 Something between LTE Sahara and Corona and moreover, if you build it correctly, it gives you Sahara LTE at high densities and Corona at low densities. So, I think the collision-related model was introduced in the early 60s, more than 50 years ago. And the basic idea is, it sounds very natural. Let's take into account all important processes that affect populations of atomic levels and try to include them as accurately as we can. So, if this is energy scheme for one ionization stage with the proper organization limit and continue maybe organizing states sitting above the organization limit. And this is the energy scheme for the next ion. And, of course, we can include low ions and high ions as well. We will take into account all possible collisions between different states within each ion excitation, de-excitation. And this may be, of course, first of all, electrons are important that are the fastest particles in plasma, but in some cases you may want to include heavy particle collisions, for instance, protons. Then we take into account all possible radiative transitions that are important for this problem. And then we include different processes that connect atomic states in one ion and the other. So, it would be, of course, ionization by electron impact, or maybe by proton impact. Of course, various kinds of recombination. Would it be radiative recombination of three-body or de-electronic recombination that may be included just as one process or we can explicitly include its component, that is, the electronic capture into organizing states and organization. In some models, you have to take into account other processes, for instance, charge exchange. If you have neutrals penetrating your plasma and therefore your charge ions can grab electrons from neutrals that pass by. So, let's consider which processes are important, build a rate equation, and solve it and find level populations. So, what we do here actually, we start with the construction of vector of atomic state population. And this vector can be, small can be large. In some cases, you may have just a few dozen of levels. In some cases, you can have hundreds of thousands. And then you build a rate matrix, which is, of course, square matrix. It may depend on different parameters. It may maybe time dependent. It certainly would depend on electron density and temperature. It may depend on ion density and ion temperature if those are important. It may be nonlinear, for instance, for opacity calculations when radiative probabilities are modified by the already known populations of atomic levels. You may have some source function when particles come into your plasma volume. In general case, you would solve time dependent first order equation. In many cases, steady state approximation when you have zero on the left is sufficient. Now, in the rate matrix, the off diagonal elements correspond to total rates of all processes connecting these particular two levels for the matrix element ij. Those will be connections between level i and level j. And diagonal elements describe the total destruction rates for a particular level. So far it looks not too scary if we write down actually what goes inside. There's a little bit more than that. So in here, this formula says that for this particular model, we include electron impact, excitation, heavy ion excitation, photon excitation, electron impact, de-excitation, heavy ion de-excitation, spontaneous radiative emission, stimulated radiative emission, all kinds of recombination and ionization, and other processes and so on and so forth. So basically the goal here is to calculate all rates for all processes that are important, put them into one big computer code, try to find the level populations, and then calculate line intensities, and we allow the problem more or less is solved. Now, collision with data model is certainly the most general approach to analysis of population kinetics in atomic system. Similar to coronal model, it does depend on detailed atomic data and really requires a lot. Again, if your model includes hundreds of thousands of levels, you may have hundreds of millions, if not billions of transitions, and for each of them you have to calculate rate, or rate coefficient for collision, of course you have to take integrals of over Maxwellian distribution, so it may take quite a while to get the solution. A properly built collision-related model should reach SAHA LT conditions at higher densities, and coronal conditions and flow, and again it may vary in size very much. Now, when you start building a collision-related model, there are very important questions that you must ask yourself before this task. Which state description is the most relevant to the problem you are trying to solve? Remember, we had a picture on Monday of different representation of atomic states going from average atoms to levels, and this is one of the first questions to ask when you are working with collision-related model. Then what are the most important and not so important physical processes? In some cases again collisions with protons would be crucial for the spectrum that you are trying to analyze. In some cases you can completely neglect them. How you try to calculate the rates? What is the source of data? Which level of data accuracy is sufficient for this particular problem? In some cases you may use very approximate atomic formulas, for instance, for excitation or ionization, simple kinds like lots formula for ionization or one regimortive formula for excitation. In some cases you have to go much, much farther to get real good and accurate data, and all this is relevant to construction of your CR model. Finally, which plasma effects would be important? Is opacity important? Is ionization potential depression? That's what we discussed on Monday is important. All together all this question probably clearly tells you that there is no universal collision-related model for all cases. One model that you develop may be applicable to some cases, but certainly not for all. For instance, if you need a collision-related model that would be working online with some radiation hydrodynamic code, you don't want to include hundreds of thousands of levels. Simply it will never finish the run. You need something much more simpler, but still a model that picks up the most important physical processes that are related to your problem. Again, as I mentioned that you remember this picture from Monday, you can build collision-related model for an average atom, super configuration like FlightCheck code that you will learn today and Cretin that you will learn tomorrow and Friday. You can use configurations. You can use terms to describe atomic state. You can use, of course, levels if you need very detailed spectra coming out of your model, but the story doesn't end here. For instance, if you have external field, your states may not be normal spherically symmetric states. One of the examples is the motionless stark effect in magnetic fusion devices. There you have, let's say, this is the wall of a tokamak or stellarator, and you have neutral hydrogen flying in. Of course, all this plasma is in magnetic field, and therefore, as soon as hydrogen crosses the line of the magnetic field, it immediately sees the induced electric field that splits all the level because of the stark effect. After a few centimeters, spherical normal states like S, P, D, and F are no more relevant. It is the parabolic states that are the eigenfunctions of hydrogen in electric field that become the core of your simulation. Therefore, in this case, you simply must build your collision-related model, not in terms of S, P, D, F, and other states, but in terms of parabolic states. Once again, before doing collision-related model, you have to look at the problem and understand both of the most important features. Speaking of the motionless stark effect, there were many attempts to explain the results of the experiment building collision-related model with the spherical S, P, D states. They always had very significant disagreement, and only after attempts were made to build parabolic collision-related model, the results were more or less explained from Tokamak and stellar-related experiments. Again, the whole story of collision-related model sounds reasonably simple logically. You understand what you want. You understand where data comes from. You calculate rate coefficients, collision-related optimization. You build the rate matrix. For steady state condition, you basically diagonalize all these metrics. The rate metrics find eigenvalues, and those would be the level populations, and then with these level populations, you can easily calculate line intensities. Now, how can we use line intensities for analysis of plasma parameters? There are certainly different methods to do this. I will say a few words in what follows about electron density and electron temperature diagnostics. Probably the main idea of diagnostics with electron temperature is that in a plasma that has just one temperature, and I'm not talking about non-Maxwellian, different parts of this Maxwellian distribution, in other words, different electron energy, would be responsible for population of different levels that result in specific line. If we have some transitions that have small energy difference, some transitions that have large energy difference, of course, you need low-energy electrons here, large-energy electrons here, and this would be similar to the Boltzmann factor minus delta E over T that would be sensitive if delta E is large and temperature is not so large. So, this is the basic idea for diagnostics with electron temperature. For electron density, there are also several methods to look at how lines, line intensities depend on density. Certainly, some lines may feel collisional dumping, which means we have a line that emits, comes from the upper level emitting photon, and something prevents this relative transition to happen through collisions. We'll see in the next slide what it actually means. But also, it may happen that whatever comes to this level, this population influx, is also affected by density dependent, by processes dependent on electron density. These are two different factors, but I'll show you examples of both. So, it's often mentioned that forbidden lines may be sensitive to electron density as compared to resonance line, and the physics is very simple. So, imagine that our electron sits on a level, upper level J and have to go down to level I. Now, we remember that the lifetime of a relative transition is about one over A. So, if we have no collisions in our plasma, let's say we have isolated atom, we put electron onto the upper level, no matter how, maybe through collisions or some other effect, and then without collisions, this electron lives at this level as long as necessary, meaning one over A, in order to jump down a meter photon. So, basically, without any external particles, external collisions, any electron put in the higher level will emit a photon. Now, if we are in a plasma, this electron feels particles that come and try to hit it off. So, it needs the same time to reach the end of the bar to jump down. But if collisions are very fast, it may not have enough time, and some will kick it off, and therefore, electron will not leave long enough to produce a photon. Now, obviously, if this electron moves very slowly, and this means that the lifetime is very long and A coefficient is very small, the probability of get hit by electrons is much larger if your line is strong. If your line is strong, A is huge, and therefore, the electron will run very rapidly and produce a photon anyway. Now, let's put all this story into a formula. Let's say we have two levels in an atom that are connected with the ground state, and let's look at a situation when one of the level has very strong radiative transitions. So, we excite electron from the ground state to both level one and level two, and then from level one, it's only the case radiatively, meaning that all other possible processes are completely negligible because the transition probability is so huge. For level two, this is not the case. Let's say that transition probability here is not as overwhelmingly large as the transition for level one, and therefore, we take into account possible collisions. It doesn't matter where it goes down or to some other places, but we will say, okay, there are some collisions that also try to depopulate level two. So, in equilibrium under steady state equilibrium, the balance equation for level one would be simply this. Excitation from the ground state with the rate coefficient g1, and of course, we take into account electron density is exactly what comes down through radiative transition. For level two, again, excitation from the ground state populates the level, but then the level can decay through radiation with A2 transition probability, and some extra contribution due to collisions. Again, we only consider electrons. So, the population for level one can be the right from here. The population for level two comes from this equation, and if we look at the ratio of line intensities, I mean the energy factor. What is left from this ratio is just ratio of rate coefficients, all electron density factors here, and Ng, the population of the ground state, cancel out, and you are left with this ratio, A1 goes away, only A2 is left. So, let's see what happens. If we have very low density, coronal regime, we can neglect this factor, this ratio is one, and ratio of two line intensities is simply ratio of the corresponding excitation rate coefficient. This ratio doesn't depend on density. This is what we would have for, let's say, two strong transitions. Moreover, again, like in corona, the ratio does not depend on transition probabilities at all. Everything is determined by the collisions. Now, if we increase the density, at some point when this term is on the order of magnitude of A2 transition probability, the ratio starts feeling dependence on density. This is exactly the mechanism that is responsible for density diagnostics when you compare, for instance, resonance and inter-combination lines in helium-like ion through this simple formula. Of course, there are other processes, and in helium-like ion, you have other types of collisions, but again, when you try to qualitatively figure out what happens, pick up the most important processes and look at what happens with them. Let me talk a little bit about diagnostics with the electronic resonances. You heard from Stefan Fritsche and Achim Kunze about helium-like ions, and I think, yeah, Konor Balans also mentioned helium-like ions, and I think John Sealy will be talking about them as well, because this is so, not only, this is simple, but extremely powerful method of diagnostics. I will say a few general words without going much into details, maybe more to give you a feeling of what happens here. So, there are, of course, strong resonance lines in each ion. Normally, you call resonance line transition between the ground state and the first excited state, or serious, normally the lowest excited state gives rise to a principal resonance line, but in any case, 1s squared 1s to p is the resonance line in helium-like ion, and the satellite transitions are transitions that are similar. Again, 2p falls down to 1s, but you have some extra electron sitting somewhere higher. Somewhere higher means anything from n equals 2 to infinity, so you may have situation when you have nl is 2s or 2p, and obviously, since here we have 3 electrons, not 2, this atomic configuration belongs to lithium-like ion, but if your n is not, certainly when you have an extra electron, the wavelengths of this transition from 2 to 1 is slightly shifted because of extra screening, and if n is low, like 2 or 3, this shift in wavelengths is significant, so that you can easily resolve these satellite lines with regard to the resonance line, and we'll see it in a minute. Now, the upper level of this transition, as you see, actually has two excited electrons. The ground state for lithium-like is 1s squared 2s, so that technically both 2p and nl are excited compared to the ground state, and therefore they can be produced by different mechanisms. One simply sounds in is double electron excitation when you hit your ground state of lithium with electron and two electron excited, but probability of this process is extremely small compared to the main population mechanism, which is the delectronic capture, and delectronic capture is the opposite of authorization. That is, you have ground state of lithium-like ion, free electron passes by. One of the 1s electron is excited by some delta E, but to conserve the energy, the free electron must lose exactly the same delta E, and if this loss of delta E on the energy scheme that we started with brings the electron below ionization limit, the electron gets captured into bound state, and such authorizing state 1s2l and nl prime is produced. Then the state can certainly authorize back with the inverse process, but especially if ion charge is large, and therefore, relative transitional probabilities are strong, one of the electrons can fall down. In this case, I drew that 2l, because it better be 2p, goes down to 1s in the means of photon, but in principle, also the spectator out of most electron can also fall into 1s producing a photon with a different energy. Now, these delectronic satellites are produced only at specific values of electron energy, because this is a resonance process. Again, you have to match the exact excitation of atomic electron from one level to another, which is a fixed value, with the drop of energy of the free electron by this exactly same amount. If the free electron has very high energy, then dropping by delta E here may not bring it at all below the ionization limit, and therefore, electron will not be captured, and will not produce an ionized state. Now, if in some cases, this electron can lose exactly the same delta E, but there will not be an atomic state at this particular energy. Again, the state is not produced. All this means that delectronic capture is a truly resonant process that happens only at specific energies of free electrons, and therefore, this rate coefficient, or its dependence on temperature, is different from collisional excitation rate coefficient that is required to produce resonance lines in hidden-like ions. In this case, with excitation, we have just a straight ionization, excitation energy that is required to bring electron from 1s to 2p, and therefore, if we have electrons with all energies in the Maxwellian, all electrons with energies higher than this delta E can excite electron. This is very obvious. You have very high energies. It gives delta E, loses a little bit. Everything is possible, not so much for delectronic capture. So, all this is illustrated here. So, let's take a look again at the energy scheme and the diagram. So, this is the ground state of the hidden-like ion. This is the energy of the 1s to p electron. These are higher excited states. Now, here you can see the Maxwellian distribution. Normally, you would draw it here, but put in Maxwellian right at the ground state of hidden-like ion helps you better understand which part of free electrons can be responsible for excitation of 1s to p. And of course, if this is the energy threshold, then all electrons with energies higher than this delta E can excite 1s to p. Now, when you calculate the excitation rate for 1s to p, although the Maxwellian formula has 1 over t to the third half factor in the denominator, the integral from delta E to the infinity leaves you with square root of t in the denominator. So, the excitation rate for 1s to p, generally we can say that the main factors dependent on temperature is this exponent 1 and also square root of t in the denominator. Now, if we look at energy dependence or temperature dependence for the electronic capture rate for the electronic satellites like 1s to p to p or 1s to s to p or 1s to p3l, again, because only electrons with a specific energy can produce these electronic satellites, only electrons here for the Maxwellian distribution can do it. And the part of electrons with this energy is proportional to 1 over t to the third half power. Again, if we look at the corona, we remember that from the formula that we got a few slides back, it's really the ratio of excitation rates that gives you actually dependence on temperature. Here we have similar ratio of excitation rate for 1s to p and the electronic capture rate for the electronic satellite. And therefore, when you take the ratio that eventually comes to ratio of these two factors, square root of t canceled out and you're left with temperature. This immediately tells you that if you compare the intensities of resonance and the electronic satellite lines, it would be inversely proportional to temperature and therefore it is really sensitive to temperature. This type of diagnostic for with the electronic satellites is used in many, many applications. In solar corona, in various terrestrial plasmas, in z-pinges, in laser produced plasmas and probably on Friday we'll hear more about this from John Silly. Here's an example not for helium-like but for hydrogen-like ion. Hydrogen-like is, can produce the same type of the electronic satellites. Generally, you can produce them for any atom or ion. It's simply that for a few electron systems like hydrogen and helium, the spectral pattern becomes very clear and very easy to work with and therefore they use a lot for diagnostics. The difference from helium-like ion would be that we have not 1 squared 1s2p but 1s2p transition which is certainly Lyman-alpha and the satellite transitions would be 1snl2pnl and so on. What you see here is the result of calculation for hydrogen-like neon-10. This is Lyman-alpha, transition both 1 1⁄2 and 3⁄2, they're very close and therefore were not resolved in this CERA model and the other lines are the satellites and you see that the resolution that is required to resolve the satellites from the Lyman-alpha line is not that high. I mean the wavelength is 12.1 approximately and the strongest the electronic satellite sits probably 0.25 angstrom to the longer side. So again you remember that the previous slide we got that the ratio is approximately proportional to 1 over temperature as indeed as you see here you go from 100 to 130, 160 not exactly 1 over T but it gives you the flavor that this ratio is really sensitive to electron temperature and can be used for diagnostic. Now these satellites where this spectator electron sits at n equal 2 they pushed the most away from Lyman-alpha if you are thinking about satellites with n equal 3, 4 and higher well first of all their intensity drops because the probability of the electronic capture drops with n substantially typically 1 over n cube but more importantly when you take your electrons farther and farther away from the electron that produces the photon this electron is affected less and less and therefore its behavior becomes more and more similar to what you would have without spectator electron and that's why the wavelength for this transition for higher ends becomes closer and closer to Lyman-alpha and at one point you already really cannot resolve higher end satellites from Lyman-alpha in hydrogen like ions or helium alpha in helium like ions. Now the same the same the electronic satellites can be used also for density diagnostics and we already mentioned the density diagnostic ratio of forbidden line and forbidden line and strong like resonance line where level cannot live or electron cannot live long enough because of collision here we have a different mechanism a collisional redistribution of population so the same neon 10 Lyman-alpha and we have three satellites here that as you see change their relative intensities with density so we have A, B, C I will explain second what they are and B obviously increases with increase of density and I think A slightly decreases so what happens here now of these three satellites the satellites B that corresponds to radiate decay of the 2p squared triplet P term and here I'm talking not about levels but about terms has very small intensity at lower densities and the reason is rather simple it is the selection rules like you have selection rules for radiative transitions for instance for electric dipole transitions the total momentum cannot change more than by one and you don't have zero to zero transitions and you do not have transition between states of the same parity for electric dipole transition the same is valid for authorization and inverse process of the electronic capture so the selection rule basically tells us that this level 2p squared 3p of this term is very weakly populated by the electronic capture there rarely an action comes to this level and therefore very rarely we have a photon that is emitted by from this level down therefore the intensity of this line is now if we increase population the populations of the nearby 2s to p triplet p term increases because generally the electronic capture rate is proportional to electron density now of course there are collisions between this term and this term but when population of this term is low collisions from here to here are not important but at one point at one density more and more electrons are transformed from this level from this term to this one and you see that this is good strong transition the spin doesn't change it's actually optical large transition so cross section is large and therefore at high densities more and more electrons are transformed from 2s to p triplet p to 2p squared triplet p and therefore more and more population is accumulated here and therefore this line becomes stronger and stronger and therefore the ratio for instance of this satellite to this satellite to see which is one of the strongest so-called jc satellite becomes sensitive to density for instance this method was used for density diagnostic in in z pinch and and some other plasmas as well okay so a few words about helium like lines satellites or better to say some some example so this is an example of resonance line in helium like iron I think this is argon produced on I think texture tokamak in Germany this tells you that working with helium like lines you can if you if your spectroscopic equipment is good enough you can easily resolve lines within the helium like iron which are w x y z you can easily resolve satellites and everything if everything looks nice you can use these lines for for diagnostics so let's talk a little bit we still have seven minutes or so let's talk a little bit about helium like iron so the ground state is one squared the only level is possible one is zero you have two subsystems of terms singlets and triplets radiative transition within each subsystem are strong between triplet triplets and singlet not so strong for lower members of the electronic sequence you go higher in z spin orbit mixing becomes stronger stronger singlets mixed with triplets redation transition probabilities increase and everything becomes stronger so if we look again at one s2p levels and this is the scheme that you already saw from from kunzi's lecture we have six excited states within n equal two for triplets two singlets and there are different types of radiative transitions that are possible first of all of course the resonance slide from singlet p to the ground state which is electric dipole very strong with transition probabilities so the order of 10 to the 14 for helium like argon that is the 15 for helium like krypton uh the level one s zero does not have e1 transition actually no one one photon transition is allowed for this level going into the ground state because you have j equals zero to j equals zero so the the first order of transition that can happen here is a mission of two electric dipole transition simultaneously which is very process now looking at triplets some of them do have forbidden transition first of all the intercombination line from triplet p1 to the ground it is an e1 transition that appears because of mixing in the wave function of singlet p and triplet p triplet p2 can have magnetic dipole including magnetic quadruple transition these two levels have different parities delta j equals to so m2 is the lowest possible uh triplet s1 can have magnetic dipole transition and these three lines are normally designated as x, y, z and the resonance line is w and the line from 3p0 to the ground state is also j equals zero to j equals zero transition that can only with two photons but here we have e1 and m1 as the strongest transition now the probabilities for these lines strongly depend on ion uh charges spectroscopic charge so if for for instance for x line for neutral helium this line has probability of only about two times to the two for argon it's stronger by 10 orders of magnitude for helium like argon excuse me this is for y for x the change in in transitional probability is also like nine orders of magnitude here 10 others of magnitude for z so once again the forbidden lines depend very strongly on on ion charges uh their dependence uh is again z z to the four this is what we already saw for typical e1 transitions the intercombination line changes as z to the 10 for low z z to the eight for some larger and z to the fourth for very large one it's basically because of very strong mixing with w becomes became like resonant line and these two lines depend z to the eight z to the 10th now we can certainly calculate population of these levels as a function of uh electron density now when density is low and again you remember that 10 to the 15 10 to the 16 is not low density for neutral hydrogen but it is low for helium like argon because the ion charge is much higher so this behavior of population linear dependence of population with electron density is typical for corona we remember that excitation is proportional to n e and then you see that the populations of many levels do change except for the strongest the strongest e1 resonance transition still is in corona for very high density you see it's 10 to the 22nd 2023rd still goes linearly with with density which means that this particular level is in corona with the ground state but the others do show some deviation for densities and then looking at uh ratio of population for these levels can tell us a lot about density and this is the example of the calculated line ratios for several uh for several uh pairs or groups of lines conor bell's was mentioned in g and uh r ratios that are often used in astrophysics for ratios of line intensities and helium like ions here's an example this is ratio of inter-cabination to resonance stays constant for a very large range of electron density obviously for low densities when everything is in corona then it starts increasing and you see that many ratios change in think which simply means that you have transfer of population between different levels some lines decrease in intensity ratios some increase which means that basically levels are talking which each other population is transferred because of collisions and therefore uh uh this this line ratios can be used uh for diagnostics of them density as well again this example for argon for iron certainly you will have different ranges of densities for helium for helium like neon you will have different range of density but uh as far as helium like ions are concerning you can always find something interesting about them in different regimes and in different classrooms okay uh the the conclusion of all this story is that there are certainly many methods to diagnose plasma with line intensities but um and certainly there are different ways to build collision-related models be careful about your models try to think in advance what you're trying to to find out what is important for a collision-related model how extensive it should be how accurate the data should be again in some cases you're doing well with simple estimates in some cases especially if your experimental spectra are very very accurate you may want to do uh uh something serious about the quality of data uh and enjoy this is this is really fun thing when when you start feeling what happens in plasma in many cases uh we we get experimental spectrum that oh god what does it mean and then little by little little by little you start feeling and you start to understand what happens with densities temperatures plasmas okay thank you