 the other types of equilibrium and basically how the line intensities can be understood to some degree. So in a normal story of plasma spectroscopy we understand that a photon that we eventually register on our device is somehow created in plasma, then it may propagate with reabsorption, remission, which is certainly the subject of opacity and relation transport that I am not going to touch today. However, we'll do this. And then it leaves plasma volume, comes to our experimental device is somehow registered. Of course, you have some instrumental effect, some photons are lost here and there, but I'll try to talk about what actually happens here, how photons are produced and how they are emitted. So as Ahim Kunso mentioned on the first day, there are different words that they use to define this particular quantity, which is here is just the energy emitted due to a specific transition in atom or ion from unit volume per unit time. And here I just defined this product of three quantities. One is the photon energy. Then this is the transition probability of Einstein coefficients. Photon energy of course has some energy units, let's say Joule. This one has units of inverse second. And the third factor here is the upper state density or normally we call just level population. Now of these three parameters that tell us about line spectral line intensity in opticality in plasma, the two parameters on the right are almost purely atomic parameters. That is, they do depend slightly on plasma properties because both energy and transition probability as we perfectly understand the result, for instance, calculation. You take two wave functions and some operator in between and of course the wave function as we understand can be modified by plasma, but generally we can more or less safely assume that these two remain constant for very large range of plasma parameters such as densities and this is what we'll assume for the future, but the level population is very strongly dependent on what actually goes on in plasma and this is will be the focal point of everything that follows. Now before we come to discussion of level populations, let me remind you that atomic processes are characterized by their corresponding rates, basically number of processes per unit time and of course we can split them into collisional and radiative or non-radiative like authorization. Collisional rate is normally defined for a projectile heating, let's say our atom and ion and result in some reaction as first of all integral over distribution of velocities or energies for the projectile with some distribution function. For electrons it's normally Maxwellian but not necessarily in many cases electron energy or velocity distribution plasma deviates sometimes very strongly from Maxwellian distribution. Of course then you have the cross section which is an effective area and that's why its units are centimeter squared for this particular elementary process. Of course you need to get into taking into account the velocity because the faster they move the more collisions we have and of course everything depends also on the projectile density, electron density for electrons and so on so forth. The integral itself is called the rate coefficient, it has units of centimeter cubed per second. The radiative rates certainly have just units of inverse second and of course the lifetime associated with a particular radiative process is just one over a and this is something that we will be using later on. Now another important thing that we certainly should remember before discussing what happens with population distributions and other quantities is the scaling of various atomic parameters. Let's take a look at z-scaling where z is the spectroscopic charge. It's ion charge plus one and of course for very large ion charges for highly ionized system they're almost the same but generally it's physically more reasonable to use to discuss scaling related to spectroscopic charge. So starting with radiative processes the transition probability is proportional to the product of the oscillator strands and energy difference squared. For transitions within the same principle quantum numbers the energies, energy differences normally scale as you remember we had that expansion for non-rotivistic system of energies. It starts with z squared then z, z or z minus one. z squared is responsible for energy differences with delta n larger than zero but the z term is responsible for what happens within a principle quantum number. So delta e goes at z, the oscillator strength can be shown to actually decrease as one over z and that's why the transition probability for delta n equals z transition from this formula obviously goes as just z. You go to transitions between different ends delta e looks like hydrogenic and therefore it's z squared. We know that for hydrogenic ions oscillator strands remain the same basically z does not depend on z and therefore the radiative transition probability for delta n equals z transition shows very strong behavior z to the fourth. Now if we look at how a value changes with the principle quantum number let's say we consider transition from high end to some lower end and then we go higher and higher it drops like one over n cube and if you look at total radiative decays from a principle quantum number within hydrogenic ion it still has the same z to the fourth dependence but the total sum of all possible radiative rates from hydrogenic level decrease even stronger as one over n to the four and a half. Now these dependencies are written for electric dipole typically the strongest transitions in atoms forbidden transitions other multiples which are magnetic dipole, electric quadruple, magnetic quadruple, electric octopole, magnetic octopole and so on so forth. They even have stronger dependencies than z to the fourth and obviously this means that you go to higher charge ions and forbidden transition becomes stronger and stronger and stronger and therefore much easier to see and measure. Now if we're talking about collisional transitions for which we have several types dipole allow or optically allowed optically forbidden spin forbidden for the strongest of those which are optically allowed and they happen between transitions that can be connected radiatively the cross section behaves differently from the a value and normally it goes like f over delta e squared. Now again looking at already known dependencies of f delta e on z we can figure out that cross section actually decreases with z and the gate pretty strongly z minus cube the rate coefficients behaves certainly differently. Now v of course is the velocity or in this case let's say average velocity of the electrons that come to heat our atom but to produce higher ions we need certainly electrons with higher energy and therefore we can kind of assume that we goes approximately as z or maybe a little weaker than z so therefore the rate coefficient changes maybe slightly a little less strongly with z than the cross section so let's say z minus squared for transitions between different ends cross section goes as one over z to the fourth the rate coefficient z minus cube and cross sections collision cross section strongly depend on on principal quantum number. This is very important point because this simply means that you go to higher ends and you try to figure out how balance between radiative and collisional processes changes you may see that going higher collisions become more and more important and on the other hand radiative process become less and less important and this will change the whole population dynamics for higher ends. Okay so starting from the complete thermodynamic equilibrium this is situation when all components of our physical systems are in equilibrium which means that principal detail balance is valid here in other words each direct process is exactly balanced by its inverse so if we have excitation by some kind of particle it's exactly compensated by the excitation by the same part and the same goes for all other physical processes such as ionization it's inverse three body recombination, photonization, photo recombination, ottonization, dielectric capture, radiative decay, spontaneous flux, stimulated is compensated by photo excitation and this was Einstein's logic when he basically defined all these Einstein coefficients in his analysis of radiation in atomic systems. Now typically in plasma we have four or maybe you can say three systems those are of course photons, electrons, atoms and ions. Thermodynamic equilibrium means that all of them have the same temperature characterized which is that is radiation temperature and electron temperature and ion temperature is the same and we know of course the equilibrium distribution for each of these subsystems. For photons it's Planck, for electrons Maxwell, populations within atoms, ions are distributed according to Boltzmann Exponent Formula, populations between atoms and ions are distributed according to Saha equations. Now it's generally useful to have this energy scheme in mind when you consider all these physical processes so let's say this is our atomic system and we have a number of bound states and then of course the ionization potential for an ion and then we have continuum state with let's say positive energy so this is the energy scale. Basically all these distributions and all these processes are related to electron movements within subsystems of this energy scheme that is Boltzmann of course corresponds to what happens here between bound states, the formula is very well known of course, I'm not going to discuss it. Saha simply corresponds to situation when electron goes between bound and continuum states and of course Maxwell distribution describes what happens in the continuum so all these parts of thermodynamic equilibrium Maxwell, Saha and Boltzmann corresponds simply to processes that happen within the same unifying energy scale simply by talking differently but bound free and free free so to say transition for electrons not for photon. Now talking about photons of course we very well know how the Planck distribution looks like and probably the easiest object to see Planck distribution with photon is our sun so what you see here is two curves the gray one is the black body spectrum for the temperature of 5777 Kelvin and the yellow is the extraterrestrial solar spectrum irradiance which looks very much like black body and certainly we know that sun gives us more or less close to perfect black body spectrum. McSullin is also known it's rather common in real calculation to move from velocity representation to the energy representation and of course when you take into account the velocity in the rate coefficient you remember it's V times sigma times cross section and when you move to energy representation it becomes square root of E times McSullin distribution for energy it's also useful to remember that actually this parameter peaks at the electron temperature and therefore this is where we have the largest number of electrons for particular McSullin here's temperature 200 eV and this is where most of the reaction actually happens because you have the largest number of electrons here. Okay, Saha distribution as I said before Saha corresponds to station when electron moves between bound and continuum state so of course for instance most common physical processes that are associated with establishment of Saha distribution are ionization and three body recombination which corresponds to this formula with parenthesis draw but also you can consider another pair of processes that is authorization and electronic capture you heard yes about that organization from Stefan Friedscher and electronic capture is just the inverse process when electron approaches charge ion and get captured coming to some relatively higher orbit and one of the atomic electron is excited so the loss of energy here is exactly compensated by increase of energy by the bound electron therefore energy conserves. The formula for Saha distribution that is in this particular case the ratio of total populations for the next ion z plus one to the ion with charge z is of course the ratio of total partition function which are defined as sum of statistical weight with exponents here of course you have some factors that are related to the continuum characteristics of course electron temperature must be sitting there you have the standard Boltzmann factor of ionization potential of a temperature and importantly you have this factor one over electron density and this factor is certainly important because if you have a Saha distribution and your electron density increases with all other parameters then you will have a kept constant you see that basically it results in increased recombination and it goes up you have more and more ions with charge z as compared to charge z plus one so in many plasmas which are not in Saha equilibrium the situation is opposite you increase depth of energy density and your ionization actually increases but inside it is the other way and you can certainly understand it from for instance comparing ionization and three body recombination. Electron impact ionization is of course proportional to the first power of electron density you have just one electron coming in and kicking another one out but in three body recombination you have two electrons coming close to your atom and therefore the probability or rate for this process is proportional to electron density squared so you increase the density this is first power this is second recombination becomes stronger ionization balance shift to low ionization stages. Now one of the primary questions that we normally want to ask is which ion is the most abundant under this particular conditions in plasma. In other case let's say we have situation when this ion with charge z and z plus one have about the same populations of course the others will be going down it simply follows from all the Saha formulas it turns out that for Saha equilibrium typically this happens when electron temperature is much smaller well much maybe a factor of five or even a little higher smaller than the ionization potential of this particular ion and we'll see some examples that this is really the case. Now so the complete thermal or thermodynamic equilibrium is reached when we have complete balance between photons, electrons and let's say ions and this situation almost never happened. There are some indication that there may be complete thermodynamic equilibrium in the interiors of white dwarfs but on our planet in terrestrial conditions this is not the case and the reason is that photons decouple easily they easily escape they very quickly interact with other particles and therefore more common situation is that photons are becoming completely separate part and the plasma is characterized with the rest of our particles electrons and ions and this situation is called local thermodynamic equilibrium which simply means that our matter photons are the field but electrons and ions are matter distribution here follows Saha between ions Boltzmann for bound state within ion and Maxwell of course for electron. Now the typical now if photons decouple basically means that radiation effects are much smaller than particle and certainly you can think that this would happen when density of electrons is much higher and therefore collision rates are much stronger than radiative rates and there are conditions derived by different scientists they may differ a little bit but this criteria basically describes what density corresponds to situation when you have Boltzmann distribution established within your ion for instance and Hans Green derived this condition long long ago assuming that collision rates are at least a factor of magnitude factor of ten order of magnitude stronger than radiative rates so if you consider something like hydrogenic ion and look at the strongest radiative transition which is the resonance line between the first excited level to the ground and then carefully write down formulas for a value for collision rates you can derive it yourself that you need pretty high densities which follows from this formula to reach Boltzmann distribution so for instance if we are talking about neutral hydrogen and electron temperature which enters here is to EV you need electron densities on the order of 10 to the 17 or higher in order to provide Boltzmann distribution of population between the atomic states you go to a little higher ions so for instance we are talking about carbon 5 which is 4 times ionized carbon with typical temperature of 80 electron volts in this case the density increases by 5 orders of magnitude so you just ionize your matter a little bit and to reach Boltzmann distribution you really need higher and higher density and you see that this density changes per strongest z to the 7th. Saha criterion shows about the same dependence on z again just from these formulas it follows that for hydrogen 10 to the 17 carbon 5 3 10 to the 21 to get the bottom line here is to get LT you really need higher densities especially for charged ion and especially for highly charged ion. Again for neutrals 10 to the 17 4 times ionized 10 to the 22 you go another order of magnitude for your charge from 4 to 40 and it goes through the roof becomes exceedingly high. Now what's good about local thermodynamic equilibrium regarding calculation of ion intensities good point is that we need practically no atomic data well only energies statistical weights to calculate the populations. We just take the Boltzmann formula and this is what done. Now from here it follows that the intensity ratio of two lines is a very nice function of temperature only and therefore if our plasma is an LT which means dense plasma we can get electron temperature just looking at the ratios of two lines or you can take more than two lines and plot intensities of several measured lines on a plot that is normally called Boltzmann plot so basically since our intensity is product of population transition probability and energy which goes like ratio of statistical weight to partition function Ae and exponent if you take log of this ratio it becomes a linear function of the level energy and just looking at the slope you can easily derive the temperature. So this is an example from one of the recent papers and there are many papers that use this technique guys measured I think that was I think neon. Several lines took this ratio log and here you have nice linear distribution that immediately tells you that the temperature in this particular case is 14 and a half thousand Kelvin's with relatively small error of plus minus 3 Kelvin, 300 Kelvin's only. So if we look back at what happens with Saha LT again we can see that the temperature can reach Saha LT condition when collisions are much stronger than radiative properties. Saha is established between ions, Boltzmann within ions and since collisions again decrease with Z you remember cross section goes one over Z to the fourth and radiative process increase with Z it means that high and high densities are needed for ions to reach Saha LT conditions. Now can we find some tools to easily calculate Saha LT spectra? Yes we have such a tool at our database let me show you an example. So again you go to the NIST Atomic Spectra database line form that we saw on Monday. I was talking about crocheting diagrams, other criteria but here is this little part on the left that is called dynamic plots. These are plots that you can generate and Saha LT spectrum is one of them. Now again to generate Saha LT spectrum you need only energies, statistical weights which are 2J plus 1 and transition probabilities and this is exactly what we have in database. We do not have cross sections because from the beginning the NIST Atomic Spectra database was about spectroscopy not collisions but this is enough to get some spectrum. So let's take a look. Carbon, okay let's start from scratch. So let's take something like iron first five ionization stages and let's take the range of equivalent particles comes from 300 to 500 nanometers because we need not too high temperature. Okay we select Saha LT spectrum. We need not too high temperature to really have these ions most above so let's take maybe three electron volts. The density 10 to the 20 should be alright. If we want to build not spectrum as simple bars but widened profiles we can use this option for Doppler broaden spectrum but here we should put relatively high ion temperature let's say 200,000, 20,000 electron volts so the profiles are really wide. Okay let's see what we get now. Iron has quite a few lines so it takes a little while to get the data so you see it shows that we have 2057 lines and of course you need lines that have information about transition probability so otherwise you will get no results and here at the bottom at the end of this long output table you see different options for the plots so if we look at the PNG file you see the Saha LT generated spectrum. In addition here on the right you see the calculated distribution of population over ionization stages so at this particular density and this particular temperature iron 2 is the most populated, iron 3 has 26% of the total population sensor 4 and you can look at the spectra and use them if your plasma is really dense and close to Saha LT condition. Okay so let's move on. Now when we look at the spectrum one can we expect deviations from Saha LT? One is obvious we already were talking about the densities we really want density to be pretty high to reach Saha LT but you remember that Saha LT basically corresponds to the situation when you have a collaboration between direct and inverse process and if our plasma is non-maxwellian that is you have let's say bulk of electron have maxwellian distribution but you may have some hot tails or hot electrons then the distribution of electrons is non-equilibrium and therefore the distribution of populations in your atoms ions becomes non-equilibrium as well so you have deviation from LT in non-maxwellian plasmas if you have unbalanced processes if you have an isotropy in your plasmas external field so all these things results in various types of deviation from Saha LT. So Saha LT is high density what happens at the other side of densities. The other limiting case is of course the coronal equilibrium that corresponds to low electron densities of course the name comes from the studies of solar corona and if by any chance you are in the United States on August 21 you have a chance to see the total solar eclipse. The timing is certainly very good it will happen around 1 p.m. the shadow of the moon will enter the west coast I think around Oregon and then crosses the whole country the peak of the eclipse will happen around Tennessee Kentucky it's not too long only two minutes and forty seconds at the peak but still it's very worthwhile scene. Okay so what's the difference between coronal model and Saha LT approach. In coronal model we think that the electron density is so low that the collisions happen very early and therefore excitations and ionization only happen from the ground state so if this is the scheme of energy level in particular ion and solid lines show you possible all possible collisional excitation or de-excitation we let's say neglect everything except for direct excitation from the ground state and sometimes you also include metastable state that they're pretty well populated in atoms and ions. We remember that radiative properties do not depend on the electron density and collisional processes go as n e or n is squared and of course if your density is very small then collisional properties collisional process become less and less probable than the radiative one. Now because collisions are indeed important here in the sense that this is the real way to move electron from the ground state to excited state coronal model does require a complete set of collisional cross-sections and this is the difference from Saha LT where we don't need cross-section we need just Boltzmann formula and that's it. Now generally speaking here we only need excitations and we can neglect the excitation but in general case do we have to calculate separately excitation processes and de-excitation and the answer is no if we know excitation we can immediately derive de-excitation and this derivation of connection between these two types of cross-sections can be illustrated by the principle of detail balance. So let's say we have a two level system excitation de-excitation and in equilibrium what comes up is exactly equilibrated but what comes down. So we neglect all radiation just for illustration of this connection between excitation and de-excitation. In thermodynamic equilibrium these two levels are connected by Boltzmann formula or their populations and therefore substituting Nj and Ni in here we get connection between rate coefficients for excitation and de-excitation they are just connected with the exponent. Now each rate coefficient is an integral over Maxwell distribution the difference is that for excitation we have excitation threshold and therefore the integral starts from the threshold delta E. For de-excitation electron with any antigen as small as it can only be can approach the atom atomic electron goes down gives the excess of energy to the passing by electron therefore here the integral starts from zero. We can do substitution of the integration variable in this form so that finally we have connection between excitation and de-excitation cross-section in this formula and this should be valid for any value of temperature there was no real indication what kind of temperature it is in this derivation which means that the integrants must be equal also and therefore finally we come to this very nice connection between excitation and de-excitation cross-section and therefore it means that if we have excitation we can immediately determine de-excitation cross-section or even rates if we rate coefficient if we need them to be calculated. Now what happens with line intensities on the coronal equilibrium? Again we can rather easily calculate the intensity for two level system in the balance equation what comes from the let's say ground state to the excited level and this is population of the ground state times rate times excitation rate is exactly equilibrated by the radiative decay from the upper level which would be its population times radiative rate. From here it follows first of all that the population of the upper level is proportion to electron density and because coronal equilibrium is characterized by low density it means that the populations of excited states in corona are typically very, very small. You remember in Boltzmann you just have delta E over T and T can be different which means that the difference between let's say ground state and first excited is not that small but here we have basically very small factor electron density which tells us that if we look at the total population of an ion it's really only the ground state under coronal equilibrium that has any substantial population all other levels have very small population but from here even more important conclusion follows. Now the intensity of the line is a product of level population, radiative transition probability and energy and these two factors are exactly what we have here therefore we can just substitute them with ground state population and excitation. Basically this formula tells you that under coronal equilibrium the intensities of spectral lines actually do not depend on radiative transition probabilities and this sounds quite natural. Let's say we have two levels and collisions happen very early because the electron density is low so electron goes from the ground state to the upper level and the next collision will happen nobody knows when in very large time so therefore this long, long time between collision is actually larger than the typical radiative lifetime so the electron will always have a chance to drop down in the midi-forten therefore practically for each collisional excitation we will have one photon and therefore the number of photons which actually is the intensity is directly related to the number of collisions. Now this is of course simplified a scheme of just two levels. If we have more than one possible radiative transition from the upper level of course we have to take them into account. At the end we have a branching ratio entering the intensity of each particular line from the upper level but the idea basically is the same. It's really the collisions that are important for determination of line intensities under coronal equilibrium. Now in SAHA LTE the most abundant ion is produced when temperature is significantly lower than ionization potential for coronal equilibrium lower densities the ratio is less, the ratio is smaller so you need temperatures that are typically a factor of three only smaller than the ionization potential and this condition is actually valid for light and medium elements in the periodic system like nuclear charts let's say smaller than 30. Probably we'll have a chance to look at what happens at higher z's but for astrophysical purposes at least from where the coronal equilibrium comes this ratio is generally quite valid. Now the ionization balance in coronal equilibrium does not depend on electron density. You remember again in SAHA we had this factor one over n e density increases we have more and more recombination shift ionization balance to lower ionization stage this is not the case in corona why? The main processes that establish ionization balance in coronal equilibrium are electron impact ionization which is proportional to electron density and photorecombination and dielectronic recombination both of which are also proportional to the electron density. The three body recombination which generally is present here is proportional to n e squared and because n e is small we can completely neglect it for coronal equilibrium and of course n e simply cancels out and therefore the ratio of population between two neighboring ions and also the total ionization distribution becomes completely independent on electron density. Now we already know what happens for SAHA at high dense so let's say we are looking at the log of the ratio of population between two ions z plus one and z again the ionization proceeds mainly through electron impact we have both photorecombination photorecombination dielectronic recombination three body recombination pushing balance to low z we remember that SAHA has one over n e dependence at higher electron densities corona gives us concentration at low density question is what happens in between now certainly you would at first thing that ok these two curves are connected by smooth curve here but this is not really the case actually when you increase density moving from coronal condition to higher electron densities. First you produce more ionization and this happens simply because the excited states become more and more populated. You remember in corona the population of each excited state is proportional to electron density and of course it's negligible if density is very small and therefore if we consider ionization in corona it's only direct ionization from the ground state maybe metastable that contributes but density increases you excited states become more and more populated and suddenly they start contributing to ionization particularly because the ionization potential from those states is much smaller than ionization potential from the ground state and therefore cross section SAHA therefore rate coefficients and rate are higher and therefore there is always range of electron densities where excited state become contributing to ionization and then of course at the end at higher densities you reach SAHA condition and the whole ionization balance changes to typical SAHA behavior ok we have few more minutes before the break so let's discuss few more things. Now we already learned that radiative processes and collisional processes depend differently within an ion. Radiation transition probabilities drop when we go high in N typical is one over N cube and on the other hand collisions become stronger and stronger when we move high in N again this is bound state continuum so if we even start from coronal condition when radiative processes are much stronger than collisional for lower bound state moving up in this tower of atomic states we will definitely reach situation when collisions become as strong as radiation and then collision will become much stronger than radiation this simply means that in any plasma if you go sufficiently high in principal quantum numbers you will reach situation when distribution of population between this level switches from corona to LTE that is to Boltzmann. Now where it actually happens depends on several factors of course it depends on electron density because you always have electron density factor for collisions it depends on ion charge it's very weakly depends on electron temperature because certainly the main collisional transition that establish Boltzmann distribution here are transition between neighboring levels and for them energy difference is much smaller than electron temperature so you can for instance use this formula again there are several different formulas that try to describe transition from coronal distribution for lower bound states to partial LTE p means partial here for higher bound state this is one of the formulas also derived by Grimm and you see that the dependence on temperature is really weak it's 1 over 17 power for electron density for electron temperature for density is just slightly higher but still you can use it to see but the bottom line here is that there's always part in your atom ion that will be in LTE and those are the higher levels okay it's exactly 950 and probably this is a good place to stop for 10 minute break and we will continue with collisional radiative model after that.