 Atomic physics is well aware of the vast research contribution Steve Manton has made to this field, but today we have the opportunity to discuss the results with him, and he will be making a presentation on photoanization of free and confined atoms in the recent times. Steve. Thank you. I have to give a disclaimer to begin with. I found this presentation and put it together about seven o'clock this morning, so it may be a little bit rough, because I hadn't had time to go through it as much as I would have liked to before making the presentation. In any case, what I'm going to talk about today is, as the title suggests, some relatively recent results in the area of photoabsorption by a number of different kinds of targets, and you know, it will not be an exhaustive study, but I will try to be representative, give you representative results in various areas. Now to begin with, up until probably the 1960s, it was thought that photoabsorption, photoanization of atoms was pretty much a one electron process, and there may have been correlations that did a little, but they weren't terribly, terribly important. This turned out not to be true, and over the years we have found all sorts of interesting things which occur owing to the fact that the world is not a single particle world. Wave functions are not single particle wave functions, but really, really do include many body correlations. Evaluation is just a general term we use to say that the electrons of an atom or molecule or solid or anything do not move independently, but they are correlated with each other, thus the term correlation. Okay, to begin with, that's what this says. I said it in other words, it's not a one electron process, and this correlation could take many different forms, you can be in the, remember to calculate photoanization you have an initial state wave function, and a final state wave function, and you'll take an integral with something in the middle. Something in the middle is general, the initial state wave function, the final state wave function have to do with the particular case you're dealing with. All right, now, take a look at this. In the final continuum state wave function, correlation or configuration interaction generally goes under the rubric of inter-channel coupling. One of the reasons we use that term, which is really sexy, it sounds like something very, very special, and it confuses experimentalists, which is always good. The essence of this is that when you have degenerate channels, what I mean by degenerate channel is when you could have photoanization from more than one sub-shell at a given energy, you know, at high energy, you can get all the different sub-shells. Well, it turns out that due to this, what is called inter-channel coupling, if you have a channel with a large cross-section, and a little of it is mixed in with another channel, with a weak channel, it can make a big change. The other way doesn't do anything. You mix a little of a weak channel with something as strong, you know, it's like if you have something weighing a thousand pounds or something weighing one pound, and you get a tenth of a percent mixing, well, a tenth of a percent, a thousand pounds is one pound, it'll double the one pound weight, a tenth of a percent of the one pound added to the thousand ain't going to do nothing. So, this is kind of the idea of this. And what has been found, for example, at high energies, this can actually change the asymptotic form of the wave function, of the, not the wave function of the cross-section. It turns out in the single particle model, you get one asymptotic form, and when you put in inter-channel coupling, it just changes in many of the channels. The details I don't want to go into right now. Another thing that happens, and this is interesting, this is a calculation for the photoionization of krypton in the range, you can say, about 600 to 1200 eV. The thresholds of these, 3S, 3T and 3V, are several hundred eV. And this is really well, well above threshold. By the way, up until this work was done, it was thought that, well, there are correlation effects, but they're mostly around thresholds. This is not around threshold. This is very high above threshold. And look at what happens. We have, look at around 600. We have these dashed lines. These dashed lines are essentially the single particle model. But when we put in this inter-channel coupling, look at what happens. With the biggest one, nothing. With the one that's the middle-sized one, something. And the smallest one, a lot. And this just continues on. And look what happens here. When you go up to high energy, high-ish energy, this one is the biggest one. And it doesn't change. These two change. This is a consequence of what we call inter-channel coupling. And I guess I don't have it in this slide, but there's also experiment. And while the experiment isn't perfect, it does a pretty good job. But in every case, the solid line is closer to experiment than the dashed line, showing that we have some idea what we're doing. Oh, this is a nice one. This is, instead of the cross-section, it's the so-called angular distribution parameter beta. This was kind of cool. Because the independent particle model gives this. The coupled model gives this. Don't worry about all the variations there. And this is experiment. As you can see, when you put in the inter-channel coupling, every point is within error bars of the experiment. And this led to some consternation. When we submitted this as a paper, we had a big fight. We were right, but we lost anyway. Because there had been some earlier experiments that basically had been misinterpreted because they thought things were single particle. However, we also looked at the measurement of the cross-sections of the... See, at this energy, you can have the 2s and the 2p of neon. By the way, this is neon. I should have mentioned that. And we're well, well above threshold. Threshold, the neon for the 2p is about 0, 20, Rdv, and for the 2s is about 40dv. And here we're talking about a kilovolt, 1,000dv. We know we need a threshold. So what we looked at then was the ratio of the two cross-sections, the 2p and the 2s cross-sections. Now, from an experimental point of view, this is a very good thing to do because many of the experimental errors are canceled out that way when you just take a ratio of two cross-sections. And here's what we got. This is the ratio. The solid line is our calculation, and these are the experimental points. And all of these experimental points were done before the experiment, before the calculation. And the experiment said, OK, if you think you're so smart, we're going to measure a point and we're going to do a really, really good job of it and get tiny error bars. That's that one right here. That was done after the calculation. That was a really, really good calculation. In other words, all these equations that we write on the board and things, they really work if you do them right. OK, however, now we have found something new. This is in palladium. It says so right here, pd. One of the things you learn in this business, if you're in it long enough, you learn where all the chemical symbols mean. It's something. And here's what we found. In this case, this is where the two p-thresholds are. That's a different kind of notation. It's called x-ray notation. And we have two of them because now these are deep. Look, 3.2, 3.3 kilovolts. And relativity is important here. So the spin orbit effect is really, really large and you get rather than a single two p level, you get the two p one half and the two p three halves. That's what three halves and one half, what these are. And the difference is almost 200 EV. That's a big spin orbit splitting. You have to remember that the spin orbit splitting goes as z to the fourth. There's a z in it from the nuclear charge. And then there's another z cubed that comes from the one over r cubed in it. So z to the fourth. So the effect of z for a two p in palladium is really large. And to the fourth, that makes it really big. So, but look at what we find. We find that in the 3d cross section, due to this inter-channel coupling, we get actual structure, significant structure. And when we leave out the two p, we don't put in coupling with it, we get this. So not only do we get a change in magnitude. By the way, I didn't show the two p cross sections here, but they're much bigger. They're well above the ceiling here. So not only do we get a change of magnitude, like we saw in an earlier slide, but we also get structure. Yes, I'm sorry. Because there was an experiment done in silver, which is the next element. Show that in a moment. This is the experiment along with the theory. This is the actual experimental results. We figured that move changing z by one for such an inner shell wouldn't matter much. And the experiment was only relative, so it had to be normalized at one point. But as you see, the agreement is really awfully good. But to do better, again, we take the ratio of the two channels. And this is it. This is the ratio. So here, nothing is normalized to anything else. That's a pretty remarkable agreement. In other words, we're pretty sure we understand what's going on. Because again, there's nothing normalized to anything else here. Actually, how we got to this was a funny story. I was at a conference, and I gave a talk that included something about inter-channel coupling. And after the talk, one of the people at the conference talked to me. He said he had some experimental results to show me and asked me if I understood them. And he showed me the results, and I said, yes, I understand them, and we can do that. And then we did it. Actually, the calculations were done here. I mean, not in this room, but, I mean, here at IIT. And OK, enough about that. What about ions? Free ions. And here, we are aimed at calculations that approach exactness. And owing to the fact that there are really good possibilities for doing experiment now, there's been a big push to do really good calculations on ions, on the photoionization of ions. See, ordinarily, it's hard to get positive ions here because they don't remain around very long. Let's see what we get here. This is scandium. Actually, scandium double plus. Why are we interested in scandium double plus? How on earth did we pick something like this? Well, it is the simplest ionic system that has an open D-shell. The structure of this is argon plus a 3D electron. What turns out that once you get D-electrons there in open shells, it becomes a far more difficult calculation. Having been involved in the calculation, I can tell you, it's really difficult. Anyway, here's our results. This is non-relativistic, this is relativistic, and this is experiment. As you can see, the agreement isn't bad. You see what relativity does. Here you have a single resonance that breaks into two and you really do have two. The agreement isn't perfect, but it's pretty good. We think we have a pretty good handle on how things work here. I'm not going to go into detail about the calculation, but this was a relativistic R-matrix calculation. We did some... This is the next one in the isoelectronic sequence. Again, argon plus a single electron. Let's see now. Trying to remember which is experiment and which is theory. I believe this is theory, this experiment, and what's this? That was contaminants. That's why we didn't see it in the theory. Anyway, so you get the idea, and then we went on to a whole bunch of different ones. They just get very messy. I'm going to... Oops, go back here. That's enough of that. All right, another thing. Up until relatively recently, it had been taken as gospel that for relatively low energies, the dipole approximation was essentially perfect. You didn't have to worry about anything else. Turns out that's not true. There are many examples of non-dipole effects being important at low energy. However, not in the total cross-section, but in angular distributions. And here's an example for magnesium, 3S. And the example is interesting because in the dipole channel, which is called E1, you have what is called the Cooper minimum. It goes down to a very low cross-section. And in that region, you see the E2, the quadrupole cross-section, which is really very small, right around here it gets larger. That has really interesting consequences. That's a close-up. You see, the photoelectron angular distribution when you include the next order looks something like this. The red and the violet are the non-dipole part. So if you don't have dipole, if you don't have just dipole, you just have beta, 1 plus beta times P2. That's the second Legendre polynomial. And the rest of this drops out. So as you can see, you would only get a dependence on theta, theta being the angle that the photoelectron direction makes with the polarization. But here, you get that and phi, the angle that it makes with the photon direction as well. And what we find then is just to give you an idea. Don't worry about that. Instead of a simple sine squared, a sine squared theta, but it's b cosine squared theta distribution, you get something very, very complicated as a function of theta as a function. I mean, clearly it's not constant as a function of phi, function of theta. You get a really, really complicated angular distribution. Another thing that has been of interest lately, about 20 years ago, Buckminster Fullerene was discovered. The C60 that was shaped like a soccer ball was also shaped like a geodesic. Geodesic is a kind of structure. And it got its name from an architect whose name was Buckminster Fuller, who designed buildings like that. And so they are not exactly but close to being circular. They are called Fullerene's Bucky Balls and some other things as well. The actual full name is Buckminster Fullerene. Okay, not only can you create these structures, but you can put things inside of them. You can confine atoms inside. And this has been done experimentally. And okay, you can ask yourself, how does this affect the property of the atoms? Or another way to think about this is, if we put the atoms in there and it changes the properties, are this system, it's actually a molecule I guess, is it useful for anything? The answer is maybe. For example, people have talked about using a structure like this for drug delivery to a specific location by putting it inside the Bucky Ball and putting something at the location that will dissolve the Bucky Ball. And so the shell keeps the drug from interacting anywhere else other than the location where you want it. And you can imagine how this would be useful for something like chemotherapy. I don't know if you know exactly how chemotherapy works, but chemotherapy is basically poison. And the idea of it is it is supposed to be used to kill the tumor before it kills the patient, but it's close very often. If you can get it directed to the tumor and not to the rest of the patient, that would be really good for the patient. Not so good for the tumor, but good for the patient. But really chemotherapy is just poison. And we hope that it's more toxic to the tumor than to the person, because there's no way to, at present, to really make sure that it works there. The human system being what it is. I mean, think of it. You get a pain in your head. You take an aspirin. It goes to your stomach. How does that work? You know, it works everywhere. You would like to get something directly to your head, but there were no good outlets. Anyway, so that's one possible application. So what we're interested in now is how the existence of a cage around an atom changes its properties. And as you can see, this is a relatively complicated system. I mean, if we take the ordinary one, the one that's most common is C60. So to do this calculation, you have 61 atoms. That seems really hard. So you make some approximations. First of all, we work with systems where the atom is located in the center. So we maintain spherical symmetry. Although the C60 is not exactly symmetric, it's very close. Not exactly spherical, but it's pretty close. Now the first thing that we found was confinement resonances in the photoionization cross-section when you put something in there. And the physical explanation is real simple. You can have the photoelectron going right out when it's ionized by radiation, or it can go backwards, hit the wall, and then go out. So you have two different waves going out. Actually, you get lots of them, and they interfere. And the interference is based on the actual geometry of the situation. So actually, from the interferences, you can learn something about the geometry. And we've done numerous calculations of this sort of thing. And here's an example. This is in xenon 4D, and the dashed line is the free atom, and the solid line is what we think it actually is inside a fullerine making a model of the C60, and the dotted line is a very simple model, which we think is wrong. But anyway, just to give you an idea, this is what these confinement oscillations look like, and they are ubiquitous. We find them everywhere. We find them in data, in the angular distribution parameter. Here, this is for neon 2p, and the dashed line, or it's actually a dotted line, I guess, is for the free atom, and you see our calculation for neon at C60. It bounces around. It oscillates, or resonates, whatever you'd like. Now, one can ask, what happens if you move off-center? Well, here, without going into the detailed calculations, you can guess, because as long as you're on the center, then the phase difference is the same no matter which way you go. But if you're off-center, you get one phase here, but if you're going further here, you get a different phase. So probably this confinement oscillation business would get much less, and in fact it does. You've got to look at this carefully. Here, the solid line is basically at the center, and this, I don't know what color, the violet, it's well off the center, you see. It's almost gone. It's just basically the geometry of the situation. So we have some idea what happens if it's off-center. So if you do an experiment, and you see these oscillations, you know it's pretty much close to the center. So here's an experiment that was done. This is in xenon 4D, and as you can see, you really do see these oscillations. They are actually larger than the error bars. There's a new experiment that has smaller error bars that also sees this. So it really does exist, and this is one of the phenomena that was predicted well before it was measured. All right, moving right along. C60 is not the only fullerine, and as you can see here, we show a series of calculations of argon in C60 and argon in C240. You see the oscillations differ. By the way, this is the 1s of argon. The reason we took the 1s is 1s of argon is really, really tiny. It doesn't know about the wall around it. So everything you are seeing here is due to the final state. Nothing because the initial state doesn't change. And that's why we picked this calculation to do so we could focus on just as what is happening in the final state. And you see with the 240, you get more. With 540, you get still more in the way of these oscillations. And if you get a really big fullerine, you can not only put an atom inside, but you can put another fullerine inside. And that's where the bottom two curves are. You have argon in C60 in C240, and the bottom we have argon in C60 in C240 in C540. All of these are possible, at least in principle, and this is the kind of things our relatively simple calculations show. However, oh, let's not talk about that. In the calculations that I've just shown you, the electrons of the fullerine itself are considered inert. They're just sitting there, and it's just the atomic electrons that see a different potential. What about, is it possible to have interactions between the atomic electrons and the shell electrons? The answer is yes. But in order to do that, we have to consider all of them. And so we have a, use a more accurate model of the system, and we treat the system in the following way. We smear out the, you see the carbon atoms, they have two S and two P electrons that are relatively weakly bound, and then they have one S electron that are bound by about 300 eV, so they're very different. So what we do, we consider the fullerine as a bunch of carbon plus fours, that is carbon nuclei with the one S is still there, so the net charge is just plus four, and we take all of the other electrons and assume they're delocalized. That's 240 electrons there. And then we take the positive charge, and rather than having 60 points, we smear it out. It's called Jellium. That's from the solid state physics. It's an approximation to be sure, but it's easier to solve. So this is a Jellium approximation. And all sorts of things happen. Now, first, let me tell you, supposing I were doing xenon inside of a C60. Well then, I would be doing a calculation of the 240 electrons of the C60 plus the 54 electrons of the xenon, which is 294 electrons. That's a big, messy calculation. And so I have to do some approximation. And the approximation that I do is called the time-dependent local density approximation. And, you know, I have to... So what I have here, then, is the atomic potential plus a well representing the shell potential. And that's here. Atomic potential and a well potential. And then I do the calculation for those 294 electrons in this potential. It gets messy. One of the things we see, supposing you have an atomic level that is bound by some energy here. Supposing you have a well level that's also bound by some energy here. And then you need degenerate. What happens? Well, the degenerate perturbation theory tells me what happens. I'm going to get a wave function that's a linear combination of the two. And that's exactly what we see. This is the 5s of free xenon. However, in here, it is mixed. And I get two wave functions, one that looks like this and the other that looks like this. In other words, positive and negative mixing of this wave function and this one. This is an atomic wave function. This is a shell wave function. And they mix. This is called hybridization. The molecular physicists knew about it for years. Not in this particular case. I didn't know about it, though. Now, remember, when doing photoionization, this would be my initial state. If I have a very different initial state, I'm going to get a very different cross-section. That's the importance of hybridization. It changes things very greatly. For example, this black line, solid black line, I want you to focus on that. That is the 5s of xenon, free atom. This is one of the combined levels that looks like this. Now, you say, well, it's not terribly different. Yes, it is. This is a logarithmic scale, just as much as a factor of 10. It completely changes the cross-section. There's no relation at all. So hybridization, which the simple model doesn't take into account, is very, very important. That's what we find. This is something that should be noted. It's also interesting. The 5p of xenon is not hybridized. But look at what we find. This is the 5p of free xenon. This is the cross-section of free C-60. Again, this is a factor of 100. It's about 100 times larger. Remember what we talked about in the first example I showed? Not in this business, with free atoms. If you have a cross-section, which is degenerate with a much larger cross-section, you can get very strong inter-channel coupling. That's what happens here. Look at this. This yellow, kind of a nice yellow, kind of mustard, I think. Not just yellow. Look what happens to the 5p. Owing to this inter-channel coupling, the cross-section goes from about 50 to about, what, 800 or so. It increases by about an order of magnitude because it mixed with something that's really very large. Little piece of that. So what we find is very significant inter-channel coupling here. There is something called a sum rule, which says that the integral over the cross-section, suitably scaled, should be the number of electrons in that state. So 5p should be about 6. And this comes, if I integrate this over this particular range, actually you need to integrate out to infinity, but just over that range, you get about 545 in neon, in xenon. Here, in this case, with the inter-channel coupling, you get something like 50. In other words, strength from the shell is transferred for some reason to the atom. And furthermore, when the cross-section for the shell goes down and is no longer much larger, you see, you get pretty much the same cross-section, except for these confinement oscillations. So this makes a consistent picture here. And however, these are non-trivial calculations. These require a lot. What comes next? Do I remember? You see, you remember they were too different. You know, when you had the atomic level and the shell level, it mixed into two different cross-sections, into two different states. So two different cross-sections were too messy to show all of them on previous slides. This one is the other one. Looks like this. Anyway, this is the cross-section for the free C60. But you get all sorts of interesting phenomenology here. And that's about it. That gives you sort of a flavor of the kinds of things that we have been doing and the kinds of things that we are capable of doing. Now, I glossed over the details of the calculation. A lot of them took a long time and are very messy, but they can be done. And they can give you all sorts of information and not only explain experiment, but give a guide to how experiment might be done. You know, what experiments to do, what might be interesting. Because usually experiment comes first, not always. Sometimes we predict things first. And I will stop here and thank you for that. Thank you for your attention.