Study of single electron capture in O6+ + He collisions

Research on electron capture (EC) process are undoubtedly helpful for maturing theoretical models on ion-induced collision especially for low-energy region. In this work, a two-active-electron semiclassical asymptotic-state close-coupling method was used to calculate the total and l-solved state-selective single EC cross sections of O6+ + He collisions in the energy range of 0.3–100 keV u−1, accompanied with experimental measurements in the energy range of 2.63–37.5 keV u−1 with an uncertainty of 11% in good agreement. Above 4.5 keV u−1, the state-selective cross section of n = 5 was reported experimentally for the first time. Calculations with multiple theoretical methods were gathered and compared with present calculations. The importance of two-active-electrons correlation and large basis sets in theoretical calculations was found, and discrepancies between previous theoretical and experimental results can be explained by the present results.


Introduction
In the course of ion-atom/molecule collisions, electron capture (EC), ionization and other inelastic processes may occur alone or simultaneously. Among these processes, their absolute and relative cross sections depend strongly on the impact energy and are often closely coupled with each other. Therefore, the close coupling method was established as a standard method for the description of these inelastic processes, which was based on the impact parameter approximation allowing the relative motion between the projectile-target to be regarded as straight-line trajectories. With this approximation, a time-dependent Schrödinger equation for the electrons called the eikonal equation could be obtained, and then solved by the electronic wavefunction expanded on atomic orbitals or molecular orbitals by Ferguson and McCarroll firstly [1,2]. This method has been used involving one active electron for a long time [3][4][5]. Fritsch and Lin [6] developed the close-coupling method based on the atomic orbital expansion (AOCC) scheme to study one-electron transfer process for the (quasi-)two-electron collision system O 6+ + He and C 6+ + He, and the electron correlation was taken into account in their calculation. However, double-EC channels were not included in this calculation, and the excitation channels were also not enough due to a fact that a very small atomic orbital expansion basis had been used for He or He + ions. The discrepancies with the experimental data had been observed for their partial charge transfer cross sections in the considered energy region. Recently, a two-active electrons close-coupling method has been developed by [7][8][9][10][11][12] which includes the fully correlation between two electrons. Moreover, many different competing reaction channels for the collision systems can be included as the same time. In order to validate the accuracy of these methods and compare their respective superiority, more studies integrated with both experiments and theoretical calculations are desired currently. When the collision happens at impact velocities less than one atomic unit, the EC process is usually a dominant collision mechanism [13] which becomes a key research object in collision studies especially for low-energy region.
In this work, the single EC (SEC) process between O 6+ and He is concerned in consideration of its great importance in diverse plasma environments. For example, oxygen is observed as one of the most abundant impurities in present tokamak, and helium-like O 6+ ions are found in considerable quantities at plasma edge and divertors [14][15][16][17]. These ions may have EC process with neutral helium atoms which are rich as the products of D-T fusion reactions. Studies on their state-selective cross sections can be exploited for interpreting the measured emission spectra to diagnose the impurity density [18] and assess their influence on plasma operations. In addition, O 6+ and Helium are of great abundant in the solar wind [19,20] and the exospheres of some astrophysical objects such as Jupiter [21], respectively. Charge exchange between these kinds of ions and neutral gas materials is confirmed as an important source of extreme ultraviolet and soft x-ray emissions from astrophysical objects in the solar system [20][21][22][23]. To better understand the phenomenon of x-ray emission in astrophysics, the cross section data of EC is required for modeling the hot plasma impacting cold media.
To the best of our knowledge, cross section calculations for the SEC of O 6+ -He have been carried out by the atomic-orbital expansion [6,24] and molecular orbital expansion [25] methods in the energy region of 0.5-300 keV u −1 and 0.1-7 keV u −1 , respectively. Machacek [26] also provided theoretical results by classical trajectory Monte Carlo (CTMC) method at 1.17 keV u −1 and 2.33 keV u −1 . On the experimental side, a few measurements on the SEC cross sections between O 6+ and He were reported in a low-energy region below 7 keV u −1 [26][27][28][29]. Nevertheless, discrepancies between these experimental and theoretical results exist. In corresponding energy region, the transferred electron is mainly captured into the n = 3 state of O 5+ ion, while for the cross section of 3d state, there is an apparent difference between the previous theoretical results and experimental results (figure 5). As for the n = 4 state, completely different theoretical curves can be found (figure 6). To explain these discrepancies, we present both experimentally and theoretically the total and state-selective SEC cross sections for O 6+ colliding with He. The reaction process can be written as Experimental n and l measurements are extracted from the recoil momentum spectra of He + using the technique of cold target recoil ion momentum spectroscopy (COLTRIMS) in the impact energy range of 2.63-37.5 keV u −1 . The results are compared with those from the calculation by a two-active-electron semiclassical asymptotic-state close-coupling (SCASCC) method from 0.3 to 100 keV u −1 and show a good consistency. The SCASCC method is a semiclassical non-perturbative method where eikonal equation were solved based on the asymptotic-state expansion of the electronic wavefunction, therefore it is suitable for the considered energies. It should be also mentioned that the correlation of two active electrons was taken into account completely in this method compared to the previous AOCC calculation of [24]. For another AOCC calculation of Fritsch and Lin [6], although electron correlation was considered, double-electron channels were excluded in their calculation and the use of small expansion basis set on He (or He + ion) make the excitation channels cannot be fully included. Furthermore, with the improvement of computing power, the basis sets that couple more open channels were calculated larger than before. All these advantages can be helpful for explaining the discrepancies mentioned above.
The present article is organized as follows: The experimental setup and data acquisition are described in section 2. A brief overview of the two-electron SCASCC method is given in section 3, and the basis sets and model potential are also reported. Experimental and theoretical electron-capture cross sections are presented and compared with other available data in section 4. Conclusions are given at last. Atomic units will be used throughout this paper, unless explicitly indicated.

Experimental method
The present experiments were performed on the highly charged ions (HCI) Collision Platform at Fudan University [30,31] using a 14.5 GHz electron cyclotron resonance (ECR) ion source which could provide HCI beam in an acceleration filed from 5 kV to 150 kV, the beam line A with a gas cell for measuring total EC cross sections and the beam line B with a COLTRIMS for state-selective EC cross section measurements. Figure 1 describes the experimental set-up.
In short, oxygen ions were extracted from the ECR at the desired energy. Mass-charge selected by a 90 • magnetic analyzer, a collimated beam of O 6+ ions entered the beam line A or B by initiating a 45 • magnetic analyzer or not. The background vacuum of beam lines was better than 10 −6 Pa.

Measurement of the total EC cross sections
Details of the total EC cross sections measurements in the beam line A have been reported elsewhere [31]. Briefly, selected O 6+ ions underwent EC collisions with neutral He gas in the gas cell, where the pressure inside was controlled below 0.04 Pa to fulfill the single collision condition. After the interaction, the charge state of the projectile beam was analyzed with an electric deflector and finally detected with a position-sensitive detector (PSD) composed of an 80 mm-diameter microchannel plate and a delay line anode. Based on the growth-rate method [32], the total SEC cross sections were given by the equation: Here F q−1 stands for the fraction of ejected ions with charge of q−1 to the total. σ q,q−1 is the total SEC cross sections. l and P are the length and pressure of the gas cell, respectively. k is the Boltzmann constant, and T is the temperature of the gas cell in units of kelvin. As l, k and T were constant during the experiments, σ q,q−1 could be derived from the linear relationship of F q−1 as a function of P.
The total error of σ 6,5 was about 11% deduced from the measurement error of T, l, P and F q−1 based on the equation (2), which had been analyzed in detail in our previous work [31].

Measurement of the state-selective EC cross sections
The beam line B with a COLTRIMS [33,34] was built for performing state-selective EC measurements [30]. In brief, the target helium gas was ejected through the supersonic gas jet perpendicular to the direction of projectiles. After the EC interaction, recoil ionized He was extracted by a 3 V cm −1 focused electric field along the time of flight (TOF) axis and collected by an 80 mm in diameter PSD. Downstream of the TOF spectrometer, scattered charge-changed oxygen ions were analyzed by electric deflectors and detected by another PSD. The coincidence signal between the scattered ion and the recoil ion served as a common stop for the multi-hit time-to-digital converter. With the TOF and coincident position information, it is feasible to identify the SEC events and reconstruct three-dimensional momenta of recoil ions. The resolution of used COLTRIMS was approximately 0.25 a.u. in momentum limited by the Gaussian distribution of supersonic gas jet and instrumental broadening.
After SEC, the recoil momentum of He + ions along the projectile direction, which is defined as the X-dimension momentum p X , has a specific relation with the state selectivity of captured electrons. According to the conservation of energy and momentum, p X gained by recoil ions follows [35,36] where v is the velocity of O 6+ ion, and Q is the binding energy difference, which is defined as Q = ε f −ε i (ε i and ε f are binding energy of the captured electron in the initial ground-state target and the final excited ion, respectively).
In O 6+ + He system, ε i equals 24.59 eV, and ε f is determined by the quantum numbers n and l of the transferred electron in O 5+ . All the data of binding energies was found from the NIST Atomic Spectra Database [37]. In other words, by measuring the p X distribution of recoil He + , the branch ratio of state-selective SEC in different final states can be acquired.
Combined with the absolute total SEC cross sections, the state-selective SEC cross sections of O 6+ colliding with He can then be normalized. However, the σ 6,5 measured in beam line A contained not only the true SEC process, but also the transfer ionization process and autoionizing double EC process as the growth-rate method could not distinguish the final state of recoil He ions. To get the σ 6,5 revised, the proportion of true SEC process in σ 6,5 was acquired with COLTRIMS by comparing the counts of recoil He + and He 2+ both with a coincident selection on scattered O 5+ . All the given experimental results of state-selective SEC cross sections below have been revised in that way.
The uncertainty of measured state-selective SEC cross sections mainly resulted from the error of σ 6,5 and statistical error of counting recoil ions. In order to raise the resolution and eliminate the ground noise, Gaussian fittings were used to get the area of all peaks and calculated their relative proportion. As for the result of state-selective SEC cross sections, the total error was calculated by the error transfer formula with the error of corresponding σ 6,5 and uncertainties of each peak area given by Gaussian fittings.

Theoretical method
In the present work, a two-active-electron SCASCC method [7][8][9][10][11][12] was used to investigate the collisions of O 6+ with He. Here introduce the main features of this method briefly.
The two electrons time-dependent Schrödinger equation under the impact parameter approximation can be written as ⃗ r i and⃗ r p i =⃗ r i − ⃗ R (t) are the position vectors of the electrons with respect to the target and the projectile, respectively. The relative projectile-target position ⃗ R (t) is defined as ⃗ R (t) = ⃗ b +⃗ vt, where ⃗ b and ⃗ v are the impact parameter and velocity, respectively. V T (V P ) is the potential between the active electron and target (projectile) [38,39] which includes inner electrons. For the collisions of O 6+ with He, they are expressed as The Schrödinger equation is solved by expanding the wave function accordingly on a set of electronic states of isolated collision partners where the superscripts T and TT (P and PP) denote the states and corresponding energies for one or two electrons are on the target (projectile), respectively. For both electrons, the projectile states contain plane-wave electron translation factors e i⃗ v·⃗ r i −i 1 2 v 2 t in order to ensure Galilean invariance of the results. After inserting the wave function into the Schrödinger equation equation (4), a first-order coupled differential equations can be obtained, which are written in matrix form as where c is the column vector of the time-dependent expansion coefficients, S and M are the overlap and coupling matrices, respectively. Equation (9) could be solved by predictor-corrector, variable-time-step Adams-Bashford-Moulton method for a given impact parameter b and projectile velocity v, the probability for a transition from the initial state i to a final state f could be given by equation (10) The corresponding cross sections can be calculated from the above probabilities as equation (11) σ In calculations, A set of 77 GTOs (12 for l = 0, 8 × 3 for l = 1, 4 × 5 for l = 2, and 3 × 7 for l = 3) was used on the projectile (O) center, and a set of 28 GTOs (10 for l = 0 and 6 × 3 for l = 1) was used on the target (He) center. These allowed the inclusion of 147 TT (two electrons on He), 1144 TP (one electron on He + and the other electron on O 5+ ) and 1583 PP (two electrons on O 4+ ) states in the present calculation. The convergence of the GTOs was also checked by calculating the cross sections with different size of GTO basis sets, difference was evaluated less than 10% in the energy range considered for total and dominant 3l state-selective cross sections. Table 1 shows the energies for the important states of O 5+ and compared with the corresponding experimental data from NIST [37]. The overall agreement of calculated energies with the NIST data was within 1%.

Total SEC cross sections
The present experimental and theoretical total SEC cross sections are shown as a function of impact energy in figure 2 together with other available data for comparison. A clear tendency is shown that σ 6,5 gradually rises when Energy <2 keV u −1 , stays at about 14 × 10 −16 cm 2 for 2-30 keV u −1 , and then decays rapidly above 30 keV u −1 . In 0.3-30 keV u −1 , the present SCASCC results agree better with all the experimental data compared with AOCC and CTMC results, while for higher impact energies, the SCASCC curve decreases faster than the present measurements. This disagreement may originate from a fact that the reaction channels of 5g and n ⩾ 6 states of O 5+ ion have not been considered in the present SCASCC calculations, but with the increase of incident energy, their contribution becomes gradually important. Figure 2 also shows that the present SCASCC results are in general agreement with those of AOCC method [6] in the energy range below 8 keV u −1 , but for the energies above, the AOCC results are significantly larger. The discrepancy is mainly from the neglect of double-EC process and the use of small AO expansion basis in Fritsch's [6] calculation. With good agreement among all the data mentioned above, this work successfully extends the research on σ 6,5 in O 6+ + He to the higher impact energies at 37.5 keV u −1 experimentally and 100 keV u −1 theoretically.

N-and nl-state-selective SEC cross sections
The total SEC cross sections give the probability that one electron is captured, but without the information of state (n, l) for the captured electron. The state-selective distribution in SEC process for O 6+ + He are measured with a COLTRIMS, shown in figure 3. In the energy range of 2.63-37.5 keV u −1 , the transferred electron from the target He is mainly captured into the n = 3 state of O 5+ ion. And with the increase of incident energies, the branch ratio of n = 4 and n = 5 rises obviously, which means more reaction channels are opened as the collision energy increases. For convenience of application, all the measured and calculated data including the total and state-selective SEC cross sections are listed in tables 2 and 3, respectively. Figure 4 shows the n-resolved SEC cross sections as a function of impact energies together with the data reported in [6,24,25,28,29] for the collisions of O 6+ with He. For n = 3 channel, the SEC cross sections increase with increasing impact energies, and then decrease dramatically after reaching a maximum at E ≈ 4 keV u −1 . The present theoretical results and experimental data agree well with other data in the overlapping energy region. It is notable that the present SCASCC results are generally smaller than those calculated by Fritsch [6], and the reason should be the same as mentioned in the section 4.1. A similar phenomenon also exists in the molecular orbital close coupling (MOCC) results of Shimakura [25] below 4 keV u −1 , but a better fit can be found among the present calculation and other data.
For the capture channel n = 4, the MOCC results of Shimakura [25] are significantly different from other results due to insufficient reaction channels included in the MOCC calculations. The present SCASCC results are in better agreement with the present experimental data compared with other AOCC results [6,24] for the impact energies above 10 keV u −1 . However, below 10 keV u −1 , the SCASCC results are significantly larger than the present measurements, while better agreement is found with AOCC results from Zhao et al [24] and the experimental data from Dijkkamp [28], which needs more experimental data to assess this discrepancy. The obvious difference between the present experimental data and the measurements from Dijkkamp [28] in the overlapping energy range probably come from the VUV spectrometer detection used by Dijkkamp [28] which might bring more complex ground noises and wrong judgments on sources of collected photons.
With regard to n = 5, an overall upward trend of cross sections with the increase of impact energy is demonstrated both in the present experimental and theoretical results except for a trough at nearly 15 keV u −1 within the SCASCC calculation, which shows a considerable distinction compared with calculations by Zhao et al [24]. It should be noted the close encounter of two theoretical results at 10 keV u −1 is as an illustration of an accidental coincidence, the lack of correlation of electrons lead to the divergence in the low energy. It is difficult to obtain precise results since the cross sections of n = 5 are smaller than 1 × 10 −16 cm 2 , and the lack of cross sections of 5g also causes the trough. Thus, the fact that present theoretical results are smaller than the experimental data in comparatively higher energy region can be understood.     The state-selective cross sections for electron captured to the 3l states of O 5+ ions are presented in figure 5. Similarly, the available data from previous research is also shown for comparison. As can be seen from figures 5, 3p is the main reaction channel for O 6+ + He. For the 3s and 3p capture cross sections, except for the CTMC results of Machacek et al [26], all the theoretical and experimental results show the same tendency in the considered energy range. And for the 3d capture channel, the present theoretical results agree better with others experimental data. Figure 6 shows that the cross sections for electron captured to the 4l states of O 5+ ion are significantly less than 3l states, therefore, it becomes more difficult to obtain the accurate results. For the 4s channel, no experimental data can be compared and unfortunately, there exist discrepancies among the theoretical values in the entire energy region. For 4p, the results from Fritsch and Lin [6] are totally different from other results, since only the ground state, 2s of He and n = 3, 4 states of O 5+ were included in its expansion. The present SCASCC results are in better agreement with the experimental data than those of the other theoretical  methods [6,[24][25][26] for 4l (l = d, f ) captured channels in the overlapping energy region. The discrepancies reveal that sufficient AO bases are needed for weak reaction channels.

Conclusion
In the present work, the investigation of SEC process in collisions of O 6+ with He is performed both experimentally and theoretically. The total cross sections for the electron captured to O 5+ (1s 2 nl) from He atoms have been measured with an uncertainty of 11% in the energy range of 2.63-37.5 keV u −1 , and the branch ratios of different state-selective channels (n = 3, 4, 5 and partially 3s) have been obtained from the target recoil momentum distribution measured by the COLTRIMS. Total and state-selective cross sections are also calculated by the two-active-electron SCASCC method in the energy range of 0.3-100 keV u −1 . General agreement between the present experimental and theoretical results is obtained for both the total SEC cross sections and the state-selective ones in the overlapping energy range. The importance of electronic correlations and the use of large expansion basis sets considered in present theoretical method for EC process is found in improving the calculation of cross sections significantly compared with other methods. With increasing of the collision energy, the electron tends to be captured to the higher excited states of the projectile. Thus, large basis sets are needed to obtain more accurate calculations of state-selective cross sections, especially for weak reaction channels. These energy-dependent total and state-selective SEC cross sections can also provide essential data for the analysis of spectral emission and plasma simulations in the present tokamaks, astronomical objects, etc.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).