 Okay. Hello, hello. Okay, let's start. So what I'm going to talk about is certainly not detailed introduction to atomic structure theory or not an attempt at that. But I'll just show you a few things that may be important for plasma spectroscopy in general, and I hope we'll have time to look at some online tools, in particular the atomic structure database that we maintain and develop at NISDAT is the major source of evaluated atomic structure and spectra data. It's probably the only database in the world that contains truly evaluated atomic spectroscopy data, and also we will look at online interface that develop at Los Alamos that allows you to quickly calculate wave functions, energies, oscillator strength, and so on and so forth. So certainly we understand why atomic structure is important. We know that whenever we're talking about atomic physics there is always some kind of matrix element that we have to calculate or analyze, and of course we have initial state characterized by a set of quantum numbers of various nature. The final state characterized by a different, maybe the same, when function, then operator in between. So whenever we're talking about calculations of wavelengths, energies, transition probabilities doesn't matter, radiative or non-radiative, like organization transition probabilities that Stefan Fritsch will be talking about tomorrow. Collisional cross-sections or anything else that may be relevant for instance, amplitude, we always are facing these metric elements where of course these and these are characteristics of particular atomic states. So let me start with before going into atomic structure, let me mention a few text books generally related to atomic processes in plasmas and some of them are more on atomic structure. So probably plasma spectroscopy as a field of plasma science originated already more than 50 years ago when Hans Krim published his famous text book on plasma spectroscopy back in 1964. Since then he published another book, Principles of Plasmas Spectroscopy in 1967, but this is when it started and still this book remains probably one of the most authoritative sources on information and everything related to plasma spectroscopy. This list is in chronological order, so in 81 Bob Cowan published a fundamental very thick, extremely detailed and a pleasure to read book that's called Theoretomic Structure in Spectra and many of you have heard about Cowan's code, it's very well described in this book. Then there are a few other books with very similar names, atomic physics for hot plasmas, atomic physics in hot plasmas and atomic properties in hot plasmas, published by different authors, but each of these is also very, very interesting and detail. In particular, I would like to mention this last book that contains many aspects of plasma spectroscopy developed over the recent probably 20 years, maybe 30 years or so, that are not so much touched in any of the other books. Then Fujimoto's Plasmas Spectroscopy text book contains a lot on collision-related modeling in particular. Then we saw today in Professor Kunze's talk references to his extremely readable introduction to plasma spectroscopy. Of course, you all are pretty much into plasma spectroscopy, but if you were just novices, this would be the book to start with, absolutely. Last year, a bunch of lecturers here, Hume, Hoverscod and myself, we compiled together a few chapters in the book that is called Modern Method and Collision Reliative Model of Plasmas, a little bit advanced, but still it gives you up-to-date information of what happens in collision-related model. Units. One of the mostly used units in the atomic spectroscopy and atomic structure is inverse centimeter, which is related to one Riedberg through this 10 to the 5th coefficient, and one EV is about 8,000 inverse centimeters. You will see both of these, mainly EV and inverse centimeter later on. Of course, the typical unit of lens is Bohr radius, which is just radius of hydrogen atom, and accordingly the typical area is the cross-section, which is pi zero squared, about 10 to the minus 16, and this is the characteristic size of collisional cross-sections for neutral atoms. Now, a lot of information on units, in general, you can find at the NIST website, I hope you can see it says, physics.nist.gov.cu.u.slash units with capital U, but I'd like to mention that there is a common redefinition of the international system of units that will most likely be approved in the next year, 2018. We know that right now the seven fundamental units of SI based on different, some of them based on artifacts. For instance, still one kilogram is a little piece of platinum meridium that sits somewhere on the ground in Paris. But the problem with this thing is that its mass is not constant. It loses it a little bit, which is natural. So the new international system will be completely redefined in terms of fundamental constants. So right now we all know that the speed of light is a fixed value, 299792458. Some of the fundamental constants will also be given exact values, and from them the whole system of units will be derived. Now, this is the latest addition of the periodic table that we published at NIST. It still has old names for the four heaviest elements and the names were signed just a couple of months ago, but at least all numericals in the tables are supposed to be good, so please take it and use it. Okay, so what kind of atomic structure are we generally interested in in plasma spectroscopy? Simple answer is more or less everything. There are some models that use notion of an average atom characterized by a bunch of electrons sitting together around the nucleus. For instance, if we're looking at 16 electron ion, which would be sulfur like ion. This is very picturesque representation of average atom, but if you want to look at more details, you may go with the super configuration representation, and we'll talk about super configurations as well a little bit, where you simply specify how many electrons sit in different principle, in different shell principle quantum numbers. If you're interested in even more detailed representation, you may use configuration. So for instance, this super configuration may consist of bunch of configuration where you use NL distributions for your electrons. Then certainly you may want to go even the next level represented by atomic terms characterized by total spin and total angle of momentum values, which you're certainly familiar with. And of course, you may go all the way down to the fundamental level, atomic levels representation characterized by J values. Of course, the story doesn't end here if you have electrocomagnetic fields. You go to components, but the point here is that generally, you may be interested in different level of representation for atomic structure. And this tells you, for instance, that there is no universal collision-related model that can be used in all cases. Simply in, well, you can't build it in terms of atomic levels, but it may become so huge that no computer in the world can work with it. Now, fortunately, modern nature gave us hydrogen and hydrogen-like ions, exactly solvable problem in quantum mechanics, with which we can learn so much about atomic structure in general, the scalings, how far electrons sit from the nuclear and so on and so forth. Unfortunately, there is no hydrogen atom for plasmas. Each plasma is unique and specific, and if you can solve more or less your problem for one plasma, well, unless you're working with something extremely similar, you can hardly apply this knowledge directly to some other plasma cases. But hydrogen atom certainly gives us a lot regarding the atomic structure of all atoms and ions. And of course, we know that moving from hydrogen atom where the ground state sits at one ring below the ionization potential, the energy scales as one over n squared. The same structure is more or less valid for any hydrogen-like ion that is ion with electron. Again, in units of energy z squared read work, the first, the ground state sits at one, this two sits at one fourth and so on and so forth. We understand how radius of an electron orbit changes with principal quantum number goes like n squared. We understand that the orbit's collapse is one over z with the increase of ion charge. We know how the energy scales and this information remains very, very important for future analysis of other atoms and ions. Now, we know that in general atomic states are characterized by number of quantum numbers, but there are only, probably it's fair to say that only two that are exact. One is total angle momentum and the other is relative. Of course, they have energy of the state, which is just a number. But as far as discrete numbers are concerned, only two are truly exact and their conservation of causes comes from the most fundamental properties of our world. Everything else, including total angle momentum, total spin is really not exact. We will see, of course, the cases when atomic state is a mix of different L's or S's. Now, when we're moving from hydrogen atom to complex atoms, generally, it's fair to say that we probably know all important interactions that enter our Hamiltonian. And, of course, we have a kinetic pattern here. I just use atomic units, emitting all H bars and so on and so forth, which is just a lesson here. Then electrons interact with the nucleus with the Coulomb potential. Of course, I hear enumerates different electrons in the atom. Of course, electrons interact between them again with the Coulomb law. Then we know that there is spin orbit interaction that basically can be easily described when you move to the reference frame of the electron that sees nucleus charge induced magnetic field due to movement of the nucleus charge and some other corrections that I will omit. They're not so much important for this story. Of course, with this Hamiltonian, we want to solve the Schrodinger equation. And, of course, we know that we cannot solve it exactly. Therefore, some approximation must be introduced. And the mostly used approximation is the so-called central field approximation that basically tries to mimic the effects of Coulomb repulsion among the electrons. So, basically, you start with the modified Hamiltonian, of course, keeping the kinetic part, interaction with nucleus, and then we assume that each electron, again, some here goes from over all electrons, moves in a potential independently of all other electrons. Now, the problem, of course, is to probably choose the potential. And here we can find different methods and techniques, including Hartree-Forg, Thomas Fermi, some types of model potentials. But in any case, almost all realistic approaches to calculation of atomic structure in multi-electron systems starts with some kind of central field approximation. Then again, for non-rotivistic case, you find configuration state functions for this Hamiltonian characterized by total Ls and other quantum numbers. Of course, you must take into account the anti-symmetry of your wave function because we know of the poly-exclusion principle that is valid for electrons. So, these configuration states are characterized by the principal quantum number and orbital angle momentum. And now energy depends on both. For hydrogen, we remember that simply due to accidental degeneracy of clone potential, there's no dependence on L, but here there is. Then we assume that the atomic state function is a linear combination of the configuration state function. And this is the final atomic state function that we want to find. We represent it as a sum of all this found configuration function with some weighting coefficients. And then, of course, the sum goes not over the electrons, but number of the configuration states that we choose as a basis. Then you build Schrodinger equation for this new basis, eventually coming to a solution for the mixing coefficient. And as soon as you find the mixing coefficient and you know the configuration state functions, you know your atomic state function. And then you can try to do calculation of energies, oscillators, transfer, creative transition probabilities and so on and so forth. Of course, you must include other effects primarily through perturbation theory like spin orbit and others if you want really good accuracy. But this is standard procedure that is followed in many, many codes. When you move from non-relativistic theory to relativistic theory especially if you're dealing with heavy ions and actually for not too heavy because relativistic approach certainly is much more general than non-relativistic which should follow from this naturally. Here the Hamiltonian, of course, looks slightly different because you start not from Schrodinger equation, but from Dirac equation and therefore you have all these terms that are built with the standard Dirac matrices. In calculations for relativistic atomic theory it is generally important to include nuclear charge not as a point charge, but as some extended distribution and therefore you have this potential. And, of course, you have the Coulomb interaction 1 over R again. Of course, in higher orders you must include corrections to the Coulomb potential and this was done by Bright and therefore this particular additional term that is then added to Dirac Coulomb Hamiltonian is normally called generalized Bright which includes magnetic interaction and retardation effects. Then, again, especially if you're working with really heavy elements, it's good to add quantum electrodynamic effects like self-energy, vacuum polarization which are standard things that are taught in QED classes and then you combine all this together and this is new Hamiltonian that you want to work with and there are codes I'll mention in a minute that basically do all relativistic atomic structure very well and Stefan Friedrich will be tomorrow talking about grasp and rate if that use relativistic approach to analysis of atomic structure. Now, because relativistic notations are used not so frequently generally in the literature as non-relativistic LS notations, I just want to mention that generally in this notation use J value of course because S and L couple together to J and normally the J value comes on the right-low side of the L. So if you have S electron it's the only J value of course is one half you add zero and spin one half you get one half then for P you have two options one half three halves and very often you use P minus P plus to designate them and so on for other electrons D F G and so on and so forth. So which method and codes are available and which are used nowadays? There's another approximation developed by Bates and Darmgart in the late 40s basically you try to again mimic your Coulomb interaction introducing effective principle quantum numbers probably nobody uses it today but still some codes are available and in simple cases this this is this this approximation should work. There are some codes based on Thomas Fermi statistical approach and there's superstructure and then auto structure which is development of superstructure that have Thomas Fermi option as option to calculate this central field potential used at the first step. Then you can certainly use different kinds of Hartree-Fock methods single configuration which is used in COUNS code but then COUNS code doesn't stop with this it uses configuration interaction that we'll talk about a little bit later and COUNS code has online interface that I'll show you in the second in the second lecture. There are few codes that use model potentials that basically based on some knowledge and of how what what the potential of atom is there were some recommendations were developed to use some artificial potentials to describe what electrons in different ions produce as a center of central field and most widely used codes for relativistic model potentials are Hewlett and flexible atomic code. Evgeny Stambulchik will be given lecture on on FAC implementation. Then there are a few codes that use multi-configuration approach. You remember here oops we talk about this expansion and then in this procedure use Schrodinger equation just to find the mixing coefficients assuming that the configuration state functions have already been determined. In the multi-configuration approach both which actually is variational approach both mixing coefficients and the wave functions are varied and probably it's fair to say that for general atomic systems multi-electron atoms and ions multi-configuration methods provide the best accuracy. There are non relativistic htfoc codes and you can go to this site nlte.news.gov.mchf and you can download multi-configuration htfoc code developed by Charlotte Fraser Fisher. We also have their multi-configuration Dirac folk codes which are fully relativistic. Grasp2k and then there's another code developed by Dick Law in France also multi-configuration Dirac folk and there are different flavors of grasp. Original grasp, grasp96, grasp2k and relative Stefan's code is based on grasp. So these multi-configuration type codes are extremely widely used in research. Then there are various perturbation theory method that probably best work when you have just one, two electron outside of closed cells but they can be really very powerful and recently there are new methods developed based on B-splines that also take ideas of multi-configuration htfoc or Dirac folk as the basis and then build everything on B-splines. Now one of the important ideas about atomic structure is idea of how everything scales when you change the ion charge more or less keeping the same number of electrons because we know that this is called isoelectronic sequence when we keep the same number of electrons but simply change the charge of the nucleus. Now the primary characteristic is actually not the ion charge here but rather ion charge plus one which is normally called spectroscopic charge and of course you know notation like h1, argon 15 and 15 and 1 are the spectroscopic charges. So this spectroscopic charge is actually what is seen by outermost valence electron and therefore at large distances certainly this electron what it sees nucleus some electrons here so altogether it's more or less like Coulomb field and or better to say more or less like hydrogenic ion and this is exactly where the main part of this scaling comes from. So in principle you can derive spectroscopic charge scaling for one electron energies in non relativistic case and the leading term is E0 where E0 is nothing but hydrogenic term 1 over n squared and here you have z squared so of course this is this is purely hydrogenic ion term but then you have of course the other terms lower powers of zc that describe different effects for instance this term is mainly telling you about the terms splitting within your n shell and of course relativistic effect slightly modified this depends but general trend remains valid that you go to higher ion charges of course zc is practically the same and this term becomes more and more important compared to all other terms which means that going from low on charges to high on charges your atomic structure becomes more and more hydrogenic so if you have levels of terms for the same principal quantum number or different quantum numbers that even may overlap you increase the charge and everything starts shrinking into groups of elements that look very much like hydrogenic systems so from this it follows that the whole energy structure of an ion has part that is related to primarily electron nucleus interaction with energy splitting small less proportional to z squared then we have electron electron interactions proportional to z that are responsible for terms and then we have spin orbit splitting that gives levels from terms if we're talking about non relativistic LS coupling and this is where we start from so how can we easily see that indeed going along isotonic sequence we become more and more hydrogenic well I'll show you in the second lecture the interface to the NIST atomic spectrum database but one of the things that we do have there is this ability to build grotesque diagrams that simply show you the positions of all levels that we have in the database for a particular ion the x-axis simply denotes here different configuration or different series of levels in this case so if we look at singly ionized aluminum which is magnesium like because of course it has 12 electrons like magnesium and let's find out where the levels belonging to configurations where both electrons have unequal 3R so what we have in the database is of course the ground state 3S squared then these two groups of level there are more than one here belong to 3S 3P these two are 3P squared then we have 3S 3D then we have 3P 3D and of course there must be also 3D 3D but they are sitting above ionization potential and then actually for aluminum to those levels have not been determined experimentally as of today so you see that three three levels are distributed all over the whole range of energies from the ground state all the way to the ionization potential this purple line shows you the position of the ionization potential for aluminum 2 and these are units of inverse centimeters now if we look at another magnesium like ion but with much higher charge and try to find the same levels we'll find that they're sitting much much closer to the ground state with regard to the ionization potential so now we're looking at 26 times ionized transom again magnesium like and these levels are more or less the same 3S squared 3S 3P 3P squared 3S 3D 3P 3D and even 3D squared and you see they're all compressed which means that the energy differences between all the signs they do increase but not as fast as ionization potential which increases as z squared now spin orbit interaction is certainly one of the most interesting thing in atomic structure for hydrogenic ion the parameter that characterizes it is proportional to the fourth power of the nuclear charge so it's really very strong effect and it's inversely proportional to the third power of the principal quantum number so it dies out quickly with increase of n and of course you see that the l value it also decreases with with l for hydrogenic if you move from hydrogenic ion to general atom there is no simple formula like this one that is derived from Dirac equation you can use some semi theoretical one the formula where which looks more or less the same but there are some difference of course the fine structure constant is the same indeed and here is replaced by effective principal quantum number which is derived through hydrogenic formula from the energy of this particular state that we're trying to characterize L is of course the same but then Z to the fourth is split into two parts one part is the spectroscopic charge or effective charge that the electron sees and the other is effective nuclear charge and which slightly changes depending on whether the orbit of your electron penetrates much or it doesn't so for instance for NP orbitals you can replace Z tilde with a nuclear charge minus L and then this is this spin orbit parameter enters our Hamiltonian right here and can be used to calculate the energy splitting through perturbation here for instance so there are different types of couplings that you certainly heard about LS coupling JJ some other intermediate and more or less the whole story is based between comparison of Coulomb interaction between electrons and spin orbit interaction of electrons for particular electron so if electron electron interaction is much stronger than spin orbit then this is what we have for LS coupling which is typical for light elements in that case you get the total angular momentum just summing up all else of particular electrons the same for the total spin and your final total or total angular momentum J is simply some of two vectors and for JJ coupling the situation is inverse spin orbit is much stronger than electron electron interaction therefore you must start with taking each electron and summing up its L and S value into a J and you do this for all electrons of course not those that are in the closed shell because in closed shells everything is zeroed out you just forget about closed shell for for your angular momentum consideration and final J is of course the sum of total J so you're used to LS coupling notation like singlet P or triplet D or something in JJ coupling slightly different notations I use to use those relativistic thin that I showed you before for instance to S to P would be represented as to S one half to P one half or to S to P minus for one half if there were three halves here then here you would put plus if you have more complex configuration like 3d5 you may split this into dependent on really how the quantum numbers are calculated and so on so forth now of course these are extreme cases one one of the types of interaction is much stronger than the other but of course you may meet cases when neither of these is stronger and this is what is normally referred to as the intermediate coupling what is the energy structure for different types of coupling for LS let's consider for instance simplest case of SP configuration so we know that if it's an LS coupling then electrostatic is stronger than spin orbit and naturally your SP is split by electrostatic into terms so you have only one possible value of L which is P of course you add 0 and 1 get only 1 and then you have singlet and triplet triplet lights below than singlet and this is the Hoon's rule that we will talk about again and then with spin orbit you split all these terms into the levels and of course if spin orbit interaction is not too strong this splitting's would not be very large compared to the difference in energies of these two terms and of course if you put your system in magnetic field your J values which is 0 1 2 here will split into into simplest states so this is the standard terminologies J values are levels these are terms and these M sub levels are normally called states going to JJ coupling you get the same number of levels simply you get them slightly differently you have SP splitting by spin orbit of the P electron into two JJ terms again remember that here one half is related to S three halves is related to P and then through electrostatic interaction these two summed up so you take J of one electron one half J of the other electron three halves sum them up get value of 1 and 2 and the same here you get value of 0 and 1 and finally the states are produced if you put the same system into a magnetic field now looking at this simple structure you can ask a natural question can we find some simple formulas that can describe what happens here and the answer is yes for for simpler system if you forget about everything else you can write simple formulas there are few very nice books that describe the details starting from classical continent shortly theory of atomic spectra of 1935 counts book has a lot of it so basically the non-central part of the Coulomb potential can generally be represented as some of two types of contributions one is related to direct Coulomb interaction and the other is directed to exchange Coulomb interaction with certainly is specific for quantum mechanics not for classical mechanics these two parameters which are called slatter integrals introduced through with factors excuse me the here it should be G G so F F G and lowcase F and G are simply numbers that can be derived from angular and algebra so if we are looking at situation when we have one S electron and another electron that has orbital angle momentum L we can write formulas that describe how the levels will be split so F zero is direct Coulomb slatter integral G L in just one in this series the series is not infinite it ends up depending on quantum numbers and then you have contribution from the spin orbit now let's see what happens if spin orbit becomes zero that is we only expected to have singlet and triple triplet term and all levels in triplet will have the same energy so if zeta spin orbit parameter goes to zero this becomes F minus G this becomes F minus G and triplet level 3 L L which corresponds to the lower sine becomes let's see zero zero zero square root of G square G also becomes F zero minus G so from here you see that at least this formulas really give us what we expect to have for such simple system and the last transparency before we end up for break as we already know the spin orbit very strongly depends on on iron charge so we can expect that if for lower members of iso electronic sequence with low iron charges spin orbit is small and therefore we have a less coupling for higher members with high iron charges because of such transmit orbit we may expect transition from LS to JJ and indeed this is exactly what what happens what you see here is the energy structure of the 1s to P configuration in helium like ion alone practically all sequence of nuclear charges from neutral helium to probably ending with tungsten which is 74 now the energies are rescaled so what you see here the energy of the highest level which is singlet P1 for low members of the sequence is one and it kept one everywhere and energies of the triplet levels are rescaled keeping J equal zero at zero all the time what we see at lower members of the other electronic sequence there is a singlet there's a triplet and all triplets sitting together basically this means that this is indeed a less coupling the electrostatic interaction splits terms far and spin orbit is too small to produce something more than just tiny split in here now we go to high as these and immediately you see that the whole structure changes we have two levels here doublet two levels here doublet and they exactly correspond to what we would expect for JJ coupling pictures as we saw previously so let me stop at this point we continue in 10 or 20 minutes 10 minutes so we'll back at 2.15