From dynamic to static nuclear chirality: the microscopic conditions for genuine spontaneous breaking of handedness in triaxial nuclei

Nuclear chirality — near-degenerate $\Delta I=1$ partner bands of the same parity expected when the angular momenta of an odd proton, an odd neutron, and a rotating triaxial core form a left- or right-handed aplanar geometry — has been sought in hundreds of candidate nuclei since its 1997 proposal, yet lifetime, electromagnetic-transition, and $g$-factor tests of the archetypes $^{134}$Pr, $^{128}$Cs, $^{102}$Rh, and $^{130}$La consistently favor weak, fluctuation-dominated 'dynamic' chirality: in the interacting boson-fermion-fermion model (IBFFM) wave functions the left- and right-handed sectors are not well separated, and the difference between the two observed phenotypes of chiral candidates is attributed to $\beta$ and $\gamma$ shape fluctuations induced by the exchange boson–fermion interaction. The open problem is whether, and under what systematic microscopic conditions — triaxial deformation stiffness, fermion occupation numbers, exchange interaction strengths, pairing — nuclear chirality in realistic nuclei can evolve from this dynamic character into genuine static spontaneous breaking of handedness, and how that transition can be established and tested experimentally. Despite intense activity through 2025 (multiple chiral doublet bands, self-consistent covariant density functional chiral dynamics, pairing effects on chiral rotation, semiclassical chiral-vibration models), no nucleus has been confirmed as statically chiral, and the quantitative mapping of how occupations and exchange strengths control the chiral phenotypes across isotopic chains remains incomplete.

Background

In an odd-odd nucleus with a triaxially deformed (non-axial, three unequal principal axes) even-even core, the angular momenta of an unpaired high-$j$ proton, an unpaired high-$j$ neutron, and the collective core rotation can arrange themselves so that no plane contains all three vectors. The two mirror-related arrangements are then left- and right-handed, and since the nuclear Hamiltonian has no handedness preference, the laboratory-frame eigenstates must be combinations of the two. Frauendorf and Meng proposed in 1997 that for a $\pi h_{11/2}\otimes\nu h_{11/2}^{-1}$ configuration (an $h_{11/2}$ proton particle coupled to an $h_{11/2}$ neutron hole) in a maximally triaxial core, this aplanar geometry becomes favorable at intermediate spin; the experimental signature would be a pair of near-degenerate $\Delta I=1$ rotational bands of the same parity, called chiral doublet or twin bands. If the left- and right-handed configurations were each long-lived — the static chirality limit — quantum tunneling between them would be small and the two bands would be degenerate in energy with correlated electromagnetic decay rates: similar $B(E2)$ values for transitions from analog states and a characteristic staggered (odd–even) pattern of the magnetic $B(M1)$ values. If instead the system oscillates rapidly between the two handednesses (a chiral vibration) or the aplanar geometry is only occasionally realized among large shape fluctuations, the bands can still be near-degenerate in energy but the decay rates need not follow the static pattern — this is dynamic chirality. Reduced transition probabilities $B(M1)$ and $B(E2)$ are extracted from level lifetimes measured by the recoil-distance Doppler-shift (RDDS) and Doppler-shift attenuation (DSAM) methods, and $g$-factor measurements probe the relative orientation of the proton, neutron, and core angular momenta. The main theoretical frameworks are the tilted-axis cranking (TAC) mean-field approach, the two-quasiparticle-plus-triaxial-rotor and particle-rotor models (PRM), covariant density functional theory (CDFT, a relativistic self-consistent mean-field method), and the interacting boson-fermion-fermion model (IBFFM), in which the even-even triaxial core is described by interacting $d$-bosons and the odd proton and neutron by fermions coupled to it; the exchange boson–fermion interaction is a specific IBFFM term, arising from the microscopic particle–core interaction, that mixes boson and fermion configurations and is controlled by the occupation probabilities of the valence nucleons. Lifetime measurements in $^{134}$Pr (Tonev et al. 2006, 2007) found transition probabilities inconsistent with static chirality, and the IBFFM analysis of Brant, Tonev, de Angelis, and Ventura (2008), summarized in the source conference paper (Tonev et al. 2009), showed for both $^{128}$Cs and $^{134}$Pr that the favorable orthogonal angular-momentum geometry is 'present, but not dominant': in both types of chiral-candidate nuclei — those where energies and electromagnetic decay properties both display the chiral pattern ($^{128}$Cs-like) and those where only the energies of analog levels are close ($^{134}$Pr-like) — the chirality is weak and dynamic, with the difference between the two phenotypes attributed to different $\beta$ (triaxial-softness) and $\gamma$ (asymmetry-angle) fluctuations induced by the exchange boson–fermion interaction. What is not established is whether any realistic nucleus crosses into the static limit, and which microscopic knobs would drive it there.

Problem Statement

For realistic triaxial nuclei exhibiting chiral doublet bands, determine whether the observed dynamic (fluctuation-dominated) chirality can evolve into genuine static spontaneous breaking of handedness — a regime in which the left- and right-handed sectors of the nuclear wave function are well separated and the twin bands satisfy the full static fingerprint (near-degenerate energies, equal analog $B(E2)$ values, staggered $B(M1)$ values, chiral electromagnetic selection rules) — and, if so, identify the microscopic conditions that produce it. This objective has three linked parts: (i) determine whether well-separated left- and right-handed sectors (the static regime) are attainable in realistic chiral-candidate nuclei and, if so, which microscopic conditions produce them — the $\beta$ and $\gamma$ shape fluctuations induced by the exchange boson–fermion interaction, the valence fermion occupation numbers that control that interaction, triaxial-deformation stiffness, and pairing — explicitly admitting the answer that no realistic nucleus reaches the static limit, given that in the calculated IBFFM wave functions the two sectors are not well separated; (ii) establish how the occupation numbers and the strength of the exchange boson–fermion interaction control which of the two observed phenotypes of chiral candidates appears (nuclei with the chiral pattern in both energies and electromagnetic decays versus nuclei with near-degenerate analog energies only), for which systematic evidence across isotopic chains would be valuable; and (iii) specify the observables — lifetimes yielding $B(M1)$/$B(E2)$ and $g$ factors — that discriminate the static from the dynamic regime and hence test the conclusion in the chiral-candidate nuclei. All three parts are required to resolve the task.

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