Magic numbers of cylindrical symmetry

TL;DR

This paper proposes a new perspective on nuclear magic numbers based on symmetry and selection rules, explaining phenomena that challenge traditional definitions and predicting two distinct sets of magic numbers from harmonic oscillator symmetries.

Contribution

It introduces a symmetry-based framework using quadrupole interaction selection rules to define two sets of magic numbers, addressing limitations of the traditional energy gap approach.

Findings

Identifies two sets of magic numbers from harmonic oscillator symmetries.

Explains phenomena like the island of inversion and shape coexistence.

Predicts new magic numbers based on cylindrical symmetry.

Abstract

In nuclear physics a magic number is defined as the nucleon number, which is separated by a significantly large single-particle energy gap from the next nucleon. Magic numbers define the nuclear shells, which are considered to be active, only if they are partially occupied by nucleons. As a consequence the single particle interactions of the valence nucleons lead to the description of the collective properties of the whole nucleus in the shell model theory. But phenomena as the island of inversion, the shape coexistence and the break down of the N=20 magic number reveal that the above definition of a magic number is deficient. A complementary definition should rely on the selection rules of the single particle interactions. Specifically the selection rules of the quadrupole-quadrupole interaction lead to two sets of magic numbers, namely the harmonic oscillator magic numbers 2, 8 20, 40, 70, 112, ... and the spin-orbit SO-like magic numbers 2, 6, 14, 28, 50, 82, 126, ... The underlying symmetries are respectively the spherical symmetry of the 3D isotropic harmonic oscillator and the cylindrical symmetry of the 3D anisotropic harmonic oscillator with two frequencies equal. The above two sets of magic numbers along with the Elliott SU(3) symmetry framework predict long standing and puzzling phenomena in nuclear physics.