Berakdar, J., Lahmam-Bennani, A., Dal Cappello, C.

Over the past decades impressive progress has been made in the theoretical and the experimental study of the multiple excitation and of the complete fragmentation of four-body Coulomb systems. The double ionization of atoms by charged particle impact is employed routinely to prepare and to explore the Coulomb four-body excited states (the two ionized electrons and the scattered charged projectile moving in the field of the residual ion). The spectrum of this four-body system can be determined experimentally by resolving simultaneously the momentum vectors of all particles. Such a multi-coincidence measurement entails however low counting rates which makes the experimental realization a challenging task. This work gives a brief overview on recent achievements in multi- detection techniques and outlines the various methods to carry out the double ionization experiments induced by electron impact. The advantages and the limits of the various experimental approaches are pointed out. On the theoretical side, serious difficulties are encountered which are prototypical for the theoretical treatment of many-body correlated systems: (A) With increasing number of interacting particles (and hence of degrees of freedom) a direct numerical evaluation of the four-body Green's function, which encompasses the entire spectrum of the system, becomes a challenge. (B) Due to the non-integrable character of interacting many particle systems, an analytical approach can only be approximate. In this report we discuss in detail the various methods that have been put forward to deal with the four-body problem, including: perturbative many-body treatments (first and second order theories) and non-perturbative methods as well as pure numerical approaches. Due to the complicated structure of the four-particle continuum spectrum we present and discuss simple qualitative arrangements to explain the main features (peaks and dips) that are observed in the experiments. The limitations of these simple methods are illustrated by contrasting the predictions with full numerical calculations and with experimental data. Future directions and possible applications are also discussed.

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