Effective collision strengths for fine-structure forbidden transitions among the 2s22p3 levels of S x

Bell, K. L.; Ramsbottom, C. A.

doi:10.1046/j.1365-8711.1999.02732.x

Abstract

Effective collision strengths for electron-impact excitation of the N-like ion S x are calculated in the close-coupling approximation using the multichannel R-matrix method. Specific attention is given to the 10 astrophysically important fine-structure forbidden transitions among the ⁴S^o, ²D^o and ²P^o levels in the 2s²2p³ ground configuration. The total (e⁻+ion) wavefunction is expanded in terms of the 11 lowest LS eigenstates of S x, and each eigenstate is represented by extensive configuration-interaction wavefunctions. The collision strengths obtained are thermally averaged over a Maxwellian distribution of velocities, for all 10 fine-structure transitions, over the range of electron temperatures log T(K)=4.6–6.7 (the range appropriate for astrophysical applications). The present effective collision strengths are the only results currently available for these fine-structure transition rates.

atomic processes, line: formation

1 Introduction

Emission lines from nitrogen-like ions with 2s²2p³ ground configurations have long been known to be useful as density diagnostics of solar and nebular plasmas. Aller, Uffer & Van Vleck (1949) suggested using the intensity ratio of forbidden lines that arise from transitions within the ground state of O ii as density diagnostics for gaseous nebulae. The lines that they considered corresponded to the transitions and Since then there have been numerous applications of the technique to planetary nebulae and other astrophysical objects, and the necessary electron collision atomic data have recently been calculated to high accuracy by McLaughlin & Bell (1998). However, the O ii ratio is not a useful diagnostic for the solar atmosphere, and instead other nitrogen-like ions can be used to determine electron densities; in particular, Wilhelm et al. (1995) have suggested that the 1213 Å/1196 Å line pair in S x is useful for electron density or temperature diagnostic studies with SUMER (Solar Ultraviolet Measurements of Emitted Radiation) flown on the Solar and Heliospheric Observatory. Indeed, Feldman et al. (1997), in a coronal spectrum of the solar high-temperature atmosphere, have detected the above S x lines as well as the 1552-Å line and the 1575-Å line.

Surprisingly little attention has been paid to this ion. Blaha (1969) used a weak coupling approximation, with exchange, but only considered the transition He tabulated the collision strength at a single energy value and suggested that the result could be in error by as much as a factor of 2. The only other investigation is that carried out by Bhatia & Mason (1980). They employed the distorted wave method, together with a transformation of the R-matrices obtained in LS coupling, into pair coupling using algebraic re-coupling coefficients which allow for spin—orbit interaction and other relativistic effects. 13 fine-structure levels were considered, namely those arising from the seven LS-coupling states associated with the 2s²2p³ and 2s2p⁴ configurations, and limited configuration-interaction was included in the target state representation. They presented collision strengths only at three values of the electron energy between 15 and 30 Ryd(where 1 Ryd = 2.180 × 10⁻¹⁸ J).

Clearly there is a need for a large and sophisticated close-coupling calculation that includes extensive configuration-interation wavefunctions to represent the target states and which takes full account of resonance effects.

2 Methods

The calculation was performed within the framework of the R-matrix method, using the computer codes described by Berrington et al. (1987), with the associated theory being given by Seaton (1987). The wavefunction for the total system of electron plus ion is expanded in terms of a finite number of target state wavefunctions. For the present calculation, we have included the lowest 11 target eigenstates: 2s²2p³⁴S^o, ²D^o, ²P^o; 2s2p⁴⁴P^e, ²D^e, ²S^e, ²P^e; 2p⁵²P^o and 2s²2p²3s ⁴P^e, ²P^e, ²D^e. Each of these target states is represented by sophisticated configuration-interaction wavefunctions constructed from one-electron orbitals. The radial part of these one-electron functions is given by a linear combination of Slater-type orbitals:

(1)

The parameters c_jnl, I_jnl and ζ_jnl for the 1s, 2s and 2p functions were taken to be the Hartree—Fock values obtained by Clementi & Roetti (1974) for the 2s²2p³⁴S^o S x ground state. Spectroscopic 3s and 3d functions were obtained by optimizing the energies of the 2s²2p²(¹D)3s ²D and 2s²2p²3d ⁴P states, respectively, using the configuration-interaction code civ3 (Hibbert 1975). A 3p pseudo-orbital was included to allow for the different 2p functions occurring in the different target states, and the parameters were obtained by optimizing the energy of the 2s²2p²3s ⁴P state. Further correlation orbitals, 4s, 4d and 4f, were obtained by optimization of the energies of the 2s2p⁴²S, 2s²2p³⁴S^o and 2s²2p³²D^o states, respectively. The parameters for the orbitals are given in Table 1.

Table 1.

The orbital parameters (c, I, ζ) of the radial wavefunctions.

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The configuration-interaction wavefunctions of the S x target states were then constructed from this orbital set using configurations obtained from one electron replacement in the basis reference set 1s²2s²2p³, 1s²2s2p⁴, 1s²2p⁵ and 1s²2s²2p²3s, the 1s shell being kept close, together with the configurations 2s²2p3p², 2s²2p4s², 2s²2p4d², 2p³3s3p, 2p³3s4s, 2p³3p3d and 2p³3p4d. A total of 330 configurations formed in this way were retained in the wavefunction expansion for both odd and even parity states. Table 2 presents the calculated energy levels in rydbergs relative to the 2s²2p³⁴S^o ground state. Comparisons are made with the observed data of Martin, Zalubas & Musgrove (1990), the theoretical predictions of the Opacity Project (obtained from the TOPbase data bank: Cunto et al. 1993), the stationary second-order many-body perturbation theory (MBPT) results of Merkelis et al. (1997) and the intermediate-coupling calculations of Bhatia & Mason (1980) using the code sstruct. The agreement between the present results and experiment is excellent with the largest difference of 3 per cent occurring for the 2s²2p³²D^o state. A significant improvement over the work of Bhatia & Mason (1980) is apparent, particularly for the highest state considered by them, namely the 2s2p⁴²P state. Similarly, it is seen that the present data agree better in general with experiment than do the Opacity Project data, but not quite so well as the results of Merkelis et al. (1997).

Target state energies (in rydbergs) relative to the 2s22p34So ground state of S x.

Table 2.

Target state energies (in rydbergs) relative to the 2s²2p³⁴S^o ground state of S x.

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Although the agreement of the present energy levels with experiment is excellent, when calculating effective collision strengths for use in astrophysical applications it is essential to utilize experimental energy differences (Martin et al. 1990) to ensure that the thresholds lie in their exact positions. Hence the present R-matrix calculation has been carried out with the energies shifted to the experimental values of Martin et al. (1990).

The calculation of the collision strength, Ω_if, between an initial target state i and a final state f was then performed using the R-matrix close-coupling codes (Berrington et al. 1987). The (N+1)-electron configurations were generated by allowing one electron excitation from the two basis distributions 1s²2s2p⁴ and 1s²2s²2p²3s, with the restrictions that the 1s shell remains closed, that the 2p shell retains at least one electron and that there is a maximum of only one electron in the 4f shell. Experience has shown that this achieves a correct balance in the various correlations. In constructing the total wavefunction in the R-matrix internal region, 20 continuum orbitals were used for each incident electron orbital angular momentum. The R-matrix boundary was found to be 7.6 Ryd. We included all partial waves with total angular momentum , for both even and odd parities and singlet, triplet and quintet multiplicities, which was found to be sufficient to ensure convergence of the cross-sections for the forbidden transitions considered in this paper.

The cross-section between an initial target state i and a final state f is related to the collision strength (Ω_if) by the expression

(2)

where Ω_i is the statistical weight of the initial target state,

the incident electron energy in rydbergs and a₀ the Bohr radius. Having obtained collision strengths within the framework of the LS-coupling scheme, we utilized the program of Saraph (1978) to transform to a jj-coupling scheme and thus produced the collision strengths between the j-resolved levels.

Finally, for many astrophysical and plasma applications it is often more convenient to deal with the thermally or Maxwellian averaged effective collision strength (ϒ_if). For excitation from level i to level f, the dimensionless effective collision strength at electron temperature T_e (in kelvin) is given by

(3)

where E_f is the final free electron energy after excitation and k is Boltzmann's constant. Both the collision strength Ω_if and the corresponding effective collision strength ϒ_if are presented in this paper for all 10 fine-structure forbidden transitions between the 2s²2p³ levels in S x.

3 Results and discussion

In order to evaluate accurately the effective collision strengths, it is imperative to know the energy dependence of the collision strength over a wide energy region and for a large number of energies. Therefore, to delineate properly the complex autoionizing resonances in the cross-sections, we have utilized a sufficiently fine mesh of incident impact energies (typically 0.005 Ryd) across the range 0-129.6 Ryd. Pseudo-resonances arising from our inclusion of pseudo-orbitals in the wavefunction representation (Burke, Sukumar & Berrington 1981) have been smoothed out for energies above the highest lying threshold included (2s²2p²3s ²D^e). Figs 1–10 graphically present the collision strength and the corresponding Maxwellian averaged effective collision strength for the forbidden fine-structure transitions of interest. Comparison is made with the earlier results of Bhatia & Mason (1980) for the collision strengths.

Figure 1.

The collision strength as a function of incident electron energy in rydbergs (a), and the effective collision strength as a function of log temperature in kelvin (b), for the fine-structure transition. Solid line: present results; filled circles: Bhatia & Mason (1980).