Oblate Distorted Harmonic Oscillator Symmetry

Abstract

Harmonic oscillator states have shown advantageous for nuclear structure. Nuclear physics experts have developed sophisticated group theory-based mathematical techniques to handle n-particle states in the harmonic oscillator (ho) potential as a result of this. The deformed harmonic oscillator (HO)\'s oblate shell closure has been studied, and the results demonstrate that the accompanying magic numbers are connected to a universal sequence of the triangular numbers that identically characterize oblate, spherical, and prolate. This offers a distinct framework for comprehending the distorted HO. There is discussion of the effects on oblate nuclei in the present study. The current work examines the symmetries of the deformed HO on the oblate side of deformation and demonstrates that these spherical degeneracies are equally important on this side of deformation as they are on the prolate side. The significance of these symmetries for comprehending nuclear structure is highlighted.

Introduction

I. INTRODUCTION

The nuclear structure has benefited from harmonic oscillator states. This has prompted experts in nuclear physics to create complex group theory-based mathematical methods for handling n-particle states in the harmonic oscillator (ho) potential. The potential can be distorted and oscillation frequencies can vary along the three-axis of Cartesian coordinate directions thanks to the three-dimensional HO. When the system is constrained, the deformation along one of the axis is changed but the deformation in the other remaining Cartesian directions is kept constant and equal, this is known as axially symmetric deformation. The ensuing deformations are either of the oblate or prolate types, depending on whether the deformation along one axis is longer or shorter than in the other directions. The distorted HO's solution as a result of this distortion is shown in Figure 1.

Conclusion

The current study uncovers a fresh framework for comprehending the distorted HO in reference to a single order of triangular numbers that are, 2, 6, 12, 20, 30… which connects the structures throughout all different deformations, from oblate to prolate. The stacking of -particle triangular structures and the subsequent degeneracy sequences 2, 6, 12, 20, 30, can be used to explain the symmetries of spherical nuclei. While nuclei connected to prolate shell closures during deformations in order of N:1 are associated with N spherical clusters, oblate nuclei during deformations in order of 1: N may be described in terms of the assembling of spherical-shaped clusters into a square-shaped N×N matrix. Symmetries are still important for nuclei in more unusual deformations, but these structures will only be seen at high excitation energies since their associated states will fragment due to coupling with the background of other states. The question of whether the symmetries discovered in the current computations endure the stimulus of the spin-orbit interactions and Coulomb consequently have any impact on actual nuclei is another concern. Some solace may be taken in the fact that the shell structures seen in one are also seen in the other, and that the DHO and Nilsson level schemes are relatively equivalent for light nuclei. The AMD method, which employs genuine nucleon-nucleon interactions as demonstrated for 12C and 28Si, yields outcomes that are equivalent to those of more basic nuclear models, demonstrating once more that the ideas discussed here do in fact guide the behavior in these more complex nuclear models. It would be desirable to do a more thorough investigation of how the discovered symmetries affect microscopic nuclear models. However, it should not be interpreted that the ideas presented here interpret nuclei in respect of a crystalline structure of the clusters. Instead, they display a number of symmetries that have a big impact on how nuclei are structured. These symmetries may also be useful for group theory approaches to the distorted HO\'s energy level scheme.

References

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Copyright

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