Single-particle potential and quasiparticle properties of protons in 208Pb

Abstract

The real part V( r; E) of the proton- 208Pb mean field is approximated by a Woods-Saxon potential with diffuseness equal to 0.70 fm. Its depth U v ( E) and radius R v ( E) are extrapolated from positive to negative energy by using a previously developed iteration procedure based on the dispersion relation which connects the real and imaginary parts of the mean field. It is shown that these extrapolated values are quite stable with respect to changes of the main input of the calculation, namely a set of empirical optical-model potentials. Near the Fermi energy the potential radius R v ( E) has a characteristic energy dependence which is qualitatively similar to that recently found in the case of neutrons in 208Pb. However, interesting differences exist. For nucleon energies larger than 10 MeV, the mean field felt by protons has approximately the same root mean square radius as that felt by neutrons. In contradistinction, the root mean square radius of the proton potential at the Fermi energy is sizeably smaller than that of the neutron potential. Furthermore, the potential depth U v ( E) approximately has a linear energy dependence from −20 MeV to +50 MeV in the case of protons, while in the case of neutrons it displays a plateau at small energy. The calculated potential 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 is due to excitations of the 208Pb core. If V HF( r; E) is assumed to have a Woods-Saxon shape, ΔV( r; E) is found to be surface peaked, even at energies as large as 30 MeV at which in the case of neutrons ΔV( r; E) has a Woods-Saxon shape. The proton single-particle energies calculated from the full potential V( r; E) are in good agreement with the empirical values, while those obtained from the Hartree-Fock type component V FH( r; E) alone are too widely spaced. The energy dependence of V( r; E) is characterized by the effective mass m ∗(r; E) , whose dependence upon r and E is calculated. At the nuclear centre, the effective mass is almost independent of energy in contrast to the neutron case; this difference is ascribed to the Coulomb barrier. At the nuclear surface, m ∗ displays a sharp enhancement peak for E close to the Fermi energy, as in the neutron case. The spectroscopic factors of single-particle excitations in 207Tl and 209Bi are calculated from the ratio between m ∗ and its Hartree-Fock type approximation; they are in fair agreement with empirical values deduced from recent electron inelastic scattering data.