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A Study of Shell Model Neutron States in $^{207,209}Pb$ Using the Generalized Woods-Saxon plus Spin-Orbit Potential

The experimental binding energies of single-particle and single-hole neutron states belonging to neutron shells that extend from N = 126 to 184 and 82 to 126 respectively, have been reproduced by solving the Schrödinger equation with a potential that has two components: the generalized Woods-Saxon (GWS) potential and the spin-orbit (SO) coupling term. The GWS potential contains the traditional WS potential plus a term (SU) whose intensity reaches a maximum in the nuclear surface. Our results indicate the existence of a explicit relationship between the strength of the SU potential and the orbital angular momentum quantum number $\ell$ of the state. This dependence has been used to make reasonable predictions for the excitation energy centroids of states located inside and outside the neutron shells investigated. Comparisons are made with results reported in previous investigations.