KAPPA: A Package for the Synthesis of Optically Thin Spectra for the Non-Maxwellian κ-Distributions. III. Improvements to Ionization Equilibrium and Extension to κ < 2

The KAPPA package is designed for calculations of optically thin spectra for the non-Maxwellian κ-distributions. This paper presents an extension of the database to allow calculations of the spectra for extreme values of κ < 2, which are important for accurate diagnostics of the κ-distributions in the outer solar atmosphere. In addition, two improvements were made to the ionization equilibrium calculations within the database. First, the ionization equilibrium calculations now include the effects of electron impact multi-ionization (EIMI). Although relatively unimportant for Maxwellian distribution, EIMI becomes important for some elements, such as Fe and low values of κ, where it modifies the ionization equilibrium significantly. Second, the KAPPA database now includes the suppression of dielectronic recombination at high electron densities, evaluated via the suppression factors. We find that at the same temperature, the suppression of dielectronic recombination is almost independent of κ. The ionization equilibrium calculations for the κ-distributions are now provided for a range of electron densities.

Detection of the κ-distributions in the low solar corona or transition region (including source regions of the solar wind) relies on analyses of emission spectra.In principle, electron κdistributions can be detected from ratios of emission-line intensities, while ion κ-distributions can be detected by analysis of well-resolved emission-line profiles.Indications of the presence of electron κ-distributions from line intensity ratios have been obtained, for example, by Dudík et al. (2015), Lörinčík et al. (2020), andDel Zanna et al. (2022), who showed that in active regions, the distribution is likely strongly non-Maxwellian with a very low value of κ ≈ 2. Similarly, emission lines from flaring plasma also show strong departures from Maxwellian distributions (Dzifčáková et al. 2018).Contrary to that, the quiet Sun or even bright-point plasma are Maxwellian (Lörinčík et al. 2020;Del Zanna et al. 2022;Savage et al. 2023).In Del Zanna et al. (2022), the quiet Sun was observed within the same data set directly in the vicinity of the active region.In Lörinčík et al. (2020), the quiet Sun was observed at a date similar to the active region.Therefore, in both cases, the degradation of the instrument sensitivity with time could have been neglected.This is important, since the spatial variations of κ are therefore independent of the instrument calibration used.Thus, the veracity of detections of κ-distributions in active region coronae can be established.Emission-line profiles from the transition region, corona, and flaring plasma are also consistent with being strongly non-Maxwellian, with low values of κ (Jeffrey et al. 2016(Jeffrey et al. , 2017(Jeffrey et al. , 2018;;Dudík et al. 2017a;Polito et al. 2018).In addition, some continuum bremsstrahlung emission from flaring plasma has also been found to be consistent with κ-distributions in some instances (Kašparová & Karlický 2009;Oka et al. 2013Oka et al. , 2015;;Battaglia et al. 2015Battaglia et al. , 2019)).Finally, other indications of the presence of κ-distributions in solar flares have been obtained from hydrodynamic modeling of various spectral properties (see Allred et al. 2022).
Recently, Mondal et al. (2020) reported the presence of weak radio bursts (in the mSFU and sub-picoflare range) originating in the solar corona.Such weak bursts are likely associated with small EUV brightenings (Mondal 2021).Subsequently, Sharma et al. (2022) showed that such radio events are ubiquitous, occur both in the quiet Sun and active regions, carry sufficient energy to heat the solar corona, and are associated with the acceleration of electrons to 0.4-4 keV.The more energetic events were found in the vicinity of active regions.We note that these electron energies are similar to the electron energies possessed by the high-energy tail of the κ-distributions detected from analysis of emission-line spectra (see, e.g., the top panel of Figure 10 in Del Zanna et al. 2022).We also note that the energies of the bursts detected by Mondal et al. (2020) are possibly too low to be detected with the dedicated hard X-ray instrumentation used to observe energies only above 2-4 keV, depending on the instrument (see, e.g., Hannah et al. 2010Hannah et al. , 2016;;Marsh et al. 2017;Buitrago-Casas et al. 2022;Paterson et al. 2023), while focusing on constraining the nonthermal electrons in the quiet Sun.
The spectroscopic diagnostics of κ-distributions in the outer solar atmosphere rely on the availability of spectral synthesis, which in turn relies on the availability of the atomic data sets containing rates for individual processes for κ-distributions.This task is accomplished by the KAPPA database and software. 1In Paper I (Dzifčáková et al. 2015), the basic concept was established and presented.In Paper II (Dzifčáková et al. 2021), the database was updated to be compatible with the latest release of CHIANTI, version 10 (Dere et al. 1997;Del Zanna et al. 2021), including additional processes such as the two-ion model.Here, we provide improvements to the existing database and software by including calculations for values of κ < 2 (Sections 3.1 and 4), as well as processes such as electron impact multi-ionization (EIMI; Section 3.2) and density suppression of dielectronic recombination (Section 3.3).Finally, we note that since 2020, the synthetic spectra for non-Maxwellian κ-distributions can also be calculated using the AtomDB project and its extensions (see Smith et al. 2001;Foster et al. 2012;Cui et al. 2019;Foster & Heuer 2020).The AtomDB project relies on the Maxwellian decomposition method (Hahn & Savin 2015a; see also our Section 3.3), while the KAPPA package relies on its own calculations of the individual non-Maxwellian rates (see Papers I and II for details) and follows the CHIANTI database formats and procedures.

The Non-Maxwellian Electron κ-Distributions
The KAPPA database allows for calculations of the synthetic optically thin spectra for the non-Maxwellian electron κdistributions (Olbert 1968;Vasyliunas 1968aVasyliunas , 1968b;;Livadiotis & McComas 2009).We note that several definitions of the κ-distributions exist (see, e.g., Livadiotis & McComas 2013;Lazar et al. 2016;Livadiotis 2017;Lazar & Fichtner 2021).At present, the KAPPA database uses a simple formulation for the isotropic electron distribution f κ (E)dE, which depends only on the electron kinetic energy E and has two parameters, κ and T: where k B is the Boltzmann constant, A κ is a normalization constant, and the mean kinetic energy á ñ E is given by the expression á ñ = E k T 3 2 B .The 3/2 < κ < ∞ is an independent parameter describing the degree of departure from Maxwellian, which corresponds to the limit of κ → ∞ (see Figure 1).We also note that Equation (1) corresponds to a "Kappa-A," distribution as defined and discussed by Lazar et al. (2016).
At present, such a definition of the κ-distribution is sufficient to evaluate the effects of the non-Maxwellian electron distributions (NMEDs) characterized by a high-energy powerlaw tail on the optically thin spectra.This is because for the κdistributions, the tail is relatively strong (see Figure 1), compared, for example, to the distributions composed of a core Maxwellian and a power-law tail (Dzifčáková et al. 2011), used to describe the electron distribution derived from X-ray bremsstrahlung in flares.Still, even with a relatively strong tail, the changes in intensity of most emission lines with κdistributions are of the order of several tens of percent, and rarely more than a factor of 2 compared to the Maxwellian (see, e.g., Dzifčáková & Kulinová 2010;Mackovjak et al. 2013;Dudík et al. 2015;Lörinčík et al. 2020;Del Zanna et al. 2022;  Savage et al. 2023).The reasons for this choice are chiefly twofold.First, the observational uncertainties in the emission-line intensities are relatively large, due to the radiometric calibration and its degradation, with the precision of the calibration of spectroscopic instruments such as Hinode's Extreme-Ultraviolet Imaging Spectrometer being about 20% (see, e.g., Culhane et al. 2007;BenMoussa et al. 2013;Del Zanna 2013).Second, the κdistributions (and any other NMEDs) are usually detectable using line intensity ratios involving two lines with different sensitivity to κ, such as an allowed and a forbidden line (see, e.g., Dudík et al. 2017b;Dzifčáková et al. 2018;Lörinčík et al. 2020;Del Zanna et al. 2022).Forbidden lines are usually much weaker than allowed ones, which means their photon noise uncertainty also limits the determination of κ only to a range of values.Therefore, resolving different types of κ-distributions or indeed differentiating between κ-distributions and other NMEDs with a high-energy power-law tail in the outer solar atmosphere is very difficult at present.
Finally, we note that we do not repeat here the equations for the calculation of the line intensities or individual ionization, recombination, and excitation coefficients, as these have been presented and discussed at length in previous literature, and are summarized in the previous Papers I and II.In the following, we describe the upgrades to the existing database and software.

Ionization and Recombination Rates for κ < 2
Previous versions of the KAPPA database contained individual rates for discrete values of κ = 2, 3, 4, 5, 7, 10, 15, 25, and 33 (see Paper I).However, recent diagnostics of electron distribution in the solar corona showed that the value of κ can be lower than 2 (see Dudík et al. 2015Dudík et al. , 2017a;;Dzifčáková et al. 2018;Polito et al. 2018;Lörinčík et al. 2020;Del Zanna et al. 2022).Previously, the value of κ = 2 was considered a relatively extreme one and was the lowest one available in the KAPPA database.We have now extended our calculations of individual rates, as well as ionization equilibria for values of κ < 2. Namely, the newly available values are κ = 1.9, 1.8, and 1.7.We note that the asymptotic value of κ is κ → 1.5.However, our choice of κ = 1.7 is about the lowest detected (Dudík et al. 2017a;Polito et al. 2018) and should be sufficient to illustrate the effects of such low κ-distributions on the spectra, taking into account the increasing uncertainty of the approximations used for the calculation of the recombination and excitation rates (Dzifčáková et al. 2015).The value of κ = 2.5 was also added to bridge the relatively large gap between κ = 2 and 3.This value of κ also corresponds approximately to one of the critical κ indices in nonextensive thermodynamics (see Livadiotis & McComas 2010).
We calculated the ionization rates directly from the cross sections, similar to other values of κ (see Paper II).The cross sections we used are those from the compilation of Hahn et al. (2017), similar to the latest version 10.1 of CHIANTI (Dere et al. 2023).The recombination rates were calculated using the approximate method of Dzifčáková (1992) and Dzifčáková & Dudík (2013), and we subsequently obtained the ionization equilibria.We note that the ionization equilibria for such low κ values were for the first time calculated by Hahn & Savin (2015a), and we checked that our calculations are in good agreement with those using the method of Hahn & Savin (2015a).
The changes in the ionization equilibrium of iron for low κ in comparison with κ = 2 and other values of κ are shown in Figure 2.For transition-region ions such as Si IV and Fe VIII (top row)-that is, for temperatures below about log(T [K]) ≈ 5.7-the changes in the peaks of the relative ion abundance for κ = 1.7 are small compared to κ = 2.The peaks are widened and shifted to slightly higher temperatures.Note that this shift is in the reverse direction to the shifts for higher κ values (see Dzifčáková & Dudík 2013).That is, with decreasing κ, the peaks are first shifted to progressively lower T until about κ = 2; then, for κ < 2, the peaks shift to slightly higher T.This effect is well visible for Fe VIII (see the top right panel of Figure 2).
At coronal temperatures and higher degrees of ionization, a small change of κ means substantial changes in the ionization peaks (bottom row of Figure 2).The shape of the peaks is wider, but the relative ion abundances are shifted significantly to higher temperatures.For κ = 1.7, the peak temperatures can be up to a factor <2 higher than for κ = 2.This happens for both Fe XII, where the shift of the log (T max [K]) is from 6.35 to 6.55 (a factor of ≈1.6 higher), as well as for Fe XVII, where the ionization peak shifts from log (T max [K]) = 6.65 for κ = 2 to 6.85 for κ = 1.7 (again a factor of ≈1.6).This means that the peak of the relative ion abundance continues its shift to higher temperatures with decreasing κ (a fact previously described up to κ = 2 by Dzifčáková 1992Dzifčáková , 2002;;Dzifčáková & Dudík 2013).Reliable detection of such extremely low values of κ in the solar corona would have significant consequences for the thermal energy content of the solar corona and thus coronal heating requirements.

Electron Impact Multi-ionization
A single electron-ion collision can lead not only to ionization or excitation, but also to multiple ionization of the target ion.For example, it is possible to produce Fe XIV directly from Fe XII if the impacting electron has sufficient energy.Triple ionizations are also possible, although the cross sections fall rapidly with each additional ejected electron (see Hahn et al. 2017).Such EIMIs are typically neglected for equilibrium (Maxwellian) plasmas, because the EIMI becomes important only at very high temperatures, where the abundance of the target ion is already small.For iron, EIMI changes the ionization equilibrium (charge state distribution) by less than about 5%; see Figure 3 of Hahn & Savin (2015b).However, EIMI becomes important both in situations when the plasma is rapidly heated (where EIMI reduces the time plasma needs to reach ionization equilibrium) or when the electron distribution is non-Maxwellian with high-energy tails (Hahn & Savin 2015b).For a κ-distribution with extremely low κ, the EIMI changes the relative ion abundances of Fe by a much larger amount, up to a factor of 2-6, depending on the ion and T (see Figure 9 of Hahn & Savin 2015a).Clearly, EIMI becomes an important process for such NMEDs and needs to be taken into account in calculating the ionization equilibrium.
Denoting I km and R mk as the ionization and recombination rate coefficients for ions in the kth and mth ionization state, k < m Z, and N k = N(X + k )/N(X), with the relative abundance of the ion X + k and Z being the proton number, the ion populations in equilibrium should satisfy the set of linear equations: Ionization rates for single ionization and for EIMI were calculated using the approximation formulae for ionization cross sections provided by Hahn et al. (2017).The new ionization equilibria in the KAPPA database now include the effects of EIMI for all of elements and κ.
As pointed out by Hahn et al. (2017), the effect of multiple ionization on the ionization equilibrium for a Maxwellian distribution is small, but it can become important for the elements with high Z and low κ-values or high T.These effects are shown in Figure 3, where the ionization equilibrium of iron, including EIMI, is plotted for a Maxwellian distribution and several values of κ.These Fe ionization equilibria are compared to calculations without EIMI and in the low-density limit (see Section 3.3).The effects of EIMI are of importance for coronal Fe ions such as Fe X-Fe XVIII between log(T [K]) ≈ 6-7 (see Figure 9 of Hahn & Savin 2015a), although the details depend on the ion.Generally, for the affected ions, EIMI shifts the peak formation temperature T max toward lower values.These shifts of T max toward lower values of T occur since additional ionization leads to the increase of the ionization state at a given electron kinetic energy.For some ions, the changes in the relative ion abundances occur at > T T ; max for others, they occur at all temperatures (for example, Fe XIV; see Figure 3), and for other ions still, such as Fe XVII, the EIMI dominantly affects temperatures below T max .Therefore, the EIMI is also a potentially significant process affecting the X-ray spectra, such as those observed by MaGIXS (Savage et al. 2023), should the X-ray lines of Fe XVII-Fe XVIII be formed in non-Maxwellian conditions.We note that for coronal ions, the shifts in the ionization equilibrium with κ occur in the opposite direction to the shifts due to the inclusion of EIMI.That is, with decreasing κ, the peaks are shifted toward higher T compared to Maxwellian calculations.This shift with κ is a result of two competing processes.The increase of the ionization rate due to highenergy electrons pushes the ionization peaks toward lower T, while the increase of the radiative recombination rates due to the excess of electrons at low energies and low κ (see Figure 1) pushes them to higher temperatures.In addition, changes in dielectronic recombination rates with κ also affect the shift.The resulting net effect of κ-distributions on coronal Fe ions is the shift of their peaks toward higher temperatures, as the changes in the total recombination rate are larger than the changes in the ionization rate at the temperatures where the ion peak occurs (see Figure 2 of Dzifčáková & Dudík 2013).When EIMI is added, the total ionization rate is increased, resulting in the shift of the ionization peak to lower T, i.e., in the opposite direction to the shifts with κ.

Density Suppression of Dielectronic Recombination
Recombination in the outer solar atmosphere consists of two processes, radiative and dielectronic recombination.Burgess (1964) has shown that the latter one can be the more important one.Therefore, the accurate implementation of dielectronic recombination is critical for analysis of the observed transition region and coronal spectra.The KAPPA database contains recombination rates for κ-distributions calculated using the approximate methods developed by Dzifčáková (1992) and summarized in Section 3.1 of Dzifčáková & Dudík (2013).These rates are valid, similar to the corresponding Maxwellian ones in CHIANTI, only in the limit of low electron densities log(N e [cm −3 ]) → 0.
However, in relatively high-density plasmas, additional electrons within the plasma can lead to electron-ion collisions and ionization from the doubly excited resonance states.Thus, the radiative rates from the doubly excited states are diminished and the total dielectronic recombination rate R DR is suppressed.This suppression of dielectronic recombination was initially studied by Burgess & Summers (1969) and Summers (1972Summers ( , 1974)), then later by Summers & Hooper (1983), Badnell et al. (1993Badnell et al. ( , 2003)), Nikolić et al. (2013Nikolić et al. ( , 2018)) In the last three of these works, the generalized collisionalradiative modeling was employed for carbon (Dufresne & Del Zanna 2019), oxygen (Dufresne et al. 2020), and then for low charge states generally (Dufresne et al. 2021).The generalized collisional-radiative modeling incorporates not only density suppression of dielectronic recombination, but also other processes affecting the ion population, such as photoinduced processes and charge transfer.Although these processes can be of importance for transition-region ions, such modeling relies on a vast quantity of reliable atomic data that are not yet readily available.Implementing the generalized collisional-radiative modeling would also mean significant divergence from the methods of calculating synthetic spectra employed in the present version 10.1 of the CHIANTI database, on which the KAPPA package is based.Therefore, for the present, we focus on studying the behavior of the density suppression of dielectronic recombination with κ.Although this process is important, we caution the reader that the calculations presented below are not a substitute for generalized collisional-radiative modeling.
Generally, the suppression of dielectronic recombination can be expressed through a dimensionless factor of S M (T, N e , q) (Nikolić et al. 2013(Nikolić et al. , 2018)): where R DR (T) is the dielectronic recombination rate in the log(N e [cm −3 ]) → 0 limit, R DR (N e , T, q, M) is the densitysuppressed rate, q is a parameter depending on the ion, and M is the isoelectronic sequence.Nikolić et al. (2013) used earlier generalized radiative-collisional models to develop an approximation formula for the suppression factor as a function of isoelectronic sequence, charge, electron density, and temperature.Nikolić et al. (2018) presented improved fits to calculate the suppression of dielectronic recombination at intermediate electron densities, but only for the Maxwellian electron distribution.These authors have shown that the suppression factor depends on the atomic parameters q of ion, on the electron density N e , and through activation density on T 1/2 (Equation (3) of Nikolić et al. 2018).Therefore, the effect of electron density dominates over the effect of electron temperature.
As no attempt to calculate the suppression factors of dielectronic recombination for any other distribution than Maxwellian has been made so far, we calculated the suppression of dielectronic recombination ( ) for κdistributions.To do that, we employed the Maxwellian decomposition approach of Hahn & Savin (2015a).These authors approximated κ-distributions by a sum of several Maxwellians with different temperatures T j : This allowed them to use the linearity property and thus calculate the non-Maxwellian rates R κ (T) for any collisional process as a weighted sum of the corresponding Maxwellian rates (see Equation (12) of Hahn & Savin 2015a): where the coefficient a j and temperatures T j are tabulated in Hahn & Savin (2015a).It follows that the suppression factor for κ-distributions can then also be calculated as a weighted sum of the Maxwellian suppression factors: The suppressed dielectronic recombination rate for κ-distributions is then simply given by the expression analogous to Equation (3): where R DR,κ is the dielectronic recombination rate for κdistributions in the low-density limit.We note that although the present method is based on the fits of Nikolić et al. (2018), and thus not exact, it allows us to estimate the effect of the electron density on the ionization equilibrium for κ-distributions.
We found that the suppression factors ( ) for κdistributions are usually very close to the Maxwellian ones at the same temperature.The differences reach only about 10% in most cases, although larger differences can occur, as discussed below.Generally, the differences are largest for low values of κ.Examples of the behavior of the suppression factors ( ) with κ are shown in Figure 4.The cases shown include dielectronic recombination from Si IV to Si III, as well as Fe VIII → Fe VII, Fe X → Fe IX, and Fe XV → Fe XIV.Where the Maxwellian suppression factor increases with temperature, the changes with κ are small, and the suppression factor for low κ is slightly smaller than for a Maxwellian distribution.This behavior with κ is independent of electron density.For ions such as Fe VIII and Fe X, this behavior of the suppression factor occurs in the entire range of T (see Figure 4), while for ions such as Fe XV, this behavior occurs at temperatures log(T [K]) 6, where the ion is formed.Clearly, the suppression factor in these cases depends primarily on log(N e [cm −3 ]) and only weakly on κ.These small differences in the suppression factors with κ are however comparable with the precision of the fits to the suppression factors themselves as obtained by Nikolić et al. (2018).
Larger differences with κ in the suppression factors occur in cases where the Maxwellian suppression factors decrease with T. With decreasing κ, the suppression factor progressively increases, i.e., the dielectronic recombination rate becomes progressively less suppressed (see Equation ( 7)).This behavior of the suppression factor is important for transition-region ions, such as Si IV, at temperatures where the ion is formed, and it occurs for a range of electron densities (see the top panel of Figure 4).It also occurs at high densities for ions such as Fe XV, but only at relatively low temperatures below log(T [K]) 6, where the ion is not formed in ionization equilibrium.
Figure 5 shows the resulting density-suppressed dielectronic recombination rates for Maxwellian (left panels) and κdistributions with κ = 2 (right panels).Other κ-distributions are not shown, as in most cases the suppression coefficient is not strongly sensitive to κ.It can be seen that the density suppression strongly depends on log(N e [cm −3 ]), the atomic parameters of ions (with Fe VIII being much more strongly affected than Fe X), and only slightly on T.
How the resulting relative ion abundances are affected varies depending on all parameters, mostly on the ion and N e , but also on T and κ, as the changes in the low-density ionization equilibrium with T and κ (see Dzifčáková & Dudík 2013) are compounded or reduced by changes with N e for the individual ion.Generally, the peaks of the relative ion abundances are shifted to lower log(T [K]) for higher electron densities.Elements with higher Z are typically more affected, and the largest changes can occur at transition-region temperatures, i.e., for log(T [K])  6.
A comparison of the resulting ionization equilibria (i.e., relative ion abundances) for Si and Fe are shown in Figures 6,   7, and 8, respectively.Figure 6 shows the behavior of the relative ion abundance for several important ions, Si IV, Fe VIII, and Fe X, while Figures 7 and 8 show multiple ions of Si and Fe, respectively.Aside from the density-dependent dielectronic recombination, these ionization equilibria also contain the effects of EIMI (Section 3.2).As already noted, however, other processes could still influence the ionization equilibrium, especially for low charge states (see, for example, Dufresne et al. 2020).
For the elements Si and Fe, the resulting ionization equilibrium in the transition region is sensitive to the electron density regardless of the electron distribution, with some ions, such as Si IV and Fe VI-Fe VIII, being affected more than others.For example, at electron densities log(N e [cm −3 ]) ≈ 10-11, typical of the transition region, the peak of Fe VII is shifted to lower T by a factor of 0.7 both for the Maxwellian and κ = 2, and the peaks get wider with progressively lower κ (a well-known effect; see Figure 8 and Dzifčáková & Dudík 2013).The relative abundance of Si IV is very sensitive to N e , but more so for Maxwellian distributions than for κ = 2.For the Maxwellian distribution, the peak of Si IV is shifted slightly, to about log(T [K]) = 4.8, and at densities of log(N e [cm −3 ]) = 10, it is nearly 2 times higher than for the low-density limit (see also Figure 13 of Polito et al. 2016).For κ = 2, however, the peak of Si IV is almost independent of N e .This is due to the increase in the suppression factor for such low κ (see the top panel of Figure 4).
At temperatures and electron densities typical of the solar corona, the effects of suppression of dielectronic recombination at higher N e are much more subtle.Regardless of the value of κ, for log(N e [cm −3 ]) = 9-10, the shift of the ionization peaks with N e is small and the peaks almost do not change their shapes (see Figure 9).This is important for diagnostic purposes, as the previous diagnostics of κ (Dudík et al. 2015;Lörinčík et al. 2020;Del Zanna et al. 2022) were based on the low-density ionization equilibrium calculations.For flare temperatures and densities, the density suppression of dielectronic recombination is negligible and thus does not affect the diagnostics of flaring plasma (see Dzifčáková et al. 2018).
We note that the CHIANTI software and database as yet does not contain ionization equilibria with density-suppressed dielectronic recombination or EIMI.Therefore, the respective Maxwellian ionization equilibria are also included in the present version of the KAPPA database.The naming conventions for the respective ionization equilibrium file names are detailed in Appendix A. Finally, we caution the reader that the effects of finite density due to the suppression of dielectronic recombination and the effects of κ-distributions may not be readily distinguishable, especially in the transition region, and that caution should be exercised in interpreting the observed spectra arising from plasma at electron densities where these effects play a role.

Excitation Rates
We endeavor to maintain the database compatible with the latest version of CHIANTI, currently in version 10.1.2Following that, there are no major changes in the excitation data within KAPPA since Paper II.The only exception is that the KAPPA database now includes the requisite containing collisional excitation and deexcitation rates for the respective values of κ = 1.7, 1.8, 1.9, as well as 2.5 (see also Section 3.1), so that the respective synthetic spectra can be calculated.The naming conventions for the corresponding files are described in Appendix B.
We note that the atomic data sets are huge, with some ions containing hundreds of energy levels.Therefore, maintaining compatibility with CHIANTI is a huge task.As the atomic data within CHIANTI can change at any time, KAPPA contains since Paper II its own branch of Maxwellian excitation cross sections for spectral synthesis.This is done so that the corresponding Maxwellian calculations are always available.Should the compatibility of the NMED data within KAPPA with respect the atomic data in CHIANTI be broken at any time, users are encouraged to contact the KAPPA team with a request to update our atomic data for the κ-distributions.
As an example of the synthetic spectrum calculated for low values of κ for multiple ions, in Figure 10 we show a portion of the X-ray spectrum at 14-18 Å observable by the MaGIXS instrument (Savage et al. 2023), currently scheduled for second launch in 2024.There, three spectra are shown: a Maxwellian one in black, a κ = 2 one in red, and κ = 1.7 spectrum in violet.The spectra are calculated for a constant log(T [K]) = 6.6 and scaled in emission measure so that the ratio of the two Fe XVII lines, 15.01 Å/15.26Å, is kept constant.This approach follows the example spectrum provided in Figure 3 of Dudík et al. (2019), and enables one to immediately recognize which spectral lines are sensitive to κ if the value of log(T [K]) is held constant.Figure 10 shows that at this temperature typical of active region cores, the O VII and O VIII lines become strongly enhanced with respect to the neighboring Fe XVII ones, especially for extremely low κ = 1.7.However, we note that the present spectra are calculated for a simple case of constant log(T [K]) = 6.6; a value where the Fe XVII abundance is relatively low for κ = 1.7 (see Figure 2).The temperature is typically a parameter determined from observations using a range of synthetic spectrum calculations where the log(T [K]) can vary (see, for example, Figures 14 and 16 of Savage et al. 2023 and the discussion therein).Nevertheless, our example calculations demonstrate that at least in principle, it could be possible to distinguish the extreme case of κ = 1.7 even from κ = 2 using the optically thin spectra of the solar corona.

Summary
We performed an update of the ionization equilibrium calculations in the KAPPA database together with adding several improvements.These include: 1. extension of the calculations toward low values of κ < 2 and adding κ = 2.5; 2. addition of EIMI; and 3. addition of density suppression of dielectronic recombination.
The extension of the KAPPA database to extremely low values of κ < 2 was also done for the excitation rates, so that full calculations for such values of κ can now be performed.This extension of the database was prompted by the recent results indicating that the value of κ in the solar transition region, corona, and flares can be quite low (Dudík et al. 2017a;Dzifčáková et al. 2018;Lörinčík et al. 2020;Del Zanna et al. 2022).
The process of EIMI is generally not important for the Maxwellian electron distribution, but it becomes important for strongly NMEDs, such as those with low values of κ.Therefore, its inclusion is necessary for proper diagnostics of the electron distribution using emission-line intensities formed in neighboring ionization stages (see Lörinčík et al. 2020;Del Zanna et al. 2022).In agreement with Hahn & Savin (2015a), we find that the ionization balance for Fe is significantly modified at log(T [K]) ≈ 6-7, where multiple ions are affected if the value of κ is low.
To evaluate the density suppression of dielectronic recombination for κ-distributions, we followed the approach of Nikolić et al. (2018) by calculating the respective suppression factors for κ-distributions.To do that, we employed the Maxwellian decomposition method of Hahn & Savin (2015a).We find that the suppression factors are in most cases very close to Maxwellian.At the same temperature, they are within 10% of the respective Maxwellian ones, although larger differences can occur, especially for transition-region ions such as Si IV.The ionization equilibria were subsequently calculated for all values of κ and a range of electron densities, and made available within the KAPPA database.Notably, the suppression of dielectronic recombination affects the coronal iron ions, as well as several transition-region ions, such as Si IV.For the latter, the density suppression of dielectronic recombination becomes relatively unimportant for extremely low values of κ.
We note that the present improvements are not a substitute for the generalized collisional-radiative modeling that is required for some emission lines, such as those from the transition region (see, e.g., Dufresne et al. 2020 and references therein for a detailed discussion).Nevertheless, the processes listed above should serve to increase the accuracy of diagnostics of κ-distributions, especially low values of κ, in the solar corona and possibly in other astrophysical environments.

Figure 1 .
Figure 1.Electron κ-distributions as a function of electron kinetic energy E, plotted for various values of κ and log(T [K]) = 6.6.

Figure 2 .
Figure 2. Relative ion abundances and their changes with decreasing κ.The individual colors denote the values of κ.Violet stands for κ = 1.7, while red denotes κ = 2.A variety of ions are shown, from the transition-region Si IV and Fe VIII to the coronal Fe XII and Fe XVII.

Figure 3 .
Figure3.The effect of multi-ionization on the Fe ionization equilibrium for a Maxwellian distribution (top) and κ-distributions with κ = 5, 2, and 1.7 (rows 2-4).Calculations for all distributions, including the Maxwellian, were based on the ionization cross sections ofHahn et al. (2017) and recombination rates from CHIANTI version 10.1.

Figure 4 .
Figure 4. Examples of the behaviors of the suppression factors for dielectronic recombination for a selection of ions.Individual values of κ and log(N e [cm −3 ]) are indicated.

Figure 5 .
Figure 5. Dielectronic recombination rate coefficients, where the effects of finite density on dielectronic recombination are shown for Maxwellian distributions (left) and κ-distributions with κ = 2 (right).Several ions are shown: Si V, i.e., recombination from Si V to Si IV (top), Fe VIII (middle), and Fe X (bottom).

Figure 6 .
Figure 6.Relative ion abundances of Si IV (top), Fe VIII (middle), and Fe X (bottom), where the effects of finite density on dielectronic recombination are shown for Maxwellian distributions (left) and κ-distributions with κ = 2 (right).The calculations for all distributions, including the Maxwellian, were based on the ionization cross sections of Hahn et al. (2017) and recombination rates from CHIANTI version 10.1.

Figure 7 .
Figure 7. Silicon relative ion abundances (ionization equilibrium) for a Maxwellian distribution (top) and κ-distributions with κ = 5, 2, and 1.7 (rows 2-4).The individual line styles correspond to different values of log(N e [cm −3 ]).The calculations for all distributions, including the Maxwellian, were based on the ionization cross sections ofHahn et al. (2017) and recombination rates from CHIANTI version 10.1.Note that the effect of EIMI was included.

Figure 8 .
Figure8.The same as Figure7, but for iron.

Figure 9 .
Figure 9. Fe Maxwellian ionization equilibrium (top) and κ = 2 (bottom) in the vicinity of T = 10 6 K for the low electron density (full lines) and density-dependent equilibria for log(N e [cm −3 ]) = 9 (dotted-dotted-dotted-dashed lines) and 10 (dashed lines).Calculations for all distributions, including the Maxwellian, were based on the ionization cross sections of Hahn et al. (2017) and recombination rates from CHIANTI version 10.1.

Figure 10 .
Figure10.Simulated MaGIXS spectra in the 14-18 Å range showing multiple Fe XVII, Fe XVIII, O VII, and O VIII lines at constant log(T [K]) = 6.6.Maxwellian spectra are denoted by the black lines, while κ = 2 and 1.7 are denoted by red and violet, respectively.Note the spectra are scaled to keep the Fe XVII 15.01 Å/ 15.26 Å ratio constant, to highlight changes in other spectral lines.