Deuteron-nucleus elastic scattering: Solution of the standard model via the finite element method

The standard, three-body model of deuteron-nucleus elastic scattering and breakup is described by a Hamiltonian consisting of the neutron and proton binding interaction and their kinetic energy operators, while the interaction of each nucleon with the (unexcited) target nucleus is represented by an absorptive, optical-type potential at fixed energy. The standard method for solving this model involves expanding the three-body wave function in states of the neutron-proton system and then truncating some or all of the continuum states in the expansion. Within such an approximation technique, it is not possible to determine the importance of (the neglected) high-lying continuum states. Their contribution can in principle be estimated, however, by employing a solution algorithm which avoids the eigenstate expansion technique. This is done in the present paper by means of the finite element method, applied to the solution in coordinate space. Two different models for the potentials were investigated: that of Farrell, Vincent, and Austern, in which all form factors are of Gaussian type; and a second in which Woods-Saxon form factors were used for the absorptive potentials. The only stable results obtained were for elastic S-matrix elements of the model of Farrell, Vincent, and Austern. These results were in good agreement with the elastic S-matirx elements $S_L$ as calculated using an $L^2$ discretization (‘‘variational’’) procedure and via the continuum, discretized, coupled-channels method, at an incident energy of 22.9 MeV. This agreement confirms that neglect of the high-lying, neutron-proton continuum states is a valid approximation for determining elastic S-matrix elements. The persistent instability of the numerically determined, finite-element-method elastic $S_L$’s for the Woods-Saxon case and of the breakup S-matrix elements is an example of the inappropriate application of the asymptotic boundary conditions, recently discussed by Kuruoglu and Levin.