# Semi-spheroidal Quantum Harmonic Oscillator

## TL;DR

This paper introduces a new shell model based on a semi-spheroidal potential well, revealing unique magic numbers for oblate and prolate shapes, aligning with known superdeformed and spherical harmonic oscillator results.

## Contribution

It derives a semi-spheroidal quantum harmonic oscillator model, identifying new magic numbers for oblate and prolate shapes, extending shell model understanding.

## Key findings

- Identifies new magic numbers for oblate semi-spheroids: 2, 6, 14, 26, 44, 68, 100, 140.
- Finds superdeformed prolate magic numbers match spherical harmonic oscillator: 2, 8, 20, 40, 70, 112, 168.
- Shows only negative parity states are allowed for the Z(z) wave function component.

## Abstract

A new single-particle shell model is derived by solving the Schr\"odinger equation for a semi-spheroidal potential well. Only the negative parity states of the $Z(z)$ component of the wave function are allowed, so that new magic numbers are obtained for oblate semi-spheroids, semi-sphere and prolate semi-spheroids. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... The superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator: 2, 8, 20, 40, 70, 112, 168 ...