Effect of deuteron breakup on elastic deuteron-nucleus scattering

The properties of the transition matrix elements $V_{ab}\left(R\right)$ of the breakup potential $V_N$ taken between states φₐ(r⃗) and ${\phi }_b\left(r\right)$ are examined. Here φₐ(r⃗) are eigenstates of the neutron-proton relative-motion Hamiltonian, and the eigenvalues of the energy εₐ are positive (continuum states) or negative (bound deuteron); $V_N(\overrightarrow{r},\overrightarrow{R})$ is the sum of the phenomenological proton nucleus $V_{p-A}\left(\right|\overrightarrow{R}-\frac{1}{2}\overrightarrow{r}\left|\right)$ and neutron nucleus $V_{n-A}\left(\right|\overrightarrow{R}+\frac{1}{2}\overrightarrow{r}\left|\right)$ optical potentials evaluated for nucleon energies equal to half the incident deuteron energy. The bound-to-continuum transition matrix element for relative neutron-proton angular momenta l=2 are found to be comparable in magnitude to the ones for l=0 for values of εₐ larger than about 3 MeV, and both decrease only slowly with εₐ, suggesting that a large breakup spectrum is involved in deuteron-nucleus collisions. The effect of the various breakup transitions on the elastic phase shifts is estimated by numerically solving a set of coupled equations. These equations couple the functions χₐ(R⃗) which are the coefficients of the expansion of the neutron-proton-nucleus wave function in a set of the φₐ(r⃗)'s. The equations are rendered manageable by performing a (rather crude) discretization in the neutron-proton relative-momentum variable kₐ. Numerical results for 21.6-MeV deuterons incident on Ni and Ca which include only the first momentum bin (εₐ≤10 MeV) and l=0 and 2 show that the effects on the elastic phase shifts are similar in several respects to those found by Johnson and Soper.[NUCLEAR REACTIONS Elastic deuteron-nucleus scattering theory. Effect of deuteron breakup. Numerical applications to E=21.6 MeV d-Ca and d-Ni.]