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

S. Ohkubo, J. Takahashi, Y. Yamanaka

Abstract

We show for the first time that the low-spin ($J \le 4^+$) six-$α$ condensate candidate states in $^{24}$Mg, recently reported by Fujikawa et al. [Phys. Lett. B 848, 138384 (2024)], are well described by the superfluid $α$-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-$α$ 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-$α$ 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 $E_{\rm c.m.} = 32.5$ MeV in inelastic $^{12}$C+$^{12}$C scattering. This result provides a unified description of both the low-spin six-$α$ 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.

Six-$α$ 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 () six- condensate candidate states in Mg, recently reported by Fujikawa et al. [Phys. Lett. B 848, 138384 (2024)], are well described by the superfluid -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- system. We find that a roton rotational band with a large moment of inertia is built on the first excited NG state, analogous to the roton bands observed in three-, four-, and five- condensates in C, O, and Ne, respectively. Remarkably, our calculated roton band reproduces the well-known molecular resonance with a C()+C() structure () observed at MeV in inelastic C+C scattering. This result provides a unified description of both the low-spin six- condensate states and the high-spin C()+C() molecular resonance. Analysis of the wave functions reveals a large overlap between the SCM states and a geometrical C()+C() configuration. This dual nature -the coexistence of superfluidity and crystallinity- identifies these states as a signature of a supersolid.
Paper Structure (6 sections, 7 equations, 7 figures, 2 tables)

This paper contains 6 sections, 7 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The field theoretical six-$\alpha$ superfluid cluster model for $^{24}$Mg
  3. Six-$\alpha$ condensation in $^{24}$Mg and roton rotational band
  4. Six-$\alpha$ condensation and $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) molecular resonance in $^{24}$Mg
  5. Supersolidity with $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) dinuclear Bose-Einstein condensate
  6. Summary

Figures (7)

  • Figure 1: The experimental energy levels of the six-$\alpha$ condensate candidate states in $^{24}$Mg Fujikawa2024 are compared with the SCM calculations with a condensation rate of 70%. The vertical axis shows the energy from the $^{12}$C+$^{12}$C threshold in $^{24}$Mg
  • Figure 2: The rotational roton band $K_{\rm rot}$ (closed circles), built on the NG mode $0^+$ of the six-$\alpha$ condensate and calculated in the SCM with a condensation rate of 70%, is compared with the experimental six-$\alpha$ candidate states (squares) in $^{24}$Mg Fujikawa2024. This band is excited from the vacuum $0^+$ state near 19 MeV. For comparison, the band of molecular resonances with a $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) structure (open diamonds), calculated using the coupled-channel method Hirabayashi1995 based on the crystallinity picture, is also presented. The lines serve as a guide for the eye for the rotational bands. The vertical axis represents the energy from the $^{12}$C+$^{12}$C threshold in $^{24}$Mg
  • Figure 3: The SCM calculations (closed circles) with a 70% condensation rate are compared with the experimental data (squares) of the molecular resonance with a high-spin $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) cluster structure from Refs. Wuosmaa1992Szilner1997, as well as the low-spin six-$\alpha$ condensate candidates from Ref. Fujikawa2024 in $^{24}$Mg. For comparison, the theoretical molecular resonances with the $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) cluster structure (diamonds), calculated using the coupled-channel method Hirabayashi1995, are also presented. The lines are a guide for the eye for the rotational roton band $K_{\rm rot}$. The vertical axis represents the energy from the $^{12}$C+$^{12}$C threshold in $^{24}$Mg
  • Figure 4: The BdG wave functions, ${\mathcal{U}}_{n\ell}(r)$ and ${\mathcal{V}}_{n\ell}(r)$, for the roton band states of the six-$\alpha$ condensate in $^{24}$Mg with $n=1$ and $\ell=2$ - 16 under a condensation rate of 70%
  • Figure 5: The squares of the wave functions for the NG zero-mode states, $|\Psi_\nu(q)|^2$ (for $\nu$=0, 1 and 2), of the six $\alpha$ clusters in $^{24}$Mg, calculated with a condensation rate of 70%
  • ...and 2 more figures