Interplay of quadrupole and octupole degrees of freedom in the Gd isotopes

Technical Summary: Interplay of Quadrupole and Octupole Degrees of Freedom in the Gd Isotopes

Problem Statement
The atomic nucleus in the rare-earth region presents a complex landscape where collective and single-particle degrees of freedom interact. Specifically, Gadolinium (Gd, Z=64) isotopes exhibit a diverse range of structural phenomena, including a sub-shell closure at Z=64 and a transition from spherical to rotational shapes as the neutron number (N) increases. A significant challenge for current theoretical models is the simultaneous description of quadrupole (β₂) and octupole (β₃) collectivity across an entire isotopic chain. Recent experimental data has revealed that the maximum octupole collectivity in this region may be located at ¹⁵⁰Gd (N=86), challenging previous assumptions that placed octupole magic numbers near N=88 or N=90. Furthermore, while the X(5) critical point symmetry (associated with the shape-phase transition from vibrational to rotational) has been proposed for nuclei like ¹⁵⁴Gd, other theoretical studies suggest the best realization might be ¹⁵⁰Gd or ¹⁵²Gd. The simultaneous reproduction of quadrupole deformation evolution and octupole collectivity trends remains a difficult task for existing approaches.

Methodology
The authors employ a quadrupole-octupole axially symmetric collective model based on the Bohr Hamiltonian. The theoretical framework utilizes a Hamiltonian (Eq. 1) comprising a collective part ($\hat{H}_{QO}$) and a rotational term ($C\hat{L}^2$). The collective potential $U(\beta_2, \beta_3)$ is treated using a new set of variables: an overall deformation measure $\tilde{\beta}$ and an angular variable $\phi$ representing the relative contribution of quadrupole and octupole deformations.

Key features of the methodology include:

• Potential Construction: The angular potential $u_L(\phi)$ is modified to allow interaction between $\pm\phi$ (or $\pm\beta_3$) configurations by introducing a parabolic inner region, allowing for single or double minima consistent with octupole stability or vibration.
• Parameter Fitting: The model parameters ($|\phi_c|$, $w_0$, and $c$) are optimized by fitting to the experimental energy levels of yrast states with alternating parity and the lowest positive-parity excited ($\beta$) band for Gd isotopes with mass numbers $A = 148$ to $160$.
• Electromagnetic Observables: Transition probabilities (E0, E1, E2, E3) are calculated using specific integrals over the $\tilde{\beta}$ and $\phi$ variables. Scaling parameters ($s$ and $x$) are determined by fitting to available experimental transition strengths to convert theoretical results into absolute values (in Weisskopf units or appropriate units).

Key Contributions and Results
The study provides a systematic investigation of the $84 \le N \le 96$ Gd isotopic chain, yielding the following results:

1. Energy Level Reproduction: The model successfully reproduces the experimental energy levels for both positive and negative parity states across the chain. It captures the transition from an alternating parity sequence in lighter nuclei (¹⁴⁸Gd, ¹⁵⁰Gd) to a structure where negative-parity states shift to higher energies in heavier isotopes, consistent with the evolution from octupole deformation to octupole vibration.
2. Quadrupole Deformation Evolution: The results show a smooth evolution of quadrupole deformation ($\beta_2$), highlighted by a distinct jump at $N=90$. This jump aligns with the X(5) critical point symmetry, marking the transition from spherical to rotational shapes. The calculated deformation increases from $\approx 0.01$ in ¹⁴⁸Gd to $\approx 0.35$ in ¹⁵⁸Gd and ¹⁶⁰Gd.
3. Octupole Deformation and Collectivity:
   • The model predicts non-zero static octupole deformation ($\beta_3$) only for the lightest isotopes, ¹⁴⁸Gd and ¹⁵⁰Gd.
   • For heavier isotopes ($N \ge 88$), the potential exhibits a single minimum at zero octupole deformation, indicating an octupole soft mode or vibration rather than static deformation.
   • E3 Strength: The calculated $B(E3)$ transition strengths show an increasing trend in lighter isotopes, peaking at ¹⁵²Gd ($N=88$), and then decreasing for heavier nuclei. While the absolute values show some quantitative deviation from experiment, the model qualitatively reproduces the experimental trend of a maximum octupole collectivity around $N=88$.
4. Transition Probabilities:
   • E1 Transitions: The model achieves a notably high level of agreement with experimental E1 transition strengths, a result the authors describe as rarely achieved by existing theoretical models.
   • E2 Transitions: The model reproduces the general trends of in-band and inter-band E2 transitions, though it tends to overestimate absolute values. It captures the sharp decrease in $B(E2)$ values at $N=92$ observed in some data.
   • E0 Transitions: The model reproduces the trend of E0 strengths, including a maximum at $N=90$ (consistent with the critical nature of ¹⁵⁴Gd), though it overestimates the absolute magnitudes.

Significance and Claims
The paper claims that this work offers a "unified description" of quadrupole and octupole collectivity across the Gd isotopic chain using a phenomenological model with a small number of adjustable parameters. The authors emphasize that their approach successfully reproduces the complex interplay of shape-phase transitions and octupole correlations simultaneously.

Specifically, the paper highlights:

• The reproduction of the "octupole driving numbers" behavior, predicting a peak in E3 strength at $N=88$ (¹⁵²Gd), which aligns with theoretical predictions of octupole magic numbers and recent experimental indications.
• The ability to describe the transition from reflection-asymmetric (static octupole deformation) to reflection-symmetric (octupole vibration) nuclei within the same formalism.
• The successful reproduction of E1 transition strengths, which are typically difficult to model due to their sensitivity to wave function admixtures.

The authors conclude that while more experimental data is needed to fully test these predictions, particularly for heavier isotopes and high-spin states, the model provides a robust framework for understanding the structural evolution in this region. They suggest that similar systematic studies are needed for neighboring isotopic chains (Nd, Sm, Dy) and that the model warrants a revisit of heavy nuclei previously used only as proof-of-concept examples.