 Having seen how this coupling occurs in practice, we are in a very favorable situation that we can now discuss phenomenologically a number of processes which all based on this coupling. In addition, we will see that we, of course, we have also something like electron-electron coupling, which leads to the originization. Just if we go towards more inertial processes, then, of course, we know that the frequencies of the transition increases. We go towards x-rays. And the main difference is here, if we talk about excitation processes and decay processes, the notation that one talks then instead of a 1s or 2s2p electron from k-shell electrons, l-shell electrons, you are aware on this topic. In particular, if you observe after inner shell, then often you have these so-called characteristic lines, which are associated with k-alpha. You see it here from either the l-shell or the m-shell k-alpha k-vita decay, which is on top of a bremsstrahlung continuum, basically a scattering of the photons within the continuum. What is characteristic in this case, that one uses a proper notation and for the photo-excitation and subsequent fluorescence? This is given simply by this line that an photon excites, interacts with an atom. It leads to an excited atom, a star, and which subsequently decays under the emission of another photon on either the same or different frequencies. And of course, it may remain with some excitation and may lead later to further. One can also discuss with regard to polarization, but I will not go into the details, just typically. If you make angle-resolved type of measurements, then you need, of course, also to include angular distribution and the polarization properties of the light into account. In order to characterize this transition, you are all aware of something like k-alpha, k-alpha 1, k-alpha 2 transitions, k-vita. But if you go down, I guess, as I by myself would have to look it up in most of you, likely also. The more obvious notation is this IUPAC notation, which simply tells between which of the shells you have a transition. So basically, in L3, which means 2p, 3 half, 2p, 3 half going down to k refers to a k-alpha transition. And you see it also for the L and for the m type of transitions that you always can associate the advantage here is that we will see a very similar notation later for the outerization, like a KLL or Shea spectra means that one of the L-shell electron fills the K-shell hole under the emission of another L-shell electron. If we talk about characteristic X-ray radiation, then one should give both the notation with respect to the X-rays as well as sometimes also explicitly which type of transition occurs here. If we have, for instance, an X-ray fluorescence following electron impact, then again, of course, here, the excitation is different. It will also lead to a different occupation, typically of the magnetic substates with regard to this excited system. And therefore, also with regards to the angular and polarization properties of emitted fluorescence, but in this simplified notation, we would simply have the situation that due to this electron impact, excitation or ionization process, we get an electron with different energy, but eventually, of course, also this fluorescence photon where we found back what is called Mosley's law, basically, that becomes hydrogenic if we go deeper in the core. This is the same message which Yuri gave yesterday, that if you go either deeper in the core or if you go to higher charged ions, in both case, the system becomes more hydrogenic. And people have used this in order to characterize a different characteristic X-ray absorption to make spectroscopy, which are specially devoted to this type of edges, like exafts or xanas, which helps you, basically, to understand further detail. I want to focus myself on the central processes. And the next one here is Rayleigh and Compton scattering. This means the elastic or the inelastic scattering of photons, typically, hard photons, as of UV or X-ray photons, sometimes also gamma ray photons on atoms and ions. And in this case, the process is a bit hidden. You need, in order to describe these processes, second order perturbations here, you have to sum once over the complete spectrum. So this makes the treatment of this process much, much harder. I will not explain this from a technical and mathematical viewpoint, but from a phenomenological viewpoint. What you should keep in mind is, of course, this competition. If you increase, for instance, the incident photon energy, you see that, basically, the photoelectric, the photo ionization process for a long range up to 10 to the 4, perhaps even 10 to the 5 here, it is clearly dominating. Of course, it depends also a little bit on the element, on the particular element, which you consider over something like the Compton scattering or even the pair production, which, of course, can only start if you have photon energies which are sufficient in order to create electron proton pairs. So you need to beyond 1 MeV, 511 KeV. You need, in order to create the rest, mass for a single electron or positron. So you need twice this energy as the threshold energy to create an electron photon pair. But you see here, this interaction as this competition of the different processes for a long time, photoelectric effect is a dominant one. But if you go now really in the region of hard x-rays or even beyond, then, of course, also this Compton effect and eventually pair creation becomes basically the dominant process. The notation for this elastic in the terms of Rayleigh or the inelastic part is simply given here in the same notation that basically you have a scattering. You have a scattering process where, however, in order to describe this scattering process, you once have to interact with an electron photon interaction to get it excited, and it's a second step of the process to get it de-excited. But this is a consequence of perturbation theory. You can simply consider it in this process. You need only this more complex one if you really want to analyze this quantitatively. I also took this picture from some recent publication where you can now have basically all these competitive processes which may appear like photon scattering, as it was explained here, photoionization, which leads to the release of electrons, which subsequently may, of course, depend if you are in a plasma condition, may interact with other atoms in order to make electron impact excitation or electron impact ionization. In this case, you have the filling of shells via orcher decay, and the fluorescence and all this in a competition. In particular, you will then also have premsstrahlung, which, again, is basically an electron-photon couplings process. But in this particular case, the electron is nicer, is not captured, but it simply changes a transition. It makes a so-called continuum transition. We saw just characteristic lines as bound-bound transition. One bound state goes into a lower-lying bound state under the emission of a photon. Premsstrahlung can be, in this context, considered to be as a continuum-continuum process where basically the electron either gives away, releases some of its energy, and goes to a continuum state of lower-lying energy, or it may also capture something, which is, of course, less likely. And in this sense, you can easily determine what is the maximal kinetic energy of this premsstrahlung, basically in terms of the accelerator that you have here, this standard scheme in mind that you accelerate an electron beam and you hit an atomic target. And in this case, of course, this minimal wavelength is just given by this ratio to divide by the voltage with which you accelerate your electrons. OK. Another process refers to relative recombination, or also in competition to dielectronic processes. Since I want to discuss at this point here, I said, if you have already in a multiply charged ions, in particular in astrophysical environments, then, of course, you can inverse this argumentation. So far, we say we make photo ionization. We come within radiation field and we kick out or excite one of the electrons of the atomic system. Due to this inherent interaction with the radiation field, of course, it may also act in the other way, that you basically capture one of the free electrons. This is, of course, a process which in particular occurs with multiple or even highly charged ions and under the release of the photons, just the inverse of the photo electric effect. And in this case, in our notation, we would say an electron interacts with a q plus charged ion. And it becomes eventually a q minus 1 plus charged ions in a particular final state, not necessarily the ground state of this ion under the emission of one of the, here I'm here at the moment, one of the greater free combination photon. Of course, we have energy conservation, the electron energy plus the binding energy together gives us the photon energy in this particular case. If it ends up still in an excited state, for instance, a capture into a bare ion can proceed directly into the case shell. This would be the ground state, or it can proceed, for instance, into the 2p shell, which is then subsequently within a very short time for highly charged ions, typically 10 to the minus 15, 10 to the minus 16 seconds decays under the emission of a characteristic photon, because this 2p falls back into the 1s shell. And this would give us then what is called this radiative stabilization of this particular ion. This is a known resonant process in the sense that for every energy of the electron, it may occur and it competes with the so-called dielectronic recombination process, which can only happen if the energy which is gained due to the capture of the electron is just appropriate in order to excite the atom to an excited state, as often one has a picture that one has a capture of a one electron helps you to excite a second electron. And this is nothing else than the inverse of the well-known or shape processes. Instead of one electron falling out and releasing an electron with well-defined energy here, one electron comes in with this well-defined energy. Therefore, a resonant process allows us to make a whole shell state, a double excited state of the initial ion which subsequently may stabilize, for instance, under the emission of photons. And this is a very frequent process in astrophysical plasmas that basically is a recombination occurs either via a radiative or if you have wide energies also via a dielectronic recombination because these highly charged ions, they prefer to emit photons. One can still go one step further and consider also two photon absorption and two photon emission processes. I always try to use a simplified language. In this case, you come in with two photons on the same or even this different one would talk about single color or two color, two photon absorption processes in order to excite the atom in an excited state. And the same may happen due to this coupling which we understood in this talk manner from the spontaneous emission may also decay again under the emission of a photon continuum. While the characteristic lines, therefore, the name, have a characteristic energy, in the case of these two photon processes, of course, these two photons together share the total de-excitation energies. And therefore, these processes are much harder to observe and to analyze because you don't have this characteristic line spectral anymore. You have always continuum which you have to understand and to separate from the background. Again, because we have here two photons with respect to theoretical treatment, it requires at least a second order process. And this always means, in the sense of perturbation theory, one summation over the complete spectrum for hydrogen. This is where you are all aware of what it means once over all the possible states. The same you can say, of course, for every multi electron system, but then it becomes quite a bit more sophisticated because one has to distinguish between basically different continua due to the symmetries, how you couple the electrons. And this may become really a very complex and a very cumbersome thing to do in practice. But again, you can derive selection rules. And I also show you a few examples here for this two photon absorption or two photon emission processes for people like us who study often also highly charged ions, the process of highly charged ions. One of these prototype examples is the decay of the 1S2S singlet S0. Remember the hierarchy of the interaction? If the nucleus has no nuclear spin, then we have here a 0, 0 transition. And this is nothing possible than a two photon decay in order to bring it down. If you make this decomposition of the electron photon coupling operator, you see because of the symmetry of the wave functions that you can have either two dipole photons or two electric dipole photons, two magnetic dipole photons, and so on in order to fulfill the angular momentum conservation. And then you remember you immediately know this was the M1 is suppressed by alpha squared with respect to E1. So you have here this part is already suppressed by a factor alpha to the fourth. And you can imagine this is a pretty tiny number. So this would be by far the dominant process in order. If you think back on this hierarchy, however, as soon as you have a nuclear spin and a hyperfine structure, then, of course, you often or you typically have a so-called hyperfine quenching that due to this coupling with a nuclear spin, it is weak to this coupling, but it is known 0. You have the possibility that, basically, you get one photon transition down. And for this type of couplings, this means that lifetime would typically decrease by many orders of magnitude because one photon transition is, of course, much more prominent than such a two-photon or even higher photon transition. Another nice example, which I already mentioned before, is this beryllium-like ions, this triplet P0, as the lowest lying excited state it has for neutral beryllium due to calculation, several billion years of lifetime. Of course, there will be many other processes which deter this process, but there has been some recent interest at GSI in Darmstadt to measure something like this for some Mitzi region. Even for Z equivalent 50, we expect here a lifetime of seconds, which, of course, on the atomic scale is arbitrarily long, almost infinite long in comparison to what we used for this type of beryllium-like ions somewhere else. Of course, this was basically the one and two-photon decay processes. The same coupling, this A dot P terms, either in the one electron or A dot sum over all Pi, in the many electron case, applies also for the photoinitiation process, in particular in the electric dipole approximation. Instead of a rate of a probability per time unit, then, of course, what one determines is the ionization probability per atom and time unit, but divided over the photon flux. And this gives you, of course, what you expect from a nice cross-section, you get here centimeters squared by this division. And the procedure is very similar as before. You, again, have to evaluate with respect to this electric dipole operator, which comes from the expansion of the multiple field or from the photon field into multiples. You can, first of all, see that the electric dipole is again the strongest, and it is usually always possible. So in photoinitiation, higher multiples play only a role under very, very selected condition. That, for instance, in the angular distribution, you have sometimes zeroes in the cross-section. Then you can hope that you see small deviation in all other cases. It's basically hopeless to see any of the higher multiples. It is, of course, we are now dealing with wave function in the continuum. Instead of a nice bound state, as Yuri explained yesterday, you have to now construct so-called scattering states. I think it will be one topic of the next lecture, so I can stay short. One has quite a bit of additional mathematical complications in this case. But with respect to the physics, it's the same. You make it now a bound-free transition in this simplified language. One issue, what was in our own interest, and I will show you a little bit from an example, at the end of my talk is photoinitiation with excitation, or with direct double-photoinitiation, that you come in with a single photon, but eventually you release two electrons. This can happen in different ways. Even so, not all of them are uniquely possible to distinguish from an experimental viewpoint. The simplest way is simply you knock out one electron. This is a standard photoelectric effect, which you all know. Of course, you can leave the atom in a state where it further alternizes under the emission of the second electron. But you have, of course, also possibility if you have two electron, one photon, two electron interaction, that basically you emit this electron simultaneously. Again, this would require the next step in the electron electron interaction. And this is quite closely related to what we call also shake-off, that you also lead to a double-ionized system, but typically under the assumption that the fast photoelectron, it has nearly the same energy as before. And you get typically a second pretty slow shape of electrons, which is just released and leaves the electron in the excited state, in the ionized state. And of course, the same could happen, of course, via such a two-step or shake cascade. And this will a little bit go into the direction of what we would like to show you this afternoon with respect to this tutorial and the computations. Coming for a short time now to the electron-electron coupling, so far most of the processes, apart from the last slide, we are related to the electron-photon coupling, either to an existing field. And we use this vehicle in order to understand if there is no field, but quantum electrodynamically, if we would have fluctuations in the field that help us to allow this transition, that we have now, of course, also in many electron systems, always electron-electron couplings. And as soon as the system is excited beyond the ionization threshold of the next higher charge state, it has the tendency, or it has at least the opportunity, to outerize, to de-excite itself under the emission of a single electron, which is then called either the O'Shea electron, or the outerizing electron, or the outerization is often used if the energy of the released electron is very tiny. If you have well-defined lines, then you talk about the O'Shea process, but from a physics viewpoint, and from a mathematical viewpoint, is regard to the interaction operator, the electron-electron interaction. This is exactly the same. So in single and two-step O'Shea decay, basically, you start with an ion system, and you basically outerize, you release one electron due to the de-excitation of the system. And of course, if you still, after this step, in the continuum of the next higher charge state, you can continue this process as long as you are below. And for light and medium elements, this is typically the first line, which the atom like to do. Only if you are once under the last ionization limits, and of course, the de-excitation towards the ground state must occur via photon emission, but on a much, much longer time scale, than I showed you at the beginning, this time scale. This is a topic of O'Shea electron spectroscopy, which itself has been found a very versatile tool in order to understand different things, also in solid state physics. This is not the topic of my talk here. I want to make you aware that we typically distinguish between so-called normal O'Shea transition, where basically you fill from a different shell, and you have basically emission. In some cases, you can have, for instance, in argon, the L1, L2, 3M transitions. You can have the situation that the electron, a 2P electron, falls to 2S, and releases one of the additional electrons. Then you have a strong overlap of the wave function of this 2P to S, because they are localized in the same shell. And this means that these special O'Shea transition, which are called costerchronic, if it is comes one from the same shell, or what we call super costerchronic, if both comes from the same shell, they are much more likely. And instead of a well-defined O'Shea peak, often you have a very wide structure, which in the spectra, you cannot anymore resolve. But theoretically, this is always the same. In this case, we need to include the electron-electron interaction. This was the Coulomb interaction. As you saw yesterday, also in Eurydorff, it determines the structure, but it determines at the same time also the coupling to the continuum, if it is energetically possible, if it is laying energetically in the continuum of the next higher charge state. This is a proper phrase which one should keep in mind every time, if your atom is energetically still in the continuum of the next higher charge state, then agonization may happen, and typically also does happen. And you see it here, that basically it's a so-called O'Shea inflorescence yield. This is just in the sum to guess a one for a wide range of atomic numbers, as for all elements beyond set equivalent 20 or 25 or shares by far, the dominant part, and only if you go really to the highly charged ions, then you have the situation where the coupling to the radiation field, this is omega to the third, which we saw in the relation between spontaneous and stimulated processes, becomes so important that basically it starts to dominate. Here just for the agonization, as I said, this is exactly this type of matrix elements, which you have to calculate also in a self-consistent field calculation. But if they appear between as transition operator between a bound state and the so-called scattering state, where you couple a bound state of the ion after the agonization with the outgoing electron in a proper manner, then of course you will have the so-called O'Shea rates, which again per time unit. The selection rules here are different. Selection rules are simple. The electron-electron interaction is a scalar. It is part of the Hamiltonian. Hamiltonian is the scalar quantity. It represents the energy. Every part of the Hamiltonian is of course also a scalar operator. It cannot change neither parity nor the total angular momentum. But this refers, as you should keep in mind, it refers to the total scattering state. So the initial state and the total scattering state, they have the same parity and the same total angular momentum. Typically, what you have in your notation analysis are the parity and angular momentum of the agonized ion. They can be different because they still require the coupling of these three electron. And one can now decompose this in a manner that one can discuss in such a process what is carried away. So after a photo-ionization, for the excitation or the photo-ionization, here for the ionization, you end up with an excited atom, with an excited singly charged ions, plus a photo-electron. And if this typically lays in the continuum of the next higher charge state, for all inner shell and sub-valence shell holes, this will apply. Then typically, you end up with a double-charged ion and two electrons. And depending now what is the relative energy of these two electrons, if you have a fast electron, then it will not be affected by the subsequent slow electron. But sometimes, you have a photo-electron, which is pretty slow, and then a cha-electron, which is pretty fast. And then it may take over. And you can imagine, then, of course, it sees different potentials. And this is the origin, what is called the so-called post-collision interaction, that this line structure will affect each other. And that basically, this is just of the electron conservation, that if you take over the photo-electron, of course, you start to see a different potential. And you have to take into account which particular fraction it's doing. The same as in the photo-ionization, you can also have, in the case of ionization, shake processes that due to the de-excitation, you not only release a second electron into the continuum, you ionize it. But you can excite a second electron or also ionize. This would be a shake-up or a shake-off process, which is associated with the O'Shady k. And this leads to quite a good complexity, as I might show you with a few transparencies in a second, that this becomes quite a complex thanks, but which is in the focus of our present work, and in particular of Randolph's work, who will help me this afternoon. Just to summarize this last slide on this file, apart from this dominant electron photon, single photon interaction processes, as well as electron-electron interactions, single electron emission ionization processes, we have quite a number of weaker, but nevertheless known negligible processes. It depends very much on the circumstances. You look at your particular system. I discussed already in good details this two photon decay of an excited state, simultaneous emission of two photons under the release of two electrons, which have to share the total energy. But you can also have now mixed type of transition, like the radiative O'Shady, that this energy under the de-excitation is shared by a photon, as well as by an O'Shady electron again. This leads to continua, which are not so easy, possible to observe. You can have two photon double ionization, as the two photons together, lead to basically the release of two electrons. This can happen sequentially. So basically one photon at a time releases one electron, but it can, of course, also, in particular, if the energies of the photons in such a manner that the second photon is not anymore able to ionize, but then still may happen that the sum of the two photons may ionize two electrons. And this is then sequential or direct. You have also two-color single ionization, as well as two color double ionization under the release of one photon or two photons, as well as then the whole process of strong field physics, where you first go into the multi-photon regime. And later, even if you further increase, which is typically possible only by infrared or, actually, infrared lasers increase the photon intensity so much that you enter the tunnel regime, or perhaps even in the future, with these high-power laser sources. Also, the pair creation regime. This I, depending a little bit how much time I have, I show in a second. So this was what I wanted to make you aware here, this competition of electron photon from a mathematical viewpoint. You can take a very simple view, either electron photon coupling operator P.A. or the Coulomb operator. This allows you to describe all possible processes, which happens in your plasma or your atomic environment. But when it comes to detail, one has to distinguish between these different processes. And one has to develop also a little bit an own intuition when is which process important and which of them are negligible. It is impossible to try to do everything into your theory and to say, I will calculate. This will not be a successful pass, but you should start in atomic physics always from this initial intuition. In the remaining time, I would like to show you a few issues, which is in our own interest and which we partially discuss in this afternoon. As I just mentioned, this shake up and shake off transitions, this has raised some recent interest, also in particular for ourselves. This is the work of Randolph, who is here. I just showed you the competitive process. We are focusing now on this what we call a shake cascade. That's the initial excitation of the inner shell hole leads you eventually to several, to the emission of two or even several electrons. And there has been a quite wide interest in this field, which comes from this community or experimental community of this magnetic bottle. Original or shea electron spectroscopy meant always that you sit somewhere on a particular range in electron energy and you try to understand the individual lines, which erase from a single electron authorization process. In this case, now they have the opportunity to observe basically simultaneously more or less all the electrons which are emitted in such a cascade. And they can draw so-called coincidence maps, which in the first instance looks pretty complex, where you, for instance, display the sum of the energy of the second and third in terms of energy as function of the slowest electron of E3. In this case, you have here these dark spots. They refer really to a shea lines, but you have also these diagonal stripes, which basically arises from the summation. And you can now take projection either in this, in the vertical, or in the horizontal line to recognize different features of the spectra. One has to get induced a little bit in order to read this type of maps, but the question is, do we understand something like this also theoretically? And what is the physics in particular behind this case of three times or the double ionization of an initially photonized krypton atom that you have excitation? Basically, you create a 3D hole, but you have in addition also a good or not too small probability that you have a shake up. In addition to this 3D hole, you excite one of the 4P wavelength electron to some NL shell. And you see already from this Gaussian diagram here that of course you populate different states, and it will affect eventually which of the final states you can reach and which you cannot reach. And the idea is this will be the focus of this afternoon. We want to use grasp, and in this sense, multi-configuration calculation, where we build up atomic bound states in terms of a superposition, linear superposition of configuration states. Like Yuri explained yesterday, we want to create such wave function in order to describe simple fluorescence as well as a shady case. From a physics viewpoint, what we have to fight is on the one hand with the many particle character, often called electronic correlations. This is an electron-electron interaction operator. It looks simple in the first instance. In practice, unfortunately, it brings in quite a bit of sophistication in order to deal it. But at least to a good extent, one can deal with this electronic correlation and everything which is beyond it. And in particular, with relativistic effect. And the idea in the relativistic atomic structure theory is that one try to generalize everything which we know from one, from the known relativistic one and few electron atoms towards the relativistic regime. And only in this sense that we need more from our hierarchy is that we need to include relativistic corrections to the electron-electron interaction or that we would need to include QED. One should do it in practice. And for this, in order to deal with it, we developed what we call this RATIP program. Stands for relativistic atomic transition and ionization properties in which we will use this afternoon in order to calculate some photoionization cross-section as well as the fluorescence. So basically, it stands for a suit of programs which we developed over the last, meanwhile, 20 years, which has the opportunity to calculate or shade type properties, relative free combination, dielectronic recombination properties, but also Einstein coefficients for the ionization cross-section. So these are typical what I explained from a physics view point between different bound states, bound continuum states. It helps you to calculate this type of properties. And do I have, yeah, I have exact time for one example. Very nice example which one of calculated last autumn was, and it's associated to ongoing experiment at this so-called pipe facility, photon ion spectrometer at Petra III, a very modern type of synchrotron in Hamburg, where you see, managed to have an ion beam and to overlay it with the synchrotron, with the beam from the synchrotron, and to make very dedicated experiments for ion, both for negative ions as well as for positive ions. The main motivation is to improve the data knowledge like photo ionization, like recombination processes for ions, which are typically under normal conditions, are not so easy. So this is work of Stephan Schubert's group in Gies and also people in Frankfurt and Hamburg are involved in this business. And what was the example which I want to show is the so-called 1S minus 1 2P6 resonance. We have negatively charged oxygen. This has a fluoron-like structure. So everything is filled in the ground state up to 1 2P electron. This is the left side. And if one comes in with a gamma on the right photon energy, something like 525 electron volt, you can excite this 1 electron to this doublet as a half state. This is exactly the same electronic structure like a neon atom with a 1S whole state, but now, of course, negatively charged. And if you leave it alone, then you see that you have double or even triple photo detachment. So you have a release of two or three electrons from the system to this ionization. And the question was from the experimental side, do we understand this also theoretically? And here is just a summary of what you see, eventually with respect to energies and the width of this particular resonance state. We get a good agreement, but it needs quite a bit of effort in order to come to this point. And the reason is that if you look with simplified eyes, that you simply say, I know my configuration and I make a calculation for this type of configuration, you see you can reach, of course, 0 plus as well as 1 plus energetically, even though this 1 plus is only partially populated, but you will not come to doubly charged. Because energetically, there's no possibility from here to go somewhere to here. But this is only the picture if you take the normal or shade transition into account. If you include in addition also this shake up transition, and then you see that you start to populate in different steps higher lying levels due to this shake up of the decay. And now it becomes understandable why you might see also double charge, 2 plus, 3 electron emissions. And this is seen here just as a final point here. If you have a simple model, this is not completely because we already include some shake up here or some virtual excitation to 3S, 3P, you see the width is not good, but at least you recognize something. But if you take the ratio between single to doubly charged oxygen, it is basically known existent because you divide by a 0 in comparison to what they observe. And only if you really start to include a series of additional excitation which represents this type of shake up transition, you see that both the widths becomes very nice as well as also this ratio of oxygen plus or oxygen 2 plus. Actually, it has become a bit too good from our taste. But you see, if you have a tremendous effort, we will not be able to show you an example like this. This calculation is much larger, but we want to give you in the afternoon a feeling what can be done in order to go towards such a direction. In this sense, you understand correlated things. My time is over. We would have a few more topics like the interaction with more strong fields. And you come to sequential and direct double ionization as well as a more recent interest is that you can, of course, also consider the interaction with twisted lights. This is a very fashionable topic at the moment that you can have photon beams where each photon has also a well-defined projection of the angular momentum. And this gives rise to new features. Mathematically, always the same, I say, in an oversimplifying way, electron photon or electron-electron interaction operator. But in practice, of course, you have to think still for a good while in order to see how nature behaves because you cannot describe all of them. But with this, I'd like to thank you. And I hope I gave you a small impression what electrons and photons can do together. Many thanks.