Maybe that yeah, we're missing a lot of people yet. How's the volume? Good. All right. We'll give people another couple minutes to get in But in the meantime, there's a couple of couple of web addresses here This one the home NFS 3 SMR 3105 this these are directories. We all have access to So you have Kenny has put his slides for Unix on here under his directory Istanbul and I imagine that some of the rest of us will also put our slides in similar places So I'll put mine there By the time we're done here There's another address here and actually There's some more material here. Hewn. Which material is this? Oh from the posters Abstracts. Oh poster abstracts. Wonderful. Thank you Okay, there are actually a number of directories under this one the home NFS 3 3105 and Those will be useful in in the laboratories as well. Okay. It looks like we have a quorum so time Time to switch topics a bit now. We're gonna talk about radiation transport So here's a here's a brief outline of what we'll cover in the lecture today and the lecture tomorrow See if I got this All right, so we'll start basically defining what we mean by radio how we're gonna describe the radiation field The assumptions that we're making in describing the radiation Talk about you know get the terminology down so that we know what we're talking about Because there's lots of different ways to approach this topic We'll then move on to talk about the radiation transport equation in a fairly simple way Talk about a simple solution, which gives you some ideas of the effects that you can see from radiation transport Which will be useful when we get in further into the topic Describing how we actually get a radiation transport solution and then tie that to the material radiative properties the absorption and mission and Scattering which we've been hearing about so far this week So then finally after we do that which are all the preliminaries to doing the radiation transport We'll talk about doing coupled systems Coupled systems meaning either an LTE or a non-LTE atomic system plus the radiation So at that point we'll make up we'll make a connection back to the collisional radiative models that we've just been talking about So we'll get through most of this maybe all of this part today And in the next lecture we'll do the rest of it where we'll talk about some more details about the line radiation And how you actually get a solution to To this system of equations and what that solution might look like so that's where a lot of the information will be So first Basic assumptions how we're going to describe the radiation So this is sort of a classical semi classical description We'll be talking about a radiation field Described either in terms of the specific intensity and I'll describe what that is or we'll talk about the photon distribution function And these two will have exactly the same information in there So for simplicity we'll make the assumption that we're talking about unpolarized radiation We're going to neglect for right now at least index of refraction effects Which means that the index of refraction will assume it's one Photons will travel in straight lines Okay, scattering we'll talk about scattering some but actually the scattering that will be concerned about here is not really true Scattering like Thompson scattered or Compton scattering Rayleigh scattering those things But it'll be sort of an effective scattering which is very important for non LTE plasmas So we'll get into that and for right now at least We'll be talking about material which has no significant velocities So one difference between this topic and what we've talked about so far is we've been talking about Plasmas in the form of you know a plasmid a single point in space So we've got a single piece of material and we've been talking about the properties of that material So now we're going to put together properties of different pieces of material So we have to worry about one the spatial distribution and the velocity distribution of the material So first the radiation description. So two different descriptions So macroscopic description is in terms of the specific intensity, which is the energy traveling per unit area per unit solid time per unit Per time within a given frequency range And so this so the energy which is cross the unit surface So here's a unit surface going in a direction denoted by omega the normal to the surface is denoted by this this unit vector n is This is the area crossing this surface in a particular amount of time So it's the area solid angle the frequency interval the amount of time and then the The direction the normal to the area in Terms of a microscopic description the same information is here in the photon distribution function So we talked about the photon distribution function then the photons have positions in space in momentum and of course, it's time-dependent as well So the intensity is related to the photon distribution function The intensity is it's the sum over spins of the photon distribution function instead of a distribution function number particles We're talking about energy. So each of the photons gets multiplied by its energy h nu And so there's this relationship between the photon intensity the specific intensity and the distribution function because again since we're talking about Index of refraction one the momentum is just h nu over c times the direction a couple moments of the distribution function Are important at various times here So again the zeroth moment will be talked about is just integrating the specific intensity at a given point in space Actually averaging it over all solid angles So this in a sense is the energy density Carried by the radiation actually in this definition as the energy density times the factor c over 4 pi So when you read books and articles about photon intensities and moments You have to be careful about exactly how they're defined There are a few different common definitions and they differ in what they do with the factors of 4 pi and What they do with the with the factors of c as to whether this is actually an energy density or It's an energy density times the speed of light So we've got the zeroth moment which gives us the energy density. We've got the first moment which gives us the flux And we've got the second moment which looks like a pressure tensor for those of you who are familiar with those terms for isotropic radiation Then the pressure tensor is turns out to be diagonal with a factor of one-third here Which looks like the pressure is one-third the energy density and so in that sense radiation looks like an ideal gas of particles With a gamma of four-thirds Now we've already heard about what the thermal distribution the equilibrium distribution of radiation is It's the plank function, which was derived in previous lectures We can view this in another way is that the photon distribution function photons or bosons And so this is the Bose Einstein distribution And again the difference between the photon distribution function and the plank function is just this factor of 2h nu cubed over c squared So we can view this in either either direction and then the total energy density Integrating this overall frequencies if the photons have a distribution Characterized by a temperature t sub r is just this at to the fourth now to give you some feeling for what kind of energy density There is in radiation if we take an electron temperature equal to a radiation temperature and For hydrogen say the temperature or the density is 10 to 23rd per cubic centimeter, which is roughly solid density for hydrogen Then the energy and radiation and the energy and matter are about the same at a temperature of 300 eV Which is a reasonably high temperature for most laboratory plasmas or it wasn't till recently We now routinely produce laboratory plasmas with temperatures 10 times this And the important thing about this radiation is that the material energy density increases linearly with radiation Whereas the radiation energy density increases is the force power of radiation So as this temperature increases the amount of energy density in the radiation increases very fast So we'll see later on that the radiation has two important effects when we're talking about its effect on the plasma One reason we're talking about radiation transport here is that of course the radiation Carries the information we see about the plasma. We've talked about the spectral lines that come out of a plasma and the Emission distributions out of a plasma, but for that information to get to us We need to see it come through radiation. We need to receive the radiation as the radiation is going through the plasma It has an effect on the plasma in two different ways So one is that the radiation will affect the plasma because it affects the transition rates The second when the temperature of the density gets up high enough is the radiation actually carries a lot of energy And so that will also change the temperature of the plasma. So we'll have to be concerned about both those effects Now we've already talked a few times about what LTE means is That we will mostly be assuming That the radiation is not an equilibrium that the radiation does not have this Distribution this equilibrium distribution function But it could be possible that the plasma is an LTE that the particles have thermal distributions So there are two main branches of radiation transport which deal with Cases where the material is an LTE and when the material is not an LTE So here is the basic simplest form of the radiation transport equation Which is Maybe a little more understandable if we look at it as an equation for the photon distribution function Which case the radiation transport equation is just a Boltzmann equation for the photon distribution function There's a temperature. There's a time derivative term There's a spatial derivative term There's no momentum derivative term because there's no body force on the photons and then on the right hand side There's a collision term Now in the Boltzmann equation Formulation we would talk about a collision term in the radiation transport equation The collision term is the interactions of the photons with the plasma And so we've already talked about some of these the the emission We've talked about line emission a fair amount which is excited states the exciting producing photons Redate of recombination produces photons. So those things will show up as emission There's also photon absorption photons will get absorbed as they go through the plasma. So these transition these transitions will Attenuate the absorb the radiations that goes through so we'll talk more in detail about these in a few slides so basically The absorption coefficient and the emissivity the emission coefficient will relate to the cross sections that we've talked about earlier and Then the transport equation this left-hand side is just describing radiation flowing in phase space Completely conservative just photons making their way through phase space the right-hand side talks about interactions with the material So absorption and emission and this depends on all the atomic physics and note that photon number in these Transitions in these interactions is not conserved except for the case of true scattering Now there's a very simple form for the radiation equation radiation transport equation which is called the characteristic form and if we take if we go back to the Radiation transport equation Divide the equation through by the absorption coefficient And which is different for each frequency and write this along the characteristic a straight line So photons going along a straight line in a particular direction Then the radiation transport becomes just this is now we've defined We've defined a distance not in terms of physical distance, but in distance times absorption coefficient Which is the optical depth? The interactions with the radiation Interactions with the material are now the absorption is is hidden in terms of the optical depth The emission however is hidden in terms of what's called the source function Which is the emission coefficient divided by the absorption coefficient and The characteristic radiation transport equation has this very simple form now there's one more complication hidden in here and that's the time derivative and The time the time dependence is hidden in here because along the characteristic form then everything along that in The solution of this equation has to be evaluated at retarded times Radiation travels pretty fast and in most cases it will be sufficient to just Ignore that time dependence and treat this instead in steady state meaning that the photons travel with infinite speed effectively Now two things about this one in LTE the source function the The ratio of the emissions the absorption coefficient just gives you the plank function now. This is true That this comes about by detailed balance, and it's true in each radiative transition independently Which means that it's also true in total So this will be very useful later on now the simple form of the equation That's I mean each of us could solve this equation very simply. It's a first-order linear differential equation and so we can just write down the solution here which is We can we can talk about this characteristic solution and get some properties of what happens through radiation transport So there's there's a few important features here One this tells you explicitly that the intensity that you see at some point and we'll call this this point zero or Actually the the intensity that we're looking for here, which could be some detector is Given by the intensity which is starting at some point zero, which could be say the other side of our plasma and Then what we actually see is an integral over the emission going through the plasma the emission from each point of the plasma attenuated by the optical depth of the observation point so These diagnostics that we see from from plasmas in the low temperature limit They're they're very nice or rather than low density limit. We know what they mean Now and if the plasma is optically thin Meaning that the optical depth is very small then that should be pretty much what the what we see in the very limit of very Small optical depth this just becomes The source function times the optical depth the optical depth then is just a constant absorption coefficient times times the width of the plasma and That then gives you the emissivity of the plasma emission coefficient times the depth So it's gotten exactly the same form as the emission from a point in space So those are reasonably easy to interpret once you go through plasma, which has a finite optical depth And the different points in the plasma might have different physical conditions temperatures and densities It becomes more it becomes more difficult to interpret the spectrum you see So the important things here is that you know what we see coming out is We see radiation from different points in the plasma weighted differently according to the optical depth Most of the radiation we see comes from within an optical depth about one of the surface of the plasma Because the radiation coming from deeper into the plasma Is attenuated quite a bit Now that optical depth one happens at different points in the plasma for different frequencies So the spectrum we see actually is sampling different points in the plasma very differently now This looks like a fairly simple equation to solve But the other complication here is that the source function the properties of the plasma itself Depend on the radiation now This is a lot simpler if the radiation is an LTE in which case the source function only depends on temperature if the source function depends not just on temperature but on density and the intensity and We have to fold that in and we have to get a self-consistent solution For the properties of the plasma according to the radiation field at that point in the plasma and the physical conditions End this radiation transport equation. So this is the hard part of doing radiation transport is getting a self-consistent solution For the source function and the intensity So let's look at a couple limiting cases here. I think I've already said this in words if it's optically thin The optical depth is much less than one the intensity We see is just an integral over the source function, which then just becomes The emission coefficient times the length of the plasma the depth of the plasma and then what we see just reflects the conditions of the plasma Independent of what the absorption is if it's optically thick Then we see something just coming from within an optical depth of the surface So what we see is the source function at an optical depth of Roughly one so if the source so what we see then doesn't reflect The plasma over the entire length the depth of the plasma we see it just near the surface Now if this is happens to be LTE we know what the source function is So what we'll see is just something that looks like a playing function at that temperature The other another important limiting case that we use in the laboratory a lot is if the plasma is really not emitting or Rather if we're hitting it with an intensity Which is much larger than the self emission in other words We're backlighting a plasma and we're just getting characteristics of the absorption So in that case there is no integral here and what we see is an attenuation of the backlighting intensity Which again is different at every at every frequency and so this is giving the characteristics of the of the absorption coefficient So this gives us one view on the plasma which is what all the absorption transitions are Whereas this limit gives this the emission transitions so we can get some very different information off the plasma there So in that case then this ratio of the emission the intensity we see divided by the backlight intensity Tells us what the absorption characteristics are So here's here's an example if we take a uniform plasma and now it's a uniform sphere and I've chosen Krypton at 200 eV and a fixed density fairly low density and the important thing here now is that this is an LTE So the emission and absorption characteristics of the plasma Do not depend on the radiation intensity. They only depend on the intensity and density and I have fixed those So at 200 eV This black curve here is the flux we would get from a black body. This is the Plink distribution But what what is the intensity we see? For this amount of plasma. So the intensity we see is given here by this blue line. So we see some very strong transitions 500 eV, okay It's a it's a the strong transitions will be at a couple times the temperature most likely the if we looked at the ionization balance of this plasma We would probably find that it's ionized up to something which is probably a couple kilovolts of ionization energy and This is an LTE plasma. So I mean people use the term roughly that this is emitting as a black body But that doesn't mean that What you see is a black body because this is not an optically thick plasma So the optical depths are displayed over here notice. This is a log scale. So the optical depth here This is four orders of magnitude difference from transitions. These are transitions in the K shell These are transitions in the L shell This may well be Neon like krypton here trans strong transitions which are giving this is this emission here And of course the emission in each one of these transitions since this is an LTE From the from the absorption coefficient, which is this divided by the distance If we multiply that by the plank function at that energy, we will get the emission the emission coefficient So this is not optically thick in any transition And notice that the intensity that we see comes nowhere close to the black body limit If we now increase the size of the sphere by an order of magnitude So conditions are just the same all I've increased is the size of the sphere. I was saying okay now What do we see? So the intensity distribution is quite different One there's a lot more intensity because just because we've got a lot more plasma to look at But now the optical depths are a hundred times larger So particularly the optical depths up here in the K shell are pretty significant They vary from one up to nearly a hundred and if we look up here in the kilovolt range Then we see the intensity actually is in some cases limiting to the black body intensity So we're seeing optically thick emission here for these thick lines here in in whatever Transitions these are we also see emission up here, which is limiting to the black body emission but most of the plasma is still not optically thick and So we we see a very different Spectral output from different regions of the plasma depending on exactly what the optical depth is Now if we take this larger by another factor of a hundred now we're getting some really significant optical depths here So optical depths up about a thousand up to nearly ten thousand here and we have a Large amount of black body emissions so that the flux we actually see does is starting to look Like that from a black body now if there is still Still deviations from black body here because a lot of the transitions there's still holes here in the transmission spectrum in the optical depth spectrum There are some regions here which still have optical depth roughly one The lesson from this is that if you expect to see black body Emission a black body flux you really need pretty significant optical depths You know a thousand would be good good optical depth before you can expect to see really black body emission here So what we see in laboratory plasmas particularly the plasmas that were heating up to kilovolt energies is They will be nowhere near black body distributions and what we'll see what we'll be seeing is emission which has Characteristics some some of the emission will be optically thick some of the emission will be optically thin so the total spectrum We'll have to interpret very carefully now this is also This is one reason why in planning experiments You must take into account the optical depth that you expect to get in the experiment Because the lower the optical depths the easier it will be to interpret the experiment in terms of Plasma conditions temperature and density However, the lower the optical depth the less emission you will also get because the less plasma that you will be looking at And so there's there's a balance between the two that we try to reach The ideal experiment would have zero optical depths, but again be large enough so that we could we could See the emission So now let's go talk about the absorption and emission coefficients Which we actually have already done through the other lectures So again what these describe? Macroscopically is how the energy in a frequency range changes as the radiation passes through material The energy is removed from the radiation by absorption and So this this alpha sub new the absorption coefficient just describes that absorption and The material also emits radiation The microscopic description of this is the radiative transitions That some of the other lectures have already talked about and these absorption emission coefficients are constructed from the atomic populations and the cross sections So again, we we have radiative excitations and de-excitations ionizations radiative recombinations and anything here Which can produce so as Yuri was talking about collisional excitations ionizations will give you excited states the excited states most likely will De-excite radiatively or radiative recombination that electronic recombination can also produce radiation So figuring out How much emission you've got? Depends critically on what the populations of of these levels are so Here are expressions for the absorption coefficient and and the emission coefficient in terms of the cross sections And this is now summed over each of the transitions now The absorption is given is given by Absorption of a photon as it goes from a lower state to an upper state We also traditionally include stimulated stimulated emission in here For the reason that this is stimulated emission is also proportional to the photon field And so it comes into the radiation transport equation as a term multiplying the intensity So we lump these together as as a total absorption of Coefficient the emission is given by the spontaneous spontaneous recombination spontaneous emission so the spontaneous rates here So again to calculate these quantities we need the We need the all the transition Cross-sections as a function of energy and we also need the populations of all the levels So just a quick review of the various Transitions that are in there. So this is stuff that you've already seen is we've got bound bound transitions in there And these are the ones just described by the Einstein coefficients Uh proportional to it the absorption and stimulated emission are both proportional to an integral over the radiation field and These these Einstein coefficients are just simple rated related by this this quantity There's one other thing that comes in here is we've been describing so far these transitions as having a single discrete energy Now we know that that's not true. There will there will be several factors which gives some with some energy width to this transition And actually that's been mentioned a couple times already so that line profile We need to just fully describe this transition. We need a line profile Five in which measures the probability absorption as a function of frequency and Then the transition rate will be the integral of the radiation field over this probability so this this this Quantity we term J bar is that integral over the line profile and We'll talk a little bit later about exactly what goes into that line profile So we've got the bound bound transitions The cross section for absorption then looks like this So we've got the cross section in terms of this is in terms of the Einstein coefficient We also determine Can describe this cross section in terms of the oscillator strength So there's a characteristic size here pi e squared over mc, which is actually the classical absorption If you're doing classical E&M and looking at a harmonic oscillator This would be the absorption cross section per frequency And the oscillator strength then relates the transition the quantum mechanical transition to this classical treatment So a strong transition should have an oscillator strength of roughly order unity There are many weak transitions So the oscillator strings will get very small for the weak transitions and then there are some rules which describe what the total oscillator strength is The most important of which is that the sum over all the transitions gives you just the number of bond electrons And from that we get the absorption and emission coefficients Again, the absorption coefficient has this absorption piece and it also has the stimulated emission piece Whereas in the emission will put spontaneous spontaneous emissions part and for right now we'll assume that The absorption and emission use the same line profile That's the simplification that we'll talk about Maybe in the next lecture, but it's it's usually a pretty good approximation to assume that There's only one line profile here. Now, then there's the bound free absorptions And and here, of course, we're doing an ionization And so you need a photon of some threshold energy and largely the cross section for these Falls off as a function of frequency above that transition and in hydrogenic transitions It's it's one over new cubed once you get away from the threshold And so roughly you can describe the boundary cross section as some some threshold cross section times this one over new cube factor Times a factor which talks about so this is the absorption and this is the stimulated emission piece of it there's a this gone factor takes up the Difference between this description and the exact quantum mechanical description and there's some approximate expressions for this which actually work pretty well And then it is roughly dependent on the principal quantum number so this again is a Continuum transition start to some threshold and and then goes on to high intent to high frequency Also free free absorption or Bremshaw and and again here. There's an absorption cross section per ion Which again goes roughly is one over new cubed and in both these cases I've been assuming that the electrons have a Maxwellian distribution that's described by a temperature So that I can take account of this spot or the stimulated emission term Just by an exponential either minus h nu over kT and the same thing shows up in in the free free absorption coefficient And then there's scattering So There's true scattering and that's been mentioned before so that's an interaction in which photon energy is mostly considered But really it's it's conserving the photon number So scattering by bound electrons gives you Rayleigh scalar scattering free electrons is either Thompson or Compton scattering Compton scattering if the if the photon energy is high enough So that it's essentially relativistic and a significant amount of energy gets given to the to the electron At lower energies. It's a constant cross section So there's a frequency shift from the scattering which is of order the you know the photon energy to the electron rest mass energy in this case For most laboratory plasmas These are negligible the cross section is pretty small You know 10 to the minus 24th per square centimeter So to get an optical depth the reasonable optical depth of that you need something Which has a pretty high density and a pretty significant size So in most cases in laboratory plasmas won't won't neglect this Except for there's there's that there's a field of x-ray Thompson scattering where now they're scattering off on acoustic waves and plasma oscillations which actually turns out to be a very useful diagnostic for plasmas, but this is kind of a separate topic But there's been a lot of work on that in like the last 10 or 20 years now there's another type of scattering Well first let's talk about how the scattering goes into the radiation transport equation So the simple radiation transport equation I gave you before Again had some more details hidden in it and those details were hidden in What I was calling the emission coefficient here because This absorption coefficient is the thing which is proportional to the amount of energy Taken out of the incident radiation and anything else gets tossed into the the emission coefficient and so I Wasn't explicitly displaying something like scattering here which has a form Like this which now looks will probably look more familiar to plasma physicists something like a Boltzmann collision term So now we've got a cross-section. We've got a redistribution function, which says okay Here's the probability of taking a photon from this direction and this frequency to this direction and this frequency So we get scattering both in and out of the Intensity which now is headed in a specific which we're considering a specific intensity at a specific direction And we're getting contributions into and out of that intensity from radiation going different directions and at different frequencies and there are there are both Straightforward scattering terms and stimulative scattering terms in here. So this is a non-linear part of the radiation transport equation and this redistribution function describes the scattering So there are actually a few different forms of this redistribution function describing different forms of scattering and again, we will We'll talk some about this when we talk about When we talk about line profile functions But we'll mostly neglect this for true scattering now if we First make the first approximation that the photons are not changing energy. We can simplify this To this form so if we make the Approximation that this redistribution function is a delta function in terms of frequencies We can integrate over the frequencies it comes down to this So it looks now it looks like a phase factor, which is sketch just scattering radiation from one direction into another direction and Again, if this is isotropic we can integrate over this phase function And just get a form like this which just says okay now We've got something a little bit different for the radiation. We've got two additional terms This was the absorption we had before This was the emission we had before and now we've got a term which says we're scattering radiation out of our Out of our frequency and direction actually out of our direction since we've said it's the same frequency And we're scattering radiation in from other directions. No, it's not not now this form is the is the radiation averaged over all angles Now if we keep the phase function in here for Thompson scattering it has the form of 1 plus cos squared theta So we could have something slightly more complicated here and still put that into the radiation transport equation so we can We can write the transport equation either Explicitly displaying These terms the scattering terms or we can kind of implicitly include these in the absorption and the emission and Depending on how you're treating the scattering you'll see very different treatments in the literature If the scattering is very important to you such as it is in in Compton scattering high-energy Compton scattering an astrophysics Confidentization they'll treat this Explicitly in great detail because that's the physics they care about if you're seeing this in a laboratory plasma You might not see scattering included at all if you see it in Scattering in solar physics Where there certainly is enough density in an optical depth and scattering for this to be important Then you'll see it probably in this kind of an approximation But there's a different kind of scattering that we need to be concerned about even if we have a plasma Which does not have enough depth and enough density to worry about true scattering and that is effective scattering So that comes about via Transitions absorption and emission in the plasma so consider a a two-level case So to energy level system where this upper level we know it can decay Radiatively or it can be disrupted collisionally So if we get if we get an absorption From the lower level to the upper level And this is a low-density system most often what is going to happen is this system is going to radiatively de-excite and emit another photon and This might happen a numerous times before that that photon actually gets absorbed now occasionally then This photon or the system Will not radiatively de-excite, but it will be collisionally de-excited either collisionally de-excited or it will be collisionally excited to some other level So there's a fraction of the photons that will be destroyed now When this happens the photons that are destroyed are They're basically talking to the thermal electron background and they will be innocent thermalized However, the rest of the photons They're just going going through the plasma. They're being absorbed Re-emitted absorbed remitted each time they come out They can have a slightly different frequency within the line profile and they can have a different they can have a different direction So these photons are being scattered So this is the effect of scattering which is going on and this is a very strong effect in low-density plasmas So these photons can undergo very many of these scattering before being thermalized given by this ratio and Again, this ratio is here's here's this ratio C to 1 is the collisional Exitation rate and it's proportional to the electron density So the lower the density the lower this fraction is and the more scattering this photon will go through before Manages to to exit the plasma or before it gets thermalized So the photon can go actually a very long distance then Before it becomes thermalized. So this is the condition for a strongly non LTE transition and Again, it's easily satisfied for low density because this is proportional to the electron density or for very high energy photons Because this radiative transition probability increases as the energy to the energy squared or energy cubed so this This number the transit the radiative transition rate Increases strongly with energy. So if we are looking at high energy transitions here This is a very strong effect. Okay, so now if we want to actually start looking at radiation transport We we know how to calculate some transitions We know what the radiation transport equation is but we need to talk about the population distributions and Yuri talked about this in his lectures if we're an LTE And we know that the population distributions follow Saha Boltzmann Equation so the excited states within an ionization state Follow a Boltzmann distribution according to their excitation energy and the ionization stages obey the Saha equation which is that the The population of one ionization stage related to the next one is just given by a ratio of the partition functions of the two charged states and Actually the electron density times this factor is the partition function for the thermal electrons. So this is Straightforward thermodynamics. It's a ratio of partition functions Now if the material is not an LTE so the case we're mostly concerned about non LTE Then we get the population distributions from the collisional radiative model So we take the rate equations here Which Yuri talked about take this rate equation and here I'm just talking about I've got a simpler rate equation here I'm collecting other sources or other physics which might come in here So in my transitions here, I've got collisional transitions radiative transitions these other transitions like auto ionizations You've seen this expression for the collisional Excitations where we take the electron distribution function integrated over velocities to get the rates here We do a similar integral over rates and cross sections for the radiation field and now we have a non zero radiation field So to get the radiative rates, we take the radiation field for an excitation or ionization We take the cross section as a function of frequency Multiplied by the radiation field at that frequency and integrate over frequency Now notice what I've gotten here is not the specific intensity, but it's the it's the It's the angle integrated intensity In most cases this is what couples to the plasma so we're talking We just need the number of photons at that frequency and we don't really care what direction they're going If there's a strong magnetic field on then we would probably care about the direction But in most cases we can use the angle angle integrated the angle average intensity So again just like for the electrons we take the cross section we integrate over the number of photons Which is why this is j new divided by h new to get back to the distribution function a number of photons and integrate that for the inverse process We take the inverse Cross-section again times the photon distribution and now we've got the spontaneous and this or the spontaneous term here and the stimulated term And we integrate over this distribution to get the inverse rate. So now we've got a Collisional radiative model where the rates here depend explicitly on the radiation density and we've got the Radiation transport equation where the coefficients depend on this. So in summary, so now we can write down an expression For all the aspects of the radiation transport equation, we've got The absorption coefficients and now we sum these over all the populations and all the radiative transitions Including the stimulated terms and again the stimulated terms here For both the bound free and the free free terms in this case We've assumed that the electrons have max welling distributions So the stimulated terms look like e to the minus h nu over kt here for the bound free transitions Then we have to explicitly put in the populations of those states so we've got the absorption coefficient and then the emission coefficient here is all the all the spontaneous terms and in LTE Then these are kind of complex absorption and emission coefficients, but we know what the We know what the populations are as a function of temperature and density and we know that the source function is Equal to the Planck function at that temperature So that simplifies the radiation transport equation quite a bit and not LTE. It's quite a bit more complex So now the source function in the radiation transport equation It depends explicitly on the electron density and temperature and also the populations Which themselves depend on the radiation intensity? So this is this is a highly coupled system here that we need to solve and actually this is This is probably a good a good place to stop for the first lecture because after this we get into how do you treat this coupled system? So we'll pick it up tomorrow at this point