Effect of deuteron breakup on elastic deuteron - nucleus scattering

Abstract

The properties of the transition matrix elements ${V}_{\mathrm{ab}}(R)$ of the breakup potential ${V}_{N}$ taken between states ${\ensuremath{\varphi}}_{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and ${\ensuremath{\varphi}}_{b}(r)$ are examined. Here ${\ensuremath{\varphi}}_{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are eigenstates of the neutron-proton relative-motion Hamiltonian, and the eigenvalues of the energy ${\ensuremath{\epsilon}}_{a}$ are positive (continuum states) or negative (bound deuteron); ${V}_{N}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is the sum of the phenomenological proton nucleus ${V}_{p\ensuremath{-}A}(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}\ensuremath{-}\frac{1}{2}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|)$ and neutron nucleus ${V}_{n\ensuremath{-}A}(|\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}+\frac{1}{2}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|)$ 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 ${\ensuremath{\epsilon}}_{a}$ larger than about 3 MeV, and both decrease only slowly with ${\ensuremath{\epsilon}}_{a}$, 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 ${\ensuremath{\chi}}_{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ which are the coefficients of the expansion of the neutron-proton-nucleus wave function in a set of the ${\ensuremath{\varphi}}_{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$'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 (${\ensuremath{\epsilon}}_{a}\ensuremath{\le}10$ MeV) and $l=0 \mathrm{and} 2$ 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 $d\ensuremath{-}\mathrm{Ca}$ and $d\ensuremath{-}\mathrm{Ni}$.]