Nuclear Theory Seminar
Asymptotic Scattering wave function for three charged particles and Astrophysical capture processes

Fakhriddin Pirlepesov

March 04, 2005, 2pm
Cyclotron Conference Room #221

The asymptotic behavior of the three charged particles scattering wave functions has been investigated. There are two different types of the three-body scattering wave functions. The first type of the scattering wave function evolves from the incident three-body wave of three charged particles in the continuum. The second type of the 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 asymptotic Schrödinger equation up to terms of order $1/\rho_{\alpha}^{ 3}$, where $\rho_{\alpha}$ is the distance between the center of mass of two particles and the third particle.

The derived asymptotic three-body incident wave matches 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 three-body scattered wave, which corresponds to the breakup processes, has been derived in all the asymptotic regions.

In the asymptotic region, where all three particles are well separated, the derived asymptotic scattered wave coincides with the Peterkop asymptotic wave.

In the asymptotic region, 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 appearing in the Hamilton-Jacobi approach used previously.

The derived asymptotic scattered wave resolves the long-standing phase-amplitude ambiguity. Based on this results the expression for the exact prior- and post-breakup amplitudes for charged particles 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 the surface integral in the six dimensional hyperspace rather than the volume integral with the transition operator expressed in terms of the interaction potentials. Obtained asymptotic forms and surface integrals open up a possibility for a determination of the breakup amplitude from direct calculations of the Schrödinger equation (ab-initio calculations).

It is shown how to derive a generalized distorted-wave-Born approximation amplitude (DWBA) for the breakup from the exact prior form amplitude. The three-body Coulomb incident wave is used to calculate the reaction rates of nonradiative triple collisions in stellar environments ranging from the Sun to X-ray bursts. We estimate reaction rates of $^{7}\mathrm{Be}(ep,e)^{8}\mathrm{B}$ and $^{7}\mathrm{Be}(pp,p)^{8}\mathrm{B}$ triple collisions leading to the nonradiative formation of $^{8}\mathrm{B}$ in the temperature range from $1.4\times 10^{7}\; \mathrm{K}$ to $10^{9} \; \mathrm{K}$.