Triaxial Shapes in Nuclei

• Triaxial shapes in nuclei are deformed ellipsoids with three unequal axes, characterized by deformation parameters β and γ.
• Modeling methods like the Bohr-Mottelson model, IBM, DFT, and configuration interaction capture the complex rotational and vibrational behaviors of these systems.
• Experimental techniques, including analysis of rotational spectra and electromagnetic transitions, confirm the role of triaxial deformation in nuclear stability and shape coexistence.

Triaxial shapes in atomic nuclei refer to a geometrical configuration where a nucleus is deformed such that its shape can be described as an ellipsoid with three unequal axes. This contrasts with axial shapes, which possess rotational symmetry about one principal axis. Triaxial deformation introduces complexity into the rotational dynamics of a nucleus and is critical for understanding various nuclear structure phenomena.

Fundamental Concepts of Triaxial Nuclei

Triaxial shapes are characterized by two deformation parameters, $\beta$, which describes the overall deformation magnitude from a spherical shape, and $\gamma$, the triaxiality parameter which quantifies the degree of asymmetry between the three axes of the nucleus. The $\gamma$ parameter varies between $0°$ (prolate, or cigar-like shape) to $60°$ (oblate, or discus-like shape) with intermediate values indicating triaxiality.

The emergence of triaxial shapes is often attributed to specific components of nuclear interactions, such as monopole interactions or high multipole components, which favor binding at nonzero $\gamma$ values (Otsuka et al., 2023). Also, the restoration of symmetries that are broken in mean-field approximations contributes to non-zero triaxiality.

Approaches to Modeling Triaxial Nuclei

Various theoretical frameworks have been developed for modeling triaxial nuclei:

• Bohr-Mottelson Model: Extends the classic collective model to include triaxial shapes using parameters $\beta$ and $\gamma$. Solutions to the Bohr Hamiltonian with potentials such as the Davidson potential can describe a range of collective behaviors from triaxial vibrational to rigid rotors (Yigitoglu et al., 2010).
• Interacting Boson Model (IBM): Dynamical symmetry explorations within the IBM provide insight into triaxial shapes. Introduction of symmetry-breaking terms, like Majorana interactions, can stabilize triaxial minima (Ganev, 2011).
• Density Functional Theory (DFT): The self-consistent application of DFT, often involving the Skyrme energy density functional, aids in mapping the potential energy surfaces that characterize triaxial deformations (Zhang et al., 14 Aug 2025).
• Configuration Interaction Calculations: These computational approaches, such as those employing Monte Carlo Shell Models, include triaxial deformations by using proxy-SU(3) symmetries to account for broken axial symmetry effects (Bonatsos et al., 26 May 2025).

Experimental Signatures and Measurements

Experimentally, triaxiality is investigated through observables such as rotational spectra and electromagnetic transition rates. The $\gamma$ band, traditionally viewed as a vibrational mode, can be recast as a set of $K=2$ rotational states in a triaxial description, providing better consistency with data (Otsuka, 10 Sep 2025).

Advanced experimental techniques, such as high-energy heavy-ion collisions, provide novel ways to glean information about intrinsic nuclear shapes by analyzing final-state particle flows (Jia, 2021). Furthermore, simpler experimental methodologies using only a few E2 matrix elements allow for triaxial determinations without the complexity of exhaustive sum-rule analyses (Lawrie et al., 12 Nov 2024).

Implications and Future Directions

Triaxial shapes have broad implications for understanding nuclear stability, fission pathways, and exotic phenomena such as shape coexistence and rotational bands in neutron-rich nuclei (Zhang et al., 14 Aug 2025). They offer a framework for interpreting complex nuclear behaviors outside the scope of simple axial models, impacting fields from nuclear astrophysics to the paper of superheavy elements.

Ongoing developments in theory, such as those combining the symmetry-respecting generator coordinate methods with projection techniques, are poised to bridge gaps in our understanding of the transition between soft and rigid triaxial configurations (Chen et al., 2017). These innovations hold promise for enhancing predictive power across diverse nuclear regimes, highlighting the ongoing importance of triaxiality in nuclear structure research.