Model Three-Body Problem

Abstract

Solutions are obtained for a three-dimensional model three-body problem involving a spinless $D$ particle and a spinless $n$ particle with coupling $D\ensuremath{\rightleftarrows}n+n$. $D\ensuremath{-}n$ scattering and $D\ensuremath{-}n$ bound states are studied. The model is soluble in the sense that one obtains a linear, one-dimensional Fredholm equation for each partial wave in $n\ensuremath{-}D$ scattering. We have solved the equations numerically on a high-speed computer for different values of the interaction strength and for different values of a size parameter used in the interaction form factor. In particular, we have studied the interaction-strength limit which corresponds to making the $D$ a bound state of the $n'\mathrm{s}$. In this limit there are two three-body bound $s$ states. The $n\ensuremath{-}D$ scattering phase shifts obey a Levinson's theorem and also show the expected kink at the threshold for $n+D\ensuremath{\rightarrow}3n$. The angular distribution for $n\ensuremath{-}D$ scattering has considerable variation and shows the backward peak characteristic of an exchange mechanism. When parameters are chosen in the model to make the $D$ fit the deuteron, the major features of nucleon-deuteron scattering are reproduced except at very low energies when the three-particle bound states dominate and our neglect of spin is important.