Dispersion relation approach to the extrapolation towards negative energy of the optical potential in 40Ca and 208Pb

Accurate empirical information on the nuclear mean field V(r; E) is available at positive energy but is lacking at negative energy. A dispersion relation approach is used to extrapolate the radial moments [rq]V(E)=(4π/A)∫V(r; E)rq dr(q=0.8, 2, 4) of V(r; E) from positive to negative energies. The quantity V(r; E) is the sum of a Hartree-Fock type contribution VHF(r; E) and of a dispersive contribution ΔV(r; E) which arises from the coupling between the elastic and nonelastic channels. A dispersion relation is used to evaluate [rq]ΔV(E) from the radial moments [rq]W(E) of the imaginary part of empirical optical-model potentials, in the case of protons and neutrons on 208Pb and of protons on 40Ca. It is assumed that the radial moments of VHF are either linear or quadratic functions of energy, whose parameters are determined by fitting the empirical values of [rq]V(E) associated with optical-model potentials which yield very good fits to the experimental cross sections; close agreement is found between the calculated and empirical values of [rq]V(E) for E>0. The dispersive corrections give rise to a characteristic energy dependence of the ratios [rq]V(E)/[rq]V(E) at low energy in particular of the mean square radius. The reliability of the extrapolation of the calculated [rq]V(E) towards negative energy is discussed by studying its sensitivity to the inputs of the model. This sensitivity can be reduced when one introduces a constraint at the Fermi energy EF. The meaningfulness of this type of constraint is discussed. (orig.)