Six-$\alpha$ cluster Bose-Einstein condensation and supersolid ${}^{12}$C($0_2^{+})$+${}^{12}$C($0_2^{+})$ molecular structure in ${}^{24}$Mg

Abstract

We show for the first time that the low-spin ($J\le 4^{+}$) six-$\alpha$ condensate candidate states in ${}^{24}$Mg, recently reported by Fujikawa et al. [Phys. Lett. B 848, 138384 (2024)], are well described by the superfluid $\alpha$-cluster model (SCM). This is achieved by a rigorous treatment of the Nambu-Goldstone (NG) zero mode as the order parameter of condensation in the finite six-$\alpha$ system. We find that a roton rotational band with a large moment of inertia is built on the first excited NG $0^{+}$ state, analogous to the roton bands observed in three-, four-, and five-$\alpha$ condensates in ${}^{12}$C, ${}^{16}$O, and ${}^{20}$Ne, respectively. Remarkably, our calculated roton band reproduces the well-known molecular resonance with a ${}^{12}$C($0_2^{+}$)+${}^{12}$C($0_2^{+}$) structure ($16^{+}$) observed at $E_{c.m.}=32.5$ MeV in inelastic ${}^{12}$C+${}^{12}$C scattering. This result provides a unified description of both the low-spin six-$\alpha$ condensate states and the high-spin ${}^{12}$C($0_2^{+}$)+${}^{12}$C($0_2^{+}$) molecular resonance. Analysis of the wave functions reveals a large overlap between the SCM states and a geometrical ${}^{12}$C($0_2^{+}$)+${}^{12}$C($0_2^{+}$) configuration. This dual nature -the coexistence of superfluidity and crystallinity- identifies these states as a signature of a supersolid.

16 pages, 7 figures

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