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

Saturnin Enzonga Yoca^1,2

¹Faculty of Sciences and Techniques, Marien Ngouabi University, Brazzaville, Congo.
²African and Malagasy Council for Higher Education—CAMES, Ouagadougou, Burkina Faso.

DOI: 10.4236/jamp.2021.911182                

Abstract

This paper presents a review about the radiative properties (transition probabilities and oscillator strengths) of two xenon ions (Xe⁹⁺, Xe¹⁰⁺) and three members of Er I isoelectronic sequence (Lu³⁺, Hf⁴⁺, Ta⁵⁺) 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.

Keywords

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].

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

$H={\displaystyle \underset{i}{\overset{N}{\sum }}}\left(c{\alpha }_i\cdot p_i+\left({\beta }_i-1\right)c^2+\frac{Z}{r_i}\right)+{\displaystyle \underset{i<j}{\overset{N}{\sum }}}\frac{1}{r_{ij}},$ (1)

where c is the speed of light, $\alpha$ and $\beta$ are the Dirac matrices.

The atomic state function (ASF), $\Psi$, is represented by a superposition of configuration state functions (CSF), $\Phi$, with the same parity, $\pi$, total angular momentum and total magnetic quantum numbers, J and $M_J$, forming a basis set of the representation, $\left\{{\Phi }_k\right\}$, as

$\Psi \left(\Pi J{M}_J\right)={\displaystyle \underset{k}{\sum }}\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}{c}_k\Phi \left({\gamma }_k\Pi J{M}_J\right),$ (2)

where $c_k$ are the mixing coefficients, ${\gamma }_k$ represent all the other quantum numbers needed to uniquely specify CSF that are jj-coupled Slater determinants built from one-electron spin-orbitals, ${\varphi }_{n\kappa m}\left(r\mathrm{,}\theta \mathrm{,}\phi \right)$, of the form:

${\varphi }_{n\kappa m}\left(r,\theta ,\phi \right)=\frac{1}{r}\left(\begin{array}{c}{P}_{n\kappa }\left(r\right){\chi }_{\kappa m}\left(\theta ,\phi \right)\\ i{Q}_{n\kappa }\left(r\right){\chi }_{\kappa m}\left(\theta ,\phi \right)\end{array}\right)$ (3)

$P_{n\kappa }\left(r\right)$ and $Q_{n\kappa }\left(r\right)$ are, respectively, the large and the small radial components of the wave functions, and the angular functions ${\chi }_{\kappa m}\left(\theta ,\phi \right)$ are the spinor spherical harmonics [39]. The quantum number $\kappa$ is given by:

$\kappa =\pm \left(j+\frac{1}{2}\right)=a\left(j+\frac{1}{2}\right)$ (4)

so that

$l=j-\frac{1}{2}a$ (5)

The radial functions $P_{n\kappa }\left(r\right)$ and $Q_{n\kappa }\left(r\right)$ are numerically represented on a logarithmic grid and are required to be orthonormal within each $\kappa$ 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. $0.9\le \text{B}/\text{C}\le 1.1$, 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]:

$\text{CF}={\left(\frac{\left|{\displaystyle {\sum }_k{\displaystyle {\sum }_i{c}_k\langle {\Phi }^{\prime }\left({\gamma }_k{\Pi }^{\prime }{J}^{\prime }{M}_{J^{\prime }}\right)|P^{\left(1\right)}|\Phi \left({\gamma }_i\Pi J{M}_J\right)\rangle c_i}}\right|}{{\displaystyle {\sum }_k{\displaystyle {\sum }_i\left|\langle {\Phi }^{\prime }\left({\gamma }_k{\Pi }^{\prime }{J}^{\prime }{M}_{J^{\prime }}\right)|P^{\left(1\right)}|\Phi \left({\gamma }_i\Pi J{M}_J\right)\rangle c_i\right|}}}\right)}^2$ (6)

where $P^{\left(1\right)}$ is the electric dipole operator and $c_{i\left(k\right)}(\text{'})$ and ${\Phi }_{i\left(k\right)}(\text{'})$ 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]:

$0.90\le \text{B}/\text{C}\le 1.10\text{and}\text{CF}\ge 0.05$ (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 $LSJ{M}_J\pi$ 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.

$H_{ab}=\langle {\alpha }_a{L}_a{S}_{a}J{M}_J\pi |H|{\alpha }_b{L}_b{S}_{b}J{M}_J\pi \rangle ={\displaystyle \underset{i}{\sum }}{c}_i^{a\mathrm{,}b}{I}_i^{a\mathrm{,}b}\mathrm{,}$ (8)

where $c_i^{a\mathrm{,}b}$ and $I_i^{a\mathrm{,}b}$ 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 ( ${\zeta }_{nl}$ ) and, if necessary, the effective interaction parameters ( $\alpha$, $\beta$, $\gamma$ ) [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, Xe⁹⁺ and Xe¹⁰⁺

3.1.1. Ion Xe⁹⁺

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.

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 Xe¹⁰⁺

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

Figure 1. Comparison between our HFR and MCDHF oscillator strengths (log gf) for Xe XI spectral lines [30]. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 are shown in the figure.

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

Figure 2. Comparison between the gA-values obtained from our HFR model [30] and the one by Churilov et al. [28] for Xe XI spectral lines. Only lines with weak cancellation effects (CF > 0.05) have been selected.

Figure 3. Comparison between the gA-values obtained in our MCDHF/RCI model [30] and the HFR model by Churilov et al. [28] for Xe XI spectral lines. Only lines with CF > 0.05 and 0.9 ≤ B/C ≤ 1.1 have been selected.

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.

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: Lu³⁺, Hf⁴⁺ and Ta⁵⁺

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 Lu³⁺

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.

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 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.

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 Hf⁴⁺

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.

Figure 6. Comparison of our HFR and MCDHF/RCI oscillator strengths (log gf) for Hf V spectral lines [38]. Only transitions with log gf > 0, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been retained.

Figure 7. Comparison between our MCDHF/RCI gA-values [38] with the available data [36] [37] for Hf V spectral lines. Only transitions with log gf > 00, CF ≥ 0.5 and 0.9 ≤ B/C ≤ 1.10 have been retained.

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 Ta⁵⁺

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.

^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].

Figure 8. Comparison of our HFR and MCDHF/RCI oscillator strengths (log gf) for Ta VI spectral lines [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.

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.

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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