Shell Structure of 40Ca from the Woods-Saxon Potential by Solving the Time Independent Schr¨odinger Equation Using PINNs

Abstract

Background: The nuclear shell model provides a fundamental framework for understanding nuclear structure and magic numbers by describing nucleons moving in an average mean-field potential. Conventional grid-based solutions of the radial time-independent Schrodinger equation often require careful discretization and parameter tuning. Physics Informed Neural Networks (PINNs) offer an alternative by embedding physical laws directly into the learning process.

Purpose: The aim of this work is to compute single-particle neutron and proton energy levels of the doubly magic nucleus ⁴⁰Ca by solving the radial time-independent Schrodinger equation using PINNs and to compare the results with experimental data.

Methods: The effective potential includes the Woods-Saxon central term, centrifugal barrier, spin-orbit interaction, and Coulomb potential for protons. Separate PINNs are trained for the 1s, 1p, and 1d orbitals, with the energy eigenvalue treated as a trainable parameter. The loss function incorporates the Schrodinger equation residual, wavefunction normalization, and weak experimental guidance, and the network is optimized using the Adam optimizer.

Results: The PINNs framework accurately reproduces bound-state energy levels of neutrons and protons in ⁴⁰Ca. Neutron energies for the 1s, 1p, and 1d shells show close agreement with experimental values, with deviations below 0.3 MeV. The model correctly captures spin-orbit splitting, with j = ℓ+1/2 states more deeply bound than their j = ℓ − 1/2 counterparts. Proton energies exhibit systematic upward shifts due to Coulomb repulsion, with deviations of about 0.5-1.3 MeV for higher orbitals.

Conclusions: These results demonstrate that PINNs provide a robust and mesh-free framework for solving nuclear shell-model problems, offering a reliable alternative to traditional numerical methods for nuclear structure calculations.