EDP Sciences
Free Access
Issue
A&A
Volume 600, April 2017
Article Number A85
Number of page(s) 10
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/201630027
Published online 05 April 2017

© ESO, 2017

1. Introduction

He-like ions are abundant over a wide temperature range in astrophysical and laboratory plasmas because of their closed-shell ground state. Emission lines from the spectra of helium-like 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 helium-like ions are also prominent features in the X-ray spectra of tokamak and laser-produced 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(Te) = (z + x + y) /w and R(ne) = z/ (x + y) of the four prominent x-ray transitions w(1sS0−1s2p1P1), x(1sS0−1s2p3P2), y(1sS0−1s2p3P1) and z(1sS0−1s2s3S1) 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 2lnl(n′ ≤ 6,l′ ≤ (n′−1)) configurations of He-like 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 second-order many-body 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 electron-impact excitation (EIE) process of He-like ions as a continuation of our EIE study of K-shell ions (Chen et al. 2010; Li et al. 2015).

Many calculations on electron-impact excitation of He-like 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 R-matrix theory. For example, Griffin & Ballance (2009) performed radiatively damped Dirac R-matrix (DRM) calculations of the electron-impact excitations for all transitions between the 49 lowest levels of 1snl (n ≤ 5, l ≤ (n−1)) configurations for Fe24+ and Kr34+, but only provided the results for excitations from the ground state to the first 30 excited levels. Electron-impact excitation collision strengths for the transitions between the 49 lowest levels of Ar16+ and Fe24+ were carried out using a radiation-damped intermediate coupling frame transformation (ICFT) R-matrix approach by Whiteford et al. (2001), another set of ICFT R-matrix results with Z = 6−36 was also posted on the UK APAP website (Whiteford 2005). However, the background cross sections from this semi-relativistic 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 low-lying autoionizing levels that could be followed by autoionization cascades (DAC). Aggarwal et al. (2005, 2008, 2009, 2010, 2011, 2012a–d, 2013a, b) provided DRM electron-impact excitation collision strengths among the 49 lowest levels for He-like 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 1s6lnl 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 R-matrix theory, they often exhibit large discrepancies among themselves, even if they use the same R-matrix 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 R-matrix results.

In addition to the R-matrix 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 distorted-waves (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 R-matrix 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 electron-impact excitation calculations on He-like ions based on the IPIRDW theory. This justifies the systematic study of electron-impact excitation we present here.

The FAC package (Gu 2008) is adopted throughout this work. First we calculate two small-scale 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 electron-impact excitation collision strengths show excellent agreement. Then we carry out a large-scale 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 self-consistently 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 self-energy 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 distorted-wave (RDW) approximation. The DE cross section σij (in unit of cm2) from the initial state i to the final state j can be expressed in terms of the collision strength Ωij as (1)where gi is the statistical weight of the initial state, a0 is the Bohr radius, and ki is the relativistic kinetic momentum of the incident electron, which is related to the incident energy Ei (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 Ef is the scattered electron energy, k is the Boltzmann constant, and Te is the electron temperature in K.

RE contributions to collision strengths are included using the IPIRDW approximation, (4)where Eid 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 gd is the statistical weight of state d. The plasma RE rate coefficients for transition from level i to j of He-like ion are obtained by the summation of the contribution through individual autoionizing level d of Li-like ions, (5)The corresponding effective collision strength can be obtained from (6)where Eij is the energy difference between levels i and j.

3. Results and discussions

3.1. Small-scale exploratory calculations

In this section, we take Fe24+ and Kr34+ as examples and perform two small-scale 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 1s2, 1s2s, and 1s2p configurations are included in both the CI expansion of the target and the close-coupling 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 R-matrix boundary r0. r0 is chosen such that the exchange between the incident and target electrons is negligible when r>r0. The present r0 is chosen to enclose the 1s, 2s, and 2p orbital wavefunctions with an amplitude larger than 10-6. In the inner region (rr0), 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 R-matrix 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 top-up 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 107.8 K for dipole-allowed transitions and at a temperature of up to 108.3 K for forbidden transitions.

thumbnail Fig. 1

Comparison of the IPIRDW and DRM cross sections and effective collision strengths (Υ) for line z (1 1S0−2 3S1) of Fe24+a) and Kr34+b).

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thumbnail Fig. 2

Comparison of the IPIRDW and DRM effective collision strengths (Υ) for lines y (1 1S0−2 3P1), x (1 1S0−2 3P2), and w (1 1S0−2 1P1) transitions of Fe24+a) and Kr34+b).

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In order to assess the channel coupling effects, we carried out a same-scale 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 partial-wave expansion to 100. Higher partial-wave contributions are included using the Coulomb-Bethe approximation (Burgess et al. 1970; Burgess & Sheorey 1974). The partial waves with l> 10 are treated in a quasi-relativistic 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 Li-like doubly excited configurations 1s2lnl (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 Fe24+ and Kr34+. 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 He-like ions is negligible, which gives us confidence to move to the next large-scale IPIRDW calculation.

3.2. Large-scale IPIRDW results

In this section, we present electron-impact excitation effective collision strengths between all the singly excited levels of 1snl(n ≤ 5,l ≤ (n−1)) configurations for He-like ions with Z = 20−42 over a wide temperature range from 103 × (Z−1)2 K to 2 × 106 × (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.

Table 1

Transition energy differences ΔE (eV), transition rates Aji (s-1), and effective collision strengths Υ for transitions ji of He-like 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 He-like ions peaks at about 5 × 104 × (Z−1)2 K (Bryans et al. 2006), and therefore population modeling up to 106 × (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 high-temperature Υ values, especially for dipole-allowed transitions. However, it is computationally challenging and overambitious to explicitly calculate the collision strength at such a high energy with both the distorted-waves and the R-matrix approaches. In general, Bethe’s form (Bethe 1930; Inokuti 1971) of the Born approximation is sufficient and was employed instead. In modern R-matrix calculations (for example, see Whiteford et al. 2001), the C-plot scaling method (Burgess & Tully 1992) for dipole-allowed transitions and the extension work (Burgess et al. 1997) for high-energy limits of the Born approximation for dipole-forbidden transitions are usually used to estimate the needed high-energy collision strengths.

thumbnail Fig. 3

Reduced excitation cross sections(see text) of Fe24+ for some dipole-allowed and forbidden transitions, plotted against ln [ β2/ (1−β2) ] −β2. The scattered symbols are connected with spline functions. a) For resonance E1 transitions, b) for intercombination E1 transitions, c) for non-dipole electric multipole transitions, and d) for pure magnetic multipole transitions.

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However, the above approximations for the high-energy 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 me is the rest mass of the electron, vi is the velocity of incident electron, and Eij is the excitation energy. In the relativistic region, a Fano-plot of the reduced cross sections Qij for dipole-allowed transition against ln [ β2/ (1−β2) ] −β2 (in which β = vi/c, and c is the light velocity) will become a straight line, with a slope that corresponds to the optical oscillator strength fij. For dipole-forbidden transition, Qij, on the other hand, it will become nearly a constant against ln [ β2/ (1−β2) ] −β2 (Inokuti 1971). We thus have (8)for dipole-allowed transitions, and (9)for dipole-forbidden transitions. The parameters A and B could be estimated from the Fano-plot or obtained directly from the relativistic plane-waves approximation (Fontes & Zhang 2007).

For instance, we show some Fano-plots for both dipole-allowed and forbidden transitions of Fe24+ 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 high-energy cross sections at additional three scattered energies of (Z−0.75)2 Ry times 10, 30, and 100, employing the relativistic plane-waves (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 Li-like doubly excited configurations 1s2ln′′l′′ (l′ ≤ 1,n′′ ≤ 75,l′′ ≤ 8) and 1snln′′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 He-like ions, the electron correlations among 1snl (n ≤ 5,l ≤ (n−1)) and 1s6l (l ≤ 5) configurations are considered. For Li-like ions, configuration interaction within the same complex are taken into account. The RS damping transitions from the doubly excited configurations 1snln′′l′′ toward 1s2n′′l′′ and 1s2nl are taken into account. All possible DAC transitions nl′ → 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 small-scale calculation in Sect. 3.1 and most earlier R-matrix 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 104(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 Mo40+, 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 104(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 1P1−3 1S0 and 2 1P1−3 3S1 of Fe24+ 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 1P1−3 1S0, both RS and DAC have a significant reducing effect. Although the effect of RS damping for the Υ values of 2 1P1−3 3S1 is weak, DAC reduces them dramatically.

thumbnail Fig. 4

Effect of RS (top) and DAC (bottom) radiative dampings on effective collision strengths at the temperature of 104(Z−1)2 K along the isoelectronic sequence.

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thumbnail Fig. 5

Effects of radiative damping for 2 1P1−3 1S0a) and 2 1P1−3 3S1b) in Fe24+.

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We included the RE contribution from 1s6lnl 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 1S0−5 1G4 and 1 1S0−5 3G3,4,5. At the temperature of 104(Z−1)2 K, 60% and 45% of the transitions to and within the 1s5l configurations for Ca18+ and Mo40+ are enlarged by a factor of more than 10%, respectively.

thumbnail Fig. 6

Various Υ values for 1 1S0−4 3F4a) and 3 3D2−3 3D3b) in Fe24+.

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3.3. Comparison with other theoretical results

Since the ICFT R-matrix calculations performed by Whiteford et al. (2001, 2005) did not include the DAC effect and RE contributions from 1s6lnl 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 high-energy collision strengths, as stated in Sect. 3.2. Our IPIRDW Υ values at the low-temperature 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 1S0−4 3F4 of Fe24+ 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 3D2−3 3D3 of Fe24+ 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.

thumbnail Fig. 7

Differences of Υ values at the temperature of Te = 104(Z−1)2 K from our IPIRDW calculation and earlier R-matrix 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 1s6lnl, 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 high-Z 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 3P2−4 3F3 transition as a function of Z at the temperature of 104(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 high-Z end are probably responsible for the deviations.

thumbnail Fig. 8

Υ values from Aggarwal et al. (2012a, 2012b, 2012d, 2013a,b), Whiteford (2005), and our IPIRDW calculation at the temperature of 104(Z−1)2 K for 4 3P2−4 3F3 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 R-matrix values are also made, and show poorer agreement in the low-temperature 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 1s6lnl, as can be seen in Fig. 9b.

thumbnail Fig. 9

Various Υ values for 4 3S1−4 1S0a) and 5 3S1−5 3F4b) in Kr34+.

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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 Ti20+. 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 high-energy region. Here by considering more sufficient resonance excitation and radiation damping effects as discussed above, and including more radiative cascade contributions from high-lying 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.

thumbnail Fig. 10

Comparison of our cross sections (solid lines) with the experimental values (circles with error bars) for lines w (1 1S0−2 3S1), y (1 1S0−2 3P1), x (1 1S0−2 3P2), w (1 1S0−2 1P1) transitions of Ti20+.

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3.5. R and G ratios of Fe24+ and Kr34+

It is well known that the line intensity ratios G(Te) = (z + x + y) /w and R(ne) = 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 Fe24+ and Kr34+ to show the effects of resonance excitation and radiative damping on G(Te) and R(ne) 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 emission-line intensities. The resulting G(Te) and R(ne) ratios are shown in Fig. 11. The G(Te) ratio at the low-temperature 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(ne) ratio at the low-density end by about 20%, the radiative damping lowers it by a factor of 10%.

thumbnail Fig. 11

G(Te) for Fe XXV a) and Kr XXXV b) at ne = 1017 cm-3 and ne = 1018 cm-3. R(ne) for Fe XXV c) and Kr XXXV d) at Te = 107 K and Te = 3.162 × 107 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 He-like 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 high-energy collision strengths. Resonances attached to the 1snlnl(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 high-Z 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. 2015-04842). 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.

References

All Tables

Table 1

Transition energy differences ΔE (eV), transition rates Aji (s-1), and effective collision strengths Υ for transitions ji of He-like ions with Z = 20−42.

All Figures

thumbnail Fig. 1

Comparison of the IPIRDW and DRM cross sections and effective collision strengths (Υ) for line z (1 1S0−2 3S1) of Fe24+a) and Kr34+b).

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In the text
thumbnail Fig. 2

Comparison of the IPIRDW and DRM effective collision strengths (Υ) for lines y (1 1S0−2 3P1), x (1 1S0−2 3P2), and w (1 1S0−2 1P1) transitions of Fe24+a) and Kr34+b).

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In the text
thumbnail Fig. 3

Reduced excitation cross sections(see text) of Fe24+ for some dipole-allowed and forbidden transitions, plotted against ln [ β2/ (1−β2) ] −β2. The scattered symbols are connected with spline functions. a) For resonance E1 transitions, b) for intercombination E1 transitions, c) for non-dipole electric multipole transitions, and d) for pure magnetic multipole transitions.

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In the text
thumbnail Fig. 4

Effect of RS (top) and DAC (bottom) radiative dampings on effective collision strengths at the temperature of 104(Z−1)2 K along the isoelectronic sequence.

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In the text
thumbnail Fig. 5

Effects of radiative damping for 2 1P1−3 1S0a) and 2 1P1−3 3S1b) in Fe24+.

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In the text
thumbnail Fig. 6

Various Υ values for 1 1S0−4 3F4a) and 3 3D2−3 3D3b) in Fe24+.

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In the text
thumbnail Fig. 7

Differences of Υ values at the temperature of Te = 104(Z−1)2 K from our IPIRDW calculation and earlier R-matrix 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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In the text
thumbnail Fig. 8

Υ values from Aggarwal et al. (2012a, 2012b, 2012d, 2013a,b), Whiteford (2005), and our IPIRDW calculation at the temperature of 104(Z−1)2 K for 4 3P2−4 3F3 along the isoelectronic sequence.

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In the text
thumbnail Fig. 9

Various Υ values for 4 3S1−4 1S0a) and 5 3S1−5 3F4b) in Kr34+.

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In the text
thumbnail Fig. 10

Comparison of our cross sections (solid lines) with the experimental values (circles with error bars) for lines w (1 1S0−2 3S1), y (1 1S0−2 3P1), x (1 1S0−2 3P2), w (1 1S0−2 1P1) transitions of Ti20+.

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In the text
thumbnail Fig. 11

G(Te) for Fe XXV a) and Kr XXXV b) at ne = 1017 cm-3 and ne = 1018 cm-3. R(ne) for Fe XXV c) and Kr XXXV d) at Te = 107 K and Te = 3.162 × 107 K.

Open with DEXTER
In the text

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