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Article

Detailed Analysis of Configuration Interaction and Calculation of Radiative Transition Rates in Seven Times Ionized Tungsten (W VIII)

1
Astrophysique et Spectroscopie, Université de Mons, Mons B-7000, Belgium
2
IPNAS, Université de Liège, Liège B-4000, Belgium
*
Author to whom correspondence should be addressed.
Atoms 2015, 3(3), 299-319; https://doi.org/10.3390/atoms3030299
Received: 21 May 2015 / Revised: 15 June 2015 / Accepted: 23 June 2015 / Published: 30 June 2015
(This article belongs to the Special Issue Atomic Data for Tungsten)

Abstract

:
A new set of oscillator strengths and transition probabilities for EUV spectral lines of seven times ionized tungsten (W VIII) is reported in the present paper. These results have been obtained using the pseudo-relativistic Hartree-Fock (HFR) method combined with a semi-empirical optimization of the radial parameters minimizing the discrepancies between computed energy levels and available experimental data. The final physical model considered in the calculations has been chosen further to a detailed investigation of the configuration interaction in this atomic system characterized by complex configurations of the type 4f145s25p5, 4f145s25p4nl, 4f145s5p6, 4f135s25p6, 4f135s25p5nl and 4f125s25p6nl (nl = 5d, 6s).

1. Introduction

It is now well established that tungsten plays an important role in the development of fusion reactors (see e.g., [1,2,3,4,5]). Indeed, this element has been chosen to be the main component of the divertor of the International Thermonuclear Experimental Reactor (ITER) so that spectral lines of W ions sputtered from the wall to the core plasma provide a key information for plasma emission analysis and diagnostic purposes. As a consequence, a detailed knowledge of the atomic structure and radiative properties of almost each ionization stage of tungsten is required. Over the past few years, several of our works were focused on the determination of spectroscopic data for neutral to moderately ionized tungsten. More precisely, oscillator strengths and transition probabilities were calculated for a large number of lines in W I [6], W II [7], W III [8], W IV [9], W V [10] and W VI [11]. In all these studies, the pseudo-relativistic Hartree-Fock (HFR) method including a large amount of intravalence and core-valence electronic correlation effects was combined with a semi-empirical process minimizing the discrepancies between calculated and available experimental energy levels. For W I, W II and W III, the accuracy of this approach was assessed through detailed comparisons with experimental radiative lifetimes measured with the time-resolved laser-induced fluorescence (TR-LIF) technique while, for W IV, W V and W VI, our new HFR results were supported by a detailed comparison with transition probabilities obtained using different theoretical methods. In all cases, it was shown that the methodology used for modeling the atomic structure and computing the radiative parameters of lowly ionized tungsten was able to provide reliable spectroscopic data of great interest in fusion research. Let us also mention here that one of our recent papers [12] was dedicated to a critical evaluation of the transition rates available in the literature for electric dipole lines in W I, W II and W III.
The main goal of the present work is to extend all our previous studies related to tungsten ions to the seven times ionized species (W VIII) for which 187 spectral lines were very recently observed leading to the first experimental identification of energy levels in this ion [13]. As we did for the first W ions, we also used here the pseudo-relativistic Hartree-Fock method putting the emphasis on the sensitivity of the radiative rates to electronic correlation effects in this particularly complex atomic system characterized by interacting configurations of the type 4f145s25p5, 4f145s25p4nl, 4f145s5p6, 4f135s25p6, 4f135s25p5nl and 4f125s25p6nl (nl = 5d, 6s). The final theoretical model was then optimized through a semi-empirical adjustment of the radial energy parameters to compute the oscillator strengths and transition probabilities for a set of 227 lines involving experimentally known levels with gf-values larger than 0.0001 in the extreme ultraviolet (EUV) wavelength region from 160.9 to 347.0 Å of the W VIII spectrum.

2. Available Atomic Data in W VIII

Up until very recently, nearly nothing was known about the atomic structure of seven times ionized tungsten. This lack of knowledge was underlined by Kramida and Shirai [14] who compiled all the classified energy levels and spectral lines of multiply ionized tungsten atoms from W2+ to W73+. In this compilation, it was reminded that it was by the way uncertain whether the ground state of W VIII was 4f135s25p6 27/2 or 4f145s25p5 23/2, making this ion the only known case of p and f orbitals competing for the ground state, as previously noted by Sugar and Kaufman [15]. An isoelectronic study was not even of great help to solve the problem since it was found that the ground configuration was 4f115s25p66s2 for the first members of the sequence, Ho I and Er II, 4f135s25p6 for Hf VI and Ta VII, and 4f145s25p5 for Re IX and the rest of the sequence. By analyzing the 4f13(27/2)5s25p6ns and 4f145s25p5(23/2)ns series of W VII, Sugar and Kaufman [15] asserted that the ground state of W VIII was probably 4f135s25p6 27/2, the 4f145s25p5 23/2 level being predicted 800 ± 700 cm−1 above it. This assumption was in agreement with the calculations performed earlier by Carlson et al. [16] who showed that 4f was the least-bound orbital of W VII. Later on, an experimental observation of W VIII spectrum was performed by Veres et al. [17] in emission of tokamak plasma but the low spectral resolution did not allow them to identify the observed broad peaks. However, using the weighted average energies of sub-configurations based on the 4f135s25p6.27/2,5/2 and 4f145s25p5 23/2,1/2 in W VII, taken from [15], Kramida and Shirai [14] predicted the position of the four corresponding energy levels in W VIII at 0 cm−1, 17,440 ± 60 cm−1, 800 ± 700 cm−1 and 87,900 ± 300 cm−1, respectively.
Two years ago, the first extensive analysis of the W VIII spectrum was reported by Ryabtsev et al. [13] who used two experimental setups installed at the Institute of Spectroscopy in Troitsk (Russia) and at the Observatory of Paris-Meudon (France) for obtaining tungsten ion spectra. In their work, a total of 187 W VIII lines in the region 160–271 Å were identified as transitions from the interacting excited even 4f125s25p65d + 4f135s25p5(5d + 6s) + 4f145s25p4(5d + 6s) + 4f145s5p6 configurations to the low-lying odd configurations 4f135s25p6 and 4f145s25p5. This gave rise to the establishment of the energy values of 4 odd- and 98 excited even-parity levels up to 622,123 cm−1 with estimated uncertainties ranging from 5 to 18 cm−1. It was also firmly established that the ground state of W VIII was 4f135s25p6 27/2 and the first excited 4f145s25p5 23/2 level was located 1233 ± 3 cm−1 above it, in agreement with the value 800 ± 700 cm−1 predicted by Kramida and Shirai [14]. The level identifications reported in [13] were supported by calculations performed using the pseudo-relativistic Hartree-Fock (HFR) method of Cowan [18] combined with a semi-empirical adjustment of the energy parameters. In the HFR model considered by these latter authors, also used for providing transition probabilities corresponding to the experimentally observed spectral lines, the 4f135s25p6 and 4f145s25p5 odd-parity configurations and the 4f125s25p6(5d + 6s + 6d), 4f135s25p5(5d + 6s + 6d + 7s), 4f145s25p4(5d + 6s + 6d + 7s), 4f145s5p6 and 4f145s5p55f even-parity configurations were included. Furthermore, these calculations revealed a very strong mixing in the eigenvector compositions for many excited even-parity states, neither the LS-coupling nor the jj-coupling appearing to give a good overall description of the energy levels.

3. Configuration Interaction Analysis

As mentioned in the previous section, the only available radiative rates in W VIII were reported by Ryabtsev et al. [13] who used the HFR method with a rather unbalanced physical model since it included only 2 odd-parity configurations (4f135s25p6 and 4f145s25p5) for 13 configurations (4f125s25p6(5d + 6s + 6d), 4f135s25p5(5d + 6s + 6d + 7s), 4f145s25p4(5d + 6s + 6d + 7s), 4f145s5p6 and 4f145s5p55f) in the even parity. In order to estimate the effects of configuration interaction on the radiative parameters of W VIII, different physical models based on the pseudo-relativistic Hartree-Fock method were considered in the present work. In all of them, the electrostatic interaction Slater integrals, Fk, Gk and Rk were scaled down by a factor 0.85, as suggested by Cowan [18] while the spin-orbit parameters were kept to their ab initio values. The first model used, HFR(A), was the same as the one considered in [13] for computing their transition probabilities. The second model, HFR(B), was built to rebalance configuration interaction within both parities by adding to the HFR(A) model the 4f125s25p6(5f + 6p) + 4f135s25p5(5f + 6p) + 4f145s25p4(5f + 6p) + 4f145s5p5(5d + 6s + 6d + 7s) odd-parity configurations, leading to 12 and 13 configurations in each parity, respectively. In the third model, HFR(C), we proceeded in a more systematic way since all the configurations of the type (4f + 5s + 5p)knl (k = 20 or 21, nl = 5d, 5f, 6s, 6p, 6d, 7s) with one or two holes in the 4f, 5s and 5p subshells were included in the multiconfiguration expansions. This gave rise to the 18 odd-parity configurations 4f135s25p6, 4f145s25p5, 4f145s25p4(5f + 6p), 4f145s5p5(5d + 6s + 6d + 7s), 4f135s25p5(5f + 6p), 4f135s5p6(5d + 6s + 6d + 7s), 4f145p6(5f + 6p), 4f125s25p6(5f + 6p) and the 21 even-parity configurations 4f145s5p6, 4f145s25p4(5d + 6s + 6d + 7s), 4f145s5p5(5f + 6p), 4f135s25p5(5d + 6s + 6d + 7s), 4f135s5p6(5f + 6p), 4f145p6(5d + 6s + 6d + 7s), 4f125s25p6(5d + 6s + 6d + 7s). Finally, a fourth model, HFR(D), was tested. In this one, the same set of interacting configurations as the one considered in model HFR(C) was included with the additional permission that one electron could be excited on the 6f or the 7p subshell, giving rise to a total of 26 odd-parity and 25 even-parity configurations. Using the four HFR models presented hereabove, we computed and compared the radiative lifetimes for the 250 lowest even-parity energy levels with τ-values smaller than 100 ns. These comparisons are summarized in Figure 1 showing the ratios τ(B)/τ(A), τ(C)/τ(B) and τ(D)/τ(C), respectively. When looking at Figure 1a, it is clear that the numerous missing odd-parity configurations in the HFR(A) model, similar to the one used by Ryabtsev et al. [13], make without doubt this latter model insufficient to provide a reliable set of transition probabilities, the mean deviation between the HFR(B) and HFR(A) lifetimes being found to be within about 30%, with notable discrepancies reaching a factor of 2–3 in some cases. Moreover, Figure 1b shows that the HFR(B) model does not either include enough configuration interaction to give a reasonable accuracy of the radiative lifetime calculations, the differences between the data computed with models HFR(C) and HFR(B) still reaching a factor of 1.5–2 in many cases. However, as shown in Figure 1c, the excitations of one electron to the 6f or 7p subshell included in model HFR(D) do not really change the results obtained in model HFR(C), the mean deviation between both sets of lifetimes not exceeding 2%. We can therefore conclude that the one single excitation from 4f, 5s and 5p to nl orbitals with nl = 5d, 5f, 6s, 6p, 6d and 7s, as considered in the configuration interaction expansions of model HFR(C), should form a good basis for computing the spectroscopic data in W VIII.
Furthermore, it is also interesting to estimate the influence of the double excitations on the radiative parameters. Indeed, for transitions of the type 4f145s25p5 – 4f145s25p45d and 4f135s25p6 – 4f135s25p55d, the 5p2 → 5d2 double excitation in the lower odd-parity state leads to an allowed transition to the upper even-parity state with an electric dipole matrix element which is equal in magnitude to that for the primary transition. In order to evaluate this effect on the decay rates, we extended the multiconfiguration expansion of model HFR(C) with the two additional odd-parity configurations 4f145s25p35d2 and 4f135s25p45d2, giving rise to model HFR(E). Finally, as similar speculations can be made for the double excitation 4f2 → 5d2 in the case of the 4f145s25p5 – 4f135s25p55d and 4f135s25p6 – 4f125s25p65d transitions and for the double excitation 5p2 → 6s2 in the case of the 4f145s25p5 – 4f145s25p46s and 4f135s25p6 – 4f135s25p56s transitions, the odd-parity configurations 4f125s25p55d2 and 4f115s25p65d2 were added to model HFR(E) to give model HFR(F), while 4f145s25p36s2 and 4f135s25p46s2 were added to model HFR(F) to give model HFR(G). The radiative lifetimes calculated with these latter models are compared to those obtained with HFR(C) in Figure 2. As observed in this figure, if the 5p2 → 6s2, considered in model HFR(G) do not change the final results (see Figure 2c), it is not the same for the 5p2 → 5d2 and 4f2 → 5d2 excitations, respectively included in models HFR(E) and HFR(F), which both lead to non negligible changes in the computed radiative lifetimes by about 20% (see Figure 2a,b). This is nevertheless not very surprising since this kind of effect was already highlighted by Quinet and Hansen [19] who pointed out the influence of 3p2 → 3d2 core excitation on the 3p63dN – 3p53dN+1 transition rates in iron group elements.
Figure 1. Comparison between radiative lifetimes obtained in the present work using different HFR models for short-lived even-parity energy levels (τ < 100 ns) in W VIII. In each panel, the y-axis gives the ratio of τ -values computed with two successive models including only single excitations (see text) while the x-axis corresponds to the level indexes, assigned according to the order of increasing energies.
Figure 1. Comparison between radiative lifetimes obtained in the present work using different HFR models for short-lived even-parity energy levels (τ < 100 ns) in W VIII. In each panel, the y-axis gives the ratio of τ -values computed with two successive models including only single excitations (see text) while the x-axis corresponds to the level indexes, assigned according to the order of increasing energies.
Atoms 03 00299 g001
Figure 2. Comparison between radiative lifetimes obtained in the present work using different HFR models for short-lived even-parity energy levels (τ < 100 ns) in W VIII. In each panel, the y-axis gives the ratio of τ-values computed with two successive models including both single and double excitations (see text) while the x-axis corresponds to the level indexes, assigned according to the order of increasing energies.
Figure 2. Comparison between radiative lifetimes obtained in the present work using different HFR models for short-lived even-parity energy levels (τ < 100 ns) in W VIII. In each panel, the y-axis gives the ratio of τ-values computed with two successive models including both single and double excitations (see text) while the x-axis corresponds to the level indexes, assigned according to the order of increasing energies.
Atoms 03 00299 g002

4. Radiative Parameter Calculations

Further to the detailed discussion presented in the previous section, HFR(F) was chosen as the final model to compute the radiative parameters in W VIII. To summarize, the following configurations were explicitly included in the calculations: 4f135s25p6 + 4f145s25p5 + 4f145s25p4(5f + 6p) + 4f145s5p5(5d + 6s + 6d + 7s) + 4f135s25p5(5f + 6p) + 4f135s5p6(5d + 6s + 6d + 7s) + 4f145p6(5f + 6p) + 4f125s25p6(5f + 6p) + 4f145s25p35d2 + 4f135s25p45d2 + 4f125s25p55d2 + 4f115s25p65d2 (odd parity) and 4f145s5p6 + 4f145s25p4(5d + 6s + 6d + 7s) + 4f145s5p5(5f + 6p) + 4f135s25p5(5d + 6s + 6d + 7s) + 4f135s5p6(5f + 6p) + 4f145p6(5d + 6s + 6d + 7s) + 4f125s25p6(5d + 6s + 6d + 7s) (even parity). This model was then combined with a semi-empirical adjustment of the radial energy parameters in order to minimize the discrepancies between calculated and available experimental energy levels. The strategy followed in the fitting process was exactly the same as the one developed by Ryabtsev et al. [13] to the extent that, starting with the numerical values given in their paper, the same parameters were adjusted with the same constraints as those used by these latter authors. This allowed us simply to adapt and optimize the final radial parameters to our physical model including a much larger number of interacting configurations than the model used in [13].
The numerical values of the radial parameters adopted in our work are reported in Table 1 while the calculated energy levels are compared to the experimental data in Table 2. For the 98 even-parity levels, the standard deviation of the fit was found to be equal to 438 cm−1 which is comparable to the value of 443 cm−1 obtained by Ryabtsev et al. [13]. When looking at the table, it appears that the ordering of experimental and calculated energy levels can be different in a few cases. This is simply due to the fact that some experimental level values are close to each other, with a difference of the same order of magnitude as the standard deviation mentioned above. However, in any case, the ordering of the calculated energies always corresponds to the observed one within a J-matrix. Table 2 also lists the first three LS-components for each level. We can note that, as expected, most of the even-parity states are very strongly mixed and, as already pointed out by Ryabtsev et al., the jj-coupling scheme given by these authors appears a bit more appropriate than the LS one, with average eigenvector purities of 45% and 32%, respectively. It is also worth mentioning that, if LS purities of 100% were reported by Ryabtsev et al., for the four levels belonging to the 4f135s25p6 and 4f145s25p5 odd-parity configurations, it is no more the case in our extended configuration interaction model which gives slightly reduced purities of 97%–98% for those levels. In spite of their very strong mixing, the first LS-component of each level is given in boldface in Table 2.
Table 1. Numerical values (in cm−1) of the radial energy parameters adopted in the Hartree-Fock (HFR) calculations.
Table 1. Numerical values (in cm−1) of the radial energy parameters adopted in the Hartree-Fock (HFR) calculations.
ConfigurationParameterFitUnc.Note aFit/HFR
Odd parity
4f135s25p6Eav29,0530
ζ (4f)49710 0.988
4f145s25p5Eav51,9680
ζ(5p)59,1480 1.003
Even parity
4f145s5p6Eav377,512 f
4f135s25p55dEav403,242249
ζ (4f)502248r110.995
ζ(5p)60,705127r20.984
ζ(5d)4716109r100.988
F2(4f,5p)52,9091606r120.785
F2(4f,5d)35,3452124r40.780
F4(4f,5d)17,1651031r40.780
F2(5p,5d)63,9842700r70.815
G2(4f,5p)28,1611313r31.025
G4(4f,5p)23,4351092r31.025
G1(4f,5d)13,991413r50.876
G3(4f,5d)13,111387r50.876
G5(4f,5d)10,482309r50.876
G1(5p,5d)67,786607r10.711
G3(5p,5d)42,728383r10.711
4f135s25p56dEav775,400 f
4f135s25p56sEav542,642139
ζ(4f)503348r110.995
ζ(5p)61,895130r20.984
F2(4f,5p)53,4291622r120.785
G2(4f,5p)28,3521322r31.025
G4(4f,5p)23,6641103r31.025
G3(4f,6s)68732032 1.294
G1(5p,6s)9385742r130.881
4f135s25p57sEav813,134 f
4f145s25p45dEav420,745400
F2(5p,5p)77,0352775r140.834
ζ(5p)58,775123r20.984
ζ(5d)4465103r100.988
F2(5p,5d)62,8652653r70.815
G1(5p,5d)66,467595r10.711
G3(5p,5d)41,892375r10.711
4f145s25p46dEav778,313 f
4f145s25p46sEav550,299268
F2(5p,5p)77,9322807r140.838
ζ(5p)59,932126r20.984
G1(5p,6s)9397743r130.881
4f145s25p47sEav813,793 f
4f145s5p56pEav992,590 f
4f145s5p55fEav1092,223 f
4f125s25p65dEav429,283138
F2(4f,4f)149,0652553 0.843
F4(4f,4f)111,5006763 0.991
F6(4f,4f)77,0443283 0.947
α22 f
β−1000 f
γ−70 f
ζ(4f)520049r110.995
ζ(5d)4977115r100.988
F2(4f,5d)36,3001467r60.785
F4(4f,5d)17,592711r60.785
G1(4f,5d)13,832408r50.876
G3(4f,5d)13,147388r50.876
G5(4f,5d)10,564312r50.876
4f125s25p66dEav819,431 f
4f125s25p66sEav579,803 f
F2(4f,4f)147,636 f0.833
F4(4f,4f)110,752 f0.982
F6(4f,4f)78,585 f0.963
α22 f
β−1000 f
γ−70 f
ζ(4f)5161 f0.985
G3(4f,6s)3974 f0.758
4f135s25p55d – 4f145s25p45dD2(4f,5p;4f,4f)−566051r80.864
D4(4f,5p;4f,4f)−3093r80.864
D2(5p,5p;4f,5p)−34,864312r80.864
D2(5p,5d;4f,5d)−27,246244r80.864
D4(5p,5d;4f,5d)−17,855160r80.864
E1(5p,5d;4f,5d)−24,107216r80.864
E3(5p,5d;4f,5d)−18,067162r80.864
4f135s25p55d – 4f125s25p65dD2(4f,4f;4f,5p)−37928r90.827
D4(4f,4f;4f,5p)8582r90.827
D2(4f,5p;5p,5p)−32,55668r90.827
D2(4f,5d;5p,5d)−25,60454r90.827
D4(4f,5d;5p,5d)−16,84535r90.827
E1(4f,5d;5p,5d)−22,39347r90.827
E3(4f,5d;5p,5d)−16,98236r90.827
4f145s25p45d – 4f125s25p65dD2(4f,4f;5p,5p)24,59252r90.827
D4(4f,4f;5p,5p)20,29143r90.827
4f135s25p56s – 4f145s25p46sD2(4f,5p,4f,4f)−6391 f1.000
D4(4f,5p;4f,4f)−244 f1.000
D2(5p,5p;4f,5p)−40,487 f1.000
a f: fixed parameter from [13]; r n : parameters linked by their corresponding HFR ratios.
Table 2. Comparison between the energy levels computed in the present work and the experimentally known values available in seven times ionized tungsten (W VIII). Energies are given in cm−1.
Table 2. Comparison between the energy levels computed in the present work and the experimentally known values available in seven times ionized tungsten (W VIII). Energies are given in cm−1.
IndexEexp aEcalc bΔE cJPercentage Composition in LS-Coupling d
Odd Parity
0007/298 4f135p6 2F
1233123303/298 4f145p5 2P
17,41017,41005/298 4f135p6 2F
89,12389,12301/297 4f145p5 2P
Even Parity
1377,119377,140−219/228 4f135p5(3F)5d 2H + 23 4f135p5(1D)5d 2G + 8 4f135p5(3F)5d 2G
2377,288376,9922963/225 4f135p5(1D)5d 2P + 18 4f135p5(3D)5d 4F + 15 4f135p5(3D)5d 4P
3377,867377,7141535/221 4f135p5(3D)5d 4F + 12 4f145p4(1S)5d 2D + 9 4f135p5(3F)5d 2F
4380,899381,127−2289/218 4f135p5(1G)5d 2G + 18 4f135p5(3G)5d 2G + 14 4f135p5(3F)5d 2G
5382,019382,285−2667/222 4f135p5(1D)5d 2F + 15 4f135p5(3D)5d 2G + 13 4f135p5(3F)5d 2F
6383,133383,535−4025/218 4f135p5(3D)5d 2F + 17 4f135p5(1D)5d 2D + 10 4f145p4(1S)5d 2D
7384,400383,6227789/224 4f135p5(1G)5d 2H + 20 4f135p5(3G)5d 4H + 11 4f135p5(3D)5d 4G
8386,704386,4972075/216 4f135p5(3F)5d 2D + 11 4f135p5(3G)5d 4D + 9 4f135p5(3F)5d 4P
9391,541392,282−7419/214 4f135p5(1D)5d 2G + 13 4f125p6(3H)5d 2G + 5 4f135p5(3G)5d 2H
10393,992393,5394537/218 4f125p6(3H)5d 2F + 14 4f135p5(3G)5d 2F
11395,276395,201755/219 4f135p5(3D)5d 2F + 17 4f135p5(3D)5d 4D + 16 4f135p5(3D)5d 4F
12395,471395,624−153 7/214 4f125p6(3F)5d 4F + 11 4f135p5(3F)5d 2G + 8 4f125p6(3F)5d 4D
13395,474395,1243501/238 4f145p4(3P)5d 4P + 17 4f135p5(3D)5d 2S + 8 4f145p4(3P)5d 2P
14396,505396,1473587/220 4f135p5(3D)5d 4F + 14 4f135p5(3F)5d 2G + 8 4f135p5(3D)5d 2F
15396,894396,67122 3/217 4f135p5(3F)5d 2P + 17 4f145p4(3P)5d 4P + 16 4f145p4(3P)5d 4F
16397,612397,709−975/219 4f125p6(1G)5d 2D + 19 4f125p6(3F)5d 4D
17398,707398,2694389/215 4f135p5(1G)5d 2G + 15 4f135p5(3G)5d 4G + 13 4f135p5(1G)5d 2H
18400,203400,110939/219 4f125p6(3F)5d 4G + 14 4f125p6(1G)5d 2H + 13 4f125p6(3F)5d 4F
19401,984402,399−4155/220 4f145p4(3P)5d 4F + 17 4f135p5(3G)5d 2D + 8 4f145p4(1D)5d 2F
20405,907405,7571509/241 4f125p6(3H)5d 4G + 25 4f125p6(3H)5d 4H + 8 4f125p6(3H)5d 4F
21408,086407,9481387/242 4f125p6(3H)5d 2F + 7 4f125p6(3F)5d 2F + 7 4f125p6(1G)5d 2F
22408,833409,199−3663/221 4f125p6(1G)5d 2D + 17 4f125p6(1D)5d 2D + 8 4f125p6(3F)5d 2D
23409,362409,0473155/238 4f125p6(3H)5d 4F + 25 4f125p6(3F)5d 4F + 10 4f125p6(1G)5d 2D
24409,676409,619575/218 4f125p6(3F)5d 4P + 16 4f125p6(3F)5d 4G + 10 4f125p6(3H)5d 2F
25410,258409,8853733/217 4f125p6(3F)5d 2D + 13 4f135p5(3G)5d 4F + 12 4f135p5(1G)5d 2D
26410,654410,2703847/213 4f125p6(3F)5d 4G + 9 4f125p6(3H)5d 2G + 8 4f135p5(3D)5d 2G
27411,819412,499−680 5/218 4f125p6(3H)5d 2F + 10 4f135p5(3G)5d 4G + 10 4f125p6(3F)5d 4G
28411,832411,4993337/223 4f125p6(3F)5d 4D + 21 4f125p6(3F)5d 4H + 14 4f125p6(1G)5d 2G
29413,450413,503−539/228 4f125p6(3H)5d 2G + 13 4f125p6(3F)5d 4H + 12 4f125p6(3H)5d 4I
30414,888414,85335 7/239 4f125p6(3F)5d 4G + 11 4f125p6(3F)5d 2F + 10 4f125p6(3F)5d 4H
31415,852415,874−225/227 4f125p6(3F)5d 4G + 16 4f125p6(1G)5d 2F + 13 4f125p6(3F)5d 4P
32416,481416,737−2569/229 4f125p6(3F)5d 2G + 18 4f125p6(3H)5d 4I + 10 4f125p6(1G)5d 2G
33417,394417,333613/242 4f125p6(3F)5d 2P + 16 4f125p6(1G)5d 2D
34418,403418,2741299/228 4f125p6(3F)5d 4H + 28 4f125p6(3H)5d 2G + 18 4f125p6(3H)5d 4G
35419,585419,953−3687/235 4f125p6(3H)5d 4G + 11 4f125p6(3F)5d 4F + 10 4f125p6(3H)5d 4F
36419,811419,829−185/225 4f125p6(3H)5d 4F + 14 4f125p6(3H)5d 2F
37424,781424,77655/226 4f125p6(3H)5d 4G + 19 4f125p6(3F)5d 4P + 15 4f125p6(1G)5d 2D
38425,843425,913−703/238 4f125p6(3H)5d 4F + 27 4f125p6(3F)5d 2P + 11 4f125p6(3F)5d 4P
39428,199427,9002993/242 4f125p6(3F)5d 4P + 11 4f125p6(3H)5d 4F + 6 4f125p6(3F)5d 2P
40428,216428,371−1557/230 4f125p6(3F)5d 2G + 10 4f125p6(3H)5d 2F + 7 4f125p6(3H)5d 2G
41428,777428,727507/226 4f125p6(3F)5d 2F + 17 4f125p6(3F)5d 4F + 11 4f125p6(3H)5d 4H
42430,708430,640687/220 4f145p4(3P)5d 4F + 16 4f125p6(1G)5d 2F + 10 4f125p6(3H)5d 4H
43432,963433,564−6015/213 4f125p6(3P)5d 4D + 10 4f145p4(3P)5d 4F + 9 4f145p4(1D)5d 2F
44435,561435,879−318 5/215 4f125p6(1D)5d 2F + 14 4f125p6(1G)5d 2D + 8 4f145p4(1D)5d 2F
45435,658435,5161427/214 4f125p6(3H)5d 2G + 12 4f125p6(1G)5d 2G
46437,149437,658−509 5/214 4f125p6(3H)5d 2F + 14 4f125p6(3H)5d 4G + 11 4f125p6(1G)5d 2D
47439,561439,3182437/215 4f125p6(3H)5d 2G + 13 4f125p6(3P)5d 2F + 12 4f125p6(1G)5d 2F
48445,286445,209775/215 4f125p6(3P)5d 2D + 13 4f125p6(1D)5d 2D + 5 4f145p4(1D)5d 2D
49447,909447,860495/219 4f125p6(1G)5d 2F + 14 4f125p6(3P)5d 4D + 13 4f125p6(3F)5d 2F
50452,821452,5113103/224 4f125p6(3P)5d 4P + 11 4f135p5(1G)5d 2D + 10 4f125p6(3P)5d 2P
51454,067454,016517/266 4f125p6(1I)5d 2G + 7 4f135p5(3G)5d 4H + 5 4f125p6(3H)5d 2G
52457,652457,2454075/217 4f135p5(1G)5d 2F + 14 4f135p5(3G)5d 2F + 9 4f125p6(1D)5d 2D
53457,815457,776399/264 4f125p6(1I)5d 2H + 7 4f135p5(1G)5d 2G + 6 4f135p5(3D)5d 2G
54458,380458,337433/216 4f125p6(3P)5d 2P + 12 4f125p6(3P)5d 4P
55459,570460,304−7345/222 4f125p6(3P)5d 2D + 6 4f125p6(3P)5d 2F
56462,927463,121−1943/224 4f125p6(3P)5d 2D + 14 4f135p5(1D)5d 2D + 10 4f135p5(3F)5d 4F
57466,219466,223−45/233 4f125p6(3P)5d 2F + 10 4f125p6(1D)5d 2D + 7 4f125p6(3P)5d 4F
58468,034467,6204145/223 4f125p6(3P)5d 2F + 9 4f135p5(1D)5d 2F + 6 4f135p5(3F)5d 4G
59468,523468,3122115/252 4f125p6(3P)5d 4P + 5 4f125p6(1D)5d 2D + 5 4f135p5(3G)5d 4F
60475,117475,279−1623/215 4f145p4(1D)5d 2P + 9 4f145p4(3P)5d 4F
61481,035480,4415945/226 4f145p4(3P)5d 2D + 16 4f145p4(1D)5d 2D + 9 4f135p5(1D)5d 2D
62481,473481,921−4485/265 4f145p4(3P)6s 4P + 23 4f145p4(1D)6s 2D
63483,243483,385−1423/218 4f135p5(1D)5d 2P + 16 4f145p4(1D)5d 2P
64485,175486,300−11255/222 4f135p5(1D)5d 2D + 11 4f135p5(1D)5d 2F + 7 4f135p5(3F)5d 4F
65487,901487,0648373/250 4f145p4(3P)6s 2P + 27 4f145p4(1D)6s 2D + 13 4f145p4(3P)6s 4P
66492,337493,284−947 1/232 4f145p4(1D)5d 2P + 21 4f145p4(1D)5d 2S + 16 4f145p4(3P)5d 2P
67495,690495,0416499/227 4f135p5(3D)5d 2G + 18 4f135p5(3F)5d 2G + 10 4f135p5(3G)5d 2G
68498,037499,096−10595/229 4f135p5(3G)5d 2D + 14 4f135p5(1G)5d 2D + 9 4f125p6(3P)5d 2D
69498,541498,826−2853/226 4f145p4(1D)5d 2D + 13 4f125p6(1S)5d 2D + 10 4f145p4(3P)5d 2D
70498,792499,045−2537/218 4f135p5(3G)5d 2F + 15 4f135p5(3F)5d 2F + 13 4f135p5(1G)5d 2F
71500,313500,0133005/256 4f135p5(1F)6s 2F + 19 4f135p5(3D)6s 2D + 11 4f135p5(3D)6s 4D
72503,071502,6054667/247 4f135p5(3F)6s 2F + 16 4f135p5(1F)6s 2F + 10 4f135p5(3F)6s 4F
73504,615504,586299/271 4f135p5(3G)6s 2G + 24 4f135p5(3G)6s 4G
74504,691505,845−11543/232 4f125p6(1S)5d 2D + 14 4f135p5(3G)5d 2D + 7 4f135p5(3F)5d 2D
75512,790512,6281627/217 4f135p5(1D)5d 2G + 15 4f135p5(3F)5d 2G + 14 4f135p5(3G)5d 2G
76514,063514,203−1405/249 4f135p5(1D)6s 2D + 26 4f135p5(3D)6s 4D + 10 4f135p5(3D)6s 2D
77514,413514,677−2647/237 4f135p5(3F)6s 4F + 22 4f135p5(3F)6s 2F + 15 4f135p5(1F)6s 2F
78514,628515,153−5255/250 4f135p5(3F)6s 4F + 16 4f135p5(3G)6s 4G + 14 4f135p5(1F)6s 2F
79516,493515,7257685/224 4f135p5(3G)5d 2F + 14 4f135p5(3F)5d 2F + 11 4f135p5(1D)5d 2F
80521,876522,299−4235/235 4f125p6(1S)5d 2D + 26 4f135p5(3F)6s 2F + 10 4f135p5(3D)6s 4D
81522,610521,48711235/236 4f125p6(1S)5d 2D + 25 4f135p5(3F)6s 2F + 10 4f135p5(3D)6s 4D
82522,881522,986−1053/241 4f125p6(1S)5d 2D + 26 4f135p5(3G)5d 2D + 7 4f125p6(3P)5d 2D
83527,376527,853−4777/246 4f135p5(1G)6s 2G + 29 4f135p5(3G)6s 4G + 23 4f135p5(3G)6s 2G
84528,462528,388741/228 4f145p4(1D)5d 2S + 17 4f145p4(3P)5d 2P + 8 4f135p5(3D)5d 2S
85528,652528,582703/265 4f135p5(3D)6s 2D + 30 4f135p5(3D)6s 4D
86567,191565,98512063/228 4f145p4(3P)5d 2D + 28 4f145p4(1S)5d 2D + 10 4f145p4(3P)5d 2P
87568,644567,8727723/277 4f145p4(3P)6s 4P + 13 4f145p4(3P)6s 2P
88572,004572,684−6801/266 4f145p4(3P)6s 2P + 32 4f145p4(3P)6s 4P
89581,635582,142−5075/260 4f145p4(1D)6s 2D + 18 4f145p4(3P)6s 4P + 7 4f125p6(3F)6s 2F
90583,560583,744−1843/260 4f145p4(1D)6s 2D + 23 4f145p4(3P)6s 2P + 8 4f125p6(3F)6s 4F
91594,941594,876657/256 4f135p5(3D)6s 4D + 23 4f135p5(3F)6s 4F + 11 4f135p5(1F)6s 2F
92597,436597,803−3675/249 4f135p5(3D)6s 2D + 14 4f135p5(3F)6s 4F + 13 4f135p5(3D)6s 4D
93598,904598,949−457/239 4f135p5(1G)6s 2G + 21 4f135p5(3F)6s 2F + 20 4f135p5(3G)6s 4G
94599,423599,2741499/244 4f135p5(1G)6s 2G + 25 4f135p5(3G)6s 4G + 19 4f135p5(3F)6s 4F
95609,206609,1041025/273 4f135p5(3G)6s 4G + 18 4f135p5(1F)6s 2F
96611,283611,555−2727/250 4f135p5(3G)6s 2G + 27 4f135p5(3G)6s 4G + 16 4f135p5(1F)6s 2F
97621,343620,9314125/239 4f135p5(1D)6s 2D + 19 4f135p5(3F)6s 2F + 14 4f135p5(3D)6s 4D
98622,123621,6075163/240 4f135p5(1D)6s 2D + 28 4f135p5(3F)6s 4F + 9 4f135p5(3D)6s 4D
a From [13]. b This work : model HFR(F). c ΔE = Eexp − Ecalc. d Only the first three LS components larger then 5% are given.
The calculated oscillator strengths (log gf) and transition probabilities (gA) obtained in the present work are given in Table 3 for all the W VIII spectral lines with log gf-values greater than −4. These parameters are only given in the length form, the HFR code of Cowan not allowing the calculation of radiative decay rates in the velocity form. Observed wavelengths (ëexp) taken from the work of Ryabtsev et al. [13] are also given in the same table together with the 'Ritz' wavelengths (λRitz) deduced from the experimental energy levels identified by the same authors. When looking in detail at the table, one can see that forty listed lines were not observed in [13]. If most of those lines are characterized by rather weak transition probabilities, three of them appear to be strong enough, i.e., with gA-values greater than 1010 s−1, to be experimentally observed, according to our calculations. These are located at λ = 191.700 Å (gA = 1.46 × 1010 s−1), λ = 229.402 Å (gA = 1.16 × 1010 s−1) and λ = 244.249 Å (gA = 1.33 × 1011 s−1) and correspond respectively to transitions from the lower odd level at 1233 cm−1 (J = 3/2), to the upper even level at 522,881 cm−1 (J = 3/2), from the lower odd level at 1233 cm−1 (J = 3/2) to the upper even level at 437,149 cm−1 (J = 5/2), and from the lower odd level at 89,123 cm−1 (J = 1/2) to the upper even level at 498,541 cm−1 (J = 3/2).
Table 3. Oscillator strengths and transition probabilities in W VIII (log gf > −4). A table entry 6.81E + 08 means 6.81×108.
Table 3. Oscillator strengths and transition probabilities in W VIII (log gf > −4). A table entry 6.81E + 08 means 6.81×108.
Indexes aWavelength bLower Odd Level cUpper Even Level c
λexp (Å)λRitz (Å)E (cm−1)JE (cm−1)Jlog gf dgA (s-1) dCF e
1° − 97160.940160.94203.5621,3432.5−2.586.81E + 080.003
2° − 98161.057161.05912331.5622,1231.5−2.411.01E + 090.045
2° − 97161.260161.26212331.5621,3432.5−1.261.40E + 100.333
1° − 96163.596163.59003.5611,2833.5−2.301.24E + 090.007
1° − 95164.143164.14803.5609,2062.5−2.784.15E + 080.014
2° − 95164.479164.48112331.5609,2062.5−3.291.27E + 080.226
3° − 98165.369165.36817,4102.5622,1231.5−0.261.34E + 110.854
3° − 97165.583165.58117,4102.5621,3432.5−0.429.25E + 100.609
1° − 94166.827166.82703.5599,4234.50.153.39E + 110.872
1° − 93166.971166.97203.5598,9043.5−0.221.44E + 110.558
1° − 92167.382167.38203.5597,4362.5−0.042.16E + 110.721
1° − 91168.084168.08403.5594,9413.5−0.566.57E + 100.820
3° − 96168.381168.38617,4102.5611,2833.50.052.62E + 110.730
3° − 95168.980168.97717,4102.5609,2062.5−0.596.03E + 100.778
2° − 90171.727171.72512331.5583,5601.5−0.803.56E + 100.131
1° − 89 171.92903.5581,6352.5−2.763.98E + 080.038
3° − 93171.973171.97117,4102.5598,9043.5−1.704.48E + 090.013
2° − 89172.295172.29412331.5581,6352.5−0.191.46E + 110.641
3° − 91 173.15117,4102.5594,9413.5−2.674.74E + 080.026
2° − 88175.199175.20212331.5572,0040.5−0.585.76E + 100.632
2° − 87176.237176.23912331.5568,6441.5−0.803.38E + 100.039
3° − 90176.630176.63217,4102.5583,5601.5−1.271.15E + 100.701
2° − 86176.694176.69212331.5567,1911.5−0.753.77E + 100.030
3° − 89177.232177.23417,4102.5581,6352.5−1.369.27E + 090.591
3° − 87181.410181.41117,4102.5568,6441.5−1.071.74E + 100.134
3° − 86181.888181.89117,4102.5567,1911.5−0.674.28E + 100.176
4° − 98187.608187.61789,1230.5622,1231.5−1.505.93E + 090.256
2° − 85 189.60312331.5528,6521.5−2.446.70E + 080.083
1° − 83189.616189.61803.5527,3763.5−1.171.25E + 100.452
2° − 84189.667189.67112331.5528,4620.5−0.594.81E + 100.030
1° − 81191.348191.34703.5522,6102.5−0.673.89E + 100.082
1° − 80191.617191.61603.5521,8762.5−0.466.38E + 100.102
2° − 82 191.70012331.5522,8811.5−1.101.46E + 100.069
2° − 81 191.80012331.5522,6102.5−3.001.78E + 080.001
2° − 80192.070192.07012331.5521,8762.5−2.251.02E + 090.007
1° − 79193.614193.61303.5516,4932.5−0.961.95E + 100.029
2° − 79194.077194.07712331.5516,4932.5−1.377.51E + 090.095
1° − 78194.315194.31503.5514,6282.5−2.942.05E + 080.011
1° − 77194.397194.39603.5514,4133.5−0.594.58E + 100.047
1° − 76194.527194.52903.5514,0632.50.092.19E + 110.840
2° − 76194.998194.99612331.5514,0632.5−1.466.02E + 090.403
1° − 75195.021195.01203.5512,7903.5−1.644.01E + 090.003
3° − 85195.598195.60217,4102.5528,6521.50.102.19E + 110.790
3° − 83196.093196.09217,4102.5527,3763.50.202.76E + 110.866
3° − 82197.835197.83517,4102.5522,8811.50.637.37E + 110.514
3° − 81197.941197.94117,4102.5522,6102.5−1.912.07E + 090.005
1° − 73198.171198.17103.5504,6154.50.464.87E + 110.850
3° − 80198.229198.22917,4102.5521,8762.50.495.31E + 110.659
1° − 72198.779198.77903.5503,0713.50.677.94E + 110.742
1° − 71199.875199.87503.5500,3132.5−1.021.58E + 100.098
3° − 79200.367200.36717,4102.5516,4932.51.041.82E + 120.555
2° − 71 200.36912331.5500,3132.5−2.318.23E + 080.183
1° − 70200.483200.48403.5498,7923.51.142.31E + 120.467
1° − 68200.787200.78803.5498,0372.51.051.90E + 120.572
2° − 69201.079201.08312331.5498,5411.50.092.05E + 110.358
3° − 78201.119201.11917,4102.5514,6282.5−0.852.33E + 100.219
3° − 77201.205201.20617,4102.5514,4133.50.831.13E + 120.562
2° − 68201.288201.28712331.5498,0372.50.122.17E + 110.196
3° − 76 201.34817,4102.5514,0632.5−2.011.61E + 090.026
1° − 67201.739201.73903.5495,6904.51.293.22E + 120.577
3° − 75201.864201.86517,4102.5512,7903.50.981.56E + 120.446
4° − 90202.250202.25089,1230.5583,5601.5−0.071.39E + 110.835
2° − 66203.623203.62312331.5492,3370.50.626.72E + 110.563
3° − 74205.221205.22017,4102.5504,6911.50.535.37E + 110.474
2° − 65205.479205.47912331.5487,9011.50.323.29E + 110.754
3° − 72 205.90517,4102.5503,0713.5−3.564.31E + 070.000
1° − 64 206.11103.5485,1752.5−1.209.98E + 090.028
2° − 64206.634206.63612331.5485,1752.50.293.08E + 110.518
3° − 71 207.08117,4102.5500,3132.5−1.643.56E + 090.020
4° − 88207.092207.09089,1230.5572,0040.5−0.317.59E + 100.784
2° − 63207.466207.46512331.5483,2431.50.272.86E + 110.266
1° − 62207.690207.69603.5481,4732.5−0.693.20E + 100.364
3° − 70207.736207.73517,4102.5498,7923.5−0.802.49E + 100.027
3° − 69207.850207.84417,4102.5498,5411.5−1.239.15E + 090.025
1° − 61207.884207.88503.5481,0352.5−0.376.55E + 100.216
3° − 68 208.06217,4102.5498,0372.5−3.792.56E + 070.000
2° − 62208.227208.22912331.5481,4732.5−0.219.49E + 100.160
2° − 61208.420208.41912331.5481,0352.50.831.04E + 120.638
4° − 87208.543208.54189,1230.5568,6441.50.424.04E + 110.534
4° − 86209.175209.17589,1230.5567,1911.50.626.31E + 110.502
2° − 60211.027211.02212331.5475,1171.50.282.87E + 110.369
1° − 59213.436213.43703.5468,5232.5−2.911.81E + 080.001
1° − 58213.661213.66003.5468,0342.5−3.041.33E + 080.000
3° − 64213.785213.78317,4102.5485,1752.5−1.031.36E + 100.024
2° − 59214.001214.00012331.5468,5232.5−2.504.61E + 090.013
2° − 58214.229214.22412331.5468,0342.5−0.091.17E + 110.428
1° − 57214.488214.49103.5466,2192.5−2.317.13E + 080.002
2° − 57215.055215.06012331.5466,2192.5−0.218.80E + 100.439
3° − 62215.496215.48817,4102.5481,4732.5−2.228.60E + 080.023
3° − 61215.692215.69217,4102.5481,0352.5−1.752.57E + 090.012
2° − 56216.596216.59412331.5462,9271.5−0.712.79E + 100.187
1° − 55217.601217.59503.5459,5702.5−2.395.78E + 080.001
2° − 55218.174218.18012331.5459,5702.5−2.257.95E + 080.004
1° − 53218.429218.42903.5457,8154.5−0.931.66E + 100.028
3° − 60218.477218.48017,4102.5475,1171.5−2.257.91E + 080.003
1° − 52218.507218.50703.5457,6522.5−1.248.04E + 090.011
2° − 54218.747218.74812331.5458,3801.5−0.881.85E + 100.063
2° − 52219.097219.09712331.5457,6522.5−2.891.76E + 080.002
1° − 51220.239220.23203.5454,0673.5−3.131.02E + 080.000
2° − 50221.443221.44112331.5452,8211.5−0.494.43E + 100.075
3° − 59 221.67417,4102.5468,5232.5−2.131.01E + 090.005
3° − 58221.908221.91517,4102.5468,0342.5−0.802.14E + 100.030
3° − 57222.818222.81217,4102.5466,2192.5−1.159.46E + 090.018
1° − 49223.260223.26003.5447,9092.5−1.336.17E + 090.028
2° − 49 223.87612331.5447,9092.5−1.593.39E + 090.048
3° − 56 224.45817,4102.5462,9271.5−2.633.12E + 080.002
1° − 48224.573224.57503.5445,2862.5−2.385.55E + 080.002
2° − 48225.203225.19812331.5445,2862.5−1.613.22E + 090.026
3° − 55 226.16217,4102.5459,5702.5−3.041.20E + 080.000
3° − 54 226.77317,4102.5458,3801.5−2.011.27E + 090.003
3° − 52 227.14817,4102.5457,6522.5−2.642.96E + 080.000
1° − 47227.497227.50003.5439,5613.5−0.891.64E + 100.091
4° − 85227.519227.51689,1230.5528,6521.5−2.267.04E + 080.057
4° − 84227.617227.61589,1230.5528,4620.50.574.79E + 110.552
3° − 51229.011229.01317,4102.5454,0673.5−0.901.58E + 100.023
2° − 46 229.40212331.5437,1492.5−1.041.16E + 100.072
1° − 45229.541229.53803.5435,6583.5−2.021.21E + 090.007
1° − 44229.590229.58903.5435,5612.5−1.071.07E + 100.047
3° − 50229.666229.66817,4102.5452,8211.5−1.533.67E + 090.007
2° − 44230.246230.24112331.5435,5612.5−1.011.23E + 100.070
4° − 82230.544230.54389,1230.5522,8811.5−0.603.17E + 100.107
1° − 43230.964230.96703.5432,9632.5−1.444.60E + 090.019
2° − 43231.629231.62612331.5432,9632.5−0.742.28E + 100.064
1° − 42232.176232.17603.5430,7083.5−1.188.25E + 090.049
3° − 49232.288232.28917,4102.5447,9092.5−0.841.80E + 100.084
1° − 41233.225233.22103.5428,7773.5−0.811.89E + 100.073
1° − 40 233.525233.52703.5428,2163.5−0.493.93E + 100.161
3° − 48233.709233.71317,4102.5445,2862.5−0.642.77E + 100.122
2° − 39 234.21112331.5428,1991.5−2.991.25E + 080.001
1° − 37235.418235.41503.5424,7812.5−1.237.03E + 090.055
2° − 38235.509235.51012331.5425,8431.5−2.071.02E + 090.126
2° − 37 236.10112331.5424,7812.5−2.881.59E + 080.007
3° − 47236.884236.88217,4102.5439,5613.5−1.011.16E + 100.040
1° − 36 238.20203.5419,8112.5−3.325.62E + 070.000
3° − 46238.243238.24317,4102.5437,1492.5−0.951.32E + 100.063
1° − 35238.330238.33103.5419,5853.5−2.286.19E + 080.010
2° − 36 238.90412331.5419,8112.5−3.138.56E + 070.011
1° − 34239.004239.00403.5418,4034.5−1.031.08E + 100.057
3° − 45239.089239.09317,4102.5435,6583.5−0.593.01E + 100.156
3° − 44239.142239.14817,4102.5435,5612.5−1.217.14E + 090.035
1° − 32240.107240.10703.5416,4814.5−0.473.94E + 100.240
2° − 33 240.29212331.5417,3941.5−3.246.63E + 070.004
1° − 31240.468240.47003.5415,8522.5−2.404.62E + 080.007
4° − 74240.634240.63589,1230.5504,6911.5−0.355.14E + 100.150
1° − 30241.037241.02903.5414,8883.5−1.217.09E + 090.024
2° − 31241.183241.18512331.5415,8522.5−2.364.97E + 080.065
1° − 29241.867241.86703.5413,4504.5−0.276.16E + 100.193
3° − 42 241.95617,4102.5430,7083.5−2.167.91E + 080.007
1° − 28242.819242.81703.5411,8323.5−1.364.97E + 090.071
1° − 27242.829242.82503.5411,8192.5−2.021.07E + 090.006
3° − 41243.088243.09217,4102.5428,7773.5−2.118.72E + 080.007
3° − 40243.426243.42417,4102.5428,2163.5−1.543.23E + 090.012
3° − 39243.434243.43417,4102.5428,1991.5−1.513.45E + 090.121
1° − 26243.518243.51403.5410,6543.5−2.011.11E + 090.004
2° − 27243.551243.55412331.5411,8192.5−1.961.23E + 090.073
1° − 24 244.09503.5409,6762.5−3.961.24E + 070.000
4° − 69 244.24989,1230.5498,5411.50.081.33E + 110.183
1° − 23244.281244.28303.5409,3622.5−1.305.53E + 090.088
2° − 25 244.48412331.5410,2581.5−2.543.21E + 080.005
2° − 24244.833244.83212331.5409,6762.5−1.364.86E + 090.473
3° − 38244.839244.83817,4102.5425,8431.5−1.197.21E + 090.142
1° − 21245.046245.04603.5408,0863.5−0.811.71E + 100.041
2° − 22245.334245.33912331.5408,8331.5−2.001.11E + 090.079
3° − 37245.474245.47617,4102.5424,7812.5−1.266.09E + 090.060
1° − 20246.362246.36203.5405,9074.5−0.732.06E + 100.359
4° − 66248.007248.00789,1230.5492,3370.5−1.011.07E + 100.016
3° − 36248.508248.50817,4102.5419,8112.5−0.712.11E + 100.080
3° − 35248.649248.64817,4102.5419,5853.5−0.612.65E + 100.130
1° − 19248.765248.76603.5401,9842.5−0.374.61E + 100.069
2° − 19249.533249.53212331.5401,9842.50.031.16E + 110.551
1° − 18249.873249.87303.5400,2034.5−0.652.39E + 100.152
3° − 33250.010250.01017,4102.5417,3941.5−1.137.92E + 090.141
4° − 65250.76689,1230.5487,9011.5−2.702.11E + 080.002
1° − 17250.811250.81103.5398,7074.5−1.069.19E + 090.031
3° − 31250.978250.97817,4102.5415,8522.5−1.011.03E + 100.057
1° − 16251.500251.50103.5397,6122.5−1.255.87E + 090.041
3° − 30251.584251.58617,4102.5414,8883.5−0.433.94E + 100.150
1° − 14252.203252.20403.5396,5053.5−0.344.82E + 100.116
2° − 16252.285252.28412331.5397,6122.5−1.059.30E + 090.413
2° − 15252.740252.74212331.5396,8941.50.081.27E + 110.330
1° − 12252.862252.86303.5395,4713.5−1.543.04E + 090.001
1° − 11252.989252.98803.5395,2762.5−1.503.31E + 090.027
3° − 28253.534253.53617,4102.5411,8323.5−0.921.23E + 100.110
3° − 27253.541253.54417,4102.5411,8192.5−0.049.56E + 100.131
2° − 13253.653253.65212331.5395,4740.5−0.443.73E + 100.140
4° − 63253.726253.73089,1230.5483,2431.5−1.493.35E + 090.013
2° − 11253.779253.77912331.5395,2762.5−2.404.08E + 080.024
1° − 10253.812253.81203.5393,9923.50.091.26E + 110.101
3° − 26254.294254.29517,4102.5410,6543.50.011.04E + 110.143
3° − 25254.551254.55117,4102.5410,2581.5−0.029.92E + 100.190
3° − 24254.928254.92917,4102.5409,6762.5−0.921.23E + 100.039
3° − 23255.140255.13317,4102.5409,3622.5−2.039.57E + 080.013
1° − 9255.401255.40103.5391,5414.50.322.16E + 110.165
3° − 22255.479255.47817,4102.5408,8331.5−0.991.05E + 100.076
3° − 21255.967255.96717,4102.5408,0863.5−0.404.09E + 100.115
1° − 8258.592258.59603.5386,7042.50.051.10E + 110.210
4° − 60259.069259.07189,1230.5475,1171.5−0.861.39E + 100.048
2° − 8259.419259.42312331.5386,7042.5−0.245.66E + 100.293
3° − 19260.027260.02817,4102.5401,9842.5−1.632.29E + 090.015
1° − 7260.146260.14603.5384,4004.5−1.542.85E + 090.009
1° − 6261.002261.00603.5383,1332.5−1.453.49E + 090.029
1° − 5261.767261.76703.5382,0193.5−0.931.14E + 100.027
2° − 6261.849261.84912331.5383,1332.5−2.186.41E + 080.004
1° − 4262.537262.53703.5380,8994.5−1.285.12E + 090.005
3° − 16 263.01817,4102.5397,6122.5−2.225.79E + 080.004
3° − 15263.521263.51617,4102.5396,8941.5−2.891.24E + 080.001
3° − 14263.787263.78617,4102.5396,5053.5−1.413.74E + 090.014
3° − 12264.508264.50817,4102.5395,4713.5−1.048.63E + 090.014
1° − 3264.644264.64303.5377,8672.5−1.078.00E + 090.032
3° − 11264.644264.64417,4102.5395,2762.5−1.225.67E + 090.016
1° − 1265.168265.16803.5377,1194.5−1.028.91E + 090.013
2° − 3265.510265.51012331.5377,8672.5−1.265.18E + 090.031
2° − 2265.919265.91912331.5377,2881.5−1.473.16E + 090.055
4° − 56267.518267.52089,1230.5462,9271.5−2.821.42E + 080.002
3° − 8270.794270.78717,4102.5386,7042.5−2.532.66E + 080.001
4° − 54270.816270.81489,1230.5458,3801.5−1.344.11E + 090.019
3° − 5 274.26617,4102.5382,0193.5−2.841.28E + 080.000
4° − 50 274.95389,1230.5452,8211.5−1.077.47E + 090.022
3° − 3 277.42617,4102.5377,8672.5−1.921.03E + 090.004
4° − 39 294.91989,1230.5428,1991.5−3.086.41E + 070.003
4° − 38 296.98389,1230.5425,8431.5−2.641.75E + 080.029
4° − 33 304.62689,1230.5417,3941.5−2.928.72E + 070.008
4° − 25 311.39689,1230.5410,2581.5−2.234.06E + 080.008
4° − 22 312.78389,1230.5408,8331.5−3.214.21E + 070.004
4° − 15 324.91789,1230.5396,8941.5−2.781.04E + 080.000
4° − 13 326.42389,1230.5395,4740.5−2.442.28E + 080.002
4° − 2 347.02389,1230.5377,2881.5−3.143.94E + 070.002
a Indexes of levels as given in Table 2. b λexp are taken from [13] while λRitz are deduced from experimental energy levels given by the same authors. c From [13]. d This work : model HFR(F). e Cancellation factor as defined in Equation (6).
On the other hand, one spectral line observed at 198.625 Å by Ryabtsev et al. [13] is not present in Table 3. This can be explained by the fact that, for this transition, our calculated oscillator strength is unexpectedly found to be much smaller than the cut-off chosen for drawing up the table (log gf > −4). The reason could be found in the strong cancellation effects affecting the calculation of the line strength corresponding to this transition for which our HFR values are log gf = −4.44 and gA = 6.16E + 06 s−1. As a reminder, in order to calculate gA or gf for a transition between the atomic states γJ and γ′J′, we have to compute the value of the line strength
S = | γ J P ( 1 ) γ J | 2
or that of its square root
S 1 / 2 = γ J P ( 1 ) γ J
where P(1) is the electric dipole operator. Because of intermediate coupling and configuration interaction mixing, the wavefunctions are expanded in terms of basis functions:
| γ J = β y β J γ | β J
| γ J = β y β J γ | β J
We may then write Equation (2) in the form
S 1 / 2 = β β y β J γ β J P ( 1 ) β J y β J γ
This sum thus represents a mixing of amplitudes rather than line strengths themselves with the consequence that the effect of mixing is not necessarily a tendency to average out the various line strengths. There are frequently destructive interference effects that cause a weak line to become still weaker. In this context, the cancellation factor is given by
C F = [ | β β y β J γ β J P ( 1 ) β J y β J γ | β β | y β J γ β J P ( 1 ) β J y β J γ | ] 2
According to Cowan [18], very small values of this factor (typically when CF is smaller than about 0.02) indicate that the corresponding transition rates may be expected to show large percentage errors. In Table 3, CF-factors are given for each line in order to give an idea of the reliability of the corresponding transition rates. It is clear that many lines with computed gA-values smaller than 109 s−1 are affected by very small values of CF indicating that the corresponding transition rates must be taken with caution. On the contrary, most of the strongest transitions listed in Table 3, in particular those with gA > 1010 s−1, do not appear to be affected by cancellation effects.
Finally, in Figure 3, we compare our transition probabilities with those reported by Ryabtsev et al. [13]. As expected, a rather large scatter is observed between both sets of results. However, as already discussed in Section 3, in view of the much more extended configuration interaction model adopted here in comparison with the rather limited physical model used by Ryabtsev et al., in particular in the odd parity, the decay rates obtained in the present work should indisputably be more accurate.
Figure 3. Comparison between the transition probabilities (gA in s−1) computed in the present work and those obtained by Ryabtsev et al. [13] for experimentally identified spectral lines in W VIII.
Figure 3. Comparison between the transition probabilities (gA in s−1) computed in the present work and those obtained by Ryabtsev et al. [13] for experimentally identified spectral lines in W VIII.
Atoms 03 00299 g003
For plasma diagnostic purposes, it is sometimes useful to know the decay rates corresponding to forbidden lines. In the present work, such parameters were thus also computed for magnetic dipole (M1) and electric quadrupole (E2) transitions involving the four experimentally known levels within the odd parity of W VIII. It was found that only three transitions could be considered as having rather strong transition probabilities. These lines are the following: (1) λvac = 1137.79 Å, Elow = 1233 cm−1 (J = 3/2) − Eup = 89,123 cm−1 (J=1/2), gA(M1 + E2) = 2.55 × 104 s−1, (2) λvac = 1394.45 Å, Elow = 17,410 cm−1 (J = 5/2) − Eup = 89,123 cm−1 (J=1/2), gA(E2) = 6.54 × 101 s−1, and (3) λair = 5742.26 Å, Elow = 0 cm−1 (J = 7/2) − Eup = 17,410 cm−1 (J=5/2), gA(M1) = 4.90 × 102 s−1.

5. Conclusions

A detailed analysis of configuration interaction effects in seven times ionized tungsten allowed us to give a new reliable set of transition probabilities and oscillator strengths for lines of this ion in the spectral range from 160 to 347 Å. The final results were obtained within the framework of an extended physical model based on the pseudo-relativistic Hartree-Fock approach combined with a semi-empirical optimization of the radial energy parameters. Just like our previous studies related to the lowest ionization stages of W, it is expected that the radiative data reported in the present work for the W VIII spectrum will be useful for plasma diagnostics in future fusion reactors where tungsten will be used as a plasma facing material.

Acknowledgments

Pascal Quinet is Research Director of the Belgian National Fund for Scientific Research F.R.S.-FNRS. Financial support from this organization is acknowledged.

Author Contributions

Both authors were equally involved in the calculations reported in the present paper as well as in the the writing of the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Federici, G.; Skinner, C.H.; Brooks, J.N.; Coad, J.P.; Grisolia, C.; Haasz, A.A.; Hassanein, A.; Philipps, V.; Pitcher, C.S.; Roth, J.; et al. Plasma-material interactions in current tokamaks and their implications for next step fusion reactors. Nucl. Fusion 2001, 41, 1967–2138. [Google Scholar] [CrossRef]
  2. Neu, R.; Dux, R.; Kallenbach, A.; Pütterich, T.; Balden, M.; Fuchs, J.C.; Herrmann, A.; Maggi, C.F.; O’Mullane, M.; Pugno, R.; et al. Tungsten: An option for divertor and main chamber plasma facing components in future fusion devices. Nucl. Fusion 2005, 45, 209–218. [Google Scholar] [CrossRef]
  3. Pospieszczyk, A. Nuclear Fusion Research; Springer: Berlin, Germany, 2006. [Google Scholar]
  4. Skinner, C.H. Applications of EBIT to magnetic fusion diagnostics. Can. J. Phys. 2008, 86, 285–290. [Google Scholar] [CrossRef]
  5. Skinner, C.H. Atomic physics in the quest for fusion energy and ITER. Phys. Scr. 2009, T134, 014022. [Google Scholar] [CrossRef]
  6. Quinet, P.; Palmeri, P.; Biémont, E. Spectroscopic data for atomic tungsten transitions of interest in fusion plasma research. J. Phys. B At. Mol. Opt. Phys. 2011, 44, 145005. [Google Scholar] [CrossRef]
  7. Nilsson, H.; Engström, L.; Lundberg, H.; Palmeri, P.; Fivet, V.; Quinet, P.; Biémont, E. Lifetime measurements and transition probability calculations in singly ionized tungsten (W II). Eur. Phys. J. D 2008, 49, 13–19. [Google Scholar] [CrossRef]
  8. Palmeri, P.; Quinet, P.; Fivet, V.; Biémont, E.; Nilsson, H.; Engström, L.; Lundberg, H. Lifetime measurements and calculated transition probabilities in W III. Phys. Scr. 2008, 78, 015304. [Google Scholar] [CrossRef]
  9. Enzonga Yoca, S.; Quinet, P.; Biémont, E. Configuration interaction and radiative decay rates in trebly ionized tungsten (W IV). J. Phys. B At. Mol. Opt. Phys. 2012, 45, 035001. [Google Scholar]
  10. Enzonga Yoca, S.; Quinet, P.; Palmeri, P.; Biémont, E. Comparative semi-empirical and ab initio atomic structure calculations in Yb-like tungsten W4+. J. Phys. B Mol. Opt. Phys. 2012, 45, 065001. [Google Scholar] [CrossRef]
  11. Enzonga Yoca, S.; Palmeri, P.; Quinet, P.; Jumet, G.; Biémont, E. Radiative properties and core-polarization effects in the W5+ ion. J. Phys. B At. Mol. Opt. Phys. 2012, 45, 035002. [Google Scholar] [CrossRef]
  12. Quinet, P.; Vinogradoff, V.; Palmeri, P.; Biémont, E. Radiative decay rates for W I, W II and W III allowed and forbidden transitions of interest for spectroscopic diagnostics in fusion plasmas. J. Phys. B At. Mol. Opt. Phys. 2010, 43, 144003. [Google Scholar] [CrossRef]
  13. Ryabtsev, A.N.; Kononov, E.Y.; Kildiyarova, R.R.; Tchang-Brillet, W.-Ü.L.; Wyart, J.-F. The spectrum of seven times ionized tungsten (W VIII) relevant to tokamak divertor plasmas. Phys. Scr. 2013, 87, 045303. [Google Scholar] [CrossRef]
  14. Kramida, A.E.; Shirai, T. Energy levels and spectral lines of tungsten, W III through W LXXIV. At. Data Nucl. Data Tables 2009, 95, 305–474. [Google Scholar] [CrossRef]
  15. Sugar, J.; Kaufman, V. Seventh spectrum of tungsten (W VII); resonance lines of Hf V. Phys. Rev. A 1975, 12, 994–1012. [Google Scholar] [CrossRef]
  16. Carlson, T.A.; Nestor, C.W., Jr.; Wasserman, N.; McDowell, J.D. Calculated ionization potentials for multiply charged ions. At. Data Nucl. Data Tables 1970, 2, 63–99. [Google Scholar] [CrossRef]
  17. Veres, G.; Bakos, J.S.; Kardon, B. Energy levels and the vacuum ultraviolet spectrum of W VIII. J. Quant. Spectrosc. Rad. Transfer 1996, 56, 295–301. [Google Scholar] [CrossRef]
  18. Cowan, R.D. The Theory of Atomic Structure and Spectra; University of California Press: Berkeley, CA, USA, 1981. [Google Scholar]
  19. Quinet, P.; Hansen, J.E. The influence of core excitations on energies and oscillator strengths of iron group elements. J. Phys. B At. Mol. Opt. Phys. 1995, 28, L213–L220. [Google Scholar] [CrossRef]

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Deprince, J.; Quinet, P. Detailed Analysis of Configuration Interaction and Calculation of Radiative Transition Rates in Seven Times Ionized Tungsten (W VIII). Atoms 2015, 3, 299-319. https://doi.org/10.3390/atoms3030299

AMA Style

Deprince J, Quinet P. Detailed Analysis of Configuration Interaction and Calculation of Radiative Transition Rates in Seven Times Ionized Tungsten (W VIII). Atoms. 2015; 3(3):299-319. https://doi.org/10.3390/atoms3030299

Chicago/Turabian Style

Deprince, Jérôme, and Pascal Quinet. 2015. "Detailed Analysis of Configuration Interaction and Calculation of Radiative Transition Rates in Seven Times Ionized Tungsten (W VIII)" Atoms 3, no. 3: 299-319. https://doi.org/10.3390/atoms3030299

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