1 Introduction

Collisions between bare ions and multi-electron targets is an area of particular interest within atomic physics. Developing a detailed understanding of the various processes that take place in these scattering systems has essential applications in areas such as astrophysics, plasma physics, and medical physics. The simplest scattering problem of this kind is the four-body system presented by collisions of protons with helium atoms. While the study of the electron capture, elastic scattering and target excitation processes have progressed, few attempts have been made to probe ionisation in a non-perturbative manner [1,2,3,4]. For a recent review of energetic ion-atom and ion-molecule collisions, see Refs. [5, 6].

In this work, we consider single ionisation in the intermediate energy region where the projectile speed is either comparable to, or somewhat larger than, the electron’s orbital speed. In this energy region coupling between various channels cannot be ignored. The difficulty in modelling ionisation in this energy region is caused by the movement of the ejected electron in the Coulombic field of both the target nucleus and the projectile. When the projectile ionises the atom, it may be ionised into the continuum of the target, the process hereafter called direct ionisation (DI), or the continuum of the projectile, called electron capture into the continuum (ECC). Hence, any theoretical calculation requires two-centre effects to be accounted for [7].

Furthermore, upon the recent collection of data by Schulz et al. [8] of the fully differential cross section for single ionisation in the proton–helium collision system at the intermediate projectile energy of 75 keV, it has been suggested that a possible higher-order effect is influencing the cross section. Peak structures present for projectile scattering in the forward direction and electron ejection into the perpendicular plane indicate the existence of some intricate two-electron mechanisms that are not well understood. Other experiments [9, 10] have also gone on to expose a lack of theoretical understanding of two-electron processes, like transfer excitation, occurring in proton collisions with helium. This indicates that the full proton–helium problem is far from being solved.

The doubly differential cross section that is a function of the energy and angle of the ejected electron has been considered to study the ionisation process both experimentally [11,12,13,14,15,16,17,18,19] and theoretically [16, 19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. The difficulty of describing interactions in the exit channel means that, as with the singly differential ionisation cross sections, the theoretical approaches that have been employed are all lower-order perturbative ones.

Born-type approximations are typically only valid at high energies, so, when applied to low and intermediate energy collisions, discrepancies with experimental data tend to arise. The first Born approximation (FBA) results presented by Rudd et al.  [16] at an incident-projectile energy of 100 keV and Crooks and Rudd [21], Salin [24], Band [25], Bell and Kingston [26] and Godunov et al. [33] at 300 keV were overall unable to describe experimental data, particularly for small ejection energies and small ejection angles. This is due to an inadequate representation of the final-state wave functions which is demonstrated by the lack of the ECC peak in the FBA results. Agreement with experiment improved with increasing ejection angle, but the disagreement at ejection energies near zero persisted. Improved results were produced by Salin [24], who introduced correct boundary conditions into the Born approximation (BC-FBA) at a proton energy of 300 keV and ejection angle of 10\(^{\circ }\). Very good agreement between this method and the experimental data was found across the ejection-energy range and it notably reproduced both the ECC and binary peaks. Godunov et al. [33] also performed calculations using the second Born approximation at 300 keV for forward ejection, finding little difference in this particular type of doubly differential cross section compared to first-order Born calculations.

The perturbative continuum-distorted-wave eikonal-initial-state (CDW-EIS) approach was applied to 100 keV [19] and to 300 keV [28] collisions at selected electron ejection angles between 0 and 90\(^{\circ }\). This method was able to successfully reproduce the ECC and binary peaks. Agreement with experiment was generally good for emission energies up to 280 eV at a collision energy of 100 keV and up to 1000 eV at a projectile energy of 300 keV. Miraglia and Macek [32] employed the distorted-wave formalism with both an impulse approximation (IA) and an eikonal initial state (EIS) approximation. In general, the EIS method underestimated the IA and the experimental results. The IA found agreement with experimental data at both 0 and 160\(^{\circ }\) for electron ejection energies below 30 eV.

Classical methods were first applied to the proton–helium problem by Bonsen and Vriens with their binary-encounter (BE) theory [20] and Bonsen and Banks employing the classical trajectory Monte Carlo (CTMC) method [23]. The BE approach was only applied at an electron-ejection angle of 10\(^{\circ }\) with limited success at both 100 and 300 keV. Generally the results underestimated experimental data at small ejection energies and did not display an ECC peak but did reproduce the binary peak and found fair agreement at large ejection energies. The CTMC method, however, produced results in overall agreement with experimental data at both 100 and 300 keV projectile energies at the studied ejection angles. However, those ejection angles were all less than 90\(^{\circ }\). Moreover, the method began to falter somewhat at 50\(^{\circ }\) and 70\(^{\circ }\) at 100 and 300 keV, respectively. Reinhold and Olson [30] presented results for a proton impact energy of 100 keV at selected electron-ejection angles between 1 and 90\(^{\circ }\) and ejection energies less than 100 eV. Generally good agreement was found with the experimental data by Gibson and Reid [18] at all ejection angles and with the data by Bernardi et al. [19] at the larger ejection angles. However, these CTMC results also tended to underestimate experimental data presented by Rudd and Jorgensen [11], Rudd et al. [12] and Rudd and Madison [15].

An adaptation of a molecular approach, based on linear combinations of atomic orbitals (LCAO), was presented by Band [25]. Calculations were only completed at a projectile energy of 300 keV and an electron-ejection angle of 10\(^{\circ }\). This approach produced results in generally good agreement with experiment [12] within their chosen energy range. Macek [22] implemented the first term in the Neumann expansion of the Faddeev-Merkuriev (FM) equations for 0\(^\circ \) and 1.4\(^\circ \) ejection angles at a proton energy of 300 keV. The results agreed with experimental data in shape but overestimated the magnitude for both ejection angles. Moreover, they could not consider the low-energy electron emissions due to intrinsic problems specific to the Faddeev approach when it is applied to ionisation.

The close-coupling formalism is yet to be applied to doubly differential ionisation in proton–helium collisions. This is mainly due to the two-centre nature of the problem, challenges related to rearrangement matrix elements and adequate representation of the continuum. The convergent close-coupling approach has been developed to circumvent these difficulties. In particular, the fully quantum-mechanical [34, 35], standard semiclassical [36], and wave-packet [37,38,39] variations have been implemented for the fundamental proton–hydrogen collision system.

The wave-packet convergent close-coupling (WP-CCC) method has been used to study differential ionisation [40, 41] in the proton–helium system at sufficiently high energies that electron-capture channels can be neglected. Furthermore, the integrated cross sections have also been calculated for various processes using the two-centre WP-CCC approach in a wide energy range [4]. The obtained results agree very well with experiment. Calculations of the singly differential cross sections using the WP-CCC method have also been performed for direct scattering and electron capture processes in Ref. [42] and ionisation in Ref. [43] (hereafter referred to as Paper I and Paper II, respectively). Agreement with experiment was excellent for all binary processes and ionisation leading to the conclusion that the WP-CCC approach can provide a complete and accurate singly differential picture of proton collisions with helium. In the present work, the two-centre four-body WP-CCC method is used to calculate the doubly differential cross section for ionisation as a function of the ejected-electron energy for selected ejection angles at intermediate projectile energies. Also used, for the purposes of comparison, is a recently developed approach [44] that reduces the two-electron helium atom to an effective single-electron system convenient for scattering calculations. We report results from both methods at projectile energies from 70 to 300 keV.

Unless specified otherwise, atomic units (a.u.) are used throughout this manuscript.

2 Two-centre wave-packet convergent close-coupling method

The two-centre wave-packet convergent close-coupling method has been developed in detail in our earlier works [4, 38, 40]. The approach was first applied to calculate double differential cross sections for ionisation in Ref. [38]. The extension to multiply-charged projectiles is given in Refs. [45, 46] and to two-electron targets in Ref. [4]. Furthermore, a single-centre implementation of the WP-CCC approach to the two-centre rearrangement collision problem has been developed in [47]. Below, a brief summary of the two-centre WP-CCC method is provided with emphasis on the parts relevant to the present calculations.

The exact nonrelativistic time-independent Schrödinger equation for the total scattering wave function \(\varPsi _i^+\) is written as

$$\begin{aligned} (H - E)\varPsi _i^+ = 0, \end{aligned}$$
(1)

where H is the full four-body Hamiltonian and E is the total energy of the collision system. The total scattering wave function develops from initial channel i representing a proton incident on ground-state helium and is subject to the outgoing-wave boundary conditions. It is expanded in terms of N target-centred and M projectile-centred pseudostates as

$$\begin{aligned} \varPsi _i^+ =&\sum _{\alpha =1}^{N}F_{\alpha }\psi _\alpha ^{\textrm{He}}({\varvec{r}}_{1},{\varvec{r}}_{2})e^{{ i}{\varvec{k}}_\alpha {\varvec{\sigma }}} \nonumber \\&+ \frac{1}{\sqrt{2}}\sum _{\beta =1}^{M}G_{\beta } \Big [ \psi ^{\textrm{H}}_{\beta }({\varvec{x}}_{1}) \psi ^{\mathrm{He^+}}_{1s}({\varvec{r}}_{2}) e^{{ i}{\varvec{k}}_{1\beta }{\varvec{\rho }_1}} \nonumber \\&+ \psi ^{\textrm{H}}_{\beta }({\varvec{x}}_{2}) \psi ^{\mathrm{He^+}}_{1s}({\varvec{r}}_{1}) e^{{ i}{\varvec{k}}_{2\beta }{\varvec{\rho }_2}}\Big ], \end{aligned}$$
(2)

where the indices \(\alpha \) and \(\beta \) denote the full set of quantum numbers that represent a quantum state in the direct and rearrangement channels, respectively. The position vectors in the above equation are as in Fig. 1 of Ref. [4]. Accordingly, the projectile momentum relative to the target in the \(\alpha \) channel is denoted as \({\varvec{k}}_{\alpha }\), and the momentum of the formed hydrogen atom relative to the residual helium ion in the \(1\beta \) channel is \({\varvec{k}}_{1\beta }\). The \(2\beta \) channel is the same as the \(1\beta \) channel where the electron of the hydrogen atom and that of the residual helium ion are exchanged. Thus, our method accounts for electron-exchange effects. The wave functions \(\psi _\alpha ^{\textrm{He}}\), \(\psi _{1s}^{\mathrm{He^+}}\), and \(\psi ^{\textrm{H}}_{\beta }\) represent the helium atom, the ground-state of the helium ion and the hydrogen atom, respectively. We apply the frozen-core approximation for the helium atom [4]. Finally, the expansion coefficients \(F_{\alpha }\) and \(G_{\beta }\), represent the impact-parameter space transition amplitudes into the corresponding final channel as \(\sigma \) or \(\rho \) \(\rightarrow \infty \). By substituting Eq. (2) into Eq. (1) and using the semiclassical approximation, a set of coupled first-order differential equations for the expansion coefficients is obtained:

$$\begin{aligned}&i \dot{F}_{\alpha ^{\prime }} +i\sum _{\beta =1}^M \dot{G}_{\beta } K^T_{\alpha ^{\prime }\beta } = \sum _{\alpha =1}^N F_{\alpha } D^T_{\alpha ^{\prime }\alpha }+\sum _{\beta =1}^M G_{\beta } Q^T_{\alpha ^{\prime }\beta }, \nonumber \\&i\sum _{\alpha =1}^N \dot{F}_{\alpha } K^P_{\beta ^{\prime }\alpha }+ i\sum _{\beta =1}^M \dot{G}_{\beta } L^P_{\beta ^{\prime }\beta } = \sum _{\alpha =1}^N F_{\alpha } Q^P_{\beta ^{\prime } \alpha }+\sum _{\beta =1}^M G_{\beta } D^P_{\beta ^{\prime } \beta }, \nonumber \\&\alpha ^{\prime }=1,2,...,N, ~~ \beta ^{\prime }=1,2,...,M. \end{aligned}$$
(3)

There are three types of doubly differential cross sections (DDCS) for ionisation; the DDCS differential in the ejected-electron energy and angle, the DDCS differential in the projectile-scattering angle and electron-ejection energy, and finally, the DDCS differential in the projectile-scattering angle and electron-ejection angle. In this work, we calculate the DDCS differential in the energy and in the angle of the ejected electron. The other types of DDCS will be considered elsewhere. The DDCS differential in the ejected-electron energy and angle can be calculated from the fully differential cross section as

$$\begin{aligned} \frac{d^2\sigma _{\textrm{ion}} }{dE_{\textrm{e}}d\varOmega _{\textrm{e}}} = \int \frac{d^3\sigma _{\textrm{ion}}}{dE_{\textrm{e}}d\varOmega _{\textrm{e}}d\varOmega _f} d\varOmega _f, \end{aligned}$$
(4)

where \(\varOmega _{\textrm{e}}\) is the solid angle of \(\varvec{\kappa }\), the electron momentum in the lab frame, into which the electron is ejected, \(E_{\textrm{e}}=\varvec{\kappa }^2/2\) is the ejected-electron energy, and \(\varOmega _{f}\) is the solid angle of the scattered projectile in the final channel f.

In the current two-centre approach, the fully differential cross section for ionisation consists of the incoherent combination of the direct ionisation and electron capture into continuum components. In the laboratory frame, it is written as

$$\begin{aligned} \frac{d^3\sigma _{\textrm{ion}} }{dE_{\textrm{e}}d\varOmega _{\textrm{e}}d\varOmega _f} =&\frac{\mu _T^2}{(2 \pi )^2}\frac{q_f \kappa }{q_i}\big (\big |T^{\textrm{DI}}_{fi}(\varvec{\kappa },\varvec{q}_f,\varvec{q}_i) \big |^2\nonumber \\&+\big |T^{\textrm{ECC}}_{fi}(\varvec{\kappa }-\varvec{v},\varvec{q}_f,\varvec{q}_i)\big |^2\big ). \end{aligned}$$
(5)

Choosing the target to be initially in its ground state, we set \(\varvec{q}_i=\varvec{k}_{\alpha =1}\) in both terms, while \(\varvec{q}_f= {\varvec{k}}_{\alpha '}\) in the DI term and \(\varvec{q}_f={\varvec{k}}_{1\beta '}\equiv {\varvec{k}}_{2\beta '}\) in the ECC term. The reduced mass in the direct channel is denoted as \(\mu _T\).

Generalising the approach given in Ref. [38] to proton-impact ionisation of helium, the direct ionisation amplitude can be written in terms of the direct-scattering (DS) amplitude as

$$\begin{aligned} T_{fi}^{\textrm{DI}}(\varvec{\kappa },\varvec{q}_{f},\varvec{q}_i) =&\langle {\varphi }^{\textrm{He}}_{\varvec{\kappa }} | \psi ^{}_{f} \rangle T_{fi}^{\textrm{DS}}(\varvec{q}_{f},\varvec{q}_i), \end{aligned}$$
(6)

where \(\varphi ^{\textrm{He}}_{\varvec{\kappa }}\) is the wave function representing the true continuum state of the ejected electron with momentum \(\varvec{\kappa }\) relative to the target residual \(\hbox {He}^+\) ion. The amplitude for electron capture into the continuum of the formed hydrogen atom is, accordingly, written in terms of the electron-capture (EC) amplitude as

$$\begin{aligned} T_{fi}^{\textrm{ECC}}(\pmb {\varkappa },\varvec{q}_{f},\varvec{q}_i) =&\langle {\varphi }^{\textrm{H}}_{\pmb {\varkappa }} | \psi ^{}_{f} \rangle T_{fi}^{\textrm{EC}}(\varvec{q}_{f},\varvec{q}_i), \end{aligned}$$
(7)

where \(\varphi ^{\textrm{H}}_{\pmb {\varkappa }}\) is the true Coulomb wave representing the continuum state of the ejected electron with momentum \(\pmb {\varkappa }\) relative to the projectile nucleus. For further details of the direct-scattering and electron-capture amplitudes, see Papers I and II.

The full treatment of bare-ion collisions with multi-electron targets, as described above, is very involved theoretically and time-consuming computationally. On the other hand, treating the target as hydrogen-like can simplify the structure-related complications. Recently, a method was developed within our wave-packet convergent close-coupling formalism [44] that allows one to reduce the multi-electron target to an effective single-electron system. This enables us to make use of existing numerical methods implemented for collisions between bare-ions and one-electron targets, thereby reducing the amount of required computational resources when calculating cross sections. Below we use both two-electron and effective one-electron methods.

3 Energy and angular distribution of emitted electrons

3.1 Details of calculations

Below we present the results obtained for the doubly differential cross section for ionisation calculated using Eq. (4) at five intermediate projectile energies: 70, 75, 100, 150, and 300 keV. The final four energies are the same as used in Papers I and II, where the results on the angular differential cross section calculations of the concurrent binary processes and singly differential ionisation cross sections were reported, respectively. Unfortunately, at a collision energy of 75 keV there are no data to compare to. However, this incident energy is important because the latest FDCS measurements have been performed at this energy. Therefore, in this work, we add results at 70 keV, where there are experimental data to verify our calculations. The results are presented in the laboratory frame as a function of ejected-electron energy at selected ejection angles ranging from the near-forward direction through to near-backward ejection. As in Papers I and II, our main results obtained using the correlated two-electron wave-packet convergent close-coupling approach are denoted as WP-CCC. For comparison, we also present results obtained using the effective one-electron treatment of the helium target, hereafter denoted as E1E WP-CCC. In both approaches the number of included negative- and positive-energy pseudostates are increased until adequate convergence is achieved in the DDCS of interest.

The set of equations (3) for the expansion coefficients, representing the transition amplitudes, is solved using the Runge-Kutta method by incrementing the position of the projectile along the z-axis from \(z_{\textrm{min}}=-100 \) to \(z_{\textrm{max}}=100 \ \mathrm{a.u.}\) for all impact parameters. The exponential z-grid, with denser discretisation around \(z=0\), contains 300 points. The impact parameters ranged from 0 up to 40 a.u., which was sufficient to allow for the probability of all of the collision processes to fall off several orders of magnitude.

For discretisation of the continuum, the maximum momentum of the ejected electron \(\kappa _{\textrm{max}}\) was increased systematically until no further change in the results was observed. The required maximum momentum \(\kappa _{\textrm{max}}\) ranged from 8 a.u. at the lowest three energies to 10 a.u. at the highest two.

To achieve convergence in the two-electron WP-CCC calculations, a basis containing bound states up to the principal quantum number 5 and 20 bin states for each orbital angular momentum l on both centres was used at 70 and 75 keV. At larger energies, the number of bins was increased to 23. Within the E1E WP-CCC calculations, the basis contained bound states with the principal quantum number \(n \le 5\) and 25 bins for 70 and 75 keV, and 28 bins for all higher energies. The maximum orbital angular momentum for both negative-energy and positive-energy states in all cases was 3.

3.2 Results of calculations

Fig. 1
figure 1

Doubly differential cross section for ionisation of helium by 70 keV protons as a function of ejected-electron energy. The electron is ejected at angles of 10\(^\circ \), 50\(^\circ \), 90\(^\circ \) and 130\(^\circ \). Experimental data are by Rudd and Madison [15]. The present two-electron and E1E WP-CCC calculations are shown by straight lines connecting small dots. The dots correspond to the energies of the wave packets. The present FBA results are also shown

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Results obtained for a proton energy of 70 keV within the WP-CCC approach are presented in Fig. 1 (upper panels) in comparison with the experimental data [15] and the E1E WP-CCC calculations. Also shown are the FBA results calculated within the E1E method. An FBA calculation performed using the two-electron WP-CCC approach would produce similar results. In the lower panels, we present the DI and ECC components of the DDCS obtained within the two-electron WP-CCC method. In all panels, our results are shown by small dots connected by straight lines. The dots correspond to the energies of the wave-packet pseudostates.

Agreement between the WP-CCC and E1E WP-CCC results and the experiment by Rudd and Madison [15] is very good at all ejection angles and across the ejected-electron energy range shown. However, we see small deviations between the two methods. As we will see below, agreement between the two-electron and E1E methods improves with increasing collision energy. The FBA results only agree with experiment and our WP-CCC method in the peak near zero energy at 10\(^\circ \). Everywhere else, the FBA significantly overestimates both experimental data and WP-CCC results. There are no other calculations performed for this proton energy and ejection angles.

The lower panels show the interplay between the DI and ECC components of the corresponding DDCS. In particular, the 10\(^\circ \) panel emphasises the importance of the two-centre model as the peaks of the DI and ECC cross sections near 0 eV combine to form a ridge that is essential for replicating the experiment in this region. At 10\(^\circ \) and 50\(^\circ \), the dominant channel is ECC everywhere except for very small emission energies. When electrons are emitted into the perpendicular and backward directions, the magnitude of the ECC cross section decreases and the DI channel dominates across the entire energy range.

Fig. 2
figure 2

The same as in Fig. 1 but at a projectile energy 75 keV

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The results obtained for a projectile energy of 75 keV are presented in Fig. 2. One can notice that the agreement between the two-electron and E1E methods has slightly improved. Otherwise, the results are very similar to those at 70 keV shown in Fig. 1. No experiments or other theoretical calculations are available at this projectile energy.

The upper panels in Fig. 3 show the present DDCS for a proton energy of 100 keV, obtained using the two-electron WP-CCC and E1E WP-CCC methods in comparison with experiment [11, 12, 15, 17, 19] and other calculations [12, 19, 23, 30]. Overall, our results agree very well with the experimental data. It should be noted that there is a consistent discrepancy between the experimental data of Gibson and Reid [17] and Rudd et al.  [11, 12, 15, 16] at all ejected-electron angles. Gibson and Reid [18] suggested that the Rudd et al.  measurements may be inaccurate due to the influence of electrons reflected by the chamber walls or multiple scattering in the target gas. Both methods quoted comparable uncertainties. To be specific, these were 80% below 5 eV, 40% below 10 eV and 13% above 10 eV for Gibson and Reid [17], and as high as 100% below 10 eV, 40% below 20 eV and 20% above 20 eV for Rudd et al.  [11, 12, 15, 16]. For clarity, error bars have not been included in the figures. There is currently no consensus on which experimental method is more accurate, however, as with the SDCS of ionisation presented in Paper II, we find better agreement with the measurements of Rudd et al.  [11, 12, 15, 16].

At the smallest ejection angle, 10\(^\circ \), the WP-CCC results are in excellent agreement with the experiments of Rudd et al.  [11, 12, 15] and Bernardi et al. [19] up to 100 eV. Our results slightly overestimate the binary peak though, once again, display very good agreement at ejection energies above 300 eV. At this angle we find that the WP-CCC cross sections slightly underestimate the Gibson and Reid [17] data available only below 100 eV. Furthermore, the E1E WP-CCC calculations coincide with the full WP-CCC at this angle. The CDW-EIS approach [19] agrees with experiment for large ejection energies, however, completely misrepresents the shape for small ones. This method predicts that the cross section peaks as ejection energy approaches 0 eV. The experimental data of Rudd et al.  [11, 12, 15], however, shows a clear downward turn, which is reproduced by the WP-CCC method. The CTMC [30] was only applied up to an ejection energy of 100 eV with moderate success. The results at 10\(^\circ \) overestimate all experiments at small energies and underestimate them near 100 eV. At 50\(^\circ \) and 90\(^\circ \), agreement for small energies is generally good, however, the CTMC calculations still underestimate the experimental measurements as ejection energy increases. The BE approach [20] underestimates all the experimental data as well as other theoretical results below 150 eV. However, the approach reproduces the binary peak well.

Fig. 3
figure 3

Doubly differential cross section for ionisation of helium by 100 keV protons as a function of ejected-electron energy. The electron is ejected at angles of 10\(^\circ \), 50\(^\circ \), 90\(^\circ \) and 130\(^\circ \). Experimental data are by Rudd and Jorgensen [11] (a), Rudd et al. [12] (b), Rudd and Madison [15] (c), and Gibson and Reid [17]. The lines with points represent the present WP-CCC calculations. Other theoretical calculations are the BE of Bonsen and Vriens [20], Bonsen and Banks [23], the CTMC by Reinhold and Olson [30], and the CDW-EIS by Bernardi et al. [19]

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The WP-CCC and E1E WP-CCC methods at 50\(^\circ \) show very good agreement with the experimental data of Rudd et al.  [11, 12, 15] for all ejection energies. The results also agree well with the Gibson and Reid [17] data in the somewhat narrower energy range where it is available. Our calculations agree well with the experimental measurements by Bernardi et al. [19] for emission energies below 100 eV but underestimate them at higher energies, agreeing better with the other available experimental data. The WP-CCC and CDW-EIS results both agree well with the data of Rudd et al.  [11, 12, 15] up to 200 eV. However, the CDW-EIS calculations begin to deviate from the experiment at higher emission energies, unlike the WP-CCC calculations.

For ejection in the perpendicular direction, the WP-CCC results are in perfect agreement with the experimental data by Rudd et al.  [11, 12, 15]. Noteworthy is the small bump, which can be seen in the ECC component in the lower panel around 140 eV. The presence of this feature demonstrably improves the agreement of the WP-CCC approach with experiment. As expected, our results do not agree with the Gibson and Reid [17] experiment or the Bernardi et al. [19] experiment except for the last few points around 100 eV. The same can be said for the E1E WP-CCC results except for ejection energies above 200 eV, where this approach falls below the WP-CCC results. This discrepancy is the manifestation of the electron–electron correlation effects. The electron–electron correlations may appear smaller than expected, however, it should be emphasised that the E1E WP-CCC method [44] is quite different from traditional effective-potential methods and includes the electron–electron correlations, albeit partially. The other theories applied at this angle, the CTMC method of Reinhold and Olson [30] and the CDW-EIS method of Bernardi et al. [19], are limited to energies below 100 eV. They underestimate the Rudd et al.  [11, 12, 15] experimental data. However, they agree with the Gibson and Reid [17] and Bernardi et al. [19] experiments below 75 eV. Above 75 eV, the CDW-EIS results sharply deviate from all other calculations and experimental data.

The WP-CCC method again demonstrates excellent agreement with the experimental data by Rudd et al.  [11, 12, 15] for the backward ejection angle of 130\(^\circ \) across the entire ejection energy range presented here. It should be noted that a small feature in the Rudd et al. [12] measurement around 35 eV is due to an auto-ionisation peak. Autoionisation is a two-electron process so is not considered in the present work. Hence, this feature is not reproduced by the WP-CCC method. The E1E WP-CCC results are also in very good agreement, though they tend to slightly underestimate the full WP-CCC results and the experimental data. No other calculations are available at this angle.

In the bottom panel, we can see how the relative importance of the DI and ECC components changes as the ejection angle increases. In general, the DI channel is fairly consistent in magnitude at all ejection angles presented, while the ECC cross section decreases as ejection angle increases. Again, for 10\(^\circ \), we see that both the DI and ECC cross sections peak at small energies. The DI component of the cross section peaks close to zero emission energy, whereas the ECC component peaks at a slightly higher emission energy creating the ridge, which is supported by experimental findings. ECC remains the dominant channel beyond the ridge. At 50\(^\circ \), the ECC cross section is not as large as for small energies such that the total DDCS maintains a steady decline between the regions where DI and ECC dominate, which is in perfect agreement with experiment. At 90\(^\circ \) and 130\(^\circ \), DI becomes the dominant channel regardless of the energy of the emitted electron.

Fig. 4
figure 4

Doubly differential cross section for ionisation of helium by 150 keV protons as a function of ejected-electron energy. The electron is ejected at angles of 10\(^\circ \), 50\(^\circ \), 90\(^\circ \), and 130\(^\circ \). Experimental data are by Rudd and Jorgensen [11]. The lines with points represent the present WP-CCC calculations

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The results obtained within the two-electron and one-electron WP-CCC approaches at a projectile energy of 150 keV are presented in the upper panels of Fig. 4 in comparison with the experimental data [11]. Excellent agreement between the WP-CCC method and the experiment is found at an ejection angle of 10\(^\circ \) for all emission energies except in the region of the binary-encounter peak, where our results slightly overestimate the experiment. We also see very good agreement between our results and the experimental data at emission angles of 50\(^\circ \), 90\(^\circ \) and 130\(^\circ \). The FBA does not agree with experiment or the present WP-CCC results anywhere. In particular, at 10\(^\circ \), where the FBA had been adequate in predicting the low-energy peak at 70 and 75 keV, the shape of the results is highly inaccurate. No other theoretical calculations are available at this projectile energy. In the lower panels we see that the DI and ECC contributions to the ionisation cross section follow a similar behaviour as observed at smaller projectile energies.

Fig. 5
figure 5

Doubly differential cross section for ionisation as a function of ejected-electron energy at a projectile energy of 300 keV. The electron is ejected with angles of 10\(^\circ \), 50\(^\circ \), 90\(^\circ \) and 130\(^\circ \). Experimental data are by Rudd and Jorgensen [11], Toburen [16], Stolterfoht [16] and Gibson and Reid [17]. The lines with points represent the present WP-CCC calculation. Other theoretical calculations are the FBA by Bell and Kingston [26], the BC-FBA by Salin [24], the BE by Bonsen and Vriens [20], the FM approach by Macek [22] (extracted from Band [25]), the LCAO method of Band [25], and the CDW-EIS by Fainstein et al. [28]

Full size image

Finally, the results obtained at a projectile energy of 300 keV are presented in Fig. 5. The present two approaches are in perfect agreement at all ejected-electron energies and angles except for very large angles at very high energies. Comparison is also made with the experimental data [12, 16] and other calculations [20, 22, 24,25,26, 28].

For an ejection angle of 10\(^\circ \), the WP-CCC approach agrees very well with the only available experiment [12] for smaller ejection energies. However, it again overestimates the binary peak, despite reproducing its shape very well. The LCAO method of Band [25] also gives good agreement for large ejection energies, but does not extend to small energies. Despite the fact that it is a lower-order method, the BC-FBA of Salin [24] agrees well with the experimental data. The CDW-EIS results by Fainstein et al. [28] agree well in terms of shape but underestimate the experimental data for ejection energies below 200 eV. The BE method of Bonsen and Vriens [20] expectedly does not agree with the experiment except at the binary peak. The Faddeev-Merkuriev approach by Macek [22] was applied only within the ejected-electron energy range of 100–700 eV. Within this range, it reproduces the shape but consistently overestimates experimental measurements.

At 50\(^\circ \), results from the WP-CCC method are in excellent agreement with the experimental data from Rudd et al.  [12, 16]. The CDW-EIS method by Fainstein et al. [28] also agrees very well with experiment. However, the two methods deviate slightly beyond 200 eV. Interestingly, our results align very well with the measurements of Stolterfoht [16], whereas the CDW-EIS follows the data of Toburen [16] and Rudd et al. [12].

The WP-CCC results are in excellent agreement with the experiments of Rudd et al. [12] and Stolterfoht [16] for 90\(^\circ \) emission, but slightly overestimate the experiment of Toburen [16] which falls slightly below the other experimental data. The other theoretical methods available at this angle agree with experiment up to 50 eV. However, above 50 eV, the FBA of Bell and Kingston [26] overestimates and the CDW-EIS calculations by Fainstein et al. [28] underestimates all the experiments.

Finally, for the backward ejection angle of 130\(^\circ \), we find excellent agreement between the experiment by Rudd et al. [12] and the results of the WP-CCC method.

In the bottom panel, we see a similar pattern as for the other proton energies. For forward ejection, the DI cross section peaks near 0 eV, then falls rapidly such that the DDCS is dominated by the ECC channel at larger ejection energies. For perpendicular and backward ejection angles, the ECC cross section falls significantly, making DI the dominant channel across the entire ejection energy range considered here.

Overall, Figs. 1, 2, 3, 4 and 5 have demonstrated that the two-electron and E1E WP-CCC approaches agree very well with the available experimental data. This study is the first to systematically apply a single method to calculating DDCS for a wide range of ejection angles in this scattering system. For electrons ejected into small angles around the forward direction, our results reproduced the ECC peak near zero energy. However, while the WP-CCC method demonstrated agreement in terms of shape, our results consistently overestimated the binary peak when electrons are ejected at 10\(^\circ \). The discrepancy becomes more pronounced as the projectile energy increases and the binary-encounter process becomes more prominent. At this stage, we have no explanation for this discrepancy. Calculations of the FDCS, that are currently in progress, will hopefully shed more light on the situation. At 50\(^\circ \), 90\(^\circ \) and 130\(^\circ \), where the contribution of the binary-encounter mechanism becomes small, agreement with experimental measurements is better.

4 Conclusion

We have extended our investigation of the four-body proton–helium collision problem using the two-centre wave-packet convergent close-coupling approach to doubly differential ionisation. The WP-CCC method uses a correlated two-electron wave function for the helium target. Positive-energy wave-packet pseudostates are used to discretise the continuum. The DDCS for ionisation as a function of the energy and angle of the ejected electron has been calculated at intermediate projectile energies. The obtained results agree very well with experimental data where available.

Also used was an alternative, simpler method that reduces the target to an effective single-electron system. The high level of agreement between the two-electron and effective one-electron methods leads us to conclude that the effective one-electron treatment of the target we have developed can be an acceptable alternative for calculating the doubly differential cross section for ionisation in proton–helium collisions. It remains to be seen whether this finding holds for the fully differential cross section for single-ionisation.

In this work, we used the WP-CCC approach to calculate one type of the doubly differential cross section for ionisation. In Paper I, the method was used to calculate angular differential cross sections for elastic-scattering, target excitation, and electron-capture processes. In Paper II, the method was applied to all three types of the singly differential cross sections for ionisation. Here, we have shown the method to also work well for the DDCS considered. The results presented in these three works demonstrate that a realistic differential picture of all processes taking place in collisions of intermediate-energy protons with helium atoms can be obtained by the WP-CCC method. Thus, one can conclude that the approach is capable of providing accurate integrated [4] and differential cross sections for all the reaction channels from a single calculation in a unitary fashion. This positions the WP-CCC method well to investigate the helium ionisation process further. There are a number of studies presenting the DDCS in the energy and angle of the electron in the high energy regime [48,49,50] and the forward direction [17,18,19, 21, 22, 26, 28, 32]. We also intend to apply the WP-CCC method to the DDCS as a function of the projectile scattering angle and ejected-electron energy as well as the fully differential ionisation cross section. A kinematically complete description of ionisation remains one of the most challenging problems in atomic collision physics. In particular, the fully differential cross section of single ionisation of helium in collisions with 75 keV protons measured by Schulz et al. [8] in various kinematic regimes is yet to see a satisfactory description. There is little to no agreement between the experiment and theoretical results or even between various theoretical results themselves. The WP-CCC approach developed here may shed some light on the situation.

Furthermore, we intend to apply the two-centre WP-CCC approach to other cases where discrepancies still remain. In the high-energy regime, where electron-capture channels can be neglected, the single-centre WP-CCC method has been applied to the proton–helium ionisation process [40, 41] and the results compared with the ultrahigh-resolution experiment by Chuluunbaatar et al. [51]. Agreement was excellent. There was, however, a small deviation identified in the position of the binary peak at the smallest considered momentum transfer [41]. Application of the two-centre WP-CCC method to this problem may clarify whether the double-scattering mechanism, which is important for calculations of the the angular differential cross section for electron capture [52, 53], affects the ionisation cross section. For single ionisation of helium by the impact of \(\hbox {C}^{6+}\) ions at 100 MeV/amu investigated by Schulz et al. [54], we previously employed a single-centre quantum-mechanical implementation of the CCC method [55]. Results obtained did not support the features reported by Schulz et al. [54]. This problem remains a controversial topic and a challenge for theoretical models. Again, the two-centre WP-CCC should allow us to investigate possible subtle high-order effects that could not be seen with a single-centre approach.