WEBVTT 00:00:00.000 --> 00:00:00.966 align:middle line:90% 00:00:00.966 --> 00:00:02.415 align:middle line:90% [SQUEAKING] 00:00:02.415 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:10.150 align:middle line:90% [CLICKING] 00:00:10.150 --> 00:00:11.020 align:middle line:90% JACK HARE: Right. 00:00:11.020 --> 00:00:15.070 align:middle line:84% So we talked already about free-free radiation, 00:00:15.070 --> 00:00:17.650 align:middle line:84% bremsstrahlung, electron, cyclotron radiation. 00:00:17.650 --> 00:00:19.450 align:middle line:84% We talked a little bit about free bounds 00:00:19.450 --> 00:00:21.770 align:middle line:90% or recombination radiation. 00:00:21.770 --> 00:00:24.490 align:middle line:84% And these are all relatively simple 00:00:24.490 --> 00:00:27.970 align:middle line:84% compared to the complexity of bound-bound radiation, which 00:00:27.970 --> 00:00:31.090 align:middle line:84% is what we're going to be dealing with today and maybe 00:00:31.090 --> 00:00:33.430 align:middle line:90% for the next few classes. 00:00:33.430 --> 00:00:44.770 align:middle line:90% 00:00:44.770 --> 00:00:47.500 align:middle line:84% The bound-bound radiation is the radiation 00:00:47.500 --> 00:00:53.380 align:middle line:84% that we get when electrons move between discrete energy levels 00:00:53.380 --> 00:00:58.170 align:middle line:84% within what we're going to call it Eta 00:00:58.170 --> 00:01:02.060 align:middle line:84% So we have maybe four energy levels like this. 00:01:02.060 --> 00:01:06.560 align:middle line:84% We might have an electron in some excited state 00:01:06.560 --> 00:01:09.830 align:middle line:84% up here that drops down to this energy level, 00:01:09.830 --> 00:01:12.680 align:middle line:90% and it will emit a photon. 00:01:12.680 --> 00:01:16.850 align:middle line:84% Or there could be an electron in the same energy level here 00:01:16.850 --> 00:01:19.520 align:middle line:84% that drops down all the way to the ground state, 00:01:19.520 --> 00:01:22.040 align:middle line:84% and emits a photon of a different energy. 00:01:22.040 --> 00:01:27.540 align:middle line:84% And what we would see on our spectrometer 00:01:27.540 --> 00:01:32.370 align:middle line:84% would be some distinct lines corresponding 00:01:32.370 --> 00:01:35.680 align:middle line:90% to these different photons. 00:01:35.680 --> 00:01:38.070 align:middle line:84% And so one of the things we might want to know 00:01:38.070 --> 00:01:40.020 align:middle line:84% is how strong should these lines be? 00:01:40.020 --> 00:01:41.070 align:middle line:90% Where should they be? 00:01:41.070 --> 00:01:43.170 align:middle line:84% If I see these lines, what does it tell me 00:01:43.170 --> 00:01:44.940 align:middle line:84% about the temperature of my plasma, 00:01:44.940 --> 00:01:49.130 align:middle line:84% about the density of the plasma, things like this. 00:01:49.130 --> 00:01:52.610 align:middle line:84% And so, again, bound-bound radiation here, 00:01:52.610 --> 00:01:55.000 align:middle line:84% we might also call this line radiation. 00:01:55.000 --> 00:01:57.407 align:middle line:90% 00:01:57.407 --> 00:01:59.740 align:middle line:84% The reason being that when people first did spectroscopy 00:01:59.740 --> 00:02:02.110 align:middle line:84% on films, you would get these peaks showing up 00:02:02.110 --> 00:02:05.120 align:middle line:84% as very discrete lines on their spectra. 00:02:05.120 --> 00:02:07.300 align:middle line:84% So they're called spectroscopic lines. 00:02:07.300 --> 00:02:09.340 align:middle line:84% And just a note on notation here, 00:02:09.340 --> 00:02:11.230 align:middle line:84% or at least on the [INAUDIBLE], we're 00:02:11.230 --> 00:02:15.070 align:middle line:84% going to be using the word atom a lot here. 00:02:15.070 --> 00:02:17.230 align:middle line:84% And for the purposes of this section, 00:02:17.230 --> 00:02:29.350 align:middle line:84% an atom is a nucleus with one or more electrons. 00:02:29.350 --> 00:02:34.110 align:middle line:84% So this could be 73rd states ionized tungsten. 00:02:34.110 --> 00:02:35.910 align:middle line:84% As long as it's still got an electron, 00:02:35.910 --> 00:02:38.720 align:middle line:84% we're still going to have spectroscopic lines coming out 00:02:38.720 --> 00:02:39.220 align:middle line:90% of here. 00:02:39.220 --> 00:02:42.250 align:middle line:84% So this is not an atom as in an unionized ion. 00:02:42.250 --> 00:02:45.280 align:middle line:84% This is a slightly different definition here. 00:02:45.280 --> 00:02:49.590 align:middle line:84% So, yeah, this could be something like argon 5 plus. 00:02:49.590 --> 00:02:51.060 align:middle line:90% That still counts as an atom. 00:02:51.060 --> 00:02:54.380 align:middle line:90% 00:02:54.380 --> 00:02:58.510 align:middle line:84% So in order to get a feel for some of the physics that's 00:02:58.510 --> 00:03:00.010 align:middle line:84% involved in these transitions, we're 00:03:00.010 --> 00:03:01.930 align:middle line:84% going to go to a very simple system. 00:03:01.930 --> 00:03:03.850 align:middle line:84% We're going to go to a two level system 00:03:03.850 --> 00:03:07.460 align:middle line:84% and consider the processes which exist in that two level system. 00:03:07.460 --> 00:03:11.500 align:middle line:84% So in this two level system, we just have a level L lower, 00:03:11.500 --> 00:03:15.590 align:middle line:90% and another level U for upper. 00:03:15.590 --> 00:03:18.800 align:middle line:84% The upper level is an energy EU, and the lower level 00:03:18.800 --> 00:03:21.610 align:middle line:90% is energy at EL. 00:03:21.610 --> 00:03:28.650 align:middle line:84% So this is a two level system, and we're 00:03:28.650 --> 00:03:32.190 align:middle line:84% going to consider three processes that go on 00:03:32.190 --> 00:03:34.350 align:middle line:90% inside this two level system. 00:03:34.350 --> 00:03:35.010 align:middle line:90% OK. 00:03:35.010 --> 00:03:41.140 align:middle line:90% So two levels, three processes. 00:03:41.140 --> 00:03:43.838 align:middle line:84% And some of you may have seen this before already, we're 00:03:43.838 --> 00:03:45.880 align:middle line:84% going to be looking at things called the Einstein 00:03:45.880 --> 00:03:46.870 align:middle line:90% coefficients. 00:03:46.870 --> 00:03:54.260 align:middle line:90% 00:03:54.260 --> 00:03:56.240 align:middle line:84% Can't be bothered to write out coefficients. 00:03:56.240 --> 00:03:56.810 align:middle line:90% OK. 00:03:56.810 --> 00:03:59.420 align:middle line:84% So the three processes that we're going to be looking at, 00:03:59.420 --> 00:04:01.940 align:middle line:84% the first process I'll just draw on here, 00:04:01.940 --> 00:04:04.700 align:middle line:84% is what is called spontaneous emission. 00:04:04.700 --> 00:04:06.320 align:middle line:84% This is probably the one that we're 00:04:06.320 --> 00:04:08.195 align:middle line:84% most often thinking about when we're thinking 00:04:08.195 --> 00:04:09.590 align:middle line:90% about spectroscopy here. 00:04:09.590 --> 00:04:12.890 align:middle line:84% We have an electron in the upper level, 00:04:12.890 --> 00:04:19.500 align:middle line:84% and it drops spontaneously the lower level and emits a photon. 00:04:19.500 --> 00:04:21.600 align:middle line:84% There's a frequency H nu, which is 00:04:21.600 --> 00:04:25.330 align:middle line:84% equal to the difference in energy between the upper level 00:04:25.330 --> 00:04:28.660 align:middle line:90% and the lower level here. 00:04:28.660 --> 00:04:31.350 align:middle line:90% So this is spontaneous emission. 00:04:31.350 --> 00:04:44.350 align:middle line:90% 00:04:44.350 --> 00:04:47.500 align:middle line:84% Just want to note as an aside here that Hutchinson calls 00:04:47.500 --> 00:04:56.290 align:middle line:84% these levels I and J. This is his notation, 00:04:56.290 --> 00:04:58.630 align:middle line:84% and I've swapped it around because then you 00:04:58.630 --> 00:05:02.410 align:middle line:84% start writing like N subscript I for the number of states. 00:05:02.410 --> 00:05:04.420 align:middle line:84% But then we also use the ion density, 00:05:04.420 --> 00:05:05.537 align:middle line:90% and I found it confusing. 00:05:05.537 --> 00:05:07.870 align:middle line:84% And I just found it easier to think of them as the lower 00:05:07.870 --> 00:05:09.460 align:middle line:90% and upper states instead. 00:05:09.460 --> 00:05:10.360 align:middle line:90% OK. 00:05:10.360 --> 00:05:13.340 align:middle line:84% And this process here has a rate, 00:05:13.340 --> 00:05:15.580 align:middle line:84% which it occurs at an Einstein coefficient 00:05:15.580 --> 00:05:20.170 align:middle line:84% that we call A subscript U subscript L. So this is 00:05:20.170 --> 00:05:23.770 align:middle line:84% the rate of spontaneous emission going from level U 00:05:23.770 --> 00:05:26.330 align:middle line:84% to level L. Of course, it's a two level system 00:05:26.330 --> 00:05:29.390 align:middle line:84% so that's the only spontaneous emission process we can have. 00:05:29.390 --> 00:05:31.690 align:middle line:84% But when we expand this to a multi-level system, 00:05:31.690 --> 00:05:35.367 align:middle line:84% we have A subscript, sub number subscript, some other number, 00:05:35.367 --> 00:05:37.450 align:middle line:84% and that will be the spontaneous emission from one 00:05:37.450 --> 00:05:40.090 align:middle line:90% level to another level. 00:05:40.090 --> 00:05:40.690 align:middle line:90% OK. 00:05:40.690 --> 00:05:43.060 align:middle line:84% If we have spontaneous emission, we 00:05:43.060 --> 00:05:47.190 align:middle line:84% should have a process which looks the exact opposite, 00:05:47.190 --> 00:05:49.710 align:middle line:84% and this is spontaneous absorption. 00:05:49.710 --> 00:05:54.420 align:middle line:84% We tend to drop the spontaneous bit and just call it absorption. 00:05:54.420 --> 00:05:59.650 align:middle line:90% 00:05:59.650 --> 00:06:02.680 align:middle line:84% And indeed, it looks just like the mirror of this process. 00:06:02.680 --> 00:06:05.740 align:middle line:84% We have some electron in the ground state, 00:06:05.740 --> 00:06:10.390 align:middle line:84% some photon that happens to have exactly the right energy comes 00:06:10.390 --> 00:06:10.930 align:middle line:90% in. 00:06:10.930 --> 00:06:17.730 align:middle line:84% The electron is excited up, and the photon is absorbed here. 00:06:17.730 --> 00:06:21.590 align:middle line:84% And this has a rate, which is the Einstein E 00:06:21.590 --> 00:06:26.030 align:middle line:84% coefficient between the lower and the upper level. 00:06:26.030 --> 00:06:28.700 align:middle line:84% But this is now multiplied, as well, 00:06:28.700 --> 00:06:32.420 align:middle line:84% by the density of photons with this specific energy. 00:06:32.420 --> 00:06:36.140 align:middle line:84% So we'll use RHO to be the density, the number of photons 00:06:36.140 --> 00:06:42.590 align:middle line:84% per meter cubed, of photons with exactly the right energy RHO UL, 00:06:42.590 --> 00:06:43.808 align:middle line:90% which can do that excitation. 00:06:43.808 --> 00:06:45.350 align:middle line:84% There's an important difference here. 00:06:45.350 --> 00:06:46.730 align:middle line:90% This happens spontaneously. 00:06:46.730 --> 00:06:49.670 align:middle line:84% It doesn't matter how many photons or electrons are around 00:06:49.670 --> 00:06:50.390 align:middle line:90% in this system. 00:06:50.390 --> 00:06:54.150 align:middle line:84% This one has to happen only when this photon comes in. 00:06:54.150 --> 00:06:56.480 align:middle line:84% And so the rate at which this process occurs 00:06:56.480 --> 00:06:58.838 align:middle line:84% is going to depend on the number of photons. 00:06:58.838 --> 00:07:00.755 align:middle line:84% If there aren't any photons around to do this, 00:07:00.755 --> 00:07:03.590 align:middle line:90% this process just won't happen. 00:07:03.590 --> 00:07:05.290 align:middle line:90% OK. 00:07:05.290 --> 00:07:13.050 align:middle line:84% And then the final process that we're going to consider, 00:07:13.050 --> 00:07:16.185 align:middle line:84% instead of spontaneous emission, is stimulated emission. 00:07:16.185 --> 00:07:28.410 align:middle line:90% 00:07:28.410 --> 00:07:31.320 align:middle line:84% Stimulated emission is a very interesting process, 00:07:31.320 --> 00:07:34.650 align:middle line:84% because here, we have electron in the excited 00:07:34.650 --> 00:07:36.750 align:middle line:84% state up here in the upper state, 00:07:36.750 --> 00:07:43.680 align:middle line:84% and some photon comes through with energy that is 00:07:43.680 --> 00:07:45.760 align:middle line:90% the same as this band gap here. 00:07:45.760 --> 00:07:48.280 align:middle line:84% Now, of course, this photon can't be absorbed. 00:07:48.280 --> 00:07:49.530 align:middle line:90% There's only this upper level. 00:07:49.530 --> 00:07:51.630 align:middle line:84% There's nowhere for that energy to be absorbed. 00:07:51.630 --> 00:07:55.065 align:middle line:84% What it does, instead, is it encourages this electron 00:07:55.065 --> 00:08:01.100 align:middle line:84% to drop down, and it stimulates the emission of a second photon. 00:08:01.100 --> 00:08:05.080 align:middle line:84% So the first photon comes through still at this energy, 00:08:05.080 --> 00:08:09.190 align:middle line:84% and we have a second photon at exactly the same energy. 00:08:09.190 --> 00:08:13.520 align:middle line:84% And this is the process by which lasers work. 00:08:13.520 --> 00:08:19.360 align:middle line:84% So this has Einstein coefficient BUL. 00:08:19.360 --> 00:08:23.260 align:middle line:84% So we have a transition from the upper state to the lower state. 00:08:23.260 --> 00:08:26.440 align:middle line:84% And again, the rate at which this process occurs 00:08:26.440 --> 00:08:28.510 align:middle line:84% depends on the density of photons. 00:08:28.510 --> 00:08:31.600 align:middle line:84% We have to have some photons to stimulate the emission so it 00:08:31.600 --> 00:08:35.289 align:middle line:84% has the same RHO density of photons 00:08:35.289 --> 00:08:37.270 align:middle line:90% with that specific energy. 00:08:37.270 --> 00:08:40.120 align:middle line:84% So note these are actually different coefficients. 00:08:40.120 --> 00:08:44.860 align:middle line:84% The rate of absorption, the LU, is not the same, in general, 00:08:44.860 --> 00:08:48.250 align:middle line:90% as the rate of emission BUL. 00:08:48.250 --> 00:08:49.680 align:middle line:90% OK. 00:08:49.680 --> 00:08:52.140 align:middle line:84% So those are the three processes we're going to consider, 00:08:52.140 --> 00:08:56.360 align:middle line:84% and we'll then start deriving what these coefficients actually 00:08:56.360 --> 00:08:56.860 align:middle line:90% are. 00:08:56.860 --> 00:08:59.110 align:middle line:84% So any questions before we keep going? 00:08:59.110 --> 00:09:00.110 align:middle line:90% Yes. 00:09:00.110 --> 00:09:08.259 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:09:08.259 --> 00:09:10.640 align:middle line:84% JACK HARE: Basically, every subscript is going to be U 00:09:10.640 --> 00:09:14.340 align:middle line:90% and L. So UL here. 00:09:14.340 --> 00:09:19.940 align:middle line:84% So this new UL is just to say it's 00:09:19.940 --> 00:09:22.580 align:middle line:84% a wave with energy which is equal, the difference, 00:09:22.580 --> 00:09:24.330 align:middle line:84% between the energy of the two levels here. 00:09:24.330 --> 00:09:26.288 align:middle line:84% So the total energy in the system is conserved. 00:09:26.288 --> 00:09:29.730 align:middle line:84% Yeah, I guess I could write these as uppercase L 00:09:29.730 --> 00:09:33.760 align:middle line:90% and uppercase V like that. 00:09:33.760 --> 00:09:38.680 align:middle line:90% 00:09:38.680 --> 00:09:40.840 align:middle line:90% Yes. 00:09:40.840 --> 00:09:41.340 align:middle line:90% Yeah. 00:09:41.340 --> 00:09:43.260 align:middle line:84% So the important thing about stimulated emission 00:09:43.260 --> 00:09:45.820 align:middle line:84% is that you stimulate a photon with exactly the same energy. 00:09:45.820 --> 00:09:49.750 align:middle line:84% And, in fact, the photon in the same mode of your cavity. 00:09:49.750 --> 00:09:54.280 align:middle line:84% So polarization and things like that are also the same. 00:09:54.280 --> 00:09:55.462 align:middle line:90% Yes. 00:09:55.462 --> 00:10:00.082 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:10:00.082 --> 00:10:01.390 align:middle line:90% JACK HARE: Excellent question. 00:10:01.390 --> 00:10:02.973 align:middle line:84% We will get on to that when we do line 00:10:02.973 --> 00:10:04.930 align:middle line:90% broadening in a few classes. 00:10:04.930 --> 00:10:08.955 align:middle line:84% The rough answer is, roughly, there's 00:10:08.955 --> 00:10:11.080 align:middle line:84% a sort of uncertainty principle type thing involved 00:10:11.080 --> 00:10:13.450 align:middle line:84% if you do-- if the process happens quickly, 00:10:13.450 --> 00:10:16.600 align:middle line:84% you can be less sure about the energy. 00:10:16.600 --> 00:10:19.040 align:middle line:90% So, yeah. 00:10:19.040 --> 00:10:21.710 align:middle line:84% But these lines can be extraordinarily narrow, right? 00:10:21.710 --> 00:10:24.740 align:middle line:84% And so if you look at a spectra, then these lines 00:10:24.740 --> 00:10:26.000 align:middle line:90% could be extremely narrow. 00:10:26.000 --> 00:10:29.780 align:middle line:84% That would correspond to needing very precisely the same energy. 00:10:29.780 --> 00:10:30.540 align:middle line:90% Yeah. 00:10:30.540 --> 00:10:31.040 align:middle line:90% OK. 00:10:31.040 --> 00:10:32.370 align:middle line:90% Any other questions? 00:10:32.370 --> 00:10:35.005 align:middle line:90% 00:10:35.005 --> 00:10:35.505 align:middle line:90% Yeah. 00:10:35.505 --> 00:10:40.363 align:middle line:90% 00:10:40.363 --> 00:10:42.530 align:middle line:84% Is what we're going to get onto, you're quite right. 00:10:42.530 --> 00:10:43.613 align:middle line:90% But, no, no, you're right. 00:10:43.613 --> 00:10:46.063 align:middle line:84% They can't just all be random because in steady state, 00:10:46.063 --> 00:10:47.480 align:middle line:84% they're going to have to match up. 00:10:47.480 --> 00:10:48.890 align:middle line:84% Otherwise, our system will be driven 00:10:48.890 --> 00:10:50.160 align:middle line:90% in one direction or the other. 00:10:50.160 --> 00:10:52.850 align:middle line:84% And so we are, indeed, thank you for the transition, 00:10:52.850 --> 00:10:55.430 align:middle line:84% going to use thermodynamic arguments, which 00:10:55.430 --> 00:10:58.830 align:middle line:84% we did once before already to match 00:10:58.830 --> 00:11:01.540 align:middle line:84% the emissivity and the absorption. 00:11:01.540 --> 00:11:07.930 align:middle line:84% So now, we're going to use thermodynamics, 00:11:07.930 --> 00:11:15.270 align:middle line:84% and we're going to do that to link these coefficients AUL, 00:11:15.270 --> 00:11:18.762 align:middle line:90% BLU, and BUL. 00:11:18.762 --> 00:11:22.090 align:middle line:90% 00:11:22.090 --> 00:11:22.590 align:middle line:90% OK. 00:11:22.590 --> 00:11:30.022 align:middle line:90% 00:11:30.022 --> 00:11:32.230 align:middle line:84% I think it's best if I just cover this off and start. 00:11:32.230 --> 00:11:37.200 align:middle line:90% 00:11:37.200 --> 00:11:37.740 align:middle line:90% OK. 00:11:37.740 --> 00:11:41.720 align:middle line:84% So one thing we want to be able to write down 00:11:41.720 --> 00:11:46.600 align:middle line:84% is the number of electrons in the upper state 00:11:46.600 --> 00:11:49.720 align:middle line:84% versus the number of electrons in the lower state here. 00:11:49.720 --> 00:11:54.510 align:middle line:84% And to do this, we're going to use a Boltzmann distribution 00:11:54.510 --> 00:11:55.010 align:middle line:90% for that. 00:11:55.010 --> 00:11:59.070 align:middle line:84% So this is a Boltzmann type argument 00:11:59.070 --> 00:12:04.083 align:middle line:84% where we say that the number of electrons in the upper state-- 00:12:04.083 --> 00:12:06.000 align:middle line:84% imagine we have an ensemble of these two level 00:12:06.000 --> 00:12:09.210 align:middle line:84% systems non-interacting a large number of them, on average, 00:12:09.210 --> 00:12:12.240 align:middle line:84% the number of these systems, which are in the upper state, 00:12:12.240 --> 00:12:17.910 align:middle line:84% is going to be proportional to the exponential of minus 00:12:17.910 --> 00:12:21.270 align:middle line:84% the energy of that upper state divided by temperature. 00:12:21.270 --> 00:12:22.770 align:middle line:84% And again, we're writing temperature 00:12:22.770 --> 00:12:25.150 align:middle line:84% in energy units, like, joules, or EV. 00:12:25.150 --> 00:12:27.690 align:middle line:84% I folded the Boltzmann constant inside the T 00:12:27.690 --> 00:12:30.100 align:middle line:84% here so I don't have to keep writing. 00:12:30.100 --> 00:12:32.090 align:middle line:84% This doesn't have any normalization on it. 00:12:32.090 --> 00:12:35.450 align:middle line:84% But for a two level system, the normalization is very simple. 00:12:35.450 --> 00:12:40.490 align:middle line:84% We'll just divide an upper by and lower, 00:12:40.490 --> 00:12:48.170 align:middle line:84% and we will get exponential of minus E upper minus E 00:12:48.170 --> 00:12:52.638 align:middle line:84% lower upon temperature like that. 00:12:52.638 --> 00:12:54.430 align:middle line:84% And you'll notice, of course, straight away 00:12:54.430 --> 00:13:01.950 align:middle line:84% that this quantity is just Planck's constant NU UL. 00:13:01.950 --> 00:13:04.740 align:middle line:84% There's a slight subtlety to this. 00:13:04.740 --> 00:13:07.020 align:middle line:84% If you have systems with degeneracy so 00:13:07.020 --> 00:13:08.880 align:middle line:84% that there are multiple different ways 00:13:08.880 --> 00:13:10.860 align:middle line:84% the energy can be in these states. 00:13:10.860 --> 00:13:13.750 align:middle line:84% So it could be spin, or something like that. 00:13:13.750 --> 00:13:17.220 align:middle line:84% Then you have an extra factor due to degeneracy at the front 00:13:17.220 --> 00:13:17.760 align:middle line:90% here. 00:13:17.760 --> 00:13:24.820 align:middle line:84% This degeneracy factor is like G of U over G of L like that. 00:13:24.820 --> 00:13:27.490 align:middle line:84% So these could be numbers like 1 or 2, things like that. 00:13:27.490 --> 00:13:30.660 align:middle line:84% So this is just making this slightly more generic 00:13:30.660 --> 00:13:33.450 align:middle line:84% for a broader range of two level systems. 00:13:33.450 --> 00:13:35.430 align:middle line:84% For now, you can just consider this factor 00:13:35.430 --> 00:13:37.210 align:middle line:90% to be one for what we're doing. 00:13:37.210 --> 00:13:41.485 align:middle line:84% But this factor is due to degeneracy. 00:13:41.485 --> 00:13:50.220 align:middle line:90% 00:13:50.220 --> 00:13:53.490 align:middle line:84% Then we also need to have an expression 00:13:53.490 --> 00:13:58.500 align:middle line:84% for the density of photons with a given frequency here. 00:13:58.500 --> 00:14:01.875 align:middle line:84% And here, we're going to go back to our blackbody spectrum. 00:14:01.875 --> 00:14:05.720 align:middle line:90% 00:14:05.720 --> 00:14:10.520 align:middle line:84% And that density of photons is going to be the standard result 00:14:10.520 --> 00:14:23.740 align:middle line:84% 8pi H nu cubed over C cubed, 1 over the exponential of H nu 00:14:23.740 --> 00:14:29.820 align:middle line:90% on T minus 1. 00:14:29.820 --> 00:14:32.700 align:middle line:84% So this is for a system at temperature T. 00:14:32.700 --> 00:14:35.730 align:middle line:84% What is the density of photons with frequency nu, 00:14:35.730 --> 00:14:39.300 align:middle line:84% or what is the density of photons of energy H nu. 00:14:39.300 --> 00:14:42.360 align:middle line:90% 00:14:42.360 --> 00:14:44.910 align:middle line:84% And finally, we're going to use the thermodynamic argument 00:14:44.910 --> 00:14:47.730 align:middle line:84% that these systems should be in equilibrium. 00:14:47.730 --> 00:14:54.670 align:middle line:90% 00:14:54.670 --> 00:14:59.670 align:middle line:84% And so that says that the rate at which atoms 00:14:59.670 --> 00:15:03.030 align:middle line:84% fall from their top level to the bottom level, which 00:15:03.030 --> 00:15:16.730 align:middle line:84% is AUL plus BUL RHO of nu UL times by the number of atoms 00:15:16.730 --> 00:15:21.440 align:middle line:84% in the top level must be equal to the inverse process, which 00:15:21.440 --> 00:15:29.780 align:middle line:84% is the rate of absorption BLU of photons with energy nu UL times 00:15:29.780 --> 00:15:32.360 align:middle line:84% the number of atoms in the lower level. 00:15:32.360 --> 00:15:37.010 align:middle line:90% 00:15:37.010 --> 00:15:40.300 align:middle line:84% So this just says the rate of de-excitation 00:15:40.300 --> 00:15:42.220 align:middle line:84% is equal to the rate of excitation 00:15:42.220 --> 00:15:44.230 align:middle line:90% our system is in steady state. 00:15:44.230 --> 00:15:48.040 align:middle line:84% And if you put all of these together, 00:15:48.040 --> 00:15:51.040 align:middle line:84% these effectively form a set of simultaneous equations, 00:15:51.040 --> 00:15:54.820 align:middle line:84% you find that, indeed, our Einstein coefficient 00:15:54.820 --> 00:16:07.120 align:middle line:84% for spontaneous emission AUL is equal to 8pi H nu UL cubed upon 00:16:07.120 --> 00:16:14.270 align:middle line:90% V cubed BUL, like this. 00:16:14.270 --> 00:16:16.930 align:middle line:84% This means that we have something 00:16:16.930 --> 00:16:24.390 align:middle line:84% linking the spontaneous emission to the stimulated emission. 00:16:24.390 --> 00:16:27.070 align:middle line:90% 00:16:27.070 --> 00:16:36.816 align:middle line:84% And more simply, we also find that GLBLU is equal to GUBUL. 00:16:36.816 --> 00:16:42.080 align:middle line:90% 00:16:42.080 --> 00:16:45.505 align:middle line:84% So again, this is the two different types of-- 00:16:45.505 --> 00:16:49.380 align:middle line:90% 00:16:49.380 --> 00:16:50.220 align:middle line:90% sorry. 00:16:50.220 --> 00:16:53.230 align:middle line:84% This is linking the processes going up and down 00:16:53.230 --> 00:16:55.650 align:middle line:90% and this is the absorption. 00:16:55.650 --> 00:17:02.020 align:middle line:90% 00:17:02.020 --> 00:17:05.520 align:middle line:90% And this is stimulated emission. 00:17:05.520 --> 00:17:11.630 align:middle line:90% 00:17:11.630 --> 00:17:13.430 align:middle line:84% So similar to how we did last time, 00:17:13.430 --> 00:17:16.790 align:middle line:84% we have used a system in complete thermodynamic 00:17:16.790 --> 00:17:20.780 align:middle line:84% equilibrium with Boltzmann occupation, steady state, 00:17:20.780 --> 00:17:22.460 align:middle line:90% and blackbody photons. 00:17:22.460 --> 00:17:25.667 align:middle line:84% And we've used that to pin these different coefficients 00:17:25.667 --> 00:17:26.250 align:middle line:90% to each other. 00:17:26.250 --> 00:17:28.099 align:middle line:84% So if we calculate even one of them, like, 00:17:28.099 --> 00:17:32.720 align:middle line:84% BUL, we now immediately know BLU and AUL. 00:17:32.720 --> 00:17:35.820 align:middle line:84% And so that's good because we get all of them for free. 00:17:35.820 --> 00:17:39.560 align:middle line:84% But it also means that all the processes are linked together 00:17:39.560 --> 00:17:42.020 align:middle line:90% in this system. 00:17:42.020 --> 00:17:42.900 align:middle line:90% OK. 00:17:42.900 --> 00:17:45.310 align:middle line:90% Any questions on that? 00:17:45.310 --> 00:17:47.520 align:middle line:84% I then tell you in a moment how we actually calculate 00:17:47.520 --> 00:17:49.790 align:middle line:84% even one of these so that we can get the [INAUDIBLE]. 00:17:49.790 --> 00:17:52.290 align:middle line:84% But now, it's important just to realize that those three are 00:17:52.290 --> 00:17:53.274 align:middle line:90% [INAUDIBLE]. 00:17:53.274 --> 00:18:13.932 align:middle line:90% 00:18:13.932 --> 00:18:16.140 align:middle line:84% Also, the cool thing is, this has no material physics 00:18:16.140 --> 00:18:17.700 align:middle line:84% in the relationship between these, 00:18:17.700 --> 00:18:19.900 align:middle line:90% has no materials physics in it. 00:18:19.900 --> 00:18:21.280 align:middle line:90% This is related to lasers. 00:18:21.280 --> 00:18:24.940 align:middle line:84% For laser, you need a three or a four level system 00:18:24.940 --> 00:18:27.920 align:middle line:84% because you have to pump the excited state. 00:18:27.920 --> 00:18:32.270 align:middle line:84% And then you have to have a metal stable state halfway down. 00:18:32.270 --> 00:18:34.577 align:middle line:84% You can't actually get lasing in a two level system. 00:18:34.577 --> 00:18:36.660 align:middle line:84% But in a three level, four level system, you have. 00:18:36.660 --> 00:18:40.130 align:middle line:84% So the systems which are favorable to lasing 00:18:40.130 --> 00:18:42.920 align:middle line:84% have specific energy levels that are spaced in a way 00:18:42.920 --> 00:18:46.150 align:middle line:84% that you can get stimulated emission. 00:18:46.150 --> 00:18:48.520 align:middle line:84% I just mentioned stimulated emission as that sort of-- 00:18:48.520 --> 00:18:52.540 align:middle line:84% I think when you look at this diagram here, 00:18:52.540 --> 00:18:55.410 align:middle line:84% you initially think to yourself, do I need stimulated emission? 00:18:55.410 --> 00:18:57.813 align:middle line:84% It seems a bit niche, like excessive bit of physics 00:18:57.813 --> 00:18:58.980 align:middle line:90% that I could throw in there. 00:18:58.980 --> 00:19:01.530 align:middle line:84% Can't I just get away with balancing spontaneous emission 00:19:01.530 --> 00:19:02.700 align:middle line:90% with absorption? 00:19:02.700 --> 00:19:03.810 align:middle line:90% They look quite similar. 00:19:03.810 --> 00:19:06.400 align:middle line:84% But it turns out that for the thermodynamic balance, 00:19:06.400 --> 00:19:08.350 align:middle line:84% and this is the thing that Einstein realized, 00:19:08.350 --> 00:19:09.810 align:middle line:84% which is why the Einstein coefficients, 00:19:09.810 --> 00:19:11.920 align:middle line:84% you have to include the stimulated emission in there. 00:19:11.920 --> 00:19:13.378 align:middle line:84% Otherwise, the theory doesn't work. 00:19:13.378 --> 00:19:15.690 align:middle line:84% You can't get them balanced properly. 00:19:15.690 --> 00:19:17.220 align:middle line:84% So I mentioned this as a thing which 00:19:17.220 --> 00:19:19.500 align:middle line:84% is important that lasers in the two level system. 00:19:19.500 --> 00:19:22.050 align:middle line:84% It doesn't play an important role 00:19:22.050 --> 00:19:25.030 align:middle line:84% and, also, this is material independent, right? 00:19:25.030 --> 00:19:26.730 align:middle line:84% Once we've calculated one of these, 00:19:26.730 --> 00:19:30.030 align:middle line:84% the other one depends purely on the energy level gaps. 00:19:30.030 --> 00:19:33.000 align:middle line:84% It has nothing to do with the crystal structure or something 00:19:33.000 --> 00:19:33.940 align:middle line:90% fun like that. 00:19:33.940 --> 00:19:34.440 align:middle line:90% Yes. 00:19:34.440 --> 00:19:40.090 align:middle line:90% 00:19:40.090 --> 00:19:41.860 align:middle line:84% Any reason it isn't stimulated absorption? 00:19:41.860 --> 00:19:44.090 align:middle line:90% What would that look like? 00:19:44.090 --> 00:19:46.660 align:middle line:84% So at the moment for spontaneous absorption, 00:19:46.660 --> 00:19:50.070 align:middle line:84% we have one photon come in, and the electron goes up. 00:19:50.070 --> 00:19:53.237 align:middle line:84% So with stimulated absorption with two photons coming in, 00:19:53.237 --> 00:19:55.070 align:middle line:84% one getting absorbed and one going forwards, 00:19:55.070 --> 00:19:56.528 align:middle line:84% like, the opposite of this process. 00:19:56.528 --> 00:19:59.280 align:middle line:90% 00:19:59.280 --> 00:20:01.740 align:middle line:84% I don't have a good answer for why that doesn't occur. 00:20:01.740 --> 00:20:03.280 align:middle line:90% But it's an interesting one. 00:20:03.280 --> 00:20:06.960 align:middle line:84% It would go as the square of the photon density field 00:20:06.960 --> 00:20:09.450 align:middle line:84% because you'd have to have two photons in the same volume 00:20:09.450 --> 00:20:10.200 align:middle line:90% to do it. 00:20:10.200 --> 00:20:13.440 align:middle line:84% So possibly, for most reasonable systems 00:20:13.440 --> 00:20:15.480 align:middle line:84% where we don't have that much radiation density, 00:20:15.480 --> 00:20:17.310 align:middle line:90% that wouldn't be a big-- 00:20:17.310 --> 00:20:20.382 align:middle line:84% that wouldn't be a very important process. 00:20:20.382 --> 00:20:21.590 align:middle line:90% But I see what you're saying. 00:20:21.590 --> 00:20:22.730 align:middle line:90% I hadn't thought about that. 00:20:22.730 --> 00:20:23.230 align:middle line:90% Yeah. 00:20:23.230 --> 00:20:24.730 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:20:24.730 --> 00:20:29.011 align:middle line:90% 00:20:29.011 --> 00:20:32.510 align:middle line:84% JACK HARE: Not in this, because in order for that to happen, 00:20:32.510 --> 00:20:38.290 align:middle line:84% you need to have some ill-defined, metastable level 00:20:38.290 --> 00:20:39.820 align:middle line:90% halfway up like this. 00:20:39.820 --> 00:20:41.590 align:middle line:90% But there has to be some level. 00:20:41.590 --> 00:20:43.600 align:middle line:84% It has to-- it doesn't have to be exactly there. 00:20:43.600 --> 00:20:45.310 align:middle line:84% Because of the uncertainty, it could 00:20:45.310 --> 00:20:47.600 align:middle line:84% be like this, or something like that. 00:20:47.600 --> 00:20:49.480 align:middle line:84% But it has vaguely possibly exist. 00:20:49.480 --> 00:20:51.220 align:middle line:84% But in this model, I've said very clearly 00:20:51.220 --> 00:20:52.413 align:middle line:90% I only have two levels. 00:20:52.413 --> 00:20:54.080 align:middle line:84% And so I don't have to think about that. 00:20:54.080 --> 00:20:56.380 align:middle line:84% But you're right, if you-- in a real system, 00:20:56.380 --> 00:20:58.570 align:middle line:84% you can have two photon absorption where 00:20:58.570 --> 00:21:00.070 align:middle line:90% one photon takes up to there. 00:21:00.070 --> 00:21:02.590 align:middle line:84% And before the atom has time to realize that this energy 00:21:02.590 --> 00:21:05.290 align:middle line:84% level doesn't really exist, the second photon 00:21:05.290 --> 00:21:07.480 align:middle line:84% comes in and takes it up even further 00:21:07.480 --> 00:21:08.890 align:middle line:90% and then everything is OK. 00:21:08.890 --> 00:21:10.360 align:middle line:90% So there's like-- yeah. 00:21:10.360 --> 00:21:12.920 align:middle line:84% But in this, I've been very clear I only have two levels 00:21:12.920 --> 00:21:14.420 align:middle line:84% so I don't have to worry about that. 00:21:14.420 --> 00:21:16.070 align:middle line:90% But, in general, [INAUDIBLE]. 00:21:16.070 --> 00:21:17.305 align:middle line:90% Any other questions? 00:21:17.305 --> 00:21:21.760 align:middle line:90% 00:21:21.760 --> 00:21:23.245 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:21:23.245 --> 00:21:29.680 align:middle line:90% 00:21:29.680 --> 00:21:30.610 align:middle line:90% JACK HARE: Yeah. 00:21:30.610 --> 00:21:33.637 align:middle line:84% So actually, none of this is about ionization at this point. 00:21:33.637 --> 00:21:35.470 align:middle line:84% So these are just two levels within an atom. 00:21:35.470 --> 00:21:37.330 align:middle line:84% The number of electrons is conserved. 00:21:37.330 --> 00:21:39.350 align:middle line:84% And you're right, they're just energy levels. 00:21:39.350 --> 00:21:40.402 align:middle line:90% So this could be-- 00:21:40.402 --> 00:21:41.860 align:middle line:84% actually, the simplest way to think 00:21:41.860 --> 00:21:43.777 align:middle line:84% about this, maybe the simplest quantum system, 00:21:43.777 --> 00:21:46.540 align:middle line:84% would be a spin which is flipping up and down. 00:21:46.540 --> 00:21:49.132 align:middle line:84% So these could be the hyperfine splitting of the ground state, 00:21:49.132 --> 00:21:50.590 align:middle line:84% or the fine splitting of the ground 00:21:50.590 --> 00:21:54.140 align:middle line:84% state and the magnetic field, or something like that. 00:21:54.140 --> 00:21:54.640 align:middle line:90% Yeah. 00:21:54.640 --> 00:21:56.590 align:middle line:84% So that's why this is quite generic. 00:21:56.590 --> 00:21:58.930 align:middle line:84% The way I'm thinking about it is about energy levels 00:21:58.930 --> 00:22:01.720 align:middle line:84% within an atom that isn't undergoing ionization. 00:22:01.720 --> 00:22:03.670 align:middle line:84% And we'll talk about all the ionization 00:22:03.670 --> 00:22:05.050 align:middle line:90% processes in a moment. 00:22:05.050 --> 00:22:06.520 align:middle line:84% But you're right, this applies more 00:22:06.520 --> 00:22:09.340 align:middle line:84% generally to other systems that have discrete energy levels. 00:22:09.340 --> 00:22:10.990 align:middle line:84% They've got two discrete energy levels. 00:22:10.990 --> 00:22:14.040 align:middle line:90% All of this math still works. 00:22:14.040 --> 00:22:16.280 align:middle line:90% Cool. 00:22:16.280 --> 00:22:17.580 align:middle line:90% OK. 00:22:17.580 --> 00:22:21.930 align:middle line:84% So, unfortunately, if you want to work out any of these, 00:22:21.930 --> 00:22:25.010 align:middle line:84% you're going to have to do some quantum, right? 00:22:25.010 --> 00:22:28.160 align:middle line:84% And the reason for that is, if you go into Hutchinson's book 00:22:28.160 --> 00:22:36.820 align:middle line:84% and you look at an expression, for example, BUL here, 00:22:36.820 --> 00:22:50.360 align:middle line:84% you find out that it has 8pi cubed over 3H squared 4pie 00:22:50.360 --> 00:22:51.200 align:middle line:90% epsilon zero. 00:22:51.200 --> 00:22:55.270 align:middle line:84% Don't ask me where any of these factors come from. 00:22:55.270 --> 00:23:02.320 align:middle line:84% And this is times a quantity called SUL. 00:23:02.320 --> 00:23:05.360 align:middle line:84% SUL is like the dimensionless rate, 00:23:05.360 --> 00:23:07.100 align:middle line:84% and then this is all the constants 00:23:07.100 --> 00:23:09.290 align:middle line:84% that you need in order to get all of your equations 00:23:09.290 --> 00:23:10.780 align:middle line:90% to match up properly. 00:23:10.780 --> 00:23:14.880 align:middle line:84% SUL is the thing that depends on quantum. 00:23:14.880 --> 00:23:18.960 align:middle line:84% And that is to do with the transition between your two 00:23:18.960 --> 00:23:19.630 align:middle line:90% states. 00:23:19.630 --> 00:23:22.890 align:middle line:84% So we're calculate an overlap integral, 00:23:22.890 --> 00:23:29.470 align:middle line:84% the integral between your upper state and your lower state. 00:23:29.470 --> 00:23:32.738 align:middle line:90% 00:23:32.738 --> 00:23:35.280 align:middle line:84% But we're not just doing the overlap between those two things 00:23:35.280 --> 00:23:38.490 align:middle line:84% because there is a operator, which acts on the upper state 00:23:38.490 --> 00:23:40.590 align:middle line:84% to transform it into the lower state. 00:23:40.590 --> 00:23:43.440 align:middle line:84% And that, for most electromagnetic radiation, 00:23:43.440 --> 00:23:48.090 align:middle line:84% is the dipole operator, which is the charge times the position 00:23:48.090 --> 00:23:49.230 align:middle line:90% vector here. 00:23:49.230 --> 00:23:53.170 align:middle line:84% We integrate that, as we always do, over all space, 00:23:53.170 --> 00:23:56.680 align:middle line:84% and we square it because we're interested in the power 00:23:56.680 --> 00:23:59.310 align:middle line:84% here, rather than the electric field. 00:23:59.310 --> 00:24:03.160 align:middle line:84% So this-- you can think of this as representing a transition 00:24:03.160 --> 00:24:04.940 align:middle line:90% probability. 00:24:04.940 --> 00:24:10.050 align:middle line:84% How likely is it that the emission of a photon 00:24:10.050 --> 00:24:14.440 align:middle line:84% can take you from the upper level to the lower level? 00:24:14.440 --> 00:24:18.360 align:middle line:84% And so levels which are more alike, 00:24:18.360 --> 00:24:20.670 align:middle line:84% you're more likely to have transitions between them. 00:24:20.670 --> 00:24:22.253 align:middle line:84% Of course, there are some levels which 00:24:22.253 --> 00:24:24.780 align:middle line:84% are very alike, which we can't have a transition to, 00:24:24.780 --> 00:24:29.790 align:middle line:90% where we have SUL equals zero. 00:24:29.790 --> 00:24:32.580 align:middle line:84% And these are called forbidden transitions. 00:24:32.580 --> 00:24:36.860 align:middle line:90% 00:24:36.860 --> 00:24:40.720 align:middle line:84% And these forbidden transitions don't do things 00:24:40.720 --> 00:24:44.350 align:middle line:84% like conserve angular momentum, other important things 00:24:44.350 --> 00:24:45.190 align:middle line:90% like that. 00:24:45.190 --> 00:24:48.010 align:middle line:84% And the way that that manifests itself in the mathematics 00:24:48.010 --> 00:24:52.300 align:middle line:84% is that under the operation R on the lower energy level, 00:24:52.300 --> 00:24:56.920 align:middle line:84% the PSI of U is orthogonal to R of PSI L. 00:24:56.920 --> 00:24:59.660 align:middle line:84% And so when you do the integral, you get zero out. 00:24:59.660 --> 00:25:02.950 align:middle line:84% So effectively, this keeps track of angular momentum 00:25:02.950 --> 00:25:05.350 align:middle line:84% conservation, or momentum conservation for you. 00:25:05.350 --> 00:25:07.870 align:middle line:84% So you have SUL there equals zero. 00:25:07.870 --> 00:25:11.170 align:middle line:84% So you might have, for example, a system with three energy 00:25:11.170 --> 00:25:17.990 align:middle line:84% levels where you can have transitions like this 00:25:17.990 --> 00:25:21.700 align:middle line:84% that both produce photons with different energies. 00:25:21.700 --> 00:25:27.190 align:middle line:84% But for some reason, this transition here, as SUL, 00:25:27.190 --> 00:25:28.450 align:middle line:90% equals zero. 00:25:28.450 --> 00:25:31.460 align:middle line:84% And you can't have a transition like that at all. 00:25:31.460 --> 00:25:32.920 align:middle line:84% So these are forbidden transitions, 00:25:32.920 --> 00:25:35.700 align:middle line:84% and sometimes people call these selection rules. 00:25:35.700 --> 00:25:42.410 align:middle line:90% 00:25:42.410 --> 00:25:45.820 align:middle line:84% So if you've done any amount of quantum physics 00:25:45.820 --> 00:25:48.302 align:middle line:84% or atomic physics, I think you'll probably 00:25:48.302 --> 00:25:49.510 align:middle line:90% have come across this before. 00:25:49.510 --> 00:25:52.090 align:middle line:84% If you haven't done a class on that before, 00:25:52.090 --> 00:25:55.090 align:middle line:84% I'm just trying to give you a very brief flavor of it. 00:25:55.090 --> 00:25:57.160 align:middle line:84% In general, this is a difficult problem. 00:25:57.160 --> 00:26:00.690 align:middle line:84% For a two level system, you only have to calculate SUL once. 00:26:00.690 --> 00:26:03.060 align:middle line:84% But if you imagine for a three level system, 00:26:03.060 --> 00:26:05.910 align:middle line:84% now, we have to calculate the overlap between this level 00:26:05.910 --> 00:26:08.190 align:middle line:84% and this level, this level and this level, 00:26:08.190 --> 00:26:10.230 align:middle line:90% and this level and this level. 00:26:10.230 --> 00:26:13.180 align:middle line:84% And for a four level system-- and then you get the idea. 00:26:13.180 --> 00:26:15.510 align:middle line:84% So this quickly becomes very, very challenging. 00:26:15.510 --> 00:26:17.400 align:middle line:84% If you want to actually do this, you 00:26:17.400 --> 00:26:19.620 align:middle line:84% have to calculate this overlap integral for all 00:26:19.620 --> 00:26:20.940 align:middle line:90% the different wave functions. 00:26:20.940 --> 00:26:22.590 align:middle line:84% To make it worse, you also need to know 00:26:22.590 --> 00:26:24.040 align:middle line:90% what the wave functions are. 00:26:24.040 --> 00:26:26.373 align:middle line:84% And so you have to solve the Schrodinger equation, first 00:26:26.373 --> 00:26:29.067 align:middle line:84% of all, to know what the states are allowed in your system. 00:26:29.067 --> 00:26:31.150 align:middle line:84% But, of course, if you have more than one electron 00:26:31.150 --> 00:26:32.890 align:middle line:84% in your system, not just one more 00:26:32.890 --> 00:26:34.060 align:middle line:84% than one level, more than one electron, 00:26:34.060 --> 00:26:35.470 align:middle line:84% this is now a many body problem and so 00:26:35.470 --> 00:26:37.480 align:middle line:84% you can't even calculate the wave functions properly. 00:26:37.480 --> 00:26:39.355 align:middle line:84% So very quickly you can see that spectroscopy 00:26:39.355 --> 00:26:41.057 align:middle line:84% is going to be a horrible mess compared 00:26:41.057 --> 00:26:42.640 align:middle line:84% to all of the other types of radiation 00:26:42.640 --> 00:26:43.780 align:middle line:90% we've dealt with before. 00:26:43.780 --> 00:26:45.197 align:middle line:84% Well, line emission is going to be 00:26:45.197 --> 00:26:48.700 align:middle line:84% a horrible mess compared to all these other types of radiation. 00:26:48.700 --> 00:26:52.520 align:middle line:84% But say you have managed to overcome all of these problems, 00:26:52.520 --> 00:27:01.840 align:middle line:84% the main takeaway from this is, if we know BUL 00:27:01.840 --> 00:27:04.360 align:middle line:84% by calculating this overlap integral, 00:27:04.360 --> 00:27:08.830 align:middle line:84% then we also get the other two coefficients 00:27:08.830 --> 00:27:14.730 align:middle line:90% AUL and BLU for free. 00:27:14.730 --> 00:27:17.270 align:middle line:84% And that means that even in a system that is not 00:27:17.270 --> 00:27:19.760 align:middle line:84% in thermodynamic equilibrium where we don't-- 00:27:19.760 --> 00:27:21.800 align:middle line:84% where we can't assume Boltzmann occupation, 00:27:21.800 --> 00:27:24.200 align:middle line:84% we can still calculate now the occupation 00:27:24.200 --> 00:27:26.090 align:middle line:84% of the upper and the lower energy levels 00:27:26.090 --> 00:27:29.360 align:middle line:84% through a rate equation where these are the rates. 00:27:29.360 --> 00:27:32.340 align:middle line:84% This is effectively a steady state rate equation here. 00:27:32.340 --> 00:27:34.287 align:middle line:84% And that means we now know how many electrons 00:27:34.287 --> 00:27:36.620 align:middle line:84% are in the upper level, how many are in the lower level, 00:27:36.620 --> 00:27:38.930 align:middle line:84% and we can make a prediction of how many photons 00:27:38.930 --> 00:27:40.230 align:middle line:90% we're going to get per second. 00:27:40.230 --> 00:27:44.360 align:middle line:84% And so that means that we can predict the intensity of one 00:27:44.360 --> 00:27:45.890 align:middle line:90% of these spectral lines here. 00:27:45.890 --> 00:27:48.228 align:middle line:84% Again, this is an extremely simple system. 00:27:48.228 --> 00:27:50.270 align:middle line:84% There's already quite a lot of work to get there, 00:27:50.270 --> 00:27:51.510 align:middle line:90% but we can do that. 00:27:51.510 --> 00:27:54.440 align:middle line:84% So that's a very thorough explanation 00:27:54.440 --> 00:27:56.090 align:middle line:84% of just the two level atom before we 00:27:56.090 --> 00:28:00.620 align:middle line:84% go on to more general systems with multiple levels 00:28:00.620 --> 00:28:01.950 align:middle line:90% and multiple electrons. 00:28:01.950 --> 00:28:04.820 align:middle line:84% So any questions on that before we move on? 00:28:04.820 --> 00:28:05.753 align:middle line:90% Yes, go ahead. 00:28:05.753 --> 00:28:06.920 align:middle line:90% Pointed to the wrong person. 00:28:06.920 --> 00:28:07.840 align:middle line:90% There you go. 00:28:07.840 --> 00:28:10.318 align:middle line:90% 00:28:10.318 --> 00:28:12.110 align:middle line:84% You get slightly different frames, I think. 00:28:12.110 --> 00:28:22.430 align:middle line:90% 00:28:22.430 --> 00:28:22.930 align:middle line:90% Yeah. 00:28:22.930 --> 00:28:25.090 align:middle line:84% So the question was, do you have to do this every time, 00:28:25.090 --> 00:28:26.390 align:middle line:90% or is it tabulated somewhere? 00:28:26.390 --> 00:28:31.060 align:middle line:84% So the answer is, someone may have already done 00:28:31.060 --> 00:28:33.400 align:middle line:84% the tabulation for you and made that available in what's 00:28:33.400 --> 00:28:34.720 align:middle line:90% called an atomic code. 00:28:34.720 --> 00:28:38.720 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:28:38.720 --> 00:28:43.450 align:middle line:84% JACK HARE: Oh, generally, these are experimentally-- 00:28:43.450 --> 00:28:45.010 align:middle line:84% these are theoretically predicted, 00:28:45.010 --> 00:28:46.780 align:middle line:84% and then experimentally confirmed. 00:28:46.780 --> 00:28:48.340 align:middle line:84% For some of the transitions, they're 00:28:48.340 --> 00:28:50.470 align:middle line:84% so weak it's really hard to do it experimentally. 00:28:50.470 --> 00:28:54.130 align:middle line:84% If you go on the NIST website, then NIST 00:28:54.130 --> 00:28:57.137 align:middle line:84% has a huge table of all the different transitions here 00:28:57.137 --> 00:28:58.720 align:middle line:84% and they are like oscillator strength. 00:28:58.720 --> 00:29:00.595 align:middle line:84% And sometimes, they have a star next to them, 00:29:00.595 --> 00:29:03.178 align:middle line:84% being like this was theoretically predicted, 00:29:03.178 --> 00:29:04.970 align:middle line:84% or they have something else being like, no, 00:29:04.970 --> 00:29:06.310 align:middle line:84% this has actually been experimentally checked, 00:29:06.310 --> 00:29:07.710 align:middle line:84% or like someone tried to check it 00:29:07.710 --> 00:29:09.460 align:middle line:84% but we don't think it's a very good check. 00:29:09.460 --> 00:29:11.500 align:middle line:84% Because these are just-- there are so many of these lines, 00:29:11.500 --> 00:29:13.060 align:middle line:84% it's impossible to check all of them. 00:29:13.060 --> 00:29:14.602 align:middle line:84% And, of course, you can theoretically 00:29:14.602 --> 00:29:17.310 align:middle line:84% predict these four states of matter, 00:29:17.310 --> 00:29:19.610 align:middle line:84% which are very hard to reach in a real plasma. 00:29:19.610 --> 00:29:22.310 align:middle line:84% So for very, very hot very, very dense plasmas made out 00:29:22.310 --> 00:29:24.038 align:middle line:84% of plutonium, or something like that. 00:29:24.038 --> 00:29:26.330 align:middle line:84% We probably do actually have good measurements of that, 00:29:26.330 --> 00:29:27.590 align:middle line:90% but not unclassified ones. 00:29:27.590 --> 00:29:29.822 align:middle line:90% 00:29:29.822 --> 00:29:31.280 align:middle line:84% So there are things that maybe it's 00:29:31.280 --> 00:29:32.590 align:middle line:90% hard to experimentally verify. 00:29:32.590 --> 00:29:34.340 align:middle line:84% But actually, you can still theoretically. 00:29:34.340 --> 00:29:37.460 align:middle line:84% So, yeah, but if you are doing something 00:29:37.460 --> 00:29:39.980 align:middle line:84% where that hasn't already been tabulated, 00:29:39.980 --> 00:29:41.542 align:middle line:90% you may have to do it yourself. 00:29:41.542 --> 00:29:44.000 align:middle line:84% So you may actually have to do these calculations yourself. 00:29:44.000 --> 00:29:45.583 align:middle line:84% And there are people who build careers 00:29:45.583 --> 00:29:48.080 align:middle line:84% out of doing these calculations more and more precisely. 00:29:48.080 --> 00:29:50.760 align:middle line:84% Because as I mentioned, this is a very hard calculation to do. 00:29:50.760 --> 00:29:52.830 align:middle line:84% So there's always going to be some approximation. 00:29:52.830 --> 00:29:55.245 align:middle line:84% And so then, it's like you could make your career out 00:29:55.245 --> 00:29:57.620 align:middle line:84% of doing a slightly better approximation than previously, 00:29:57.620 --> 00:29:59.245 align:middle line:84% and you'll get slightly better results. 00:29:59.245 --> 00:30:01.730 align:middle line:84% And that will be worthwhile to some people. 00:30:01.730 --> 00:30:03.130 align:middle line:90% Yeah. 00:30:03.130 --> 00:30:05.476 align:middle line:90% You have a question? 00:30:05.476 --> 00:30:06.670 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:30:06.670 --> 00:30:12.900 align:middle line:90% 00:30:12.900 --> 00:30:15.840 align:middle line:84% JACK HARE: Yes, this is-- you should think of it as a number 00:30:15.840 --> 00:30:16.710 align:middle line:90% density of photons. 00:30:16.710 --> 00:30:20.430 align:middle line:84% We're using RHO here, which often is used for mass density, 00:30:20.430 --> 00:30:23.700 align:middle line:84% but I'm using it here for number density because that's what 00:30:23.700 --> 00:30:24.540 align:middle line:90% [INAUDIBLE]. 00:30:24.540 --> 00:30:25.320 align:middle line:90% So, yeah. 00:30:25.320 --> 00:30:26.781 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:30:26.781 --> 00:30:43.076 align:middle line:90% 00:30:43.076 --> 00:30:46.250 align:middle line:84% JACK HARE: We will actually talk about that later, you're right. 00:30:46.250 --> 00:30:49.260 align:middle line:84% And unless your spectrometer is absolutely calibrated, 00:30:49.260 --> 00:30:52.910 align:middle line:90% it's very, very hard to get-- 00:30:52.910 --> 00:30:54.750 align:middle line:84% it's very, very hard to get it just 00:30:54.750 --> 00:30:56.580 align:middle line:84% from a single measurement of a line. 00:30:56.580 --> 00:30:58.110 align:middle line:84% What's interesting is, if you have 00:30:58.110 --> 00:31:02.890 align:middle line:84% two lines here, this isn't the best drawing, 00:31:02.890 --> 00:31:04.060 align:middle line:90% actually, to show this. 00:31:04.060 --> 00:31:05.980 align:middle line:84% A better drawing would be if I had 00:31:05.980 --> 00:31:08.320 align:middle line:84% one of these transitions coming from this state here. 00:31:08.320 --> 00:31:11.280 align:middle line:90% 00:31:11.280 --> 00:31:13.730 align:middle line:84% So then you'll have two lines which 00:31:13.730 --> 00:31:16.110 align:middle line:84% have a downward transition, which are both emitting 00:31:16.110 --> 00:31:17.360 align:middle line:90% photons at different energies. 00:31:17.360 --> 00:31:20.193 align:middle line:90% 00:31:20.193 --> 00:31:22.110 align:middle line:84% There are two things you'll need to know then. 00:31:22.110 --> 00:31:25.170 align:middle line:84% You'll need to know what the strength of this transition 00:31:25.170 --> 00:31:28.980 align:middle line:84% is, which is the strength of the spontaneous emission AUL, which 00:31:28.980 --> 00:31:31.385 align:middle line:84% will be different for these two different lines. 00:31:31.385 --> 00:31:32.760 align:middle line:84% But then you'll also want to know 00:31:32.760 --> 00:31:36.460 align:middle line:84% what the occupation probability is of these two upper states. 00:31:36.460 --> 00:31:39.400 align:middle line:84% And that will, in general, be a function of temperature. 00:31:39.400 --> 00:31:40.980 align:middle line:84% So it will be-- this state will more 00:31:40.980 --> 00:31:43.830 align:middle line:84% likely to be occupied than this state just from a Boltzmann 00:31:43.830 --> 00:31:44.353 align:middle line:90% argument. 00:31:44.353 --> 00:31:46.770 align:middle line:84% If you happen to be able to use a Boltzmann type argument, 00:31:46.770 --> 00:31:48.160 align:middle line:90% that's what you predict. 00:31:48.160 --> 00:31:50.910 align:middle line:84% So that means that the ratio between the amount of light 00:31:50.910 --> 00:31:53.670 align:middle line:84% in this blue line and the amount of light in the orange line 00:31:53.670 --> 00:31:57.300 align:middle line:84% here will tell you, once you've taken out the factor of AUL 00:31:57.300 --> 00:31:59.400 align:middle line:84% that you've precalculated, it will tell you 00:31:59.400 --> 00:32:02.370 align:middle line:84% the occupation of this upper state versus this state. 00:32:02.370 --> 00:32:04.100 align:middle line:84% And that will be related to temperature. 00:32:04.100 --> 00:32:05.850 align:middle line:84% And then you have your plasma temperature. 00:32:05.850 --> 00:32:07.213 align:middle line:90% And it's a relative ratio. 00:32:07.213 --> 00:32:09.130 align:middle line:84% You don't need to know the absolute intensity. 00:32:09.130 --> 00:32:12.990 align:middle line:84% So in reality in spectroscopy, we normally look at line ratios. 00:32:12.990 --> 00:32:14.400 align:middle line:84% They're much better to work with. 00:32:14.400 --> 00:32:15.960 align:middle line:84% There'll be many, many lines, not 00:32:15.960 --> 00:32:17.470 align:middle line:90% like in this two level system. 00:32:17.470 --> 00:32:19.095 align:middle line:84% There'll be many, many lines out there. 00:32:19.095 --> 00:32:22.380 align:middle line:84% We look at line ratios like a differential measurement again. 00:32:22.380 --> 00:32:25.540 align:middle line:84% The thing that we love to do with experimental. 00:32:25.540 --> 00:32:26.680 align:middle line:90% Does that make sense? 00:32:26.680 --> 00:32:27.280 align:middle line:90% Good. 00:32:27.280 --> 00:32:30.880 align:middle line:84% We will now go through lots of other processes 00:32:30.880 --> 00:32:34.210 align:middle line:84% which can excite and de-excite your electrons. 00:32:34.210 --> 00:32:37.210 align:middle line:84% Not just radiative processes like this. 00:32:37.210 --> 00:32:40.570 align:middle line:84% And then we will use a subset of those 00:32:40.570 --> 00:32:42.790 align:middle line:84% to define different sorts of equilibrium, which 00:32:42.790 --> 00:32:45.877 align:middle line:84% help you to calculate what this occupation probability is 00:32:45.877 --> 00:32:47.210 align:middle line:90% for the different energy levels. 00:32:47.210 --> 00:32:49.240 align:middle line:84% So it's quite an involved series of steps, 00:32:49.240 --> 00:32:51.750 align:middle line:84% and I'm trying to go through it quite slowly. 00:32:51.750 --> 00:32:54.750 align:middle line:84% Any questions before we move on to all the other processes 00:32:54.750 --> 00:32:58.294 align:middle line:84% out there that aren't just these simple radiative [INAUDIBLE]? 00:32:58.294 --> 00:33:00.840 align:middle line:90% 00:33:00.840 --> 00:33:01.340 align:middle line:90% OK. 00:33:01.340 --> 00:33:22.310 align:middle line:90% 00:33:22.310 --> 00:33:23.750 align:middle line:90% OK. 00:33:23.750 --> 00:33:27.650 align:middle line:84% So the first set of processes we're going to look at 00:33:27.650 --> 00:33:36.770 align:middle line:84% are processes which excite or de-excite electrons. 00:33:36.770 --> 00:33:46.590 align:middle line:84% So in this case, we're looking at a change in the electron 00:33:46.590 --> 00:33:56.870 align:middle line:84% energy, but, or and, the number of electrons is constant. 00:33:56.870 --> 00:33:58.780 align:middle line:84% So we are not dealing with ionization here. 00:33:58.780 --> 00:34:01.150 align:middle line:84% We're going to get on to ionization and recombination 00:34:01.150 --> 00:34:02.658 align:middle line:90% in a moment. 00:34:02.658 --> 00:34:05.200 align:middle line:84% So the first set of processes we've actually already covered. 00:34:05.200 --> 00:34:06.825 align:middle line:84% But I'm going to draw them on the board 00:34:06.825 --> 00:34:08.580 align:middle line:84% here so that we have a nice complete set. 00:34:08.580 --> 00:34:10.687 align:middle line:84% These are the radiative processes. 00:34:10.687 --> 00:34:13.429 align:middle line:90% 00:34:13.429 --> 00:34:17.620 align:middle line:84% So these are processes where we have, for example, some energy 00:34:17.620 --> 00:34:20.409 align:middle line:90% levels, again, lower and upper. 00:34:20.409 --> 00:34:24.730 align:middle line:84% We have different processes going between them. 00:34:24.730 --> 00:34:28.210 align:middle line:84% We have spontaneous emission, absorption, 00:34:28.210 --> 00:34:30.490 align:middle line:90% and stimulated emission. 00:34:30.490 --> 00:34:33.177 align:middle line:90% 00:34:33.177 --> 00:34:35.719 align:middle line:84% And again, we just covered this, but I'm just writing it down 00:34:35.719 --> 00:34:38.546 align:middle line:84% here so that we've got a full set of [INAUDIBLE]. 00:34:38.546 --> 00:34:43.739 align:middle line:90% 00:34:43.739 --> 00:35:02.460 align:middle line:84% Continuous emission, absorption, and stimulated emission. 00:35:02.460 --> 00:35:07.850 align:middle line:90% 00:35:07.850 --> 00:35:10.850 align:middle line:84% And I'll put the density of photon states 00:35:10.850 --> 00:35:15.470 align:middle line:84% on here, as well, well, RHO of new UL. 00:35:15.470 --> 00:35:25.490 align:middle line:90% 00:35:25.490 --> 00:35:27.470 align:middle line:84% Competing with these processes, or ongoing 00:35:27.470 --> 00:35:31.880 align:middle line:84% with these processes, a set of processes which are collisional. 00:35:31.880 --> 00:35:33.960 align:middle line:84% And in some cases, the collisional processes 00:35:33.960 --> 00:35:35.520 align:middle line:84% may dominate, and in other cases, 00:35:35.520 --> 00:35:37.530 align:middle line:84% the radiative processes will dominate. 00:35:37.530 --> 00:35:39.840 align:middle line:84% And we'll talk about those in a moment 00:35:39.840 --> 00:35:42.360 align:middle line:84% once we've got through this whole zoo. 00:35:42.360 --> 00:35:45.990 align:middle line:84% So the collisional processes, again, 00:35:45.990 --> 00:35:49.320 align:middle line:84% between the lower and the upper levels like this, 00:35:49.320 --> 00:35:50.565 align:middle line:90% are called things like-- 00:35:50.565 --> 00:35:54.610 align:middle line:90% 00:35:54.610 --> 00:35:59.200 align:middle line:84% this one here has a coefficient CLU, 00:35:59.200 --> 00:36:03.160 align:middle line:84% which is proportional to the density of electrons. 00:36:03.160 --> 00:36:08.260 align:middle line:84% Because this happens when an electron comes nearby the atom. 00:36:08.260 --> 00:36:12.358 align:middle line:84% Maybe it's an electron which is a free electron so it's 00:36:12.358 --> 00:36:13.525 align:middle line:90% up here at some high energy. 00:36:13.525 --> 00:36:16.620 align:middle line:90% 00:36:16.620 --> 00:36:19.740 align:middle line:84% It interacts with the electric field of the ion, and then 00:36:19.740 --> 00:36:20.970 align:middle line:90% the atom and flies off. 00:36:20.970 --> 00:36:23.040 align:middle line:84% But that interaction with the electric field 00:36:23.040 --> 00:36:27.930 align:middle line:84% promotes an electron from a lower energy level to a higher 00:36:27.930 --> 00:36:33.190 align:middle line:84% energy level, taking some energy from the free electron here. 00:36:33.190 --> 00:36:38.410 align:middle line:84% And this process is called electron impact ionization. 00:36:38.410 --> 00:36:44.160 align:middle line:84% So I'm just going to write that as P minus impact-- 00:36:44.160 --> 00:36:47.730 align:middle line:84% not ionization, excitation-- excitation. 00:36:47.730 --> 00:36:54.960 align:middle line:90% 00:36:54.960 --> 00:36:58.650 align:middle line:84% And actually, exactly the same process can happen in reverse. 00:36:58.650 --> 00:37:01.320 align:middle line:84% This electron can come in, its electric field 00:37:01.320 --> 00:37:03.420 align:middle line:84% can interact with the electric field and the wave 00:37:03.420 --> 00:37:04.950 align:middle line:90% functions of the atom. 00:37:04.950 --> 00:37:08.620 align:middle line:84% And this can drive a downwards transition, as well. 00:37:08.620 --> 00:37:12.930 align:middle line:84% And this has a coefficient CUL, which 00:37:12.930 --> 00:37:16.270 align:middle line:84% is proportional to the density of electrons, as well. 00:37:16.270 --> 00:37:18.450 align:middle line:84% And so this would be the same name 00:37:18.450 --> 00:37:20.850 align:middle line:90% as above, but the excitation. 00:37:20.850 --> 00:37:24.140 align:middle line:90% 00:37:24.140 --> 00:37:26.980 align:middle line:84% So you can imagine in a very dense plasma 00:37:26.980 --> 00:37:29.710 align:middle line:84% where there's a lot of electrons flying around, 00:37:29.710 --> 00:37:33.310 align:middle line:84% you may well end up with these processes being very dominant. 00:37:33.310 --> 00:37:35.680 align:middle line:84% And they will dominate the population 00:37:35.680 --> 00:37:37.400 align:middle line:84% of these different excited states. 00:37:37.400 --> 00:37:39.160 align:middle line:84% And so when you're trying to calculate the excited state 00:37:39.160 --> 00:37:40.870 align:middle line:84% population, these are the ones you're 00:37:40.870 --> 00:37:42.040 align:middle line:90% going to have to focus on. 00:37:42.040 --> 00:37:44.320 align:middle line:84% We'll talk a little bit about how we actually 00:37:44.320 --> 00:37:46.953 align:middle line:90% calculate these in a moment. 00:37:46.953 --> 00:37:48.870 align:middle line:84% So those two-- well, we've seen these already. 00:37:48.870 --> 00:37:49.840 align:middle line:90% Hopefully, they make sense. 00:37:49.840 --> 00:37:50.715 align:middle line:90% Does this make sense? 00:37:50.715 --> 00:38:07.150 align:middle line:90% 00:38:07.150 --> 00:38:08.700 align:middle line:90% OK. 00:38:08.700 --> 00:38:11.670 align:middle line:84% These were the excitation, and the de-excitation. 00:38:11.670 --> 00:38:14.610 align:middle line:84% Now, we're going to have processes which actually 00:38:14.610 --> 00:38:16.240 align:middle line:90% change the number of electrons. 00:38:16.240 --> 00:38:18.940 align:middle line:90% So these are ionization. 00:38:18.940 --> 00:38:21.280 align:middle line:84% And we don't tend to call it de-ionization. 00:38:21.280 --> 00:38:22.990 align:middle line:84% We call it recombination, instead. 00:38:22.990 --> 00:38:30.748 align:middle line:90% 00:38:30.748 --> 00:38:32.290 align:middle line:84% So in general, you can think of these 00:38:32.290 --> 00:38:39.790 align:middle line:84% as processes that give an electron enough energy 00:38:39.790 --> 00:38:45.880 align:middle line:84% to be free, so delta E, and that makes it free, 00:38:45.880 --> 00:38:55.260 align:middle line:84% or they absorb free electrons energy 00:38:55.260 --> 00:38:58.080 align:middle line:84% so that our electron becomes bound. 00:38:58.080 --> 00:39:00.450 align:middle line:84% So that first process is an ionization process 00:39:00.450 --> 00:39:02.340 align:middle line:84% where we lose an electron from our atom. 00:39:02.340 --> 00:39:06.690 align:middle line:84% And the second one is a recombination process where we 00:39:06.690 --> 00:39:09.340 align:middle line:90% gain an electron onto our atom. 00:39:09.340 --> 00:39:11.685 align:middle line:84% So I'm going to draw this as one long sort 00:39:11.685 --> 00:39:14.350 align:middle line:84% of [INAUDIBLE] tapestry of various different processes 00:39:14.350 --> 00:39:14.850 align:middle line:90% here. 00:39:14.850 --> 00:39:18.660 align:middle line:90% 00:39:18.660 --> 00:39:21.090 align:middle line:84% This is the lower energy level and this 00:39:21.090 --> 00:39:24.585 align:middle line:84% is the-- now the ionization energy level I. 00:39:24.585 --> 00:39:26.460 align:middle line:84% And I'm going to shade the region above here, 00:39:26.460 --> 00:39:29.310 align:middle line:84% because if an electron is in this region, 00:39:29.310 --> 00:39:30.690 align:middle line:90% it's now a free electron. 00:39:30.690 --> 00:39:36.990 align:middle line:90% 00:39:36.990 --> 00:39:39.060 align:middle line:84% But you can think of this as the ionization 00:39:39.060 --> 00:39:40.750 align:middle line:90% energy you get to this point. 00:39:40.750 --> 00:39:44.040 align:middle line:84% And I've called this L. This may not necessarily be the lowest 00:39:44.040 --> 00:39:45.690 align:middle line:90% energy state in the system. 00:39:45.690 --> 00:39:47.430 align:middle line:90% It's just the lower state. 00:39:47.430 --> 00:39:49.770 align:middle line:84% There could be other states down below. 00:39:49.770 --> 00:39:51.630 align:middle line:84% There could be states up above it. 00:39:51.630 --> 00:39:55.970 align:middle line:84% We're just considering one level L at the moment here. 00:39:55.970 --> 00:39:56.470 align:middle line:90% OK. 00:39:56.470 --> 00:39:57.970 align:middle line:84% So the first process, we're going 00:39:57.970 --> 00:40:02.410 align:middle line:84% to consider is where we have an electron in the ground state-- 00:40:02.410 --> 00:40:04.630 align:middle line:84% oh, sorry, in the lower state down here, 00:40:04.630 --> 00:40:07.240 align:middle line:84% and we have a free electron that swings by. 00:40:07.240 --> 00:40:10.970 align:middle line:90% 00:40:10.970 --> 00:40:13.250 align:middle line:84% And it gives up enough of its energy 00:40:13.250 --> 00:40:16.490 align:middle line:84% to excite this electron, not to the upper energy level, 00:40:16.490 --> 00:40:20.570 align:middle line:84% but actually, up so that it's ionized, and then two electrons 00:40:20.570 --> 00:40:21.170 align:middle line:90% come out. 00:40:21.170 --> 00:40:27.180 align:middle line:90% 00:40:27.180 --> 00:40:31.550 align:middle line:84% And this we give a coefficient CLI. 00:40:31.550 --> 00:40:35.870 align:middle line:90% 00:40:35.870 --> 00:40:40.550 align:middle line:84% And we give it a superscript CI here. 00:40:40.550 --> 00:40:43.900 align:middle line:84% And that is for collisional ionization. 00:40:43.900 --> 00:40:50.540 align:middle line:84% And we call this electron impact ionization. 00:40:50.540 --> 00:40:55.240 align:middle line:90% 00:40:55.240 --> 00:41:00.730 align:middle line:84% You can see this is analogous to electron impact excitation here. 00:41:00.730 --> 00:41:03.700 align:middle line:84% I'm just going to make it clear that this is a lowercase l. 00:41:03.700 --> 00:41:05.380 align:middle line:84% But there's another coefficient here. 00:41:05.380 --> 00:41:08.420 align:middle line:84% What's this coefficient proportional to? 00:41:08.420 --> 00:41:10.220 align:middle line:84% Remember previously, we've had density, 00:41:10.220 --> 00:41:14.180 align:middle line:84% we've had photon density, all sorts of things like that. 00:41:14.180 --> 00:41:16.021 align:middle line:84% What's this coefficient proportional to? 00:41:16.021 --> 00:41:18.970 align:middle line:90% 00:41:18.970 --> 00:41:20.620 align:middle line:84% Yeah, exactly the same as previously. 00:41:20.620 --> 00:41:21.920 align:middle line:90% There's nothing special. 00:41:21.920 --> 00:41:22.420 align:middle line:90% Yeah. 00:41:22.420 --> 00:41:25.710 align:middle line:90% 00:41:25.710 --> 00:41:28.250 align:middle line:90% OK. 00:41:28.250 --> 00:41:31.050 align:middle line:84% Now, we have a slightly different one. 00:41:31.050 --> 00:41:33.720 align:middle line:84% We actually have two electrons coming in. 00:41:33.720 --> 00:41:39.760 align:middle line:90% 00:41:39.760 --> 00:41:44.260 align:middle line:84% As they get close to the atom, one of the electrons 00:41:44.260 --> 00:41:49.300 align:middle line:84% runs off with a load of energy, and the other electron 00:41:49.300 --> 00:41:51.460 align:middle line:84% has lost so much energy to the first one 00:41:51.460 --> 00:41:54.325 align:middle line:84% that it drops down into this energy state here. 00:41:54.325 --> 00:41:58.390 align:middle line:90% 00:41:58.390 --> 00:42:00.360 align:middle line:84% So this is a recombination process, 00:42:00.360 --> 00:42:05.490 align:middle line:84% and this is called three body recombination. 00:42:05.490 --> 00:42:13.480 align:middle line:90% 00:42:13.480 --> 00:42:21.150 align:middle line:84% And we give it a rate coefficient CIL, 00:42:21.150 --> 00:42:26.900 align:middle line:84% and a superscript three body recombination. 00:42:26.900 --> 00:42:28.370 align:middle line:90% What is this proportional to? 00:42:28.370 --> 00:42:35.050 align:middle line:90% 00:42:35.050 --> 00:42:35.900 align:middle line:90% Someone else. 00:42:35.900 --> 00:42:36.670 align:middle line:90% But I will come back to you. 00:42:36.670 --> 00:42:37.170 align:middle line:90% Yeah. 00:42:37.170 --> 00:42:40.100 align:middle line:90% 00:42:40.100 --> 00:42:41.790 align:middle line:90% What are the neutrals in this? 00:42:41.790 --> 00:42:42.965 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:42:42.965 --> 00:42:48.185 align:middle line:84% JACK HARE: Why didn't the atoms come in here, in that case? 00:42:48.185 --> 00:42:52.060 align:middle line:90% 00:42:52.060 --> 00:42:52.880 align:middle line:90% It's a good point. 00:42:52.880 --> 00:42:54.400 align:middle line:84% In fact, all of these are proportional to the atom 00:42:54.400 --> 00:42:55.108 align:middle line:90% density, as well. 00:42:55.108 --> 00:42:58.610 align:middle line:84% I've just left it off because every single one of these 00:42:58.610 --> 00:43:00.110 align:middle line:84% is proportional to the atom density. 00:43:00.110 --> 00:43:01.475 align:middle line:90% So we don't need [INAUDIBLE]. 00:43:01.475 --> 00:43:01.975 align:middle line:90% Yeah. 00:43:01.975 --> 00:43:03.040 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:43:03.040 --> 00:43:05.390 align:middle line:90% JACK HARE: Why is that? 00:43:05.390 --> 00:43:06.810 align:middle line:90% Yeah, exactly. 00:43:06.810 --> 00:43:10.870 align:middle line:84% So we've actually got to get two electrons in the same place 00:43:10.870 --> 00:43:11.870 align:middle line:90% close to the atom. 00:43:11.870 --> 00:43:13.960 align:middle line:84% And so you're going to square the density in order 00:43:13.960 --> 00:43:15.877 align:middle line:84% to work out the probability of that happening. 00:43:15.877 --> 00:43:18.490 align:middle line:84% So this three body recombination can 00:43:18.490 --> 00:43:20.740 align:middle line:84% be a really important process in dense plasma. 00:43:20.740 --> 00:43:23.170 align:middle line:84% It's actually particularly important in dense cold plasma. 00:43:23.170 --> 00:43:25.660 align:middle line:84% So plasmas with very low ionization 00:43:25.660 --> 00:43:30.940 align:middle line:84% tend to have dominant three-body recombination. 00:43:30.940 --> 00:43:31.450 align:middle line:90% OK. 00:43:31.450 --> 00:43:34.570 align:middle line:90% The next one is also quite odd. 00:43:34.570 --> 00:43:37.140 align:middle line:90% 00:43:37.140 --> 00:43:40.710 align:middle line:84% We imagine that there's some intermediate energy 00:43:40.710 --> 00:43:44.460 align:middle line:84% level like this in which there are two electrons. 00:43:44.460 --> 00:43:49.990 align:middle line:90% 00:43:49.990 --> 00:43:52.470 align:middle line:84% And there is some spontaneous process 00:43:52.470 --> 00:43:58.440 align:middle line:84% by which this electron drops down to this energy level, which 00:43:58.440 --> 00:44:01.890 align:middle line:84% provides exactly enough energy for this electron 00:44:01.890 --> 00:44:04.500 align:middle line:90% to bump up and be free. 00:44:04.500 --> 00:44:06.270 align:middle line:84% In reality, this will happen as long 00:44:06.270 --> 00:44:10.200 align:middle line:84% as this intermediate level is somewhere above halfway, 00:44:10.200 --> 00:44:12.370 align:middle line:84% because then one electron dropping down, 00:44:12.370 --> 00:44:14.100 align:middle line:84% will have enough energy to boost this off 00:44:14.100 --> 00:44:16.830 align:middle line:84% and it will be not just a free electron but a free electron 00:44:16.830 --> 00:44:21.200 align:middle line:84% with some bonus kinetic energy, as well. 00:44:21.200 --> 00:44:25.150 align:middle line:84% This is still written as a coefficient from lower to upper, 00:44:25.150 --> 00:44:27.005 align:middle line:84% even though we should probably think 00:44:27.005 --> 00:44:28.880 align:middle line:84% of a name for this intermediate energy level. 00:44:28.880 --> 00:44:31.150 align:middle line:90% So we call it CLI. 00:44:31.150 --> 00:44:37.100 align:middle line:84% And its superscript is AI because this is a process called 00:44:37.100 --> 00:44:38.402 align:middle line:90% autoionization. 00:44:38.402 --> 00:44:46.750 align:middle line:90% 00:44:46.750 --> 00:44:49.420 align:middle line:84% Does this have any dependence on the electron density? 00:44:49.420 --> 00:44:52.340 align:middle line:90% 00:44:52.340 --> 00:44:53.270 align:middle line:90% I saw a shaking head. 00:44:53.270 --> 00:44:55.562 align:middle line:84% It doesn't have any dependence on the electron density. 00:44:55.562 --> 00:44:57.990 align:middle line:84% We don't require any free electrons to do this. 00:44:57.990 --> 00:45:00.800 align:middle line:84% This is something the atom has just decided to do by itself. 00:45:00.800 --> 00:45:02.750 align:middle line:84% Of course, for this to happen, there 00:45:02.750 --> 00:45:06.240 align:middle line:84% has to be an unoccupied lower energy state. 00:45:06.240 --> 00:45:09.470 align:middle line:84% So this has to be some excited atom, right? 00:45:09.470 --> 00:45:14.060 align:middle line:84% So some process like electron impact excitation 00:45:14.060 --> 00:45:17.120 align:middle line:84% has moved an electron from the lower level 00:45:17.120 --> 00:45:20.630 align:middle line:84% to this intermediate level here, leaving 00:45:20.630 --> 00:45:22.760 align:middle line:84% a state for this electron to fall back into. 00:45:22.760 --> 00:45:26.983 align:middle line:90% So this can't just happen in us. 00:45:26.983 --> 00:45:29.150 align:middle line:84% We're not going to suddenly start emitting electrons 00:45:29.150 --> 00:45:30.650 align:middle line:90% by autoionization. 00:45:30.650 --> 00:45:32.570 align:middle line:84% Because in general, most of our electrons, 00:45:32.570 --> 00:45:35.810 align:middle line:84% if we're well behaved, stay in their ground state, OK. 00:45:35.810 --> 00:45:38.690 align:middle line:84% But if in a plasma where these processes are happening, 00:45:38.690 --> 00:45:42.380 align:middle line:84% we very well may have a hole in a lower energy level, which 00:45:42.380 --> 00:45:43.730 align:middle line:90% allows this process to happen. 00:45:43.730 --> 00:45:44.720 align:middle line:90% Pretty cool. 00:45:44.720 --> 00:45:45.530 align:middle line:90% OK. 00:45:45.530 --> 00:45:48.205 align:middle line:90% I saw Nicola. 00:45:48.205 --> 00:45:49.690 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:45:49.690 --> 00:45:58.105 align:middle line:90% 00:45:58.105 --> 00:45:59.047 align:middle line:90% JACK HARE: Yeah. 00:45:59.047 --> 00:46:01.005 align:middle line:84% So basically, it needs to be more than halfway. 00:46:01.005 --> 00:46:03.640 align:middle line:90% 00:46:03.640 --> 00:46:09.010 align:middle line:84% It needs to be greater than energy lower ionization 00:46:09.010 --> 00:46:11.260 align:middle line:90% divided by 2, right, as you say. 00:46:11.260 --> 00:46:13.750 align:middle line:84% Otherwise, there won't be enough energy for-- 00:46:13.750 --> 00:46:17.650 align:middle line:84% I don't know if there's auto excitation that can happen. 00:46:17.650 --> 00:46:19.570 align:middle line:90% That wasn't in the book. 00:46:19.570 --> 00:46:21.760 align:middle line:84% But it seems like it might be possible, right, 00:46:21.760 --> 00:46:24.308 align:middle line:84% where you could have excitation of this electron 00:46:24.308 --> 00:46:25.600 align:middle line:90% up to some higher energy level. 00:46:25.600 --> 00:46:28.970 align:middle line:84% Maybe that doesn't happen for some reason, I don't know. 00:46:28.970 --> 00:46:30.780 align:middle line:84% Did you have a question, as well? 00:46:30.780 --> 00:46:31.280 align:middle line:90% OK. 00:46:31.280 --> 00:46:58.330 align:middle line:90% 00:46:58.330 --> 00:47:00.872 align:middle line:90% Yeah, we're coming to that. 00:47:00.872 --> 00:47:02.830 align:middle line:84% That's a question I'm about to ask in a moment. 00:47:02.830 --> 00:47:04.788 align:middle line:84% You're absolutely right, that can still happen. 00:47:04.788 --> 00:47:07.090 align:middle line:84% But it's subtly different from these four. 00:47:07.090 --> 00:47:08.682 align:middle line:84% I'm up to the fourth process in a way 00:47:08.682 --> 00:47:09.890 align:middle line:90% which I think is informative. 00:47:09.890 --> 00:47:13.230 align:middle line:84% And then we will go on and cover what you're talking about, yes. 00:47:13.230 --> 00:47:17.650 align:middle line:84% Any other questions on this before I do the full process? 00:47:17.650 --> 00:47:18.220 align:middle line:90% OK. 00:47:18.220 --> 00:47:20.890 align:middle line:84% You may have noticed there's a sort of symmetry 00:47:20.890 --> 00:47:22.910 align:middle line:90% to these processes so far. 00:47:22.910 --> 00:47:27.340 align:middle line:84% And so this is the symmetric partner of this process. 00:47:27.340 --> 00:47:30.310 align:middle line:84% This is where we have some intermediate energy level. 00:47:30.310 --> 00:47:37.350 align:middle line:84% We have an electron drop down out of the unbound states, 00:47:37.350 --> 00:47:44.480 align:middle line:84% and this spontaneously promotes an electron 00:47:44.480 --> 00:47:47.570 align:middle line:84% for this intermediate state here. 00:47:47.570 --> 00:47:48.290 align:middle line:90% OK. 00:47:48.290 --> 00:47:50.780 align:middle line:90% And this is called-- 00:47:50.780 --> 00:47:52.025 align:middle line:90% well, this is, again, from-- 00:47:52.025 --> 00:47:55.330 align:middle line:90% 00:47:55.330 --> 00:47:56.620 align:middle line:90% got LI written here. 00:47:56.620 --> 00:47:57.620 align:middle line:90% I'm going to go with it. 00:47:57.620 --> 00:48:00.160 align:middle line:90% But maybe it should be IL. 00:48:00.160 --> 00:48:04.570 align:middle line:84% And this has a superscript DR, and this is called 00:48:04.570 --> 00:48:07.090 align:middle line:90% dielectronic recombination. 00:48:07.090 --> 00:48:19.370 align:middle line:90% 00:48:19.370 --> 00:48:21.110 align:middle line:90% What is it proportional to? 00:48:21.110 --> 00:48:28.400 align:middle line:90% 00:48:28.400 --> 00:48:30.350 align:middle line:90% Yeah. 00:48:30.350 --> 00:48:32.090 align:middle line:84% Then see electrons, again, it relies 00:48:32.090 --> 00:48:34.007 align:middle line:84% on there being electrons here, free electrons. 00:48:34.007 --> 00:48:35.673 align:middle line:84% When we're writing any, obviously, we're 00:48:35.673 --> 00:48:37.160 align:middle line:84% talking about free electrons here, 00:48:37.160 --> 00:48:40.200 align:middle line:84% ones which are not bound by atoms lies on that. 00:48:40.200 --> 00:48:42.290 align:middle line:84% So a couple of interesting things here. 00:48:42.290 --> 00:48:44.750 align:middle line:84% Notice that these reverse processes do not 00:48:44.750 --> 00:48:46.430 align:middle line:84% have the same scaling with density, 00:48:46.430 --> 00:48:48.140 align:middle line:84% and so we don't necessarily expect these 00:48:48.140 --> 00:48:49.310 align:middle line:90% processes to balance. 00:48:49.310 --> 00:48:52.910 align:middle line:84% In fact, in an arbitrary plasma, they will not balance. 00:48:52.910 --> 00:48:56.090 align:middle line:84% In steady state overall, you have 00:48:56.090 --> 00:48:58.205 align:middle line:84% to have balance between the upward processes 00:48:58.205 --> 00:48:59.330 align:middle line:90% and the downward processes. 00:48:59.330 --> 00:49:00.872 align:middle line:84% But it doesn't mean that each process 00:49:00.872 --> 00:49:04.130 align:middle line:84% has to be balanced by its mirror because they are not symmetric. 00:49:04.130 --> 00:49:09.560 align:middle line:84% These recombination methods here require one more electron 00:49:09.560 --> 00:49:12.650 align:middle line:84% than the ionization mechanism here. 00:49:12.650 --> 00:49:16.820 align:middle line:84% What else is interesting about all of these ionization 00:49:16.820 --> 00:49:18.710 align:middle line:84% and recombination processes from the point 00:49:18.710 --> 00:49:20.425 align:middle line:90% of view of a spectroscopist? 00:49:20.425 --> 00:49:27.220 align:middle line:90% 00:49:27.220 --> 00:49:28.240 align:middle line:90% How would-- yes, go on. 00:49:28.240 --> 00:49:31.730 align:middle line:90% 00:49:31.730 --> 00:49:34.940 align:middle line:90% They are non-radiative. 00:49:34.940 --> 00:49:44.730 align:middle line:90% 00:49:44.730 --> 00:49:47.400 align:middle line:84% Does that mean, given this is a diagnostics course, 00:49:47.400 --> 00:49:49.890 align:middle line:84% that I just wasted your time by telling you about them 00:49:49.890 --> 00:49:51.540 align:middle line:90% because we cannot diagnose them? 00:49:51.540 --> 00:49:56.790 align:middle line:90% 00:49:56.790 --> 00:49:58.440 align:middle line:90% I've wasted your time, OK. 00:49:58.440 --> 00:50:00.510 align:middle line:84% Any other votes for wasting your time? 00:50:00.510 --> 00:50:02.805 align:middle line:90% I've been known to do it before. 00:50:02.805 --> 00:50:03.305 align:middle line:90% Yeah. 00:50:03.305 --> 00:50:10.880 align:middle line:90% 00:50:10.880 --> 00:50:13.610 align:middle line:84% They are absolutely key in some plasmas. 00:50:13.610 --> 00:50:16.790 align:middle line:84% But understanding the ionization state of the plasma, 00:50:16.790 --> 00:50:20.330 align:middle line:84% and also for understanding, in the case of these ones, 00:50:20.330 --> 00:50:22.640 align:middle line:84% the excitation state of each ion. 00:50:22.640 --> 00:50:24.680 align:middle line:84% And if you don't know that, you can't 00:50:24.680 --> 00:50:27.210 align:middle line:84% predict how many photons you're going to get out. 00:50:27.210 --> 00:50:29.840 align:middle line:84% So even if your signature is a radiated signature, 00:50:29.840 --> 00:50:32.570 align:middle line:84% it will be strongly modified by this. 00:50:32.570 --> 00:50:43.670 align:middle line:84% So these are still important for the population. 00:50:43.670 --> 00:50:47.760 align:middle line:90% 00:50:47.760 --> 00:50:54.470 align:middle line:84% And in this case, this is the population of excited states 00:50:54.470 --> 00:50:55.815 align:middle line:90% and ionization states. 00:50:55.815 --> 00:51:05.000 align:middle line:90% 00:51:05.000 --> 00:51:06.450 align:middle line:84% Eh, I definitely ran out of room. 00:51:06.450 --> 00:51:07.730 align:middle line:90% That says states. 00:51:07.730 --> 00:51:08.540 align:middle line:90% OK. 00:51:08.540 --> 00:51:09.650 align:middle line:90% Cool. 00:51:09.650 --> 00:51:11.248 align:middle line:84% Any questions on these before we go on 00:51:11.248 --> 00:51:13.790 align:middle line:84% to some [? radiative ?] ones that we can directly [INAUDIBLE] 00:51:13.790 --> 00:51:15.300 align:middle line:90% and also [INAUDIBLE]. 00:51:15.300 --> 00:51:18.115 align:middle line:90% 00:51:18.115 --> 00:51:18.615 align:middle line:90% Yes. 00:51:18.615 --> 00:51:58.345 align:middle line:90% 00:51:58.345 --> 00:51:58.845 align:middle line:90% Yeah. 00:51:58.845 --> 00:52:02.445 align:middle line:90% I mean, you-- what I did-- 00:52:02.445 --> 00:52:10.260 align:middle line:90% 00:52:10.260 --> 00:52:13.110 align:middle line:84% what I did here without making it very explicit, 00:52:13.110 --> 00:52:15.030 align:middle line:84% and we will go on to this in a moment, 00:52:15.030 --> 00:52:17.050 align:middle line:90% is write a rate equation. 00:52:17.050 --> 00:52:21.240 align:middle line:84% And this is just a rate equation with two levels. 00:52:21.240 --> 00:52:25.350 align:middle line:84% But instead, you would write a set of coupled rate equations 00:52:25.350 --> 00:52:29.282 align:middle line:84% which has the rate equation for each pair of energy levels. 00:52:29.282 --> 00:52:31.740 align:middle line:84% And you'd have to write that for each pair of energy levels 00:52:31.740 --> 00:52:33.870 align:middle line:84% within the hydrogen or the deuterium, 00:52:33.870 --> 00:52:35.315 align:middle line:84% and each pair within the tritium. 00:52:35.315 --> 00:52:37.440 align:middle line:84% Now, deuterium and tritium are close enough that we 00:52:37.440 --> 00:52:38.523 align:middle line:90% don't have to worry about. 00:52:38.523 --> 00:52:40.832 align:middle line:84% But you will have tungsten in your reactor, 00:52:40.832 --> 00:52:41.790 align:middle line:90% or something like that. 00:52:41.790 --> 00:52:44.220 align:middle line:84% And then you will want to solve the tungsten rate 00:52:44.220 --> 00:52:47.880 align:middle line:84% equation so you know how much ionizing radiation is coming out 00:52:47.880 --> 00:52:50.073 align:middle line:84% of the tungsten that can then reionize hydrogen 00:52:50.073 --> 00:52:51.240 align:middle line:90% and you'll scrape off layer. 00:52:51.240 --> 00:52:52.657 align:middle line:84% I don't know, something like that. 00:52:52.657 --> 00:52:53.890 align:middle line:90% This stuff may be important. 00:52:53.890 --> 00:52:56.370 align:middle line:84% And so, yes, of course, as soon as you 00:52:56.370 --> 00:52:58.872 align:middle line:84% do anything more interesting than these very simple systems, 00:52:58.872 --> 00:53:01.080 align:middle line:84% you're going to have even more complicated equations. 00:53:01.080 --> 00:53:05.440 align:middle line:90% And I will write down the full-- 00:53:05.440 --> 00:53:08.590 align:middle line:84% think, the full-rate equation towards the end of this class 00:53:08.590 --> 00:53:09.820 align:middle line:90% if we get to it. 00:53:09.820 --> 00:53:12.983 align:middle line:84% But it looks like this, but with lots of some symbols 00:53:12.983 --> 00:53:15.400 align:middle line:84% to indicate the fact that you have to do it lots of times. 00:53:15.400 --> 00:53:16.480 align:middle line:90% Yeah. 00:53:16.480 --> 00:53:17.951 align:middle line:90% Any other questions? 00:53:17.951 --> 00:53:22.035 align:middle line:90% 00:53:22.035 --> 00:53:22.535 align:middle line:90% OK. 00:53:22.535 --> 00:53:25.690 align:middle line:90% 00:53:25.690 --> 00:53:26.190 align:middle line:90% Right. 00:53:26.190 --> 00:53:32.083 align:middle line:84% Now, we're getting on to processes that, I think, 00:53:32.083 --> 00:53:33.250 align:middle line:90% Grant was asking about that. 00:53:33.250 --> 00:54:13.570 align:middle line:90% 00:54:13.570 --> 00:54:18.510 align:middle line:84% So these are, again, ionization and recombination processes. 00:54:18.510 --> 00:54:21.450 align:middle line:84% So again, I have this shaded region 00:54:21.450 --> 00:54:23.970 align:middle line:90% with free electrons in it. 00:54:23.970 --> 00:54:27.390 align:middle line:84% I'm going to call the highest energy level the ionization 00:54:27.390 --> 00:54:28.030 align:middle line:90% level. 00:54:28.030 --> 00:54:30.930 align:middle line:84% And the lower level L. And now, I'm 00:54:30.930 --> 00:54:34.050 align:middle line:84% going to look at processes, for example, which involve protons. 00:54:34.050 --> 00:54:41.670 align:middle line:84% So here is a photon coming in and it's got energy H nu IL. 00:54:41.670 --> 00:54:43.470 align:middle line:84% So it's got an energy which is matched 00:54:43.470 --> 00:54:46.740 align:middle line:84% to this gap between the lower state and the excited state. 00:54:46.740 --> 00:54:50.800 align:middle line:84% And as you might expect, this takes an electron 00:54:50.800 --> 00:54:52.990 align:middle line:90% and ionizes it. 00:54:52.990 --> 00:54:53.560 align:middle line:90% OK. 00:54:53.560 --> 00:54:58.090 align:middle line:84% And so this process here is called photoionization. 00:54:58.090 --> 00:55:08.170 align:middle line:90% 00:55:08.170 --> 00:55:15.330 align:middle line:84% It has a rate coefficient C lower to ionized state. 00:55:15.330 --> 00:55:21.790 align:middle line:84% It has a superscript Vi And what is it proportional to? 00:55:21.790 --> 00:55:27.490 align:middle line:90% 00:55:27.490 --> 00:55:30.050 align:middle line:90% Yeah, thank you. 00:55:30.050 --> 00:55:31.730 align:middle line:90% With that specific energy. 00:55:31.730 --> 00:55:42.370 align:middle line:90% So RHO of nu [? LIL. ?] OK. 00:55:42.370 --> 00:55:44.590 align:middle line:84% Now, we have the opposite of that process. 00:55:44.590 --> 00:55:48.100 align:middle line:84% We have some electron, which is in some excited state. 00:55:48.100 --> 00:55:51.650 align:middle line:90% And then it drops down to here. 00:55:51.650 --> 00:55:53.390 align:middle line:84% We've already covered this, right? 00:55:53.390 --> 00:55:57.260 align:middle line:84% This is spontaneous radiative recombination. 00:55:57.260 --> 00:55:59.150 align:middle line:84% We talked about this briefly, but now we're 00:55:59.150 --> 00:56:04.160 align:middle line:84% putting it into our more quantum picture of what's going on. 00:56:04.160 --> 00:56:06.810 align:middle line:84% And we will give it a rate, as well, 00:56:06.810 --> 00:56:09.210 align:middle line:84% so that we can include it in our rate equation here. 00:56:09.210 --> 00:56:17.420 align:middle line:84% So this is CIL radiative recombination. 00:56:17.420 --> 00:56:24.920 align:middle line:84% I'll write that up here, CIL radiative recombination. 00:56:24.920 --> 00:56:31.770 align:middle line:84% So technically, this is spontaneous radiative 00:56:31.770 --> 00:56:32.910 align:middle line:90% recombination. 00:56:32.910 --> 00:56:47.440 align:middle line:90% 00:56:47.440 --> 00:56:51.300 align:middle line:90% What's this proportional to? 00:56:51.300 --> 00:56:51.960 align:middle line:90% Thank you. 00:56:51.960 --> 00:56:55.430 align:middle line:90% 00:56:55.430 --> 00:56:58.927 align:middle line:84% Note again, big asymmetry between these two processes 00:56:58.927 --> 00:57:00.510 align:middle line:84% which look like mirrors of each other. 00:57:00.510 --> 00:57:02.760 align:middle line:84% One of them depends on the photon density, one of them 00:57:02.760 --> 00:57:04.100 align:middle line:90% depends on the electron density. 00:57:04.100 --> 00:57:06.100 align:middle line:84% There's no good reason to believe that these two 00:57:06.100 --> 00:57:07.760 align:middle line:90% processes balance each other. 00:57:07.760 --> 00:57:08.630 align:middle line:90% OK. 00:57:08.630 --> 00:57:12.560 align:middle line:84% The final process is, again, you can see these 00:57:12.560 --> 00:57:17.230 align:middle line:84% if you want to as the ionization recombination 00:57:17.230 --> 00:57:20.470 align:middle line:84% analogs of the excitation and de-excitation processes. 00:57:20.470 --> 00:57:22.780 align:middle line:84% And indeed, we do have a process where 00:57:22.780 --> 00:57:26.030 align:middle line:84% we come in with a photon with energy HIL. 00:57:26.030 --> 00:57:28.540 align:middle line:90% 00:57:28.540 --> 00:57:33.670 align:middle line:84% We have an electron drop down, and then we get two photons 00:57:33.670 --> 00:57:38.150 align:middle line:90% out here, H nu IL. 00:57:38.150 --> 00:57:44.140 align:middle line:84% This is stimulated radiative recombination. 00:57:44.140 --> 00:57:46.640 align:middle line:84% You see, I'm getting lazier and lazier as the class goes on. 00:57:46.640 --> 00:57:50.720 align:middle line:84% I can't be bothered to write out any full words anymore. 00:57:50.720 --> 00:57:57.410 align:middle line:84% And this has a coefficient CIL stimulated radiative 00:57:57.410 --> 00:57:58.820 align:middle line:90% recombination. 00:57:58.820 --> 00:58:02.280 align:middle line:84% That is proportional not only to the electron density, 00:58:02.280 --> 00:58:07.100 align:middle line:84% but also, the density of photon states with frequency 00:58:07.100 --> 00:58:09.440 align:middle line:90% nu IL, like that. 00:58:09.440 --> 00:58:12.030 align:middle line:90% 00:58:12.030 --> 00:58:17.930 align:middle line:84% And these processes, as we discussed, are radiative. 00:58:17.930 --> 00:58:20.180 align:middle line:84% So we can look out for them, but there 00:58:20.180 --> 00:58:21.800 align:middle line:90% are nice diagnostic signature. 00:58:21.800 --> 00:58:24.680 align:middle line:84% But they are also important for setting 00:58:24.680 --> 00:58:31.560 align:middle line:84% the occupation of the different energy and ionization states. 00:58:31.560 --> 00:58:32.750 align:middle line:90% Question. 00:58:32.750 --> 00:58:38.550 align:middle line:84% AUDIENCE: You mentioned that the process of [INAUDIBLE] there 00:58:38.550 --> 00:58:50.420 align:middle line:84% is some [INAUDIBLE] by [INAUDIBLE] but like [INAUDIBLE] 00:58:50.420 --> 00:58:56.710 align:middle line:84% far above [INAUDIBLE] having [INAUDIBLE] was this really 00:58:56.710 --> 00:58:58.990 align:middle line:90% like a quality [? type? ?] 00:58:58.990 --> 00:59:00.400 align:middle line:90% JACK HARE: Yeah, absolutely. 00:59:00.400 --> 00:59:03.190 align:middle line:84% The process up to some energy state I, which 00:59:03.190 --> 00:59:05.600 align:middle line:90% could be a free state up here. 00:59:05.600 --> 00:59:09.420 align:middle line:84% So this could be some other state like that. 00:59:09.420 --> 00:59:12.570 align:middle line:84% We can think of the free states as a continuum of states. 00:59:12.570 --> 00:59:14.640 align:middle line:84% So once you get above that ionization energy, 00:59:14.640 --> 00:59:17.265 align:middle line:84% you don't have discrete states, you have a continuum of states. 00:59:17.265 --> 00:59:20.970 align:middle line:84% And that would depend on this energy density 00:59:20.970 --> 00:59:23.820 align:middle line:84% here, which of course, for very high energy 00:59:23.820 --> 00:59:24.930 align:middle line:90% photons will be very low. 00:59:24.930 --> 00:59:27.720 align:middle line:84% We don't have very many gamma rays from a black body 00:59:27.720 --> 00:59:29.917 align:middle line:90% unless it's very, very hot. 00:59:29.917 --> 00:59:31.750 align:middle line:84% The only thing I would say, and I'm not sure 00:59:31.750 --> 00:59:36.280 align:middle line:84% exactly how to make this not hand-wavy, is that your gamma 00:59:36.280 --> 00:59:38.050 align:middle line:84% rays, although, they are good at ionizing, 00:59:38.050 --> 00:59:40.253 align:middle line:84% they tend to just go through stuff, right? 00:59:40.253 --> 00:59:42.670 align:middle line:84% I mean, the point is if you put a bit of paper in the way, 00:59:42.670 --> 00:59:43.803 align:middle line:90% you're like great. 00:59:43.803 --> 00:59:45.220 align:middle line:84% The gamma ray hits a bit of paper. 00:59:45.220 --> 00:59:46.690 align:middle line:84% It's got all the energy it needs. 00:59:46.690 --> 00:59:48.693 align:middle line:84% It should just be absorbed and ionize something. 00:59:48.693 --> 00:59:50.110 align:middle line:84% But in fact, it goes through that, 00:59:50.110 --> 00:59:51.650 align:middle line:84% and it goes through lots of other things. 00:59:51.650 --> 00:59:53.770 align:middle line:84% So I think there is a quantum mechanical effect 00:59:53.770 --> 00:59:55.780 align:middle line:84% about the interaction of this very high energy 00:59:55.780 --> 00:59:58.510 align:middle line:84% radiation with these energy levels, which makes 00:59:58.510 --> 01:00:00.620 align:middle line:90% that a less likely process. 01:00:00.620 --> 01:00:02.710 align:middle line:84% So the gamma ray's, in general, not going 01:00:02.710 --> 01:00:05.410 align:middle line:84% to have very high cross-section for that interaction. 01:00:05.410 --> 01:00:08.710 align:middle line:84% But I don't know how to take that hand-wavy argument 01:00:08.710 --> 01:00:10.712 align:middle line:84% and turn it into something mathematically. 01:00:10.712 --> 01:00:14.537 align:middle line:90% 01:00:14.537 --> 01:00:15.037 align:middle line:90% Yeah. 01:00:15.037 --> 01:00:31.110 align:middle line:90% 01:00:31.110 --> 01:00:32.980 align:middle line:90% Yeah. 01:00:32.980 --> 01:00:34.150 align:middle line:90% Yeah. 01:00:34.150 --> 01:00:36.130 align:middle line:84% And so I'd have to average over-- 01:00:36.130 --> 01:00:40.530 align:middle line:84% integrate over all the energies, which is what I'm about to do. 01:00:40.530 --> 01:00:41.780 align:middle line:90% So, no, it's a great question. 01:00:41.780 --> 01:00:42.697 align:middle line:90% It's a great question. 01:00:42.697 --> 01:00:47.810 align:middle line:84% So for the collisional processes, 01:00:47.810 --> 01:00:54.450 align:middle line:84% so these non-radiative ones, these ones. 01:00:54.450 --> 01:00:59.720 align:middle line:84% And the ones on this side, as well, at least this set here. 01:00:59.720 --> 01:01:01.710 align:middle line:84% We actually-- we have a different way. 01:01:01.710 --> 01:01:04.670 align:middle line:84% We don't tend to write these coefficients in terms like this. 01:01:04.670 --> 01:01:07.310 align:middle line:84% We tend to think about them in terms of reactivities, right? 01:01:07.310 --> 01:01:10.760 align:middle line:84% So we have some cross section for the interaction which 01:01:10.760 --> 01:01:14.030 align:middle line:90% we could write as sigma IJ. 01:01:14.030 --> 01:01:16.640 align:middle line:84% So that is the cross section for an interaction going 01:01:16.640 --> 01:01:19.070 align:middle line:90% from state I to state J. 01:01:19.070 --> 01:01:24.200 align:middle line:84% We multiply that by the velocity of the free electron 01:01:24.200 --> 01:01:27.860 align:middle line:84% that we're working with multiplied by our distribution 01:01:27.860 --> 01:01:30.140 align:middle line:90% function of electrons. 01:01:30.140 --> 01:01:35.330 align:middle line:84% And then we integrate that up over all the electrons 01:01:35.330 --> 01:01:37.010 align:middle line:90% in our distribution function. 01:01:37.010 --> 01:01:38.960 align:middle line:84% And that will give us something that 01:01:38.960 --> 01:01:47.700 align:middle line:84% looks like, for example, any sigma i j v some reactivity 01:01:47.700 --> 01:01:49.140 align:middle line:90% here, times the density. 01:01:49.140 --> 01:01:53.440 align:middle line:90% 01:01:53.440 --> 01:01:59.250 align:middle line:84% If we're dealing with a process that involves two electrons, 01:01:59.250 --> 01:02:02.640 align:middle line:84% like three-body recombination, we'd 01:02:02.640 --> 01:02:05.880 align:middle line:84% actually have to put the distribution function in twice 01:02:05.880 --> 01:02:09.600 align:middle line:84% and integrate over the interactions between particles 01:02:09.600 --> 01:02:13.020 align:middle line:84% going from distribution 1 V1, and particles going 01:02:13.020 --> 01:02:14.768 align:middle line:90% from distribution 2 at V2. 01:02:14.768 --> 01:02:17.310 align:middle line:84% So you don't just square this, you actually have to integrate 01:02:17.310 --> 01:02:23.468 align:middle line:84% and then do a double integral over these two distributions. 01:02:23.468 --> 01:02:24.914 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:02:24.914 --> 01:02:28.288 align:middle line:90% 01:02:28.288 --> 01:02:29.560 align:middle line:90% JACK HARE: Per nucleus, yeah. 01:02:29.560 --> 01:02:33.340 align:middle line:84% So the total rate would have an additional times 01:02:33.340 --> 01:02:38.020 align:middle line:84% N. I'm going to go with I here, as in whatever. 01:02:38.020 --> 01:02:42.430 align:middle line:84% This is an electron causing a change in the atom state 01:02:42.430 --> 01:02:45.610 align:middle line:84% from I to J, and so we need to know how many atoms there 01:02:45.610 --> 01:02:49.360 align:middle line:84% were initially in state I. And in this case, when we're writing 01:02:49.360 --> 01:02:54.640 align:middle line:84% this in terms of I and J, so this is transition from I to J, 01:02:54.640 --> 01:03:00.920 align:middle line:84% and that could be excitation or de-excitation, 01:03:00.920 --> 01:03:04.680 align:middle line:84% or it could be ionization, as well. 01:03:04.680 --> 01:03:07.260 align:middle line:84% The reaction rates, the reactivities, 01:03:07.260 --> 01:03:11.330 align:middle line:90% have the same form as that. 01:03:11.330 --> 01:03:13.580 align:middle line:84% So you'll still end up with something like that. 01:03:13.580 --> 01:03:14.570 align:middle line:90% OK. 01:03:14.570 --> 01:03:16.220 align:middle line:84% And so then, we're now in a position 01:03:16.220 --> 01:03:18.335 align:middle line:90% to write down our full balance. 01:03:18.335 --> 01:03:21.670 align:middle line:90% 01:03:21.670 --> 01:03:29.620 align:middle line:84% So we can say the change in particles, atoms in state I-- 01:03:29.620 --> 01:03:32.080 align:middle line:84% for this example is an excited state, 01:03:32.080 --> 01:03:36.130 align:middle line:84% or it could be an ionization state with respect to time 01:03:36.130 --> 01:03:44.220 align:middle line:84% is going to be equal to the sum over states not equal to I 01:03:44.220 --> 01:03:44.720 align:middle line:90% states. 01:03:44.720 --> 01:03:47.465 align:middle line:84% J not equal to I, because if we have a transition from state I 01:03:47.465 --> 01:03:50.090 align:middle line:84% to state I, we don't change the number of particles in state I, 01:03:50.090 --> 01:03:52.340 align:middle line:84% so we don't want to double count. 01:03:52.340 --> 01:03:55.400 align:middle line:84% This is going to be the number of particles in I, 01:03:55.400 --> 01:04:01.450 align:middle line:84% the spontaneous emission process from I to J 01:04:01.450 --> 01:04:03.970 align:middle line:84% minus the number of particles in J, 01:04:03.970 --> 01:04:08.290 align:middle line:84% the spontaneous emission from some other J to I. 01:04:08.290 --> 01:04:11.138 align:middle line:84% These coefficients may be zero if spontaneous emission 01:04:11.138 --> 01:04:11.680 align:middle line:90% is forbidden. 01:04:11.680 --> 01:04:13.210 align:middle line:84% For example, if spontaneous emission 01:04:13.210 --> 01:04:16.190 align:middle line:84% would involve going up an energy level. 01:04:16.190 --> 01:04:19.210 align:middle line:84% So in this case, we're looking at spontaneous emission 01:04:19.210 --> 01:04:24.650 align:middle line:84% from I downwards and from a different J back upwards here. 01:04:24.650 --> 01:04:32.260 align:middle line:84% So this is I. This is I like that. 01:04:32.260 --> 01:04:34.750 align:middle line:84% Then we would also have a term that's 01:04:34.750 --> 01:04:40.790 align:middle line:84% to do with our absorption and stimulated emission. 01:04:40.790 --> 01:04:54.350 align:middle line:84% So that is ni Bij minus nj Bji density of photons with energy, 01:04:54.350 --> 01:04:57.140 align:middle line:90% or with frequency IJ like that. 01:04:57.140 --> 01:04:59.800 align:middle line:90% 01:04:59.800 --> 01:05:03.740 align:middle line:84% So that covers these processes, as well. 01:05:03.740 --> 01:05:05.890 align:middle line:84% Then we have the two-body processes, 01:05:05.890 --> 01:05:10.120 align:middle line:84% which are these collisions, impact ionization, 01:05:10.120 --> 01:05:12.460 align:middle line:84% autoionization, this sort of thing. 01:05:12.460 --> 01:05:16.170 align:middle line:84% Actually, autoionization is not a two-body one. 01:05:16.170 --> 01:05:16.670 align:middle line:90% OK. 01:05:16.670 --> 01:05:18.740 align:middle line:90% We'll get that back flat. 01:05:18.740 --> 01:05:21.830 align:middle line:84% Well, we're going to have the density of electrons now, 01:05:21.830 --> 01:05:27.170 align:middle line:84% the density of ions in state I, and then this sigma 01:05:27.170 --> 01:05:31.940 align:middle line:84% IJV that we just derived, or at least handwaved, 01:05:31.940 --> 01:05:33.530 align:middle line:90% into existence up here. 01:05:33.530 --> 01:05:35.330 align:middle line:84% And that's also going to be balanced 01:05:35.330 --> 01:05:38.780 align:middle line:84% by the opposite process driving the system 01:05:38.780 --> 01:05:40.250 align:middle line:90% in the opposite direction. 01:05:40.250 --> 01:05:43.080 align:middle line:90% 01:05:43.080 --> 01:05:46.790 align:middle line:90% So sigma JI instead of sigma IJ. 01:05:46.790 --> 01:05:55.090 align:middle line:84% That should be times V. These processes are two-body. 01:05:55.090 --> 01:06:01.680 align:middle line:84% These processes on the top line here are one-body, maybe 01:06:01.680 --> 01:06:04.140 align:middle line:84% plus 1 photon, but that doesn't count. 01:06:04.140 --> 01:06:07.170 align:middle line:84% And then we have the three-body processes at the end here. 01:06:07.170 --> 01:06:10.170 align:middle line:90% 01:06:10.170 --> 01:06:18.340 align:middle line:90% And E squared and I sigma IJV. 01:06:18.340 --> 01:06:20.230 align:middle line:84% And the meaning of the angle brackets 01:06:20.230 --> 01:06:24.340 align:middle line:84% here is actually an average over both the distribution functions, 01:06:24.340 --> 01:06:26.560 align:middle line:84% or at least over the distribution function 01:06:26.560 --> 01:06:27.970 align:middle line:90% with itself. 01:06:27.970 --> 01:06:30.790 align:middle line:84% And then, of course, there'll be the opposite process here. 01:06:30.790 --> 01:06:34.480 align:middle line:84% And E squared and J is an opposite process. 01:06:34.480 --> 01:06:40.345 align:middle line:90% 01:06:40.345 --> 01:06:41.470 align:middle line:90% I'm not convinced there is. 01:06:41.470 --> 01:06:43.822 align:middle line:84% I'm going to put it in anyway because it's my notes. 01:06:43.822 --> 01:06:45.280 align:middle line:84% I can think about that in a moment. 01:06:45.280 --> 01:06:49.070 align:middle line:90% 01:06:49.070 --> 01:06:51.333 align:middle line:84% And these are the three-body processes. 01:06:51.333 --> 01:06:59.932 align:middle line:90% 01:06:59.932 --> 01:07:01.390 align:middle line:84% Now, in general, we allow ourselves 01:07:01.390 --> 01:07:03.490 align:middle line:84% a small simplification, and we tend 01:07:03.490 --> 01:07:06.970 align:middle line:84% to look for plasmas in steady state. 01:07:06.970 --> 01:07:08.890 align:middle line:84% And that steady state doesn't have 01:07:08.890 --> 01:07:10.390 align:middle line:90% to be an absolute steady state. 01:07:10.390 --> 01:07:12.910 align:middle line:84% It can be on some time scale of interest. 01:07:12.910 --> 01:07:17.060 align:middle line:84% The plasma does not change its occupation state very much. 01:07:17.060 --> 01:07:20.950 align:middle line:84% And so this is a reasonable approximation, 01:07:20.950 --> 01:07:22.240 align:middle line:90% quasi steady state. 01:07:22.240 --> 01:07:24.220 align:middle line:84% Because even with this approximation, 01:07:24.220 --> 01:07:26.800 align:middle line:84% I think it's pretty clear that this is a very long equation 01:07:26.800 --> 01:07:30.220 align:middle line:84% that you would have to solve in order to work out the-- 01:07:30.220 --> 01:07:32.080 align:middle line:84% and this, I believe, is just what 01:07:32.080 --> 01:07:36.440 align:middle line:84% we need to get the excitation state of a single atom. 01:07:36.440 --> 01:07:41.470 align:middle line:84% So this is the excitation, for example, of an atom at state Z, 01:07:41.470 --> 01:07:42.280 align:middle line:90% like that. 01:07:42.280 --> 01:07:44.560 align:middle line:84% We'd have to have another coupled equation, which 01:07:44.560 --> 01:07:51.720 align:middle line:84% would tell us whether we're going from NI of Z to NIZ, 01:07:51.720 --> 01:07:56.220 align:middle line:90% plus 1, or NIZ minus 1. 01:07:56.220 --> 01:08:00.150 align:middle line:84% So that'll be our ionization balance equation, as well. 01:08:00.150 --> 01:08:03.140 align:middle line:84% And you have to couple all of those equations together. 01:08:03.140 --> 01:08:07.410 align:middle line:84% And this is where I start thinking that I've mixed up 01:08:07.410 --> 01:08:09.420 align:middle line:84% the excited states in the ionization states, 01:08:09.420 --> 01:08:11.340 align:middle line:84% because if I'm talking about excited states, 01:08:11.340 --> 01:08:13.530 align:middle line:84% I don't have these three-body processes. 01:08:13.530 --> 01:08:16.220 align:middle line:90% 01:08:16.220 --> 01:08:17.350 align:middle line:90% So it's a bit of a mess. 01:08:17.350 --> 01:08:17.859 align:middle line:90% OK. 01:08:17.859 --> 01:08:22.359 align:middle line:84% But my general point is a very, very large number of equations. 01:08:22.359 --> 01:08:26.050 align:middle line:84% What we will do in the next lecture is, 01:08:26.050 --> 01:08:28.990 align:middle line:84% we will work out some types of equilibrium 01:08:28.990 --> 01:08:31.479 align:middle line:84% where some of these processes are unimportant 01:08:31.479 --> 01:08:32.950 align:middle line:90% and, therefore, we drop them. 01:08:32.950 --> 01:08:35.529 align:middle line:84% And that simplification allows us 01:08:35.529 --> 01:08:38.350 align:middle line:84% to make a calculation of the occupation states 01:08:38.350 --> 01:08:41.680 align:middle line:84% in a much simpler way that is actually tractable. 01:08:41.680 --> 01:08:43.838 align:middle line:84% But, in general, for some arbitrary plasma 01:08:43.838 --> 01:08:45.880 align:middle line:84% where we can't make some of the assumptions we'll 01:08:45.880 --> 01:08:47.620 align:middle line:84% make in the next class, you will need 01:08:47.620 --> 01:08:50.229 align:middle line:84% to solve this full-rate equation in order 01:08:50.229 --> 01:08:55.330 align:middle line:84% to work out the occupation of excited states, the ionization 01:08:55.330 --> 01:08:57.880 align:middle line:84% states in your plasma and, therefore, 01:08:57.880 --> 01:09:01.810 align:middle line:84% what radiative signatures you're going to get out of it so 01:09:01.810 --> 01:09:03.160 align:middle line:90% that you can do spectroscopy. 01:09:03.160 --> 01:09:06.319 align:middle line:90% So all gets a bit complicated. 01:09:06.319 --> 01:09:08.630 align:middle line:90% We will leave it there. 01:09:08.630 --> 01:09:10.189 align:middle line:90% Does anyone have any questions? 01:09:10.189 --> 01:09:10.760 align:middle line:90% Yeah. 01:09:10.760 --> 01:09:12.242 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:09:12.242 --> 01:09:21.628 align:middle line:90% 01:09:21.628 --> 01:09:22.470 align:middle line:90% JACK HARE: Yeah. 01:09:22.470 --> 01:09:27.030 align:middle line:90% 01:09:27.030 --> 01:09:27.530 align:middle line:90% Oh. 01:09:27.530 --> 01:09:30.370 align:middle line:90% 01:09:30.370 --> 01:09:31.060 align:middle line:90% Yeah. 01:09:31.060 --> 01:09:33.010 align:middle line:84% Thank you, that's great, good point. 01:09:33.010 --> 01:09:34.240 align:middle line:90% You were paying attention. 01:09:34.240 --> 01:09:34.899 align:middle line:90% Very good. 01:09:34.899 --> 01:09:37.779 align:middle line:90% 01:09:37.779 --> 01:09:40.655 align:middle line:90% Fixed it. 01:09:40.655 --> 01:09:41.155 align:middle line:90% Yeah. 01:09:41.155 --> 01:09:48.840 align:middle line:90% 01:09:48.840 --> 01:09:51.270 align:middle line:90% NI in this case, yes. 01:09:51.270 --> 01:09:53.640 align:middle line:84% I would prefer to think of it as atoms 01:09:53.640 --> 01:09:56.280 align:middle line:84% in a certain electronic configuration. 01:09:56.280 --> 01:09:56.790 align:middle line:90% OK. 01:09:56.790 --> 01:09:59.782 align:middle line:84% So imagine you've got two electrons here. 01:09:59.782 --> 01:10:01.740 align:middle line:84% They're indistinguishable, so it doesn't really 01:10:01.740 --> 01:10:03.365 align:middle line:84% matter what state the electrons are in, 01:10:03.365 --> 01:10:06.900 align:middle line:84% as long as the configurations are indistinguishable. 01:10:06.900 --> 01:10:07.680 align:middle line:90% Yeah. 01:10:07.680 --> 01:10:11.230 align:middle line:84% So this is an atom in some state which has some energy. 01:10:11.230 --> 01:10:13.890 align:middle line:84% And so, for example, in a three-level system 01:10:13.890 --> 01:10:19.490 align:middle line:84% with two electrons, this would be your ground state, 01:10:19.490 --> 01:10:24.880 align:middle line:90% and this would be another state. 01:10:24.880 --> 01:10:29.260 align:middle line:84% But because we're not tracking which electron is which, 01:10:29.260 --> 01:10:32.050 align:middle line:84% this would be an identical state. 01:10:32.050 --> 01:10:33.940 align:middle line:84% So this would actually be like a degeneracy. 01:10:33.940 --> 01:10:36.670 align:middle line:84% When we're talking about that G factor before, 01:10:36.670 --> 01:10:38.900 align:middle line:84% these two states would be degenerate here. 01:10:38.900 --> 01:10:41.900 align:middle line:84% So this is-- so these are all different-- 01:10:41.900 --> 01:10:45.610 align:middle line:84% or this is, for example, N0, and this could 01:10:45.610 --> 01:10:47.950 align:middle line:90% be N3, or something like that. 01:10:47.950 --> 01:10:50.080 align:middle line:84% And maybe I would label my states 01:10:50.080 --> 01:10:52.558 align:middle line:84% in terms of their increasing overall energy. 01:10:52.558 --> 01:10:54.850 align:middle line:84% And so you can imagine there's some intermediate states 01:10:54.850 --> 01:10:56.660 align:middle line:84% with different energies like this. 01:10:56.660 --> 01:10:58.030 align:middle line:90% OK. 01:10:58.030 --> 01:10:59.227 align:middle line:90% Pick three out of a hat. 01:10:59.227 --> 01:11:00.310 align:middle line:90% I think it's probably two. 01:11:00.310 --> 01:11:03.150 align:middle line:90% Anyway, yeah, Nicola. 01:11:03.150 --> 01:11:06.594 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:11:06.594 --> 01:11:07.810 align:middle line:90% JACK HARE: Which one? 01:11:07.810 --> 01:11:10.696 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:11:10.696 --> 01:11:13.501 align:middle line:84% JACK HARE: Stimulated radiative recombination. 01:11:13.501 --> 01:11:35.110 align:middle line:90% 01:11:35.110 --> 01:11:38.420 align:middle line:84% This process is possible for any of those, but it doesn't-- 01:11:38.420 --> 01:11:40.520 align:middle line:84% there's a proportionality here, but there'll 01:11:40.520 --> 01:11:43.280 align:middle line:84% be a constant that depends on quantum in front of it. 01:11:43.280 --> 01:11:45.800 align:middle line:84% And I would have thought for very high energy electrons, 01:11:45.800 --> 01:11:48.870 align:middle line:84% this is not going to be a very significant process. 01:11:48.870 --> 01:12:25.240 align:middle line:90% So-- yeah, that could work. 01:12:25.240 --> 01:12:27.930 align:middle line:84% But the trouble is, of course, if you make your photon energy 01:12:27.930 --> 01:12:30.510 align:middle line:84% density field very large in order for this process 01:12:30.510 --> 01:12:33.690 align:middle line:84% to be big, then your photoionization also 01:12:33.690 --> 01:12:34.630 align:middle line:90% goes back up. 01:12:34.630 --> 01:12:37.740 align:middle line:84% And so if you have a radiation dominated plasma like that, 01:12:37.740 --> 01:12:39.750 align:middle line:84% it will still come to some equilibrium, 01:12:39.750 --> 01:12:42.630 align:middle line:84% and that equilibrium will not be everything in the ground state. 01:12:42.630 --> 01:12:43.922 align:middle line:90% It will be something different. 01:12:43.922 --> 01:12:45.030 align:middle line:90% And we'll talk about that. 01:12:45.030 --> 01:12:47.280 align:middle line:84% But, yes, you're-- I don't know anyone who's done this 01:12:47.280 --> 01:12:50.670 align:middle line:84% in the lab, just pumping in photons and somehow reducing 01:12:50.670 --> 01:12:52.983 align:middle line:84% the total energy of the system and getting light out. 01:12:52.983 --> 01:12:54.900 align:middle line:84% I think it's just one process that takes place 01:12:54.900 --> 01:12:55.733 align:middle line:90% with all the others. 01:12:55.733 --> 01:12:58.140 align:middle line:84% So you're unlikely to be able to engineer a plasma that 01:12:58.140 --> 01:12:59.790 align:middle line:90% is dominated by this. 01:12:59.790 --> 01:13:02.940 align:middle line:84% Do any of the other ones have a proportionality to both electron 01:13:02.940 --> 01:13:05.100 align:middle line:90% density and photon density? 01:13:05.100 --> 01:13:05.850 align:middle line:90% No. 01:13:05.850 --> 01:13:09.900 align:middle line:84% So there is scope here for making a very dense, very photon 01:13:09.900 --> 01:13:11.790 align:middle line:84% dominated plasma where you could imagine 01:13:11.790 --> 01:13:15.040 align:middle line:84% that this coefficient could be larger than any of the others. 01:13:15.040 --> 01:13:19.560 align:middle line:84% But I don't know how dense and how energetic, 01:13:19.560 --> 01:13:22.660 align:middle line:84% or how dense your photon field and your electron density 01:13:22.660 --> 01:13:24.670 align:middle line:90% needs to be for that to happen. 01:13:24.670 --> 01:13:26.400 align:middle line:90% Cool question. 01:13:26.400 --> 01:13:29.000 align:middle line:90%