Deuteron-Pickup Reaction in an Optical-Model Approximation

Abstract

A theory of the ($p, d$) pickup reaction is described in which the nuclear interactions of the incoming and outgoing particles are considered. Two different formal expressions that give the transition amplitude are derived, and the wave functions in this amplitude are approximated by an optical-model procedure in which it is assumed that the initial-and final-state particles scatter elastically in the nucleus. Several closed forms for these optical-model wave functions are derived on the basis of a WKB approximation for a complex square-well scattering potential. The use of these wave functions, along with an approximation that gives the form of the transition amplitude in terms of Gaussian functions, allows a closed-form solution for the differential cross section.It is found that the elastic-scattering processes are not negligible, since they affect considerably the magnitude and the shape of the differential cross section. By comparing the theory with recent pickup experiments on ${\mathrm{C}}^{12}$ at 95 and 145 Mev, one obtains a nuclear-momentum distribution that, unlike the Born approximation analysis, is in good agreement with the results of other determinations of momentum distributions. It is found that a neutron number of from 4 to 6 neutrons and a momentum distribution of $\mathrm{exp}(\ensuremath{-}\frac{E}{14})$ are required to fit the data.