Mutual neutralisation in Li^++H^-/D^- and Na^++H^-/D^- collisions: Implications of experimental results for non-LTE modelling of stellar spectra
Paul S. Barklem, Anish M. Amarsi, Jon Grumer, Gustav Eklund, Stefan Rosén, MingChao Ji, Henrik Cederquist, Henning Zettergren, Henning T. Schmidt
DDraft version December 23, 2020
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Mutual neutralisation in Li + +H − /D − and Na + +H − /D − collisions: Implications of experimentalresults for non-LTE modelling of stellar spectra Paul S. Barklem, Anish M. Amarsi, Jon Grumer, Gustav Eklund, Stefan Ros´en, MingChao Ji, Henrik Cederquist, Henning Zettergren, and Henning T. Schmidt Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden Department of Physics, Stockholm University, Stockholm, 10691, Sweden (Received soon; Revised later; Accepted even later)
Submitted to ApJABSTRACTAdvances in merged-beams instruments have allowed experimental studies of the mutual neutrali-sation (MN) processes in collisions of both Li + and Na + ions with D − at energies below 1 eV. Theseexperimental results place constraints on theoretical predictions of MN processes of Li + and Na + withH − , important for non-LTE modelling of Li and Na spectra in late-type stars. We compare exper-imental results with calculations for methods typically used to calculate MN processes, namely thefull quantum (FQ) approach, and asymptotic model approaches based on the linear combination ofatomic orbitals (LCAO) and semi-empirical (SE) methods for deriving couplings. It is found that FQcalculations compare best overall with the experiments, followed by the LCAO, and the SE approaches.The experimental results together with the theoretical calculations, allow us to investigate the effectson modelled spectra and derived abundances and their uncertainties arising from uncertainties in theMN rates. Numerical experiments in a large grid of 1D model atmospheres, and a smaller set of 3Dmodels, indicate that neglect of MN can lead to abundance errors of up to 0.1 dex (26%) for Li atlow metallicity, and 0.2 dex (58%) for Na at high metallicity, while the uncertainties in the relevantMN rates as constrained by experiments correspond to uncertainties in abundances of much less than0.01 dex (2%). This agreement for simple atoms gives confidence in the FQ, LCAO and SE modelapproaches to be able to predict MN with the accuracy required for non-LTE modelling in stellaratmospheres. INTRODUCTION1.1.
Non-LTE modelling of stellar spectra
The uncertainty regarding the effects of low-energyhydrogen atom collisions on non-local thermodynamicalequilibrium (non-LTE) modelling of the spectra of late-type stars like the Sun has been a long-standing prob-lem (e.g. Plaskett 1955; Gehren 1975; Steenbock & Hol-weger 1984; Lambert 1993; Asplund 2005; Barklem 2012,2016a). Many initial studies both of the astrophysicalmodelling, and of the collision physics, focussed on Liand Na, due to their relative simplicity from an atomicphysics point of view as well as their astrophysical im-portance. However, the focus of initial studies was onexcitation processes. For this reason, Fleck et al. (1991)
Corresponding author: Paul S. [email protected] performed an experimental study of the processNa(3 s ) + H → Na(3 p, s ) + H (1)at moderately low collision energy (down to ∼
10 eV) andmade an analysis in terms of the avoided curve crossingsin NaH potentials and the Landau-Zener model for dy-namics at the crossings, showing that the data couldbe reasonably explained through such a model. Belyaevet al. (1999) later performed full quantum calculations(FQ, quantum chemistry with quantum dynamics) downto the threshold (2.1 eV), finding excellent agreementwith the available experimental results above ∼
10 eV.It should be noted here that the relevant collision ener-gies in late-type stellar atmospheres are of order 0.2 –0.7 eV.Later, Belyaev & Barklem (2003) made de-tailed quantum scattering calculations for the excita- a r X i v : . [ a s t r o - ph . S R ] D ec Barklem et al. tion/deexcitation processesLi( nl ) + H (cid:29) Li( n (cid:48) l (cid:48) ) + H (2)including the 9 lowest states of Li up to 4 f . They alsoincluded the charge transfer processesLi( nl ) + H (cid:29) Li + + H − (3)based on the results of Croft et al. (1999a) and Croftet al. (1999b). This process naturally arises from theavoided curve crossing mechanism and was found toshow large cross sections compared to the excitation anddeexcitation processes. The forward reaction in equa-tion 3 is referred to as ion-pair production, and the re-verse process mutual neutralisation (MN). A completeset of rate coefficients for excitation/dexcitation and ion-pair production/mutual neutralisation processes involv-ing Li were produced by Barklem et al. (2003), and ap-plied to non-LTE modelling of stellar spectra. This workshowed that excitation/deexcitation processes were infact unimportant in the astrophysical modelling of theLi resonance line, but the charge transfer processes hadsignificant effects, a result later confirmed by a largerstudy across many different stellar atmosphere models(Lind et al. 2009). MN processes were later also seento be important for Na and its spectrum (Belyaev et al.2010; Barklem et al. 2010; Lind et al. 2011).At the time of these studies, it was difficult to esti-mate the uncertainties in the MN cross sections in theenergy regime relevant to late-type stellar atmospheres,roughly 0.2 – 0.7 eV, and thus in the rate coefficients forrelevant temperatures, roughly 2000 – 7000 K. Peart &Foster (1987) had measured total cross sections for MNin Li + +H − collisions in an inclined-beams experimentcarried out in Newcastle, but at energies above 33 eV.Peart & Hayton (1994) later measured MN in Li + +D − collisions down to ∼ . + +H − and Li + +D − , which allowed directcomparison with the experimental results for Li + +D − ofPeart & Hayton (1994) at low energy. After a correctionfor the finite aperture of the detectors in the experiment,the results were found in reasonable agreement, differ-ing by about 10% at energies in the range 10 – 50 eV,and the deviation increased with decreasing energy toreach 20% at the lowest measured energy of around 0.7eV. However, this comparison does not probe the energyrange of most interest in stellar atmospheres, a few 0.1eV. Croft et al. (1999b) extended their calculations forLi + +H − to lower energies (meV energies), allowing cal-culations of rate coefficients for temperatures of a fewthousand degrees. Based on the poorer agreement with experiment forLi + +D − at lower collision energies, Barklem et al.(2003) performed numerical experiments for three starsassuming an error of 50%. For example, in the classi-cal metal-poor star HD140283 it was found that includ-ing the calculated MN processes, compared to ignoringthem, changed the Li I resonance line strength by 20%,while if half of the MN rates are used the line strengthchanged by 18%. Thus, the MN rates do not influencethe modelled line strength linearly (due to the generalnon-linear behaviour of the statistical equilibrium equa-tions and competition with other processes), and thelatter was found to be relatively insensitive to the uncer-tainty in the MN rate. However, this uncertainty wouldstill give uncertainties in line strengths and abundancesof a few per cent, and was not well constrained. Giventhat spectral lines are regularly measured to better thanper cent level, and that more accurate abundances willyield more information, it is useful to constrain this un-certainty further. Further, this numerical experimentonly probes the effect of uncertainty in the total rate,but not the possible effect of changes in the branchingfractions between final states.1.2. Measurements of mutual neutralisation andcomparisons with atomic theory
Recent advances in merged-beams instruments haveopened the way for experimental studies of MN at lowenergies ( < + + D − → Li(3 s, p, d ) + D , (4)total cross sections and branching fractions have beenmeasured in Louvain (Launoy et al. 2019), and branch-ing fractions in Stockholm (Eklund et al. 2020) at theDESIREE infrastructure (Thomas et al. 2011; Schmidtet al. 2013). Results for branching fractions have alsobeen obtained forNa + + D − → Na(4 s, d, p ) + D (5)reactions at DESIREE (Eklund et al. in prep.). Wenote here that the low energy experimental results usingmerged-beams instruments from Louvain, DESIREE, aswell as the earlier results from Newcastle, all considerMN predominantly with D − , despite that it is H − thatis of interest for astrophysics. The maximum mass ratiofor any given merged-beams instrument is determined bythe ratio of the highest to the lowest energy ion beamsthat can be handled. For the existing instruments thevalue of this ratio is of the order of 10 for low (nearzero eV) relative collision energies. For the specific caseof DESIREE, the limit is a factor of 20 as reported in i + +H − /D − and Na + +H − /D − collisions + +H − (in fact such an experiment was also performedfor this case in Louvain for one energy as a check on theLi + +D − branching fraction results) there are variousadvantages to using D − , including that if the massesare more similar it is easier to produce beams of thesame velocity (leading to low relative velocity and thuslow energy collisions). In addition, heavier isotopes areless affected by stray magnetic fields.The new experimental results at low energies allowus to constrain the uncertainties in the rate coefficientsused in non-LTE modelling of stellar spectra with muchgreater precision than before. Further, since the the-oretical and astrophysical modelling studies mentionedabove in the period 2003–2011, two asymptotic modelapproaches to the problem of hydrogen collisions havebeen developed, both based on Landau-Zener dynam-ics of the avoided ionic curve crossing mechanism, butwith different methods for calculating the couplings atthe avoided crossings. The first method, developed byBelyaev (2013), uses semi-empirical (SE) couplings fromOlson et al. (1971). The second method, developed byBarklem (2016b, 2017), employs a two-electron linearcombination of atomic orbitals (LCAO) method, extend-ing the method of Grice & Herschbach (1974) and Adel-man & Herschbach (1977). These asymptotic model ap-proaches have allowed estimates of hydrogen collisionrates, including MN, for many astrophysically interest-ing and complex atoms, such as Al (Belyaev 2013), Ca(Barklem 2016b), Mn (Belyaev & Voronov 2017; Grumer& Barklem 2020), Fe (Barklem 2018a), C, N, (Amarsi &Barklem 2019) and O (Barklem 2018b), among others.In view of these theoretical developments, as well asthe new experimental results, we now approach the ques-tion of the uncertainty in MN rates, and the direct im-plications for modelling Li and Na spectra. We will dis-cuss how the new information may shed light on theaccuracies of the asymptotic model approaches in gen-eral (Sect. 2). Further, with increased computing power,as well as continued development of radiative transfercodes, it is now feasible to perform non-LTE spectrumsynthesis across very large grids of 1D or small grids of3D stellar atmospheres to assess the impact of the un-certainties in MN rates more thoroughly (Sect. 3). Fi-nally the conclusions are presented and in particular itis found that though there are some differences betweenthe theoretical predictions from the FQ, LCAO and SEmethods, and some discrepancies with the experimentalresults, the methods are all sufficiently accurate (basedon comparison with experiment) and sufficiently simi-lar that the choice of theoretical MN description among these methods for non-LTE modelling of Li and Na spec-tra in late-type stars has only small effects on the mod-elled spectral lines, and thus small effects on derivedabundances. EXPERIMENTAL RESULTS ANDCOMPARISON WITH THEORYAs discussed above, the present experimental resultsare for MN with D − , rather than the case of actual as-trophysical interest H − , and this has consequences forour comparisons. The isotopic differences between theXH and XD (here X is Li or Na) interaction potentialsand couplings due to the mass (e.g. nuclear motion,mass polarisation) and nuclear spin differences (e.g spinorbit coupling) are expected to be small. These effectsare often neglected in molecular structure calculations,as was done by Croft et al. (1999a) in their calculationsfor LiH and LiD where they used the same potentials forLi + +H − and Li + +D − . Regarding the collision dynam-ics, at high energies cross sections are expected only tobe a function of the relative collision velocity, and thusisotope effects are also negligible. However, trajectoryeffects arise due to the mass difference at low energies.In particular, Coulomb focussing causes the cross sec-tions for processes with H − to be larger than those forD − , since for given values of the relative energy andimpact parameter the lighter anion will have a shorterdistance of closest approach to the cation. Since partialcross sections into different final states are affected dif-ferently, this affects the branching fractions. Note thatquantum dynamical calculations will also show interfer-ence effects that may differ between masses due to thetrajectory effects.As it is H − that is of interest in astrophysics, the FQcalculations that have been performed so far have oftenfocussed on MN with H − . In the case of Li, while Croftet al. (1999a) calculated for Li + +D − down to 0.68 eV tocompare with the experimental results of Peart & Hay-ton (1994), Croft et al. (1999b) calculated for Li + +H − down to meV energies to enable rate coefficients to becalculated. For Na + +H − , Dickinson et al. (1999) havemade FQ calculations down to meV energies, while noFQ calculations are available for Na + +D − . For thisreason, it is useful to be able to present experimen-tal and theoretical results for MN with D − and H − together, and to present the comparisons on the re-duced energy scale, rather than with respect to abso-lute collision energy. The reduced energy is defined as E R = E CM /µ = v , where E CM is the collision energyin the centre-of-mass frame, µ is the reduced mass of Barklem et al. the system , and v is the relative velocity. On this scalethe collisions for both D − and H − have the same rela-tive velocity v at the same E R , and this scaling has anadditional advantage of being independent of referenceframe. At high E R the results for H − and D − shouldagree on this scale, while at low E R , trajectory effectsgive rise to differences.Comparison of calculations for Li + +H − and Li + +D − with the LCAO model show the effect of Coulomb fo-cussing on the total cross section is 4, 25, and 54 percent at E R =10, 1, and 0.1 eV, respectively (see belowand Fig. 2), while the effects on branching fractions atlow energy are typically 5 per cent (in absolute terms) orless (see below and Fig. 1). For Na + +H − and Na + +D − the effects are similar: 4, 25, and 61 per cent at E R =10,1, and 0.1 eV, respectively (see below and Fig. 4), andfor branching fractions are also typically 5 per cent orless (see below and Fig. 3).Our comparisons will focus on available FQ results, aswell as the LCAO and SE asymptotic models, as theseare the approaches that have been used to generate ratecoefficients and applied in astrophysics, and for whichtotal and partial FQ cross sections as a function of col-lision energy are available or can be calculated by uswith existing codes based on LCAO or SE methods. Re-sults from the SE model presented here are calculatedby us with the same code as used for the LCAO model,such that the only difference between the two sets ofcalculations is that the couplings are calculated usingthe semi-empirical formula of Olson et al. (1971) in theSE case. The SE model approach used by Belyaev andcollaborators uses this formula for the couplings, butnaturally their descriptions and codes will differ slightlyin other ways. In any case, to our knowledge they havenot made any calculations for Li and Na, and furtherthis approach allows us to separate out the key effectsof different ways to calculate the coupling strength.2.1. Li + + H − /D − Launoy et al. (2019) have performed merged-beamsexperiments for Li + +D − in Louvain, obtaining totalMN cross sections for a large number of collision en-ergies, E CM , between 3.9 meV and 1.1 eV, as well asbranching fractions into the 3 s , 3 p and 3 d states atthree energies, E CM = 3.9, 20, and 200 meV. Theyalso obtained branching fractions for Li + +H − at E CM =3 meV, which are in agreement with their Li + +D − val-ues, though no error estimates are given. Eklund et al.(2020) have with DESIREE obtained branching frac- The reduced masses are for Li+H 0.88, Li+D 1.56, Na+H 0.97,and Na+D 1.85 amu. tions into the 3 s state for Li + +D − MN at E CM = 78,262, and 630 meV. The 3 p and 3 d states are not resolved,but the measured 3 s branching fraction (57.8 ± s production are in reasonable agree-ment, and together imply a rather flat trend at low en-ergy, in good agreement with the theoretical predictions.The FQ calculation of Croft et al. (1999a) for Li + +D − agrees very well with the DESIREE result at E CM = 623meV ( E R = 404 meV/amu), but unfortunately the cal-culations do not extend to lower energies. The Li + +H − branching fraction for 3 s from Croft et al. (1999b) isgenerally of order 0.05 higher than the calculation forLi + +D − , the result of Coulomb focussing.In Fig. 2, the experimental total cross sections ofLaunoy et al. (2019), and those at higher energies fromPeart & Hayton (1994), are compared with calculations.It should be noted that the Peart & Hayton (1994) re-sults plotted do not include the finite aperture correctionimplied by Croft et al. (1999a), which would increasethe total cross section by roughly 30% at the lowestmeasured energy. The FQ results for Li + +D − of Croftet al. (1999a) are generally larger than both sets of ex-perimental results in the region E R = 0 . − E R (cid:46) . E − . Correction of the Li + +H − FQ results for Coulomb focussing at E R (cid:46) . i + +H − /D − and Na + +H − /D − collisions s b r an c h i ng f r a c t i on Theory - FQTheory - FQ (H)Theory - LCAOTheory - LCAO (H)Theory - SETheory - SE (H)Experiment - LouvainExperiment - DESIREE b r an c h i ng f r a c t i on -3 -2 -1 b r an c h i ng f r a c t i on Reduced collision energy E R [eV/amu] Figure 1.
Branching fractions for the MN reaction Li + +D − → Li( nl ) + D as a function of reduced collision energy.The 3 s , 3 p and 3 d channels are shown in separate panels.Experimental results from Louvain (Launoy et al. 2019) andfrom DESIREE (Eklund et al. 2020) are shown, with esti-mated errors (1 σ ). Theoretical results are shown from FQcalculations (Croft et al. 1999a), and from our calculationsusing both the LCAO method (Barklem 2016b) and semi-empirical (SE) couplings from Olson et al. (1971). Theoret-ical results are also shown for the case where D − is replacedby H − , marked by (H) in the legend. For Li + +H − the FQresults are from Croft et al. (1999b). results at low energy, though the FQ results deviate themost. Based on the scatter between the different cal-culations, and the general agreement with experiment,it seems reasonable to conclude that the uncertainty inthe total cross section from any given theoretical calcu-lation is not worse than 30%. Perhaps even more im-portantly, the comparisons indicate that the uncertaintyis roughly constant in the region of E − behaviour; the -15 -14 -13 -12 -11 -10 -3 -2 -1 T o t a l c r o ss s e c t i on [ c m ] Reduced collision energy E R [eV/amu] Theory - FQTheory - FQ (H)Theory - LCAOTheory - LCAO (H)Theory - SETheory - SE (H)Experiment - LouvainExperiment - Newcastle
Figure 2.
Total cross sections for the MN reaction Li + +D − → Li+D as a function of reduced collision energy. Exper-imental results from Louvain (Launoy et al. 2019) and fromNewcastle (Peart & Hayton 1994) are shown, with estimatederrors (1 σ ). Theoretical results are shown from FQ calcu-lations (Croft et al. 1999a), and from our calculations usingboth the LCAO method (Barklem 2016b) and semi-empirical(SE) couplings from Olson et al. (1971). Theoretical resultsare also shown for the case where D − is replaced by H − ,marked by (H) in the legend. For Li + +H − the FQ resultsare from Croft et al. (1999b). growing difference between theory and experiment seenin Croft et al. (1999a) was a result of the lowest mea-sured energies being in the “knee” region between the E − behaviour of the cross section at low energy, andthe flatter behaviour at high energy. The FQ resultsagree best with the 3 s and 3 p branching fractions, fol-lowed by the LCAO model, while the SE model resultsperform worst, often disagreeing by of order 0.2. For3 d none of the theoretical models match the experimen-tal results. Considering all the comparisons together, wesuggest that the strongest experimental constraint is the3 s branching fraction, as there are two independent ex-perimental studies, and the experimental uncertaintiesare significantly smaller than the differences between themodels. For Li + +H − / D − we conclude that the physi-cally most advanced FQ theory performs best while theless sophisticated LCAO and SE models are somewhatless successful. 2.2. Na + + H − /D − For Na + +D − MN reactions, branching fractions havebeen obtained with DESIREE for the production of4 s , 3 d , and 4 p final states at three energies, E CM =79 , ,
392 meV (Eklund et al. in prep.). All threestates are resolved, though we note that at higher en-ergy the error bars are larger due to the decreased reso-
Barklem et al. lution and thus increased difficulty to resolve 3 d , and4 p . The obtained branching fractions are comparedwith FQ, LCAO and SE theoretical calculations for bothNa + +D − and Na + +H − in Fig. 3. For Na + +D − thereare no FQ calculations, while there are FQ calculationsfor Na + +H − (Dickinson et al. 1999). On the assumptionthat the differences between Na + +D − and Na + +H − branching fractions for LCAO and SE calculations re-flect the size of the Coulomb focussing effect, one expectsthat FQ calculations for Na + +D − would lead to a low-ering of the 4 s branching fraction compared to that seenfor Na + +H − , and an increase in the 3 d and 4 p branchingfractions. In the 4 s and 4 p cases the expected correc-tion is not sufficient to bring the theoretical predictionsinto agreement with the experimental results. In par-ticular, for the 4 s branching all theoretical predictionsare significantly higher than the experimental results,the lowest energy value for 4 s differing by several stan-dard deviations from any of the theoretical values. Theexperiment, however, confirms the ordering of relativepopulations of states predicted by all three theoreticalcalculations, namely MN predominantly populates the4 s channel, followed by 3 d , and finally 4 p . Though thereare significant discrepancies for the FQ and the LCAOresults, they clearly perform better than those from theSE model for all branching fractions measured. Janev& Radulovic (1978), using the Landau-Herring method,obtained the surprising result that 3 p has the secondlargest branching fraction among final channels, with abranching fraction of order 0.1, and this result is ruledout by the experimental results from DESIREE, whereno capture into 3 p is detected.While there are no experimental results for the totalcross section for MN involving Na, the theoretical resultsare shown in Fig. 4. Here one sees that the generalrelationship between the different models is similar tothe case of Li. In particular, the effect on the totalcross section due to Coulomb focussing predicted by theLCAO and SE models is roughly the same, and the FQcalculations provide larger total cross sections than theLCAO and SE models.In general, the comparisons for branching fractions inNa are in line with the overall results for Li, with theFQ results (correcting for Coulomb focussing) and theLCAO results generally performing better than thosefrom the SE model. Considering the Li and Na resultstogether, but excluding the Li 3 d branching fraction re-sults which are not well explained by any of the theoret-ical calculations, the FQ results match experiments thebest, followed by the LCAO model, while the SE modelperforms least well of the three theoretical approaches,sometimes giving branching fractions discrepant by of s b r an c h i ng f r a c t i on Theory - FQ (H)Theory - LCAOTheory - LCAO (H)Theory - SETheory - SE (H)Experiment - DESIREE b r an c h i ng f r a c t i on -2 -1 b r an c h i ng f r a c t i on Reduced collision energy E R [eV/amu] Figure 3.
Branching fractions for the MN reaction Na + +D − → Na( nl ) + D as a function of reduced collision en-ergy. Experimental results from DESIREE (Eklund et al. inprep.) are shown, with estimated errors (1 σ ). Theoreticalresults are shown from our calculations using both the LCAOmethod (Barklem 2016b) and semi-empirical (SE) couplingsfrom Olson et al. (1971). Theoretical results are also shownfor the case where D − is replaced by H − , marked by (H)in the legend. FQ results for H − are from Dickinson et al.(1999). order 0.05-0.2. Finally, we note that though the experi-mental results for branching fractions in Li(3 p ), Li(3 d ),Na(3 d ), and Na(4 p ) show some hint of a possible trendwith energy, noting that the error bars are 1 σ , the ex-perimental results are also compatible with a flat trendas predicted by theory. More, or more precise, resultswould be required to study this question further. IMPLICATIONS FOR NON-LTE MODELLINGOF STELLAR SPECTRA i + +H − /D − and Na + +H − /D − collisions Table 1.
Model atoms for Li and Na. Describes the rate coefficient data used for MN involving the { s , 3 p , 3 d } states of Li, and for MN involving the { s , 3 d , 4 p } states of Na.Element ID Description ReferenceLi/Na FQ Quantum chemistry potentials + Full quantum scattering 1, 2, 3, 4, 5, 6LCAO Theoretical LCAO couplings + Landau-Zener model 7SE Semi-empirical couplings + Landau-Zener model 8SE-sFQ SE scaled to have FQ total rate -None Zero rates -Li BV18 Quantum chemistry potentials + Quantum current probability method 9, 10L19 Quantum chemistry potentials + Landau-Zener model 11 References —(1) Croft et al. (1999b); (2) Belyaev & Barklem (2003); (3) Barklem et al. (2003); (4) Dickinsonet al. (1999); (5) Belyaev et al. (2010); (6) Barklem et al. (2010); (7) Barklem (2016b); (8) Olson et al. (1971);(9) Croft et al. (1999a); (10) Belyaev & Voronov (2018); (11) Launoy et al. (2019). -15 -14 -13 -12 -11 -10 -3 -2 -1 T o t a l c r o ss s e c t i on [ c m ] Reduced collision energy E R [eV/amu] FQ (H)LCAOLCAO (H)SESE (H) Figure 4.
Total cross sections for the MN reaction Na + +D − → Na + D as a function of reduced collision energy.Theoretical results are shown from our calculations usingboth the LCAO method (Barklem 2016b) and semi-empirical(SE) couplings from Olson et al. (1971). Theoretical resultsare also shown for the case where D − is replaced by H − ,marked by (H) in the legend. FQ results for H − are fromDickinson et al. (1999). Based on the study of fundamental MN data from dif-ferent theoretical models and experiments carried outin the previous section, we now turn to stellar spec-trum modelling and the implications for the choice ofMN data to be used. We do this using 1D and 3D non-LTE radiative transfer calculations, based on differentdescriptions of MN motivated by the discussion above,finally comparing the shapes and strengths of the result-ing line profiles as well as the effects on inferred Li andNa abundances. The discussion is centred on the Li I . I . I . . I multiplets at 515 nm, 568 nm, 615 nm,and 818 nm.3.1. Spectrum synthesis was carried out with the 3D non-LTE radiative transfer code
Balder (Amarsi et al.2018b), a modified version of the
Multi3D code(Leenaarts & Carlsson 2009). Calculations were per-formed across a grid of 103 1D hydrostatic
MARCS modelatmospheres (Gustafsson et al. 2008) that span effectivetemperatures 4000 ≤ T eff / K ≤ ≤ log g/ cm s − ≤ − ≤ [Fe / H] ≤ ,as well as the extremely metal-rich case [Fe / H] = 0 . ≤ log (cid:15) Li ≤ − ≤ [Na / Fe] ≤
1, in steps of 0 . v mic = 1 and 2 km s − .In addition, 3D non-LTE spectrum synthesis calcula-tions were performed on four 3D hydrodynamic STAGGER model atmospheres (Magic et al. 2013). The four mod-els were chosen close to the edges of the 1D grid dis-cussed above: namely, a giant star with T eff ≈ g = 2, and a turn-off star with T eff ≈ g = 4; in both cases, this was done for both The logarithmic abundance of an element A is defined with re-spect to hydrogen: log (cid:15) A = log N A N H + 12, where N A is the num-ber density of A. The standard notation for the abundance ratioA/B relative to the solar ratio is used [A / B] = (log (cid:15) A − log (cid:15) (cid:12) A ) − (log (cid:15) B − log (cid:15) (cid:12) B ), where (cid:12) denotes the solar value, and all loga-rithms are to base 10. Barklem et al. F l u x λ Air / nm−0.050.000.05 F l u x − F Q FQLCAONone3D LTE
Li I 670.8nmlog ε Li =36500/4/−3 F l u x λ Air / nm−0.050.000.05 F l u x − F Q FQLCAONone3D LTE
Na I 589.0nm[Na/Fe]=04500/2/0
Figure 5.
Example synthetic 3D non-LTE profiles andbisectors (upper subpanels) for the Li I . I . STAGGER atmospheres are labelled by { T eff / log g/ [Fe / H] } . Rotational and instrumental broaden-ing have been neglected. solar-metallicity ([Fe / H] = 0), and for low metallicity([Fe / H] = − (cid:15) Li = 3 (close to the primordial value),and for [Na / Fe] = 0.3.2.
Model atoms
The different non-LTE model atoms considered hereare summarised in Table 1. The model atoms are allbased on those of Wang et al. (2020) and Lind et al.(2011), for Li and Na respectively. The only differencesbetween the model atoms are the rate coefficients forMN involving the { s , 3 p , 3 d } states of Li, and the { s , 3 d , 4 p } states of Na. For each system, we alsotested the importance of these three transitions, rela-tive to other processes involving hydrogen collisions. Inbrief, we found that: • Out of all the MN processes included in the mod-els, those involving the { s , 3 p , 3 d } states of Li,and the { s , 3 d , 4 p } states of Na, play the dom-inating role in setting the statistical equilibria,across the entire parameter space considered here. • For Li, out of all processes involving hydrogencollisions included in the models, MN plays thedominating role in setting the statistical equilibriaacross the entire parameter space considered here. • For Na, out of all processes involving hydrogencollision included in the models, MN usually playsthe dominating role in setting the statistical equi-libria. The exception is in cool metal-poor dwarfs,where excitation by hydrogen collisions also playsa significant role.The first three model atoms listed in Table 1 arebased on rate coefficients predicted by the theoreticalFQ, LCAO and SE models. We saw in Sect. 2 thatthese roughly span the range of uncertainty in the ex-perimental data, and in light of this it is interesting toconsider how this translates to differences in Li and Naabundances. Next, the SE-sFQ model atom adopts theFQ total rate, but with the SE branching fractions. InFigs 2 and 4 we saw that FQ gives the highest totalcross section while SE gives the lowest; also, in Figs 1and 3 we saw that FQ matches the experimental branch-ing fractions best, whereas SE is less successful. There-fore, comparison of results from the SE-sFQ model withthose from the FQ and SE models, may give some hintsregarding the importance of branching ratios into differ-ent final states, versus the importance of the total ratecoefficient, in the context of non-LTE modelling. Fi-nally, to illustrate their overall importance, model atoms A “model atom” refers to a collection of all required fundamentaldata on the atom of interest needed for non-LTE modelling itsspectrum, including energy levels and descriptions of all radiativeand collisional processes on the atom. Overviews of the basicstructure and spectral lines of Li I and Na I are available atNIST (Kramida et al. 2019) and Grotrian diagrams in Bashkin& Stoner (1975). i + +H − /D − and Na + +H − /D − collisions −1.0−0.8−0.6−0.4−0.20.00.2−1.0−0.8−0.6−0.4−0.20.00.2 A bundan c e − D L T E −4 −3 −2 −1 0[Fe/H]−0.2−0.10.00.10.2 A bundan c e − F Q FQLCAOSESE−sFQBV18L19None
Li I 670.8nm0 ≤ log ε Li ≤
4, 1 ≤ v mic ≤ −0.10.00.1−0.10.00.1 − l og W / W D L T E
3D model atmosphere−0.050.000.05 − l og W / W F Q FQLCAOSESE−sFQBV18L19None
Li I 670.8nmlog ε Li =3 −1.0−0.8−0.6−0.4−0.20.00.2−1.0−0.8−0.6−0.4−0.20.00.2 A bundan c e − D L T E −4 −3 −2 −1 0[Fe/H]−0.2−0.10.00.10.2 A bundan c e − F Q FQLCAOSESE−sFQNone
Na I 589.0nm−1 ≤ [Na/Fe] ≤
1, 1 ≤ v mic ≤ −0.10.00.1−0.10.00.1 − l og W / W D L T E
3D model atmosphere−0.050.000.05 − l og W / W F Q FQLCAOSESE−sFQNone
Na I 589.0nm[Na/Fe]=0
Figure 6.
Impact of different MN descriptions (Table 1) on strengths of Li I . I . MARCS atmospheres using different abundances and microturbulences;solid lines delineate the smallest and largest differences. Right: logarithmic 3D non-LTE equivalent width ratios compared to3D LTE (upper subpanels) and compared to the FQ model (lower subpanels), a proxy for the abundance differences; the 3D
STAGGER atmospheres are labelled by { T eff / log g/ [Fe / H] } . were constructed wherein MN involving the { s , 3 p , 3 d } states and the { s , 3 d , 4 p } states were switched off, forLi and Na respectively (model “None”).Two more model atoms shown in Table 1 were con-structed specifically for Li. First, the BV18 model atomis based on the recent calculations by Belyaev & Voronov(2018). This set of calculations uses the same LiH poten-tials (Croft et al. 1999a) as the FQ model, albeit adopt-ing the quantum probability current method instead offull quantum dynamical calculations. Finally, the L19model atom is based on the rate coefficient fits for MNinvolving the { s , 3 p , 3 d } states provided by Launoyet al. (2019). his data seems to be based on their theo- retical calculations which best match the experimentalresults (ACV5Z+G quantum chemical calculations withLandau-Zener dynamics).To quantify the relative differences between the mod-els, Tables 2 and 3 compare the total MN rate coef-ficients and branching fractions between final states at6000 K. Note, due to high-energy contributions, branch-ing fractions may not correspond exactly with those inthe low-energy cross sections. As expected from the to-tal cross section comparisons, for Li the FQ model givesa rate larger than the L19 model, at 6000 K 10% higher.Note that at 6000 K, the most probable reduced collisionenergy for Li + +H − is 0.59 eV/amu, which falls around0 Barklem et al.
Table 2.
Comparisons of Li + +H − MN rate coefficients at 6000 K from various sources used in different modelatoms. FQ LCAO SE SE-sFQ BV18 L19Total rate coefficient [cm /s] 1.21 × − × − × − × − × − × − Ratio to FQ 1.00 0.81 0.72 1.00 1.14 0.91Ratio to L19 1.10 0.89 0.79 1.10 1.26 1.00Branching fraction 3 s p d Table 3.
Comparisons of Na + +H − MN rate coefficients at 6000 K from varioussources used in different model atoms.FQ LCAO SE SE-sFQTotal rate coefficient [cm /s] 9.47 × − × − × − × − Ratio to FQ 1.00 0.82 0.78 1.00Branching fraction 4 s d p the bend in the cross sections. In general, the FQ modelgives larger total rate coefficients than the asymptoticLCAO and SE models; the data from BV18 are largerstill. 3.3. Implementation of rates
The non-LTE calculations with
Balder take rate co-efficients as inputs, which result from convolving colli-sional cross-sections with the Maxwell-Boltzmann dis-tribution at a given temperature. For the FQ casesthe rate coefficients were taken directly from the arti-cles referenced in Table 1, and are based on MN crosssections covering an energy range ( E CM ) of roughly 10 − to 10 eV in the case of Li, and 10 − to 100 eV in the caseof Na. For LCAO and SE calculations, cross sections arecalculated between 10 − and 100 eV. Outside of theseregions cross sections are log-log extrapolated in energywhen calculating the rate coefficients. In the case of FQdata, the reverse rate coefficients for ion pair production(IPP) (Li / Na( nl ) + H → Li / Na + + H − ), are calculatedfrom detailed balance relations, while for LCAO and SEthey are calculated directly from cross sections, but con-firmed to fulfil detailed balance with the correspondingMN calculation.For all inelastic collisions, Balder takes as input therate coefficients for the process in one direction only.The code calculates the rate coefficients of the reverse processes on the fly, via the detailed balance relation.In this way, spurious non-LTE effects in the deep at-mosphere that may occur from errors due to stipulat-ing the rate coefficients to finite precision, or due tointerpolation errors, are avoided. Since MN rate co-efficients have a smoother behaviour with temperaturethan IPP,
Balder uses the logarithmic MN rate coeffi-cients on a grid of temperatures, and interpolates at theatmospheric gas temperatures of interest. For consis-tency, identical temperature grids were adopted for eachmodel: they span 2000 K to 8000 K in steps of 2000 K.Using a set of LCAO data calculated on a finer grid, theinterpolation error at 5000 K when interpolating in stepsof 2000 K was found to be smaller than 1% (i.e. smallerthan the nominal accuracy in the rate coefficients them-selves). 3.4.
Results
The main result of this section is that the differentMN descriptions result in nearly identical line strengthsand derived abundances for both Li and Na. This can beseen in the 3D non-LTE profiles of the Li I . I . . i + +H − /D − and Na + +H − /D − collisions MARCS and 3D
STAGGER model atmospheres. These abundance differ-ences are based on equivalent widths of the spectrallines. For example, for a given 1D LTE abundance,the 1D non-LTE versus 1D LTE abundance differenceis defined as log (cid:15)
1D NLTE − log (cid:15)
1D LTE such that the 1Dnon-LTE equivalent width is equal to that in 1D LTE(see for example Sect. 2.6 of Amarsi et al. 2016). In 1D,the abundance differences of the LCAO model with re-spect to the FQ model are at most 0 .
006 dex for the Li I . .
003 dex for the Na I . . . / H], the abundance corrections for Li becomemore severe, whereas for Na they become less severe onthe whole. CONCLUSIONSComparisons have been made between experimentaland theoretical results for MN processes in Li + +H − /D − and Na + +H − /D − collisions at energies below 1 eV, to-tal cross sections for Li, and branching fractions for bothLi and Na. The studies were made with theoretical approaches that have been used to calculate MN datafor astrophysical applications in non-LTE modelling ofstellar spectra, in particular the FQ approach, and theLCAO and SE asymptotic model approaches. Generally,the comparisons support the expectation that the FQcalculations are superior to the model approaches. Scat-ter among theoretical calculations and general agree-ment with experimental results for the Li + +D − totalcross section would seem to indicate that total MN ratesare not uncertain by more than 30%. Just as impor-tantly, the comparisons indicate the uncertainty doesnot vary strongly with collision energy i.e. it is likelyto be constant in the low energy regime. The com-parisons also indicate that the LCAO model providesmore reliable branching fractions than the SE model.Again, this is perhaps in line with expectations, as thesemi-empirical couplings are averaged over different ex-perimental and theoretical results available at the time(Olson et al. 1971) and not just for H − /D − , while theLCAO calculations are specific to the case at hand.The experiments together with the various sets of the-oretical calculations allow us to place constraints on theaccuracy of the rates used in non-LTE modelling. Thisin turn constrains the uncertainty in modelled spectraand derived abundances for Li and Na arising from thissource. Sets of model atoms for Li and Na that span thevarious uncertainties in total cross section and branchingfractions were constructed, and used to compute non-LTE spectra and abundance corrections with respect toLTE across a large grid of 1D model atmospheres forlate-type stars. For Li the uncertainties do not exceed0.006 dex (1.4%), while for Na the uncertainties do notexceed 0.003 dex (0.7%). Thus, the most important con-clusion of this work is that uncertainties in Li and Naabundances in late-type stars due to uncertainties in MNprocesses are not larger than these values.In complex atoms, often of great astrophysical inter-est such as Fe, presently the LCAO and SE approachesare the only viable option for estimating hydrogen col-lision processes. FQ approaches are in general too com-putationally demanding to deal with the many excitedstates of complex atomic systems required for non-LTEmodelling. It is unclear if the results found here can beextrapolated to complex atoms, but this level of agree-ment for simple atoms gives some confidence in the ap-proach for the MN processes. Excitation/deexcitationprocesses due to hydrogen were found here to be unim-portant for Li, yet they were seen to be important forNa in cool metal-poor dwarfs. Further, Amarsi et al.(2018a) found that some excitation processes on O areimportant to reproduce the centre-to-limb variation of Olines in the Sun, and that the LCAO method alone may2 Barklem et al. underestimate these processes; similar results were laterobtained for C (Amarsi et al. 2019). From a physicspoint of view, this would not be surprising, as excita-tion processes between near-lying states could occur viamechanisms other than the avoided ionic crossing mech-anism. From an astrophysics point of view, the fact thateven relatively similar atoms like Li and Na have differ-ent sensitivities, and different non-LTE effects, means itis difficult to draw general conclusions. ACKNOWLEDGMENTSWe thank Xavier Urbain for providing the Lou-vain and Newcastle experimental results in electronicform. We thank Ella Wang, Thomas Nordlander, andKarin Lind for providing the base model atoms usedin this study. Significant parts of this work were per-formed at the Swedish National Infrastructure, DE-SIREE (Swedish Research Council contract No. 2017-00621). This work is a part of the project “Prob-ing charge- and mass- transfer reactions on the atomiclevel”, supported by the Knut and Alice WallenbergFoundation (2018.0028). The non-LTE computationswere performed on resources provided (through projectSNIC 2019/3-532) by the Swedish National Infrastruc-ture for Computing (SNIC) at the MultidisciplinaryCenter for Advanced Computational Science (UPP-MAX) and at the High Performance Computing Cen-ter North (HPC2N), partially funded by the SwedishResearch Council through grant agreement no. 2016-07213. Furthermore, Paul Barklem, Henrik Cederquist,Henning Zettergren, and Henning Schmidt thank theSwedish Research Council for individual project grants(with contract Nos. 2016-03765, 2019-04379, 2016-04181, and 2018-04092). Paul Barklem, Anish Amarsi,and Jon Grumer would like to acknowledge financialsupport from the project grant “The New Milky Way”(2013.0052) from the Knut and Alice Wallenberg Foun-dation.REFERENCES
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