Unambiguous identification and characterization of chiral doublet bands in triaxial nuclei: distinguishing static chirality, chiral vibration, and non-chiral alternatives

Rotating triaxial nuclei with high-j valence particles and holes can realize a left- or right-handed arrangement of three angular-momentum vectors, a spontaneous breaking of chiral symmetry in the body-fixed frame whose laboratory-frame signature is a near-degenerate pair of same-parity $\Delta I=1$ rotational bands (chiral doublet bands). Since the 1997 prediction, dozens of candidate doublet bands have been reported in the $A\sim 80$, $100$, $130$, and $190$ mass regions, but their interpretation has remained contested: band near-degeneracy alone is not decisive, proposed electromagnetic fingerprints such as odd-even staggering of $B(M1)/B(E2)$ values are model-dependent, and distinguishing static chirality from chiral vibration and from non-chiral explanations (shape coexistence, wobbling, octupole or pseudospin effects) lacks consolidated criteria. The problem, posed by Meng, Zhang, Qi, and Wang (2010) and restated in the companion 'Open Problems in Understanding the Nuclear Chirality' review, asks for: (i) definitive experimental and theoretical criteria that discriminate static chirality, chiral vibration, and non-chiral alternatives across configurations and mass regions; (ii) a systematic understanding of how electromagnetic fingerprints depend on the triaxiality parameter $\gamma$ and configuration mixing; (iii) self-consistent cranked mean-field theories that produce fully aplanar solutions with time-odd fields and yield laboratory-frame tunneling splittings; and (iv) lifetime and configuration-assigned data, including odd-$A$ and multi-quasiparticle candidates, to test the models.

Background

A triaxial nucleus is one whose mass distribution is deformed with three unequal principal axes, conventionally labeled short ($s$), intermediate ($i$), and long ($l$); the quadrupole shape is parametrized by an overall deformation $\beta$ and a triaxiality angle $\gamma$ (with $\gamma=30^\circ$ the maximally triaxial shape). In such a nucleus containing a few high-$j$ valence nucleons - particles and/or holes occupying large-$j$ shell-model orbitals such as $h_{11/2}$ or $g_{9/2}$ - the collective rotation favors alignment of its angular momentum along the intermediate axis (the axis with the largest moment of inertia), while valence particles tend to align along the short axis and valence holes along the long axis. These three mutually perpendicular angular momenta can then form a handed, or chiral, arrangement: the two mirror-related (left- and right-handed) configurations are interchanged by the chiral operator, a $180^\circ$ rotation combined with time reversal, and cannot be connected by any spatial rotation. This spontaneous breaking of chiral symmetry in the body-fixed frame was proposed for nuclei by Frauendorf and Meng in 1997. In the laboratory frame the broken symmetry is restored by quantum tunneling between the two handed configurations, so the predicted observable signature is a pair of nearly degenerate $\Delta I=1$ rotational bands with the same parity and similar intra-band electromagnetic transition rates - the chiral doublet bands.

The first candidates were reported in 2001 in the odd-odd $N=75$ isotones (e.g. $^{128}$Cs) and were interpreted as chiral vibrations - a soft oscillation between left- and right-handed geometries rather than a statically aplanar rotation. Static chirality instead denotes the regime in which the angular momentum is stably aplanar and the doublet splitting is produced by tunneling between the two degenerate minima. Two standard theory frameworks are used. The tilted-axis cranking (TAC) model is a mean-field treatment in which the rotational axis is allowed to tilt away from the principal axes; it can find the two degenerate aplanar (chiral) solutions but, being semiclassical, cannot describe the tunneling that splits them. The particle-rotor model (PRM) couples valence nucleons to a rotating triaxial core and is diagonalized with total angular momentum $I$ as a good quantum number, so it yields laboratory-frame spectra, energy splittings, and transition probabilities directly; its predictions, however, depend on the assumed core deformation and configuration. Electromagnetic observables are quantified by the reduced transition probabilities $B(M1)$ and $B(E2)$ for magnetic-dipole and electric-quadrupole radiation; the odd-even staggering of $B(M1)$ values (their systematic alternation with spin) and of the $B(M1)/B(E2)$ ratio was proposed in 2004 as a chirality fingerprint. Relativistic (covariant) density functional theory (CDFT) provides a self-consistent microscopic mean field, in which rotating solutions are obtained by three-dimensional tilted-axis cranking; the functional's time-odd components - terms that are odd under time reversal and vanish without rotation - enter the chiral solutions but are weakly constrained by data.

The identification problem crystallized in 2006, when lifetime measurements in $^{134}$Pr showed very different intra-band $B(E2)$ strengths for the two 'partner' bands, in disagreement with the chiral picture (the 'chiral conundrum'), and a companion analysis warned against interpreting any nearly degenerate pair as chiral partners; the same year, electromagnetic data in $^{128}$Cs gave strong support to a chiral interpretation, and in 2007 $^{135}$Nd was shown to evolve from chiral vibration to static chirality with increasing spin. Thus the interpretation of doublet bands can be contradictory from nucleus to nucleus, and the criteria for deciding the chiral character - static, vibrational, or absent - were not consolidated. The source study of this candidate (Meng, Zhang, Qi, Wang 2010) developed a triaxial multi-quasiparticle PRM for the odd-$A$ nucleus $^{135}$Nd with the configuration $\pi h_{11/2}^{2}\otimes\nu h_{11/2}^{-1}$, showed that $B(M1)$ staggering is weak in the chiral-vibration region and strong in the static-chirality region, and concluded with the open tasks stated above. A related development is the prediction (2006) and subsequent observation of multiple chiral doublets (M$\chi$D) - more than one pair of chiral doublet bands in a single nucleus, e.g. in $^{133}$Ce, $^{103}$Rh, $^{78}$Br, and $^{135}$Nd - which multiplies the cases any identification scheme must handle.

Problem Statement

The problem, as posed by the source study and its companion review, is to establish the unambiguous identification and characterization of chiral doublet bands in triaxial nuclei - that is, to answer, with validated and mutually consistent criteria, whether a given observed pair of near-degenerate same-parity $\Delta I=1$ bands arises from static chirality (stable aplanar rotation whose laboratory-frame doublet structure is produced by tunneling between left- and right-handed configurations), from chiral vibration (soft oscillation between the two handed geometries below the onset of static chirality), or from a non-chiral alternative, and to do so consistently across different quasiparticle configurations and mass regions. Its four linked components are: (i) consolidate the experimental and theoretical criteria that definitively distinguish static chirality, chiral vibrations, and non-chiral explanations such as shape coexistence, wobbling, octupole correlations, or band mixing, and establish their validity across configurations and mass regions; (ii) determine systematically how the electromagnetic fingerprints - the odd-even staggering of $B(M1)$ strengths and of $B(M1)/B(E2)$ ratios - depend on the triaxiality parameter $\gamma$ and on configuration mixing, and validate these dependences against measured lifetimes and other observables; (iii) develop self-consistent cranked mean-field approaches that locate fully aplanar (static chiral) solutions including time-odd fields and yield the laboratory-frame tunneling splittings between the doublet partners, so that chiral vibration, tunneling, and static chirality are described within one microscopic framework; and (iv) obtain the further lifetime measurements and configuration-assigned data needed to test and refine the models - lifetime measurements being essential to extract the absolute $B(M1)$ and $B(E2)$ transition probabilities, with the odd-$A$ and multi-quasiparticle candidates that the source paper emphasizes among the cases requiring such data. Components (i)-(iii) constitute the required research result; component (iv) is the experimental input that would test and refine it and is not itself a required deliverable. The components share one scientific objective - deciding which nuclei actually exhibit chiral symmetry breaking and of which kind - and the problem is complete when the criteria, their microscopic basis, and their demonstrated application to the candidate nuclei are jointly sufficient for that decision.

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