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 ⁹⁺, Xe ¹⁰⁺) and three members of Er I isoelectronic sequence (Lu ³⁺, Hf ⁴⁺, Ta ⁵⁺) 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.

• Atomic Spectra
• Atomic Data
• Transition Probabilities
• Oscillator Strengths
• Heavy Elements

1. 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 knowledge of spectroscopic parameters from all ionization stages of xenon, including Xe⁹⁺ and Xe¹⁰⁺, would be of key importance in order to model the plasma and facilitate the analysis of the spectra used for the estimation of physical conditions inside the fusion reactors such as densities and temperatures.

Up to now, experimental and theoretical investigations on spectroscopic properties of xenon ions have been performed. Recently, a review by Almandos and Raineri [4] reported the extensive use of Pulsed discharges in La Plata (Argentina) to produce spectra of Xe III-IX falling in ultraviolet (UV), visible and infrared regions [5] - [11], so as to identify the corresponding lines. In those studies, time-resolved experiments and relativistic Hartree-Fock calculations were also carried out to obtain radiative lifetimes and transition probabilities. Biémont’s team [12] [13] [14] [15] realized large-scale calculations of lifetimes, oscillator strengths and transition probabilities in moderately charged xenon ions (Xe V-Xe IX) by combining often theory (HFR/HFR+CPOL [16], MCDF [17] [18] [19] [20] ) and experiment (Beam Foil Spectroscopy [21] ). Saloman [22] compiled the energy levels and observed spectral lines of the xenon atom in all stages of ionization for which experimental data are available before 2004, i.e. Xe I-Xe XI, Xe XIX, Xe XXV-Xe XXIX, Xe XLIII-Xe XLV, and Xe LI-Xe LIV. In that compilation, data on Xe⁹⁺ and Xe¹⁰⁺ 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 Xe⁹⁺ and Xe¹⁰⁺ 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 Xe⁹⁺ and Xe¹⁰⁺, 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 interaction. Recently, we reported in Ref. [38] extensive calculations of transition probabilities and oscillator strengths in Lu³⁺, Hf⁴⁺ and Ta⁵⁺ using the same methods as in the case of Xe⁹⁺ and Xe¹⁰⁺ [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 Xe⁹⁺, Xe¹⁰⁺, Lu³⁺, Hf⁴⁺ and Ta⁵⁺, 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.

2. Theoretical Methods

Xe⁹⁺, Xe¹⁰⁺, Lu³⁺, Hf⁴⁺ and Ta⁵⁺ 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 ]

Ion	Physical modelsa
	HFR	MCDHFc
Xe9+	Even parity: 4d9, 4d85s, 4d86s, 4d87s, 4d85d, 4d86d, 4d75s2,	4s, 4p,
	4d75p2, 4d75d2, 4d75s6s, 4d75s5d, 4d75s6d, 4p54d95p	4d, 4f,
	Odd parity: 4d85p, 4d86p, 4d87p, 4d84f, 4d85f, 4d86f,	5 s, 5p,
	4d75s5p, 4d75s6p, 4d75p5d, 4d75p6s, 4p54d10	5d, 5f
Xe10+	Even parity: 4d8, 4d7ns (n = 5 - 7), 4d7nd (n = 5 - 6), 4d65s2,	4p, 4d,
	4d65p2, 4d65d2, 4d65s6s, 4d65s5d, 4p54d85p	4f, 5 s,
	Odd parity: 4d75p, 4d7np (n = 6 - 7), 4d74f, 4d7nf (n = 5 - 6),	5p, 5d,
	4d65snp (n = 5 - 6), 4d65p5d, 4p54d9	5f
Lu3+, Hf4+,	Even parity: 4f14, 4f136p, 4f137p, 4f135f, 4f13nf (n = 6 - 7),	4f, 5p, 5d,
Ta5+	4f125d2, 4f126s2, 4f126p2, 4f125d6s, 5p56pb	5f, 6s, 6p,
	Odd parity: 4f135d, 4f136d, 4f137d, 4f136s,	6d, 6f, 7s,
	4f137s, 4f125d6p, 4f126s6p, 5p54f145db, 5p54f146sb	7p, 7d, 7f

a: Underlined configurations are spectroscopic ones, used as reference configurations in MCDHF. b: spectroscopic configurations in Ta⁵⁺ in addition to the ones in Lu³⁺ and Hf⁴⁺. c: Active set of orbitals to which there are single and double electron excitations in MCDHF.

2.1. 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

H = ∑ i N ( c α i ⋅ p i + ( β i − 1 ) c 2 + Z r i ) + ∑ i < j N 1 r i j , (1)

where c is the speed of light, α and β are the Dirac matrices.

The atomic state function (ASF), Ψ , is represented by a superposition of configuration state functions (CSF), Φ , with the same parity, π , total angular momentum and total magnetic quantum numbers, J and M J , forming a basis set of the representation, { Φ k } , as

Ψ ( Π J M J ) = ∑ k   c k Φ ( γ k Π J M J ) , (2)

where c k are the mixing coefficients, γ k represent all the other quantum numbers needed to uniquely specify CSF that are jj-coupled Slater determinants built from one-electron spin-orbitals, ϕ n κ m ( r , θ , φ ) , of the form:

ϕ n κ m ( r , θ , φ ) = 1 r ( P n κ ( r ) χ κ m ( θ , φ ) i Q n κ ( r ) χ κ m ( θ , φ ) ) (3)

P n κ ( r ) and Q n κ ( r ) are, respectively, the large and the small radial components of the wave functions, and the angular functions χ κ m ( θ , φ ) are the spinor spherical harmonics [39]. The quantum number κ is given by:

κ = ± ( j + 1 2 ) = a ( j + 1 2 ) (4)

so that

l = j − 1 2 a (5)

The radial functions P n κ ( r ) and Q n κ ( r ) are numerically represented on a logarithmic grid and are required to be orthonormal within each κ symmetry. In the MCDHF variational procedure, the radial functions and the expansion coefficients c k 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]:

CF = ( | ∑ k ∑ i c k 〈 Φ ′ ( γ k Π ′ J ′ M J ′ ) | P ( 1 ) | Φ ( γ i Π J M J ) 〉 c i | ∑ k ∑ i | 〈 Φ ′ ( γ k Π ′ J ′ M J ′ ) | P ( 1 ) | Φ ( γ i Π J M J ) 〉 c i | ) 2 (6)

where P ( 1 ) is the electric dipole operator and c i ( k ) ( ' ) and Φ i ( k ) ( ' ) have the same meanings as in Equation (2) for the initial (non-primed symbols) and final (primed symbols) states of the transition. Computed line strength with a small value of the CF, e.g. less than 0.05, is strongly affected by destructive interference effects resulting from intermediate-coupling and interaction-configuration mixing of basis states. The GRASP2K and GRASP2018 packages have been modified in order to implement the calculation of this latter accuracy indicator [38] [43]. Computed line strength may thus be expected reliable if it simultaneously fulfills these conditions [43]:

0.90 ≤ B / C ≤ 1.10     and     CF ≥ 0.05 (7)

2.2. Relativistic Hartree-Fock Method

In the Hartree-Fock method with relativistic corrections (HFR) of Cowan [44], a set of orbitals is obtained for each electronic configuration by solving the Hartree-Fock equations for the spherically averaged atom. The equations resulting from the application of the variational principle to the configuration average energy. Relativistic corrections are included in this set of equations, i.e. the Blume-Watson spin-orbit, mass-velocity and one-body Darwin terms. The Blume-Watson spin-orbit term comprises the part of the Breit interaction that can be reduced to a one-body operator.

The multiconfiguration Hamiltonian matrix is constructed and diagonalized in the L S J M J π representation within the framework of the Slater-Condon theory [45]. Each matrix element is a sum of products of Racah angular coefficients and radial integrals, i.e.

H a b = 〈 α a L a S a J M J π | H | α b L b S b J M J π 〉 = ∑ i     c i a , b I i a , b , (8)

where c i a , b and I i a , b stand for the angular coefficients and the radial parameters, respectively. The radial parameters correspond to the configuration average energies (E_av), the mono-configuration (F^k, G^k) and configuration interaction (R^k) Slater integrals, the spin-orbit parameters ( ζ n l ) and, if necessary, the effective interaction parameters ( α , β , γ ) [44]. These parameters can be adjusted to fit the eigenvalues of the Hamiltonian to the available observed energy levels in a least-squares approach. Note that this approach is linked more strongly to the quantity and the quality of the experimental energy levels. The eigenvalues and the eigenstates resulting from this way (abinitio or semi-empirically) are used to compute the wavelength, the transition probability and the oscillator strength for each possible transition. Concerning an allowed line (E1), the cancellation factor (CF) as described in Equation (6) constitutes a reliable indicator for its computed line strength.

3. Radiative Transitions

3.1. Xenon ions, Xe 9+ and Xe 10+

3.1.1. 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 Xe⁹⁺ 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 4d⁹ − (4d⁸5p + 4d⁸4f + 4p⁵4d¹⁰) 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 ágA_MCDHF)/gA_HFR)ñ ~ 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 ágA(MCDHF)/gA( [24] )ñ = 1.14 ± 0.77 and ágA(HFR)/gA( [24] )ñ =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 these researchers was the term analysis of the Xe⁹⁺ ion. In this work, we have adopted the MCDHF transition probabilities reported by Bokamba et al. [27].

We report in Table 2 the adopted transition probabilities (column 3), and column 4 contains other available data.

Table 2

Adopted transition probabilities (gA) in Xe X, as well as other available gA-values

λ (nm)a	Transition	Adopted gA (s−1)b	Other gA (s−1)
11.0133	16,7253/2 - 924,721 1 / 2 o	3.10E+12	3.84E+12c, 3.80E+12d
11.2714	05/2 - 887,203 5 / 2 o	3.68E+10	1.03E+11c, 9.70E+10d, 1.20E+11e
11.3438	05/2 - 881,539 3 / 2 o	5.96E+11	2.28E+12c, 2.26E+12d, 4.41E+12e
114.313	05/2 - 874,794 3 / 2 o	5.48E+12	5.02E+12c, 4.94E+12d, 2.95E+12e
11.4879	16,7253/2 - 887,203 5 / 2 o	9.24E+12	1.08E+13c, 1.07E+13d, 1.07E+13e
11.4880	05/2 - 870,470 7 / 2 o	1.24E+13	1.47E+13c, 1.44E+13d, 1.45E+13e
11.5632	16,7253/2 - 881,539 3 / 2 o	5.32E+12	4.76E+12c, 4.68E+12d, 2.88E+12e
11.5661	05/2 - 864,592 5 / 2 o	8.76E+12	9.98E+12c, 9.80E+12d, 9.82E+12e
13.5729	16,7253/2 - 753,489 1 / 2 o	5.20E+10	6.78E+10c, 6.20E+10d
14.1545	16,7253/2 - 723,216 1 / 2 o	1.76E+10	1.27E+10c, 1.10E+10d
14.2046	05/2 - 703,997 7 / 2 o	4.52E+10	4.11E+10c, 4.20E+10d
14.3477	05/2 - 696,975 3 / 2 o	2.15E+10	1.98E+10c, 1.40E+10d
14.4654	05/2 - 691,306 3 / 2 o	5.80E+10	3.04E+10c, 2.50E+10d
14.4769	05/2 - 690,757 5 / 2 o	2.47E+10	1.38E+10c, 1.40E+10d
14.5150	16,7253/2 - 705,669 1 / 2 o	2.00E+11	2.07E+11c, 2.01E+11d
14.5323	05/2 - 688,121 3 / 2 o	1.02E+11	9.35E+10c, 9.40E+10d
14.5983	16,7253/2 - 701,735 5 / 2 o	2.65E+11	1.89E+11c, 1.81E+11d
14.6148	05/2 - 684,240 7 / 2 o	2.82E+10	1.13E+10c, 1.00E+10d
14.6413	05/2 - 682,998 3 / 2 o	3.89E+10	4.01E+10c, 2.40E+10d
14.6448	05/2 - 682,838 5 / 2 o	5.03E+09	1.46E+10c, 1.30E+10d
14.7381	16,7253/2 - 695,239 1 / 2 o	1.98E+10	1.82E+10c, 1.90E+10d
14.7619	05/2 - 677,421 3 / 2 o	2.68E+11	2.37E+11c, 2.26E+11d
14.7638	16,7253/2 - 694,056 5 / 2 o	5.58E+10	6.56E+10c, 6.60E+10d
14.7956	05/2 - 675,878 7 / 2 o	4.19E+11	3.45E+11c, 3.21E+11d
14.8005	05/2 - 675,652 5 / 2 o	2.33E+11	1.53E+11c
14.8240	16,7253/2 - 691,306 3 / 2 o	7.92E+10	2.08E+10c, 1.60E+10d
14.8333	05/2 - 674,159 3 / 2 o	7.60E+09	4.95E+10c, 5.20E+10d
14.8361	16,7253/2 - 690,757 5 / 2 o	1.16E+11	1.31E+11c, 1.29E+11d
14.8707	16,7253/2 - 689,190 3 / 2 o	1.99E+06	4.65E+10c, 4.80E+10d
14.8943	16,7253/2 - 688,121 3 / 2 o	1.41E+11	1.47E+11c, 1.33E+11d
14.9021	05/2 - 671,045 5 / 2 o	1.82E+11	1.47E+11c, 1.33E+11d
14.9358	05/2 - 669,531 7 / 2 o	1.13E+11	1.93E+11c, 1.95E+11d

14.9583	05/2 - 668,525 5 / 2 o	4.40E+10	6.87E+10c;, 6.60E+10d
15.0089	16,7253/2 - 682,998 3 / 2 o	2.71E+11	2.71E+11c, 2.42E+11d
15.0125	16,7253/2 - 682,838 5 / 2 o	4.83E+10	5.94E+10c, 5.60E+10d
15.0544	05/2 - 664,256 5 / 2 o	2.03E+11	2.44E+11c, 2.38E+11d
15.1021	05/2 - 662,160 3 / 2 o	6.88E+09	1.29E+10c, 1.30E+10d
15.1291	16,7253/2 - 677,704 1 / 2 o	5.54E+09	8.27E+09c, 8.00E+09d
15.1747	05/2 - 658,993 7 / 2 o	8.56E+10	8.94E+10c, 8.60E+10d
15.1762	16,7253/2 - 675,652 5 / 2 o	3.86E+10	4.46E+10c, 4.00E+10d
15.2058	05/2 - 657,645 5 / 2 o	9.84E+10	1.12E+11c, 1.10E+11d
15.2830	16,7253/2 - 671,045 5 / 2 o	2.00E+10	1.37E+10c, 1.30E+10d
15.4433	16,7253/2 - 664,256 5 / 2 o	1.52E+10	1.31E+10c, 1.20E+10d
15.4588	05/2 - 646,880 7 / 2 o	1.82E+09	3.07E+09c, 3.00E+09d
15.4680	05/2 - 646,494 5 / 2 o	1.65E+10	2.05E+10c, 1.90E+10d
15.4934	16,7253/2 - 662,160 3 / 2 o	1.22E+10	1.41E+10c, 1.30E+10d

a: Ritz wavelengths calculated employing the experimental energy level values from [22]. Transitions are given by values (in cm⁻¹) of involved energy levels where subscripts denote their J-values. b: MCDHF values from [27]. pE + q = p.10^q. c: HFR values from [27]. d: Values taken from [24]. e: Values taken from [26].

3.1.2. 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 4d⁸ − (4p⁵4d⁹ + 4d⁷5p + 4d⁷4f) transition arrays in the 105 - 157 Å spectral range, established all the 9 levels of the 4d⁸ configuration and 123 levels of the 4p⁵4d⁹ + 4d⁷5p + 4d⁷4f 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 4d⁸ − (4p⁵4d⁹ + 4d⁷5p + 4d⁷4f) transition arrays in the EUV range 102-157 Å [29]. 87 out of those E1 transitions (about 15%) satisfy the reliability criteria (7).

Figure 1 displays the comparison between our HFR and MCDHF/RCI log

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 ágf(MCDHF)/gf(HFR)ñ 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 ágA( [28] )/gA(HFR)ñ equal to 1.00 ± 0.02. Therefore, as expected we also observe from Figure 3 an about 20% systematic decrease on our MCDHF rates, for the strongest lines (gA > 10¹² s⁻¹), due to missing 4p subshell core-excited configurations in the Churilov et al.’s HFR model, with the average ratio ágA(MCDHF)/gA( [28] )ñ = 0.85 ± 0.20.

In comparing the FAC transition probabilities by Shen et al. [29] with respect to our HFR and MCDHF data [30] for the 31 trong lines presented in their table B, we have found respectively the average rate ratios ágA( [29] )/gA(HFR)ñ = 0.89 ± 0.26 and ágA( [29] )/gA(MCDHF)ñ = 1.41 ± 0.94. We conclude that the values of [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

of ágA( [29] )/gA(MCDHF)ñ. The large discrepancy observed with our MCDHF/RCI model results mainly from missing 4p subshell single, double and triple core-holes configurations in their physical model. The possible explanation of the small difference with our HFR model is the limitation of the correlations to the shell with the principal quantum number n = 5.

In the present work, the adopted transition probabilities are the MCDHF ones from Bokamba et al. [30]. Table 3 reports the adopted transition probabilities (column 3), and column 4 contains other available data [28] [29].

Table 3

Adopted transition probabilities (gA) in Xe XI, as well as other available gA-values

λ (nm)a	Transition	Adopted gA (s−1)b	Other gA (s−1)
10.3912	26,6702 - 989,020 1 o	1.20E+11	1.39E+11c
10.5695	42,9002 - 989,020 1 o	8.33E+11	1.15E+12c, 1.14E+12d
10.6125	15,2053 - 957,488 3 o	7.20E+11	1.00E+12c, 1.01E+12d
10.6770	15,2053 - 951,795 2 o	8.35E+11	9.72E+11c
10.7585	15,2053 - 944,705 2 o	5.09E+11	7.47E+11c
10.8094	26,6702 - 951,795 2 o	7.36E+11	9.54E+11c
10.8579	26,6702 - 947,660 1 o	2.38E+11	3.11E+11c
10.8928	26,6702 - 944,705 2 o	1.39E+12	1.57E+12c
10.9029	34,6101 - 951,795 2 o	2.61E+12	3.05E+12c, 3.03E+12d
10.9093	40,8354 - 957,488 3 o	8.96E+12	1.07E+13c, 1.06E+13d, 1.19E+13e
10.9236	32,2100 - 947,660 1 o	4.71E+11	7.21E+11c
10.9339	42,9002 - 957,488 3 o	1.17E+12	1.36E+12c; 1.34E+12d
10.9577	04 - 912,600 3 o	2.24E+11	2.24E+11c
10.9655	13,1402 - 925,010 1 o	2.06E+11	2.86E+11c
10.9879	34,6101 - 944,705 2 o	2.03E+12	2.56E+12c, 2.56E+12d
11.0024	42,9002 - 951795 2 o	3.92E+12	4.79E+12c; 4.72E+12d
11.0085	04 - 908,390 3 o	2.12E+11	4.49E+11c
11.0528	26,6702 - 931,420 1 o	8.36E+11	1.53E+12c
11.0889	42,9002 - 944,705 2 o	3.72E+12	4.38E+12c, 4.40E+12d
11.1001	88,1300 - 989,020 1 o	4.16E+12	4.50E+12c, 4.48E+12d
11.1178	13,1402 - 912,600 3 o	1.02E+12	1.97E+12c, 2.01E+12d
11.1209	32,2100 - 931,420 1 o	7.37E+11	2.05E+11c
11.1268	34,6101 - 933,343 0 o	1.58E+12	1.91E+12c, 1.90E+12d
11.1294	13,1402 - 911,665 2 o	1.27E+12	3.17E+12c, 3.49E+12d
11.1316	26,6702 - 925,010 1 o	4.26E+11	4.75E+11c
11.1380	26,6702 - 924,500 2 o	5.39E+12	6.60E+12c
11.1384	40,8354 - 938,628 5 o	1.82E+13	2.16E+13c, 2.14E+13d, 2.43E+13e
11.1434	15,2053 - 912,600 3 o	8.97E+12	9.86E+12c
11.1435	04 - 897,383 5 o	1.79E+13	2.11E+13c, 2.08E+13d, 2.36E+13e
11.1506	34,6101 - 931,420 1 o	2.86E+12	3.64E+12c
11.1550	15,2053 - 911,665 2 o	4.90E+12	4.21E+12c; 3.92E+12d
11.1622	15,2053 - 911,082 4 o	1.45E+13	1.69E+13c, 1.67E+13d, 1.88E+13e
11.1654	266,4702 - 922,295 3 o	1.03E+13	1.18E+13c, 1.17E+13d, 1.30E+13e

11.1701	13,1402 - 908,390 3 o	9.78E+12	1.05E+13c, 1.03E+13d, 7.76E+12e
11.1739	04 - 894,941 3 o	9.84E+12	1.13E+13c, 1.12E+13d, 1.20E+13e
11.1785	13,1402 - 907,711 2 o	6.18E+12	5.79E+12c, 5.44E+12d
11.1832	40,8354 - 935,035 3 o	7.94E+11	9.07E+11c, 8.95E+11d
11.1959	15,2053 - 908,390 3 o	7.88E+11	1.56E+12c, 1.59E+12d
11.2007	32,2100 - 925,010 1 o	3.37E+12	4.16E+12c, 4.05E+12d
11.2044	15,2053 - 907,711 2 o	1.33E+12	2.80E+12c, 3.03E+12d
11.2055	04 - 892,420 4 o	1.37E+13	1.59E+13c, 1.57E+13d, 1,75E+13e
11.2091	42,9002 - 935,035 3 o	9.94E+12	1.15E+13c, 1.14E+13d, 1.19E+13e
11.2373	34,6101 - 924,500 2 o	2.17E+12	2.09E+12c, 2.07E+12d
11.2431	13,1402 - 902,577 1 o	4.11E+12	4.61E+12c, 4.60E+12d
11.2713	26,6702 - 913,877 1 o	2.89E+12	2.81E+12c, 2.84E+12d
11.3448	40,8354 - 922,295 3 o	4.34E+11	7.72E+11c
11.5266	40,8354 - 908,390 3 o	2.57E+11	3.05E+11c
12.7610	40,8354 - 824,474 3 o	1.59E+11	1.33E+11c, 1.36E+11d
13.0443	15,2053 - 781,822 2 o	6,59E+10	7.72E+10c, 7.80E+10d
13.1360	04 - 761,266 3 o	7.03E+10	2.43E+10c
13.1978	34,6101 - 792,311 0 o	2.29E+10	2.43E+10c, 4.90E+10d
13.2573	40,8354 - 795,135 3 o	4.97E+11	4.82E+11c, 4.92E+11d, 3.07E+11e
13.2664	34,6101 - 788,396 1 o	8.49E+10	6.68E+10c
13.2708	34,6101 - 788,145 2 o	4.85E+10	2.20E+10c
13.2785	42,9002 - 795,995 2 o	9.21E+10	5.96E+10c, 6.10E+10d
13.2984	34,6101 - 786,580 2 o	7.11E+10	1.72E+11c, 1.69E+11d, 1,09E+11e
13.3361	15,2053 - 765,052 2 o	8.19E+10	9.37E+10c, 8.70E+10d
13.3518	13,1402 - 762,105 2 o	1.12E+11	8.91E+10c, 8.60E+10d
13.3655	40,8354 - 789,029 5 o	3.15E+11	2.82E+11c, 2.73E+11d, 2.59E+11e
13.3887	15,2053 - 762,105 2 o	4.38E+10	4.53E+10c, 4.60E+10d
13.3968	04 - 746,445 3 o	7.56E+10	3.63E+10c, 4.20E+10d
13.4037	15,2053 - 761,266 3 o	4.67E+10	3.01E+11c, 2.98E+11d, 2.26E+11e
13.4184	42,9002 - 788,145 2 o	6.44E+10	1.06E+11c
13.4584	13,1402 - 756,170 2 o	1.59E+11	1.86E+11c, 1.95E+11d, 1.35E+11e
13.4622	32,2100 - 775,030 1 o	3.54E+10	2.47E+10c, 2.50E+10d
13.4747	88,1300 - 830,260 1 o	1.76E+11	2.06E+11c, 2.02E+11d, 1.71E+11e
13.4843	13,1402 - 754,745 1 o	1.12E+11	1.46E+11c, 1.44E+11d, 1.13E+11e
13.4928	42,9002 - 784,035 1 o	2.03E+11	1.67E+11c, 1.64E+11d, 1.90E+11e
13.4987	15,2053 - 756,016 4 o	2.86E+11	5.61E+11c, 5.81E+11d, 3.80E+11e
13.5072	04 - 740,348 5 o	3.37E+11	6.31E+11c, 6.29E+11d, 5.96E+11e

13.5198	15,2053 - 754,860 3 o	2.25E+11	3.88E+10c, 3.40E+10d
13.5315	13,1402 - 752,155 1 o	7.28E+10	9.7E+10c, 9.60E+10d
13.5614	04 - 737,388 4 o	4.76E+11	3.20E+11c, 3.15E+11d, 4.17E+11e
13.5998	15,2053 - 750,512 2 o	1.19E+11	9.54E+10c
13.6025	32,2100 - 767,369 1 o	3.17E+10	7.72E+10c, 7.50E+10d
13.6213	15,2053 - 749,351 3 o	7.17E+10	2.79E+11c, 2.68E+11d, 2.24E+11e
13.6401	40,8354 - 773,968 5 o	5.60E+11	9.54E+10c, 8.70E+10d
13.6401	40,8354 - 773,968 4 o	1.38E+11	5.07E+11c, 3.71E+11d
13.6588	42,9002 - 775,030 1 o	7.04E+10	9.03E+10c, 8.60E+10d
13.6674	26,6702 - 758,337 1 o	5.63E+10	9.41E+10c, 9.90E+10d
13.6713	04 - 731,458 4 o	1.58E+11	4.30E+11c, 4.22E+11d
13.6734	15,2053 - 746,552 2 o	1.07E+11	2.72E+10c, 2.50E+10d
13.7238	13,1402 - 741,800 3 o	8.08E+10	1.54E+11c, 1.55E+11d, 6.09E+10e
13.8174	34,6101 - 758,337 1 o	4.92E+10	9.65E+10c, 9.30E+10d
13.8889	88,1300 - 808,130 1 o	5.2E+10	2.66E+10c, 2.50E+10d
13.8899	32,2100 - 752,155 1 o	4.32E+10	4.36E+10c, 4.30E+10d
13.9684	34,6101 - 750,512 2 o	9.46E+10	3.98E+10c, 3.70E+10d
14.3698	88,1300 - 784,035 1 o	8.24E+09	8.47E+09c
14.4881	88,1300 - 778,350 1 o	2.85E+09	2.27E+09c

a: Ritz wavelengths calculated employing the experimental energy level values from [28]. Transitions are given by values (in cm⁻¹) of involved energy levels where subscripts denote their J-values. b: MCDHF values from [30]. pE + q = p.10^q. c: HFR values from [30]. d: Values taken from [28]. e: Values taken from [29].

3.2. 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 Bureau of Standards (NBS) [47] [48] [49] [50] [51] by means of, on the one hand, 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 4f¹³ns-4f¹³6p (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 4f¹⁴-4f¹³5d and 4f¹³6p-4f¹³5d 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 MCDHF/RCI [38].

3.2.1. Ion Lu 3+

As for Lu IV, Sugar and Kaufman [47] classified 180 lines falling in the region 877 - 2128 Å, determined 57 energy levels of 4f¹⁴, 4f¹³5d, 4f¹³6s, 4f¹³6p, 4f¹³6d and 4f¹³7s configurations. Nine years later, Wyart et al. [51], in analyzing the 4f¹³5f configuration in the isoelectronic sequence of Yb III, classified 97 lines and established 13 energy levels of this configuration.

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 ágf(MCDHF)/gf(HFR)ñ = 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 > 10⁹ 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 average ratios ágA(ANI1)/gA(MCDHF)ñ = 1.21 ± 0.48, ágA(ANI2)/gA(MCDHF)ñ = 1.20 ± 0.45, ágA(LOG1)/gA(MCDHF)ñ = 0.96 ± 0.24, ágA(LOG2)/gA(MCDHF)ñ = 0.97 ± 0.24 [38], which shows the necessity to take into account the configuration interaction in calculations. Figure 5 illustrates these comparisons.

In making the similar comparisons employing here our HFR model, we have ágA(ANI1)/gA(HFR)ñ = 1.09 ± 0.26, ágA(ANI2)/gA(HFR)ñ = 1.09 ± 0.25, ágA(LOG1)/gA(HFR)ñ = 0.91 ± 0.14, ágA(LOG2)/gA(HFR)ñ = 0.89 ± 0.22 [38], which is also illustrated in Figure 5. In this case, we emphasize the importance of CI in calculations, as well.

In the present work, the recommended transition probabilities in Lu³⁺ are the MCDHF data from Bokamba et al. [38], and they are reported in Table 4 (column 3) along with other available data (column 4) [36] [37].

Table 4

Adopted transition probabilities (gA) in Lu IV, as well as other available gA-values

λ (nm)a	Transition	Adopted gA (s−1)b	Other gA (s−1)
60.8995	98,505.0 1 o - 262,710.12	3.26E+08	6.65E+08c
61.0993	98,505.0 1 o - 262,173.02	2.99E+09	1.88E+09c
62.5764	102,216.6 4 o - 262,021.35	8.20E+09	4.25E+09c
62.5959	90,432.9 2 o - 250,187.82	3.15E+08	1.36E+10c
64.2169	94,768.1 5 o - 250,490.36	6.99E+10	7.79E+10c
64.2506	106,380.6 4 o - 262,021.35	5.00E+10	5.08E+10c
64.5875	107,986.6 2 o - 262,815.43	1.03E+10	9.81E+09c
65.5141	98,239.3 6 o - 250,878.25	5.16E+08	7.67E+08c
65.6514	98,558.5 4 o - 250,878.25	2.95E+10	3.88E+10c
65.6810	98,239.3 6 o - 250,490.36	1.13E+09	2.07E+09c
65.6854	98,558.5 4 o - 250,799.34	1.47E+10	2.27E+10c
65.8536	110,858.0 1 o - 262,710.12	6.21E+09	3.73E+09c
65.9221	111,377.8 5 o - 263,072.05	1.08E+10	1.02E+10c
66.0635	99,660.0 2 o - 251,029.63	1.43E+10	1.62E+10c
66.0873	110,858.0 1 o - 262,173.02	8.41E+09	1.17E+10c
66.2441	111,377.8 5 o - 262,334.76	7.08E+10	7.24E+10c
66.2700	111,812.3 3 o - 262,710.12	2.94E+09	3.60E+09c
66.3819	111,377.8 5 o - 262,021.35	1.12E+09	1.05E+09c
66.5855	112,632.5 2 o - 262,815.43	1.33E+10	1.23E+10c
66.8715	112,632.5 2 o - 262,173.02	1.04E+10	1.25E+10c
67.1524	102,216.6 4 o - 251,131.75	2.86E+10	3.65E+10c
67.2669	102,216.6 4 o - 250,878.25	1.01E+10	6.96E+09c
67.2673	114,049.4 1 o - 262,710.12	2.88E+09	3.34E+09c
67.4596	103,070.4 3 o - 251,307.34	2.25E+10	2.84E+10c
67.5112	114,049.4 1 o - 262,173.02	1.37E+09	9.12E+08c
67.5323	114,632.8 3 o - 262,710.12	8.05E+07	4.99E+08c
67.7491	115,468.5 4 o - 263,072.05	4.82E+10	5.22E+10c

67.7781	114,632.8 3 o - 262,173.02	1.30E+09	1.85E+09c
67.9729	103,070.4 3 o - 250,187.82	2.87E+09	1.35E+09c
68.1702	103,798.7 5 o - 250,490.36	2.41E+09	9.44E+08c
68.2348	115,468.5 4 o - 262,021.35	1.02E+09	8.95E+08c
68.3649	116,798.2 4 o - 263,072.05	9.78E+08	2.36E+08c
68.8596	116,798.2 4 o - 262,021.35	3.04E+07	6.27E+07c
69.0004	106,380.6 4 o - 251,307.34	2.76E+09	1.76E+09c
69.0317	117,312.0 3 o - 262,173.02	5.39E+08	2.71E+07c
69.0317	117,312.0 3 o - 262,173.02	5.39E+08	2.71E+07c
69.0841	106,380.6 4 o - 251,131.75	2.97E+09	2.18E+09c
69.1329	106,380.6 4 o - 251,029.63	3.39E+08	2.14E+08c
69.2053	106,380.6 4 o - 250,878.25	4.75E+09	1.96E+09c
74.4923	128,573.3 2 o - 262,815.43	4.89E+07	8.37E+07c
74.6295	117,312.0 3 o - 251,307.34	1.68E+08	6.45E+07c
81.7138	128,929.0 3 o - 251,307.34	1.77E+07	1.98E+07c
114.3092	98,505.0 1 o - 185,987.02	1.92E+08	2.04E+08c, 2.25E+08d, 2.40E+08e
116.6854	98,505.0 1 o - 184,205.51	1.65E+08	1.28E+08c, 1.44E+08d, 1.59E+08e
124.7521	106,380.6 4 o - 186,539.63	4.62E+08	6.20E+08c, 5.32E+08d, 5.81E+08e
124.8100	94,768.1 5 o - 174,889.94	6.12E+08	7.57E+08c, 7.56E+08d, 6.84E+08e
126.4913	105,148.7 0 o - 184,205.51	7.67E+08	8.17E+08c, 8.01E+08d, 7.98E+08e
126.6156	106,380.6 4 o - 185,359.84	2.90E+08	4.54E+08c
127.3026	107,986.6 2 o - 186,539.63	7.14E+08	6.52E+08c, 5.81E+08d, 5.46E+08e
127.4770	94,768.1 5 o - 173,213.65	4.63E+08	6.48E+08c, 5.61E+08d, 5.94E+08e
127.6543	98,505.0 1 o - 176,841.62	4.01E+08	2.64E+08c, 2.75E+08d, 2.70E+08e
128.9386	97,333.6 3 o - 174,889.94	1.06E+09	1.03E+09c, 8.82E+08d, 8.73E+08e
130.3856	99,660.0 2 o - 176,355.63	3.43E+07	3.15E+07c, 3.85E+07d, 4.06E+07e
131.0077	98,558.5 4 o - 174,889.94	1.28E+09	1.10E+09c, 8.64E+08d, 1.05E+09e
132.3019	98,558.5 4 o - 174,143.23	2.80E+08	3.37E+08c, 3.15E+08d, 1.96E+08e
133.1044	110,858.0 1 o - 185,987.02	3.71E+08	3.81E+08c, 3.55E+08d, 2.95E+08e
133.3790	98,239.3 6 o - 173,213.65	8.74E+09	9.59E+09c, 8.84E+09d, 8.83E+09e
133.4946	98,505.0 1 o - 173,414.42	1.41E+09	1.70E+09c, 1.57E+09d, 1.55E+09e
133.8199	111,812.3 3 o - 186,539.63	4.84E+08	3.99E+08c, 4.97E+08d, 3.08E+08e
135.1680	111,377.8 5 o - 185,359.84	6.89E+09	7.76E+09c, 7.06E+09d, 7.09E+09e
135.3741	90,432.9 2 o - 164,302.33	2.71E+09	2.74E+09c, 2.60E+09d, 2.65E+09e
135.5517	164,728.54 - 238,501.1 4 o	1.60E+09	1.44E+09c
135.5851	99,660.0 2 o - 173,414.42	2.02E+09	2.29E+09c, 2.10E+09d, 2.13E+09e

135.7875	164,302.3 3 o - 237,946.82	7.68E+08	9.27E+08c
135.8725	176,355.63 - 249,954.0 2 o	4.52E+08	1.42E+08c
135.9666	111,812.3 3 o - 185,359.84	2.72E+08	3.54E+08c
136.3243	112,632.5 2 o - 185,987.02	2.10E+09	2.15E+09c, 1.87E+09d, 1.88E+09e
136.3373	110,858.0 1 o - 184,205.51	1.29E+09	1.47E+09c, 1.34E+09d, 1.32E+09e
136.3756	176,355.63 - 249,682.5 3 o	5.71E+09	5.58E+09c
137.1498	164,302.3 3 o - 237,215.33	1.03E+10	1.15E+10c
137.2855	176,841.62 - 249,682.5 3 o	5.45E+09	3.58E+09c
137.5357	164,728.54 - 237,436.9 4 o	5.03E+09	9.08E+09c
137.6021	102,216.6 4 o - 174,889.94	2.15E+09	2.46E+09c, 2.23E+09d, 2.02E+09e
137.6114	176,841.62 - 249,510.0 1 o	5.37E+09	5.86E+09c
137.7215	176,355.63 - 248,965.9 3 o	1.78E+09	8.36E+08c, 1.81E+09d, 1.86E+09e
138.1632	164,728.54 - 237,215.3 3 o	1.30E+09	3.97E+09c
137.9562	164,728.5 4 o - 237,106.73	5.55E+09	4.48E+09c, 4.35E+09d, 4.37E+09e
138.3185	164,728.5 4 o - 237,025.4 4 o	4.65E+09	3.86E+09c, 2.37E+09d, 2.36E+09e
138.3798	176,355.63 - 248,620.5 2 o	4.18E+09	1.27E+09c
138.9850	176,355.63 - 248,305.8 4 o	1.69E+10	2.05E+10c
139.0074	164,728.54 - 236,667.1 5 o	2.11E+10	2.59E+10c
139.0094	114,049.4 1 o - 185,987.02	1.50E+08	2.63E+08c, 2.05E+08d, 1.45E+07e
139.0306	102,216.6 4 o - 174,143.23	2.67E+09	3.22E+09c, 2.87E+09d, 2.98E+09e
139.0689	114,632.8 3 o - 186,539.63	1.30E+09	1.92E+09c, 1.68E+09d, 1.84E+09e
139.7175	112,632.5 2 o - 184,205.51	8.12E+08	9.75E+08c, 9.24E+08d, 9.03E+08e
140.1318	164,302.3 3 o - 235,663.72	9.37E+09	1.13E+10c
140.1459	114,632.8 3 o - 185,987.02	1.96E+09	2.51E+09c, 2.24E+09d, 2.07E+09e
140.6644	103,798.7 5 o - 174,889.94	4.20E+09	4.86E+09c, 4.15E+09d, 4.23E+09e
140.7042	115,468.5 4 o - 186,539.63	3.41E+09	4.06E+09c, 3.56E+09d, 3.54E+09e
142.1585	103,070.4 3 o - 173,414.42	1.06E+09	1.26E+09c, 1.01E+09d, 9.95E+08e
142.9082	106,380.6 4 o - 176,355.63	3.95E+09	4.49E+09c, 4.15E+09d, 4.12E+09e
142.9380	94,768.1 5 o - 164,728.54	5.38E+09	5.65E+09c, 5.16E+09d, 5.18E+09e
143.0793	115,468.5 4 o - 185,359.84	1.44E+09	1.73E+09c, 1.50E+09d, 1.52E+09e
144.0613	103,798.7 5 o - 173,213.65	1.88E+09	2.18E+09c, 1.93E+09d, 1.93E+09e
145.2327	107,986.6 2 o - 176,841.62	1.76E+09	1.86E+09c, 1.68E+09d, 1.67E+09e
145.9656	106,380.6 4 o - 174,889.94	1.56E+08	9.90E+07c, 6.03E+07d, 9.90E+07e
151.1259	98,558.5 4 o - 164,728.54	1.81E+09	2.06E+09c, 1.81E+09d, 1.75E+09e
152.0947	184,205.51 - 249,954.0 2 o	3.98E+09	3.26E+09c
152.0947	184,205.51 - 249,954.0 2 o	3.98E+09	3.26E+09c

152.2207	173,213.65 - 238,907.7 5 o	7.09E+09	7.04E+09c
152.8358	185,359.84 - 250,789.5 4 o	5.14E+09	4.26E+09c
153.0782	173,414.44 - 238,740.5 3 o	3.56E+09	3.30E+09c
153.1288	184,205.51 - 249,510.0 1 o	2.87E+08	1.74E+09c
153.1687	173,213.65 - 238,501.1 4 o	5.39E+08	4.19E+08c
154.9609	173,414.42 - 237,946.8 2 o	6.08E+09	6.30E+09c
155.1588	173,213.65 - 237,663.7 6 o	2.44E+10	2.56E+10c
155.2433	184,205.51 - 248,620.5 2 o	1.48E+09	2.12E+08c
155.3779	185,359.84 - 249,719.0 5 o	2.05E+10	2.16E+10c
155.4661	185,359.84 - 249,682.5 3 o	1.15E+09	1.30E+08c
155.6423	186,539.63 - 250,789.5 4 o	1.10E+10	9.29E+09c
155.7067	173,213.65 - 237,436.9 4 o	3.88E+09	2.04E+09c
156.3306	185,987.02 - 249,954.0 4 o	3.95E+09	4.95E+09c
156.6156	174,889.94 - 238,740.5 3 o	1.14E+09	1.44E+09c
156.7108	173,213.65 - 237,025.4 4 o	4.75E+09	5.54E+09c, 5.64E+09d, 5.64E+09e
156.7310	174,143.23 - 237,946.8 2 o	2.38E+09	2.60E+09c
157.0049	173,414.42 - 237,106.7 3 o	3.58E+09	2.78E+09c, 2.49E+09d, 2.49E+09e
157.2050	174,889.94 - 238,501.1 4 o	5.09E+09	6.01E+09c
157.2176	185,359.84 - 248,965.9 3 o	5.64E+09	5.75E+09c, 4.56E+09d, 4,56E+09e
157.5957	173,213.65 - 236,667.1 5 o	1.28E+09	1.50E+09c
157.6929	186,539.63 - 249,954.0 2 o	9.80E+08	1.02E+09c
158.5349	111,812.3 3 o - 174,889.92	7.91E+07	3.39E+07c, 3.60E+06d, 3.60E+06e
158.8222	174,143.23 - 174,889.9 2 o	3.95E+09	3.10E+09c, 2.77E+09d, 2.85E+09e
158.8663	185,359.84 - 248,305.8 4 o	1.03E+09	1.20E+09c
158.9481	185,987.02 - 248,900.6 2 o	2.80E+08	2.22E+09c, 2.09E+09d, 2.11E+09e
159.8558	110,858.0 1 o - 173,414.42	7.11E+07	6.52E+07c, 6.00E+07d, 4.70E+07e
160.1889	186,539.63 - 248,965.9 3 o	3.62E+09	9.75E+08c, 2.13E+09d, 2.12E+09e
160.3566	186,539.63 - 248,900.6 2 o	2.38E+09	1.01E+09c, 1.23E+09d, 1.23E+09e
160.4482	174,889.94 - 237,215.3 3 o	3.14E+09	1.90E+09c
160.7283	174,889.94 - 237,106.7 3 o	1.09E+09	2.03E+09c, 1.51E+09d, 1,51E+09e
160.9386	174,889.94 - 237,025.4 4 o	6.46E+09	4.21E+09c, 2.75E+09d, 2,76E+09e
161.0801	186,539.63 - 248,620.5 2 o	3.32E+08	1.58E+09c
161.7186	111,377.8 5 o - 173,213.65	7.85E+07	4.65E+07c, 4.29E+07d, 3.08E+07e
161.9009	186,539.63 - 248,305.8 4 o	1.11E+09	1.08E+09c
162.3609	176,355.63 - 237,946.8 2 o	6.61E+07	9.30E+07c
162.5474	174,143.23 - 235,663.7 2 o	1.15E+09	1.27E+09c

165.9556	114,632.8 3 o - 174,889.94	7.44E+08	8.71E+07c, 1.80E+06d, 2.70E+06e
167.9053	116,798.2 4 o - 176,355.63	3.62E+07	4.02E+07c, 1.01E+08d, 5.81E+07e
169.3664	117,312.0 3 o - 176,355.63	1.01E+08	2.16E+08c, 3.97E+08d, 2.50E+08e
170.1213	114,632.8 3 o - 173,414.42	1.13E+09	1.09E+08c, 5.00E+05d, 1.00E+06e
172.1416	116,798.2 4 o - 174,889.94	5.04E+09	5.64E+09c, 7.62E+09d, 7.63E+09e
172.5140	128,573.3 2 o - 186,539.63	2.21E+09	2.62E+09c, 3.33E+09d, 3.35E+09e
173.1749	115,468.5 4 o - 173,213.65	1.59E+08	4.96E+07c, 9.90E+05d, 8.80E+05e
173.5792	128,929.0 3 o - 186,539.63	4.00E+09	4.31E+09c, 5.66E+09d, 5.64E+09e
174.1745	128,573.3 2 o - 185,987.02	3.75E+09	4.10E+09c, 5.39E+09d, 5.43E+09e
174.3831	116,798.2 4 o - 174,143.23	7.42E+08	1.01E+09c, 1.20E+09d, 1.20E+09e
175.2603	128,929.0 3 o - 185,987.02	5.11E+08	7.04E+08c, 9.00E+08d, 8.50E+08e
175.9597	117,312.0 3 o - 174,143.23	4.23E+09	5.55E+09c, 6.97E+09d, 7.13E+09e
177.2082	128,929.0 3 o - 185,359.84	7.51E+09	8.51E+09c, 1.08E+10d, 1.08E+10e
178.2455	117,312.0 3 o - 173,414.42	2.89E+09	4.49E+09c, 6.80E+09d, 5.83E+09e
179.7520	128,573.3 2 o - 184,205.51	2.49E+09	2.79E+09c, 3.45E+09d, 3.45E+09e
199.6179	114,632.8 3 o - 164,728.54	8.64E+08	8.11E+07c, 8.10E+05d, 1.08E+06e
207.1092	128,573.3 2 o - 176,841.62	2.58E+08	3.85E+08c, 5.55E+08d, 5.25E+08e
208.5699	116,798.2 4 o - 164,728.54	1.42E+09	1.78E+09c, 2.33E+09d, 2.32E+09e
208.6470	128,929.0 3 o - 176,841.62	2.31E+09	2.45E+09c, 3.09E+09d, 3.12E+09e
209.2160	128,573.3 2 o - 176,355.63	2.38E+09	2.50E+09c, 3.05E+09d, 3.12E+09e
210.4414	116,798.2 4 o - 164,302.33	3.20E+09	3.43E+09c, 4.27E+09d, 4.29E+09e
210.7853	128,929.0 3 o - 176,355.63	1.22E+09	1.42E+09c, 1.74E+09d, 1.79E+09e
210.8302	117,312.0 3 o - 164,728.54	2.22E+09	3.24E+09c
212.7427	117,312.0 3 o - 164,302.33	2.95E+08	5.18E+08c, 7.07E+08d, 6.85E+08e
219.3745	128,573.3 2 o - 174,143.23	5.44E+07	8.02E+07c, 1.57E+08d, 9.87E+07e
682.8623	236,667.1 5 o - 251,307.34	1.21E+07	6.83E+06c
689.0088	248,305.8 4 o - 262,815.43	7.84E+06	4.59E+06c
704.2842	248,620.5 2 o - 262,815.43	2.77E+06	6.94E+07c
707.4089	236,667.1 5 o - 250,799.34	2.75E+06	7.12E+06c
723.2222	236,667.1 5 o - 250,490.36	5.59E+08	4.38E+08c
730.0030	237,436.9 4 o - 251,131.75	5.12E+07	2.74E+07c
730.1203	237,106.7 3 o - 250,799.34	3.85E+06	5.49E+05c
735.4864	237,436.9 4 o - 251,029.63	2.08E+07	9.65E+06c
735.9575	237,215.3 3 o - 250,799.34	9.92E+07	1.14E+08c
742.2963	237,663.7 6 o - 251,131.75	9.03E+06	7.13E+06c
764.1520	237,946.8 2 o - 251,029.63	9.77E+07	8.71E+07c

777.3064	249,954.0 2 o - 262,815.43	9.05E+07	5.64E+07c
779.4154	237,663.7 6 o - 250,490.36	7.12E+06	1.01E+07c
791.5103	238,501.1 4 o - 251,131.75	1.89E+08	1.86E+08c
792.4451	249,719.0 5 o - 262,334.76	4.63E+08	3.57E+08c
800.3883	249,682.5 3 o - 262,173.02	2.68E+06	1.63E+06c
812.6327	249,719.0 5 o - 262,021.35	6.85E+06	4.98E+06c
812.9036	238,501.1 4 o - 250,799.34	6.13E+07	1.58E+07c
813.9427	250,789.5 4 o - 263,072.05	2.89E+08	1.76E+08c
818.1726	249,954.0 2 o - 262,173.02	5.96E+07	5.35E+07c
890.0849	250,789.5 4 o - 262,021.35	6.64E+06	2.55E+06c

a: Ritz wavelengths calculated employing the experimental energy level values from [47, 51]. Transitions are given by values (in cm⁻¹) of involved energy levels where subscripts denote their J-values. b: MCDHF values from [38]. pE + q = p.10^q. c: HFR values from [38]. d: Values from Newton method taken in [36] [37]. e: Values from least-squares method taken in [36] [37].

3.2.2. Ion Hf 4+

As regards Hf V, Sugar and Kaufmann [48] firstly classified 173 lines in the region 545 - 1793 Å and determined 59 energy levels of 4f¹⁴, 4f¹³5d, 4f¹³6s, 4f¹³6p, 4f¹³6d and 4f¹³7s configurations, and secondly [50] classified 5 resonnance transitions 5p⁶-5p⁵ (5d, 6s) in the spectral range 257 - 373 Å. Wyart et al. [51] classified 102 lines falling in the domain 459 - 510 Å, established 22 energy levels of the 4f¹³5f configuration.

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 ágf(MCDHF)/gf(HFR)ñ 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.

When comparing our MCDHF/RCI and HFR gA-values with respect to the data available in the literature [36] [37], for the strongest lines (gA > 10⁹ s⁻¹), we obtain these average ratios ágA/gA(HFR or MCDHF)ñ: 1.03 ± 0.19 (Newton method set in [36] ), 1.03 ± 0.19 (least-squares method set in [36] ), 0.89 ± 0.25 (Newton method set in [37] ) and 0.87 ± 0.27 (least-squares method set in [37] ) with respect to HFR model; 1.34 ± 0.17 (Newton method set in [36] ), 1.35 ± 0.17 (least-squares method set in [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 Hf⁴⁺ 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].

3.2.3. Ion Ta 5+

Concerning Ta VI, Kaufman and Sugar [49] classified 169 lines in the region 218 - 1587 Å and deduced 71 energy levels of 4f¹⁴, 4f¹³5d, 4f¹³6s, 4f¹³6p, 4f¹³6d and 4f¹³7s, 5p⁵5d, 5p⁵5d, 5p⁵6s and 5p⁵6p configurations. Wyart et al. [51] classified 96 lines appearing in the range 335 - 409 Å, and determined 26 levels of the 4f¹³5f configuration.

We used the two independent theoretical methods HFR and MCDHF/RCI to determine a set of radiative properties (transition probabilities and oscillator strengths) for 1101 Ta VI allowed spectral lines falling in the range 200 Å - 90 μm [38]. 196 of those E1 transitions (about 22%) satisfy reliability criteria (7).

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 ágf(MCDHF)/

Table 5

Adopted transition probabilities (gA) in Hf V, as well as other available gA-values

λ (nm)a	Transition	Adopted gA (s−1)b	Other gA (s−1)
45.8280	166,667.05 1 o - 384,874.42	8.16E+08	1.38E+09c
45.9847	166,667.05 1 o - 384,130.72	5.20E+09	4.97E+09c
46.0721	166,667.05 1 o - 383,718.11	1.70E+09	2.35E+09c
46.3085	166,667.05 1 o - 382,610.01	6.68E+09	5.88E+09c
47.1355	171,240.86 4 o - 383,395.35	1.96E+10	1.83E+10c
47.6577	175,073.22 4 o - 384,903.04	1.67E+10	2.21E+10c
47.7817	174,433.1 0 o - 383,718.11	8.22E+09	8.92E+09c
48.0026	161,705.88 5 o - 370,028.16	1.45E+11	1.89E+11c
48.0026	175,073.22 4 o - 383,395.35	1.01E+11	1.40E+11c
48.0361	174,433.1 0 o - 382,610.01	8.61E+09	9.88E+09c
48.0385	176,850.41 2 o - 385,017. 3	1.71E+10	1.33E+10c
48.1185	176,850.41 2 o - 384,670.53	1.86E+10	3.41E+10c
48.2439	176,850.41 2 o - 384,130.72	3.52E+09	2.66E+09c
48.4071	164,621.73 3 o - 371,203.24	3.37E+10	4.92E+10c
48.6004	176,850.41 2 o - 382,610.01	3.76E+09	3.34E+09c
48.7445	166,051.67 4 o - 371,203.24	3.89E+10	5.76E+10c
48.8127	166,780.4 6 o - 371,645.26	3.14E+10	3.38E+10c
48.8903	166,780.4 6 o - 371,319.85	1.85E+09	2.08E+09c
49.0247	181,271.17 3 o - 385,250.14	3.89E+09	3.97E+09c
49.0605	181,044.27 1 o - 384,874.44	1.26E+10	1.55E+10c
49.0808	181,271.17 3 o - 385,017.3	2.03E+10	1.40E+10c
49.1082	181,271.17 3 o - 384,903.04	6.46E+10	8.63E+10c
49.1151	181,271.17 3 o - 384,874.42	6.10E+09	8.60E+09c
49.1185	166,780.4 6 o - 370,369.87	1.79E+11	2.14E+11c
49.1269	166,667.05 1 o - 370,221.41	1.24E+10	1.74E+10c
49.1397	181,867.07 5 o - 38,5368.75	2.25E+10	2.47E+10c
49.1644	181,271.17 3 o - 384,670.53	1.36E+10	2.90E+10c
49.2011	166,780.4 6 o - 370,028.16	2.90E+09	4.58E+09c
49.2401	181,044.27 1 o - 384,130.72	1.84E+10	2.03E+10c
49.2524	181,867.07 5 o - 384,903.04	7.67E+08	9.79E+08c
49.2952	181,271.17 3 o - 384,130.72	3.93E+08	4.24E+08c
49.3404	181,044.27 1 o - 383,718.11	1.12E+10	1.54E+10c
49.4248	181,867.07 5 o - 384,194.66	1.50E+11	1.81E+11c
49.5648	183,261.02 2 o - 385,017.3	3.49E+10	5.38E+10c
49.6116	181,044.27 1 o - 382,610.01	4.92E+09	4.78E+09c

49.6208	181,867.07 5 o - 383,395.35	2.66E+09	3.80E+09c
49.6210	183,346.90 1 o - 384,874.42	5.46E+09	5.28E+09c
49.6501	183,261.02 2 o - 384,670.53	1.86E+10	8.76E+09c
49.7835	183,261.02 2 o - 384,130.72	2.31E+10	2.83E+10c
49.8048	183,346.90 1 o - 384,130.72	3.73E+09	6.27E+09c
49.8659	171,240.86 4 o - 371,778.94	2.01E+10	3.46E+10c
49.9074	183,346.90 1 o - 383,718.11	6.39E+09	8.03E+09c
50.0974	185,638.77 3 o - 385,250.14	6.61E+10	8.05E+10c
50.1374	172,327.17 3 o - 371,778.94	5.48E+10	7.00E+10c
50.1559	185,638.77 3 o - 385,017.3	5.73E+09	1.54E+10c
50.1633	183,261.02 2 o - 382,610.01	3.07E+09	3.75E+09c
50.1846	185,638.77 3 o - 384,903.04	1.16E+10	1.23E+10c
50.1849	183,346.90 1 o - 382,610.01	8.22E+09	1.08E+10c
50.1903	171,240.86 4 o - 370,482.73	6.85E+09	7.42E+09c
50.1918	185,638.77 3 o - 384,874.42	5.51E+08	8.87E+08c
50.2189	172,327.17 3 o - 371,455.63	2.48E+10	3.45E+10c
50.2433	185,638.77 3 o - 384,670.53	2.32E+10	1.90E+10c
50.3072	186,590.15 4 o - 385,368.75	1.03E+11	1.24E+11c
50.3373	186,590.15 4 o - 385,250.14	3.06E+10	3.51E+10c
50.3799	185,638.77 3 o - 384,130.72	3.85E+09	4.75E+09c
50.3949	173,212.31 5 o - 371,645.26	1.16E+11	1.42E+11c
50.4254	186,590.15 4 o - 384,903.04	3.73E+09	4.84E+09c
50.4654	172,327.17 3 o - 370,482.74	1.51E+10	1.34E+10c
50.4846	186,590.15 4 o - 384,670.53	4.60E+09	5.99E+09c
50.8117	186,590.15 4 o - 383,395.35	2.17E+09	1.84E+09c
50.9563	175,073.22 4 o - 371,319.85	5.00E+09	4.30E+09c
57.8587	212,533.85 4 o - 385,368.75	1.16E+07	5.06E+06c
58.0150	212,533.85 4 o - 384,903.04	5.44E+05	6.20E+05c
58.0782	213,068.38 3 o - 385,250.14	3.79E+07	2.05E+07c
58.2743	213,068.38 3 o - 384,670.53	1.92E+07	1.27E+07c
58.4582	213,068.38 3 o - 384,130.72	5.23E+06	4.78E+06c
62.9027	226,041.41 2 o - 385,017.3	2.16E+07	9.90E+06c
62.9779	212,533.85 4 o - 371,319.85	7.26E+06	2.13E+06c
63.1149	226,461.70 3 o - 384,903.04	8.67E+05	1.23E+06c
63.3116	212,533.85 4 o - 370,482.73	6.11E+06	2.88E+06c
76.8520	166,667.05 1 o - 296,787.222	4.37E+08	4.98E+08c, 5.20E+08d, 5.55E+08e

78.1461	166,667.05 1 o - 296,787.222	3.10E+08	2.75E+08c, 3.00E+08d, 3.24E+08e
78.6846	156,785.87 2 o - 283,875.573	1.71E+08	7.07E+08c, 6.93E+08d, 5.95E+08e
80.1299	171,240.86 4 o - 296,038.214	3.30E+08	3.20E+08c, 1.89E+08d, 2.79E+08e
81.7047	175,073.22 4 o - 297,465.233	1.09E+09	1.44E+09c, 1.21E+09d, 1.27E+09e
82.6686	175,073.22 4 o - 296,038.214	7.41E+08	1.12E+09c, 1.15E+09d, 1.04E+09e
82.9086	176,850.41 2 o - 297,465.233	1.51E+09	1.55E+09c, 1.33E+09d, 1.27E+09e
83.1951	174,433.1 0 o - 294,632.561	1.83E+09	2.02E+09c, 1.91E+09d, 1.91E+09e
84.9025	176,850.41 2 o - 294,632.561	1.83E+09	4.35E+08c
85.0796	166,667.05 1 o - 284,204.002	6.98E+08	2.17E+08c, 2.30E+08d, 2.60E+08e
85.6316	166,051.67 4 o - 282,830.933	9.71E+08	1.01E+09c, 8.89E+08d, 7.35E+08e
86.0629	181,271.17 3 o - 297,465.233	1.10E+09	1.22E+09c, 1.11E+09d, 8.12E+08e
86.3091	168,341.35 2 o - 284,204.002	8.59E+07	3.82E+08c, 4.55E+08d, 4.00E+08e
86.3984	181,044.27 1 o - 296,787.222	8.40E+08	7.24E+08c, 7.05E+08d, 5.35E+08e
86.5544	168,341.35 2 o - 283,875.573	3.99E+08	9.94E+08c, 9.52E+08d, 9.45E+08e
86.5681	181,271.17 3 o - 296,787.222	8.51E+08	1.01E+09c, 8.35E+08d, 1.15E+09e
86.7248	166,780.4 6 o - 282,087.735	2.20E+10	2.44E+10c, 2.19E+10d, 2.19E+10e
87.1330	181,271.17 3 o - 296,038.214	6.75E+08	8.82E+08c, 6.84E+08d, 6.12E+08e
87.3442	168,341.35 2 o - 282,830.933	1.81E+09	1.45E+09c, 1.21E+09d, 1.33E+09e
87.5625	183,261.02 2 o - 297,465.233	6.24E+08	8.80E+08c, 7.70E+08d, 8.26E+08e
87.5878	181,867.07 5 o - 296,038.214	1.77E+10	2.00E+10c, 1.76E+10d, 1.78E+10e
87.7401	268,637. 1 o - 382,610.01	8.61E+05	5.11E+08c
87.7872	168,341.35 2 o - 282,253.222	5.18E+09	5.62E+09c, 4.97E+09d, 5.05E+09e
88.0372	181,044.27 1 o - 294,632.561	3.39E+09	3.77E+09c, 3.39E+09d, 3.27E+09e
88.0854	183,261.02 2 o - 296,787.222	5.03E+09	5.53E+09c, 4.74E+09d, 4.75E+09e
88.1521	183,346.90 1 o - 296,787.222	5.62E+08	9.68E+08c, 7.40E+08d, 9.00E+07e
88.5583	156,785.87 2 o - 296,787.222	6.71E+09	6.93E+09c, 6.40E+09d, 6.52E+09e
88.5801	171,240.86 4 o - 284,133.064	5.44E+09	6.35E+09c, 5.74E+09d, 5.36E+09e
89.4243	185,638.77 3 o - 297,465.233	4.54E+09	5.16E+09c, 4.43E+09d, 4.68E+09e
89.4407	172,327.17 3 o - 284,133.064	1.08E+09	1.47E+09c, 1.22E+09d, 1.22E+09e
89.6137	171,240.86 4 o - 282,830.933	7.72E+09	8.02E+09c, 6.56E+09d, 7.14E+09e
89.6472	172,327.17 3 o - 283,875.573	7.04E+08	2.62E+09c, 2.40E+09d, 2.11E+09e
89.7896	183,261.02 2 o - 294,632.561	2.29E+09	2.55E+09c, 2.33E+09d, 2.27E+09e
89.8588	183,346.90 1 o - 294,632.561	6.85E+06	7.64E+07c, 3.30E+07d, 1.38E+08e
89.9698	185,638.77 3 o - 296,787.222	5.82E+09	6.65E+09c, 5.70E+09d, 5.40E+09e
90.1545	173,212.31 5 o - 284,133.064	1.12E+10	1.28E+10c, 1.08E+10d, 1.11E+10e
90.1916	186,590.15 4 o - 297,465.233	9.47E+09	1.08E+10c, 9.31E+09d, 9.31E+09e

90.2146	171,240.86 4 o - 282,087.735	2.89E+08	3.80E+08c, 4.18E+08d, 3.63E+08e
90.4947	172,327.17 3 o - 282,830.933	2.89E+08	3.80E+08c
90.5802	185,638.77 3 o - 296,038.214	2.79E+08	3.66E+08c, 2.97E+08d, 3.51E+08e
90.9703	172,327.17 3 o - 282,253.222	2.76E+09	2.99E+09c, 2.49E+09d, 2.51E+09e
91.3675	186,590.15 4 o - 296,038.214	4.03E+09	4.56E+09c, 3.89E+09d, 3.91E+09e
91.6928	175,073.22 4 o - 284,133.064	6.44E+08	5.53E+08c, 3.33E+08d, 4.50E+08e
91.8481	173,212.31 5 o - 282,087.735	5.11E+09	5.82E+09c, 4.99E+09d, 4.97E+09e
91.9098	175,073.22 4 o - 283,875.573	1.07E+10	1.01E+10c, 8.54E+09d, 8.89E+09e
92.1665	161,705.88 5 o - 270,205.154	1.41E+10	1.48E+10c, 1.31E+10d, 1.31E+10e
93.1501	176,850.41 2 o - 284,204.002	4.61E+09	4.66E+09c, 4.06E+09d, 4.08E+09e
93.4453	175,073.22 4 o - 282,087.735	7.80E+06	7.49E+06c, 5.50E+06d, 8.80E+06e
95.1619	164,621.73 3 o - 269,705.843	5.97E+09	6.38E+09c, 5.56E+09d, 5.57E+09e
96.0122	166,051.67 4 o - 270,205.154	5.16E+09	5.70E+09c, 4.92E+09d, 4.82E+09e
96.4747	166,051.67 4 o - 269,705.843	3.37E+09	3.85E+09c, 3.40E+09d, 3.40E+09e
96.9371	181,044.27 1 o - 284,204.002	4.13E+08	6.20E+08c, 4.10E+08d, 6.75E+08e
97.1507	181,271.17 3 o - 284,204.002	3.14E+09	3.47E+09c, 2.79E+09d, 2.80E+09e
97.4617	181,271.17 3 o - 283,875.573	3.48E+09	2.84E+09c, 2.46E+09d, 2.65E+09e
98.4642	181,271.17 3 o - 282,830.933	3.33E+08	1.34E+09c, 1.14E+09d, 1.00E+09e
98.8055	181,044.27 1 o - 282,253.222	1.62E+08	1.75E+08c, 1.70E+08d, 1.20E+08e
99.1502	183,346.90 1 o - 284,204.002	2.51E+09	2.92E+09c, 2.49E+09d, 7.50E+06e
99.5103	269,705.843 - 370,198.0 3 o	1.08E+09	1.80E+09c, 1.55E+09d, 1.53E+09e
101.1058	183,346.90 1 o - 282,253.222	7.71E+08	4.78E+08c
105.8544	270,205.154 - 364,674.5 3 o	6.29E+08	3.54E+08c
106.4041	269,705.843 - 363,687.2 2 o	7.38E+08	1.16E+09c
106.6467	282,830.933 - 376,598.5 3 o	1.88E+09	6.29E+09c
107.8425	269,705.843 - 362,433.7 3 o	1.36E+10	2.15E+10c
107.8482	283,875.573 - 376,598.5 3 o	1.01E+10	8.98E+09c
108.4263	270,205.154 - 362,433.7 3 o	3.38E+09	6.12E+09c
109.7280	269,705.843 - 360,840.3 2 o	1.55E+10	2.03E+10c
116.2872	284,204.002 - 370,198.0 3 o	1.39E+08	5.02E+08c, 3.43E+08d, 3.57E+08e
120.8880	282,087.735 - 364,808.9 5 o	1.18E+10	1.22E+10c
121.3279	282,253.222 - 364,674.5 3 o	5.61E+09	5.56E+09c
121.5949	296,038.214 - 378,278.5 4 o	8.80E+09	9.13E+09c
121.7006	294,632.561 - 376,801.4 1 o	7.06E+09	7.54E+09c
121.8977	296,038.214 - 378,074.2 3 o	5.43E+08	6.41E+08c

122.1843	282,830.933 - 364,674.5 3 o	1.03E+10	8.76E+09c
122.4620	282,830.933 - 364,488.9 4 o	1.21E+10	9.28E+09c
122.7989	282,253.222 - 363,687.2 2 o	1.03E+10	1.02E+10c
123.0209	296,787.222 - 378,074.2 3 o	1.15E+10	1.23E+10c
123.2030	282,087.735 - 363,254.6 6 o	4.13E+10	4.39E+10c
123.3589	296,038.214 - 377,102.5 5 o	3.48E+10	3.70E+10c
123.3991	294,632.561 - 375,670.4 2 o	3.34E+08	5.57E+08c
123.6763	282,830.933 - 363,687.2 2 o	3.80E+09	2.82E+09c
123.9529	284,133.064 - 364,808.9	2.28E+10	2.47E+10c
124.0490	283,875.573 - 364,488.9 4 o	1.94E+09	5.82E+09c
124.1306	296,038.214 - 376,598.5 3 o	1.26E+09	1.30E+09c
124.2691	284,204.004 - 364,674.5 3 o	1.14E+08	4.61E+08c
124.9345	297,465.233 - 377,507.2 2 o	1.03E+09	1.39E+09c
124.9779	296,787.222 - 376,801.4 1 o	1.41E+09	1.69E+09c
125.2950	283,875.573 - 363,687.2 1 o	7.75E+08	2.08E+09c
125.2956	296,787.222 - 376,598.5 3 o	1.77E+09	1.86E+09c
125.5829	282,087.735 - 361,716.4 5 o	2.38E+09	2.62E+09c
125.8128	284,204.002 - 363,687.2 2 o	2.02E+08	8.93E+08c
126.5371	296,038.214 - 375,066.4 4 o	1.93E+09	2.07E+09c
127.1105	284,133.064 - 362,804.8 4 o	3.21E+09	3.21E+09c
127.8688	297,465.233 - 375,670.4 2 o	2.50E+09	2.35E+09c
128.8640	297,465.233 - 375,066.4 4 o	1.85E+09	1.94E+09c
128.8937	284,133.064 - 361,716.4 5 o	2.24E+09	2.34E+09c
140.0093	226,041.41 2 o - 297,465.233	3.59E+09	4.29E+09c, 5.22E+09d, 5.23E+09e
140.8381	226,461.70 3 o - 297,465.233	6.02E+09	6.80E+09c, 8.51E+09d, 8.96E+09e
142.1959	226,461.70 3 o - 296,787.222	9.16E+08	1.20E+09c, 1.42E+09d, 1.40E+09e
143.7267	226,461.70 3 o - 296,038.214	1.17E+10	1.36E+10c, 1.65E+10d, 1.65E+10e
143.7734	212,533.85 4 o - 282,087.735	1.44E+10	1.68E+10c, 2.02E+10d, 2.02E+10e
144.5403	213,068.38 3 o - 282,253.222	6.32E+09	6.93E+09c, 8.39E+09d, 8.38E+09e
145.7914	226,041.41 2 o - 294,632.561	3.83E+09	4.44E+09c, 5.27E+09d,5.27E+09e
171.9318	226,041.41 2 o - 284,204.002	4.76E+08	6.22E+08c, 8.05E+08d, 7.95E+08e
172.9082	226,041.41 2 o - 283,875.573	2.87E+09	2.23E+09c, 2.53E+09d, 2.64E+09e
173.1833	226,461.70 3 o - 284,204.002	3.24E+09	3.38E+09c, 4.11E+09d, 4.10E+09e
173.3965	212,533.85 4 o - 270,205.154	2.28E+09	2.72E+09c, 3.36E+09d, 3.38E+09e
176.0888	226,041.41 2 o - 282,830.933	5.20E+08	1.53E+09c, 1.96E+09d, 1.86E+09e

176.5616	213,068.38 3 o - 269,705.843	6.56E+08	8.42E+08c, 1.07E+09d, 1.06E+09e
179.2387	226,461.70 3 o - 282,253.222	5.41E+07	2.93E+08c, 3.20E+08d, 3.30E+08e
690.7751	370,198.0 3 o - 282,253.222	4.17E+04	1.54E+05c
818.6751	371,455.63 - 383,667.1 3 o	4.01E+03	5.44E+03c
990.4160	361,716.4 5 o - 371,810.45	1.92E+06	6.18E+05c
1016.3329	375,066.4 4 o - 384,903.04	2.99E+07	1.65E+07c
1036.8021	360,840.3 2 o - 370,482.73	1.27E+07	8.99E+06c
1041.0126	361,716.4 5 o - 371,319.85	3.73E+07	2.03E+07c
1086.1866	375,670.4 2 o - 384,874.42	3.06E+07	1.88E+07c
1110.7946	375,670.4 2 o - 384,670.53	1.91E+07	1.46E+07c
1140.0037	362,433.7 3 o - 371,203.24	2.93E+07	2.01E+07c
1155.5392	376,598.5 3 o - 385,250.14	3.35E+06	1.42E+06c
1191.4838	363,254.6 6 o - 371,645.26	2.79E+07	1.43E+07c
1203.8370	376,598.5 3 o - 384,903.04	6.01E+07	3.18E+07c
1209.4147	377,102.5 5 o - 385,368.75	1.94E+07	9.94E+06c
1238.3581	376,801.4 1 o - 384,874.42	1.00E+07	4.26E+06c
1239.5557	363,254.6 6 o - 371,319.85	1.54E+06	6.73E+05c
1242.0505	362,433.7 3 o - 370,482.73	2.52E+07	1.54E+07c
1286.9143	363,687.2 2 o - 371,455.63	2.08E+07	1.12E+07c
1331.2293	377,507.2 2 o - 385,017.3	2.33E+07	1.46E+07c
1364.0138	376,801.4 1 o - 384,130.72	1.07E+07	5.95E+06c
1365.4670	364,488.9 4 o - 371,810.45	5.10E+07	2.56E+07c
1395.6231	377,507.2 2 o - 384,670.53	9.84E+06	1.84E+06c
1407.1937	364,674.5 3 o - 371,778.94	2.83E+07	1.72E+07c
1445.3811	376,801.4 1 o - 383,718.11	5.76E+06	2.32E+06c
1462.3799	364,808.9 5 o - 371,645.26	6.40E+07	3.16E+07c
1463.5359	364,488.9 4 o - 371,319.85	3.01E+06	1.78E+06c
1471.1600	363,687.2 2 o - 370,482.73	2.19E+06	2.82E+05c
1488.9516	364,488.9 4 o - 371,203.24	9.94E+06	2.41E+06c
1509.3633	377,507.2 2 o - 384,130.72	9.98E+06	5.16E+06c
1535.4664	364,808.9 5 o - 371,319.85	5.15E+06	1.81E+06c
1721.2337	364,674.5 3 o - 370,482.73	3.95E+06	1.68E+06c

^aRitz wavelengths calculated employing the experimental energy level values. Transitions are given by values (in cm⁻¹) of involved energy levels where subscripts denote their J-values. b: MCDHF values from [38]. pE + q = p.10^q. c: HFR values from [38]. d: Values from Newton method taken in [36] [37]. e: Values from least-squares method taken in [36] [37].

gf(HFR)ñ 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 > 10⁹ s⁻¹), gives these average ratios ágA/gA (HFR or MCDHF)ñ: 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 Ta⁵⁺ 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].

Table 6

Adopted transition probabilities (gA) in Ta VI, as well as other available gA-values

λ (nm)a	Transition	Adopted gA (s−1)b	Other gA (s−1)
21.8259	0.00 - 458,172.0 1 o	1.43E+10	7.38E+10c
26.0048	0.00 - 384,545.2 1 o	6.28E+10	5.16E+10c
37.3512	242,614.2 1 o - 510,343.01	1.09E+10	1.50E+10c
37.5722	230,924.4 2 o - 497,079.03	2.27E+10	3.50E+10c
37.7279	230,924.4 2 o - 495,980.02	5.01E+10	7.84E+10c
38.1708	251,742.8 4 o - 513,723.04	2.33E+10	4.19E+10c
38.3359	242,614.2 1 o - 503,466.00	1.61E+10	2.57E+10c
38.4766	253,591.0 2 o - 513,489.02	4.93E+10	8.08E+10c
38.4840	236,438.8 5 o - 496,287.06	2.32E+11	3.60E+11c
38.5888	239,599.4 3 o - 498,742.02	4.20E+09	5.49E+09c
38.6792	251,806.0 0 o - 510,343.01	1.09E+10	1.40E+10c
38.9012	241,235.5 4 o - 498,297.03	5.99E+09	6.25E+09c
39.0045	239,599.4 3 o - 495,980.02	3.25E+10	5.20E+10c
39.0430	242,614.2 1 o - 498,742.02	1.54E+10	2.30E+10c
39.0864	241,235.5 4 o - 497,079.03	1.51E+10	2.73E+10c
39.1394	243,309.0 6 o - 498,806.06	5.13E+10	6.47E+10c
39.1998	258,566.4 3 o - 513,670.06	2.12E+10	1.59E+10c
39.2400	243,309.0 6 o - 498,151.05	3.18E+09	4.01E+09c
39.2757	258,566.4 3 o - 513,177.05	3.34E+10	6.43E+10c
39.3855	260,457.1 5 o - 514,358.05	3.70E+10	4.75E+10c
39.4454	243,309.0 6 o - 496,824.07	2.94E+11	4.05E+11c
39.4842	260,457.1 5 o - 513,723.04	1.47E+09	1.70E+09c
39.5291	243,309.0 6 o - 496,287.06	5.34E+09	9.68E+09c

39.5554	258,774.2 1 o - 511,584.01	2.32E+10	4.18E+10c
39.6630	260,457.1 5 o - 512,581.06	2.48E+11	3.43E+11c
39.7347	244,923.7 2 o - 496,593.01	1.73E+10	2.53E+10c
39.8189	260,457.1 5 o - 511,594.05	4.63E+09	7.63E+09c
39.8716	248,055.0 4 o - 498,860.05	1.08E+11	1.63E+11c
39.9608	260,097.8 1 o - 510,343.01	1.75E+10	3.36E+10c
39.9847	248,055.0 4 o - 498,151.05	3.91E+09	1.33E+10c
40.1029	249,484.5 5 o - 498,843.04	9.46E+10	1.31E+11c
40.1400	264,541.9 3 o - 513,670.03	1.88E+10	3.91E+10c
40.1692	264,541.9 3 o - 513,489.02	1.15E+09	4.08E+09c
40.1909	249,484.5 3 o - 498,297.03	4.56E+10	6.92E+10c
40.2204	265,728.2 4 o - 514,358.05	1.73E+11	2.35E+11c
40.2585	261,948.0 2 o - 510,343.01	5.05E+09	7.97E+09c
40.2788	250,536.4 5 o - 498,806.06	1.95E+11	2.70E+11c
40.3031	264,541.9 3 o - 512,662.02	6.08E+09	7.01E+09c
40.4666	251,742.8 4 o - 498,860.05	3.71E+10	2.74E+10c
40.5687	249,484.5 3 o - 495,980.02	6.29E+09	8.98E+09c
40.7604	251,742.8 4 o - 497,079.03	1.34E+09	7.56E+08c
40.8677	258,774.2 1 o - 503,466.00	1.13E+09	1.64E+09c
43.1166	230,924.4 2 o - 462,853.73	3.94E+07	5.31E+07c
44.9425	244,923.7 2 o - 467,430.12	2.65E+07	2.27E+07c
45.1227	241,235.5 4 o - 462,853.73	2.69E+08	3.60E+08c
45.7639	244,923.7 2 o - 463,436.41	2.80E+08	3.00E+08c
45.8863	244,923.7 2 o - 462,853.73	1.86E+07	1.22E+07c
46.5552	248,055.0 4 o - 462,853.73	3.27E+08	4.68E+08c
46.8671	249,484.5 3 o - 462,853.73	1.43E+08	1.34E+08c
47.3685	251,742.8 4 o - 462,853.73	4.49E+07	2.68E+07c
47.9258	258,774.2 1 o - 467,430.12	2.40E+05	4.68E+06c
48.1498	306,511.8 3 o - 514,197.04	2.42E+08	6.98E+08c
48.8610	258,774.2 1 o - 463,436.41	1.91E+07	1.90E+08c
49.2238	242,614.2 1 o - 445,767.91	2.16E+08	2.10E+08c
49.7746	261,948.0 2 o - 462,853.73	1.35E+07	1.54E+07c
49.9994	296,590.5 1 o - 496,593.01	9.00E+07	1.80E+08c
50.7291	265,728.2 4 o - 462,853.73	5.16E+07	8.15E+07c
51.5461	320,195.9 3 o - 514,197.04	1.04E+08	2.52E+07c
51.5565	251,806.0 0 o - 445,767.91	1.32E+08	1.42E+08c
56.0043	335,112.2 2 o - 513,670.03	3.39E+07	1.04E+07c

56.5791	242,614.2 1 o - 419,358.02	8.11E+08	9.26E+08c, 9.55E+08d, 1.03E+09e
57.3954	242,614.2 1 o - 416,844.31	5.73E+08	4.50E+08c, 4.92E+08d, 5.37E+08e
58.5344	296,590.5 1 o - 467,430.12	1.54E+09	1.63E+09c
58.6656	248,055.0 4 o - 418,512.64	1.01E+09	8.45E+08c, 5.49E+08d, 6.93E+08e
59.9629	251,742.8 4 o - 418,512.64	1.21E+09	2.16E+09c, 2.17E+09d, 2.04E+09e
60.0311	253,591.0 2 o - 420,171.33	2.81E+09	2.90E+09c, 2.45E+09d, 2.31E+09e
60.4561	239,599.4 3 o - 405,008.84	4.37E+09	4.66E+09c, 3.88E+09d, 3.75E+09e
60.5920	251,806.0 0 o - 416,844.31	3.52E+09	3.88E+09c, 3.66E+09d, 3.66E+09e
61.2545	253,591.0 2 o - 416,844.31	5.08E+08	9.80E+08c, 8.55E+08d, 1.01E+09e
61.3745	241,235.5 4 o - 404,169.73	5.05E+08	1.22E+09c, 1.16E+09d, 8.19E+08e
61.8793	258,566.4 3 o - 420,171.33	2.08E+09	1.97E+09c, 1.93E+09d, 1.69E+09e
61.9652	241,235.5 4 o - 402,616.45	1.81E+09	2.48E+09c, 2.18E+09d, 2.26E+09e
62.1923	258,566.4 3 o - 419,358.02	1.82E+09	2.39E+09c, 1.78E+09d, 2.00E+09e
62.7717	243,309.0 6 o - 402,616.45	4.34E+10	4.84E+10c, 4.27E+10d, 4.27E+10e
62.7903	260,097.8 1 o - 419,358.02	1.62E+09	3.25E+09c, 2.63E+09d, 1.11E+09e
63.2631	258,774.2 1 o - 416,844.31	6.50E+09	5.36E+09c, 5.13E+09d, 2.78E+09e
63.2689	260,457.1 5 o - 418,512.64	3.51E+10	3.99E+10c, 3.47E+10d, 3.48E+10e
63.7000	306,450.5 2 o - 463,436.41	1.21E+09	1.05E+09c
63.7741	244,923.7 2 o - 401,727.13	7.79E+08	4.48E+08c, 3.36E+08d, 4.69E+08e
63.7973	260,097.8 1 o - 416,844.31	2.53E+08	2.40E+09c, 1.66E+09d, 3.96E+09e
63.9373	306,450.5 2 o - 462,853.73	2.21E+09	2.41E+09c
64.0555	248,055.0 4 o - 404,169.73	2.88E+09	8.64E+09c, 8.19E+09d, 7.21E+09e
64.2552	264,541.9 3 o - 420,171.33	9.32E+09	1.07E+10c, 8.96E+09d, 9.24E+09e
64.2986	249,484.5 3 o - 405,008.84	2.28E+09	2.98E+09c, 2.50E+09d, 2.62E+09e
64.3585	230,924.4 2 o - 386,304.13	1.32E+10	1.37E+10c, 1.24E+10d, 1.26E+10e
64.5168	307,855.3 4 o - 462,853.73	2.45E+10	3.35E+10c
64.5593	261,948.0 2 o - 416,844.31	4.52E+09	5.10E+09c, 4.56E+09d, 4.41E+09e
64.5928	264,541.9 3 o - 419,358.02	1.16E+10	1.30E+10c, 1.12E+10d, 1.10E+10e
64.6474	249,484.5 3 o - 404,169.73	9.95E+09	1.30E+10c, 1.13E+10d, 1.09E+10e
64.7365	250,536.4 5 o - 405,008.84	2.28E+10	2.64E+10c, 2.19E+10d, 2.21E+10e
64.7488	265,728.2 4 o - 420,171.33	1.93E+10	2.21E+10c, 1.87E+10d, 1.87E+10e
65.0736	248,055.0 4 o - 401,727.13	1.36E+10	8.91E+09c, 6.60E+09d, 7.98E+09e
65.4433	345,938.0 1 o - 498,742.02	6.73E+07	9.34E+07c
65.4865	249,484.5 3 o - 402,187.72	4.98E+09	3.08E+09c, 2.45E+09d, 2.85E+08e
65.6052	251,742.8 4 o - 404,169.73	1.38E+10	8.98E+09c, 6.60E+09d, 7.91E+09e
65.7549	250,536.4 5 o - 402,616.45	1.03E+10	1.18E+10c, 9.96E+09d, 9.85E+09e
65.9853	296,590.5 1 o - 448,139.52	1.49E+09	1.53E+09c

66.4105	253,591.0 2 o - 404,169.73	6.94E+08	5.21E+08c, 3.64E+08d, 4.34E+08e
66.4699	236,438.8 6 o - 386,882.94	2.82E+10	2.95E+10c, 2.57E+10d, 2.58E+10e
66.5689	313,216.2 2 o - 463,436.41	9.77E+09	1.19E+10c
66.6737	251,742.8 4 o - 401,727.13	6.82E+09	1.41E+10c, 1.37E+10d, 1.23E+10e
67.0343	296,590.5 1 o - 445,767.91	5.98E+09	6.78E+09c
67.8963	239,599.4 3 o - 386,882.94	3.38E+09	3.95E+09c, 3.38E+09d, 3.38E+09e
68.1641	239,599.4 3 o - 386,304.13	1.22E+10	1.30E+10c, 1.12E+10d, 1.12E+10e
68.2862	258,566.4 3 o - 405,008.84	7.51E+07	7.50E+07c, 3.69E+07d, 3.96E+07e
68.6798	258,566.4 3 o - 404,169.73	2.51E+09	1.53E+09c, 1.18E+09d, 1.58E+09e
68.9329	241,235.5 4 o - 386,304.13	6.96E+09	8.04E+09c, 6.95E+09d, 6.92E+09e
69.0376	258,566.4 3 o - 403,415.02	6.00E+09	3.88E+09c, 2.82E+09d, 6.05E+09e
69.3740	323,283.8 3 o - 467,430.12	1.39E+10	1.62E+10c
69.8516	258,566.4 3 o - 401,727.13	5.53E+09	7.35E+09c, 6.26E+09d, 5.94E+09e
69.9285	319,850.5 4 o - 462,853.73	4.30E+09	7.79E+08c
70.7312	244,923.7 2 o - 386,304.13	9.91E+08	9.46E+08c, 7.84E+08d, 6.02E+08e
71.7785	306,450.5 2 o - 445,767.91	4.47E+09	5.26E+09c
73.9973	251,742.8 4 o - 386,882.94	7.48E+08	4.20E+08c, 2.61E+08d, 3.51E+08e
74.1162	313,216.2 2 o - 448,139.52	2.67E+09	2.66E+09c
75.8874	335,655.9 3 o - 467,430.12	1.73E+09	1.03E+08c
76.6224	401,727.13 - 532,237.2 3 o	4.35E+09	7.30E+09c
76.7134	386,304.13 - 516,659.5 4 o	1.39E+10	1.66E+10c
76.7311	401,727.13 - 532,052.4 2 o	8.50E+09	1.16E+10c
76.8938	402,187.72 - 532,237.2 3 o	2.03E+09	7.96E+09c
76.9532	386,882.94 - 516,832.1 3 o	1.25E+10	1.48E+10c
77.0555	386,882.94 - 516,659.5 4 o	5.80E+09	8.96E+09c
77.6264	403,415.02 - 532,237.2 3 o	7.63E+09	4.08E+09c
78.1967	404,169.73 - 532,052.4 2 o	1.80E+09	9.29E+08c
85.1075	345,938.0 1 o - 463,436.41	1.63E+09	1.51E+09c
86.7995	416,844.31 - 532,052.4 2 o	7.21E+09	6.05E+09c
86.8101	386,882.94 - 502,076.9 3 o	5.31E+08	4.94E+08c
87.2262	402,187.72 - 516,832.1 3 o	8.42E+09	3.20E+09c
87.6862	402,616.45 - 516,659.5 4 o	2.41E+10	2.11E+10c
87.9317	418,512.64 - 532,237.2 3 o	1.93E+10	1.72E+10c
87.9347	306,450.5 2 o - 420,171.33	1.31E+06	3.11E+06c
87.9821	306,511.8 3 o - 420,171.33	3.07E+07	5.66E+07c
88.0655	401,727.13 - 515,279.0 3 o	1.51E+10	2.24E+10c
88.1701	403,415.02 - 516,832.1 3 o	1.32E+09	4.91E+09c

88.2924	386,304.13 - 499,564.2 4 o	1.55E+10	2.16E+10c
88.5681	306,450.5 2 o - 419,358.04	7.50E+06	1.52E+07c
88.5903	419,358.02 - 532,237.2 3 o	1.70E+09	1.48E+09c
88.7356	419,358.02 - 532,052.4 2 o	8.07E+09	7.69E+09c
88.8970	404,169.73 - 516,659.5 4 o	2.02E+09	1.99E+09c
89.0345	307,855.3 4 o - 420,171.33	1.49E+07	1.28E+07c
89.1271	386,882.9 4 o - 499,082.23	6.71E+09	9.69E+09c
89.3806	420,171.33 - 532,052.4 2 o	5.22E+09	4.90E+09c
89.3943	403,415.02 - 515,279.0 3 o	1.19E+10	7.88E+09c
89.4268	405,008.84 - 516,832.1 3 o	5.94E+09	5.44E+09c
89.5062	401,727.13 - 513,451.2 4 o	3.50E+10	5.29E+10c
90.0015	404,169.73 - 515,279.0 3 o	2.70E+09	1.60E+09c
90.3691	307,855.3 4 o - 418,512.64	1.74E+07	1.53E+07c
91.5068	404,169.73 - 513,451.2 4 o	8.78E+09	4.89E+09c
100.2938	402,616.45 - 502,323.5 5 o	1.70E+10	1.83E+10c
100.7982	401,727.13 - 500,935.2 2 o	1.10E+09	5.54E+08c
100.8529	418,512.64 - 517,666.9 4 o	1.23E+10	1.34E+10c
101.5259	306,511.8 3 o - 405,008.84	7.93E+08	1.78E+08c
102.2083	402,616.45 - 500,455.8 6 o	5.84E+10	6.63E+10c
102.3234	418,512.64 - 516,242.0 5 o	4.91E+10	5.59E+10c
102.3983	306,511.8 3 o - 404,169.73	1.11E+09	2.49E+08c
102.7585	419,358.02 - 516,673.6 2 o	1.33E+10	1.51E+10c
102.7594	405,008.84 - 502,323.5 5 o	3.20E+10	3.76E+10c
103.0205	405,008.84 - 502,076.9 3 o	3.11E+09	3.96E+09c
103.3417	418,512.64 - 515,279.0 3 o	1.75E+09	1.89E+09c
103.3426	404,169.73 - 500,935.2 2 o	5.10E+09	7.12E+09c
103.8266	307,855.3 4 o - 404,169.73	2.06E+08	2.15E+07c
104.5471	402,616.45 - 498,267.1 5 o	3.40E+09	4.12E+09c
105.0103	323,283.8 3 o - 418,512.64	2.12E+09	2.03E+08c
105.0251	306,511.8 3 o - 401,727.13	2.60E+08	2.92E+07c
105.3312	418,512.64 - 513,451.2 4 o	2.72E+09	3.29E+09c
105.5285	307,855.3 4 o - 402,616.45	2.83E+09	2.98E+08c
106.3000	405,008.84 - 499,082.2 3 o	4.90E+09	5.25E+09c
106.4240	420,171.33 - 514,135.1 2 o	3.22E+09	3.45E+09c
107.2042	420,171.33 - 513,451.2 4 o	2.53E+09	2.77E+09c
107.2291	405,008.84 - 513,451.2 4 o	3.08E+09	3.39E+09c
114.9487	380,434.82 - 467,430.1 2 o	6.70E+08	5.69E+09c

117.5653	335,112.2 2 o - 420,171.33	5.02E+09	6.34E+09c
117.9066	320,195.9 3 o - 405,008.84	5.17E+09	6.80E+09c
118.5970	319,850.5 4 o - 404,169.73	1.45E+09	2.53E+09c
118.7003	335,112.2 2 o - 419,358.02	7.93E+09	9.44E+09c
119.0848	320,195.9 3 o - 404,169.73	7.90E+09	1.09E+10c
120.6492	384,545.2 1 o - 467,430.12	3.71E+09	5.59E+09c
120.6903	335,655.9 3 o - 418,512.64	1.41E+10	1.97E+10c
122.1350	319,850.5 4 o - 401,727.13	2.88E+08	1.77E+08c
122.3510	335,112.2 2 o - 416,844.31	5.22E+09	6.43E+09c
122.6524	320,195.9 3 o - 401,727.13	1.76E+09	8.38E+08c
124.4228	306,511.8 3 o - 386,882.94	6.61E+08	9.22E+07c
124.7953	323,283.8 3 o - 403,415.02	3.78E+08	3.74E+08c
126.7365	323,283.8 3 o - 402,187.72	2.06E+08	5.71E+08c
127.4717	307,855.3 4 o - 386,304.13	8.60E+08	1.10E+08c
127.4806	323,283.8 3 o - 401,727.13	3.17E+08	1.87E+08c
146.4069	335,112.2 2 o - 403,415.02	6.11E+08	5.01E+08c
147.5817	335,655.9 3 o - 403,415.02	3.14E+09	1.87E+09c
147.7002	380,434.8 2 o - 448,139.52	1.67E+08	2.47E+09c
149.9543	320,195.9 3 o - 386,882.94	4.97E+09	5.97E+09c
150.3041	335,655.9 3 o - 402,187.72	5.28E+08	3.03E+09c
151.3519	335,655.9 3 o - 401,727.13	1.56E+09	2.68E+09c
2759.3907	498,297.03 - 501,920.0 4 o	1.80E+03	7.94E+04c
2997.7729	498,742.02 - 502,076.9 3 o	5.25E+03	1.14E+05c
7889.2632	498,297.03 - 499,564.2 4 o	2.69E+03	4.39E+03c
10,434.4792	513,177.03 - 514,135.1 2 o	6.45E+04	2.10E+04c

a: Ritz wavelengths calculated employing the experimental energy level values from [50] [51]. Transitions are given by values (in cm⁻¹) of involved energy levels where subscripts denote their J-values. b: MCDHF values from [38]. pE + q = p.10^q. c: HFR values from [38]. d: Values from Newton method taken in [37].

4. Conclusions

We Critically evaluated available dipole-transition rates in Xe⁹⁺, Xe¹⁰⁺, Lu³⁺, Hf⁴⁺ and Ta⁵⁺ with respect to our recent results obtained through large-scale calculations using 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). The adopted data would allow plasma physicists to diagnose and model fusion plasmas in tokamaks where xenon, lutetium, hafnium and tantalum could be used.

In literature, transition probabilities and oscillator strengths of the studied 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 Lu³⁺, Hf⁴⁺ and Ta⁵⁺.

Acknowledgements

Our own work, discussed in the framework of this review, was carried out in collaboration with P. Quinet and P. Palmeri (Atomic Physics and Astrophysics, Mons University, Belgium), and E. Bokamba Motoumba (Marien Ngouabi University, Congo). I would like to thank them very much. The Author is a Senior Lecturer at Marien Ngouabi University of Congo whose financial support is gratefully acknowledged.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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