 Hello everyone. Okay. We will talk today about line shapes. Well, I probably do not have to tell you that spectroscopy is really one of the very unique diagnostics tool, plasma diagnostics tools because it allows for probing plasma non-intrusively. Well, in general, what is spectroscopy? Just looking for radiation that's coming from plasma. But you can look at that at different levels. You say just, okay, there is a light. And you say, okay, there is a plasma. Then you can look a bit in more detail like using the toy spectrometer Yuri gave everybody to test on the first day of this school. And then you see there is some discrete spectrum, which already says you can infer probably more or less roughly the temperature, electron temperature of the plasma. Well, of course, the chemical composition. Looking even at more details, you now start to see that this discrete spectrum actually consists of many lines with each having its own width and shift. And that already may give you some information about plasma density. Looking even with more details, you no longer see the separate lines as just something which has only width and then shifted. But there is a real line profile, line shape. And line shape gives you provided you have tools and you know how to calculate the interpreted line shape data. You can get a lot of more information about plasma, including density like it was before, but also electromagnetic field, which could be just spontaneous fields like plasma waves that have been created in the plasma, also macroscopic external fields. So what is line shape? Actually, it tells you what is the probability to emit or to absorb a photon at a given energy or wave usually one uses normalized to unity line shape. Sometimes in some books you will see and also you saw it in the previous lectures that people sometimes define it with a phi. I prefer to use L to designate normalized line shape, area normalized line shape. Howard already mentioned yesterday that absorption and emission line shapes are not necessarily the same. That's usually the case, but not always. Just please keep it in mind. We will not talk about this here because it's a rather advanced topic indeed, but one needs to keep it in mind in general. Many units you will see people use in different rocks. That could be frequency, some use angular frequency, wave number, which is just frequency over the speed of light and expressed in units of inverse centimeters, or in units of read-bursts. Again, it could be expressed in wave number, units, or in units of energy. People say both values read-burst, so strictly speaking read-burst is just the energy unit. And finally there is wave length, which is the inverse value of the wave number. For most of them you can easily convert one to another. For example, EV, just multiplication by some number, fixed number, you can convert it to inverse centimeters, or hertz, or angular frequency. When you want to convert to wave lengths, you have something different. You need to make to calculate inverse of that value. Usually people in line shapes talking about not absolute values of energies or frequencies, but only a deviation, which for example one, if one talks about width, right? And these values are usually much smaller than the value itself, that the unperturbed energy or unperturbed wave lengths. And for most of these units and entities the conversion is exactly the same, except for wave lengths, which now makes even more complex. And you often see in literature expressions which involve something like delta lambda over lambda squared, which what I want to say is they are just unnecessary complex. In my view they are also ugly. And come on, when we all now know quantum mechanics, and in quantum mechanics there is no wave length really, except for the Brault, okay? The experimentalists have always used wave lengths, anstrom, or nanometers lately, because the spectrometers itself are based on the wave nature of the light. That's okay. They are based on interference or diffraction and all wave optics phenomena. But physics is really about energy, not about wavelengths. So I will use only energy or angular frequency units. In atomic units they are the same. And I actually urge you to always try to use, to work with one of these good units and non-end wave lengths, especially when you are talking about line broadening in plasma. Okay. In plasma there are several phenomena that affect line radiation. These are the motion of the radiator, which is the Doppler effect and magnetic field, which gives rise to the effect of Zeeman splitting. And finally the electric field, which is the reason for stark effect. One can mention also natural broadening and some other types of natural decays, not only radiative like Kozhe. But in principle I would say all this is due to the interaction with magnetic or electric fields of spontaneous radiation. So basically it's just those three. Okay. Let's start with the Doppler effect, the simplest one. So instead of original energy or frequency, the light is an emitter. You detect the light, which is shifted by a constant value, which is proportional to the velocity, to the projection of velocity towards the observer. So in plasma there are a huge number of particles, a huge number of emitters, and each one emits at a, in each one has a different velocity. So one we need to go from a distribution of particle velocities, which is defined by e p of v, to the distribution of frequencies or energies that you absorb. So that's just a trivial variable substitution. And you can get the line profile, which is expressed through distribution of velocities. And the conversion factor here is the derivative of frequency shift with respect to velocity. And one needs to take the derivatives as a point when the velocity gives the frequency that one is interested here. So in case of thermal Doppler broadening, we will know what is the distribution of velocities, right? That's the Maxwellian. It's one-dimensional in the projection of a lone one axis. And so we immediately convert one to another and get a distribution of the probability of observing photon at given energy. And well, of course, one immediately says that's a Gaussian with standard deviation expressed by that formula. And that's a convenient numerical expression when you use temperature and electron volts and mass in atomic mass units. And you get full width half max of the line profile. Interesting property is that a product of peak intensity on the full width half max is a value close to one so. Yes, please. The same, the same. OK, full width of half minimum. What about the absorption profile? There is no maximum. I mean the maximum is zero. OK, so you just inverted. As I said, in most situations you can safely assume that absorption and emission profiles are the same. So we'll talk here when I say line profile, it will be line profile for emission, but you know it is the same, just inverted. Just a few examples of Gaussian profiles with different value of sigma of standard deviation parameter. And well, an important point here is that the winds of Gaussian decreases very fast. So it goes basically in practical terms to zero. So if the line profile is Gaussian, you can measure it in its entirety and don't care about far winds. We will see that it's not always the case with other types of line broadening. Actually, what the simple exercise that we've done with Doppler, with thermal Doppler broadening, is a good example of what's called quasi-static or statistical broadening. People usually do not use this term for Doppler, but in fact it's exactly the same. So what we have done, there is a huge ensemble of different emitters, different radiators. Each one emits at a just single given frequency. It's a delta function if we talk about its spectrum. But when we do averaging over a huge ensemble of independent or dependent radiators, we see a continuous entity, which is the line shape. And that's not the result of line broadening that comes from a single emitter, single radiator that is broadened by some effect. It's just pure mathematical results of mathematical averaging, averaging over an ensemble. Well, here we did it, we converted from probability to find a radiator with a given velocity, but in principle it could be very similar when we consider other effect which influences radiation of energy of radiation of one emitter. We will soon talk about stark broadening in which case this parameter will be electric field. So what are the distributions of microfield in plasma? Obviously in plasma we have a large number of charged particles, and if we just sit at one specific point or we consider just one plasma particle and look at different, at all other particles, charged particles, obviously there is a different probability to find different electric field. Each charged particle creates some field and we have an ensemble of such particles. So at each given moment, instant of time, there is a given, there is some probability to find one specific electric field. For another radiator, at the same time, there will be a different field because the combination of its neighbors will be different. At first, such a distribution was derived by Johann Holzmark in 1919 and it's expressed if one neglects interaction between different particles that it can be expressed in a nice analytic form. One usually use not absolute value of the electric field but normalize it to constant which is called the Holzmark normal field. If one forgets about this numerical pre-factor, the rest is actually obvious. If you want to know what is the typical electric field, you need to know the typical distance to the perturber which is density to the power minus third. That's the typical interparticle distance. And then what you see here is just Coulomb law expressed in this unit. You have a charge of the perturber and you divide it over r squared. So it's just Coulomb law and we get a typical electric field. The specific numerical factor is really not important. It comes from some mathematical considerations. So here I plot this Holzmark distribution in the units of this normalized electric field strength. So you see that it peaks at about a value just a bit smaller than 2. Actually it's 1.6 something and important to notice right now is the asymptotics of this distribution. For the weak fields it is quadratic. In the very strong fields its dependence is the field in the power minus 2.5. Actually this dependence is very easy. You can just derive it with one line expression on the back of envelope and just assume just a single perturber. It's in binary approximation. Again it's just the Coulomb law. Just express the radius distance to the back express distance from the field using the Coulomb law and you immediately get the asymptotics. Two notes. One is of course there is nothing special about electric field for gravitation. It's exactly the same Coulomb law and the distribution of one doesn't say micro fields in astrophysics of course but the distribution of gravitational fields by random distribution of stars follows exactly the same law and there is a very nice paper at Chandrasekhar and von Neyman from 42 which of course they quote Holzmark but they continue this to quite great length. It's very interesting paper by the way. The fact that it's the same distribution for plasma micro fields and astrophysical star fields, gravitational star fields either produces some confusion. Well because first of all Chandrasekhar and von Neyman are much more known to average physicists and especially not physicists than Holzmark. So I've had many times people even in the plasma spectroscopy community they're saying oh of course but Holzmark first derived it for gravitational fields. Not only that I just recently saw a large textbook and in the introduction it was written there was a sentence something like that in 1919 a famous Danish astronomer discovered distribution of gravitational fields. It's a textbook so there are three mistakes in one sentence. He is not Danish, he is Norwegian, he is not astronomer, he is plasma physicist and not gravitational fields he considered but plasma fields. By the way Holzmark was a postdoc of Peter Dubai so good school. Okay let's talk a bit about stark effect about static stark effect. We will consider a simplest two-level system that you all learn in the introductory course of quantum mechanics and actually you must know all this of course by heart but let me repeat to show you that there is nothing very difficult here. So the perturbation due to the electric field is just the dipole electric moment times electric field right we will assume the field which is directed say along the z direction so in the two-level Hamiltonian that's of course will be off diagonal terms because dipole because of symmetry properties cannot be must be zero between the same diagonal elements in the same way functions. Okay so it's only off diagonal terms now we add the unperturbed Hamiltonian with just diagonal unperturbed values of energies e1 and e2 both zero labeled zero now we want to find eigenvalues so we demand the determinant of h Hamiltonian minus e is zero and we immediately find that's a simple quadratic equation roots of the quadratic equations like that I've plotted the dependence of energies of both e1 and e2 levels as a function of electric field normalized to some value to obvious value right you will see it here asymptotics when the field in the weak field limit when the field is much smaller than the initial when the perturbation is much smaller than the initial split splitting between the levels for whatever reason it is the effect is quadratic in the strong field limit when the perturbation is much stronger than the initial splitting the effect becomes linear and you see it's here and I plotted asymptotics both for the weak the parabola and the asymptotics for the strong field limit which are just the straight lines in general so when people say that this level of this line has a quadratic or linear stark effect one should understand that's always a question of how strong a field is because in principle there are no fully degenerate levels there is there is always some distance between the levels at least between the level that talk to each other via dipole electric dipole interaction right even if you consider the simplest hydrogen neutral hydrogen one say okay all levels with principle quantum numbers have the same energy but we know if you take into account relativistic effects so you get fine structure and there is already split in between levels then okay even in that approximation to s say one half and to p one half have the same energy but then quantum electrodynamics corrections gives you the the lamp shift so in in practice there is there are no fully degenerate atomic systems so it is always the question how strong a field is whether you can talk about quadratic or linear effect should be kept in mind by the way when you say when one says linear effect it's also not exactly true because well you can continue expanding this expression in Taylor series and you will see that will be only even powers of electric field there is no linear truly linear term okay okay that's a stark effect of the simplest atomic transition which is limon alpha that's transition from n equal to n equal one in hydrogen or hydrogen like atom ion so the static stark effect is just a splitting when you have three components now we will talk assuming the effect is linear that's neglecting all fine structure and lamp shifts and whatever okay so there are three components a central component and there are side components which are split shifted proportional to the electric field so that the static picture have three components now when there is a distribution of of electric field like a micro field distribution and plasma instead of one single position which is in terms of line profile it's a delta function now we have a distribution of such an infinite number of delta functions but practically what we see of course it's just a smooth wing both wings red and blue wing okay the central component remains in this simple approximation and shifted which is of course not true but that's okay to demonstrate how the quasi static broadening works we next want to consider time dependent perturbation in general it doesn't matter what kind of what kind of phenomena we are talking about if something changing in time one can define what's called spectrum power spectrum of this entity okay for that if there is some entity of physical quantity evolving with time one takes Fourier transformative takes absolute value raises to the second power and that's a power spectrum okay so I just wrote it wrote here implicitly Fourier transform in the case of dipole radiation the electric field as you know is proportional to the dipole moment of radiator that's true also in classical mechanics and quantum mechanics quantum electrodynamics as well okay so we do not want to to to take with us all the numerical proportionality coefficient let's just focus on the main important proportionality so the electric field as observed by some distant observer is proportional to the dipole electric dipole moment of the radiator okay so we are following just this prescription now the dipole is oscillating okay because that's the in well that's omega is just the energy difference between the initial and final state let's consider natural broadening so we have a relative decay from state i to state f and the characteristic time of course is the Einstein coefficient if we neglect we do not consider induced radiation decay rate is just the inverse of the Einstein coefficient sorry it's proportional of course so when we say a state decays with the rate gamma it means that the wave function which describes the upper state absolute value squared decaying like exponent to the minus gamma t which means the wave function itself must follow that time dependence and we can neglect the any time dependence of the ground state assuming it's ground of the lower state assuming it's a ground state so that can be written as the wave function of the of the upper state and I introduced here what's called Helicite Helicite in step function that means that for time before zero it is constant it's radiate somehow nothing prevents us to radiate and then we start to decay so taking Fourier transform of this expression with just a simple exercise you get that now we are taking absolute value squared and what we see up to a constant Lorentzian line shape right if there are by the way some questions just ask right in the middle don't wait until the end of the two hours okay some examples of Lorentzians with different also standard deviations contrary to Gaussians profile that we saw before Lorentzian has very long tails long wings okay there could be the often our situation when a line is brought in by different mechanism one is has a Lorentzian line shape and another Gaussian for example Lorentzian could be natural line shape and natural broadening and there is a Doppler broadening and one needs to do is a convolve both line shapes assuming they are independent of course so the convolution of Lorentzian and Gaussian is a is what's called the Void profile Void distribution Void function let me first show you some examples so the actually the both Lorentzian and Gaussian is extreme limiting cases of Void profile for example here Sigma that's Gaussian part is one and Lorentzian broadening is zero you have just a normal Gaussian in the opposite case there is no Gaussian contribution only Lorentzian we have well we have Lorentzian and that's a convolution example of convolution when both contribute let's say equally and of course you have a broader profile important point here that the wings are Lorentzian wings so Gaussian broadening contributes to extra broadening in the core of the line but the wings remains the same as they were just due to the whatever phenomenon that caused Lorentzian line shape go back and that's for practical terms there is a convenient expression when you know the contribution the Gaussian and Lorentzian contributions so full with half marks of each one to get a full with half marks of the total Void profile just use this simple expression and you get with pretty good accuracy typically about one to percent okay now we want to do some more mathematical exercise but I promise to it be quite straightforward and again please now this is something which you will see if you ever want to see some theoretical work you will see the expression that we will derive in the end of five minutes I guess so please if something is not clear just ask right now I think it is important to understand it's simple stuff but even in textbook sometimes people just start from the final expression because it was derived you know 50 years ago in some another textbook or paper and then continue on and you don't really understand what the roots of this equation number one okay so that's we saw if we want to calculate intensity line shape of dipole we will talk for now only about dipole emission dipole radiation so we take dipole momentum take Fourier transform of it at a frequency omega the same omega absolute value squared nice and that's again Fourier transform okay now we can use cross correlation theorem that I believe you you've had about it and learn and maybe even proved it on a some exam so what it says that product of Fourier transforms of two functions is up to a normalization factor which is different for different by the way for different Fourier transforms some put here one over square root of 2 pi let's forget about this fine unnecessary details so again the product of two Fourier transforms of two functions is just Fourier transform of product of a convolution of this function okay so we can express this value Fourier transform of of our dipole function okay in our case by the way both both f and g are the same so instead of correlation we are talking about autocorrelation okay so c is just the correlation function right that's the definition of convolution of two functions that's exactly the definition of convolution or correlation up to a minus sign okay so f in my case and g in my case are the same d as a function of time so that's just autocorrelation function in if I take its Fourier transform according to this cross correlation theorem I must get Fourier transform of if the same function squared that's exactly what I want is it okay at this point I can repeat it on explain with more details if it's okay let's go on of course correlation autocorrelation function is symmetric so instead of just take an integral from minus infinity to plus I can do it from zero to plus infinity and take only the real part okay that should be well next that was for a single radiator but plasma as we know contains a practically infinite number of radiators so we need to average our ensemble of plasma radiators so we let's designate it by this average symbol and of course averaging taking averaging or integral commuting operations so I move this averaging stuff inside of the integral and I want just to average the the correlation autocorrelation function over the same ensemble now if you actually look at what is autocorrelation function that's this integral but integral is basically a sum infinite sum of pairs like that now one can say why do I need to do this operation twice if we believe in the ergodicity arguments that's the time average is equal to ensemble average we can substitute time average with an ensemble average of vice-versa that's actually very important point but it should be quite simple let's say we want to do some statistical experiment what is the probability what is the distribution of getting different values when you throw dice okay let's do the experiment I can give each of you one dice and each of you will throw it up and get some value then we can combine and calculate average standard deviation or whatever other statistical property alternatively I may pick up just one person out for example Yuri Ralschenko who is not listening to me give him the same one dice and ask to do it we are now 60 people about here so to do this this operation 60 times and again write down and do the same average and standard deviation and of course we will get the same well within statistical errors the same results well that's the error could be significant when we are talking about 60 people but when we are talking about billions of atoms in even small amount of plasma that's the averaging on time and averaging on ensemble should be exactly the same so what I say that in this infinite sum if I talk let's take just one one term for example tau is equal one million and consider correlation term for t equal to whatever units so what is this term for one radiator so correlation between time equal one million and one million and two I claim that I can find on the same term in the similar summation when I will do on some another radiator it will not be at t equal tau equal million it will be tau equal whatever other numbers but when I have infinite number of radiators to play with I'm sure I will find another one which gives for the same difference tau and tau plus t will give exactly the same result that just statistics okay so if so why do I need to take with me all this infinite sum let's restrict myself to just one single value of of tau and let's take tau equal zero why not so I can write if I do the average I can do it average over just a single term in this infinite sum or in this integral right so it's dipole moment at time zero times dipole moment at time t and I am going to of course average over ensemble of infinite ensemble of statistically representative ensemble of emitters okay so next as you remember once we have the autocorrelation function we need we take its Fourier transform and that's all that the line intensity that's the the line profile okay so we've done up to now we considered just a single dipole but in principle of course one atom can radiate a different frequency there are different transitions so instead of talking about dipole just dipole momentum one's talk about operator of the dipole momentum but the rest is exactly the same in addition we need to take into account probable distribution of population on different states of the radiator for example it could be Bolshevian but it could be any of course so that's a dense thematic throw so we multiply everything and take trace because we have all many possible transition even in single atom and that's again the same autocorrelation function an important note here is that you sometimes I would say even often see in the literature line shape broadening in literature expression like that when you the autocorrelation function is just this term without the important averaging over ensemble so of course in textbook even in textbook sometimes but in papers when people write it they understand what they are doing they assume that it is all is done in the context of averaging over an ensemble afterwards but it's not always clear for new cameras let's say to the field so please pay attention because obviously such an expression is invalid because what's that it's just it's a constant and that some function in principle arbitrary function so when you take Fourier transform of an arbitrary function you can get anything including negative values which is of course a nonsense for line profile but of course when you take this line profile which contains negative value and do averaging over ensemble there is no negative values anymore but please keep that important point in mind another note is that sometimes this autocorrelation this dipole dipole autocorrelation function is called autocorrelation function of the light amplitude or light magnitude and in fact I believe that a more correct name for this value because the transition is not necessarily dipole we can consider we can use exactly the same derivation for example quadrupole radiation in which case of course there will be no dipole dipole autocorrelation function there will be quadrupole quadrupole whatever autocorrelation function but in terms of light of light magnitude amplitude it's the same so I believe physically it's more correct way to call it okay so that's the final expression of the formal theory of line broadening and it looks very simple right but as it's always in the case in physics the simplest equations are most difficult to solve so what we see here is time evolution of dipole momentum we are still talking about dipole radiation okay and we have dense thematics and we have operation over over of averaging over ensemble and all of these three simple operations are actually infinitely complex why because plasma is basically is dynamics time evolution of its n body where n goes to infinity number of interacting particles each one may serve both as a perturber and as an emitter okay and all interact on each other there is also back and from from radiator to the bus which is the rest of the plasma particles because there is no rest all particles both perturber and radiator in general okay so in the remaining a few minutes till the break I want to show you some movie well this is the result of computer simulation of molecular dynamics of plasma particles okay what you see in this short movie is a life of plasma during just one picosecond I'm plotting here projection of of electric field onto some arbitrarily selected axis the color the color designates all designates absolute value of the direction of the projection and the intensity of the color is the the magnitude of the field okay and that's we are taking just very small volume inside of plasma so you see by the way it looks like something like biological stuff now we see why why why plasma in physics is called after plasma the blood plasma okay it looks like that so you see here the result of motion of different particles and some are obviously are very slow for example here right it's almost not moving but if you wait long enough you will see that slowly it does move and the field changes so this slow points are of course ions and the fast moving stuff feels due to electrons so I guess you realize that's quite a complex dynamics with very complex interaction between different plasma particles so coming back to this simple expression so coming back to this simple expression which is the basically the describe the line broadening is not easy okay so we will continue after the break