State-Selective Differential Cross Sections for Single-Electron Capture in Slow He⁺–He Collisions

Abstract

A combined experimental and theoretical study is carried out on the single-electron capture process in He⁺–He collisions at energies ranging from 0.5 keV/u to 5 keV/u. Using cold target recoil ion momentum spectroscopy, we obtain state-selective cross sections and angular differential cross sections. Within the entire studied energy range, the dominant channel is the electron captured into the ground-state, and the relative contribution of the dominant channel shows a decreasing trend with increasing energy. The angular differential cross sections of ground-state capture exhibit obvious oscillatory structures. To understand the oscillatory structures of the differential cross sections, we also performed theoretical calculations using the two-center atomic orbital close-coupling method, which well reproduced the oscillatory structures. The results indicate that these structures are strongly correlated to the oscillatory structures of the impact parameter dependence of electron probability.

1. Introduction

Charge exchange is not only a fundamental process in atomic physics but also possesses extensive applications in fields such as astrophysics and fusion plasma physics. In terms of applications, astrophysical studies have shown that the soft X-rays detected from comets can be attributed to the charge exchange processes between solar wind ions and the cometary atmosphere [1,2,3,4,5,6]. As a representative ion-atom interaction system containing three electrons, the collision between He⁺ ions and He atoms has long been a research focus in both experimental and theoretical fields. This is owing to its capacity to display the key inelastic processes, such as electron capture, excitation, or ionization, which take place in ion-atom collisions [7,8,9,10,11]. This collision system holds significant application value in astrophysics (e.g., the interaction between solar wind and planetary atmospheres) and laboratory plasma studies (e.g., diagnostic work for tokamak fusion devices). Additionally, its cross-section data serve as critical foundations for the modeling of radiation processes and the analysis of plasma evolution [12,13].

Early studies were predominantly focused on the measurement of total cross sections [14,15,16]. For example, Hegerberg et al. measured the total cross section for symmetric charge transfer in He⁺–He collisions within the energy range of 1–10 keV [17], and DuBois reported the total cross sections for single electron capture in the range of 3–500 keV [18]. However, they offered only limited information regarding detailed reaction dynamics, which is included in differential cross sections such as angular distributions and state-selective cross sections. Along with the continuous development of experimental technologies, the employment of high-resolution measurement approaches, with Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) as a prominent example, has enabled the measurement of state-selective differential cross sections and angular differential cross sections. This in turn offers a powerful tool for investigating the dynamics of ion-atom collisions [19,20].

From a theoretical standpoint, early coupled-channel calculation methods such as the three-electron model developed by Sural et al. only took a limited number of channels (6 channels) into account and overlooked momentum transfer phases, which rendered them inadequate for describing complex many-body interactions [21]. Later on, extended models including the 128-channel calculation proposed by Hildenbrand et al. enhanced computational precision yet still had constraints, specifically failing to incorporate the coupling effect between s- and p-states [22]. In recent decades, the adoption of approaches like the two-center atomic orbital close-coupling (TC-AOCC) and four-body continuum distorted-wave (CDW-4B) has substantially improved the capability to characterize state-selective cross sections and angular distributions [23,24]. Nevertheless, despite the large volume of relevant research, data on state-selective differential cross sections in the low-energy region are still insufficient, and there exist considerable discrepancies among different theoretical models in terms of angular distribution predictions [23]. As an illustration, the classical trajectory Monte Carlo (CTMC) method is limited in its ability to depict quantum effects in the low-energy region [25,26]; in contrast, quantum mechanical methods [27], while achieving higher accuracy, require substantial computational resources.

Since the angular differential cross section contains more abundant collision information, it can effectively verify the accuracy of the theoretical models for electron capture processes. A distinctive feature of He⁺-He collisions is the unique dynamic behavior arising from their symmetry. In particular, owing to resonance properties, ground-state transfer processes predominate in the low-energy region, and their angular differential cross sections frequently display oscillatory structures, which makes this collision system an ideal object for testing theoretical models [7,8,10]. Using a COLTRIMS setup, this study systematically measured state-selective single-electron capture in the 0.5–5 keV/u He⁺–He collision system. Fully differential cross sections for the reaction process are obtained.

2. Experimental Setup

As illustrated in Figure 1, the present experiment was carried out on the Electron Beam Ion Source (EBIS) platform affiliated with the Institute of Modern Physics, Chinese Academy of Sciences. Comprehensive details concerning the experimental setup have been documented in previous studies [28,29]; thus, only a concise description is provided herein. In brief, helium gas was injected into the EBIS to produce helium ions with multiple charge states. Subsequently, these ions undergo acceleration via a high-voltage acceleration platform. After subsequent processing by Wien-filter, the He⁺ ion beam that meets the energy requirements of this experiment is finally screened out. The ions then traversed a two-dimensional beam-limiting diaphragm before entering a target chamber, where they underwent perpendicular collision with a supersonic helium target. This supersonic helium target was situated on the central axis of a Time-of-Flight (TOF) spectrometer. After the collision reaction, scattered projectile ions with different charge states were analyzed by an electrostatic analyzer placed downstream of the reaction site. Projectile ions that exhibited a reduced charge state were identified by an additional position-sensitive detector positioned behind the incident beam path. In contrast, unreacted projectile ions were collected using a Faraday Cup. It is noteworthy that the TOF spectrometer, which consists of an acceleration region and a drift region, was employed to measure the TOF of the recoil ions. In ion collision experiments, the time signal is measured relative to the time zero of the detector’s clock. However, the absolute time zero (t₀) for the particle’s TOF should be the moment when the collision reaction occurs. Since the distance from the reaction zone to the scattering detector is fixed, and the velocity of the projectile remains approximately unchanged before and after the collision (with minimal energy loss), the time difference (Δt) between the moment the collision reaction occurs and the moment the scattered projectile reaches the scattering detector is a fixed value. Taking the recoil ion as an example, as shown in Figure 2, let t_P denote the time when the scattered projectile arrives at the scattering detector, and t_R denote the time when the recoil ion arrives at the ion detector. Then, the absolute flight time T_R of the recoil ion is T_R = t_R − t_P + Δt. This experiment adopts a dual-coincidence measurement technique between scattered ions and recoil ions, in combination with measurements conducted in an event-recording mode. By analyzing the two-dimensional correlation spectrum between the ion positions on the scattering detector and the TOF of recoil ions, the identification of reaction channels such as single-electron capture and multiple-electron capture can be achieved. For specific reaction channels, the three-dimensional momentum of recoil ions can be reconstructed through analyzing the positions of recoil ions on the recoil detector and measuring their TOF values.

As shown in Figure 1, the momentum of recoil ions measured parallel to the beam direction corresponds to their longitudinal momentum (P_long), whereas the momentum measured perpendicular to this direction corresponds to their transverse momentum (P_trans). The core principle of the COLTRIMS technique is based on constructing a correlation between the recoil ion momentum and two key parameters: the Q value (which characterizes the change in binding energy before and after the collision) and the scattering angle θ of the projectile ion [19,20]. From a kinetic perspective, this electron capture process can be conceptually interpreted as an inelastic two-body collision. Therefore, by adhering to the laws of energy conservation and momentum conservation, a straightforward relational expression can be derived under the condition of small scattering angles:

$$P_{long}=-{\displaystyle \frac{Q}{v_o}}-{\displaystyle \frac{n}{2}}·v_o$$

(1)

$$\theta ={\displaystyle \frac{P_{trans}}{P_o}}$$

(2)

Here, Q embodies information related to the variation in binding energy, providing insights into the specific state populated on the projectile as a result of the capture process. The variables v_o, n, P_o, and θ respectively represent the velocity of the projectile ion, the number of electrons captured by the projectile, the initial momentum of the projectile ion, and the scattering angle of the projectile ion. It should be emphasized that all physical quantities in the aforementioned equations are expressed in atomic units.

3. Results and Discussion

3.1. State-Selective Electron-Capture Process

The longitudinal recoil ion momentum distributions are shown in Figure 3 and the relative cross sections are listed in

Table 1. Relative cross sections (%) of the He⁺–He single-electron capture process as a function of collision energy.

E (keV/u)	n = 1		n ≥ 2
	Experiment	Theory	Experiment	Theory
0.5	100	99.27		0.73
1.25	99.13 ± 0.66	98.62	0.87 ± 0.45	1.38
2.5	97.75 ± 0.51	97.71	2.25 ± 0.33	2.29
3.75	97.32 ± 0.68	96.87	2.68 ± 0.43	3.13
5	96.89 ± 0.42	96.19	3.11 ± 0.27	3.81

. The absolute scale of momentum was calibrated using the well-defined single electron capture process in He⁺–He collisions [21]. Different peaks correspond to different final capture states (where n represents the principal quantum number of the captured electron), with the recoil ions being in the ground state. Within the measured energy range, the experimental results indicate that the process of single electron capture into the ground state (n = 1) of the projectile is dominant, which is consistent with the consideration of “resonant electron capture” [30]. This is a common feature of symmetric ion–atom charge transfer processes, as the ground state transfer in He⁺–He collisions is a resonant charge transfer process with a binding energy difference of 0 before and after the reaction. With the increase of projectile energy, in addition to the dominant reaction channel, contributions from excited state transfer (i.e., target electrons captured into states with n ≥ 2) can be identified, and the relative contribution of this process increases with increasing energy.

3.2. Angular Differential Cross Sections

In Figure 4, we present the projectile angular distributions of ground-state transfer in the single-electron capture process of He⁺–He collisions at energies ranging from 0.25 keV/u to 5 keV/u. The blue solid dots in the figure represent experimental values, the red solid line represents the theoretical calculation results using the TC-AOCC method [23], and the black solid line represents the theoretical calculation results of Fraunhofer-type diffraction [31,32,33]. In the figure, R is the maximum collision parameter calculated (equivalent to the Fraunhofer diffraction aperture radius) [33]. Within the entire studied energy range, the angular differential cross sections of ground-state capture (1s − 1s) exhibit obvious oscillatory structures. In the past, various mechanisms have been proposed to explain the oscillatory structures observed in angular differential cross sections, such as the interference between gerade and ungerade scattering amplitudes in the quasi-molecular description [34], dynamic effects caused by projectile-electron scattering [35], and diffraction phenomena of matter waves on atoms [33,36]. These methods have been applied to explain the oscillatory structures of angular differential cross-sections in electron capture processes within ion–atom collision systems during low-energy collisions.

It can be observed in Figure 4 that within the energy range of this study, the calculation results of the Fraunhofer diffraction theory agree well with the experimental results only at the first maximum; the degree of agreement between the first minimum, the second maximum, and the experimental results gradually improves as the collision energy increases. Within this energy range, the Fraunhofer diffraction theory shows an obvious energy dependence and cannot describe the situation of large scattering angles. This characteristic has also been confirmed in the study by Guo et al. [7]. In the study by Gao et al. [10], there was also a deviation between the experimental value and the second minimum, which they attributed to the limitations of the Fraunhofer-type model: first, the interaction region where electron transfer occurs is not an ideal circular aperture; second, at larger scattering angles, the contribution of hard (small impact parameter) collisions becomes dominant, and there exists a significant internuclear repulsive interaction, which will mask the diffraction pattern.

In Figure 4, we also present the angular-differential cross sections calculated by the TC-AOCC method and compare them with the experimental measurement data at the same collision energy. Within the energy range of this study, both the experimental results and the TC-AOCC theoretical results exhibit rich irregular oscillatory structures within the selected angular range. Specifically, for the collision energy E = 0.5 keV/u, the TC-AOCC results agree well with the experimental data when the scattering angle is less than 1.5 mrad; for E = 1.25 keV/u, 2.5 keV/u, 3.75 keV/u, and 5 keV/u, the TC-AOCC results show good agreement with the experimental data for scattering angles of less than 1 mrad. For larger scattering angles, the TC-AOCC results underestimate the cross sections.

Within this energy range, the TC-AOCC calculations also have a good ability to predict the positions of the cross section maxima and minima (especially when θ < 2 mrad). In the angular range of θ < 1 mrad, the calculated oscillation maxima are also in good agreement with the experimental results, but for larger deflection angles the calculated values tend to be smaller, which is consistent with the research results of Zhao et al. [23]. Compared with the Fraunhofer diffraction theory, the TC-AOCC theory is more suitable for calculating angular-differential cross sections at lower collision energies.

Figure 5 shows the calculated differential cross sections for He⁺–He collisions at an energy of 1.25 keV/u, as well as the experimental and theoretical results of Gao et al. [34]. For easier comparison with the experimental data of Gao et al., we adjusted the second maximum of our own experimental data to correspond to the position of the second maximum reported in the study by Gao et al. At this collision energy, our experimental results, along with the experimental and theoretical results of Gao et al., all exhibit rich irregular oscillatory structures within the selected angular range. Among them, the structure of our experimental results at the minima is clearer, which indicates that the resolution of this experiment (approximately 0.01° [37]) is higher than that of Gao et al.’s experiment. It can be seen from the figure that our experimental and theoretical results are in good agreement with those of Gao et al. over the entire studied angular range. For this energy point, when the scattering angle θ < 1 deg, the TC-AOCC calculation results are in fairly good agreement with the experimental data and also quite well predict the positions of the cross-section maxima and minima. For larger scattering angles, the TC-AOCC results underestimate the cross section, while the calculations by Gao et al. continue to agree well with the experimental data [23].

4. Theoretical Method

4.1. Integrated Cross Sections

In its application to the investigation of ion–atom collisions, the TC-AOCC method requires the determination of single-center electronic states over which the total scattering wavefunction is expanded. Inserting the total wave function into the time-dependent Schrödinger equation, one can obtain the coupled equations of the scattering amplitudes [38]. To determine the bound electronic state with Debye–Hückel potential on either of the two centers, Liu et al. used the variational method with even-tempered trial functions by taking the trial functions in the form [39,40]

$$\begin{array}{cc}\hfill {\chi }_{klm}\left(\overrightarrow{r}\right)& =N_l\left({\xi }_k\right)r^l{\mathrm{e}}^{-{\xi }_{k}r}{Y}_{lm}\left(\widehat{\overrightarrow{r}}\right),\hfill \\ \hfill {\xi }_k& =\alpha {\beta }^k,k=1,2,\dots ,N\hfill \end{array}$$

(3)

where $Y_{lm}\left(\overrightarrow{r}\right)$ denotes a normalization constant, $Y_{lm}\left(\overrightarrow{r}\right)$ represents the spherical harmonics, and α and β are variational parameters. The atomic states ${\varphi }_{nlm}\left(\overrightarrow{r}\right)$ are subsequently constructed as a linear combination of these trial functions:

$${\varphi }_{nlm}\left(\overrightarrow{r}\right)=\sum _k{c}_{nk}{\chi }_{klm}\left(\overrightarrow{r}\right),$$

(4)

where the expansion coefficients c_nk are determined by diagonalization of single-centre Hamiltonian. This diagonalization yields the energies $E_{nl}\left(\alpha \right)$ of atomic states. In the present calculation, the projectile contains all bound atomic states of n ≤ 6(l ≤ 5), while the target includes all bound states of n ≤ 4(l ≤ 3).

Under the semiclassical approximation, the electron wavefunction of the He⁺+He(1s²) collision system, denoted as $\Psi (\overrightarrow{r},t)$, obeys the following equation [23]:

$$\left(H-i{\displaystyle \frac{\partial }{\partial t}}\right)\Psi (\overrightarrow{r},t)=0,$$

(5)

where

$$H=-{\displaystyle \frac{1}{2}}{\nabla }_r^2+V_A\left(r_A\right)+V_B\left(r_B\right).$$

(6)

V_A(r_A) and V_B(r_B) are the interactions of the active $\overline{\mathrm{e}}$ electron with the projectile (He⁺) and target (He) ion core, respectively. A model potential $V\left(r\right)=-{\displaystyle \frac{1}{r}}-{\displaystyle \frac{1}{r}}{e}^{-2.69697r}$ is used to describe this interaction. We present the energies of ground and excited states of He atoms obtained by diagonalization of the single-center Hamiltonian with the above model potential. The corresponding data from the National Institute of Standards and Technology (NIST) table are also given in

Table 2. Eigenenergies (in a.u.) of He atom obtained by diagonalization of a single-center atomic Hamiltonian compared with the NIST table [41].

State	Present	NIST
1s2	−0.90349	−0.90395
1s2s	−0.15735	−0.16797
1s2p	−0.12749	−0.13087
1s3s	−0.06428	−0.06686
1s3p	−0.05638	−0.05736
1s3d	−0.05557	−0.05565

, showing good agreement with our calculated energies. The relative motion of the nuclei is described classically by a rectilinear trajectory with a constant velocity v ($R\left(t\right)=\overrightarrow{b}+\overrightarrow{v}t$, with b being the impact parameter). The time-dependent two-center electron wave function is expanded as

$$\Psi (\overrightarrow{r},t)=\sum _i{a}_i\left(t\right){\varphi }_i^A(\overrightarrow{r},t)+\sum _j{b}_j\left(t\right){\varphi }_j^B(\overrightarrow{r},t),$$

(7)

where ${\varphi }_i^A(\overrightarrow{r},t)$ and ${\varphi }_j^B(\overrightarrow{r},t)$ are traveling (i.e., containing plane-wave electron translational factors) atomic orbitals centered on the target (A) and the projectile (B). Substituting these into Equation (5), one obtains the system of coupled equations for the amplitudes a_i(t) and b_j(t) [38],

$$\begin{array}{cc}\hfill \mathrm{i}(\dot{A}+S\dot{B})& =HA+KB,\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill \mathrm{i}(\dot{B}+S^{†}\dot{A})& =\overline{K}A+\overline{H}B.\hfill \end{array}$$

(8b)

where A and B are vectors representing the amplitudes a_i ($i=1,2,\dots ,N_A$) and b_j ($j=1,2,\dots ,N_B$), respectively. S is the overlap matrix (S^† is its transposed form), H and $\overline{H}$ are direct coupling matrices, and K and $\overline{K}$ are the electron-exchange matrices. Equations (8a) and (8b) are solved under the initial conditions

$$a_i(-\infty )={\delta }_{1i},b_j(-\infty )=0.$$

(9)

Upon solving the coupled Equation (6), the cross section for the 1 → j electron capture transition is computed as:

$${\sigma }_{cx,j}=2\pi {\int }_0^{\infty }{\left|b_j(+\infty )\right|}^{2}b\mathrm{d}b.$$

(10)

The sum of σ_cx,j over j gives the corresponding total electron capture cross section.

4.2. Differential Cross Sections

The angular-differential cross sections associated with the inelastic transition from the initial state i to the final state j may be expressed as:

$${\displaystyle \frac{\mathrm{d}{\sigma }_{ji}}{\mathrm{d}\theta }}=2\pi sin\theta {\left|A_{ji}\right|}^2,$$

(11)

where A_ji is the scattering amplitude at a given angle θ. In the eikonal approximation, the scattering amplitude is given by

$$A_{ji}{\left(\theta \right)=\mu \nu (-i)}^{|m_j-m_i|+1}{\int }_0^{+\infty }bF\left(b\right)J_{|m_j-m_i|}\times (2b\mu \nu sin{\displaystyle \frac{\theta }{2}})\mathrm{d}b,$$

(12)

$$F\left(b\right)=C_{ji}(b,+\infty ){\mathrm{e}}^{2(i/\nu )Z_T{Z}_{P}lnb},$$

(13)

where μ is the reduced mass, ν is the relative collision velocity, and $m_j\left(m_i\right)$ is the magnetic quantum number of the final (initial) state. $J_{|m_j-m_i|}$ represents the Bessel function of the first kind, and C_ji denotes the semiclassical transition amplitude for a given impact parameter b. For the electron capture process under consideration, C_ji = b_ij = b_j (with (i = 1)) in the notation of the previous subsection. The term ${\mathrm{e}}^{2(i/\nu )Z_T{Z}_{P}lnb}$ corresponds to the eikonal phase, where Z_T and Z_P are the effective ion-core charges of the projectile and target, respectively. It should be noted that the eikonal approximation is valid for small scattering angles (i.e., when collision energies are significantly higher than the interaction energy).

5. Conclusions

In this work, we conducted a combined experimental and theoretical investigation on the single-electron capture process in He⁺–He collisions within the energy range of 0.5–5 keV/u. Using the cold target recoil ion momentum spectroscopy (COLTRIMS) technique, we successfully obtained state-selective cross sections and angular differential cross sections for the collision process. Experimental results show that the ground-state resonant transfer (1s → 1s) is the dominant channel throughout the studied energy range, which is a typical feature of symmetric ion–atom charge transfer reactions. With the increase of collision energy, the relative contribution of excited-state transfer channels (electrons captured to n ≥ 2 states) gradually increases, reflecting the evolution of collision dynamics from resonant to non-resonant mechanisms.

The angular differential cross sections of ground-state capture exhibit obvious oscillatory structures, which are closely related to the quantum interference effect in the collision process. Theoretical calculations based on the two-center atomic orbital close-coupling (TC-AOCC) method well reproduce the oscillatory characteristics of the angular differential cross sections, especially in the small scattering angle range (θ < 1 mrad). The TC-AOCC results not only accurately predict the positions of the cross section maxima and minima but also effectively describe the trend of the cross sections with collision energy, confirming that the oscillatory structures originate from the correlation between the impact parameter dependence of electron capture probability and the matter wave diffraction of the projectile. A comparison with the Fraunhofer diffraction model shows that although the latter can explain the first maximum of the angular differential cross section, it has limitations in describing the oscillatory structures at larger angles.

Overall, this work provides reliable experimental data for the single-electron capture process in He⁺–He collisions at low energies and verifies the applicability of the TC-AOCC method in describing such collision systems. The results deepen the understanding of the microscopic mechanisms of ion–atom charge transfer, especially the quantum interference and diffraction effects in low-energy collisions, and can provide important reference for related studies in astrophysics and plasma physics.