 00:50:34.190 --> 00:50:42.760 align:middle line:84% and that is divided by dI d0, like this. 00:50:42.760 --> 00:50:50.670 align:middle line:84% And so if we look again at our plot of our ID characteristic 00:50:50.670 --> 00:50:56.490 align:middle line:84% for this probe, where we have the floating potential here, 00:50:56.490 --> 00:51:04.080 align:middle line:84% and we have the plasma potential up here 00:51:04.080 --> 00:51:08.310 align:middle line:84% and the electron saturation and the iron saturation, 00:51:08.310 --> 00:51:13.900 align:middle line:84% we're working with the slope of the curve in this point here. 00:51:13.900 --> 00:51:16.020 align:middle line:84% And what we want to do is a two-step process 00:51:16.020 --> 00:51:21.420 align:middle line:84% where, first of all, we fit the natural logarithm of I minus I 00:51:21.420 --> 00:51:28.130 align:middle line:84% Is versus V0, and that is going to give us the electron 00:51:28.130 --> 00:51:29.242 align:middle line:90% temperature. 00:51:29.242 --> 00:51:31.700 align:middle line:84% So you can probably just stare at the equation for electron 00:51:31.700 --> 00:51:33.630 align:middle line:84% temperature and see where that comes from. 00:51:33.630 --> 00:51:35.930 align:middle line:84% And then once you've got the electron temperature, 00:51:35.930 --> 00:51:41.430 align:middle line:84% we can measure the ion saturation as well. 00:51:41.430 --> 00:51:44.960 align:middle line:84% And we know that the ion saturation current is equal 00:51:44.960 --> 00:51:55.550 align:middle line:84% to exponential of minus 1/2 e A0 n infinity Te over mi 00:51:55.550 --> 00:51:56.600 align:middle line:90% to the 1/2. 00:51:56.600 --> 00:51:59.730 align:middle line:84% We've just measured Te from this, 00:51:59.730 --> 00:52:02.100 align:middle line:90% so now we can measure density. 00:52:02.100 --> 00:52:04.340 align:middle line:84% So we need to measure the slope here, 00:52:04.340 --> 00:52:07.280 align:middle line:84% and we need to measure the ion saturation current. 00:52:07.280 --> 00:52:14.890 align:middle line:84% And we can do that just by sweeping V0 in this narrow range 00:52:14.890 --> 00:52:17.200 align:middle line:84% here from the ion saturation current up to just 00:52:17.200 --> 00:52:19.090 align:middle line:90% above the floating potential. 00:52:19.090 --> 00:52:20.650 align:middle line:84% And we never draw very much current 00:52:20.650 --> 00:52:23.060 align:middle line:84% because the ion saturation current is very, very low. 00:52:23.060 --> 00:52:25.790 align:middle line:84% And that gives us out the temperature and the density, 00:52:25.790 --> 00:52:27.670 align:middle line:84% which is pretty remarkable for something 00:52:27.670 --> 00:52:29.440 align:middle line:84% that we've just stuck in the plasma that's 00:52:29.440 --> 00:52:32.400 align:middle line:90% just a little rod of metal. 00:52:32.400 --> 00:52:32.900 align:middle line:90% OK. 00:52:32.900 --> 00:52:34.820 align:middle line:84% So that's how you use a single Langmuir probe. 00:52:34.820 --> 00:52:38.100 align:middle line:90% Any questions on that? 00:52:38.100 --> 00:52:40.710 align:middle line:84% AUDIENCE: Do we have a name for the voltage at which the ion 00:52:40.710 --> 00:52:43.370 align:middle line:90% current saturates? 00:52:43.370 --> 00:52:44.915 align:middle line:90% JACK HARE: No, we don't. 00:52:44.915 --> 00:52:49.090 align:middle line:90% 00:52:49.090 --> 00:52:49.750 align:middle line:90% AUDIENCE: OK. 00:52:49.750 --> 00:52:50.417 align:middle line:90% JACK HARE: Yeah. 00:52:50.417 --> 00:52:52.840 align:middle line:90% 00:52:52.840 --> 00:52:54.440 align:middle line:90% I'm trying to think what it is. 00:52:54.440 --> 00:53:01.220 align:middle line:84% Effectively, we want to have this exponent be very small. 00:53:01.220 --> 00:53:05.740 align:middle line:84% So you could write something to do with this being less 00:53:05.740 --> 00:53:08.350 align:middle line:90% than 0.61, and it has to be-- 00:53:08.350 --> 00:53:09.100 align:middle line:90% I don't know. 00:53:09.100 --> 00:53:10.510 align:middle line:90% Let's say it's 10 times less. 00:53:10.510 --> 00:53:13.300 align:middle line:84% And then you could define some voltage at which it's 00:53:13.300 --> 00:53:17.030 align:middle line:84% 10 times less, and we could call that the ion saturation 00:53:17.030 --> 00:53:17.530 align:middle line:90% potential. 00:53:17.530 --> 00:53:19.280 align:middle line:84% But I haven't seen that in the literature. 00:53:19.280 --> 00:53:21.190 align:middle line:84% But you could work out what it needs to be. 00:53:21.190 --> 00:53:23.690 align:middle line:84% So you could say, if I think I know what the temperature is, 00:53:23.690 --> 00:53:25.890 align:middle line:84% I know how much I have to bias my probe negative. 00:53:25.890 --> 00:53:27.640 align:middle line:84% The other thing is just to bias your probe 00:53:27.640 --> 00:53:30.820 align:middle line:84% negative until it starts to roughly asymptote. 00:53:30.820 --> 00:53:33.610 align:middle line:84% But as I said before, it never actually asymptotes. 00:53:33.610 --> 00:53:36.110 align:middle line:84% What happens is it just keeps going down. 00:53:36.110 --> 00:53:39.960 align:middle line:84% So it's a real pain in the ass to measure this properly. 00:53:39.960 --> 00:53:40.740 align:middle line:90% OK, good. 00:53:40.740 --> 00:53:42.180 align:middle line:90% Daniel, I saw your hand first. 00:53:42.180 --> 00:53:43.050 align:middle line:90% AUDIENCE: Yeah. 00:53:43.050 --> 00:53:46.210 align:middle line:84% So I thought I had an idea of this until a moment ago, 00:53:46.210 --> 00:53:48.360 align:middle line:90% and then realized maybe I don't. 00:53:48.360 --> 00:53:51.540 align:middle line:84% But where's the return current in this? 00:53:51.540 --> 00:53:53.370 align:middle line:90% Because you can put-- 00:53:53.370 --> 00:53:55.920 align:middle line:84% my intuition is you put a voltage on this 00:53:55.920 --> 00:53:57.330 align:middle line:90% and it'll do stuff initially. 00:53:57.330 --> 00:53:58.920 align:middle line:84% But if you wait long enough, it'll 00:53:58.920 --> 00:54:01.440 align:middle line:84% settle to some other charge where you've just 00:54:01.440 --> 00:54:03.563 align:middle line:90% charged the plasma slightly. 00:54:03.563 --> 00:54:04.230 align:middle line:90% JACK HARE: Yeah. 00:54:04.230 --> 00:54:06.330 align:middle line:84% So the plasma is going to be drawing electrons 00:54:06.330 --> 00:54:10.830 align:middle line:84% back off from the vacuum chamber or from the vast infinity 00:54:10.830 --> 00:54:11.908 align:middle line:90% of the universe. 00:54:11.908 --> 00:54:12.450 align:middle line:90% AUDIENCE: OK. 00:54:12.450 --> 00:54:16.417 align:middle line:84% So it is from just other places where electrons can come from. 00:54:16.417 --> 00:54:18.000 align:middle line:84% JACK HARE: Now, you've got to remember 00:54:18.000 --> 00:54:20.070 align:middle line:84% that all the way around the vacuum chamber, there 00:54:20.070 --> 00:54:21.750 align:middle line:90% is a sheath. 00:54:21.750 --> 00:54:25.140 align:middle line:84% And that sheath obviously has balanced and electron currents. 00:54:25.140 --> 00:54:27.900 align:middle line:84% But the area there is so huge in the vacuum chamber 00:54:27.900 --> 00:54:30.330 align:middle line:84% that if this probe is locally drawing a few more 00:54:30.330 --> 00:54:34.440 align:middle line:84% electrons then the rest of the vacuum chamber can just push-- 00:54:34.440 --> 00:54:37.395 align:middle line:84% slightly fewer electrons, and it'll all work itself out. 00:54:37.395 --> 00:54:39.520 align:middle line:84% And as we know, quasi neutrality is pretty strictly 00:54:39.520 --> 00:54:41.812 align:middle line:84% enforced by plasma because the electric fields get big. 00:54:41.812 --> 00:54:44.530 align:middle line:84% So it's no trouble at all to get those electrons back 00:54:44.530 --> 00:54:45.928 align:middle line:90% from somewhere else, yeah. 00:54:45.928 --> 00:54:46.720 align:middle line:90% AUDIENCE: OK, cool. 00:54:46.720 --> 00:54:47.290 align:middle line:90% That makes sense. 00:54:47.290 --> 00:54:48.610 align:middle line:90% JACK HARE: Good question, yeah. 00:54:48.610 --> 00:54:51.090 align:middle line:90% And Nigel, I see your hand. 00:54:51.090 --> 00:54:53.000 align:middle line:84% AUDIENCE: So the practice of actually 00:54:53.000 --> 00:54:55.850 align:middle line:84% doing this, besides just sweeping the ion saturation 00:54:55.850 --> 00:54:58.580 align:middle line:84% current regime near the floating potential-- you also 00:54:58.580 --> 00:55:01.190 align:middle line:84% need to then sweep positive enough to figure out 00:55:01.190 --> 00:55:02.810 align:middle line:90% what the plasma potential is? 00:55:02.810 --> 00:55:04.040 align:middle line:90% Is that correct? 00:55:04.040 --> 00:55:05.737 align:middle line:90% JACK HARE: Oh, no, sorry. 00:55:05.737 --> 00:55:07.820 align:middle line:84% With the previous technique, we were talking about 00:55:07.820 --> 00:55:09.360 align:middle line:84% if you just want the temperature, 00:55:09.360 --> 00:55:11.492 align:middle line:84% then you would have to get the plasma potential. 00:55:11.492 --> 00:55:12.950 align:middle line:84% The beauty of this technique is you 00:55:12.950 --> 00:55:17.660 align:middle line:84% don't have to go anywhere close to the plasma potential. 00:55:17.660 --> 00:55:19.070 align:middle line:90% You're just sweeping. 00:55:19.070 --> 00:55:22.070 align:middle line:84% You just need the slope at this point. 00:55:22.070 --> 00:55:28.120 align:middle line:84% So you need di, d, and d0 at V0 equals Vf. 00:55:28.120 --> 00:55:29.870 align:middle line:84% And so of course, to determine that slope, 00:55:29.870 --> 00:55:31.245 align:middle line:84% you just need to sweep far enough 00:55:31.245 --> 00:55:33.933 align:middle line:84% that you can fit a straight line and you're happy with that. 00:55:33.933 --> 00:55:36.350 align:middle line:84% AUDIENCE: And so the benefit of this, like we were saying, 00:55:36.350 --> 00:55:38.480 align:middle line:84% is that we don't need to reach high currents, which 00:55:38.480 --> 00:55:39.950 align:middle line:90% could screw with theory? 00:55:39.950 --> 00:55:40.790 align:middle line:90% JACK HARE: Yeah. 00:55:40.790 --> 00:55:41.990 align:middle line:90% So that's one thing. 00:55:41.990 --> 00:55:43.820 align:middle line:84% And the other thing is the theory-- yeah, so actually, 00:55:43.820 --> 00:55:44.060 align:middle line:90% sorry. 00:55:44.060 --> 00:55:44.880 align:middle line:90% You said it there. 00:55:44.880 --> 00:55:46.630 align:middle line:84% We don't want to reach high currents where 00:55:46.630 --> 00:55:48.650 align:middle line:84% our theory is invalid or high potentials 00:55:48.650 --> 00:55:49.850 align:middle line:90% where our theory is invalid. 00:55:49.850 --> 00:55:51.780 align:middle line:84% And we don't want to reach the high currents 00:55:51.780 --> 00:55:54.150 align:middle line:84% as high potentials, where we draw lots of current 00:55:54.150 --> 00:55:55.302 align:middle line:90% and melt our probe. 00:55:55.302 --> 00:55:56.010 align:middle line:90% AUDIENCE: Got it. 00:55:56.010 --> 00:55:56.640 align:middle line:90% Thank you. 00:55:56.640 --> 00:55:57.690 align:middle line:84% JACK HARE: The melting of the probe thing 00:55:57.690 --> 00:55:58.860 align:middle line:84% probably sounds a bit funny, but this 00:55:58.860 --> 00:56:01.320 align:middle line:84% is genuinely a huge problem, which is why I keep saying it. 00:56:01.320 --> 00:56:03.792 align:middle line:84% So we must operate at the lowest currents possible. 00:56:03.792 --> 00:56:04.500 align:middle line:90% AUDIENCE: Oh, no. 00:56:04.500 --> 00:56:05.610 align:middle line:90% I work with bias probes. 00:56:05.610 --> 00:56:06.485 align:middle line:90% JACK HARE: OK, great. 00:56:06.485 --> 00:56:07.470 align:middle line:90% Cool. 00:56:07.470 --> 00:56:09.188 align:middle line:90% Nice. 00:56:09.188 --> 00:56:10.980 align:middle line:84% Any other questions on this before we go on 00:56:10.980 --> 00:56:13.415 align:middle line:84% to some more advanced Langmuir probes? 00:56:13.415 --> 00:56:14.040 align:middle line:90% AUDIENCE: Yeah. 00:56:14.040 --> 00:56:16.505 align:middle line:90% 00:56:16.505 --> 00:56:17.380 align:middle line:90% JACK HARE: Go for it. 00:56:17.380 --> 00:56:20.925 align:middle line:90% 00:56:20.925 --> 00:56:21.550 align:middle line:90% AUDIENCE: Sure. 00:56:21.550 --> 00:56:26.200 align:middle line:84% So how do you measure the ion saturation current 00:56:26.200 --> 00:56:30.213 align:middle line:84% if it keeps on going down, like you said? 00:56:30.213 --> 00:56:30.880 align:middle line:90% JACK HARE: Yeah. 00:56:30.880 --> 00:56:32.838 align:middle line:90% So it drops for a bit-- 00:56:32.838 --> 00:56:34.630 align:middle line:84% sorry, you can't see my mouse on my screen. 00:56:34.630 --> 00:56:36.920 align:middle line:84% It drops for a bit, and then levels off. 00:56:36.920 --> 00:56:40.430 align:middle line:84% And so you just eyeball it as like, it's here. 00:56:40.430 --> 00:56:43.330 align:middle line:84% Maybe you would fit some sort of curve. 00:56:43.330 --> 00:56:47.230 align:middle line:84% But it's not easy to do, so that gives you a lot of error. 00:56:47.230 --> 00:56:50.560 align:middle line:84% And that error won't affect your interpretation of Te, 00:56:50.560 --> 00:56:56.220 align:middle line:84% but it will affect your interpretation of the density. 00:56:56.220 --> 00:56:58.920 align:middle line:84% AUDIENCE: Why wouldn't it affect your interpretation of Te if it 00:56:58.920 --> 00:57:00.323 align:middle line:90% depends on Iis ? 00:57:00.323 --> 00:57:01.740 align:middle line:84% JACK HARE: Oh, you're quite right. 00:57:01.740 --> 00:57:03.450 align:middle line:90% Yes, you're quite right. 00:57:03.450 --> 00:57:07.340 align:middle line:84% I guess it's inside a logarithm, so it's less important there. 00:57:07.340 --> 00:57:09.598 align:middle line:84% But it will still have some effect. 00:57:09.598 --> 00:57:10.640 align:middle line:90% You're quite right, yeah. 00:57:10.640 --> 00:57:13.537 align:middle line:90% 00:57:13.537 --> 00:57:15.620 align:middle line:84% I just want to point out-- once when a student did 00:57:15.620 --> 00:57:18.930 align:middle line:84% this exercise, they forgot to subtract the ion saturation 00:57:18.930 --> 00:57:19.430 align:middle line:90% current. 00:57:19.430 --> 00:57:21.263 align:middle line:84% And this doesn't really work properly there. 00:57:21.263 --> 00:57:22.730 align:middle line:90% So this is a common pitfall. 00:57:22.730 --> 00:57:24.200 align:middle line:84% Make sure when you're doing this, 00:57:24.200 --> 00:57:27.030 align:middle line:84% you subtract the ion saturation current from your current curve. 00:57:27.030 --> 00:57:32.780 align:middle line:84% So it should look like that, right? 00:57:32.780 --> 00:57:35.570 align:middle line:84% The thing that you're working with that you're trying to find. 00:57:35.570 --> 00:57:37.700 align:middle line:84% It doesn't matter for the slope, of course-- 00:57:37.700 --> 00:57:41.140 align:middle line:84% or it does matter a little bit for the slope, so yeah. 00:57:41.140 --> 00:57:41.770 align:middle line:90% OK. 00:57:41.770 --> 00:57:43.180 align:middle line:90% We're going to move on. 00:57:43.180 --> 00:57:44.570 align:middle line:90% How fast are these mechanisms? 00:57:44.570 --> 00:57:45.853 align:middle line:90% Oh, thank you. 00:57:45.853 --> 00:57:48.520 align:middle line:84% My notes didn't update, and I'd actually made a new note on this 00:57:48.520 --> 00:57:49.603 align:middle line:90% that I didn't want to say. 00:57:49.603 --> 00:57:51.220 align:middle line:90% So let's talk about limitations. 00:57:51.220 --> 00:57:54.440 align:middle line:90% 00:57:54.440 --> 00:58:02.550 align:middle line:84% So one limitation is you might draw I greater than the ion 00:58:02.550 --> 00:58:03.660 align:middle line:90% saturation current. 00:58:03.660 --> 00:58:05.988 align:middle line:84% So you don't know where Vf is to start with. 00:58:05.988 --> 00:58:07.530 align:middle line:84% So when you put your probe in and you 00:58:07.530 --> 00:58:09.655 align:middle line:84% start adjusting the voltage, you might accidentally 00:58:09.655 --> 00:58:12.120 align:middle line:84% draw too much potential and melt the probe. 00:58:12.120 --> 00:58:15.240 align:middle line:84% And we'll look at some other designs which are safe and don't 00:58:15.240 --> 00:58:16.000 align:middle line:90% ever do that. 00:58:16.000 --> 00:58:19.420 align:middle line:84% And so this is a disadvantage of this design. 00:58:19.420 --> 00:58:22.960 align:middle line:84% The other disadvantage is the sweep time. 00:58:22.960 --> 00:58:27.660 align:middle line:84% So we're talking about literally sweeping the potential. 00:58:27.660 --> 00:58:31.250 align:middle line:84% I wish I could draw straight lines. 00:58:31.250 --> 00:58:32.120 align:middle line:90% There we go. 00:58:32.120 --> 00:58:35.620 align:middle line:84% So if we've got time here, and we've got voltage here, 00:58:35.620 --> 00:58:37.690 align:middle line:84% and this is the floating potential, 00:58:37.690 --> 00:58:43.330 align:middle line:84% maybe we have a sweep pattern that looks like this. 00:58:43.330 --> 00:58:48.330 align:middle line:84% But it takes some time, delta t, to sweep. 00:58:48.330 --> 00:58:53.340 align:middle line:84% And that might be limited by how quickly your voltage source can 00:58:53.340 --> 00:58:54.195 align:middle line:90% sweep voltage. 00:58:54.195 --> 00:58:56.820 align:middle line:84% But it might also be limited by how quickly your digitizing can 00:58:56.820 --> 00:58:58.560 align:middle line:84% digitize data, because effectively, 00:58:58.560 --> 00:59:02.152 align:middle line:84% as you do this sweep, you're digitizing this curve here. 00:59:02.152 --> 00:59:04.110 align:middle line:84% And you want to have enough data points on that 00:59:04.110 --> 00:59:05.220 align:middle line:90% to do all of your fitting. 00:59:05.220 --> 00:59:06.540 align:middle line:90% And so you are limited. 00:59:06.540 --> 00:59:08.640 align:middle line:84% You're very much temporarily limited. 00:59:08.640 --> 00:59:10.380 align:middle line:84% If there's an event which takes place 00:59:10.380 --> 00:59:12.420 align:middle line:84% on a time scale that's faster than this, 00:59:12.420 --> 00:59:14.010 align:middle line:84% you won't be able to resolve that. 00:59:14.010 --> 00:59:16.870 align:middle line:84% And in fact, it will do horrible things to your data analysis. 00:59:16.870 --> 00:59:19.830 align:middle line:84% So you want to make sure that your plasma is not fluctuating 00:59:19.830 --> 00:59:21.180 align:middle line:90% on rapid time scales. 00:59:21.180 --> 00:59:22.060 align:middle line:90% Yeah, Daniel. 00:59:22.060 --> 00:59:23.135 align:middle line:90% I see your hand. 00:59:23.135 --> 00:59:23.760 align:middle line:90% AUDIENCE: Yeah. 00:59:23.760 --> 00:59:25.710 align:middle line:84% Do you also run into situations where 00:59:25.710 --> 00:59:28.500 align:middle line:84% the actual response of the plasma 00:59:28.500 --> 00:59:33.900 align:middle line:84% is ever a limiting factor for your measurement time there? 00:59:33.900 --> 00:59:37.315 align:middle line:84% JACK HARE: So response being this whole setting up the sheath 00:59:37.315 --> 00:59:38.190 align:middle line:90% and things like that? 00:59:38.190 --> 00:59:39.030 align:middle line:90% AUDIENCE: Yeah. 00:59:39.030 --> 00:59:39.900 align:middle line:90% JACK HARE: Yeah. 00:59:39.900 --> 00:59:41.910 align:middle line:84% I think that happens really quickly. 00:59:41.910 --> 00:59:44.070 align:middle line:84% I think that's going to happen on electron time 00:59:44.070 --> 00:59:45.700 align:middle line:90% scales over a Debye length. 00:59:45.700 --> 00:59:49.150 align:middle line:84% And so those are probably very small time scales. 00:59:49.150 --> 00:59:53.740 align:middle line:84% So I would guess that delta t of the perturbation 00:59:53.740 --> 00:59:58.985 align:middle line:84% is going to be on the order of lambda d over VTe. 00:59:58.985 --> 01:00:00.860 align:middle line:84% That's the sort of time scale I can think of. 01:00:00.860 --> 01:00:04.690 align:middle line:84% And that seems like a really short time scale. 01:00:04.690 --> 01:00:06.730 align:middle line:84% In fact, I know what that time scale is. 01:00:06.730 --> 01:00:12.870 align:middle line:84% It is 1 over the electron plasma frequency. 01:00:12.870 --> 01:00:14.710 align:middle line:90% And so that's really fast. 01:00:14.710 --> 01:00:17.822 align:middle line:84% So maybe in some plasmas, you could reach that timescale. 01:00:17.822 --> 01:00:20.030 align:middle line:84% But probably that's all going to happen very quickly. 01:00:20.030 --> 01:00:22.030 align:middle line:84% So I think from the point of view of the plasma, 01:00:22.030 --> 01:00:24.400 align:middle line:84% it responds instantaneously to the change 01:00:24.400 --> 01:00:26.140 align:middle line:84% in the potential on your probe, and we 01:00:26.140 --> 01:00:27.560 align:middle line:90% don't have to worry about that. 01:00:27.560 --> 01:00:29.143 align:middle line:84% But if you have a really funky plasma, 01:00:29.143 --> 01:00:31.180 align:middle line:90% that could be problematic. 01:00:31.180 --> 01:00:32.170 align:middle line:90% AUDIENCE: OK, cool. 01:00:32.170 --> 01:00:33.790 align:middle line:84% JACK HARE: I think omega p is probably 01:00:33.790 --> 01:00:36.280 align:middle line:84% the fastest timescale-- one of the fastest, 01:00:36.280 --> 01:00:39.927 align:middle line:84% probably the fastest timescale we see in most plasmas. 01:00:39.927 --> 01:00:40.885 align:middle line:90% Did I see another hand? 01:00:40.885 --> 01:00:45.740 align:middle line:90% 01:00:45.740 --> 01:00:46.240 align:middle line:90% OK. 01:00:46.240 --> 01:00:47.290 align:middle line:84% I'm going to keep moving because I've 01:00:47.290 --> 01:00:50.020 align:middle line:84% got a few more things to get through in the next 20 minutes 01:00:50.020 --> 01:00:50.620 align:middle line:90% or so. 01:00:50.620 --> 01:00:53.310 align:middle line:90% So let's power on. 01:00:53.310 --> 01:00:55.070 align:middle line:84% So the next probe we can consider 01:00:55.070 --> 01:00:57.380 align:middle line:90% is called the double probe. 01:00:57.380 --> 01:00:59.960 align:middle line:84% The previous one, we're going to retroactively rename 01:00:59.960 --> 01:01:00.695 align:middle line:90% the single probe. 01:01:00.695 --> 01:01:03.390 align:middle line:90% 01:01:03.390 --> 01:01:05.100 align:middle line:84% And the double probe-- once again, 01:01:05.100 --> 01:01:07.270 align:middle line:90% we've got some sort of plasma. 01:01:07.270 --> 01:01:11.140 align:middle line:84% But now we have two probes sticking into the plasma. 01:01:11.140 --> 01:01:14.770 align:middle line:84% And the clever thing is that the two probes are attached 01:01:14.770 --> 01:01:19.390 align:middle line:84% together, and they are biased with respect to one another 01:01:19.390 --> 01:01:21.970 align:middle line:90% by a bias potential, VB. 01:01:21.970 --> 01:01:25.780 align:middle line:84% And again, we measure the current. 01:01:25.780 --> 01:01:27.270 align:middle line:90% This is just like a resistor. 01:01:27.270 --> 01:01:29.530 align:middle line:84% So I can measure the voltage across this resistor, 01:01:29.530 --> 01:01:32.200 align:middle line:84% and I'm going to get out the current here. 01:01:32.200 --> 01:01:35.070 align:middle line:84% And that goes to an oscilloscope. 01:01:35.070 --> 01:01:37.020 align:middle line:84% Now, the cool thing about this setup 01:01:37.020 --> 01:01:45.980 align:middle line:84% is that if probe 1 draws a current I1, 01:01:45.980 --> 01:01:50.630 align:middle line:84% then probe 2 is going to have to discharge a current, I2. 01:01:50.630 --> 01:01:52.790 align:middle line:84% And from Kirchhoff's law, we know straight 01:01:52.790 --> 01:01:56.870 align:middle line:84% away that I1 plus I2 is equal to 0. 01:01:56.870 --> 01:01:59.420 align:middle line:84% So any current that comes in one probe 01:01:59.420 --> 01:02:01.130 align:middle line:84% has to be ejected out the other probe. 01:02:01.130 --> 01:02:03.602 align:middle line:84% There's no other place for it to go. 01:02:03.602 --> 01:02:05.310 align:middle line:84% This probably addresses Daniel's question 01:02:05.310 --> 01:02:07.950 align:middle line:84% about the plasma accidentally charging up. 01:02:07.950 --> 01:02:11.550 align:middle line:84% It turns out-- and Hutchinson talks about this in great depth 01:02:11.550 --> 01:02:12.570 align:middle line:90% in his book-- 01:02:12.570 --> 01:02:15.180 align:middle line:84% that almost every probe is really a double probe. 01:02:15.180 --> 01:02:17.400 align:middle line:84% It's just that either you have a double probe 01:02:17.400 --> 01:02:19.800 align:middle line:84% or you have a single probe, and the entire vacuum vessel 01:02:19.800 --> 01:02:21.180 align:middle line:90% as the other probe surfaces. 01:02:21.180 --> 01:02:23.260 align:middle line:84% And I'm not going to go into that in much detail, 01:02:23.260 --> 01:02:25.740 align:middle line:84% but that's a reasonable thing to think about. 01:02:25.740 --> 01:02:28.458 align:middle line:84% Yeah, I see a question from Nigel. 01:02:28.458 --> 01:02:29.000 align:middle line:90% AUDIENCE: OK. 01:02:29.000 --> 01:02:31.110 align:middle line:84% This might be a little out there. 01:02:31.110 --> 01:02:33.590 align:middle line:84% But effectively, because they're in a plasma, 01:02:33.590 --> 01:02:35.922 align:middle line:84% there is a pseudo resistor that we 01:02:35.922 --> 01:02:37.880 align:middle line:84% could draw on a circuit diagram between the two 01:02:37.880 --> 01:02:39.290 align:middle line:90% tips of the probe. 01:02:39.290 --> 01:02:43.925 align:middle line:84% Is there a problem with that creating a closed loop now? 01:02:43.925 --> 01:02:46.880 align:middle line:84% And that magnetic flux could go through and then make 01:02:46.880 --> 01:02:49.775 align:middle line:84% Kirchhoff's law not entirely valid. 01:02:49.775 --> 01:02:51.650 align:middle line:84% JACK HARE: What do you mean by magnetic flux? 01:02:51.650 --> 01:02:54.080 align:middle line:84% Because we haven't really talked about magnetic fields very much. 01:02:54.080 --> 01:02:56.205 align:middle line:84% AUDIENCE: I guess I've been assuming that these are 01:02:56.205 --> 01:02:57.650 align:middle line:90% on a tokamak this whole time. 01:02:57.650 --> 01:03:02.560 align:middle line:90% 01:03:02.560 --> 01:03:05.710 align:middle line:84% JACK HARE: So is your question, is there a problem 01:03:05.710 --> 01:03:08.475 align:middle line:84% if there's a current flowing inside the plasma here? 01:03:08.475 --> 01:03:10.600 align:middle line:84% AUDIENCE: No, it's more like if those are connected 01:03:10.600 --> 01:03:12.250 align:middle line:84% and there's a resistance between them, 01:03:12.250 --> 01:03:14.125 align:middle line:84% if we're just looking at this like a circuit, 01:03:14.125 --> 01:03:15.740 align:middle line:90% we've created a closed loop. 01:03:15.740 --> 01:03:17.860 align:middle line:84% And so now if you get a change in magnetic flux 01:03:17.860 --> 01:03:20.200 align:middle line:84% through the loop closed by that, you 01:03:20.200 --> 01:03:22.907 align:middle line:84% could induce an EMF that could affect-- 01:03:22.907 --> 01:03:23.740 align:middle line:90% JACK HARE: Yeah, OK. 01:03:23.740 --> 01:03:24.560 align:middle line:90% I see what you're saying. 01:03:24.560 --> 01:03:25.602 align:middle line:90% I see what you're saying. 01:03:25.602 --> 01:03:28.690 align:middle line:84% We're not actually measuring the potential on the probes, though. 01:03:28.690 --> 01:03:32.590 align:middle line:84% So we're measuring the current through the probes 01:03:32.590 --> 01:03:33.950 align:middle line:90% through this resistor. 01:03:33.950 --> 01:03:36.670 align:middle line:84% So if we induce a voltage around the whole loop, 01:03:36.670 --> 01:03:39.040 align:middle line:84% I don't know whether we would see that show up 01:03:39.040 --> 01:03:41.728 align:middle line:90% across this resistor. 01:03:41.728 --> 01:03:43.270 align:middle line:84% That's a really interesting question. 01:03:43.270 --> 01:03:44.840 align:middle line:84% I'm going to say that I don't know the answer, 01:03:44.840 --> 01:03:46.080 align:middle line:84% and I haven't thought about it much. 01:03:46.080 --> 01:03:47.360 align:middle line:84% So I'll certainly have to think about it 01:03:47.360 --> 01:03:50.090 align:middle line:84% and get back to you if I have any interesting thoughts on it. 01:03:50.090 --> 01:03:51.710 align:middle line:90% But I get what you're saying. 01:03:51.710 --> 01:03:54.950 align:middle line:84% You're saying, potentially, this could have some sort 01:03:54.950 --> 01:03:58.925 align:middle line:90% of plasma current channel here. 01:03:58.925 --> 01:04:00.800 align:middle line:84% And then that looks like a little V dot probe 01:04:00.800 --> 01:04:04.370 align:middle line:84% that we've stuck in the plasma, and something could induce-- 01:04:04.370 --> 01:04:06.330 align:middle line:90% yeah, I see what you're saying. 01:04:06.330 --> 01:04:06.830 align:middle line:90% Cool. 01:04:06.830 --> 01:04:08.580 align:middle line:90% I'll have a think about it. 01:04:08.580 --> 01:04:10.500 align:middle line:90% Cool. 01:04:10.500 --> 01:04:11.880 align:middle line:90% OK. 01:04:11.880 --> 01:04:16.280 align:middle line:84% And so there's other things that we can say about the currents 01:04:16.280 --> 01:04:16.780 align:middle line:90% here. 01:04:16.780 --> 01:04:19.890 align:middle line:84% So we know that the current going into-- 01:04:19.890 --> 01:04:23.790 align:middle line:84% excuse me-- probe 1 is going to be 01:04:23.790 --> 01:04:27.420 align:middle line:84% equal to the ion saturation current plus 01:04:27.420 --> 01:04:34.150 align:middle line:84% the electron current modified by a factor e V1 over Te. 01:04:34.150 --> 01:04:36.130 align:middle line:84% So that's just our Boltzmann factor. 01:04:36.130 --> 01:04:41.320 align:middle line:84% And the current through probe 2 looks very similar, 01:04:41.320 --> 01:04:47.830 align:middle line:84% except that we modify this now by e V2 over Te, like that. 01:04:47.830 --> 01:04:49.540 align:middle line:84% So we haven't used Kirchhoff's law yet. 01:04:49.540 --> 01:04:51.760 align:middle line:84% This is just saying if we had two random probes, 01:04:51.760 --> 01:04:54.690 align:middle line:84% this is the current that we would draw. 01:04:54.690 --> 01:04:58.140 align:middle line:84% Now we use Kirchhoff's law here, and we 01:04:58.140 --> 01:04:59.760 align:middle line:84% note the nice thing about this is 01:04:59.760 --> 01:05:01.410 align:middle line:84% that the current through any probe 01:05:01.410 --> 01:05:05.820 align:middle line:84% is now strictly less than the ion saturation current here. 01:05:05.820 --> 01:05:11.010 align:middle line:84% So we can't accidentally draw the electron saturation current 01:05:11.010 --> 01:05:12.910 align:middle line:90% and melt probe. 01:05:12.910 --> 01:05:16.635 align:middle line:84% So this is a definite advantage of this system. 01:05:16.635 --> 01:05:19.010 align:middle line:84% I'll do this a little bit more mathematically in a moment 01:05:19.010 --> 01:05:20.330 align:middle line:90% if you're not convinced yet. 01:05:20.330 --> 01:05:24.760 align:middle line:84% So then what we do is we continue and we say, 01:05:24.760 --> 01:05:28.480 align:middle line:84% we've got these equations separately for I1 and I2. 01:05:28.480 --> 01:05:32.810 align:middle line:84% We've got this equation that links I1 and I2 together. 01:05:32.810 --> 01:05:35.770 align:middle line:84% And we're also going to operate this 01:05:35.770 --> 01:05:43.915 align:middle line:84% so that the probes are floating near the floating potential. 01:05:43.915 --> 01:05:45.290 align:middle line:84% In fact, that will happen as soon 01:05:45.290 --> 01:05:46.915 align:middle line:84% as we put these probes into the plasma. 01:05:46.915 --> 01:05:49.575 align:middle line:84% We're not biasing the probes themselves. 01:05:49.575 --> 01:05:51.450 align:middle line:84% We're biasing them with respect to each other 01:05:51.450 --> 01:05:55.190 align:middle line:84% so the whole setup will float towards the floating potential. 01:05:55.190 --> 01:05:58.250 align:middle line:84% And that means that our ion current 01:05:58.250 --> 01:06:05.040 align:middle line:84% is going to be equal to our electron current, like this. 01:06:05.040 --> 01:06:06.800 align:middle line:84% Then we can put all of this together, 01:06:06.800 --> 01:06:18.670 align:middle line:84% and we can say that I1 is equal to Isi 1 minus 01:06:18.670 --> 01:06:22.660 align:middle line:90% exponential of e V1 over Te. 01:06:22.660 --> 01:06:27.380 align:middle line:84% So that's just using this fact here. 01:06:27.380 --> 01:06:30.470 align:middle line:84% And then that is going to be equal to minus I2 01:06:30.470 --> 01:06:31.770 align:middle line:90% from Kirchhoff's law. 01:06:31.770 --> 01:06:37.170 align:middle line:84% And so that's going to be equal to minus Isi 1 minus 01:06:37.170 --> 01:06:43.520 align:middle line:90% exponential of e V2 over Te. 01:06:43.520 --> 01:06:46.160 align:middle line:84% And then we're going to say that this bias voltage that we're 01:06:46.160 --> 01:06:49.640 align:middle line:84% applying is simply, by definition, the difference 01:06:49.640 --> 01:06:52.400 align:middle line:90% between V1 and V2. 01:06:52.400 --> 01:06:54.985 align:middle line:84% And if we do a little bit of magic on this-- and this 01:06:54.985 --> 01:06:57.110 align:middle line:84% is actually an exercise in Hutchinson's book, which 01:06:57.110 --> 01:07:00.030 align:middle line:84% I had to do to try and work out where on Earth this came from. 01:07:00.030 --> 01:07:02.398 align:middle line:84% You find out that the current that you're measuring, 01:07:02.398 --> 01:07:04.940 align:middle line:84% which is the current flowing into one probe and out the other 01:07:04.940 --> 01:07:09.860 align:middle line:84% probe, is going to be equal to the ion saturation current times 01:07:09.860 --> 01:07:17.910 align:middle line:84% by the hyperbolic tangent of e times the bias voltage over 2Te, 01:07:17.910 --> 01:07:18.820 align:middle line:90% like that. 01:07:18.820 --> 01:07:24.371 align:middle line:84% And so now if I sketch the IV curve for this, 01:07:24.371 --> 01:07:28.690 align:middle line:84% this is B bias and this is I. I end up with something 01:07:28.690 --> 01:07:30.968 align:middle line:90% that looks like-- 01:07:30.968 --> 01:07:32.260 align:middle line:90% if I could draw straight lines. 01:07:32.260 --> 01:07:32.968 align:middle line:90% Give me a moment. 01:07:32.968 --> 01:07:38.050 align:middle line:90% 01:07:38.050 --> 01:07:38.590 align:middle line:90% OK. 01:07:38.590 --> 01:07:44.320 align:middle line:84% And so this is asymptoting at the ion saturation current, 01:07:44.320 --> 01:07:48.400 align:middle line:84% or at minus the ion saturation current here. 01:07:48.400 --> 01:07:50.970 align:middle line:90% 01:07:50.970 --> 01:07:53.630 align:middle line:84% So this is sort of the proof of what I said earlier, 01:07:53.630 --> 01:07:56.750 align:middle line:84% that we can't draw more than the ion saturation current here. 01:07:56.750 --> 01:07:59.390 align:middle line:84% And this is very nice and symmetric. 01:07:59.390 --> 01:08:00.890 align:middle line:84% And I'll show you in a moment how 01:08:00.890 --> 01:08:03.170 align:middle line:84% we use that to actually measure all 01:08:03.170 --> 01:08:04.580 align:middle line:90% the properties of the plasma. 01:08:04.580 --> 01:08:07.513 align:middle line:84% So I'm happy to take questions on this, but if you're like, 01:08:07.513 --> 01:08:08.930 align:middle line:84% I don't understand the derivation, 01:08:08.930 --> 01:08:11.150 align:middle line:84% the reason is I didn't really do the derivation. 01:08:11.150 --> 01:08:12.710 align:middle line:84% There's lots of steps missing here. 01:08:12.710 --> 01:08:14.540 align:middle line:84% I'm just giving you a couple of intermediate steps. 01:08:14.540 --> 01:08:15.964 align:middle line:84% And if you want to go get it, you 01:08:15.964 --> 01:08:18.339 align:middle line:84% should go work through the exercise in Hutchinson's book. 01:08:18.339 --> 01:08:29.747 align:middle line:90% 01:08:29.747 --> 01:08:31.580 align:middle line:84% I'm going to try and write this on this page 01:08:31.580 --> 01:08:33.350 align:middle line:84% because I think it'll be more useful. 01:08:33.350 --> 01:08:39.390 align:middle line:84% If we have for small bias voltages-- so 01:08:39.390 --> 01:08:43.170 align:middle line:84% we're operating just in this little regime here. 01:08:43.170 --> 01:08:48.540 align:middle line:84% We find similarly to the probe that we had before that di d 01:08:48.540 --> 01:09:01.049 align:middle line:84% the bias around B bias equals 0 is e times Isi over 2Te. 01:09:01.049 --> 01:09:04.022 align:middle line:90% 01:09:04.022 --> 01:09:05.939 align:middle line:84% And we're also going to be able to measure Isi 01:09:05.939 --> 01:09:09.910 align:middle line:84% by sweeping just a little bit larger, like this. 01:09:09.910 --> 01:09:15.420 align:middle line:84% And so we'll get, again, Isi is equal to whatever our equation 01:09:15.420 --> 01:09:21.640 align:middle line:84% we had before, which has a proportionality to ne and Te 01:09:21.640 --> 01:09:23.020 align:middle line:90% to the 1/2. 01:09:23.020 --> 01:09:25.630 align:middle line:84% So from this probe characteristic here, 01:09:25.630 --> 01:09:27.760 align:middle line:84% we can now get out both the density 01:09:27.760 --> 01:09:29.470 align:middle line:84% and the temperature-- again, from looking 01:09:29.470 --> 01:09:32.050 align:middle line:84% at the gradient near the bias voltage of 0 01:09:32.050 --> 01:09:35.450 align:middle line:84% and looking at the ion saturation current 01:09:35.450 --> 01:09:36.590 align:middle line:90% as we did before. 01:09:36.590 --> 01:09:39.680 align:middle line:84% But the advantage of this is that we can no longer draw 01:09:39.680 --> 01:09:41.569 align:middle line:90% a large current by accident. 01:09:41.569 --> 01:09:46.960 align:middle line:84% But the limitation is still the sweep time. 01:09:46.960 --> 01:09:50.850 align:middle line:84% So we still need to sweep this bias voltage up and down 01:09:50.850 --> 01:09:52.710 align:middle line:84% so that we can trace out this tank 01:09:52.710 --> 01:09:55.300 align:middle line:90% function so that we can fit it. 01:09:55.300 --> 01:09:58.260 align:middle line:84% So you still can't resolve very fast moving things, 01:09:58.260 --> 01:10:02.880 align:middle line:84% but this at least is a nice and safe probe. 01:10:02.880 --> 01:10:04.830 align:middle line:90% So any questions on this? 01:10:04.830 --> 01:10:17.640 align:middle line:90% 01:10:17.640 --> 01:10:20.790 align:middle line:84% So no prizes for anyone who guessed after the double probe 01:10:20.790 --> 01:10:23.400 align:middle line:90% we go to the triple probe. 01:10:23.400 --> 01:10:26.490 align:middle line:84% I promise I will stop after this. 01:10:26.490 --> 01:10:33.400 align:middle line:84% So a triple probe is simply a double probe 01:10:33.400 --> 01:10:34.690 align:middle line:90% whilst a floating probe. 01:10:34.690 --> 01:10:37.045 align:middle line:84% So let me sketch that and explain what that means. 01:10:37.045 --> 01:10:39.830 align:middle line:90% 01:10:39.830 --> 01:10:44.690 align:middle line:90% That's meant to read floating. 01:10:44.690 --> 01:10:51.830 align:middle line:84% So we have our double probe as before, like this. 01:10:51.830 --> 01:10:55.330 align:middle line:84% And we bias it with some bias potential. 01:10:55.330 --> 01:10:59.370 align:middle line:84% And we measure the current across a resistor. 01:10:59.370 --> 01:11:01.400 align:middle line:90% But we also have-- 01:11:01.400 --> 01:11:07.600 align:middle line:84% stuck somewhere nearby, we have our single probe. 01:11:07.600 --> 01:11:10.930 align:middle line:90% So that is by floating. 01:11:10.930 --> 01:11:12.335 align:middle line:90% There we go. 01:11:12.335 --> 01:11:12.835 align:middle line:90% OK. 01:11:12.835 --> 01:11:17.040 align:middle line:90% 01:11:17.040 --> 01:11:18.882 align:middle line:90% Sorry. 01:11:18.882 --> 01:11:21.090 align:middle line:84% For this probe, we actually don't measure the current 01:11:21.090 --> 01:11:23.257 align:middle line:84% through it because we're going to leave it floating. 01:11:23.257 --> 01:11:27.160 align:middle line:84% So we actually measure the voltage that it's at here. 01:11:27.160 --> 01:11:36.320 align:middle line:84% So for the double probe, we set our bias voltage 01:11:36.320 --> 01:11:42.353 align:middle line:84% to be greater than a few electron temperatures. 01:11:42.353 --> 01:11:44.270 align:middle line:84% And maybe this takes a bit of trial and error, 01:11:44.270 --> 01:11:45.910 align:middle line:90% but we find it in the end. 01:11:45.910 --> 01:11:50.790 align:middle line:84% And then we find out that one of the probes 01:11:50.790 --> 01:11:56.740 align:middle line:84% is going to draw positive ion saturation current. 01:11:56.740 --> 01:12:05.840 align:middle line:84% Second probe is going to draw a negative ion saturation 01:12:05.840 --> 01:12:08.900 align:middle line:84% current because it's at a potential 01:12:08.900 --> 01:12:13.140 align:middle line:84% greater than a floating potential. 01:12:13.140 --> 01:12:16.590 align:middle line:84% This one is at a potential less than the floating potential. 01:12:16.590 --> 01:12:24.980 align:middle line:84% And finally, the floating probe is 01:12:24.980 --> 01:12:28.790 align:middle line:84% going to allow us to measure the floating potential itself 01:12:28.790 --> 01:12:29.630 align:middle line:90% directly. 01:12:29.630 --> 01:12:32.270 align:middle line:84% And the reason that this is neat and fundamentally 01:12:32.270 --> 01:12:34.730 align:middle line:84% different from the other probes that we've looked at 01:12:34.730 --> 01:12:39.500 align:middle line:84% is if I redraw our IV characteristic-- 01:12:39.500 --> 01:12:44.270 align:middle line:90% 01:12:44.270 --> 01:12:51.510 align:middle line:84% so I and V. And again, we have a floating potential. 01:12:51.510 --> 01:12:54.580 align:middle line:90% We have a plasma potential here. 01:12:54.580 --> 01:12:58.240 align:middle line:84% We have the ion saturation current 01:12:58.240 --> 01:13:01.720 align:middle line:84% and we have the electron saturation current. 01:13:01.720 --> 01:13:04.420 align:middle line:84% These three probes are actually representing three 01:13:04.420 --> 01:13:06.680 align:middle line:90% different places on this curve. 01:13:06.680 --> 01:13:11.230 align:middle line:84% So probe number one is representing a place down here, 01:13:11.230 --> 01:13:15.490 align:middle line:84% probe number three is representing this point, 01:13:15.490 --> 01:13:22.170 align:middle line:84% and probe number two is representing the opposite point 01:13:22.170 --> 01:13:27.930 align:middle line:84% up here, where we're drawing minus the ion saturation current 01:13:27.930 --> 01:13:28.750 align:middle line:90% here. 01:13:28.750 --> 01:13:32.520 align:middle line:84% And this means we're measuring three 01:13:32.520 --> 01:13:37.040 align:middle line:90% points on an exponential curve. 01:13:37.040 --> 01:13:40.223 align:middle line:90% 01:13:40.223 --> 01:13:41.890 align:middle line:84% And three points on an exponential curve 01:13:41.890 --> 01:13:43.400 align:middle line:90% is enough to specify it. 01:13:43.400 --> 01:13:48.500 align:middle line:84% And so we can just fit the IV curve, 01:13:48.500 --> 01:13:51.320 align:middle line:84% and we can get out straight away what the density is 01:13:51.320 --> 01:13:53.520 align:middle line:90% and what the temperature is. 01:13:53.520 --> 01:13:55.850 align:middle line:84% And we can do that without sweeping any voltage here. 01:13:55.850 --> 01:13:58.017 align:middle line:84% Remember, we are not sweeping the bias voltage here. 01:13:58.017 --> 01:14:00.470 align:middle line:84% We've just set it to a few times eTe. 01:14:00.470 --> 01:14:09.070 align:middle line:84% So the whole system floats near the floating potential. 01:14:09.070 --> 01:14:17.500 align:middle line:84% So we can't draw the electron saturation current by design. 01:14:17.500 --> 01:14:21.660 align:middle line:84% We also have fast time resolution, 01:14:21.660 --> 01:14:23.910 align:middle line:84% because now we're no longer sweeping anything but just 01:14:23.910 --> 01:14:26.170 align:middle line:90% digitizing this in time. 01:14:26.170 --> 01:14:29.340 align:middle line:84% So we can resolve any short-lived phenomena. 01:14:29.340 --> 01:14:31.050 align:middle line:84% But the only problem with this is 01:14:31.050 --> 01:14:39.460 align:middle line:84% that it assumes implicitly that your IV curve model is good. 01:14:39.460 --> 01:14:43.600 align:middle line:90% 01:14:43.600 --> 01:14:45.592 align:middle line:84% And I think that is a big assumption. 01:14:45.592 --> 01:14:47.300 align:middle line:84% So you've only got three points on there. 01:14:47.300 --> 01:14:50.170 align:middle line:84% So if it starts deviating from exponential, 01:14:50.170 --> 01:14:52.090 align:middle line:84% you're effectively not overfitting, 01:14:52.090 --> 01:14:53.870 align:middle line:90% but precisely fitting it. 01:14:53.870 --> 01:14:56.103 align:middle line:84% And so you won't be able to see any deviations. 01:14:56.103 --> 01:14:57.520 align:middle line:84% And that's when people start using 01:14:57.520 --> 01:14:59.770 align:middle line:84% quadruple probes, and all sorts of exciting things 01:14:59.770 --> 01:15:01.603 align:middle line:84% like that, which we're not going to go into. 01:15:01.603 --> 01:15:03.160 align:middle line:90% But you can keep adding probes. 01:15:03.160 --> 01:15:04.840 align:middle line:84% And one way to think about adding probes 01:15:04.840 --> 01:15:08.080 align:middle line:84% is that you keep adding more points along this line 01:15:08.080 --> 01:15:11.150 align:middle line:84% by biasing probes in different places. 01:15:11.150 --> 01:15:13.600 align:middle line:84% And if you measure their current and their potential, 01:15:13.600 --> 01:15:17.530 align:middle line:84% you can reproduce that curve instantaneously. 01:15:17.530 --> 01:15:20.410 align:middle line:84% Of course, if you try and crowd too many probes together 01:15:20.410 --> 01:15:23.150 align:middle line:84% in the same place, you're going to cause lots of perturbations 01:15:23.150 --> 01:15:25.400 align:middle line:84% and the probes are going to interfere with each other. 01:15:25.400 --> 01:15:28.360 align:middle line:84% So this is not flawless, but people do all sorts 01:15:28.360 --> 01:15:30.610 align:middle line:90% of clever things with this. 01:15:30.610 --> 01:15:31.110 align:middle line:90% OK. 01:15:31.110 --> 01:15:31.740 align:middle line:90% So questions? 01:15:31.740 --> 01:15:33.243 align:middle line:90% And I see Nigel. 01:15:33.243 --> 01:15:34.910 align:middle line:84% AUDIENCE: Yeah, I wanted to actually ask 01:15:34.910 --> 01:15:38.165 align:middle line:84% about the last point you brought up about-- 01:15:38.165 --> 01:15:40.130 align:middle line:84% could you talk more about the trade-off 01:15:40.130 --> 01:15:43.670 align:middle line:84% between having the probes close together 01:15:43.670 --> 01:15:45.230 align:middle line:84% so you ensure that they're measuring 01:15:45.230 --> 01:15:50.072 align:middle line:84% the same plasma versus them interfering with each other 01:15:50.072 --> 01:15:52.280 align:middle line:84% like they're sheets overlapping and things like that? 01:15:52.280 --> 01:15:53.180 align:middle line:90% JACK HARE: Yeah. 01:15:53.180 --> 01:15:53.932 align:middle line:90% It's fair enough. 01:15:53.932 --> 01:15:56.390 align:middle line:84% So I can talk a little bit about it, but I'm not an expert. 01:15:56.390 --> 01:15:59.690 align:middle line:84% So one thing I'll point out is our sheath thickness 01:15:59.690 --> 01:16:01.700 align:middle line:84% is going to be a few Debye lengths. 01:16:01.700 --> 01:16:03.410 align:middle line:84% So you might think, well, that's OK. 01:16:03.410 --> 01:16:05.870 align:middle line:84% I can keep my probes more than 60 microns apart 01:16:05.870 --> 01:16:06.950 align:middle line:90% or something like that. 01:16:06.950 --> 01:16:09.470 align:middle line:84% The trouble is that there's actually a region here called 01:16:09.470 --> 01:16:12.590 align:middle line:84% the presheath, and although we tried our absolute hardest 01:16:12.590 --> 01:16:16.040 align:middle line:84% not to model the presheath very much, if the presheath starts 01:16:16.040 --> 01:16:18.270 align:middle line:84% to overlap, it can be a big deal. 01:16:18.270 --> 01:16:21.290 align:middle line:84% And the presheath is much larger than the Debye length. 01:16:21.290 --> 01:16:23.480 align:middle line:84% It's maybe hundreds of Debye lengths. 01:16:23.480 --> 01:16:26.340 align:middle line:90% In fact, it's all a bit subtle. 01:16:26.340 --> 01:16:28.340 align:middle line:84% And if you start looking at Hutchinson's book, 01:16:28.340 --> 01:16:30.200 align:middle line:84% you realize that I haven't mentioned collisions 01:16:30.200 --> 01:16:31.700 align:middle line:84% or magnetic fields at all, and these 01:16:31.700 --> 01:16:34.320 align:middle line:84% make this extremely subtle and complicated to work out. 01:16:34.320 --> 01:16:36.862 align:middle line:84% So you really want to have your probe spaced far enough apart 01:16:36.862 --> 01:16:39.620 align:middle line:90% the presheaths don't interfere. 01:16:39.620 --> 01:16:41.930 align:middle line:84% So if I was doing this in an experiment, 01:16:41.930 --> 01:16:45.590 align:middle line:84% I would have one probe, and then I'd put another probe in next 01:16:45.590 --> 01:16:48.600 align:middle line:84% to it and see if my signal on the first probe changed. 01:16:48.600 --> 01:16:51.440 align:middle line:84% And if it changed significantly, by 10% or more, 01:16:51.440 --> 01:16:53.885 align:middle line:84% I would say those probes are too close together. 01:16:53.885 --> 01:16:55.760 align:middle line:84% And so you might have to do this a little bit 01:16:55.760 --> 01:16:58.117 align:middle line:84% quasi experimentally in order to work out how close you 01:16:58.117 --> 01:16:59.075 align:middle line:90% can put all the probes. 01:16:59.075 --> 01:17:02.157 align:middle line:90% 01:17:02.157 --> 01:17:02.990 align:middle line:90% AUDIENCE: Thank you. 01:17:02.990 --> 01:17:03.780 align:middle line:90% JACK HARE: You're welcome. 01:17:03.780 --> 01:17:05.060 align:middle line:90% Any other questions on this? 01:17:05.060 --> 01:17:11.000 align:middle line:90% 01:17:11.000 --> 01:17:15.080 align:middle line:84% I have just got two bullet points before we finish up. 01:17:15.080 --> 01:17:18.170 align:middle line:84% And these are just due to what we 01:17:18.170 --> 01:17:20.210 align:middle line:90% would call added complications. 01:17:20.210 --> 01:17:25.500 align:middle line:90% 01:17:25.500 --> 01:17:28.812 align:middle line:84% So the first complication are magnetic fields. 01:17:28.812 --> 01:17:30.770 align:middle line:84% And you definitely might want to use a Langmuir 01:17:30.770 --> 01:17:32.060 align:middle line:90% probe in a magnetic field. 01:17:32.060 --> 01:17:33.643 align:middle line:84% You might want to put it in the scrape 01:17:33.643 --> 01:17:35.853 align:middle line:84% off layer of a tokamak or something like that. 01:17:35.853 --> 01:17:37.520 align:middle line:84% And the big problem with magnetic fields 01:17:37.520 --> 01:17:39.500 align:middle line:90% is they alter particle orbits. 01:17:39.500 --> 01:17:44.720 align:middle line:90% 01:17:44.720 --> 01:17:47.910 align:middle line:84% If you have very strong magnetic fields, 01:17:47.910 --> 01:17:50.440 align:middle line:84% this may not seem like too big of a deal. 01:17:50.440 --> 01:17:52.440 align:middle line:90% Here's my probe here. 01:17:52.440 --> 01:17:56.880 align:middle line:84% I've got a magnetic field that's oriented like this. 01:17:56.880 --> 01:17:59.010 align:middle line:84% And I'll say, OK, well, all my particles 01:17:59.010 --> 01:18:03.550 align:middle line:84% are going to gyrate around this magnetic field. 01:18:03.550 --> 01:18:06.330 align:middle line:84% And so what you end up with is a sheath 01:18:06.330 --> 01:18:10.110 align:middle line:84% that's very close in like this, but you end up 01:18:10.110 --> 01:18:13.980 align:middle line:84% with a presheath that is effectively swept out 01:18:13.980 --> 01:18:19.790 align:middle line:84% by the projection of the probe area along this magnetic field. 01:18:19.790 --> 01:18:23.450 align:middle line:84% And you have particles which enter the presheath 01:18:23.450 --> 01:18:26.780 align:middle line:84% and then travel down the magnetic fields 01:18:26.780 --> 01:18:28.340 align:middle line:90% until they hit our probe. 01:18:28.340 --> 01:18:31.530 align:middle line:90% 01:18:31.530 --> 01:18:33.260 align:middle line:84% So I've probably drawn too many lines, 01:18:33.260 --> 01:18:35.640 align:middle line:84% but this is the rough picture here. 01:18:35.640 --> 01:18:42.550 align:middle line:84% So this is the presheath, this is the sheath, 01:18:42.550 --> 01:18:43.695 align:middle line:90% and this is the probe. 01:18:43.695 --> 01:18:46.415 align:middle line:90% 01:18:46.415 --> 01:18:48.790 align:middle line:84% It turns out that if you assume your plasma is completely 01:18:48.790 --> 01:18:51.010 align:middle line:84% collisionless, this all falls apart 01:18:51.010 --> 01:18:54.400 align:middle line:84% because you need some process to knock the particles from being 01:18:54.400 --> 01:18:59.080 align:middle line:84% outside the presheath to inside the presheath to keep fueling 01:18:59.080 --> 01:19:00.800 align:middle line:90% the particle flux onto it. 01:19:00.800 --> 01:19:03.190 align:middle line:84% And so even if the collisionality is very, very 01:19:03.190 --> 01:19:05.620 align:middle line:84% weak, this presheath keeps expanding 01:19:05.620 --> 01:19:08.200 align:middle line:84% until it reaches a length scale where the collisionanilty is 01:19:08.200 --> 01:19:08.750 align:middle line:90% important. 01:19:08.750 --> 01:19:10.167 align:middle line:84% So the exact size of the presheath 01:19:10.167 --> 01:19:11.620 align:middle line:90% is very hard to calculate. 01:19:11.620 --> 01:19:13.360 align:middle line:84% And this is only in the case that I've 01:19:13.360 --> 01:19:15.550 align:middle line:84% sketched here for very strong magnetic fields. 01:19:15.550 --> 01:19:17.538 align:middle line:84% Of course, if your magnetic fields are weaker, 01:19:17.538 --> 01:19:19.330 align:middle line:84% your particles are going to be doing things 01:19:19.330 --> 01:19:20.890 align:middle line:84% like this, where they spend some of their time 01:19:20.890 --> 01:19:23.050 align:middle line:84% inside the presheath and some of the time outside. 01:19:23.050 --> 01:19:26.170 align:middle line:84% And then you have to start doing single particle orbit 01:19:26.170 --> 01:19:29.110 align:middle line:84% calculations to see how many particles hit your probe, 01:19:29.110 --> 01:19:30.738 align:middle line:90% and it becomes very complicated. 01:19:30.738 --> 01:19:32.530 align:middle line:84% So probes in magnetic fields are very hard, 01:19:32.530 --> 01:19:36.650 align:middle line:84% and people spend their whole careers trying to work on that. 01:19:36.650 --> 01:19:42.100 align:middle line:84% The other thing we haven't discussed is collisions here. 01:19:42.100 --> 01:19:45.310 align:middle line:84% So in the sheath theory that I presented, 01:19:45.310 --> 01:19:46.880 align:middle line:90% I didn't derive it, as I said. 01:19:46.880 --> 01:19:49.457 align:middle line:84% But I had assumed that there were no collisions here 01:19:49.457 --> 01:19:51.290 align:middle line:84% so all the particles would just be streaming 01:19:51.290 --> 01:19:52.970 align:middle line:90% inwards without interacting. 01:19:52.970 --> 01:19:54.590 align:middle line:84% In a real plasma, especially the sorts 01:19:54.590 --> 01:19:56.132 align:middle line:84% of temperatures the plasma is at when 01:19:56.132 --> 01:19:57.860 align:middle line:84% you tend to work with Langmuir probes, 01:19:57.860 --> 01:20:00.020 align:middle line:84% collisions are actually quite important. 01:20:00.020 --> 01:20:02.480 align:middle line:84% And these collisions are going to modify 01:20:02.480 --> 01:20:04.520 align:middle line:84% your flux, which, if you remember, 01:20:04.520 --> 01:20:06.080 align:middle line:84% was the first thing we wrote down-- 01:20:06.080 --> 01:20:07.220 align:middle line:90% gamma. 01:20:07.220 --> 01:20:09.650 align:middle line:84% And they're going to modify it to include 01:20:09.650 --> 01:20:11.610 align:middle line:90% particle diffusion here. 01:20:11.610 --> 01:20:14.780 align:middle line:84% And so you need to go back and rederive all these results 01:20:14.780 --> 01:20:16.940 align:middle line:84% with this modified gamma in this case. 01:20:16.940 --> 01:20:19.190 align:middle line:84% And again, Hutchinson spends a very long time 01:20:19.190 --> 01:20:21.290 align:middle line:84% on this in the book because this is something 01:20:21.290 --> 01:20:22.468 align:middle line:90% that he cares about deeply. 01:20:22.468 --> 01:20:24.260 align:middle line:84% And if you're working with Langmuir probes, 01:20:24.260 --> 01:20:26.840 align:middle line:84% then one very good resource is to go look at his book. 01:20:26.840 --> 01:20:29.300 align:middle line:84% And then Lieberman and Lichtenberg 01:20:29.300 --> 01:20:31.910 align:middle line:84% also deal with magnetized sheaths and collisional sheaths 01:20:31.910 --> 01:20:33.000 align:middle line:90% in a great deal of detail. 01:20:33.000 --> 01:20:35.430 align:middle line:84% So there's more information there. 01:20:35.430 --> 01:20:37.080 align:middle line:84% So I'm very happy to take questions. 01:20:37.080 --> 01:20:39.372 align:middle line:84% But I actually have to go teach another class virtually 01:20:39.372 --> 01:20:41.225 align:middle line:84% in 4 minutes, so please keep them brief. 01:20:41.225 --> 01:20:51.990 align:middle line:90% 01:20:51.990 --> 01:20:55.200 align:middle line:84% AUDIENCE: Are there games to play when you sample? 01:20:55.200 --> 01:20:58.140 align:middle line:84% Instead of doing a perfect sweep, 01:20:58.140 --> 01:21:03.697 align:middle line:84% if you're clever about trying to change 01:21:03.697 --> 01:21:06.280 align:middle line:84% the frequency with which you do that, you can get a little bit 01:21:06.280 --> 01:21:09.730 align:middle line:84% better idea of how consistent your response 01:21:09.730 --> 01:21:11.350 align:middle line:90% is in time or something? 01:21:11.350 --> 01:21:13.713 align:middle line:84% Make sure you're not having sampling effect. 01:21:13.713 --> 01:21:15.130 align:middle line:84% JACK HARE: I can imagine you could 01:21:15.130 --> 01:21:17.680 align:middle line:84% do all sorts of clever things with some fast sweeps 01:21:17.680 --> 01:21:20.920 align:middle line:84% and some slow sweeps and not sweeping the whole range, 01:21:20.920 --> 01:21:23.720 align:middle line:84% if you only need a few points to fit the characteristic. 01:21:23.720 --> 01:21:26.770 align:middle line:84% So yeah, I imagine there are some things that you can 01:21:26.770 --> 01:21:28.540 align:middle line:90% do in order to optimize that. 01:21:28.540 --> 01:21:33.990 align:middle line:90% 01:21:33.990 --> 01:21:35.220 align:middle line:90% Any other questions? 01:21:35.220 --> 01:21:39.760 align:middle line:90% 01:21:39.760 --> 01:21:40.260 align:middle line:90% Cool. 01:21:40.260 --> 01:21:41.760 align:middle line:84% Well, thank you very much, everyone. 01:21:41.760 --> 01:21:43.650 align:middle line:84% This is likely to be online on Thursday 01:21:43.650 --> 01:21:46.560 align:middle line:84% as well, as I won't be able to stop self-isolating by then. 01:21:46.560 --> 01:21:49.470 align:middle line:84% And we'll be talking about refractive index diagnostics. 01:21:49.470 --> 01:21:51.952 align:middle line:84% So have a good day, and I'll see some of you 01:21:51.952 --> 01:21:53.910 align:middle line:84% again in a couple of minutes at the next class. 01:21:53.910 --> 01:21:55.910 align:middle line:90% So bye for now. 01:21:55.910 --> 01:21:59.000 align:middle line:90%