CDW and CDW-EIS calculations for FDCSs in highly charged ion impact ionization of helium

Brazilian Journal of Physics, vol. 36, no. 2B, June, 2006 524 CDW and CDW-EIS Calculations for FDCSs in Highly Charged Ion Impact Ionization of Helium M. Ciappina and W. R. Cravero Av. Alem 1253, (8000) Bahı́a Blanca, Argentina Received on 29 July, 2005 In this work we present fully differential cross sections (FDCS) calculations using CDW and CDW-EIS theories for helium single ionization by 100 MeV/amu and 2 MeV/amu C6+ and 3.6 MeV/amu Au24+ and Au53+ ions. We performed our calculations for different momentum transfers and ejected electron energies. We study the influence of the internuclear potential on the ejected electron spectra. We discuss different regimes where the internuclear interaction can or cannot be neglected. We compare our calculations with experimental data available. It is shown that for high impact energy and small momentum transfer, internuclear potential effects can be neglected in FDCSs. Keywords: CDW; CDW-EIS; FDCS I. INTRODUCTION The study of electron emission spectra in ion-atom collisions has been a field of intense activity for years [1]. For intermediate to high energy single ionization there has been considerable theoretical efforts focused in the so-called two centre electron emission (TCEE) [2]. Improvement in the description of the ionized electron moving in the presence of both residual target and projectile fields after the collision (final state) has been a key aspect for the correct description of experimental data [3]. Within distorted wave approximations, it has been shown that, at least for high impact energy and multiply charged projectiles, the CDW theory of Belkić [4], used together with an appropiate description of the initial bound and final continuum electron states, yields best results for doubly differential cross sections (DDCSs) [5]. However, when the projectile impact velocity decreases, the CDW-EIS theory of Crothers and McCann [6] gives better results, its only difference being the choice of the initial state. Moreover the CDW-EIS is formally free of criticisms regarding the initial state proper normalization, and the transition amplitudes have not the divergent behavior that CDW exhibits (although it has been demonstrated that the CDW amplitudes are integrable and its DDCSs are well behaved [7]). The field has now a renewed interest as a result of the development of the COLTRIMS (cold target recoil ion momentum spectroscopy) technique [8]. With COLTRIMS, the projectile tiny scattering angle can be obtained indirectly by measuring the ionized electron and recoil ion momenta [9]. Fully differential cross sections for ion impact ionization can now be measured and constitute a challenging ground for existing theories [10]. Fischer et al. [11] have reported absolute experimental measurements for 2 MeV/amu C6+ single ionization of helium in the scattering plane, i.e., that defined by the initial and final projectile momenta, for various momentum transfers and ejected-electron energies. Theoretical results reported by these authors using a CDW-EIS model exhibited differences between experiment and theory on an absolute scale for emis- sion in the scattering plane [11]. Their calculations were made using the active electron approximation and hydrogenic wavefunctions for the initial and final states of the active electron [11]. Indeed, the simplest description for the He bound initial state is to assume it has one ‘active’ and one ‘passive’ electron and that the ‘active’ electron can be described as moving in the effective Coulomb field of the atomic core with an effective charge chosen (a) to reproduce the ionization energy or (b) so that the continuum wave is orthogonal to the initial state. A more sophisticated way is to apply a Hartree–Fock description for the initial state or to both initial and final states of the active electron [12]. However, Hartree–Fock wave functions do not include proper angular correlation between the two electrons in the helium target. Furthermore, for large perturbations, the incoming projectile may interact with more than one electron in a single collision event. An explicit two-electron description, i.e., a four-body theory might be necessary in that case. We have shown that by using the prior version of the usual CDW-EIS approximation together with an appropiate Roothan–Hartree–Fock description of the initial state and an effective charge coulomb wave function for the target electron continuum, we can get similar results to those obtained by using numerical Hartree–Fock wave functions in both channels [13], at least for DDCSs. The aim of this paper is to present prior CDW and CDWEIS calculations with and without internuclear interaction taken into account for ion helium single ionization FDCSs at different perturbation regimes. Atomic units are used throughout unless otherwise stated. II. THEORIES We regard He single ionization as a single electron process and assume that in the final state the ‘active’ target electron moves in the combined Coulomb field of the target core with an effective charge Ze f f = 1.345 and the projectile field as considered within the CDW-EIS approach. For the initial bound state a Roothan–Hartree–Fock description is used. N-N interaction is treated as a pure Coulomb interaction between Brazilian Journal of Physics, vol. 36, no. 2B, June, 2006 525 90 0.2 Eele = 6.5 eV 6 10 FDCS (10 a.u.) Exp. Data CDW CDW N-N 2 1 60 E=4.0eV q=0.45 a.u. 8 |q| = 0.88 a.u. 0.1 120 10 4 2 30 150 4 0.0 0.2 4 2 2 1 0.1 2 Exp. Data CDW-EIS NN CDW-EIS 0 180 0 2 4 0.0 0 90 180 270 0 90 180 270 0 90 180 270 6 Emission angle (deg.) Experimental data CDW-EIS prior CDW-EIS NN prior CDW prior CDW prior NN 360 210 8 10 330 240 300 270 FIG. 1: FDCS for 100 MeV/amu C6+ on He. Calculations in CDW and prior CDW-EIS,for different momentum transfer. Experimental data is from Ref. [9]. 90 0.4 120 60 0.3 0.2 E=4.0eV q=1.5 a.u. 150 30 0.1 0.0 0 180 90 0.0 14 120 12 60 E=1eV q=0.45u.a. 10 8 0.1 150 210 0.2 30 Experimental data CDW-EIS prior CDW-EIS N-N prior CDW prior (divided by 200) CDW N-N prior (divided by 40) 6 4 0.3 2 0 180 0 240 0.4 300 270 0 2 Exp. data CDW-EIS (prior) CDW-EIS N-N (prior) CDW (prior) 330 CDW N-N (prior) 4 6 210 FDCS (a.u.) 8 330 FIG. 3: The same as in Figure 2 for electron emission energy 4 eV. 10 the projectile with a charge Z p and the true target core charge, ZT = 1. N-N interaction can be taken into account in the transition amplitude ai f (ρ), in an eikonal approximation, through its multiplication by the corresponding phase factor [14], which for pure coulomb internuclear interaction results in [6] 12 240 300 14 270 90 0.25 120 0.20 0.15 150 60 E=1eV q=1.5u.a. 30 0 0.10 0.05 0.00 180 Exp. data 0 CDW-EIS N-N CDW-EIS CDW N-N x 0.025 CDW x 0.25 0.05 0.10 0.15 210 330 ai f (ρ) = i(ρv)2iν a0i f (ρ), (1) ³ ´ where ν0 = ZPvZT , ai f (ρ) a0i f (ρ) is the transition (...truncated)