Phenomenological analysis of dispersion corrections for neutron and proton scattering from ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$

Elastic scattering of protons and neutrons from ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ in the energy range up to 61.4 MeV is analyzed in order to establish a consistent phenomenology with which to examine the recent results of dispersion theory. The present analysis avoids conventional assumptions about the energy dependences of potential depths or geometrical parameters and attempts to include estimates of the uncertainty in our knowledge of the derived potential parameters. Recently reported evidence for energy-dependent geometrical parameters in the n${+}^{208}$Pb potential is supported by the present analysis, but no comparable effect is observed for phenomenological proton potentials in the range of available data ($E_p$>9 MeV). The present analysis shows that, unlike the situation for heavy-ion scattering, the Coulomb potential does not cut off the imaginary potential for p${+}^{208}$Pb. Consequently, no rapid excursions in the parameters of the p${+}^{208}$Pb potential are expected to occur until the incident energy is well below the Coulomb barrier where the nuclear potential is essentially unobservable.