Onset conditions for static chirality in triaxial nuclei: mapping the boundary between fluctuation-dominated and static-chiral regimes in the A≈100 and A≈130 regions

Odd-odd triaxial nuclei such as $^{102}$Rh and $^{134}$Pr display pairs of nearly degenerate $\Delta I=1$ bands that are candidates for spontaneous chiral symmetry breaking: the angular momenta of a high-$j$ valence proton, a high-$j$ valence neutron of opposite particle-hole character, and the triaxial core can form a left- or a right-handed system, and in a static symmetry-broken mean field with triaxiality $\gamma\approx30^{\circ}$ the two chiral partners would be exactly degenerate with characteristic equal intraband $B(E2)$ values and out-of-phase odd-even staggering of $B(M1)$. Lifetime measurements in $^{102}$Rh and $^{134}$Pr, compared with two-quasiparticle + triaxial rotor (TQPTR) and interacting boson-fermion-fermion model (IBFFM) calculations, do not support static chirality in either nucleus: core shape fluctuations and dynamical coupling appear to reduce or eliminate the static-chiral fingerprints, leaving at most dynamical chirality. The unresolved problem is to determine whether, and under precisely what conditions, static nuclear chirality is realized: to quantify the boundary in deformation-triaxiality-pairing-configuration space separating fluctuation-dominated (chiral-vibration) regimes from a genuine static-chiral regime, to delineate the roles of dynamical core-quasiparticle coupling and the residual proton-neutron interaction, and to validate this map through systematic lifetime, polarization, and $g$-factor measurements across isotopic chains together with microscopic calculations that treat shape fluctuations and tunneling beyond the rigid-rotor approximation. Case-by-case progress exists (the $^{128}$Cs critical spin, the chiral-vibration-to-static-chirality transition in $^{135}$Nd, resolution of the sibling conundrum in $^{106}$Ag), but no validated quantitative boundary or unified microscopic account is known to date.

Background

Chirality in atomic nuclei was predicted by Frauendorf and Meng in 1997 within the tilted-axis cranking (TAC) model, a mean-field approach in which the rotation axis of a triaxial nucleus need not coincide with any principal axis. When the Fermi levels place the unpaired proton and the unpaired neutron in high-$j$ shells with opposite particle-hole character — in the $A\sim130$ region the standard configuration is a $\pi h_{11/2}$ proton particle coupled to a $\nu h_{11/2}$ neutron hole, while in the $A\sim100$ region it is a $\pi g_{9/2}$ proton hole coupled to a $\nu h_{11/2}$ neutron particle — the particle angular momenta favor the short and the long axes respectively, while the triaxial core rotates about its intermediate axis (largest moment of inertia); the three angular-momentum vectors are then mutually perpendicular and can form either a left- or a right-handed system. Because the two handednesses are degenerate in the intrinsic frame, the quantum-mechanical restoration of the broken symmetry in the laboratory frame — treated, e.g., in the particle-rotor model (PRM), which couples the valence particles to a rotating triaxial core — yields a pair of nearly degenerate $\Delta I=1$ bands of the same parity, called chiral doublet bands. The quadrupole deformation is conventionally parametrized by $\epsilon$ (or $\beta$) and a triaxiality angle $\gamma$, with $\gamma=0$ corresponding to prolate and $\gamma=30^{\circ}$ to maximal asymmetry; the ideal static chiral geometry is associated with $\gamma$ around $30^{\circ}$. Candidate chiral doublet bands have been reported in the $A\sim80$, $\sim100$, $\sim130$, and $\sim190$ mass regions — 59 pairs in 47 nuclei by a 2019 compilation, with further candidates reported since.

Ideal static chirality has electromagnetic fingerprints derived by Koike, Starosta, and Hamamoto: for transitions de-exciting analog states of the two partner bands, the intraband reduced electric-quadrupole transition probabilities $B(E2)$ should be nearly equal, the intraband reduced magnetic-dipole probabilities $B(M1)$ should show a pronounced odd-even spin staggering that is out of phase between the two bands, and — by their selection rule for the ideal chiral geometry — interband $E2$ transitions should vanish at high spin. If instead the chiral mode fluctuates between left and right (a 'chiral vibration', i.e., broken symmetry only dynamically), the degeneracy-based fingerprints are washed out to a degree that depends on the fluctuation amplitude and on the tunneling splitting between the two wells — a quantity absent from mean-field TAC solutions altogether.

The problem arises from a sustained discrepancy. For $^{134}$Pr — whose near-degenerate band pair had been suggested as a chiral-doublet candidate already in 1997 (Frauendorf and Meng), with the $N=75$ isotone chiral candidates proposed in 2001 — Doppler-shift lifetime measurements (Tonev et al., 2006) found large differences between the in-band $B(E2)$ values of the two nominally chiral bands, disagreeing with the static-chiral picture; Petrache et al. (2006) warned generally against misinterpreting nearly degenerate band pairs as chiral partners (the 'chiral conundrum'). Meanwhile $^{128}$Cs (Grodner et al., 2006) was described as the best example supporting the chiral interpretation, and $^{135}$Nd (Mukhopadhyay et al., 2007) was shown to evolve from chiral vibration to static chirality with increasing spin. Tonev et al. (2016) measured lifetimes in $^{102}$Rh via the Doppler-shift attenuation method (DSAM) with the Indian National Gamma Array and in $^{134}$Pr via DSAM and the recoil-distance Doppler-shift method (RDDS) with EUROBALL IV, established spin-parity assignments with polarization and angular-correlation measurements, and compared the resulting $B(M1)$ and $B(E2)$ values with two model families: the two-quasiparticle + triaxial rotor model (TQPTR), whose Hamiltonian contains the rotational core energy, quasiparticle energies, and a residual proton-neutron interaction $V_{\pi\nu}$, with a rigid core of fixed deformation $\epsilon$ and triaxiality $\gamma$; and the interacting boson-fermion-fermion model (IBFFM), which extends the interacting boson model of the even-even core and lets the core deformation fluctuate. For $^{102}$Rh the best TQPTR fit gave $\gamma=20^{\circ}$ — away from the $\approx30^{\circ}$ of the static-chiral geometry — and the data lack the required $B(M1)$ staggering; for $^{134}$Pr the IBFFM, in which fluctuations of $\gamma$ around its mean of $30^{\circ}$ admix near-axial (achiral) shapes, reproduces the data far better than the rigid-rotor TQPTR. The authors concluded that the measurements do not support static chirality in either nucleus, that any chirality in $^{102}$Rh is mainly dynamical, and that coupling to shape fluctuations plays a central role — while leaving open under what precise conditions a genuinely static chiral regime could be realized, a question that in the intervening literature has been addressed only case by case.

Problem Statement

Determine whether and under what precise conditions static nuclear chirality — a spontaneously symmetry-broken mean field in which the left- and right-handed triaxial configurations of a high-$j$ proton and a high-$j$ neutron of opposite particle-hole character, together with a rotating core with $\gamma\approx30^{\circ}$, are degenerate and static, as opposed to a dynamical chiral vibration restored by fluctuations and tunneling — is realized in odd-odd triaxial nuclei of the $A\sim100$ and $A\sim130$ regions (with $^{102}$Rh and $^{134}$Pr as the motivating fluctuation-dominated counterexamples). Concretely, the unresolved objective is to quantify the boundary in the physical parameter space — quadrupole deformation ($\epsilon$ or $\beta$), triaxiality $\gamma$, pairing correlations, and valence particle-hole occupations — at each fixed spin (the deliverable being the resulting spin-resolved phase map rather than a single spin-independent boundary), a boundary that separates fluctuation-dominated regimes from a genuine static-chiral regime; to delineate the roles of dynamical core-quasiparticle coupling, of the tunneling splitting between the chiral wells, and of the residual proton-neutron interaction $V_{\pi\nu}$ in stabilizing or destroying the chiral geometry; and to validate this boundary against systematic measurements (lifetimes giving $B(M1)/B(E2)$, linear polarizations, and $g$ factors of chiral-band states) across isotopic chains, using microscopic calculations that treat shape fluctuations and coupling beyond the rigid-rotor (TAC/TQPTR) approximation. These three components serve one common objective — establishing the onset conditions of static chirality — and an answer is complete when the stated boundary is quantitatively specified, the roles of dynamical core-quasiparticle coupling, of the tunneling splitting, and of the residual interaction $V_{\pi\nu}$ are delineated, and the result is shown to be consistent with the body of experimental data on the chiral candidates of these mass regions, including the nuclei for which static chirality fails. Here the boundary is the locus, at each fixed spin, at which fluctuation-dominated character gives way to static-chiral character — operationally, where the static-chiral fingerprints defined above sharpen rather than wash out, that is, where the tunneling splitting between the chiral wells and the fluctuation amplitude become comparable. The spin-resolved contour of that crossover must be reported over all four axes named above — quadrupole deformation ($\epsilon$ or $\beta$), triaxiality $\gamma$, pairing correlations, and valence particle-hole occupations — and validated against the measured observables of the nuclei that bracket the two regimes, the fluctuation-dominated $^{102}$Rh and $^{134}$Pr and the cases in which static character has been claimed, such as $^{135}$Nd and $^{128}$Cs; a boundary fixed along only some of the axes or for a single nucleus is a partial result rather than a complete answer.

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