Electron-impact fine-structure excitation of Fe II at low temperature
Yier Wan, C. Favreau, S. D. Loch, B. M. McLaughlin, Yueying Qi, P. C. Stancil
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r MNRAS , 1–8 (2019) Preprint 12 March 2019 Compiled using MNRAS L A TEX style file v3.0
Electron-impact fine-structure excitation of Fe ii at lowtemperature
Yier Wan, C. Favreau, S. D. Loch, B. M. McLaughlin, Yueying Qi and P. C. Stancil ⋆ Department of Physics and Astronomy, Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA Department of Physics, Auburn University, Auburn, AL 36849, USA Centre for Theoretical Atomic and Molecular Physics (CTAMOP), School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK College of Mathematics, Physics and Information, Jiaxing University, Jiaxing, Zhejiang, 314001, PRC
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT Fe ii emission lines are observed from nearly all classes of astronomical objects overa wide spectral range from the infrared to the ultraviolet. To meaningfully interpretthese lines, reliable atomic data are necessary. In work presented here we focusedon low-lying fine-structure transitions, within the ground term, due to electron im-pact. We provide effective collision strengths together with estimated uncertainties asfunctions of temperature of astrophysical importance ( − , K). Due to theimportance of fine-structure transitions within the ground term, the focus of this workis on obtaining accurate rate coefficients at the lower end of this temperature range,for applications in low temperature environments such as the interstellar medium. Weperformed three different flavours of scattering calculations: i) a intermediate cou-pling frame transformation (ICFT) R -matrix method, ii) a Breit-Pauli (BP) R -matrixmethod, and iii) a Dirac R -matrix method. The ICFT and BP R -matrix calculationsinvolved three different AUTOSTRUCTURE target models each. The Dirac R -matrixcalculation was based on a reliable 20 configuration, 6069 level atomic structure model.Good agreement was found with our BP and Dirac R -matrix collision results comparedto previous R -matrix calculations. We present a set of recommended effective collisionstrengths for the low-lying forbidden transitions together with associated uncertaintyestimates. Key words: atomic data – atomic processes – scattering
Electron-impact is a dominant populating mechanism for theexcited fine-structure levels of Fe + . The energy gap betweenthe ground level of Fe + and its next four excited levels inthe ground term is less than 1500 K (see Figure 1), so thesefine-structure levels can be easily excited and the subsequentemission lines appear in the mid-infrared (mid-IR), there-fore falling within the detector windows of many telescopes,namely the Spitzer Space Telescope , the Stratospheric Obser-vatory for Infrared Astronomy (SOFIA), and the up-coming
James Webb Space Telescope (JWST) . Neufeld et al. (2007)reported the detection of Fe ii (25.99 µ m and 35.35 µ m)emissions in supernova remnants. Perlman et al. (2007) de-tected the Fe ii (25.99 µ m) emission from M87 (the dominantgalaxy in the Virgo Cluster). Green et al. (2010) reportedthe detection of Fe ii (25.99 µ m) emission in the proto-stellar ⋆ E-mail: [email protected] outflow GGD 37. More recently, Harper et al. (2017) investi-gated SOFIA-EXES mid-IR observations of forbidden Fe ii emissions in the early-type M super-giants and spectrallyresolved the Fe ii (25.99 µ m) emission line from Betelgeuse.The Fe ii fine-structure lines can serve as diagnosticsof the local physical conditions of many cool plasma envi-ronments. For example, solving for the thermal balance andchemistry self-consistently, Gorti & Hollenbach (2004) mod-eled the IR spectra from intermediate-aged disks around Gand K stars and found that the Fe ii (25.99 µ m) emissionis among the strongest features. Assuming thermal pres-sure balance, Kaufman et al. (2006) calculated Fe ii (25.99 µ m) emission that may arise from H ii regions and/or pho-todissociation regions (PDRs) in massive star-forming envi-ronments. Of objects for which local thermodynamic equi-librium (LTE) is not valid, the physical conditions can beextracted from the spectra only when the collisional ratesare known. For example, Bautista et al. (1996) studied theexcitation of Ni ii and Fe ii based on collisional data from © Yier Wan et al.
Bautista & Pradhan (1996). Verner et al. (1999) performednumerical simulations of Fe ii emission spectra with the sim-ulation code CLOUDY using the collision strengths fromZhang & Pradhan (1995). The same data set was also usedby Hartigan et al. (2004) to model Fe ii (25.99 µ m) emis-sion as a diagnostic of shocked gas in stellar jets and morerecently by Lind et al. (2017) to study non-LTE line forma-tion of Fe in late-type stars.There has been considerable effort and resources dedi-cated to the computation of electron-impact fine-structureexcitation rates for Fe + . Such calculations are challengingand demanding for many reasons. First, the competition be-tween filling of the d and l shells makes it a non-trivialexercise to obtain a sufficiently accurate atomic structuretarget model. Second, the number of closely coupled chan-nels increases dramatically as more configurations are addedto the target model. A compromise has to be made betweenaccuracy and computational resources. Third, the existenceof Rydberg resonance series below each excitation thresholdrequires a very fine energy mesh in order to obtain reliableeffective collision strengths.Previous calculations can be naturally divided into twogroups according to the different choice of target mod-els. The first group considers only even-parity configu-rations. Nussbaumer & Storey (1980) calculated electron-ion collision strengths for the lowest four terms of Fe + with d s and d in the target model. Berrington et al.(1988) and Keenan et al. (1988) extended this work byincluding a ¯4 d pseudo-orbital and applying the BP ap-proximation. Recently Bautista et al. (2015) reported neweffective collision strengths applying the ICFT R -matrixmethod (Griffin et al. 1998) and the Dirac Atomic R -matrixcode (DARC) (Norrington & Grant 1981; Dyall et al. 1989;Norrington 2004). For the excitation from the ground levelto the first excited level, most of their ICFT calculations give Υ (10 K) of about 2, while their DARC calculation gives Υ (10 K) about 5. However, the electron configurations intheir ICFT target are all of even parity, while the target forDARC contained d p . The usage of different target mod-els as well as R -matrix method makes it difficult to attributethe variation of the collision results to a particular reason.The second group of calculations considered the d p configuration in the target model. Pradhan & Berrington(1993) carried out two sets of close-coupling calculations.The first set included 38 quartet and sextet terms belong-ing to the d s , d and d p using the non-relativistic(NR) LS coupling R -matrix package (Berrington et al. 1987)with ¯4 d correlation orbital included in some of the config-urations. The second set was carried out using the semi-relativistic Breit-Pauli R -matrix package (Scott & Taylor1982) with only 41 fine-structure levels included, primarilydue to the limit of computation capability. These LS cou-pling calculations were then extended by Pradhan & Zhang(1993) and Zhang & Pradhan (1995) to obtain fine-structureeffective collision strengths using a recoupling method.Ramsbottom et al. (2007) presented new fine-structure cal-culations using a parallel Breit-Pauli (BP) R -matrix pack-age (Ballance & Griffin 2004). Their target model con-tained d s , d , d p with additional correlation ef-fects incorporated via the d ¯4 d configuration. In addi-tion, Bautista & Pradhan (1996) and Bautista & Pradhan(1998) studied the influence of including doublets arising E ne r g y ( R y d ) Fe II ( D)4s D J F J ( D)4s D J Figure 1.
Energy diagram for Fe II. from d s and found the collision strengths of the D / - D / transition have similar background values to thosewithout doublets. On average, calculations from the secondgroup tend to give similar effective collision strengths to eachother, and show larger differences with those from the firstgroup.The primary aim of this paper is to evaluate accuratelow temperate rate coefficients for fine-structure transitionswithin the ground term of Fe ii , while previous work pri-marily focused on high temperatures ( ≥ , K). Low-temperature collision data, required for cool plasma envi-ronments, can only be extrapolated from the available high-temperature data, which would inevitably generate largeuncertainties and compromise the reliability of astronomi-cal spectra analysis. A second aim of this work is elucidatethe reason for the inconsistency reported by Bautista et al.(2015). Similar work was performed by Badnell & Ballance(2014) who discussed the differences (about a factor ofthree) between ICFT and DARC calculations for Fe + re-ported by Bautista et al. (2010). Excellent agreement ( < R -matrix codes. For theselection of ions, namely Cr ii , Mn v and Mg viii , which areimportant iron-peak species, the relativistic and LS trans-formed R -matrix approaches all produce rates of a similaraccuracy. We will check if this conclusion is also valid forFe ii . Finally, we want to investigate the sensitivity of theeffective collision strengths to the choice of target models aswell as the adopted R -matrix method. This will allow us toevaluate the uncertainties of our results.The rest of this paper is structured as follows. In Sec-tion 2 we provide a brief guide to the three (ICFT, BP,and DARC) R -matrix methods used in this work. In Sec-tion 3 we built several target models and discuss the resultsof the atomic structure calculations. In Section 4, we gavedetails of six independent R -matrix calculations. Collisionstrengths and effective collision strengths are presented inSection 5. The reliability of the methods and rationale forchoosing recommended effective collision strengths, includ-ing uncertainty estimates, are also addressed. Our findingsare summarized in Section 6. MNRAS , 1–8 (2019) lectron-impact excitation of Fe + The theory behind the R -matrix method has been well docu-mented in the literature (Eissner et al. 1974; Hummer et al.1993; Burke 2011) and many versions of computer pack-ages adopting the R -matrix approach have been developedin the past decades. Generally speaking, for collisions in-volving heavy atoms, the relativistic effects are expected tobe important and have to be included in the scattering cal-culation. In the DARC R -matrix collision program, the rel-ativistic effects are introduced via the Dirac Hamiltonian.In another commonly used semi-relativistic BP R -matrix(BPRM) approach, one-body relativistic terms (relativis-tic mass-correction, one-electron Darwin, and the spin-orbitterm) are considered in the Hamiltonian. Good agreementbetween these two methods was found by Berrington et al.(2005) in the study of Fe + in collision with electrons, whentarget states of the two methods are in agreement and res-onances are resolved adequately.However, computational challenges arises when usingthe DARC or BPRM approaches. The inclusion of thespin-orbit term requires j j (for DARC) and jK (for BP)coupling; the size of the Hamiltonian matrices that needto be diagonalized can become very large. Many frame-transformation methods have been developed to make thecalculations possible and less time-consuming. One such pro-cedure is called intermediate coupling frame transformation(ICFT) (Griffin et al. 1998). The ICFT R -matrix approachcarries only the non-fine-structure terms, mass-correctionand Darwin terms in the Hamiltonian operators of the in-terior region. The spin-orbit term is only considered in theHamiltonian operators for the exterior and asymptotic re-gions. On the boundary, multi-channel quantum defect the-ory (MQDT) is employed to generate LS -coupled ‘unphys-ical’ K -matrices and those matrices are then transformedinto a jK coupling representation. Since the Hamiltonianmatrix in the interior region is written in LS coupling, thediagonalization of the ICFT R -matrix method is an order ofmagnitude more efficient than BPRM.In this work we use all three R -matrix methods de-scribed above, with the calculations for each one optimizedfor the case of fine-structure excitation within the groundterm of Fe ii . For fine-structure excitation of Fe ii within the ground term,low-energy electron-ion collisions are dominated by reso-nance structures. If the ionic states themselves are not accu-rately represented in the target model, this inaccuracy willaffect the collision strengths by shifting the resonance peaksto wrong positions. However, obtaining a sufficiently accu-rate atomic structure for Fe ii is a non-trivial exercise.The atomic structure program AUTOSTRUCTURE(Badnell 1997, 2011) includes one-body relativistic correc-tions and was used to generate targets for the BP and ICFT R -matrix methods. We built three small-scale target models(see Table 1). What we refer to as the 3-even target modelcontains only even-parity configurations, so that the follow-ing scattering calculations exclude dipole transitions, whichshould be much stronger than the fine-structure transitions Table 1.
Breit-Pauli/ICFT target models for Fe ii .Model 3-even 3-mix 4-mixTarget d s , d d s , d d s , d d s d p d p , d d Scaling λ s =1.0000 λ s =1.0000 λ s =1.00000parameter λ s =0.9000 λ s =0.9000 λ s =1.27407 λ p =1.0360 λ p =1.0360 λ p =1.11361 λ s =1.1000 λ s =1.1000 λ s =1.09525 λ p =1.0050 λ p =1.0050 λ p =1.05904 λ d =1.0381 λ d =1.0381 λ d =1.04657 λ s =0.9400 λ s =0.9400 λ s =0.89000 λ p =0.8000 λ p =0.98955 λ d =1.34726target levels 119 262 538target terms 48 100 204 ( N + ) d d d bound d s d { s , p } d { s , p , d } system d s d { s , p } d { s , p , d } d s p d { s p , s d , p d } RA(BP) 12.11523 18.17773 16.92773RA(ICFT) 12.86523 18.17773 -
Notes.
RA (in units of a.u.) represents the R -matrix boundary. in which we are interested. However, as mentioned in Section1, it was found that the d p configuration played a vitalrole in the transitions among the low-lying fine-structurelevels, which is possibly due to its coupling with d s . Inthe 3-mix target model, we include the d p configurationand the same scaling parameters as for target 3-even. Thespectroscopic configuration d d is retained in the 4-mixtarget model.While there is an iterative variational procedure imple-mented in AUTOSTRUCTURE, satisfactory level energiesof the first excited term a F cannot be obtained withoutthe inclusion of d orbitals. To improve the target structurefurther, we developed a code to vary scaling parameters asso-ciated with the Thomas-Fermi-Dirac-Amaldi potential andthen compared the resulting energies until a minimum wasfound in the differences with the NIST (Kramida et al. 2018)level energies. In the code that was developed for this opti-mization, a grid of λ nl parameters was chosen, followed bya comparison with NIST level energies for the levels of theground term. A subset of these, which gave the closest agree-ment with NIST, was then chosen and the level energies ofthe first excited term. A subset of these was then examined,comparing the level energies of the higher excited levels. Inaddition, a comparison with NIST A-values for the transi-tions within ground term was also performed, to sub-selecton the the set of λ nl that were closest to NIST A-values. Itwas found that this method gave better agreement for theenergies of the low lying terms and associated A-values thatthe existing optimization procedure within AUTOSTRUC-TURE, when it was optimized on just the first few terms.This variation method is used for 3-even and 3-mix targetmodels, and the built-in AUTOSTRUCTURE variation pro-cedure is used for 4-mix target model.The target model for the DARC calculation was ob-tained via the multi-configuration Dirac-Fock method us- MNRAS , 1–8 (2019)
Yier Wan et al.
Table 2.
Level energies (in Ry) of Fe + .No. Term/Level Observed a GRASP b B88 b d s D / d s D / d s D / d s D / d s D / d F / d F / d F / d F / d s D / d s D / d s D / d s D / d P / d P / d P / d p D o / d p D o / d p D o / d p D o / d p D o / a Kramida et al. (2018). b R07 = Ramsbottom et al. (2007) and B88= Berrington et al. (1988).
Table 3.
Einstein A coefficient (in s − ) for Fe ii .Transition D / − D / D / − D / D / − D / D / − D / D / − D o / Type M1 M1 M1 M1 E1Wavelength( µ m) 25.988 35.349 51.301 87.384 259.940NIST a b − − − − Notes. a Kramida et al. (2018). b Units of Q × s − . ing the computer package GRASP (Dyall et al. 1989;Parpia et al. 1996). The Fe + target has been in-vestigated by Smyth et al. (2018). We adopt their20 configuration target model: d ; d s , p , d , s , p ; d s , p , d , s p , s d , s , p ; p d , d ; p ed d ; p d d ; and p d d . The full DARC target gives 6069levels. Extended Average Level (EAL) optimization optionwere used in the GRASP structure calculation to optimizethe level energies upon all of the levels. Better overall atomicstructure should be obtained through this method.A selection of fine-structure level energies are presentedin Table 2. Compared with previous work (Berrington et al.1988; Ramsbottom et al. 2007), significant improvements are achieved in the first 9 levels (term D and F ). Target4-mix estimates P term/level energies better than the tar-get 3-mix and 3-even models, but gives worse D term/levelenergies. None of the three small-scale BP/ICFT targetscan predict the D o term/level energies well, mainly be-cause of the limited target size. The DARC target givesvery good D o term/level energies, but overestimates the D term/level energies.Radiative rates (A-values) for fine-structure transitionsin the ground term as well as the first dipole transitionare presented in Table 3 and compared to NIST values(Kramida et al. 2018). The three AUTOSTRUCTURE tar-gets give better M1 transitions, while the DARC target gives MNRAS , 1–8 (2019) lectron-impact excitation of Fe + better E1 transitions. The difference in the results comesfrom the different target wavefunctions and computationalmethods. In our GRASP calculation, energies are optimizedupon all of the level energies included in the calculation, sobetter overall atomic structure should be obtained, whilethe lowest few levels are not so well optimized as com-pared to our AUTOSTRUCTURE targets. We believe thatthis explains some of the differences between the GRASP and NIST M1 A-values. As will be shown later, except forthe 3-even calculation all other targets give similar collisionstrengths. So the final results are not highly sensitive to theA-value differences. We performed two ICFT R -matrix calculations with the 3-even and 3-mix targets, three BP R -matrix calculations with3-even, 3-mix and 4-mix targets, and one DARC calculation.In the 3-mix BP, 4-mix BP, and DARC calculations, the fullconfiguration target was taken through until the Hamilto-nian diagonalization, and then the first 100 levels are shiftedto NIST values and retained in the rest of the calculation.This process will still include all of the important resonancecontributions to the fine-structure excitations reported onin the paper, given the low temperature focus of this work.The scattering calculation included J Π partial wavesfrom J = to J = 30 with 20 continuum basis terms foreach value of angular momentum. Total angular momenta L ≤ and ≤ ( S + ) ≤ were used for both the even andodd parities. The contributions from higher J were obtainedfrom the top-up procedure. The R -matrix boundaries for thedifferent collision calculations were automatically selectedby the R -matrix code. The ( N +1) bound configurations in-cluded in the scattering computations are listed targets inTable 1. Convergence checks on the size of the continuumbasis used was determined by identifying the most dominantpartial waves ( J = . × − Ryd up to 0.1035 Ry and then − Rydup to 0.6035 Ry. Coarse meshes with an interval of − Ryd with different numbers of energy points are tested upto 2.6035 Ryd. Adding more data points within the coarsemesh doesn’t show any noticeable differences in final effec-tive collision strengths.
In Figures 2 and 3 we present the collision strengths as func-tions of the incident electron energy for the fine-structuretransition from the ground level d s D / to the first ex-cited level d s D / . The first point to notice is that whenthe 3-even target is used, either the BP or ICFT collisionapproaches give a set of sharp resonances at energies from0.03 to threshold (0.0089 Ryd). The average value of the col-lision strengths at low energies (below 0.003 Ryd) is about4. This is significantly smaller than previous calculations.Second, with the inclusion of the d p configuration, theICFT, BP, and DARC collision approaches all give much Electron energy (Ryd) C o lli s i on s t r eng t h Figure 2.
Collision strengths for the d s D / - d s D / transition. Top panel: 3-even target model + BP R -matrixmethod; middle panel: 3-even target model + ICFT R -matrixmethod; bottom panel: 3-mix target model + ICFT R -matrixmethod. Electron energy (Ryd)
DARC4-mix BP3-mix BP C o lli s i on s t r eng t h Figure 3.
Collision strengths for the d s D / - d s D / transition. Top panel: DARC calculation; middle panel: 4-mixtarget model + BP R -matrix method; bottom panel: 3-mix targetmodel + BP R -matrix method. Table 4.
Effective collision strengths for select transitions of Fe ii calculated by the DARC approach. The uncertainty % ∆ was es-timated by using the two BP calculations.Temperature(K) D / − D / D / − D / D / − D /
10 3.29 (17%) 4.92 (31%) 3.44 (10%)20 3.81 (18%) 5.60 (33%) 3.70 (12%)100 4.30 (20%) 5.79 (31%) 3.82 (13%)200 4.09 (16%) 5.62 (27%) 4.08 (19%)500 3.72 ( 7%) 5.29 (19%) 4.38 (24%) , 1–8 (2019) Yier Wan et al.
Electron energy (Ryd) C o lli s i on s t r eng t h This work DARCThis work 4-mix BPThis work 3-mix BP
Figure 4.
Collision strengths for the d s D / - d s D / transition. Temperature (K) E ff e c t i v e c o lli s i on s t r eng t h DARC4-mix BP3-mix BP3-even BP3-mix ICFT3-even ICFTR07Z95B88K88
Figure 5.
Effective collision strengths for the d s D / - d s D / transition. Standard deviations are marked as errorbars for the recommended DARC computations. Notation R07,Z95, B88 and K88 denote the results from Ramsbottom et al.(2007), Zhang & Pradhan (1995), Berrington et al. (1988) andKeenan et al. (1988), respectively. broader background features. Third, in the comparison ofthe 3-mix ICFT and 3-mix BP results, the profiles of thetwo curves are generally similar except that the 3-mix ICFTresults at very low energies are enhanced. In Figure 4, wepresent the collision strengths for this transition from thecurrent DARC, 4-mix BP and 3-mix BP calculations, butto electron impact energies as large as 0.5 Ryd. We see thatthere is generally good agreement between the three calcu-lations.The collision strength ( Ω ) tends to vary widely from thenon-resonant background value. Therefore, the Maxwellianaveraged effective collision strength ( Υ ) is preferred in astro-physics, instead of employing the collision strength. We com-puted the thermally averaged effective collision strengths us-ing, Υ ij ( T e ) = ∫ ∞ Ω ij ( E j ) exp (− E j / kT e ) d ( E j / kT e ) , (1)where Ω ij is the collision strength for the transition from Temperature (K) E ff e c t i v e c o lli s i on s t r eng t h Figure 6.
Effective collision strengths for the d s D / - d s D / (top panel), d s D / - d s D / (middlepanel) and d s D / - d s D / (bottom panel) transitions.Black circles are recommended DARC results. Standard devia-tions are marked as error bars. Red and blue curves are results ofRamsbottom et al. (2007) and Zhang & Pradhan (1995), respec-tively. Temperature (K) E ff e c t i v e c o lli s i on s t r eng t h Figure 7.
Effective collision strengths for the d s D / - d s D / (top panel), d s D / - d s D / (middle panel) and d s D / - d s D / (bottom panel) transitions. Symbolsare the same as for Figure 6. level i to j . E j is the final energy of the electron, T e is theelectron temperature in Kelvin and k is Boltzmann’s con-stant. It was shown to be a good approximation for a posi-tive ion that if Ω varies with energy much more slowly thandoes the exponential in equation 1, one may equate Υ to thethreshold value of Ω (Seaton 1953).The effective collision strength for the transition d s D / - d s D / is presented in Figure 5. We clearlysee that applying the 3-even target, the ICFT and BP R -matrix approaches yield good agreement and the results arein reasonable agreement with the previous calculated valuesof Keenan et al. (1988) and Berrington et al. (1988). Theirtarget models only included configurations of even parity aswell. The inclusion of the d p configuration enhances thecalculated effective collision strength and thus our 3-mix, 4- MNRAS , 1–8 (2019) lectron-impact excitation of Fe + Temperature (K) E ff e c t i v e c o lli s i on s t r eng t h Figure 8.
Effective collision strengths for the d s D / - d s D / (top panel), d s D / - d s D / (middle panel) and d s D / - d s D / (bottom panel) transitions. Symbolsare the same as for Figure 6. mix results are more consistent with previous calculationsof Zhang & Pradhan (1995) and Ramsbottom et al. (2007)and with our DARC results (see below).As stated in Section 1, there have been several Fe ii fine-structure data sets calculated, but discrepancies existsmainly due to the different target models and R -matrixmethods adopted. Similar trends occur in this work as well.First, among all the target models, the GRASP target isconsidered to be the best, as generally it predicts the ener-gies of the lowest 16 levels well and for other highly excitedlevels it gives correct relative positions (see Table 2). Amongthe three models used in the BP calculations, target 3-evenis not sufficient. The comparison between the 3-mix BPRMand 3-mix ICFT models shows that the ICFT approach can-not give reliable fine-structure transitions for Fe ii . This maybe due to the fact that the ICFT method solves the innerregion problem in LS coupling for a configuration-mixed tar-get, and thus our ICFT calculations were not shifted to NISTlevel energies. Previous works comparing the BPRM andICFT methods showed very good agreement between thecollision cross sections and rates for both methods, whenthe same target description was used (Badnell & Ballance2014). We expect that the difference between our ICFT andBPRM results was primarily due to these differences in tar-get energies. Thus, while we do not use the ICFT resultswhen calculating our uncertainties, they are an indicationof the likely differences between previous unshifted ICFTcalculations and calculations that shifted to NIST energies.We note that shifting to target energies in an ICFT calcu-lation is described in detail in Del Zanna & Badnell (2014).It is evident in Figure 5 that the 3-mix BP, 4-mix BP, andDARC calculations agree overall. Therefore, we adopted theeffective collision strengths from the DARC calculations asour recommended values. Results from the 3-mix BP and 4-mix BP models are used to calculate the standard deviationfrom the recommended values at each temperature point. Wepresent the recommended effective collision strengths andstandard deviation (marked as error bars) for the other ninefine-structure transitions within the ground term in Figures6, 7, and 8. The results from Ramsbottom et al. (2007) and Zhang & Pradhan (1995) are also plotted for comparison.Part of our results is tabulated in Table 4. In this work we studied the electron-impact fine-structureexcitation of Fe ii . Two ICFT calculations with the 3-evenand 3-mix targets, three BPRM calculations with 3-even,3-mix and 4-mix targets, and one DARC calculation basedon a reliable 20 configuration atomic structure model weretested and small-scale computations were performed. Thefull configuration target was taken through until the Hamil-tonian diagonalization, and then the first 100 levels wereshifted to NIST values and retained in the rest of the calcula-tion. The effective collision strengths for low-lying forbiddentransitions are presented. In this paper, we are mostly inter-ested in the rates at low temperatures, from 10 to 2,000 K,but we also include high-temperature results up to 100,000K to compare with the plethora of previous calculations. Itturns out that our results yield good agreement with somelarge-scale calculations even at high temperatures.We found the inclusion of d p is essential for reliablefine-structure transition data. In our 3-even BPRM/ICFTcalculations when d p was not included, the dominantfine-structure transition D / − D / was underestimatedcompared to other calculations, which is similar to the find-ings in Bautista et al. (2015). For the excitation from theground level to the first excited level, most of their ICFTcalculations give Υ (10 K) of about 2, while their DARCcalculation gives Υ (10 K) about 5. However, the electronconfigurations in their ICFT target are all of even parity,while their DARC target contained d p . When d p wastaken into consideration, our 3-mix/4-mix BPRM calcula-tions are in good agreement with DARC calculations. It is anindication that the discrepancy in the work of Bautista et al.(2015) likely depends on the difference in configuration ex-pansion.The resulting level energies as well as Einstein A coef-ficients from the atomic structure calculations were used toevaluate the reliability of the target model. The GRASP ,3-mix and 4-mix AUTOSTRUCTURE target models couldgive good overall atomic structure. The effective collisionstrengths from the DARC calculations were adopted asthe recommended values. The uncertainties were evaluatedby calculating the standard deviation of 3-mix and 4-mixBPRM results from the recommended values. The completedata set is available online in favor of astrophysical envi-ronment modeling. ACKNOWLEDGEMENTS
We would like to thank Dr. Connor Ballance for his as-sistance with the Dirac and BP R-matrix codes and cal-culations and Dr. Manuel Bautista for assistance with theAUTOSTRUCTURE package. BMMcL thanks the Univer-sity of Georgia for the award of an adjunct professorship,and Queen’s University for a visiting research fellowship. , 1–8 (2019) Yier Wan et al.
Computing resources were provided by the Georgia Ad-vanced Computing Resource Center, the UNLV NationalSuper Computing Institute, and the UGA Center for Sim-ulational Physics. This work was funded by NASA grantNNX15AE47G.
REFERENCES A TEX file prepared bythe author. MNRAS000