© ESO, 2012

1. Introduction

Emission lines of boron-like ions have been recorded in both earlier spacecraft (Malinovsky & Heroux 1973; Huber et al. 1973; Acton et al. 1985) and more recent Hinode/EIS (Brown et al. 2008; Landi & Young 2009; Del Zanna 2012) solar observations. The diagnostic usefulness of B-like n = 2  →  n = 2 transition lines was first noted by Flower & Nussbaumer (1973) for several ions, viz. sodium, silicon, and sulphur. Later, Peng & Pradhan (1995) performed a systematic investigation for line intensity ratios of the boron-like iso-electronic sequence (including C II, N III, O IV, Ne VI, Mg VIII, Al IX, Si X, and S XII), and illustrated some line ratios that are sensitive to electron temperature or electron density. The authors used data from an R-matrix calculation by Zhang et al. (1994), which included just the n = 2 configurations. However, Keenan et al. (2000, 2002) noticed a slight error by Zhang et al. (1994) when including term coupling coefficients in their R-matrix calculation. Keenan et al. (2002) re-calculated the excitation data amongst the fine-structure levels of the n = 2 configurations for Si X and S XII. The diagnostic line ratios involving those levels were re-examined by them also and applied to solar observations. Liang & Zhao (2008) furthermore applied the electron density diagnostic line ratio of two 3d−2p transition lines to stellar observations. However, the available excitation data for n = 2  →  n = 3 transitions in the boron-like iso-electronic sequence are mainly from distorted-wave (DW) calculations, which omit the contribution from resonances. In addition to the resonances attached to them, more highly excited states also contribute to lower-lying line ratios via cascades – see for example Del Zanna et al. (2012) and Foster et al. (2012), where R-matrix data were additionally supplemented by DW calculations.

For the astrophysically important Fe²¹⁺ ion, Badnell et al. (2001) performed a 204-level intermediate-coupling frame transformation (ICFT) R-matrix calculation, which included a much larger configuration interaction (CI) expansion. Recently, Liang et al. (2009a, 2011) extended R-matrix calculations up to n = 3 and 4 for Si⁹⁺ and S¹¹⁺ ions, respectively. Ludlow et al. (2010) performed Breit-Pauli R-matrix calculations for all ions of the argon isonuclear sequence. The boron-like case used a close-coupling expansion comprising the 2s^x2p^y (x + y = 3), 2s²{3,4,5}l configurations. For other ions along this iso-electronic sequence, the distorted-wave calculations performed by Zhang & Sampson (1994a,b) are still the main source of data for various modelling databases, e.g. Chianti v7 (Landi et al. 2012), AtomDB v2¹.

Here, we report on calculations for the electron-impact excitation of the boron-like iso-electronic sequence from C⁺ to Kr³¹⁺ ions that were made using the ICFT R-matrix method. This paper is part of our series of works on iso-electronic sequences: Li-like, Liang et al. (2011); F-like, Witthoeft et al. (2007); Ne-like, Liang & Badnell (2010); and Na-like, Liang et al. (2009a,b). This work is part of the UK Atomic Processes for Astrophysical Plasmas (APAP) network².

The remainder of this paper is organized as follows. In Sect. 2, 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. 3. The excitation results themselves are discussed in Sect. 4.

2. Sequence calculation

The aim of this work is to perform R-matrix calculations employing the ICFT method (see Griffin et al. 1998) for all boron-like ions from C⁺ to Kr³¹⁺. The close-coupling (CC) expansion we used consists of the 2s^x2p^y (x + y = 3), 2s²{3,4}l, 2s2p{3,4}l, and 2p²3l (92 LS terms, 204 fine-structure levels) configurations. The additional configuration interaction (CI) from the 2p²4l, 2s3l3l′ and 2p3s3l (71 LS terms, 160 fine-structure levels) configurations was included for the target structure used in the collision calculation.

2.1. Structure: level energies

The target wavefunctions (1s−4f) were obtained from autostructure (AS, Badnell 1986) using the Thomas-Femi-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. The radial scaling parameters, λ_nl (n = 1−4; l ∈ s,p,d, and f, were obtained separately for each ion by a three-step optimization procedure. In the first step, the weighted sum of all term energies of the 1s²2s^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 1s²2s²3l and 1s²2s²4l configurations were minimized by varying the λ_3l and λ_4l scaling parameters, respectively. The resultant scaling parameters are listed in Table 1. For lower charged ions, λ_4f is far away from unity. Tests show that it is insensitive to optimization and the atomic structure itself is insensitive to it.

A comparison of level energies is made with the experimentally derived data available from the compilation of NIST v4³, and with other theoretical results, for four specific ions (Ne⁵⁺, Ar¹³⁺, Fe²¹⁺, and Kr³¹⁺) that span the sequence, so as to assess the accuracy of our present structure over the entire iso-electronic sequence – see Tables 2−5. For many excited levels of the n = 3 and 4 complexes, the present AS calculation agrees to better than within 1% with the NIST v4³ recommended values for the four ions. For levels of n = 2 configurations, the energy difference is less than 4%. For highly charged ions, this discrepancy becomes smaller. Therefore, we performed a calculation with level energy corrections to the diagonal of the Hamiltonian matrix before diagonalization for the 15 fine-structure levels of the n = 2 configurations of almost all ions over the iso-electronic sequence (except for Z = 29−35) and iterated to convergence. For those levels missing in the NIST compilation, we adopted the mean value of differences between our level energies and corresponding NIST values of the same configuration. The resulting e-vectors and e-energies were used to calculate the oscillator strengths and archived energies.

Fig. 1 Comparison of line strengths (S) of electric-dipole transitions for ions spanning the sequence. For Ne5+, MGB01 corresponds to the MCHF calculation of Mitnik et al. (2001) for transitions among the lowest 20 levels, while “up-triangle” symbols correspond to the data from the MCHF/MCDF collection6. For Ar13+, comparisons are made with the grasp calculation by Aggarwal et al. (2005) and with the autostructure calculation by Ludlow et al. (2010, hereafter LBL10) for all transitions amongst levels of the n = 2 configurations. The “ × ” symbols denote transitions among the lowest 20 levels. For Fe21+, a comparison is made with the MCDF calculation (Jonauskas et al. 2006) and that of Badnell et al. (2001, hereafter BGM01) for all transitions amongst levels of the n = 2 configurations. The “ × ” symbols are the same as for Ar13+. For Kr31+, we compare with the grasp calculation by Aggarwal et al. (2008). The horizontal dashed lines correspond to an agreement within 20%.

For Ne⁵⁺, Mitnik et al. (2001) performed a multi-configuration Hartree-Fock (MCHF) calculation with three pseudo-orbitals ($\overline{\mathrm{5}\mathrm{s}}$, $\overline{\mathrm{5}\mathrm{p}}$, and $\overline{\mathrm{5}\mathrm{d}}$) included, which shows a better agreement with NIST v4 data for the levels of n = 2 configurations, see Table 2. For higher excited levels of n = 3 and 4 configurations, the two different sets of predictions agree better, to within 1%.

For Ar¹³⁺, a grasp calculation is available, which included the 2s^x2p^y (x + y = 3), 2s²{3,4,5}l, 2s2p{3,4,5}l, 2p²{3,4,5}l (l < f), and 2s²{6,7,8}s configurations (Aggarwal et al. 2005). The present AS calculation agrees well with the grasp calculation, to within 1%, and even better than 0.5% for most levels, see Table 3. In the work of Ludlow et al. (2010)⁴, the orbital scaling parameters were adjusted by hand to bring the calculated n = 2 level energies into close agreement with the NIST v4 experimental values. But our ab initio results agree much better with the NIST values for the n = 3 and 4 levels. For compactness, we do not present their results in Table 3. We do note that the Ludlow et al. (2010) R-matrix calculations covered the whole argon isonuclear sequence and so they were constrained to use a simplified structure. With an isoelectronic sequence, once we have determined the best approach to the structure for one ion, the same one works with little change for every other one of the sequence (excluding near neutral).

For Fe²¹⁺, Jonauskas et al. (2006) performed an unprecedented large-scale calculation using the multi-configuration Dirac-Fock (MCDF) approach with the promotion of one, two, and three electrons from n = 2 to all possible combinations of one, two, or three electrons in the shells up to n = 3. The present AS level energies agree within 0.5% for all levels of n = 3 and 4 configurations. For low-lying levels of n = 2 complex, the difference is still within 2%, see Table 4. Badnell et al. (2001) performed a similar calculation to the present one, but without inclusion of 2s3l3l′ and 2p3s3l configurations, again using the autostructure code⁵. Their resulting level energies show an excellent agreement with the present ones. Due to the similarity of the two approaches, their results are not listed in Table 4.

For the highly charged case of Kr³¹⁺, the most recent theoretical work is the fully relativistic grasp calculation performed by Aggarwal et al. (2008). Table 5 shows that the present AS calculation agrees to better than 1% with the grasp calculation.

2.2. Structure: line strength S

Another test of our structure calculation is to compare line strengths (S_ij for a given i ← j transition). In terms of the transition energy E_ji (Ryd) for the j → i transition, the absorption oscillator strength, f_ij, can be written as $\begin{array}{ccc}{\mathit{f}}_{\mathit{ij}}& \mathrm{=}\frac{{\mathit{E}}_{\mathit{ji}}}{\mathrm{3}{\mathit{g}}_{\mathit{i}}}\mathit{S,}& \end{array}$(1)and the transition probability or Einstein’s A-coefficient, A_ji, as $\begin{array}{ccc}{\mathit{A}}_{\mathit{ji}}\mathrm{\left(}\mathrm{au}\mathrm{\right)}& \mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\mathit{\alpha }}^{\mathrm{3}}\frac{{\mathit{g}}_{\mathit{i}}}{{\mathit{g}}_{\mathit{j}}}{\mathit{E}}_{\mathit{ji}}^{\mathrm{2}}{\mathit{f}}_{\mathit{ij}}\mathit{,}& \end{array}$(2)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 illustrates a graphical comparison of line strengths with previous calculations for electric-dipole transitions for the four representative ions. For the lower charged Ne⁵⁺ ion, 76% of all possible transitions among the lowest 20 levels agree with the MCHF calculation (Mitnik et al. 2001) to within 20%. Mitnik et al. (2001) demonstrated that the average difference is 24% between their results and the data from MCHF/MCDF collection database⁶. We illustrate this comparison again by “up-triangle” symbols in the top-left panel in Fig. 1, which reveals a comparable level of agreement with that in Table 2 of Mitnik et al. (2001).

For Ar¹³⁺, the 460-level grasp calculation performed by Aggarwal et al. (2005) is the largest-scale work for this ion to our best knowledge. We provide a comparison with their results, again in Fig. 1. There are 63% of all electric-dipole transitions to levels of n = 2 configurations which agree to within 20% between the two different calculations. For transitions among the lowest 20 levels, an even better (84%) agreement is found. The data from Ludlow et al. (2010) show a worse agreement with ours. There are only 32% of all available transitions to n = 2 levels to agree within 20%. Even for transitions within n = 2 levels, the percentage is 53%. Their lack of an optimized structure is likely the cause. Additionally, their radiative decay rate (A_ij) values in adf04 file were scaled according the theoretical and NIST level energies. That might also contribute to some of the low percentage in the differences of dipole line strengths.

For Fe²¹⁺, the work of Jonauskas et al. (2006) is likley the largest-scale calculation. The present AS calculation agrees with their results to within 20% for 79% of all electric-dipole transitions to levels of n = 2 complex. In comparison with the results of Badnell et al. (2001), 85% of all electric-dipole transitions to n = 2 levels agree within 20%. This difference is attributed to the CI effect from the 2s3l3l′ and 2p3s3l configurations included in our AS calculation.

For the highly charged case of Kr³¹⁺, a comparison is made with the grasp calculation of Aggarwal et al. (2008). There are 64% of all transitions to levels of n = 2 that agree to within 20%. For transitions within n = 2 levels, the percentage is 92%.

The scatter plots show that the deviations between the present AS calculation and various previous results are much stronger and more widespread for the weaker transitions than the strong ones, as expected.

Overall, the atomic structure of the ions spanning the sequence is reliable, and the uncertainty in collision strengths (Ωs) due to inaccuracies in the target structure is correspondingly small.

3. Scattering

The present parallel ICFT R-matrix calculations employed 40 continuum basis orbitals per angular momentum to represent the (N + 1)th-electron over the sequence. All partial waves from J = 0 to 41 were included explicitly and the contribution from higher J-values was included using a “top-up” procedure (Burgess 1974; Badnell & Griffin 2001). The contribution from partial waves up to J = 12 included electron exchange while those from J = 13 to 41 were non-exchange, calculated using the exchange R-matrix code in its non-exchange mode. For the exchange calculation, a fine energy mesh was used to resolve the majority of narrow resonances below the highest excitation threshold, which has been tested to be sufficient for the convergence of the effective collision strength, see Fig. 2. From just above the highest threshold excitation to a maximum energy of ten times the ionization potential for each ion, a coarse energy mesh (1.0 × 10^-3   z² Ryd, where z = Z − 5 is the residual charge of ion) was employed. For the non-exchange calculation, a step of 1.0 × 10^-3   z² 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 highly charged sulphur ions (Liang et al. 2011). The lowest-lying 8 LS terms (15 IC levels) of the n = 2 complex were corrected for almost all ions over the iso-electronic sequence, as explained in detail in the structure section.

Fig. 2 Fine energy mesh employed in the outer region (exchange) R-matrix calculation for each ion.

As mentioned in the introduction, we wish to determine excitation data for the 204 CC levels of the 2s^x2p^y (x + y = 3), 2s²{3,4}l, 2s2p{3,4}l, and 2p²3l configurations. But for some ions, these do not correspond to the lowest energy 204 levels (92 LS terms). When this occurs, we include additional terms in the close-coupling expansion so that all 2p²3l levels are still in the close-coupling expansion and there are no “gaps” below them. For example, for N²⁺ and O³⁺, the 2p²4s   ⁴P/²P, 2p²4p ${\mathrm{2}}^{}\mathrm{S}{\mathit{/}}^{\mathrm{4}}\mathrm{P}{\mathit{/}}^{\mathrm{4}}\mathrm{D}$ terms were included.

To take the contribution from higher electron energies to high temperature effective collision strengths into account, the infinite energy Born limits (non-dipole allowed) and line strengths (dipole allowed) from autostructure were used to obtain collision strengths (Ω) at higher energies according to the procedure defined by Burgess & Tully (1992). The effective collision strengths at 13 electron temperatures ranging from 2 × 10²   (z + 1)² K to 2 × 10⁶(z + 1)² K are produced as the end product with ADAS adf04 format (Summers 2004).

4. Results and discussions

4.1. Comparison with previous calculations

As in our other iso-electronic sequence work, we selected several transitions to test the original collision strength for four ions (Ne⁵⁺, Ar¹³⁺, Fe²¹⁺ and Kr³¹⁺) spanning over the sequence. An extensive comparison for effective collision strength will be given by a scatter plot to test how far the reliability reaches.

• Ne⁵⁺ Mitnik et al. (2001) performed a 180-level ICFT R-matrix calculation for this ion with the same CC expansion as we did except for the 2s2p4f configuration. As mentioned above, the MCHF approach was used to describe the target structure and three pseudo-orbitals were included to partially correct spectroscopic orbitals in this previous calculation. Our 204-level collision strength shows an excellent agreement with this previous one, see left panel in Fig 3. An extensive comparison of effective collision strength is given in the right panel of Fig. 3 for all excitations from levels of n = 2 configurations at a temperature of peak fraction in ionization equilibrium. About 73% of the excitations agree within 20%.

  Fig. 3 (Effective) collision strengths for Ne5+. Left: excitation from the 2s22p   2P1/2 ground level to the 2s22p   2P3/2 level (1–2). Right: excitations amongst all 15 levels of the n = 2 configurations at the temperature (Te = 4.0 × 105 K) of peak fractional abundance in ionization equilibrium. The (red) “ × ” marks the 1–2 excitation shown in the lefthand panel. Double-horizontal lines correspond to agreement within 20%. Notes: the collision strength of Mitnik et al. (2001) is a scanned picture.

  Fig. 4 Comparison of effective collision strengths for Ar13+. Left: all excitations among the 15 levels of the n = 2 configurations at the temperature (Te = 4.0 × 106 K) of peak fractional abundance in ionization equilibrium. The “ × ” symbol corresponds to the 2s22p   2P3/2 − 2p3   2P3/2 (1–15) excitation shown in the righthand panel, which is linked for the two different previous calculations. Double-horizontal lines correspond to agreement within 20%. Right: the effective collision strength of the 1–15 transition as a function of temperature (K). ZGP94 and LBL10 correspond to the Breit-Pauli R-matrix works of Zhang et al. (1994) and Ludlow et al. (2010), respectively.

• Ar¹³⁺ The small-scale (n = 2) R-matrix calculation performed by Zhang et al. (1994) is extensively adopted by various databases, e.g. Open-ADAS⁷ and Chianti v7 (Landi et al. 2012). Comparison with these data for all excitations among the 15 lowest-lying levels reveals that the previous R-matrix data are systematically lower than ours, and the deviations are stronger for weaker excitations, see Fig. 4. However, the data from the work of Ludlow et al. (2010) agree well with ours even though only 36 IC levels of (1s²)2s^x2p^y (x + y = 3), 2s²{3,4,5}l configurations were included in their close-coupling expansion. Ninety percent of all transitions within n = 2 levels agree to within 20%. For the 2s²2p   ²P_3/2 − 2p^3   2P_3/2 (1–15) excitation, there is an obvious bump in the present ICFT R-matrix calculation at temperatures of 10⁶−10⁷ K, with the difference being up to  ~3.5 when compared with Zhang et al.’s data. This is an obvious enhancement due to resonances attached to n = 3 that were not included in the previous small-scale calculation. The good agreement with Ludlow et al.’s data also supports this.

• Fe²¹⁺ After carrying out a level mapping procedure according to LSJ^π and configurations, an extensive comparison was made with a previous (204-level) ICFT R-matrix calculation (Badnell et al. 2001) at three temperatures of 10⁵,   10⁶, and 10⁷ K. Almost all excitations (93%) agree to within 20% for the two different calculations with a different target structure, see Fig. 5. The differences between them and the widespread agreement do not change much with increasing temperature. This results from the consistent resonance structure in the two calculations. Badnell et al. (2001) made a detailed assessment for their calculation with previous available data, therefore we will not repeat the comparison here.

  Fig. 5 Comparison of effective collision strength with the results of Badnell et al. (2001) for Fe21+ for all excitations from the lowest 15 levels at temperatures of Te = 1.0 × 106,107 and 108 K. Double-horizontal lines correspond to agreement within 20%.

  Fig. 6 (Effective) collision strengths for Kr21+. Left: present collision strengths for 2s22p   2P1/2–2s2p2   4P3/2 (1–4), 2s22p   2P1/2–2s2p2   2P3/2 (1–10), and 2s22p   2P1/2–2p3   2P3/2 (1–15) transitions along with the distorted-wave calculation by Zhang & Sampson (1994a, ZS94). Right: ratio of effective collision strength between the present ICFT R-matrix calculation and DW results available from the Open-ADAS database7 at a temperature of 1.0 × 108 K. Double-horizontal lines correspond to agreement within 20%.

• Kr³¹⁺ To our best knowledge, there are no R-matrix excitation data available for this ion. Therefore, we compared the background of the present ICFT R-matrix calculation with that from a distorted-wave calculation performed by Zhang & Sampson (1994a,b) for excitations from the ground level 2s²2p   ²P_1/2. Figure 6 illustrates some excitations from this comparison. The background of the collision strengths agrees well with the DW calculation. As expected, the present effective collision strengths Υ are systematically higher than those from the distorted-wave approach due to the inclusion of resonances.

From the above comparison for the four specified ions (Ne⁵⁺, Ar¹³⁺, Fe²¹⁺ and Kr³¹⁺) spanning the iso-electronic sequence, we believe the present ICFT R-matrix results (Ω and Υ) to be reliable. For ions near neutral (below O³⁺), R-matrix with pseudostates calculations are needed to consider ionization loss in the excitation, but ours are the best data available to date.

Fig. 7 Level ordering with the original level index (ID) relative to the ordering of Fe21+ by mapping according to the “good” quantum numbers – configuration, total angular momentum J, and energy ordering for ions spanning the entire sequence. The spikes and dips are due to the shift of a given level, for example, 2p23s   4P (62–65) levels in Fe21+ move to levels above 120 in N2+.

Fig. 8 Effective collision strength (Υ) for excitations from the ground level to all 22 lowest-lying excited levels at temperatures of Te = 5 × 102(q + 1)2 and 1 × 104(q + 1)2 K (here q = Z − 5) along the iso-electronic sequence. Notes: the index number refers to the ID number in the reference ion – Fe.

4.2. Trends of iso-electronic sequence

As in our previous sequence works (Witthoeft et al. 2007; Liang et al. 2009c, 2010, 2011), we take configuration, total angular momentum J, and energy ordering as a good quantum number for level matching in the comparison between different calculations and the investigation of Υ along the iso-electronic sequence, see Fig. 7. This satisfactorily eliminates uncertainty originating from the non-continuity of level-ordering along the sequence. We used Fe as the arbitrary reference ion for the level-ordering.

In Fig. 8, we show the effective collision strength Υ at temperatures of T_e/(q + 1)² = 5 × 10² and 10⁴ K along the iso-electronic sequence for excitations from the ground level to the lowest-lying (in Fe²¹⁺) 22 excited levels. At the low temperature of 5 × 10²(q + 1)² K, spikes and/or dips are observed at low charges for some transitions, e.g. 2s²2p   ²P_1/2  →  2s²2p   ²P_3/2 (1−2). With increasing threshold energy, that is to higher excited levels, this irregularity becomes weaker and eventually disappears. At the high temperature of 1 × 10⁴(z + 1)² K, the spikes and/or dips disappear, as expected, because the resonance contribution becomes weaker and eventually negligible.

5. Summary

We have performed 204-level ICFT R-matrix calculations for the electron-impact excitation of all ions of the boron-like iso-electronic sequence from C⁺ to Kr³¹⁺.

Good agreement with the available experimentally derived data and the results of others for level energies and line strengths S for several specific ions (Ne⁵⁺, Ar¹³⁺, Fe²¹⁺, and Kr³¹⁺) spanning the iso-electronic sequence, supports the reliability of our R-matrix excitation data. This was confirmed specifically by detailed comparisons of Ω and/or Υ with previous R-matrix calculations, where available, for the four specific reference ions.

Our R-matrix excitation data are expected to be an important improvement on the current data (from relativistic distorted-wave approach), which are extensively used by the spectroscopic diagnostic modelling communities in astrophysics and magnetic fusion.

By excluding the level-crossing effects on the Υ, we examined the iso-electronic trends of the effective collision strengths. As expected, a complicated pattern of spikes and dips of Υ at low temperatures was noted again along the sequence as shown in our other series works.

The data are made available in the ADAS adf04 format (Summers 2004) at the archives of the APAP², OPEN-ADAS⁷ and will be included in the CHIANTI⁸ database. At the APAP-network website², the original collision strength also can be made available.

In conclusion, we have generated an extensive set of reliable excitation data with the ICFT R-matrix method for spectroscopy/diagnostic research within the astrophysical and fusion communities. This will replace the data from DW and small-scale R-matrix calculations presently used by these communities, and it is expected to identify new lines and may overcome some shortcomings in present astrophysical modelling.

Acknowledgments

We thank Connor Ballance at Auburn University for helpful comments. The work of the UK APAP Network is funded by the UK STFC under grant No. ST/J000892/1 with the University of Strathclyde. GYL acknowledges the support from the One-Hundred-Talents programme of the Chinese Academy of Sciences (CAS). G.Z. acknowledges the support from National Natural Science Foundation of China under grant No. 10821061.

References

All Tables

All Figures

Fig. 1 Comparison of line strengths (S) of electric-dipole transitions for ions spanning the sequence. For Ne5+, MGB01 corresponds to the MCHF calculation of Mitnik et al. (2001) for transitions among the lowest 20 levels, while “up-triangle” symbols correspond to the data from the MCHF/MCDF collection6. For Ar13+, comparisons are made with the grasp calculation by Aggarwal et al. (2005) and with the autostructure calculation by Ludlow et al. (2010, hereafter LBL10) for all transitions amongst levels of the n = 2 configurations. The “ × ” symbols denote transitions among the lowest 20 levels. For Fe21+, a comparison is made with the MCDF calculation (Jonauskas et al. 2006) and that of Badnell et al. (2001, hereafter BGM01) for all transitions amongst levels of the n = 2 configurations. The “ × ” symbols are the same as for Ar13+. For Kr31+, we compare with the grasp calculation by Aggarwal et al. (2008). The horizontal dashed lines correspond to an agreement within 20%.

In the text

	Fig. 2 Fine energy mesh employed in the outer region (exchange) R-matrix calculation for each ion.
In the text

Fig. 3 (Effective) collision strengths for Ne5+. Left: excitation from the 2s22p   2P1/2 ground level to the 2s22p   2P3/2 level (1–2). Right: excitations amongst all 15 levels of the n = 2 configurations at the temperature (Te = 4.0 × 105 K) of peak fractional abundance in ionization equilibrium. The (red) “ × ” marks the 1–2 excitation shown in the lefthand panel. Double-horizontal lines correspond to agreement within 20%. Notes: the collision strength of Mitnik et al. (2001) is a scanned picture.

In the text

Fig. 4 Comparison of effective collision strengths for Ar13+. Left: all excitations among the 15 levels of the n = 2 configurations at the temperature (Te = 4.0 × 106 K) of peak fractional abundance in ionization equilibrium. The “ × ” symbol corresponds to the 2s22p   2P3/2 − 2p3   2P3/2 (1–15) excitation shown in the righthand panel, which is linked for the two different previous calculations. Double-horizontal lines correspond to agreement within 20%. Right: the effective collision strength of the 1–15 transition as a function of temperature (K). ZGP94 and LBL10 correspond to the Breit-Pauli R-matrix works of Zhang et al. (1994) and Ludlow et al. (2010), respectively.

In the text

	Fig. 5 Comparison of effective collision strength with the results of Badnell et al. (2001) for Fe21+ for all excitations from the lowest 15 levels at temperatures of Te = 1.0 × 106,107 and 108 K. Double-horizontal lines correspond to agreement within 20%.
In the text

Fig. 6 (Effective) collision strengths for Kr21+. Left: present collision strengths for 2s22p   2P1/2–2s2p2   4P3/2 (1–4), 2s22p   2P1/2–2s2p2   2P3/2 (1–10), and 2s22p   2P1/2–2p3   2P3/2 (1–15) transitions along with the distorted-wave calculation by Zhang & Sampson (1994a, ZS94). Right: ratio of effective collision strength between the present ICFT R-matrix calculation and DW results available from the Open-ADAS database7 at a temperature of 1.0 × 108 K. Double-horizontal lines correspond to agreement within 20%.

In the text

	Fig. 7 Level ordering with the original level index (ID) relative to the ordering of Fe21+ by mapping according to the “good” quantum numbers – configuration, total angular momentum J, and energy ordering for ions spanning the entire sequence. The spikes and dips are due to the shift of a given level, for example, 2p23s   4P (62–65) levels in Fe21+ move to levels above 120 in N2+.
In the text

	Fig. 8 Effective collision strength (Υ) for excitations from the ground level to all 22 lowest-lying excited levels at temperatures of Te = 5 × 102(q + 1)2 and 1 × 104(q + 1)2 K (here q = Z − 5) along the iso-electronic sequence. Notes: the index number refers to the ID number in the reference ion – Fe.
In the text