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

We discuss resonance properties in three-body systems, with examples on ${}^{12}\mathrm{C}$ and ${}^{24}\mathrm{M}\mathrm{g}$. 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 ${}^{24}\mathrm{M}\mathrm{g}$ 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 ${}^{24}\mathrm{M}\mathrm{g}$. 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 $^{24}{\rm Mg}$,weidentifytwo$0^+$ resonances slightly below the three-body threshold.