 So now we come to some diagnostic applications. I can only talk about some very simple examples. And for the diagnostic applications, there are prerequisites. You need excellent atomic data. You can obtain them from experiments and from atomic calculations. And you can obtain them, they are supplemented by simulations of the atomic behavior. And you will learn a lot during this week about this part. To get the data, you need a plasma which is well-diagnosed. To get atomic data, you have to know very well your plasma. And you can analyze your plasma by Thomson scattering, employing lasers. It's usable for a broad range of spectrographs of atomic species. You can use electric probes. They are used in low-temperature plasmas. You use interferometry and polarimetry on small and big machines. Or you use other spectroscopic data, which are very well-based to transfer the results to unknown data. Now, let us start again. We assume we have measured the spectral radians of a line and have obtained the absolute local emission coefficient. I put it on the word also absolute. This has been done with calibrated instruments. Now, what you get is we have to start with the fundamental equation, I call it, of the emission. The emission is proportional to the density of the atoms in the upper level of the line. And it is proportional to the probability that is down, which is defined as the experimental decay rate. Then it's proportional to the photon energy of the electron going down divided by 4 pi into solid angle. So the best thing is always to look at the dimension of the line, because they are words differently used. I think most of the scientists still use instead of absolute radians, the use intensity, which according to some standards should be the radians. So each time you look what is meant, and I usually prefer then looking at the dimension, then I know what is meant. What's per cubic meter per steradian? Now, so if you have obtained the emission coefficient, the quantity do you obtain is given here. So from these measurements, you obtain the density of the atoms in the upper level. And one would say to an outsider, well, this is all. This is trivial. You get only one quantity in the plasma from a line, the upper density. And all the fast writing books and papers, well, that's the work scientists have to do. What can you do with the single quantity? Or many single quantities like that. So that's what we're all, what's all about it. So from spectroscopy of a line, we obtain the absolute particle density of the atomic species in the upper level and nothing else. OK, now we have to look if we have local thermodynamic equilibrium. Local thermodynamic equilibrium means there is collisional equilibrium between all levels without equilibrium with radiation. I don't go into that. This will take too much time. Then you can use thermodynamic relations. And you can relate the upper level density to the total density. But you must know the statistical weight of the upper level. You must know the partition function. And you must know the electron density. So you see, you can get the total density from one measurement if you know the plasma quite well, the temperature. And you know the atomic properties of the ion. And this you can reach in high density plasmas or the hot dense matter. Then you can reach this equilibrium. Now, if this equilibrium, local thermodynamic is done by collisions, they are so fast that they keep the electrons in the population densities in equilibrium. However, if now the density goes down, at first collisions between the lower levels will become weaker. And they will not be in equilibrium. So collisions of the upper levels only will be in equilibrium. And this we call partial thermodynamic equilibrium. So then the upper levels are in equilibrium. They are Boltzmann distribution. And they are connected to the next ion, the highest level of the lower element and the lowest level of the next element. And they are given by the thermodynamic equation called the accurate equation. And then we go to low densities. At low densities, collisions between the levels are so weak. Collisions between the level can be neglected. So you have only collisions from lower level to the upper level and radiative decay. And this is called corona equilibrium because this was where the conditions of the corona. And you see you have collisional excitation from the lower level and radiative decay. And this is called the rate coefficient cross sections times velocity distribution function averaged. OK. And so you need other atomic quantities here. You need the temperature. And you need the cross sections or rate coefficient for collisional excitation. So you see now comes in, it blows up what you need to do spectroscopy. And here you also can get the total density because collisions are so weak that you get collisional excitation and radiative decay. But it is so fast that most of the atoms or ions stay in the ground state. So the ground state equals very good or is very approximate to the total density. But what you need now, you see in the corona, you need the electron density. You need the rate coefficient. And you need the electron temperature to obtain the total density of this ion species. OK. So that's you need good experimental data and very good theoretical data. And the best, they must coincide. Now there is one exception where one can have also essentially corona equilibrium, but the lower states are so closely spaced that they are that collisions, electron collisions, and in Tokamak proton collisions are so fast that they make the lower levels equalized. And they are in a kind of PLE, the lower levels. The transitions are so slow because these are magnetic dipole transitions at the lower level that this equilibrium is reached. And these lines have the advantage. They are in the visible. And so they are used in Tokamak to measure the density. You get the total density of the lower level. The transition probability is very low. But since all ions are in the lower level, you still get enough signal. OK. So this sentence at Tokamak was that these magnetic dipole transitions in the ground state are in the visible or near UV. Now what you can do is if you don't have ions in the plasma, you inject ion beams. And people have done lithium, helium beams. I think I come to that later. Then you get also from the emission of these ions a particle density is one method. And one method coming up is the actinometry. This is usually used on technical plasmas where they are very complex. And there are poisonous materials sometimes. There they do the following. They add an atomic species, which is known very well properties, and which has a wavelength close to a higher wavelength in the bad species. But you don't have enough atomic data. And you compare both. And from the comparison, you get the density of this second material. But this is a comparison which can be very uncertain. But if you have no chance to get measurements otherwise, you do it. And in the literature, I find this technique mostly on technical plasmas. So here you see the ratio of both lines. The actinomer element is the emission coefficient of that times the excitation rate coefficient. And although this is unknown. And even if they are nearly equal, even that props. If they are, for example, both typo lines or something like that. But to do very good measurements, you need a collisional radiative model for both species. Then you can also compare both species if you had proper collisional models. So temperature measurements. Now you have temperatures. You have atomic temperatures, the ion temperatures. We have molecular temperatures. And where do you get temperatures? You get the temperature from line broadening of spectral lines. And at first you have, everybody knows, we have natural broadening each line due to its lifetime has a natural width, which is usually very small and negligible. Then we have pressure broadening by neutral particles. This is important in very weakly ionized plasmas. But in many, many cases negligible. Then you have broadening by Stark and Siemens effect. And there is included the broadening by electrons and plasma broadening is in here. Then you have Doppler broadening and broadening by the instrument function. This is also very important to account for. Because you have a line characterized by a delta function. It does not give the output of your spectra to have a delta function. It gives a broadened line. And this line is called instrumental function. And this can be very strong. Then people forget they open the slit because you have no line. Or I make it a little bit, the entrance slit wider to have more intensity. And your line gets broader. And it's usually a mistake. Now the Doppler broadening is very easy. You have the Doppler effect. And though if you have a Maxwellian velocity distribution function, the Doppler broadening is a Gaussian. And the Doppler width is given by this simple relation here, where m a is a mass in atomic unity and temperature of the gas. And in some plasmas, you also observe Doppler shifts. Like I mentioned with the crystal spectrograph on the talker mark, there from the line shifts we can measure rotation velocities. Different line shifts. Oh, sorry. Now we come to the electron temperature, which is one of the two most important quantities of a plasmas. And what we need, we need one technique is you use two lines whose upper level are in partial equilibrium. Because then the upper levels are coupled and described by a Boltzmann factor. And the ratio of the two lines here is the ratio of the emission coefficients. You see the ratio of the wavelength ratio of transition probability statistical weights. And here, together here, the Boltzmann factor. Now the question is how accurate can these measurements be? And if you look delta t over t, and here's the ratio of the two lines, delta r, the uncertainty, you see the accuracy depends on the energy difference of the upper levels. So even if you do a very good measurement, very accurate measurements of the intensity ratio of the two lines, the point is the difference of the energy of the upper levels. And if this energy is too small, you know, the uncertainty gets too big. So very important is this ratio compared to the electron comparison to the electron temperature. Now what one does is one takes not two lines, one takes many lines. And one plots, does not calculate the temperature, but one plots the upper densities, which I get, as a function of p of the upper level energy. And then you get, for a Boltzmann distribution, a straight line. And this kind of technique is called using the Boltzmann plot. And this gives high, good energy. Now what do you have to watch? Is a line optically thick? Because if the concentration of a line is too high, it is reabsorbed, it gets optically thick. So forget about it. So you have to check another line, maybe one line, which has a lower transition probability, and so on. Usually a Boltzmann plot helps because if a line is optically thick, it will fall out of the straight line. It will be a large deviation. So this is one possibility. Now what one does with molecules? In molecules, one can study the rotational and vibrational lines. And they are very close. So usually they are in Boltzmann distribution, PLTE. And one can then define a temperature. But one should be careful on this temperature I get is the electron temperature. No, it is a rotational temperature. And it is a vibrational temperature because the equilibrium may not be done by electron collision, can be done by molecular collisions and neutral collisions. So one should clearly state, I measure the rotational temperature or the vibrational temperature. They can be in some cases equal to the electron temperature, but one should not mislead the reader. Then in other plasmas where one cannot do much, one defines in technical plasmas, people do it. They describe the upper level by a Boltzmann distribution. And as if the LTE holds, independently holds or not. And this is correctly termed excitation temperature. It must not be the electron temperature because if we don't have LTE, we just describe the unknown temperature as excitation temperature. Interpretation is different. Now, as I said, if you want to do temperature measurement with two lines, you take the ratio of the lines. Now is one trick. Here you take the ratio of two lines from different ionization states or consecutive ionization stages. So you see, you take this line here, is in LTE with set one. This line is in LTE with set ground state. And the ions are connected via the Sahay-Eggert equation. And you can get a ratio of both lines where now this difference of the energies of the upper level is very large. So you get a high accuracy. But usually you get nothing for nothing. You get dependence on the electron density because due to the Sahay-Eggert equation, you introduce the electron density. It's easier at low densities. You take the ratio and the ratio in the corona equilibrium. Remember, ionization by collisions, electron collisions, deaccitation by radiative decay. Here is the ratio. And you see it is the ratio of the rate coefficients. And this is a sole function of the electron density. So you need very good collision. Now this is even more convenient if the both lines are dipole transitions. Then you can use a very simple formula for the excitation rate coefficient, the gone factor approximation. So also important, large spacing of the energy of the upper levels. A nice example. I show you only a few very old and common examples. What you can do is lithium like ions are nice ions for diagnostics too. You take two lines, one line between the upper level and this line. So this line is excited by electron collisions. This also in the ratio of these two lines is a strong function of the electron temperature, large spacing. And these lines, the upper lines are in the visible. So it's convenient to take these studies, these lines. Here I have two examples. This was also the case which we use for branching ratio calibration, as I mentioned in the first lecture. Now when you increase the density, the situation becomes more complex and in most cases you need a collisional radiative model. Of interest are ions or elements which have metastable levels because metastable levels have high population densities. And an interesting case here is helium. And I show you one example. Helium, as you remember, has a singlet system as a triplet system. And here you see the excitation rate. No, it's a cross-section. Here you see the cross-section for excitation to the 2p singlet level. And here you see the cross-section for excitation to the 2p triplet level. And you see this is much smaller because of spin exchange. And here you see when you go up in the energy, it's really different. So this can be used for spectroscopy. And I think there was a first paper in the 1950s who used this ratio of tingling and triplet level propagating as a temperature measurement, easy temperature measurement. But unfortunately it was forgotten that this holds only for electron density that tends to the 8th by cubic centimeter. Once you go up, collisions between the levels make it completely different. But one uses nowadays a helium, not helium, but helium like iron for diagnostics. And I can briefly mention that too. So when you go up in the density, and then from the corona limit, collisions become dominant. And of course you find ratios which can be used for diagnostics and then radiated collisional models you will learn this week will also contain transport problems. And so tokamak people have used now helium beams at higher densities for injecting them into a plasma and really modeled the rate and emission. And so they find ratios in a certain regime which depend only on the electron density or mostly and others only on the temperature. Here I see, show you some cases for line ratios and on the right here where you can measure. You also have to watch self-absorption because the singlet to S and to triplet is a meta stable and so they mix up everything most of the time. Now helium like ions, did I only mention here, helium like ions are heavily mentioned used in tokamak diagnostics. Okay, now can one in, if you have corona equilibrium, can one not measure the distribution function? And I know of one experiment done in Greifwald excitation when you have excitation you have excitation also by all electrons above the excitation energy. So usually excitation follows by the tail. Now when you go to another level you take another part with the tail and so when you take different levels and do this systematically the excitation is different because you use different parts of the energy distribution function of the electrons. So this has been done and so was done in a neon glow discharge and one could get information on the distribution function but this is not a very general easy use technique, okay? That one can use part of the distribution function. I think I don't have time to go about that. It's also in helium like ions when you have dielectronic recombination because dielectronic recombination and you have dielectronic capture into one level. This also takes only part of the distribution function so you get different information. Oh excuse me now if you have optically thick lines what happens? Here the lines grow to the black body radiation and if you have a Gaussian profile like a Doppler profile you see it gets very narrow. On the right side we have a Laurentian and you see when it gets optically thick it gets a broad tail. So different profiles and if you have inhomogeneous plasmas you get lines here with a central self-reversal dip. Okay, another technique is using recombination edges. Recombination radiation has edges depending upon which level the recombination occurs and this gives these continuities which I have shown here. From the levels recombination into first discontinuity recombination into all levels, higher levels and so on. And you can use this continuity you can use as temperature this depends upon the temperature the temperature diagnostic however it has limited applicability. Then you can use a continuum. Now I repeat the picture from the first lecture here in the lower part of the region both the Bremstrahlung and recombination radiation has an exponential function so if you plot the logarithm of this part you see it's proportional to the frequency and this gives a straight line and you get the temperature. This has been done in hot machines very often but another technique is let me go back to this picture here using thin foils. You take a detector and you take a thin foil metallic foil, it cuts off at this part of the spectrum this is not transmitted then you take another foil which cuts off here and the ratio of the two is only a function of the electron temperature this is known as a two-foil technique and this has been used since early 1960s especially by groups in Los Alamos and it's now really known as a two-foil technique. Okay, then comes the ionization equilibrium and as I mentioned already that when you are in equilibrium let's say you have mostly lines from neon 4 here then you have this kind of temperature so by looking at the lines strong lines, how they are you know this ionization state exists and you have a rough temperature estimate is not too bad okay, only from the existence now if ions go through ionization stages you can also get information from the lifetime of the ions and here we will see also with the electron density well the electron density is most commonly used from line profiles and they are characterized by the shape by the width and the shift and the width is usually determined by the electron density although the shift path is less accurate and what one measures the full half width of the line after one has accounted for the Doppler width and the instrumental width okay, if you know the Doppler temperature you just deconvolve the line profile with the instrument function and the air Gaussian function and you get the function due to broadening by the electron density or by the electric micro field and I will not go into that too much because one expert gives you an x-lectures well they are usually worked functions and so you have mostly convolution between Gaussian and worked function so okay, the main problem is when you do line profiles that you subtract correctly the background this is sometimes very difficult you get a line and you don't have much background information and then you get easily error by determining the half width if the background is taken wrong so well they are patobelius rather let me see the hydrogen like ones are the best known lines because they are broadened by the linear stark effect and the bilma line, the beta line is a very nice line to be used characteristic uncertainties are below 10% and the temperature dependence is very weak so you don't need to know the temperature it only is important if you have temperatures below half an AV then the broadening is influenced and high principal quantum number bilma lines are useful candidates in the radio frequency range for edge plasmas infusion devices or the ionosphere and so on or radio lines between liquid levels in hydrogen are used for studies of interstellar plasmas here I see one another element which has been used quite studied quite extensively is a helium lines and here you see an example of the passion alpha line in helium 2 and here you see the background has to be determined accurately without helium otherwise if you take one would take a straight background here you see would be really wrong so this has to be determined by best fit curves and this is the true profile giving here and this is done with a kind of C pinch gas liner pinch we called it and you see also a small shift shifts are less accurate so to measure the shift of a line it has not the accuracy it is measuring the width of a line here okay this plasma was analyzed or characterized by Thomson scattering so I give you two or three formula now another simple technique is using the English teller limit when you go up in the lines they become broader and broader and suddenly the lines essentially merge in the continuum and this you cannot see and there were English and teller looked at this and the last line where you can see you take the upper level this is an estimate of the electron density and here you see such theories done in the same pinch we could identify here the line this is the last line and upper level 6 and this gives an electron density of this value I would not say it's more accurate than a factor of 2 but it was more accurate in our case but in general one can use it if you have no other possibility to get to do measurements this gives you at least an order of magnitude of your plasma okay then we have isolated lines of atoms and ions they have forbidden components so you have one line a loud line and the neighboring line is forbidden because of delta L equals 0 or 2 but when the density goes up the field electric field mixes the upper levels and this ratio becomes density dependent okay I see one example okay I mentioned again helium like lines they are very high used for high V elements have been used in argon they have been studied and published in nitrogen oxygen and so on up to tanks like then another technique is if you look excuse me it's a bremstrahlung the bremstrahlung goes like N squared before the emission and it goes like N squared so if you know roughly the temperature you can get the electron density a very good estimate of the electron density just from the absolute measurement of the continuum bremstrahlung okay because of the N squared scaling and little temperature dependence then another estimate you can get of the electron density when you have ions and you look one ion comes next ion and so on and you study the decay time of the ion this decay time of the ion is a characteristic time one calls lifetime of the ion and this decay time of an ion is given by one over N over the ionization rate now if you have high temperatures the ionization rate coefficients level off so they have a constant so by measuring looking at the decay time you get and excuse me here you know theoretically roughly the ionization rate coefficient and the decay time gives you the electron density I think it can be done up to a factor of 2 very quick measurements you look how the lines decay okay then you do magnetic field measurements from broadening of the lines okay and from the magnetic field you get the current distribution then the emotional strike effect if an ion moves or hydrogen moves then if it has high energy the Lorentz electric field is so big that you get strike shifts stronger than the beach splitting and so people in fusion plasma so use hydrogen beams for heating they study the structure of the electric fields induced broadening it's also a very modern field of measuring electric fields you can get strikes splitting I think okay high electric fields you get of course high magnetic fields you get of course the semen I think what is the ultimate goal and the ultimate goal is if you had collisional radiative model for all the atoms and ions and this you the experts are in this hall you could do all the measurements by measuring many lines describing them by collisional radiative models and you get accurate data thank you for your attention