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Physical Review A

Parametrization of radiative-recombination cross sections

Andrea Erdas, Giuseppe Mezzorani, P. Quarati

Published Jul 1, 1993• 9 citations
Publisher

Abstract

Using exact analytical expressions of the nonrelativistic dipole recombination cross sections for a pure Coulomb potential summed over the quantum numbers j and l, we numerically compute the recombination cross sections in the various n states for 1\ensuremath{\le}n\ensuremath{\le}500 and the total cross section for all possible values of the incoming electron energy. We present a parametrization of the recombination cross sections in the various n states, of the total cross section, and of the energy emission cross section. A few low- and high-energy approximations are also derived and discussed.

DOI
10.1103/physreva.48.452

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Using exact analytical expressions of the nonrelativistic dipole recombination cross sections for a pure Coulomb potential summed over the quantum numbers j and l, we numerically compute the recombination cross sections in the various n states for 1\ensuremath{\le}n\ensuremath{\le}500 and the total cross section for all possible values of the incoming electron energy. We present a parametrization of the recombination cross sections in the various n states, of the total cross section, and of the energy emission cross section. A few low- and high-energy approximations are also derived and discussed.
We illustrate the possibility for calculation of bremsstrahlung tip angular destributions and polarization correlations from corresponding photoeffect results with an extension of the method for connecting bremsstrahlung tip and direct-radiative-recombination (DRR) cross sections developed by Lee and Pratt. The agreement between results obtained with this method and results from our numerical partial-wave calculations is, in general, satisfactory. We note features of the distributions and correlations, and once again point out the inadequacies of the nonrelativistic dipole approximation for the evaluation of bremsstrahlung or DRR angular distributions.
Although dielectronic and radiative recombination are usually treated as distinct noninterfering processes, Alber, Cooper, and Rau [Phys. Rev. A 30, 2845 (1984)] have recently presented a scattering- or S-matrix analysis which provides a unified description of these processes. This description employs a diagonalization of the atom plus radiation field Hamiltonian using a limited basis set consisting of one discrete autoionizing state, a single-electron continuum, and a single-photon continuum. In the present work we extend Mo?ller scattering operator and resolvent operator techniques, which have previously been used to discuss the decay of prepared systems, in order to provide an S-matrix analysis of the electron-ion photorecombination process near an isolated autoionizing resonance. We explicitly allow for degenerate magnetic sublevels of the atomic system and for multiple angular momentum contributions in the partial-wave expansion of the electron-continuum eigenstate. After the introduction of the pole approximation, in which only the \ensuremath{\delta}-function term is retained in the evaluation of the various self-energies that occur in the diagonalization of the Hamiltonian for the combined many-electron radiation-field system, we obtain the total electron-ion photorecombination cross section as the sum of the radiative and dielectronic recombination contributions together with the conventionally ignored interference term.The radiative and dielectronic recombination cross sections reduce to the familiar forms when the continuum-continuum coupling effects are neglected. Alternatively, the combined cross section for the entire electron-ion photorecombination process may be represented by a modified Fano line profile, which is shifted and broadened as a result of the coupling between the autoionization and radiation continua. Recombination processes that involve more than a single state of the initial ion, of the autoionizing resonance, or of the final system can be treated by appropriately augmenting the unperturbed basis set. It is anticipated that the effects of the interference between radiative and dielectronic recombination and of the continuum-continuum coupling will be most important for individual transitions involving low-lying autoionizing levels and will probably be negligible for the total dielectronic recombination rates due to the highly excited levels.
One-photon free-free absorption cross sections are evaluated by means of an integral relationship involving electron-atom off-shell scattering amplitudes. Off-shell effects are explicitly calculated and compared to various on-shell approximations. This calculation shows that off-shell effects become small with increasing energy as expected. One of the on-shell approximations agrees with the exact result over the entire energy range under consideration to within a few percent. The only integrals which appear in the calculations are absolutely convergent in contrast to methods used in previous calculations.
The analytical formulae for the cross-sections with arbitrary change {triangle}n=n{prime}-n and {triangle}1=1{prime}-1 has been obtained in the Born limit. No spin exchange {triangle}S=0: although the total cross-sections n->n are dominated by dipole transitions, the non-dipole transitions {triangle}1>1 can contribute up to 30-50%ER for large {triangle}n>>1. Intercombination transitions {triangle}S=1: the large transfer transitions {triangle}1>>1 are of major importance, as opposed to the previous case.
The intercombination transitions between l-resolved Rydberg levels due to collisions with fast electrons are studied within the Ochkur approximation for principal quantum numbers n\ensuremath{\le}20, where l is the angular momentum. The use of the Heisenberg correspondence principle for radial integrals enables one to obtain analytic expressions for any term in the multipole expansion of the cross section for collisions with an arbitrary n and l change. For transitions involving angular momenta that are small compared to the principal quantum numbers, the semiclassical and the exact results are found to be fairly close, within the accuracy consistent with the Born approximation. As a result, the semiclassical approach furnishes a ready estimate for cross sections, and its validity range proves broad enough to cover the region of large momenta transferred to the atom. The numerical results also confirm that dipole transitions have no dominance over the collisions with other \ensuremath{\Delta}l values, in accord with the general model of intercombination scattering.
Formulas for the cross section and event rate constant describing recombination of N particles are derived in terms of general S-matrix elements. Our result immediately yields the generalized Wigner threshold scaling for the recombination of N bosons. A semianalytical formula encapsulates the overall scaling with energy and scattering length, as well as resonant modifications by the presence of N-body states near the threshold collision energy in the entrance channel. We then apply our model to the case of four-boson recombination into an Efimov trimer and a free atom.