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

Abstract

We discuss resonance properties in three-body systems, with examples on ${}^{12}\mathrm{C}$ and ${}^{24}\mathrm{Mg}$. We use a microscopic cluster model, where the generator coordinate is defined in the hyperspherical formalism. The ${}^{12}\mathrm{C}$ nucleus is described by an $\alpha +\alpha +\alpha$ structure, whereas ${}^{24}\mathrm{Mg}$ is considered as an ${}^{16}\mathrm{O}+\alpha +\alpha$ 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 ${}^{12}\mathrm{C}$ and ${}^{24}\mathrm{Mg}$. 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 ${}^{12}\mathrm{C}$) whose radius has been calculated or measured in several works. In ${}^{24}\mathrm{Mg}$, we identify two $0^{+}$ resonances slightly below the three-body threshold.

Comments: 8 pages, 6 figures, accepted for publication at EPJA