Introduction

The problem of electron–atom scattering is one of the oldest problems in atomic physics dating back to the early part of this century. This problem has often been studied and there were periods for which the problem was considered solved if one defined solved to mean that there was agreement between the existing experimental data and theoretical calculations. The early experimental measurements were typically total cross sections—that is cross sections summed over spins, summed over magnetic sub levels and integrated over scattering angles. For the case of atomic excitation, the measurements were normally performed for the allowed transitions. These cross sections were generally in reasonable agreement with the highly popular Born approximation which treats the projectile as a plane wave.

In the late 1960's and early 1970's, improved technology permitted the measurement of cross sections differential in the electron scattering angle following the collision. These more detailed cross sections revealed that the Born approximation was totally incorrect for large scattering angles. It was not possible to see this fact from the total cross sections which are dominated by the small angle differential cross sections (DCS). These DCS measurements revealed that an improved theoretical model was required. Most of the efforts in this direction since the early 1970's fall either into the category of a perturbation series approach or a non-perturbative close-coupling approach.

Hugh Padraic Kelly was a pioneer in many-body perturbation theory (MBPT) and its application to atomic systems. He was the first to apply the new diagrammatic technique, developed mainly in field theory and nuclear physics, to problems in atomic physics. He introduced many new ideas, which have been widely used in different areas. In the first years, his work was concerned with the correlation energy of simple atoms and with the hyperfine interaction. In the last two decades he concentrated his efforts on the photoabsorption and photoionization problem, a field where he played a dominant role for a long time. In this chapter I shall give a brief review of the development of atomic MBPT and of the role played by Kelly. I shall also discuss later developments, including relativistic MBPT and QED calculations. I do not intend to give a full account of Kelly's work in different areas, since much of that will be covered in other chapters.

Background

Perturbation theory, which has been used for a long time in mathematical physics and astronomy, was introduced into quantum physics shortly after the advent of quantum mechanics in the 1920s. The most well-known of these schemes are the Rayleigh-Schrödinger and the Brillouin-Wigner expansions. In principle, these schemes can be used to any order, but they are, in actual atomic and molecular applications, almost prohibitive beyond fourth order, which may be insufficient to achieve the desired accuracy in many cases.

Introduction

Transfer of mass is a quasi-forbidden process. It does not occur very simply. So new insights into how masses interact in our environment may be discovered by considering this challenging little puzzle of how a quasi-forbidden process operates. The simple pickup of a stationary mass, M2, by a moving mass, M1, is forbidden by conservation of energy and momentum. Specifically, if M1 < M2 then M1 rebounds, if M1 = M2 then M1 stops and M2 continues on, and if M1 > M2 then M2 leaves faster than M1. In none of these cases do M1 and M2 leave together. Thus, mass transfer may not occur in a single elastic collision. L.H. Thomas understood this in 1927 and further realized that transfer of mass occurs only when a third mass is present and all three masses interact. The simplest allowable process is a two-step process now called a Thomas process. It was not until the work of Shakeshaft and Spruch 52 years later that the significance of Thomas processes was properly understood. There is evidence, for example, that Bohr did not understand the significance of the failure of the first Born approximation to reduce to the correct classical limit.

The resolution of Bohr's dilemma lies in the unusual feature that the second Born term for mass transfer is larger than the first Born term at high collision velocity, v.

Introduction

In the thirty years since Kelly's seminal papers appeared in print, many-body perturbation theory has become one of the most important and widely used methods for determining energy levels and properties of atoms. The theory has developed in two directions during recent years. Firstly, it has been extended to include infinite classes of terms in the perturbation expansion using coupled-cluster and other all-order methods. These extensions were discussed in the first chapter of this book. Secondly, the theory has been modified to include relativistic corrections in an ab initio way starting from the Dirac equation. Kelly contributed to both of these extensions; indeed, two of his recent papers were devoted to relativistic all-order methods.

Relativistic many-body theory is based on the no-pair Hamiltonian, an approximate Hamiltonian derived from QED by ignoring the effects of virtual electron-positron pairs, The no-pair Hamiltonian accounts for the electron-electron Coulomb interaction and the Breit interaction, but does not include QED effects such as the electron self-energy and vacuum-polarization. Relativistic atomic structure calculations based strictly on QED are difficult if not intractable; they have been carried out only for the simplest many-electron atom, helium. The dominant QED corrections to energies of many-electron atoms have been evaluated in a few cases; however, a systematic account of QED corrections to the no-pair Hamiltonian has not as yet been given for atoms other than helium.

Multiconfiguration Dirac-Fock (MCDF) methods are also extensively used to determine energy levels of relativistic atomic systems.

Introduction

The R-matrix theory was first introduced by Wigner and Wigner and Eisenbud to describe nuclear resonance reactions. This theory was subsequently developed and applied by many workers in nuclear physics, and comprehensive reviews have been written by Lane and Thomas, Breit, and Barrett et al.

In recent years, it has been increasingly realised that the R-matrix method can also be used as an ab initio approach to describe a broad range of atomic and molecular structure and collision processes. These include: electron–atom and –ion excitation and ionization processes, electron–molecule scattering, positron–atom and –molecule scattering, atomic and molecular photoionization processes, free–free transitions, atomic and molecular bound state energies and oscillator strengths, atomic polarizabilities, atom–molecule reactive scattering, dissociative attachment and recombination processes and atomic multiphoton processes. Reviews of the theory and applications to atomic and molecular processes have been written by Burke and Robb and Burke and Berrington.

The present chapter is organised as follows. In the next section the basic R-matrix theory of atomic and molecular processes is summarised. In the following section some recent applications of this basic theory are presented. Three examples are discussed: (i) low energy electron scattering by ions of Fe; (ii) electron scattering by atoms and ions at intermediate energies and (iii) multiphoton processes.

Basic theory

We summarise in this section R-matrix theory describing low energy electron impact excitation of light atoms and ions where relativistic effects can be neglected.

Hugh Padraic Kelly is generally recognized by the scientific community as a pioneer in the application of diagrammatic many-body perturbation theory to problems in atomic physics. His work began in the late 1950's and early 1960's with the study of correlation energies in light atoms. From that time, with numerous colleagues and students, he led the development of many-body methods for the treatment of a wide variety of problems in atomic structure, photoionization of atoms, and electron-scattering from atoms. On April 15–17, 1993, some of the leading atomic theorists in the world attended a workshop at the Institute for Theoretical Atomic and Molecular physics at the Harvard-Smithsonian Center for Astrophysics in order to review the application of many-body theory to atomic physics in memory of Hugh Kelly. The organizers of the workshop took advantage of this setting in order to ask the participants to contribute chapters for a textbook on theoretical many-body atomic physics – one that would promote the field to prospective students, as well as would pay tribute to the memory of the field's pioneer. This book is the result of that effort.

This book can be used as a supplement to seminar courses in atomic physics, many-body physics, or quantum chemistry. Each chapter is self-contained and there are cross references between the chapters. Where possible, consistent notation is used in the the various chapters and a key to this notation is included in the Appendix.

There are three main topics with a total of fifteen chapters.

Introduction

Beginning with the pioneering experiments of Ehrhardt et al. and Amaldi more than 30 years ago, a field of considerable activity has developed in the area of measuring and theoretically predicting the angular and energy correlations of two electrons leaving an ion following ionization by electron impact, the so-called (e, 2e) problem. This field is daunting both experimentally and theoretically; the experiments require coincidence measurements with typically small counting rates while the adequate description of two light charged particles with modest kinetic energy moving in the Coulomb potential of a heavy ion, one version of the Coulomb three body problem, has provided a continuing challenge to theorists.

The underlying goal of this research has been to achieve a better understanding of the ubiquitous process of ionization by studying the basic process of an incoming electron transferring a measured amount of momentum to a target atom with the result that a secondary electron is ejected, its momentum being measured also. Such experiments then measure all kinematic variables and provide the most detailed information possible about the event. The only observable averaged over is the spin. A practical application of this activity is that, if theories can be developed to get these differential cross sections right, then integration should produce reliable total ionization cross sections which are in constant demand in other areas, astrophysics and the controlled fusion program for example.

In its several decade existence the (e, 2e) work has already spread into several subfields.

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.