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

Abstract

Single particle energy states as described by the nuclear shell model are obtained for doubly magic nuclei using a Gnumeric worksheet environment. The Numerov method rephrased in matrix form is utilized to solve the Time-Independent Schrödinger Equation (TISE) within mean-field approximation, described by Woods-Saxon (WS) potential along with the 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 ₂₀Ca. 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 J ^π configurations for ground states of nuclei neighboring 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.