Numerical Simulation of Shell Model Single Particle Energy States using Matrix Numerov Method in Gnumeric Worksheet

TL;DR

This paper demonstrates a matrix Numerov method implementation in Gnumeric to simulate single particle energy states in nuclear shell models, optimizing parameters to match experimental data and engaging students in nuclear physics learning.

Contribution

It introduces a matrix Numerov algorithm within Gnumeric for solving the Schrödinger equation in nuclear shell models, enhancing educational engagement and parameter optimization.

Findings

Optimized step size and matrix size for accurate energy levels.

Matrix method reproduces experimental single particle energies.

Engages students in nuclear physics through guided inquiry.

Abstract

Single particle energy states as described by nuclear shell model are obtained for doubly magic nuclei using Gnumeric worksheet environment. Numerov method rephrased in matrix form is utilised to solve time-independent Schrodinger equation (TISE) within mean-field approximation, described by Woods-Saxon (WS) potential along with spin-orbit term, to obtain the single particle energies for both neutron and proton states. The WS model parameters are chosen from previous simulation results performed using matrix methods technique involving sine basis, where optimization was done w.r.t available experimental single particle energies for 208Pb and 48Ca. In this paper, only the algorithm parameters, step size h and matrix size N are optimized to obtain the expected energy level sequence obtained using matrix methods. An attempt is made, by incorporating this tool within the framework of guided inquiry strategy (a constructivist approach to learning), to actively engage the students in assigning appropriate spin-parity configurations for ground states of nuclei neighbouring the doubly magic ones. It has been observed that the ground state configurations could be better predicted when energy level sequences are known for all nuclei as compared to what is usually obtained from that of 208Pb alone.