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2.3. Electron-Ion Recombination

Recombination rates are usually calculated for two processes, radiative recombination (RR) and di-electronic recombination (DR), and summed together to obtain the total. It is assumed that the RR values are derived from photoionization cross sections for the ``background'' cross sections (i.e. without autoionizing resonances), and the DR values correspond to the contribution from resonances as they converge to excited states of the recombining ion, at electron temperatures sufficiently high to affect excitation of the ion states. The rates for RR and DR are calculated separately, the former using ``background'', resonance-free photoionization cross sections obtained via approximations such as the Central Field (e.g. Reilman and Manson 1979), and the latter using the Burgess (1965) General Formula, or more recent works by Jacobs et al. (1977 onwards) and others (references are given later).

However the two processes, RR and DR, are in principle unified. With the availability of the Opacity Project photoionization cross sections, including resonances, it might at first appear that a straightforward integration over the photoionization cross sections, using the Milne relation (Osterbrock 1989), should yield the effective (e+ion) recombination rate coefficients. However, a careful consideration shows that the treatment is rather difficult and laborious. For example, low-lying resonances close to ionization threshold may significantly enhance the recombination rate, i.e. low-temperature DR first discussed by Nussbaumer and Storey (1983). A unified approach to electron-ion recombination has been developed by Nahar and Pradhan (1992, 1994b), that subsumes both RR and DR processes in an ab initio manner within the framework of the close coupling approximation, thus extending excitation and photoionization calculations to recombination. The method has been applied to a number of ions. Fig. 5 shows the total electron-ion recombination rate coefficient alphaR(T) for recombination to several ions in the carbon isoelectronic sequence. Results in Fig. 5 are illustrative; numerical data - unified recombination rates - for over 40 atoms and ions may be obtained from the first author (references at:

Owing to the paucity of atomic data, ionization balance calculations have heretofore employed radiative and collisional data that is inconsistent in the sense that different physical approximations are used to calculate all the data. With the new Opacity Project data, and ongoing recombination calculations, it is now possible to redress the situation in the case of radiative ionization, using photoionization and recombination rates that are self-consistent.

Figure 5

Figure 5. Total electron-ion recombination rate coefficients for C-like ions (Nahar and Pradhan 1994b).

During the previous decade, experimental measurements of DR cross sections for some ions showed large enhancements due to weak, external electric fields. Theoretical works indicate that while for neutral atoms the enhancement could be several factors, the effect is much smaller for mutiply charged ions due to stronger intrinsic Coulomb field. Badnell et al. (1993) discuss the problem and find that the maximum field enhancement for C IV is about 40%. Further work is needed to determine the precise extent of the influence of plasma microfields on DR.

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