Refining the isovector component of the Woods-Saxon potential

TL;DR

This paper refines the isovector component of the Woods-Saxon potential, demonstrating improved accuracy with Lane's formula and highlighting the importance of a surface-peaked form factor for better modeling of nuclear properties.

Contribution

It reexpresses Lane's isospin dependence in terms of parameters suited for bound states, improving the phenomenological modeling of nuclear potentials and aiding in the discrimination of shell-model calculations.

Findings

Lane's formula outperforms traditional symmetry terms in data fitting.

A surface-peaked form factor is favored for the isovector component.

Results assist in reducing uncertainties in superallowed beta decay corrections.

Abstract

We investigate the isovector component in the phenomenological mean field model of nuclei. Lane's isospin dependence, initially proposed for the nuclear optical potential, is reexamined within the context of bound states using the Woods-Saxon potential. We demonstrate that the original parametrization can be reexpressed in terms of parameters associated with the compound nucleus, enhancing its suitability for bound states. Comparisons with the conventional symmetry term are performed to assess how well each approach fits experimental data on single-particle/hole energies and reproduces charge-radius systematics. Our results indicate that Lane's formula provides better accuracy compared with the traditional approach to the nuclear potential. Additionally, we find that the isovector component of the nuclear potential favors a surface-peaked form factor, especially one described by the first derivative of the Fermi like function divided by the radial coordinate. This consideration is crucial for open-shell nuclei where Woods-Saxon eigenfunctions serve as a realistic basis for other many-body methods. Our findings also enable discrimination among various shell-model calculations of the isospin-symmetry breaking correction to superallowed $0^+\rightarrow0^+$ nuclear $\beta$ decays [I.~S. Towner and J.~C. Hardy, Phys. Rev. C {\bf 77}, 025501 (2008)]. This disparity currently constitutes the main source of theoretical uncertainty in subsequent tests of the standard model.