On the Model Space Solution of a Three-Body Lippmann-Schwinger Equation

Abstract

In order to settle the question on the model-space dependence of a model-space solution to a single three-body Lippmann-Schwinger (LS) equation, we carry out a carefully conducted calculation on a simplified deuteron + nucleus ,scattering. In this calculation, real potentials of surface-delta forms are used for nucleon-nucleus interactions, and no bin-averaging procedure of the CDCC approach is employed. The resulting elastic and breakup t-matrix elements show rapid convergence regardless of the presence or absence of rearrangement channels. They show no sign of suspected sustained dependence on the model space. Thus, the model-space solution appears to be truely unique. The conclusion drawn is quite general despite the use of surface delta interactions. We further prove that the model-space solution satisfies automatically the boundary conditions at the origins of rearrangement channels imposed by the two homogeneous members of the LS triad. § 1. Introd:uction Projectile breakup reactions are treated in the CDCC approachl),2) as three-body problems under model three-body Hamiltonians. While a rigorous treatment of three-body scattering requires formulations based on the Faddeev theory3) or its equivalents,4) in the CDCC approach an approximate method of solution is developed which amounts to solving a single three-body Lippmann-Schwinger(LS) equation under the model Hamiltonian. In its simplest version (referred to as the model-space approach in this work), the three-body Schrodinger equation is subjected to an angular momentum decomposition together with an expansion in terms of a judicious­ ly truncated basis set (=a model space) of two-body internal wave functions of the projectile pair. This leads to a set of coupled-channel equations for the radial wave functions F of the relative motion between the center-of-mass of the projectile pair and the target nucleus. This is solved by assuming the ordinary outgoing asymptotic boundary conditions on F. The above procedure is entirely equivalent to subjecting the original single three-body LS equation to the same treatment and to solve the resulting set of coupled-channel radial'LS equations for F. As has been pointed out many times, for example in Ref. 1) and elsewhere,5),6) in this model-space approach there can be no rearrangement components in the asymptotic region. Nevertheless, this approach has been amply proven to be practical by actual numerical calculations to successfully account for the experimental data on elastic and breakup cross sections for variety of cases. Using the model-space solution thus obtained, even rearrangement cross sections are computed in first order. 2 )