Extended Atomic Structure Calculations for W 11 + and W 13 +

: We report an extensive and elaborate theoretical study of atomic properties for Pm-like and Eu-like Tungsten using Flexible Atomic Code (FAC). Excitation energies for 304 and 500 ﬁne structure levels are presented respectively, for W 11 + and W 13 + . Properties of the 4 f -core-excited states are evaluated. Di ﬀ erent sets of conﬁgurations are used and the discrepancies in identiﬁcations of the ground level are discussed. We evaluate transition wavelength, transition probability, oscillator strength, and collisional excitation cross section for various transitions. Comparisons are made between our calculated values and previously available results, and good agreement has been achieved. We have predicted some new energy levels and transition data where no other experimental or theoretical results are available. The present set of results should be useful in line identiﬁcation and interpretation of spectra as well as in modelling of fusion plasmas.


Introduction
There has been strong interest in the spectroscopy of tungsten as it is planned to be used in plasma facing components of future fusion devices, such as International Thermonuclear Experimental Reactor (ITER) due to its favourable physical and chemical properties, e.g., high energy threshold of sputtering, low sputtering yield, low tritium retention, and high melting temperature [1,2]. Since tungsten is a high-Z element (Z = 74), where Z is the atomic number, it will contribute a large fraction of energy carried out from the plasma, which leads to plasma cooling. Atomic data such as energy levels, radiative transition rates, and photoionization cross sections for low-charged and medium-charged ions are of great importance in the ITER plasma diagnostics [3]. In the past few decades, atomic data for several highly charged tungsten ions have been determined using different experimental and theoretical methods [4][5][6][7][8], but still there is demand for more accurate atomic data, especially for low and medium ionization states of tungsten.
In the present work, spectra of moderately charged states of tungsten (W 11+ and W 13+ ) are theoretically investigated. Several observations and theoretical calculations have been performed for Pm-like W but, for Eu-like W, only a few experimental data are available in the literature. In fact, only one theoretical energy value can be found for Eu-like W in the Atomic Spectra Database of the National Institute of Standards and Technology (NIST) [9]. These ionized states of tungsten are complex due to an open 4f shell, and obtaining accurate atomic data for these ions is a largely unsolved problem. For example, by inclusion of different configuration sets in the calculations, one will obtain a different ground state. The accuracy of a calculation can be estimated by considering (i) the convergence rate, (ii) the agreement between experimental measurements and theoretical calculations, and (iii) the

Theoretical Method
In spite of calculations performed by different authors for Pm-like and Eu-like W, there are no medium-scale calculations for these ions, and the shortage of complete and accurate data for these ions motivates this work. Most past calculations included very limited CI with an arbitrary choice of configurations. Therefore, in the present work, extensive calculations for W 11+ and W 13+ have been performed within the framework of FAC, which was developed by Gu [35]. FAC is a fully relativistic program used to compute the atomic structure, photoionization cross sections, and other atomic data. It is based on the Dirac-Hartree-Fock-Slater (DHFS) method, which uses perturbation theory. Optimization of orbitals is performed in a self-consistent-field iterative procedure in which the average energy of a fictitious mean configuration is minimized. This mean configuration represents the average electron cloud of the configurations retained in the CI expansion. In FAC, the Hamiltonian and configuration atomic state functions are similar to those of the MCDF (multi-configuration Dirac-Fock) method, including relativistic effects and higher-order QED effects, e.g., the Breit interaction in the zero-energy limit for the exchanged photon, and hydrogenic approximations for self-energy and vacuum polarization effects.
The effective Hamiltonian for an N-electron system is given by: whereĤ i is the Dirac one-particle operator for the ith-electron and V ij represents effective electron-electron interactions. An atomic state function (ASF) with total angular momentum J, its z-projection M, and parity p is assumed in the following form.
where φ γ m JM P are configuration state functions (CSF), c m (s) are configuration mixing coefficients for the states, and γ m represents all information required to uniquely define a certain CSF. A detailed description of this theoretical approach can be found in the literature [35].
In Table 2, we present a comparison of wavelengths calculated with FAC with other experimental and theoretical wavelengths [30] for the (4f 13 5s 2 5p 2 ) 5/2 -(4f 13 5s5p 3 ) 3/2,5/2,7/2 transitions in Eu-like W, which is also shown in Figure 1. For all transitions, our calculated transition wavelengths agree with the experimental results of Li et al. [30] within 4.7%, while the theoretical results of Li et al. [30] deviate from their experimental results by up to 3.1%. Li et al. [30] also used FAC as in the present calculations, but with a different number of configurations. They have included the 4f 14 5s 2 5p, 4f 14 5s5p 2 , 4f 14 5s5p5d, 4f 13 5s 2 5p 2 , 4f 13 5s 2 5p5d, 4f 13 5s5p 3 , 4f 13 5s5p 2 5d, 4f 12 5s 2 5p 3 , and 4f 12 5s 2 5p 2 5d configurations, which generate 2538 fine structure levels. Table 2. Comparison of calculated wavelengths (in nm) using FAC with other wavelengths for various transitions in Eu-like W. i and j represent the sequential numbers assigned in Table 1 to the lower and upper levels, respectively.  Table 3 presents transition data for some strong electric dipole (E1) transitions (transition probability A > 10 8 s −1 ), respectively, from the ground state to various levels among the lowest 304 levels. We present a transition wavelength λ (in nm), a weighted oscillator strength gf (dimensionless) (both the length and velocity forms), and a transition rate A ji (s −1 ) calculated using FAC for Eu-like W. In Figure 2, a comparison of the 'length' and 'velocity' forms of gf (actually, results in the Babushkin and Coulomb gauges) is made for a few of the strongest transitions given in Table 3. One can see that the plot has the usual regular behaviour of increasing scatter with a decreasing line strength. However, for the strongest transitions, although the scatter is small, there is a systematic offset. The velocity form is smaller than the length form by 30%. Thus, all E1 transitions in Table 3 are estimated to have a common uncertainty of 30%.
{[(4f 13 )5/25s]2(5p 2 1/25p3/2)3/2}7/2→{[(4f 13 )5/25s 2 ]5/2(5p 2   th / exp  th (nm) Figure 1. Comparison of theoretical wavelengths with the experimental wavelengths of Reference [30]. Table 3 presents transition data for some strong electric dipole (E1) transitions (transition probability A > 10 8 s −1 ), respectively, from the ground state to various levels among the lowest 304 levels. We present a transition wavelength λ (in nm), a weighted oscillator strength gf (dimensionless) (both the length and velocity forms), and a transition rate Aji (s −1 ) calculated using FAC for Eu-like W. In Figure 2, a comparison of the 'length' and 'velocity' forms of gf (actually, results in the Babushkin Figure 1. Comparison of theoretical wavelengths with the experimental wavelengths of Reference [30]. Table 3. Oscillator strengths (length and velocity form) gf L and gf v , vacuum wavelengths λ (in nm), and transition probabilities A ji (in s −1 ) for some strong electric dipole (E1) transitions from the ground state to various levels of Eu-like W. Tables 4 and 5 present transition data for magnetic dipole (M1) and magnetic quadrupole (M2) transitions from the ground state to some of the lowest 304 levels. We have presented transition wavelength λ (in nm), weighted oscillator strength gf (dimensionless), and transition rate A ji (s −1 ) calculated using FAC for Eu-like W. We predict new oscillator strength and transition probability data, where no other theoretical or experimental results are available, which will form the basis for future experimental work.    In Table 6, we provide collisional excitation cross-sections of Eu-like W from the ground state for the incident electron energy range of 65 to 125 eV. To the best of our knowledge, there are no other data points for collisional cross sections of Eu-like W in the literature within the given energy range. In Table 6, we provide collisional excitation cross-sections of Eu-like W from the ground state for the incident electron energy range of 65 to 125 eV. To the best of our knowledge, there are no other data points for collisional cross sections of Eu-like W in the literature within the given energy range.

Pm-Like W
Realizing the importance of Pm-like W and considering the paucity of atomic data for this ion, in the present work, we have calculated energy levels and radiative transition rates using FAC. To check the convergence of results, we have calculated results with different sets of configurations. Table 7 shows the configurations used in various calculations and the number of levels generated using these configurations. In the present work, we have increased the number of configurations in the sets in a systematic way to study the CI effect. In INP1, we have included 4f 14 5s, 4f 13 5s 2 , 4f 13 5s5p, 4f 14 5p, and 4f 13 5p 2 configurations, which generate 59 fine structure levels. In INP2, we have added the 4f 12 5s 2 5p, 4f 12 5s5p 2 , and 4f 12 5p 3 configurations. These three configurations generate 621 levels. Furthermore, in order to check the effect on energies, INP3 forms a complex system by adding 4f 11 5s 2 5p 2 , 4f 11 5s5p 3 , 4f 10 5s 2 5p 3 , and 4f 10 5s5p 4 to INP2, which generates a total of 7790 fine structure levels. We observe that the energies of these additional configurations are distributed inside the interval between the 4f 14 5s and 4f 145 p energies, which shows the importance of adding these configurations. Finally, in the INPF column of Table 7, we have considered a larger CI for Pm-like W, which includes 4f 14 5s, 4f 13 5s 2 , 4f 13 5s5p, 4f 14 5p, 4f 13 5p 2 , 4f 12 5s 2 5p, 4f 12 5s5p 2 , 4f 12 5p 3 , 4f 11 5s 2 5p 2 , 4f 11 5s5p 3 , 4f 10 5s 2 5p 3 , 4f 10 5s5p 4 , 4f 12 5s 2 5d, 4f 12 5s5d 2 , 4f 13 5d 2 , 4f 13 5p5d, 4f 13 5d 2 , 4f 13 5p5d, 4f 12 5p5d 2 , 4f 12 5p 2 5d, and 4f 13 5s5d configurations. This set generates 13,160 levels. Table 8 shows the list of configurations included and the number of fine structure levels arising from each configuration of Pm-like W. The ground state configuration in each case is the same. Table 9 lists the energy values from each input for the 4f 13 5s5p configuration. To check convergence, the difference between various output levels (Out1, Out2, Out3) and OutF is plotted in Figure 3. The different colors correspond to the different levels of Table 9. For levels 19 to 24, the difference is very large. Although the differences do not tend to zero with an increasing size of the calculation, the level splitting is almost the same in each output using different configuration sets for the first 18 levels, as shown in Figure 4. The root-mean-square (rms) difference from OUTF for the intervals ∆E j is 0.16 eV for OUT1, 0.10 eV for OUT2, and 0.08 eV for OUT3.  4f 14 5s, 4f 13 5s 2 , 4f 13 5s5p, 4f 14 5p, 4f 13 5p 2 , 4f 12 5s 2 5p, 4f 12 5s5p 2 , 4f 12 5p 3 4f 14 5s, 4f 13 5s 2 , 4f 13 5s5p, 4f 14 5p, 4f 13 5p 2 , 4f 12 5s 2 5p, 4f 12 5s5p 2 , 4f 12 5p 3 , 4f 11 5s 2 5p 2 , 4f 11 5s5p 3 , 4f 10 5s 2 5p 3 , 4f 10 5s5p 4 4f 14 5s, 4f 13 5s 2 , 4f 13 5s5p, 4f 14 5p, 4f 13 5p 2 , 4f 12 5s 2 5p, 4f 12 5s5p 2 , 4f 12 5p 3 , 4f 11 5s 2 5p 2 , 4f 11 5s5p 3 , 4f 10 5s 2 5p 3 , 4f 10 5s5p 4 , 4f 12 5s 2 5d, 4f 12 5s5d 2 , 4f 13 5d 2 , 4f 13 5p5d, 4f 13 5d 2 , 4f 13 5p5d, 4f 12 5p5d 2 , 4f 12 5p 2 5d, 4f 13 5s5d         Table 10 presents the energy levels (in Rydberg) for the lowest 500 levels calculated with FAC for Pm-like W. As can be seen from this table, we report results for many new levels that are not listed in the NIST tables [9]. The format of Table 10 is the same as in Table 1. Since 85 levels out of 500 of Table 8 have the same '2J' label as some other levels, we report the eigenvector compositions and  Table 10 presents the energy levels (in Rydberg) for the lowest 500 levels calculated with FAC for Pm-like W. As can be seen from this table, we report results for many new levels that are not listed in the NIST tables [9]. The format of Table 10 is the same as in Table 1. Since 85 levels out of 500 of Table 8 have the same '2J' label as some other levels, we report the eigenvector compositions and mixing coefficients of W 13+ in Table S2 of the supplementary material. In this table, we report the composition of the levels given in Table 10. In the column, the 'Comp. No.' of Table S2 contains sequential numbers of the basis states in decreasing order of contribution. We have included only a few basis states with the largest contributions. For each basis state, Table S2 gives an electronic configuration in the JJ coupling scheme. In these designations, '4f13' represents that there are 13 electrons in the 4f subshell. In '4f+7(7)7,' the '+' sign denotes the larger of the two possible values of the angular momentum for the f electron, i.e., '4f+' corresponds to 4f7/2. '7' after the '+' sign shows the number of electrons in the relativistic subshell 4f7/2, '(7),' which means that the total 2J value of the 4f7/2 subshell is 7, and '7' at the end denotes the final 2J value of the configuration.  (7)8.5p-1(1)9 11.036302 12 4f 12 5s 2 5p 1 4f-5(5)5.4f+7 (7)6.5p-1(1)5 11.306358 In Table 11, some of the calculated energies are compared with those from critically evaluated data compiled by NIST [9], which are commonly used as a reference set for atomic results. The calculated values of Safronova et al. [28] and Zhou et al. [32] are also presented for comparison in Table 11. For the level 4f 13 5s 2 2 F 5/2 , our calculated value matches the results of Zhou et al. [32]. For this level, the NIST value is from Vilkas et al. [24] whose results have been proven wrong by many authors in the recent past. They have used the MR-MP method for their calculation. For the 4f 13 5s5p levels, the maximum difference between our calculation using FAC and the calculation of Safronova et al. [28] is 3.1%.  (3) 3.878 a-Reference [28]. b-Reference [32].
In Table 12, we present the transition data for the 4f 13 5s 2-4f 13 5s5p and 4f 14 5s-4f 14 5p transitions including transition wavelengths (in nm), transition probabilities (A ji in s −1 ), and oscillator strengths (gf ij , dimensionless). We give the oscillator strengths both in the velocity and length forms to check the accuracy of the calculated results, as it is one of the criteria used to assess the accuracy. The ratio of the velocity and length forms of the oscillator strength should be close to unity in accurate calculations. It is a necessary condition, even though it is not sufficient. As one can see in Table 12, this ratio is close to unity for most of the strong transitions. It is plotted in Figure 5 against line strength. For the Pm-like W, the systematic offset between the length and velocity forms is significantly smaller, at only about 10%. Therefore, the strongest E1 transitions with the line strength S V > 0.017 a.u. can be assigned an uncertainty of 10%. Those with 0.004 a.u. < S V < 0.017 a.u. can be assigned an uncertainty of 15%, and weaker transitions can be estimated as accurate to 30%. In Table 12, we also present the transition rates from the OUT3 results for comparison since, for magnetic transitions, the velocity form is not available. From Table 12, one can see that transition rates of OUTF have large differences from OUT3 for many levels, which is due to additional configurations in OUTF. In Table S1 of the supplementary material, we report M1 and E2 transition data for the transitions to various levels for 4f 12 5s 2 5p of Pm-like W, as these transitions could help to analyze experimental EBIT and tokamak spectra.   In the 4f 13 5s 2 2 F7/2 and 4f 13 5s 2 2 F5/2 levels, the mixing is very low, and their labeling is unambiguous. For both levels, the dominant eigenvector component contributes 98.3% and 98.4%, respectively. We would also like to mention that mixing among some of the levels is strong in our calculations. We found that level 250 is strongly coupled with level 317 with percentages of the corresponding components of 71% and 23%, respectively. We found that the level 277 is composed of 52% of the basis state used in its label mixed with 39% of the state used to label the level 357. Similarly, mixing In the 4f 13 5s 2 2 F 7/2 and 4f 13 5s 2 2 F 5/2 levels, the mixing is very low, and their labeling is unambiguous. For both levels, the dominant eigenvector component contributes 98.3% and 98.4%, respectively. We would also like to mention that mixing among some of the levels is strong in our calculations.
We found that level 250 is strongly coupled with level 317 with percentages of the corresponding components of 71% and 23%, respectively. We found that the level 277 is composed of 52% of the basis state used in its label mixed with 39% of the state used to label the level 357. Similarly, mixing among some other levels is very strong. Hence, the labeling of a particular level is not always based on the dominant eigenvector component. The configuration and J values given in Table 8 are definite, but the labels are not unique and can be interchanged. Table S1 provides the means to estimate the uncertainties of the M1 and E2 transitions in Pm-like W. For example, Figure 6 plots the natural logarithm of the ratio A outF /A out3 of the E2 transition rates calculated in the two largest calculations. From this figure, one can see that, except for a few strongly discrepant transitions, the two data sets agree with each other fairly well. This figure is consistent with a typical behavior of calculated transition rates. For the strongest transitions (in this case, S outF > 0.94 a.u.), the mean (i.e., root-mean-square, rms) of the logarithmic ratio is 0.014, corresponding to an agreement within 1.4% on average. For weaker transitions, the scatter of the data points increases, but, even for the weakest transitions with S < 0.00212 a.u., the mean disagreement is only 52%.
Atoms 2020, 8, x FOR PEER REVIEW 32 of 39 corresponding to an agreement within 1.4% on average. For weaker transitions, the scatter of the data points increases, but, even for the weakest transitions with S < 0.00212 a.u., the mean disagreement is only 52%. For M1 transitions, if three strongly discrepant transitions are excluded, the comparison of out3 and outF looks qualitatively similar, as shown in Figure 7. Here again, the strongest transitions (SoutF > 2.8 a.u. in this case) exhibit a mean discrepancy of 3.2%, while, for weaker transitions, it increases, but, even for the weakest transitions with S < 0.1 a.u., the mean discrepancy is only 15%. The three transitions excluded from this plot are 47  38, 65  38, and 74  38. For them, the discrepancies amount to a factor between 10 and 100. For M1 transitions, if three strongly discrepant transitions are excluded, the comparison of out3 and outF looks qualitatively similar, as shown in Figure 7. Here again, the strongest transitions (S outF > 2.8 a.u. in this case) exhibit a mean discrepancy of 3.2%, while, for weaker transitions, it increases, but, even for the weakest transitions with S < 0.1 a.u., the mean discrepancy is only 15%. The three transitions excluded from this plot are 47 → 38, 65 → 38, and 74 → 38. For them, the discrepancies amount to a factor between 10 and 100.

1.5
In Table 13, we compare transition wavelengths (in nm) calculated using FAC with the data compiled by NIST [9] and with experimental and theoretical results of Kobayashi et al. [31] along with theoretical results of Safronova et al. [28] and Ding et al. [33]. For the transitions 0-91, 0-263, and 0-326, our calculated transition wavelengths are comparable (maximum difference of 5%) with the data compiled by NIST. For the transitions 0-433, 0-434, and 0-451, our calculated transition wavelength agrees well with the experimental results of Kobayashi et al. [31] (error within 0.4%) and are more accurate than the theoretical results of Safronova et al. [28] and Kobayashi et al. [31]. Furthermore, our calculated transition wavelengths for all transitions are in excellent agreement with the recent results of Ding et al. [33]. This is a clear indication of accuracy of our results.
For M1 transitions, if three strongly discrepant transitions are excluded, the comparison of out3 and outF looks qualitatively similar, as shown in Figure 7. Here again, the strongest transitions (SoutF > 2.8 a.u. in this case) exhibit a mean discrepancy of 3.2%, while, for weaker transitions, it increases, but, even for the weakest transitions with S < 0.1 a.u., the mean discrepancy is only 15%. The three transitions excluded from this plot are 47  38, 65  38, and 74  38. For them, the discrepancies amount to a factor between 10 and 100.  In Table 14, we have compared transition wavelength (in nm) calculated with FAC for the transition 4f 13 5s 2 2 F 5/2-2 F 7/2 with other results for Pm-like W. The theoretical wavelengths are calculated in vacuum, while the experimental wavelengths are measured in standard air. Li et al. [30] and Zhao et al. [32] in Shanghai and Kobayashi et al. [31] in Tokyo observed similar spectra, but the charge state assignments differ by one. Recently, the Shanghai group confirmed [36] that their assignment made in References [30,32] was wrong and confirmed the assignment in Reference [31]. In addition, recent theoretical calculations by Ding et al. [33] also support the assignment in Reference [31]. Our theoretical result agrees within 2% with the experimental wavelengths of Kobayashi et al. [31]. This is better than the result of Safronova et al. [28], who used the COWAN code, and Vilkas et al. [24]. The differences of those two results from the measurement of Kobayashi et al. are larger (4% and 30%, respectively). The maximum discrepancy of 29% is between the experimental wavelength and the results of Vilkas et al. [24], which are responsible for discrepancies between our calculated results and the NIST data [9] quoted from Vilkas et al. [24]. Furthermore, our transition wavelength is in good agreement with the recent results of Ding et al. [33] as well as with other theoretical results listed in Table 14. In Table 15, we give the collisional excitation cross-sections of Pm-like W from the ground state for incident electron energy ranging from 45 to 75 eV. The collision cross section is given in units of 100 Mb.  [32]. b-Reference [28]. c-Reference [24]. d-Reference [38]. e-Reference [33]. f-Reference [31].

Conclusions
Motivated by the need of atomic data for fusion plasma research, in the present paper, we have reported energy levels and radiative data, such as transition wavelengths and oscillator strengths, as well as collisional excitation cross sections for W 11+ and W 13+ . For the calculations, the fully relativistic FAC has been adopted with the inclusion of CI. The effect of CI has been examined and investigated systematically by including different configurations. The present calculations provide new energy level data and are useful for accuracy assessments. Based on several comparisons with NIST and other available results, our listed energies are accurate (error <1%) for most levels of both ions. Energies have been listed for the lowest 304 and 500 levels of W 11+ and W 13+ , respectively, but the remaining data for higher levels can be obtained from the authors on request. We have also presented radiative transition rates for both ions that are expected to be highly useful in analysis and modeling of plasmas. We have presented oscillator strengths in both velocity and length forms for both ions. We have compared our calculated data with the other available theoretical and experimental data and no (major) discrepancies have been found, except for some levels of Pm-like W calculated by Vilkas et al. [24]. Furthermore, we observed discrepancies in the ground state of Eu-like W, which depends on the configurations included. There is scope for further work in these complex systems.
Author Contributions: All authors have contributed equally in preparation of the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by IAEA grant number 23244.