Effect of deuteron breakup on elastic deuteron - nucleus scattering

Abstract

The properties of the transition matrix elements $V_{\mathrm{ab}}\mathrm{}\left(R\right)\mathrm{}$ of the breakup potential $V_N$ taken between states ${\phi }_a\left(\overrightarrow{\mathrm{r}}\right)$ and ${\phi }_b\mathrm{}\left(r\right)\mathrm{}$ are examined. Here ${\phi }_a\left(\overrightarrow{\mathrm{r}}\right)$ are eigenstates of the neutron-proton relative-motion Hamiltonian, and the eigenvalues of the energy ${\epsilon }_a$ are positive (continuum states) or negative (bound deuteron); $V_N(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{R}})$ is the sum of the phenomenological proton nucleus $V_{p-A}\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{R}}-\frac{1}{2}\overrightarrow{\mathrm{r}}\mathrm{}\left|\right)\mathrm{}$ and neutron nucleus $V_{n-A}\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{R}}+\frac{1}{2}\overrightarrow{\mathrm{r}}\mathrm{}\left|\right)\mathrm{}$ 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 $l=2\mathrm{}$ are found to be comparable in magnitude to the ones for $l=0\mathrm{}$ for values of ${\epsilon }_a$ larger than about 3 MeV, and both decrease only slowly with ${\epsilon }_a$, 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 ${\chi }_a\left(\overrightarrow{\mathrm{R}}\right)$ which are the coefficients of the expansion of the neutron-proton-nucleus wave function in a set of the ${\phi }_a\left(\overrightarrow{\mathrm{r}}\right)$'s. The equations are rendered manageable by performing a (rather crude) discretization in the neutron-proton relative-momentum variable $k_a$. Numerical results for 21.6-MeV deuterons incident on Ni and Ca which include only the first momentum bin (${\epsilon }_a\le 10$ MeV) and $l=0\mathrm{} \mathrm{and} 2$ show that the effects on the elastic phase shifts are similar in several respects to those found by Johnson and Soper.