Reliable identification of chiral doublet bands in real nuclei: disentangling the chirality fingerprints and establishing model-independent criteria

The standard experimental fingerprints of nuclear chirality - near-degenerate same-parity Delta(I)=1 doublet bands, odd-even spin staggering of intra-band B(M1) strengths, and the absence of energy staggering - were derived for the idealized limit of strongly broken chiral symmetry in triaxial nuclei. Particle-rotor calculations with realistic multi-orbital configurations showed that all of them can fail or mislead for real chiral systems, creating a risk of both false positives and false negatives. This problem asks for the reliable identification of chirality in real nuclei through three linked results: a microscopic disentangling of the mechanisms (chiral geometry, angle staggering, Coriolis effects, residual proton-neutron interaction, configuration mixing, and mixing with planar (coplanar) configurations) that generate the observed staggering patterns, including the origin of the phase change of the B(M1) staggering between ideal and realistic descriptions; robust, model-independent identification criteria with demonstrated discriminative power against non-chiral look-alike explanations of candidate doublet bands; and the implementation and validation of those criteria for candidate chiral doublet bands, for which Shirinda and Lawrie (2012) suggest an angular-momentum-geometry analysis as one possible route.

Background

In a nucleus with a triaxially deformed shape - a quadrupole deformation in which the short, intermediate, and long principal axes all differ, commonly parameterized by the deformation angle $\gamma$ (maximal triaxiality at $\gamma=30^\circ$) - three angular momenta can orient along three different axes: the high-$j$ valence proton particle (a proton occupying a valence orbital of large angular momentum $j$) tends to align its angular momentum with the short axis, a valence neutron hole with the long axis, and the collective rotation of the even-even core with the intermediate axis, which has the largest moment of inertia. When these three vectors are not coplanar they define a handed (left- or right-handed) screw; this configuration is called nuclear chirality. The two handed configurations are related by time reversal, so an intrinsic-frame mean-field solution that breaks this chiral symmetry comes in two mirror images, and the exact laboratory-frame states appear as pairs of near-degenerate $\Delta I=1$ rotational bands of the same parity, called chiral doublet bands, as predicted by tilted-axis cranking (a mean-field method that rotates the nucleus about an axis tilted away from every principal axis) in Frauendorf and Meng (1997). Here $I$ is the nuclear spin and $\Delta I=1$ means consecutive levels of a band differ by one unit of angular momentum.

After the first experimental candidates were reported in the odd-odd $N=75$ isotones (nuclei with neutron number $N=75$ in which both valence proton and valence neutron numbers are odd; Starosta et al. 2001), candidate chiral doublet bands have been proposed in dozens of nuclei across the $A\approx80$, 100, 130, and 190 mass regions (compiled in the data tables of Xiong and Wang 2019). To judge these candidates the community adopted chirality fingerprints derived for the ideal limit of strongly broken chirality, in which the three angular momenta are mutually orthogonal and each valence nucleon occupies a single fixed magnetic substate (a single-particle state of definite angular-momentum projection on the symmetry axis): (i) near-degeneracy of the partner bands; (ii) an odd-even spin staggering of the intra-band reduced magnetic-dipole transition probabilities $B(M1)$ (the strength of an $M1$ gamma-ray transition, determined experimentally from level lifetimes) with the same phase in both bands, together with chiral electromagnetic selection rules (rules that, in the ideal chiral geometry, allow one subset of intra-band transitions in one band and the complementary subset in the partner band; Koike, Starosta, and Hamamoto 2004); and (iii) a smooth spin dependence of the energy staggering parameter $S(I)=[E(I)-E(I-1)]/(2I)$, that is, no energy staggering. Koike et al. (2004) also distinguished static chirality (strongly broken symmetry, degenerate bands) from chiral vibration (dynamical spontaneous symmetry breaking, in which the system tunnels between left- and right-handed shapes). Cautionary results accumulated early: Petrache et al. (2006) warned that nearly degenerate pairs can be misinterpreted as chiral partners, and Tonev et al. (2006) found unequal $B(E2)$ values for the supposed partners in $^{134}$Pr - the start of the so-called chiral conundrum. Qi et al. (2009) showed in a particle-rotor model (a model coupling the valence nucleons to a rotating triaxial core) with restricted configurations that the $B(M1)$ staggering is sensitive to $\gamma$ and weak in the chiral-vibration region.

Shirinda and Lawrie (2012) then examined these fingerprints with realistic two-quasiparticle-plus-triaxial-rotor calculations (a quasiparticle being a dressed single-particle excitation of the nuclear mean field) for the $\pi h_{11/2}\otimes\nu h_{11/2}^{-1}$ configuration (an odd $h_{11/2}$ proton particle coupled to an odd $h_{11/2}$ neutron hole, the archetypal configuration of the $A\approx130$ region), in which each valence nucleon can occupy any of the five $h_{11/2}$ magnetic substates rather than one fixed orbital. They found that the fingerprints fail qualitatively: the $B(M1)$ staggering persists over $\gamma=20^\circ$-$50^\circ$ and even for axial shapes, so it does not indicate triaxiality or strong chirality breaking; the side-band staggering weakens and changes phase at high spin; realistic chiral systems can show vanishing $B(M1)$ staggering; an odd-even staggering of the mutual angles between the three angular momenta (angle staggering) appears and correlates with the $B(M1)$ staggering; and a smooth $S(I)$ does not imply orthogonal, well-broken chiral angular momenta. Applied to $^{128}$Cs, the same calculations reproduce the measured energies and transition strengths. The standard fingerprints therefore cannot reliably identify real chiral systems.

Problem Statement

The problem, as posed by Shirinda and Lawrie (2012) and sharpened by the subsequent literature, is to place the identification of nuclear chirality on a reliable footing. Its overarching objective is a validated way to decide, from experimental data, whether a candidate pair of doublet bands in a real nucleus possesses a chiral (aplanar, handed) angular-momentum geometry. Three linked results are requested:

1. Mechanism disentanglement. Quantitatively disentangle how the various phenomena - the chiral geometry itself, angle staggering, Coriolis effects (the coupling of the valence particles to the rotating core), the residual proton-neutron interaction (the proton-neutron force beyond the average mean field), configuration mixing (quantum mixing between different single-particle configurations), and mixing with planar (coplanar) configurations - generate the observed $B(M1)$ and energy staggering patterns in realistic chiral systems; in particular, establish what causes the phase change of the $B(M1)$ staggering between the ideal (single-orbital) and realistic (multi-orbital) descriptions. The observable at issue is the odd-even spin staggering of the absolute intra-band $B(M1)$ strengths: in the ideal case the staggering has a definite phase (high values for even spins), whereas in realistic multi-orbital calculations the yrast-band intra-band staggering can appear with the opposite phase (low values for even spins) and, as a distinct effect, the side-band staggering can disappear or weaken and change phase at high spin. Shirinda and Lawrie (2012) associated this changed behaviour with the strong staggering of the mutual angles between the angular momenta (angle staggering) and with the multiplicity of factors that influence $B(M1)$ rates, but left the definitive microscopic explanation open.

2. Robust identification criteria. Develop criteria, applicable to experimental data and as model-independent as possible, that reliably identify chiral geometry in real nuclei - going beyond the unreliable standard fingerprints - with demonstrated discriminative power against non-chiral explanations that mimic a doublet band for a given near-degenerate pair (planar tilted rotation, in which the angular momenta are coplanar but tilted away from the principal axes; gamma-soft shape coexistence, rapid fluctuations between shapes of different deformation angle $\gamma$; and other look-alike mechanisms). Octupole correlations (reflection-asymmetric nuclear shapes) and pseudo-spin doublets (near-degenerate pairs built on the pseudo-spin symmetry of the nuclear shell model) are documented to coexist with genuine chirality in the same nuclei, so their presence in a nucleus does not by itself make it a look-alike or exclude a chiral geometry. The observables raised by Shirinda and Lawrie (2012) and the follow-up literature include absolute transition strengths and the chiral electromagnetic selection rules, decay patterns of multi-quasiparticle bands (bands built on more than two quasiparticles), $g$ factors (the dimensionless ratio of a state's magnetic moment to its angular momentum), and static quadrupole moments (the intrinsic electric-quadrupole moments of the band states).

3. Implementation and validation. Implement the criteria above as a procedure applicable to experimental data for candidate chiral doublet bands, and demonstrate that the procedure discriminates a genuine chiral (aplanar, handed) angular-momentum geometry from the non-chiral look-alike explanations of the same near-degenerate pair listed in result 2. Shirinda and Lawrie (2012) suggest, as one possible route toward this goal, the use of realistic calculations to evaluate the orientations of the individual angular momenta and test whether a chiral geometry is present; this route is suggested rather than mandatory.

The problem is complete when the fingerprint failures are microscopically explained (result 1) and a criterion or set of criteria, applicable to experimental data, has demonstrated that it correctly identifies chiral geometry and discriminates it from the non-chiral look-alike explanations (results 2 and 3). Criteria that are proposed but not validated, or mechanisms that are listed but not quantified, do not complete it.

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