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Journal of Physics B Atomic and Molecular Physics

Time-dependent theory of overlapping resonances interacting with electron and photon continua: dielectronic recombination

Eleonore Trefftz

Published Jun 1, 1970• 14 citations
Publisher

Abstract

Davies and Seaton (1969) generalized the radiation damping theory of Wigner and Weisskopf to the case of an optical continuum. Their techniques suffice to solve the problem of overlapping resonances, met in dielectronic recombination. The result justifies the 'stationary' theory developed by Shore (1967) and by Trefftz (1967) though it is not identical with the former result. The discrepancy is discussed.

DOI
10.1088/0022-3700/3/6/004

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Davies and Seaton (1969) generalized the radiation damping theory of Wigner and Weisskopf to the case of an optical continuum. Their techniques suffice to solve the problem of overlapping resonances, met in dielectronic recombination. The result justifies the 'stationary' theory developed by Shore (1967) and by Trefftz (1967) though it is not identical with the former result. The discrepancy is discussed.
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.
The problem of an atom interacting simultaneously with the quantized modes of the vacuum electromagnetic field and with a classical time dependent field is investigated utilizing the general theory of damping devised by Heitler. Spectral lineshapes are obtained as a function of the variables of the classical field and are studied in detail for the case of spontaneous emission from coupled levels of dissimilar lifetimes. In the limit that the time dependent coupling vanishes, the resulting spectral line is the same as that derived by Weisskopf and Wigner.
A coupled-channel analysis of electron-positive ion recombination is carried out, with full treatment of the coupling between radiation and autoionization continua. The cross section for this process reduces in the appropriate limits to the expressions for radiative and dielectronic recombination. The coupling to the radiation continuum leads to a modified Fano profile for the autoionizing resonances. The more complete, combined expressions derived here may be of interest for recent experimental studies of recombination and their comparison with theoretical results.
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The problem considered is that of two identical two-level atoms a fixed distance rrr apart, one of which is excited at t=0t=0t=0. A simple improvement on the usual on-shell approximation (which in the single-atom case is part of the Weisskopf-Wigner approximation) yields new solutions for the various probability amplitudes in the form of infinite series involving all the retardation times nrc\frac{\mathrm{nr}}{c}cnr​. The truncated solutions involving only the single retardation time rc\frac{r}{c}cr​ are compared with previously published results both when all photon modes are allowed and when only photons propagating along the interatomic axis are allowed. When the retardation times are neglected, the series are summed to give the well-known results of Stephen and others.
Studies the coupling of N discrete levels to a common continuum. The discrete levels, which can include both genuine bound states and autoionisation states, are assumed to be coupled to the continuum and to each other in such a way as to permit a rotating wave approximation in the Schrodinger equation. A general solution for the model is derived with the aid of Laplace transforms and the Weisskopf-Wigner approximation. The author discusses general properties of the discrete level populations and the electron spectrum like population trapping, zeros, crossings, resonances and confluences. As examples the author treats cases where two or three discrete levels interact with a continuum.