Doubly differential cross sections for the proton- impact double ionization of helium

A time-dependent close-coupling approach is used to investigate 6 MeV proton-impact double ionization of helium. We extract doubly differential (in the emission angles of both outgoing electrons) cross sections for comparison with recent experimental measurements. Good qualitative agreement is found between our calculations and the measurements. We also present differential cross sections for cases where the energy available to the outgoing electrons is unrestricted, which allows further information about the double escape process to be inferred. (Some figures in this article are in colour only in the electronic version) The few-body Coulomb problem has interested atomic physicists for the past several decades. Substantial theoretical progress has been made in understanding the importance of electronic correlations in atomic collision processes such as double photoionization of helium and electron impact ionization of hydrogen [1–5]. More complex processes, such as ion-impact single and double ionization of helium, remain significant theoretical challenges. On the experimental front, Schulz et al measured unique higher-order structures in fully differential cross sections for 100 MeV/u C6+ single ionization of helium [6]. The kinematically complete experiments [6] were able to obtain three-dimensional images of the collisional breakup process. Since these pioneering experiments, other ion–atom collision experiments have been produced that examine double ionization of helium by 6 MeV proton-impact [7–9]. These experiments measured doubly differential cross sections as a function of both electrons’ solid angles. However, the double ionization experiments had to be integrated over a range of momentum transfers to provide adequate statistics. Few attempts have been made to theoretically understand the complex electron dynamics described by these experiments, due to the difficulty of describing the electron– electron correlations under the influence of a two-centre potential formed by the impact ion and target nucleus. A correlation function has been introduced [9–11], which suppresses the effects of the two-centre potential, in an effort to better understand the role of electron correlation in the double ionization processes. Only one attempt, to our knowledge, has been made to examine theoretically the differential cross sections which result from 6 MeV proton double ionization of helium [12]. This approach used a simple perturbative twoCoulomb-wave (2C) model to describe the double ionization of helium by the fast proton impact. Some investigations have been conducted to better understand the correlations of four-body collisions from the viewpoint of the frozencorrelation approximation [13, 14]. Dynamic correlation effects were neglected, but initialand final-state correlations were taken into account on the basis of different models. The introduction of the model dynamic screening improved the situation, but was unable to achieve quantitative agreement with the experimental measurements. Continuum-DistortedWave (CDW) theory has been applied to multi-ionization of helium and lithium by ion impact. While CDW theory has been successful in reproducing single ionization collisions, double ionization presents a unique challenge because of the complex nature of the electron–electron interactions [14, 15]. Electron emission spectra have been studied for four-body effects using Continuum-Distorted-Wave-Eikonal-Initial-State (CDW-EIS) theories in single ionization collisions yielding mixed results compared to experimental measurements [16, 17]. CDW-EIS 0953-4075/08/111002+06$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 111002 Fast Track Communication models have also been used to better understand the four-body effects in single ionization collisions where fully differential cross sections were measured [18]. A sophisticated treatment of the correlated continuum electrons is required in order to understand the complete double ionization dynamics involved in the 6 MeV proton experiments. Non-perturbative time-dependent methods [19– 21] have recently been applied to heavy ion collisions, and have been shown to produce accurate double ionization total cross sections for α-particle [20] and protonand antiprotonimpact [21] double ionization of helium. Up until now, these numerical approaches have not been applied to differential cross sections. In this communication, we present timedependent close-coupling (TDCC) calculations of the angular doubly differential cross sections for 6 MeV proton-impact double ionization of helium. The ion–atom TDCC method calculates the fully correlated electronic dynamics while approximating the ion’s motion as a straight-line trajectory for all given impact parameters. The fully correlated wavefunction, LM , for the double ionization of a two-electron target atom by ion collision is obtained by the evolution of the time-dependent Schrödinger equation in real time: