Introduction

Many-electron atoms differ from H in an essential respect: when they are excited up to and above the first ionisation potential, they exhibit structure which is not simply due to the excitation of one valence electron. The clearest manifestation of this behaviour occurs in the ionisation continuum. For H, the continuum is clean, i.e. exempt from quasidiscrete features. In any many-electron atom, there will be autoionising resonances of the type discussed in chapter 6. Autoionisation is therefore a clear manifestation of the many-electron character of nonhydrogenic atoms.

In the present chapter, the questions: why does this extra structure occur and how does one set about interpreting it? are addressed. Thus, we will not be so concerned about the lineshapes or even (in first approximation) about interseries perturbations (although they do turn out in some cases to play a crucial role), but rather with the configurations of the inner-shell and doubly-excited states, and their relation in energy to the valence spectrum.

Inner-shell excitation

Even within the independent electrom approximation, it is obvious that there must exist inner-shell excitation spectra, and that their energy must extend well above the first ionisation potential. This arises from the simple fact that one can choose which electron is excited: it does not necessarily have to be the valence electron, and the inner electrons, being more strongly bound, require photons of higher energy to excite them.

Introduction

A system is considered as chaotic in classical mechanics if the orbits, instead of remaining confined to a specific region, invade the whole of available phase space. A simple example is a pendulumn with a magnet below the bob: if a sheet of paper is placed between the magnet and the bob and a pen is attached, the pendulumn will write all over the page within the range accessible to it. More exactly, if we examine phase space, it will seem completely disordered, with interwoven tracks throughout its volume. If we magnify the volume, the disorder will persist, and so on ad infinitum no matter how great the magnification, because classical mechanics imposes no limit on the resolution which can be achieved.

Chaotic behaviour can arise in any system whose motion is described by a nonlinear differential equation. Whether or not it is prevalent depends on the details of the problem, but it is a general theorem that any system described by a nonlinear differential equation possesses some chaotic regime.

In quantum mechanics, by contrast, chaos does not occur. We may see this in several ways. First, note that we cannot magnify ad infinitum the volume to be analysed in phase space: eventually, we reach the elementary volume ħ3 within which trajectories lose their meaning. Another way of reaching the same conclusion is to note that any Schrödinger type equation is linear: its solutions obey the superposition theorem. Under these circumstances genuine chaos is excluded by fundamental principles.

Beutler–Fano resonances

The photoionisation continuum of H is clean and featureless. Its intensity declines monotonically with increasing energy. Many-electron systems, in general, always exhibit structure embedded in the continuum. Such features are neither purely discrete nor purely continuous, but of mixed character, and are referred to as autoionising resonances. They were discovered experimentally by Beutler [254], and the asymmetric lineshape which they can give rise to follows a simple analytic formula derived by Fano [256]. For this reason, they are often referred to as Beutler–Fano resonances. A typical autoionising resonance is shown in fig. 6.1

Autoionisation is a correlation effect. It occurs for all many-electron atoms in highly-excited configurations which lie above the first ionisation threshold. Many spectra used as illustrations in the present volume provide examples of autoionising lines (see in particular chapter 7).

The origin of autoionising structure can be either of the following mechanisms or a combination of both. First, it is possible to excite more than a single electron at a time. Although forbidden in the independent particle model of the atom, many-electron excitation is physically possible, and indeed likely. It provides tangible evidence that the independent particle model is only an approximation. The fact that double excitation can give rise to very intense resonances shows that the breakdown of the independent particle model is by no means a small or negligible effect. The magnitude of this breakdown depends on the proximity in energy between single and double excitations.

Introduction

Quantum defect theory (QDT) was developed by Seaton [111] and his collaborators, from ideas which can be traced to the origins of quantum mechanics, through the work of Hartree and others. They relate to early attempts to extend the Bohr theory to many-electron systems (see e.g. [114]).

In chapter 2, we saw how the quantum defect is defined from a slight modification of the Rydberg formula for H. It is found experimentally to be nearly constant for different series members, especially for unperturbed series in atoms with a compact core. The first task of QDT is to ‘explain’ this fact, and to extract from this empirical observation an appropriate wavefunction, consistent with an effective one-electron Schrödinger equation, such that the quantum defect would turn out to be be nearly constant as the principal quantum number n is changed.

QDT is not an ab initio theory, i.e. it is not an attempt to solve the many-body problem from first principles. Rather, it is a theoreticallybased parametrisation. One seeks a form for the wavefunctions and for their dependence on n; this in turn leads to precise rules for the variation of many other quantities with n because, in quantum mechanics, once the wavefunctions are known, many observable properties of the system become calculable.

The benefits of QDT do not stop there.

Introduction

The present chapter is devoted to the comparatively new and rapidly developing subject of clusters, a field intermediate between atomic physics, chemistry and solid state physics, in which concepts borrowed from nuclear physics have also proved very important. Although the field is new, it has expanded very rapidly, and there are many different aspects beyond the scope of the present book. We therefore confine our attention to: (i) a general introduction and (ii) some aspects of cluster physics which are specifically connected with material already presented in the previous chapters.

A cluster is an assembly of identical objects whose total number can be chosen at will. An atomic cluster is therefore an assembly of atoms in which the total number is adjustable. Just as, in solid state physics, one distinguishes between cases in which the valence electrons become mobile and those in which they remain localised on individual atomic sites, so one finds different kinds of clusters, depending on the degree of localisation of the valence electrons. Broadly speaking, these differences are dictated by the periodic table: at one extreme, one has the rare-gas clusters, in which electrons remain localised, while at the other, one finds the alkali clusters, which are metallic in the sense that the valence electrons can move throughout the cluster.

The subject of atomic clusters arose only recently because it was not appreciated in earlier times that identical atoms could hang together in this way.

Introduction

The present chapter provides a summary of the basic principles of Wigner scattering or K-matrix theory, followed by examples of its application to atomic spectra, and more specifically to the study of interacting autoionising resonances, for which it happens to provide a very suitable analytic framework, within which most of the important effects can be illustrated rather simply. We concentrate on an elementary account of basic principles rather than on the most complete algebraic formulation, because the theory in its full generality becomes rather forbidding. Thus, when only a small number of channels needs to be included in order to illustrate an effect, suitable references are indicated, where the reader can find a fuller treatment. We also make the fullest possible use of analytic methods, which allow one to pick out a number of significant effects without detailed numerical computations: this turns out, rather remarkably, to be possible only for atoms, and this is a consequence of the asymptotic Coulomb potential.

Atoms therefore provide an excellent testing ground for the details of Wigner's theory. Wigner's [370] S-matrix theory postulates the existence of a Schrödinger-type equation, but actually requires no explicit knowledge of its solutions. In this sense, it is regarded as the most general formulation of scattering theory (and is more general than MQDT). One can even handle photon decay channels, although no explicit wavefunction can be written for photons. They appear in scattering theory as weaklycoupled radiative channels, and examples will be given in the present chapter.

Multiphoton spectroscopy

The subject of multiphoton excitation spectroscopy began in 1931 when Göppert-Mayer [450] wrote a theoretical paper in which she calculated the transition rate for an atom in the presence of two photons rather than just one. At the time, the process seemed rather exotic, and it was reassuring that the calculated rate was so low as to guarantee that it could not readily be observed in the laboratory with conventional sources. This conclusion was reassuring because it implies that a simple perturbative theory (one photon per transition is the weak-field limit) is adequate for most purposes.

The subject came to life with the advent of lasers, when it became easy to create intense beams of light. Since the probability of excitation by two photons grows as the square of the photon density, whereas the probability of single-photon excitation grows only linearly with photon density, two-photon transitions gain in relative strength with increasing intensity despite the small value of the rate coefficient.

The development of multiphoton spectroscopy has followed that of lasers: as the available power has increased, so has the number of photons involved in individual transitions. More significantly, it has become apparent that the physics of the interaction between radiation and matter is not the same at high laser powers as under weak illumination, i.e. that there is a qualitative change which sets in at strong laser fields. This is normally expressed by saying that perturbative approximations break down.

Motivation for this book

Atomic physics is a well-established subject with a distinguished history, and many books are available which cover its traditional applications. However, it is also a rapidly developing research area, and it is perhaps not surprising that most of the classic texts on which undergraduate courses are usually based no longer reflect its evolution. When the early texts on the subject were written, the prime concern was to demonstrate by many beautiful examples how the principles of quantum mechanics find application in atomic physics. Since numerical methods for solving the radial Schrödinger equation were known in principle but were not generally available, the emphasis was on angular momentum algebra and on formal developments involving electron spin, while the radial integrals were treated as parameters.

Such tools are of course essential in the armoury of any practising atomic physicist, but, in the author's view, lengthy developments in angular momentum algebra no longer form the best introduction to the subject. With ready access to fast computers, solving the radial equation is now a straightforward matter, and there exist many excellent codes for this purpose. Thus, a significant change of attitude has occurred amongst researchers: it is no longer sensible to concentrate on the angular part of the central field equation. Indeed, one can argue that the opposite approach is the correct one. The properties of spherical harmonics need only to be determined once, and can then be used to model all central field atoms.

Introduction

Centrifugal barrier effects have their origin in the balance between the repulsive term in the radial Schrödinger equation, which varies as 1/r2, and the attractive electrostatic potential experienced by an electron in a many-electron atom, whose variation with radius differs from atom to atom because of screening effects. In order to understand them properly, it is necessary to appreciate the different properties of short and of long range potential wells in quantum mechanics.

As the energy of the incident photon is increased above the ionisation threshold, centrifugal barrier effects often come to dominate the response of many-electron atoms, which totally alters the spectral distribution of oscillator strength in the continuum from what might be anticipated by comparison with H. This applies not only to free atoms, but also to the same atoms in molecules and in solids: many of the changes due to centrifugal effects occur within a small enough radius that they are able to survive changes in the environment of the atom.

Since the centrifugal term is present in the radial Schrödinger equation for all atoms, we must explain why centrifugal effects only dominate the inner valence spectra of fairly heavy atoms. Centrifugal barrier effects are present even in H. However, they act differently in transition elements or lanthanides.

One reason is straightforward: the ground state of H has ℓ = 0, and therefore only p states are accessible directly by a dipole transition from the ground state.

Introduction

There are many connections between the physics of free atoms and that of solids which have been noted, in passing, several times already in the present volume. One should add that many-body theory and, especially, the concept of excitations as quasiparticles in free atoms, owe much to the theory of excitations in solids [590].

The theme of the present chapter is rather more specific: the intention is to present a number of effects which are counterparts of those we have studied in previous chapters, but for atoms in the solid rather than in the gaseous phase. Also, the intention is to set the scene for the last chapter, in which atomic clusters will be used in an attempt to bridge the gap from the atom to the solid experimentally. A highly excited atom in a solid will be taken as an atom excited close to or above the Fermi energy (including, of course, core excitation). There are some solid state systems for which electrons with energies close to the Fermi level behave like those in atoms. X-ray absorption and electron energy loss spectroscopy involving core excitation to empty electronic states can then be described by initial and final states possessing L, S and J quantum numbers, and the allowed transitions follow strict dipole selection rules. Examples include the d → f transitions of Ba in high Tc superconductors, and many instances involving transition metals and lanthanides.

Line strengths of discrete transitions

In the present chapter, we consider line strengths of transitions between bound states, i.e. of lines whose only form of natural broadening is radiative, and which lie below the first ionisation potential. The simplest situation is encountered in the photoabsorption or photoexcitation of an atom, initially in its ground state | i ≥, in which case one transition is observed to each excited final state | f ≥. The price one pays for this simplicity is that all excited states cannot be reached in this way because of selection rules.

The distribution of intensities is an essential property of a spectrum. We may consider: (a) the distribution in energy over the whole spectrum; or (b) the distribution within an individual spectral line.

It was noted in the previous chapter that spectral lines of interacting channels can differ greatly in intensity within a narrow energy range. However, with some significant exceptions, if one can approach the series limit closely enough to be clear of perturbers, the intensities of successive members decrease monotonically with increasing principal quantum number n. This is described as the normal course of intensity for a Rydberg series.

In the presence of perturbations, the course of intensities becomes far less regular than that of transition energies, and it is generally more likely that intensities will exhibit fluctuations or departures from the expected behaviour.

It took a very long time to write this book, especially to bring it to a relatively consistent and complete form. The journey of the reader to these final pages was also not easy and straightforward. What are your feelings after getting through the jungle of more than 1300 many-storeyed formulas? Perhaps, twofold. At first – relief and satisfaction: it is all over now, I made it! But secondly – are all these formulas correct? The answer is not so simple. I tried to do my utmost to be able to answer ‘yes’: compared with the original papers, deduced some of them again, checked numerically, looked for special cases, symmetry properties, etc. But I cannot assert that absolutely all signs, phases, indices, etc. are correct. Therefore, if you intend to do some serious research starting with one or other formula from the book, it is worthwhile carrying out additional checks, making use of one of the above mentioned methods.

Not all aspects of the theory are dealt with in equal depth. Some are just mentioned, some even omitted. For example, the method of effective (equivalent) operators deserves mentioning. It allows one to take into account the main part of relativistic effects but at the same time to preserve the LS coupling used for classification of the energy spectra of the atoms or ions considered.