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Nuclear experimental data has suggested that like electrons, nuclei are arranged in shells. Developed by Maria Goeppert Mayer and J. Hans D. Jensen, the nuclear shell model is a phenomenological model, which uses shell closure as an explanation for the unique stability that occurs when the nucleon number is equal to: 2, 8, 20, 28, 50, 82, and 126. Nuclei are spin-half fermions that exhibit wave behavior and are thus treated quantum mechanically. In this thesis, we analytically solve the Schrodinger equation in one-dimension, two-dimensions and three dimensions for an infinite square well and harmonic oscillator potential to acquire the proper tools needed to tackle a more complicated problem. We then implement a more accurate mean field potential for nuclei, known as the Woods-Saxon potential, which must be solved numerically. We transform this continuous second-order differential equation into a difference equation, which approximates the second derivative by a quartic fit method and calculates the curvature of points in space. We build a computational model that calculates the energy eigenvalues for a specific nucleus and incorporates the spin-orbit interaction, which is responsible for energy level splitting. We construct an energy level diagram, experiment with different values for the proportionality factor in the spin-orbit interaction, and use the factors that produce the most optimal scheme, thus allowing us to derive the magic numbers. We speculate about the possibility of a new magic number as well as evaluate the strengths and weaknesses of the nuclear shell model.
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Conant, Emily Jane, "Construction of a Nuclear Shell Model" (2015). Senior Projects Spring 2015. 96.