A self-consistent three-dimensional tilted-axis-cranking covariant density functional study (Chen, Li, and Guo, 2023) traced the transition from planar to chiral rotation across 120−134^{120-134}Cs and found the critical rotational frequency for chiral rotation decreasing with neutron number, but it neglected pairing correlations and all beyond-mean-field chiral dynamics (chiral vibration and tunneling between left- and right-handed sectors), could not reproduce the partner bands of the chiral doublets, overestimated spins and B(M1)/B(E2)B(M1)/B(E2) ratios in several isotopes, and left the proton-only chirality candidate 119^{119}Cs, the experimental verification of the predicted critical frequency in 133^{133}Cs, and the boundaries of the A≈130A\approx130 chiral island open. Later work (2024-2026) has addressed pieces of this at the particle-rotor-model level (pairing lowers the critical frequency) and experimentally (new chiral doublet candidates in 133^{133}Cs and, notably, 134^{134}Cs, where the mean-field calculation had found only planar rotation), but no pairing-including or beyond-mean-field microscopic description of the cesium chain exists and the island boundary is unsettled. The problem is to build and test such a description, thereby establishing where and how chirality sets in across the cesium isotopes and where the A≈130A\approx130 chiral island ends.

Background

In a triaxial (non-axially-symmetric) nucleus, static nuclear chirality arises when the angular momenta of an odd proton, an odd neutron, and the triaxial core point along three different principal axes — conventionally the proton along the short axis, the neutron hole along the long axis, and the collective core rotation along the intermediate axis of maximal moment of inertia. The three orthogonal angular momenta then form a left- or a right-handed system, related by time reversal, and the quantum-mechanical tunneling between the two mirror images splits the chiral partner states into nearly degenerate same-parity ΔI=1\Delta I=1 'chiral doublet bands' with similar electromagnetic transition rates (Frauendorf and Meng, 1997). Nearly degenerate doublet bands of this kind were first reported in the odd-odd (both proton and neutron numbers odd) N=75N=75 isotones — nuclei sharing the neutron number N=75N=75 — of the A≈130A\approx130 mass region, where an 'island of chiral rotation' was proposed (Starosta et al., 2001); the cesium (Z=55Z=55) isotopes contain the largest number of chiral candidates in this island. Key experimental fingerprints are the energy degeneracy and staggering (alternating spacings of consecutive levels) of the doublet, similar spin sequences, and comparable ratios of magnetic-dipole to electric-quadrupole transition probabilities, B(M1)/B(E2)B(M1)/B(E2), which measure the magnetic (single-particle) versus collective rotational content of the band. When the tunneling is fast compared with the rotation, the system executes a 'chiral vibration' (dynamic chirality) rather than static chiral rotation; distinguishing these regimes is a central difficulty of the field (Meng and Zhang, 2010).

The microscopic workhorse for these questions is the covariant density functional theory (CDFT, the relativistic mean-field framework) combined with tilted-axis cranking (TAC): instead of rotating about a symmetry axis, the mean field is minimized with the rotational angular momentum along a self-consistently determined tilted direction. In its fully three-dimensional form (3DTAC-CDFT) it yields, per isotope and configuration, the polar and azimuthal angles of the total angular momentum J\mathbf{J} as functions of the rotational frequency ℏω\hbar\omega; the azimuth angle becoming nonzero marks the transition from planar to aplanar (chiral) rotation above a critical frequency ωcrit\omega_{\mathrm{crit}}. The 2023 study of Chen, Li, and Guo (Eur. Phys. J. A 59, 142 (2023)) applied 3DTAC-CDFT to 120−134^{120-134}Cs with the standard two-quasiparticle configuration πh11/21⊗νh11/2−1\pi h_{11/2}^{1}\otimes\nu h_{11/2}^{-1} (an h11/2h_{11/2} proton particle coupled to an h11/2h_{11/2} neutron hole; quasiparticles are the single-particle excitations of the mean field) for the odd-odd isotopes and the corresponding three-quasiparticle configurations πh11/21⊗νh11/2−1(gd)−1\pi h_{11/2}^{1}\otimes\nu h_{11/2}^{-1}(gd)^{-1} and πh11/21⊗νh11/2−1(sd)1\pi h_{11/2}^{1}\otimes\nu h_{11/2}^{-1}(sd)^{1} for the odd-AA isotopes, finding planar-to-chiral transitions in the odd-AA isotopes 121−133^{121-133}Cs with ωcrit\omega_{\mathrm{crit}} decreasing with neutron number, and only planar rotation in the odd-odd isotopes 120,134^{120,134}Cs. Being a mean-field method, it describes a static orientation: it neither includes chiral vibrations nor tunneling between the left- and right-handed sectors, so only the lower band of each doublet is reproduced, and pairing correlations (the residual interaction that couples nucleons into correlated pairs, important for moments of inertia and transition rates) were neglected. The particle-rotor model (PRM), which couples quasiparticles to a quantized triaxial rotor and treats the total angular momentum as a good quantum number, is the standard independent framework against which such mean-field conclusions are checked. Beyond these, 119^{119}Cs presents a qualitatively new case: candidate 'revolving' chiral doublet bands built on a configuration of three protons only (one strongly coupled g9/2g_{9/2} proton plus two aligning h11/2h_{11/2} protons in the backbending regime, where the rotational frequency temporarily decreases as quasiparticles align), with no neutron participation (Zheng et al., 2022) — a geometry the usual proton-neutron chiral picture does not cover. A technical precedent exists for including pairing in tilted-axis-cranking CDFT via a separable pairing force, a pairing interaction written in separable form so that the tilted-axis-cranking equations remain numerically tractable (Wang, 2017, for 109^{109}Ag), and multiple-chiral-doublet phenomena were predicted in odd-A cesium isotopes by the same group earlier (Li, 2018).

Problem Statement

Construct a quantitative, microscopic account of the onset and evolution of nuclear chirality across the cesium isotope chain 119−134^{119-134}Cs, and thereby determine where the A≈130A\approx130 chiral island ends. Relative to the pairing-free static mean-field description of Chen, Li, and Guo (2023), resolving this objective requires all of the following components together, none of which alone settles the task: (i) incorporating pairing correlations into the self-consistent three-dimensional tilted-axis-cranking covariant density functional treatment (or an equivalent microscopic framework) and quantifying their effect on the critical rotational frequency ωcrit\omega_{\mathrm{crit}}, on the calculated spins I(ℏω)I(\hbar\omega) — including the backbending isotopes 128,132,134^{128,132,134}Cs — and on B(M1)/B(E2)B(M1)/B(E2) ratios, removing the present overestimate of I(ℏω)I(\hbar\omega) for 124−134^{124-134}Cs and of B(M1)/B(E2)B(M1)/B(E2) for 120,122,124^{120,122,124}Cs; (ii) going beyond the static mean field to describe chiral vibrations and tunneling between the left- and right-handed sectors, so that the partner bands and the energy splitting of the chiral doublets in the cesium isotopes are reproduced, not only the lower band of each doublet; (iii) clarifying, with an independent approach such as the particle-rotor model, the rotational structure of the proton-only 'revolving' chiral candidate bands in 119^{119}Cs, whose geometry differs from the usual proton-neutron chiral picture; (iv) confronting the predicted critical frequency (the kink in I(ℏω)I(\hbar\omega)) for the chiral band in 133^{133}Cs with the published data; and (v) establishing, in confrontation between such calculations and experiment, the boundaries of the A≈130A\approx130 chiral island — including whether 134^{134}Cs, now experimentally identified with chiral doublet bands where the mean-field calculation found only planar rotation, marks the neutron-rich edge toward the closed neutron shell at N=82N=82, and which configurations sustain chirality across the related isotones. The pairing treatment (i) and the beyond-mean-field treatment (ii) are required for every isotope of the chain 119−134^{119-134}Cs; components (iii)-(v) are required only at the isotopes each of them names.

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