Astronomy. Astrophysics. Electron impact excitation for He-like ions with Z = 20 42

Transcription

1 A&A, A85 (17 DOI: 1.151/-31/137 c ESO 17 Astronomy & Astrophysics Electron impact excitation for He-like ions with Z = R. Si 1,, S. Li 3, K. Wang, X. L. Guo 5, Z. B. Chen, J. Yan 3, C. Y. Chen, T. Brage 1, and Y. M. Zou 1 Division of Mathematical Physics, Department of Physics, Lund University, Box 118, 1 Lund, Sweden tomas.brage@fysik.lu.se Shanghai EBIT Lab, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, 33 Shanghai, PR China chychen@fudan.edu.cn 3 Institute of Applied Physics and Computational Mathematics, 188 Beijing, PR China Hebei Key Lab of Optic-electronic Information and Materials, The College of Physics Science and Technology, Hebei University, 71 Baoding, PR China 5 Department of Radiotherapy, Shanghai Changhai Hospital, Second Military Medical University, 33 Shanghai, PR China College of Science, National University of Defense Technology, 173 Changsha, PR China Received 8 November 1 / Accepted 8 January 17 ABSTRACT Aims. Spectral lines of He-like ions are among the most prominent features in X-ray spectra from a large variety of astrophysical and high-temperature 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 electron-impact excitation (EIE process. Methods. We adopted the independent process and isolated resonances approximation using distorted waves (IPIRDW. Resonant stabilizing transitions and decays to lower-lying autoionizing levels from the resonances are included as radiative damping. To verify the applicability of the IPIRDW approximation, an independent Dirac R-matrix 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 9 levels of the 1snl(n 5, l (n 1 configurations in He-like ions with Z. 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 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. ; Dere et al. 1; Ness et al. 3; Paerels & Kahn 3; Phillips ; Landi & Phillips 5; Güdel & Nazé 9; 1. 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, 1, 15; Beiersdorfer et al An analysis of spectral lines provides information on the temperature, density, and chemical composition of the plasma. For example, the line intensity ratios G( = (z + x + y/w and R(n e = z/(x + y of the four prominent x-ray transitions w(1s 1 S 1sp 1 P 1, x(1s 1 S 1sp 3 P, y(1s 1 S 1sp 3 P 1 and z(1s 1 S 1ss 3 S 1 are useful tools in the diagnostics of the plasma density and temperature (Gabriel & Jordan 199a,b; Gabriel 197; Porquet & Dubau ; Porquet et al. 1, 1. These diagnostics have been widely used for solar plasmas Full Table 1 is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr ( or via (Doschek & Meekins 197; Doyle 198; McKenzie et al. 198; Pradhan & Shull 1981; McKenzie & Landecker 198; Wolfson et al. 1983; Keenan et al. 198, 1987; Doyle & Keenan 198 and tokamak plasmas (Doyle & Schwob 198; Källne et al. 1983; Keenan et al 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. 1; Kallman & Palmeri 7; Smith & Brickhouse 1; Beiersdorfer 15. In our recent work (Si et al. 1, we provided energy levels for the 1snl (n, l (n 1 and ln l (n, l (n 1 configurations of He-like ions with Z = 1 3, as well as the radiative rates for all electric dipole (E1, magnetic dipole (M1, electric quadrupole (E, and magnetic quadrupole (M transitions among these levels, by using the second-order many-body perturbation theory (MBPT implemented in the flexible atomic code (FAC; Gu 8. 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. 1; Li et al. 15. Article published by EDP Sciences A85, page 1 of 1

2 A&A, A85 (17 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 198, 1985; Zhang & Sampson 1987; Nakazaki et al Most of the more recent studies have used the R-matrix theory. For example, Griffin & Ballance (9 performed radiatively damped Dirac R-matrix (DRM calculations of the electron-impact excitations for all transitions between the 9 lowest levels of 1snl (n 5, l (n 1 configurations for Fe + and Kr 3+, but only provided the results for excitations from the ground state to the first 3 excited levels. Electron-impact excitation collision strengths for the transitions between the 9 lowest levels of Ar 1+ and Fe + were carried out using a radiation-damped intermediate coupling frame transformation (ICFT R-matrix approach by Whiteford et al. (1, another set of ICFT R-matrix results with Z = 3 was also posted on the UK APAP website (Whiteford 5. However, the background cross sections from this semi-relativistic approach were found to be about 1% lower than the fully relativistic results (Malespin et al. 11. Additionally, Whiteford et al. (1, 5 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. (5, 8, 9, 1, 11, 1a d, 13a, b provided DRM electron-impact excitation collision strengths among the 9 lowest levels for He-like ions with Z = 3 3 (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 1sln 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 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 ; Chen et al. 1; Wang et al. 1, 11. 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 8 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, 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 9 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.. Calculation FAC (Gu 8 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 ( and resonance excitation (RE to the total EIE rate coefficients are obtained independently. collision strengths are straightforward to calculate employing the relativistic distorted-wave (RDW approximation. The cross section σ i j (in unit of cm from the initial state i to the final state j can be expressed in terms of the collision strength Ω i j as σ i j = πa ki g Ω i j, (1 i where g i is the statistical weight of the initial state, a 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 k i = E i ( 1 + α E i /, ( where α is the fine structure constant. effective collision strengths (Υ are obtained after integrating Ω over a Maxwellian distribution of electron velocities, Υ( = Ω(E f exp( E f /k d(e f /k, (3 where E f is the scattered electron energy, k is the Boltzmann constant, and is the electron temperature in K. RE contributions to collision strengths are included using the IPIRDW approximation, Ω RE i j = πg d A a di BRE d j δ(e E id, ( d where E id is the resonant energy, A a di is the Auger rate from d to i, B RE d j 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 He-like ion are obtained by the summation of the contribution through individual autoionizing level d of Li-like ions, α RE i j (k = (π3/ 3 (m e k 3/ d g d g i A a di BRE d j exp ( E id (5 k The corresponding effective collision strength can be obtained from α i j = Υ exp( E g i Te 1/ i j /k, ( where E i j is the energy difference between levels i and j. A85, page of 1

3 R. Si et al.: Electron impact excitation for He-like ions with Z = 3. Results and discussions 3.1. Small-scale exploratory calculations In this section, we take Fe + and Kr 3+ 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 ( has included a reimplementation of the same theory within the FAC package. In the present DRM calculation, the seven lowest levels of the 1s, 1ss, and 1sp 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 r. r is chosen such that the exchange between the incident and target electrons is negligible when r > r. The present r is chosen to enclose the 1s, s, and p orbital wavefunctions with an amplitude larger than 1. In the inner region (r r, the exchange effects are taken into account for l < 1, whereas they are discarded in the outer region. Partial waves up to l = 7 and 3 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.1 ev is used in the resonance range, while collision strengths are calculated with a coarse mesh of 1 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 K for dipole-allowed transitions and at a temperature of up to K for forbidden transitions. 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 collision strengths, we set the maximum of orbital angular momentum (l for the partial-wave expansion to 1. Higher partial-wave contributions are included using the Coulomb- Bethe approximation (Burgess et al. 197; Burgess & Sheorey 197. The partial waves with l > 1 are treated in a quasirelativistic approximation (Zhang et al The collision strengths are calculated at eight scattered electron energies, that is, (Z.75 Ry times.1,.,.1,.,.1,.3,.75,.5. RE contributions through the relevant Li-like doubly excited configurations 1sln 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 + and Kr 3+. The cross sections are convoluted with a.35 ev Gaussian. As shown in the figure, both the background ( cross sections and resonance structures from the IPIRDW and DRM calculations are in good agreement. The resulting effective collision strengths agree within 1%. For the other three important lines y, x, and w, two sets of effective collision strengths also agree very well, as Fig. 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. Cross section (1 - cm Cross section (1 - cm DRM (K Incident electron energy (ev..1. (a (b (K Incident electron energy (ev DRM Fig. 1. Comparison of the IPIRDW and DRM cross sections and effective collision strengths (Υ for line z (1 1 S 3 S 1 of Fe + a and Kr 3+ b. 3.. 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 = over a wide temperature range from 1 3 (Z 1 K to 1 (Z 1 K in Table 1. For the sake of completeness, the energy differences E and transition rates A are also listed. However, we replace the CI values with our more elaborate MBPT results (Si et al. 1 and extend the calculation to Z =. We calculate collision strengths up to the scattered electron energy of (Z.75.5 Ry as in Sect However, the fractional abundance of He-like ions peaks at about 5 1 (Z 1 K (Bryans et al., and therefore population modeling up to 1 (Z 1 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 193; Inokuti 1971 of the Born approximation is sufficient and was employed instead. In modern R-matrix calculations (for example, see Whiteford et al. 1, the C-plot scaling method (Burgess & Tully 199 for dipole-allowed transitions and the extension work (Burgess et al for high-energy limits of A85, page 3 of 1

4 A&A, A85 (17 Table 1. Transition energy differences E (ev, transition rates A ji (s 1, and effective collision strengths Υ for transitions j i of He-like ions with Z =. Z i Level j Level E A ji S 3 S S 3 3 P S 3 P S 5 1 S S 3 P S 7 1 P S S S P S S Notes. Electron temperatures are in units of (Z 1 K. Transition rate A ji stands for sum of E1, M1, E, and M transition rates for transition j i. The notations are n 1,3 L J 1snl 1,3 L J ; a ± b a 1 ±b. (The full table is available at the CDS. Υ/ (a DRM w (1 1 S - 1 P 1 (b y (1 1 S - 3 P 1 x (1 1 S - 3 P (K DRM w (1 1 S - 1 P 1 y (1 1 S - 3 P 1 x (1 1 S - 3 P (K Fig.. Comparison of the IPIRDW and DRM effective collision strengths (Υ for lines y (1 1 S 3 P 1, x (1 1 S 3 P, and w (1 1 S 1 P 1 transitions of Fe + a and Kr 3+ b. the Born approximation for dipole-forbidden transitions are usually used to estimate the needed high-energy collision strengths. 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 193. According to the discussions of Inokuti (1971 and Bartiromo et al. (1985, we may define the reduced cross section as Q i j (E i = m ev i E i j Ry Ry 1 σ πa i j (E i, (7 where m e is the rest mass of the electron, v i is the velocity of incident electron, and E i j is the excitation energy. In the relativistic region, a Fano-plot of the reduced cross sections Q i j for dipole-allowed transition against ln[β /(1 β ] β (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 i j. For dipole-forbidden transition, Q i j, on the other hand, it will become nearly a constant against ln[β /(1 β ] β (Inokuti We thus have Q i j (E i = f i j { ln[β /(1 β ] β } + A (8 for dipole-allowed transitions, and Q i j (E i B (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 7. For instance, we show some Fano-plots for both dipoleallowed and forbidden transitions of Fe + 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.75 Ry times 1, 3, and 1, 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.75 Ry A85, page of 1

5 R. Si et al.: Electron impact excitation for He-like ions with Z = Reduced cross section Reduced cross section Reduced cross section Reduced cross section 1 8 (a 1-7 (1 1 S - 1 P ( 1 P D 17-31(3 1 D - 1 F ( 1 F G ln[ /(1- ] (b 1-(1 1 S - 3 P ( 1 P D 15-3(3 1 P 1-3 D 17-(3 1 D - 3 F ln[ /(1- ] (c -8 ( 3 S S 1-13( 3 S D -17( 3 S D 3-1( 3 P -3 3 P ln[ /(1- ] (d 1- (1 1 S - 3 S 1 1- (1 1 S - 3 P 3-13 ( 3 P -3 3 D 3-17 ( 1 P -3 1 D ln[ /(1- ]- Fig. 3. Reduced excitation cross sections(see text of Fe + for some dipole-allowed and forbidden transitions, plotted against ln[β /(1 β ] β. 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. times 3 for the Maxwellian integration. In the scattered electron energy range of (Z.75 Ry times.1 to 3, the Ω values are interpolated by splines. The RE contributions through the relevant Li-like doubly excited configurations 1sl n l (l 1, n 75, l 8 and 1sn l n l (n, l (n 1, n 5, l 8 are included. The higher n contributions are included up to n = by using the n 3 scaling law (Shen et al. 7a,b. For He-like ions, the electron correlations among 1snl (n 5, l (n 1 and 1sl (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 1sn l n l toward 1s n l and 1s n l are taken into account. All possible DAC transitions n l n l (n < n and n l n l (n < 1 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. 7a,b, 9; Zhang et al. 9; Chen et al. 1; Wang et al. 11, 1; Li et al. 15, when we use different approaches to include the radiative decay processes of the autoionizing state d, the calculated Auger decay branching ratio B RE d j 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 shows the effect of RS and DAC on effective collision strengths at the temperature of 1 (Z 1 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 +, with the inclusion of the RS and DAC dampings, about 7% and 35% of the effective collision strengths are reduced by more than 1% at the temperature of 1 (Z 1 K, respectively. The maximum RE damping effect is about 5%, and the maximum DAC damping effect is about 55%. The Υ values as a function of electron temperature with different treatments of B RE d j for 1 P S and 1 P S 1 of Fe + 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 9% at maximum. For the Υ values of 1 P S, both RS and DAC have a significant reducing effect. Although the effect of RS damping for the Υ values of 1 P S 1 is weak, DAC reduces them dramatically. We included the RE contribution from 1sln 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 5 1 G and 1 1 S 5 3 G 3,,5. At the temperature of 1 (Z 1 K, % and 5% of the transitions to and within the 1s5l configurations for Ca 18+ and Mo + are enlarged by a factor of more than 1%, respectively Comparison with other theoretical results Since the ICFT R-matrix calculations performed by Whiteford et al. (1, 5 did not include the DAC effect and RE contributions from 1sln 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. (1, 5 show excellent agreement in the medium temperature range, but show poor agreement at lower and higher temperatures. A85, page 5 of 1

6 A&A, A85 (17 +RS / -1 (% (a _RE+RS +RS+DAC Z (K -1 (% +RS+DAC / +RS Z Fig.. Effect of RS (top and DAC (bottom radiative dampings on effective collision strengths at the temperature of 1 (Z 1 K along the isoelectronic sequence. Only about 1% of the Υ values at the high temperature end agree within %, the deviations are mainly due to the different treatments of the high-energy collision strengths, as stated in Sect. 3.. Our IPIRDW Υ values at the low-temperature end generally agree with those from Whiteford et al. (1 to within a factor of, but the differences are higher by up to a factor of 8 for some forbidden transitions to or within the 1sl levels, for example, for the Υ values for 1 1 S 3 F of Fe + that we show in Fig. a. The Υ values from Whiteford et al. (1 differ greatly from our results, but closely agree with our 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 (5 included the RE contribution successfully for these transitions. Some of the effective collision strengths from Whiteford (5 still differ from the present results by up to orders of magnitude, however. Most of these are for forbidden transitions within the same complex, such as 3 3 D 3 3 D 3 of Fe + in Fig. b. As stated in Aggarwal & Keenan (13a, the differences are not due to the resonances, but arise from the limitation of the approach adopted by Whiteford (5. Additionally, Figs. a and b shows that our IPIRDW results agree very well with those from Aggarwal & Keenan (1b, 13a. With increasing Z, the differences between the present results and those from Whiteford (5 generally increase (see Fig. 7. This is due to.1.5. (b _RE+RS +RS+DAC (K Fig. 5. Effects of radiative damping for 1 P S a and 1 P S 1 b in Fe +. the approach adopted by Whiteford et al. (1, 5, which is performed in the LS coupling scheme. The applicability of this model will be reduced with increasing Z. As Aggarwal et al. (1a, b, d, 13a, b did not include any radiative damping or the RE contribution from 1sln 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. (1a, b, d, 13a, 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 = complex of high-z ions, values from Aggarwal et al. (1a, b, d, 13a, b are higher than our results by nearly a factor of three. Figure 8 shows various Υ values for the 3 P 3 F 3 transition as a function of Z at the temperature of 1 (Z 1 K. The effective collision strengths vary smoothly along the isoelectronic sequence, except for the values from Aggarwal et al. (1a, b, d, 13a, b; the unusual behavior of the Υ values from Aggarwal et al. (1a, b, d, 13a, b in the high-z end are probably responsible for the deviations. 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 A85, page of 1

7 R. Si et al.: Electron impact excitation for He-like ions with Z = (a +RS +RS+DAC Whiteford et al.(1 Whiteford (5 Aggarwal & Keenan (13 Difference (% S S 1 1 S P 1 3 S S 1 3 P - 3 P 1 3 P P (K (b +RS +DAC Whiteford et al.(1 Whiteford (5 Aggarwal & Keenan ( (K Atomic Number Fig. 7. Differences of Υ values at the temperature of = 1 (Z 1 K from our IPIRDW calculation and earlier R-matrix calculations. The solid lines show percentage differences with those from Whiteford (5. The open lines show percentage differences with those from Aggarwal et al. (1a, 1b, 1d, 13a, b. Comparisons are made on the basis of the same computed model IPIRDW Whiteford(5 Aggarwal et al. Fig.. Various Υ values for 1 1 S 3 F a and 3 3 D 3 3 D 3 b in Fe +..5 from the inclusion of the RE contribution from 1sln l, as can be seen in Fig. 9b. 3.. Comparison with experimental observations To our knowledge, electron impact excitation experimental measurements are rare (Chen & Beiersdorfer 8. Chantrenne et al. (199 obtained the measurement of electron impact excitation cross sections as a function of energy for w, x, y, and z lines of Ti +. The experimental measurements have been extensively compared with theoretical values (Chantrenne et al. 199; Gorczyca et al. 1995; Zhang & Pradhan 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. 199 in Fig. 1. The good agreement over the entire energy range is obvious, which confirms the reliability of the present results Atomic Number Fig. 8. Υ values from Aggarwal et al. (1a, 1b, 1d, 13a,b, Whiteford (5, and our IPIRDW calculation at the temperature of 1 (Z 1 K for 3 P 3 F 3 along the isoelectronic sequence R and G ratios of Fe + and Kr 3+ It is well known that the line intensity ratios G( = (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 199a,b; Gabriel 197; Porquet & Dubau ; Porquet et al. 1, 1. We performed several calculations for Fe + and Kr 3+ to show the effects of resonance excitation and radiative damping on G( and R(n e ratios, employing the collisional radiative model. Collisional excitation and deexcitation as well as spontaneous radiative transitions among the 9 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( and R(n e ratios are shown in Fig. 11. The G( ratio at the low-temperature end is increased by about 5% by the resonance excitations and is lowered by radiative damping by about %. The inclusion A85, page 7 of 1

8 A&A, A85 (17..1 (a _RS +RS+DAC Whiteford(5 Aggarwal & Keenan (1b Cross section (1 - cm 8 z (1 1 S - 3 S (K Electron Energy(eV 5 y (1 1 S - 3 P (b +RS +RS+DAC Whiteford (5 Aggarwal & Keenan (1b Cross section (1 - cm Electron Energy(eV (K x (1 1 S - 3 P Fig. 9. Various Υ values for 3 S 1 1 S a and 5 3 S F b in Kr 3+. of resonance excitations raises the R(n e ratio at the low-density end by about %, the radiative damping lowers it by a factor of 1%. Cross section (1 - cm. Conclusion We have presented collision strengths and effective collision strengths for all transitions among the 9 lowest levels belonging to the 1snl (n 5, l (n 1 configurations of He-like ions with Z =, employing the IPIRDW approximation. 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 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 1sl 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. Cross section (1 - cm Electron Energy(eV 1 w (1 1 S - 1 P Electron Energy(eV Fig. 1. Comparison of our cross sections (solid lines with the experimental values (circles with error bars for lines w (1 1 S 3 S 1, y (1 1 S 3 P 1, x (1 1 S 3 P, w (1 1 S 1 P 1 transitions of Ti +. A85, page 8 of 1

9 R. Si et al.: Electron impact excitation for He-like ions with Z = G ( (a +RS +RS+DAC Acknowledgements. The authors acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 117, 1173, 1137 and 1151, the project was funded by the China Scholarship Council (Grant No , the China Postdoctoral Science Foundation (Grant No. 1M59319 and the Swedish Research Council (Grant No 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. G ( R (n e R (n e (K (b (K +RS +RS+DAC +RS +DAC n e (cm n e (cm -3 (d (c +RS +RS+DAC Fig. 11. G( for Fe XXV a and Kr XXXV b at n e = 1 17 cm 3 and n e = 1 18 cm 3. R(n e for Fe XXV c and Kr XXXV d at = 1 7 K and = K. References Aggarwal, K. M., & Keenan, F. P. 8, A&A, 89, 1377 Aggarwal, K. M., & Keenan, F. P. 5, A&A, 1, 831 Aggarwal, K. M., & Keenan, F. P. 1, Phys. Scr., 8, 53 Aggarwal, K. M., & Keenan, F. P. 1a, Phys. Scr., 85, 53 Aggarwal, K. M., & Keenan, F. P. 1b, Phys. Scr., 8, 353 Aggarwal, K. M., & Keenan, F. P. 1c, Phys. Scr., 85, 535 Aggarwal, K. M., & Keenan, F. P. 1d, Phys. Scr., 85, 531 Aggarwal, K. M., & Keenan, F. P. 13a, Phys. Scr., 87, 553 Aggarwal, K. M., & Keenan, F. P. 13b, Phys. Scr., 87, 53 Aggarwal, K. M., Keenan, F. P., & Heeter, R. F. 9, Phys. Scr., 8, 531 Aggarwal, K. M., Kato, T., Keenan, F. P., & Murakami, I. 11, Phys. Scr., 83, 153 Bartiromo, R., Bombarda, F., & Giannella, R. 1985, Phys. Rev. A, 3, 531 Beiersdorfer, P. 15, J. Phys. B, 8, 117 Beiersdorfer, P., Osterheld, A. L., Phillips, T. W., et al. 1995, Phys. Rev. E, 5, 198 Bethe, H. 193, Ann. Phys. Lpz, 397, 35 Bryans, P., Badnell, N. R., Gorczyca, T. W., et al., ApJS, 17, 33 Burgess, A., & Sheorey, V. B. 197, J. Phys. B, 7, 3 Burgess, A., & Tully, J. A. 199, A&A, 5, 3 Burgess, A., Hummer, D. G., & Tully, J. A. 197, Phil. Trans. R. Soc. London Ser. A,, 5 Burgess, A., Chidichimo, M. C., & Tully, J. A. 1997, J. Phys. B, 3, 33 Burke, P. G., Hibbert, A., & Robb, W. D. 1971, J. Phys. B,, 153 Chang, J. J. 1977, J. Phys. B, 1, 3335 Chantrenne, S., Beiersdorfer, P., Cauble, R., & Schneider, M. B. 199, Phys. Rev. Lett., 9, 5 Chen, H., & Beiersdorfer, P. 8, Can. J. Phys., 8, 55 Chen, C. Y., Wang, K., Huang, M., Wang, Y. S., & Zou, Y. M. 1, J. Quant. Spectr. Rad. Transf., 111, 83 Dere, K. P., Landi, E., Young, P. R., & Zanna, G. D. 1, ApJS, 13, 331 Doschek, G. A., & Meekins, J. F. 197, Solar Physics, 13, Doyle, J. G. 198, A&A, 87, 183 Doyle, J. G., & Keenan, F. P. 198, A&A, 157, 11 Doyle, J. G., & Schwob, J. L. 198, J. Phys. B, 15, 813 Dufton, P. 1977, Comput. Phys. Commun., 13, 5 Fano, U. 193, Ann. Rev. Nucl. Part. Sci., 13, 1 Feldman, U., Curdt, W., Landi, E., & Wilhelm, K., ApJ, 5, 58 Fontes, C. J., & Zhang, H. L. 7, Phys. Rev. A, 7, 73 Gabriel, A. H. 197, MNRAS, 1, 99 Gabriel, A. H., & Jordan, C. 199a, MNRAS, 15, 1 Gabriel, A. H., & Jordan, C. 199b, Nature, 1, 97 Gorczyca, T. W., Robicheaux, F., Pindzola, M. S., & Badnell, N. R. 1995, Phys. Rev. A, 5, 385 Griffin, D. C., & Ballance, C. P. 9, J. Phys. B,, 351 Gu, M. F., Phys. Rev. A, 7, 7 Gu, M. F. 8, Can. J. Phys., 8, 75 Güdel, M., & Nazé, Y. 9, A&ARv, 17, 39 Hsuan, H., Bitter, M., Hill, K. W., et al. 1987, Phys. Rev. A, 35, 8 Inokuti, M. 1971, Revs. Mod. Phys., 3, 97 Kallman, T. R., & Palmeri, P. 7, Revs. Mod. Phys., 79, 79 Källne, E., Källne, J., & Pradhan, A. K. 1983, Phys. Rev. A, 7, 17 Keenan, F. P., Tayal, S. S., & Kingston, A. E. 198, Sol. Phys., 9, 85 Keenan, F. P., McCann, S. M., Kingston, A. E., & McKenzie, D. L. 1987, ApJ, 318, 9 Keenan, F. P., McCann, S. M., Barnsley, R., et al. 1989, Phys. Rev. A, 39, 9 Landi, E., & Phillips, K. J. H. 5, ApJS, 1, 8 Li, S., Yan, J., Li, C. Y., et al. 15, A&A, 583, A8 Malespin, C., Ballance, C. P., Pindzola, M. S., et al. 11, A&A, 5, A115 McKenzie, D. L., Broussard, R. M., Landecker, P. B., et al. 198, ApJ, 38, L3 McKenzie, D. L., & Landecker, P. B. 198, ApJ, 59, 37 Nakazaki, S., Sakimoto, K., & Itikawa, Y. 1993, Phys. Scr., 7, 359 Ness, J.-U., Brickhouse, N. S., Drake, J. J., & Huenemoerder, D. P. 3, ApJ, 598, 177 Norrington, P. H., & Grant, I. P. 1987, J. Phys. B,, 89 A85, page 9 of 1

10 A&A, A85 (17 Paerels, F. B. S., & Kahn, S. M. 3, ARA&A, 1, 91 Phillips, K. J. H., ApJ, 5, 91 Porquet, D., & Dubau, J., A&AS, 13, 95 Porquet, D., Mewe, R., Dubau, J., Raassen, A. J. J., & Kaastra, J. S. 1, A&A, 37, 1113 Porquet, D., Dubau, J., & Grosso, N. 1, Space Sci. Revs., 157, 13 Pradhan, A. K. 1983, Phys. Rev. A, 8, 113 Pradhan, A. K. 1985, ApJS, 59, 183 Pradhan, A. K., & Shull, J. M. 1981, ApJ, 9, 81 Rice, J. E., Marmar, E. S., Källne, E., & Källne, J. 1987, Phys. Rev. A, 35, 333 Rice, J. E., Fournier, K. B., Safronova, U. I., et al. 1999, New J. Phys., 1, 19 Rice, J. E., Reinke, M. L., Ashbourn, J. M. A., et al. 1, J. Phys. B, 7, 7571 Rice, J. E., Reinke, M. L., Ashbourn, J. M. A., et al. 15, J. Phys. B, 8, 113 Sampson, D. H., Goett, S. J., & Clark, R. E. 1983, Atomic Data and Nuclear Data Tables, 9, 7 Seely, J. F., & Feldman, U. 1985, Phys. Rev. Lett., 5, 11 Shen, T. M., Chen, C. Y., Wang, Y. S., Zou, Y. M., & Gu, M. F. 7a, Phys. Rev. A, 7, 73 Shen, T.-M., Chen, C.-Y., Wang, Y.-S., Zou, Y.-M., & Gu, M.-F. 7b, J. Phys. B,, 375 Shen, T. M., Chen, C. Y., Wang, Y. S., & Zou, Y. M. 9, Eur. Phys. J. D, 53, 179 Si, R., Guo, X. L., Wang, K., et al. 1, A&A, 59, A11 Smith, R. K., & Brickhouse, N. S. 1, Adv. At. Mol. Opt. Phys., 3, 71 Sylwester, J., Sylwester, B., Phillips, K. J. H., & Kuznetsov, V. D. 1, ApJ, 71, 8 Tayal, S. S., & Kingston, A. E. 198, J. Phys. B, 17, 1383 Tayal, S. S., & Kingston, A. E. 1985, J. Phys. B, 18, 983 Wang, K., Chen, C. Y., Huang, M., et al. 1, J. Phys. B, 3, 175 Wang, K., Chen, C., Huang, M., Wang, Y., & Zou, Y. 11, Atomic Data and Nuclear Data Tables, 97, Wang, K., Yan, J., Huang, M., et al. 1, Atomic Data and Nuclear Data Tables, 98, 779 Whiteford, A. D. 5, Whiteford, A. D., Badnell, N. R., Ballance, C. P., et al. 1, J. Phys. B, 3, 3179 Wolfson, C. J., Leibacher, J. W., Doyle, J. G., & Phillips, K. J. H. 1983, ApJ, 9, 319 Zhang, H., & Sampson, D. H. 1987, ApJS, 3, 87 Zhang, H. L., & Pradhan, A. K. 1995, Phys. Rev. A, 5, 33 Zhang, H. L., Sampson, D. H., & Mohanty, A. K. 1989, Phys. Rev. A,, 1 Zhang, Y., Chen, C. Y., Wang, Y. S., & Zou, Y. M. 9, J. Quant. Spectr. Radiative Transfer, 11, 18 A85, page 1 of 1