The electron energy-loss rate due to radiative recombination
aa r X i v : . [ a s t r o - ph . H E ] D ec Astronomy & Astrophysicsmanuscript no. rrwelf_arxiv c (cid:13)
ESO 2018October 8, 2018
The electron energy-loss rate due to radiative recombination
Junjie Mao , , Jelle Kaastra , and N. R. Badnell SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands Leiden Observatory, Leiden University, Niels Bohrweg 2, 2300 RA Leiden, the Netherlands Department of Physics, University of Strathclyde, Glasgow G4 0NG, UKReceived date / Accepted date
ABSTRACT
Context.
For photoionized plasmas, electron energy-loss rates due to radiative recombination (RR) are required for thermal equilib-rium calculations, which assume a local balance between the energy gain and loss. While many calculations of total and / or partial RRrates are available from literature, specific calculations of associated RR electron energy-loss rates are lacking. Aims.
Here we focus on electron energy-loss rates due to radiative recombination of H-like to Ne-like ions for all the elements up toand including zinc ( Z = Methods.
We use the AUTOSTRUCTURE code to calculate the level-resolved photoionization cross section and modify the ADASRRcode so that we can simultaneously obtain level-resolved RR rate coe ffi cients and associated RR electron energy-loss rate coe ffi cients.The total RR rates and electron energy-loss rates of H i and He i are compared with those found in literature. Furthermore, we utilizeand parameterize the weighted electron energy-loss factors (dimensionless) to characterize total electron energy-loss rates due to RR. Results.
The RR electron energy-loss data are archived according to the Atomic Data and Analysis Structure (ADAS) data class adf48 . The RR electron energy-loss data are also incorporated into the SPEX code for detailed modelling of photoionized plamsas.
Key words. atomic data – atomic processes
1. Introduction
Astrophysical plasmas observed in the X-ray band can roughlybe divided into two subclasses: collisional ionized plasmas andphotoionized plasmas. Typical collisional ionized plasmas in-clude stellar coronae (in coronal / collisional ionization equilib-rium), supernova remnants (SNRs, in non-equilibrium ioniza-tion) and the intracluster medium (ICM). In a low-density,high-temperature collisional ionized plasma, e.g. ICM, colli-sional processes play an important role (for a review, see e.g.Kaastra et al. 2008). In contrast, in a photoionized plasma, pho-toionization, recombination and fluorescence processes are alsoimportant in addition to collisional processes. Both the equationsfor the ionization balance (also required for a collisional ionizedplasma) and the equations of the thermal equilibrium are used todetermine the temperature of the photoionized plasma. Typicalphotoionized plasmas in the X-ray band can be found in X-raybinaries (XRBs) and active galactic nuclei (AGN).For collisional ionized plasmas, various calculations oftotal radiative cooling rates are available in the litera-ture, such as Cox & Daltabuit (1971), Raymond et al. (1976),Sutherland & Dopita (1993), Schure et al. (2009), Foster et al.(2012) and Lykins et al. (2013). These calculations take advan-tage of full plasma codes like SPEX (Kaastra et al. 1996) andAPEC (Smith et al. 2001), and do not treat individual energy-loss (cooling) processes separately. Total radiative cooling ratesinclude the energy-loss of both the line emission and the contin-uum emission. The latter includes the energy-loss due to radia-tive recombination (RR). Even more specifically, the energy-lossdue to RR can be separated into the electron energy-loss and theion energy-loss. Send o ff print requests to : J. Mao, e-mail: [email protected]
On the other hand, for photoionized plasmas, the electronenergy-loss rate due to RR is one of the fundamental parametersfor thermal equilibrium calculations, which assume a local bal-ance between the energy gain and loss. Energy can be gainedvia photoionization, Auger e ff ect, Compton scattering, colli-sional ionization, collisional de-excitation and so forth. Energy-loss can be due to radiative recombination, dielectronic recom-bination, three body recombination, inverse Compton scatter-ing, collisional excitation, bremsstrahlung, etc., as well as theline / continuum emission following these atomic processes. Infact, the energy-loss / gain of all these individual processes needto be known. The calculations of electron energy-loss rates dueto RR in the Cloudy code (Ferland et al. 1998, 2013) are basedon hydrogenic results (Ferland et al. 1992; LaMothe & Ferland2001). In this manuscript, we focus on improved calculationsof the electron energy-loss due to radiative recombination, espe-cially providing results for He-like to Ne-like isoelectronic se-quences.While several calculations of RR rates, including the totalrates and / or detailed rate coe ffi cients, for di ff erent isoelectronicsequences are available, e.g. Gu (2003) and Badnell (2006), spe-cific calculations of the associated electron energy-loss rate dueto RR are limited. The pioneering work was done by Seaton(1959) for hydrogenic ions using the asymptotic expansion ofthe Gaunt factor for photoionization cross sections (PICSs).By using a modified semi-classical Kramers formula forradiative recombination cross sections (RRCSs), Kim & Pratt(1983) calculated the total RR electron energy-loss rate for a fewions in a relatively narrow temperature range.Ferland et al. (1992) used the nl -resolved hydrogenic PICSsprovided by Storey & Hummer (1991) to calculate both n - Article number, page 1 of 9 & Aproofs: manuscript no. rrwelf_arxiv resolved RR rates ( α RRi ) and electron energy-loss rates ( L RRi ).Contributions up to and include n = nl -resolved hydrogenic PICSs provided byStorey & Hummer (1991), Hummer (1994) calculated the RRelectron energy-loss rates for hydrogenic ions in a wide tem-perature range. In addition, Hummer & Storey (1998) calcu-lated PICSs of He i (photoionizing ion) for n ≤
25 with theirclose-coupling R -matrix calculations. Together with hydrogenic(Storey & Hummer 1991) PICSs for n >
25 (up to n =
800 forlow temperatures), the RR electronic energy-loss rate coe ffi cientof He i (recombined ion) was obtained.Later, LaMothe & Ferland (2001) used the exact PICSs fromthe Opacity Project (Seaton et al. 1992) for n <
30 and PICSs ofVerner & Ferland (1996) for n ≥
30 to obtain n -resolved RRelectron energy-loss rates for hydrogenic ions in a wide temper-ature range. The authors introduced the ratio of β/α (dimension-less), with β = L / kT and L the RR electron energy-loss rate. Theauthors also pointed out that β/α changes merely by 1 dex in awide temperature range meanwhile α and β change more than 12dex.In the past two decades, more detailed and accurate cal-culations of PICSs of many isoelectronic sequences have beencarried out (e.g. Badnell 2006), which can be used to calculatespecifically the electron energy-loss rates due to RR.Currently, in the SPEX code (Kaastra et al. 1996), the as-sumption that the mean kinetic energy of a recombining elec-tron is 3 kT / code (v24.24.3, Badnell 1986), the electron energy-loss rates dueto RR are calculated for the H-like to Ne-like isoelectronic se-quences for elements up to and including Zn ( Z =
30) in a widetemperature range. Subsequently, the electron energy-loss ratecoe ffi cients ( β = L / kT ) are weighted with respect to the totalRR rates ( α t ), yielding the weighted electron energy-loss factors( f = β/α t , dimensionless). The weighted electron energy-lossfactors can be used, together with the total RR rates, to update thedescription of the electron energy-loss due to RR in the SPEXcode or other codes.In Sect. 2, we describe the details of the numerical calcula-tion from PICSs to the electron energy-loss rate due to RR. Typ-ical results are shown graphically in Sect. 3. Parameterizationof the weighted electron energy-loss factors is also illustrated inSect. 3.1. The detailed RR electron energy-loss data are archivedaccording to the Atomic Data and Analysis Structure (ADAS)data class adf48 . Full tabulated (unparameterized and parameter-ized) weighted electron energy-loss factors are available in CDS.Comparison of the results for H i and He i can be found in Sec-tion 4.1. The scaling of the weighted electron energy-loss factorswith respect to the square of the ionic charge of the recombinedion can be found in Section 4.2. We also discuss the electron andion energy-loss due to RR (Section 4.3) and the total RR rates(Section 4.4).Throughout this paper, we refer to the recombined ion whenwe speak of the radiative recombination of a certain ion, sincethe line emission following the radiative recombination comesfrom the recombined ion. Furthermore, only RR from the groundlevel of the recombining ion is discussed here. http: // amdpp.phys.strath.ac.uk / autos /
2. Methods
The AUTOSTRUCTURE code is used for calculating level-resolved non-resonant PICSs under the intermediate coupling(“IC") scheme (Badnell & Seaton 2003). The atomic and numer-ical details can be found in Badnell (2006), we briefly state themain points here. We use the Slater-Type-Orbital model poten-tial to determine the radial functions. PICSs are calculated first atzero kinetic energy of the escaping electron, and subsequently ona z -scaled logarithmic energy grid with three points per decade,ranging from ∼ z − to z ryd, where z is the ionic chargeof the photoionizing ion / atom. PICSs at even higher energies areat least several orders of magnitude smaller compared to PICSsat zero kinetic energy of the escaping electron. Nonetheless, itstill can be important, especially for the s - and p -orbit, to derivethe RR data at the high temperature end. We take advantage ofthe analytical hydrogenic PICSs (calculated via the dipole radialintegral, Burgess 1965) and scale them to the PICS with the high-est energy calculated by AUTOSTRUCTURE to obtain PICSsat very high energies. Note that fully nLS J -resolved PICSs forthose levels with n ≤
15 and l ≤ n PICSs for n =
16, 20, 25, 35, 45, 55, 70, 100,140, 200, 300, 450, 700, 999 are also calculated specifically inorder to derive the total RR and electron energy-loss rates (inter-polation and quadrature required as well).The inverse process of dielectronic and radiative recombina-tion is resonant and non-resonant photoionization, respectively.Therefore, radiative recombination cross sections (RRCSs) areobtained through the Milne relation under the principle of de-tailed balance (or microscopic reversibility) from non-resonantPICSs.
The RR rate coe ffi cient is obtained by α i ( T ) = Z ∞ v σ i ( v ) f ( v, T ) d v , (1)where v is the velocity of the recombining electron, σ i is theindividual detailed (level / term / shell-resolved) RRCS, f ( v, T ) isthe probability density distribution of the velocity of the recom-bining electrons for the electron temperature T . The Maxwell-Boltzmann distribution for the free electrons is adopted through-out the calculation, with the same quadrature approach as de-scribed in Badnell (2006). Accordingly, the total RR rate perion / atom is α t ( T ) = X i α i ( T ) . (2)Total RR rates for all the isoelectronic sequences, taking con-tributions up to n = into account (see its necessity in Sec-tion 3).The RR electron energy-loss rate coe ffi cient is defined as(e.g. Osterbrock 1989) β i ( T ) = kT Z ∞ m v σ i ( v ) f ( v, T ) d v , (3)The total electron energy-loss rate due to RR is obtained simplyby adding all the contributions from individual captures, L t ( T ) = X i L i = kT X i β i , (4) Article number, page 2 of 9ao et al.: RR electron energy-loss rate which can be identically derived via L t ( T ) = kT α t ( T ) f t ( T ) , (5)where f t ( T ) = P i β i ( T ) α t ( T ) , (6)is defined as the weighted electron energy-loss factor (dimen-sionless) hereafter.The above calculation of the electron energy-loss rates is re-alized by adding Equation (3) into the archival post-processorFORTRAN code ADASRR (v1.11). Both the level-resolvedand bundled- n / nl RR data and the RR electron energy-loss dataare obtained. The output files have the same format of adf48 withRR rates and electron energy-loss rates in the units of cm s − and ryd cm s − , respectively. Note that ionization potentials ofthe ground level of the recombined ions from NIST (v5.3) areadopted to correct the conversion from PICSs to RRCSs at lowkinetic energy for low-charge ions. We should point out that al-though the level-resolved and bundled- nl / n RR data are, in fact,available on OPEN ADAS , given the fact that we use the latestversion of the AUTOSTRUCTURE code and a modified versionof the ADASRR code, here we re-calculate the RR data, whichare used together with the RR electron energy-loss data to de-rive the weighted electron energy-loss factor f t for consistency.In general, our re-calculate RR data are almost identical to thoseon OPEN ADAS, except for a few many-electron ions at the thehigh temperature end, where our re-calculated data di ff er by afew percent. Whereas, both RR data and electron energy-lossdata are a few orders of magnitude smaller compared to those atthe lower temperature end, thus, the above mentioned di ff erencehas negligible impact on the accuracy of the weighted electronenergy-loss factor (see also in Section 4.4).For all the isoelectronic sequences discussed here, the con-ventional ADAS 19-point temperature grid z (10 − ) K isused.
3. Results
For each individual capture due to radiative recombination, when kT ≪ I , where I is the ionization potential, the RR elec-tron energy-loss rate L i is nearly identical to kT α i , since theMaxwellian distribution drops exponentially for E k & kT , where E k is the kinetic energy of the free electron before recombina-tion. On the other hand, when kT ≫ I , the RR electron energy-loss rate is negligible compared with kT α i . As in an electron-ioncollision, when the total energy in the incident channel nearlyequals that of a closed-channel discrete state, the channel in-teraction may cause the incident electron to be captured in thisstate (Fano & Cooper 1968). That is to say, those electrons with E k ≃ I are preferred to be captured, thus, L i ∼ I α i . Figure 1shows the ratio of β i /α i = L i / ( kT α i ) for representative nLS J -resolved levels (with n ≤
8) of He-like Mg xi .In terms of capturing free electrons into individual shells(bundled- n ), due to the rapid decline of the ionization potentialsfor those very high- n shells, the ionization potentials can be com-parable to kT , if not significantly less than kT , at the low tem-perature end. Therefore we see the significant di ff erence betweenthe top panel (low- n shells) and middle panel (high- n shells) of http: // amdpp.phys.strath.ac.uk / autos / ver / misc / adasrr.f http: // physics.nist.gov / PhysRefData / ASD / ionEnergy.html http: // open.adas.ac.uk / adf48 T (eV) −1 b i / a i , S P D F G H I K He−like Mg XI
Fig. 1.
For He-like Mg xi , the ratio between level-resolved electronenergy-loss rates L i and the corresponding radiative recombination ratestimes the temperature of the plasma, i.e. β i /α i (not be confused with β i /α t ), where i refers to the nLS J -resolved levels with n ≤ Figure 2. In order to achieve adequate accuracy, contributionsfrom high- n shells (up to n ≤ ) ought to be included. Themiddle panel of Figure 2 shows clearly that even for n = β n = and α n = does not drop to zero. Nevertheless, thebottom panel of Figure 2 illustrates the advantage of weightingthe electron energy-loss rate coe ffi cients with respect to the to-tal RR rates, i.e. β i /α t , which approaches zero more quickly. Atleast, for the next few hundreds shells following n = − , thus, their contribution to the total electron energy-loss rateshould be less than 1%.The bottom panels of Figure 3 and 4 illustrate the weightedelectron energy-loss factors for He-like isoelectronic sequences(He, Si and Fe) and Fe isonuclear sequence (H-, He-, Be- andN-like), respectively. The deviation from (slightly below) unityat the lower temperature end is simply due to the fact that theweighted electron energy-loss factors of the very high- n shellsare no longer close to unity (Figure 2, middle panel). The de-viation from (slightly above) zero at the high temperature endis because the ionization potentials of the first few low- n shellscan still be comparable to kT , while sum of these n -resolved RRrates are more or less a few tens of percent of the total RR rates.Due to the non-hydrogenic screening of the wave functionfor low- nl states in low-charge many-electron ions, the charac-teristic high-temperature bump is present in not only the RR rates(see Figure 4 in Badnell 2006, for an example) but also in theelectron energy-loss rates. The feature is even enhanced in theweighted electron energy-loss factor. Article number, page 3 of 9 & Aproofs: manuscript no. rrwelf_arxiv b i / a i T (eV) b i / a t −8 −6 −4 −2 b i / a i Be−like Fe XXIII
Fig. 2.
Ratios of β i /α i for Be-like Fe xxiii (upper and middle panel) andratios of β i /α t (bottom panel) where i refers to the shell number. Low-and high- n shell results are shown selectively in the plot. The upperpanel shows all the shells with n ≤
8. The middle panel shows shellswith n = n = , , , , , , We parameterize the ion / atom-resolved radiative recombinationelectron energy-loss factors using the same fitting strategy de-scribed in Mao & Kaastra (2016), with the model function of f t ( T ) = a T − b − c log T + a T − b + a T − b ! , (7)where the electron temperature T is in units of eV, a and b are primary fitting parameters, c , a , and b , are additionalfitting parameters. The additional parameters are frozen to zeroif they are not used. Furthermore, we constrain b − to be within-10.0 to 10.0 and c between 0.0 and 1.0. The initial values of thetwo primary fitting parameters a and b are set to unity togetherwith the four additional fitting parameters a , and b , if theyare thawn. Conversely, the initial value of c , if it is thawn, is setto either side of its boundary, i.e. c = . c = . r ). We started with r = . . a t ( c m s − ) −15 −14 −13 −12 −11 −10 −9 He−like isoelectronic sequence L t (r y d c m s − ) −14 −13 −12 −11 −10 T / z (eV) −3 −2 −1 f t He ISi XIIIFe XXV
Fig. 3.
The total RR rates α t (top), electron energy-loss rates L t (mid-dle) and weighted electron energy-loss factors f t (bottom) of He-likeisoelectronic sequences for ions including He i (black), Si xiii (red) andFe xxv (orange). The temperature is down-scaled by z , where z is theionic charge of the recombined ion, to highlight the discrepancy be-tween hydrogenic and non-hydrogenic. The captures to form the He i shows non-hydrogenic feature in the bottom panel. statistics adopted here are χ = N X i = n i − m i r n i ! , (8)where n i is the i th numerical calculation result and m i is the i thmodel prediction (Equation 7).For the model selection, we first fit the data with the sim-plest model (i.e. all the five additional parameters are frozen tozero), following with fits with free additional parameters step bystep. Thawing one additional parameter decreases the degrees offreedom by one, thus, only if the obtained statistics ( χ ) of themore complicated model improves by at least 2.71, 4.61, 6.26,7.79 and 9.24 for one to five additional free parameter(s), respec-tively, the more complicated model is favored (at a 90% nominalconfidence level).Parameterizations of the ion / atom-resolved RR weightedelectron energy-loss factors for individual ions / atoms in H-liketo Ne-like isoelectronic sequences were performed. A typical fitfor non-hydrogenic systems is shown in Figure 5 for N-like iron(Fe xx ). The fitting parameters can be found in Table 2. Again,the weighted energy-loss factor per ion / atom is close to unity at Article number, page 4 of 9ao et al.: RR electron energy-loss rate a t ( c m s − ) −14 −13 −12 −11 −10 −9 Fe isonuclear sequence L t (r y d c m s − ) −11 −10 −9 T / z (eV) −3 −2 −1 f t H−likeHe−likeBe−likeN−like
Fig. 4.
Top panel is total RR rates α t of the Fe isonuclear sequence,including H- (black), He- (red), Be- (orange) and N-like (blue); Middlepanel is the RR electron energy-loss rates L t ; And bottom panel is theweighted electron energy-loss factors f t . The temperature of the plasmais down-scaled by z , as in Figure 3. low temperature end and drops towards zero rapidly at the hightemperature end.In Figure 6 we show the histogram of maximum deviation δ max (in percent) between the fitted model and the original cal-culation for all the ions considered here. In short, our fitting ac-curacy is within 4%, and even accurate ( . . f A = β t /α t ) and Case B (Baker & Menzel 1938, f B = β n ≥ /α n ≥ ) RRweighted electron energy-loss factors of H i (Figure 7) and He i (Figure 8). Typical unparameterized factors ( f A and f B ) and fit-ting parameters can be found in Table 1 and 2, respectively.
4. Discussions i and He i Figure 9 shows a comparison of RR rates ( α RR t ), electron energy-loss rates ( L RR t ), weighted electron energy-loss factors ( f RR t )from this work, Seaton (1959, blue), Ferland et al. (1992, or-ange) and Hummer (1994, red). Since both Ferland et al. (1992)and Hummer (1994) use the same PICSs (Storey & Hummer f t T (eV) d ( % ) −2−101 N−like Fe XX
Fig. 5.
The RR weighted electron energy-loss factor for N-like iron(Fe xx ). The black dots in both panels (associated with artificial errorbars of 2.5% in the upper one) are the calculated weighted electronenergy-loss factor. The red solid line is the best-fit. The lower panelshows the deviation (in percent) between the best-fit and the originalcalculation. d max (%) C oun t s H to Ne−likeH/He/Ne−like
Fig. 6.
The histogram of maximum deviation in percent ( δ max ) for all theions considered here, which reflects the overall goodness of our param-eterization. The dashed-histogram is the statistics of the more importantH-like, He-like and Ne-like isoelectronic sequences, while the solid oneis the statistics of all the isoelectronic sequences. ∼
5% (underes-timation). For the high temperature end ( T & . i is rather low (almost completely ionized),the present calculation is still acceptable. A similar issue to-wards to the high temperature end is also found in the Case Aresults of Seaton (1959), with a relatively significant overesti-mation ( & Article number, page 5 of 9 & Aproofs: manuscript no. rrwelf_arxiv
Table 2.
Fitting parameters of RR weighted electron energy-loss factors for H i , He i and Fe xx . For the former two, both Case A and Case B resultsare included. s Z Case a b c a b a b δ max +
00 5.432E-01 0.000E +
00 1.018E +
01 5.342E-01 0.000E +
00 0.000E +
00 1.2%1 1 B 2.560E +
00 4.230E-01 0.000E +
00 2.914E +
00 4.191E-01 0.000E +
00 0.000E +
00 2.1%2 2 A 2.354E +
00 3.367E-01 0.000E +
00 6.280E +
01 8.875E-01 2.133E +
01 5.675E-01 1.5%2 2 B 1.011E +
04 1.348E +
00 4.330E-03 1.462E +
04 1.285E +
00 0.000E +
00 0.000E +
00 3.5%7 26 A 2.466E +
01 4.135E-01 0.000E +
00 2.788E +
01 4.286E-01 0.000E +
00 0.000E +
00 2.1%
Notes. s is the isoelectronic sequence number of the recombined ion, Z is the atomic number of the ion, a − , b − and c are the fitting parametersand δ max is the maximum deviation (in percent) between the “best-fit" and original calculation. Case A and Case B refers to β t /α t and β n ≥ /α n ≥ RRweighted electron energy-loss factors, respectively. Machine readable fitting parameters and maximum deviation (in percent) for the total weightedelectron energy loss factors for all the ions considered here are available on CDS. f A / B Case ACase B T (eV) −3 −2 −1 d ( % ) −2−101 H I
Fig. 7.
The Case A (solid line, filled circles) and Case B (dashed line,empty diamonds) RR weighted electron energy-loss factor ( f A / B ) forH i . The black dots in both panels (associated with artificial error bars inthe upper one) are the calculated weighted electron energy-loss factor.The red solid line is the best-fit. The lower panel shows the deviation(in percent) between the best-fit and the original calculation. Table 1.
Unparameterize of RR weighted electron energy-loss factorsfor H i , He i and Fe xx . For the former two, both Case A and Case Bresults are treated seperately. T / z H i H i He i He i Fe xx K Case A Case B Case A Case B Case A10 Notes.
Machine readable unparameterized Case A factors for all theions considered here are available on CDS.
Likewise, the comparison for He i between this work andHummer & Storey (1998) is presented in Figure 10. The CaseA and Case B results from both calculations agree well (within2%), at the low temperature end ( T . . T & i are not available in Hummer & Storey (1998). f A / B Case ACase B T (eV) −3 −2 −1 d ( % ) −3−2−1012 He I
Fig. 8.
Similar to Figure 7 but for He i . z Previous studies of hydrogenic systems, Seaton (1959);Ferland et al. (1992); Hummer & Storey (1998), all use z scal-ing for α RRt . That is to say, α X t = z α Ht , where z is the ioniccharge of the recombined ion X . The same z scaling also ap-plies for β RRt (or L RRt ). LaMothe & Ferland (2001) also pointedout that the shell-resolved ratio of f RRn ( = β RR n /α RR n ) can also bescaled with z / n , i.e. f Xn = z n f H n with n refers to the principlequantum number.In the following, we merely focus on the scaling for theion / atom-resolved data set. We show in the top panel of Fig-ure 11 the ratios of f t / z for H-like ions. Apparently, from thebottom panel of Figure 11, the z scaling for the H-like iso-electronic sequence is accurate within 2%. For the rest of theisoelectronic sequences, for instance, the He-like isoelectronicsequence shown in Figure 12, the z scaling applies at the lowtemperature end, whereas, the accuracies are poorer toward thehigh temperature end. We also show the z scaling for the Feisonuclear sequence in Figure 13. We restrict the discussion above for the RR energy-loss of theelectrons in the plasma only. The ion energy-loss of the ionsdue to RR can be estimated as P RR ∼ I i α i , where I i is the ion-ization potential of the level / term the free electron is capturedinto, and α i is the corresponding RR rate coe ffi cient. Whether to Article number, page 6 of 9ao et al.: RR electron energy-loss rate a A / B ( c m s − ) −20 −18 −16 −14 −12 −10 Seaton 1959Ferland et al. 1992Hummer 1994This work L A / B (r y d c m s − ) −17 −16 −15 −14 −13 Case ACase B f A / B f o t he r A / f p r e s en t A T (eV) −3 −2 −1 f o t he r B / f p r e s en t A Fig. 9.
The comparison of the RR data for H i among results from thiswork (black), Seaton (1959, blue), Ferland et al. (1992, orange) andHummer (1994, red). Both results of case A (solid lines) and case B(dashed lines) are shown. The total RR rates ( α RRA / B ) and electron energy-loss rates ( L RRA / B ) are shown in the top two panels. The RR weightedelectron energy-loss factors ( f A / B ) are shown in the middle panel. Theratios of f A / B from this work and previous works with respect to the fit-ting results (Equation 7 and Table 2) of this work, i.e. f otherA / B / f presentA / B , areshown in the bottom two panels. include the ionization potential energies as part of the total in-ternal energy of the plasma is not critical, as long as the entirecomputation of the net energy gain / loss is self-consistent (see adiscussion in Gnat & Ferland 2012). On the other hand, wheninterpreting the emergent spectrum due to RR, such as the ra-diative recombination continua (RRC) for a low-density plasma,the ion energy-loss of the ion is essentially required. The RRCemissivity (Tucker & Gould 1966) can be obtained via dE RRC dt dV = Z ∞ n e n i I + m v ! v σ ( v ) f ( v, kT ) d v = n e n i I (1 + f t kT / I ) α t , (9)where n e and n i are the electron and (recombining) ion num-ber density, respectively. Generally speaking, the ion energy-loss of the ion dominates the electron energy-loss of the elec-trons, since f t is of the order of unity while kT . I holds forthose X-ray photoionizing plasmas in XRBs (Liedahl & Paerels1996), AGN (Kinkhabwala et al. 2002) and recombining plas-mas in SNRs (Ozawa et al. 2009). Figure 14 shows the threshold a A / B ( c m s − ) −15 −14 −13 −12 −11 Hummer&Storey 1998This work L A / B (r y d c m s − ) −15 −14 −13 Case ACase B f A / B f o t he r A / f p r e s en t A T (eV) −3 −2 −1 f o t he r B / f p r e s en t A Fig. 10.
Similar to Figure 10 but for He i between this work (black) andHummer & Storey (magenta 1998). The latter one only provides datawith T ≤ . K. temperature above which the electron energy-loss via RR cannotbe neglected compared to the ion energy-loss. For hot plasmaswith kT & Z >
5. It is necessary to emphasize that werefer to the electron temperature T of the plasma here, which isnot necessarily identical to the ion temperature of the plasma, inparticular, in the nonequilibrium ionization scenario. Various calculations of (total or shell / term / level-resolved) RRdata are available from the literature. Historically, di ff erent ap-proaches have been used for calculating the total RR rates, in-cluding the Dirac-Hartree-Slater method (Verner et al. 1993) andthe distorted-wave approximation (Gu 2003; Badnell 2006). Ad-ditionally, Nahar and coworkers (e.g. Nahar 1999) obtained thetotal (unified DR + RR) recombination rate for various ions withtheir R -matrix calculations. Di ff erent approaches can lead to dif-ferent total RR rates (see a discussion in Badnell 2006), as wellas the Individual term / level-resolved RR rate coe ffi cients, evenamong the most advanced R -matrix calculations. Nevertheless,the bulk of the total RR rates for various ions agrees well amongeach other. As for the detailed RR rate coe ffi cients, consequently,the detailed RR electron energy-loss rate, as long as the di ff er-ence among di ff erent methods are within a few percent and given Article number, page 7 of 9 & Aproofs: manuscript no. rrwelf_arxiv f t H−like isoelectronic sequenceT (eV) −3 −2 −1 ( f t / z ) X / ( f t / z ) H H OAr Ni
Fig. 11.
The z scaling for the H-like isoelectronic sequence (Case A),including H i (black), O viii (red), Ar xviii (orange) and Ni xxviii (green).The top panel shows the ratios of f t / z as a function of electron temper-ature ( T ). The bottom panel is the ratio of ( f t / z ) X for ion X with respectto the ratio of ( f t / z ) H for H. f t He−like isoelectronic sequenceT (eV) −3 −2 −1 ( f t / z ) X / ( f t / z ) H e HeOSiFe
Fig. 12.
Similar to Figure 11 but for the z scaling for the He-like iso-electronic sequences. the fact that each individual RR is .
10% of the total RR rate fora certain ion / atom, the final di ff erence in the total weighted elec-tron energy-loss factors f t are still within 1%. In other words,although we used the re-calculated total RR rate (Section 2.2)to derive the weighted electron energy-loss factors, we assumethese factors can still be applied to other total RR rates. f t Fe isonuclear sequenceT (eV) ( f t / z ) X − li k e / ( f t / z ) H − li k e H−likeHe−likeLi−likeBe−likeB−like C−likeN−likeO−likeF−likeNe−like
Fig. 13.
The z scaling for the Fe isonuclear sequence. The top panelshows the ratios of f t / z as a function of electron temperature ( T ). Thebottom panel is the ratio of ( f t / z ) X − like for X -like Fe with respect to theratio of ( f t / z ) H − like for H-like Fe xxvi . H−like
Element
He C OBe Ne Mg Si S Ar Ca Ti Cr Fe Ni Zn T ( e V ) He−like kT f t > I kT f t > 0.1 I Fig. 14.
The threshold temperature above which the electron energy-loss via RR cannot be neglected, compared to the ion energy-loss, forH-like (solid lines) and He-like ions (dashed lines).
Acknowledgements.
J.M. acknowledges discussions and support from M.Mehdipour, A. Raassen, L. Gu and M. O’Mullane. We thank the referee, G.Ferland, for the valuable comments on the manuscript SRON is supported fi-nancially by NWO, the Netherlands Organization for Scientific Research.
Article number, page 8 of 9ao et al.: RR electron energy-loss rate
References