Electron impact excitation for Helike ions with Z = 20–42^{⋆}
^{1} Division of Mathematical Physics, Department of PhysicsLund University, Box 118, 221 00 Lund, Sweden
email: tomas.brage@fysik.lu.se
^{2} Shanghai EBIT Lab, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, 200433 Shanghai, PR China
email: chychen@fudan.edu.cn
^{3} Institute of Applied Physics and Computational Mathematics, 100088 Beijing, PR China
^{4} Hebei Key Lab of Opticelectronic Information and Materials, The College of Physics Science and Technology, Hebei University, 071002 Baoding, PR China
^{5} Department of Radiotherapy, Shanghai Changhai Hospital, Second Military Medical University, 200433 Shanghai, PR China
^{6} College of Science, National University of Defense Technology, 410073 Changsha, PR China
Received: 8 November 2016
Accepted: 8 January 2017
Aims. Spectral lines of Helike ions are among the most prominent features in Xray spectra from a large variety of astrophysical and hightemperature fusion plasmas. A reliable plasma modeling and interpretation of the spectra require a large amount of accurate atomic data related to various physical processes. In this paper, we focus on the electronimpact excitation (EIE) process.
Methods. We adopted the independent process and isolated resonances approximation using distorted waves (IPIRDW). Resonant stabilizing transitions and decays to lowerlying autoionizing levels from the resonances are included as radiative damping. To verify the applicability of the IPIRDW approximation, an independent Dirac Rmatrix calculation was also performed. The two sets of results show excellent agreement.
Results. We report electron impact excitation collision strengths for transitions among the lowest 49 levels of the 1snl(n ≤ 5,l ≤ (n−1)) configurations in Helike ions with 20 ≤ Z ≤ 42. The line ratios R and G are calculated for Fe XXV and Kr XXXV.
Conclusions. Compared to previous theoretical calculations, our IPIRDW calculation treats resonance excitation and radiative damping effects more comprehensively, and the resulting line emission cross sections show good agreement with the experimental observations. Our results should facilitate the modeling and diagnostics of various astrophysical and laboratory plasmas.
Key words: atomic data / atomic processes
Full Table 1 is only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/600/A85
© ESO, 2017
1. Introduction
Helike ions are abundant over a wide temperature range in astrophysical and laboratory plasmas because of their closedshell ground state. Emission lines from the spectra of heliumlike ions are often observed in the spectra of solar, stellar, and other astrophysical plasmas (Seely & Feldman 1985; Feldman et al. 2000; Dere et al. 2001; Ness et al. 2003; Paerels & Kahn 2003; Phillips 2004; Landi & Phillips 2005; Güdel & Nazé 2009; 2010). Spectral lines of heliumlike ions are also prominent features in the Xray spectra of tokamak and laserproduced plasmas (Hsuan et al. 1987; Rice et al. 1987, 1999, 2014, 2015; Beiersdorfer et al. 1995). An analysis of spectral lines provides information on the temperature, density, and chemical composition of the plasma. For example, the line intensity ratios G(T_{e}) = (z + x + y) /w and R(n_{e}) = z/ (x + y) of the four prominent xray transitions w(1sS_{0}−1s2p^{1}P_{1}), x(1sS_{0}−1s2p^{3}P_{2}), y(1sS_{0}−1s2p^{3}P_{1}) and z(1sS_{0}−1s2s^{3}S_{1}) are useful tools in the diagnostics of the plasma density and temperature (Gabriel & Jordan 1969a,b; Gabriel 1972; Porquet & Dubau 2000; Porquet et al. 2001, 2010. These diagnostics have been widely used for solar plasmas (Doschek & Meekins 1970; Doyle 1980; McKenzie et al. 1980; Pradhan & Shull 1981; McKenzie & Landecker 1982; Wolfson et al. 1983; Keenan et al. 1984, 1987; Doyle & Keenan 1986) and tokamak plasmas (Doyle & Schwob 1982; Källne et al. 1983; Keenan et al. 1989). Reliable line interpretation and plasma modeling require a large amount of accurate atomic data, including energy levels, radiative rates, and collisional rate coefficients related to states up to the n = 5 configurations (Porquet et al. 2010; Kallman & Palmeri 2007; Smith & Brickhouse 2014; Beiersdorfer 2015).
In our recent work (Si et al. 2016), we provided energy levels for the 1snl(n ≤ 6,l ≤ (n−1)) and 2ln′l′(n′ ≤ 6,l′ ≤ (n′−1)) configurations of Helike ions with Z = 10−36, as well as the radiative rates for all electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), and magnetic quadrupole (M2) transitions among these levels, by using the secondorder manybody perturbation theory (MBPT) implemented in the flexible atomic code (FAC; Gu 2008). The accuracy of the MBPT level energies is expected to be a few tens of a ppm, the line strengths for strong transitions among singly excited levels, and their lifetimes are assessed to be accurate to within 1%. We here mainly focus on the electronimpact excitation (EIE) process of Helike ions as a continuation of our EIE study of Kshell ions (Chen et al. 2010; Li et al. 2015).
Many calculations on electronimpact excitation of Helike ions have been published a few decades ago (Sampson et al. 1983; Pradhan 1983, 1985; Tayal & Kingston 1984, 1985; Zhang & Sampson 1987; Nakazaki et al. 1993). Most of the more recent studies have used the Rmatrix theory. For example, Griffin & Ballance (2009) performed radiatively damped Dirac Rmatrix (DRM) calculations of the electronimpact excitations for all transitions between the 49 lowest levels of 1snl (n ≤ 5, l ≤ (n−1)) configurations for Fe^{24+} and Kr^{34+}, but only provided the results for excitations from the ground state to the first 30 excited levels. Electronimpact excitation collision strengths for the transitions between the 49 lowest levels of Ar^{16+} and Fe^{24+} were carried out using a radiationdamped intermediate coupling frame transformation (ICFT) Rmatrix approach by Whiteford et al. (2001), another set of ICFT Rmatrix results with Z = 6−36 was also posted on the UK APAP website (Whiteford 2005). However, the background cross sections from this semirelativistic approach were found to be about 10% lower than the fully relativistic results (Malespin et al. 2011). Additionally, Whiteford et al. (2001, 2005) only included the resonant stabilizing (RS) damping source, but ignored decays from the resonances into lowlying autoionizing levels that could be followed by autoionization cascades (DAC). Aggarwal et al. (2005, 2008, 2009, 2010, 2011, 2012a–d, 2013a, b) provided DRM electronimpact excitation collision strengths among the 49 lowest levels for Helike ions with Z = 3−36 (except for Ne IX), but discarded all the radiative damping effects. Furthermore, both Whiteford et al. and Aggarwal et al. ignored resonance excitation contributions from the 1s6ln′l′ states, which we will show contribute significantly to the collision strengths for transitions to and within the n = 5 levels. Moreover, although there have been many calculations based on the Rmatrix theory, they often exhibit large discrepancies among themselves, even if they use the same Rmatrix code. It is therefore necessary to treat the resonance excitation and radiative damping effects more comprehensively, and it is very useful to apply another independent theory to assess the accuracy for various Rmatrix results.
In addition to the Rmatrix approach in which the resonances and the interactions among them are naturally taken into account, the resonances can be treated with a completely different method, namely the independent processes and isolated resonances approximation using distortedwaves (denoted the IPIRDW approximation). In general, the interference effects, for instance, between resonances or between resonances and continua, are ignored in the IPIRDW approximation. However, the method gives results in good agreement with those from Rmatrix for most of the highly charged ions (Gu 2004; Chen et al. 2010; Wang et al. 2010, 2011). More importantly, various physical processes can be taken into account separately in the IPIRDW approach, which facilitates studying their contributions. To our knowledge, there are no electronimpact excitation calculations on Helike ions based on the IPIRDW theory. This justifies the systematic study of electronimpact excitation we present here.
The FAC package (Gu 2008) is adopted throughout this work. First we calculate two smallscale IPIRDW and DRM to further verify the applicability of the former approximation, in which the target expansion includes the seven lowest levels among the 1snl (n ≤ 2, l ≤ (n−1)) configurations. These two sets of electronimpact excitation collision strengths show excellent agreement. Then we carry out a largescale IPIRDW calculation with a more comprehensive treatment of resonance excitation and radiative damping effects, in which the target expansion includes the 49 lowest levels of 1snl (n ≤ 5, l ≤ (n−1)) configurations. Extensive comparisons are made with experimental and some other theoretical values to assess the quality and reliability of our final IPIRDW values.
2. Calculation
FAC (Gu 2008) is a fully relativistic program computing both structure and scattering data. The atomic structure can be obtained using the relativistic configuration interaction (CI) method or the MBPT approach. The basic wavefunctions are derived from a local central potential, which is selfconsistently determined to represent electronic screening of the nuclear potential. Relativistic effects are taken into account using the Dirac Coulomb Hamiltonian. Breit interaction in the zero energy limit for the exchanged photon and hydrogenic approximations for selfenergy and vacuum polarization effects are also included. The CI wavefunctions can then be used to obtain the scattering data using the IPIRDW approximation or DRM theory.
Within the IPIRDW approximation, contributions from direct excitation (DE) and resonance excitation (RE) to the total EIE rate coefficients are obtained independently. DE collision strengths are straightforward to calculate employing the relativistic distortedwave (RDW) approximation. The DE cross section σ_{ij} (in unit of cm^{2}) from the initial state i to the final state j can be expressed in terms of the collision strength Ω_{ij} as (1)where g_{i} is the statistical weight of the initial state, a_{0} is the Bohr radius, and k_{i} is the relativistic kinetic momentum of the incident electron, which is related to the incident energy E_{i} (in units of Ry) by (2)where α is the fine structure constant. DE effective collision strengths (Υ) are obtained after integrating Ω over a Maxwellian distribution of electron velocities, (3)where E_{f} is the scattered electron energy, k is the Boltzmann constant, and T_{e} is the electron temperature in K.
RE contributions to collision strengths are included using the IPIRDW approximation, (4)where E_{id} is the resonant energy, is the Auger rate from d to i, is the Auger decay branching ratio from state d to state j, and g_{d} is the statistical weight of state d. The plasma RE rate coefficients for transition from level i to j of Helike ion are obtained by the summation of the contribution through individual autoionizing level d of Lilike ions, (5)The corresponding effective collision strength can be obtained from (6)where E_{ij} is the energy difference between levels i and j.
3. Results and discussions
3.1. Smallscale exploratory calculations
In this section, we take Fe^{24+} and Kr^{34+} as examples and perform two smallscale calculations using both the IPIRDW and DRM methods, to verify the applicability of the IPIRDW approximation.
The theoretical basis of the DRM method was described in Chang (1977), and a numerical implementation was developed by Norrington & Grant (1987). Gu (2004) has included a reimplementation of the same theory within the FAC package.
In the present DRM calculation, the seven lowest levels of the 1s^{2}, 1s2s, and 1s2p configurations are included in both the CI expansion of the target and the closecoupling expansion of the subsequent scattering calculations. As in the nonrelativistic case (Burke et al. 1971), the configuration space is partitioned into two regions that are separated by the Rmatrix boundary r_{0}. r_{0} is chosen such that the exchange between the incident and target electrons is negligible when r>r_{0}. The present r_{0} is chosen to enclose the 1s, 2s, and 2p orbital wavefunctions with an amplitude larger than 10^{6}. In the inner region (r ≤ r_{0}), the exchange effects are taken into account for l< 12, whereas they are discarded in the outer region. Partial waves up to l = 70 and 30 radial basis functions per partial wave are included in the Rmatrix calculation. To map out the fine details of the resonance structure, a mesh of 0.001 eV is used in the resonance range, while collision strengths are calculated with a coarse mesh of 10 eV up to three times the maximum threshold energy. A topup procedure is often used to obtain the convergence of collision strengths at high energies. This is not applied here since we focus on the resonance contribution and close coupling effects. The comparison with our IPIRDW calculation shows that the present DRM Υ values are converged at a temperature of up to 10^{7.8} K for dipoleallowed transitions and at a temperature of up to 10^{8.3} K for forbidden transitions.
Fig. 1 Comparison of the IPIRDW and DRM cross sections and effective collision strengths (Υ) for line z (1 ^{1}S_{0}−2 ^{3}S_{1}) of Fe^{24+}a) and Kr^{34+}b). 

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Fig. 2 Comparison of the IPIRDW and DRM effective collision strengths (Υ) for lines y (1 ^{1}S_{0}−2 ^{3}P_{1}), x (1 ^{1}S_{0}−2 ^{3}P_{2}), and w (1 ^{1}S_{0}−2 ^{1}P_{1}) transitions of Fe^{24+}a) and Kr^{34+}b). 

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In order to assess the channel coupling effects, we carried out a samescale IPIRDW calculation. The target expansion also includes the seven lowest levels. To ensure the convergence of the DE collision strengths, we set the maximum of orbital angular momentum (l) for the partialwave expansion to 100. Higher partialwave contributions are included using the CoulombBethe approximation (Burgess et al. 1970; Burgess & Sheorey 1974). The partial waves with l> 10 are treated in a quasirelativistic approximation (Zhang et al. 1989). The DE collision strengths are calculated at eight scattered electron energies, that is, (Z−0.75)^{2} Ry times 0.001, 0.002, 0.01, 0.04, 0.12, 0.30, 0.75, 2.25. RE contributions through the relevant Lilike doubly excited configurations 1s2ln′l′ (l ≤ 1,n′ ≤ 75,l′ ≤ 8) are included using the IPIRDW approximation.
Figure 1 shows the comparison of the IPIRDW and DRM cross sections and effective collision strengths for line z in Fe^{24+} and Kr^{34+}. The cross sections are convoluted with a 2.35 eV Gaussian. As shown in the figure, both the background (DE) cross sections and resonance structures from the IPIRDW and DRM calculations are in good agreement. The resulting effective collision strengths agree within 10%. For the other three important lines y, x, and w, two sets of effective collision strengths also agree very well, as Fig. 2 shows.
The excellent agreement between our DRM and IPIRDW results shows that the interference effects in highly charged Helike ions is negligible, which gives us confidence to move to the next largescale IPIRDW calculation.
3.2. Largescale IPIRDW results
In this section, we present electronimpact excitation effective collision strengths between all the singly excited levels of 1snl(n ≤ 5,l ≤ (n−1)) configurations for Helike ions with Z = 20−42 over a wide temperature range from 10^{3} × (Z−1)^{2} K to 2 × 10^{6} × (Z−1)^{2} K in Table 1. For the sake of completeness, the energy differences ΔE and transition rates A are also listed. However, we replace the CI values with our more elaborate MBPT results (Si et al. 2016) and extend the calculation to Z = 42.
Transition energy differences ΔE (eV), transition rates A_{ji} (s^{1}), and effective collision strengths Υ for transitions j−i of Helike ions with Z = 20−42.
We calculate DE collision strengths up to the scattered electron energy of (Z−0.75)^{2} × 2.25 Ry as in Sect. 3.1. However, the fractional abundance of Helike ions peaks at about 5 × 10^{4} × (Z−1)^{2} K (Bryans et al. 2006), and therefore population modeling up to 10^{6} × (Z−1)^{2} K is usually needed. Thus the Ω values at high energy up to hundreds of keV are required to obtain convergence of the hightemperature Υ values, especially for dipoleallowed transitions. However, it is computationally challenging and overambitious to explicitly calculate the collision strength at such a high energy with both the distortedwaves and the Rmatrix approaches. In general, Bethe’s form (Bethe 1930; Inokuti 1971) of the Born approximation is sufficient and was employed instead. In modern Rmatrix calculations (for example, see Whiteford et al. 2001), the Cplot scaling method (Burgess & Tully 1992) for dipoleallowed transitions and the extension work (Burgess et al. 1997) for highenergy limits of the Born approximation for dipoleforbidden transitions are usually used to estimate the needed highenergy collision strengths.
Fig. 3 Reduced excitation cross sections(see text) of Fe^{24+} for some dipoleallowed and forbidden transitions, plotted against ln [ β^{2}/ (1−β^{2}) ] −β^{2}. The scattered symbols are connected with spline functions. a) For resonance E1 transitions, b) for intercombination E1 transitions, c) for nondipole electric multipole transitions, and d) for pure magnetic multipole transitions. 

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However, the above approximations for the highenergy collision strengths do not include relativistic effects. When the velocity of the incident electron increases, eventually approaching relativistic energies, the corresponding modification of the cross section should be included (Fano 1963). According to the discussions of Inokuti (1971) and Bartiromo et al. (1985), we may define the reduced cross section as (7)where m_{e} is the rest mass of the electron, v_{i} is the velocity of incident electron, and E_{ij} is the excitation energy. In the relativistic region, a Fanoplot of the reduced cross sections Q_{ij} for dipoleallowed transition against ln [ β^{2}/ (1−β^{2}) ] −β^{2} (in which β = v_{i}/c, and c is the light velocity) will become a straight line, with a slope that corresponds to the optical oscillator strength f_{ij}. For dipoleforbidden transition, Q_{ij}, on the other hand, it will become nearly a constant against ln [ β^{2}/ (1−β^{2}) ] −β^{2} (Inokuti 1971). We thus have (8)for dipoleallowed transitions, and (9)for dipoleforbidden transitions. The parameters A and B could be estimated from the Fanoplot or obtained directly from the relativistic planewaves approximation (Fontes & Zhang 2007).
For instance, we show some Fanoplots for both dipoleallowed and forbidden transitions of Fe^{24+} in Fig. 3. As described before, we calculate the cross sections (or collision strengths) at eight scattered energies, employing the RDW approximation. We also compute the highenergy cross sections at additional three scattered energies of (Z−0.75)^{2} Ry times 10, 30, and 100, employing the relativistic planewaves (RPW) approximation. The results from the RDW and RPW calculations are connected by spline functions. It can be seen that the RDW results can be connected smoothly with the RPW values, and the relativistic asymptotic behavior (Eqs. (8) and (9)) of the reduced cross sections is reached. We linearly extrapolated the collision strength nearby threshold and used Eqs. (8) and (9) to extrapolate the cross section at scattered energy above (Z−0.75)^{2} Ry times 300 for the Maxwellian integration. In the scattered electron energy range of (Z−0.75)^{2} Ry times 0.001 to 300, the Ω values are interpolated by splines.
The RE contributions through the relevant Lilike doubly excited configurations 1s2l′n′′l′′ (l′ ≤ 1,n′′ ≤ 75,l′′ ≤ 8) and 1sn′l′n′′l′′ (n′ ≤ 6,l′ ≤ (n′−1), n′′ ≤ 50,l′′ ≤ 8) are included. The higher n′′ contributions are included up to n′′ = 200 by using the n^{3} scaling law (Shen et al. 2007a,b). For Helike ions, the electron correlations among 1snl (n ≤ 5,l ≤ (n−1)) and 1s6l (l ≤ 5) configurations are considered. For Lilike ions, configuration interaction within the same complex are taken into account. The RS damping transitions from the doubly excited configurations 1sn′l′n′′l′′ toward 1s^{2}n′′l′′ and 1s^{2}n′l′ are taken into account. All possible DAC transitions n′l′ → n^{′′′}l^{′′′} (n^{′′′}<n′) and n′′l′′ → n^{′′′}l^{′′′} (n^{′′′}< 10) are also included. We here only considered E1 transitions, but all possible autoionization channels of the doubly excited states were taken into account.
As discussed in our recent work (Shen et al. 2007a,b, 2009; Zhang et al. 2009; Chen et al. 2010; Wang et al. 2011, 2012; Li et al. 2015), when we use different approaches to include the radiative decay processes of the autoionizing state d, the calculated Auger decay branching ratio and the subsequent RE rate coefficients will differ. When we disregard all the radiative transitions, as is done in our smallscale calculation in Sect. 3.1 and most earlier Rmatrix calculations, we obtain the undamped RE rate coefficients. These rates could be radiatively damped by the RS transitions, taking additionally the DAC transitions into account, the RE rates will be further changed. Figure 4 shows the effect of RS and DAC on effective collision strengths at the temperature of 10^{4}(Z−1)^{2} K along the isoelectronic sequence. The RS and DAC effects both increase with increasing Z, and the DAC effect is stronger than RS. For Mo^{40+}, with the inclusion of the RS and DAC dampings, about 7% and 35% of the effective collision strengths are reduced by more than 10% at the temperature of 10^{4}(Z−1)^{2} K, respectively. The maximum RE damping effect is about − 45%, and the maximum DAC damping effect is about − 55%. The Υ values as a function of electron temperature with different treatments of for 2 ^{1}P_{1}−3 ^{1}S_{0} and 2 ^{1}P_{1}−3 ^{3}S_{1} of Fe^{24+} are shown in Fig. 5 as examples. The damping effects are usually stronger at low temperatures where RE enhancements may play an important role, the Υ values could be reduced by more than 90% at maximum. For the Υ values of 2 ^{1}P_{1}−3 ^{1}S_{0}, both RS and DAC have a significant reducing effect. Although the effect of RS damping for the Υ values of 2 ^{1}P_{1}−3 ^{3}S_{1} is weak, DAC reduces them dramatically.
Fig. 4 Effect of RS (top) and DAC (bottom) radiative dampings on effective collision strengths at the temperature of 10^{4}(Z−1)^{2} K along the isoelectronic sequence. 

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Fig. 5 Effects of radiative damping for 2 ^{1}P_{1}−3 ^{1}S_{0}a) and 2 ^{1}P_{1}−3 ^{3}S_{1}b) in Fe^{24+}. 

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We included the RE contribution from 1s6ln′l′ levels here, in contrast to the previous theoretical works. This inclusion shows significant enhancement over the Υ values of transitions to and within the 1s5l configurations, the largest enhancement is up to two orders of magnitude for such as 1 ^{1}S_{0}−5 ^{1}G_{4} and 1 ^{1}S_{0}−5 ^{3}G_{3,4,5}. At the temperature of 10^{4}(Z−1)^{2} K, 60% and 45% of the transitions to and within the 1s5l configurations for Ca^{18+} and Mo^{40+} are enlarged by a factor of more than 10%, respectively.
Fig. 6 Various Υ values for 1 ^{1}S_{0}−4 ^{3}F_{4}a) and 3 ^{3}D_{2}−3 ^{3}D_{3}b) in Fe^{24+}. 

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3.3. Comparison with other theoretical results
Since the ICFT Rmatrix calculations performed by Whiteford et al. (2001, 2005) did not include the DAC effect and RE contributions from 1s6ln′l′ levels, we removed the above contributions from our IPIRDW results before the comparison. The resulted IPIRDW Υ values and those from Whiteford et al. (2001, 2005) show excellent agreement in the medium temperature range, but show poor agreement at lower and higher temperatures. Only about 10% of the Υ values at the high temperature end agree within 20%, the deviations are mainly due to the different treatments of the highenergy collision strengths, as stated in Sect. 3.2. Our IPIRDW Υ values at the lowtemperature end generally agree with those from Whiteford et al. (2001) to within a factor of 2, but the differences are higher by up to a factor of 8 for some forbidden transitions to or within the 1s4l levels, for example, for the Υ values for 1 ^{1}S_{0}−4 ^{3}F_{4} of Fe^{24+} that we show in Fig. 6a. The Υ values from Whiteford et al. (2001) differ greatly from our DE+RE results, but closely agree with our DE results. We therefore attribute this deviation to them not including the RE contribution for this type of transitions, and the later set of values from Whiteford (2005) included the RE contribution successfully for these transitions. Some of the effective collision strengths from Whiteford (2005) still differ from the present results by up to 2 orders of magnitude, however. Most of these are for forbidden transitions within the same complex, such as 3 ^{3}D_{2}−3 ^{3}D_{3} of Fe^{24+} in Fig. 6b. As stated in Aggarwal & Keenan (2013a), the differences are not due to the resonances, but arise from the limitation of the approach adopted by Whiteford (2005). Additionally, Figs. 6a and b shows that our IPIRDW results agree very well with those from Aggarwal & Keenan (2012b, 2013a). With increasing Z, the differences between the present results and those from Whiteford (2005) generally increase (see Fig. 7). This is due to the approach adopted by Whiteford et al. (2001, 2005), which is performed in the LS coupling scheme. The applicability of this model will be reduced with increasing Z.
Fig. 7 Differences of Υ values at the temperature of T_{e} = 10^{4}(Z−1)^{2} K from our IPIRDW calculation and earlier Rmatrix calculations. The solid lines show percentage differences with those from Whiteford (2005). The open lines show percentage differences with those from Aggarwal et al. (2012a, 2012b, 2012d, 2013a, b). Comparisons are made on the basis of the same computed model. 

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As Aggarwal et al. (2012a, b, d, 2013a, b) did not include any radiative damping or the RE contribution from 1s6ln′l′, the above contributions in our IPIRDW values were also removed to verify the same computational model. The results show good agreement with values from Aggarwal et al. (2012a, b, d, 2013a, b) throughout the temperature range of their calculations, and the difference decreases with increasing Z for most transitions, as can be seen in Fig. 7. However, for some forbidden transitions within the n = 4 complex of highZ ions, values from Aggarwal et al. (2012a, b, d, 2013a, b) are higher than our results by nearly a factor of three. Figure 8 shows various Υ values for the 4 ^{3}P_{2}−4 ^{3}F_{3} transition as a function of Z at the temperature of 10^{4}(Z−1)^{2} K. The effective collision strengths vary smoothly along the isoelectronic sequence, except for the values from Aggarwal et al. (2012a, b, d, 2013a, b); the unusual behavior of the Υ values from Aggarwal et al. (2012a, b, d, 2013a, b) in the highZ end are probably responsible for the deviations.
Fig. 8 Υ values from Aggarwal et al. (2012a, 2012b, 2012d, 2013a,b), Whiteford (2005), and our IPIRDW calculation at the temperature of 10^{4}(Z−1)^{2} K for 4 ^{3}P_{2}−4 ^{3}F_{3} along the isoelectronic sequence. 

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It should be mentioned that the above comparisons are made on the basis of the same model. Comparisons between our final IPIRDW results and the above Rmatrix values are also made, and show poorer agreement in the lowtemperature range. Some of the differences are due to the different treatment of radiative damping, as shown in Fig. 9a. Some of the differences result from the inclusion of the RE contribution from 1s6ln′l′, as can be seen in Fig. 9b.
Fig. 9 Various Υ values for 4 ^{3}S_{1}−4 ^{1}S_{0}a) and 5 ^{3}S_{1}−5 ^{3}F_{4}b) in Kr^{34+}. 

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3.4. Comparison with experimental observations
To our knowledge, electron impact excitation experimental measurements are rare (Chen & Beiersdorfer 2008). Chantrenne et al. (1992) obtained the measurement of electron impact excitation cross sections as a function of energy for w, x, y, and z lines of Ti^{20+}. The experimental measurements have been extensively compared with theoretical values (Chantrenne et al. 1992; Gorczyca et al. 1995; Zhang & Pradhan 1995). However, because of the insufficient treatment of resonance excitation, radiation damping, and radiative cascade effects, the above mentioned theoretical results are not entirely within the experimental error bars, especially in the highenergy region. Here by considering more sufficient resonance excitation and radiation damping effects as discussed above, and including more radiative cascade contributions from highlying levels (all the levels of 1snl(n ≤ 5,l ≤ (n−1)) configurations), the obtained line emission cross sections are compared with the experimental observations (Chantrenne et al. 1992) in Fig. 10. The good agreement over the entire energy range is obvious, which confirms the reliability of the present results.
Fig. 10 Comparison of our cross sections (solid lines) with the experimental values (circles with error bars) for lines w (1 ^{1}S_{0}−2 ^{3}S_{1}), y (1 ^{1}S_{0}−2 ^{3}P_{1}), x (1 ^{1}S_{0}−2 ^{3}P_{2}), w (1 ^{1}S_{0}−2 ^{1}P_{1}) transitions of Ti^{20+}. 

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3.5. R and G ratios of Fe^{24+} and Kr^{34+}
It is well known that the line intensity ratios G(T_{e}) = (z + x + y) /w and R(n_{e}) = z/ (x + y) are useful tools in the diagnostics of the plasmas density and temperature (Gabriel & Jordan 1969a,b; Gabriel 1972; Porquet & Dubau 2000; Porquet et al. 2001, 2010). We performed several calculations for Fe^{24+} and Kr^{34+} to show the effects of resonance excitation and radiative damping on G(T_{e}) and R(n_{e}) ratios, employing the collisional radiative model. Collisional excitation and deexcitation as well as spontaneous radiative transitions among the 49 lowest levels are included in the model. Using our radiative and collisional atomic data in conjunction with the statistical equilibrium code of Dufton (1977), we can obtain relative level populations and emissionline intensities. The resulting G(T_{e}) and R(n_{e}) ratios are shown in Fig. 11. The G(T_{e}) ratio at the lowtemperature end is increased by about 50% by the resonance excitations and is lowered by radiative damping by about 20%. The inclusion of resonance excitations raises the R(n_{e}) ratio at the lowdensity end by about 20%, the radiative damping lowers it by a factor of 10%.
Fig. 11 G(T_{e}) for Fe XXV a) and Kr XXXV b) at n_{e} = 10^{17} cm^{3} and n_{e} = 10^{18} cm^{3}. R(n_{e}) for Fe XXV c) and Kr XXXV d) at T_{e} = 10^{7} K and T_{e} = 3.162 × 10^{7} K. 

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4. Conclusion
We have presented collision strengths and effective collision strengths for all transitions among the 49 lowest levels belonging to the 1snl(n ≤ 5,l ≤ (n−1)) configurations of Helike ions with Z = 20−42, employing the IPIRDW approximation.
DE collision strengths were calculated employing the RDW approximation in conjunction with the RPW approximation. The relativistic asymptotic behaviors were used to evaluate the highenergy collision strengths. Resonances attached to the 1snln′l′(n ≤ 6) levels were taken into account by the IPIRDW approach. Inclusion of the RS and the DAC radiative damping significantly reduced the total effective collision strengths at low electron temperatures for a number of transitions, especially for highZ ions. Resonances attached to the 1s6l levels were found to significantly enhance the effective collision strengths for transitions to and within the 1s5l levels at low temperatures. The resulting line emission cross sections are in excellent agreement with the experimental values. Compared to the previously reported theoretical works, we present a more comprehensive and accurate set of results. Our data are expected to be helpful in plasma modeling and diagnostics.
Acknowledgments
The authors acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 11674066, 11474034, 11374062 and 11504421), the project was funded by the China Scholarship Council (Grant No. 201608130201), the China Postdoctoral Science Foundation (Grant No. 2016M593019) and the Swedish Research Council (Grant No. 201504842). R. Si would especially like to acknowledge the International Exchange Program Fund for Doctorate Students of Fudan University Graduate School. K. Wang, S. Li, and X. L. Guo express their gratitude for the support from the visiting researcher program at Fudan University.
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All Tables
Transition energy differences ΔE (eV), transition rates A_{ji} (s^{1}), and effective collision strengths Υ for transitions j−i of Helike ions with Z = 20−42.
All Figures
Fig. 1 Comparison of the IPIRDW and DRM cross sections and effective collision strengths (Υ) for line z (1 ^{1}S_{0}−2 ^{3}S_{1}) of Fe^{24+}a) and Kr^{34+}b). 

Open with DEXTER  
In the text 
Fig. 2 Comparison of the IPIRDW and DRM effective collision strengths (Υ) for lines y (1 ^{1}S_{0}−2 ^{3}P_{1}), x (1 ^{1}S_{0}−2 ^{3}P_{2}), and w (1 ^{1}S_{0}−2 ^{1}P_{1}) transitions of Fe^{24+}a) and Kr^{34+}b). 

Open with DEXTER  
In the text 
Fig. 3 Reduced excitation cross sections(see text) of Fe^{24+} for some dipoleallowed and forbidden transitions, plotted against ln [ β^{2}/ (1−β^{2}) ] −β^{2}. The scattered symbols are connected with spline functions. a) For resonance E1 transitions, b) for intercombination E1 transitions, c) for nondipole electric multipole transitions, and d) for pure magnetic multipole transitions. 

Open with DEXTER  
In the text 
Fig. 4 Effect of RS (top) and DAC (bottom) radiative dampings on effective collision strengths at the temperature of 10^{4}(Z−1)^{2} K along the isoelectronic sequence. 

Open with DEXTER  
In the text 
Fig. 5 Effects of radiative damping for 2 ^{1}P_{1}−3 ^{1}S_{0}a) and 2 ^{1}P_{1}−3 ^{3}S_{1}b) in Fe^{24+}. 

Open with DEXTER  
In the text 
Fig. 6 Various Υ values for 1 ^{1}S_{0}−4 ^{3}F_{4}a) and 3 ^{3}D_{2}−3 ^{3}D_{3}b) in Fe^{24+}. 

Open with DEXTER  
In the text 
Fig. 7 Differences of Υ values at the temperature of T_{e} = 10^{4}(Z−1)^{2} K from our IPIRDW calculation and earlier Rmatrix calculations. The solid lines show percentage differences with those from Whiteford (2005). The open lines show percentage differences with those from Aggarwal et al. (2012a, 2012b, 2012d, 2013a, b). Comparisons are made on the basis of the same computed model. 

Open with DEXTER  
In the text 
Fig. 8 Υ values from Aggarwal et al. (2012a, 2012b, 2012d, 2013a,b), Whiteford (2005), and our IPIRDW calculation at the temperature of 10^{4}(Z−1)^{2} K for 4 ^{3}P_{2}−4 ^{3}F_{3} along the isoelectronic sequence. 

Open with DEXTER  
In the text 
Fig. 9 Various Υ values for 4 ^{3}S_{1}−4 ^{1}S_{0}a) and 5 ^{3}S_{1}−5 ^{3}F_{4}b) in Kr^{34+}. 

Open with DEXTER  
In the text 
Fig. 10 Comparison of our cross sections (solid lines) with the experimental values (circles with error bars) for lines w (1 ^{1}S_{0}−2 ^{3}S_{1}), y (1 ^{1}S_{0}−2 ^{3}P_{1}), x (1 ^{1}S_{0}−2 ^{3}P_{2}), w (1 ^{1}S_{0}−2 ^{1}P_{1}) transitions of Ti^{20+}. 

Open with DEXTER  
In the text 
Fig. 11 G(T_{e}) for Fe XXV a) and Kr XXXV b) at n_{e} = 10^{17} cm^{3} and n_{e} = 10^{18} cm^{3}. R(n_{e}) for Fe XXV c) and Kr XXXV d) at T_{e} = 10^{7} K and T_{e} = 3.162 × 10^{7} K. 

Open with DEXTER  
In the text 