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Contribution of high-nl shells to electron-impact ionization processes

The contribution to electron-impact ionization cross sections from excitations to high-nl shells and a consequent autoionization is investigated. We perform relativistic subconfiguration-average and detailed level-to-level calculations for this process. Ionization cross sections for the W27+ ion are presented to illustrate the large influence of the high shells (n>= 9) and orbitals (l>= 4) in the excitation-autoionization process. The obtained results show that the excitations to the high shells (n>= 9) increase cross sections of the indirect ionization process by a factor of 2 compared to the excitations to the lower shells (n<= 8). The excitations to the shells with orbital quantum number l = 4 give the largest contribution comparedwith the other orbital quantum numbers l. Radiative damping reduces the cross sections of the indirect process approximately twofold in the case of the level-to-level calculations. Determined data show that the excitation-autoionization process contributes approximately 40% to the total ionization cross sections.


I. INTRODUCTION
Energy losses from heavy elements, such as tungsten, due to radiative emission are one of the crucial problems to be overcome for the successful performance of themonuclear reactors. Nevertheless, tungsten is used as the plasma-facing component in the modern fusion facilities because of its essential properties, like high-energy threshold for sputtering, low sputtering yield, and excellent thermal features. Even small concentration of tungsten ions (∼ 10 −4 ) relative to the electron density prevents ignition of a deuterium-tritium plasma [1]. Theoretical modeling provides information about processes in such harsh conditions. However, fusion plasma modeling requires a significant amount of atomic data. The ionization and recombination processes determine the charge-state distribution in plasma. The electron-impact single ionization is the strongest one among the ionization processes. Contribution of the double ionization as a rule is much weaker compared with the single ionization.
Electron-impact ionization for singly charged tungsten ions was previously studied by using the crossed-beam technique [5]. These data were supplemented by the electron-impact single, double, and triple ionization measurements of W q+ ions in the charge states q = 1 − 10, q = 1 − 6, and q = 1 − 4, respectively [6,7]. The configuration-average distorted-wave (CADW) calculations provided good agreement with experimental measurements for higher ionization stages (q 4) [8]. The DW approach succeeded in getting a very good agreement with experiment for the ionization stages up to W 9+ ion and therefore it was applied to all isonuclear sequence [9]. Further level-to-level studies [10] included W atom and W 2+ ion in the binaryencounter-dipole model as well as W 4+ and W 6+ ions in the DW approach. For the tungsten isonuclear sequence, cross sections were calculated using semirelativistic and relativistic DW methods for configurations and subconfigurations, respectively [9]. Influence of the excitation-autoionization (EA) process, radiative damping, and relativistic effects were analyzed. The work of Loch et al. [9] considered the electronimpact excitations to all shells with n 8 and l 3. The investigation has determined that the EA contribution is relatively small compared to direct ionization (DI) for the W 11+ to W 27+ ions. However, later experimental measurements for the W 17+ ion have demonstrated a significant role of indirect process [11]. Theoretical study of cross sections using the Dirac-Fock-Slater approach confirmed these findings [12].
Furthermore, the authors found that Maxwellian rate coeffcients are larger than the CADW rate coefficients by about 16%. The discrepancy was attributed to the EA channels originating from the high-n shells up to n = 38.
Therefore, the main aim of our work is to determine the influence of excitations to the high-n shells which lead to the EA process. For the study, we have chosen the W 27+ ion which has only one 4f electron in the valence shell. At first, we will present calculated results of the EA contribution for subconfigurations. Later, levelto-level distorted-wave (LLDW) data in a single configuration approach are analyzed.
In addition, the influence of the radiative damping is studied in both cases. The contribution of the direct and indirect processes is compared for the W 27+ ion.

II. THEORETICAL APPROACH
Direct and indirect processes contribute to the total electron-impact singleionization cross sections from the level i of A q+ ion to the level j of A (q+1)+ ion: where σ DI ij (ε) is the direct ionization cross section at the electron energy ε, σ exc ik is the electron-impact excitation cross section to the level k of the A q+ ion. Autoionization branching ratio B kj is determined by the expression: where A a and A r are the Auger and radiative transition probabilities, respectively. We do not consider higher order indirect ionization process such as a resonantexcitation double-autoionization. Previous analysis of this process for the W 17+ ion determined a negligible contribution for the ionization cross sections [12].
The electron-impact excitation and ionization cross sections are determined in the DW approximation using the Flexible Atomic Code (FAC) [13] which implements the Dirac-Fock-Slater method. For moderate and highly charged ions, the direct The ionization process affecting the 4s, 4p, 4d, and 4f shells contribute to the DI process. On the other hand, study of the W 17+ ion has demonstrated that ionization from the 4s shell lies above the double-ionization threshold and contributes to the indirect double-ionization process [12].  orbital is the largest compared to other orbitals. It is worth to note that excitations to l = 5 orbital lead to the strong EA channel for which the relative contribution increases for 9 n 25 shells (Fig. 3b). For these shells, the largest increase of the relative contribution occurs for the excitations to a l = 2 orbital compared to the n 8 case. This can be explained by the fact that the strongest excitations to the l = 2 orbital take place from the 4d shell. However, the configurations 4d 9 4f nd with n = 5, 6, 7 are below the ionization threshold, and they do not contribute to the EA process. Only some levels of the 4d 9 4f 8d configuration are above the ionization threshold (Fig. 1). On the other hand, all levels of 4d 9 4f 8g configuration are above the ionization threshold. Furthermore, the energy levels of the 4d 9 4f 7g configuration straddle the ionization threshold. It explains much larger relative contribution to the EA process of the excitations to the l = 4 orbital for n 8 shells.
Other interesting result occurs when the EA channel originating from excitations to the l = 3 orbital starts to dominate at high electron energies for the 9 n 25 shells. Contribution of the EA channel to the l = 4 orbital is slightly smaller at the higher energies when the excitations to the large principal quantum numbers n are considered (Fig. 3b).
The EA channel for the excitations from the 4d shell is predicted to produce the largest contribution compared to the EA channels from the 4s and 4p shells because the excitation cross sections are larger for the 4d shell. However, as it is mentioned above, many configurations produced by the promotion from the 4d shell to the nl shell for n 8 are below the ionization threshold. Therefore, these configurations do not contribute to the EA process. Figure 4a demonstrates that the contribution of the excitations from the 4d shell to the EA process for n 8 is more than two times smaller compared to the EA channel for the excitations from the 4p shell. The similar result was obtained for the W 17+ ion when the excitations only up to n 8 were considered [12]. However, the situation drastically changes for the excitations from the 4d to the higher shells because all arising configurations are above the ionization threshold (Fig. 4b). Contribution of this EA channel is much larger compared to the EA channel from the 4p shell. Furthermore, the contribution of the EA channel from the 4d shell for the excitations to 9 n 25 shells is comparable to the total cross sections of the EA process for the excitations to n 8 shells (Fig. 4a).
Calculations of the EA process for subconfigurations can differ from those for the detailed level-to-level studies. In our SCADW calculations, the energy levels are grouped into the subconfigurations. For the energy levels that straddle the ionization threshold, the corresponding energy of the subconfiguration can be above or below the ionization threshold. The subconfiguration with the energy above the ionization threshold contributes to the EA process. Therefore, all levels of the subconfiguration, even those located below the ionization threshold, are included when the cross sections are calculated. On the other hand, for the subconfiguration below the ionization threshold, the contributions from the corresponding energy levels above the ionization threshold are neglected.
It is interesting to note that the EA cross sections obtained using SCADW and LLDW approaches are in good agreement when the autoionization branching ratios are not included in calculations (Fig. 5). Figure 5 shows that the radiative damping has the large impact on the ionization cross sections. However, it was found in the previous study that the autoionization branching ratios are close to 1 for this ionization stage [9]. The difference can be explained by the fact that the previous study used the configuration-average quantities while the current SCADW calculations employ the subconfigurations. The higher values of the cross sections for the SCADW calculations compared to the LLDW results suggest that the autoionization branching ratios for the CADW data are closer to 1. Therefore, the radiative decay paths are restricted for some excited configurations in these two averaged approaches. Figure 5 demonstrates that the level-to-level study is crucial for the EA process in the W 27+ ion. Figure 6 presents the total ionization cross sections for the direct and indirect ionization parts where the indirect part accounts for the radiative damping. In this case, the level-to-level calculations are presented. The contributions of DI from the 4s, 4p, 4d, and 4f shells are highlighted. The largest contribution comes from the ionization of the 4d electron. As it is shown above, the EA channel for the excitations from the 4d shell has also the largest impact compared to the 4s and 4p contributions (Fig. 4). For the DI process, the influence of the 4s shell is very small. To ensure our data can be easily utilized in modeling studies, we present the total ionization cross sections Table II.
It can be seen that the EA process increases the total cross section at the peak nearly by 40 % (Fig. 6). This contribution would be about two times smaller if the excitations to the higher shells (n 9) were not considered. The EA channels which include only the excitations with n 8 and l 3 provide for the EA process approximately 10% of the total cross section at the peak.
The ionization cross sections determined for the metastable [Kr]4d 10 5s configuration are approximately by 60 % higher for the EA process compared to the ground configuration when the radiative damping is included for these configurations. The contribution of the indirect process from the high-nl shells to the total cross sections increases by a factor of two. The DI from the 5s shell is about two times higher for the metastable configuration compared to the ionization from the 4f shell of the ground configuration.

IV. CONCLUSIONS
The contribution of the excitations to high-nl shells has been studied for the electron-impact excitation-autoionization process. Calculations for the W 27+ ion illustrate that the excitations to the high-nl shells (9 n 40) increase the cross section values by a factor of two for the indirect part of the ionization process. The largest contribution to the indirect part comes from the excitations to l = 4 orbital.
Surprisingly, large contribution also arises from the excitations to l = 5 orbital.
The EA cross sections for excitations from the 4d shell to the shells with n