Convergence issues in existing numerical methods for describing the double ionization of helium by high energy electron impact

L.U. ANCARANI, G. GASANEO, AND D.M. MITNIK

Hefei

Conferencia; International Symposium on (e,2e), Double Photo-Ionization and Related Topics; 2013

ICPEAC

It has been stated that the three-body Coulomb problem has been solved numerically, the proof being given through calculations of the single ionization ((e,2e) process) of hydrogen by electron impact [1]. Several numerical recipes like the Convergent-Close-Coupling [2], the Exterior-Complex-Scaling [3], the J-matrrix [4], among others, managed with great success to numerically approach the solution of the problem. A similar success is also obtained when the same methods are applied to the double ionization of helium by photon impact ((γ,2e) process). Except in some minor details, it can be said that the same methods agree remarkably well with each other, and with the experimental observations. However, for the double ionization of helium by impact of high energy electrons, the (e,3e) process, the same is not observed. In some sense, this brings some doubts on whether the three-body problem can be considered as solved in all cases. On one hand, no method is able to satisfactorily reproduce absolute experimental data [5,6]. On the other hand, the available numerical methods do not yield agreement between each other [7]. When dealing with high energy projectiles the four-body problem corresponding to the (e,3e) process can be reduced to a three-body one. Within such a First Born Approximation, the various numerical methods ? which are in such beautiful agreement for (e,2e) and (γ,2e) processes - do not agree with each other when applied to the (e,3e) case. The aim of this presentation is to pinpoint the reasons behind such a failure, whether it is related to numerical convergence issues, the way the cross sections are extracted or to limitations of the methods themselves. For this purpose we study the three-body Schrödinger equation corresponding to high impact energy (e,3e) processes, using the technique based on Generalized Sturmian functions [8]. Through a careful investigation of the scattering solution, convergence issues are investigated in details and compared with those for (e,2e) and (γ,2e) processes. The main idea is to try to understand the origin of the disagreement observed between existing numerical methods, to see if they possess intrinsic limitations in applicability, and/or to find out if there exists any additional hidden - not yet understood - difficulty within the Coulomb three-body.