Modification of the Nuclear Landscape in the Inverse Problem Framework using the Generalized Bethe-Weizs\"{a}cker Mass Formula

Abstract

The dependence on the structure functions and Z, N numbers of the nuclear binding energy is investigated within the inverse problem(IP) approach. This approach allows us to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. The IP was formulated for the numerical generalization of the semi-empirical mass formula of BW. It was solved in step by step way based on the AME2012 nuclear database. The established parametrization describes the measured nuclear masses of 2564 isotopes with a maximum deviation less than 2.6 MeV, starting from the number of protons and number of neutrons equal to 1. The set of parameters $\{a_{i}\}$, $i=1,\dots, {\mathcal{N}}_{\rm{param}}$ of our fit represent the solution of an overdetermined system of nonlinear equations, which represent equalities between the binding energy $E_{B,j}^{\rm{Expt}}(A,Z)$ and its model $E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\})$, where $j$ is the index of the given isotope. The solution of the overdetermined system of nonlinear equations has been obtained with the help of the Aleksandrov's auto-regularization method of Gauss-Newton(GN) type for ill-posed problems. The efficiency of the above methods was checked by comparing relevant results with the results obtained independently. The explicit form of unknown functions was discovered in a step-by-step way using the modified least $\chi^{2}$ procedure, that realized in the algorithms which were developed by Aleksandrov to solve nonlinear systems of equations via the GN method, lets us to choose between two functions with same $\chi^{2}$ the better one. In the obtained generalized model the corrections to the binding energy depend on nine proton (2, 8, 14, 20, 28, 50, 82, 108, 124) and ten neutron (2, 8, 14, 20, 28, 50, 82, 124, 152, 202) magic numbers as well on the asymptotic boundaries of their influence.