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Spectrum and electromagnetic properties of $^{24}\mathrm{Mg}$ in the Geometric $α$-cluster Model with $\mathcal{D}_{4h}$ symmetry at leading order

Gianluca Stellin, Karl-Heinz Speidel

TL;DR

This work investigates the low-energy spectrum and electromagnetic properties of $^{24}\mathrm{Mg}$ within a geometric $\alpha$-cluster framework enforcing $D_{4h}$ symmetry. By treating the nucleus as six $\alpha$-particles at the vertices of a square bipyramid and employing a Watson-type Hamiltonian, the authors construct a rotation–vibration description at leading order, yielding closed-form rovibrational energies and symmetry-dictated selection rules. They identify nine singly excited rotational bands, classify them by $\mathcal{D}_{4h}$ irreps, and compute transition form factors and intraband $E2$ strengths, showing generally good agreement with available data and highlighting where further measurements are needed. The results demonstrate the predictive power of a symmetry-based, macroscopic $\alpha$-cluster approach for sd-shell nuclei and outline planned extensions (NLO) and experimental tests (e.g., VEGA at ELI-NP) to further validate the framework.

Abstract

The relevance of the point-symmetry group $\mathcal{D}_{4h}$ for the prediction of spectrum and electromagnetic properties of the $^{24}\mathrm{Mg}$ nucleus is discussed in the framework of the geometric $α$-cluster model at leading order. The latter represents a macroscopic $α$-cluster framework wherein nuclear excitations are described in terms of rotations and vibrations of $^4\mathrm{He}$ clusters about their equilibrium positions, at the vertices of a square bipyramid. The finite group associated with the latter regulates the composition of the rotational bands as well as the transitions between the energy levels, by means of additional selection rules, of molecular nature. A sample of reduced electric multipole transition probabilities of intraband nature is provided.

Spectrum and electromagnetic properties of $^{24}\mathrm{Mg}$ in the Geometric $α$-cluster Model with $\mathcal{D}_{4h}$ symmetry at leading order

TL;DR

This work investigates the low-energy spectrum and electromagnetic properties of within a geometric -cluster framework enforcing symmetry. By treating the nucleus as six -particles at the vertices of a square bipyramid and employing a Watson-type Hamiltonian, the authors construct a rotation–vibration description at leading order, yielding closed-form rovibrational energies and symmetry-dictated selection rules. They identify nine singly excited rotational bands, classify them by irreps, and compute transition form factors and intraband strengths, showing generally good agreement with available data and highlighting where further measurements are needed. The results demonstrate the predictive power of a symmetry-based, macroscopic -cluster approach for sd-shell nuclei and outline planned extensions (NLO) and experimental tests (e.g., VEGA at ELI-NP) to further validate the framework.

Abstract

The relevance of the point-symmetry group for the prediction of spectrum and electromagnetic properties of the nucleus is discussed in the framework of the geometric -cluster model at leading order. The latter represents a macroscopic -cluster framework wherein nuclear excitations are described in terms of rotations and vibrations of clusters about their equilibrium positions, at the vertices of a square bipyramid. The finite group associated with the latter regulates the composition of the rotational bands as well as the transitions between the energy levels, by means of additional selection rules, of molecular nature. A sample of reduced electric multipole transition probabilities of intraband nature is provided.

Paper Structure

This paper contains 14 sections, 25 equations, 3 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Low-energy spectrum and $\gamma$-transitions
  4. Unexcited bands
  5. First excited $A_{1g}$ band
  6. Second excited $A_{1g}$ band
  7. First excited $A_{2u}$ band
  8. First excited $B_{1g}$ band
  9. First excited $B_{2g}$ band
  10. First excited $B_{2u}$ band
  11. First excited $E_g$ band
  12. First excited $E_u$ band
  13. Second excited $E_u$ band
  14. Conclusion

Figures (3)

  • Figure 1: Equilibrium $\alpha$-cluster configuration of ${}^{24}\mathrm{Mg}$ in the intrinsic reference frame (left) with the underlying microscopic structure in terms for protons (red) and neutrons (blue) with realistic charge radii (right). The structure parameters $(\beta_1,\beta_2)$, highlighted in red, are evaluated at $(2.380, 3.857)$ fm, corresponding to a prolate shape, consistently with the measured charge radius of the $0_1^+$ state and the electric quadrupole moment of the $2_1^+$ state. The charge distribution of the $\alpha$-particles is assumed to be pointlike.
  • Figure 2: Normal vibrations of a square bipyramidal configuration with $\mathcal{D}_{4h}$ symmetry. The oriented segments with arrows denote the displacements of the $\alpha$-clusters with respect to their equilibrium positions. The 'prime' superscripts in the modes with frequencies $\omega_1$ and $\omega_2$ are suppressed.
  • Figure 3: Squared charge form factor of the ground state $0^+$ of the ^24Mg nucleus. The measured dataset in Ref. MaM89 is superimposed by its fit in Ref. IsF25 (thin orange curve). The theoretical results in the static limit in Eq. \ref{['eqn:StaticFormFactor_LabFrame']} for pointlike (thin purple curve) and spherical Gaussian (thin magenta curve) $\alpha$-clusters are superimposed, together with the G$\alpha$CM counterpart at LO with the parameter set $(\beta_1,\beta_2) \approx (2.380,3.808)~\mathrm{fm}$ (thick light blue curve).