Influence of the post-collision interaction on interference effects in ionization of H₂ by proton impact

The reaction dynamics of ionization of simple atoms by ion-impact has been studied extensively in kinematically complete experiments over the last decade [e.g. 1–5, for a recent review see 6]. The features in the three-dimensional electron ejection angle dependence of the resulting fully differential cross sections (FDCS) are in most cases remarkably simple. These data tend to be dominated by a pronounced peak structure approximately in the direction of the momentum transfer q (defined as the difference between the initial and scattered projectile momenta) known as the binary peak. For certain kinematic conditions a second less pronounced maximum, dubbed the recoil peak, is observed in the direction of –q. The qualitative shape of this basic double lobe pattern is not even strongly altered when higher-order contributions are large. In spite of this simplicity the theoretical quantitative description of the reaction dynamics has proven to be quite challenging [e.g. 7–14].

One might suspect that the qualitative features in the FDCS for ionization of homo-nuclear molecules are essentially the same as for the corresponding atoms. However, one important difference between atomic and molecular targets is that for the latter the electron can be emitted from and the projectile scattered off either of the atomic centers in the molecule. The coherent sum of both contributions can lead to observable interference effects. Such structures were indeed reported in the ejected electron spectra [e.g. 15, 16] and in double differential cross sections (DDCSs) as a function of projectile scattering angle [17]. However, these experiments were not kinematically complete and as a result the spectra were partially averaged over the phase angle in the interference term thereby somewhat restricting the depth in the information extracted from the data.

More recently, we reported fully differential studies on contributions from interference to ionization in p + H₂ collisions [18, 19]. One interesting result was that, apart from molecular two-center interference, single-center interference between first- and higher-order transition amplitudes, involving for example the interaction between the projectile and the target nuclei, also plays an important role and at small electron energies is, in fact, more important than two-center interference. The impact parameters contributing to a specific scattering angle usually differ between first- and higher-order processes. Single-center interference can thus also be interpreted as interference between different impact parameters leading to the same scattering angle.

These findings of [18, 19] represent just one example of how fully differential data can provide more insight into the reaction dynamics than less differential data. However, there are many more questions relating to the reaction dynamics of ionization of molecular targets on which less differential studies so far did not provide answers. For example, in [17] we observed that the interference structures appeared to be much less pronounced for ejected electron energies corresponding to electron speeds v_e close to the projectile speed v_p. In that work we discussed a possible explanation which is based on two hypothetical assumptions: first, the apparent suppression of interference may be related to a secondary interaction between the outgoing projectile and the electron lifted to the continuum by a preceding primary interaction. It is well established that such effects maximize at v_e = v_p [20, 21]. A well-known manifestation of such multiple projectile-electron interactions is the occurrence of the so-called cusp peak in the energy spectrum of electrons emitted in collisions with ionic projectiles at an energy corresponding to v_e = v_p [22]. However, the cusp peak is only observed in the forward direction within a very narrow angular range. In the literature on ion–atom scattering the underlying process leading to cusp electron production is often referred to as post-collision interaction (PCI). Another manifestation observed in the electron ejection angle dependence of the FDCS is a shift of the binary peak in the backward (forward) direction relative to the direction of q for electron (ion) impact. This feature in the FDCS is what is regarded as PCI in the literature on electron-impact ionization [e.g. 23]. Here, we adopt the latter (more general) notation on PCI.

The second assumption entering in the possible explanation for the dampened interference structure for v_e ≈ v_p is that the coherence required for observable interference may be lost (or at least reduced) for this electron speed. The coherence requirement means that the length, over which the phase fluctuation of the incoming projectile wave is sufficiently small to not smear out the interference pattern, i.e. the coherence length, must be larger than the dimension of the diffracting object. However, neither could we provide conclusive evidence for this explanation nor could we offer reasons as to why interference would be somehow linked to PCI (or lack thereof) or why coherence would be reduced in the presence of strong PCI. Another factor which could lead to a damping of the interference, which was not discussed in [17], but will be addressed in this article, is that one of the interfering amplitudes is significantly larger than the other(s).

The difficulties in performing a more conclusive analysis of the observations in [17] stem from two sources: first, the experiment was not kinematically complete, thereby compromising the level of detail in the data. Second, at the time it was not clear how the role of coherence could be experimentally tested. Since then both of these problems have been addressed: we have performed kinematically complete experiments [18, 19, 24] and we demonstrated that the coherence length in the direction transverse to the projectile beam direction Δx can experimentally be varied by changing the geometry of a collimating slit in front of the target [25]. Indeed, for small Δx interference structures were found to be much less pronounced than at large Δx.

Performing kinematically complete experiments for different projectile coherence properties we could demonstrate that single- and two-center interference can be separated by analyzing the FDCS either for fixed momentum transfers or for fixed recoil-ion momenta [19]. However, a possible link between interference effects and PCI, as suggested in [17], was not addressed in that work. Here, we present the results of a kinematically complete experiment on ejection of target electrons with an energy corresponding to v_e = v_p. The data were analyzed under kinematic conditions either favoring or suppressing PCI. A comparison between multiple differential momentum spectra for these kinematic conditions suggests that single-center interference is significantly affected by PCI.

The details of the experiment were described previously [18]. In brief, a proton beam was generated by a hot cathode ion source and accelerated to an energy of 75 keV. The beam was collimated by an aperture with a diameter of 1.5 mm at the exit of the accelerator terminal and by a vertical and horizontal slit with width a = 150 μm placed at a distance of L₁ = 50 cm or L₂ = 6.5 cm before the target region. The transverse coherence length, given by Δx = ½ Lλ/a, was about Δx = 3.3 a.u. for L₁ and less than 1 a.u. for L₂. The proton beam, propagating in the z-direction, was then intersected with a very cold (T ≈ 1–2 K) neutral H₂ beam, propagating in the y-direction, from a supersonic gas jet.

Protons which were not charge-exchanged in the collision with the target were selected by a switching magnet and energy-analyzed by an electrostatic parallel-plate analyzer [26]. The projectiles which suffered an energy loss of  = 57 eV were detected by a position-sensitive channel-plate detector. The position in the x-direction, i.e. the horizontal component, determines the scattering angle and thereby the x-component of the momentum transfer q. Because of the very narrow width of the entrance and exit slits of the analyzer in the y-direction (75 μm) q_y = 0 within the experimental resolution. The z-component of q is to a very good approximation given by q_z = /v_p.

The recoiling H₂⁺ ions produced in the collisions with the projectiles were extracted by a weak and uniform electric field of 8 V cm⁻¹ pointing in the x-direction, then drifted in a field-free region twice as long as the extraction region, and were detected by a two-dimensional position-sensitive channel-plate detector. The recoil-ion and projectile detectors were set in coincidence. From the position information the y- and z-components of the recoil-ion momentum p_rec could be determined. The x-component was obtained from the time-of-flight from the collision region to the detector which, in turn, is contained in the coincidence time. Finally, the ejected electron momentum was deduced by p_el = q − p_rec using momentum conservation.

In analogy to classical optics the cross section for a coherent beam (i.e. Δx > D, where D is the dimension of the diffracting object) dσ_coh can be expressed as a product of the cross section for an incoherent beam (Δx < D) dσ_inc and the interference term I. In other words, I is given by the ratio R = dσ_coh/dσ_inc, which is plotted for  = 57 eV as a function of the projectile scattering angle θ in figure 1. For comparison, the same data previously reported for  = 30 eV [27] are shown as open symbols. Earlier, we attempted to extract the interference term as a ratio between the experimental cross sections (taken for a coherent beam) and twice the theoretical cross sections for ionization of atomic hydrogen dσ_H [17]. Later, large discrepancies between the theoretical and experimental dσ_H were found [28] so that twice the theoretical values do not represent a good approximation for dσ_inc. As a result these ratios from [17] are not the same as those plotted in figure 1. This comparison illustrates the importance of directly measuring the incoherent cross sections for obtaining reliable information about the interference term.

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The θ-dependence of R plotted in figure 1 is not constant. A difference in the cross sections measured for the two slit distances can only be due to either experimental resolution effects (where contributions from the beam divergence are particularly important) or due to projectile coherence effects. Previously, we demonstrated already that the experimental resolution was identical for both slit distances [27]. Therefore, the data of figure 1 confirm the importance of coherence effects and show that some kind of interference is present in the coherent cross sections with a phase angle which depends on q_x = p_osinθ. Earlier, we demonstrated that FDCS ratios as a function of q_x can be well described under the assumption that interference is dominated by single-center interference with a q_x-dependent phase angle [18, 19]. It is therefore reasonable to assume that the structure in R(θ) of figure 1 is due to the same type of interference. On the other hand the structure is much less pronounced than for  = 30 eV, at least for θ < 0.8 mrad. This is consistent with the observation in [17] that interference structures in DDCSs for  = 57 eV as a function of θ are much weaker, if present at all, than for  = 30 eV. In [19] we also reported signatures of two-center interference in the p_rec-dependent FDCS ratios for  = 57 eV, however, they were not observed in the q_x-dependence of these ratios.

In the following we will discuss to what extent there may be a link between the apparent disappearance of interference structures in the DDCS at  = 57 eV and PCI, as contemplated in [17]. To this end we analyzed the ratios of figure 1 with additional kinematic conditions suitable to either enhance or suppress the effect of PCI on the reaction dynamics. For positively charged ion impact PCI has a tendency of focusing the ejected electrons in the forward direction. Since  = 57 eV corresponds to an electron speed which is almost the same as the projectile speed, one would therefore expect that in the longitudinal electron momentum spectrum (plotted in the top panel of figure 2 for the coherent beam) PCI leads to an enhancement of the intensity near p_ez ≈ v_p (indicated by the vertical dashed line in figure 2). This is indeed confirmed by the top panel of figure 2, where a pronounced maximum is seen at p_ez ≈ v_p. In the longitudinal recoil-ion momentum spectrum, shown in the bottom panel of figure 2, p_ez ≈ v_p corresponds to p_rz = q_z − p_ez ≈ /v_p − v_p = −0.54 a.u., which again is indicated by a dashed vertical line. Here, too, a pronounced maximum is observed at this value of p_rz. Therefore, it should be possible to enhance the effect of PCI on R by setting a condition on p_rz near −0.54 a.u. and to suppress PCI by setting a condition near significantly more positive values of p_rz⁴ . However, it should be noted that these conditions are not very effective in enhancing or suppressing contributions from cusp electrons because, as noted in the introduction, these occur only within a very narrow angular range in the forward direction.

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In figure 3 the ratio between the cross sections with a condition on p_rz = −1.0 to −0.1 a.u. (to which we refer as the 'PCI on' condition) and the cross sections with a condition on p_rz = −0.1 to +1.0 a.u. ('PCI off' condition) is plotted as a function of θ for the coherent beam. It can clearly be seen that the 'PCI on' condition strongly favors small scattering angles, which is exactly the behavior expected for PCI [21]. This can be explained in terms of a focusing effect due to PCI, in which the scattered projectile and the ejected electron attract each other towards the initial beam axis. At θ larger than approximately 0.4 mrad, on the other hand, the cross sections for 'PCI off' become larger than for 'PCI on'. This illustrates that indeed a condition on the longitudinal recoil-ion momentum is an effective method to either enhance or suppress effects due to PCI.

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In figure 4 we present the ratios between cross sections measured with coherent and incoherent beams with the 'PCI on' condition (closed symbols) and the 'PCI off' condition (open symbols) as a function of θ. The 'PCI on' ratios look very similar to those without any condition on p_rz plotted in figure 1. This is not surprising considering that the majority of events contained in figure 1 lie within the 'PCI on' condition. The θ-dependence of the 'PCI off' ratios, on the other hand, is quite different. More specifically, the interference minimum, seen at about 0.3 mrad in the 'PCI on' case, is shifted to 0.5−0.6 mrad.

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Without elaborate theoretical calculations it is difficult to analyze quantitatively what role PCI may play in suppressing interference structures by a loss of coherence, as suggested in [17]. In the following, we will therefore qualitatively discuss to what extent the present data are consistent with this hypothesis and to what extent other factors may contribute to the damping of the interference. As a first step we attempted to quantify how pronounced the interference structure is for the 'PCI on' condition compared to the 'PCI off' condition. To this end we fitted the ratios plotted in figure 4 by the model interference term, given by I = 1 + αcos(q_xΔb), which we reported for single-center interference in [27], using α and Δb as fitting parameters. Here, Δb is the impact parameter separation between the interfering amplitudes and α describes the damping of the interference structure by incomplete coherence even for the large slit distance or experimental resolution effects. For the 'PCI on' condition the best fit to the measured ratios (solid curve in figure 4) yields α = 0.2 ± 0.04 and Δb = 3.5 ± 0.5 a.u. while for the 'PCI off' condition we obtain α = 0.3 ± 0.03 and Δb = 2 ± 0.3 a.u. (dashed curve in figure 4).

This difference in α seems to support the hypothesis of [17] under consideration. However, for two reasons conclusions should be drawn cautiously. First, within the uncertainties the fitted values for α are not very far apart and it is thus not clear just how significant the difference in α is. Second, even for the 'PCI off' condition α is significantly smaller than for the measured ratios for  = 30 eV (α = 0.45 ± 0.03), which are compared in figure 5 (open symbols) to the 'PCI off' ratios for  = 57 eV (closed symbols). Therefore, even if the difference in the fitted values of α for the 'PCI on' and 'PCI off' conditions signify an important role of PCI it seems likely that PCI is not the only factor leading to the damping of the interference structure. Therefore, with the present data we cannot conclusively determine the role of PCI in interference effects. On the other hand, it is certainly fair to state that the present data are not inconsistent with the hypothesis of [17] in so far as they provide some support that PCI may contribute to the damping of the interference structure.

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Apart from the different damping in the ratios for  = 30 eV and the 'PCI off' data for  = 57 eV the shape of the θ-dependence of both data sets in figure 5 is remarkably similar. More specifically, the period and position of the extrema in the interference oscillation are the same and fitting the ratios by the model interference term yields the same value of Δb (2 a.u.). This similarity provides a clue for what else, apart from PCI, may contribute to the larger damping of the interference structure at  = 57 eV ('PCI off') compared to  = 30 eV. It suggests that for both the same transition amplitudes (in e.g. a perturbation expansion) interfere with each other, although we cannot entirely rule out that different transition amplitudes could coincidentally yield the same Δb. However, the relative importance of the interfering amplitudes is likely to change. For example, increasing the ejected electron energy presumably requires increasingly closer collisions between the projectile and the electron. The interaction of the projectile with the target nucleus then has a smaller effect. Therefore, one might expect that the importance of the first-order amplitude, relative to higher-order amplitudes involving the nucleus–nucleus interaction, increases with increasing electron energy. Since the present collision system corresponds to a relatively large perturbation parameter q_p/v_p ≈ 0.6, at small electron energies the higher-order amplitudes could be of similar magnitude as the first-order amplitude, which is a favorable condition for pronounced interference structures. At larger electron energies, on the other hand, the higher-order amplitudes could be substantially smaller leading to a damping of the interference structure.

The additional damping of the interference structure for the 'PCI on' condition, relative to the 'PCI off' condition', might be due to the coherence properties of the projectile beam with respect to Δb. In the comparison between  = 57 eV ('PCI off') and  = 30 eV this does not play any role because both Δx and Δb are identical for both cases. But for the 'PCI on' condition our fit of the model interference term to the measured ratios yields Δb similar to Δx so that the beam is only marginally coherent (although the observation of a residual interference structure shows that the beam is not completely incoherent yet). At present, we cannot offer a qualitative explanation as to why Δb is significantly larger for the 'PCI on' condition. PCI is a rather complex process which in some sense goes beyond second order. This can be understood using a classical analogy: after two particles collided with each other they cannot collide for a second time, because they depart from each other, unless one of them gets redirected by an interaction with a third particle. In the case of PCI that third particle is the target nucleus, which can redirect either the electron or the projectile. Another complication is that Δb is measured in three-dimensional space. For example, the ionization process may be selective on impact parameter vectors pointing in opposite directions for the interfering amplitudes under some kinematic conditions, but in the same direction for other conditions. It is therefore not easy to make even a qualitative prediction as to how Δb should depend on the relative importance of PCI without elaborate theoretical calculations.

Nevertheless, we obtained some support for our interpretation for the additional damping for the 'PCI on' condition by a classical trajectory Monte Carlo (CTMC) calculation [29]. In this approach an event file (similar to the data files generated in the experiment) is obtained in which for each ionization event the momentum components of all collision fragments are recorded. It is therefore possible to apply the 'PCI on' and 'PCI off' conditions to the calculated cross sections in exactly the same manner as was applied to the experimental data. The ratio between the calculated 'PCI on' and 'PCI off' probabilities are plotted in figure 6 for energy losses of 30 eV (dashed line) and 57 eV (solid line) as a function of impact parameter. The calculation was actually performed for an atomic hydrogen target, however, we believe that there are no significant differences to molecular hydrogen in the impact parameter dependence, which is not affected by single-center interference. Figure 6 shows that indeed for the 'PCI on' condition large impact parameters play a much more important role compared to the 'PCI off' condition. Furthermore, this effect is even more pronounced for  = 57 eV than for  = 30 eV, which is consistent with PCI effects generally maximizing for v_e ≈ v_p. Of course, the impact parameter dependence plotted in figure 6 do not directly reflect the impact parameter separation Δb between the interfering amplitudes, but it is reasonable to assume that Δb also increases with an increasing importance of large impact parameters. The CTMC calculation is thus not inconsistent with Δb for  = 57 eV being on average much closer to (or even larger than) Δx than for  = 30 eV. This would explain the larger damping of the interference structure for the larger energy loss and the apparent link between single-center interference and PCI.

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We have presented a kinematically complete experimental study of interference effects in ionization of H₂ by intermediate energy proton impact for electrons emitted with a speed equal to the projectile speed. Several years ago, we observed that for this electron energy interference structures in the projectile scattering angle dependence of the ionization cross sections were significantly less pronounced than at smaller electron energies [17]. More recently we demonstrated that single-center and molecular two-center interference can experimentally be separated [19]. Furthermore, in that work we found that scattering-angle dependent cross sections are dominated by single-center interference and recoil-ion momentum dependent cross sections by molecular two-center interference. Considering the combination of the results obtained from [17] and [19] it was the aim of the present study to better understand a potential link between single-center interference and the PCI, which is known to be very important for ejected electron speeds close to the projectile speed.

We demonstrated that setting a condition on the longitudinal recoil-ion momentum is an effective method to either enhance (by selecting ions recoiling in the backward direction) or to suppress (by selecting all other recoil ions) effects due to PCI. Thereby we were able to analyze the interference term for a given electron speed (here v_e = v_p) under conditions of weak or strong PCI. Our present results confirm our earlier observation [17] that, independent of whether PCI is weak or strong, interference is less pronounced than at smaller electron energies. A possible explanation, calling for theoretical confirmation, is that for small electron energies the interfering amplitudes may be of similar magnitude (due to the large perturbation parameter) while for larger electron energies the first-order amplitude may be significantly larger. In addition, for strong PCI the interference structure is even less pronounced than for weak PCI. Furthermore, in the former case the impact parameter separation between the interfering amplitudes is significantly larger than in the latter case, for which it is essentially the same as for small electron energies. Therefore the fixed coherence length corresponds to a less coherent projectile beam for strong PCI and a more coherent beam for weak PCI. This offers a plausible explanation for the stronger damping of the interference structure in the case of strong PCI. This interpretation is supported by a CTMC calculation investigating the effect of PCI on the impact-parameter dependent ionization probabilities.

This work was supported by the National Science Foundation under grant no. PHY-1401586.