This book is concerned mainly with the interactions of positrons and positronium with individual atoms and molecules in gases. Brief mention is also made of positrons interacting with bulk matter but this is in the context of describing the slowing down of positrons in solids and the subsequent ejection of low energy positrons and positronium from the surface of the solid. A technique using the angular correlation of annihilation radiation, which is widely used in studies of electron momentum distributions and defects in condensed matter, is also described but again the emphasis is mainly on positron annihilation in gases.

Theoretical studies of positron collisions with atomic and molecular systems have been made for many years, as also have both theoretical and experimental studies of the lifetimes of positrons diffusing in gases. Only since the development of energy-tunable monoenergetic positron beams in the early 1970s, however, has it been possible to make detailed comparisons between theoretical predictions and the increasingly accurate experimental measurements of total, partial and differential scattering cross sections. These experimental developments have in turn stimulated renewed interest in theoretical studies of systems containing positrons. In this book we have attempted to integrate both theoretical and experimental aspects of the field into a reasonably coherent whole, although some sections are predominantly either experimental or theoretical.

Positron physics has undergone very rapid development during the past several years.

We now consider several of the more exotic systems in which one or more positrons may be involved, some of which were introduced in subsection 1.2.3. The positronium negative ion (e–e+e–), Ps–, has been observed in the laboratory (Mills, 1981) and its lifetime against annihilation determined experimentally (Mills, 1983b). We discuss these experiments and the relevant theory in section 8.1. Observation of the positronium molecule, Ps2, and other systems containing more than one positron or positronium atom (as yet unrealized) depends upon the generation of large instantaneous densities of positrons. The situation here is more encouraging than might be expected, owing to progress in developing very intense brightness-enhanced and time-focussed beams, as summarized in subsection 1.4.4. Many-positron systems and how they may be observed are described in section 8.2.

Antihydrogen, as discussed in subsection 1.2.3, has recently been observed in the laboratory, although only at relativistic speeds. However, progress with the trapping of cold antiprotons and positrons, and the production of positronium in a cryogenic environment, leads us to anticipate the synthesis of antihydrogen atoms with very low kinetic energies (or temperatures); thus it may be possible to trap them, and perform precision spectroscopy upon them. The motivation for the production of low temperature antihydrogen is described in section 8.3, along with the mechanisms and methodologies involved in some of the proposed formation processes.

In this chapter we consider the physics of the positronium atom and what is known, both theoretically and experimentally, of its interactions with other atomic and molecular species. The basic properties of positronium have been briefly mentioned in subsection 1.2.2 and will not be repeated here. Similarly, positronium production in the collisions of positrons with gases, and within and at the surface of solids, has been reviewed in section 1.5 and in Chapter 4. Some of the experimental methods, e.g. lifetime spectroscopy and angular correlation studies of the annihilation radiation, which are used to derive information on positronium interactions, have also been described previously. These will be of most relevance to the discussion in sections 7.3–7.5 on annihilation, slowing down and bound states. Techniques for the production of beams of positronium atoms were introduced in section 1.5. We describe here in more detail the method which has allowed measurements of positronium scattering cross sections to be made over a range of kinetic energies, typically from a few eV up to 100–200 eV, and the first such studies are summarized in section 7.6.

Important advances continue to be made in measurements of the intrinsic properties of the positronium atom, e.g. its ground state lifetimes (Rich, 1981; Al-Ramadhan and Gidley, 1994; Asai, Orito and Shinohara, 1995) and various spectroscopic quantities (Berko and Pendleton, 1980, Mills, 1993; Hagena et al., 1993). These are reviewed in section 7.1.

Introduction

In this chapter we describe the elastic scattering of positrons by atoms and molecules over the kinetic energy range from zero to several keV, concentrating mainly on the angle-integrated cross section, σel. However, reference is also made to differential cross sections, dσel/dΩ, which have recently become amenable to experimental measurement using crossed gas and positron beams.

Particular attention is given to relatively simple targets, e.g. atomic hydrogen, helium, the alkali and heavier rare gas atoms and small molecules, and some comparisons are made with the corresponding data for electron impact. This again highlights the differences and similarities in the scattering properties of the two projectiles, which have already been mentioned in subsection 1.6.1 and in Chapter 2.

At energies below the lowest inelastic threshold, elastic scattering is the only open channel (except for electron–positron annihilation, which is always possible but which usually has a negligibly small cross section). For all atoms, the lowest inelastic threshold is that for positronium formation, at an energy EPs, but for the alkali atoms positronium formation is possible even at zero incident energy. Molecular targets usually have thresholds for rotational and vibrational excitation at energies below EPs, although the elastic scattering cross section is nevertheless expected to dominate over the cross sections for these inelastic channels.

We continue this chapter with a detailed description of the theoretical models applied to the elastic scattering of positrons by atoms and molecules.

The preceding chapters have been concerned primarily with explicating the physical features of scattering electromagnetic plane waves from dielectric spheres, particularly for large size parameters. Much of the theoretical development is strongly dependent upon the maximal symmetry of this scenario, as well as upon the idealization of an incident plane wave. Although the spherical target is a good approximation to many of those encountered in important physical problems, there exist many other situations in which departures from sphericity cannot be ignored. Moreover, an infinite plane wave is clearly a fiction, albeit a very useful one, and in reality we have only the approximation of a locally plane wave. In many applications the incident radiation is provided by a tightly focused laser beam that may, but need not, satisfy this criterion. Thus, while the bulk of the work presented here provides a sound basis for understanding the basic scattering problem, there is a large body of physical applications in which one or more of the idealizations inherent in our model may fail to be realized. In this final chapter we shall attempt a brief and necessarily incomplete survey of some of the ways in which the fundamental model and its analysis must be altered in these situations. For the most part derivations and extensive mathematical expressions are omitted.

Appendices A–D define and describe properties of the various special functions employed in the main text, although we have included for the most part only those properties directly relevant to present needs. Authoritative general references to the behavior of these functions include the Handbook of Mathematical Functions edited by Abramowitz and Stegun (1964), which we usually abbreviate as HMF; Gradshteyn and Ryzhik (1980), Table of Integrals, Series, and Products; Watson's Theory of Bessel Functions (1995); Whittaker and Watson, A Course of Modern Analysis (1963); the Bateman-manuscript project's Higher Transcendental Functions (Erdélyi et al. 1953); and Szegö's Orthogonal Polynomials (1959). The reader is referred to these sources for all proofs – none is given here, though occasionally some are suggested. Additional appendices provide reference to mathematical and numerical techniques of value in studying scattering processes.

Almost all we see and perceive comes to us indirectly by the scattering of light from various objects; that is, by the scattering of electromagnetic radiation over a very restricted interval of the frequency spectrum. Much of this merely illuminates our world and helps us move about, while some exceptional natural scattering phenomena such as rainbows, glories, and halos touch our aesthetic sense. On a technical level, a very large portion of what we have learned about the physical world over the past four millennia has come to us via scattering experiments with both particles and waves, so that a study of scattering theory is an integral part of physics itself.

Classically the most familiar type of scattering is that among particles, such as balls on a pool table – or, more deeply, among gas molecules in the room where we work. Equally evident, however, are the results of scattering of electromagnetic and sound waves, and at first these appear to be entirely different phenomena. Just as modern quantum theory has compelled us to view all matter in terms of a particle–wave dichotomy, however, so have we also learned to view scattering processes as both particle-like and wave-like. That is, at high frequencies and short wavelengths even intrinsically wave-like classical phenomena exhibit particle-like scattering behavior, whereas on the quantum level particle scattering usually must be viewed in terms of waves.

Although the scattering of plane waves from spheres is an old subject, there is little doubt that it is still maturing as a broad range of new applications demands an understanding of finer details. The classical theory of electromagnetic scattering from dielectric spheres is due to Lorenz, Mie, and Debye, and has proved to be enormously rich; it is still being developed and continues to yield new insights. Much of this development has been motivated by the availability of small silicon spheres that can be probed precisely with laser light, as well as by new techniques in acoustics, in atmospheric physics, and in the study of biological molecules.

The classic treatise in the subject has long been van de Hulst's Light Scattering by Small Particles (1957), supplemented in later years by the application-oriented works of Kerker, The Scattering of Light and Other Electromagnetic Radiation (1969), and Bohren and Huffman, Absorption and Scattering of Light by Small Particles (1983). These volumes, and others, have contributed greatly to the subject, while concerning themselves primarily (though not exclusively) with scattering from particles whose dimensions are on the order of an incident wavelength or less. Among my reasons for writing the present book, however, is a long-time interest in understanding the detailed physics of the rainbow and glory in terms of modern scattering theory, and these phenomena arise from water droplets whose dimensions are a great deal larger than optical wavelengths. Thus, the time seems ripe for a theoretical exposition extending the earlier works to encompass a broader range of phenomena.

In discussions of the type undertaken in this monograph it is common to deal with functions represented by integrals that are resistant to exact evaluation, yet whose integrands contain one or more parameters that approach specific values in the problem of interest. In such cases it is often possible to find asymptotic representations of the function in a series of terms rapidly decreasing in value as z → z0, say. Even if such series do not converge, they can provide representations of the function for those parameter values to any desired degree of accuracy. With sufficient attention to detail, such expansions can often be differentiated and integrated term by term.

Many asymptotic developments pursued here arise after continuation into the complex plane, in which case additional difficulties emerge because we must insist that asymptotic relations be unique and independent of the path of approach of z to z0.