Projectile angular distributions in He⁺ + H collisions

The He⁺(1s)+H(1s) collision is the simplest system, where two electrons are involved and the projectile has a structure. The integrated cross sections (ICSs) for the excitation to H(n = 2) states have been measured by Young et al [1], Mckee et al [2], Geddes et al [3], and Aldag et al [4, 5]. The ICSs for electron transfer, He⁺(1s) + H(1s) → He + H⁺, have been measured by Lockwood et al [6], Olson et al [7], and Hvelplund et al [8]. Hsu et al [9] have reported the ICS data for the ionization of H atom. Within the one-electron approximation where He⁺(1s) is treated as a core or the electron in He⁺(1s) is frozen, the ICSs were calculated for the H(n = 2) excitation [10–14], ionization of H [11, 15], and electron transfer [11–13, 15]. The calculations based on the molecular orbital close-coupling (MOCC) method have been used to calculate the ICSs for the excitation of H and electron transfer [13, 16, 17]. While, the atomic-orbital close-coupling (AOCC) method with considering two electrons have been also reported [11, 18, 19] for the ICSs of H(n = 2) excitation and electron transfer.

Present work is concerned with the differential cross sections (DCSs) with respect to the projectile scattering angle for the He⁺(1s)+H(1s) collision. Fayeton et al [20] have measured the DCSs for the elastic scattering, H(n = 2) excitation, and electron capture into He(1s2p) with the incident energy (E) of He⁺ ion for 0.1875 ≤ E ≤ 0.625 keVu⁻¹. Full quantal calculations based on a molecular treatment [20–22] were applied to analyze the experimental data of Fayeton et al [20]. Bordnave-Montesqiue et al [23] measured the DCSs for elastic scattering and H(n = 2) excitation at large scattering angles for 0.375 ≤ E ≤ 6.25 keVu⁻¹. While, the DCSs for H(n = 2) excitation at small angles were measured by Aldag et al [5] for 3.75 ≤ E ≤ 25 keVu⁻¹. Within the frozen He⁺(1s) model, Flannery et al [10] calculated the DCSs for the elastic scattering and excitation to H(n = 2) states with close-coupling (CC) calculations, where the channels of He⁺(1s) + H(n ≤ 2) were coupled. The DCSs for the H(n = 2) excitation were also calculated within the frozen He⁺(1s) model by Maidagan et al [24] using the two-state CC method and by Ramírez et al [14] using the symmetric eikonal (SE) approximation. Franco used Glauber approximation for the excitation [25], but a further approximation was adopted to make the calculation tractable. In these calculations [10, 14, 24, 25], the electron-exhange effect is neglected. Errea et al [26] have used MOCC method to analyze the experimental data of Fayeton et al [20], Bordnave-Montesqiue et al [23], and Aldag et al [5]. Errea et al [26] also presented the DCSs for the electron-transfer below 6.25 keVu⁻¹. For the electron transfer, there exists no experimental work above 1 keVu⁻¹ and no theoretical work above 6.25 keVu⁻¹. The experimental data of DCSs for H(n = 2) excitation are limited below 25 keVu⁻¹, and the same is true for the calculations except for reference [10]. Thus, the DCSs of He⁺(1s) + H(1s) collision have not been explored adequately.

Recently, we have calculated the ICSs for elastic scattering, H(n = 2) excitation, and electron transfer using a two-electron version of the AOCC method with considering both the electron-transfer and ionization channels [27]. The atomic basis are prepared using Gaussian-type orbitals which allow analytic evaluation of matrix elements. Similar AOCC calculations have been applied to ion-atom and ion-molecule collisions involving multi-electrons at intermediate enegy region [28–33]. In this work, the calculation [27] is applied to calculate the DCSs for 2.5 ≤ E ≤ 200 keVu⁻¹.

Atomic units are used throughout this paper except when otherwise stated.

We extend the previous work [27] for the ICSs in He⁺(1s)–H(1s) collision to the calculation of the DCSs. The calculation is based on the impact parameter method, and the position vector of He²⁺ nucleus from H⁺ is classically described by R = (b, Z = vt) = b + v t, where b is the impact parameter vector and v (⊥b) the incident velocity of He⁺ ion. The z-axis is taken parallel to v.

The electronic wavefunction Ψ^(S) for spin state S(= 0 or 1) at an impact parameter b is expanded as

$$\begin{eqnarray}&&{{\rm{\Psi }}}^{(S)}=\displaystyle \sum _{\nu =1}^{{N}_{1}+{N}_{2}}{C}_{\nu }^{(S)}(b,t){{ \mathcal B }}_{\nu }^{(S)}\end{eqnarray} \tag{ 1 }$$

where ${{ \mathcal B }}_{\nu }^{(S)}$ are the channel functions for He⁺ + H arrangement (ν = 1–N₁) and He + H⁺ arrangement (ν = N₁+1–N₁+N₂), respectively (see equations (7) and (8) of [27]).

The expansion coefficients ${\bf{C}}=\{{C}_{\nu }^{(S)}\}$ satisfy

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\bf{C}}=-{\rm{i}}\,{{\bf{O}}}^{-1}(t)\,{\bf{V}}(t)\,{\bf{C}}\end{eqnarray} \tag{ 2 }$$

with ${O}_{{\nu }_{1}{\nu }_{2}}=\int \int {\rm{d}}{{\bf{r}}}_{1}{\rm{d}}{{\bf{r}}}_{2}\,{{{ \mathcal B }}_{{\nu }_{1}}^{(S)}}^{* }{{ \mathcal B }}_{{\nu }_{2}}^{(S)}$ and ${V}_{\mu \nu }=\int \int {\rm{d}}{{\bf{r}}}_{1}{\rm{d}}{{\bf{r}}}_{2}\,{{{ \mathcal B }}_{{\nu }_{1}}^{(S)}}^{* }\left(H-{\rm{i}}\tfrac{\partial }{\partial t}\right){{ \mathcal B }}_{{\nu }_{2}}^{(S)}$, where H is the electronic Hamiltonian of the system.

The off-diagonal couplings become negligible for large ∣t∣ ≥ t₁(> 0). The coupled equations in (2) are solved from t = − t₁ to t = t₁ under the initial condition ${C}_{\nu }^{(S)}(b,t=-\mbox{--}{t}_{1})={\delta }_{\nu {\nu }_{0}}$, where ν₀ denotes the initial channel, He⁺(1s) + H(1s). Considering the internuclear interaction, V_NN = Z_P Z_T /R = 2/R (Z_P and Z_T are the nuclear charges of projectile and target, respectively), the transition amplitude is written as

$$\begin{eqnarray}&&{{ \mathcal T }}_{\nu }^{(S)}(b)={{\rm{e}}}^{{\rm{i}}\delta }{C}_{\nu }^{(S)}(b,{t}_{1})-{\delta }_{\nu {\nu }_{0}}\end{eqnarray} \tag{ 3 }$$

with $\delta =-{\int }_{-{t}_{1}}^{+{t}_{1}}{V}_{N}(t){dt}=\eta (\mathrm{log}b-\mathrm{log}[R({t}_{1})+{{vt}}_{1}])$ with η = 2Z_P Z_T /v. The ICS for the transition to channel ν for spin state S is calculated by

$$\begin{eqnarray}&&{\sigma }_{\nu }^{(S)}=2\pi \int {\rm{d}}b\,b| {{ \mathcal T }}_{\nu }^{(S)}(b){| }^{2}.\end{eqnarray} \tag{ 4 }$$

The DCS for the projectile scattering angle is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{\nu }^{(S)}}{{\rm{d}}{{\rm{\Omega }}}_{L}}={\left({{ \mathcal M }}_{P}v\right)}^{2}{\left|\int {db}\,{{bJ}}_{{M}_{\nu }}({q}_{\perp }b){{ \mathcal T }}_{\nu }^{(S)}(b)\right|}^{2}\end{eqnarray} \tag{ 5 }$$

with ${q}_{\perp }=2{{ \mathcal M }}_{P}v\sin ({\theta }_{L}/2)$, where ${{ \mathcal M }}_{P}$ is the mass of projectile nucleus (⁴He²⁺) and θ_L is the projectile scattering angle in the laboratory system. In equation (5), M_ν (≥ 0) is the magnitude of the z-component of the angular momentum of two electrons for channel ν, and J_M is the Bessel function of the first kind. For large q_⊥, the main contribution in the integral in equation (5) comes from small b due to the rapidly oscillating function b^iη in e^iδ , and the DCS may be approximated by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\sigma }_{\nu }^{(S)}}{{\rm{d}}{{\rm{\Omega }}}_{L}} & \simeq & {\left({{ \mathcal M }}_{P}v\right)}^{2}{\left|\displaystyle \int {db}\,{{bJ}}_{{M}_{\nu }}({q}_{\perp }b){b}^{{\rm{i}}2{Z}_{P}{Z}_{T}/v}{C}_{\nu }^{(S)}(0)\right|}^{2}\\ & = & {\left({{ \mathcal M }}_{P}v\right)}^{2}\,\displaystyle \frac{{\eta }^{2}}{{q}_{\perp }^{4}}{\left|{C}_{\nu }^{(S)}(0)\right|}^{2}={\left|{C}_{\nu }^{(S)}(0)\right|}^{2}\,{\left(\displaystyle \frac{{\rm{d}}\sigma }{{\rm{d}}{\rm{\Omega }}}\right)}_{{\rm{C}}}\end{array}\end{eqnarray} \tag{ 6 }$$

with ${\left(\tfrac{{\rm{d}}\sigma }{{\rm{d}}{\rm{\Omega }}}\right)}_{{\rm{C}}}$=${\left({Z}_{P}{Z}_{T}\right)}^{2}/(16{E}_{L}^{2}\,{\sin }^{4}({\theta }_{L}/2))$ and ${E}_{L}={{ \mathcal M }}_{P}{v}^{2}/2$. The notation ${\left(\tfrac{{\rm{d}}\sigma }{{\rm{d}}{\rm{\Omega }}}\right)}_{{\rm{C}}}$ corresponds to the DCS for the Coulomb scattering of He²⁺ by H⁺.

The atomic basis used in equations (1) are the sames as the previous work [27], and the numbers of channels become 70 and 69 for the spin states S = 0 and S = 1, respectively. The calculations are called CC-full. The results in this work are averaged over spin states with weight 1/4 for S = 0 and 3/4 for S = 1, unless otherwise stated.

We have calculated the DCSs in four types of CC methods; CC4-wo (four-channel calculation [He⁺(1s) + H(1s, 2s, 2p_0,1)] without the electron-exchange effect), CC4 (four-channel calculation [He⁺(1s) + H(1s, 2s, 2p_0,1)]), CC8 (eight-channel calculation [He⁺(1s) + H(1s,2s,2p_0,1), H(1s², 1s2s, 1s2p_0,1) + H⁺]), and CC-full. ${{ \mathcal B }}_{\nu }^{(S)}$ is replaced as $({{ \mathcal F }}^{(1)}({{\bf{r}}}_{1},{{\bf{r}}}_{2},t){\psi }_{\nu }^{(1)}({{\bf{r}}}_{1P},{{\bf{r}}}_{2T})$ + ${\left(-1\right)}^{S}({{\bf{r}}}_{1}\iff {{\bf{r}}}_{2}))/\sqrt{2}$ →${{ \mathcal F }}^{(1)}({{\bf{r}}}_{1},{{\bf{r}}}_{2},t){\psi }_{\nu }^{(1)}({{\bf{r}}}_{1P},{{\bf{r}}}_{2T})$ in CC4-wo (see equation (7) of [27]), and the electron-exchange effect is neglected in the He⁺ + H channels. The electron transfer channels are considered in CC8, but not in CC4-wo and CC4. CC-full includes the continuum channels, but not in the other three calculations. Figure 1 compares the elastic DCSs in the four CC calculations. For E = 6.25–50 keVu⁻¹, the DCSs in CC4-wo are smaller than those in the other calculations at small q_⊥ owing to the lack of electron-exchange effect. Above 25 keVu⁻¹, CC-full gives larger (smaller) DCSs at small q_⊥ (large q_⊥) due to the effect of continuum channels. All the DCSs come to agree for higher energies.

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Bordnave-Montesqiue et al [23] have reported the DCSs of elastic scattering and H(n = 2) excitation for E = 0.425–6.25 keVu⁻¹ and θ_L > 1 mrad. In figure 2, the experimental data for elastic scattering at E = 2.5, 5, and 6.25 keVu⁻¹ are compared with the DCSs in CC-full and MOCC of Errea et al [26], and the Rutherford-type DCSs in equation (6) within CC-full (R-CC-full). The DCSs in R-CC-full are much larger than those in CC-full at small θ_L , but they are in better agreement with the experimental data than those in CC-full and MOCC for θ_L > 1 mrad. At 2.5 keVu⁻¹, the DCSs of Errea et al and CC-full are fairly close and they show a kink near θ_L = 5 mrad. At 5 and 6.25 keVu⁻¹, the DCSs in CC-full and MOCC are close at θ_L = 0.5 mrad, but the kink structures of MOCC are stronger than those of CC-full for 1 ≤ θ_L ≤ 5 mrad. The experimental data do not show any kink. Figure 3 shows the DCSs of H(n = 2) excitation calculated with the four CC calculation. The DCSs in CC4-wo and CC4 differ considerably for q_⊥ > 0.5 at 6.25 keVu⁻¹ owing to the electron-exchange effect, and they become similar for E > 25 keVu⁻¹. The differences between CC4 and CC8 are seen below 100 keVu⁻¹, where the electron-transfer channels affect the excitation DCSs. At 200 keV, the DCS in CC-full shows a slight deviation from the others due to the contribution from the continuum channels. Flannery et al [10] used the CC method corresponding to CC4-wo to calculate the DCSs for the elastic scattering and the H(2s) and H(2p) excitation at E = 6.25, 25, 100, and 225 keVu⁻¹. Their DCSs are generally in good agreement with those in the present CC4-wo at the overlapping energies (E = 6.25, 25, and 100 keVu⁻¹, not shown in the figures 1 and 3), but one exception is seen for the elastic DCS at E = 25 keVu⁻¹ (v = 1) [10]: their DCS shows a deep dip near θ_L = 0.04 mrad (not shown).

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The experimental data of Bordnave-Montesqiue et al [23] for H(n = 2) excitation are compared in figure 4 with the DCSs in CC-full and MOCC of Errea et al [26]. The DCSs in MOCC are in fairly good agreement with those in CC-full. The experimental data are generally in accord with the DCSs in CC-full and MOCC. The DCS of R-CC-full converges to that of CC-full at large angles. The DCSs for H(2s) and H(2p) excitations are shown in figure 4, where the DCSs of H(2p) are larger than those of H(2s).

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Aldag et al [5] measured the DCSs for the H(n = 2) excitation and obtained the ICSs of H(n = 2) excitation for E = 3.75–25 keVu⁻¹. Before comparing the DCS data with theoretical results, we revisit the ICSs for H(n = 2) excitation (see figure 5, and also figure 4 in reference [27]). The experimental data of Aldag et al have large uncertainties and considerably larger than the results in CC-full, which are in better agreement with the data measured by Geddes et al [3]. The results of Errea et al [17] are much larger than those of CC-full for E > 5 keVu⁻¹. Figure 6 shows the experimental data of Aldag et al [5] and the theoretical DCSs in CC-full, MOCC of Errea et al [26] (3.75 and 6.25 keVu⁻¹), and SE approximation of Ramírez et al [14] (15, 18.75, and 25 keVu⁻¹). The DCSs in CC-full are overall smaller than the experimental data as expected from the comparison of the ICSs, but the angular distributions are similar. The DCSs in CC-full show structures for θ_L < 0.5 mrad and E ≤ 12.5 keVu⁻¹, where the experimental data are rather smooth. The DCSs in MOCC of Errea et al [26] are in accordance with the experimental data at 3.75 and 6.25 keVu⁻¹ (the DCSs at small angles are not shown in reference [26]). Aldag et al [5] derived the DCSs with numerical extraction method (E = 3.75–25 keVu⁻¹) , and forward modeling approach to consider the angular width of incident beam (E = 12.5–25 keVu⁻¹). The DCSs derived with the forward modeling approach are closer to the DCSs in CC-full for E = 15–25 keVu⁻¹. The result of Ramírez et al is close to that in CC-full at 15 keVu⁻¹, but they are larger than those in CC-full at large angles at 18.75 and 25 keVu⁻¹. For the energy region in figure 6 (E < 25 keVu⁻¹), the DCSs of H(2s) and H(2p) are comparable near the forward direction, and the DCS of H(2p) dominates at large angles. For higher energy, the DCS of H(2p) (H(2s)) becomes dominant at small (large) angles.

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Figure 7 show the DCSs for electron transfer in CC8 and CC-full, and partial DCSs to He(1s²,1s2s,1s2p) in CC-full. Note that the electron transfer to He(1s²) is not allowed for the triplet spin state. The DCSs in CC8 and CC-full are generally close, indicating that the continuum channels do not affect the electron transfer so much. For the partial contributions at 6.25 and 12.5 keVu⁻¹, the electron transfer to He(1s²) is largest near the forward angle, and that to He(1s2p) is dominant overall. With increasing energies, the electron transfer to He(1s²) becomes dominant at all angles.

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The experimental DCSs have not been reported for the electron transfer process above 1 keVu⁻¹. The theoretical DCSs for electron transfer is reported by Errea et al for E ≤ 6.25 keVu⁻¹ [26]. We compare the DCSs in CC-full with those in MOCC of Errea et al at E = 2.5, 3.75, and 6.25 keVu⁻¹ in figure 8. The DCSs in CC-full and MOCC are in good agreement at these low collision energies. The DCSs in R-CC-full converge to those in CC-full at large angles.

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We show the DCSs of electron-transfer He⁺(1s) + H(1s) → He(1s²) + H⁺ for singlet spin state at E = 60, 100, and 150 keVu⁻¹ in figure 9. The DCSs for the time reversal have been measured by Schöffler et al [34], and calculated by Zapukhlyak et al [35] with the two-center extension of the basis generator method (TC-BGM) and by Spicer et al [36] using the wave-packet convergent close-coupling (WP-CCC) approach. After conversion, they are in good agreement with the DCSs in CC-full in figure 9.

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In the present work, we have applied the AOCC method to calculate the DCSs for the elastic, H(n = 2) excitation, and electron-transfer in He⁺ + H collision for E = 2.5–200 keVu⁻¹.

For the elastic scattering (figure1), the electron-exchange effect is important for small angle scattering below 50 keVu⁻¹, and the continuum channels affect the DCS up to 200 keVu⁻¹. The elastic DCSs in CC-full (figure 2) are similar to those in MOCC by Errea et al [26], but the experimental DCS of Bordnave-Montesqiue et al [23] are closer to R-CC-full than those in CC-full and MOCC.

For the H(n = 2) excitation, the electron-exchange effect is seen at large angles for E ≤ 6/25 keVu⁻¹, the electron-transfer channels affect the excitation DCSs below 100 keVu⁻¹, and the continuum channels are important for E = 12.5-200 keVu⁻¹ (figure 3). The DCSs for H(n = 2) excitation in CC-full and MOCC are generally in accord with the experimental data of Bordnave-Montesqiue et al [23] for θ_L > 1 mrad and E < 6.25 keVu⁻¹ (figure 4). The DCSs in CC-full are overall smaller than the experimental data of Aldag et al [5] for E = 3.75–25 keVu⁻¹ (figure 6), but the angular distributions are similar.

For the DCSs of electron transfer, the continuum channels do not affect the electron transfer so much (figure 7). The DCSs in CC-full and MOCC of Errea et al are in good agreement for E = 2.5–6.25 keV (figure 8). The DCSs in CC-full for the electron transfer to the ground state of helium are in good agreement with the results converted from the reported works for the time reversal process [34–36] (figure 9).

This work is performed with the support and under the auspices of the NIFS Collaboration Research Program (NIFS23KIPF006). The computation was performed using Research Center for Computational Science, Okazaki, Japan (Project: 23-IMS-C179).

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