Issue 
A&A
Volume 528, April 2011



Article Number  A69  
Number of page(s)  15  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201016417  
Published online  03 March 2011 
Rmatrix electronimpact excitation data for the Lilike isoelectronic sequence including Auger and radiation damping^{⋆}
Department of PhysicsUniversity of Strathclyde, Glasgow, G4 0NG, UK
email: guiyun.liang@strath.ac.uk
Received: 28 December 2010
Accepted: 27 January 2011
We present results for the electronimpact excitation of all Lilike ions from Be^{+} to Kr^{33+} which we obtained using the radiation and Augerdamped intermediatecoupling frame transformation Rmatrix approach. We have included both valence and coreelectron excitations up to the 1s^{2}5l and 1s2l4l′ levels, respectively. A detailed comparison of the target structure and collision data has been made for four specific ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence so as to assess the accuracy for the entire sequence. Effective collision strengths (Υs) are presented at temperatures ranging from 2 × 10^{2}(z + 1)^{2} K to 2 × 10^{6}(z + 1)^{2} K (where z is the residual charge of the ions, i.e. Z − 3). Detailed comparisons for the Υs are made with the results of previous calculations for several ions which span the sequence. The radiation and Auger damping effects were explored for coreexcitations along the isoelectronic sequence. Furthermore, we examined the isoelectronic trends of effective collision strengths as a function of temperature.
Key words: atomic data / atomic processes / plasmas
These data are made available in the archives of APAP via http://www.apapnetwork.org, OPENADAS via http://open.adas.ac.uk, as well as anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsweb.ustrasbg.fr/cgibin/qcat?J/A+A/528/A69
© ESO, 2011
1. Introduction
Lilike ions are of importance both in astrophysics and in theoretical and experimental atomic physics. Satellite lines arising from transitions of the type 1s2lnl′ → 1s^{2}2l (where the generation of 1s2lnl′ states could be from dielectronic recombination of Helike ions or innershell excitation of Lilike ions), were observed in solar flare spectra by the Rentgenowsky Spektrometr s Izognutymi Kristalami (RESIK) instrument on the Russian CORONASF mission, launched on 2001 July 31 (Phillips et al. 2006). These satellite lines complicate the analysis of the spectrum around the Helike transition lines but they are important diagnostics of the electron density and temperature of a plasma (Phillips et al. 2006; Nahar et al. 2009; Oelgoetz et al. 2009, and references therein). Atomic data for the dielectronic recombination process populating the 1s2lnl′ levels has been reported by Bautista & Badnell (2007).
For the case of outershell transitions, some n = 2 → 2 transition lines in Mg^{9+}–Ni^{25+} were recorded in the early solar flare observation by Widing & Purcell (1976). These have been extensively used in a variety of diagnostic applications, for which accurate atomic data are needed.
The group of Sampson carriedout early work to provide comprehensive atomic data for Lilike ions using the nonresonant relativistic distortedwave (DW) method. Zhang et al. (1990) calculated collision strengths (Ω) for n = 2 → n = 3,4,5 excitations for the 85 ions with nuclear charge number Z: 8 ≤ Z ≤ 92. These data are still extensively used by current astrophysical modelling codes, e.g. CHIANTI v6 (Dere et al. 2009). Sampson and coauthors (1985a,b; Zhang et al. 1986; Goett & Sampson 1983) reported the collision strengths of innershell (n ≤ 3) excitations in the Lilike ions with Z: 6 ≤ Z ≤ 74. These were obtained using a CoulombBornExchange (CBE) method. Their data is the main source used by CHIANTI v6 to model the satellite lines of Helike ions.
We turn next to Rmatrix calculations, which take account of resonances normally omitted by the DW and CBE methods. Berrington & Tully (1997) calculated the valenceelectron impact excitation of Fe^{23+} (up to n = 5) using the BreitPauli Rmatrix method. Gorczyca & Badnell (1996) demonstrated significant reduction of the resonance contribution due to radiation damping in the Kshell excitation of Fe^{25+} and Mo^{41+}. Subsequently, Ballance et al. (2001) performed a BreitPauli Rmatrix calculation including radiation damping for the innershell excitations (1s^{2}nl → 1s2ln′l′, where n,n′ ≤ 3) in Fe^{22+} and Fe^{23+}. Furthermore, taking Auger damping into account, Whiteford et al. (2002) performed new calculations for these (to n = 3) innershell excitations in Ar^{15+} and Fe^{23+}, using an intermediatecoupling frame transformation (ICFT) Rmatrix method. A complete excitation dataset for the Ar^{15+} and Fe^{23+} ions was presented in their work by incorporating data from a separate calculation for outershell excitations up to levels of the n = 5 complex. There appears to be no previous innershell electronimpact excitation work using the closecoupling method for all other ions in this isoelectronic sequence.
For valenceelectron excitations, Griffin et al. (2000) carriedout LScoupling Rmatrix with pseudostates (RMPS) calculations for C^{3+} and O^{5+}. Similar calculations for Be^{+} (n ≤ 5) were performed by Ballance et al. (2003). Aggarwal and coauthors (2004a,b, 2010) reported results for n ≤ 5 for N^{4+}, O^{5+}, F^{6+}, Ne^{7+}, Na^{8+}, Ar^{15+} and Fe^{23+} which were obtained by using the Dirac Rmatrix method as implemented in darc.
Here, we reporton calculations for the electronimpact excitation of the Lilike isoelectronic sequence ions from Be^{+} to Kr^{33+} which were made using the ICFT Rmatrix method. The main focus of the present work is on the innershell transitions to n = 4 which contribute to the population of the upper levels (1s2lnl′) of the Helike satellite lines. Separate outershell calculations were made which went up to n = 5.
This paper is one of our series of works on isoelectronic sequences: Flike, Witthoeft et al. (2007), Nelike, Liang & Badnell (2010), Nalike, Liang et al. (2009a,b). This work is part of the UK Atomic Processes for Astrophysical Plasmas (APAP) network^{1}.
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 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 Rmatrix calculations employing the ICFT method (see Griffin et al. 1998) for all Lilike ions from Be^{+} to Kr^{33+}. Our approach to the valence and coreexcitation data is to perform the two calculations independently and later merge the effective collision strengths (Υ) back together into a single dataset for each ion.
The reason for this approach is that the number of Rmatrix continuum basis orbitals required increases with box size (which scales as n^{2}) and also with scattering energy. Innershell excitations require a large scattering energy and n = 5 gives rise to a large box size. The two together resultin too large an (N + 1)electron Hamiltonian for Auger plus radiation damped Rmatrix calculations along an entire sequence. There are also numerical stability issues when the number of continuum basis orbitals exceeds 100 per orbital l.
The closecoupling (CC) and configuration interaction (CI) expansions used consist of the 1s^{2} { 2,3,4,5 } l (14 LS terms, 24 finestructure levels) and 1s^{2} { 2,3,4 } l, 1s2l { 2,3,4 } l′ (89 LS terms, 195 finestructure levels) configurations for the valence and coreexcitation calculations, respectively.
2.1. Structure: level energies
The target wavefunctions (1s − 5g) were obtained from autostructure (AS, Badnell 1986) using the ThomasFemiDiracAmaldi model potential. Relativistic effects were included perturbatively from the onebody BreitPauli operator (viz. massvelocity, spinorbit and Darwin) without valenceelectron twobody finestructure operators. This is consistent with the operators included in the standard BreitPauli Rmatrix suite of codes. The radial scaling parameters, λ_{nl} (n = 1 − 5; l ∈ s,p,d, and f, were obtained separately for each ion by a twostep optimization procedure. In the first step, the energy of the 1s^{2}2l was minimized by varying the λ_{1s}, λ_{2s} and λ_{2p} scaling parameters. Then, the energy of the 1s^{2} { 3,4,5 } l configurations was minimized by varying the λ_{ { 3,4,5 } l} scaling parameters. In the calculation including doublyexcited configurations, the energy of the 1s^{2} { 3,4 } l configurations was minimized by varying the λ_{ { 3,4 } l} scaling parameters. In order to maintain consistency and so as not to introduce arbitrary changes along the sequence, the optimization procedure is done automatically in autostructure without any manual readjustment. The resultant scaling parameters are listed in Table 1. We took λ_{5g} to be unity since it is insensitive to optimization and the atomic structure itself is insensitive to it.
ThomasFermi potential scaling factors used by autostructure for the outershell calculations (see text for details).
Comparisons of the level energies (Ryd) of 1s^{2} {2,3,4,5} l (l ∈ s,p,d,f and g), 1s2l^{2} and 1s2s2p configurations in O^{5+}, Ar^{15+}, Fe^{23+}, and Kr^{33+} ions.
A comparison of level energies is made with the experimentally derived data available from the compilation of NIST v3^{2} and with other theoretical results for several specific ions (O^{5+}, Ar^{15+}, Fe^{23+}, and Kr^{33+}) spanning the sequence so as to assess the accuracy of our present structure over the entire isoelectronic sequence – see (the composite) Table 2. The lowlying level energies of all ions agree to within about 1%. The present calculations also show a good agreement (1%) with previous results obtained by using grasp (Aggarwal et al. 2004a,b, 2010) and the DiracFockSlater method (Zhang et al. 1990) for O^{5+}, Ar^{15+} and Fe^{23+} ^{3}. It should be noted that there can be large factor differences in energy separations between closelyspaced levels arising from different configurations, e.g. 5f_{j} − 5g_{j′} etc. For higherlying levels, our energies differ from NIST’s by at most 0.5% (see Table 2). Checks with AS calculations including both the twobody finestructure and quantumelectrodynamic (QED) effects revealed the twobody finestructure contribution to be negligible when compared with those of QED for these Lilike ions. (Twobody finestructure and QED operators have not yet been incorporated into the present Rmatrix codes, but such work is in progress – Eissner, private communication.)
2.2. Structure: line strength S
A further 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 (1)and the transition probability or Einstein’s Acoefficient, A_{ji}, as (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. Table 3 shows a comparison of line strengths for the most prominent satellite lines (Gabriel 1972) for several ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence.
Comparisons of line strengths S for Kshell transitions in ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning sequence.
Todate, the Lilike innershell linestrength data (n = 2p → 1s) calculated by Goett & Sampson (1983) is the main source of data used in astrophysical modelling. So, a comparison with their results, and others, has been made to assess the present structure results for S over the isoelectronic sequence – see Table 3. For O^{5+}, results for most (76%) of the listed transitions agree to within 15% . For Ar^{15+}, there is good agreement with Goett & Sampson (1983) – about 86% of transitions are within 15%. The present results also show excellent agreement with the data of Whiteford et al. (2002) also obtained using autostructure. For Fe^{23+}, almost all transition lines show excellent agreement (to within 5%) with the results of Whiteford et al. (2002) except for the sline – there is somewhat of a spread of results for this line. Most of transitions (64%) show agreement to within 15% between the present AS and the multiconfiguration DiracFock (MCDF) calculation of Chen (1972), in which the Breit and QED contributions have been included in the transition energy. Bautista et al. (2003) took the relativistic twobody operator and termenergy corrections (TECs) into account in their autostructure calculation, resultingin their line strengths being slightly larger than the present AS ones. Yet, they are still within 15% for most transitions (63%). For the highlycharged ion, Kr^{33+}, 73% of transitions agree to within 15% with the results of Goett & Sampson (1983).
Furthermore, a comparison of the line strength has been done here for outershell (dipole) transitions of ions spanning the sequence, see Fig. 1. For O^{5+}, the data of Zhang et al. (1990) is still the main source for astrophysical modelling and 66% of available transitions agree to within 5% of the present results. An excellent agreement is obtained between the present results and those of the grasp calculation by Aggarwal & Keenan (2004a): 97% of available dipole transitions agree to within 5%. For Ar^{15+}, a comparison with the previous AS (Whiteford et al. 2002) and grasp calculations (McKeown et al. 2004b) has been made: around 93% and 81%, respectively, of available transitions agree to within 5%. For Fe^{23+}, 98% of available outershell transitions from the ADAS database^{4} (Whiteford et al. 2002) show agreement to within 5%. The present AS calculation also shows good agreement with the grasp calculation performed by McKeown et al. (2004b) – 75% of available transitions show agreement to within 5%. For Kr^{33+}, somewhat worse agreement appears with the results of Zhang et al. (1990) which were obtained using DiracSlater atomicstructure approach. However, there are still about 57% of available transition showing agreement to within 10%.
Fig. 1 Comparison of the line strength (S) of outershell dipole transitions for ions spanning the sequence. MAK04 corresponds to the grasp calculation done by McKeown et al. (2004b). The horizontal dashed lines correspond to agreement within 5%. [Colour online] 
Thus, we believe that the atomic structure of the ions spanning the sequence is reliable, and expect the uncertainty in collision strengths (Ωs) due to inaccuracies in the target structure to be correspondingly small.
3. Scattering
As demonstrated for innershell excitations of the Nalike isoelectronic sequence, including the astrophysically abundant Fe^{15+}, (Liang et al. 2008; 2009b) and the Lilike ions Ar^{15+} and Fe^{23+} (Ballance et al. 2001; Whiteford et al. 2002), the radiation and Augerdamping effects significantly reduce the resonant enhancement of the collision strengths. In the Nalike isoelectronic sequence, the Auger damping effect was found to be the dominant damping mechanism over the entire sequence although the radiation damping increased quickly with increasing nuclear charge. The radiation and Augerdamping effects have been incorporated into the present ICFT and BreitPauli Rmatrix suite code via a complex optical potential as detailed by Gorczyca & Robicheaux (1999) and Robicheaux et al. (1995). For clarity, we give a brief description of the two damping effects for the specific case of Lilike ions. Over the sequence Be^{+}–Kr^{33+}, the fourelectron resonance configurations are of the form 1s [ 2s,2p ] [ 2s − 4f ] nl, and they can decay via the following channels: The participator KLn/KMn/KNn Auger channel (Eq. (3)) scales as n^{3} and is automatically described in the Rmatrix method by the contribution to the closecoupling expansion from the righthand side of Eq. (3). However, the spectator KLL/KLM/KLN Auger pathway (Eq. (4)) is independent of n and only lown resonances (n ≤ 4 here) are included explicitly in the closecoupling expansion. The highern are accountedfor by the complex optical potential which acts as a loss mechanism. The last two channels, Eqs. (5) and (6), represent radiation damping.
Our ICFT Rmatrix calculations employed 40 (coreelectron excitation) or 60 (valenceelectron excitation) continuum basis orbitals per angular momentum to represent the (N + 1)thelectron, over most of the sequence. For lowercharged ions, the number of continuum basis orbitals was increased. For example, the values were 46 and 65 in B^{2+} for the valence and coreelectron excitations, respectively. All partial waves from J = 0 to 41 were included explicitly and contributions from higher Jvalues were included using a “topup” procedure (Burgess 1974; Badnell & Griffin 2001). The contributions from partial waves up to J = 10 were included in the exchange Rmatrix, while those from J = 11 to 41 where included via a nonexchange Rmatrix calculation. For the exchange calculation, a fine energy mesh (less than 0.005 Ryd, and even finer to 0.0002 Ryd for lowercharged ions) 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. 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^{2}z^{2} Ryd, where z = Z − 3 is the residual charge of ion) was employed. For the nonexchange calculation, a step of 1.0 × 10^{2}z^{2} Ryd was used over the entire energy range.
We then used the infinite energy Born limits (nondipole allowed) and line strengths (dipole allowed) from autostructure so that higher energy reduced collision strengths (Ω), as defined by Burgess & Tully (1992), can be found from interpolation in BurgessTully space for all additional higher energies. The effective collision strengths at 13 electron temperatures ranging from 2 × 10^{2}(z + 1)^{2} K to 2 × 10^{6}(z + 1)^{2} K are produced as the end product. The data were stored in the ADAS adf04 format (Summers 2004).
Fig. 2 Comparison of collision strengths Ω from the present ICFT Rmatrix calculation in with those of Goett & Sampson (1983). a) O^{5+}: 1s^{2}2s ^{2}S_{1/2} → 1s2s2p ^{4}P_{3/2} excitation (u line in Table 3); b) Ar^{15+}: 1s^{2}2s ^{2}S_{1/2} → 1s2p^{2} ^{2}P_{1/2} excitation (137, ID is as list in Table 2); c) Fe^{23+}: 1s^{2}2s ^{2}S_{1/2} → 1s2s2p ^{2}P_{1/2} excitation (r line in Table 3). ASDW corresponds to the present BreitPauli distortedwave calculation using autostructure; d) Kr^{33+}: 1s^{2}2p ^{2}P_{1/2} → 1s2p^{2} ^{2}D_{3/2} excitation (k line in Table 3). [Colour online] 
Fig. 3 Comparison of the effective collision strength Υ between the present ICFT Rmatrix calculation and previous results of Goett & Sampson (1983) and Whiteford et al. (2002). a) O^{5+}: 1s^{2}2s → 1s2s2p ; b) Ar^{15}: 1s^{2}2s → 1s2p^{2}; c) Fe^{23+}: 1s^{2}2s → 1s2s2p ; d) Kr^{33+}: 1s^{2}2p → 1s2p^{2}. [Colour online] 
A separate BreitPauli DW calculation has been done for Fe^{23+} to study the importance of the effect of resonance enhancement. We find that it is still significant and widespread after Augerplusradiation damping has been taken into account. The DW approach has recently been incorporated into the autostructure code by Badnell (2011, see also autostructure codelog in APAP website^{1}). It is selfconsistent with the present Rmatrix ICFT method in its calculation of the target structure.
4. Results and discussions
4.1. Comparisons with previous calculations for coreexcitations
The present ICFT Rmatrix results are compared with those of previous works for four ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) which span the range of calculated data for this isoelectronic sequence. Here we select a few innershell transition lines to test the accuracy of the present ICFT Rmatrix calculations. An extensive comparison (all available excitation data from the ground level 1s^{2}2s ^{2}S_{1/2}) between the present ICFT Rmatrix and previous calculations has been made for the four ions to check the broad accuracy and the resonant enhancement (when compared with distortedwave data) of the present ICFT Rmatrix data. Earlier sequence calculations of coreelectron impact excitation from the 1s^{2} {2,3} l states were by Goett & Sampson (1983, hereafter GS83) and Sampson et al. (1985b) using the CoulombBorn exchange method.

O^{5+} Todate, there is no Rmatrix calculation available for this ion. The background of the ordinary collision strength Ω of the present ICFT Rmatrix calculation agrees well with that from the CoulombBorn exchange method by Goett & Sampson (1983), as shown in Fig. 2 for the 1s2s2p ^{4}P_{3/2} → 1s^{2}2s ^{2}S_{1/2} transition line (the u line in Table 3). The resultant effective collision strengths (Υ) are in agreement to within 10% over the entire temperature range, which is due to the scarce and weak resonances left after the Augerplusradiation damping has been taken into account. The undamped Υ is higher than the results of Goett & Sampson (1983) by ~20–35% for the temperature range 4 × 10^{2}(z + 1)^{2}–4 × 10^{3}(z + 1)^{2} K.

Ar^{15+} A weak nondipole transition line, due to 1s2p^{2} ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2}, was selected for this ion. Figure 2b demonstrates that the background of the present calculation agrees well with the results of Goett & Sampson (1983). Strong resonances appear as expected for forbidden transitions, which significantly enhances the effective collision strengths over the temperature range 10^{3}(z + 1)^{2}–10^{5}(z + 1)^{2} K. We also find strong Auger and radiation damping effects on the Ω/Υ for this transition, that will be discussed in detail later. At higher temperatures, the resonance contribution to the present Rmatrix results becomes negligible, which leadsto good agreement with those of Goett & Sampson (1983). In a comparison with previous Rmatrix data (Whiteford et al. 2002), our results with Augerplusradiation damping also show a good agreement over the entire temperature range, see Fig. 3b. A complete dataset from Whiteford et al. (2002) is available from the OPENADAS database^{5}. Thus we have made an extensive comparison (all coreexcitation data from the 1s^{2}2l states) with them at the temperature (5.0 × 10^{6} K) of peak Ar^{15+} fractional abundance in equilibrium (Bryans et al. 2009), as well as at a lower (5.0 × 10^{5} K) and a higher (2.0 × 10^{7} K) temperature. In this comparison, we adopt the configuration, total angular momentum, and energy ordering as the “good” quantum numbers to match the transitions from the two different calculations. Recall, Whiteford et al. (2002) did not include any n = 4 states. We find that the Υ results for 73% of coreexcitations agree to within 20% at T_{e} = 5.0 × 10^{6} K. This slightly large difference between the two calculations with same method, is attributed to the different atomic models (n = 3 vs. n = 4). There are 88% of coreexcitations to n = 2 levels showing agreement to within 20%, whereas the percentage is about 70% for coreexcitations to n = 3 levels. This is consistent with the expectation that the influence of n = 4 states is stronger for the coreexcitations to n = 3 than to n = 2. At the low and high temperatures, the percentage is 59% and 78%, respectively.

Fe^{23+} A strong transition (the r line), due to 1s2s2p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2}, was selected for this ion. The ordinary collision strength Ω of Goett & Sampson (1983) shows a good agreement with the background of the present ICFT Rmatrix calculation, as well as the present BreitPauli DW calculation using autostructure (ASDW), see Fig. 2. For this ion, earlier Rmatrix excitation data is available (Ballance et al. 2001; Whiteford et al. 2002). The present radiation damped calculation agrees well with the smallerscale calculation by Ballance et al. (2001, see Fig. 7a in their work). Their corresponding Maxwellianaveraged Υ is also in agreement with the present Augerplusradiation damping results and with those likewise of Whiteford et al. (2002). Due to the weak resonance contribution for this excitation, the calculation from the CoulombBorn exchange method (GS83) also agrees well with the result of Rmatrix calculation, see Fig. 3. A complete dataset of Whiteford et al. (2002) is available from the OPENADAS database^{5}. Thus we make an extensive comparison as just done for Ar^{15+} at three temperatures (2.0 × 10^{6} K, 2.0 × 10^{7} K and 1.0 × 10^{8} K). The percentage agreements are 68%, 78% and 81% at the low, middle (corresponding to peak abundance) and high temperatures, respectively.

Kr^{33+} Another satellite line (the k line in Table 3) is selected to test the accuracy of the present calculation. Figure 2 displays that the background of the present ICFT Rmatrix results is slightly lower than the CoulombBorn exchange results of Goett & Sampson (1983) by ~10%. This mirrors the reduction of the line strength demonstrated in Table 3. The resulting effective collision strengths agree with those from the CoulombBorn exchange approach to within ~10% over the entire temperature range, which is due to the scarcity of resonances for this satellite (k) line following damping. Augerplusradiation damping is significant for this line (see Fig. 2) and so the final damped resonance contribution is not significant for the effective collision strength at any temperature.
From the above comparison for the four specified ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence, we believe that the present ICFT Rmatrix results (Ω and Υ) are reliable. For ions near neutral (below O^{5+}), Rmatrix with pseudostates calculations are needed to consider ionization loss in the excitation, but the present are the best data to be made available todate for these innershell transitions.
Fig. 4 Collision strengths Ω from the present ICFT Rmatrix calculation and DW data (Zhang et al. 1990) for the dipole 1s^{2}4p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2} transition (left) and nondipole 1s^{2}4d ^{2}D_{3/2} → 1s^{2}2s ^{2}S_{1/2} transition (right) of ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence. Note: The scattered electron energy is in unit of z^{2} Ryd (z is the residual charge). [Colour online] 
4.2. Comparisons with previous calculations for valenceexcitations
As mentioned in the introduction, most available valenceexcitation data is from the (relativistic) distortedwave method (Zhang et al. 1990). We select one dipole transition line (1s^{2}4p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2}) and one nondipole transition line (1s^{2}4d ^{2}D_{3/2} → 1s^{2}2s ^{2}S_{1/2}) to test the accuracy of the present ICFT Rmatrix calculations, see Fig. 4. The background of the collision strength Ω agrees well with the distortedwave calculation for ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence.
For the cosmic abundant ion Fe^{23+}, various Rmatrix calculations including BreitPauli (Berrington & Tully 1997), ICFT (Whiteford et al. 2002) and darc (McKeown 2005) are available. In the results of Whiteford et al. (2002), McKeown (2005, see Figs. 7.13–7.16) noticed that there are strong resonances, resonance shifts and background enhancement around thresholds for some outershell transitions when compared with her darc calculation. The ordinary and effective collision strengths for 1s^{2}4d ^{2}D_{3/2} → 1s^{2}4d ^{2}D_{5/2} are shown in Fig. 5 to illustrate this problem. A hardcopy comparison of Ω indicates that the present ICFT Rmatrix calculation agrees well with the darc calculation by McKeown (2005) – electronic results for McKeown (2005) are not available – which is confirmed by the effective collision strength Υ, see the inset panel. The numerical problem in the work of Whiteford et al. (2002) has been traced to the treatment of the innerregion exchange integrals for highL partial waves. The problem and solution is discussed in detail by Berrington (2006) and was incorporated into the BreitPauli Rmatrix codes at the time (2006) – the ICFT method uses the standard innerregion BreitPauli codes. Moreover, an extensive comparison (see Table 4^{6}) has been made for transitions with a large disagreement as reported by McKeown (2005) and they show good agreement between the present ICFT and darc calculations. This demonstrates that the present results for outershell excitations are reliable for diagnostic modelling application.
Fig. 5 Collision strength Ω for the 1s^{2}4d ^{2}D_{3/2} → 1s^{2}4d ^{2}D_{5/2} excitation, as well as the comparison of resultant effective collision strength Υ with previous works of McKeown (2005) using darc Rmatrix suite code and Whiteford et al. (2002). [Colour online] 
A complete set of effective collision strengths for O^{5+} obtained using darc (Aggarwal & Keenan 2004a) is available electronically. Comparison with the present results shows agreement to within 20% for 97% of all excitations at the temperature (T_{e} = 3.2 × 10^{5} K) of peak fractional abundance in equilibrium (Bryans et al. 2009). This provides further support that the present results are reliable over the sequence. For ions near neutral, viz. C^{3+}, Griffin et al. (2000) performed an LScoupling Rmatrix with 32 pseudostates (5s, 5p, 5d, 5f ··· 12s, 12p, 12d and 12f) calculation to determine ionization loss in the valenceexcitations, and found the reduction of the effective collision strengths to be less than 10% for excitations up to n = 3. For excitations to the n = 4 shell, the pseudostate results are typically 20% smaller, with one transition (2s−4d) being 30%. However, the present are the best Jresolved data to be made available todate.
Finally, we note that dipole transitions between closelyspaced levels, e.g. 5f_{j} − 5g_{j′}, are dominated by contributions from high angular momentum and these come mainly from the CoulombBethe “topup”. The topup is inversely proportional to the energy separation. Our sequence work makes use of calculated energies. This can give rise to large errors in the topup. This has no practical consequences. The excitation rates are very large and so, along with proton collisions, establish statistical values for these level populations independently of the precise value of the excitation rate.
4.3. Damping effects along the sequence
As shown in Fig. 2, radiation damping significantly reduces the resonance strength of innershell transitions. The inclusion of Auger damping further reduces the resonance strength. Correspondingly, the resonance enhancement of the effective collision strengths is reduced, see Fig. 3. For some strong excitations, the resonance contribution to the Υ decreases to the level of 10%, e.g. the 1s^{2}2s ^{2}S_{1/2} → 1s2s2p ^{2}P_{1/2} excitation line of Fe^{23+} shown in Fig. 3. However, for some weak transition lines, the damped resonance contribution to the effective collision strength is still nonnegligible and leads to significantly higher results than those without resonances (see Fig. 3 for Ar^{15+}), e.g. the work of Goett & Sampson (1983) using the CoulombBorn exchange method.
In the innershell excitations for the Nalike isoelectronic sequence, Liang et al. (2009b) clearly demonstrated that radiation damping increases steadily with increasing of nuclear charge Z, but the Auger damping effect still plays an important role on the reduction of the resonance enhancement of Υ over the isoelectronic sequence.
Here we illustrate the radiation and Augerplusradiation damping effects along the sequence by a scatter plot of the ratios of damped to undamped (or only radiation damped) Υ at T_{e} = 10^{4}(z + 1)^{2} K for dipole transitions of O^{5+} (Fig. 6a), Ar^{15+} (Fig. 6c), Fe^{23+} (Fig. 6e) and Kr^{33+} (Fig. 6g). The widespread effect of the radiation and Augerplusradiation damping effects is illustrated. For the lowcharge ion (O^{5+}) the radiation damping is small, being less than 10% for 99% of all dipole coreexcitations at the temperature of 10^{4}(z + 1)^{2} K. The Auger damping is the prominent factor for the reduction of resonance enhancement of Υ. It can up to a factor of 5 for a few dipole transitions. For Ar^{15+}, the radiation damping is less than 10% for most (94%) of the dipole transitions. The Auger damping is stronger and larger than a factor of 2 for 30% of the illustrated dipole transitions. For Fe^{23+}, the radiation damping increases, but the Auger damping is still the dominant and stronger damping effect in the reduction of resonance enhancement of Υ. About 30% of dipole transitions show an Auger damping reduction of over a factor of 2 when compared with the radiation damped Υ. For highercharge ions, e.g. Kr^{33+}, the radiation damping is greater than 30% for 10% of the dipole transitions. But, Fig. 6g demonstrates that Auger damping is still the dominant resonance damping reduction of Υs at T_{e} = 10^{4}(z + 1)^{2} K. Additionally, we notice that there are a few weak transitions with the Υ_{R}/Υ_{U} ratio being slightly larger than unity. This was found to be due to the low resolution of the undamped resonances. Recall, damping both broadens and reduces the height of resonance profiles.
Fig. 6 Lefthand panels: scatter plots showing the ratio of the effective collision strength Υ with Augerplusradiation damping (Υ_{A + R}) or radiation damping (Υ_{R}) to without damping (Υ_{U}) or with radiation damping alone, as a function of line strength for dipole coreexcitation of O^{5+}a), Ar^{15+}c), Fe^{23+}e) and Kr^{33+}g) at temperature of 10^{4}(z + 1)^{2} K, where z is the ionic charge. Righthand panels: percentage of the corresponding transitions where the effect of damping exceeds 10%, 20% and 30%. [ Colour online] 
An illustrative way to quantify the information in the scatter plot is to count how many transitions differ by more than a given value. In Figs. 6b, d, f and h, we show the percentage of the dipole transitions where the Augerplusradiation damping effect, or radiation or Auger damping effects alone, are at least 10%, 20% and 30% for the four ions. Here, 0%, < 1%, 3% and 10% of dipole transitions show a radiation damping effect of more than 30% at T_{e} = 10^{4}(z + 1)^{2} K for O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}, respectively. The percentage for highercharge ions, e.g. Fe^{23+} and Kr^{33+}, is higher than that of lowercharge ions. This illustrates that the radiation damping is more widespread for highercharge ions, as one would expect. There are about 36%, 43%, 41% and 37% of dipole transitions showing a further Auger damping of more than 30% at the temperature of 10^{4}(z + 1)^{2} for O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}, respectively. This means that the Auger damping is still the dominant damping effect over the entire sequence.
4.4. Resonant enhancement for coreexcitations
The Augerplusradiation damping effect significantly reduces the resonance enhancement of the effective collision strength for coreexcitations. Does this mean that one can use nonresonant excitation data such as DW? Here we investigate it statistically by comparing distortedwave (ASDW) and Rmatrix (ICFT) calculations for Fe^{23+}. The exact same atomic structure was used for the ASDW calculation as has been described and used in the Rmatrix calculation. This eliminates differences in the collision data due to the use of different atomic structures. The scatter plot in Fig. 7 demonstrates that the resonant enhancement can be larger than a factor of 2 for some coreexcitations. The percentage is about 66% and 59% of all coreexcitations at T_{e} = 2.0 × 10^{5} and 2.0 × 10^{7} K, respectively. So a complete sequence calculation with resonances is still necessary. The nonresonant DW results could be supplemented by resonances calculated perturbatively. This is just the complement of dielectronic recombination. It is beyond the scope of the present work.
Fig. 7 The ratio of the effective collision strengths Υ for all coreexcitations in Fe^{23+} between the present ICFT Rmatrix and BreitPauli DW (autostructure) calculations at the temperature of 2.0 × 10^{5} and 2.0 × 10^{7} K. The filled and dashed regions correspond to a agreement of 20% and a factor 2, respectively. [Colour online] 
4.5. Isoelectronic trends
As noted in the previous sequence works (Witthoeft et al. 2007; Liang et al. 2009b; 2010), the level mixing effect for higher excited levels strongly affects the behaviour of the Υ along the sequence. Here, we also take configuration, total angular momentum J and energy ordering for level matching in the comparison between different calculations and the investigation of Υ along the isoelectronic sequence. This satisfactorily eliminates uncertainty originating from the noncontinuity of levelordering along the sequence. The choice of reference ion, Fe here, is irrelevant of course.
Fig. 8 Effective collision strengths Υ at temperatures of T_{e} = 5 × 10^{2}(z + 1)^{2}, 10^{3}, 10^{4} K (here, z = Z − 3) along the isoelectronic sequence. Lefthand panels: 1s2s2p ^{4}P_{3/2} → 1s^{2}2s ^{2}S_{1/2} (u line), 1s2s2p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2} (r line) and 1s2s2p ^{4}P_{1/2} → 1s^{2}2p ^{2}P_{3/2} (326) transitions. Righthand panels: 1s2p^{2} ^{4}P_{1/2} → 1s^{2}2p ^{2}P_{3/2} (h line), 1s2p^{2} ^{2}D_{3/2} → 1s^{2}2p ^{2}P_{1/2} (k line) and 1s2s^{2} ^{2}S_{1/2} → 1s^{2}2p ^{2}P_{3/2} (325) transitions. [Colour online] 
In Fig. 8, we show the effective collision strength Υ at T_{e}/(z + 1)^{2} = 5 × 10^{2}, 10^{3} and 10^{4} K along the sequence for a few satellite lines in Lilike ions: at the low temperature of 5 × 10^{2}(z + 1)^{2} K, spikes and/or dips are observed at low charges for some transitions, e.g. 1s^{2}2p ^{2}P_{3/2} → 1s2s2p ^{4}P_{1/2} (3 − 26). With increasing temperature, the spikes and/or dips disappear, as expected, because the resonance contribution becomes weaker and eventually negligible. For Kshell excitations, the behaviour of Υ along the sequence differs from the Lshell, for example Flike (Witthoeft et al. 2007) and Nelike sequences (Liang & Badnell 2010). The irregularity appears only for the lowcharged ions. Above Z ~ 14, the effective collision strengths show a smooth behaviour. This is due to the high coreexcitation energy. KMn resonances are all positioned well above threshold while KLn can only Auger to the finalstate for highn at highcharge and so there is little variation. Only at lowcharge do lown KLn resonances come into play. The “precise” nvalue can be estimated from the Rydberg formula: , where f is the final state and j is an intermediate parent resonant state to which the Rydberg series of resonances converges. For example, for O^{5+} (z = 5) from Table 2 we have E_{fj} = 2.36 Ryd for the 40 → 25 (j → f) coreAuger, giving n = 4 as the lowest KLn resonance for any excitation transition i − 25. The variation in n and energy spacing is relatively largest for the smallest nvalues. This variation can then be seen in the Maxwellaveraged results.
5. Summary
We have performed 195level (24level) ICFT Rmatrix calculations the of core (valence) electronimpact excitation of all ions of the Lilike isoelectronic sequence from Be^{+} to Kr^{33+}.
Good agreement with the available experimentally derived data and the results of others for level energies and line strengths S for several specific ions (O^{5+}, Ar^{15+}, Fe^{23+}, and Kr^{33+}) spanning the isoelectronic sequence supports the reliability of the present Rmatrix excitation data. This was confirmed specifically by detailed comparisons (including previously available Rmatrix calculations) of Ω and Υ for O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}.
A problem for some Fe^{23+} outershell excitation transitions in the earlier ICFT calculation by Whiteford et al. (2002) was checkedfor in the present calculations by comparison with the fully relativistic darc calculation by McKeown (2005) and not found. It had been solved by a previous correction (Berrington 2006) to the innerregion BreitPauli codes which the ICFT method makes use of.
The present Rmatrix excitation data is expected to be an important improvement on the current data (from CoulombBorn exchange approximation) extensively used by the spectroscopic diagnostic modelling communities in astrophysics and magnetic fusion.
The Augerplusradiation damping effect along the sequence was examined, it is significant and widespread over the entire sequence and moreso for the highercharge ions. For some innershell transitions (39% of available DW data to 1s^{2}2l levels in Ar^{15+}), the damped effective collision strengths are still larger than those without the inclusion of resonances (by 20%). The Auger damping effect was found to be dominant in the reduction of resonance enhancement on the electronimpact excitation over the entire sequence, whereas the radiation damping is small for lowercharge ions but increases with increasing nuclear charge.
By excluding the level crossing effects on the Υ, we examined the isoelectronic trends of the effective collision strengths. A complicated pattern of spikes and dips of Υ at low temperatures was noted again along the sequence for some transitions with strong resonances, which precludes the generality of interpolation in Z. With increasing temperature, the resonance effects decrease as expected. Such irregular effects are only seen at lowcharges for innershell transitions since lown resonances cannot straddle thresholds in highcharge ions.
The data are made available in the ADAS adf04 format (Summers 2004) at the archives of the APAP^{1}, OPENADAS^{5} and and will be included in CHIANTI^{7} database.
In conclusion, we have generated an extensive set of reliable excitation data, utilizing the ICFT Rmatrix method, for spectroscopy/diagnostic research within the astrophysical and fusion communities. This will replace data from DW and CoulombBorn exchange approaches presently used by these communities and its use can be expected to identify new lines and may overcome some shortcomings in present astrophysical modelling, as we have seen for Mg^{8+} (Del Zanna et al. 2008), Fe^{6+}, Fe^{7+} and Fe^{10+} (Del Zanna 2009a,b, 2010), Si^{9+} (Liang et al. 2009c) and Fe^{7+}–Fe^{15+} (Liang & Zhao 2010).
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.
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All Tables
ThomasFermi potential scaling factors used by autostructure for the outershell calculations (see text for details).
Comparisons of the level energies (Ryd) of 1s^{2} {2,3,4,5} l (l ∈ s,p,d,f and g), 1s2l^{2} and 1s2s2p configurations in O^{5+}, Ar^{15+}, Fe^{23+}, and Kr^{33+} ions.
Comparisons of line strengths S for Kshell transitions in ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning sequence.
All Figures
Fig. 1 Comparison of the line strength (S) of outershell dipole transitions for ions spanning the sequence. MAK04 corresponds to the grasp calculation done by McKeown et al. (2004b). The horizontal dashed lines correspond to agreement within 5%. [Colour online] 

In the text 
Fig. 2 Comparison of collision strengths Ω from the present ICFT Rmatrix calculation in with those of Goett & Sampson (1983). a) O^{5+}: 1s^{2}2s ^{2}S_{1/2} → 1s2s2p ^{4}P_{3/2} excitation (u line in Table 3); b) Ar^{15+}: 1s^{2}2s ^{2}S_{1/2} → 1s2p^{2} ^{2}P_{1/2} excitation (137, ID is as list in Table 2); c) Fe^{23+}: 1s^{2}2s ^{2}S_{1/2} → 1s2s2p ^{2}P_{1/2} excitation (r line in Table 3). ASDW corresponds to the present BreitPauli distortedwave calculation using autostructure; d) Kr^{33+}: 1s^{2}2p ^{2}P_{1/2} → 1s2p^{2} ^{2}D_{3/2} excitation (k line in Table 3). [Colour online] 

In the text 
Fig. 3 Comparison of the effective collision strength Υ between the present ICFT Rmatrix calculation and previous results of Goett & Sampson (1983) and Whiteford et al. (2002). a) O^{5+}: 1s^{2}2s → 1s2s2p ; b) Ar^{15}: 1s^{2}2s → 1s2p^{2}; c) Fe^{23+}: 1s^{2}2s → 1s2s2p ; d) Kr^{33+}: 1s^{2}2p → 1s2p^{2}. [Colour online] 

In the text 
Fig. 4 Collision strengths Ω from the present ICFT Rmatrix calculation and DW data (Zhang et al. 1990) for the dipole 1s^{2}4p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2} transition (left) and nondipole 1s^{2}4d ^{2}D_{3/2} → 1s^{2}2s ^{2}S_{1/2} transition (right) of ions (O^{5+}, Ar^{15+}, Fe^{23+} and Kr^{33+}) spanning the sequence. Note: The scattered electron energy is in unit of z^{2} Ryd (z is the residual charge). [Colour online] 

In the text 
Fig. 5 Collision strength Ω for the 1s^{2}4d ^{2}D_{3/2} → 1s^{2}4d ^{2}D_{5/2} excitation, as well as the comparison of resultant effective collision strength Υ with previous works of McKeown (2005) using darc Rmatrix suite code and Whiteford et al. (2002). [Colour online] 

In the text 
Fig. 6 Lefthand panels: scatter plots showing the ratio of the effective collision strength Υ with Augerplusradiation damping (Υ_{A + R}) or radiation damping (Υ_{R}) to without damping (Υ_{U}) or with radiation damping alone, as a function of line strength for dipole coreexcitation of O^{5+}a), Ar^{15+}c), Fe^{23+}e) and Kr^{33+}g) at temperature of 10^{4}(z + 1)^{2} K, where z is the ionic charge. Righthand panels: percentage of the corresponding transitions where the effect of damping exceeds 10%, 20% and 30%. [ Colour online] 

In the text 
Fig. 7 The ratio of the effective collision strengths Υ for all coreexcitations in Fe^{23+} between the present ICFT Rmatrix and BreitPauli DW (autostructure) calculations at the temperature of 2.0 × 10^{5} and 2.0 × 10^{7} K. The filled and dashed regions correspond to a agreement of 20% and a factor 2, respectively. [Colour online] 

In the text 
Fig. 8 Effective collision strengths Υ at temperatures of T_{e} = 5 × 10^{2}(z + 1)^{2}, 10^{3}, 10^{4} K (here, z = Z − 3) along the isoelectronic sequence. Lefthand panels: 1s2s2p ^{4}P_{3/2} → 1s^{2}2s ^{2}S_{1/2} (u line), 1s2s2p ^{2}P_{1/2} → 1s^{2}2s ^{2}S_{1/2} (r line) and 1s2s2p ^{4}P_{1/2} → 1s^{2}2p ^{2}P_{3/2} (326) transitions. Righthand panels: 1s2p^{2} ^{4}P_{1/2} → 1s^{2}2p ^{2}P_{3/2} (h line), 1s2p^{2} ^{2}D_{3/2} → 1s^{2}2p ^{2}P_{1/2} (k line) and 1s2s^{2} ^{2}S_{1/2} → 1s^{2}2p ^{2}P_{3/2} (325) transitions. [Colour online] 

In the text 
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