Projectile and target excitation in He⁺ + He collisions at intermediate energies

He⁺ + He is one of the simplest atomic collisional systems containing only three electrons. It is, however, sufficiently complex to give rise to the main types of inelastic reactions (electron capture, excitation, ionization, and their bielectronic counterparts) observed in ion–atom collisions, since both target and projectile carry active electrons. It has therefore attracted a great deal of interest for several decades.^1–22 The majority of these studies have concentrated on electron capture (charge transfer) processes, since these are generally the dominant channels at intermediate impact energies. By contrast, electronic excitations have not usually been included in these studies owing to the fact that they are comparatively less likely to be active at intermediate impact energies. For target excitations occurring in the course of He⁺ and He collisions

some experimental results can be found in Refs. 9, 12, 16, and 19 for impact energies ranging from 5 keV/u to 75 keV/u and for nl = 2s, 2p, and 3s, while for the projectile excitation

only a single series of experimental data, in the energy range 6.25 keV/u–35 keV/u, has been published.¹⁶ 

From a theoretical point of view, the investigation of He⁺ + He collisions in the intermediate impact energy region still remains a challenge: perturbative approaches or approximate calculations using a model potential with only one (or two) active electrons may be inadequate owing to electronic correlation effects and the strong coupling between various channels; see, e.g., Ref. 4. Early in 1967, the calculated cross sections for capture and target excitation into He*(1s2s^1,3S) were first reported by Sural et al.²⁰ using a three-electron coupled-channel method. These authors considered only six channels and neglected all of the momentum transfer phases. This model allowed the description of single-target excitation and transfer into the first excited states at low impact energies of 0.15 keV/u–10 keV/u. Note that direct projectile excitation was excluded. Later, both projectile and target excitation processes in He⁺ + He collisions were investigated by Hildenbrand et al.⁷ using an extended three-electron coupled-channel calculation for impact energies of 2.5 keV/u–150 keV/u. Although they included more channels (up to 128) in their calculations, the couplings between S and P states were not taken into account. To the best of our knowledge, no other theoretical results for the processes in Eqs. (1) and (2) are available. However, large discrepancies still exist between the available experimental and calculated results. This is particularly unfortunate from the viewpoint of applications, since there is a growing awareness that accurate excitation cross sections are important elements in databases for modeling of astrophysical^23–25 and laboratory plasmas.^26–28 

In the present work, we investigate both projectile and target excitation processes presented in Eqs. (1) and (2) using a three-electron semiclassical asymptotic-state close-coupling (3eASCC) method. Focusing on the energy domain 1 keV/u–225 keV/u, we compare our cross sections with the experimental data and discuss the possible reasons for the disagreement with previous calculations. Furthermore, we find that the cross sections for target excitation to singlet states and projectile excitation show clear oscillations as a function of impact energy, which we further discuss and interpret.

The remainder of the paper is organized as follows. In Sec. II, we briefly outline the 3eASCC method used in the present calculations. Section III is devoted to a detailed analysis of state-selective projectile and target excitation cross sections and direct comparisons with available experimental and theoretical results. This is followed by our conclusions in Sec. IV. Atomic units are used throughout, unless explicitly indicated otherwise.

In the present work, a three-electron semiclassical asymptotic-state close-coupling approach is employed to calculate the cross section of the electronic processes in Eqs. (1) and (2) occurring during He⁺ + He collisions. Only the main features of the method are outlined here, and more details can be found in our previous works.^4,29–31

The n_e-electron time-dependent Schrödinger equation (TDSE) is written as

where

is the electronic Hamiltonian, and $r⃗i$ and $r⃗ip=r⃗i−R⃗(t)$ are the position vectors of the electrons with respect to the target and the projectile, respectively. The relative projectile–target position $R⃗(t)$ defines the trajectory, with $R⃗(t)=b⃗+v⃗t$ in the usual straight-line, constant-velocity approximation (⁠$b⃗$ and $v⃗$ are the impact parameter and velocity; see Fig. 1 in Ref. 4). V_T and V_P are the electron–target and electron–projectile nuclear potentials, respectively.

The Schrödinger equation is solved by expanding the wavefunction on a basis set composed of spin-adapted states of the isolated collision partners,

with

where $N(nT,nP)$ denotes the number of states (and corresponding energies) for which n_T and n_P (n_T + n_P = n_e) electrons are on the target and projectile, respectively. The multielectron states $Φ(nT,nP)$ are expressed as linear combinations of spin-adapted products of Gaussian-type orbitals (GTOs) centered on isolated collision partners. The operator $P^$ in Eq. (6) ensures the permutation of any two electrons among n_e and the full antisymmetry of the total wavefunctions as well as of the asymptotic target and projectile states. Note that for all electrons, the projectile states $ϕjPPnP$ contain plane-wave electron translation factors (ETFs), $expiv⃗·r⃗−i12v2t$⁠, ensuring Galilean invariance of the results. Inserting Eqs. (5) and (6) into Eq. (3), we obtain a set of first-order coupled differential equations that can be written in matrix form as

where a(t) is the column vector of time-dependent expansion coefficients, and S and M are the overlap and coupling matrices, respectively. These equations are solved for a set of initial conditions (initial state i, b, and v) using a robust predictor–corrector time-step variable method developed by Shampine and Gordon.³² Using the orthogonality properties of the asymptotic states, the probability of a transition i → f is given by the coefficients a_f (⁠$≡aJ(nT,nP)$⁠) as

The corresponding integral (total) cross sections for the considered transition are calculated as

In the present calculations, a set of 19 GTO basis sets⁴ (10 for l = 0 and 3 × 3 for l = 1) are used on both projectile and target centers, and a total of 1260 states (states of two electrons on target and one electron on projectile, and vice versa) are included in the calculations. These states can describe elastic, excitation, single electron capture, and double electron capture channels, as well as ionization through the inclusion of pseudostates with energy lying above ionization thresholds. The energies of the relevant bound states of He⁺ and He included in the calculations are shown in Table I, in which good agreement (within 1%) with available data can be seen. Convergence tests on the cross sections are performed by comparing the present results with those from a smaller basis set (12 GTOs on each center, i.e., 6 for l = 0 and 2 × 3 for l = 1), which allows the inclusion of 582 states in total. The convergence is evaluated to be better than 1% for the total single electron transfer, about 10% for the state-resolved electron transfer cross sections, and about 20% for the cross sections of projectile and target excitation.

We first investigate the excitation of the He target atom; see Eq. (1). Figures 1(a)–1(c) show the cross sections for target excitation (TE) to singlet He(1s2s¹S, 1s2p¹P, and 1s3s¹S) excited states, respectively. Previous experimental^9,12,16,19 and theoretical^7,20 results are also displayed for comparison. Our results show that TE to He(1s2p¹P) is dominant in the entire energy region we considered. Furthermore, the cross sections for both TE to He(1s2s¹S) and TE to He(1s2p¹P) show similar structures with a clear valley centered at about 25 keV/u and extending from 8 keV/u to 70 keV/u. Similarly, for TE to He(1s3s¹S), the cross sections exhibit impact-energy-dependent oscillatory structures with two valleys located at 1–8 and 8 keV/u–70 keV/u, respectively. Before commenting on this complex behavior, we compare our results with the existing data. As can be observed in Fig. 1, our results show the best overall agreement with available experimental results.^9,12,16,19 It is only for TE to He(1s3s¹S) at E < 3 keV/u that our results lie above the experimental data of Ref. 9, with a different slope. This disagreement might be due to the lack of higher excited states correlating with He(1s3d) in the present GTO basis sets. Since only one series of measurements exists in this energy region, and the values of the cross sections are quite small (<5 × 10⁻¹⁸ cm²), we cannot draw conclusions from that deviation or confirm the structure of the first valley around 1 keV/u–8 keV/u.

From a comparison with the theoretical calculations, it can be seen that the results of Sural et al.²⁰ are about two times larger than our results as well as the experimental data.¹⁶ The calculations by Hildenbrand et al.⁷ show better agreement with our results, except that their results lie sometimes lower or higher at different overlapping impact energies. The discrepancies between our results and the other two calculations may be due to the fact that only 6 and 128 channels, respectively, were used in these treatments, whereas we included 1260 channels in our calculations, for which the convergence has also been checked, as already mentioned.

To gain insight into the valley structure exhibited by the cross sections for TE to singlet He excited states at about 25 keV/u, we present in Fig. 2 the cross sections for two resonant processes, namely, TE and electron transfer (ET) to a singlet He excited state,

with nl = 2s, 2p, and 3s. Note that the validity of our calculations for ET processes is supported by the comparison with experimental data, as shown in our previous work (see Fig. 9 in Ref. 4). It can be observed that the magnitudes of the cross sections for populating He(1s2s¹S, 1s2p¹P, and 1s3s¹S), respectively, by either excitation or capture processes are comparable for E < 8 keV/u but that their convergence due to symmetry is not reached at our lowest considered energy. However, each of the ET cross sections exhibits a maximum at an energy in the range from around 8 keV/u to 70 keV/u, followed by a rapid decrease for increasing energies. These maxima are located in the same energy region as the valley structure observed in the cross sections for TE to singlet He excited states, indicating the existence of strong competition between TE and ET resonant processes.

In Figs. 3(a)–3(c), the cross sections for TE to triplet He(1s2s³S, 1s2p³P, and 1s3s³S) excited states, respectively, are presented, together with theoretical^7,20 and experimental^9,16 results for comparison. Note that since the Hamiltonian does not contain spin-dependent interactions, triplet states can only be excited by an exchange of electrons between target and projectile. From a comparison with the results for TE to singlet He excited states shown in Figs. 1(a)–1(c), it is found that (i) the cross sections for TE to triplet He excited states have about the same values at low energy and are sometimes higher than those for TE to singlet He excited states, although TE to triplet states requires an exchange of electrons with the projectile; (ii) a rapid decrease appears at lower energies for TE to He(1s2s and 1s3s¹S) excited states; (iii) there are no valley structures in the cross sections for TE to triplet He excited states. We mention here that we have investigated ET processes to triplet states in Ref. 4. Although the corresponding cross sections are of the same order of magnitude as those for TE processes, no competition between ET and TE is observed in the case of triplet states. A possible reason for the contrast between the singlet and triplet cases may come from the different mechanisms for TE and ET to singlet and to triplet states. In the triplet case, TE can only take place by an exchange of electrons between target and projectile, as indicated above. ET to triplet states is, however, possible through direct electron transfer. In the singlet case, both TE and ET can be achieved by direct electron transition.

For TE to triplet He(1s2s³S and 1s2p³P), our results are in very good agreement with experimental results¹⁶ in the overlapping energy region. Note that it was indicated in Ref. 16 that the experimental data for the two lowest-energy points may be too low owing to scattering outside the angular range of the detector, which might explain the small disagreement with our results. The theoretical calculations of Hildenbrand et al.⁷ overestimate the experimental data¹⁶ by a factor of two, which again may be due to the limited number of channels included in their calculations. For TE to the higher triplet He(1s3s³S) excited state shown in Fig. 3(c), our results are in less satisfactory agreement with the only available series of data.⁹ Similarly to the TE process to He(1s3s¹S) at E < 3 keV/u, the disagreement is probably due to the lack of higher excited states in the present calculations. Further theoretical and experimental investigations are needed to allow definite conclusions to be drawn.

We next investigate the excitation of the projectile ion He⁺. In comparison with TE processes, investigations of projectile excitation (PE) have been much scarcer. In Fig. 4, we present the cross sections for PE to the He⁺(n = 2) level in order to provide a comparison with the only existing experimental¹⁶ and theoretical⁷ results. To the best of our knowledge, no theoretical or experimental investigations are available for other PE processes. As can be seen from Fig. 4, the cross sections for PE to He⁺(n = 2) increase steadily to a maximum located at about 12 keV/u. The cross sections decrease at higher energies and exhibit a shallow valley around 100 keV/u. From a comparison with the existing data, it can be seen that our results are in good agreement with the experimental results of Pol et al.,¹⁶ while the calculations by Hildenbrand et al.⁷ again overestimate the experimental data¹⁶ by a factor of two. It can also be observed that the cross sections for PE to He⁺(2p) excited states are dominant for impact energies lower than 50 keV/u. For E > 50 keV/u, the cross sections for PE to He⁺(2s) excited states become comparable to those for PE to He⁺(2p) excited states. Furthermore, the cross sections for PE to He⁺(2p) excited states exhibit an oscillatory energy dependence, which has been attributed to strong competition with a two-electron process, namely, ET and TE.⁴ 

In this paper, the processes of projectile and target excitation (PE and TE) occurring in the course of He⁺ + He collisions have been theoretically investigated. We have used a three-electron semiclassical asymptotic-state close-coupling approach. Furthermore, we have studied a wide collision energy region ranging from 1 keV/u to 225 keV/u, which extends previous predictions to high energies. Overall, our present calculations agree well with available measurements for both PE and TE in the respective overlapping energy regions. However, for TE to He(1s3s³S) [and also TE to He(1s3s¹S) at E < 3 keV/u], our results are in less satisfactory agreement with the only available series of data,⁹ which may be due to the lack of well-described high excited states in the basis set used in the present calculations. Further theoretical and experimental investigations would be useful to allow definite conclusions to be drawn.

A comparison of our results with those of other theoretical calculations further demonstrates the importance of nonperturbative calculations in which a sufficient number of channels are taken into account. Furthermore, the cross sections for all the considered TE to singlet He excited states show a valley structure located at about 25 keV/u. We attribute this to competition between the TE and ET to the resonant singlet excited states centered on the projectile. The oscillatory energy dependence of the cross sections for PE to He⁺(2p) excited states is due to competition with a two-electron process, namely ET and TE.⁴ Our work brings insights into electronic excitation processes and provides new data that are beneficial for many applications, such as modeling of astrophysical^23–25 and laboratory^26–28 plasmas.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11934004 and 11704040), the National Key Research and Development Program of China (Grant No. 2017YFA0402300) and the CAEP Foundation (Grant No. YZJJLX2017010). Z.H. is supported by the CAEP Foundation (Grant No. CX2019022) and the National Nature Science Foundation of China (Grant No. 11675158). This project has received funding from the LABEX Plas@par (Grant No. ANR-11-IDEX-0004-02).