Electron Impact Excitation of Extreme Ultra-Violet Transitions in Xe 7 + – Xe 10 + Ions

: In the present work, a detailed study on the electron impact excitation of Xe 7 + , Xe 8 + , Xe 9 + and Xe 10 + ions for the dipole allowed (E1) transitions in the EUV range of 8–19 nm is presented. The multi-conﬁguration Dirac–Fock method is used for the atomic structure calculation including the Breit and QED corrections along with the relativistic conﬁguration interaction approach. We have compared our calculated energy levels, wavelengths and transition rates with other reported experimental and theoretical results. Further, the relativistic distorted wave method is used to calculate the cross sections from the excitation threshold to 3000 eV electron energy. For plasma physics applications, we have reported the ﬁtting parameters of these cross sections using two different formulae for low and high energy ranges. The rate coefﬁcients are also obtained using our calculated cross sections and considering the Maxwellian electron energy distribution function in the electron temperature range from 5 eV to 100 eV. 9 5p + 4d 9 4f + 4d 9 6p + 4d 9 5f + 4d 9 7p + 4d 9 6f) for Xe 8 + , 4p 6 4d 9 –(4p 6 4d 8 5p + 4p 6 4d 8 4f + 4p 5 4d 10 ) for Xe 9 + and 4d 8 –(4d 7 5p + 4d 7 4f + 4p 5 4d 9 ) for Xe 10 + . These arrays result into 9, 18, 75 and 57 E1 transitions in Xe 7 + through Xe 10 + in EUV range. We have used multiconﬁguration Dirac–Fock method within RCI approximation to calculate the energy levels, wavelengths and transition rates. These results are compared in detail with the previously reported measurements and theoretical calculations. The target ion wavefunctions are further implemented in the evaluation of the transition ( T − ) matrix amplitude using relativistic distorted wave (RDW) approximation and excitation cross sections are obtained up to 3000 eV electron energy. The analytical ﬁtting of the electron excitation cross sections is also performed as it is more convenient to feed the analytical expression with ﬁtting parameters for plasma modeling. Further, assuming electron energy distribution to be


Introduction
Spectroscopic and collisional data of highly charged xenon ions in the extreme ultraviolet (EUV) spectral range play a vital role in several research areas. For example, laser produced xenon plasma exhibits [1] the possibility to become an EUV source for the next generation lithography. Xenon ions are detected in the UV spectrum of the astrophysical objects viz., hot DO-type white dwarf [2] and planetary nebula [3]. In the next generation fusion reactor ITER, xenon is expected to be used as edge plasma coolant. Xenon ions being used in ion thruster for electric propulsion [4] plays key role in making the modern space exploration cheaper. Since emissions from various charged species of xenon ions carry information about the plasma parameters and impurities, their atomic structure and dynamical properties in the EUV range are essential for the accurate diagnostics of the aforementioned plasmas. Therefore, in this work we have focused on the electron impact excitation of the electric dipole (E1) transitions in Xe 7+ , Xe 8+ , Xe 9+ and Xe 10+ ions in the EUV region 8-19 nm. We consider excitation of E1 transitions that are responsible for the most intense lines of the spectra.
To determine the emission properties of xenon ions, experiments have been performed with either laser or gas discharge-produced plasmas. Churilov and Joshi [5] recorded xenon spectra in the 7-17 nm region on a 10.7 m grazing incidence spectrograph and analyzed the 4p 6 4d 9 -(4p 6 4d 8 5p + 4p 6 4d 8 4f + 4p 5 4d 10 ) transition array of Rh-like Xe 9+ . They also identified the resonance transitions arising from the excited 4d 9 (6p + 5f + 7p + 6f) states of Pd-like Xe 8+ and 4d 10 5s 2 S-4d 9 5s4f 2 P transitions for Ag-like Xe 7+ . Churilov et al. [6] observed the transition array 4d 8 -(4d 7 5p + 4d 7 4f + 4p 5 4d 9 ) of Xe 10+ using a low inductance vacuum spark and a 10.7 m grazing incidence photograph in the EUV region 10.5-15.7 nm. These lines were analyzed using Hartree-Fock (HFR) calculations in relativistic mode with the help of the Cowan suite of codes [7]. Fahy et al. [8] reported the EUV spectra of Xe 6+ to Xe 41+ in the wavelength region of 4.5 to 20 nm using the electron beam ion trap (EBIT) facility at NIST while varying electron beam energy from 180 eV to 8 keV. They also calculated the transition probabilities and wavelengths using the HF approximation with the Cowan code [7]. Ali and Nakamura [9] observed the EUV spectra of Rh-like Xe 9+ -Cd-like Xe 6+ and Cu-like Xe 25+ -Se-like Xe 20+ using a compact electron beam ion trap (CoBIT) and a flat-field grazing incidence spectrometer in the wavelength range of 15-20 nm with an uncertainty of 0.05 Å. The electron beam energy was varied between 200-890 eV during these measurements. Ali and Nakamura [10] also used their experimental facilities to record EUV spectra of highly charged Xe 8+ -Xe 11+ and Ba 18+ -Ba 21+ ions in the wavelength range 9-13 nm. Merabet et al. [11] studied spectra of various xenon ions (Xe 2+ -Xe 10+ ) in the EUV region 10-16 nm using a compact electron cyclotron resonance ion source (CECRIS) equipped with a grazing monochromator operating in 4-90 nm.
Various theoretical studies have been carried out to report energy levels, wavelengths, oscillator strengths and transition probabilities of xenon ions. Safronova et al. [12] calculated the atomic properties of Pd-like ions Xe 8+ with nuclear charge ranging from Z = 47 to 100 using relativistic many-body perturbation theory (RMBPT) with Breit correction. Ivanova [13] used the relativistic perturbation theory with a model potential to calculate the energy levels of Ag-, Pd-and Rh-like ions with Z = 52-86. Motoumba et al. [14] reported transition probabilities and oscillator strengths for the transition array 4d 8 -(4p 5 4d 9 + 4d 7 5p + 4d 7 4f) of Xe 10+ in the EUV spectral range of 10.2-15.7 nm. These results were obtained using two different methods viz., the semi-empirical pseudo-relativistic Hartree-Fock (HFR) method and the relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) theory within the relativistic configuration interaction (RCI) approximation. Motoumba et al. [15] also employed the above two methods to report transition probabilities and oscillator strengths for 92 spectral lines of Xe 9+ ion in the range of 11-16.4 nm. Shen et al. [16] used Flexible Atomic Code (FAC), based on a fully relativistic approach, to calculate the energy levels, oscillator strengths, electron impact collision strengths as well as effective collision strengths for Xe 10+ .
It is clear from the above discussion that most of the previous experimental or theoretical studies on Xe 7+ -Xe 10+ ions have focused on their spectroscopic properties, while the electron impact cross section data are scarcely reported. However, various studies in the past have clearly demonstrated that using accurate cross section results in a collisional radiative model provides a better agreement with the measurements on the plasma parameters, viz., electron temperature and density [17][18][19][20]. Therefore, reliable cross sections are essential for the success of any plasma model. In general, suitable theoretical methods are employed to carry out cross section calculations due to limitations, such as accurate identification of the fine-structure levels for open shell ions, in performing the scattering experiments.

Theory
In order to calculate the energy levels, wavelengths and transition probabilities, we have obtained MCDF wavefunctions of Xe 7+ -Xe 10+ ions using GRASP2K code [21]. In the MCDF method, the atomic state functions (ASFs) are written as linear combination of configuration state functions (CSFs) having same parity P and angular momentum quantum number J, as follows: Here a i refers to the mixing coefficient of the CSF Φ i (PJ M) which are anti-symmetrized products of a common set of orthonormal orbitals. In our calculations, we take as many CSFs as are having at least 0.001% value of the mixing coefficient. The configurations that are included in the atomic-structure calculations of xenon ions are listed in Table 1. These configurations are shown here in their non-relativistic notations. The MCDF method implements a self-consistent field procedure for obtaining the radial functions and the mixing coefficients. Further, we performed RCI calculations by considering the Breit and quantum electrodynamic (QED) corrections in the Dirac-Coulomb Hamiltonian. The transition probabilities are computed from the matrix element of dipole operator of the electromagnetic field. Table 1. Configurations of the initial and final states and the CSFs in non-relativistic notations.

Atomic-Structure Calculations
We have used GRASP2K code [21] to perform MCDF and RCI calculations to obtain energy levels, wavelengths and transition rates of Xe 7+ -Xe 10+ ions. Our energy values are presented and compared with other theoretical and experimental results through Tables 2-5 for the four ions. The fine-structure states are represented in the relativistic j − j coupling scheme in which all shells, excluding s, split into two subshells with j = l ± 1/2. For example, a p shell will be broken asp with j = 1/2 and p with j = 3/2. In order to identify the levels, their indices are assigned in each table. This will help to clearly recognise the states for which wavelengths, transition rates, electron impact cross sections and excitation rate coefficients will be presented.        [23]. In addition to the j − j coupling representation, we have also included the notations of the states used in the NIST database to make the comparison convenient between the two sets of the results. We find from Table 2 that our calculated energies show an average deviation of nearly 1.5% with the corresponding energies from the NIST database [23]. A maximum variation of nearly 3% is found for the 5s4f 3 P 1/2,3/2 levels. We have listed only those levels in Table 2 that are reported to be involved in emitting intense lines in the EBIT measurements of Fahy et al. [8] and Ali and Nakamura [9].
For Xe 8+ , in our calculations we got two levels with leading contribution from 4d 9 7p 1 P 1 , one at 138.7018 eV (53.65% 4d 5 5/2 7p 3/2 1 P + 37.44% 4d 3 3/2 7p 1/2 3 P + 8.79% 4d 3 3/2 7p 3/2 3 D) and another at 140.2617 eV (44.29% 4d 5 5/2 7p 3/2 1 P + 30.70% 4d 3 3/2 7p 1/2 3 P + 24.90% 4d 3 3/2 7p 3/2 3 D). Considering the maximum contribution, we have classified the level at 138.7018 eV as 4d 9 7p 1 P 1 , and 140.2617 eV as 4d 9 7p 3 P 1 . This changed the energy order of 1 P 1 and 3 P 1 in our calculations with respect to those reported by Churilov and Joshi [5]. As can be seen from Table 3, the agreement between the measurements [5] and our results is within 0.8% for most of the cases. The maximum difference of nearly 3 eV is found for the 4d 9 4f 1 P 1 level. The energy levels of Xe 9+ are listed in Table 4 and are compared with the measurements [5] as well as HFR and MCDHF calculations of Motoumba et al. [15]. The open-shell structure of Xe 9+ leads to the formation of a large number of closely spaced fine-structure levels for its ground and excited state configurations. Consequently, it becomes extremely difficult to correctly identify these states as well as to label them uniquely in LS coupling notations. Churilov and Joshi [5] reported Xe 9+ levels with the wavenumbers (in cm −1 ) which are also included in Table 4 to guide us in right recognition of the states. From our calculations, we found that the inclusion of the triple excitation 4d 6 4f 3 improves the match between the present energies and measurements for the higher 4d 8 4f levels, while it deteriorates the agreement for other levels. Thus we have considered two sets of calculations for Xe 9+ , one with and the other without including the CSF 4d 6 4f 3 . The energies marked with * in Table 4 indicate the inclusion of the CSF 4d 6 4f 3 . For 4p 5 4d 10 levels, our energy results overestimate the measurements [5] and theoretical results [15] by nearly 2 eV. Except for this transition, in most of the cases our energies show better agreements with the experimental results than the MCDHF calculations [15]. Table 5 presents a comparison of the present energies with the experimental energies from Churilov et al. [6] and RCI and HFR calculations of Motoumba et al. [14] for Xe 10+ . Similar to Xe 9+ , Xe 10+ has an open shell structure and hence, we have included the wavenumbers reported in [6] so that the small spaced levels can be rightly identified. We learnt that adding the CSF 4p 4 4d 10 improves the energy of the 4d 8 levels, while including the CSF 4d 5 4f 3 with triple excitation improves the energy of the higher 4d 7 4f levels. The order of a few levels from 4d 8 , 4d 7 5p and 4d 7 4f configurations are not as per the order reported in the measurements [6]. Similar cases are also observed in the RCI results [14]. Our reported energies show a deviation of nearly 2-4 eV for the 4p 5 4d 9 levels, however, they are in good agreement with the RCI calculations by Motoumba et al. [14].
The comparison of our calculated wavelengths and transition rates with other theoretical and experimental results is shown through Tables 6-9. For Xe 7+ , Table 6 includes the measurements from NIST EBIT and Cowan code calculations reported by Fahy et al. [8], compact EBIT results from Ali and Nakamura [9] as well as HFR calculations of Churilov and Joshi [5]. Though Table 6 shows a maximum deviation of 3.5 Å for levels of 4d 9 5s4f configurations with indices 9 and 10, a good agreement is found between our reported transition rates and the calculated results from Cowan code [8].
Wavelengths and transition rates for Xe 8+ from the present work are reported and compared in Table 7 with the measurements and other calculations [5,[8][9][10]13]. Overall, our calculations are in good agreement with other results. However, a maximum deviation of 3.4 Å is found in the wavelength corresponding to 1 → 15 (4d 10 1 S 0 → 4d 9 4f 1 P 1 ) transition. This is because from Table 3 our calculated energy of the 4d 9 4f 1 P 1 level is overestimated by nearly 3 eV in comparison to the result reported by Churilov and Joshi [5]. It is further noticed for the above transition that our calculated wavelength shows a better match with that from Ivanova [13] and there is a good agreement among various values of the transition rate.   [5], a * -calculated wavelengths from the energy levels [5], b-Fahy et al. [8], d-Ali and Nakamura [10], e-Ali and Nakamura [9]. Theoretical results: c-Fahy et al. [8], f -Ivanova [13].   For Xe 9+ , our wavelengths and transition rates are compared with the measurements [5] and HFR and MCDHF results [15] in Table 8. Our reported wavelengths show a good match with the experimental results [5] with an average difference of 0.5 Å. The two transitions 1 → 4 and 2 → 4, where 1, 2 and 4 refer to the indices assigned to the states of Xe 9+ , show a maximum difference of nearly 5 Å. However, their transition rates are in good agreement with the reported results from Churilov and Joshi [5].
In Table 9, measurements and theoretical results from Churilov et al. [6] as well as HFR and RCI results of Motoumba [14] are included for Xe 10+ along with our calculated wavelengths and transition rates. Previous studies [6,24] showed that there are two possible strong transition arrays of Xe 10+ in 11.1 nm-11.3 nm and 13 nm-14 nm regions with possible applications in EUV Lithography [1]. Thus, we have reported results only for the transitions that fall in these ranges for Xe 10+ . The HFR and RCI wavelengths are calculated from the energy levels provided in [14]. Our results show a maximum deviation of nearly 3.5 Å from measurements and HFR calculations. This discrepancy is found for the transitions from the 3rd, 7th and 8th states to the 57th state. Overall, a better match is seen between the present results and the RCI calculations. Our calculated transition rates agree well with the corresponding values from Churilov et al. [6] except for a few cases, that is, 3 → 25, 3 → 21 and 1 → 11 transitions. However, the present transition rates are in reasonable agreement with the RCI calculations for these transitions.

Cross Sections and Rate Coefficients
The atomic wavefunctions of the four ions are used in our RDW program to calculate the electron impact excitation cross sections for the E1 transitions in Xe 7+ -Xe 10+ ions. In the previous subsection, we have given a detailed comparison of our calculated results for energy levels, wavelengths and transition rates with other experimental and theoretical results and found an overall satisfactory agreement. This ensures the quality of the target ion's wavefunctions that are crucial in determining the accuracy of the scattering parameters. Moreover, the RDW method has been successfully implemented in the previous work on a variety of targets from closed to open-shell systems and neutral atoms to multiply or highly charged ions atoms/ions [25][26][27][28][29][30]. It has also been found that using RDW cross sections in a collisional radiative (CR) model provides plasma parameters that are in better agreement with the measurements [31][32][33][34]. Therefore, the success of a CR model depends heavily on the accuracy of the collision cross sections being fed to the model. In this connection, we have calculated cross sections for 9, 18, 75 and 57 transitions, respectively, for Xe 7+ , Xe 8+ , Xe 9+ and Xe 10+ . Their excitation energies, as discussed earlier, lie in the EUV region. For the sake of simplicity in presenting our results, we have shown only a few transitions graphically through Figure 1 for Xe 8+ . However, cross sections for all the transitions considered in the four ions are provided in the supplementary file through Tables S1-S4 in the incident electron energy range 200-3000 eV. We notice the usual behaviour of the cross sections from Figure 1, that is, they decrease with increasing electron energies and their magnitudes follow the increasing order of the transition rates. Transitions which involve the change of the spin of the state have lesser cross sections as compared to those with the same spin.
Further, to make available our cross sections in a convenient manner, we have performed the fitting of our cross sections Equation (4) with two analytical forms. The first form is a rational fit and suitable for low energy, given by: where σ a→b is the excitation cross section from the initial level a to final level b and E is the energy of the incident electron. Both the cross section and the energy are considered in atomic units. x i s and y i s are fitting coefficients. The second fitting, appropriate for high energy, is performed using the Bethe-Born formula, that is, The Bethe-Born fitting is valid for energy above 2000 eV in the present case. The fitting parameters are provided in Tables 10-13 for Xe 7+ , Xe 8+ , Xe 9+ and Xe 10+ ions, respectively. The fitted and calculated cross sections agree within 5%. We have also obtained the rate coefficient k a→b at an electron temperature T for a transition from initial level a to final level b. For this purpose, our calculated excitation cross sections are used in the following expression: where m e represents the mass of electron, k B is the Boltzmann constant, E ab denotes the excitation threshold energy for the transition from a to b and σ a→b (E) is the calculated cross section at the incident electron energy E. The rate coefficients are provided through Tables 14-17 for Xe 7+ -Xe 10+ ions in the electron temperature range 5-100 eV. The values of rate coefficients rise rapidly at first and then there is a slower logarithmic increase. In order to clearly demonstrate this trend, Figure 2 displays rate coefficients for the transitions reported in Table 7 for Xe 8+ . The same behaviour has been noticed in our previous work on excitation of highly charged xenon ions [25].

Conclusions
We employed the MCDF approach within the framework of the Dirac-Coulomb Hamiltonian, including the Breit and QED corrections using the GRASP2K program [21] and calculated the energy levels, wavelengths and transition rates for the electric dipole allowed transitions of Xe 7+ , Xe 8+ , Xe 9+ and Xe 10+ ions in the EUV range of 8-19 nm. These results are compared with other reported experimental and theoretical results and, overall, a good agreement is found. After confirming the reliability of our ionic wavefunctions, we used them in the RDW method to calculate the excitation cross sections for a total of 159 transitions in the four ions. To make our cross sections conveniently available for plasma modelling, we obtained the fitting parameters for these cross sections for both low and high incident electron energies. The maximum error in fitted cross sections is found to be well within 5% for most of the cases. Further, these cross sections are used to calculate the excitation rate coefficients for several electron temperatures ranging from 5 to 100 eV, assuming a Maxwellian electron energy distribution. Our cross sections and rate coefficients are reported for the first time, as no other experimental or theoretical results are available. We hope our results will be useful for the successful interpretation of EUV emissions from various sources.
Author Contributions: Both the authors have contributed equally in performing calculations and preparing the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available in the article or supplementary material here.