⇦Chinese Physics Letters, 2021, Vol. 38, No. 11, Article code 113401 Ab Initio Study of Single- and Double-Electron Capture Processes in Collisions of He$^{2+}$ Ions and Ne Atoms Xiao-Xia Wang (王小霞)1, Kun Wang (王堃)2, Yi-Geng Peng (彭裔耕)3, Chun-Hua Liu (刘春华)4, Ling Liu (刘玲)5, Yong Wu (吴勇)5, Heinz-Peter Liebermann6, Robert J. Buenker6, and Yi-Zhi Qu (屈一至)1* Affiliations 1College of Material Sciences and Optoelectronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China 2Institute of Environmental Science, Shanxi University, Taiyuan 030006, China 3Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China 4School of Physics, Southeast University, Nanjing 210094, China 5Data Center for High Energy Density Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 6Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universitat Wuppertal, D-42097 Wuppertal, Germany Received 2 August 2021; accepted 29 September 2021; published online 27 October 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11774344 and 11474033), and the National Key Research and Development Program of China (Grant No. 2017YFA0402300).
*Corresponding author. Email: yzqu@ucas.ac.cn Citation Text: Wang X X, Wang K, Peng Y G, Liu C H, and Liu L et al. 2021 Chin. Phys. Lett. 38 113401    Abstract The single- and double-electron capture (SEC, DEC) processes of He$^{2+}$ ions colliding with Ne atoms are studied by utilizing the full quantum-mechanical molecular-orbital close-coupling method. Total and state-selective SEC and DEC cross sections are presented in the energy region of 2 eV/u to 20 keV/u. Results show that the dominant reaction channel is Ne$^{+}$(2$s2p^{6}$ $^{2}\!S$) + He$^{+}$(1$s$) in the considered energy region due to strong couplings with the initial state Ne(2$s^{2}2p^{6}$ $^{1}\!S$) + He$^{2+}$ around the internuclear distance of 4.6 a.u. In our calculations, the SEC cross sections decrease initially and then increase whereby, the minimum point is around 0.38 keV/u with the increase of collision energies. After considering the effects of the electron translation factor (ETF), the SEC cross sections are increased by 15%–25% nearby the energy region of keV/u and agree better with the available results. The DEC cross sections are smaller than those of SEC because of the larger energy gaps and no strong couplings with the initial state. Due to the Demkov-type couplings between DEC channel Ne$^{2+}$(2s$^{2}2p^{4}$ $^{1}\!S$) + He(1$s^{2}$) and the dominating SEC channel Ne$^{+}$(2$s2p^{6}$ $^{2}\!S$) + He$^{+}$(1$s$), the DEC cross sections increase with increasing impact energies. Good consistency can also be found between the present DEC and the experimental measurements in the overlapping energy region. DOI:10.1088/0256-307X/38/11/113401 © 2021 Chinese Physics Society Article Text The electron capture processes involving collisions between ions and atoms or molecules not only are of significance in fundamental atomic physics[1] but also can be directly applied to magnetic fusion plasma diagnostics for planetary research.[2] In the astronomy environment, the charge exchange in solar wind ions (e.g., H, He, C, N, O, Ne) and neutrals has contributed to understanding the processes that produce x-rays from the solar system, supernova remnant, and starburst galaxies.[3] As reported by Cravens[4] and Bochsler,[5] the abundance of helium in solar wind ions is second only to hydrogen. In addition, compared with the difficulties in the detection of neutral hydrogen the relative concentrations of He$^{2+}$ and He$^{+}$ can monitor the charge exchange process very accurately because no measurable He$^{+}$ is expected from the comet.[6] Research[7] of cometary noble gas abundances and their isotopic compositions is helpful to investigate the compositional links between comets, the solar nebula, primitive meteorites and the atmospheres of Earth, Mars, and Venus. Neon[5] is an excellent tracer since the types of isotopic compositions in various solar system materials are distinctive. The composition of solar energetic particles varies significantly from event to event. In view of data completeness in spectral modeling tools, details of the collisions of He$^{2+}$ ions with Ne atoms are very necessary for modelers. Many earlier works have studied the charge exchange process by the He$^{2+}$–Ne collisions. For the experimental work, e.g., Rudd et al.,[8] Dubois et al.[9,10] and Baragiola et al.,[11] have measured the electron capture and ionization cross sections of He$^{2+}$–Ne in the energy range above 5 keV/u. Afrosimov et al.[12] observed the single-electron capture processes in the impact energies of 0.68–10 keV/u. Hanaki et al.[13] measured the single- and double-electron capture (SEC, DEC) of He$^{2+}$ from Ne gas targets in the energy range of 0.7–4.5 keV (0.18–1.13 keV/u). In 2007, Abdelrahman and co-workers[14] observed the total cross sections for SEC in He$^{2+}$–Ne collisions in the 260–2000 eV (0.065–0.5 keV/u). During the same year, the absolute cross sections for SEC and DEC in collisions energies of 0.1–10 keV (0.025–2.5 keV/u) for He$^{2+}$–Ne were given by Silva et al.[15] For the theoretical aspect, Afrosimov et al.[16] and Kirchner et al.[17,18] have investigated the total and multiple cross sections of electron capture, loss and ionization by the continuum distorted-wave (CDW) approximation and the basis generator method (BGM) above 10 keV/u. Hong et al.[19] calculated the SEC and DEC based on the time-dependent density functional theory (TDDFT) in the energy range of 1–1000 keV/u. As the impact energies above 1.0 keV/u, the available theoretical data agree well with the experimental results. However, when the impact energies are below 0.5 keV/u, obvious discrepancies remain in the experimental data of Silva et al.[15] and Abdelrahman et al.[14] One increases with the increasing impact energies but the other one decreases. Though the Landau–Zener theory (LZ) results of Abdelrahman et al.[14] also have a falling trend, the data are about 2–3 times larger than in the measurements. More investigations are expected for the He$^{2+}$ and Ne collisions. In this Letter, the full quantum-mechanical molecular-orbital close-coupling (QMOCC) method[20,21] is used to study the electron capture processes by the He$^{2+}$ and Ne collisions in the energy region of 2 eV/u to 20 keV/u. The QMOCC could accurately deal with the correlation effect among electrons and nucleus in the collisions between low-energy ions and atoms,[22,23] which has been developed to treat systems with higher precisions,[24] more electrons and states (even pseudo-states)[25,26] as well as more complex reactions (dissociation and dissociative capture processes, and so on)[27] with the development of the computing capacity and technology. The corresponding electronic structures (potential energies, radial and rotational coupling matrices) needed in the dynamics are calculated from the multi-reference single- and double-excitation configuration interaction (MRDCI) package.[28,29] With this calculation, the discrepancies of the experimental measurements[14,15] are clarified in the low-energy region. The present dominant electron capture channel with an exothermic energy defect of about 5.9 eV with the initial state is consistent with the experimental conclusion of Abdelrahman et al.,[14] Silva et al.[15] and Austin et al.[30] Atomic units are used throughout unless otherwise stated. The electron capture cross sections for He$^{2+}$–Ne collision are performed by the QMOCC method. This method has been described in detail in the literature.[21,31] In the perturbed stationary state model (PSS),[20–22] after expanding the total wave functions by adiabatic electric wave functions, the radial function satisfies a set of second-order differential equations, in which the radial and rotational coupling matrices $A_{ij}^{R}$ and $A_{ij}^{c}$ connect the adiabatic states with the same and different symmetry and cause transitions between them. For numerical convenience, a unitary transformation of the adiabatic representation is made to an adiabatic one. In the diabatic potentials and couplings, the coupled set of second-order differential equations is solved by the multichannel log-derivative algorithm of Johnson.[32] By matching the plane-wave boundary conditions at a large nuclear separation, the $K$ matrix is obtained and the $S$ matrix can be written as $$ S_{J} =\frac{I+{i}K_{J} }{I-{i}K_{J} },~~ \tag {1} $$ where $I$ represents the unit matrix and $J$ is the partial wave. Then the cross section $\sigma$ for states $i$ to $j$ can be defined as $$ \sigma_{({i\to j})} =\frac{\pi }{k_{i}^{2} }\sum\limits_{_{\scriptstyle J}}{({2J+1})|{S_{J}}|_{i,j}^{2}},~~ \tag {2} $$ with $k_{i}$ being the initial momentum in the center-of-mass coordinate. In the QMOCC calculations, the effects of electron translation factor (ETF) were made by choosing the reaction coordinates and switching function from Gargaud,[33] which is identical to the semi-classical CTF adopted by Errea,[34] the radial and rotational interaction can be replaced by[35] $$\begin{align} &\Big\langle {\psi_{i} \vert \partial /\partial R-(\varepsilon_{i} -\varepsilon_{j})z^{2}/2R\vert \psi_{j} } \Big\rangle, \\ &\Big\langle {\psi_{i} \vert {i}L_{y} +(\varepsilon_{i} -\varepsilon_{j})zx\vert \psi_{j} } \Big\rangle,~~ \tag {3} \end{align} $$ where $R$ is the internuclear distance, $L_{y}$ is the component of the electronic angular momentum, $z^{2}$ and $zx$ are the components of the quadrupole moment tensor, $\varepsilon_{i}$ and $\varepsilon_{j}$ are the electronic energies of states $\psi_{i}$ and $\psi_{j}$. The adiabatic potential energies of the HeNe$^{2+}$ system are obtained from the MRDCI package in the region of 1.0 to 50.0 a.u. In this calculation, the lowest nineteen molecular states: eight $^{1}\!\varSigma^{+}$, four $^{1}\!\varDelta$ ($A_{1}$ symmetry) and seven $^{1}\!\varPi$ ($B_{1}$ symmetry) are included. For neon atoms, the effective core potential (ECP)[36] and the Aug-cc-pVQZ (augmented correlation consistent polarization valence quadruple $\zeta$) basis[37] diffused by a set of (1$s$,1$p$) functions are employed, i.e., (14$s$,8$p$,3$d$,2$f$) contracted to [7$s$,6$p$,3$d$,2$f$]. The basis set of (7$s3p$)/[5$s3p$] of Romelt et al.[38] by adding a $d$-type function for He is used here. The $g$ functions are discarded and the spin-orbit interactions are not considered for their small effects. The asymptotic atomic energies corresponding to the molecular states for HeNe$^{2+}$ are listed in Table 1 and compared with the experimental results.[39] Except for the initial state 7$^{1}\!\varSigma^{+}$ asymptotically corresponding to the Ne(2$s^{2}2p^{6}$ $^{1}\!S$) + He$^{2+}$ atomic state, i.e., [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$], the differences between our calculation and the experimental data are below 0.1 eV in the asymptotic region. The adiabatic potential curves of the nineteen molecular states above are plotted in Fig. 1 ranged from 1.0 a.u. to 20.0 a.u., the lowest two states [Ne$^{+}(^{2}\!P^{\rm o})$ + He$^{+}$; 1$^{1}\!\varSigma^{+}$ and 1$^{1}\!\varPi$] are partly shown for their energies are about 30 eV lower than the entrance channel. Besides, two sharp avoided crossings, between [Ne$^{+}(^{2}\!D$) + He$^{+}$; 5$^{1}\!\varSigma^{+}$], [Ne$^{+}(^{2}\!P^{\rm o})$ + He$^{+}$; 6$^{1}\!\varSigma^{+}$] and initial state [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$] around 10.6 a.u. and 18.4 a.u., have been treated diabatically. From Fig. 1, it is known that a strong crossing exists between the initial state [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$] and [Ne$^{+ }(^{2}\!S$) + He$^{+}$; 4$^{1}\!\varSigma^{+}$] at internuclear distance about 4.6 a.u., which will play a crucial role in the low-energy collision dynamics. Strong avoided crossing can also be found between initial state [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$] and [Ne$^{+}(^{2}\!D$) + He$^{+}$; 5$^{1}\!\varSigma^{+}$] as well as [Ne$^{+}(^{2}\!P^{\rm o})$ + He$^{+}$; 6$^{1}\!\varSigma^{+}$] and [Ne$^{+}(^{2}\!D$) + He$^{+}$; 5$^{1}\!\varSigma^{+}$] at internuclear distance around 2.3 a.u. and 4.0 a.u., respectively, which will be important for SEC process if enough impact energies are reached. With the reducing internuclear distances $R$, the single-electron capture (SEC) channel [Ne$^{+ }(^{2}\!S$) + He$^{+}$; 4$^{1}\!\varSigma^{+}$] and the double-electron capture (DEC) channel [Ne$^{2+ }(^{1}\!S$) + He; 3$^{1}\!\varSigma^{+}$] begin to approach each other. It is suggested that the electron can flux between the two states by the Demkov-type mechanism,[40,41] which would be the main gateway of electron capture to the DEC channels. Furthermore, compared with the DEC channels [Ne$^{2+}(^{1}$D) + He; 2$^{1}\!\varSigma^{+}$, 2$^{1}\!\varPi$ and 1$^{1}\!\varDelta$] and [Ne$^{2+}(^{1}\!S$) + He; 3$^{1}\!\varSigma^{+}$], the SEC channels, e.g., [Ne$^{+}(^{2}\!S$) + He$^{+}$; 4$^{1}\!\varSigma^{+}$], [Ne$^{+}(^{2}\!D$) + He$^{+}$; 5$^{1}\!\varSigma^{+}$, 4$^{1}\!\varPi$ and 2$^{1}\!\varDelta$] and [Ne$^{+}(^{2}\!D$$^{\rm o})$ + He$^{+}$; 5$^{1}\!\varPi$ and 3$^{1}\!\varDelta$], are energetically closer to the initial state. It can be deduced that the SEC should be the dominant electron capture processes in the low energy region.
Table 1. Asymptotic atomic energies for the HeNe$^{2+}$ molecular ions.
Molecular states Asymptotic atomic states Energy and error (eV)
NIST MRDCI Error
1$^{1}\!\varSigma^{+}$, 1$^{1}\!\varPi$ Ne$^{+}$(2$s^{2}2p^{5}$ $^{2}\!P^{\rm o})$ + He$^{+}$(1$s$) 0 0 0
2$^{1}\!\varSigma^{+}$, 2$^{1}\!\varPi$, 1$^{1}\!\varDelta$ Ne$^{2+}$(2$s^{2}2p^{4}$ $^{1}\!D$) + He(1$s^{2}$) 19.579 19.665 0.086
3$^{1}\!\varSigma^{+}$ Ne$^{2+}$(2$s^{2}2p^{4}$ $^{1}\!S$) + He(1$s^{2}$) 23.288 23.207 0.080
4$^{1}\!\varSigma^{+}$ Ne$^{+}$(2$s2p^{6}$ $^{2}\!S$) + He$^{+}$(1$s$) 26.910 26.872 $-0.038$
3$^{1}\!\varPi$ Ne$^{+}$(2$s2p^{4}3s$ $^{2}\!P$) + He$^{+}$(1$s$) 27.783 27.790 0.007
5$^{1}\!\varSigma^{+}$, 4$^{1}\!\varPi$, 2$^{1}\!\varDelta$ Ne$^{+}$(2$s^{2}2p^{4}3s$ $^{2}\!D$) + He$^{+}$(1$s$) 30.549 30.533 $-0.016$
5$^{1}\!\varPi$, 3$^{1}\!\varDelta$ Ne$^{+}$(2$s^{2}2p^{4}3p$ $^{2}\!D$$^{\rm o})$ + He$^{+}$(1$s$) 31.121 31.076 $-0.045$
6$^{1}\!\varSigma^{+}$, 6$^{1}\!\varPi$ Ne$^{+}$(2$s^{2}2p^{4}3p$ $^{2}\!P^{\rm o})$ + He$^{+}$(1$s$) 31.512 31.453 $-0.059$
7$^{1}\!\varSigma^{+}$ Ne(2$s ^{{\boldsymbol 2}}2p^{{\boldsymbol 6}}$ $^{1}\!S$) + He$^{{\boldsymbol 2+}}$ 32.853 33.018 0.165
8$^{1}\!\varSigma^{+}$, 7$^{1}\!\varPi$, 4$^{1}\!\varDelta$ Ne$^{+}$(2$s^{2}2p^{4}3p$ $^{2}\!F$$^{\rm o})$ + He$^{+}$(1$s$) 34.017 34.096 0.079
cpl-38-11-113401-fig1.png
Fig. 1. Adiabatic potential curves of HeNe$^{2+}$. The solid lines, dotted lines and dashed lines represent the $^{1}\!\varSigma^{+}$, $^{1}\!\varPi$ and $^{1}\!\varDelta$ states, respectively.
The radial and rotational coupling matrix elements represent the couplings between the states in identical or different symmetry. Some important radial coupling matrix elements of $^{1}\!\varSigma^{+}$ and $^{1}\!\varPi$ states calculated from the finite-difference approximation are shown in Fig. 2 by considering the ETF effects.[34] There is almost a one-to-one correlation between the position of a peak in the radial couplings and an avoided crossing of the adiabatic potentials in Fig. 1. As the internuclear distances $R > 3.0$ a.u. in Fig. 2(a), a strong peak centered around 4.6 a.u. appears between the 4$^{1}\!\varSigma^{+}$ state and the initial state 7$^{1}\!\varSigma^{+}$. This interaction contributes to the main electron capture in the low-energy collision. As the internuclear distance $R$ decreases, the couplings of 4$^{1}\!\varSigma^{+}$–$7^{1}\!\varSigma^{+}$ approach zero until $R \sim 1.8$ a.u. The obvious strong couplings can also be found between 5$^{1}\!\varSigma^{+}$ and 7$^{1}\!\varSigma^{+}$ in internuclear distance about 2.30 a.u., 1.67 a.u. and 1.40 a.u., which will be crucial to the SEC process as impact energies increase. As the states 6$^{1}\!\varSigma^{+}$ and 8$^{1}\!\varSigma^{+}$ are adjacent to the initial state 7$^{1}\!\varSigma^{+}$, the typically Demkov-type couplings featured with the broad and shallow peaks appear between them nearly 1.8 a.u. Similarly, the coupling between 3$^{1}\!\varSigma^{+}$ and 4$^{1}\!\varSigma^{+}$ can also be classified to the Demkov-type, which is responsible for the population of the double-electron capture channel. The electrons transferred between states of different symmetry are driven by rotational couplings. Some typical rotational coupling matrix elements are displayed in Fig. 3. These curves are not smooth as the radical near the positions of avoided crossings, because of the strong adiabatic interactions between the adjacent states. As displayed in Fig. 3(a), compared with the couplings between the same atomic configurations, e.g., 1$^{1}\!\varSigma^{+}-1^{1}\!\varPi$ and 6$^{1}\!\varSigma^{+}-6^{1}\!\varPi$, the couplings between the different atomic configurations states are much weaker and reduce to zero in the region of $R > 5.0$ a.u. In this system, the rotational couplings tend to redistribute the populations between states with the same configurations. Since there are no $^{1}\!\varSigma^{+}$ states with the same configurations with 3$^{1}\!\varPi$ and 5$^{1}\!\varPi$, which contribute to the population of Ne$^{+}(^{2}\!P$) + He$^{+}$ and Ne$^{+}(^{2}\!D^{\rm o})$ + He$^{+}$ exit channels, three prominent rotational coupling matrix elements between $^{1}\!\varSigma^{+}$ and 3$^{1}\!\varPi /5^{1}\!\varPi$ are displayed in Fig. 3(b). The broad coupling curves between $^{1}\!\varSigma^{+}$ and 3$^{1}\!\varPi /5^{1}\!\varPi$ around 8.0 a.u. will guarantee a certain amount of electron flux in the low-energy collision.
cpl-38-11-113401-fig2.png
Fig. 2. Some important radial coupling matrix elements (a) for $^{1}\!\varSigma^{+}$ states and (b) for $^{1}\!\varPi$ states.
The present QMOCC total single- and double-electron capture cross section of the He$^{2+}$–Ne collision system is displayed in Fig. 4 in the energy region from 2 eV/u to 20 keV/, and compared with the available experimental and other theoretical results. For the SEC cross sections in Fig. 4(a), it is clear that the results above 1.0 keV/u can be explained well with the existing theoretical work of TDDFT and the SEC cross sections from QMOCC can smoothly connect with the TDDFT results of Hong et al.[19] at 1.0 keV/u. In our calculations, the ETF effects contribute largely to the SEC cross sections, which are enhanced by 15%–25% as the collision energies $E > 0.5$ keV/u, e.g., the cross section is increased from 2.95 to $3.61 \times 10^{-16}$ cm$^{2}$ at collision energy of 1.2 keV/u. In this energy region, the present data are less than $1.0 \times 10^{-16}$ cm$^{2}$ larger than the measurements of Hanaki et al.[13] and Silva et al.[15] as well as lower than the results of Afrosimov et al.,[12] but lie between them in the overlapping energy region. In several keV/u, the present SEC cross sections lie between the measurements of Rudd et al.[8] and Baragiola et al.[11]
cpl-38-11-113401-fig3.png
Fig. 3. Some typical rotational coupling matrix elements between $^{1}\!\varSigma^{+}$ and $^{1}\!\varPi$ states for HeNe$^{2+}$ (a) for 1$^{1}\!\varPi$, 2$^{1}\!\varPi$, 4$^{1}\!\varPi$ and 6$^{1}\!\varPi$ states, (b) for 3$^{1}\!\varPi$ and 5$^{1}\!\varPi$ states.
While below the impact energy of 0.5 keV/u, the measurements of Silva et al.[15] and Abdelrahman et al.[14] are under debate due to the opposite tendency. The Landau–Zener theory (LZ) results are characterized by a similar downtrend with the measurements of Abdelrahman et al.,[14] but are about 2–3 times larger in quantitative values. In this work, a valley of present SEC cross sections is formed at around 0.38 keV/u, which locate in the region of 0.2–0.5 keV/u, where the observations of Abdelrahman et al.[14] are minimal. Even for the differences of around $1.0 \times 10^{-16}$ cm$^{2}$, which existed between the present data and measurements, as far as known, the valley appears in theoretical works for the first time in the collision of He$^{2+}$ and Ne. When the collision energies are below 0.38 keV/u, the SEC cross sections increase with decreasing impact energies, which support the measurements of Abdelrahman et al.[14] as well as the LZ method in trend. The present results are within the error range of measurements of Abdelrahman et al.[14] at impact energies of 66 and 100 eV/u. Compared with the LZ method, the SEC cross sections of the QMOCC are closer to the observations, for instance, at energy of 0.25 keV/u, the results of QMOCC and LZ are about 1.0 and $2.5 \times 10^{-16}$ cm$^{2}$ higher than the measurement, respectively. With the reduction of impact energies, the differences of SEC cross sections between the QMOCC and LZ are increased, which are even larger than $4 \times 10^{-16}$ cm$^{2}$ at some energy points. In the low-energy region, the QMOCC methods are more reliable due to the fact that HeNe$^{2+}$ may stay in the molecular region for a longer time and the quasimolecule treatment of present QMOCC is sufficient to perform the interactions between ions and atoms.[42] The disagreement between the measurements of Silva et al.[15] and Abdelrahman et al.[14] may result from the data processing. In their experiments, the attenuation method was used for determining the electron capture cross sections. The differences are that in the measurements of Silva et al.,[15] the attenuated cross sections are composed of SEC, DEC and transfer ionization, and the SEC cross sections are obtained by subtracting the transfer ionization. As described by Lin,[43] at low energies, the excitation or ionization processes are very weak to compete with the strong transfer channels. The underestimate of about $1.0 \times 10^{-16}$ cm$^{2}$ between the measurement of Abdelrahman et al.[14] and present QMOCC from 0.1 to 1 keV/u may ascribe to the influence of metastable fraction, contamination from ions with the same charge/mass ratio of the ion beam, or insufficient collection in scattering processes. Some further measurements should help clarify the discrepancy.
cpl-38-11-113401-fig4.png
Fig. 4. Comparisons of the present QMOCC electron capture cross sections with other experimental and theoretical results. Theory: the present QMOCC (solid line with circle), the time-dependent density functional theory (TDDFT) results of Hong et al.[19] (solid line), the Landau–Zener theory (LZ) results of Abdelrahman et al.[14] (dashed line); experiment: Silva et al.[15] (full square), Abdelrahman et al.[14] (full pentagon), Rudd et al.[8] (full hexagon), Hanaki et al.[13] (diamond), Baragiola et al.[11] (inverted full triangle), Afrosimov et al.[12] (full triangle). (a) For SEC cross sections, and (b) for DEC cross sections.
The DEC cross sections of the He$^{2+}$–Ne collision are displayed in Fig. 4(b). On the whole, the present DEC are smaller than the SEC process and are increased with increasing impact energies. These features will be explained in the state-selective cross sections part. As the impact energies $E < 66$ eV/u, the measurement of Silva et al.[15] are smaller than our calculations. This underestimate may be the result of insufficient collection due to angular scattering effects in the low-energy collision. In the energy region of 66–250 eV/u, the present DEC cross sections are in good agreement with Silva et al.[15] and about $0.7 \times 10^{-16}$ cm$^{2}$ larger than measurements of Hanaki et al.[13] With the increasing of impact energies, the experimental measurements of Silva et al.[15] and Hanaki et al.[13] successively approach each other and the present DEC cross sections lie between them in the overlapping energy region. Similar to the SEC cross section, the present DEC can connect well with the results of Hong et al.[19] In several keV/u, good agreement can be found between the present data and the experimental measurements of Rudd et al.[8] By taking the effect of ETF into account, the DEC cross sections are reduced with differences below 15%, e.g., from $2.06 \times 10^{-16}$ cm$^{2}$ to $1.75 \times 10^{-16}$ cm$^{2}$ at 4.8 keV/u.
cpl-38-11-113401-fig5.png
Fig. 5. State-selective cross section from initial state [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$] to other states with considering ETF effects. $^{1}\!\varSigma^{+}$ states: solid lines with filled symbols; $^{1}\!\varPi$ states: dotted lines with open symbols. (a) For SEC cross sections, and (b) for DEC cross sections.
To study the electron capture cross sections in detail, the state-selective cross sections of SEC and DEC are displayed in Fig. 5. As shown in Fig. 5(a), the SEC channels can be classified into four groups according to the contribution to the total SEC cross sections. Firstly, due to the strong interactions between the initial state [Ne($^{1}\!S$) + He$^{2+}$; 7$^{1}\!\varSigma^{+}$] and [Ne$^{+ }(^{2}\!S$) + He$^{+}$; 4$^{1}\!\varSigma^{+}$] around 1.8 a.u. and 4.6 a.u., the characteristic of the SEC cross sections is dominantly determined by the electron exchange between the 7$^{1}\!\varSigma^{+}$ and 4$^{1}\!\varSigma^{+}$ states. As the increasing of impact energies, the SEC cross sections decrease firstly and subsequently increase. This feature can be explained by the classical turning point, which moves to the smaller internuclear distance as the increase of impact energies. For no noteworthy couplings between 7$^{1}\!\varSigma^{+}$ and 4$^{1}\!\varSigma^{+}$ in the region of 2.0–4.0 a.u., the SEC cross sections are minimized around the energy of 0.38 keV/u. Secondly, because the obvious couplings between 5$^{1}\!\varSigma^{+}$ and 7$^{1}\!\varSigma^{+}$ about 2.30 a.u. and the Demkov-type couplings in 6$^{1}\!\varSigma^{+}-7^{1}\!\varSigma^{+}$ and 8$^{1}\!\varSigma^{+}-7^{1}\!\varSigma^{+}$, the contributions from the adjacent states 5$^{1}\!\varSigma^{+}$, 6$^{1}\!\varSigma^{+}$ and 8$^{1}\!\varSigma^{+}$ are about one order of magnitude smaller than those of 4$^{1}\!\varSigma^{+}$, respectively. Thirdly, because of no prominent rotational couplings in this system, the SEC cross sections from $^{1}\!\varPi$ states are smaller than those of the second groups. The main effect of the rotational couplings is to redistribute the cross sections between states with the same configurations. As displayed in Fig. 3(b), the relatively broad couplings between $^{1}\!\varSigma^{+}$ and 3$^{1}\!\varPi/5^{1}\!\varPi$ are enough to ensure the magnitude of cross sections. Lastly, the results of the states 1$^{1}\!\varSigma^{+}$ and 1$^{1}\!\varPi$ are the lowest in the low-energy region, since the large energy gaps between 1$^{1}\!\varSigma^{+}/1^{1}\!\varPi$ and the entrance channel 7$^{1}\!\varSigma^{+}$. With the increasing collision energy, the classical turning point moves to the smaller internuclear distance, the number of channels needed in calculations is increased dramatically in the small $R$ region. When the impact energies are higher than 20 keV/u, the state-selective cross sections from 1$^{1}\!\varSigma^{+}$ may exceed the 4$^{1}\!\varSigma^{+}$ state, but this conclusion should be certified by specialized calculations in their valid energy region. Figure 5(b) gives the state-selective cross sections for initial state 7$^{1}\!\varSigma^{+}$ to the DEC channels. Different from the SEC channels, no obvious couplings appear between the entrance channel and the DEC in the large $R$ region, thus the state-selective DEC cross section increases with the increasing energy duo to the Demkov-type couplings. In the beginning, the contributions from 2$^{1}\!\varSigma^{+}$ and 3$^{1}\!\varSigma^{+}$ are close to each other. However, because of the Demkov-type couplings between 3$^{1}\!\varSigma^{+}$ and 4$^{1}\!\varSigma^{+}$ nearby 2.2 a.u., the DEC cross sections of 3$^{1}\!\varSigma^{+}$ become 2–3 times larger than those of 2$^{1}\!\varSigma^{+}$ with increasing impact energies. The reduction of the DEC cross sections around 7.5 keV/u is mainly the result of the contributions from the 2$^{1}\!\varSigma^{+}$ state. Similarly, the cross sections of 2$^{1}\!\varPi$ are also smaller due to the weak rotational couplings with other $^{1}\!\varSigma^{+}$ states. In summary, the total and state-selective SEC/DEC cross sections in the collisions of He$^{2+}$–Ne are performed by the QMOCC method in the energy range of 2–20 keV/u. Electronic structures of eight $^{1}\!\varSigma^{+}$, four $^{1}\!\varDelta$ and seven $^{1}\!\varPi$ are calculated by the ab initio MRDCI package. In addition, two high energy states (one $^{1}\!\varSigma^{+}$ and one $^{1}\!\varPi$ states with asymptotic energies close to the 8$^{1}\!\varSigma^{+}$) are added to check the convergence of our considered channels, the good agreement proves that the states whose energy are higher than the 8$^{1}\!\varSigma^{+}$ state in the asymptotic region can be ignored in our calculations. Because of the strong interactions between the initial state 7$^{1}\!\varSigma^{+}$ and 4$^{1}\!\varSigma^{+}$, the 4$^{1}\!\varSigma^{+}$ is the dominant SEC channel. That is, after collision with He$^{2+}$ ions the Ne prefers to be in the Ne$^{+}$(2$s2p^{6}$ $^{2}\!S$) state. Although the present data are about $1.0 \times 10^{-16}$ cm$^{2}$ larger than the measurements of Abdelrahman et al.,[14] the present SEC cross sections support the falling trend below the energy of about 0.5 keV/u. When the collision energies are above 0.5 keV/u, the ETF effects can increase the SEC cross sections by 15%–25%, which are in accord with the TDDFT results of Hong et al.[19] In several keV/u, the present SEC cross sections are lower than the data of Hong et al.[19] while lie between the measurements of Rudd et al.[8] and Baragiola et al.[11] Generally, the DEC cross sections are smaller than those of SEC as larger energy gaps and no strong couplings with the initial state. The present DEC cross sections lie between the experimental data of Silva et al.[15] and Hanaki et al.[13] in the overlapping energy region. Similarly, the present DEC is around 20% smaller than the results of Hong et al., but in good agreement with the experimental measurements of Rudd et al.[8] in the region of several keV/u. References Signature of Single Binary Encounter in Intermediate Energy He 2+ —Ar CollisionsCometary x-rays: line emission cross sections for multiply charged solar wind ion charge exchangeX-rays from solar system objectsComet Hyakutake x-ray source: Charge transfer of solar wind heavy ionsSolar Wind Ion CompositionPossible cometary origin of heavy noble gases in the atmospheres of Venus, Earth and MarsIonization cross sections for 10 – 300-keV/u and electron-capture cross sections for 5 – 150-keV/u He 2 + 3 ions in gasesIonization and charge transfer in He 2 + –rare-gas collisionsIonization and charge transfer in He 2 + –rare-gas collisions. IICross sections for single and double electron capture from He++ on hydrogen, neon and argonElectron Capture of He 2+ from Gas Target Atoms at around a Few keVSingle and double electron capture by slow He 2+ from atoms and moleculesInfluence of electronic exchange on single and multiple processes in collisions between bare ions and noble-gas atomsTime-dependent screening effects in ion-atom collisions with many active electronsTheoretical investigation of the electron capture and loss processes in the collisions of He 2+ + NeCharge transfer of N 4 + with atomic hydrogenCharge transfer of multiply charged ions at thermal energiesSingle- and Double-Electron Capture Processes in Low-Energy Collisions of N 4+ Ions with He *Single- and double-charge transfer in slow He 2+ –He collisionsAb initio investigation of electron capture by Cl 7 + ions from HThe influence of pseudo states on the single-electron capture processes in low-energy collisions of N 5+ with HeMolecular-alignment dependence of collision-induced dissociation and electron capture in H 2 +   +   He collisionsA new table‐direct configuration interaction method for the evaluation of Hamiltonian matrix elements in a basis of linear combinations of spin‐adapted functionsImplementation of the table CI method: Matrix elements between configurations with the same number of open-shellsScattering of low energy He2+ ions by Ne, Ar, and Kr atomsAdvances In Atomic, Molecular, and Optical PhysicsThe multichannel log-derivative method for scattering calculationsInfluence of rotational coupling on charge transfer in low-energy C 4+ /H collisionsOn the choice of translation factors for approximate molecular wavefunctionsAb initio molecular treatment of charge transfer by Si 4 + ions in heliumAb initio relativistic effective potentials with spin‐orbit operators. 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