© ESO, 2011

1. Introduction

Many extreme ultraviolet (EUV) lines due to n = 2 → 2 transitions of highly-charged sulphur ions have been recorded in solar observations (Thomas & Neupert 1994; Brown et al. 2008). Some transition lines show diagnostic potential for electron density, e.g. the emission lines of S XII (Keenan et al. 2002). Additionally, a few soft X-ray emission lines due to n = 3 → 2 transitions of sulphur ions have also been detected in solar observations (Acton et al. 1985) and a Chandra Procyon observation (Raassen et al. 2002). However, most of the excitation data adopted in astrophysical modelling for sulphur spectra are from the distorted-wave (DW) method, which is well known to underestimate results for weaker transitions compared to those from an R-matrix calculation. Only data from a few small R-matrix calculations is available for sulphur ions to-date.

For S⁸⁺, Bhatia & Landi (2003a) reported an extensive excitation calculation (n = 3) with the DW method, which is currently used by astrophysical modelling codes, e.g. chianti v6 (Dere et al. 2009). R-matrix data is available only for transitions within the ground configuration (Butler & Zeippen 1994).

For S⁹⁺, DW excitation data for transitions of levels up to n = 3 were calculated by Bhatia & Landi (2003b), which were also incorporated into chianti v6. An R-matrix calculation has been performed by Bell & Ramsbottom (2000), but only data for transitions among levels of n = 2 complex and 2s²2p²3s configuration were reported.

For S¹⁰⁺, the newest excitation data are attributed to be the work of Landi & Bhatia (2003), who have extended a previous (n = 3) DW calculation to include n = 4 configurations (viz 2s²2p4l′, l′ = s, p and d). Their results have been incorporated into astrophysical modelling codes, e.g. chianti v6. By using R-matrix results available for some carbon-like ions (e.g. O²⁺, Ne⁴⁺, Mg⁶⁺, Si⁸⁺ and Ca¹⁴⁺), Conlon et al. (1992) derived the electron excitation data for other carbon-like ions, including S¹⁰⁺, by interpolation. These resultant data are only valid for a range of electron temperatures approximately equal to log T_max ± 0.8 dex, where T_max is the temperature of maximum fractional abundance in ionization equilibrium. However, interpolation and/or extrapolation from R-matrix calculations was proved to be invalid at low temperatures because of the complexity of effective collision strengths along the iso-electronic sequence, as shown in Li-like (Liang & Badnell 2011), F-like (Witthoeft et al. 2007), Ne-like (Liang & Badnell 2010) and Na-like (Liang et al. 2009b) iso-electronic sequences. An explicit R-matrix calculation for this ion for transitions between levels of the n = 2 complex was reported by Lennon & Burke (1994).

For S¹¹⁺, the R-matrix calculations by Zhang et al. (1994) and Keenan et al. (2002) are still the main data source for modelling – the close-coupling expansion included the lowest 8 LS terms of the n = 2 complex. However, there are potential problems for B-like ions as demonstrated by Liang et al. (2009a) for Si⁹⁺, viz. effective collision strengths of a previous R-matrix calculation (Keenan et al. 2000) do not converge to the correct high-temperature limit for a few strong transitions.

Here, we report-on calculations for the electron-impact excitation of four iso-nuclear ions of sulphur (S⁸⁺–S¹¹⁺) which were made using the ICFT R-matrix method. The remainder of this paper is organized as follows. In Sects. 2 and 3, we discuss details of the calculational method and pay particular attention to comparing our underlying atomic structure results with those of previous workers. The model for the scattering calculation is outlined in Sect. 4. The excitation results themselves are discussed in Sect. 5 and we summarise in Sect. 6.

2. Structure: level energy

The target radial wavefunctions (1s–4f) were obtained from autostructure (AS, Badnell 1986) using the Thomas-Fermi-Dirac-Amaldi model potential. Relativistic effects were included perturbatively from the one-body Breit-Pauli operator (viz. mass-velocity, spin-orbit and Darwin) without valence-electron two-body fine-structure operators. This is consistent with the operators included in the standard Breit-Pauli R-matrix suite of codes.

2.1. S IX

Configuration interaction among 30 configurations (see Table 1) was included to describe the target used to calculate level energies and weighted absorption oscillator strengths (g_if_ij, for a given i ← j transition). The model potential radial scaling parameters, λ_nl (n = 1−4; l ∈ s,p,d, and f), were obtained by a three-step optimization procedure. In the first step, the energy of the 2s^x2p^y (x + y = 6) was minimized by varying the λ_1s, λ_2s and λ_2p scaling parameters. Then, the energies of the 2s²2p³3l and 2s²2p³4l configurations were minimized by varying the λ_3l and λ_4l scaling parameters, respectively. The resultant scaling parameters are listed in Table 2.

The 92 lowest-lying fine-structure target levels of 2s^x2p^y   (x + y = 6), 2s²2p³3l and 2s2p⁴3s   (⁵P   and   ³P, only) configurations were used in the close-coupling expansion for the scattering calculation. The resultant level energies are compared with experimentally derived data from NIST v4¹ and previous calculations, see Table 3. The present AS level energies show an excellent agreement (less than 0.5% except for 2s²2p^4   3P) with those of Bhatia & Landi (2003a). This is due to the use of similar structure codes and calculations. The main difference between the two is the n = 4 configurations that we included in our CI expansion but were not used by Bhatia & Landi (2003a).

When compared with NIST data and the MCHF collection² (Tachiev & Froese Fischer 2002), the present results agree to within 1% for all levels of the n = 3 configurations. For levels of the n = 2 configurations, the energy difference is about 2–5%. So, we performed a calculation with energy corrections to the diagonal of the Hamiltonian matrix before diagonalization, for the 16 lowest-lying levels, and iterated to convergence. For the missing level (2s²2p³3s   ⁵S₂) in the NIST compilation, we adopt the mean value of differences between our level energies and corresponding NIST values of the same configuration. The resultant e-vectors and e-energies are used to calculate the oscillator strengths and archived energies.

We notice that the MCHF data shows an excellent agreement with the data compiled in the NIST database. A non-relativistic multi-configuration Hartree-Fock (MCHF) approach was used by Tachiev & Froese Fischer (2002) to generate radial orbitals for subsequent use in diagonalizing a smaller scale Breit-Pauli Hamiltonian. A large set of configurations was used in the LS-coupling calculation, for example, configuration states were included up to n = 7. Orbitals sets were optimized separately for the initial and final states and no orthonormality was imposed between the two sets. Thus, we consider this MCHF data to be the theoretical reference work.

2.2. S X

Configuration interaction among 24 configurations (see Table 1) was included to calculate the level energies and oscillator strengths between levels of the configurations 2s^x2p^y   (x + y = 5) and 2s²2p²3l. Since the same configuration interaction and a similar structure code (superstructure) was used by Bhatia & Landi (2003b), we use their scaling parameters λ_nl in the present work for this ion. We note that there is severe interaction between the 2s²2p²3l and 2s2p³3l′ configurations. So some terms of the 2s2p³3s (⁶S,   ⁴S and ⁴D) and 2s2p³3p (⁶P and ⁴D) configurations were included in the close-coupling expansion for the excitation calculation, as detailed in the next section.

The calculated energies for the 84 lowest-lying levels are listed in Table 4 along with experimentally derived data from the NIST v4 compilation as well as other predictions. Although we have used the same configuration expansion and same radial orbitals as Bhatia & Landi (2003b), the energies do not quite match because we have omitted all two-body fine-structure operators so as to be consistent with our subsequent R-matrix calculation. (If we include them for the 3 configurations of the ground complex then we reproduce their energies). The difference is much smaller than the difference with the experimentally derived NIST energies. When compared with the NIST compilation¹ and the MCHF collection² , both agree to within 0.5% for all levels of the n = 3 configurations. For the levels of n = 2 complex, the difference is up to about 5%. So, we again iterated with energy corrections to the diagonal of Hamiltonian matrix before diagonalization, for the 22 lowest-lying levels, to calculate oscillator strengths and archived energies. The data of MCHF collection (Tachiev & Froese Fischer 2002) shows an excellent agreement again with the NIST data.

2.3. S XI

As shown in Table 1, configuration interaction among 24 configurations has been taken into account to calculate the level energies and oscillator strengths. The radial scaling parameters λ_nl were obtained by a three-step optimization procedure. In the first step, the energy of the 2s^x2p^y (x + y = 4) was minimized by varying the λ_1s, λ_2s and λ_2p scaling parameters. Then, the energies of the 2s²2p3l and 2s²2p4l configurations were minimized by varying the λ_3l and λ_4l scaling parameters, respectively. The resultant scaling parameters are listed in Table 2.

Fig. 1 Comparison of weighted oscillator strengths gf of electric-dipole transitions (to the 5 lowest-lying levels) for S8+ – S11+. BL03 refers to the work of Bhatia & Landi (2003a,b), whereas LB03 and NSD07 correspond to the work of Landi & Bhatia (2003) and Nataraj et al. (2007), respectively. The dashed lines correspond to agreement within 20%. (Colour online.)

The 254 lowest-lying fine-structure levels were used in the close-coupling expansion for the scattering calculation. They are compared with those data available from the NIST compilation and other predictions in Table 5. The present calculation shows a good agreement (1%) with those experimentally determined data in NIST database and the MCHF collection² for 2s2p^3 3D, ³P and n = 3,4 levels. For other levels of the 2s2p³ configuration and those of the 2p⁴ configuration, the present results are systematically higher than NIST data by 1–2%. The present AS result shows an excellent agreement (less than 0.5%) with the result of Landi & Bhatia (2003) for all levels of the n = 3 configurations. However, both sets of results are systematically higher than the NIST data for the levels of n = 2 complex, those of Landi & Bhatia (2003) more-so than the present which are within 2% (excluding the ). So, we perform an iterated energy correction calculation again for the 23 lowest-lying excited levels.

The data of MCHF collection show better agreement again with the NIST data than other predictions. Unfortunately, there are no published papers to indicate the scale of calculations.

2.4. S XII

Configuration interaction among 32 configurations has been taken into account to calculate level energies and oscillator strengths, see Table 1. The radial scaling parameters λ_nl were obtained by a three-step optimization procedure. In the first step, the energy of the 2s^x2p^y (x + y = 3) configurations was minimized by varying the λ_1s, λ_2s and λ_2p scaling parameters. Then, the energies of the 2s²2p3l and 2s²2p4l configurations were minimized by varying the λ_3l and λ_4l scaling parameters, respectively. The resultant scaling parameters are listed in Table 2.

The 204 lowest-lying target levels were used in the close-coupling expansion for the scattering calculation. The present AS level energies are compared with those data available from the NIST compilation and other predictions, see Table 6. A good agreement (less than 1%) is obtained when compared with those experimentally determined in the NIST database for levels of the n = 3 configurations. For levels of the n = 2 complex, the difference is slightly larger, but still within  ~2%. Comparison with data in chianti v6³ demonstrates that the differences are within 1% for almost all levels except for those of the n = 2 complex. A good agreement is found when compared with calculations by Merkelis et al. (1995) with many-body perturbation theory (MBPT) and all-order relativistic many-body theory by Nataraj et al. (2007). As done for other ions, energy corrections for the levels of 2s^x2p^y (x + y = 3) configurations have been included to improve the accuracy of the oscillator strengths and archived energies.

3. Structure: oscillator strengths

A further test of our structure calculation is to compare weighted oscillator strengths gf_ij. In terms of the transition energy E_ji (Ryd) for the j → i transition, the transition probability or Einstein’s A-coefficient, A_ji can be written as (1)where α is the fine structure constant, and g_i, g_j are the statistical weight factors of the initial and final states, respectively.

Figure 1 shows such a comparison for the transitions into the five lowest-lying levels for the four iso-nuclear ions to assess the accuracy of the structure calculation. For S⁸⁺, about 86% of all available transitions in the work of Bhatia & Landi (2003a) show agreement to within 20%. When compared with the data from the MCHF collection², 61% of all available transitions agree to within 20%. For those transitions with larger differences in the two cases, the data points are linked together by a solid line. For some transitions, the present results agree better with the results of Bhatia & Landi (2003a) than with the data from MCHF collection, while for others they agree better with the data from the MCHF method. Since correlation from much higher excited configuration has been taken into account in the data of MCHF collection, their gf-values are the best transition data so-far, as demonstrated by their level energies. The present results show a better agreement with MCHF calculation (Tachiev & Froese Fischer 2002) than those of Bhatia & Landi (2003a), which indicates that we have a more accurate structure.

For S⁹⁺, 82% of transitions agree to within 20% for the present AS results and those of Bhatia & Landi (2003b). Recall, we omitted two-body fine-structure but have iterated to the observed energies, for levels of the ground complex, compared to Bhatia & Landi (2003b). We have also performed calculations with/without the two-body fine-structure and level energy corrections to study the effect of the two, see Table 7. It appears that the energy corrections play an more important on the resultant gf-value for these weak transitions. When compared with the data from the MCHF collection² , about 64% of all available transitions show agreement to within 20%. For those transitions with larger differences, the data points are linked together as done for S⁸⁺. We also notice that the level label for 2s²2p²3s (21th in the Table 4) and (22th) in MCHF collection should be exchanged because a good agreement between the MCHF calculation and the other two predictions can be obtained after such a procedure for all transitions to the five lowest-lying levels from the two levels, e.g. the 22 → 3 and 21 → 3 transitions marked in Fig. 1.

For S¹⁰⁺, most (67%) transitions are in agreement to within 20% for the present AS results and those of Landi & Bhatia (2003). When compared with calculation from MCHF method, the percentage is about 90% of available transition data.

For S¹¹⁺, the present AS results agree well (within 20%) with predictions from other sources including superstructure (the data in chianti database, 94% of available transitions), MCHF² (87%), MBPT (Merkelis et al. 1995⁴, 83%) and the relativistic coupled-cluster theory (Nataraj et al. 2007), for transitions between levels of the n = 2 complex. For transitions from higher excited levels, e.g. n = 3 configurations, only the unpublished calculation of Sampson & Zhang is available (from the chianti database). Figure 1 illustrates that only 46% of available transitions show agreement to within 20%.

Additionally, we explicitly label some transitions with large differences in Fig. 1. They are all from the 50th and 52nd levels. We recall that we take configuration, total angular momentum and energy ordering to be the “good” quantum numbers when level matching for comparisons. Exchanging the level matching for these two levels cannot eliminate the large difference now, unlike the case of S⁹⁺. Level mixing (2s2p3s   ²P contributes 86% for the 50th, 2s2p3d   ⁴P contributes 90% for the 52nd) also can not explain this discrepancy for these strong transitions.

Thus, we believe the atomic structure of the four iso-nuclear ions to be reliable, and expect the uncertainty in collision strengths due to in accuracies in the target structure to be correspondingly small.

4. Scattering

The scattering calculations were performed using a suite of parallel intermediate-coupling frame transformation (ICFT) R-matrix codes (Griffin et al. 1998). We employed 40 continuum basis orbitals per angular momentum so as to represent the (N + 1)th scattering electron for the four ions. All partial waves from J = 0 to J = 41 (S⁹⁺ and S¹¹⁺) or J = 1/2 to J = 81/2 (S⁸⁺ and S¹⁰⁺) were included explicitly and the contribution from higher J-values were included using a “top-up” procedure (Burgess 1974, Badnell & Griffin 2001). The contributions from partial waves up to J = 12 (S⁹⁺ and S¹¹⁺) or J = 23/2 (S⁸⁺ and S¹⁰⁺) were included in the exchange R-matrix, while those from J = 13 to 41 or J = 25/2 to 81/2 were included via a non-exchange R-matrix calculation. In the exchange calculation, a fine energy mesh (1.0 × 10^-5z² Ryd, where z is the residual charge of ions) was used to resolve the majority of narrow resonances below the highest excitation threshold. From just above the highest threshold to a maximum energy of eight times the ionization potential for each ion, a coarse energy mesh (1.0 × 10^-3z² Ryd) was employed. For the non-exchange calculation, a step of 1.0 × 10^-3z² Ryd was used over the entire energy range. Additionally, experimentally determined energies or adjusted energies were employed in the MQDT expressions used by the ICFT method to further improve the accuracy of the results, as was done for Si⁹⁺ (Liang et al. 2009a). The correction procedure was mainly done for levels of the n = 2 complex (needed because of the difficulty in obtaining a good structure here at the same time as describing n = 3 and 4 configurations with a unique orbital basis) and some levels of the 2s²2p^x3s (where x = 3, 2, 1 or 0 for S^{8+,9+,10+,11+}, respectively) configuration, as explained in detail in the structure section.

We make use of the infinite energy Born limits (non-dipole allowed) or line strengths (dipole) to extend the R-matrix collision strengths to higher scattering energies by interpolation of reduced variables, as described by Burgess & Tully (1992). Finally, thermally averaged collision strengths (Υ) were generated at 13 electron temperatures ranging from 2 × 10²(z + 1)² K to 2 × 10⁶(z + 1)² K. The data were stored in the ADAS adf04 format (Summers 2004) being available electronically from the OPEN-ADAS database  , APAP-network and the CDS archives.

5. Results and discussions

5.1. S IX

Fig. 2 Comparison of (effective) collision strengths (Υs, see the lower panel) Ωs from the ground state of S8+. BL03 refers to the distorted wave calculation by Bhatia & Landi (2003a), AS-DW refers to the present Breit-Pauli DW calculation using autostructure. Upper panel: scaled collision strength for a dipole transition 2s22p4   3P2  ←  2s22p33d   3P1 (1–80) with C = 2.0. The limit value is 4gifij/Eij at 1.0 for the dipole transition. Lower panel: the ratio of Υs between the results of the DW calculation by Bhatia & Landi (2003a) and the ICFT R-matrix calculation at log Te (K) = 5.1, 6.1 (corresponding to peak abundance of S8+ in ionization equilibrium) and 7.1. The dashed lines correspond to agreement within 20%. The transition marked by dotted box is the dipole transition 1–80 shown in the upper panel. (Colour online.)

In Fig. 2, we make an extensive comparison of the present effective collision strengths with the DW data of Bhatia & Landi (2003a) for excitations from the ground level 2s²2p^4   3P₂. At the low temperature (log T_e (K) = 5.1), only 27% of transitions show agreement within 20%. This can be easily explained by the omission of resonances in the DW calculation by Bhatia & Landi (2003a). At the temperature (log T_e (K) = 6.1) of peak fractional abundance in ionization equilibrium, the percentage is still low (41%). At the high temperature log T_e (K) = 7.1), the percentage increases to 58%. This is due to the reduced contributions of near threshold resonances with increasing temperature. However, we note that there are a few transitions showing a ratio Υ_BL03/Υ_ICFT > 1.3, and the ratio increases with increasing temperature, e.g. the dipole transition of 2s²2p^4   3P₂  ←  2s²2p³3d   ³P₁ (1–80) marked by the dotted box in the lower-panel of Fig. 2. In the upper-panel of Fig. 2, we show the scaled collision strength as a function of reduced energy so as to shed light on this odd behaviour. The DW calculation by Bhatia & Landi (2003a) is higher than the background of the present ICFT R-matrix calculation and the present Breit-Pauli DW (hereafter AS-DW) calculation using autostructure (Badnell 2011). And the three different calculations show a self-consistent behaviour approaching the infinite-energy limit point. So the odd behaviour is due to the higher background in the DW calculation by Bhatia & Landi (2003a). The limit value from chianti (v6) is also plotted, which shows an excellent agreement with present calculations. This inconsistency in the chianti (v6) database is due to different data sources being adopted, e.g. the structure data is from a 24 configuration calculation, whereas the scattering data is from a 6 configuration calculation⁵.

Fig. 3 Comparison of effective collision strengths of S8+ with the R-matrix results of Butler and Zeippen (1994) for transitions of the ground configuration 2s22p4. Filled symbols with solid curves are results of Butler and Zeippen (1994), while open symbols with dotted curves corresponds to the present ICFT R-matrix calculation. Note: The same symbol in the two sets of results corresponds to the same transition. (Colour online)

For S⁸⁺, an earlier R-matrix calculation for transitions within the ground configuration is available (Butler & Zeippen 1994) for which the LS-coupling K-matrices were transformed algebraically to intermediate coupling to obtain collision strengths between the fine-structure levels. A detailed comparison has been made between the two different R-matrix calculations, see Fig. 3. At the low temperature (T_e ~ 1.0 × 10⁴ K), there is a large difference between the two different R-matrix calculations. A separate ICFT R-matrix calculation with finer mesh (1.0 × 10^-6z²) near threshold confirms that the effect of resonance resolution is less than 2% for nearly all excitations, except for the 2–5 (10% at log T_e(K) = 4.1) and 3–5 (24% at log T_e(K) = 4.1) transitions. So the present effective collision strengths are generally converged with respect to resonance resolution. The large differences between the two different R-matrix calculations may be due to deficiencies in the transformational approach used by Butler & Zeippen (1994), as detailed by Griffin et al. (1998) and demonstrated by Liang et al. (2008). The adoption of observed energies for levels of n = 2 complex in the present ICFT R-matrix calculation gives better positioning of near threshold resonances than the previous ones with theoretical energies (Butler & Zeippen 1994). So, the present effective collision strengths are expected to be more reliable at low temperatures.

5.2. S X

Fig. 4 Comparison of effective collision strengths of S9+ with the jajom R-matrix results of Bell & Ramsbottom (2000) for transitions of the n = 2 complex and 2s22p23s configuration. One point marked by bold “ ↘ ” refers to the 2s22p3   4S3/2  ←  2s2p4   2P3/2 transition (1–12), that will be examined in Fig. 5. (Colour online.)

In Fig. 4, an extensive comparison has been made with previous R-matrix calculation (Bell & Ramsbottom 2000) at three temperatures: log T_e(K) = 5.2, 6.2 (corresponding to peak fraction in ionization equilibrium) and 6.7. At the low temperature (log T_e(K) = 5.2), only 27% of all available transitions show an agreement within 20%. Even at the high temperature (log T_e(K) = 6.7), the percentage is only about 34%. Ratios () less than unity can be understood in terms of the finer energy mesh used (present: 1.0 × 10^-5z² Ryd, Bell & Ramsbottom 2000:  ≥0.008 Ryd) and resonances attached to the 2s²2p²3l configurations in our present ICFT R-matrix calculation, as well as the purely algebraic jajom approach that was used by Bell & Ramsbottom (2000). However, the ratio being larger than unity requires another explanation. So, we select one transition marked by the bold “ ↘ ” in Fig. 4 to investigate the source of the difference between the two different R-matrix results.

Fig. 5 Comparison of excitation data for the 2s22p3   4S3/2  ←  2s2p4   2P3/2 transition (1–12) of S9+. Here, BR00 refers to the R-matrix calculation of Bell & Ramsbottom (2000), BL03 to the DW result of Bhatia & Landi (2003b) and ICFT to the present calculation. Upper panel: effective collision strengths – the BR00 without pseudo-resonances result was re-derived by us from the said original collision strengths provided by Ramsbottom (2011 priv. comm.). Lower panel: scaled collision strengths, with the scaling parameter C set to 2.0. (Colour online.)

Figure 5 shows a comparison of our present (effective) collision strength with the previous R-matrix results for the 2s²2p^3   4S_3/2  ←  2s2p^4   2P_3/2 (1–12) dipole transition. Around the temperature of T_e ~ 9.0 × 10⁵–6.0 × 10⁶ K, the Bell & Ramsbottom (2000) result is higher than present ICFT R-matrix calculation, and by up to a factor of 2. We note that some pseudo-orbitals (, , and ) were included in the work of Bell & Ramsbottom (2000). They stated that some pseudo-resonances are found above the highest threshold (19.682 Ryd). One of the authors (Ramsbottom, priv. comm. 2011) has provided us with collision strengths (Ω) with the pseudo-resonances at high energies removed. A comparison of the scaled collision strengths Ω reveals that the backgrounds of the two different R-matrix calculations agree well, and are consistent with the DW calculation by Bhatia & Landi (2003b). So, the large difference between the two different R-matrix calculations is not arising from the difference in their structures. We then re-derived the effective collision strengths, which shows the expected behaviour, see Fig. 5. So, it appears that the previously published R-matrix effective collision strengths of Bell & Ramsbottom (2000) were derived from their collision strengths before the pseudo-resonances were subtracted. So, the ratios greater than unity in Fig. 4 should be mostly/partly attributed to the pseudo-resonances in the previous R-matrix calculation. So, the present results are more reliable for modelling applications.

For excitations to higher levels of n = 3 configurations, only DW data is available, e.g. the latest work of Bhatia & Landi (2003b). A comparison there demonstrates that the resonance contribution is strong for some transitions and is widespread, as expected. For conciseness, the figure is not shown here.

5.3. S XI

Fig. 6 Comparison of effective collision strengths between different R-matrix calculations for transitions in the ground configuration 2s22p2 of S10+, where CKA92 refers to the interpolated data from R-matrix results for other ions in carbon-like sequence (Conlon et al. 1992), LB94 corresponds to the R-matrix work of Lennon & Burke (1994), ICFT denotes the present work. The transition is marked by i − j adjacent to the relevant set of curves. Note: the CKA92 data is extracted from the chianti v6 database. (Colour online.)

Fig. 7 Collision strengths (Ω) of the 2s22p2   3P0  ←  2s2p3   5S2 (1–6) transitions of S10+, where LB94 corresponds to the R-matrix work of Lennon & Burke (1994) from the TIPTOPbase 6, ICFT denotes the present work. (Colour online.)

As mentioned in the introduction, interpolated data from R-matrix results is available for S¹⁰⁺ (Conlon et al. 1992). These resultant data are valid over a temperature range approximately equal to T_e ~ 3.2 × 10⁵–1.3 × 10⁷ K for S¹⁰⁺. In this temperature range, the interpolated excitation data show a good agreement with present the ICFT R-matrix calculation for almost all transitions, as shown in Fig. 6, even though only partial waves of L < 9 and 12 continuum basis orbitals in each channel were included. That is, the effective collision strength is converged in this temperature range using a small range of partial waves etc. Lennon & Burke (1994, hereafter LB94) performed an R-matrix calculation which included all 12 terms of the ground complex and adjusted the diagonal elements of the LS-coupling Hamiltonian matrix to the (fine-structure averaged) observed energies before diagonalization. They provided data for transitions between fine-structure levels in the ground configuration 2s²2p² plus the 2s2p³ . At low temperatures T_e < 1.0 × 10⁵ K, the present ICFT R-matrix calculation is systematically higher than this previous small-scale R-matrix result except for the 1–6 transition, see Fig. 6. This situation can likely be attributed to the much larger close-coupling expansion (to n = 4) and associated resonances in the present calculation. We recall also that we used observed level energies in the present ICFT R-matrix calculation via multi-channel quantum defect theory (MQDT).

In case of the 1–6 transition, the original collision strength of Lennon & Burke (1994) is available from TIPTOPbase⁶. In Fig. 7, we compare the two sets of results. We see that there is a somewhat oddly high background around 0.5–2.0 Ryd in the results of Lennon & Burke (1994). This is the likely reason their effective collision strength is notably larger than the present one at lower temperatures.

Fig. 8 Comparison of effective collision strengths of the present ICFT R-matrix results with other (DW) results. a) Scatter plot showing the ratio () for S10+ at three temperature of logTe(K) = 5.3, 6.3 and 7.3, where the DW calculation refers to the work of Landi & Bhatia (2003, LB03). The bold “ ↖ ” refers to a forbidden transition 2s22p2   3P0  ←  2s22p4p   3P0 transition (1–200), that is examined in panel b). b) Comparison of scaled collision strength Ω (with scaling parameter C = 2.0) of the 2s22p2   3P0  ←  2s22p4p   3P0 transition (1–200). AS-DW (9 and 24 models) corresponds to the present Breit-Pauli DW calculation by using autostructure with 9 and 24 configurations, corresponding to that used in the scattering and structure calculations of Landi & Bhatia (2003), respectively. (Colour online.)

Comparison with the DW calculation of Landi & Bhatia (2003) demonstrates that only 22% of all available transitions show agreement within 20%. Figure 8 demonstrates that the resonance contribution is strong for some transitions, and widespread as expected again. At high temperature, uncertainties of scattering data are dominated by the accuracy of structure calculation because the resonance contribution becomes increasingly small. But only 43% of all available transitions show agreement within 20%, which is significantly lower than that in the assessment for weighted oscillator in Sect. 3. We also notice there are a few transitions showing the ratio being lower than unity. So we select one transition (2s²2p^2   3P₀  ←  2s²2p4p   ³P₀, see the bold “ ↖ ” mark in Fig. 8a) to investigate. Fig. 8 clearly demonstrates that the DW data of Landi & Bhatia (2003) is higher than the background of the present ICFT R-matrix calculation. But the present Breit-Pauli DW calculation using autostructure (Badnell 2011) shows an excellent agreement with the background of the R-matrix calculation – both use the exact same atomic structure. As stated by Landi & Bhatia (2003), a small atomic model (nine lowest configurations, 72 fine-structure levels) was adopted in their scattering calculation because of their available computer resource. So, we performed another separate AS-DW calculation with the 9 lowest configurations, in which the optimization procedure is done as mentioned above for S¹⁰⁺. The resultant data show good agreement with the DW calculation by Landi & Bhatia (2003). So ratios lower than unity and the low percentage of agreement in the scatter plot mentioned above are likely due to the use of a much larger configuration interaction expansion in the present ICFT R-matrix calculation.

5.4. S XII

As stated by Keenan et al. (2002), a small error in the previous excitation data (Zhang et al. 1994) was found for a few transitions of some boron-like ions, and those data were replaced. In Fig. 9, we compare the present ICFT R-matrix excitation data with the revised data of Keenan et al. (2002) at three different temperatures (log T_e(K) = 6.04, 6.40 and 6.78) to check the validity of the present results or improvement by including larger CI and extensive close-coupling expansions. For strong excitations (≥ 0.1), a good agreement (within 20%) is obtained for most excitations (82%). For weak excitations, the present ICFT R-matrix results are systematically larger than previous ones except for a few transitions, e.g. 8–13 and 9–13. Indeed, the weaker the excitation, the greater the difference, and by more than a factor of 2 for a group of the weakest excitations. This can be easily explained by resonances attached to n = 3 levels included in the present work, and this effect is stronger for weaker excitations. For the two above mentioned transitions (8–13 and 9–13), the previous R-matrix calculation is significantly higher than the present ones at log T_e(K) = 6.04 by a factor of 2.5 and 40%, respectively. Unfortunately, there are no previous collision strengths available to compare with – examination of the present collision strengths uncovers no untoward behaviour for these two transitions.

For excitations to higher excited levels of the n = 3 configurations, only an unpublished DW calculation (Zhang & Sampson 1995) is available – compiled in the chianti database. A comparison demonstrates that the resonance contribution is strong for some transitions, and is widespread as expected. For conciseness, the figure is not shown here.

Additionally, we checked the sensitivity of the high-T_e Υs to the top-up and find that it is greatest on the weakest (dipole) transitions but it is not significant compared to the inherent uncertainties in the atomic structure (f-values) for such transitions – the strong transitions are well converged.

6. Summary

Fig. 9 Comparison of effective collision strengths of S11+ for all excitations between levels of the n = 2 complex at three different temperatures log Te(K) = 6.04, 6.40 and 6.87. ΥICFT refers to the present ICFT R-matrix calculation, and KKR02 corresponds to the previous R-matrix results by Keenan et al. (2002). A few transitions with large difference are marked by labels around the points, and are linked together for results at different temperatures. (Colour online.)

Electron-impact excitation data for four iso-nuclear sulphur ions (S⁸⁺, S⁹⁺, S¹⁰⁺ and S¹¹⁺) have been calculated using the ICFT R-matrix method with extensive CI and large close-coupling expansions, as listed in Table 1.

Good agreement overall with the available experimentally derived data and other theoretical results for level energies and weighted oscillator strengths supports the reliability of the present R-matrix excitation data.

For excitations to levels of the n = 2 complex, an extensive assessment have been made with previous R-matrix calculations available to check the validity and improvement of the present ICFT R-matrix results. For excitations to higher excited levels of n = 3 and/or 4 configurations, only DW calculations are available to compare with. The improvement of the present calculations is illustrated as expected by including resonances. For some transitions, configuration interaction has a significant effect on the atomic structure and this carries through to the final (effective) collision strengths, as shown in the cases of S⁸⁺ and S¹⁰⁺.

In conclusion, the present ICFT R-matrix excitation data of S^8+,9+,10+ and S¹¹⁺ are assessed to be valid over an extensive temperature range, and a significant improvement is achieved over previous available ones to date due to the extensive CI and large close-coupling expansions used in the present work. This will replace data from DW and small R-matrix calculations presently used by astrophysical and fusion communities, and its use can be expected to identify new lines, improve spectral analyses and diagnostics of hot emitters or absorbers in astrophysics and fusion researches.

Acknowledgments

The work of the UK APAP Network is funded by the UK STFC under grant No. PP/E001254/1 with the University of Strathclyde. G.Y.L. acknowledges the support from the One-Hundred-Talents programme of the Chinese Academy of Sciences (CAS), and thanks Dr Cathy Ramsbottom at Queen’s University Belfast for providing her original electronic data as well as Dr Enrico Landi at University of Michigan for a helpful discussion. G.Z. and F.L.W. acknowledges the support from National Natural Science Foundation of China under grant No. 10821061 and NSAF under grant No. 10876040, respectively.

References

All Tables

All Figures

	Fig. 1 Comparison of weighted oscillator strengths gf of electric-dipole transitions (to the 5 lowest-lying levels) for S8+ – S11+. BL03 refers to the work of Bhatia & Landi (2003a,b), whereas LB03 and NSD07 correspond to the work of Landi & Bhatia (2003) and Nataraj et al. (2007), respectively. The dashed lines correspond to agreement within 20%. (Colour online.)
In the text

Fig. 2 Comparison of (effective) collision strengths (Υs, see the lower panel) Ωs from the ground state of S8+. BL03 refers to the distorted wave calculation by Bhatia & Landi (2003a), AS-DW refers to the present Breit-Pauli DW calculation using autostructure. Upper panel: scaled collision strength for a dipole transition 2s22p4   3P2  ←  2s22p33d   3P1 (1–80) with C = 2.0. The limit value is 4gifij/Eij at 1.0 for the dipole transition. Lower panel: the ratio of Υs between the results of the DW calculation by Bhatia & Landi (2003a) and the ICFT R-matrix calculation at log Te (K) = 5.1, 6.1 (corresponding to peak abundance of S8+ in ionization equilibrium) and 7.1. The dashed lines correspond to agreement within 20%. The transition marked by dotted box is the dipole transition 1–80 shown in the upper panel. (Colour online.)

In the text

Fig. 3 Comparison of effective collision strengths of S8+ with the R-matrix results of Butler and Zeippen (1994) for transitions of the ground configuration 2s22p4. Filled symbols with solid curves are results of Butler and Zeippen (1994), while open symbols with dotted curves corresponds to the present ICFT R-matrix calculation. Note: The same symbol in the two sets of results corresponds to the same transition. (Colour online)

In the text

	Fig. 4 Comparison of effective collision strengths of S9+ with the jajom R-matrix results of Bell & Ramsbottom (2000) for transitions of the n = 2 complex and 2s22p23s configuration. One point marked by bold “ ↘ ” refers to the 2s22p3   4S3/2  ←  2s2p4   2P3/2 transition (1–12), that will be examined in Fig. 5. (Colour online.)
In the text

Fig. 5 Comparison of excitation data for the 2s22p3   4S3/2  ←  2s2p4   2P3/2 transition (1–12) of S9+. Here, BR00 refers to the R-matrix calculation of Bell & Ramsbottom (2000), BL03 to the DW result of Bhatia & Landi (2003b) and ICFT to the present calculation. Upper panel: effective collision strengths – the BR00 without pseudo-resonances result was re-derived by us from the said original collision strengths provided by Ramsbottom (2011 priv. comm.). Lower panel: scaled collision strengths, with the scaling parameter C set to 2.0. (Colour online.)

In the text

Fig. 6 Comparison of effective collision strengths between different R-matrix calculations for transitions in the ground configuration 2s22p2 of S10+, where CKA92 refers to the interpolated data from R-matrix results for other ions in carbon-like sequence (Conlon et al. 1992), LB94 corresponds to the R-matrix work of Lennon & Burke (1994), ICFT denotes the present work. The transition is marked by i − j adjacent to the relevant set of curves. Note: the CKA92 data is extracted from the chianti v6 database. (Colour online.)

In the text

	Fig. 7 Collision strengths (Ω) of the 2s22p2   3P0  ←  2s2p3   5S2 (1–6) transitions of S10+, where LB94 corresponds to the R-matrix work of Lennon & Burke (1994) from the TIPTOPbase 6, ICFT denotes the present work. (Colour online.)
In the text

Fig. 8 Comparison of effective collision strengths of the present ICFT R-matrix results with other (DW) results. a) Scatter plot showing the ratio () for S10+ at three temperature of logTe(K) = 5.3, 6.3 and 7.3, where the DW calculation refers to the work of Landi & Bhatia (2003, LB03). The bold “ ↖ ” refers to a forbidden transition 2s22p2   3P0  ←  2s22p4p   3P0 transition (1–200), that is examined in panel b). b) Comparison of scaled collision strength Ω (with scaling parameter C = 2.0) of the 2s22p2   3P0  ←  2s22p4p   3P0 transition (1–200). AS-DW (9 and 24 models) corresponds to the present Breit-Pauli DW calculation by using autostructure with 9 and 24 configurations, corresponding to that used in the scattering and structure calculations of Landi & Bhatia (2003), respectively. (Colour online.)

In the text

Fig. 9 Comparison of effective collision strengths of S11+ for all excitations between levels of the n = 2 complex at three different temperatures log Te(K) = 6.04, 6.40 and 6.87. ΥICFT refers to the present ICFT R-matrix calculation, and KKR02 corresponds to the previous R-matrix results by Keenan et al. (2002). A few transitions with large difference are marked by labels around the points, and are linked together for results at different temperatures. (Colour online.)

In the text