Issue 
A&A
Volume 634, February 2020



Article Number  A7  
Number of page(s)  13  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201936931  
Published online  28 January 2020 
Rmatrix electronimpact excitation data for the Clike isoelectronic sequence
^{1}
Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
email: junjie.mao@strath.ac.uk
^{2}
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Received:
16
October
2019
Accepted:
14
December
2019
Context. Emission and absorption features from Clike ions serve as temperature and density diagnostics of astrophysical plasmas. Rmatrix electronimpact excitation data sets for Clike ions in the literature merely cover a few ions, and often only for the ground configuration.
Aims. Our goal is to obtain levelresolved effective collision strength over a wide temperature range for Clike ions from N II to Kr XXXI (i.e., N^{+} to Kr^{30+}) with a systematic set of Rmatrix calculations. We also aim to assess their accuracy.
Methods. For each ion, we included a total of 590 finestructure levels in both the configuration interaction target and closecoupling collision expansion. These levels arise from 24 configurations 2l^{3}nl′ with n = 2−4, l = 0−1, and l′ = 0−3 plus the three configurations 2s^{2}2p5l with l = 0−2. The AUTOSTRUCTURE code was used to calculate the target structure. Additionally, the Rmatrix intermediate coupling frame transformation method was used to calculate the collision strengths.
Results. We compare the present results of selected ions with archival databases and results in the literature. The comparison covers energy levels, transition rates, and effective collision strengths. We illustrate the impact of using the present results on an Ar XIII density diagnostic for the solar corona. The electronimpact excitation data is archived according to the Atomic Data and Analysis Structure (ADAS) data class adf04 and will be available in OPENADAS. The data will be incorporated into spectral codes, such as CHIANTI and SPEX, for plasma diagnostics.
Key words: atomic data / techniques: spectroscopic / Sun: corona
© ESO 2020
1. Introduction
Emission and absorption features from Clike ions serve as temperature and density diagnostics for various types of astrophysical plasmas such as (Mason et al. 1984; Mao et al. 2017; Del Zanna & Mason 2018). Plasma models built on extensive and accurate atomic databases are essential to determine plasma parameters that span several orders of magnitudes in the parameter space. For instance, the density of photoionized outflows in the vinicity of black holes can vary from ∼10^{3 − 5} cm^{−3} (C III, Gabel et al. 2005; Arav et al. 2015) to ≳10^{6 − 14} cm^{−3} (Si IX and Fe XXI, Miller et al. 2008; King et al. 2012; Mao et al. 2017). Currently, the status of levelresolved electronimpact excitation data of Clike ions is rather poor. Such data are either lacking or obtained from distorted wave calculations in plasma codes, which are widely used in the community (Mao et al. 2019).
More accurate Rmatrix electronimpact excitation data for Clike ions are available in the literature, but only for a few ions and oftentimes only for the ground configuration. This is mainly because Rmatrix calculations are rather computationally expensive. Therefore, Rmatrix electronimpact excitation data for Clike ions are needed.
Griffin et al. (1998) introduced the Rmatrix intermediatecoupling frame transformation (ICFT) method, which employs multichannel quantum defect theory. The ICFT method first calculates the electronimpact excitation in pure LScoupling, which subsequently, transforms into a relativistic coupling scheme via the algebraic transformation of the unphysical scattering or reactance matrices. Consequently, the ICFT method is inherently significantly faster than the classic BreitPauli Rmatrix (BPRM) method (Berrington et al. 1995), Bspline Rmatrix (BSR) code (Zatsarinny 2006), and Dirac atomic Rmatrix code (DARC^{1}). We refer readers to FernándezMenchero et al. (2016), Aggarwal (2017), and Del Zanna et al. (2019) for recent comparisons among different Rmatrix methods and the impact on plasma diagnostics.
In the past few decades, the Rmatrix ICFT method has been used to perform largescale calculations of electronimpact excitation data for a number of isoelectronic sequences: Liang & Badnell (2011, Lilike), FernándezMenchero et al. (2014a, Belike), Liang et al. (2012, Blike), Witthoeft et al. (2007, Flike), Liang & Badnell (2010, Nelike), Liang et al. (2009, Nalike), and FernándezMenchero et al. (2014b, Mglike). A review is presented by Badnell et al. (2016). We note that there are also other largescale Rmatrix calculations that cover individual ions in the Clike sequence, for instance, Ludlow et al. (2010) and Liang et al. (2011).
Here we present a systematic set of Rmatrix ICFT calculations of Clike ions from N II to Kr XXXI (i.e., N^{+} to Kr^{30+}) to obtain levelresolved effective collision strengths over a wide temperature range. Section 2 describes the atomic structure (Sect. 2.1) and collision calculations (Sect. 2.2). The results are summarized in Sect. 3. In Sect. 4, we present comparisons between the present results for selected ions and some previous Rmatrix calculations. This is followed by our summary in Sect. 5.
A supplementary package can be found at Zenodo (Mao 2019). This package includes the inputs of the AUTOSTRUCTURE and Rmatrix ICFT calculations, atomic data from the present work, the archival database and literature, as well as scripts used to create the figures presented in this paper.
2. Method
Following the previous case study of Clike Fe XXI (FernándezMenchero et al. 2016), for each ion, we include a total of 590 finestructure levels (282 terms) in the configurationinteraction target expansion and closecoupling collision expansion. These levels (terms) arise from 27 configurations 2l^{3}nl′ with n = 2 − 4, l = 0 − 1, and l′ = 0 − 3 plus the 3 configurations 2s^{22}p5l with l = 0 − 2 (Table 1).
Configurations used for the collision calculations.
2.1. Structure
We use the AUTOSTRUCTURE code (Badnell 2011) to calculate the target structure. The wave functions are calculated by diagonalizing the BreitPauli Hamiltonian (Eissner et al. 1974). The onebody relativistic terms, massvelocity, spinorbit, and Darwin terms are included perturbatively. The ThomasFermiDiracAmaldi model is used for the electronic potential. We adjust the nldependent scaling parameters (Nussbaumer & Storey 1978) in the following procedure without manual readjustment to avoid introducing arbitrary changes across the isoelectronic sequence. For each ion, we first optimize the scaling parameters of 1s, 2s, and 2p to minimize the equallyweighted sum of all LS term energies with n = 2 (i.e., Conf. 1–3 in Table 1. Since then, we fix the obtained scaling parameters of 1s, 2s, and 2p. Subsequently, we optimize the scaling parameters of 3s, 3p and 3d to minimize the equallyweighted sum of all LS term energies with n = 3 (Conf. 4–12). We repeat this progressive procedure for n = 4 (Conf. 13–24) and n = 5 (15 configurations in total, including Conf. 25–27 in Table 1). A similar optimization procedure was also used in Liang et al. (2011) for instance. The scaling parameters of the 13 atomic orbitals (1s–5d) listed in Table 2 are used for the structure (282 terms and 590 levels arising from 24 configurations) and the following collision calculation for all the ions (Z = 7 − 36) in the sequence.
ThomasFermiDiracAmaldi potential scaling parameters used in the AUTOSTRUCTURE calculations for the Clike isoelectronic sequence. Z is the atomic number, such as 8 for oxygen.
Since the innerregion Rmatrix codes require a unique set of nonrelativistic orthogonal orbitals (Berrington et al. 1995), we cannot exploit the full power of the general atomic structure codes. As shown later in Sect. 4, the atomic structures obtained in the present work show relatively large differences with respect to experiment values, especially for the first few ions in the isoelectronic sequence, which require Rmatrix calculations with pseudostates. In general, this inaccuracy does not significantly affect plasma diagnostics using spectroscopically and astrophysically important lines (Del Zanna et al. 2019).
2.2. Collision
The Rmatrix collision calculation consists of the inner and outerregion calculations (Burke 2011). The innerregion calculation is further split into the exchange and nonexchange calculations. Following the previous case study of Clike Fe XXI (FernándezMenchero et al. 2016), we include angular momenta up to 2J = 23 and 2J = 77 for the innerregion exchange and nonexchange calculation, respectively, for the entire isoelectronic sequence. For higher angular momenta up to infinity, we use the topup formula of the Burgess sum rule (Burgess 1974) for dipole allowed transitions, and a geometric series for the remaining nonforbidden (i.e., nondipole allowed) transitions (Badnell & Griffin 2001).
The outerregion calculation is split into a fine energy mesh exchange calculation, a coarse energy mesh exchange calculation, and a (coarse energy mesh) nonexchange calculation. A fine energy mesh is used between the first and last thresholds for the outerregion exchange calculation to sample the resonances. With an increasing ionic charge, the number of sampling points in the fine energy mesh increases from ∼3600 for N II to ∼30 000 for Kr XXXI to strike the balance between the computational cost and resonance sampling. Along the isoelectronic sequence, in the resonance region, the characteristic scattering energy increases by a factor of z^{2}, with z the ionic charge (such as z = 3 for O III and z = 20 for Fe XXI). However, the autoionization width of the resonance remains constant. That is to say, to resolve the resonance region to the same degree, the step size of the energy mesh needs to be reduced by a factor of z^{2} with increasing z. To avoid unreasonable computation cost of highz ions, following Witthoeft et al. (2007), we reduce the step size of the energy mesh by a factor of z (see also Appendix B).
A coarse energy mesh, with ∼1000 points for all the ions along the isoelectronic sequence is used from the last threshold up to ∼3I_{p}, where I_{p} is the ionization potential in units of Rydberg This allows us to determine a smooth background of the outerregion exchange calculation.
Another coarse energy mesh with ∼1400 points for all the ions along the isoelectronic sequence is used from the first threshold up to ∼3I_{p} for the outerregion nonexchange calculation. Since this coarse energy mesh covers the resonance region, it is possible that unresolved resonance(s) appear in the ordinary collision strength of the outerregion nonexchange calculation. Therefore, postprocessing to remove the unresolved resonance(s) is necessary.
The effective collision strength (Υ_{ij}) for electronimpact excitation is obtained by convolving the ordinary collision strength (Ω_{ij}) with the Maxwellian velocity distribution:
where E is the kinetic energy of the scattered free electron, k the Boltzmann constant, and T the electron temperature of the plasma.
To obtain effective collision strengths at high temperatures, ordinary collision strengths at high collision energies are required, which is inefficient to be calculated with the Rmatrix method. Hence, we use AUTOSTRUCTURE to calculate the infiniteenergy Born and radiative dipole limits. Between the last calculated energy point and the two limits, we interpolate taking into account the type of transition in the BurgessTully scaled domain (i.e., the quadrature of reduced collision strength over reduced energy, Burgess & Tully 1992) to complete the Maxwellian convolution (Eq. (1)).
3. Results
We obtain Rmatrix electronimpact excitation data for the Clike isoelectronic sequence from N II to Kr XXXI (i.e., N^{+} and Kr^{30+}). Our effective collision strengths cover a wide range of temperature (z + 1)^{2}(2 × 10^{1}, 2 × 10^{6}) K. They are to be applied to astrophysical plasmas in various conditions.
The ordinary collision strengths will be archived in OPENADAS^{2}. The effective collision strengths are archived according to the Atomic Data and Analysis Structure (ADAS) data class adf04 and will be available in OPENADAS and our UKAPAP website^{3}. These data will be incorporated into plasma codes like CHIANTI (Dere et al. 1997, 2019) and SPEX (Kaastra et al. 1996, 2018) for plasma diagnostics.
4. Discussion
We selected four ions O III, Ne V, Si XI, and Fe XXI across the isoelectronic sequence to illustrate the quality of our structure and collision calculation. These ions were selected because detailed results from archival databases (NIST^{4}, MCHF^{5}, and OPENADAS) and the literature are available for comparison purposes.
For each ion (Sects. 4.1–4.4), we first compare the energy levels. Figure 1 illustrates the deviation (in percent) of the energy levels in archival databases and previous works with respect to the present work. A histogram plot of the data shown in Fig. 1 is also shown in Appendix A. Generally speaking, the energy levels of the present work agree well (≲5%) with archival databases and previous works for highcharge ions. A larger deviation (≲15%) is found for lowcharge ions, in particular, for some of the lowest lying energy levels.
Fig. 1. Percentage deviations between the present energy levels (horizontal lines in black), the experimental ones (NIST) and other theoretical values as available in archival databases (MCHF, OPENADAS) and previous works: F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). 
Transition strengths are also compared. The oscillator strength (f_{ij}), which is related to the Avalue (i.e., the Einstein coefficient), is often used,
where m and e are the rest mass and charge of electron, respectively, c the speed of light, g_{j} and g_{i} the statistical weights of the upper (j) and lower (i) levels, respectively, and λ_{ij} the wavelength of the transition i − j.
Figure 2 shows the deviation Δlog (gf) of archival databases and previous studies with respect to the present work. A histogram plot of the data shown in Fig. 2 is shown in Appendix A. We limit the comparison to relatively strong transitions with log (gf) ≳ 10^{−6} from the lowest five energy levels of the ground configuration: 2s^{22}p^{2} (^{3}P_{0 − 2},^{1}D_{2},^{1}S_{0}) as they are metastable levels. For those weak transitions excluded in our comparison, log (gf) might differ by several orders of magnitude among archival databases, previous studies, and the present work. This is often due to the different number of configuration interaction levels included, as well as the method adopted. Nonetheless, the weak transitions are not expected to significantly impact the plasma diagnostics as the five metastable levels drive the population of all the other levels in the Clike ions, for astrophysical plasma.
Fig. 2. Comparisons of log (gf) from the present work (black horizontal line) with archival databases and previous works. F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). We note that this comparison is limited to relatively strong transitions with log (gf) ≳ 10^{−6} originating from the lowest five energy levels. 
Subsequently, we compare the collision data for Fe XXI (Sect. 4.1), S XI (Sect. 4.2), Ne V (Sect. 4.3), and O III (Sect. 4.4) with previous Rmatrix calculations. Hexbin plots^{6} (Carr et al. 1987) are used to compare the effective collision strengths of a large number of transitions. In Sect. 4.5, we compare the collision data for Ar XIII with a previous distorted wave calculation. Finally, we demonstrate the impact on the density diagnostics using these two data sets of Ar XIII.
We note that all Rmatrix calculations (including the present calculation) without pseudostates are not converged for the highlying levels, both with respect to the Nelectron target configuration interaction expansion and the (N+1)electron closecoupling expansion. Here we include configurations up to n = 4 (24 in total) in addition to three configurations with n = 5 (Table 1). Accordingly, the present effective collision strengths involving energy levels with n ≤ 3 are better converged than those with n ≥ 4. Future largerscale Rmatrix ICFT calculations or Rmatrix calculations with pseudostate calculations can improve the accuracy of transitions involving the highlying levels, especially amongst the highlying levels.
4.1. Fe XXI
The most recent calculation of Rmatrix electronimpact excitation data for Fe XXI (or Fe^{20+}) is presented by FernándezMenchero et al. (2016, F16 hereafter), including 590 finestructure levels in both the configuration interaction and closecoupling expansions. We limit our comparison to F16 and refer readers to F16 for their comparison with other previous calculations (Aggarwal & Keenan 1999a; Butler & Zeippen 2000; Badnell et al. 2001). Both F16 and the present work use the AUTOSTRUCTURE code for the structure calculation. Although both calculations include 590 finestructure levels in the configuration interaction and closecoupling expansions, different scaling parameters (data sets A and B in Fig. 3) lead to slightly different atomic structures. As shown in the topleft panel of Fig. 1, generally speaking, the energy levels of the present work and F16 agree within ≲0.1%. The first few levels have a slightly larger deviation of ≲0.5%, yet smaller than the ≲2% deviation with respect to NIST and (Ekman et al. 2014, E14 hereafter). Additionally, there is a shift of ∼0.3 Ryd between E14 and the present work (and F16) for the 2s^{22}p3s (3P_{0}).
Fig. 3. Three sets of scaling parameters for Fe XXI. The black squares (set A) correspond to those listed in Table 2. The purple diamonds (set B) correspond to the scaling parameters used in FernándezMenchero et al. (2016). The red stars (set C, see Sect. 4.1) correspond to the third set of scaling parameters which have a smaller deviation with respect to the default set. The level energies, Avalues, and effective collision strengths shown in Table 3 are very sensitive to the scaling parameters of 3s and 3d. 
Similarly, as shown in the topleft panel of Fig. 2, transition strengths agree well among NIST, F16, and the present work with merely a few exceptions. Larger deviations are found between E14 and the present work.
The scattering calculations of both the present work and F16 were performed with the Rmatrix ICFT method. We remind the readers that the atomic structures are slightly different between the two calculations. Figure 4 shows the comparison of the effective collision strengths at relatively high temperatures (4.41 × 10^{6} K and 2.20 × 10^{7} K). There are in total ∼1.3 − 1.4 × 10^{5} transitions with log (gf) > − 5 in both data sets for all three panels. In the left and middle panels, ∼5 − 8%, ∼1%, and ∼0.1% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. In the right panel, ∼3%, ∼0.2%, and 0.01% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively.
Fig. 4. Hexbin plots of the comparison of the Fe XXI (or Fe^{20+}) effective collision strengths between two sets (A and B) of calculations (Υ_{1}) and FernándezMenchero et al. (2016, Υ_{2}) at relatively high temperatures. Data set A is the default of the present work. In data set B, we use the scaling parameters of FernándezMenchero et al. (2016) (Fig. 3). The darker the color, the greater number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. The dashed lines in red highlight transitions in the “long island” (see Sect. 4.1 for more details). 
In the left and middle panels of Fig. 4, the “long island” in parallel to yet below the diagonal line in red is mainly contributed by transitions involving level #78 (2s2p^{23}s, ^{3}P_{1}) and #79 (2s2p^{23}d, ^{5}D_{1}). When we use the scaling parameters of F16 (set B) the long island is no longer present (the right panel of Fig. 4). We performed another calculation (set C) with a third set of scaling parameters^{7}, which has a smaller deviation with respect to the default calculation (set A). This shows that the level energies, Avalues, and effective collision strength in the “long island” are very sensitive to the scaling parameters of 3s and 3d (Table 3). When we compare the energies and Avalues with respect to CHIANTI and SPEX (Table 3), our default calculation (A) are comparable in terms of energies, and slightly “better” than set B yet slightly “worse” than set C in terms of Avalues.
Energy (cm^{−1}), Avalue (s^{−1}), Υ (at 4.41 × 10^{6} K), and Υ (at ∞) for levels 2s 2p^{2} 3s, ^{3}P_{1} (#78 in the present work) and 2s 2p^{2} 3d, ^{5}D_{1} (#79) of Fe XXI.
At relatively low temperatures, a large deviation is still observed even when we use the scaling parameters of F16 (the middle panel of Fig. 5. There are in total ∼1.6 × 10^{5} transitions with log (gf) > − 5 in both data sets for all three panels. In the left panel, ∼20%, ∼3%, and ∼0.1% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. In the middle panel, ∼20%, ∼2%, and ∼0.06% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. In the right panel, ∼10%, ∼1%, and 0.04% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively.
Fig. 5. Hexbin plots of the comparison of the Fe XXI (or Fe^{20+}) effective collision strengths between three sets (A, B, and D) of present calculations (Υ_{1}) and FernándezMenchero et al. (2016, F16, Υ_{2}) at a relatively low temperature. Data set A is the default of the present work. In data set B, we use the scaling parameters of FernándezMenchero et al. (2016) (Fig. 3). In data set D, we use the same scaling parameters and the same number of points (four times our default calculation) for the fine energy mesh, as in F16. The darker the color, the greater number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 
This can be attributed to the different fine energy meshes used for the outerregion exchange calculations. The number of points of the fine energy mesh in F16 is four times larger than that of the present work so that resonances are better resolved in F16. Therefore, we performed another calculation using the same scaling parameters and the same number of points of the fine energy mesh as in F16. The comparison of the effective collision strength between this calculation (data set D) and F16 at 8.82 × 10^{4} K is shown in the right panel of Fig. 5. The difference is negligible even toward the lowtemperature end. Since the resonance enhancement is more significant at lower temperatures, the deviation in the right panel of Fig. 4 is smaller than that in the middle panel of Fig. 5.
The remaining deviation in the right panel of Fig. 4 is asymmetric (Υ_{2} > Υ_{1}). This is attributed to the following two additional causes.
First, some numerical failures are found in the outerregion exchange calculation of F16. Several test calculations are performed, however, we were unable to reproduce the numerical failures. Second, some unresolved resonances (Sect. 2.2) were found in the outerregion nonexchange calculation of F16. The Perl script used by F16 bypassed the routine to remove the unresolved resonances.
The above two additional causes explain the remaining deviations in the right panel of Fig. 5. Since the numerical failures and unresolved resonances are present in the resonance region, effective collision strengths at high temperatures are less affected (cf. the right panels of Figs. 4 and 5).
4.2. S XI
Liang et al. (2011, L11 hereafter) calculated the electronimpact excitation data of S XI (or S^{10+}) in a similar approach as the present work. The configuration interaction among 24 configurations was used to calculate the structure (see their Table 1 for more details). The lowest 254 finestructure levels were included in the closecoupling expansion and the scattering calculation. For simplicity, we limit our comparison to L11 and refer readers to L11 for their comparison with other previous calculations Bell & Ramsbottom (2000, Rmatrix) and Landi & Bhatia (2003, distorted wave).
Both L11 and the present work use the AUTOSTRUCTURE code for the structure calculation. Slightly different scaling parameters are used, yet the energy levels are nearly identical. Energy levels from the present work and L11 agree well with the NIST and MCHF atomic databases, except for the lowest 20 energy levels (the topright panel of Fig. 1). As shown in the topright panel of Fig. 2, transition strengths agree well among NIST, L11, and the present work with merely a few exceptions.
Both the present work and L11 used the Rmatrix ICFT method for the collisional calculation. For relatively weaker transitions, the effective collision strengths of L11 are systematically smaller than those of the present work (Fig. 6). There are in total ∼32 000 transitions with log (gf) > − 5 in both data sets, ∼50 − 80%, ∼30 − 50%, and ∼10 − 20% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively.
Fig. 6. Hexbin plots of the comparison of the S XI (or S^{10+}) effective collision strengths between the present work (Υ_{1}) and Liang et al. (2011, Υ_{2}) at T ∼ 6.05 × 10^{4} K (left) and 2.42 × 10^{6} K (middle), and ∼2.5 × 10^{5} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 
Since the present work has a significantly larger closecoupling expansion (590 levels vs. 254 levels), the additional resonances contribute most to the asymmetric deviation. Similar results were also found by FernándezMenchero et al. (Fig. 4 of 2016), where the effective collision strength of Fe XXI as obtained by two Rmatrix ICFT calculations with different closecoupling expansions (564 levels vs. 200 levels) were compared.
4.3. Ne V
The most recent Rmatrix calculation of the electronimpact excitation data of Ne V (or Ne^{4+}) is presented in Griffin & Badnell (2000, G00 hereafter). The calculation had 261 finestructure levels in the configurationinteraction expansion and 138 levels in the closecoupling expansion. Nonetheless, only data for the lowest 49 levels are archived in OPENADAS.
The energy levels of the present calculation are less accurate (within ∼10%, the bottomleft panel of Fig. 1) compared to NIST and MCHF databases. G00 performed a single configuration MCHF calculation for their atom structure, their energy levels are comparable with the present calculation. The transition strengths also agree well between the present work and G00 (the bottomleft panel of Fig. 2).
Both G00 and the present work used the Rmatrix ICFT method for the collision calculation. The effective collision strengths in G00 had a temperature range of (10^{3}, 10^{6}) K with three points (1.00, 2.51 and 6.30) per decade. The present calculation covers a different temperature range of (5 × 10^{3}, 5 × 10^{7}) K with three points (1.25, 2.50 and 5.00) per decade. We calculate the effective collision strength of Ne V at the same temperature points of G00 and show the comparison at at T = 2.51 × 10^{4} K and 2.51 × 10^{5} K in Fig. 7. We note that the comparison is limited to effective collision strengths involving the lowest 49 levels (all the n = 2 levels and about a quarter of the n = 3 levels), which are archived in OPENADAS. There are in total ∼1170 transitions with log (gf) > − 5 in both data sets, ∼40 − 50%, ∼10%, and ∼0.09% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. The significant and asymmetric deviation shown in Fig. 6 for S XI is not found here because the results for the lowlying levels are better converged.
Fig. 7. Hexbin plots of the comparison of the Ne V (or Ne^{4+}) effective collision strengths between the present work (Υ_{1}) and Griffin & Badnell (2000, OPENADAS, Υ_{2}) at T = 2.51 × 10^{4} K (left) and 2.51 × 10^{5} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 
4.4. O III
The most recent Rmatrix calculation of the electronimpact excitation data of O III (or O^{2+}) is presented by Tayal & Zatsarinny (2017, T17 hereafter), including 202 finestructure levels in the closecoupling expansion. For simplicity, we limit our comparison to T17 and refer readers to T17 for their comparison with other previous calculations (Storey & Sochi 2015; Palay et al. 2012; Aggarwal & Keenan 1999b).
T17 used the nonorthogonal MCHF program for their structure calculation, leading to a better agreement of the level energies with respect to the NIST and MCHF atomic databases. As shown in the bottomright panel of Fig. 1, the level energies of the present calculation are less accurate (within ∼15%). As for the transition strength, strong transitions (i.e., log (gf) ≳ 10^{−1}) agree well among all the calculations and databases. A larger deviation is found for some of the weaker transitions (Fig. 2).
The scattering calculation of T17 utilized Bspline BreitPauli Rmatrix (BSR) code (Zatsarinny 2006), where an accurate target description is obtained by taking advantage of termdependent orbitals. The effective collision strengths in T17 are tabulated with a narrower temperature range: 10^{2} K, 5 × 10^{2} K, 10^{3} K, 5 × 10^{3} K, 10^{4} K, 2 × 10^{4} K, 4 × 10^{4} K, 6 × 10^{4} K, 8 × 10^{4} K, and 10^{5} K. The present calculation covers a wider temperature range of (1.8 × 10^{3}, 1.8 × 10^{7}) K with three points (1.80, 4.50 and 9.00) per decade. We calculate the effective collision strength of O III at the same temperature points of T17 and show the comparison at T = 10^{4} K and 8 × 10^{4} K in Fig. 8. There are in total ∼19 600 transitions with log (gf) > − 5 in both data sets, ∼60%, ∼20%, and ∼1.5% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. The deviation observed is mainly due to the different atomic structures and the different sizes of the closecoupling expansion. The T17 data set is recommended if it suits the purpose of the user.
Fig. 8. Hexbin plots of the comparison of the O III (or O^{2+}) effective collision strengths between the present work (Υ_{1}) and Tayal & Zatsarinny (2017) (Υ_{2})at T = 10^{4} K (left) and = 8 × 10^{4} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 
4.5. Ar XIII
The collision data of Ar XIII in the latest version of the CHIANTI database (V9.0 Dere et al. 2019) originate from Dere et al. (1979), where the collision calculation was carried out with the UCL distorted wave codes for small angular momentum values of the incoming electron and the Bethe approximation for large angular momentum values. Figure 9 compares the ordinary collision strength (Ω) of two transitions from the ground level as calculated with the present Rmatrix codes and the previous distorted wave calculation. The previous distorted wave calculation provides a good description of the “background”, while the Rmatrix calculation includes resonances that stand above the background.
Fig. 9. Ordinary collision strengths (Ω) for Ar XIII. Left panel: refers to the forbidden transition from the ground level 2s^{22}p^{2} (^{3}P_{0}) to the metastable level 2s^{22}p^{2} (^{3}P_{1}), while the right panel refers to the transition from the ground level 2s^{22}p^{2} (^{3}P_{0}) – 2s2p^{3} (^{3}D_{1}). The present work (PW) is shown in pink solid lines. The solid squares and the dashed lines are previous distorted wave approximations from Table 4 of Dere et al. (1979, D79). 
It is well known that the presence of the resonances increases significantly the effective collisions strengths for forbidden transitions, especially at low temperatures, while differences for the strong allowed transitions are often small, and mostly dominated by differences in the target structure. Figure 10 shows the effective collision strengths for the two strongest transitions in Ar XIII. It confirms that the differences for the allowed transition are minor, while those for the forbidden transition are about a factor of ≳2 at the typical formation temperature of this ion in ionization equilibrium (2.8 × 10^{6} K, equivalent to log_{10} (T/K) = 6.45). The distorted wave effective collision strengths were obtained from the CHIANTI database, and are based on the calculations reported by (D79, Dere et al. 1979).
Fig. 10. Effective collision strengths for one of the strongest forbidden (top) and allowed (bottom) transitions for Ar XIII. We show both the present values (Rmatrix) and those obtained from the CHIANTI database, which were based on the distorted wave calculations reported by (D79, Dere et al. 1979). 
The forbidden line shown in Fig. 10 is one of the two strong infrared transitions which are currently receiving much interest in the solar physics community as they are potentially very useful to measure electron densities and the chemical abundance of Ar, one of the elements for which photospheric abundances are not available. These transitions have never been observed, as they are in a relatively unexplored spectral range; however, they will be observable by the nextgeneration 4 m DKIST telescope, as discussed in detail by Del Zanna & DeLuca (2018).
The increased effective collision strengths for all the forbidden transitions we have obtained with the present calculations have a significant effect on their predicted intensities, even though the main populating mechanism for these transitions is cascading from higher levels. To estimate these effects we have considered three ion models. The first has the present effective collision strengths and Avalues, but has the Avalues for the transitions within the ground configuration from Jönsson et al. (2011). These latter values are obtained with a largescale GRASP2K calculation and should be very accurate; we note that the differences between these two sets of Avalues are small (10–20%). The second is the CHIANTI model, but with the Avalues within the ground configuration from Jönsson et al. (2011). The third one is the CHIANTI model.
Figure 11 shows the intensity ratio between the main forbidden and allowed transitions for Ar XIII, indicating that the cumulative effect of the changes in the effective collision strengths is to increase the intensity of the forbidden line by up to 40% in the lowdensity regime.
Fig. 11. Intensity ratio between the main forbidden and allowed transitions for Ar XIII, as calculated with the present Rmatrix crosssections and with the distorted wave calculations as available in CHIANTI (see text for details). 
5. Summary
We have presented a systematic set of Rmatrix intermediatecoupling frame transfer calculation of Clike ions from N II to Kr XXXI (i.e., N^{+} to Kr^{30+}) to obtain levelresolved effective collision strengths over a wide temperature range. The present calculation is the first Rmatrix calculation for many ions in the Clike isoelectronic sequence and an extension/improvement for several ions, with respect to previous Rmatrix calculations.
As we have shown for Ar XIII, the present effective collision strengths increase significantly the predicted intensities of the forbidden lines, compared to earlier calculations. Forbidden lines from Ar XIII, as well as those from other ions (such as Si IX and S XI) are prominent diagnostics for the upcoming DKIST (Rimmele et al. 2015) solar facility as discussed in Del Zanna & DeLuca (2018) and Madsen et al. (2019).
The present atomic data will allow more accurate plasma diagnostics with future highresolution spectral missions such as Athena XIFU (Nandra et al. 2013; Barret et al. 2018) and Arcus (Smith et al. 2016). For instance, as shown in Kaastra et al. (2017), Arcus has the capability to measure absorption lines from the ground and metastable levels of Si IX, which enables us to constrain the density of the photoionized outflows in active galactic nuclei.
The effective collision strengths are archived according to the Atomic Data and Analysis Structure (ADAS) data class adf04 and will be available in OPENADAS and our UKAPAP website. These data will be incorporated into plasma codes like CHIANTI (Dere et al. 1997, 2019) and SPEX (Kaastra et al. 1996, 2018) for diagnostics of astrophysical plasmas. We plan to perform the similar type of calculations for Nlike and Olike isoelectronic sequences.
To represent the relationship of two large sets of numerical variables, instead of overlapping data points in a scatter plot, the hexbin plotting window is split into hexbins, and the number of points per hexbin is counted and color coded. The supplementary package on Zenodo (Mao 2019) provides scripts to reproduce the hexbin plots presented in this paper. A simple demo of hexbin plot is also available here.
These scaling parameters were determined by following the progressive optimization procedure described in Sect. 2.1 but for configurations with n = 5, we only include the lowest three (instead of 15 in total) configurations.
Acknowledgments
The present work is funded by STFC (UK) through the University of Strathclyde UK APAP network grant ST/R000743/1 and the University of Cambridge DAMTP atomic astrophysics group grant ST/P000665/1. JM acknowledges useful discussions with L. FernándezMenchero.
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Appendix A: Comparison of the structure calculation in histograms
We present the histograms (Fig. A.1) of the energy levels (Fig. 1) for the four selected ions: Fe XXI, S XI, Ne V, and O III. Similar histograms (Fig. A.2) are presented for log (gf) (Fig. 2).
Fig. A.1. Histogram plots of the percentage deviations between the present energy levels (deviation = 0%), the experimental ones (NIST) and other theoretical values as available in archival databases (MCHF, OPENADAS) and previous works: F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). 
Fig. A.2. Histogram plots of the comparing log (gf) in the present work (Δlog (gf) = 0) with archival databases and previous works. F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). We note that this comparison is limited to relatively strong transitions with log (gf) ≳ 10^{−6} from the lowest five energy levels. 
Appendix B: Energy resolution of the resonance region
For the present calculations of the entire isoelectronic sequence (Sect. 2.2), our energy resolution of the resonance region is a factor four times larger (poorer) than that of F16 for Fe XXI. This “poorer” energy mesh is adequate for lowcharge ions like Si IX. As shown in Fig. B.1, we compare the default data set of the present work (Υ_{1}) and another calculation where we double the size of the energy mesh for the outerregion exchange calculation (Υ_{2}). The difference is negligible even toward the lowtemperature end. There are in total ∼1.4 × 10^{5} transitions with log (gf) > − 5 in both data sets, ∼0.5%, ∼0.01%, and 0% have deviation larger than 0.1 dex, 0.3 dex, and 1 dex, respectively. Thus, the energy mesh used in the present calculation is fine enough for the lowcharge ions.
Fig. B.1. Hexbin plots of the comparison of the Si IX (or Si^{8+}) effective collision strengths between the default data set of the present work (Υ_{1}) and another calculation where we double the size of the energy mesh for the outerregion exchange calculation (Υ_{2}). The effective collision strengths are compared at T ∼ 8 × 10^{4} K (left), ∼1.6 × 10^{6} K (middle), and ∼4 × 10^{7} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 
For highcharge ions like Fe XXI, at higher temperatures (cf. the middle and right panels of Fig. 4), the difference between our default data set (A, Υ_{1}) and F16 (Υ_{2}) is mainly due to the (slightly) different atomic structures. At lower temperatures, the scatter caused by different atomic structures is even larger. That is to say, a “better” atomic structure is the leading concern for highcharge ions at lower temperatures. A “finer” energy mesh requires more computation time yet only leads to a minor improvement in accuracy.
All Tables
ThomasFermiDiracAmaldi potential scaling parameters used in the AUTOSTRUCTURE calculations for the Clike isoelectronic sequence. Z is the atomic number, such as 8 for oxygen.
Energy (cm^{−1}), Avalue (s^{−1}), Υ (at 4.41 × 10^{6} K), and Υ (at ∞) for levels 2s 2p^{2} 3s, ^{3}P_{1} (#78 in the present work) and 2s 2p^{2} 3d, ^{5}D_{1} (#79) of Fe XXI.
All Figures
Fig. 1. Percentage deviations between the present energy levels (horizontal lines in black), the experimental ones (NIST) and other theoretical values as available in archival databases (MCHF, OPENADAS) and previous works: F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). 

In the text 
Fig. 2. Comparisons of log (gf) from the present work (black horizontal line) with archival databases and previous works. F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). We note that this comparison is limited to relatively strong transitions with log (gf) ≳ 10^{−6} originating from the lowest five energy levels. 

In the text 
Fig. 3. Three sets of scaling parameters for Fe XXI. The black squares (set A) correspond to those listed in Table 2. The purple diamonds (set B) correspond to the scaling parameters used in FernándezMenchero et al. (2016). The red stars (set C, see Sect. 4.1) correspond to the third set of scaling parameters which have a smaller deviation with respect to the default set. The level energies, Avalues, and effective collision strengths shown in Table 3 are very sensitive to the scaling parameters of 3s and 3d. 

In the text 
Fig. 4. Hexbin plots of the comparison of the Fe XXI (or Fe^{20+}) effective collision strengths between two sets (A and B) of calculations (Υ_{1}) and FernándezMenchero et al. (2016, Υ_{2}) at relatively high temperatures. Data set A is the default of the present work. In data set B, we use the scaling parameters of FernándezMenchero et al. (2016) (Fig. 3). The darker the color, the greater number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. The dashed lines in red highlight transitions in the “long island” (see Sect. 4.1 for more details). 

In the text 
Fig. 5. Hexbin plots of the comparison of the Fe XXI (or Fe^{20+}) effective collision strengths between three sets (A, B, and D) of present calculations (Υ_{1}) and FernándezMenchero et al. (2016, F16, Υ_{2}) at a relatively low temperature. Data set A is the default of the present work. In data set B, we use the scaling parameters of FernándezMenchero et al. (2016) (Fig. 3). In data set D, we use the same scaling parameters and the same number of points (four times our default calculation) for the fine energy mesh, as in F16. The darker the color, the greater number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 

In the text 
Fig. 6. Hexbin plots of the comparison of the S XI (or S^{10+}) effective collision strengths between the present work (Υ_{1}) and Liang et al. (2011, Υ_{2}) at T ∼ 6.05 × 10^{4} K (left) and 2.42 × 10^{6} K (middle), and ∼2.5 × 10^{5} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 

In the text 
Fig. 7. Hexbin plots of the comparison of the Ne V (or Ne^{4+}) effective collision strengths between the present work (Υ_{1}) and Griffin & Badnell (2000, OPENADAS, Υ_{2}) at T = 2.51 × 10^{4} K (left) and 2.51 × 10^{5} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 

In the text 
Fig. 8. Hexbin plots of the comparison of the O III (or O^{2+}) effective collision strengths between the present work (Υ_{1}) and Tayal & Zatsarinny (2017) (Υ_{2})at T = 10^{4} K (left) and = 8 × 10^{4} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 

In the text 
Fig. 9. Ordinary collision strengths (Ω) for Ar XIII. Left panel: refers to the forbidden transition from the ground level 2s^{22}p^{2} (^{3}P_{0}) to the metastable level 2s^{22}p^{2} (^{3}P_{1}), while the right panel refers to the transition from the ground level 2s^{22}p^{2} (^{3}P_{0}) – 2s2p^{3} (^{3}D_{1}). The present work (PW) is shown in pink solid lines. The solid squares and the dashed lines are previous distorted wave approximations from Table 4 of Dere et al. (1979, D79). 

In the text 
Fig. 10. Effective collision strengths for one of the strongest forbidden (top) and allowed (bottom) transitions for Ar XIII. We show both the present values (Rmatrix) and those obtained from the CHIANTI database, which were based on the distorted wave calculations reported by (D79, Dere et al. 1979). 

In the text 
Fig. 11. Intensity ratio between the main forbidden and allowed transitions for Ar XIII, as calculated with the present Rmatrix crosssections and with the distorted wave calculations as available in CHIANTI (see text for details). 

In the text 
Fig. A.1. Histogram plots of the percentage deviations between the present energy levels (deviation = 0%), the experimental ones (NIST) and other theoretical values as available in archival databases (MCHF, OPENADAS) and previous works: F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). 

In the text 
Fig. A.2. Histogram plots of the comparing log (gf) in the present work (Δlog (gf) = 0) with archival databases and previous works. F16 refers to FernándezMenchero et al. (2016), E14 refers to Ekman et al. (2014), L11 refers to Liang et al. (2011), G00 refers to Griffin & Badnell (2000, OPENADAS), and T17 refers to Tayal & Zatsarinny (2017). We note that this comparison is limited to relatively strong transitions with log (gf) ≳ 10^{−6} from the lowest five energy levels. 

In the text 
Fig. B.1. Hexbin plots of the comparison of the Si IX (or Si^{8+}) effective collision strengths between the default data set of the present work (Υ_{1}) and another calculation where we double the size of the energy mesh for the outerregion exchange calculation (Υ_{2}). The effective collision strengths are compared at T ∼ 8 × 10^{4} K (left), ∼1.6 × 10^{6} K (middle), and ∼4 × 10^{7} K (right). The darker the color, the greater the number of transitions log_{10}(N). The diagonal line in red indicates Υ_{1} = Υ_{2}. 

In the text 
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