Radiative rates and electron impact excitation rate coefficients for H-like Fe XXVI

ARTICLE IN PRESS Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 843–856

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Radiative rates and electron impact excitation rate coefficients for H-like Fe XXVI Chong-Yang Chen a,b,, Kai Wang a,b, Min Huang a,b, Yan-Sen Wang a,b, Ya-Ming Zou a,b a b

Shanghai EBIT Lab, Modern Physics Institute, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, China The Key Lab of Applied Ion Beam Physics, The Ministry of Education, Fudan University, Shanghai 200433, China

a r t i c l e in fo

abstract

Article history: Received 24 September 2009 Received in revised form 27 November 2009 Accepted 28 November 2009

In this paper we report on calculations on energy levels, radiative rates, collision strengths, and effective collision strengths for transitions among the lowest 36 levels of the n r 6 configurations of H-like Fe XXVI. Flexible atomic code (FAC) is adopted for the calculation. Energy levels and radiative rates are calculated within relativistic configuration-interaction method. Direct excitation collision strength is calculated using relativistic distorted-wave approximation. Resonance contributions through the relevant He-like doubly excited n0 l0 n00 l00 configurations with n0 r 7 and n00 r 75 are explicitly taken into account using the isolated process and isolated resonances approximation. We present the radiative rates, oscillator strengths, and line strengths for all electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), and magnetic quadrupole (M2) transitions among the 36 levels. Furthermore, collision strengths and effective collision strengths are reported for all the 630 transitions among the above 36 levels over a wide energy (temperature) range up to 25 keV ð109 KÞ. Extensive comparisons are made with earlier available results and the accuracy of the data is assessed. & 2009 Elsevier Ltd. All rights reserved.

Keywords: H-like iron Electron impact excitation Resonance excitation Rate coefficient Relativistic distorted-wave approximation

1. Introduction Recently, a large amount of high resolution spectra in the UV, EUV, and X-ray regions have been obtained for solar, stellar and other astrophysical sources by many space missions, such as SOHO, Chandra and XMMNewton. To interpret these spectra, a vast amount of atomic data, such as energy levels, radiative rates, electron impact excitation (EIE) rate coefficients, are needed. In last decades, many researchers such as the group at Queen’s University of Belfast (QUB) have made large efforts to provide the EIE cross sections and rates of astrophysical importance under the Iron Project [1] and

 Corresponding author at: Shanghai EBIT Lab, Modern Physics Institute, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, China. E-mail address: [email protected] (C.-Y. Chen).

0022-4073/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2009.11.028

the RmaX Network (http://amdpp.phys.strath.ac.uk/ UK_RmaX/), using the R-matrix code originally developed at QUB. Iron is an abundant element in solar and fusion plasma, and its emission lines are observed over almost all ionization stages. Emission lines involving the n ¼ 5 levels of Fe XXVI have been observed in astrophysical plasmas [2]. In very recent works, Aggarwal et al. [3] reported the radiative rates, collision strengths (O), and effective (Maxwellian-averaged) collision strengths (U), for the transitions among the lowest 25 levels of the n r 5 configurations of Fe XXVI, employing Dirac atomic R-matrix code (DARC). They presented effective collision strengths for the 300 transitions among the above 25 levels at temperatures up to 107:7 K. A comprehensive survey of prior works on EIE rates for Fe XXVI was also provided in [3]. Of particular note is the work presented by Ballance et al. [4], in which a radiation-damped Breit–Pauli R-matrix (BPRM) calculation has been carried

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out. The detailed considerations of high-energy behavior of collision strengths enabled them to tabulate effective collision strengths up to 109 K. The comparison shows that the above two sets of U values differ over entire temperature range for some transitions and the differences are up to 80% for four transitions, though they are in satisfactory agreements for a majority of transitions [3]. Apart from R-matrix approach in which the resonances and the interaction among them are naturally taken into account, the resonances may be treated with a completely different fashion, namely the independent processes and isolated resonances approximation using distorted-waves (denoted the IPIRDW approximation). In general, the interference effects, such as interference or interaction between resonances, interference between resonances and continua, are ignored in the IPIRDW approximation. However, it holds good agreement with the R-matrix results in general for most of highly charged ions, especially if only the excitation rate coefficient is concerned (for example see [5]). In addition, as shown and pointed out in [3], there are often large discrepancies among various sets of R-matrix calculations whether the same R-matrix code are employed or not. Hence, a complete independent calculation, such as the IPIRDW calculation presented here, are very useful to assess the accuracy for various atomic data obtained from different atomic computation code package. Here employing the widely used relativistic configurationinteraction (RCI) atomic code, i.e. the flexible atomic code (FAC) [6,7], we report the radiative rates, oscillator strengths, and line strengths for all electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2) and magnetic quadrupole (M2) transitions among the lowest 36 levels of the n r6 configurations of H-like Fe XXVI. Direct excitation (DE) collision strengths for all the 630 transitions among the above 36 levels are calculated at scattered electron energy up to 25 keV, employing relativistic distorted-wave (DW) approximation. Resonance contributions through He-like doubly excited states n0 l0 n00 l00 with n0 r 7 and n00 r75 are explicitly taken into account via the IPIRDW approximation. The decays to low-lying autoionizing levels from the resonances possibly followed by radiative cascades (DAC) are taken into account as well as the resonant stabilizing (RS) transitions, and are found to have large influence for some transitions. Comparing to the earlier published R-matrix works [3,4], we present here a more comprehensive data set. And the present values of U for the transitions to and among the n ¼ 5 levels should be more accurate as we include additionally the contributions from the resonances attached to the n 4 5 levels. In addition, H-like ion is the simplest one, configuration-interaction (CI) effects are expected to be small, thus the extensive comparisons presented here for various atomic parameters which are obtained from different atomic codes may reveal the large differences existing possibly between these widely used codes.

2. Energy levels and radiative rates The n r6 configurations of Fe XXVI give rise to 36 finestructure levels, listed in Table 1. In the RCI calculation, the

electron correlation among the above 36 levels are taken into account. Relativistic effects are fully considered using the Dirac Coulomb Hamiltonian. Higher order QED effects are included with Breit interaction in the zero energy limit for the exchanged photon, and hydrogenic approximations for self-energy and vacuum polarization effects. The present level energies are slightly higher (about 0.1 eV, corresponding to about 0.0015%) than the recommended ones from the NIST Atomic Spectra Database (http:// physics.nist.gov/PhysRefData/ASD/index.html). And our results are in excellent agreements with the earlier calculation within 0.05 eV, which is performed by Aggarwal et al. [3] employing the GRASP (general-purpose relativistic atomic structure package) code [8] with the inclusion of QED effects. In Table 1 the partial-total E1, M1, E2, and M2 multipole transition rates of the fine-structure levels are given together with the sum of the four types of rates and the life times (t). It can be seen that the E1 transition rates predominate the total rates for all excited levels except for level 2s1=2 (For brevity the notation nlj with j ¼ l 7 1=2 is throughout used). The detailed E1 and M2, and M1 and E2 transition rates are, respectively, listed in Tables 2 and 3 together with the corresponding wavelengths, oscillator strengths, and line strengths. The numerical conversion formulas among these atomic parameters could be found in the literature (for example see [3]). The indices used to represent the lower and upper levels of a transition in Tables 2 and 3 and elsewhere are defined and listed in Table 1. The transitions among all the lowest 36 levels yield 180 E1  , 161 M1  , 212 E2  and 151 M2 multipole ones. The NIST database tabulates the corresponding atomic parameters for 21 of the present 180 E1 transitions. The comparison of the NIST values with the present ones is listed in Table 4. It can be seen that the agreements between the NIST and FAC wavelengths are excellent within 0.0026% except for three Dn ¼ 0 transitions, i.e. 2s1=2 22p3=2 , 3s1=2 23p3=2 , and 3p1=2 23d3=2 . The differences between the other atomic parameters, such as transition rates, are somewhat larger (up to 6.5% for transition 3s1=2 24p3=2 ). This is due to that the transition rates and oscillator strengths recommended by NIST database are obtained by simply scaling the line strengths to the data tabulated for hydrogen spectra, where relativistic effects are neglected [9]. Aggarwal et al. [3] reported the results for 92 E1  , 86 M1  , 107 E2-, and 103 M2 transitions among the lowest 25 levels of the n r5 configurations of Fe XXVI, employing the GRASP code. The differences between present FAC rates and the GRASP ones [3] are less than 2% for all E1 transitions, with exceptions up to 8% for the four ns1=2 2np3=2 (n ¼ 225) transitions. The somewhat larger deviations between the rates for above Dn ¼ 0 transitions result from the differences (about 2.5%) between the FAC and GRASP wavelengths. Similar agreements between the present calculation and Aggarwal et al. are found for E2 and M2 transitions. The line strengths agree within 2% for all E2 and M2 transitions, with only three and five exceptions for E2 and M2 transitions, respectively. The deviations between line strengths are about 4% for the three 1s1=2 2nd5=2 ðn ¼ 325Þ E2 transitions. The deviations

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Table 1 Energies (in eV), partial-total E1, M1, E2, and M2 rates and total radiative rates (all Ar are in s1 ), and life time t (in s) of the lowest 36 levels nlj of Fe XXVI ðX 7 Y ¼ X  10 7 Y Þ. Index

nlj

ENIST a

EFAC b

ArE1

ArM1

ArE2

ArM2

ArTot

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1s1=2 2p1=2 2s1=2 2p3=2 3p1=2 3s1=2 3d3=2 3p3=2 3d5=2 4p1=2 4s1=2 4d3=2 4p3=2 4d5=2 4f5=2 4f7=2 5p1=2 5s1=2 5d3=2 5p3=2 5d5=2 5f5=2 5f7=2 5g7=2 5g9=2 6s1=2 6p1=2 6p3=2 6d3=2 6d5=2 6f5=2 6f7=2 6g7=2 6g9=2 6h9=2 6h11=2

0:00000000 þ 00 6:95196090 þ 03 6:95252499 þ 03 6:97317462 þ 03 8:24639175 þ 03 8:24656781 þ 03 8:25266901 þ 03 8:25268153 þ 03 8:25473095 þ 03 8:69857747 þ 03 8:69865203 þ 03 8:70122391 þ 03 8:70122528 þ 03 8:70209073 þ 03 8:70209264 þ 03 8:70252339 þ 03 8:90750417 þ 03 8:90754240 þ 03 8:90885793 þ 03 8:90885793 þ 03 8:90930053 þ 03 8:90930284 þ 03 8:90952243 þ 03 8:90952448 þ 03 8:90965727 þ 03 – – – – – – – – – – –

0:00000000 þ 00 6:95206112 þ 03 6:95261216 þ 03 6:97327500þ 03 8:24649247 þ 03 8:24665791 þ 03 8:25277064 þ 03 8:25278255 þ 03 8:25482956 þ 03 8:69867580 þ 03 8:69874582 þ 03 8:70132222 þ 03 8:70132734 þ 03 8:70219143 þ 03 8:70219143 þ 03 8:70262377 þ 03 8:90760258þ 03 8:90763848 þ 03 8:90895641 þ 03 8:90895905þ 03 8:90940147þ 03 8:90940148þ 03 8:90962293 þ 03 8:90962293 þ 03 8:90975558 þ 03 9:02093810 þ 03 9:02094017 þ 03 9:02172302þ 03 9:02172304þ 03 9:02198030 þ 03 9:02198031þ 03 9:02210848þ 03 9:02210848þ 03 9:02218527 þ 03 9:02218527 þ 03 9:02223641 þ 03

0:00 þ 00 2:86 þ 14 2:30 þ 03 2:88 þ 14 8:58 þ 13 3:11 þ 12 2:99 þ 13 8:75 þ 13 2:96 þ 13 3:66 þ 13 2:16 þ 12 1:28 þ 13 3:75 þ 13 1:27 þ 13 6:33 þ 12 6:31 þ 12 1:89 þ 13 1:38 þ 12 6:66 þ 12 1:94 þ 13 6:58 þ 12 3:27 þ 12 3:26 þ 12 1:95 þ 12 1:95 þ 12 9:07 þ 11 1:10 þ 13 1:13 þ 13 3:90 þ 12 3:85 þ 12 1:92 þ 12 1:91 þ 12 1:14 þ 12 1:13 þ 12 7:53 þ 11 7:52 þ 11

0:00 þ 00 0:00 þ 00 3:66 þ 08 4:48 þ 04 1:95 þ 06 1:62 þ 08 1:01 þ 06 1:01 þ 06 4:93 þ 01 1:23 þ 06 7:75 þ 07 6:37 þ 05 6:91 þ 05 1:06 þ 04 6:46 þ 03 4:9001 7:39 þ 05 4:19 þ 07 3:83 þ 05 4:28 þ 05 1:08 þ 04 5:93 þ 03 4:70 þ 02 2:04 þ 02 1:4702 2:50 þ 07 4:64 þ 05 2:73 þ 05 2:40 þ 05 8:39 þ 03 4:47 þ 03 5:88 þ 02 2:40 þ 02 4:14 þ 01 1:39 þ 01 8:6004

0:00þ 00 0:00þ 00 0:00þ 00 6:23 þ 00 7:42 þ 09 0:00þ 00 1:99 þ 11 7:46 þ 09 2:00 þ 11 3:92 þ 09 3:29 þ 08 1:04 þ 11 4:01 þ 09 1:05 þ 11 2:10 þ 10 2:09 þ 10 2:18 þ 09 3:06 þ 08 5:77 þ 10 2:25 þ 09 5:85 þ 10 1:32 þ 10 1:31 þ 10 3:91 þ 09 3:90 þ 09 2:33 þ 08 1:31 þ 09 1:36 þ 09 3:49 þ 10 3:54 þ 10 8:39 þ 09 8:36 þ 09 2:73 þ 09 2:72 þ 09 1:02 þ 09 1:01 þ 09

0:00 þ 00 0:00 þ 00 0:00 þ 00 1:01 þ 10 0:00 þ 00 2:47 þ 06 1:22 þ 06 3:79 þ 09 5:40 þ 07 1:20 þ 04 1:93 þ 06 7:24 þ 05 1:72 þ 09 3:21 þ 07 2:04 þ 05 2:04 þ 06 1:25 þ 04 1:25 þ 06 4:24 þ 05 9:13 þ 08 1:88 þ 07 1:52 þ 05 1:52 þ 06 3:51 þ 04 1:88 þ 05 8:16 þ 05 9:89 þ 03 5:37 þ 08 2:62 þ 05 1:16 þ 07 1:04 þ 05 1:04 þ 06 2:97 þ 04 1:59 þ 05 7:67 þ 03 2:87 þ 04

0:00þ 00 2:86 þ 14 3:66 þ 08 2:88 þ 14 8:58 þ 13 3:11 þ 12 3:01 þ 13 8:75 þ 13 2:98 þ 13 3:66 þ 13 2:16 þ 12 1:29 þ 13 3:75 þ 13 1:28 þ 13 6:35 þ 12 6:33 þ 12 1:89 þ 13 1:38 þ 12 6:72 þ 12 1:94 þ 13 6:64 þ 12 3:29 þ 12 3:28 þ 12 1:95 þ 12 1:95 þ 12 9:08 þ 11 1:10 þ 13 1:13 þ 13 3:93 þ 12 3:89 þ 12 1:92 þ 12 1:92 þ 12 1:14 þ 12 1:14 þ 12 7:54 þ 11 7:53 þ 11

– 3:5015 2:7309 3:4715 1:1714 3:2113 3:3214 1:1414 3:3514 2:7314 4:6313 7:7314 2:6714 7:8214 1:5713 1:5813 5:2914 7:2313 1:4913 5:1514 1:5113 3:0413 3:0513 5:1213 5:1313 1:1012 9:0914 8:8414 2:5413 2:5713 5:2013 5:2213 8:7813 8:8013 1:3312 1:3312

a b

ENIST : Energies from http://physics.nist.gov/PhysRefData. EFAC : Present energies calculated from the FAC code.

Table 2 ˚ transition rates (Ar , in s1 ), oscillator strengths f, and line strengths (S, in atomic unit) for E1 and M2 transitions among the The wavelengths (l, in A), lowest 36 levels of Fe XXVI ðX 7 Y ¼ X  10 7 Y Þ. Lower

Upper

l

ArE1

fE1

SE1

1 1 1 1 1 1 1 1 1 1

2 4 5 8 10 13 17 20 27 28

1:783416 þ 00 1:777991 þ 00 1:503478 þ 00 1:502332 þ 00 1:425323 þ 00 1:424888 þ 00 1:391892 þ 00 1:391680 þ 00 1:374404 þ 00 1:374285 þ 00

2:858 þ 14 2:881 þ 14 7:530 þ 13 7:731 þ 13 3:052 þ 13 3:154 þ 13 1:533 þ 13 1:590 þ 13 8:781 þ 12 9:124 þ 12

1:36301 2:73001 2:55202 5:23202 9:29403 1:92002 4:45303 9:23303 2:48703 5:16703

1:60003 3:19603 2:52604 5:17504 8:72205 1:80104 4:08105 8:46105 2:25005 4:67505

ArM2

fM2

SM2

1:006 þ 10

9:53406

4:79502

3:781 þ 09

2:55906

7:76303

1:715 þ 09

1:04406

2:70203

9:062 þ 08

5:26207

1:26903

5:332 þ 08

3:02007

7:01304

Note: Only data for the transitions involving the ground state are shown here. Table 2 is available online in its entirety in the homepage of JQSRT. A portion is shown here for guidance regarding its form and content.

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Table 3 Same as Table 2 but for M1 and E2 transitions ðX 7 Y ¼ X  10 7 Y Þ. Lower

Upper

l

ArM1

fM1

SM1

1 1 1 1 1 1 1 1 1 1 1 1 1

3 6 7 9 11 12 14 18 19 21 26 29 30

1:783275 þ 00 1:503448þ 00 1:502334þ 00 1:501959þ 00 1:425311 þ 00 1:424889 þ 00 1:424747 þ 00 1:391886 þ 00 1:391680þ 00 1:391611 þ 00 1:374405þ 00 1:374285 þ 00 1:374246 þ 00

3:662 þ 08 1:621 þ 08 9:918 þ 05

1:74607 5:49208 6:71210

1:54004 4:08405 4:98707

7:729 þ 07 6:176 þ 05

2:35408 3:76010

1:65905 2:64907

4:175 þ 07 3:684 þ 05

1:21308 2:13910

8:34706 1:47207

2:485 þ 07 2:303 þ 05

7:03709 1:30410

4:78406 8:86508

ArE2

fE2

SE2

1:823 þ 11 1:839 þ 11

1:23404 1:86604

4:98406 7:53206

1:003þ 11 1:015 þ 11

6:10605 9:26805

2:10406 3:19306

5:659 þ 10 5:737 þ 10

3:28605 4:99605

1:05506 1:60406

3:435 þ 10 3:485 þ 10

1:94505 2:96005

6:01407 9:15107

Note: Only data for the transitions involving the ground state are shown here. Table 3 is available online in its entirety in the homepage of JQSRT. A portion is shown here for guidance regarding its form and content.

Table 4 ˚ rates (Ar , in s1 ), oscillator strengths (f, dimensionless), and line strengths (S, in atomic unit) for E1 Comparisons of the present FAC wavelengths (l, in A), transitions with those from NIST database ðX 7 Y ¼ X  10 7 Y Þ. Lower

Upper

lFAC

lNIST

ArFAC

ArNIST

fFAC

fNIST

SFAC

SNIST

1s1=2 1s1=2 1s1=2 1s1=2 1s1=2 1s1=2 2s1=2 2s1=2 2s1=2 2s1=2 2p3=2 2p3=2 2p3=2 3p1=2 3s1=2 3s1=2 3s1=2 3p3=2 3p3=2

2p1=2 2p3=2 3p1=2 3p3=2 4p3=2 5p3=2 2p3=2 3p3=2 4p3=2 5p3=2 3d5=2 4d5=2 5d5=2 3d3=2 3p3=2 4p3=2 5p3=2 4d5=2 5d5=2 4f7=2 5f7=2

1:783416 þ 00 1:777991 þ 00 1:503478 þ 00 1:502332 þ 00 1:424888 þ 00 1:391680 þ 00 6:000347 þ 02 9:535999 þ 00 7:090018 þ 00 6:337536 þ 00 9:674513 þ 00 7:171210 þ 00 6:403726 þ 00 1:974845 þ 03 2:024350 þ 03 2:726909 þ 01 1:872022 þ 01 2:758828 þ 01 1:888222 þ 01 2:768776 þ 01 1:893486 þ 01

1:783442 þ 00 1:778016 þ 00 1:503496 þ 00 1:502350 þ 00 1:424905 þ 00 1:391696 þ 00 6:004200 þ 02 9:536097 þ 00 7:090077 þ 00 6:337581 þ 00 9:674501 þ 00 7:171209 þ 00 6:403725 þ 00 1:975100 þ 03 2:027000 þ 03 2:726980 þ 01 1:872050 þ 01 2:758830 þ 01 1:888220 þ 01 2:768790 þ 01 1:893490 þ 01

2:858 þ 14 2:881 þ 14 7:530 þ 13 7:731 þ 13 3:154 þ 13 1:590 þ 13 1:233 þ 08 1:019 þ 13 4:427 þ 12 2:272 þ 12 2:962 þ 13 9:466 þ 12 4:327 þ 12 1:306 þ 07 1:932 þ 07 1:376 þ 12 7:421 þ 11 3:221 þ 12 1:556 þ 12 6:312 þ 12 2:080 þ 12

2:93 þ 14 2:96 þ 14 7:83 þ 13 7:86 þ 13 3:20þ 13 1:61 þ 13 1:25 þ 08 1:08þ 13 4:61 þ 12 2:35 þ 12 2:98 þ 13 9:50þ 12 4:34 þ 12 1:31 þ 07 1:94 þ 07 1:47 þ 12 7:77 þ 11 3:27 þ 12 1:57 þ 12 6:33 þ 12 2:08þ 12

1:36301 2:73001 2:55202 5:23202 1:92002 9:23303 1:33202 2:77901 6:67302 2:73602 6:23501 1:09501 3:99002 1:52802 2:37402 3:06701 7:79702 5:51201 1:24801 9:67201 1:49101

1:4001 2:8101 2:6502 5:3202 1:9502 9:3503 1:3502 2:9401 6:9502 2:8302 6:2701 1:1001 4:0002 1:5302 2:3902 3:2801 8:1602 5:6001 1:2601 9:7001 1:4901

1:60003 3:19603 2:52604 5:17504 1:80104 8:46105 5:26102 1:74502 3:11503 1:14103 7:94402 1:03402 3:36503 1:98601 3:16401 5:50702 9:61103 2:00301 3:10202 5:29001 5:57702

1:6403 3:2803 2:6304 5:2604 1:8304 8:5705 5:3402 1:8502 3:2403 1:1803 7:9902 1:0402 3:3703 1:9901 3:1901 5:8902 1:0102 2:0301 3:1302 5:3101 5:5802

3d5=2 3d5=2

are about 3% for 1s1=2 23p3=2 ; 4p3=2 ; 5p3=2 , 4% for 4d3=2 24f7=2 , and 7% for 4d3=2 25f7=2 M2 transitions, respectively. The present transition rates also agree with GRASP ones within 2% for most E2 and M2 transitions, but with a handful of exceptions up to 12% due to the differences between the wavelengths (especially for Dn ¼ 0 transitions), or to the differences between the line strengths as mentioned above, or to both. In the case of M1 transitions, we find that the present line strengths or rates differ significantly from the GRASP ones [3], even by several orders of magnitude. We have performed yet another calculation using the GRASPVU package [10] which is one of the updated versions of GRASP code [8]. The present FAC and GRASPVU rates are found in similar good agreement for M1 transition as the above for E1, E2, and M2 transitions. For example, the M1 transition rates

between 1212 ð1s1=2 24d3=2 Þ from the present FAC and GRASPVU calculations are 6:176  105 and 6:165 105 s1 ,respectively; the corresponding value from Table 3 of [3] is 2:384  107 s1 . 3. Direct excitation collision strengths Direct excitation (DE) cross section sij (in unit of cm2 ) from the initial state i to the final state j can be expressed in terms of the collision strength Oij as [11]

sij ¼

pa20 Oij k2i gi

ð1Þ

where gi is the statistical weight of the initial state, a0 is the Bohr radius, and ki is the relativistic kinetic

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momentum of the incident electron, which is related to the incident energy Ei (in unit of Ry which is Rydberg energy) by k2i ¼ Ei ðRyÞð1 þ a2 Ei ðRyÞ=4Þ

ð2Þ

where a is the fine structure constant. Calculation of DE collision strength can be straightforwardly performed within FAC computer package [6], employing relativistic distorted-wave approximation (RDWA). One can find more details about FAC from the manual of the codes (http://sprg.ssl.berkeley.edu/  mfgu/fac/). In short, here we set the maximum of orbital angular momentum (l) for the partial-wave expansion to 100 to ensure the convergence of the collision strengths. Higher partial-wave contributions are included using the Coulomb-Bethe approximation. The partial-waves with l 4 20 are treated in quasi-relativistic approximation [12]. The correlations among all the n r6 configurations are taken into account. The collision strengths (O) for all 630 transitions among the lowest 36 levels of n r 6 configurations of Fe XXVI have been calculated at 10 scattered electron energies in the range of 10 eV–25 keV. For brevity, the O results are only electronically available from the corresponding author. Employing DARC method, Aggarwal et al. [3] listed the values of O for the 300 transitions among the lowest 25 levels of n r5 configurations of Fe XXVI at energies of 700, 800, 900, 1000, 1200, and 1500 Ry with respect to the ground state, where there is no resonance contribution. They also compared their O values to those from FAC. Here we give the comparison in some details, as shown in Fig. 1. As seen from Fig. 1(a), the two sets of O values differ significantly (by over 28%) over the entire energy range only for three transitions, namely 14215 ð4d5=2 24f5=2 Þ, 21222 ð5d5=2 25f5=2 Þ, and 23224 ð5f7=2 2 5g7=2 Þ. As discussed by Aggarwal et al. [3], the different values of transition threshold DE involved in the top-up procedure to include the higher partial waves from the Coulomb–Bethe approximation accounts for the significant deviations for the above ‘‘elastic’’ transitions. With regards to population modelling, these elastic transitions are fortunately of minor importance. For 254 of the remaining 297 transitions, the two sets of O values agree very well (within 10%) over the entire energy range while they differ by over 10% at low- or high-energy end for the other 43 transitions. For 14 of the 43 transitions, the differences between the two sets of data are over 10% at 700 Ry. Most of them are the dipole-allowed transitions towards levels 11 (4s1=2 ) and 18 (5s1=2 ) while the present values are smaller than the DARCs. And for three of them, namely 2–18, 4–11, and 4–18, the deviations are over 15% and up to 22% (see Fig. 1(b)). For another 29 transitions, the FAC values differ from the DARCs by over 10% at 1500 Ry. Fifteen of the 29 transitions are E1 ones, for which the present FAC results are all smaller than the DARC values. And for three of them, namely 5–11, 6–10, and 6–13, the deviations are about 15% (see Fig. 1(c)). The remaining 14 transitions are all non-electric-dipole allowed transitions between the n ¼ 5 levels, for which ours are all larger than the DARCs. And for 10 of them, namely 17–20, 22, 18–19, 21, 19–21, 24, 20–22, 23, 21–25, and 22–23, the

847

differences are over 15% and up to 20% (for examples, see Fig. 1(d)). It could be concluded from the above discussions that there are large discrepancies for some dipole-allowed transitions between the DARC and FAC collision strengths at low or high energy end. And the FAC values are higher than DARC results for many dipoleforbidden transitions between the n ¼ 5 levels. The detail comparison between the FAC and DARC calculations for the partial-wave contributions and the top-up procedure may reveal the reasons for above discrepancies. DE effective collision strengths (U) are obtained after integrating O over a Maxwellian distribution of electron velocities, i.e. Z 1 U ðTe Þ ¼ OðEf ÞexpðEf =kT e Þ dðEf =kT e Þ ð3Þ 0

where Ef is the scattered electron energy, k is Boltzmann constant, and Te is the electron temperature in K. The rate coefficients can be obtained from the relation between the U values (see Eq. (8)). As the peak coronal fractional abundance of Fe XXVI lies at  108 K [13], population modelling into the 107 2109 temperature regime is usually needed. It is obvious that the O values at high energy up to hundreds keV are needed to get the convergence of the high-temperature U values, especially for dipole-allowed transitions. However, it is computationally burdensome and unnecessary to calculate explicitly the collision strength at such high energy within both distorted-waves and R-matrix approaches. In general, Bethe’s form [14,15] of the Born approximation is employed instead. In modern R-matrix calculations (for example see [4,16]), the ‘‘C-plot’’ scaling method [17] for dipole-allowed transitions and the extension work [18] for high-energy limits of Born approximation for dipoleforbidden transitions are used to estimate the needed highenergy collision strengths. In short, the above extrapolation methods are in principle based on the high-energy asymptotic behaviors of O  const:  lnðEi Þ for dipoleallowed transitions and O  const: for dipole-forbidden transitions [17]. The above treatments of the high-energy collision strengths disregard relativistic effects. When the velocity of incident electron increases, eventually approaching relativistic energies, relativistic modification of the cross section should be considered [20]. According to the discussions of [15,21], we may define the reduced cross section as Qij ðEi Þ ¼

me v2i Eij 1 s ðE Þ 2Ry Ry 4pa20 ij i

ð4Þ

where me is the rest mass of electron, vi the velocity of incident electron, and Eij the excitation energy. In the relativistic region, a Fano-plot of the reduced cross sections Qij for dipole-allowed transition against 2 2 2 ln½b =ð1b Þb (in which b ¼ vi =c, c is the light velocity) will become a straight line, whose slope corresponds to the optical oscillator strength fij . And for dipole-forbidden transition, Qij will become nearly a constant against 2 2 2 ln½b =ð1b Þb [15]. We thus have 2

2

2

Qij ðEi Þ ¼ fij  fln½b =ð1b Þb g þ A

ð5Þ

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Fig. 1. Comparisons of the present FAC collision strengths with the DARC values from [3]. (a) The ratios of the FAC collision strengths for the 300 transitions among the nr 5 levels of Fe XXVI to the corresponding DARC ones are plotted against the energies relative to the ground state. For 19 of 300 transitions whose ratios are connected by lines, the FAC collision strengths may differ from the DARC ones by over 15%. The detail comparisons for 10 of the 19 transitions are shown in figures (b), (c) and (d), where the lines stand for the FAC results and the scattered symbols represent the DARC values. The results for transition 5211 plotted in (c) are multiplied by a factor of 10.

for dipole-allowed transitions, and Qij ðEi Þ  B

ð6Þ

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 [22]. For instances, we show some Fano-plots for both dipole-allowed and forbidden transitions among the 36 levels of the n r 6 configurations of Fe XXVI in Fig. 2. As mentioned before, we have calculated the cross sections (or collision strengths) at 10 scattered energies in the range of 10 eV–25 keV, employing RDWA method. We have computed yet the high-energy cross sections at additional three scattered energies of 100, 300, 1000 keV, employing the relativistic plane-waves approximation (RPWA) implemented within the FAC package. The results from both RDWA and RPWA calculations are then connected by spline functions. As shown in Fig. 2,

the RDWA results for low and intermediate energies can be connected smoothly with the RPWA values, and the relativistic asymptotic behaviors [Eqs. (5) and (6)] of the reduced cross sections does take place. In present work, we linearly extrapolate the collision strength nearby threshold, and throughout use Eqs. (5) and (6) to extrapolate the cross section at scattered energy above 1000 keV for the Maxellian integration. In the scattered electron energy range of 10 eV–1000 keV, the O values are interpolated by splines as shown in Fig. 2. The U values of direct excitation can be reliably evaluated for temperature up to and above 109 K. It should be pointed out that the full relativistic effects of Lorentz transformation of electron velocities [17,19] are considered in the present work, whereas they are disregarded in the BPRM calculation [4]. For dipole-allowed transitions, the present U results are thus larger than the BPRMs at high temperature. For dipole-forbidden transitions, from Eqs. (1), (4) and (6) one may get the relation of O  k2i =v2i , which increases with

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2

2

2

Fig. 2. Reduced excitation cross sections (see text) for some dipole-allowed and forbidden transitions are plotted against ln½b =ð1b Þb . The values at the highest three energies for each transition are obtained by relativistic plane-wave approximation. The scattered symbols are connected with spline functions. (a) For dipole-allowed transitions. (b) For dipole-forbidden transitions.

increasing energy. Hence, the resulted high-temperature (above 108 K) U values for dipole-forbidden transitions will generally increase with increasing temperature and are also generally larger than the BPRM results, as shown in Section 5. We note that it is somewhat arbitrary to choose 100 keV as the scattered energy above which the RPWA method is employed instead. We have performed yet another two calculations, choosing 50 and 200 keV rather than 100 keV. We find that the U values from different calculations agree very well (within 1%) with each other, with few exceptions up to 8% for dipole-forbidden transition.

4. Resonance excitation rate coefficients Contributions to total EIE rate coefficients from resonance excitation (RE) are included using the IPIRDW approximation. The RE contributions through the relevant He-like doubly excited configurations n0 l0 n00 l00 with l0 rn0 1 and l00 r8 are included explicitly up to n0 ¼ 7 and n00 ¼ 75. The higher n00 contributions are included up to n00 ¼ 1000 by using ðn00 Þ3 scaling law [23,24]. For H-like iron, the electron correlations among the n r6 configurations are considered. For He-like iron, configurationinteraction within the same complex are taken into account. The resonant stabilizing (RS) transitions from the doubly excited configurations n0 l0 n00 l00 towards 1sn00 l00 and 1sn0 l0 are considered as radiation damping source. Decays from the resonances into low-lying autoionizing levels possibly followed by autoionization cascade (DAC) are also taken into account. Here all possible DAC 000 000 000 transitions n0 l0 -n l (n on0 ) are included. The DAC 000 000 000 00 00 transitions from the n l electron to n l with n r 9 are also considered. Basic atomic data, such as energy levels, radiative and Auger rates, are calculated employing the

FAC code too. In present work, only E1 transitions from the doubly excited states are included. And all possible autoionization channel of the doubly excited states are taken into account. The plasma RE rate coefficients for transition from level i to j of H-like ion are obtained by the summation of the contribution through individual autoionizing level d of He-like ion:

aRE ij ðkT e Þ ¼

ð2pÞ3=2 ‘

3

3=2

ðme kT e Þ

  X gd Eid Aadi BRE dj  exp  2gi kT e d

ð7Þ

where Eid is the resonant energy, Aadi is the Auger rate from d to i, BRE dj is the Auger decay branching ratio from state d to state j, gi and gd are the statistical weights of states i and d, respectively. As discussed in our recent works on the total EIE rate coefficients for Ni-like ions [24–27], using different manners to treat the competing process with Auger decay of the autoionizing state d, i.e. radiative decay, the calculations of BRE dj and the subsequent RE rate coefficients will generally differ from each other. As done in [3] and most earlier Rmatrix calculations, disregarding any radiative transitions in the calculation of BRE dj , one obtains the undamped RE rate coefficients. These rates could be radiatively damped by the RS transitions. For instances, the ratios of the damped RE rates to the undamped ones at Te ¼ 106 K are plotted in Fig. 3(a). It shows that the RE rate coefficients are reduced by over 20% for 65 of 630 transitions. The largest reductions (by up to about 90%) occur for transitions 1–2 and 1–3. Taking additionally the DAC transitions into account, the RE rates at Te ¼ 106 K are further reduced by over 20% for 53 of 630 transitions. The largest reductions (by up to 60%) are for the excitations to levels 5 (3p1=2 ) and 6 (3s1=2 ) from the n ¼ 2 levels. The ratios of the RE rates with consideration of both RS and DAC transitions to the undamped ones at 106 K

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Fig. 3. Effects of radiative transitions on the RE rate coefficients at Te ¼ 106 K. (a) Ratios of the RE rate coefficients radiatively damped by the inclusion of RS transitions to those undamped. (b) Ratios of the RE rates considering both RS and DAC transitions to those undamped. In both figures the ratios are plotted against the final state indices of the transitions. The squares, circles, triangles, stars, and crosses are for the excitations initially from level Nos. 1, 2, 3, 4 and others, respectively.

Fig. 4. Ratios of the total of DE and RE rate coefficients at Te ¼ 106 K with considerations of both RS and DAC transitions to those without are plotted against the final state indices of the excitations. The squares, circles, triangles, stars, and crosses are for the excitations initially from level Nos. 1, 2, 3, 4 and others, respectively.

are plotted in Fig. 3(b). It shows that the undamped RE rates are finally reduced by over 20% (up to 90%) for 238 of 630 transitions. To show the overall effects of the radiative decays on the total excitation rate coefficients, in Fig. 4 we plot the ratios of the total of DE and RE rate coefficients considering both RS and DAC transitions to the undamped ones at Te ¼ 106 K. It shows that the inclusion of radiative decays reduces the total excitation rate coefficients by over 20% for 16 of 630 transitions at Te ¼ 106 K, though they could reduce significantly the RE rate coefficients for more transitions as discussed above. This is due to that resonant contributions to the total excitation rate coefficients are relatively small for many transitions (see Fig. 5). The overall largest

reductions occur for transitions 426 (2p3=2 23s1=2 ), 1–35 (1s1=2 26h9=2 ), 1–36 (1s1=2 26h11=2 ), 2–6 (2p1=2 23s1=2 ), 7–10 (3d5=2 24p1=2 ), and 8–11 (3p3=2 24s1=2 ) by 51% for the first one and about 35% for the last fives, respectively. It should be pointed out that the damping effects of radiative decays on the total rate coefficients become smaller as temperature increasing. For examples, the reductions at 107 K are over 20% for only three transitions, namely 4–6, 1–35, and 1–36. To show the overall importance of resonance excitation to the total excitation rate coefficients, in Fig. 5 we plot the ratios of the total rate coefficients to the DE contributions at Te ¼ 106 K. The RS and DAC transitions are taken into account in the calculation of RE rates. It shows that the inclusion of RE contributions may enlarge the DE

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Fig. 5. Ratios of the total of DE and RE rate coefficients at Te ¼ 106 K with considerations of both RS and DAC transitions to the DE rates are plotted against the final state indices of the excitations. The squares, circles, triangles, stars, and crosses are for the excitations initially from level Nos. 1, 2, 3, 4 and others, respectively. The dashed curve shows the RE enhancement is 50%.

rate coefficients at Te ¼ 106 K by over 50% for 388 of 630 transitions. The largest enhancements (up to two orders of magnitude) occur for the transitions towards the highlying levels with large orbital angular momentum, such as 4f5=2;7=2 , 5g7=2;9=2 , and 6h9=2;11=2 . Two factors account for these large enhancements. Firstly, the resonance states prefer to decay towards these final states because they have large statistical weights. Secondly, the background collision strengths for these transitions (especially for transitions initially from the ground state) are very small. It also shows that the resonances attached to the n ¼ 6; 7 levels contribute significantly to the total rate coefficients of transitions to and among the n ¼ 5 and 6 levels. At higher temperature, the RE enhancements become smaller, being larger than 20% (up to one order of magnitude) for 106 transitions at Te ¼ 107 K. 5. Total effective collision strengths The total effective collision strengths including both DE and RE contributions are listed in Table 5 for the 630 transitions among the lowest 36 levels of the n r 6 configurations over a wide temperature range up to 109 K. The RE contributions from the resonances attached to the n r 7 levels are taken into account, considering both RS and DAC transitions. The excitation qij and de-excitation qji rate coefficients (in cm3 s1 ) can be determined straightforwardly using U from the following relations [11]: qij ¼

8:629  106 1=2

gi Te GðkT e =me c2 Þ

UexpðEij =kT e Þ

ð8Þ

U

ð9Þ

and qji ¼

8:629  106 1=2

gj Te GðkT e =me c2 Þ

where gi and gj are the statistical weights of the initial (i) and final (j) states; Eij is the energy difference between

levels i and j; GðkT e =me c2 Þ is a relativistic corrective factor. For kT e =me c2 o 1, which is generally true for most cases of interest, it is given by [11] GðxÞ ¼ 1 þ

15 105 2 315 3 10395 4 xþ x  x þ x þ  8 128 1024 32768

ð10Þ

where x ¼ kT e =me c2 . The above factor can generally be approximated by unity when one is considering the rate coefficients at relatively low temperatures, as widely used in the literature (for example see [3]). Aggarwal et al. [3] and Ballance et al. [4] reported the effective collision strengths for all the 300 transitions among the lowest 25 levels of the n r5 configurations of Fe XXVI. The DARC code without considerations of radiation damping was used in the former. The latter employed a radiation-damped Breit–Pauli R-matrix (BPRM) approach, taking into account the radiation damping for the high-n members of a Rydberg series. To assess the reliability and accuracy of the present IPIRDW calculation, two additional FAC calculations namely undamped and damped n r 5 ones have been performed, in which the RE contributions only through the resonances attached to the n r5 levels were included without and with considerations of the RS transitions, respectively. The calculated U values are then compared, respectively, with the DARCs [3] and BPRMs [4], as shown in Fig. 6. It can be seen from Fig. 6(a) that the present undamped n r 5 values of U differ significantly (by over 28%) from the DARCs over the entire temperature range of 105:6 2107:7 K only for three elastic transitions, namely 14–15, 21–22 and 23–24. It is due to the large differences between the O values as shown in Fig. 1. At extremely low electron temperature (105:6 K), the undamped FAC values of U are smaller by over 20% (up to 52% for transition 2–18) than the DARCs for 16 transitions while the former are larger than the latter by over 20% (up to 48% for 7–8) for six transitions. Eighteen of above 22 transitions which show large discrepancies are dipole-allowed transitions,

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Table 5 Effective collision strengths for the 630 transitions among the 36 levels of n r 6 configurations of Fe XXVI (X 7 Y ¼ X  10 7 Y ). i

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

j

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Electron temperature (K) 5:00 þ 05

1:00þ 06

2:00þ 06

5:00 þ 06

1:00 þ 07

2:00 þ 07

5:00 þ 07

1:00þ 08

2:00þ 08

5:00 þ 08

1:00 þ 09

1:4903 1:1403 2:9003 3:3904 2:6204 8:7605 6:5504 1:2204 1:4504 1:1504 5:7505 2:7004 7:7205 2:2905 2:7305 7:8505 6:4405 3:4105 1:4804 4:5805 1:3305 1:6305 9:1106 1:0505 3:2005 3:9705 7:9605 1:4605 2:0605 2:2506 2:7506 2:5007 2:9407 3:5208 4:3408

1:5203 1:1603 2:9503 3:6004 2:8304 1:0904 6:8304 1:5104 1:4604 1:1704 5:7405 2:7304 7:7405 2:2405 2:6705 7:3605 6:0105 2:9505 1:4204 4:0305 9:4506 1:1605 5:7906 6:6806 2:9905 3:7605 7:5105 1:3205 1:8805 1:8806 2:3306 1:8007 2:1207 2:3008 2:8408

1:5903 1:2203 3:0703 3:6804 2:9204 1:1504 6:9404 1:5804 1:4104 1:1304 5:1605 2:6704 7:0305 1:7505 2:1005 6:8505 5:4905 2:4805 1:3404 3:4605 6:3406 7:8106 3:3806 3:9206 2:8005 3:6005 7:1405 1:2005 1:7105 1:5606 1:9606 1:2307 1:4607 1:3508 1:6608

1:6603 1:2703 3:2003 3:5704 2:8104 9:9305 6:8104 1:3704 1:3304 1:0404 4:1505 2:5604 5:7605 1:0305 1:2405 6:3705 4:9605 2:0205 1:2504 2:8605 3:6806 4:5706 1:5206 1:7606 2:6405 3:4605 6:8605 1:0705 1:5305 1:2606 1:5906 7:5808 9:0508 6:1709 7:5709

1:6803 1:2703 3:2503 3:4704 2:6704 8:1705 6:6904 1:1404 1:2904 9:8505 3:5105 2:5104 4:9305 6:4006 7:7606 6:2005 4:7305 1:7605 1:2204 2:5205 2:5106 3:1406 8:0307 9:3207 2:5805 3:4205 6:7805 9:8006 1:4105 1:0706 1:3606 5:4308 6:5008 3:3609 4:1109

1:7403 1:2503 3:3703 3:4604 2:5704 6:6905 6:7604 9:4505 1:2804 9:5305 3:0005 2:5304 4:2805 3:9606 4:8406 6:1905 4:6005 1:5605 1:2304 2:2305 1:7706 2:2206 4:2007 4:8807 2:5505 3:4505 6:8505 8:9006 1:2805 8:7907 1:1206 3:8708 4:6408 1:8309 2:2309

1:9803 1:2403 3:8703 3:7604 2:5204 5:4705 7:4604 7:8805 1:3804 9:3905 2:5705 2:7504 3:7105 2:1506 2:6706 6:6505 4:5405 1:3605 1:3304 1:9705 1:1306 1:4406 1:7707 2:0607 2:5405 3:7105 7:4405 7:9006 1:1505 6:4307 8:2607 2:3308 2:8108 8:1010 9:8410

2:3903 1:2703 4:7203 4:3904 2:5504 5:2005 8:7704 7:6005 1:5904 9:5105 2:4705 3:2004 3:6205 1:4206 1:7906 7:6305 4:6005 1:3105 1:5404 1:9305 8:3207 1:0706 9:2208 1:0807 2:5805 4:2505 8:5905 7:6806 1:1305 5:0507 6:5407 1:5308 1:8508 4:3310 5:2510

3:1303 1:3303 6:1903 5:5504 2:6604 5:5405 1:1203 8:2105 1:9904 9:8805 2:6305 4:0404 3:9105 1:0306 1:3206 9:4905 4:7805 1:4005 1:9304 2:0805 6:6307 8:6407 4:8508 5:6808 2:6805 5:2805 1:0704 8:1806 1:2205 4:2207 5:5407 1:0108 1:2308 2:3210 2:8010

4:8903 1:4903 9:7203 8:4204 2:9704 6:9505 1:7103 1:0404 3:0004 1:1004 3:3005 6:1104 4:9705 8:4507 1:1106 1:4204 5:3205 1:7505 2:9004 2:6305 5:9707 7:9307 2:2008 2:6008 2:9805 7:8905 1:6204 1:0205 1:5405 3:9607 5:2807 6:7809 8:3709 1:0510 1:2710

7:5103 1:7803 1:4902 1:2703 3:5104 9:0705 2:5803 1:3704 4:5004 1:3004 4:3105 9:2004 6:5105 8:8807 1:1906 2:1304 6:2905 2:2805 4:3704 3:4505 6:5707 8:8107 1:3708 1:6408 3:5305 1:1804 2:4304 1:3405 2:0205 4:4307 5:9607 6:1909 7:7409 6:3611 7:7111

Note: Only data for the transitions involving the ground state are shown here. Table 5 is available online in its entirety in the homepage of JQSRT. A portion is shown here for guidance regarding its form and content.

Fig. 6. Ratios of the present undamped and damped U values from the n r 5 calculations to (a) the DARCs [3] and (b) BPRMs [4] are plotted against electron temperature. (a) In the calculation of UFAC , no radiative transitions from the resonances are taken into account. (b) In the calculation of UFAC , RS transitions are considered. The dashed lines show the range where the ratios fall into 0.8–1.2.

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Fig. 7. Ratios of the present FAC effective collision strengths listed in Table 5 to the (a) DARCs [3] and (b) BPRMs [4] are plotted against final state indices of transitions, at electron temperature of 106 K. In both figures, the squares, circles, triangles, stars, and crosses are for the transitions from level Nos. 1, 2, 3, 4 and others, respectively. The dashed lines show the range where the ratios fall into 0.8–1.2.

such as 2–6, 11, 18 and 4, 5, 8, 10–18. In addition, four of them are elastic ones, namely 7–8, 12–13, 13–14 and 12–15. The resonance enhancements are found to be negligible for most of these dipole-allowed transitions. The possible large discrepancies of the background collision strengths nearby threshold between the FAC and DARC calculations, as shown in Fig. 1, could account for the above large differences of U values at low temperature. As temperature increases, the agreement become better. At 106 K, the differences exceed 20% for only 12 transitions, namely 1–24,25; 2,5–11; 7–8,17; 9–20; 12–13 and 2,4,5,8–18. At 107:7 K, the two sets of U values agree within 12% for all the 300 transitions except for 14–15, 21–22 and 23–24, which are elastic. For these and other elastic transitions, results of Aggarwal et al. [3] should be preferred. It can be seen from Fig. 6(b) that the overall agreement between the present U values at electron temperature below 108 K from the damped n r5 calculation and the BPRMs [4] is somewhat similar as the above between the undamped FAC and DARC [3] calculations. At temperature of 106 K, the two sets of U values differ by over 20% for 26 transitions, most of which are the excitations to and among the n ¼ 5 levels, such as 2, 3, 4, 6–18, 19–23 and 22–25. At 108 K, the differences are over 20% for only eight transitions, namely 1–6, 11, 18; 2–17, 3–18, 14–15, 21–22 and 23–24. As electron temperature increases further, the present FAC results become larger than the BPRMs for more and more transitions. At 109 K, the FAC values of U are larger by over 28% (up to 80% for 1–18) for all the 300 transitions except for two very weak transitions 1–24 and 1–25. This kind of discrepancy between the FAC and BPRM values of U at extremely high temperature is obviously due to the different treatments of the highenergy collision strengths as discussed in Section 3. The above comparisons show that the present undamped and damped n r5 calculations can reproduce

well the U values of the DARCs [3] and BPRMs [4] within 20% at electron temperatures below 108 K for most of the 300 transitions among the lowest 25 levels of Fe XXVI. This gives us confidence in our results from the n r 7 calculation. For instances, the U values listed in Table 5 are compared to the DARCs and BPRMs in Fig. 7, at Te ¼ 106 K. As seen from Fig. 7, it shows that the present FAC results at low temperatures are smaller by over 20% than the DARCs for 12 (namely 1–3; 4–5; 2,4–6; 7–10; 9–10; and 2, 4, 5, 7, 8, 9–11) and BPRMs for five (4–5, 7–10 and 5, 8, 9– 11) excitations, respectively. The present inclusion of both RS and DAC transitions, reducing significantly the rate coefficients for these excitations as discussed in Section 4, accounts mainly for the above differences. Our U values are larger by over 20% (up to two orders of magnitude for 1–24 and 1–25) than the DARCs and BPRMs for about 150 of the 300 transitions among the n r5 levels at low electron temperatures. Most of them are those towards and among the n ¼ 5 levels. It is obviously due to that the resonances attached to the n ¼ 6 and 7 levels are additionally included in the present n r 7 calculation, which show large enhancements for these transitions as seen in Fig. 5. At high temperature, the n r 7 results are in similar agreement as the n r5 calculations with the two R-matrix calculations, being larger by over 20% than the DARCs for 13 transitions at Te ¼ 107:7 K and the BPRMs for 298 transitions at Te ¼ 109 K. In Fig. 8, we show the detailed comparisons of the present results with the DARCs [3] and BPRMs [4] for seven transitions: 1–11, 1–18, 2–6, 3–18, 4–6, 19–23, and 22–25, for which there are large discrepancies between the DARC and BPRM results, as pointed out by Aggarwal et al. [3]. It can be seen that present n r7 U values of transitions involving the n ¼ 5 levels are all larger than the DARCs and BPRMs at low temperature. At high temperature (above 108 K) the FACs are also larger than the BPRMs for these seven transitions. The present undamped and damped U

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Fig. 8. Comparisons of the present effective collision strengths with the DARCs [squares, [3]] and BPRMs [circles, [4]] ones for transitions: (a) 1–11, (b) 1–18, (c) 2–6, (d) 3–18, (e) 4–6, (f) 19–23, and (g) 22–25. The solid lines are the present damped nr 7 results, considering both RS and DAC transitions. The dashed- and dotted-lines are the present nr 5 undamped and damped results, respectively.

values from the n r 5 calculations are also plotted in Fig. 8. The undamped n r 5 FACs agree with the DARCs over the entire temperature range of 105:6 2107:7 K for these seven

transitions to within 10%, with only one exception up to 23% at extremely low temperature of 105:6 K for transition 2–6. The BPRMs are lower than both the DARCs and the

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n r 5 FACs over the same temperature range for transitions 1–11, 1–18, and 3–18 while they are larger at relatively low temperatures for transitions 19–23 and 22–25. For transitions 2–6 and 4–6, the inclusion of RS and DAC transitions could significantly reduce the effective collision strengths at low temperature. The deviations among the FAC, DARC and BPRM results for this two transitions are mainly due to the inclusion of the RS and DAC transitions whether or not. It should be mentioned that the DARC results at their higher temperature end were possibly underestimated (by about 5%) if no account was taken of collision strengths above 1600 Ry, as discussed in [4].

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Science Foundation of China under Grant nos. 10774026, 10434050 and 10574029, the Chinese Association of Atomic and Molecular Data and National High-Tech ICF Committee in China. It is also partially supported by Chinese National Fusion Project for ITER under Grant no. 2009GB106001, and Shanghai Leading Academic Discipline Project under Grant no. B107. Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi: 10.1016/j.jqsrt.2009. 11.028.

6. Conclusion In conclusion, we report here the results for energy levels, radiative rates, life times, collision strengths, and effective collision strengths for all transitions among the lowest 36 levels belonging to the n r6 configurations of Fe XXVI, employing the FAC computer package. Comparing to the earlier works, we give many more atomic parameters. Results for E1, E2, M1 and M2 transitions are presented. Collision strengths are calculated by employing relativistic distorted-wave approximation in conjunction with relativistic plane-wave approximation. As we use the relativistic asymptotic behaviors to evaluate the high-energy collision strengths, our effective collision strengths at high temperature are generally higher than the previous R-matrix results. Resonances attached to the n r 7 levels are taken into account by the isolated process and isolated resonances approximation using distortedwaves. Inclusion of resonant stabilizing transitions and decays to lower-lying autoionizing levels from the resonances reduce significantly the total effective collision strengths at low electron temperatures for some transitions. Radiation damping effects account for the large discrepancies between the previous two R-matrix calculations for some transitions. The resonances attached to the n ¼ 6 and 7 levels are found to enlarge significantly the effective collision strengths at low temperature for transitions to and among the n ¼ 5 and 6 levels. We hope that a more large-scaled R-matrix calculation, including 36 levels of the n r 6 configurations, could be performed in future to assess the accuracy of present data for these transitions. The present results are extensively compared with the earlier published data. Based on these comparisons, we may assess the accuracies of our results. Our energy levels are assessed to be accurate to about 0.0015%. The accuracy of radiative transitions is probably better than 10%. Our effective collision strengths is estimated to be accurate to about 20%.

Acknowledgments The authors would like to thank an anonymous referee for suggestions that improved the manuscript significantly. This work is supported by the National Natural

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