Issue 
A&A
Volume 657, January 2022



Article Number  A61  
Number of page(s)  11  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202141550  
Published online  10 January 2022 
Plasma environment effects on K lines of astrophysical interest
V. Universal formulae for ionization potential and Kthreshold shifts
^{1}
Physique Atomique et Astrophysique, Université de Mons – UMONS,
7000
Mons, Belgium
email: patrick.palmeri@umons.ac.be
^{2}
Cahill Center for Astronomy and Astrophysics, California Institute of Technology,
Pasadena,
CA
91125, USA
^{3}
Department of Physics, Western Michigan University,
Kalamazoo,
MI
49008, USA
^{4}
Helmholtz Institut Jena,
07743
Jena, Germany
^{5}
Theoretisch Physikalisches Institut, Friedrich Schiller Universität Jena,
07743
Jena, Germany
^{6}
Dr. Karl RemeisObservatory and Erlangen Centre for Astroparticle Physics,
Sternwartstr. 7,
96049
Bamberg, Germany
^{7}
NASA Goddard Space Flight Center,
Code 662,
Greenbelt,
MD
20771, USA
^{8}
IPNAS,
Université de Liège,
Sart Tilman,
4000
Liège, Belgium
Received:
15
June
2021
Accepted:
9
October
2021
Aims. We calculate the plasma environment effects on the ionization potentials (IPs) and Kthresholds used in the modeling of K lines for all the ions belonging to the isonuclear sequences of abundant elements apart from oxygen and iron, namely: carbon, silicon, calcium, chromium, and nickel. These calculations are used to extend the data points for the fits of the universal formulae, first proposed in our fourth paper of this series, to predict the IP and Kthreshold lowerings in any elemental ion.
Methods. We used the fully relativistic multiconfiguration Dirac–Fock method and approximated the plasma electronnucleus and electronelectron screenings with a timeaveraged Debye–Hückel potential.
Results. We report the modified ionization potentials and Kthreshold energies for plasmas characterized by electron temperatures and densities in the ranges of 10^{5}−10^{7} K and 10^{18}−10^{22} cm^{−3}. In addition, the improved universal fitting formulae are obtained.
Conclusions. We conclude that since explicit calculations of the atomic structures for each ion of each element under different plasma conditions is impractical, the use of these universal formulae for predicting the IP and Kthreshold lowerings in plasma modeling codes is still recommended. However, their comparatively moderate to low accuracies may affect the predicted opacities with regard to certain cases under extreme plasma conditions that are characterized by a plasma screening parameter of μ > 0.2 a.u., especially for the Kthresholds.
Key words: black hole physics / plasmas / atomic data / Xrays: general
© ESO 2022
1 Introduction
Supersolar abundances have been inferred from K lines of different elements observed in the Xray spectra of Xray binaries (XRB) and active galactic nuclei (AGN) (see e.g., Kallman et al. 2009; Dong et al. 2020; Walton et al. 2020; Fukumura et al. 2021). These absorption and emission features can occur in the inner regions of the blackhole accretion disks where the plasma densities are predicted to range from 10^{15} to 10^{22} cm^{−3} (Schnittman et al. 2013). The emerging photons can be recorded with current space observatories such as XMMNewton, NuSTAR, and Chandra. In addition, synthetic spectra can provide measures of the composition, temperature, and degree of ionization of the plasma (Ross & Fabian 2005; García & Kallman 2010). Nevertheless, the great majority of the atomic parameters used for spectral modeling that involves Kshell processes do not take density effects into account and, therefore, their usefulness is comprised in abundance determinations beyond densities of 10^{18} cm^{−3} (Smith & Brickhouse 2014).
In a series of papers dedicated to plasma density effects on the atomic parameters used to model K lines in ions of astrophysical interest, the ionization potentials (IPs), Kthresholds, transition wavelengths, radiative emission rates, and Auger widths have been computed with the relativistic multiconfiguration Dirac–Fock (MCDF) method (Grant et al. 1980; McKenzie et al. 1980; Grant 1988), as implemented in the GRASP92 (Parpia et al. 1996) and RATIP (Fritzsche 2012) atomic structure packages. The plasma electron–nucleus and electron–electron screenings are approximated with a timeaveraged Debye–Hückel (DH) potential. The datasets comprise the following ionic species: O I – O VII, by Deprince et al. (2019a, hereafter Paper I); Fe XVII – Fe XXV, by Deprince et al. (2019b, hereafter Paper II); Fe IX – Fe XVI, by Deprince et al. (2020a, hereafter Paper III); and Fe II – Fe VIII, by Deprince et al. (2020b, hereafter Paper IV).
In this fifth paper of the series, the universal fitting formulae for ionization potential (IP) and Kthreshold shifts proposed in Paper IV are improved thanks to subsequent MCDF/RATIP computations of these abovementioned parameters in other representative cosmically abundant elements, namely: carbon (Z = 6), silicon (Z = 14), calcium (Z = 20), chromium (Z = 24), and nickel (Z = 28). These plasma effects are expected to be the main ones with respect to the potential alteration of the ionization balance and opacities.
2 Theoretical method used
In this section, we outline the main changes introduced in the MCDF formalism to take into account the density effects in a weakly coupled plasma. Papers I–II and IV contain more details on this, along with tests of the validity of the method.
The Debye–Hückel (DH) screened Dirac–Coulomb Hamiltonian (Saha & Fritzsche 2006) is expressed as: (1)
where r_{ij} = r_{i} −r_{j} and the plasma screening parameter μ is the inverse of the Debye shielding length λ_{D}. The screening parameter can be given in atomic units (a.u.) in terms of the plasma electron density, n_{e}, and temperature, T_{e}, as (2)
For the typical plasma conditions in blackhole accretion disks, T_{e} ~ 10^{5}−10^{7} K and n_{e} ~ 10^{18}−10^{22} cm^{−3} (Schnittman et al. 2013), the screening parameter μ falls in the interval: 0.0 ≤ μ ≤ 0.24 a.u. The DH screening theory is only valid for a weakly coupled plasma where its thermal energy dominates its electrostatic energy. This can be parameterized by the plasma coupling parameter Γ, defined as: (3)
where Z^{*} is the average plasma ionic charge or ionization: (4)
with n_{i,X} as the number density of an ion, i, of an element, X, bearing a positive charge, z_{i,X}, and d as a measurement of the interparticle distance: (5)
The plasma neutrality implies: (6)
and therefore Eqs. (4) and (5) can be rewritten as: (7) (8)
For a fully ionized hydrogen plasma with Z^{*} = 1, the plasma coupling parameter, considering typical conditions in accretion disks, is well below 1 falling in the interval 0.0003 ≤ Γ ≤ 0.3 which is characteristic of a weakly coupled plasma. Furthermore, for a more realistic cosmological plasma with a mixture of 90% hydrogen and 10% of helium by number (both fully ionized), the average plasma ionization would be Z^{*} = 0.9 + 2 × 0.1 = 1.1. Hence, the above interval limits on the plasma coupling parameter would have to be multiplied by a factor Z^{*2∕3} ~ 1.07 and would still agree with a weakly coupled plasma (see our Paper IV for further details).
In the present study, the active space (AS) method was used in order to obtain the MCDF multiconfigurational expansions for all the ionization stages of the isonuclear sequences of C, Si, Ca, Cr, and Ni, analogously to those of O and Fe in the recent past (see Papers I–IV). This method consists of exciting the electrons from the reference configurations to a given active set of orbitals. For almost all the ions studied in this work, the AS was built by considering all the single and double excitations from some reference configurations (always including the ground ones) up to the n = 3 orbitals (when only K and Lshell electrons are involved in the ground configuration) or up to the n = 3 and 4s orbitals (when K, L, and Mshell electrons are involved in the ground configuration). The reference configurations were taken as the ground one along with those obtained by considering single electron excitations leaving a single hole in the 1s, 2p, and 3p orbitals (when involved in the ground configuration). In some cases, namely for the ions close to the neutral end (i.e., for Ni I – Ni IV, Cr I – Cr II, Ca I – Ca II, Si I – Si II), only single excitations from the reference configurations were considered. In Table 1, we report for each ion grouped by number of electrons, N, the reference configurations, the active orbital sets and the final numbers of configuration state functions (CSF), Φ, generated to build the atomic state functions (ASF), Ψ, in the MCDF expansions as: (9)
where c_{k} are the mixing coefficients.
The computations were carried out with the extended average level (EAL) option, optimizing a weighted trace of the Hamiltonian using level weights proportional to 2J + 1, and they were completed with the inclusion of the relativistic twobody Breit interaction and the quantum electrodynamic corrections (QED) due to selfenergy and vacuum polarization. The MCDF ionic bound states generated by the GRASP92 code were then used in the RATIP program to compute the atomic structures diagonalizing the DH screened Dirac–Coulomb Hamiltonian (Eq. (1)), from which were then obtained the ionization potential and the Kshell threshold energies. Plasma environment effects are computed for a Debye screening parameter in the range 0 ≤ μ ≤ 0.25 a.u.
3 Results and discussion
3.1 Ionization potentials and IP shifts
In Tables A.1–A.5, the MCDF/RATIP ionization potentials (IPs) are reported for plasma screening parameters μ = 0.0 a.u., that is, in the isolated atom case, μ = 0.1 a.u., and μ = 0.25 a.u., along with the corresponding values recommended by the National Institute of Standards and Technology (Kramida et al. 2020, NIST), respectively, for carbon, silicon, calcium, chromium, and nickel. The differences between our theoretical values for μ = 0 a.u. and the NIST IPs are on average within ~±3 eV. In terms of relative differences, they can span from less than a few% for the highlycharged ions, that is, with Z_{eff} = Z − N + 1 > 10 where Z_{eff} is the effective charge and N is the number of bound electrons, up to 22% in Ni I and 38% in Cr I. The latter is the result of numerical difficulties in computing the complex atomic structures involving halffilled 3d subshell and in neutral ends of isonuclear sequences iniron group elements such as chromium and nickel, where the three lower evenparity configurations 3d^{w}, 3d^{w −1}4s, and 3d^{w −2}4s^{2} overlap and are highly mixed.
Figure 1 shows the IP, E_{0} in eV, in logarithmic scale as function of Z_{eff}, for ions belonging to nickel isonuclear sequence chosen as an example. The MCDF/RATIP IPs calculated in this work for μ = 0, 0.1 and 0.25 a.u. are plotted along with the corresponding recommended values of NIST (Kramida et al. 2020). The latter are to be compared with our calculations in the isolated atom case, namely, for μ = 0 a.u. The NIST error bars are too small to be seen in the figure. We can observe that all four curves follow the same trend with the curves for μ = 0.1 a.u. and μ = 0.25 a.u., systematically downshifted with respect to both the NIST and the isolatedatom case ones. Also, two big jumps are seen between Z_{eff} = 26 and Z_{eff} = 27 and between Z_{eff} = 18 and Z_{eff} = 19. These correspond to respectively the transition from the closure of the Kshell and the starting of the filling of the Lshell, and the transition from the closure of the Lshell and the starting of the filling of the Mshell. These features are similar to the ones observed in the oxygen (Paper I) and iron isonuclear sequences (Papers IIIV).
In Fig. 2, the MCDF/RATIP IP shift, ΔE_{0} = E_{0}(μ) − E_{0}(μ = 0) in eV, with respect to the isolated atom plasma condition (μ = 0 a.u.) is plotted as function of the effective charge in the same example (Ni I–Ni XXVII) for plasma screening parameters of 0.1 a.u. and 0.25 a.u. It can be seen that this quantity varies linearly with Z_{eff} with steeper slope for larger μ. Moreover, ithas also a linear trend with the plasma screening parameter as displayed for the nickel ions in Fig. 3. This is in agreement with the corresponding Debye–Hückel (DH) limit (Stewart & Pyatt 1966; Crowley 2014). This is also shown in Fig. 2 as solid and dashed lines as given below: (10)
where is in eV and μ in a.u. Besides, closedshell effects are clearly marked in Fig. 2 for μ = 0.25 a.u. with small departures from the DH limit at Z_{eff} = 18 and 26.
Reference configurations and active orbital sets used to build up the MCDF active space by single and double electron excitations to the corresponding active orbital sets along with the number of configuration state functions (CSFs), N_{CSF}, generated for the MCDF expansions in carbon, silicon, calcium, chromium, and nickel ions with N = 2–28 electrons.
Fig. 1
Ionization potential, E_{0} in eV, in logarithmic scale as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. The NIST error bars are too small to appear in the figure. 
Fig. 2
Ionization potential shift, ΔE_{0} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. Circles: MCDF/RATIP method for μ = 0.1 a.u. Squares: MCDF/RATIP method for μ = 0.25 a.u. Solid line: Debye–Hückel limit for μ = 0.1 a.u. Dashed line: Debye–Hückel limit for μ = 0.25 a.u. 
Fig. 3
MCDF/RATIP ionization potential shift, ΔE_{0} in eV, as a function of the plasma screening parameter μ in a.u., in some ions belonging to the nickel isonuclear sequence. 
3.2 Kthreshold energies and shifts
The Kthreshold energies, E_{K} in eV, computed in this work using the MCDF/RATIP method are reported in Tables B.1–B.5 for plasma screening parameters μ = 0 a.u. (isolated atomic system), μ = 0.1 a.u., and μ = 0.25 a.u. for respectively the carbon, silicon, calcium, chromium, and nickel isonuclear sequences. These values are plotted as function of the effective charge in Fig. 4 for our example of the nickel ions Ni I–Ni XXVII. As for the IPs, all the threecurves follow the same trend with systematic lowerings with respect to the isolated atom systems (μ = 0 a.u.). A gradient change is also seen at Z_{eff} = 19 corresponding to the closure of the Lshell and opening of the Mshell. In addition, as shown in Figs. 5–6 for the nickel isonuclear sequence, the Kthreshold energy shift, ΔE_{K} = E_{K}(μ) − E_{K}(μ = 0) in eV, with respect to the isolated atom condition (μ = 0 a.u.) displays a linear dependence on both the effective charge and the plasma screening parameter, in agreement with our results in oxygen (Paper I) and iron (Papers II–IV) ions.
Fig. 4
MCDF/RATIP Kthreshold energy, E_{K} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. 
Fig. 5
MCDF/RATIP Kthreshold energy shift, ΔE_{K} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. 
Fig. 6
MCDF/RATIP Kthreshold shift, ΔE_{K} in eV, as a function of the plasma screening parameter, μ in a.u., in some ions belonging to the nickel isonuclear sequence. 
4 Universal fitting formulae for IP and Kthreshold shifts
The aim of the present work is to refine the universal formulae first provided in our Paper IV. In order to do so, we have extended the number of different isonuclear sequences used in the fitting of the coefficients multiplying the μ Z_{eff} factor by including the MCDF/RATIP IP and Kthreshold shifts reported in the previous section for C I–C V, Si I–Si XIII, Ca I–Ca XIX,Cr I–Cr XXIII and Ni I–Ni XXVII. The resulting fitting formulae for the IP and Kthreshold lowerings, ΔE_{0} and ΔE_{K}, respectively,in eV, are given below: (11)
which are close to the Debye–Hückel limit (Stewart & Pyatt 1966; Crowley 2014), as shown in Eq. (10) and as was already the case in Paper IV. The coefficients that appear in Eqs. (11), (12) are more constrained by the fits reducing the standard deviation by a factor of ~2 with respect to those of Paper IV, where only the iron and oxygen isonuclear sequences were considered in the adjustments. Here, it has to be emphasized that the standard deviations of the fitting parameters are obtained by considering equal weights for the data points used in the fits, i.e., the calculated MCDF/RATIP values are supposed to be exact. The idea is to provide formulae that reproduce our MCDF/RATIP models almost exactly, at least at the resolution of present and future instruments.
In Figs. 7 and 8, we display the resulting fits to the MCDF/RATIP IP lowerings computed in this work and in Papers I–IV, respectively, as a function of the effective charge and of the plasma screening parameter. In Fig. 8, we restrict the plot to a sample of effective charges for the sake of clarity. Figures 9 and 10 are the equivalents for the Kthreshold energy shifts. In Figs. 11 and 12, comparisons between computed and fitted values are presented for ΔE_{0} and ΔE_{K}, respectively.The latter all show a sound reliability, although some minor scatter appears especially for μ = 0.25 a.u. In order to investigate further, the differences between the predicted values given by the formulae (11) and (12) and the MCDF/RATIP values are plotted versus the effective charge, the plasma screening parameter and the MCDF/RATIP value in the bottom panels of, respectively, Figs. 7 and 9 as well as Figs. 8 and 10 as well as Figs. 11 and 12, for the IP and Kthreshold lowerings, respectively.As we see, the differences range from up to 5 eV to less than 1 eV in the bottom panel of Fig. 7 and from up to 9 eV to less than 1 eV in the bottom panel of Fig. 9 with bigger differences for μ = 0.25 a.u. and with agreements within ~2 eV for μ = 0.1 a.u. The average of the absolute differences are 1.2 ± 1.2 eV and 1.1 ± 1.6 eV respectively for the IP and Kthreshold lowerings. Moreover, these average differences drop to 0.6 ± 0.5 eV and 0.3 ± 0.3 eV, respectively, when only the values for μ = 0.1 a.u are considered and increaseup to 1.8 ± 1.3 eV and 2.0 ± 1.9 eV, respectively, when only the values for μ = 0.25 eV are retained this time. All the figures are to be compared to the resolve energy scale of the microcalorimeter onboard the future XRISM mission, that is, 2 eV with a goal of 1 eV (Miller et al. 2020). In that respect, the IP and Kthreshold lowering differences for μ = 0.1 a.u shown in the bottom panels of Figs. 7–12 are all within ± ~ 2 eV, which is comparable to the XRISM resolve energy scale. In contrast, a large number of the values for μ = 0.25 a.u are greater than this scale, as notably seen in the bottom panels of Figs. 8 and 10.
In trying to reproduce the direct MCDF/RATIP calculations more accurately with the fitted formulae, we used a different fitted formula for each shell filling, namely: Kshell ions (C V, O VI, Si XIII, Ca XIX, Cr XXIII, Fe XXV, and Ni XXVII), Lshell ions (C I–C IV, O I–O V, Si V–Si XII, Ca XI–Ca XVIII, Cr XV–Cr XXII, Fe XVII–Fe XXIV, and Ni XIX–Ni XXVI), and Mshell ions (Si I–Si IV, Ca III–Ca X, Cr I–Cr XIV, Fe II–Fe XVI, and Ni I–Ni XVIII). The resulting fitted formulae are the following: (13)
for the Mshell ions. The corresponding differences with respect to the MCDF/RATIP values are all within 1 eV for formulae (13) and (14) for the Kshell ions. For the Lshell ions, they are also all within ~1 eV for Kthreshold lowerings given by formula (16) but they scatter more, namely, within ~3 eV for the IP lowerings (Eq. (15), essentially due to the μ = 0.25 a.u. values in C II–C IV, O IV–O VI, and Si XI–Si XII. Regarding the Mshell ions, formulae (17) and (18) predict values that differ from the MCDF/RATIP calculations by up to ~7.5 eV (in Cr I under conditions parametrized by μ = 0.25 a.u.) for the Kthreshold lowerings and up to ~3 eV (in Si IV under conditions parametrized by μ = 0.25 a.u.) for the IP lowerings. These do not therefore solve the problem of better accuracy compared to the resolve energy scale attained by future xray space telescopes. In conclusion, the universal formulae (11) and (12) are of equivalent accuracy then the shell specific formulae (13–18) but of more of practical use for the astrophysical plasma modeling codes such as XSTAR (Bautista & Kallman 2001; Kallman & Bautista 2001; Kallman et al. 2021; Mendoza et al. 2021).
Finally, we used the MCDF/RATIP values calculated in the oxygen isonuclear sequence for μ = 0.2 a.u. in Paper IVfor comparison with the predictions given by the fitting formulae, namely, Eqs. (11), (12), and we find a good agreement, as can be seen in Fig. 13. Here, the absolute differences range from ~0.1 eV (in O I) to ~1.6 eV (in O VI) for the IP lowerings and from ~0.05 eV (in O III) to ~0.9 eV (in O I) for the Kthreshold lowerings. At this point, we can conclude that our fitted formulae reproduce our MCDHF/RATIP models withan accuracy comparable to the XRISM resolve energy scale for plasma screening parameter values μ ≤0.2 a.u.
Fig. 7
Top: ionization potential lowering, ΔE_{0} in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Solid line: fitting formula (11) for μ = 0.1 a.u. Dashed line: fitting formula (11) for μ = 0.25 a.u. Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0}(formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Dashed line: straightline of equality. Dotdashed lines: straight lines of ± 1 eV differences. 
Fig. 8
Top: ionization potential lowering, ΔE_{0} in eV, as a function of the plasma screening parameter, μ in a.u., in several abundant ions. Colored lines: Fitting formula (11). Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the plasma screening parameter, μ, in several abundant ions. Dashed line: straightline of equality. Dotdashed lines: straight lines of ± 1 eV differences. 
Fig. 9
Top: Kthreshold lowering, ΔE_{K} in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Solid line: Fitting formula (12) for μ = 0.1 a.u. Dashed line: fitting formula (12) for μ = 0.25 a.u. Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K}(formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in severalabundant ions. Dashed line: straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. 
Fig. 10
Top: Kthreshold lowering, ΔE_{K} in eV, as a function of the plasma screening parameter, μ in a.u., in several abundant ions. Colored lines: fitting formula (12). Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the plasma screening parameter, μ, in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. 
Fig. 11
Top: comparison between the ionization potential shifts obtained by the fitting formula (11) and the MCDF/RATIP values computed in this work. Dashed line: straight line of equality. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of ΔE_{0}(MCDF∕RATIP) in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 
Fig. 12
Top: comparison between the Kthreshold energy shifts obtained by the fitting formula (12) and the MCDF/RATIP values computed in this work. Dashed line: straight line of equality. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of ΔE_{K}(MCDF∕RATIP) in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 
5 Summary and conclusions
We first proposed universal formulae applicable to all relevant elements in order to predict IP and Kthreshold lowerings that are due to the screening effect on atomic systems embedded in plasmas (our Paper IV). In an attempt to improve them, MCDF/RATIP calculations of IP and Kthreshold energies have been carried out in the carbon, silicon, calcium, chromium, and nickel isonuclear sequences in order to increase the number of data points that can be used in the fits. The theoretical models used are based on the multiconfiguration Dirac–Fock method that includes a Debye–Hückel model potential to take into account the effects of the plasma screening on atomic structures. In each sequence, the MCDF/RATIP IP and Kthreshold downshifts show similar linear trends on both the ion effective charge and the plasma screening parameter as the ones previously reported in oxygen and iron isonuclear sequences in, respectively, our Paper I and our Papers II–IV.
Concerning the improved universal formulae, namely those of Eqs. (11) and (12), the coefficients of proportionality closed to the Debye–Hückel limit are more constrained by the fits, reducing the standard deviations by a factor two with respect to those determined in Paper IV. Nonetheless, differences with the MCDF/RATIP values are still found to be sizable compared to the XRISM resolve energy scale (2 eV) in certain ions near the neutral end of isonuclear sequences or in atomic structures close to closedshell atoms and for extreme plasma conditions, that is, for μ = 0.25 a.u. Actually, discontinuities appear in the different trends and increase with μ, reflecting the shell structure of the different atomic systems. Considering these discontinuities by using different fitting curves for each shell filling does not solve this issue of accuracy.
In conclusion, as explicit calculations of the atomic structures for each ion of each element under different plasma conditions is impractical, we still recommend the use of universal formulae, namely, Eqs. (11–12) for predicting the IP and Kthreshold lowerings in plasma modeling codes such as XSTAR (Bautista & Kallman 2001; Kallman & Bautista 2001; Kallman et al. 2021; Mendoza et al. 2021), although their comparatively moderate to low accuracies in certain cases under extreme plasma conditions characterized by μ >0.2 a.u. (especially for the Kthresholds) may affect the predicted opacities.
Fig. 13
Top: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in the oxygen isonuclear sequence for μ = 0.2 a.u. Dashed line: straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, the oxygen isonuclear sequence for μ = 0.2 a.u. Dashed line: straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 
Acknowledgements
J.D. is Research Fellow of the Belgian American Educational Foundation Inc. (BAEF) while P.P. and P.Q. are, respectively, Research Associate and Research Director of the Belgian Fund for Scientific Research (F.R.S.FNRS). Financial supports from these organizations, as well as from the NASA Astrophysics Research and Analysis Program (grant 80NSSC17K0345) are gratefully acknowledged. J.A.G. acknowledges support from the Alexander von Humboldt Foundation.
Appendix A Computed ionization potentials for different plasma screening parameter values and comparison with the NIST recommended values
Computed ionization potentials for C I–C V as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Si I–Si XIII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Ca I–Ca XIX as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Cr I–Cr XXIII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Ni I–Ni XXVII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Appendix B Computed Kthreshold energies for different plasma screening parameter values
Computed Kthresholds for C I–C V as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Si I–Si XIII as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Ca I–Ca XIX as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Cr I–Cr XXIII as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Ni I–Ni XXVII as a function of the plasma screening parameter μ (a.u.).
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All Tables
Reference configurations and active orbital sets used to build up the MCDF active space by single and double electron excitations to the corresponding active orbital sets along with the number of configuration state functions (CSFs), N_{CSF}, generated for the MCDF expansions in carbon, silicon, calcium, chromium, and nickel ions with N = 2–28 electrons.
Computed ionization potentials for C I–C V as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Si I–Si XIII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Ca I–Ca XIX as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Cr I–Cr XXIII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed ionization potentials for Ni I–Ni XXVII as a function of the plasma screening parameter μ (a.u.). Spectroscopic values (NIST) are also listed for comparison.
Computed Kthresholds for C I–C V as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Si I–Si XIII as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Ca I–Ca XIX as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Cr I–Cr XXIII as a function of the plasma screening parameter μ (a.u.).
Computed Kthresholds for Ni I–Ni XXVII as a function of the plasma screening parameter μ (a.u.).
All Figures
Fig. 1
Ionization potential, E_{0} in eV, in logarithmic scale as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. The NIST error bars are too small to appear in the figure. 

In the text 
Fig. 2
Ionization potential shift, ΔE_{0} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. Circles: MCDF/RATIP method for μ = 0.1 a.u. Squares: MCDF/RATIP method for μ = 0.25 a.u. Solid line: Debye–Hückel limit for μ = 0.1 a.u. Dashed line: Debye–Hückel limit for μ = 0.25 a.u. 

In the text 
Fig. 3
MCDF/RATIP ionization potential shift, ΔE_{0} in eV, as a function of the plasma screening parameter μ in a.u., in some ions belonging to the nickel isonuclear sequence. 

In the text 
Fig. 4
MCDF/RATIP Kthreshold energy, E_{K} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. 

In the text 
Fig. 5
MCDF/RATIP Kthreshold energy shift, ΔE_{K} in eV, as a function of the effective charge, Z_{eff}, in ions belonging to the nickel isonuclear sequence. 

In the text 
Fig. 6
MCDF/RATIP Kthreshold shift, ΔE_{K} in eV, as a function of the plasma screening parameter, μ in a.u., in some ions belonging to the nickel isonuclear sequence. 

In the text 
Fig. 7
Top: ionization potential lowering, ΔE_{0} in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Solid line: fitting formula (11) for μ = 0.1 a.u. Dashed line: fitting formula (11) for μ = 0.25 a.u. Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0}(formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Dashed line: straightline of equality. Dotdashed lines: straight lines of ± 1 eV differences. 

In the text 
Fig. 8
Top: ionization potential lowering, ΔE_{0} in eV, as a function of the plasma screening parameter, μ in a.u., in several abundant ions. Colored lines: Fitting formula (11). Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the plasma screening parameter, μ, in several abundant ions. Dashed line: straightline of equality. Dotdashed lines: straight lines of ± 1 eV differences. 

In the text 
Fig. 9
Top: Kthreshold lowering, ΔE_{K} in eV, as a function of the effective charge, Z_{eff}, in several abundant ions. Solid line: Fitting formula (12) for μ = 0.1 a.u. Dashed line: fitting formula (12) for μ = 0.25 a.u. Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K}(formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in severalabundant ions. Dashed line: straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. 

In the text 
Fig. 10
Top: Kthreshold lowering, ΔE_{K} in eV, as a function of the plasma screening parameter, μ in a.u., in several abundant ions. Colored lines: fitting formula (12). Colored symbols: MCDF/RATIP method. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the plasma screening parameter, μ, in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. 

In the text 
Fig. 11
Top: comparison between the ionization potential shifts obtained by the fitting formula (11) and the MCDF/RATIP values computed in this work. Dashed line: straight line of equality. Bottom: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of ΔE_{0}(MCDF∕RATIP) in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 

In the text 
Fig. 12
Top: comparison between the Kthreshold energy shifts obtained by the fitting formula (12) and the MCDF/RATIP values computed in this work. Dashed line: straight line of equality. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of ΔE_{K}(MCDF∕RATIP) in severalabundant ions. Dashed line: Straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 

In the text 
Fig. 13
Top: differences between universal formula’s ionization potential lowerings and MCDF/RATIP values, (ΔE_{0} (formula) − ΔE_{0}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, in the oxygen isonuclear sequence for μ = 0.2 a.u. Dashed line: straight line of equality. Dotteddashed lines: Straight lines of ± 1 eV differences. Bottom: differences between universal formula’s Kthreshold energy lowerings and MCDF/RATIP values, (ΔE_{K} (formula) − ΔE_{K}(MCDF∕RATIP)) in eV, as a function of the effective charge, Z_{eff}, the oxygen isonuclear sequence for μ = 0.2 a.u. Dashed line: straight line of equality. Dotteddashed lines: straight lines of ± 1 eV differences. 

In the text 
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