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.

Up to this point our discussion has avoided any detailed reference to the underlying physical mechanisms producing scattered waves. One most often thinks of scattering in terms of particles undergoing elastic collision, either with other particles or with macroscopic objects such as walls. In the microscopic domain this view is even extended to light when the photon picture is appropriate. As we have seen, geometric optics permits a similar interpretation wherein rays mimic the scattering behavior of particles.

When one can no longer neglect the wave nature of light, however, this intuitive view of the scattering process is not entirely adequate and we are compelled to look beyond it for physical origins. On a microscopic level an electron, atom, or molecule will couple to an incident electromagnetic wave in an oscillating fashion, such that it re-radiates in all directions, producing a ‘scattered’ wave. It is tempting to amplify this mechanism to macroscopic targets, since they are certainly composed of these constituents, but to do so in detail would be a forbidding problem in many-body physics. A more appropriate macroscopic approach might be to envision the incident fields as inducing electric and magnetic multipoles that oscillate, and hence radiate, while maintaining definite phase relations with the incident wave. When the wavelength of the incident radiation is long compared with the dimensions of the scatterer, only the lowest-order multipoles will be important, and the re-radiation process can be approximated by invoking electric and magnetic dipoles.

Following the rather lengthy mathematical onslaught in the theoretical development of scattering from a large dielectric sphere in the preceding chapter, it is perhaps time to take a break from these labors and apply the results to prediction of some measurable quantities. As well as judging the largesphere theory against the exact expressions provided by the Mie solution, some pleasure may also be found in using the results to explore various meteorological optical phenomena.

It cannot be emphasized too strongly that the preceding theory is exact, in the same sense that the Mie solution is exact. In principle we can calculate amplitudes and cross sections to all orders in β – it is not an approximate theory. In practice, however, one need only retain a few terms in the asymptotic expansions and in the residue series to achieve reasonable accuracy, so that we can appropriately approximate the theoretical expressions and reduce the calculational labor. However, this is also in contrast with the partial-wave expansions, which must include indices l to at least O(β). The difference between approximate theories and approximations to the exact theory is that in the latter we are able to control the estimates rather precisely. A separate issue relates to how many terms must be retained in the Debye expansion itself, and this will be addressed as the need arises.