Introduction

Collision experiments provide a considerable portion of our information on the interactions between particles, as well as on the internal structure of the particles themselves, in all of the subfields of modern physics. When the interactions between particles are large, information is obtained primarily on the collision process itself; when the interactions are small, the collision probes principally the structure of the interacting particles. Since the coupling between the photon field and the electron is small, photoionization is an example of the latter type of process. Furthermore, since the photon is not thought to possess any internal structure, photoionization is clearly a probe of the target atom.

In photoionization, as in any collision process, measurements of total cross sections provide information on the absolute squares of matrix elements. Differential cross sections, angular distributions, provide information on the matrix elements themselves, including their relative phases. Thus, angular distribution studies allow us to peel back one more layer of the “onion.” Furthermore, in many cases, photoelectron angular distributions are much more sensitive to many-body effects, i.e., the effects of correlation, than are total cross sections; in fact, there are many instances in which many-body effects dominate the structure of the differential cross section.

The purpose of this chapter, then, is to provide a theoretical framework for the understanding of the physics of atomic photoelectron angular distributions. Within this framework, the separation between angular momentum geometrical factors and dynamical, wave-function-dependent, factors in the determination of the angular distribution will be highlighted.

Introduction

Photoionization of a two-electron or a divalent atom in the vicinity of a doubly excited resonance due to the strong interaction between two outer electrons represents perhaps one of the most direct and unambiguous atomic processes for a quantitative study of the many-electron effects in atomic transitions. Recent applications of high resolution intense lasers and elaborate experiments in shorter wavelength regions with more advanced applications of synchrotron radiation to highly correlated atomic systems have opened up more opportunities for such a study. For example, operating at high synchrotron radiation intensity, Domke, Remmers, and Kaindl have recently observed the doubly-excited 2pnd1P series of He below the He+N = 2 threshold with an improved energy resolution of ≅ 4 meV. Parallel to the fast growing experimental advances, significant progress have also been made in many of the existing theoretical approaches.

One of the best known doubly excited photoionization structures is perhaps the He 2s2p1P resonance shown in Fig. 5.1. This resonance was first observed in detail by Madden and Codling in He photoionization from its ground state and classified subsequently by Cooper, Fano, and Prats. Theoretically, the photoionization structure of an isolated resonance located at resonant energy Er is best described by the Fano formula in terms of a set of resonant parameters, which includes the resonance width Γ, the asymmetry parameter q, and the nonresonant background cross section σb.

Introduction

The role of atomic physics in precision tests of QED is well known. However, atomic physics has not until recently played much of a role in testing the weak interactions owing to the extremely small ratio ~ 10−11 of the energy scale of atoms to the mass of the W± and Z0 bosons. Indeed, the first discussion of parity violating effects in atoms arising from weak-neutral-current interactions between the nucleus and electrons by Zel'dovich concluded that these effects were probably too small to observe. However, after the existence of neutral-current weak interactions was established experimentally, the Bouchiats showed in 1974 that parity nonconserving (PNC) transitions in heavy atoms with atomic number Z were enhanced by a factor of Z3. While still very small, this effect has been observed in a variety of heavy atoms, specifically cesium (Z = 55), thallium (Z = 81), lead (Z = 82), and bismuth (Z = 83). These experiments deal with neutral atoms, and for this reason the electronic structure of the atoms must be understood with some precision before the experiments can be interpreted in terms of particle-physics concepts. This problem is not present for hydrogen, but PNC experiments in that atom have not proved successful.

Unfortunately, practical calculations that are accurate to the 1% level, the level of interest for particle physics, are quite difficult to carry out in heavy atoms. It is the purpose of this chapter to review the methods used in the recent highly accurate calculations of atomic PNC.

Introduction

In this chapter, we consider various theoretical methods for the calculation of electron-impact excitation and ionization in positively charged ions. First, we compare the distorted-wave and close-coupling approximations for the calculation of the non-resonant contributions to electron-impact excitation. The close-coupling method takes into account coupling effects to all orders of perturbation theory for those states included in the close-coupling expansion, while the distorted-wave method treats coupling effects by first-order perturbation theory. By comparing calculations using these two approximations along an isoelectronic sequence, we study continuum coupling as a function of ionization stage.

In many cases, excitation cross sections are dominated by strong resonance features. These occur when an electron in the continuum of an N-electron ion coincides in energy with a compound doubly-excited state of the (N + 1)–electron ion. The continuum electron can be captured into the compound state, and if this state then undergoes autoionization to an excited state of the N-electron ion, we have a resonance in the excitation cross section. Such resonances are treated in a natural way within the close-coupling method. However, using perturbation theory, it is also possible to include them within a distorted-wave formalism. We focus on the independent-processes approximation, in which interference between the individual resonances and between the resonances and the non-resonant background is ignored. In order to study the validity of this approximation, we consider transitions for which the resonant contributions are significant and compare close-coupling calculations with independent processes calculations along an isoelectronic sequence.

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.