Accuracy of Coulomb Stripping for Measuring Single-Particle Reduced Widths in Nuclei

A finite-range $n-p$ potential and the effect of deuteron stretch have been introduced into a distorted-wave Born-approximation (DWBA) analysis of the $d-p$ stripping process for the case in which both deuteron and proton have energies below the Coulomb barrier. This energy region was chosen because it permits an estimate of the experimental reduced width which does not depend significantly on assumptions regarding the target interior. The stretch of the detueron wave function is calculated with the adiabatic approximation under the electric dipole perturbation of the target. The proton and neutron wave functions are Taylor-expanded about the center-of-mass coordinate of the deuteron as a series in the relative coordinate of the deuteron. The stripping matrix element is then expressed to second order in this parameter as a sum of five terms which are introduced into a modified version of the Gibbs-Tobocman DWBA stripping program. A series of calculations was made for the case of a Bi target in order to compare these second-order terms with the first-order term, which is just the usual zero-range approximation. The shape of the proton angular distribution was only modestly affected. The change in peak magnitude varied up to about 16% in the Coulomb-stripping region, with corresponding inverse effect on the experimental reduced widths which would be inferred therefrom. The effect of the nuclear potentials producing proton and deuteron elastic scattering is significant, but it is mostly contained in the region of the integrations and therefore depends mainly on the elastic-scattering phase shifts. We conclude that the particle reduced widths can be "measured" by Coulomb stripping to better than 10% as far as theoretical uncertainties are concerned.