A novel method for determining the single particle potential directly from the measured single particle density: Application to the charge density difference between the isotones 206Pb–205Tl

Abstract

We present a novel method, based on the single particle Schroedinger equation, to determine the central potential (mean-field) directly from the single particle matter density and its first and second derivatives. As an example, we consider the experimental data for the charge density difference between the isotones ²⁰⁶Pb–²⁰⁵Tl, deduced by phase shift analysis of elastic electron scattering cross-section measurements and corresponds to the shell model 3s_1/2 proton orbit, and determine the corresponding single particle potential. We also present results of least-square fits to parametrized single particle potentials. The 3s_1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors. More accurate experimental data, with uncertainty smaller by a factor of two or more, may answer the question how well can the data be reproduced by a calculated 3s_1/2 wave function.