The p-40Ca and n-40Ca mean fields from the iterative moment approach

Abstract

Abstract The real part V ( r ; E ) of the p- 40 Ca and n- 40 Ca mean fields is extrapolated from positive towards negative energies by means of the iterative moment approach, which incorporates the dispersion relation between the real and imaginary parts of the mean field. The potential V ( r ; E ) is the sum of a Hartree-Fock type component V HF , ( r ; E ) and a dispersive correction δV ( r ; E ); the latter is due to the coupling of the nucleon to excitations of the 40 Ca core. The potentials V ( r ; E ) and V HF ( r ; E ) are assumed to have Woods-Saxon shapes. The calculations are first carried out in the framework of the original version of the iterative moment approach, in which both the depth and the radius of the Hartree-Fock type contribution depend upon energy, while its diffuseness is constant and equal to that of V ( r ; E ). The corresponding extrapolation towards negative energies is somewhat sensitive to the detailed parametrization of the energy dependence of the imaginary part of the mean field, which is the main input of the calculation. Moreover, the radius of the calculated Hartree-Fock type potential then increases with energy, in contrast to previous findings in 208 Pb and 89 Y. A new version of the iterative moment approach is thus developed in which the radial shape of the Hartree-Fock type potential is independent of energy; the justification of this constraint is discussed. The diffuseness of the potential V ( r ; E ) is assumed to be constant and equal to that of V HF ( r ; E ). The potential calculated from this new version is in good agreement with the real part of phenomenological optical-model potentials and also yields good agreement with the single-particle energies in the two valence shells. Two types of energy dependence are considered for the depth U HF ( E ) of the Hartree-Fock type component, namely a linear and an exponential form. The linear approximation is more satisfactory for large negative energies ( E E > 50 MeV). This is explained by relating the energy dependence of U HF ( E ) to the nonlocality of the microscopic Hartree-Fock type component. Near the Fermi energy the effective mass presents a pronounced peak at the potential surface. This is due to the coupling to surface excitations of the core and reflects the energy dependence of the potential radius. The absolute spectroscopic factors of low-lying single-particle excitations in 39 Ca, 41 Ca, 39 K and 41 Sc are found to be close to 0.8. The calculated p- 40 Ca and n- 40 Ca potentials are strikingly similar, although the two calculations have been performed entirely independently. The two potentials can be related to one another by introducing a Coulomb energy shift. Attention is drawn to the fact that the extrapolated energy dependence of the real part of the mean field at large positive energy sensitively depends upon the assumed behaviour of the imaginary part at large negative energy. Yet another version of the iterative moment approach is introduced, in which the radial shape of the HF-type component is independent of energy while both the radius and the diffuseness of the full potential V ( r ; E ) depend upon E . This model indicates that the accuracy of the available empirical data is probably not sufficient to draw reliable conclusions on the energy dependence of the diffuseness of V ( r ; E ).