Asymptotic scattering wave function for three charged particles and astrophysical capture processes
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The asymptotic behavior of the wave functions of three charged particles has been investigated. There are two different types of three-body scattering wave functions. The first type of scattering wave function evolves from the incident three-body wave of three charged particles in the continuum. The second type of scattering wave function evolves from the initial two-body incident wave. In this work the asymptotic three-body incident wave has been derived in the asymptotic regions where two particles are close to each other and far away from the third particle. This wave function satisfies the Schrodinger equation up to terms O(1/3pa), where pa is the distance between the center of mass of two particles and the third particle. The derived asymptotic three-body incident wave transforms smoothly into Redmond??s asymptotic incident wave in the asymptotic region where all three particles are well separated. For the scattering wave function of the second type the asymptotic threebody scattered wave has been derived in all the asymptotic regions. In the asymptotic region where all three particles well separated, the derived asymptotic scattered wave coincides with the Peterkop asymptotic wave. In the asymptotic regions where two particles are close to each other and far away from the third one, this is a new expression which is free of the logarithmically diverging phase factors that appeared in the Peterkop approach. The derived asymptotic scattered wave resolves a long-standing phase-amplitude ambiguity. Based on these results the expressions for the exact prior and post breakup amplitudes have been obtained. The post breakup amplitude for charged particles has not been known and has been derived for the first time directly from the prior form. It turns out that the post form of the breakup amplitude is given by a surface integral in the six dimensional hyperspace, rather than a volume integral, with the transition operator expressed in terms of the interaction potentials. We also show how to derive a generalized distorted-wave-Born approximation amplitude (DWBA) from the exact prior form of the breakup amplitude. It is impossible to derive the DWBA amplitude from the post form. The three-body Coulomb incident wave is used to calculate the reaction rates of 7Be(ep, e)8B and 7Be(pp, p)8B nonradiative triple collisions in stellar environments.