Resonant inelastic X-ray scattering

Transcription

[MUSIC] Stanford University. >> Thank you very much [COUGH] for the introduction and welcome back, everybody. As Margot told you, I'm an experimentalist. I think this is the first presentation today by an experimentalist. And you will see there is a different viewpoint, on various aspects. I will touch on a couple of things that, Robin, showed already this morning. X-ray interaction with, with matter. But, as I said, from an experimentalist point of view. Maybe [COUGH] one or two sentences more about, about my, myself. I actually started when I started doing science, I did atomic physics in Hamburg. And then they told me to get out of the gas phase, that's what a PhD student there told me [LAUGH]. And so I did that, I got out of the gas phase. And I do spectroscopy on many different things biocatalysis, catalysis and many, many other things. A network user in France, so as you, as you know this is supposed to be interactive. Interactive does not mean I double checked over lunch again with your organizers. Double, interactive does not mean that we watched the World Cup together and bet on the different teams. It means that you were, unfortunately. But it means that you ask me questions during the talk so whenever something is not clear, just to ask a question. It also means that I can ask you questions. There will be little warning, when when I, when I'm ready to ask a question if when I've prepared something. So the quiz, I will quiz you and and this comes up right? So you will be warned. Okay, so what okay. There's some literature actually Robin talked this morning already a couple of books, but here I show my preferences. The x-ray interaction with matter is still very nicely done in this old book by Sakurai, I like this a lot. And then the, one of the latest versions here, Winfried Schuelke. Who was one of the godfathers of of photo and [UNKNOWN] or photoscopy or photo scattering. This is a wonderful, wonderful book here. If you're interested in the x-ray absorption spectroscopy that was mentioned this morning as well. X-ray absorption [UNKNOWN]. There's a book, recent book by Grant Bunker here who's at the IAT in Chicago. A very good introduction to x-ray absorption and fine structure. And then there's another book here on, this is more [INAUDIBLE] in 3D transition metals by Frank de Groot and Akio Kotani. Also quite, quite recent and [COUGH] gives a very nice overview. Over electroscopy and also photoelectronscopy, mainly in solid state systems. So, what will I talk about? Again, there will be a little bit of an overlap with what was said this morning. I talk again about the interaction of x-rays with matter. And then I also talk a little bit about how me, as an experimentalist, how I see the, initial spectrum calculated. There are many spectroscopies, because I think we will be talking during this school here. I think then Mikael will also then go into more detail here, as a proper theoretician he will probably do this right. So I just give you a little introduction on how I see how inner shell spectra are calculated. I talk about x-ray vision spectroscopy catered towards application of free electron lasers because I think this is relevant to the community here. And I also talk a little bit if time is left about resonant inner elastic x-ray scattering. Okay. So I start with very basic aspects. So why, why do we why do we do x-ray spectroscopy? Of course we hope to get information about the electronic structure and the atomic structure and from the sample obviously right. And the, the reason why we do inner-shell spectroscopy so hard x-ray photo in and photo out spectroscopy, Its an inner shell spectroscopy. And the, the important point there its, its element specific alright. That's always the reason if you were write a proposal. And you proposed I want to do inner shell spectroscopy on a certain system. The reason why I do this because its elements select all right. That's the, that's the most important thing. And here you see the just the theoretical or the tabulated values for the cross section C and for the range 6,000 to 9,000 electron vaults,. And you see the different elements. This is for an example that contains ion and cobalt. And as you know very well you get the absorption edges at different energies for the different elements. That's why it's element selective, right? And the x-ray emissions, so the fluorescents, you can call it flourescent that comes out of the sample, is of course also element selective, right. So you, the K-alpha lines or the K-beta lines from the different elements. They also appear at different energy sources, element selective in the x-ray emission. When you do a hard x-ray, why do we do hard x-ray spectroscopy? We do this because of the large penetration depth of the of the x-rays right? So if you want to do for example, experiments under extreme conditions. So many would, people do a lot, they take a and so, with diamond anvil cells. So they are two diamonds that press the sample together. To, to hundreds of geo-pascals, so they can study conditions that you find in, inside the Earth. Then you need to have x-rays and you can penetrate through the diamond to get the x-rays out again. So this is the reason why do hard x-ray spectroscopy. So it's [UNKNOWN] sensitive for the sample, but you can also do experiments in [UNKNOWN] conditions. I want to put this a little bit with context because you will also hear speakers later this week. Who will, I think at least, talk mainly about soft x-ray spectroscopy so we have the hard x-rays and the soft x-rays. Hard x-rays start usually if it's a convention, hard x-rays start at about, let's say, 5000 electron volts. And soft x-rays are at lower energies. If you want to study the electronic structure, in principle you want to use soft x-rays. Because the spectra, the line broadening is smaller when you use soft X-Rays. And it's easier to access the electronic structure when you do, soft x-ray spectroscopy. For example, in 3D transition methods you can see the, the structure of the 3D orbitals directly by 2p to 3d absorption. And that gives it then directly be the electronic structure of the 3D orbitals. If you, however, as I told you before, if you want to do experiments with a hard x-ray under, in C2 conditions, and in extreme conditions. You want to use hard x-rays because they have a larger penetration depth. So what are the challenges in the two techniques? So, if you do soft x-rays, for example, BESSY in Berlin. Where they use mainly soft x-rays. They are they can easily study the electronic structure of the system. But they have to improve the way, how they can study a system on in C2 conditions. Right, they do very tricky things if there, if we're going to do an experiment in catalysis for example. If you want to study chemical reaction you have to flow a gas through your sample. And, and this is very difficult for soft x-rays. But they're improving those in C2 cells with various thin windows that are only like a micron or less thick. They're improving the conditioncy and normalcy here at Berkeley at the ANS as well. So the soft x-ray people there are working on improving the in sequential conditions. We have x-ray people, so I belong to this community on the right. We have to develop techniques. We have to improve, our spectroscopic techniques. To better access the the electronic structure. The information about electronic structure in the sample. And this is kind of my domain so I'm working on this. Trying to develop techniques, where we can, use hard x-rays, and learn something about the electronic structure in the system. here, so the energy range in the about 1000 electron volts and 5000 electron volts is by some people referred to as the tender x-ray range. It's kind of in between the soft x-rays and the hard x-rays. And people like this, I think they, the people in Paris, they came up with this so it shows that also scientists have a heart, romantic streak. Anyway, so what I would talk about is x-ray absorption, x-ray absorption and x-ray emission spectroscopy. So how do you measure an absorption spectrum? You all know this, I repeat it anyway. So you have your x-rays coming from the source. That maybe the free electron laser can also be a storage ring. And then you, you pick an energy so you use a, in the heart x-ray range a double-crystal monochronometer. So it's silicon, two silicon crystals here and they select an energy and this energy then hits your sample and goes through the sample. And then using the Beer-Lambert Law, this give you the absorption cross section. At the linear absorption coefficient. And you see here, this is in the chemicals sensitivity, these are two ion compounds, this is the ion cage. And you see here this is a high spin system, this is a molecular complex, and a low spin system. You see depending on the structure and the spin state of the system, you get different data. Now you don't only have element selectivity, on top you have chemical sensitivity, right? It was, Robin mentioned this already this morning, yeah? So it's very important we have chemical sensitivity so that we learn about the chemical environment of the sample. We can do this now also with the x-rays that come out from the sample, that are scattered from the sample, right? So it's the energy out and the intensity out, and in order to analyze the spectrum of the x rays that come out. I again have to analyze the x rays. What I do, I again use a single crystal. In this case we call it an analyzer crystal. It's experimentally a little bit differently done than in this case. But again, so I get a spectrum also for the emitted x rays. And again, I have a chemical sensitivity. Yeah, not only in the metal, a selectivity also, a chemical sensitivity. So a high spin compound gives me a different spectrum, than a low spin compound. Yeah. So for both absorption and emission, I have as well, a chemical sensitivity. That's important. Okay, now. How do I see spectroscopy? How do I understand spectros, spectroscopy, in a general, in a general way? Yeah, so now I give you a little, as a, as I said before, an experimentalist's, approach, [COUGH] to spectroscopy. Let's assume. I show here, on the vertical scale, I show the total energy of the system. Let's assume I have a hematomian that I can do, or I can solve the hematominan including all nuclear, all electrons in the system. So the ideal case, which is of course is not realistic, is assumed it can do this and it gives me a total energy of the system, yeah? So it lowers the energy in the ground state and it lowers the energy. And I know of the system that are excited states and higher total energy I have excited states, right. I have the sharp excited state, they are sharp in energy or a little bit broader. If you have a band formation for example, then you may have broader states. And the different total energies have all those states, yeah. And now you wonder how can I, I want to probe the, I want to learn something about the system. So I want to probe a little bit my, my excited states in the system. Okay? So, my photon comes in, so I put energy into the system. The energy's absorbed by the system. And, and then I reach those excited states. And the probability for reaching those excited states is this matrix element, so the transition matrix element. So it gives me the the velocitator strength. The probability of reaching this intermediate state I call this N here. Okay? And this is an excited state, the excited state lives for a certain time, the lifetime [UNKNOWN]. And then will decay. There are many different decay channels now, it can be a radiative decay, it can be a non radiative decay. The non radiative decay would be ray decay. I worry about radiative decay because I talk about photon-in, photon-out spectroscopy right, the photon-out is the radiative decay, yeah? So, this excited state now. The decays into those, other excited states. At lower energies relative to the original case, right. And, what is important here I may, it in some cases may be possible to reach those excited states alter directly from the common state. Okay, so then I have two ways of probing the same state. Either I give through like this, so in a two step process, or directly, okay? now, so going back, I just talked about those states in a very general. So, I solved the Schroedinger equation and I obtain all of those excited states, and then. I say I can reach with my spectroscopy, I can reach the states, with a certain probability. Now, we go to the one electron picture that was mentioned this morning as well, by Robyn, and I want to give names to those states. Just having a state is not very helpful, I want to discuss science, I want to, in a paper, I want to, I want to write something. I reach the state, I want to give a name to the state, right? And in a one, one electron picture, which is an approximation, right? So I, I'm entering, now, an approximation. I, I give names to the states. So, in this case, if I do hard x-ray spectroscopy, let's say, on an ion system. I reach an intermediate state where I take one s electron out. So, in the ground state, I have one is two. I take an electron out, and put an additional electron into the 3D [UNKNOWN]. So I start in the ground state of 3D5. And then go to an intermediate state, and have [UNKNOWN]. Okay, if I go to higher energies, I have, I take one valence electron out, and I put another electron into p state. It was also discussed this morning, but in concept, in the context of [UNKNOWN]. Where do I put my electron? So I put it into, well, it becomes maybe [UNKNOWN] electron which has p symmetry with respect to the absorbing atom, right? Relative to the absorbing atom. And then I have the lower energies. I have, I replace the 1s hole with the 2p hole. So the configuration here is 2p5, 3d5, and then the electrons in valence shell. There are kind of spectators that don't do anything. All they do here is replace the 1s hole with the 2p hole and a 3p hole. And I can go to lower total energies. In this case actually I have a hole then in and that would be molecular that I would call this molecular orbital. So you have a hole in the orbital that is [COUGH] within the valence shell. Okay. And what is interesting now, I told you, I can, I can, well first I give names now, this is the K-alpha line, and this is the K-beta line. So you heard about the fluorescence lines that come out of your sample. Okay. Alpha lines, you find those in the, in the book, for example, in this nice, orange book, a Berkeley book. You find the energies for those conditions. Okay. K-alpha line, K-beta line. And and here in this case we call this transition here valence to core transition. Because an electron from a valence orbital actually fills the 1s hole in this case. Okay? Now I told you I can reach those final states as well directly, and this would be, in this case it's the K. I'm sorry I didn't write this. This would be the K absorption edge. And lower energy is the L-edge or the M-edge. Yeah. So in principle, I have two ways of, of reaching those states. Here, either way via the K-edge or directly via the one photon process, directly. And that would be then the L-edge and the M-edge. And down here, at the lowest energies you even have UV spectroscopy, this would not be an inner shell spectroscopy. This is what you do mainly. Maybe some of you did this already in the lab. You read this spectrometer and if you have this, you probe those states down there. Yeah? So what is important about this? It's the total energy diagram that considers all multi, all electron multielectron effects in principle. The states here are or the names that I give to the states here are Mighty Electron States, so I consider all electrons of the system. That's important as well. It observes energy conservation and in principle, what is nice about a Mighty Electron picture like this, a total energy diagram like this, I can put all my spectroscopies in one diagram. Right? So I see my Cuba Vista spectroscopy, and the heart X-ray spectroscopies. Okay. And of course if I have a radioactive decay then of course the, the energy comes out. Okay, now since this is, this is important to me, this important point this mighty electron diagram. I do a little, we do, we play a little quiz we do, a little, little game now. What you're probably familiar with is this one-electron diagram. Yeah. I mean, most of you probably see this when you, I don't know, when you study. You see, if, if somebody explains to you, what, what, what are the different transitions in spectroscopy? they, usually, they draw the different orbitals here. So that's not a multi electron diagram, right? So one electron. So we have the 1S shell here and then the electron goes out. This is what you see in, in many books and yeah. So either the electron goes way above the Fermi level, this is the Fermi energy, or it'll go just above into a 3D level. And then have to decay, I have a K beta line, 3p of opera. Yeah. 3p to 1s. So I can call this non resident K data or resident K data if I excite it into a letter just above the the Fermi energy that would be resident excitation and the resident capetenine. So now what we like to do, what I would like to do with you, is we translate this diagram again into a multielectron diagram. Okay? So now I need your participation, so we have the total energy diagram. So, we start at the ground state. So where, where do I put the ground state? Do I put the ground state up here? Any comments? So I put the ground state down there, okay? Do you agree? Excellent we start at the ground site. 1 is 2, 3dn. Okay? Now I excite my system, so if I have a one edge to let's say continuum excitation. What would be, what name do I give to this excitation there? Any comments? How many, how many electron's do I have here? >> One. >> One, excellent, one is one. And, the electron goes to let's say, side kind of so 3dn and they're call this Epson-P-Epson, would mean a continuum electron, okay? And, again we have then the the resonate excitation we say we put the 1s electron and put the 1s electron in the 3d shell so 3d and plus 1. Okay? Transition maybe worry about selection rules. That exists, we can observe this. Okay? Now for this decay. So what would be my I hope doesn't show up. Now I have the k bit at k, so what is the configuration there of the final state? Where, where is my co-hole? I don't have 1 is 1, I have 3p5. Okay. At 3p5, and the electron I carry it on, right? The epson P spectator, it stays up there in 3dn. Yeah? And the same, is for the resonant excitation here down here. It goes, I replace the 1 s over the 3 p. Yeah? So this is the difference. One electron diagram and multi electron diagram. If you want to understand spectroscopy, I recommend that you use this diagram because it's, it's more physically [LAUGH] but they, it's a, as I said observes energy conservation. You can really understand processes of multi electron excitation. This is qualitative, right? So it shows you qualitatively what is going on. But some processes are very difficult to explain in a multi, in a one electron diagram like this. And I will show you a little bit later when this is the case. Okay? Okay, electromagnetic radiation, I think this was already addressed this morning. I don't have to go too much into detail. We do, scattering photon in and photon out spectroscopies. So we look at the scattering of photons, right. So we have a photon coming in with energy of frequency, a K vector and polarization and then is scattered. This angle here is my scattering angle and, the, the vector here is the so-called, is the momentum transfer. It's the difference between the incoming and the outcoming, photon K vector of your, of the photon, right. Scattering angle here and the difference between the two energies is the energy transfer, okay? So I describe my as was shown this morning, I describe my photon with a, with a vector field here. And what is important was mentioned then, I have two terms that are important. They are the A squared term, and the p.A term. Everything, the, the, the theory behind this was explained this morning. For me now, just, we just remember that we have the A squared term and the A.p term. As an experimentalist, I should not show those scary diagrams, [LAUGH], but still I find it useful. What is important is for this A squared term, I do not create intermediate states. So, I have my ground set coming here, my photon coming here, the photon is scattered by the way, without creating intermediate state. If I have the p.A term, I I have my ground state here, my photon coming and then I create an intermediate state that lives for a certain time, the lifetime tau. And then at the final state going on and a photon coming out. Yeah? What is, so let's look first at the A squared term. So actually I have to say, what, what, what I do here is maybe theoretically not entirely exact, because the two terms they interfere, the correct formulas were shown this morning, but the two terms appear within the square of the modulus, and I separate them. I consider now the case that either one of the two terms dominate, so I neglect the other one. Okay? So if I look only first at the A squared term, then my, the scattering intensity is proportional to this term here, right? So I have the ratio between the energies out and in, I have the polarization here, .product. And I have all this, this is the matrix element between your grounds and the final state. And this is the energy conservation. And this part here is called the dynamic structure factor. It tells you about, it gives you information about the electronic structure in your system. The response of your system to your preservation of the using the photon, okay? Now, I have various terms here, this is a historically, the, the different names popped up, because the different people here they discovered different aspects of the scattering cross section. So Thomson scattering and Bragg scattering are so called elastic scattering where the incoming and outcoming energies are identical, are equal, yeah? So Thomson Bragg. These two cases are inelastic scattering, is Raman scattering and Compton scattering. These are the two inelastic scattering processes that are based on this A square term, yeah? What is important now is this, this operator. I actually I have a quiz again. So the question here, I'm an experimentalist. And now I wonder, I, I can change this angle there in my experiment, right? And let's say assume I want to minimize this term because I don't want to see it. I want to see all other terms. What, what experimental configuration would you recommend to me? When, when, when does this, this term here is, is minimal. It's even zero the way it's written there. Any idea? In the case, I'm sorry I didn't tell you that, so it's a hint, well I have to tell you that. Incoming light is linear polarized like this. So the incoming light is linear polarized, so at what scattering angle theta here is my intensity zero, the scattered intensity? Any idea? Ed? >> 90 degrees. >> Ed 90 degrees. If I look here, if theta goes to 90, because of the .product here, it goes to 0, yeah? It is important, everything, all those scattering parts here they go to 0 if we look at 90 degrees. So I don't see them anymore. That's a nice way, and we do this all the time, if I measure the experiments florescence detected absorption spectroscopy whatever. We always measure 90 degrees because in this case we are not interested in this term. Other people at other experiments are interested in the term. They should not measure at 90 degrees, yeah? Okay so. Next, now we have to worry about this operator. And this operator here that connects the ground server with the factor is very interesting. Because it contains, this is the momentum transfer as I told you before. Yeah? So the, the momentum transfer is inside this operator. Which is a very interesting phenomenon. And this is the case, I'm sorry, this is the case for, for Raman scattering. By varying the momentum transfer here, I can change the selection rules. I can change my transition matrix element here by varying the scattering angles here. And, and that is very interesting. In the so-called, the, it's technique called X-ray Raman scattering. So actually I start with all the spectroscopies, I start with the most esoteric one. Then there is something called, based on this a-squared term, called the X-ray Raman spectroscopy. And I'll show you here, this is done by, by a colleague of mine, Simo Huotari, who is now in, in Helsinki. He did beautiful wrote beautiful papers on, on, on this kind of spectroscopy. So what you see here is the energy transfer, so the energy transfer to the sample when we the, when the photons are scattered. And what you see here is the Compton peak, right? And you know with increasing momentum transfer, the Compton peak moves out to higher energy transfer, yeah? You see, so what, what you see here for the different colors and momentum. Momentum plans for changes and on top of the Compton peak, you see a line here, does anybody know what that could be? What the, what the edge, the edge that you see what this could be, any idea? >> It's the carbon. What you see is the carbon KH. Yeah? So you see at a fixed energy transfer, you see the carbon KH coming up. The spectrum we measured, so just to make this clear to you. So you come in, you hit your sample, let's say with 10,000 electron bolts. A photon that has 10,000 electron volts. Then you measure the scattered x-rays at 10,000 minus, let's say roughly 200, or 280, let's say roughly 300 electron volts. So at 9,700 electron volts. You measure your scattered x-ray's. So the energy transfer is roughly 300 electron volts, which corresponds, to the 1s observed binding energy of the one is electron in a carbon atom. Yeah. So that's a, they call this a low spectroscopy, yeah. So it's a hard x-ray technique, because they go in with hard x-ray's, go out with hard x-ray's. But I measured the edge at 280 electron volts. So, I measured the edge of a low z element in the, that is usually in the soft x-ray range. Yeah? It's called x-ray raman spectroscopy. Why am I showing you this? Because it's, it's an interesting option also for free electron lasers. Because I, in order to measure the spectrum, I can scan, the emitted. I can scan the my, my, emissions spectrometer. So the, the analyzer that I showed you before. So it's not the incident energy that I have to scan. It's the or I can, I don't have to scan the incident energy. I can scan the energy of my emission spectrometer. So the energy that is scattered from the sample. Yeah. so, I think to date it has not been used at the free electron laser. It's more and more used at, at synchrotron radiation sources. because, I can, for example, study water. So there are many people, I think Mikhail was involved as well, and other speakers. [COUGH] Here, they started the oxygen KH in water using this kind of technique. Because they can use hot x-rays. well, hot x-rays in general are more suit, suitable to study liquids than soft x-rays. So it's a very beautiful technique. And another publication here by, by Seymour. He combines this x-ray ramans scattering technique together with imaging. So what he did, he put graphite, diamonds, diamond into graphite. And, as you know, both are made of carbon but the bonding is different, right? So they give you different spectroscopic signal, right? So if you look, diamond gives you, this is the carbon KH again, you see here 280, 290 electrovolts. Diamond gives you a different spectrum than graphite at the carbon H. And he used a very small beam, and built this spectrum, that he used, he had position resolution as well. And then he turned his sample, so he did something like tomography, on this sample. And he get a three dimensional image of the diamond, inside the carbon. Beautiful experiment, and, used, and it's only possible to do this if you use hard x-rays. Because if they go with 10,000 EV inside the sample, I penetrate the sample sufficiently in order to be able to do this kind of of imaging. A beautiful experiment and a nice application of this x-ray raman technique. Yeah? >> Now you presented that the hard x-ray is necessarily the method of choice. Of course, since it's, it's penetrating better if you have a really thin sample because it means you get way less signal out of nothing. >> It's true. >> Because it's not efficient. Right? >> Yeah, yeah, yeah. Of course. What x-rays they use depends on many aspects and if you have a very thin sample of course, you may want to do this imaging using soft x-rays. But they absorb quite enormously. You, the sense, is thinner than a micron right that I have to make in many cases. Well it depends on what element you look at. And now it gets very technical as well. Another aspect of this x-ray raman scattering is that, if I, again, if I change the momentum transfer here, I can expand the exponential function here. If I vary the momentum transfer, I change the selection rules within this matrix elements. So I can access different parts of my electronic structure as a function of the momentum transfer. As you see here, this is a very nice publication by the people here. So here they vary the momentum transfer in the spectrum changes dramatically. The reason is that this transition operated changes as the function of momentum transfer and it reach different final states depending on what terms dominating here in my, in my expansion. Just imagine if I, normally if I look at a transition from a 2P orbital, I brought the D density of states or D orbitals right? Using the, this technique I can instead of going to D, I can go to F directly from P, right? And I learn, I get a more complete picture of the electronic structure of my system. And so this is the the beauty of of X-ray raman spectroscopy. Okay, so we move on. We go now to the a.p term that appears here. And the Kramers-Heisenberg equation. It was shown already before. So your ground state transition operator intermediate state. It also corresponds to the diagram that I showed you before. So intermediate state, and here. Again transition operate and the final state. Okay? Again, now the transitional operator here now does not contain the momentum transfer, right? What you see here, is the photon vector the, the vector for the electron position and momentum and the prioritization of your, of your X-rays. Okay? You can expand it as well, this gives you then the dipole and quarter-pole selection rules, I don't want to go into details there, because you can find it on Wikipedia yeah, so this part here is then responsible, it gives the, is, is, is due to the A.P term in your, A treatment of the of the preservation hematonian. Okay, what you can do now, if you imagine, if this scattering angle now goes to zero. And it appears to be forward scattering, which is, absorption. So, coherent forward scattering can be viewed. As for so its clear for, for what its getting, that means that the phase is preserved, and the its elastic scattering, elastic forward scattering polarization is preserved as well, and it leads to the total absorption at the end. So you can view, you can view absorption as a photon in, photon out process but the photon hits the sample and comes out again,. And in this case yeah. If you, if you view absorption spectroscopy as a photon in and photo out process, it's elastic forward scattering. And we can treat, it on the same footing here, on the same theoretical framework. Again here now, now because I try to simplify the Kramers-Heisenberg equation here. Do you know, yeah? >> I, I didn't understand the last part with the v forwards scattering things as options because to do photon terms and the photon is kind of destroyed, it's not scattered to lower energy. >> This is a di, discussion maybe we can leave maybe on the, on the road I have been discussing a lot of this, this point with, with different theoreticians. If you, I have this from Sakurai. Sakurai treats it like this and, and I like it for several reasons. In principle, the way I argue is if you, if a photon is absorbed, you have reached an excite, an excited state that lives for a certain time and it has to decay. so, if I only observe the absorption process, I do not care about the decay afterwards, right. You can say I, I, just neglect it, yeah. Then I would say to P.A in the first order term, I hesitate a little bit to, to accept this, because I can't just forget what happens after the absorption process, right. It decays after a certain time. And if I look at the radiation decay, I have a photon out, photon in, photon out process. Okay, I neglect the non-radiat, radiat, radiated decays. And I come up, if it's in the forward direction. If it's elastic, then it becomes absorption. So then, absorption becomes the photon in, photon out process. Another view, what some people write in the books is that... Observe or consider the absorption process as the, the beams that attenuated right? In, in your sample, so it's partly, partly constructive, destructive interference between the, between two waves. >> But is it having the [LAUGH], the, >> It, exactly, that's, [LAUGH] the, that's. If you have an already decay happening, process happening, then you probably have to look at it more carefully. Still, it's a two-step process. We have an excited state and a decay sent further. If I consider the P.A term only in first order, I still don't, care about what happens after my excited state. >> And what more fundamental processes take? The photon number is conserved. >> That is if you consider, exactly if, well... >> Well, if you, if you say, it will be elastic, part of the elastic scheduling process then photon comes in. If you think about it as absorption, your picture will get shifted. But I can, I can count the number of photons that go through a sample. >> If you have less photons afterwards, the photon, the photon number is after. That is exactly. In this view, in this view I, I struggle a little bit as well to, to bring this to terms. But, in the terms of, if you look at it in a wave, the continuation of a wave, then it does make sense. It also makes sense that I do have forward scattering, right? This process is not but we can discuss this maybe later. As I said I, I, followed here the discussion by, for example Sakurai and other, people in another book. But if you view as a, as a number of count the number of photons. That suddenly disappears. Still you can not argue with my argument that you have an excited state that decays, and if it decays very actively Something has to happen to the, to the excited state. But anyway, we, we will discuss later. It may be nice, it's, it's nice exercise to think about this and to [LAUGH] many different decay paths, that's true. Okay, so let's look at the interference terms here. So what, what does interference mean? Do you know what interference means? How can I. How does interference show up here in this in this equation? Does anybody so, you have a sum within the square, right? So, that means you have the square, each term squared, plus cross terms between the different matrix elements, right? Because you have, and the sum, and then you square the sum, right? So you always sum sum deviance, yeah? So if you, let's try to simplify the Kramers-Heisenberg equation here by first neglecting interference terms. Interference would happen between if go from the, from the grounds state of decay to the same final state. And those let's say, assume those are the intermediate states that interfere, okay? Now we ignore the interference effects and then it becomes we ignore all interference terms. And then the equation becomes becomes like this. So I have the square of the matrix elements divided by this denominator that is too big for the resonance. If I, if I look at this note carefully, I observe that I just have the product between absorption, because I take the square now, between absorption and emission. Okay. If I, in the next step, that makes everything much, much easier, right? Because I can calculate my absorption cross section, and calculate the emission, and I don't need to know the phase between them. Because I take the square. And the problem is much, much easier to address theoretically. And the question is now, how far do we get by, by, by using this assumption, this simplification? If I now, and in certain cases I can do this, and this is actually what is done in many cases. When absorptions spectra are calculated, I approximate my transition metrics element by the electron density in the system. That's a very harsh approximation, and in the upright theoritation will rebel against this, but this is what I understand in many cases, just because we have to do it, we have no other way of addressing this problem. So, the transition matrix elements here is approximated by the electron density. Principal is another radial matrix element before that I'm neglecting here. You do this for the absorption process and for the for the emission process. You approximate the matrix element using the density of states of the occupi, unoccupied states and the occupied states. And then this equation reads like this. So this step, I will address this later as well, this step neglects all electron, electron interactions or many multi electron excitation's it neglects as well. Partly the cohort potentials, so there are many, many approximations there. But, it works, I will show you, in some cases, or many cases. And then I get a very simple equation here. The Kramers-Heisenberg equation. It was published already it, in 99 by those people here. Where my scattering intensity is proportionate simply to convolution between the density of occupied and unoccupied states divided by this term here. This is due to the lifetime growing. Yeah. So, the, the question is [UNKNOWN] it's kind of, we approached this, we try out how far we get by approximating our matrix solution, the absorption process, the absorption spectrum by the density of states, yeah? And this is here for, I show you for an experimental spectrum at the oxygen K-edge. in, in silicon dioxide, this is the experimental spectrum. And this is the calculated density of states. And you see in this case, it works quite nicely, so indeed I can, in certain cases, I can approximate my absorption matrix element by the density of states. One has to be very careful. It does not work. For example when,and it will be, it will be discussed later during this week. But if excitations into the 3D orbitals of 3D transition metals. So iron, manganese and then the the L-edges of 3D transition methods this doesn't work. And people will discuss this afterward later this week. Also when I look, look at rare earths four f orbitals are the valence orbitals and bare earths. Also this approach doesn't work. And also in cortical X-ray emissions spectroscopy approach doesn't work and I will address this in a little bit. Yeah. Just to, to show you a little bit, give you a feeling of how you can approach this problem of, of calculating initial spectrum. Okay, summary slides, so for the resident term, I have here the, the resident part of the Heisenberg equation but this transition operator. So, this is what we are mainly concerned with when we do element selective spectroscopy, and then we have the Thomson scattering term, and I showed you this scattering term here is due to all those Raman spectroscopy that I showed you before where you can change the transition matrix elements. Okay? The two terms interfere and neglect this, I put this here in vacant. You may wonder what is resonant? It's also an interesting question to, to contemplate it a little bit. it, I think, also this morning Robin mentioned this a little bit there was a question here. What, what is actually resonant? In principal, when you look at the literature, you say resonance are excitation's close to an absorption edge, right? They are sharp resonances in a system, right? Yeah? But, in principle, I could say also when I excite one electron from the one S-shell into the continuum, right, it become a free electron, the principles also. A resonance within this continuum state, yeah, so, a strict definition of resonance is very difficult to do. In principle you could say, the most general definition is that whenever I use this Kramers-Heisenberg term, I have a resonance excitation, and that would actually refer to all to most, to all element-selective spectroscopies. Apart from the X-ray Raman spectroscopy that I showed you before. Yeah, that would be very general definition of of resonance. Okay, so the next question, what can inertial spectral do, do for you? In principle, you want to learn what I tell, told you before. We want to learn about electronic transitions and thus the electron density and electron configuration. This will tell you, as, as Robin mentioned already before about bond distances, so you have an absolving atom here, you have bond distances, you have bond angles, and it would, may tell you about the type of number of beacon. So this is exo spectroscopy. Okay? What is important is that in principle spectroscopy, using this lecture rules, spectroscopy is symmetry selective, yeah? It will tell you about the symmetry around the ground state. So, how, how many of you are familiar with this, with term here? This is the spin orbital, 2S plus 1L, so you are familiar with multiplets, atomic multiplets, yeah, so I use this term in, in a certain coupling scheme, I use this term in order to describe the symmetry of my ground state, and based on the selection rules, I will then. Can only reach certain excited states. Yeah? And the excited state is what I probe in my spectroscopy. So that means I probe what I get information about as symmetry. The LS term of the ground state. So it's very important to remember that a, that a spectroscopy is symmetry selected on the outside. Probe symmetry of my ground state. I added this slide now because there was this discussion on what, what do I measure? Can I measure oxidation states, or rather how much does a K-edge shift as a measure of the oxidation state? That's a very fundamental question that people discuss all the time. And My reply to this question will always be the oxidation state is not an observable, it's not a quantity that I can measure. Oxidation state is a, is a chemical concept that helps you to predict certain stoichiometries in your chemical equation. But it's not really a quantity that I would I like measure. If you try to translate this into a quantity that you can measure, it would somehow relate to the charge per atom. Right? Assume if ever, if ever an ion in oxy, oxy, oxidation states three that means I have a 3d5 configuration. Yeah? That means I would translate the concept of oxidation state into a number of electrons, that I have on my atom. Yeah. But this is a tricky question, right? If you look for a time, this is a molecule, and this the electron density map for, some energies that are important. There's Manganese here in the middle, and then there's the electron density around. Now, I have to assign electron density here, to a certain atom. And you see it's a tricky business. I have to come up with a certain scheme, and then many sophisticated schemes, in order to assign the electron density. To a certain to a certain atom. But there's, its, its, there's no strict procedures. So it's not a strict observable how much charge I have on an atom, yeah. So one has to be very careful when I ask a question or when, or when, when I tell somebody I can, I can tell you if I do my spectroscopy, I can tell you how much, how many electrons I have on my absorbing atom methods not really what I, what I can measure. And a little philosophical intermezzo I found there are two provocations here, one by, by Robert Parr, who is very famous for coming up with exchange function, indensity function theory, and in a publication he was asking what is an atom in a molecule, and And what he writes then it says that a mol, an atom in a molecule is a noumenon in the sense of Kant. I'm not asking you what a noumenon is. I had to look it up myself. Wikipedia says, it's an object knowable by the mind or intellect, not by the senses. So that's bad news for the intriguer spectostrophist. Because we claim we do element selective spectrosopies. Anyway, so, but there is a point I told you, I showed you this electron density map, so this is where they're coming from. They say well I have to look at the entire molecule and my electrons are distributed over the entire molecule. It doesn't really make sense, that's the way I understand it in layman's terms; it doesn't really make sense to talk about an atom and a molecule. There was a reply right away, as you can imagine, to this paper by the group of Richard Beta. Richard Beta wrote a famous book, Atoms and Molecules, and came up with a way of projecting the electron density on the mole, on the, on the atoms. And they reply, of course, an experimentalist has no doubt the he or she is measuring the properties of a single atom. We cannot solve this issue, but I'm showing you this in order to, I dont know, sensitize you a little bit to this, to this problem. I mean, what are the properties that we actually measure? In principle, you read a publication. People like to write about 4p and 3d, and but, this doesn't really exist, right? I don't have spherical symmetry in my system. So there are no 3d orbitals in principle. yeah, it's very difficult, very difficult to find a language how we discuss electronic structure and how do I present my scientific results, spectroscopic results to a scientific community. What language do I use there? And that's tricky. So we have to find observically really observables that are probe in my spectroscopy in order to then communicate my results to the community. And that's a very difficult task that I cannot solve here. Okay, just, I want to point some more problems out to you. [COUGH] When I do intershell spectroscopy. Yeah, so I remove the electron here, that's one less electron, so this is a very simple picture of an atom. Yeah, nucleus here, I take the one 1s orbital out. What happens is that I have one 1s electron less, so the nuclear charge is screened by one electron less. So all other electrons in the field change their potential, obviously, they will relax, right? So the, the, the energy levels will relax. What can happen if I have this relaxation, that not only I remove one s electron, another electron, so when those, those shells, say they collapse a little bit, they react to the new potential, another electron is excited from an occupied level to an unoccupied level, so that would be multi-electron excitation. Yeah? So, that means. What I told you before and was, was discussed this morning. Discussing spectros we using only one electron. And assuming all other electrons are kind of spectators is a tricky business and may not work in many cases, yeah? So when I, when I calculate my, my absorption projection here I, I assume the matrix elements of these I assume that those states are mighty electron states in the most general case. I can use many electron, like a wave function. I can approximate this, and this is what is done if I then ultimately do my calculations using the electron density, using density function theory. So I approximate this multi-electron transition matrix element by using one electron wave functions as was shown this morning. And, what I can do, in order to somehow take care of the multi-electron processes, I can scale this matrix a little bit by a prefactor. But this is kind of, this is and approximation, yeah? In this case, I have one electron wave functions and then I can use density function theory in order to. Calculate my, my absorption spectra. Okay? So that's again, is my experiment with point of view how the two different ways of, of trying to calculate inner-shelf spectra. Yeah? I can start, with a nine, so I neglect all the ligands. I have an absorber, and iron absorber in my system. I neglect all the ligands, so there's no oxygen nothing. I just look, it's an ion because maybe I can remove two electrons, because I assume it's an oxidation state too, yeah? And then I have a wave function that is later determined that describes my system, right? What I can do then, artificially, I can go from the spherical symmetry here to the local symmetry, so I have six oxygen lignds around, I have octahedral symmetry. And this, I can simulate, by assigning new symmetries to my 3D objects. So, my, they're not 3D anymore, they become T2 orbiters, or T2G orbiters, EG orbiters, according to, the symmetry that I may have in the real system, yeah? And this is kind of done empirically, and I can split the orbiter, the field splitting the orbiters. I can also include covalency by mixing them in several configurations. So in this case, I, I, I treat all the, the, the effect of the ligens I treat in an empirical way. And but what's the advantage here? I can treat in principle multi electron excitations in an accurate way. And I also can include the interaction of the core wall with the valence electrons in a, in a rather accurate way. Yeah? And this is the Ligenfute Margin Rate theory. Some of you may have heard about this before. And it starts, as I say, it starts with a free ion. Alternately, I. Start with the whole structure so I am not including only the ion, but all my deletings around so the chemical environment is included and in this case it will density function theory and I calculate the electron density. Then there ways of including the core hole. I say he approximative way that maybe some people are insulted but anyway I can include some of the the cohold and include my electocitations as well. In this case I have a good treatment obviously I have a good treatment of of the legans in long range order because this is all considered in the in the in the calculations. But, it's more difficult to treat the multi electron excitations and, and treat the interactions of the valence electrons with the cohort, yeah? So these are roughly the, the two approaches, the way I see it. There's enormous work at the moment to somehow merge the two, approaches. It's kind of the holy grail of initial spectroscopy, they're called appanesio maltiplets that means you calculate the electronic structure using all ligands and have all the multiplet structure because of all the electron interactions included, so we are called appanesio multiplets and they're enormous, there's enormous progress that, I think Mikael, will talk about later, later this week. This is just to illustrate to you the fundamental problem. Yes another problem in inner shell spectroscopy. What I told you already about those multi electron expectations. So to give you an example, this is the L 3 absorption edge. So I excited 2 p 3 hops electron in serum dioxide. What you see here experimental spectrum the red line and the blue line here are one-electron calculations. So what do you see what I used here is the 5th code some of you maybe familiar with that. The 5th code is density functional theory code that approximates the obsorbtion spectrum using a one-electron transitions. And what you see, those features here are missing, they're not coming on. The rest comes on quite nicely but those features are missing, just plain missing, because it's a one electron code, yeah. Then there's another code which is called the Single impurity Anderson model, and, and this code gets those multi-electron excitations, they get it quite nicely. But one has to say that this splitting here. Which comes out in the one electron calculations. This splitting here is introduced in this code empirically, so it's kind of cheating, right? So this splitting comes from the programmer, the programmer puts the splitting in here. While in this calculations the splitting here comes out up in issue from the calculations, so that nicely illustrates the advantages and disadvantages. Of the off these different approaches. So here, the fine structure off this case in this is the 5t band. The fine structure comes out of the calculations, but I do not have them out here exertations But in this case I do have the molecular excitations, but the fine structure I have to put in by hand into the calculations, yeah? So again, what we would like to do in the future is have those two approaches combine, so I have. The correct fine structure and the multi electron excitation in one go. >> Excellent, very good, so you paid attention. I was hoping nobody would pay attention. No but, you did and these are, so I told you, this is the 5T band. So these are transitions from 2p to 4f. And in order to get them out from the calculations, er, you have to do some tricks that I have to admit that I was not aware. It's not only you have to include transitions obviously, but they also send tricks to 4f orbitals in those calculations and those on the right. Position so we have to tweak a little bit the calculations to get your four properties in the right spot. Which I didn't do back then. That's a simple explanation. Still a good question. okay. There's one, one thing I would like to mention, the multiplet theory because I'm not sure to what extent, well the other speakers might address this as well. Just to a little, to give you the idea what multiple theory means. There some, a lot of little, This I don't have to go through it. You will have the slides. So the fundamental problem is, I have my 3D orbitals that say in a 3D transition metal. If I have only one 3D electron, it's easy. My total energy is just mg of all the other electrons. The rest. And, and plus the energy of this electron that we show. If I have two 3D electrons, how do I treat this? That's a good question. It's not twice the energy, it's not that simple unfortunately. What we have to do is treat the interaction between those two electrons, there is the combined action between those two electrons. And they come up with the two electron operator. I get matrix elements for this two electron operator, and what comes out then is the so called the direct term and the exchange term. Yeah it's the direct coulomb term and the exchange term. And the exchange term is a exchange into action that is zero in the spins of the two electrons are anti parallel and it kicks in if the spin of the two electrons. It's parallel, and it enters the total energy with a minus sign here, so when the two electrons are parallel, then I get a finite value here, so the total energy, is decreased. It's Hanh's Rule, basically, so, yeah. And those terms are the Slater integrals or Racah parameters, as you may have heard. And about, okay? So, what does it mean for my, for my system where I have two, 3d electrons, right? So, I have to couple the angular momentum of the two, of the two electrons. And to couple the orbital angular momentum, and I have to couple the spin, which I didn't show here, these are the standard rules for the coupling of angular momentum which you have probably learned in quantum mechanics. So I don't have to. Had to repeat this and, so I couple them and then I have also have to observe the Heisenberg the I'm sorry the pounding exclusion principal. And what I find, if I go through all the quantum mechanics here what I find that for a 3D tool configuration, I have all those states. Yeah, that comes out of multiple theory, Atomic multiple theory. So, I don't have just a, maybe you learned, I dunno, you learned the cluster field splitting, or whatever, you have to observe also the electron elegant interactions that give you all those different terms. There are five different energy levels for 3d2 configuration, yeah. And it's very important to consider because you observe, actually, those levels, when you do spectrocity. So when you do ligenfield, mind you, the theory, the program codes you may have heard of the code that is, managed in Germany So the code does, it starts with atomic calculations, introduces then the chemical environment that I would told you before. It branches to the appropriate asymmetry. And then I can somehow consider, I shouldn't write here hybridization, sorry, it should be orbital mixing. So it considers somehow the orbits of the ligands as well, and I can, I can mix the different, by mixing different configurations. Okay? so, I told you I started with a 3d2 configuration. And then I have those five levels, just by looking at the atomic multiplets. On top of this now, I have the, the, the crystal field of the ligands. So I have an additional splitting on top. And yet, this, all this combined. It's shown in the so called Tanabe-Sugano diagram. Who has heard of a Tanabe-Sugano diagram? Okay, that's very nice. Some people at least. It's usually important and I'm surprised that this is not generally taught at universities, It's, as I said, usually important if you want to understand electronic structure in, for example, 3D transition methods, because this is what really happens in your sample, yeah? So I don't know, when you get the change look up ten other Tanabe-Sugano diagrams, because this is important to electronic structure, okay? So this was just a littler... I, I was a digressed a little bit [COUGH] to address this point and I give you now some well some examples for X-ray Emission Spectroscopy. Okay, I showed this diagram before very busy diagram I make it easier now to understand I want to look now at the capital lines, yeah. So I remove a 1s electron, have an excited state here, it decays. And then I have the capital line coming out. As I replace the 1s hole over the 3p hole. In a one electron diagram, it's a transition from 3p to 1s. And I show you here so in the context of the other fluorescence lines coming out of magnese. If the K-Alpha line's here, the K-Beta lines then, and these lines, I will address later, the core lines. So what can I learn, what is the, what is the sensitivity of those lines, and and what I can, what can I learn about my sample? Using those present lines. So first I look at the spins, right? I start out, I have a 3d5 configuration. So in the ground state. My L-S term, L a twist, was one. L term is [UNKNOWN] S. Right, it's a it's a 0 anglum momentum and I have five spins pointing up, right. And then I, photo ionize so I put one electron into into continuum with the P symmetry, and what remains, is a [UNKNOWN] or [UNKNOWN] S state. So that you have to count the spin here in the one estro together with the spins in the 3D show. So this spin here and the, and the core hole can point up or down, can be parallel to the 3D electrons or anti-parallel to the 3D electrons, okay? And, so one gives you of course, this is intercepted, and this is the quintet state. And by the way, I indicated this a little bit. So this state is a little bit lower in energy than this state. It's only 50 million [UNKNOWN], but it's still lower because of the Hans rule, because of this exchange interactions between space. Okay? So now let's take this study case. And you keep the spin orientation of the electron of the, in the core hole, no you keep it, and then you reach receptive p in the final state. And in this case, you reach a quinted p in the final state okay? And again, you have the interaction the, the energy difference here again is the exchange interaction between the 3p electron here and the 3d electron here, it's again the exchange term that I explained to you before. And the exchange energy lowers this energy, so it's lower, the septed p is lower than the quinted p, okay? So that's very nice, so we have now atomic multiplied theory, and this can explain the K beta lines to us. So this is not experimental spectrum, this is the manganese. K beta line, and you see, indeed, a lower energy, a lower emission energy. So that means the energy difference here gives you the emission energy. So the quinted P [COUGH] is at lower energies. It's here. So this corresponds to this configuration. And at higher energies, I have the septet P. So now we have the, the, the rough electronic or spectral shape we have explained, using very simple, atomic multiplets theory, arguments. Okay? And what is important is here, the orientation of the unpaired spin in the, in the 3p shell, yeah? So you realize already this spectroscopy is sensitive to the spin state, right, because there are spins interacting here. But you see there's the shoulder. The shoulder here I have not explained yet. But we can explain it. So what may happen, and that's why it's important to consider all electrons or many electrons in your system. What can happen is when you have this decay is that one spin the 3D shell flips over. You can calculate this very accurately. These are the so-called non-diagonal matrix elements. Slate already in the 60s was able to calculate this. And there's a certain probably, a very strong probability actually, that this spin in the 3d shell flips over when it has a 1s 2, 3b condition. And what you see if you count the spins now, or if you look at the LS term it's also a quinted P term, so the LS term of the entire configuration here is quinted p. Also, this, so you can realize a quintet P final state using two different configurations. And that's very important because when I have the same total LS term, those those states interfere. Yeah. And that, that is part of the reason why I get this intensity so strong for this, for this configuration here, yeah. And this spin-flip explains the shoulder here. So the shoulder B here is the spin-flip. And it shows you that the shoulder is very pronounced. And it shows you that the, the probability for this spin-flip is very high. Yeah. What I want to show you in this slide here, is that, the equivalence between the manganese K beta lines, that have a 3P whole in the final state. And 3P x ray photo emission. Because, in both techniques, you reach the same final state. In one case, I first created one as whole. And then the one as whole is replaced by a 3P hole, right? So I have a 3P hole in the final state. In the other case I do photo emission. I take the 3P hole out right away. Right? At the end I have the same final states, and I com-, compare them, and indeed I turned this around because the electron, photo electron people they like to plot it the other way round. So that's why it's here. Mirror, mirror written. And yeah, but you get the same, you get the same state. This is peak A, this is septed peak, quintet peak quintet P. Right, you get the same states in K Beta as in photo electron spectroscopy. The spec, the instrumental resolution is much better in photo electron spectroscopy. But, this is a hard... Okay. Now the chemicals entity. We explained now the spectral shape of the k beta lines. Are there questions actually because this is quite important. Questions they is all clear? I guess not. So, so what is the chemical sensitivity, where does the chemical sensitivity come from? So, I told you, it's a, it's a, it's elected to the spin, right? It's a, it's an exchange into action. If I now change the spin state in my valence shell I will get different energies, they also configurations, right? Just because I have different exchange energies. Yeah, so if I have a manganese 3+ and a manganese 4+ will give me different spectrum, yeah, so, the chemical sensitivity. And this is true,this is an experimental data now. If I change now the formula of the oxidation state of the manganese with different fluorides, I see how the K beta line changes, it moves here to lower energies with increasing oxidation state. Just because the spin state in my valence shell, the spin in the valence shell, changes as a function of the oxidation state, yeah. So, that's how I'm chemically sensitive, I'm sensitive to the spin state. >> So, what is the contribution of the over charge. Is their contribution at all of the charge? The charge, is that the atomic route but the charge moves somehow right? So how much contribution purely from the charge. If you could neglect the spin, how much contribution would you have in the energy shape? >> That is a very good question again. It's a topic of discussion. They have various publications that came out recently and another one coming out now very soon. Their people investigated to an extent K beta lines changes because of the charge. This one aspect that I haven't mentioned yet. Or I think it's Robin mentioned this morning the reason why line of flourcene line or real line may change is because of screening effects. If my valance valance electron density changes. You can say but of the electrons that are valance shell they also have a certain probability of being close to the nucleus right because the radial distribution. It's not a delta function. It has a extend, right? So also, the electrons in the valence shell, they screen electrons in the 3P shell, or even the 2P, right? So if I remove, an electron from the valence shell. Even my core levels, 2P or 3P. They will sh, they will move a little bit. The question is, by how much? Or you could ask a question by how much does my charge actually change when I change my oxidation state? People did calculations, and it's surprisingly little. The real charge, depending on how you project your charge on your ion. The charge, by how much it changes. So we, we don't have a really ionic picture. We don't have a manganese four plus ion there. That's, that's... That is not true, right? The real charge between the different oxidation state doesn't change very much. So, people recently, there's also Rubin Munheim from Maser, they investigated to what extent the K beta line shifts. Because of changes of the charge density it's very little. It's really, to a large extent, the spin state that causes the spectral changes, surprisingly. Which is good for us, because we are mainly interested in the spin state. And this has been used here around the corner, at the free electron laser at LCLS. So, what people do, they, they, looked, you saw these images probably before. If you look at protein, protein crystals. You well, you excite them with the laser that can be important to reach excitement state obviously a near-field LCLS pulse and then they use a x-ray spectrometer here that is dispersive. That means the x-rays that come from the sample, they hit this this spectrometer and they hit the position-sensitive detector. That then gives you, [COUGH] the K beta lines, right. So the sample comes here, the X-rays hit the spectrometer, the analyzer crystal here, and then on the two-dimensional detector, I get my K Beta lines. And they change as a function of oxidation state, yeah, and the people who have done this a lot in, in, in proteins in order to redox mechanisms. In proteins. But, what I wonder here, and there was an honest discussion, I think, especially in Stanford, In my free electron laser with this enormous brilliance. Am I, should I, am I able to measure, K beta lines over an intact sample. Why I'm asking this, you, you also open. You also, those, pictures, of the Coulomb explosion, right. You've all seen this before. If I have a Coulomb explosion, and this is, I, I don't know the answer, but to just to, just for the poetry maybe. If I have a Coulomb explosion, would I be able to measure. Okay. In the flourescence line, that is really reflect the ground state electronic properties or, that corresponds to flourescence line that it would measure at the sunkatoon variation source where I don't have this high brilliance. The argument there is that, I have, I have my lifetime, the lifetime of the excited state. Well, the Coulomb explosion, I'll tell you my reasoning. The Coulomb explosion happens, correct me if I'm wrong, I strip off the electrons, right? And I have many charged particles, ions, and the whole thing falls apart because they, repulsive force between the different, between the ions. Of course, the, the argument there is that the, the atoms they start moving after I measured my deflection pattern, so I detect and then destroy, but the electrons they, they react much faster no? They should, react to the to the pulse, to the electromagnetic wave for my for my free electron lasem on the on an attosecond time scale. The lifetime of the intermediate state. Is a femtosecond roughly, so my electronic structure should be destroyed, much faster than, my fluorescence decay occurs. So strictly speaking I should not be able to measure my fluorescence line. The fact that we can do, may indicate that we do not have, exposure. Maybe, under the conditions. But this is not entirely clear. >> So when, when you do this experiment, the spectrocity experiment at the LCLS, you don't need to focus the beams that hard, right? Your effective fluence would be as low as with a synchrotron? >> Well, it is the one shot that is important, right? I was one short of one molecule. >> It wasn't a molecule, but it was nano-crystals or whatever you want to, the focus, the beam was focused as much as possible. The question, of course, the question is do I have an explosion or not, and >> In principle, you asked the question, with LCLS examples, the answer is depends. [LAUGH] But if you really focus hard, or you if you focus loosely, then for all techniques you need to focus hard. So for spectroscopy, if you combine it with like a diffraction method [CROSSTALK] >> Exactly, but the goals. spectroscopy very well. >> You, exactly, you're right, but for this experiment the goal was to combine it with the refraction, right? Okay, but I am showing this to point out that that the conditions for spectroscopy, of course, different than for, diffraction in order to discuss whether I can measure an intact sample or not, right? Because in spectroscopy, I'm sensitive to the electron's, the electronic configuration that may react much faster than the Coulomb exposition occurs, right. Okay, another experiment that was done here actually some examples, here at the LCLS is the on the spin cross-over system. That can go by excitation of light, I can go from a ground state, which is a low spin term, singlet A 1 I know something funny happens here, it's not entirely clear. I may go to a triplet state, and then finally go down to a quintet state. This state here was probed at synchrotron radiation facilities already extensively. And this works. But the big question was what is happening here? Is there really a triplet state in between? And and I'm not sure what I want to do. Aya exactly. Now we raise a very obvious question, what kind of spectroscopy would you choose to study the spin state of the system? Well, it's obvious but tell me anyways [LAUGH]. Huh? EPR? Well, it's [LAUGH] it's one possibility. I would propose K Beta spectroscopy, because it's so nice and sensitive to the spin state, right, the exchange interaction. Okay, so in in's system as well, if you go from high spin to low spin, these experimental data, these are calculated data. If I go from Heisman concentration, that's the black line here, to a low spin concentration as the green line here. I'm very sensitive to the spin state using this K beta spectroscopy as I pointed out to you before. Okay, and there's a publication that just came out recently. I think the PI here is Kelly Gaffney. And they, they, these are again the K Beta lines for. Ein and different, spin states, right? Singlets, triplets, quartet, quintet, it'd be nice to see the changes then, and these are different spectra, right? Because they like to emit a transient spectra, at, at pump and prob experiments. And, these are different spectra between, you know doublet minus singlet, minus singlet quintet minus singlet. And then they compare those specter in motor systems. They compare those data then to the spectrum that they measured at a time delay between the X-ray probe of 50 femtoseconds. And what they write to you in the paper is that this spectrum here proofs really that I observes and able to observe the triplet stage. At 50 femtoseconds after the light and exertation. Okay? So that's just to, to show you an example there. So the K beta lines are very sensative, and I can study, spin crossover systems very nicely using, using this technique. Let's see. I don't have so, so much time left, so let me skip a couple of things here. And I want to there's another nice technique that has not been used yet at the free electron laser, but will, I think, very soon, will be very soon. And that's, these are those lines, right? Again, I go to my removed 1s electron. And then I look, the final step is I look at is has a hole in the valence shell. So this, as I told you I, I probe excitations that can also, also observe in principle with a uv spectrometer. I observe excitations within the valence shell. So it tells me directly about the valence-electron configuration which is a very powerful technique yeah. If you look in the one electron diagram again. You see transitions from the valence shell down to the 1s shell. And they are just here, you can imagine the Fermi energy would be here. So it, just below the Fermi level. Yeah? >> Why is it so weak? Is it because of electronic overlap? >> Em, I tell you why, in a, in a minute. You can imagine already, because, what, what, what orbit, what orbitance do I have in the valence shell of the 3D transition metal? It's a 3D transition made up of 3D orbitals in the valence , that's right? A 3D to 1s transition is not dipole allowed. It's only quadrupole allowed. So it would be from this point of view already extremely weak. But I can tell you already we don't see 3D orbitals in those transitions. And I will tell you in the next step what we see. So transitions from the valence shell down to the 1s, yeah? The transitions, where do they come from? So I do not see the 3D orbitals [COUGH]. What can happen is that the orbitals that are on the ligands. So I have here a metal center, and I have eight I'm sorry, six ligands around. So ungerade symmetry. Six ligands around. Then I have the p orbitals of the ligands in the two results. Those p orbiters or s orbiters, they can form the molecular orbiters. So I can do a linear combination of atomic orbiter to construct a molecular orbiter. And this molecular orbiter has uneven ungerade, uneven symmetry with respect to the metal side. Yeah? That means we do an inversion, I, I project the object onto itself, but with a negative sign. And so it's uneven negative parity, basically yeah, uneven symmetry. And the transition form such a molecular orbiter, to the 1s shell of the metal is dipole allowed. Yeah, and these are the transitions we see. So in this, in this we call this valence to coil mission spectroscopy. So emission lines just below the Fermi level are I am sensitive to the electrons that are on the ligand. Yeah? So I don't see my 3D orbitals in this vd transition metal. So that's bad news if I want to study my 3D levels. But if I'm interested in to, in the ligand, what kind of ligands I have, then it's good news. And as shown here if I, I, I look here for chromium, for example. So I have a chromium metal, that's a black line here. If I now put different ligands on my chromium. Oxygen, carbon, chlorine, nitrogen, I see the, the ligand comes with a tag. I can, I can see from the, from the emission line here, I see what ligand I have. That's fantastic, so I have an element selective probe that tells me what kind of ligand I have. There's, to my knowledge, no other technique that can do this. Access cannot do this, because the, the ligands are too similar in Atomic number, and it's a, it's a very nice, it's a very nice technique. It's very, weak lines, weak line as you pointed out, so very difficult to do. But I think there are experiments planned here at the LCLS where they will use those mission lines to study the ligand environment in, in proteins. So this, this is coming up. And the beauty, another wonderful thing about this, this approach, is I can model those data using ground state density theory calculations. I told you a little bit about the, the, the problems I have. When I, when I model inertial spectra, I have to worry about the core-hole, I have to worry about modulated effects, electron-electron interactions. In this case, because the transitions mainly arise from electrons that are in molecular orbiters that are, the localized on the ligands. I get very far with density function theory. And that's very nice. So, what I do, I put the electronic structure of the molecule into my DFT code. I calculate the electron density. And you may notice you have Kohn-Sham orbitals, we have something coming out, orbitals coming out of your calculations. Ground state calculations, so nothing fancy, anybody can do this. They are codes you can download from the internet. And then you just found the matrix element. One electron matrix elements, dipole operator between the 1s shell of your, of your metal, iron or manganese. And, any field molecular orbital. And, and this is what comes out to the black dots here. So this is titanium silicalite. So it's titanium in the silicalite, in the silicon oxide, okay? And so the black dots here are experimental spectrum and the red line is the did you see the six here, are the calculations. Yeah? And you see you get a very nice agreement. Main problem here is this intensity is overestimated. We don't really know why, we're struggling to find out. Still, the correspondence between the calculations and the experimental data is very nice. When initial spectroscopy this is a, you already very happy with this, yeah? As you can learn now, since the calculations work quite nice. We can learn many things enormously about your about the electronic structural system. And you can find out what ligands you have. You can distinguish between an oxygen legand and a hydrogen, hydro, hydroxide legands. So it's a very powerful, it's a very powerful technique that normal people are using. And I'm sure in the future, also at the free electron laser. Yeah. So I think five minutes, or something, just briefly, resonant inelastic X-ray scattering. I cannot say, it is a huge field. I cannot cover everything. I just want to give you a little flavor of. Okay. So what do you do? I showed you this diagram already many times before. Okay? So, I start at the ground state and I had two, let's say, discrete excitations here. I have a continuing excitation there. I have the lifetime for the, for the intermediate state and then state decays and at the final state here that is still an excited state. It also has a certain lifetime. Right? They will begin to cascade, decay, ultimately it will thermalize, to the ground statement, okay? So the, the difference between incident and emitted energy is the energy transfer, right? So let's see energy that remains in the sample if you consider the final state there. Okay now again, we play a little game. We try now to translate this energy diagram there, that I show you before. We translate this now into, into a spectrum, yeah. When I measure RIXS, I have two energies that I vary. I have the energy of the incoming light, that would be my incoming monochromator. And there also is the energy of the final state. That would be my secondary spectrometer. You remember in the beginning I showed we have an analyzer crystal you know to analyze the X-rays coming out, yeah? So we have two energy scales, the incident energy, yeah, and the energy transfer. Okay? So now we try to understand, we try to translate this energy diagram there into a RIXS flame. Okay? So I start to give you the first point. I have an absorption here, so that means that at a certain incident energy and this intermediate state now decays into a finer state there. And I chose the finer state energy here. Okay, now I have to see that if I..... I didn't want this anyway. [LAUGH] Anyway, so if you're, now you saw it anyway. But the idea is, if I, if this intermediate state now decays also to the second final state, right? So it's possible that one intermediate states decays into two final states. Okay? So the question is, where would I see this transition? Okay? So, I think you saw it already, it's at the same incident energy, but a different final state energy. It's clear? Yeah, so this is how you translate your energy diagram into. Usually you do it differently you measure this and translate it [LAUGH] okay. But as a first step, I think it's easier this way. Okay? Now in the next step let's say we go to the next higher, inter, intermediate state. And this intermediate state now it is case to the state. Where would this point be? here? Here? Here? Actually. Ahh, one line is missing, I'm sorry. That disappeared in the, in the presentation. There should be a line here, I'm sorry, it was supposed to be the high energy but for some reason it disappeared off. Yeah, so it's supposed to be a little high energy transfer. So, it's a high incident and a high energy transfer, yeah? And, and then we have also the continued excitations, and they interestingly appear as a diagonal streak. Yeah, because you vary your incident energy continuously and the final state energy varies continuously as well. So excitations into this band here, they will appear as a diagonal streak in your display, okay? >> So, why is it not, how broad is this diagonal streak? >> If you have a frescence line, if you go into a continuum, it's infinitely broad. I actually brought along [CROSSTALK] >> The width? [CROSSTALK]. >> The maybe, maybe this answer your question? >> I think this answers my question [LAUGH]. >> Yeah, so, so what I, before I only looked at the, I neglected all lifetime broadenings, not considering instrumental broadenings. But now, I consider my lifetime broadenings, right? Now I see how my lifetime broadenings extend in the RIXS plane. Now, I, oh god, I made this too quickly, now I have to correct this. So this is the intermediate stage, so this should be gamma n and gamma f. So the intermediate state lifetime broadening extends here in the horizontal direction. And the final state lifetime broadening extends in this direction, yeah? So, this is this is a model system, right? This is, this is not mattered but it tells you a little bit how you can translate an energy diagram into into a RXES plane. That you, that you actually measure. And it's really important that that, to see how the lifetime broadenings. They shape your, the intensity in the states. Yes? >> Does the diagonal engagement show some correlation between the manifolds there? >> A correlation between the two manifolds there. >> There is a diagonal allocation there? Or. So the transitions between those two states, they give rise to this diagonal there. >> The fact that they are parallel, they are going to, do some correlation between them. Can you vary this somehow, or? >> When a principle shows you, that it shows you that you replace one co-hold here, with another co-hold. In principle, if it shows up diagonally like this. It shows you that the co-hold does not really interact or it weakly interacts with a photoelectron. If it does interact, you get some nondiagonal effects. If it starts reacting it becomes non-diagonal. In principle, you can explain everything with a diagram like this. If the co-hold potentials are different in the two states. You get so called non diagonal features, in your, in your RIXS plane. And if there's a strong electron electron interaction, as well. There are many effects that can give rise to non-diagonal effect. This diagonal streak, now, is, in principle, just a fluorescence line. That means you fully ionize your system. And default electron does not interact anymore with your, with the remaining ion. Then you have the florescence line. In this step the florescence line coming out. And and this give you then, then rise to the diagonal streak. And then again the photo-electron in it's final state is not interacting with the remaining ion. That kind of works, let me show you an experimental spectrum now. So we are, this is serum dioxide, so we measure the obsorbtion spectrum and we have the emission spectrum, I showed you this before. This is another experimental RIXS plane of serum dioxide, and here it is a little bit magnified. So, in this case we have a 2p3 half whole, in this case we have 3D whole. And the life time broadenings of the, of the 2p3 updates is horizontal here and the final state is, is vertical here. Yeah. And what I can do if I take, if I take the integral of all spectral intensity I get a conventional absorption spectrum. If I take a diagonal cut here, I ca, I obtain something called high resolution absorption spectrum. It's not really an absorption spectrum, it has to be careful. It just a diagonal cut through the RIXS plane. But what is important here, if you compare the conventional absorption spectrum to the high resolution spectrum here. That you, the high resolution spectrum you see many more spectral features. For example, you see the 4F orbiters that we discussed before. We see the here in the high resolution spectrum the features here. But you don't see this or barely see them in the conventional absorption spectrum. Yeah? The reason, and that's a very important point, the reason for this sharpening effect. It's simply that I move, kind of, at 45 degrees relative to the lifetime broadening in my, in my system. In order to observe the sharpening effect in this, kind of, RIXS spectrocity, we do not need interference and there's nothing fancy going on. It's simply the effect that my lifetime broadenings, if I consider properly my lifetime broadenings right away. Is nothing fancy. In principle, you can also consider this, this here, if you, this, this, this is a cross here, you put here. And then, you consider your continuous excitations as an infinite number of discrete exaltation's infinitely close to each other. You just move, your cross here, you move it through and ultimately, and then you obtain this band. Here. And if you go through the maths, you realize that the lifetime broadening of your fluorescence line is the sum. It's the convolution of two, so it's the sum of the lifetime broadenings. Yeah? So there's nothing fancy in the RIXS process that gives rise to this sharpening effect. It's rather banal. If you just look at the Heisenberg equation and you can neglect all interference effects, it comes out correctly at the end. Okay. well, what, what is interesting in the RIXS process of course, if you [UNKNOWN]. If I reduce the, the, the energy transfer from let's say 1,000 electronvolts that I showed you before. To let's say 1 electronvolt, and I mentioned this before, I can observe in the RIXS process, low energy excitations. These can be collective excitations, modern excitations, dd excitations, charged ones for excitations. This is what many people do using this RIXS process, right? So they would use the energy transfer here. So very small numbers can be used to 50 million ev, depending on what ind of excitations you want to look at, yeah? And that's that's the idea, of the RIXS spectroscopy. And just an example, we measure here, this is Chromium, and Magnesium Chromium Oxide. This is the RIXS plane here, and the energy transfer is the, of two, three, electron balls. And we get two excitations here, that you can then, compare to this spectroscopy. And, it shows you that, that you reach, the same final states in, in RIXS or hot X-ray probe. As in UVA spectroscopy, you can see the same excitation's and with this I thank you for your attention. [NOISE]. >> For more, please visit us at stanford.edu.