Single-particle energy states for a neutron and a proton are obtained by solving the time-independent Schrödinger equation for the mean-field Woods–Saxon potential along with the spin-orbit term. The wavefunctions are expanded as a linear combination of simple sine-wave basis states, which are eigenfunctions of the infinite spherical-well potential. The requisite algorithm based on matrix diagonalization is implemented in Free Open Source Software (FOSS) Scilab. Initial values for the simulation were taken from model parameters given in the book on Nuclear Structure by Bohr and Mottelson, which were then optimized to obtain the best convergence with the available experimental energy values of various nuclei with magic proton and neutron numbers. The level scheme, as well as the energy values for doubly magic nuclei 82208Pb and 2040Ca, which are obtained using our simulation, is presented in this paper. Finally, energy level diagrams for neutrons and protons with respect to mass number A were arrived at, based on those obtained for various magic nuclei. The evaluation method, which is based on the sine-wave basis, is akin to Fourier analysis. When done with the aid of FOSS Scilab, this technique becomes easily accessible to students at the under-graduate (UG) level and may be studied through small projects.

In this paper, both Least Squares minimization (LSM) and Variational Monte-Carlo (VMC) techniques have been implemented to determine the co-efficients of Semi-Empirical Mass Formula (SEMF). First, the experimental binding energies (BEs) are determined for all the available nuclei from Atomic Mass Evaluation (AME2016) data. Then, LSM technique is implemented in Gnumeric worksheet to obtain the SEMF co-efficients by considering only the first three co-efficients which are deduced from Liquid Drop Model (LDM). The mean squared error (MSE) value, between obtained BEs from the optimized co-efficients and the experimental BEs, has been determined. Then, to emphasize the relevance of empirical terms, they have been introduced successively one after other and the procedure is repeated. A reduction in MSE-value has been observed after each iteration. This same procedure has also been employed using Monte-Carlo approach to obtain SEMF co-efficients by minimizing MSE-value as in variational principle. A comparative analysis has shown that the optimized parameters using VMC have resulted in smaller MSE-value than that of LSM technique.

In this paper, we model the nuclear potential using Woods-Saxon and Yukawa interaction as the mean field in which each nucleon experiences a central force due to rest of the nucleons. The single particle energy states are obtained by solving the time independent Schrodinger wave equation using matrix diagonalization method with infinite spherical well wave-functions as the basis. The best fit model parameters are obtained by using variational Monte-Carlo algorithm wherein the relative mean-squared error, christened as chi-squared value, is minimized. The universal parameters obtained using Woods-Saxon potential are found to be matched with literature reported data resulting a chi-square value of 0.066 for neutron states and 0.069 for proton states whereas the chi-square value comes out to be 1.98 and 1.57 for neutron and proton states respectively by considering Yukawa potential. To further assess the performance of both the interaction potentials, the model parameters have been optimized for three different groups, light nuclei up to 16O - 56Ni, heavy nuclei 100Sn - 208Pb and all nuclei 16O - 208Pb. It is observed that Yukawa model performed reasonably well for light nuclei but did not give satisfactory results for the other two groups while Woods-Saxon potential gives satisfactory results for all magic nuclei across the periodic table.

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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 Schrödinger 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 208 82 P b and 48 20 Ca.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 enquiry strategy (a constructivist approach to learning), to actively engage the students in assigning appropriate J π 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 208 82 P b alone.