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

Abstract

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 [r q] v(E) = ( 4π A ) ∝ V(r; E)r q 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 v HF ( 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 [ r q ] ΔV ( E) from the radial moments [ r q ] 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 V HF are either linear or quadratic functions of energy, whose parameters are determined by fitting the empirical values of [ r q ] 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 [ r q ] V ( E) for E > 0. The dispersive corrections give rise to a characteristic energy dependence of the ratios [r q] V(E) [r q′] V(E) at low energy in particular of the mean square radius. The reliability of the extrapolation of the calculated [ r q ] 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 E F . The meaningfulness of this type of constraint is discussed.