Single- and double-charge transfer in slow He2+–He collisions

C H Liu; J G Wang; R K Janev

doi:10.1088/0953-4075/45/23/235203

1. Introduction

The He²⁺–He collision system is the basic homonuclear two-electron ion–atom system that provides the possibility for studying the dynamics of both one- and two-electron collision processes in the most transparent way. It has been, therefore, subject to many theoretical and experimental studies in the past (see, e.g., the monographs [1, 2]). The interest of these studies was focused particularly on the one- and two-electron capture processes, at both low and high collision energies. Since in this study we shall be interested in the low-energy dynamics of these two processes, we mention here only the most extensive studies performed in the energy region below ∼20 keV/u either experimentally [3–15] or theoretically [16–26]. The theoretical description of atomic collision processes in this energy region normally requires a molecular representation of electronic states, with the nuclear motion described classically (for energies above ∼0.2 keV/u) or quantum mechanically (for energies below ∼0.2 keV/u). The semi-classical molecular orbital close-coupling (SCMOCC) method has, therefore, been most often used in theoretical cross sections of one-electron [20, 21, 24] and two-electron [16, 17, 19, 21] capture processes, often in its simplified form (such as the perturbed-stationary-state (PSS) method applied to the one-electron capture, e.g., in [20]) or in the two-state approximation (applied to two-electron capture, e.g., in [16, 17, 19]). A two-electron atomic orbital close-coupling (2e-AOCC) method has also been applied to electron capture processes in the He²⁺–He collision system [22, 23] for energies extending down to ∼1 keV/u. We also mention that in [10] the MOCC method was used with the quantum-mechanical description of nuclear motion (QMOCC) for calculation of differential cross sections for single- and double-electron capture processes, as well as for the elastic scattering, employing however a very limited expansion basis (six states for the single-electron capture and two states for the two-electron capture).

The numerous theoretical and experimental results for the total single charge transfer (SCT) cross section in the energy region of interest to us (below ∼20 keV/u) are in good mutual agreement for energies above ∼5 keV/u, while for energies below ∼3–4 keV/u the experimental results disagree among themselves, this disagreement being particularly large in the region below ∼1 keV/u. The lowest energy for which the SCT cross section has been measured is 0.14 keV/u [11], while the SCT cross-sectional measurements of [9 and 14] extend down to 0.3 keV/u. Since the SCMOCC SCT cross-sectional calculations for this collision system have been performed only for energies above 0.5 keV/u [21, 24], they cannot resolve the large experimental discrepancies observed in the energy region below 0.5 keV/u. This was one of the motivations for undertaking the present large-basis QMOCC calculations that allow the considered energy region to be extended to very low energies (confining ourselves in this work to 0.01 keV/u for the one-electron capture and to 5 × 10⁻⁴ keV/u for the two-electron capture process), but also to energies up to ∼ 17.5 keV/u.

A similar discrepancy (of a factor 1.5) is observed in the magnitudes of experimental cross sections for two-electron capture (double charge transfer (DCT)) cross sections of [11] (available for E ≤ 1 keV/u) and those of [6, 7] (available for E ≥ 1 keV/u) at the energy of 1 keV/u. Since the slopes of the two cross sections are practically the same, this discrepancy remains when the cross section of [6] (or [7]) is extrapolated in the E ≤ 1 keV/u energy region, or that of [11] is extrapolated in the E ≥ 1 keV/u region. The results of extensive SCMOCC [21] and 2e-AOCC [23] calculations support the experimental data of [6 and 7], but the experimental data of [9] show a departure from the trend of the cross sections of [6 and 7] for energies below ∼1 keV/u (in agreement with the theoretical result of [19]). Resolving these discrepancies in the total DCT cross section in the energy region below 1 keV/u was another strong motivation for this work.

In contrast to the numerous studies of integral SCT and DCT cross sections in He²⁺–He collisions, the differential cross sections of one- and two-electron capture processes in this collision system are rather scarce. Gao et al [10] have measured the SCT, DCT and the direct scattering (DS) differential cross sections for collision energies of 1.5, 2, 6 and 10 keV for ³He²⁺ colliding with ⁴He target. Afrosimov et al [6] and Bordenave-Montesquieu and Dagnac [12] (hereafter referred to as B-MD) measured the differential state-selective SCT cross sections at projectile energies of 2–8 keV in ⁴He²⁺–⁴He collision system. Zhu et al [15] measured the differential state-selective capture cross sections at the collision energy of 5–10 keV/u but at small scattering angles only (θ < 0.1°). On the theoretical side, the only studies of differential SCT, DCT and DS cross sections in He²⁺–He collisions are those performed in [10] (at ³He²⁺ energies of 1.5, 2, 6 and 10 keV) and [26] (only for SCT at ³He²⁺ energies 6 and 10 keV/u) performed by using the QMOCC method. The small MO expansion bases used in [10] (six states for SCT and two states for DCT and DS) limit the reliability of DCT and DS differential cross sections to reduced scattering angles Eθ ≤ 2.2 keV/u deg, while in the case of the SCT ones they affect the accuracy at both small and large angles and energies. Obviously, a more systematic QMOCC study of differential characteristics of SCT, DCT and DS processes in the He²⁺–He collision system, employing a large MO basis to adequately address a larger region of reduced scattering angles, is still lacking. This paper is aimed, at least partially, to respond to that need.

In this work, we shall study the processes

$\begin{eqnarray} &&\fl{\rm He}^{{\rm 2 + }} + {\rm He(1s}^{\rm 2} {\rm )} \nonumber\\ &&\fl \to {\rm He}^{\rm + } {\rm (}nl{\rm )}\;{\rm (}n = {\rm 1 - 3) + He}^{\rm + } {\rm (}1{\rm s)} \ \ {\rm SCT}\;{\rm (single}\;{\rm capture),}\nonumber\\ \end{eqnarray} \tag{ 1a }$

$\begin{eqnarray} &&\fl \to {\rm He}^{\rm + } {\rm (}1{\rm s) + He}^{\rm + } {\rm (}nl{\rm )}{\rm (}n = {\rm2, 3)}\ \ {\rm SCT (transfer excitation),}\nonumber\\ \end{eqnarray} \tag{ 1b }$

$\begin{eqnarray} &&\fl \to {\rm He(1s},nl{\rm )}\;{\rm (}n = {\rm 1, 2) + He}^{{\rm 2 + }} \quad {\rm DCT,} \end{eqnarray} \tag{ 1c }$

$\begin{eqnarray} &&\fl \to {\rm He}^{{\rm 2 + }} + {\rm He(1s}^{\rm 2} {\rm )}\quad {\rm DS} \end{eqnarray} \tag{ 1d }$

by using the QMOCC method [1, 27] with an MO basis containing 21 states for the SCT and 29 states for DCT and DS processes. As indicated in equation (1), the SCT process includes, in fact, two separate processes: a single-electron transfer from the target to the projectile with the other target electron remaining in the ground state of the residual ion and a two-electron transition process in which one electron of the target is captured to the ground state of the projectile and the other target electron in the residual ion is excited (transfer excitation). We shall separate these two processes only when we consider the production of the He⁺(2l) ion in the collision. The integral and differential cross sections for SCT, DCT and DS processes in the He²⁺–He collision will be calculated in the energy range of 0.01–17.5 keV/u (except for the integral DCT cross section for which the calculations will be made down to 5 × 10⁻⁴ keV/u).

The rest of the paper is organized as follows. In section 2, we present the ab initio molecular structure data calculated by the multireference single- and double-excitation configuration interaction (MRD-CI) method [28, 29]. In section 3, we briefly outline the QMOCC method used in the calculations. In sections 4 and 5, we present and discuss our results for the integral and differential cross sections for SCT and DCT (as well as DS) cross sections, respectively. In section 4, we give our conclusions.

Atomic units will be used throughout, unless explicitly indicated otherwise.

2. Molecular structure calculations

The potential energy curves of the He₂²⁺ molecular ion have been obtained by the ab initio MRD-CI [28, 29] method for the ten ¹Σ_g, nine ¹Σ_u, four ¹Π_g, four ¹Π_u, one ¹Δ_g and one ¹Δ_u electronic states. (The subscripts g and u refer to the symmetry of two-electron wavefunction with respect to the interchange of the positions of nuclei: the gerade wavefunctions are symmetric, while ungerade wavefunctions are anti-symmetric with respect to nuclear interchange.) The correlation-consistent, polarization valence, triple zeta (cc-pVTZ)-type basis set [30] with a diffuse (3s3p3d) set was used. The final contracted basis set was (9s, 5p, 4d)/[6s, 5p, 4d] for the helium atom. The threshold for selecting the configurations was set as 10⁻⁸ H at the internuclear distances between 0.2 and 50 au. The errors of asymptotic energies are less than 0.002 au comparing with the experimental results [31]. The energies of adiabatic electronic states have been computed with an accuracy better than 10⁻⁶ au and the same accuracy holds for the difference between g and u states of the same asymptotic states. Using the obtained electronic wavefunctions, the radial and rotational couplings were calculated by employing the finite-element differentiation and analytical approaches, respectively, (see [32]).

In the stationary-state quantum-mechanical approach, in order to ensure a correct asymptotic description of the system, it is natural to introduce the appropriate reaction coordinates [1, 33]. This approach was originally developed by Thorson and Delos [34] and Mittleman [35]. In the present QMOCC calculations, by choosing the switching function of Gauguad et al [33], the radial and rotational coupling matrix elements between the states ψ_K and ψ_L are transformed into [36]

$\begin{eqnarray} &&{ \langle {\psi _K } |\partial /\partial R - (\varepsilon _K - \varepsilon _L )z^2 /2R| {\psi _L } \rangle , \langle {\psi _K } |iL_y + (\varepsilon _K - \varepsilon _L )zx| {\psi _L } \rangle ,\hspace{16pt} } \end{eqnarray} \tag{ 2 }$

where ε_K and ε_L are the electronic energies of states ψ_K and ψ_L, and z² and zx are the components of the quadrupole moment tensor. The modification is similar in form to that resulting from the application of the common electron translation factor (ETF) method [37]. Rabli and McCarroll [38] showed that a unique connection between the reaction coordinates and the ETF requires the use of a complete basis set. Using their method of formula (31) [38], we checked our basis and found that our calculated basis are almost complete.

In figure 1, we present the calculated potential energy curves of the He₂²⁺ ion in the interval of internuclear distances R = 0–5 au. In the context of the He²⁺–He collision problem, it is important to note in figure 1 that the 2¹Σ_g and 1¹Σ_u states, corresponding to [He²⁺+He] and [He+He²⁺] asymptotic (R = ∞) configurations, are energetically degenerate in the separated atom limit, where they represent, respectively, a symmetric and anti-symmetric linear combination of two-electron wavefunctions localized on one or the other of the He²⁺ nuclei. As long as the collision energy is low enough so that the energies of 2¹Σ_g and 1¹Σ_u states remain far from the energies of other states (more specifically, the classical turning point in the potential of 2¹Σ_g remains far from the avoided 2¹Σ_g–3¹Σ_g curve-crossing), the collision dynamics can be considered as a two-(coupled) state collision problem. Under these conditions, the 'resonant' two-electron capture is the dominant inelastic process in the system, with a probability determined by the interference of the g and u scattering amplitudes [1] (see also section 3).

Figure 1 also indicates that the state 2¹Σ_g, that is mixed with the initial state 1¹Σ_u for internuclear distances larger than about 3 au, energetically lies very far (for all R) from the 1¹Σ_g state converging asymptotically to the He⁺(1s) + He⁺(1s) configuration. These two states are, therefore, very weakly coupled (see also figure 2(a)) and the one-electron capture to the He⁺ ground state is highly improbable at low collision energies. On the other hand, the 2¹Σ_g state experiences a Landau–Zener-type radial coupling with the 3¹Σ_g state (at the avoided crossing of R ≈ 1.35 au) and promotes the system in the region of internuclear distances where many of the g-states, converging asymptotically to the excited n = 2 and n = 3 states, are strongly mutually coupled. Therefore, the SCT process dominantly takes place by electron capture to n = 2, 3 states and by transfer excitation to n = 2, 3 target states. In contrast to this, the 1¹Σ_u state energetically lies far from the 2¹Σ_u state for all R and promotes the system into the group of states asymptotically converging to the He⁺(n = 2,3) + He⁺(1s) or He⁺(1s) + He(n = 2,3) configurations only via the 1¹Σ_u–1¹Π_u rotational coupling acting in the united atom region (see figures 2(b) and 3(b)). It is worthwhile to note that the potential curve of the 1¹Σ_u state possesses a very shallow potential well centred at R ≈ 3.27 au, which is close to that obtained in [39] (R = 3.25 au).

**Figure 2.** Radial coupling matrix elements between the g-states (panel (a)) and between the u-states (panel (b)) of He₂²⁺.
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**Figure 3.** Rotational coupling matrix elements between the g-states (panel (a)) and between the u-states (panel (b)).
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The radial coupling matrix elements between the lowest states of the same symmetry are shown in figures 2(a) and (b) for the gerade states and ungerade states of the system with the ETF effects included, respectively. The phases of couplings can be fixed by using the method of [40, 41]. Figure 2(a) shows that there is a series of successive radial 2¹Σ_g–3¹Σ_g, 3¹Σ_g–4¹Σ_g, 4¹Σ_g–5¹Σ_g couplings in the R < 1.4 au region which, according to the 'hidden crossing' theory [42], should continue to n → ∞ and promote the system, in the approaching stage of the collision, to the ionization limit. The observed high intensity of all these couplings, as well as the closeness of corresponding avoided crossings, make it possible to treat the 2¹Σ_g state as diabatic in the region R < 1.4 au (as was done in [10]). It should be noted also that the states 3¹Σ_g and 4¹Σ_g are coupled once again at R ≈ 3 au (during the receding stage of the collision). As anticipated earlier, the coupling 2¹Σ_g–1¹Σ_g is very weak for all R (cf, figure 2(a)). On the other hand, figure 2(b) shows that the state 1¹Σ_u is very weakly coupled with the 2¹Σ_u state for all values of R (as also anticipated from figure 1), but the couplings 2¹Σ_u–3¹Σ_u and 3¹Σ_u–4¹Σ_u, appearing at relatively large R, are very strong. The ¹Π_g and ¹Π_u states, populated at small internuclear distances by rotational couplings with the ¹Σ_g and ¹Σ_u states, respectively, exhibit somewhat weaker radial couplings with the states of the same symmetry than do the ¹Σ_g and ¹Σ_u states.

In figures 3(a) and (b), the rotational couplings between the lower g-states and u-states are shown. The sudden jumps of the rotational couplings at R = 1.3 au for g-states and at R = 3.3 au for u-states are because of the strong avoided crossings of the potential curves for the related Σ states in the adiabatic representation. After being transformed to the diabatic representation, these rotational couplings would become smooth. It can be observed that the rotational couplings between the states that converge asymptotically to the configurations He⁺(1s) + He⁺ (n = 2) are particularly large and that they extend to large internuclear distances. Similar characteristics exhibit also the rotational couplings that couple the states asymptotically converging to the He⁺(1s) + He⁺ (n = 3) configurations (not shown in figure 3). These couplings obviously play a prominent role in the promotion of the system from the states populated during the approaching stage of the collision into the system of molecular states converging asymptotically to excited reaction products. In particular, the strong rotational coupling of the initial 1¹Σ_u state with the 1¹Π_u state (cf, figure 3(b)) is the only gateway for the system to reach the SCT exit channels of u-symmetry.

Since the calculations in this work will be performed by the coupled-channel formalism employing molecular states, for which the projectile and target are indistinguishable, and since the cross-sectional results are normally presented in terms of particle distinguishing atomic states (see section 3), it is useful to have the correspondence of molecular states with the atomic states in the separated atom limit. Table 1 gives this correspondence for all molecular states used in the present calculations.

Table 1. Correlation between the molecular and atomic states at asymptotically large internuclear separations. The symbol |a(1)b(2)| = (a(1)b(2) + b(1)a(2)), (1) and (2) are the two electrons.

Molecular state	Asymptotic states
1¹Σ_g	2^−1/2\|1s_A(1)1s_B(2)\|
2¹Σ_g, 1¹Σ_u	2^−1/2(1s_A(1)1s_A(2) ± 1s_B(1)1s_B(2))
3¹Σ_g, 2¹Σ_u	2^−1/2(\|1s_A(1)2s_B(2)\|±\|2s_A(1)1s_B(2)\|)
4¹Σ_g, 3¹Σ_u	2^−1/2(\|1s_A(1)2pσ_B(2)\|∓\|2pσ_A(1)1s_B(2)\|)
1¹Π_g, 1¹Π_u	2^−1/2(\|1s_A(1)2pπ_B(2)\|∓\|2pπ_A(1)1s_B(2)\|)
5¹Σ_g, 4¹Σ_u	2^−1/2(\|1s_A(1)3s_B(2)\| ± \|3s_A(1)1s_B(2)\|)
6¹Σ_g, 5¹Σ_u	2^−1/2(\|1s_A(1)3pσ_B(2)\|∓\|3pσ_A(1)1s_B(2)\|)
7¹Σ_g, 6¹Σ_u	2^−1/2(\|1s_A(1)3dσ_B(2)\| ± \|3dσ_A(1)1s_B(2)\|)
2¹Π_g, 2¹Π_u	2^−1/2(\|1s_A(1)3pπ_B(2)\|∓\|3pπ_A(1)1s_B(2)\|)
3¹Π_g, 3¹Π_u	2^−1/2(\|1s_A(1)3dπ_B(2)\| ± \|3dπ_A(1)1s_B(2)\|)
1¹Δ_g, 1¹Δ_u	2^−1/2(\|1s_A(1)3dδ_B(2)\| ± \|3dδ_A(1)1s_B(2)\|)
8¹Σ_g, 7¹Σ_u	2^-1/2(\|1s_A(1)2s_A(2)\|±\|1s_B(1)2s_B(2)\|)
9¹Σ_g, 8¹Σ_u	2^−1/2(\|1s_A(1)2pσ_A(2)\|∓\|1s_B(1)2pσ_B(2)\|)
4¹Π_g, 4¹Π_u	2^-1/2(\|1s_A(1)2pπ_A(2)\|m\|1s_B(1)2pπ_B(2)\|)
10¹Σ_g, 9¹Σ_u	2^−1/2(\|1s_A(1)4s_B(2)\| ± \|4s_A(1)1s_B(2)\|)

3. The QMOCC method

The detailed description of the QMOCC method is given in [1, 27], and here only a brief outline is presented. The QMOCC method involves the solution of a coupled set of second-order differential equations using the log-derivative method of Johnson [43]. In order to avoid the numerical difficulties encountered in the integration of QMOCC equations based on the adiabatic representation of electronic states (related to the extremely sharp variation of some couplings with R), it is convenient to make a unitary transformation [27] to a diabatic representation of electronic states. Because the gerade and ungerade molecular states are dynamically uncoupled, the QMOCC equations are solved separately for each of these two systems of states. The radial functions for neutral channels are matched with plane-wave boundary conditions. And for the Coulomb interaction channels, the asymptotic form is modified to the Coulomb wave. By the obtained S^g,u matrices, the scattering amplitude for transition from a channel i to a channel j within the same manifold (g or u) of states can be expressed as the standard form [1]

$\begin{equation} f_{ij} ^{g,u} ( {E,\theta } ) = \frac{1}{{2ik_i }}\sum\limits_{J = 0}^\infty {( {2J + 1} )\big( {S_{ij,J}^{g,u} - \delta _{ij} } \big)P_J (\cos \theta )} , \end{equation} \tag{ 3 }$

where k_i denotes the initial momentum for centre-of-mass motion and J is the total angular momentum quantum number. P_J is the Legendre polynomial of order J.

The S matrices accounting for the identity of colliding particles are given by

$\begin{equation} S_{ij,J}^d ( {E,\theta } ) = \case{1}{2}\big[ {S_{ij,J}^g + S_{ij,J}^u } \big] \end{equation} \tag{ 4a }$

and

$\begin{equation} S_{ij,J}^{{\rm ex}} ( {E,\theta } ) = \case{1}{2}\big[ {S_{ij,J}^g - S_{ij,J}^u } \big], \end{equation} \tag{ 4b }$

corresponding to the direct and exchange reactions

$\begin{equation} A( \alpha ) + B( \beta ) \to A( \gamma ) + B( \delta ) \end{equation} \tag{ 5a }$

and

$\begin{equation} A( \alpha ) + B( \beta ) \to A( \delta ) + B( \gamma ). \end{equation} \tag{ 5b }$

The differential cross sections for direct and exchange reactions are expressed as

$\begin{equation} \frac{{{\rm d}\sigma _{i \to j}^d }}{{{\rm d}\Omega }} = \frac{1}{{4k_i^2 }}\left| {\sum\limits_{J = 0}^\infty {( {2J + 1} )\big( {S_{ij,J}^d - \delta _{ij} } \big)P_J (\cos \theta )} } \right|^2 \end{equation} \tag{ 6a }$

and

$\begin{equation} \frac{{{\rm d}\sigma _{i \to j}^{{\rm ex}} }}{{{\rm d}\Omega }} = \frac{1}{{4k_i^2 }}\left| {\sum\limits_{J = 0}^\infty {( {2J + 1} )\big( {S_{ij,J}^{{\rm ex}} - \delta _{ij} } \big)P_J (\cos \theta )} } \right|^2 , \end{equation} \tag{ 6b }$

respectively.

In order to obtain converged results, the summation in equations (6) has been performed over 10 000 J partial waves for energies about 10 keV/u, while for energies below 0.1 keV/u, about 2000 partial waves were found to be sufficient, the larger partial waves will contribute less than 1% to the total cross sections.

Finally, the integral cross sections are obtained by integration of (6) over the full scattering solid angle Ω.

In the present QMOCC calculations of SCT processes, eight ¹Σ_g and three ¹Π_g states were included in the gerade manifold, and seven ¹Σ_u and three ¹Π_u states were included in the ungerade manifold. In the DCT calculations, in order to account for the contributions from excited states to the DCT process, all 29 states shown in table 1 had to be included in the basis for energies larger than 0.1 keV/u. We mention that for energies below 0.1 keV/u, a two-state (2¹Σ_g and 1¹Σ_u states) basis was found to be quite sufficient for obtaining accurate DCT results.

4. Single charge transfer

4.1. Total SCT cross section

The present total SCT cross section that includes both the single capture and the transfer excitation cross sections (processes (1a) and (1b)) are shown in figure 4. In the same figure, the experimental results of Afrosimov et al [6], Dubois [7], Shah et al [8], Kusakabe et al [9], Okuno et al [11], Alessi et al [13], Mawhorter et al [14] and B-MD [12], as well as the results of theoretical calculations of Harel and Salin [20] obtained with the perturbed stationary state (PSS), the semiclassical MOCC (SCMOCC) results of Kimura [21], the AOCC results of Gramlich et al [22], the 2e-AOCC results of Fristch [23] and the SCMOCC results of Chaudhuri et al [24] are also shown. Our calculations are performed in the energy range of 0.01 < E < 17.5 keV/u, in which the ionization cross sections are less than 20% of charge exchange results [8]. For energies larger than 5 keV/u, the present results agree well with the experimental measurements [7, 8, 13] and other theoretical results [20–24]. All these results increase with increasing the energy up to 30 keV/u. In the energy range 1–5 keV/u, our results are in good agreement with the experimental results of Mawhorter et al [14], Kusakabe et al [9] and B-MD [12] (the latter obtained by integration of their differential cross section at the indicated energy points). Our cross section (as well as those of [14, 9, 12]) slowly decreases with energy in this energy range, while the SCMOCC results of Kimura [21] become flat for E < 3 keV/u, and the results of Afrosimov et al [6] and Chaudhuri et al [24] show a slight increase for E < 4 keV/u. Using a smaller basis with the same eight g-states and seven u-states as those used in [21], we obtained results that differ no more than 10% with the present results, and the energy dependence is the same with the 21-state calculation. The PSS cross section of Harel and Salin [20] shows a similar trend as our cross section, but their values are somewhat smaller than ours. Their claim that the lack of increase of their cross section at low energies (observed in the experimental results of [6]) is due to the absence of states in the basis correlating with the He⁺(1s) + He⁺ (n = 3) asymptotic states seems to be unsubstantiated in view of the present results that include such states. The inclusion of these states leads only to the increase of the SCT cross sections in this energy range.

**Figure 4.** Integral SCT cross sections for He²⁺–He(1s²) collisions. Experiment measurements: open rightward triangles: Afrosimov *et al* [6], open diamonds: Dubois [7], open downward triangles: Shah *et al* [8], open leftward triangles: Kusakabe *et al* [9], open circles: Okuno *et al* [11], open upward triangles: Alessi *et al* [13], open squares: Mawhorter *et al* [14], open stars: B-MD [12] (integrated); theoretical results: dash–dot–dotted line: PSS results of Harel and Salin [20], dash–dotted line: SCMOCC results of Kimura [21], dotted line: AOCC results of Gramlich *et al* [22], dashed line: AOCC results of Fristch [23], solid line: SCMOCC results of Chaudhuri *et al* [24], solid circle: present QMOCC results.
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Below 1 keV/u, the magnitude of the present SCT cross section is close to the experimental results of B-MD [12], available down to 0.4 keV/u. The results of [11 and 14] decrease rapidly with decreasing the energy, while those of Kusakabe et al [9] agree well with our cross section down to 0.4 keV/u. Only their value at the lowest measured energy (0.3 keV/u) shows a significant drop. The discrepancy between the results of present calculations with the experimental results of [11, 14] for E < 1 keV/u and the experimental results of [9] for E = 0.3 keV/u may be due to the inadequate acceptance angle in the experiment in the low-energy region, where the large angle scattering should play an important role (see the text later in this section). The largest acceptance angle is 2.50° used in the experiment of Kusakabe et al [9], by which the collection efficiency is about 0.89 at 0.5 keV/u [14]. We can see that our results are similar with their results at this energy and higher. The collection efficiency would decrease with the decreasing of energy. In the experiment of Mawhorter et al [14], the accept angle is 1.93°, by which the collection efficiency is about 0.83 at 0.5 keV/u [14]. The magnitude of their results is smaller than those of ours and those of Kusakabe et al [9] except for 0.3 keV/u. The adiabatic decrease of our SCT cross section starts only at the energies about 0.05 keV/u. The SCT cross sections fall off sharply at energies lower than about 20 eV/u. This is because the avoided crossing of 3¹Σ_g–4¹Σ_g at 1.3 au is not classically accessible in this energy region.

4.2. 2l state-selective cross sections for single capture and transfer excitation

The contributions of g- and u-channels to state-selective SCT cross sections are shown in figures 5(a) and (b), respectively. In figure 5(c), we also present the particle distinguished state-selective results denoted by 'Projectile' and 'Target' for the single capture and transfer excitation, respectively. The SCT molecular g-channels that populate the 2s, 2p_σ and 2p_π states of He⁺ are the 3¹Σ_g, 4¹Σ_g and 1¹Π_g channels, respectively (see table 1). The first two are populated by the radial 2¹Σ_g–3¹Σ_g and 3¹Σ_g–4¹Σ_g radial couplings at R ∼1.1–1.3 au, while the third one is populated by the 2¹Σ_g–1¹Π_g rotational coupling in the united atom region, as well as the rotational couplings between the 3¹Σ_g, 4¹Σ_g and 1¹Π_g states. All of these couplings are strong (cf, figures 2 and 3) and ensure large cross sections down to very low (∼0.04 keV/u) collision energies. The population dynamics of 2s, 2p_σ and 2p_π states via the g-channels is not reduced to the interactions of the above-mentioned states and includes couplings with higher g-states (e.g., the strong 4¹Σ_g–5¹Σ_g at R ∼1.1 au), especially at higher energies.

The SCT u-channels populating the 2s, 2p_σ and 2p_π states are the 2¹Σ_u, 3¹Σ_u and 1¹Π_u channels, respectively. The 1¹Σ_u initial state is not coupled with the 2¹Σ_u state (see figure 2(b)) and, consequently, the 2s and 2p_σ SCT cross sections in figure 5(b) show an adiabatic energy behaviour, i.e. decrease with the decreasing of energy. However, the rotational 1¹Σ_u–1¹Π_u coupling in the united atom region is rather strong (cf, figure 3(b)) and ensures large values of the 2p_π cross section down to ∼0.1 keV/u. For lower collision energies, the 2p_π cross section decreases adiabatically with decreasing the energy.

The rapid decrease with decreasing the energy of the g-cross section for single-electron capture to the He⁺(1s) state (cf, figure 5(a)) reflects the weak 2¹Σ_g–1¹Σ_g coupling (see figure 2(a)).

In figure 5(c), we observe that the cross sections for single-electron capture to 2s and 2p states of the projectile and the cross sections when one of target electrons is captured to the 1s state of the projectile and the other electron is excited to the 2s and 2p states of the residual target ion are essentially equal for energies below 0.3–0.4 keV/u. This indicates that in this energy region, practically, only the states 3¹Σ_g, 4¹Σ_g, 1¹Π_g and 2¹Σ_u, 3¹Σ_u, 1¹Π_u, together with the initial 2¹Σ_g and 1¹Σ_u states, are involved in the population dynamics of target or projectile 2s and 2p states (see table 1). The equality of single capture and transfer excitation 2s and 2p cross sections in this energy region results from the fact that the pairs of molecular states 3¹Σ_g–2¹Σ_u, 4¹Σ_g–3¹Σ_u and 1¹Π_g–1¹Π_u represents asymptotically symmetric and anti-symmetric combinations of identical atomic configurations (see table 1). It should also be noted that in the energy region below ∼5 keV/u, the 2p cross sections are significantly larger than the 2s cross sections, a consequence of the larger number of molecular channels leading to the population of 2p states (see table 1). For energies above 4 keV/u, our 2s and 2p single capture and transfer excitation cross sections are compared with the 2e-AOCC results of Fritsch [23]. The agreement is found to be fair, given the fact that both methods in the overlapping energy range are close to the limits of their validity.

4.3. Differential SCT cross sections

In figure 6, we display the angle-differential cross sections (DCS) for single-electron capture to n = 2 and n = 3 states at collision energies of 0.5, 1.0, 1.5, 2.0, 5.0 and 10 keV/u. The 0.5, 1.0, 1.5 and 2.0 keV/u results are compared with the experimental data of B-MD [12], Afrosimov et al [6] (for E = 1.5 keV/u) and Gao et al [10] (for E = 2.0 keV/u). It can be seen from these figures that the large-angle scattering still remains large at low energies. At the energies of 0.5, 1.0 and 1.5 keV/u, the present DCS results for capture to n = 2 and n = 3 levels are in fairly good agreement with the measurements of B-MD [12] in the overlapping range of scattering angles, except for the energy of 0.5 keV/u for which the B-MD results show a flat behaviour for scattering angles smaller than 1.5°, while ours exhibit a decrease. In figure 6(c), for the energy 1.5 keV/u, the DCSs of B-MD for capture to n = 2 and n = 3, 4 states are also plotted, showing a fair agreement with the present calculations in the overlapping range of scattering angles, except for being somewhat smaller for angles smaller than 0.5°. For the energy of 2 keV/u, comparison is made with the total (summed over all final n) DCS of Gao et al [10] and with B-MD DCS for capture to n = 2. For a correct comparison, the 6 keV ³He²⁺ colliding with ⁴He results of [10] has been transformed to the frame of 8 keV⁴He²⁺+⁴He, in order to have the same reduced coordinate of τ = E × θ and ρ = θ sin θ dσ/dΩ with our results. In [12], B-MD claimed that they have not detected any signal from the capture to n = 3, 4, and hence, their n = 2 DCS can be considered to be close to the total one. The DCS of Gao et al is larger than that of B-MD for angles larger than 0.3° and shows a maximum at 0.35° and a minimum at about 0.1°. The B-MD cross section, however, continues to increase monotonically with decreasing the scattering angle below 0.35°. Our total single capture DCS lies somewhat below those of [10] and [12] for angles in the interval 0.6°–1.5°, exhibits a maximum at 0.3° and a mild minimum at about 0.07° (at variance with much deeper minimum of [10]) and, with the further decrease of the scattering angle, agrees well with the data of [10]. For the energies of 5 and 10 keV/u, our DCS for single-electron capture to n = 2 and n = 3 projectile states are given in figures 6(e) and (f). The well pronounced maxima in the single capture n = 2 and n = 3 DCSs appearing at larger scattering angles, which shift to smaller angles and reduce their height when collision energy increases, no longer appear for E = 5 keV/u and E = 10 keV/u.

The differential cross sections shown in figure 6 all exhibit abundant oscillation structures. In order to help understanding these oscillations, we also show the contributions of g and u capture channels to the n = 2 DCS. For the sake of clarity, these contributions are multiplied by a factor of 1/5 in the figures. As the figures show, the n = 2 DCS structures appearing at reduced scattering angles τ > 0.5 keV/u deg are only due to the contribution from the g-capture channels. They result from the interference of scattering amplitudes from different reaction paths associated with the branching of incoming probability flux at the pseudo-crossings of the 2¹Σ_g potential curve with those of the upper g-states at the internuclear distance range 1.1–1.35 au (Stueckelberg oscillations). The lack of such oscillations in the contribution from the u-capture channels is obviously related to the absence of such pseudo-crossing of the 1¹Σ_u potential curve. The analyses of these oscillations, performed in [6, 12], provide a conclusive proof that they are of Stueckelberg type. For instance, the peaks in the n = 2 DCS of [6] indicated by an arrow in figure 6(c) have been interpreted by the authors as arising due to reaction path branching at the 2¹Σ_g–3¹Σ_g, 3¹Σ_g–4¹Σ_g and 4¹Σ_g–5¹Σ_g pseudo-crossings. With increasing the energy, the amplitude of n = 2 DCS oscillations tends to become smaller, but their frequency becomes larger, both effects resulting from the larger number of curve pseudo-crossings (and, hence, reaction paths) involving the higher states at higher energies. It should also be noted in figure 6 that for E = 0.5 keV/u, the contribution of u-channels to the n = 2 DCS is significantly smaller than that of g-channels for angles below 0.06°. For this energy, the steepness of the repulsive part of the 1¹Σ_u potential curve is much smaller than that of the 2¹Σ_g state (see figure 1). With increasing the energy, the steepness of the repulsive part of 1¹Σ_u potential increases faster than that of the 2¹Σ_g potential, and already at 1.5 keV/u the contribution of u-channels to n = 2 DCS becomes larger than that of g-channels for scattering angles smaller than 0.15° (see figure 6(c)). Only for energies of about 10 keV/u and above, the slopes of repulsive parts of 1¹Σ_u and 2¹Σ_g potentials become comparable and so are the contributions of g- and u-capture channels to n = 2 DCS for all angles (cf, figure 6(f)).

The n = 3 DCSs also exhibit similar oscillations as the n = 2 DCSs. For reduced scattering angles τ > 0.5 keV/u deg, they are of Stueckelberg type, but their amplitudes are much smaller than those for n = 2. This is partly due to the fact that many molecular states are involved in the population of n = 3 states (see table 1) providing many reaction paths for reaching a specific 3l state, but also due to the mutual smoothing of the oscillations in the total n = 3 result. We note that in the three-state QMOCC calculations of [10] (1¹Σ_u, 2¹Σ_u, 1¹Π_u and 2¹Σ_g, 3¹Σ_g, 1¹Π_g) no molecular states converging to the n = 3 atomic states have been included in the basis. Figure 6 shows, however, that the contribution of n = 3 channel to the total single capture DCS is large for all reduced scattering angles.

Besides the oscillations in the τ > 0.5 keV/u deg, we observe in figure 6 also oscillations of n = 2 and n = 3 DCSs at smaller reduced angles (0.1 < τ < 0.5 keV/u deg), and appearing in both the g- and u n = 2 channels (though with quite different amplitudes at smaller energies). These oscillations can be attributed to the diffraction effects induced by the repulsive parts of g- and u-potentials at small internuclear distances [10, 44, 45]. With increasing the collision energy, the amplitude of these oscillations decreases due to the increased multitude of channel couplings, particularly within the g-manifold.

5. DCT and elastic scattering

5.1. Total two-electron capture cross section

In our DCT calculations, it was found out that in the energy region below ∼0.1 keV/u the two-state (2¹Σ_g and 1¹Σ_u) approximation is sufficient for accurate calculation of the two-electron capture cross section. At E < 0.1 keV/u, the cross section difference between the 2 state and the 21 state approximations is less than 1%. In this energy region, the magnitude and energy behaviour of the cross section become highly sensitive to the accuracy of 2¹Σ_g and 1¹Σ_u adiabatic energies (more precisely to their difference, see [1, 17–19]) at very large internuclear distances. As mentioned earlier (section 2), the accuracy of potential energy curves calculated up to R = 50 au is better than 10⁻⁶ au, quite sufficient for the present DCT cross section calculation down to 5 × 10⁻⁴ keV/u. However, with increasing the collision energy above ∼0.1 eV/u, the higher states become increasingly involved in the collision dynamic and the multichannel treatment of two-electron capture becomes necessary. For E ≥ 0.1 keV/u, we have employed 15 g- and 14 u-states in our total DCT cross-sectional calculations.

In figure 7, the present QMOCC DCT cross section is shown and compared with experimental data of [6, 7, 9, 11] and theoretical results of [16, 19, 21, 23]. For energies below about 2.5 keV/u, our cross section follows the trend of experimental data of Afrosimov et al [6], as well as of theoretical predictions of [16 and 21]. The experimental results of Kusakabe et al [9] below 1 keV/u lie below our results and show a weaker increase with decreasing the energy. The two-state theoretical MOCC results of [19], although having the same slope as the experimental data of [6] and theoretical data of [16 and 21], they are about 15% smaller than those of cited references. The experimental results of Okuno et al [11], available in the energy range below 0.7 keV/u have a similar slope as the results of present calculations but are about 50–60% larger than ours. It should be noted, however, that the cross section of [11] is essentially the difference of measured projectile beam attenuation cross section reduced by the measured one-electron capture cross section, the latter being negligible in the energy region below 0.7 keV/u (see in figure 4 the data points of [11]). However, even if the attenuation of [11] is reduced by the more recent experimental data for single-electron capture, e.g., from [14], or by the results for this process of present calculations, the difference between our two-electron capture cross section and that of [11] remains very large (45–50%). For E > 2.5 keV/u, our results start to decrease faster than the decrease of other theoretical and experimental results. This may be associated with the inadequate description of electron momentum translational effects in our QMOCC calculations. For E > 0.05 keV/u, we also did the calculations by including five states (2–3¹Σ_g, 1¹Π_g and 1¹Σ_u, 1¹Π_u) and six states (2–4¹Σ_g, 1¹Π_g and 1¹Σ_u, 1¹Π_u) in the basis. In these calculations, the 3¹Σ_g (in the five-state calculations) and 4¹Σ_g (in the six-state calculations) states were considered as diabatic for R < 1.27 au and R < 1.15 au, respectively. For energies above 3 keV/u, the 5-state results agree better with the experimental results than the 6- and 29-state calculations. With the increasing of the states included in the basis, the cross sections become smaller in the high-energy region because of the transitions to upper states.

5.2. Differential DCT and DS cross sections

In figures 8 and 9, we show the differential DCT and DS cross sections, respectively, calculated by two states (2¹Σ_g and 1¹Σ_u), nine states (5 g-states and 4 u-states for E = 0.1 keV/u) and by 21 states (11 g-states and 10 u-states for E = 0.5, 2.0 and 5 keV/u) in the expansion basis. Also shown in figures 8 and 9 are the results of experimental measurements of Gao et al [10] at E = 0.5 keV/u (for DS only) and 2 keV/u. At the collision energy of 0.1 keV/u, the two-state results are identical with the nine-state results in both the DCT and the DS differential cross sections. This is because at this energy both processes take place at large internuclear distances, i.e. far from the region in which 2¹Σ_g and 1¹Σ_u interact strongly with other states of the system. With increasing the collision energy, the system enters in the region of internuclear distances where 2¹Σ_g starts to interact with the higher states and a larger basis has to be included in the close-coupled channel calculations. Thus, already for E = 0.5 keV/u, the two-state and 21-state results for both DCT and DS DCSs start to differ from each other for scattering angles larger than 1.5° (see figures 8(b) and 9(b)). The results of experimental measurements of Gao et al [10] for DS DCS at this energy, available below 1.5°, agree well with calculated results in both two-state and 21-state approximations (see figure 9(b)). For E = 2 keV/u, the disagreement of two-state and 21-state QMOCC results for DCT and DS DCSs starts already at angles about 0.3° (cf, figures 8(c) and 9(c)), reflecting the fact that for this energy the large angle scattering is affected by the repulsive parts of the potentials at small R, reached by the series of pseudo-crossings of ¹Σ_g states in the region R ≤ 1.3 au The 21-state approximation reproduces the oscillatory structure of DS DCS for θ > 0.2° quite well (cf, figure 9 (c)), but fails to do so in the case of DCT DCS for θ > 0.5° (cf, figure 8(c)). This is an indication that with increasing the energy and inclusion of an increasingly larger number of states in the basis, the collision dynamics involved in DCT and DS processes becomes increasingly different. In figures 8(c) and 9(c), we also give the results from the six-state (4 g-states and 2 u-states) calculations, in which the 4¹Σg state is considered as diabatic for R < 1.15 au. It can be seen that the six-state results agree quite well with experimental results. If we use the 'adiabatic' 4¹Σg states in the calculations, the agreement with the experiment worsens. This can be taken as an indication that the diabatic representation of ¹Σg molecular states in the region for R < 1.15 au is more appropriate than the adiabatic one for describing the DCT and DS dynamics in the considered collision system. We note that in [10] the 'diabatization' of ¹Σg states was performed starting with the 2¹Σg state (i.e. for R < 1.35 au), but the poor agreement of their differential cross-sectional results with the experiment indicates that the 2¹Σg and 3¹Σg states still keep their adiabatic character in the energy region below 3–4 keV/u. Our calculations also show that the rotational couplings influence the DCS results considerably. For E = 5 keV/u, the two-state description of DS and DCT processes becomes obviously quite inappropriate for scattering angles larger than 0.1° (cf, figures 8(d) and 9(d)).

**Figure 8.** Angular differential DCT cross sections in He²⁺–He collisions for E = 0.1, 0.5, 2 and 5 keV/u. Experimental data are those of Gao *et al.* [10].
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**Figure 9.** Angular differential DS cross sections in He²⁺–He collisions for E = 0.1, 0.5, 2 and 5 keV/u. Experimental data are those of Gao *et al* [10].
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All of the differential DCT cross sections exhibit rich oscillations due to the interference between the scattering amplitudes. The large oscillations (such as the structures for θ > 0.8° for E = 0.1 keV/u and for θ > 0.3° for E = 0.5 keV/u) result from the interference between g and u scattering amplitude. The maxima of DCT DCSs and the minima of DS DCSs are in anti-phase. The sum of DCT and DS DCSs is free from such oscillations because the interference terms are then mutually cancelled. The oscillations at smaller angles (such as structures that appear for 0.1–0.4° for E = 0.1 keV/u and for 0.04–0.2° for E = 0.5 keV/u) arise from the diffraction effects caused by the repulsive potentials at small R [10, 44, 45]. Such structures appear in the elastic differential cross sections for both g- and u-channels. The very small undulations appearing at 0.08°, 0.03° and 0.015° for E = 0.1, 0.5 and 2 keV/u DS results, respectively, are due to the rainbow scattering from the attractive 1¹Σ_u potential [10, 44]. The structures at large angles, appearing only in the 21-state approximation results, come from the interference of from the different paths through the pseudo-crossings of g-states.

6. Conclusions

In this paper, we have calculated the integral and differential SCT, DCT and DS cross sections in the He²⁺–He collision using the QMOCC method with a basis involving 21 molecular states (and 29 states for the DCT process at some energies) in the energy range of 0.0005–17.5 keV/u. The molecular wavefunctions and energies used in the calculations were calculated ab initio by the MRD-CI package. The present calculations extend the previous theoretical MOCC [20, 21] or QMOCC [10] studies of these processes to much lower energies (in the case of total SCT and DCT cross sections) and to much higher reduced scattering angles (in the case of DCS) and are found to be in good agreement with the available experimental results in the overlapping energy range. However, the present QMOCC integral SCT cross section for energies below 0.04 keV/u shows a slower decrease with decreasing the energy than the experimental ones [9, 11, 14]. The experimental total DCT cross section of [11], available in the energy range below 0.7 keV/u, is found to be inconsistent with the present QMOCC results and with the trend of other both experimental [6] and theoretical [16, 19, 21, 23] results. It was demonstrated that an accurate description of differential cross sections of SCT, DCT and DS processes for large reduced scattering angles can only be achieved with the inclusion of an adequately large expansion basis in the coupled channel calculations.

Acknowledgments

This work was supported by the International Atomic Energy Agency (Vienna, Austria) (research contract numbers 15689/R0 with RKJ and 15700/R0 with JGW) and by the National Natural Science Foundation of China (grant numbers 11004014, 11025417 and 10974021).

Single- and double-charge transfer in slow He²⁺–He collisions

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Abstract

1. Introduction

2. Molecular structure calculations

3. The QMOCC method

4. Single charge transfer

4.1. Total SCT cross section

4.2. 2l state-selective cross sections for single capture and transfer excitation

4.3. Differential SCT cross sections

5. DCT and elastic scattering

5.1. Total two-electron capture cross section

5.2. Differential DCT and DS cross sections

6. Conclusions

Acknowledgments

Single- and double-charge transfer in slow He2+–He collisions

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Dates

Abstract

1. Introduction

2. Molecular structure calculations

3. The QMOCC method

4. Single charge transfer

4.1. Total SCT cross section

4.2. 2l state-selective cross sections for single capture and transfer excitation

4.3. Differential SCT cross sections

5. DCT and elastic scattering

5.1. Total two-electron capture cross section

5.2. Differential DCT and DS cross sections

6. Conclusions

Acknowledgments

Single- and double-charge transfer in slow He²⁺–He collisions