2010 Fe 15 + dielectronic recombination and the eﬀects of conﬁguration interaction between resonances with diﬀerent captured electron principal quantum number

Dielectronic recombination (DR) of Na-like Fe 15+ forming Mg-like Fe 14+ via excitation of a 2 l core electron has been investigated. We ﬁnd that conﬁguration interaction (CI) between DR resonances with diﬀerent captured electron principal quantum numbers n can lead to a signiﬁcant reduction in resonance strengths for n ≥ 5. Previous theoretical work for this system has not considered this form of CI. Including it accounts for most of the discrepancy between previous theoretical and experimental results.


I. INTRODUCTION
Understanding the properties of astrophysical and laboratory plasmas necessitates knowing the ionization balance of the observed or modeled sources. This in turn depends on the underlying recombination and ionization processes. Of particularly importance are data for the electron-ion recombination process known as dielectronic recombination (DR) which is the dominant recombination mechanism for most ions in atomic plasmas [1,2].
The DR process can be expressed as DR is a two-step recombination process which begins when a free electron e − collides with an ion of element A with charge q+ and in initial state i. The incident electron collisionally excites a core electron of the ion with principal quantum number n c and is simultaneously captured forming a system of state j. This process is known as dielectronic capture. We use the word "core" here to distinguish initially bound electrons from the captured electron. The DR occurs when the state j radiatively decays to a state f emitting a photon of angular frequency ω. This reduces the total energy of the recombined system to below its ionization threshold. Conservation of energy requires that the energy of the initial free electron and unrecombined ion balance that of the intermediate recombined system. Thus the relative kinetic energy of the incident electron equals the excitation energy ∆E of the core electron in the recombined system in the presence of a captured electron plus the binding energy E b of this captured electron in the recombined system, i.e., ∆E = E k + E b . Because ∆E and E b are quantized, E k is quantized, making DR a resonance process.
Here we explore a particularly nagging discrepancy between theory and experiment for the simple M-shell ion Na-like Fe 15+ forming Mg-like Fe 14+ . Good agreement between experiment [3] and theory has been found for Fe 15+ (1s 2 2s 2 2p 6 3s) DR via ∆n c = 0 and 1 excitations of a 3s electron [4,5]. For DR via ∆n c = 1 core excitation of a 2l electron, previous theoretical work has shown the importance of configuration interaction (CI) within a 2s 2 2p 5 3l3l ′ nl ′′ complex for a fixed n [6]. Including this single-n CI reduced the predicted resonance structure by a factor of 2. However, that work plus other recent work [4,5], which also consider CI only within the same n complex, are still about a factor of 2 times larger than experiment [3] for resonance energies above 650 eV [7]. These resonances involve captured electron quantum numbers of n ≥ 5.
In this work we investigate the cause of this discrepancy. We use the FAC (Flexible Atomic Code) [8] which is fully relativistic and utilizes the distorted wave approximation.
We have made a more complete accounting of possible autoionization and radiative decay channels than previous theoretical works. Additionally, we pay particular attention to the effect of CI between different n configurations. This multi-n form of CI has been neglected in previous theoretical studies for this system.
The rest of this paper is organized as follows. In Sec. II we review the standard theoretical approach to calculate DR, discuss the autoionization and radiative decay channels we considered for Fe 15+ DR, and outline our approach to handling CI between different n complexes. We compare our theoretical results to experiment and previous theory in Sec. III.
Lastly, we summarize our results in Sec. IV.

A. Standard approach
We calculated DR using the independent process, isolated resonance (IPIR) approximation [9]. This method treats radiative recombination and DR separately and neglects quantum mechanical interferences between the two and between DR resonances. These interference effects have been shown to be small in general [9]. The DR cross section in the IPIR approximation for a multiply-excited intermediate state |φ j with resonance energy E j is given to lowest order in perturbation theory by [10] Atomic units are used here and throughout the paper unless otherwise noted. E is the collision energy and k is the linear momentum of incident free electron, both given in the electron-ion center-of-mass frame, g i and g j are statistical weights, |Ψ i is an initial recombining state which includes the incident free electron, and |Φ f is a final bound state. The incident free electron is not correlated with the target ion. D is the dipole radiation field interaction D = 4ω 3 3c 3 where c is the light velocity, N is the number of bound electrons before dielectronic capture and r s is the position vector of electron s from the nucleus. V is the electrostatic interaction between the N initially bound electrons and the continuum N + 1 electron The total resonance width Γ j is given by A a jk is the autoionization rate from j to any state k of A q+ and can be expressed as A r jf is the radiative decay rate from j to f which can be written as The energy integrated cross section (i.e., resonance strength) of state j is given by [11] in the approximation Γ j ≪ E j . The resonance strength can be re-written as the product of the dielectronic capture (DC) strength which is related to the autoionization rate through detailed balance, and the branching ratio B. DR channels of Fe 15+ For Fe 15+ DR via ∆n c = 1 core excitation of a 2l electron, we considered the autoionization and radiative decay channels where l ≤ 1, l ′ and l ′′ ≤ 2, and l ′′′ ≤ 5. This includes the 2l 8 3l ′ 3l ′′ radiative decay channel which was not considered by [5] and the 2l 7 3l ′ 3l ′′ autoionization channel which was not included by [6]. We also considered CI for all possible 2l 7 3l ′ 3l ′′ core configurations. Thus, unlike the previous theoretical work, 2s → 3l promotions are included.
For n > 6, the 2l 8 3l ′ nl ′′′ configuration lies in the continuum and radiative decays to autoionizing levels are possible. These can then autoionize or radiatively stabilize via The branching ratio for these radiative decays to autoionizing levels followed by radiative cascades (DAC) can be given by [12] where the final states t and t ′ are below and above the ionization threshold, respectively. B t ′ is the branching ratio for radiative stabilization of t ′ and can be determined by evaluating B j iteratively.

C. Configuration interaction between different n resonance complexes
We performed a large scale CI calculation between all 2l 7 3l ′ 3l ′′ nl ′′′ complexes from n = 3 to 14. This allows us to consider CI between resonances with different captured electron principal quantum numbers. A large orbital sensitivity of DR to the choice of the initial radial wave function has been reported in Mg 2+ [13]. For Fe 15+ this sensitivity is expected to be insignificant as a result of the high q of the ion. We explicitly investigated here the effects of optimizing radial wave functions on the 2l 7 3l ′ 3l ′′ and 2l 8 3l ′ configurations of the recombining ion as well as on the 2l 7 3l ′ 3l ′′ 3l ′′′ and 2l 8 3l ′ 3l ′′ configurations of the recombined ion. Only small differences in resonance strengths and energies were seen. In the end, radial wave functions were optimized on the 2l 8 3l ′ 3l ′′ configuration as that gave best agreement with the experimental results.
The j CI mixed stateφ n j for an n complex can be expanded in the where c n ′ j ′ denotes the mixing coefficient for the φ n ′ j ′ basis. We calculated autoionization and radiative decay rates from the wave functions obtained using this CI mixing. Past studies have not considered CI mixing between different n complexes. In those studies autoionization channels of the form 2l 7 3l ′ 3l ′′ nl ′′′ → 2l 8 nl ′′′ + e − and radiative decay channels of the form 2l 7 3l ′ 3l ′′ nl ′′′ → 2l 8 3l ′ nl ′′′ + ω were possible only between the states of same n.
However, taking into account CI mixing between different n resonance complexes allows for additional autoionization and radiative decay channels.

A. Experiment
Theoretical studies of Fe 15+ DR have been aided greatly by the merged-beams experimental results of [3] shown in Fig. 1. The measured data represent the DR cross section σ times the relative collision velocity v r convolved with the experimental energy distribution yielding a rate coefficient σv r [11]. The energy distribution is de-

B. CI within the same n complex
We performed explicit calculations of autoionization and radiative decay rates up to n = 14 and extrapolated for n from 15 to the experimental cutoff of 86. A simple hydrogenic scaling law was used for the resonance energies, the autoionization rates, and the radiative decay rates of the captured electron for n ≥ 15. The radiative decay rate of the core electron was set to the n = 14 value for all n ≥ 15. The calculated DR strengths were multiplied by v r and convolved with the experimental energy distribution of [3]. The results are shown in Fig. 1. In the figure we have also labeled some of the strong resonances based on the promotion channel gives improved agreement between theory and experiment in the collision energy range of 400-500 eV. The resonances between 400-450 eV are in the better agreement with the experiment compared to the previous FAC results [5]. Also the resonance at ∼ 470 eV does not appear unless this excitation channel is included. However including CI only within the same n complex does not remove the large discrepancy between theory and experiment for collision energies over 650 eV.

C. CI between different n complexes
Explicit calculations for autoionization and radiative decay rates were again carried out to n = 14. For higher n, the extrapolation described in Sec. III B was performed. The convolved results are shown in Fig. 1. Resonance strengths and energies are reported in Table I for the selected resonances described in Sec. III B. Figure 1 shows that above ∼ 650 eV multi-n CI dramatically reduces the theoretical results compared to single-n CI. This reduction brings theory into very good agreement with experiment. The previous factor of 2 differences have been reduced to the level of tens of percent. The remaining differences near the series limit may be due to field ionization effects in the experiment as described by [14], computational resources having limited the multi-n CI calculations to n ≤ 14, or some combination thereof.
A general sense for the importance of multi-n CI for n ≥ 5 can be gained by looking at the mixing factors for the resonances listed in Table I. The mixing factor is given by where the summation is over the j ′ basis states in the n ′ complex. The mixing occurs between levels with the same parity, symmetry, and angular momentum. The mixing factors are plotted in Fig. 2. One sees that the n = 3 and 4 resonances of Table I are largely unmixed with other n ′ complexes but the n ≥ 5 resonances can be strongly mixed. In particular the n = 5, 6, and 14 resonances are very strongly mixed with other n ′ complexes.
To gain a more quantitative understanding on how multi-n CI can affect the predicted resonance strengths, it is helpful now to re-write Eq. (8) using the expansion basis of Eq. (16) which givesσ Here A a n ′ j ′ i is the autoionization rate from the unmixed basis state φ n ′ j ′ to an initial state i and is given by A a n ′ j ′ k is given by Eq. (19) but changing i → k. A r n ′ j ′ f is the radiative decay rate from the φ n ′ j ′ to a state f and is given by The coupling (i.e., interference) terms between different basis such as Ψ k |V|φ n ′ j ′ φ n ′′ j ′′ |V|Ψ k and Φ f |D|φ n ′ j ′ φ n ′′ j ′′ |D|Φ f have been neglected just as in the IPIR approximation. The dielectronic capture strength for the CI mixing can be re-expressed as and the branching ratio for the CI mixing is given by  Table I. Each curve is labeled by the initial n configuration before mixing between n ′ configurations is included.
Now, taking the [(2p 1/2 3s) 1 3d 3/2 ] 1/2 6d 5/2 resonance level of the n = 6 complex listed in Table I as an example, we find that it mixes primarily with the basis levels listed in Table II.
Note that the autoionization rate A a n ′ j ′ i from the [(2p 1/2 3s) 1 3d 3/2 ] 1/2 6d 5/2 level to i is over a factor of 10 larger than the autionization rates from the other listed basis levels to i. This leads to a reduction in S DC by a factor of 6.5 when the values listed in Table II Table I. For the level description, relativistically closed shells with J = 0, such as 2s 2 , 2p 2 1/2 and 2p 4 3/2 , are omitted for brevity. The square of the mixing coefficient as defined in Eq. (16) is given in percent. A a n ′ j ′ i is the autoionization rate from j ′ to i where i is the initial state 2s 2 2p 6 3s 2 S 1/2 of recombining ion, k A a n ′ j ′ k + f A r n ′ j ′ f is the total autoionization and radiative decay rate of j ′ , and B j ′ is the branching ratio of j ′ . Only basis levels where |c n ′ j ′ | 2 > 2% are listed. A total of 10298 basis levels were included for this n = 6 resonance. [(2p 1/2 3s) 1 3d 3/2 ] 1/2 6d 5/2 level by a factor of 8.5, compared to the single-n CI results. This estimate agrees reasonably well with the factor of 6.5 reduction from the more complete calculation, as can be seen in Table I  resonances at this energy, as is shown in Fig. 3.

IV. SUMMARY
We have demonstrated the importance of CI between resonances with different captured electron principal quantum numbers n for DR of Na-like Fe 15+ forming Mg-like Fe 14+ via ∆n c = 1 core excitation of a 2l electron. Multi-n CI significantly reduces the theoretical resonance strengths for capture into n ≥ 5 levels which overlap in energy with other many different n levels. This brings theory into very good agreement with experiment and removes a previously existing discrepancy between the two. The n = 4 levels are largely unaffected by multi-n CI because the energy separation between the n = 4 resonances and the interacting higher n resonances is large enough to render the multi-n CI unimportant. Such is not the case for the energy separation of the n ≥ 5 resonances and those that they interact with, particularly for n = 5, 6, and 14. Additionally we have shown the importance of DR via 2s → 3l core promotions.