Differential Study of Projectile Coherence Effects on Double Capture Processes in p + Ar Collisions

atoms Article Differential Study of Projectile Coherence Effects on Double Capture Processes in p + Ar Collisions Trevor Voss 1,2 , Basu R. Lamichhane 1,3 , Madhav Dhital 1 , Ramaz Lomsadze 4 and Michael Schulz 1, * 1 2 3 4 * Department of Physics and LAMOR, Missouri University of Science & Technology, Rolla, MO 65409, USA; (T.V.); (B.R.L.); (M.D.) Biophotonics Center, Vanderbilt University, Nashville, TN 37235, USA Physics Department, Temple University, Philadelphia, PA 19122, USA Department of Exact and Natural Science, Tbilisi State University, Tbilisi 0179, Georgia; Correspondence:



 Received: 15 February 2020; Accepted: 26 March 2020; Published: 28 March 2020 Abstract: We have measured differential yields for double capture and double capture accompanied by ionization in 75 keV p + Ar collisions. Data were taken for two different transverse projectile coherence lengths. A small effect of the projectile coherence properties on the yields were found for double capture, but not for double capture plus ionization. The results suggest that multiple projectile–target interactions can lead to a significant weakening of projectile coherence effects. Keywords: ion-atom collisions; few-body problem; coherence effects; charge exchange 1. Introduction Studies of atomic collisions are particularly suitable to study the fundamentally important few-body problem (FBP) [1,2]. The essence of the FBP is that the Schrödinger equation is not analytically solvable for more than two mutually interacting particles even when the forces acting within the system under investigation are precisely known. Theory therefore has to resort to heavy modelling efforts and the assumptions and approximations entering in these models have to be tested by detailed experimental data. In most cases, the quantity that is measured in a collision experiment is the cross section for a specific process selected in the experiment. In the case of heavy projectile collisions, these cross sections are theoretically often treated within perturbative models, e.g., [3]. For electron impact, non-perturbative methods are routinely applied, e.g., [4–6], however, for ion impact such approaches, e.g., [7–10] are much more challenging and still relatively rare compared to perturbative calculations. One well-established perturbative model is represented by the Born series. There, the target is described by eigenstates of the unperturbed target Hamiltonian and the projectile by plane waves. The transition amplitude is then expanded in powers of the interaction potential. Some of the limitations of the Born series in accurately describing processes occurring in atomic collisions are well known, e.g., [3]. For example, it represents a two-state approximation because it considers only one target eigenstate for the incoming channel (typically the ground state) and one in the outgoing channel. As a result, couplings between different eigenstates in the final state, which can be very important especially at small collision velocities, are completely ignored. Furthermore, in practice, the power series has to be truncated after some term and to the best of our knowledge no calculations beyond second order have been published yet, e.g., [11–13]. Some of these shortcomings are circumvented in distorted wave approaches, e.g., [14–16]. However, recently, we Atoms 2020, 8, 10; doi:10.3390/atoms8020010 www.mdpi.com/journal/atoms Atoms 2020, 8, 10 2 of 9 demonstrated that in ionization of H2 and He by intermediate energy proton impact even distorted wave calculations do not give satisfactory results if the electron is ejected with a velocity close to the projectile velocity [17,18]. Another shortcoming of the Born series (and other methods including non-perturbative approaches) did not receive much attention until about a decade ago [19]. It results from the description of the incoming projectile in terms of a plane wave, which implies that the projectile has a sharp momentum. However, in reality the projectiles have an intrinsic momentum distribution of finite width. This means that the projectiles are not fully coherent and, in some cases (depending on the dimension of the diffracting object) may even be largely incoherent. As a result, interference effects predicted by theory are not always experimentally observable. Indeed, in the scattering angle dependence of measured double differential ionization cross sections for p + H2 collisions, interference structures were observed for a coherent projectile beam, but were absent for an incoherent beam [19]. In numerous follow-up experiments, similar projectile coherence effects on the cross sections were observed [20–30] (for a review, see [31]). Furthermore, the interpretation of the experimental data was supported by several theoretical studies, e.g., [32–36]. In measured fully differential cross sections (FDCS) for dissociative capture leading to relatively large kinetic energy releases (KER) in p + H2 collisions [28] such projectile coherence effects were found to be less pronounced than in single ionization or capture. In the case of Coulomb explosion following double capture, such effects were not discernable at all [28]. This was interpreted as a “washing out” of the phase due to multiple projectile scatterings leading to dissociative capture. For large KER values dissociative capture proceeds predominantly through excitation of the second target electron. At the relatively small projectile energy of 75 keV the capture and excitation occur mostly through two independent interactions of the projectile with each electron. Therefore, the final scattering angle observed in the experiment is due to a convolution of two scattering angles from these two interactions. The phase depends on the scattering angle in each interaction and is thus no longer unambiguously determined by the measured total scattering angle. One complication in this interpretation is that the experiment used a diatomic molecular target. As a result, both single- and two-center interference can contribute to the cross sections measured for a coherent beam [24]. In the former, different paths (impact parameters) leading to the same scattering angle and in the latter waves diffracted from the two centers of the molecule interfere with each other. For both types of interference the phases may differ from each other, which can also contribute to a “washing out” of any interference structure. Therefore, to trace the reasons for the (near) absence of coherence effects in double capture it is important to perform the experiment for an atomic target. Differential double capture cross sections have been measured for He2+ + He [37–39] and for p + He collisions [40]. However, all of these experiments were performed for only one projectile coherence length. In this article, we present measured differential cross sections for double capture (DC) and for double capture plus single ionization (DCI) for (...truncated)