Uncertainty evaluation and correlation analysis of single-particle energies in phenomenological nuclear mean field: An investigation of propagating uncertainties for independent model parameters

TL;DR

This study evaluates the uncertainties in single-particle energies in nuclear mean field models using Monte Carlo and error analysis, revealing how these uncertainties correlate and evolve with quantum numbers in $^{208}$Pb.

Contribution

It introduces a comprehensive uncertainty propagation and correlation analysis for single-particle energies in phenomenological nuclear models, using Monte Carlo methods and statistical analysis.

Findings

Monte Carlo and error propagation agree on energy uncertainties.

Uncertainties vary systematically with quantum numbers.

Correlations between levels can be positive, negative, or negligible.

Abstract

Based on Monte Carlo approach and conventional error analysis theory, taking the heaviest doubly magic nucleus $^{208}$Pb as an example, we firstly evaluate the propagated uncertainties of universal potential parameters for three typical types of single-particle energies in the phenomenological Woods-Saxon mean field. Accepting the Woods-Saxon modeling with uncorrelated model parameters, we find that the standard deviations of single-particle energies obtained by the Monte Carlo simulation and the error propagation rules are in good agreement with each other. It seems that the energy uncertaintis of the single-particle levels regularly evoluate with some quantum numbers to a large extent for the given parameter uncertainties. Further, the correlation properties of the single-particle levels within the domain of input parameter uncertainties are analyzed using the method of statistical analysis, e.g., with the aid of Pearson correlation coefficients. It is found that the positive, negative or unrelated relationship may appear between two selected single-particle levels, which will be very helpful for evaluating the theoretical uncertainty related to the single-particle levels (e.g., $K$ isomer) in nuclear structural calculations.