Resonances in $^{12}$C and $^{24}$Mg: what do we learn from a microscopic cluster theory?

TL;DR

This paper investigates resonance properties in $^{12}$C and $^{24}$Mg using a microscopic cluster model with hyperspherical formalism, focusing on how variational basis affects resonance radius calculations and identifying specific resonances.

Contribution

It introduces a method to analyze resonances in three-body nuclear systems using a microscopic cluster model with hyperspherical coordinates, highlighting the sensitivity of resonance radii to the variational basis.

Findings

Resonance radii are highly sensitive to the variational basis used.

Identification of two $0^+$ resonances in $^{24}$Mg below the three-body threshold.

Insights into the structure of the Hoyle state in $^{12}$C.

Abstract

We discuss resonance properties in three-body systems, with examples on $^{12}{\rm C}$ and $^{24}{\rm Mg}$. We use a microscopic cluster model, where the generator coordinate is defined in the hyperspherical formalism. The $^{12}{\rm C}$ nucleus is described by an $\alpha+\alpha+\alpha$ structure, whereas $^{24}{\rm Mg}$ is considered as an $^{16}{\rm O}+\alpha+\alpha$ system. We essentially pay attention to resonances. We review various techniques which may extend variational methods to resonances. We consider $0^+$ and $2^+$ states in $^{12}{\rm C}$ and $^{24}{\rm Mg}$. We show that the r.m.s. radius of a resonance is strongly sensitive to the variational basis. This has consequences for the Hoyle state ($0^+_2$ state in $^{12}{\rm C}$) whose radius has been calculated or measured in several works. In $^{24}{\rm Mg}$, we identify two $0^+$ resonances slightly below the three-body threshold.