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

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).


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 OPEN ACCESS 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 pseudorelativistic 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 4f 14 5s 2 5p 5 , 4f 14 5s 2 5p 4 nl, 4f 14 5s5p 6 , 4f 13 5s 2 5p 6 , 4f 13 5s 2 5p 5 nl and 4f 12 5s 2 5p 6 nl (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.

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 W 2+ to W 73+ . In this compilation, it was reminded that it was by the way uncertain whether the ground state of W VIII was 4f 13 5s 2 5p 6 2 F°7/2 or 4f 14 5s 2 5p 5 2 P°3/2, 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 4f 11 5s 2 5p 6 6s 2 for the first members of the sequence, Ho I and Er II, 4f 13 5s 2 5p 6 for Hf VI and Ta VII, and 4f 14 5s 2 5p 5 for Re IX and the rest of the sequence. By analyzing the 4f 13 ( 2 F°7/2)5s 2 5p 6 ns and 4f 14 5s 2 5p 5 ( 2 P°3/2)ns series of W VII, Sugar and Kaufman [15] asserted that the ground state of W VIII was probably 4f 13 5s 2 5p 6 2 F°7/2, the 4f 14 5s 2 5p 5 2 P°3/2 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 4f 13 5s 2 5p 6.2 F°7/2,5/2 and 4f 14 5s 2 5p 5 2 P°3/2,1/2 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.
Furthermore, it is also interesting to estimate the influence of the double excitations on the radiative parameters. Indeed, for transitions of the type 4f 14 5s 2 5p 5 -4f 14 5s 2 5p 4 5d and 4f 13 5s 2 5p 6 -4f 13 5s 2 5p 5 5d, the 5p 2 → 5d 2 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 4f 14 5s 2 5p 3 5d 2 and 4f 13 5s 2 5p 4 5d 2 , giving rise to model HFR(E). Finally, as similar speculations can be made for the double excitation 4f 2 → 5d 2 in the case of the 4f 14 5s 2 5p 5 -4f 13 5s 2 5p 5 5d and 4f 13 5s 2 5p 6 -4f 12 5s 2 5p 6 5d transitions and for the double excitation 5p 2 → 6s 2 in the case of the 4f 14 5s 2 5p 5 -4f 14 5s 2 5p 4 6s and 4f 13 5s 2 5p 6 -4f 13 5s 2 5p 5 6s transitions, the odd-parity configurations 4f 12 5s 2 5p 5 5d 2 and 4f 11 5s 2 5p 6 5d 2 were added to model HFR(E) to give model HFR(F), while 4f 14 5s 2 5p 3 6s 2 and 4f 13 5s 2 5p 4 6s 2 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 5p 2 → 6s 2 , considered in model HFR(G) do not change the final results (see Figure 2c), it is not the same for the 5p 2 → 5d 2 and 4f 2 → 5d 2 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 Figures 2a,2b). 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 3p 2 → 3d 2 core excitation on the 3p 6 3d N -3p 5 3d N+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 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.
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 4f 13 5s 2 5p 6 and 4f 14 5s 2 5p 5 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.    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 10 10 s −1 , to be experimentally observed, according to our calculations. These are located at λ = 191.700 Å (gA = 1.46 × 10 10 s −1 ), λ = 229.402 Å (gA = 1.16 × 10 10 s −1 ) and λ = 244.249 Å (gA = 1.33 × 10 11 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).      a Indexes of levels as given in Table 2. b λ exp are taken from [13] while λ Ritz 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 y J P J y CF y J P J y 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 10 9 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 > 10 10 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.  [13] for experimentally identified spectral lines in W VIII.
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)

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.