Extrapolation from positive to negative energy of the Woods-Saxon parametrization of the n- 208Pb mean field

Abstract

The real part V( r); E) of the nucleon-nucleus mean field is assumed to have a Woods-Saxon shape, and accordingly to be fully specified by three quantities: the potential depth U v ( E), radius R V ( E) and diffuseness a v ( E). At a given nucleon energy E these parameters can be determined from three different radial moments [ r q ] v = (4 π/ A) ∝ V( r; E) r q d r. This is useful because a dispersion relation approach has recently been developed for extrapolating [ r q ] V ( E) from positive to negative energy, using as inputs the radial moments of the real and imaginary parts of empirical optical-model potentials V( r; E) + iW( r; E). In the present work, the values of U v ( E), R v ( E) and a v ( E) are calculated in the case of neutrons in 208Pb in the energy domain −20 < E < 40 MeV from the values of [ r q ] V ( E) for q = 0.8, 2 and 4. It is found that both U V ( E) and R v ( E) have a characteristic energy dependence. The energy dependence of the diffuseness a a ( E) is less reliably predicted by the method. The radius R V ( E) increases when E decreases from 40 to 5 MeV. This behaviour is in agreement with empirical evidence. In the energy domain −10 MeV < E < 0, R V ( E) is predicted to decrease with decreasing energy. The energy dependence of the root mean square radius is similar to that of R V ( E). The potential depth U v slightly increases when E decreases from 40 to 15 MeV and slightly decreases between 10 and 5 MeV; it is consequently approximately constant in the energy domain 5 < E < 20 MeV, in keeping with empirical evidence. The depth U v increases linearly with decreasing E in the domain −10 MeV < E < 0. These features are shown to persist when one modifies the detailed input of the calculation, namely the empirical values of [ r q ] v ( E) for E > 0 and the parametrization [ r q ] w ( E) of the energy dependence of the radial moments of the imaginary part of the empirical optical-model potentials. In the energy domain −10 MeV < E < 0, the calculated V( r; E) yields good agreement with the experimental single-particle energies; the model thus accurately predicts the shell-model potential ( E < 0) from the extrapolation of the optical-model potential ( E > 0). In the dispersion relation approach, the real part V( r; E) is the sum of a Hartree-Fock type contribution V HF( r; E) and of a dispersive contribution ΔV( r; E). The latter is due to the excitation of the 208Pb core. The dispersion relation approach enables the calculation of the radial moment [ r q ] ΔV ( E) from the parametrization [ r q ] w ( E): several schematic models are considered which yield algebraic expressions for [ r q ] Δ V ( E). The radial moments [ r q ] HF( E) are approximated by linear functions of E. When in addition, it is assumed that V HF( r; E) has a Woods-Saxon radial shape, the energy dependence of its potential parameters ( U HF, R HF, a HF) can be calculated. Furthermore, the values of ΔV( r; E) can then be derived. It turns out that ΔV( r; E) is peaked at the nuclear surface near the Fermi energy and acquires a Woods-Saxon type shape when the energy increases, in keeping with previous qualitative estimates. It is responsible for the peculiar energy dependence of R V ( E) in the vicinity of the Fermi energy.