Recent Progress in Radiative-Rate Determination of Some Heavy Ions (Xe9+, Xe10+, Lu3+, Hf4+, Ta5+) of Interest in Fusion

This paper presents a review about the radiative properties (transition probabilities and oscillator strengths) of two xenon ions (Xe, Xe) and three members of Er I isoelectronic sequence (Lu, Hf, Ta) of interest in controlled thermonuclear fusion, including our recent theoretical data obtained using two independent theoretical atomic structure computational approaches (semi-empirical Hartree-Fock with relativistic corrections method (HFR) and the ab initio multiconfiguration Dirac-Hartree-Fock (MCDHF)). The tables, from the second one, summarize the recommended data expected to be useful for plasma modelling in fusion.


Introduction
There is a growing need in atomic data for elements which could be used in thermonuclear fusion installations for the fuel introduction or as plasma facing materials. Noble gases can be injected into nuclear fusion reactors, conditioned in solid pellets, for both plasma diagnostics and fuel introduction [1] [2] [3]. In particular, if xenon (Z = 54) was inserted into the international thermonuclear experimental reactor (ITER) which will be the next step towards the realization of fusion, it could be pumped out without leaving residuals on plasma facing material and would therefore be recycled in subsequent discharges. Moreover, the xenon atoms would strip to helium-like ions in the hottest part of the confined plasma. Consequently, the identification of emission lines and the know-I-Xe XI, Xe XIX, Xe XXV-Xe XXIX, Xe XLIII-Xe XLV, and Xe LI-Xe LIV. In that compilation, data on Xe 9+ and Xe 10+ were respectively based on Refs. [23] [24] and [25]. It should be noted that in Refs. [24] [25] the authors reported some radiative parameters, including transition probabilities. From 2004, the main works on Xe 9+ and Xe 10+ are those of [26] [27] and [28] [29] [30], respectively. In Refs. [27] [30], as an extension of works by Biemont et al. [12] [13] [14] [15], we used two different theoretical approaches, i.e. the semi-empirical Hartree-Fock with relativistic corrections (HFR) and the fully relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) methods, to obtain two new sets of oscillator strengths and transition probabilities of radiative transitions in Xe 9+ and Xe 10+ , in the extreme ultraviolet region.
Lutetium (Z = 71), hafnium (Z = 72) and tantalum (Z = 73) would be candidates as plasma-facing materials in controlled nuclear fusion devices [31] [32] [33] [34]. In addition, the last two of them are also produced in neutron-induced transmutation of tungsten (Z = 74) and tungsten-alloys that will compose the divertors in future tokamaks [35]. As a result, their sputtering may generate ionic impurities of all possible charge states, including the members of Er I isoelectronic sequence (Lu IV, Hf V, Ta VI), in the deuterium-tritium plasma that could contribute to radiation losses in fusion reactors. Therefore, the radiative properties of these ions have potentially important applications in this field. Unfortunately, there are very few studies devoted to the transition rates of these ions. The only data available have been computed in the Er I isoelectronic sequence by Anisimova et al. [36] (Yb III, Lu IV, and Hf V), Loginov and Tuchkin [37] (Yb III, Lu IV, Hf V, and Ta VI) and Bokamba et al. [38] (Lu IV, Hf V, and Ta VI). In the two first Refs, the authors utilized the Newton and least-squares monoconfigurational methods without taking into account that an appropriate treatment of these ions must be done in the framework of the configuration in-teraction. Recently, we reported in Ref. [38] extensive calculations of transition probabilities and oscillator strengths in Lu 3+ , Hf 4+ and Ta 5+ using the same methods as in the case of Xe 9+ and Xe 10+ [27] [30] that consider both the electron correlations and configuration interaction. The three new sets of obtained transition probabilities and oscillator strengths fall in the spectral domain from ultraviolet to infrared.
In this review, we briefly describe the methods used for obtaining the most recent radiative properties (transition probabilities and oscillator strengths) in Xe 9+ , Xe 10+ , Lu 3+ , Hf 4+ and Ta 5+ , i.e. MCDHF and HFR (Section 2). Section 3 is devoted to the discussion of the available radiative transition rates in these ions, as well as the selection of data expected reliable. Finally, the concluding remarks are given in Section 4.

Theoretical Methods
Xe 9+ , Xe 10+ , Lu 3+ , Hf 4+ and Ta 5+ being heavy ions, it is therefore important to take into account both the configuration interaction (CI) and relativistic effects for modelling their atomic structure and computing radiative rates. In the most recent radiative-rate investigations of these ions [27] [30] [38], we utilized, in view of no radiative rate measurements available in the literature, two independent theoretical methods, i.e. the semi-empirical Hartree-Fock with relativistic corrections method (HFR) and the ab initio multiconfiguration Dirac-Hartree-Fock method (MCDHF), both of them including explicitly the most important intravalence and core-valence electron correlations. Table 1 reports the HFR and MCDHF physical models used in Refs. [27] [30] [38]. Table 1. Physical models used in our work [27] [30] [38].

Multiconfiguration Dirac-Hartree-Fock Method
In the multiconfiguration Dirac-Hartree-Fock (MCDHF) method implemented in the GRASP2K and GRASP2018 computer packages [39] [40], the Hamiltonian is given by where k c are the mixing coefficients, k γ represent all the other quantum numbers needed to uniquely specify CSF that are jj-coupled Slater determinants built from one-electron spin-orbitals,

( )
, , n m r κ φ θ ϕ , of the form:  [39]. The quantum number κ is given by: so that The radial functions In the MCDHF variational procedure, the radial functions and the expansion coefficients k c are optimized to self-consistency [39] [41], which can be done employing different options: • Average Level calculation (AL), spin-orbitals are chosen to minimize the average energy of configuration state functions with different total angular momentum J; • Optimal Level calculation (OL), only the energy of an individual level is minimized; • Extended Optimal Level calculation (EOL), the minimization is extended over several selected levels; • Extended Average Level calculation (EAL), averaging of the energy expression is extended to all configuration functions, usually using statistical weights (2J + 1) as weighting factors.
In the Relativistic Configuration Interaction (RCI) step, the eigenvalue problem is solved in a CSF basis built with a fixed preoptimized orbital set [42].
The relativistic two-body Breit interaction and the quantum electrodynamic corrections due to self-energy and vacuum polarization are also considered through the implementation of the routines developed by McKenzie et al. [19].
The final transition amplitudes are computed in both the Babushkin (B) and the Coulomb (C) gauges which are respectively the relativistic equivalents of the length and velocity gauges. The gauges agreement for a given transition, i.e. 0.9 B C 1.1 ≤ ≤ , provides an indication of the accuracy of its transition probability although this condition is necessary but not sufficient [43]. Cowan proposed an independent accuracy indicator, i.e. the cancellation factor (CF), defined for the E1 transitions as below [44]: have the same meanings as in Equation (2)

Ion Xe 9+
Wavelengths of the observed lines and energy levels in the Xe X spectrum were compiled by Saloman [22] who critically evaluated the previous data published by Kaufman et al. [23] and Churilov and Joshi [24].
Churilov and Joshi [24], in their spectral analysis of Xe X, were helped by the computed transition probabilities obtained by HFR method using Cowan codes [44], and those data were the first in the literature. Fahy et al. [26] observed Xe 9+ lines in the 140 -150 Å range employing an electron beam ion trap and a flat field spectrometer, and they reported seven strongest lines along with their HFR gA-values. More recently, we used two independent theoretical approaches HFR and MCDHF to obtain a set of radiative properties (oscillator strengths and transition probabilities) for 92 Xe X allowed spectral lines belonging to the 4d 9 − (4d 8 5p + 4d 8 4f + 4p 5 4d 10 ) transition arrays, for which log gf > −4, falling in the extreme ultraviolet (EUV) range 100 -164 Å [27]. Half of those E1 transitions meet the adopted reliability criteria (7).
When comparing the expected reliable data from our two computational methods satisfying the accuracy criteria (7) [27], we have found the average ratio 〈gA MCDHF) /gA HFR) 〉 ~ 1.05 ± 0.60, showing thus a good overall agreement between the two approaches.
The weighted transition probabilities by Churilov and Joshi [24] are compared with our MCDHF and HFR values satisfying the adopted reliability criteria (7), the average rates are respectively 〈gA(MCDHF)/gA( [24])〉 = 1.14 ± 0.77 and 〈gA(HFR)/gA( [24])〉 =1.08 ± 0.13. The MCDHF calculations include more correlation than HFR technique by Churilov and Joshi [24], which could explain the difference observed between the two sets of results. As for the about 8% overall discrepancy between our HFR values with the data by Churilov and Joshi [24] obtained with a similar HFR approach [24], the authors' restricted physical model is certainly the possible explanation. In addition, the main purpose of Journal of Applied Mathematics and Physics these researchers was the term analysis of the Xe 9+ ion. In this work, we have adopted the MCDHF transition probabilities reported by Bokamba et al. [27].

Ion Xe 10+
The main works on the spectrum analysis of Xe XI are contained in Refs [25] [26] [27] [28] where the authors used, on the one hand a low-inductance vacuum spark and a 10.7 m grazing-incidence spectrograph, and on the other hand the Hartree-Fock calculations and orthogonal parameters. They classified about 200 allowed lines belonging to 4d 8 − (4p 5 4d 9 + 4d 7 5p + 4d 7 4f) transition arrays in the 105 -157 Å spectral range, established all the 9 levels of the 4d 8 configuration and 123 levels of the 4p 5 4d 9 + 4d 7 5p + 4d 7 4f configurations. These researchers reported the HFR transition probabilities of the classified lines.
Employing the RCI method and the distorted wave approximation implemented in the Flexible Atomic Code (FAC) [46], Shen et al. [29] computed the energy levels, transition probabilities and electron impact collision strengths in Xe XI. The transition rates were given for allowed lines involving the first 400 fine-structure levels of their model. These authors, in comparing their calculated rates with respect to those by Churilov et al. [28] for 31 strong lines, estimated the accuracy of their data better than 20%.
We recently utilized two independent theoretical methods HFR and MCDHF/ RCI to produce a set of radiative properties (transition probabilities and oscillator strengths) for 576 Xe XI allowed spectral lines pertaining to the 4d 8 − (4p 5 4d 9 + 4d 7 5p + 4d 7 4f) transition arrays in the EUV range 102-157 Å [29]. 87 out of those E1 transitions (about 15%) satisfy the reliability criteria (7).  gf-values (gf, weighted oscillator strength, ~gA) for the strongest lines (log gf > 0), and we can see that the MCDHF values are systematically smaller than the HFR ones. The average ratio 〈gf(MCDHF)/gf(HFR)〉 being egal to 0.78 ± 0.19, this systematics is thus about 20%. The observed trend is mainly explained by missing core-core and core-valence correlations related to missing configurations with more than one hole in the 4p core subshell in our HFR model.
In Figure 2 and Figure 3, the gA-values by Churilov et al. [28], who used a HFR approach but with smaller configuration sets, are compared respectively with our HFR and MCDHF/RCI data, in only considering the lines meeting the reliability criteria (7). Figure 2 indicates that the extension of the CI expansions in our HFR model has a marginal effect on the transition rates, and we have actually found the average ratio 〈gA( [28])/gA(HFR)〉 equal to 1.00 ± 0.02. Therefore, as expected we also observe from Figure  In comparing the FAC transition probabilities by Shen et al. [29] with respect to our HFR and MCDHF data [30] [29] appear to be overall about 10% smaller than our HFR results and 40% greater than our MCDHF data. The authors did not mention any information on the accuracy indicators (CF and B/C), the transition rates of the involved lines not satisfying the reliability criteria (7) could explain the high standard deviation  In the present work, the adopted transition probabilities are the MCDHF ones S. E. Yoca Journal of Applied Mathematics and Physics from Bokamba et al. [30]. Table 3 reports the adopted transition probabilities (column 3), and column 4 contains other available data [28] [29].

Ions of Er I Isoelectronic Sequence: Lu 3+ , Hf 4+ and Ta 5+
Investigations on the spectra of these three ions were performed at the National sliding-spark discharges and the grating spectrograph, and on the other hand semi-empirical parametric models of the corresponding atomic energy level structures. These authors did not publish any transition rates! Radiative rates in these ions are very scarce, the only few data available in the literature, to our knowledge, are those by Anisimova et al. [36], Loginov and Tuchkin [37] and Bokamba et al. [38]. The first data are E1 transition probabilities on the transition arrays 4f 13 ns-4f 13 6p (n = 6.7) in Lu IV and Hf V computed by Anisimova et al. [36], ANI, using the Newton and least-squares monoconfigurational methods and those determined by Loginov and Tuchkin [37], LOG, employing the same methods for the transition arrays 4f 14 -4f 13 5d and 4f 13 6p-4f 13 5d in Lu IV, Hf V and Ta IV that neglect the configuration interaction. More recently, we obtained sets of radiative properties (transition probabilities and oscillator strengths) in Lu IV, Hf V and Ta IV for allowed transitions using the two independent theoretical atomic structure computational approaches HFR and Journal of Applied Mathematics and Physics MCDHF/RCI [38].
Recently, we used the theoretical approaches HFR and MCDHF/RCI to obtain a set of transition probabilities and oscillator strengths for 593 allowed spectral lines in the range 400 Å -45 μm [38]. 179 out of those E1 transitions (about 30%) meet reliability criteria (7). Figure 4 displays the comparison between our HFR and MCDHF/RCI oscillator strengths, for the strongest lines (log gf > 0), with an average ratio 〈gf(MCDHF)/gf(HFR)〉 = 0.95 ± 0.22 [38], i.e. MCDHF log (gf)-values are overall about 5% smaller than those obtained by HFR, and this systematics may be attributed to missing configurations with two holes in the 5p subshell in our the HFR-model expansions.
When comparing our MCDHF/RCI gA-values [38], for the strongest lines (gA > 10 9 s −1 ), with respect to those published by Anisimova et al. [36] and Loginov and Tuchkin [37] utilizing monoconfigurational approaches (Newton and least-squares methods stand for 1 and 2, respectively), we have found these av-  [38], which shows the necessity to take into account the configuration interaction in calculations. Figure 5 illustrates these comparisons. Figure 4. Comparison between our HFR and MCDHF/RCI oscillator strengths (log gf) for Lu IV spectral lines [38]. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been selected. Figure 5. Comparison between our MCDHF/RCI gA-values [38] with the available data [36] [37] for Lu IV spectral lines. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been retained.
In the present work, the recommended transition probabilities in Lu 3+ are the MCDHF data from Bokamba et al. [38], and they are reported in Table 4

Ion Hf 4+
As regards Hf V, Sugar and Kaufmann [48] firstly classified 173 lines in the re- More recently, we utilized the two theoretical approaches HFR and MCDHF/ RCI to produce a set of radiative parameters (transition probabilities and oscillator strengths) for 820 E1 transitions appearing in the region 250 Å -40 μm [38], 219 of which (about 27%) fulfill reliability criteria (7). Figure 6 illustrates the comparison between our HFR and MCDHF log gf-values expected reliable, for the strongest lines (log gf > 0), with an average ratio 〈gf(MCDHF)/gf(HFR)〉 equal to 0.84 ± 0.20 [38], where we observe an about 15% systematics that could be explained by missing core-valence correlations in our HFR model.  [36]), 0.94 ± 0.11 (Newton method set in [37]) and 0.91 ± 0.19 (least-squares method set in [37]) in respect of our MCDHF/RCI model. Here again, we observe the importance of taking into account the configuration interaction in calculations. Figure 7 illustrates this effect.  In the present work, the adopted transition probabilities in Hf 4+ are the MCDHF data from Bokamba et al. [38] which are reported in Table 5 (column 3) along with other available data (column 4) [36] [37].
We compare in Figure 8 our HFR and MCDHF log gf-values expected reliable, for the strongest lines (gf > 0), and we can see that the MCDHF values are almost systematically weaker than the HFR ones. The average ratio 〈gf(MCDHF)/  [38]. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been retained. Figure 9. Comparison between our MCDHF/RCI and HFR gA-values [38] with those published by Loginov and Tuchkin [37] for Ta VI spectral lines. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been retained. Journal of Applied Mathematics and Physics gf(HFR)〉 being equal to 0.75 ± 0.18, the observed systematics is thus about 25% and this trend is probably caused by missing interactions with configurations having two holes in the 5p subshell in our HFR model.
The comparison between our MCDHF/RCI and HFR gA-values with the data by [37], for the strongest lines (gA > 10 9 s −1 ), gives these average ratios 〈gA/gA (HFR or MCDHF)〉: 0.94 ± 0.55 (Newton method set in [37]) and 1.01 ± 1.11 (least-squares method set in [37]) with respect to HFR model; 0.97 ± 0.49 (Newton method set in [37]) and 0.92 ± 0.45 (least-squares method set in [37]) in respect of our MCDHF/RCI model. We observe a similar trend of the effects of configuration interaction on the gA-values as in the cases of Lu IV and Hf V, which is shown in Figure 9.
In the present work, the adopted transition probabilities in Ta 5+ are the MCDHF data from Bokamba et al. [38] which are reported in Table 6 (column 3) along with other available data (column 4) [37]. ions are all theoretical, so this work is a call for additional efforts to produce experimental data in order to refine theory. Producing these ions in the laboratory for their investigations is a challenging task.
It is well known that under conditions that prevail in many astrophysical and low-density laboratory tokamak plasmas, the collisional de-excitation of metastable states is rather slow, leading to the buildup of a population of metastable levels [52]. In this context, forbidden lines resulting from electric quadrupole (E2) and magnetic dipole (M1) transitions increase in intensity and can be used to deduce information about plasma temperature and dynamics. Therefore, we intend to extend our calculations to E2 and M1 transitions in Lu 3+ , Hf 4+ and Ta 5+ .