 Okay, during the break I was asked a few questions. I guess I missed some, to mention some details or I should elaborate on some points. The first one here is that when I was talking about microfuel distribution problem I forgot to mention that this distribution is for ideal plasma and when you do take into account the interaction between plasma particles of course the distribution will be different. For the stronger plasma coupling is, the further this curve will deviate from the ideal plasma case but it's not principle difference, I mean it will look more or less like the same. There will be peak at same value and there will be tail sometimes with exponential tail when it's actually easy to understand if your radiator is charged that other perturbers cannot approach it very close, there is a say classical minimal approach distance, right? And after that the far tail will be determined only by Maxwellian tail so it will be exponential decay here. The other point I probably should mention that the natural broadening exercise line shape it's nice to show how you get Lorenzen but usually in practice natural broadening can be safely ignored except for x-ray emission when the Einstein coefficient become large and so the natural broadening could become important. And in general natural broadening includes also not only just spontaneous radiative decay but also phenomena like Auger decay in which case it could be even stronger for mid-z elements, okay? Say the shape of ka-alpha lines, I don't want to elaborate much but it's mainly determined by this natural phenomena Auger and spontaneous radiative decay. And the last point when I was talking about this ergodicity arguments of course that assumes the particles in my ensemble are independent radiators which coming back to the analogy with a throwing dice if you look at your neighbor and try to after all cheat with the dice that the result will be different. So yes we are talking about independent radiators that means no coherent plasma radiation here that's the assumption, okay? So it's nice that we talked right now about natural broadening because basically what it says that the width of a line is determined by the time of life of its level or when both levels are unstable it's of course sum of two contributions. There is probably a bit of what I will explain you that's an expression for the line broadening of isolated lines in a minute I will explain what I mean by what people mean by isolated lines. So what it says that width of a line can be calculated according to the following rules. There are two main contributions one comes through in elastic processes and if you look at one of these terms basically to see its integral f sorry for confusion here f is not the field f is distribution of velocities it's Maxwellian distribution usually. So it's just sigma and v that you know that the rate of collisional process of course when you integrate over the velocity distribution you get temperature rate for a given temperature. So it's a rate of all in elastic processes of all processes that cause electron to go from level u up a level to to some another level and the same of course for the low level of the transition we count for all transition for all collisions that cause electron to go to jump from L to to some another L prime level. So actually this part is should be for people working with collisions straight forward evident. There is something more which is less evident and here is the difference of of elastic scattering amplitudes of the upper and lower level so it can be the total broadening the total width can be written as a sum of this in elastic part and elastic part. Okay so it's here they are written explicitly like I said it's just very similar to the it's very similar to what we saw for spontaneous for natural line broadening and yeah the meaning of isolated lines is that the broadening of a given level is small compared to distance in energy to other levels with strictly speaking to only we you need to consider only those levels that cause that stark broadening that broadening okay. So once we have lines that basically do not overlap one with another in the spectrum one can assume the isolated lines case so it's just some of inverse values of time of life of upper and lower levels the inelastic part okay. In principle one can generalize it to other impact other impact processes like ionization recombination etc. So the width is just the inelastic part is sum of the population rates of upper and the population rates of the lower levels again that was derived as human isolated lines when the lines are not isolated you can no longer use this simple expression for calculating a stark line broadening a few words about what's called a standard theory of line broadening that was developed by a few famous scientists around sixties including Hans Grimm and in Russia there was Sobelman and a few more of course I don't want to limit it to just these two of course contributed to development of this theory and as I said it's sixties no computers no no computers comparable to the powerful ones we have today so people had to invent ways of doing with the problem of n-body basically n-body problem analytically so the standard theory basically says we will talk separately about consider separately ions and electrons so ions are static quasi static so for quasi static it's very simple as I showed you an example with limon alpha just take for each given value of electric field of micro field calculate the spectrum then do convolution with the field distribution and you get the line shape fine electrons are different and electrons are so fast you saw in the simulation electrons are usually much faster than ions and they are so far they can be treated like a collision the impact approximation which yeah I forgot to mention in the previous slide when I said it's very much similar to to the derivation we did for for spontaneous emission in which case we got Lorenzen so the results of this impact mechanism like impact excitation ionization etc. give rise also the Lorenzen shape and that basically what you see here if you take a real part of this complex denominator actually it's operator you will get Lorenzen modified a general Lorenzen shape in which let's not go into the details okay but the basic idea of standard theory of line broadening is strict separation of slow particles which are usually irons and fast particles which are usually electrons in principle it could be the case when irons are also so fast that they can be safely considered in the same impact approximation fine but the real problem is with intermediate cases when irons do move and the motion is important but they are not really in the impact regime and there is no easy there was no easy analytical way found by that time to deal with that the problem of inability to to to to work with ion motion was that the main driving reason behind invention of using of computer simulations for line shape broadening problem the pioneers is Roland Stamm who was by that time I guess postdoc of Waslumber, Dietrich Waslumber and the idea is actually very simple you just simulate plasma right just put and body solve and body problem puts sufficiently large number of particles in a sufficiently large box let them move and interact you can calculate the line you can calculate electric fields as a function of time just give it as a input for the Schrodinger equation time-dependent Schrodinger equation you solve it you get anything from Schrodinger equation including the evolution of the dipole operator calculate autocorrelation function take four years and that's all so very simple except that it quite time-consuming and it is still time-consuming today and you can imagine it was a real a real enterprise to do it in the end of 70s but these guys did it and the results were very impressive really so I'm just going to show you exactly the same just in the graphical way so computer simulations are still basically the same today of course computers are different and you take more particles that everything is more accurate etc etc but the main the the main structure of the computer simulation of line shape rodent is basically the same so you have something which I call particle field generator which just and body and body simulations of various complexity and in principle it may include also external magnetic or electric fields these fields also can be time-dependent what you get as a result as the output of this part of the simulation code is something like that field history has just I plotted just absolute value let's say but of course it's vector you have all projection so you have three-dimensional field and then you solve yeah you solve Schrodinger equation time-dependent equation and the result you get evolution time evolution operator and you get evolution of dipole operator taking Fourier transform that's actually the the simplest apart here you get the spectrum how do you solve it usually you one nobody solves Schrodinger equation of course in the coordinate space it's always matrix so that's the Schrodinger equation we can rewrite it in form of the time-development operator that's in the in the interaction representation the dipole operator is calculated like that when you have the evolution operator handy and then one either do the indeed autocorrelation function like I've shown you or say I in my simulation is doing just directly taking Fourier of dipole and taking square of absolute value and there is also some fine details about slight polarization and dependence on the on the well-known dependence on the energy of the transition but basically that's the same so the idea as I say is very simple there could not be a simpler model for line shape calculation except that it is very time consuming so obviously people are trying to to invent to find ways of dealing with this problem using a simpler means I don't have time to to describe everything I will be a bit biased here and show you some my results and my colleagues if you look at the yeah obviously when you solve Schrodinger equation the the larger number of states you have in it the more problematic is the more difficult is to solve difficulty in terms of computer time of course so that the higher number of components in a given transition is that the harder is to solve it and hydrogen or hydrogen like transition between high end lines read the transition I especially problematic here because the number of components goes like you know as n squared and you can easily get to a completely and manage and manageable problem even with today's computers just to put you into the some context I once calculated using computer simulations Balmer 15 line from n equal 15 to n equal two and for a single line shape for a single density single temperature it took about two months of of CPU time okay okay on the other hand high end lines are important for diagnostics I don't want to go into the details so but interesting point is if you look at a spectrum now it's static stark effect of some high end in this case limon nine between n equal nine to n equal one transition you look separately at pi and sigma components you can pay attention that each one that the components the intensities of each pi and sigma components separately form a lie on a parabola and if you do some of pi and sigma and usually that the case in plasma that the micro fields has no specific direction so you can safely average so obviously average of these two mirrors parabola is just a rectangular shape so that was the idea let's use rectangular line shape instead of very complex stark pattern well so that's it you defined a line shape static static line shape as a rectangular it's zero beyond that values and the constant value inside of course it should be normalized to have we always wanted to normalize to one of course area normalized so that's numerical coefficient which depends on the stark constant of this of given transition and in the case of hydrogen it can there is a well-known expression in all textbooks about atomic about quantum atomic physics there is simple generalization for not just limon but in principle any transition from n to n prime they're just straightforward one so again we have rectangular line shape for fixed value of electric field so next step if you want to do it in the quasi static approximation let's do a convolution with a field distribution and in the case of in the case of yes so the field distribution is given by that and I do by w of f and I do that simple convolution and you remember this normalization that is used in the that was obtained in deriving the Holtzmark field distribution so I have zero again and that's the Holtzmark constant and we can go to dimensional as units both for the field it's the same better we saw when I plotted the Holtzmark distribution and instead of using these energies we or frequencies we normalize it to let's say typical stark splitting for the same given field for Holtzmark field and then everything can be expressed in a very nice way and which becomes so there is one single numerical integration here and in fact if we talk about ideal plasma which is always good to begin with this can be done analytically and that's the result is a very simple expression and it's a universal expression that should explain in principle any line shape in the quasi static approximation for sufficiently high n what is sufficiently high is a separate question of course but in practice quite surprisingly for me even though I began all this work for a really high n transition often even you know and actually what's important is not n but delta n difference between upper and lower and so even for delta n as low as 2 the results are very nice okay so let's see how this function looks like so I plot that with a green dashed line and I custom in line shape literature if the line shape is symmetric let's just plot only one half of it the rest will be symmetric so that's this universal function that and I plot the same true calculations of two lines two high lines lineman nine and lineman ten and you see that except of the central region the rest is just extremely nice extremely nice feet it's not feet it's in analytical derivation but in fact if you do include some other broadening mechanisms which always exist like Doppler or electron impact broadening or whatever they will smooth the central part as well so the result will often be practically indistinguishable that's why it also as I said quite surprisingly work even for relatively low principle quantum numbers okay so that's quasi static and but you do want to take in touch into account dynamics that's basically the point when I said when people started to use computer simulations to to find a way to somehow deal with ion motion but there was a few other approaches to deal with ion motion and one of them that I want to briefly describe is what's called frequency fluctuation model initially it was developed by in something like 20 or even more years ago but it was recently reformulated basically it's the same but but the mathematical formulation is is a cleaner one so what it says the field the microfield fluctuates fluctuates with different frequency but we can find a some typical one well there is quite obvious choice for the typical frequency you just take typical velocity divide our typical distance inter particle distance and there must be a frequency okay so what is the typical velocity is of course maxwellian one and typical distance and just density to the power of minus three it's in denominator so here's plus one third and well forgetting about numerical factors and once you know once you assume the field fluctuates with some frequency you say okay so it fluctuates and one you have once you have electrons sitting at one component stark component of a fixed electric field there are many components like you saw in lima nine for example that's a whole forest of components so it sits with field static but then feels fluctuates and electron jumps to another component and it does it randomly that the assumption of this ffm method the jumping is random from any state from any stark component you jump randomly to another one and in mathematical terms it can be expressed by this actually it's very simple expressions so the line shape is calculated like that that's expression okay where the only computationally intensive but it's nothing compared to computational intense intensive simulation of course it just again it's just simple integral numerical integration you start from quasi static line shape which is in the case of high end I showed you can be done really simple then do this integration and you can imagine that describes something which it's kind of diffusion and that's exactly the the notion here for frequency fluctuation you diffuse the static line shape with a typical frequency which is defined by here and then there is some another operation I don't want to explain the details but if you look in this paper it has a very nice explanation how you derive it just I don't have time now but it's quite straightforward so let's try to apply this method to this I forgot to say that this substitution of complex line shape with just simple rectangular I call it quasi contiguous because you can forget about distances between there are so many components and forget about each discrete component describe it as a one a contiguous entity the rectangular one okay so if you plug in that the expression becomes very simple for that J this J integration can be done analytically and you get this universal universal expression it's called it's universal when once you do the scaling of all parameters to the to the typical frequency and the typical stark static stark effect of for a given transition but when you do the scaling all transition any transition is described by exactly the same expression I want to show you some application of this method you you've had already a few times in the previous days about continuum lowering and actually continuum lowering is a bit of confusing topics there are different ways of of thinking about continuum lowering one is what is continuum for example one can say it's once you stop to see bound spectrum another approach say no continuous when the electrons are free which is not the same it's the same when you are considering a single atom but in plasma it's different the electron could be not free you must apply some work to get it out of plasma yet the spectrum that it produces is no longer discrete okay so that's a confusion and some approaches to continuum lowering that is good when you really understand you want to define with it what is the limit of free electrons it's not necessarily the same when you see the transition from from discrete spectrum to continuum one I want to describe you some approach that was given in a book of Hans Grimm by the way he he cites in this context a paper which he co-authored but in that paper I didn't find this explanation so so I I quote is that the text book because I didn't find it explicitly explained in any other paper in any paper okay so to do let's calculate bound bound spectrum for a series we want to calculate it for a series and continue up to the some principle quantum number at which the broadening of that line will be comparable to the distance between the neighbor levels and that's the reason behind the English teller effect English teller criterion that you've heard already a few times before okay and we may continue actually if for a few more principle quantum numbers just to make sure we have a smooth transition and we will assume free bound edge at the the deposition of the last member of the series we at which we stop we can we do convolution of the free bound continuum which is in is a you know it's drawn as a sharp edge we do convolution with the same line broadening we got for the last members of the of the bound bound series and some up I want to point to stress that there is no ionization potential depression is assumed here I just want to calculate the the the line broadening of a series and that's the result for lineman series hydrogen lineman series in modestly dense plasma 10 to the 17 particles per cubic centimeter one EV temperature and I do not post here Boltzmann trivial Boltzmann factor comparison and that is the series one calculation is done with this QC FFM very fast method that took about one second the other calculation is computer simulation is actually believed one of the most massive computer simulation of line shape ever done that's almost 400 fully interacting states and it took about one month on today computer so the speed factor if you use the simple analytical expression is about 3 million and you see how nicely they compare actually there is a small difference here but there is no continuum wave function in computer simulation that was that would increase the size of my Hamiltonian to infinity which is not double so but as long as we are talking about bound bound series that the agreement is extremely nice so now we want to compare with real measurements and that's actually the only time I show you something wavelength because that's how experimental list presents a result okay so that high end series in again hydrogen it's Balmer series okay very nice experiments I do not think there are comparable experimental studies of the same accuracy so it's extremely nice data really and it's not easy in general to reproduce and now I'm trying to reproduce it using the inferred value that were given from my belief Thompson scattering so we have densities and I'm doing this it's not computer simulation it's just this simple analytical calculation which takes one second actually it can be made to milliseconds if I wish with some computer tricks program tricks and just see how similar they are really even up to a very very fine details like here for the lowest density 1.8 10 to the 16 years you have epsilon and then you have what it goes I don't remember right and then something a bit here and you do here do see something a bit here so actually the agreement is extremely nice and again that the approach is very simple and it's so simple that well I wrote a program I intended to make it public by the time of of this school I didn't have time but I I'll try to do it in the coming months so it will be distributed freely of course that as a simple program and it runs extremely fast and you can calculate and hydrogen at least hydrogen like of course it's not for for isolated lines for the lights and there are different methods but for hydrogen like especially for high end which is very difficult to calculate just analytically I promise you will have this tool soon I just need to find time to do it to publish a paper and so they will be attached extra material okay I want to to mention data source when you can find pre computed or you can compute line shapes so one is star to be database which is for isolated lines there are quite a few species and transitions it's no it's nowhere complete but but it's a large database so probably you should start if you need to find the stock brown for isolated line you should go to this point and by the way like I said there's always a question how strong field is when you go from quadratic limit to the linear and the same is true for isolated lines for low density it would be isolated the line for for any line but there is for sufficiently high density the line will no longer be isolated so this database also there is also also they they they quote the maximum value of density for which you can safely use the data okay you can safely assume that the line is isolated another source is the classic book of Hans Grimm there are appendices with least broadening and shifts and widths of of of neutral of very many neutral and single ionized ions there is also appendices in the same book about hydrogen but I do not recommend you to use those okay but for isolated lines it's it's good I from time to time I do the calculations of different lines and compare with Hans calculations I always surprised how accurate they are I mean I use today computer simulations and it's not computer simulation it's just this standard theory of line broadening and there are many problematic in principle problematic issues with that but most of the time there is also very accurate so it's not electronic form but we need to page through the book the other one that's plasma formulae that self advertisement you can calculate a lot of different things with it including the stark broadening of hydrogen like or Riedberg transition so for isolated lines you go to this this too for hydrogen like you go to this one and it's always good to compare you plot both of them as a function of density and they at which some point they will intersect and just do interpolation between the two maybe I'll show examples okay the last but not the least there is an east atomic spectral line broadening bibliographic database which has a lot of links a lot of entries on stark broadening literature okay a few notes on line shape accuracy well line shape calculation is very complex task and as always with the complex calculations to really give a correct value of theoretical uncertainty is sometimes even more difficult than do the calculations themselves so one should really be critical about using just average let's say source of published data online broadening I have a few rules of some that I use myself and you can consider them as useful something like that if there is a study online broadening and there is no mention of accuracy at all factor 2 factor 3 don't expect it to be more accurate than factor 2 okay if people do claim some accuracy about 20 40 percent give or take it must be discussed in explain in the paper if it's not go back to this assumption okay 10 to 20 percent there is really must be a separate section devoted to accuracy again if it doesn't exist go to here number one factor 2 3 to 10 that's must be a subject of the study the study must be that we want to compare that with that and accuracy is our point if not go to number one well sometimes you do see studies which claim less than better than 3 percent accuracy in particular comparison to experimental data but even without comparison to experimental data I do not list do not give you a suggestion advice for politically correct reasons but basically throw that trash in that paper okay one case pay attention some data some data base yes some published data when people say with it might mean full with at half max sometimes it means half with at half max so if you do not pay attention you get that factor 2 in addition to factor 2 that okay and that well sometimes it helps but who knows so please in particular in the book of Hans green weeds is half with at half max okay but for example in start be database is full with at half max please pay attention well there I know of at least one study published study by I don't say famous person ill famous person not to give names who claim that calculations of Hans green were wrong by a large factor about 2 which turned out to be he forgot or I don't know what that in those tables in Hans green book half weeks half max are given okay so don't make a similar mistake well I have few more minutes and well first of all I want to to give you a list of a short list of recommended books and reviews number one of course it the classical textbook of Hans green on spectral line broadening and then very nice and introduction to plasma spectroscopy really given in very simple terms and there is a nice chapter online broadening by a him who was the first lecture in this school and there is a nice nice review of of models for start broadening by Marco jigosos it's a recent review and there is a list also self promoting a bit that's a short mini review on computer simulation models in line shapes so maybe I'll show you in the remaining five or so minutes that's the interface for stark be database do you see do I need to increase font so it's okay yes or not increase okay well if I increase too much there would be a problem with you know working with that let's first keep here and when we go to the data will increase okay so you go to access to the data by the way this is part of the of the virtual observatory project European project so for example big hill okay in this case there is no there is no ionized helium because ionized helium would be hydrogenic and hydrogenic is not lines are not isolated that's why you do not see it here okay so you go to select in this case just one pick your density you can also enter just specific value but for the case of isolated lines if you remember just sigma nv so the broadening is proportional to the density so there is no need actually to pick anything then one of the standard one so transition as you see there are quite a few what do we want I don't know pick your choice whatever you can also select wavelength interval and the temperature or you can select all okay by the way the calculations are done and there by Sylvie it's based on the code by Sylvie Sahabriho and Milan Dimitriyevich and collaborators I believe that for the interface itself there are references that you can follow here the result the density the transition the wavelengths of the transition and the broadening coefficient not brown coefficient there is a width sorry width and the shift of the transition as you see it's function of function of temperature where do you see temperature yeah here's the temperature and for several values of temperature you have width full with half max in this case and shift of the transition and if you need something in between you can interpolate usually the grid is chosen sufficiently dense and sufficiently wide wide to cover any important temperature for specific transition and by the way they provide also fitting coefficients to fit the data if you do not want to interpolate just by simple interpolation also there is a simple expression for interpolation okay the other one I mentioned is the plasma formulary that the link actually will have a session afternoon devoted to resources for line for for atomic data something like that and maybe I will have more time to explain here now I just to show one simple example of calculations so it's a web based utility you don't need to install anything is select plasma parameters let's say 10 to 16 temperature ion parameters charge mass and temperature which is by default the same you select atomic system you select the same for radiator its charge mass and temperature it may be a percentage if it's used as a dopant finally you pick your select the atomic system actually in this case charges one means the core is charges zero means its hydrogen and let's look for for example H beta third transition from n equal 4 to 2 we go to explorer and here we are interested in the energy total stark width which is given in electron volts you may to calculate it in their centimeters there is no anstroms if you insist you need to do this stupid one over lambda etc. calculations yourself okay and you can well I will have more minutes to calculate to show you more capabilities of these two afternoon okay well I think that's all the time is over coffee break