γ softness and the stability of nuclear chiral geometry: a quantal, beyond-mean-field theory of chirality in soft triaxial nuclei (the ${}^{106}\mathrm{Ag}$ paradigm)

In triaxial nuclei, nuclear chirality arises when the angular momenta of valence protons, valence neutrons, and the core, aligned with the three principal axes, form a left- or right-handed system, producing nearly degenerate ΔI=1 doublet bands. The 2007 spectroscopy of ${}^{106}\mathrm{Ag}$ showed that large γ softness — a flat potential-energy surface in the triaxiality degree of freedom — can destabilize this chiral geometry: two strongly coupled negative-parity bands cross at spin I≈14 with different moments of inertia and competing axial and triaxial mean-field minima. Later work supplied the decisive lifetime measurements (which also revealed a third band) and assigned the crossing bands microscopically at the mean-field and semiclassical time-dependent level. What remains open is the fully quantal, beyond-mean-field microscopic theory of chirality in γ-soft potentials — chiral-vibration/tunneling energy splittings and absolute $B(E2)$/$B(M1)$ values and their staggering including shape mixing — with ${}^{106}\mathrm{Ag}$ as the paradigm case, determining whether and under what conditions γ softness destroys stable chiral geometry.

Background

A triaxial atomic nucleus has three unequal principal axes (short, intermediate, long), conventionally parametrized by quadrupole deformation $\beta$ and asymmetry angle $\gamma$ ($\gamma=0^{\circ}$ prolate, $\gamma=60^{\circ}$ oblate, $\gamma\approx30^{\circ}$ maximally triaxial). Nuclear chirality was predicted by Frauendorf and Meng in 1997: when such a nucleus rotates about an axis lying outside the planes spanned by the principal axes (tilted-axis rotation), the angular momenta carried by the valence protons, the valence neutrons, and the core, which preferentially align along the three different axes, can form either a left- or a right-handed triad. The two mirror images are degenerate in the intrinsic (body-fixed) frame, and restoration of the broken symmetry in the laboratory frame is expected to manifest as a pair of nearly degenerate rotational bands of the same parity connected by ΔI=1 transitions — chiral doublet bands. The standard electromagnetic fingerprints are near-equality of intraband $B(E2)$ strengths (reduced electric-quadrupole transition probabilities) in the two bands and a characteristic odd–even staggering of intraband $B(M1)$ values (reduced magnetic-dipole transition probabilities). By the mid-2000s, candidate chiral doublets had been reported in the $A\approx80$, $100$, $130$, and $190$ mass regions, but nearly all rested on energy degeneracy alone.

The nucleus ${}^{106}\mathrm{Ag}$ became a central test case because its chiral-candidate bands behave anomalously. Joshi et al. (2007) extended two strongly coupled negative-parity bands to spins $19^{-}$ and $20^{-}$ and found that they cross near $I\approx14$; across the crossing the side band has a kinematic moment of inertia roughly 40% larger than the yrast band, and the observed $B(M1)/B(E2)$ staggering does not follow the chiral pattern. A Total Routhian Surface calculation (a constrained rotating mean-field calculation) at $\hbar\omega=0.40$ MeV showed two near-degenerate minima: an axial prolate one at $(\beta,\gamma)=(0.20,0^{\circ})$ and a triaxial one at $(0.12,-28^{\circ})$. The authors proposed that the two bands, although built on similar quasiparticle configurations, correspond to different shapes — a triaxial, aplanar rotor and a planar, near-axial rotor — interconverted by the large γ softness of this nucleus, possibly via chiral-vibration dynamics, and concluded that γ softness has marked implications for nuclear chirality. They explicitly deferred the detailed theoretical analysis of this complex picture — the γ-softness-driven coexistence of axial and triaxial shapes, the possible chiral-vibration tunneling between them, and the same-configuration, different-$K$ explanation of the crossing bands — as beyond the scope of the Letter; their $A\approx100$ isotopic systematics were presented as existing supporting evidence rather than as a request for future work.

Subsequent developments define what is now at stake. Two independent 2014 Doppler-shift-attenuation lifetime experiments measured absolute transition strengths in ${}^{106}\mathrm{Ag}$ and discovered a third, close-lying negative-parity band; the data were mutually consistent but admitted conflicting phenomenological interpretations. In 2019, three-dimensional tilted-axis-cranking covariant density functional theory (a fully self-consistent microscopic mean-field method with rotation about an arbitrary axis) showed that bands 1 and 2 are built on two distinct configurations — two-quasiparticle $\pi g_{9/2}\otimes\nu h_{11/2}$ and four-quasiparticle $\pi g_{9/2}\otimes\nu h_{11/2}(gd)^{2}$ (where $\pi$ and $\nu$ denote proton and neutron quasiparticle orbitals, so these labels specify, e.g., a proton in the $g_{9/2}$ shell coupled to neutrons in the $h_{11/2}$ and $gd$ shells) — so their crossing is a configuration crossing, not a chiral partnership, while the soft Routhian curves make band 3 a chiral-vibrational partner of band 2. In 2024, time-dependent covariant density functional theory on a three-dimensional space lattice reproduced the observed band energies of ${}^{106}\mathrm{Ag}$ without free parameters and revealed a novel γ-softness-driven harmonic chiral mode, distinct from the chiral precession of rigid triaxial nuclei. These works resolved the band assignment for ${}^{106}\mathrm{Ag}$ itself, but both state — the 2019 paper explicitly, the 2024 paper by declaring chirality in soft triaxial nuclei a newly opened research area — that the quantal, beyond-mean-field level of the problem is not yet done: mean-field and semiclassical time-dependent approaches cannot compute the tunneling-induced energy splitting between chiral partners, and no microscopic quantal calculation of the full doublet-band spectroscopy in a γ-soft potential exists.

Problem Statement

Determine, in a microscopic and quantal (beyond-mean-field) framework, how γ softness affects the stability of nuclear chiral geometry, with ${}^{106}\mathrm{Ag}$ as the paradigm case — the detailed theoretical analysis that the 2007 Letter explicitly declared beyond its scope. Concretely, the target is the γ-soft nucleus ${}^{106}\mathrm{Ag}$ (the paradigm case), and a complete answer requires both of the following: (i) construct a microscopic treatment of a rotating triaxial nucleus that includes fluctuations in the γ degree of freedom and tunneling between left- and right-handed (and between triaxial and axial) intrinsic configurations, and use it to compute, for the γ-soft nucleus ${}^{106}\mathrm{Ag}$, the energies and energy splitting of the chiral doublet bands (bands 2 and 3, on the four-quasiparticle configuration $\pi g_{9/2}\otimes\nu h_{11/2}(gd)^{2}$) and the absolute intraband $B(E2)$ and $B(M1)$ values and their spin staggering, including the effects of shape mixing; and (ii) extract from the same framework the effective barrier heights and tunneling rates between the axial and triaxial minima that the 2007 Total Routhian Surface calculation found near-degenerate, and decide whether the observed band phenomenology is chiral vibration or static chiral symmetry breaking — thereby answering the source paper's question of whether and under what conditions γ softness destroys stable chiral geometry, and which observables discriminate the regimes. A complete answer must be source-faithful to this objective: a mean-field or parameter-fitted phenomenological treatment, without the quantal tunneling/shape-fluctuation physics, does not discharge it.

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