 We are very privileged to have today Mr. Steve Manson, who will give two guest lectures in this course. And today is the first one. So today he will review the topic of photoionization in which he has expertise for a very long time and anybody reading literature in atomic physics would have come across his papers for well over four decades now. So, over to you Steve. Thank you. Whether it's a privilege or not, you'll decide after the lecture and you may not agree. Well, today what I'm going to do is enlarge on what you have already learned about photoionization, starting perhaps with some things that you already know. As a matter of fact, the first invited talk I ever gave, never mind how long ago it was, the gentleman who invited me gave me some very useful advice. What he told me was, your audience is always grateful for anything that you tell them that they already know. And that's extremely good advice because it connects with the audience. If you start off too high, you've lost them to begin with. Now, I am hoping that some of the things that I say in the beginning, you've already had in this course and know something about. So, let's start. This is very, you know, from my point of view, it's very low-tech. There's a piece of chalk and a blackboard and some note scribble to keep myself more or less in order. Okay, photoionization. That is the interaction of ionizing radiation, light, ionizing electromagnetic radiation with matter. And we're going to talk specifically about photoionization in atoms because that's what I know about. And to begin with, light and matter, the coupling goes something like this. The interaction Hamiltonian is a bunch of constants here. And then e to the i k nu dot r p dot epsilon. And what these things are, this is the wave number of the light. This is the momentum of the electron. That's the polarization of the light. This is hard. And so we simplify things. And if the wave number of the light is small enough, that is if the photon energy is low enough, then for the typical size of an atom, this is a small number. What do we have when we have e to a very small exponent? You can make a Taylor series expansion, you know, like e to the x is 1 plus x plus x squared over 2, etc. And okay, how far shall we go to truncate it? You know, you don't want to go to infinity. How about 1? That makes it a lot easier. This, for historical reasons, is known as the dipole approximation. Now, there are several important things about this interaction Hamiltonian, aside from this dipole approximation. Number one is that it interacts with a single electron. It doesn't interact with two electrons. It doesn't interact with three electrons. It's a single particle interaction. That is very important. Secondly, this is small. As a matter of fact, it's of the order of alpha, the fine structure constant. Now, the fine structure constant is what? It's roughly 1 over 137. Or, taken roughly, it's 1%, 0.01. And so, remember, when you get a probability or a cross-section, which is the absolute square of a matrix element, you get alpha squared in there. So, light really doesn't perturb the system very much. So, it's an excellent probe of the properties of the target system. You see, when can you use other probes? Electron or proton. They're in the coupling, instead of being of the order of alpha, they're of the order 1. So, when you use those as a probe, you get kind of a concatenation of the properties of the target and the properties of the interaction. Here, you can get rid of the properties of the interaction because they're really, really small. That's another very important thing about photoionization. Okay. Then, using angular momentum rules, you get selection rules in photoionizing transitions. In other words, if you start it, if you are photoionizing an nL electron of an atom or molecule or anything, actually, you have certain selection rules, namely that this can go to a final state. I write epsilon, meaning it's ionized. There's some energy in the continuum. Epsilon L plus 1 and epsilon L minus 1. In other words, you can have, that's the selection rules, L to L plus or minus 1. And the cross-section, or the probability, is made up of the sum of the absolute squares. So, in a general sense, the cross-section for photoionization is, again, a bunch of constants. We don't want to worry about the constants for now. And mL minus 1 absolute square plus mL plus 1 absolute square. And if you're doing a calculation, this is what you have to do. O in many electron atoms, it gets messier. And if you put in relativity, it gets messier. But this is the fundamental idea. It's the sum of the absolute squares of the various matrix elements. We sometimes call them amplitudes. Now, at this level of approximation, I want you to understand the approximations that we've made. This is first-order perturbation theory. However, with this alpha here, first-order is really very, very good because the next order is down by another factor of alpha and alpha is smaller than 1%. So first-order perturbation theory is good. And if you're at low energy, this is very good. So while it's not absolutely exact, it's pretty close to exact. And so using this for the hydrogen atom, you can do the calculation we say exactly. However, exactly, it's kind of exactly ish. And what ish means is... But anyway, at this level, we can do hydrogen exactly, the hydrogen atom exactly. And we find a cross-section, or probability, with energy. This is the photon energy. This is the cross-section. And it looks something like this. Whoops. This is supposed to be a much straighter line. It has some threshold energy. And it falls off monotonically. It was originally thought that for all atoms, things behave this way. Not true. If you do a hydrogenic model of an atom, that is, assume an effective z, and do the calculation, well, this thing scales with z, but everything looks like this. However, with real atoms, things are different. Experiment tells us that. You see, to do even approximate what happens in a real atom, you have to take the electron, or the electrons of the atom, in some realistic potential. Now, a realistic potential, let's say you have some many electron atoms, charge z. You kick out an electron. That electron, when it's out here, sees a singly ionized system. So it sees a z of 1. When it's all the way in here, right next to the nucleus, it sees a z of capital Z. So you need a potential, v of r, which goes as capital Z over r. r goes to 0, and goes as 1 over r, actually minus its tractive, for r goes to infinity. This too is really important. If you use a potential which does not have those properties, you can get complete nonsense. And if you look back in the literature before people use this and just use the effective z's in the calculations, that's what they got. Okay, so in order to calculate this, you have to calculate an amplitude, a matrix element as it were, of this thing. I'm not going to go into detail about how you do it, but look, the matrix element from some state i to some final state is just the initial state, the interaction, the final state. Now, as I mentioned, this is nearly exact. So if you get an initial state wave function and a final state wave function that are close to exact, you get a very good answer. However, for real systems, for multi-electron systems, getting initial state wave functions and final state wave functions, which are nearly exact, is really, really difficult. So one has to make some approximations. You see, the Schrodinger equation, I'm going to do essentially everything non-relativistically. We're going to talk about the Schrodinger equation rather than the Dirac equation. Well, the Schrodinger equation is a partial differential equation. Fundamentally, we know two ways of solving partial differential equations exactly. One of them is separation of variables. The other one is guessing. If you can't do either of those two, you have to use approximations. Now, sometimes we can get very good approximations, but let me mention that for more than one electron, we know no way of separating, of separation of variables. Maybe there's some coordinate system that we haven't figured out yet, maybe you can, but nobody's figured that out. So one has to use various kinds of approximations. Anyway, we'll come to that. Now, so the electron moves in some potential, like this. However, if you remember, if you make this approximation of a single particle potential, that is just a function of scalar r, very important scalar r, the Schrodinger equation is then separable. However, you get kind of a funny, equivalent one-dimensional equation. Why is it funny? Because you have an extra term in there. How should I call it? An effective potential, which is the actual potential plus the centrifugal potential. I mean, that's just like in classical physics. If you have a rotating coordinate system, and you're just looking in the radial direction, they used to call them fictional forces. I don't think they do anymore. I'm very old, and that's what they used to call them. Anyway, so you have this v effective, which is the v of r, plus l, l plus 1, h bar squared over 2m r squared. Now, this is just like in classical physics, where the kinetic energy due to rotation is the square of the angular momentum over the one-half square of the angular momentum, one-half l squared over the moment of inertia. What's the moment of inertia of a single particle? It's just mr squared. That's what this is. That's all. However, just considering this, without doing any detailed calculations, just considering this has some consequences because this v of r is attractive. This is repulsive. Anybody who has ever swung something around knows that that force is repulsive because it tends to, you know, if you've ever seen in the Olympics, the hammer throw, would they go like this? I mean, they're just using this. Okay, clearly at small distances, since v goes as 1 over r, and this goes as 1 over r squared, that dominates. At large values of r, 1 over r, 1 over r squared, this dominates. So, what do we know? If I draw v effective versus r, we just show it at least for non-zero l. Talk about that for a moment. You get something at small r which goes like this. Because it's just this behavior. Something at large r, on the other hand, this behavior dominates. And you notice that minus sign here, I wrote it small, but it's really minus. It is attractive, so it looks like this. And how they meet in the middle is of importance. So, you know, it might be something like this. Just to give an idea. So, the electron moves in the field, a potential, an effective potential, something like this. As l gets larger and larger, this gets bigger and bigger. And so what this means is that it's some given energy. So, like this. The wave function, let's talk about the final state for the moment, the wave function of the final state, which is the classical turning point. It doesn't mean that there's no wave function inside it, of course. However, what it does mean is that in the classically forbidden region, the amplitude of the wave function is small. In other words, this angular momentum barrier pushes the wave function amplitude out. What is the consequence of that? Let us consider a particular transition, one that I happen to know about, and we're going to talk about argon 3p. Now, in argon 3p, the major transition is going to be to epsilon d. It turns out that in almost all cases, the major transition is l to l plus 1, as opposed to l to l minus 1. Both are allowed, you need to have both, but that's the major transition. Now, what happens on a graph something like this is the following. 3p wave function looks maybe something like this. However, the d potential, and so at say, right at threshold, the d epsilon d might look like this. And what you see here is very, very little overlap here. This is what we got. If the initial and final state have no region of overlap, the matrix element is going to be zero. If they have a small region of overlap, it will be small. And what happens as the energy goes up? Well, as the energy goes up, like if it's up here, it will look more like this. Ooh, big overlap here. What does that mean? It means this matrix element is going to increase with energy. Remember, well, I've erased it, but the cross-section, the sum of the absolute squares. So what that means, as far as across the cross-section or probability is concerned, it's going to be smaller. This is as a function of e. Let's say this is the threshold energy. This is h nu. It's going to look something like this. And then eventually, as it moves in, energy gets higher, it moves in further, it oscillates in this region, and it kind of oscillates itself to death. So rather than this kind of behavior, we get this. This thing is known as a delayed maximum. It was first discovered about 45 years ago. I'm old enough to remember when it was discovered. And explained. And you can have much more dramatic cases. A really dramatic case is the 4f state of mercury, because the main transition is 4f to g. And in that case, the cross-section looks something like this. This is then hg4f. It actually drops off a little from threshold, but that's due to the f to d, the l to l minus 1. But you see the g. That's l equal 4. That's really large, and it pushes the wave all the way out. And it looks something like this. And this maximum, rather than being at threshold, is I think it's about 140 eV above threshold. That's a lot. You remember when people first saw this. You know, they did, the cross-section was so small here that when they looked around here at this energy, they couldn't see anything. But when they got up here, they saw the 4f photoelectrons. Their explanation was, these were experimental, as you understand, their explanation was that the threshold energy was photon energy dependent. That's obvious nonsense. It does not... Oh, there were very well-known people who had that in their papers. It was obvious nonsense and didn't fit with quantum mechanics. As I say, these were just people who measured it. And they said, you know, how can it... Because, you know, from measuring around here, remember the Einstein relation. If you know the photon energy and you measure the photoelectron energy, then you know the binding energy. But they measured it above the binding energy here and they couldn't see anything. They say, ah, the binding energy must have changed. I want to take them and slap them. I mean, it was just... elementary quantum mechanics shows that that could not possibly be. Anyway, so this business of a... the laid maximum is ubiquitous. It's all over the periodic table. I mean, the only case you don't see it is for a wave that, like for a P to S transition, because S, L equals 0, you don't have this. And so what's interesting here is that with a very, very simple model, you could explain this phenomenology. Some phenomenology. There's another interesting thing that we find. It was actually first found experimentally. I think it was in 1928. Now that I don't actually remember. It wasn't around in 1928. And that is some cross sections were found to behave like this. As a matter of fact, it was the outer shell of the alkalis. Sodium, et cetera. And the cross section behaved something like this. It was a minimum. It wasn't quite zero, but awfully close. It was first discovered by a British experimentalist by the name of Ditchburn. In 1928, he went back to it in the 40s. And it was explained in the later 40s by a very well-known British atomic theorist by the name of David Bates. And the explanation was the following. This was the outer shell of sodium, the sodium 3S, which looked something like this. 3S has two nodes and that kind of thing. At threshold for zero energy, the continuum wave function, it's kept out a little bit because you have centrifugal repulsion, but it's only l equal 1, so it's not tremendous. But it looks something like this. What happens out here doesn't matter because the important thing is the major overlap is this region. Positive initial state, a negative final state wave function in that region. The product is negative. If you go to higher and higher energy, this moves in and eventually you get something like this. Where the main overlap is positive. Aha! You go from negative to positive, somewhere, if it's continuous, somewhere in the middle it goes through a zero. Aha! And this was the explanation. And this remains an isolated curiosity for a number of years. We were in the late 40s when Bates explained it. There were other developments about why it didn't actually go to zero and that's because there were relativistic splitings. That was explained by a man by the name of Mike Seton in 1952. But then in the early 60s, some calculations by a man by the name of John Cooper found that this was not an isolated curiosity. It happened for almost every valence level in Adam as long as the valence wave function had a node. So it didn't happen for 1s, 2p, 3d, 4d, but it happened for all the higher ones when they were valence, when they were out of shells. And then it became to be called a Cooper minimum. Sort of an interesting story because John Cooper did not have a PhD in physics. He had a master's degree in music. He had an undergraduate degree in physics. He was hired as a programmer to work with a very well-known scientist at the National Bureau of Standards in the United States by the name of Hugo Fano. And when Fano found out he knew a little physics, he gave him real physics to do, and he was the one who found all these minima and it got named after him. Anyway. So this is something which is found all over the periodic table and it's simply an overlap effect. There is no classical analog of this. It's not a resonance effect. It's just an overlap between wave functions. And this shows that they really are wave functions. They're not a figment of our imagination. They're not just a mathematical construct. They really exist and their phases and overlaps really do have consequences. Okay. All of this can be learned from a model like that. I mean you can do better and get things more exactly but qualitatively if you've got it. However, real wave functions are not simply single particle wave functions. They're more complicated than that. And this too is important in the following sense that when you know that one of the postulates of quantum mechanics says that you can always expand the wave function in any complete set and sometimes we pick a set and we try to do that. We write a wave function as a sum of terms. There's a name for this. This is called configuration interaction for historical reasons. But it merely is expanding a wave function in a complete set. Of course, complete sets are generally infinite and we can never do an infinite set so you do a finite one you truncated. And this can be done for the initial state and it can also be done for the final state. For the final state it's a little bit messy because remember, in the final state you are dealing with an unbound electron, a continuum electron. As opposed to the initial state when you're dealing with a bound electron or discrete electron. Now, discrete wave functions are normalized to unity. Continuum wave functions aren't. They are normalized to the delta function normalization. Delta function is infinite and so mathematically it's a lot messier. The idea is exactly the same but mathematically it's a lot messier and we call that interchannel coupling. Nice fancy phrase. But it really is the same thing as we have in the discrete but it's in the continuum. And again, the mathematical methods are different. The fact that you can write wave functions like that and need to is important because remember the interaction of a photon with an electron is just a single particle interaction. However, experimentally it is found that you can get ionization plus excitation. In other words, one electron ionized and the other excited with a single photon or you can get two electrons out with a single photon. As a matter of fact, about a decade or two decades ago there was a big flurry of activity in the double photoionization of helium since it only has two electrons. And trying to calculate that and do things about that. However, how can that happen if the photon only interacts with a single electron? Well, obviously then if it really does happen and it does, we measure that experimentally. Remember ultimately the only arbiter of whether or not we're doing a good job, whether or not we're right, is experiment. What we do has to agree with experiment. At least qualitatively if it's to have any validity at all. Okay, so we see say double ionization and we know the photon interacts with only a single electron. There's only one conclusion then. The electrons have to talk to each other in some general sense or their motion has to be correlated. We call this correlation. There's lots of other names for it. Multiparticle interactions, electron-electron correlation. It's called a number of different things, but that's what it is. Think of it as electrons talking to one another in electron ease, whatever that happens to be. So what we try to do is understand this electron ease, understand the language of electrons. Again, one of the ways to put this into the calculation, not the only way, is by expanding in a complete set. Or again, you can never expand in a complete set, so a truncated complete set. And that's the way you can get ionization plus excitation. Because then, let's say a simple case. How can you get ionization plus excitation in helium? Well, helium, as you know, ground state can be written 1s squared. However, it's not exactly that. If you just take two 1s type wave functions, however you pick them and take the product, you can get reasonable results for many things, but not for everything. And if you try to expand that in a complete set, you might have something, a wave function, which is at the initial state, alpha 1s squared plus beta 2s squared plus gamma 2p squared, say. Alpha, beta, and gamma are some coefficients. And you see, these are different configurations. And that's why this expansion is sometimes called configuration interaction. And typically, for the helium atom, this is close to 1, and these two are small. Now, let me show you why you need to do this if you're going to consider ionization plus excitation. Because ordinarily what you get is just from this, you get a final state 1s epsilon p. In other words, the photon comes in, hits a 1 electron. But you can also get 2s epsilon p and 2p epsilon s and 2p epsilon d. How do you do that? Well, from here to here, it's a 1 electron matrix element from 1s to 2p. From here to here, either those, it's a 1 electron matrix element. So expansion of the wave function like this, to be a more exact wave function, gives you the possibility of multi-particle. And so these beta and gamma are small, alpha is close to 1, but this beta and gamma somehow result from the electrons talking to each other. And exactly how you get these wave functions is a whole lecture in itself, and we're not going to do that today. So this, and also you can put the continuum in here so you can get double ionization. So electron correlation is part and parcel of multi-electron transitions. Without electrons talking to each other, without electron correlation, it is not possible. And so looking experimentally at multi-particle transitions with a single photon in is in effect measuring this correlation or giving a measure of this correlation. Paranthetically, there is another process which can give you multi-electron ionization having to do with inner shell. And it is called, well, it can't even be outer shell, there's another process which we call autoionization. It's not exactly the same thing, but it's related. In other words, let's say we start off with this. This can not only be depending on the photon energy knocked into the continuum, but it also be just excited. So instead of like this, you can go to a state, say, 2s, 2p. You can get there from here with the right energy. The 2s remains the same, and the 2p, 2s, the other 2s goes to 2p. That's a perfectly valid dipole transition. And this actually happens as one of the first, was one of the first ones discovered. But what happens when you have this? You see, let me show you here, this is a photon energy scale, and here is the cross-section. What you get is helium. It's binding energy is of the order of 25EV, and it goes dropping down. And all of a sudden, at about 60EV, you find something like this, a resonant. And what that is is that discrete-like state which lies well above the ionization threshold. Well, this lies well above the threshold of 1s, epsilon p. So what happens is that the same energy can get a transition between them. The 2s goes down to the 1s, and the 2p, it takes the energy of that and goes without any radiation and goes out. That's called auto-ionization. And again, this is only possible if you allow the possibility of a more complicated wave function, more complicated than a single particle. Moving right along, there are a couple of other things I would have liked to have talked about, but don't have that much time. So let's talk a little bit about photo-electron angular distributions. Because when you do a photo-ionization, you can measure the probability or the cross-section, but you can also measure the angles at which the electrons come out. And it turns out that you probably know this formula that the differential cross-section, sigma over 4pi, 1 plus beta p2 of cosine theta, where p2 is just the second order of the congeal polynomial, 3 cosine squared minus 1 over 2, I believe, and beta is the so-called asymmetry parameter. And you can work out an expression for this beta, which tells you something about the angular distribution. Now, the interesting thing about this beta is that remembering that you get l to l plus 1 and l to l minus 1 transitions, you get interference between these two amplitudes. And so if I call the amplitude for this m plus and the amplitude for this m minus, you get an expression for beta that looks something like this. Again, this just means some coefficients over something, never mind. This part we already know about is like the cross-section. But look at this. There's the magnitude of the matrix elements times the cosine of the difference in the phases of these matrix elements, phases. We didn't learn anything about them from the actual cross-sections, but they appear in the angular distribution. In other words, there is interference between these two amplitudes. Now, wait a second, though. Time out for a second. We have an l to l plus transition, l to l minus transition, l to l minus. This is angular momentum. Angular momentum are good quantum numbers. So how can you mix them? And you have to think about what you're measuring. To measure the angular distribution, what do you do? You look at the angle and the energy of the photoelectron. By the way, the angle, the way I've written it here, this is assuming linearly polarized electrons, photons, I mean, and this is the angle with the polarization vector. So the symmetry is not around the photon direction but the photon polarization vector. And so you measure the energy and the angle. Effectively, what that means, the angle is the direction. You're measuring the momentum. Okay. What are the angles? Now, you'll recall from elementary quantum mechanics the eigenfunction of a system if you measure a particular quantity. At the moment you make the measurement, you force it into an eigenfunction of that quantity, a particular one. All right. I mean, you change the wave function? Yeah. Exactly what it means. What is the eigenfunction of momentum? You know that. It's a plane wave. Aha. What's a plane wave look like? Let's say one in the z direction because it's easiest. It's not a special case. I just take my coordinate system and put it. So whatever direction the electron is moving, I call that the z direction. And what is that equal to? You may have seen this in scattering theory. It's i to the l 2l plus 1 jl of kr pl of cosine theta. Very well-known expansion. If you haven't learned about this, you will. But in any case, notice what it's assumed. So if you measure momentum, you no longer are in a fixed angular momentum state. Why? Momentum and angular momentum do not commute. I mean, quantum mechanics really, really does work. And so by measuring this, you are measuring a mixture. You are forcing the wave function into a mixture of different l states. And that is how this interference arises. But now, the study of beta is very useful from the following point of view. From the cross-section itself, you learn only about the magnitude of matrix elements or amplitudes. And fundamentally, the most fundamental information you can get about a process is the matrix element. And a matrix element has two attributes. It's magnitude and it's phase. The angular distribution allows you to get information about the phase. There are very few processes which give you information about the phase. This is one of them. And so this is a really interesting one to study. And a great deal of work has gone into over the years studying angular distributions. And unfortunately, my time is limited to a last minute. And so what happens is these betas, since the matrix elements are energy-dependent and the phases are energy-dependent, these betas are energy-dependent. And by measuring them, you learn particularly, if you measure them and the cross-sections, you know something about some of these and about the phases. And so this can give you, ultimately, as a matter of fact, there are some experiments that are called complete experiments where they measure all of the magnitudes of the matrix elements and all of the relative phases. Because you only get phase shift differences here. So you get the relative phases. And this is known as a complete experiment. And I guess since my time is pretty much up, I will stop here. And, well, I feel it's been a privilege to talk to you, whether you feel it's a privilege. I have no idea. Thank you.