Robustness of nuclear chirality: why near-perfect chiral degeneracy in $^{194}$Tl is rare and how lifetime data can establish it elsewhere

In rotating triaxial atomic nuclei, spontaneous chiral symmetry breaking is expected to produce pairs of near-degenerate rotational bands (chiral doublets) built on left- and right-handed arrangements of three angular-momentum vectors. Doppler-shift attenuation lifetime measurements in $^{194}$Tl established its negative-parity band pair as the best-known chiral candidate, degenerate not only in excitation energy and alignment but also in measured $B(M1)$ and $B(E2)$ transition probabilities, with chirality apparently persisting through the backbend where the configuration changes from two to four quasiparticles. This leaves a coherent, still-open problem: determine the nuclear-structure conditions under which such near-perfect chiral degeneracy is realized — as opposed to the approximate, possibly accidental partner-band behavior seen in most other candidates — whether chirality survives quasiparticle alignment (backbending) generically, and whether lifetime-derived electromagnetic transition strengths can serve as a general discriminator between genuine static chirality and mere energy coincidence. The problem spans experiment (extending complete chiral fingerprints to other candidate nuclei in the $A\sim80$, $100$, $130$, and $190$ regions), systematics, and theory (a quantitative account of the observed spread of degeneracy quality).

Background

A triaxial nucleus is one whose shape is deformed with three unequal principal axes (short, intermediate, and long). In certain rotating triaxial nuclei, the angular momenta of an odd proton, an odd neutron (or several unpaired quasiparticles), and the collective rotation of the core can orient along the three different axes. When the total angular momentum points away from any symmetry plane, its sense of rotation can be left-handed or right-handed; the two mirror images are related by the chiral (parity) symmetry of the underlying nuclear interaction, so spontaneous breaking of that symmetry produces two degenerate rotational sequences — a pair of $\Delta I=1$ 'chiral doublet' bands, as predicted by Frauendorf and Meng in the tilted-axis cranking framework. Starosta et al. reported the first candidate, in the $N=75$ odd-odd isotones, in 2001. If the left- and right-handed minima are well separated in the triaxial degree of freedom, 'static' chirality gives nearly identical partner bands; if the nucleus tunnels between them or oscillates about planar rotation, 'dynamic' chirality or chiral vibration gives only approximate degeneracy. Because triaxiality in nuclei is soft, most observed candidates fall somewhere in between.

The accepted fingerprints of an ideal chiral doublet are: (i) near-degeneracy of excitation energies within the partner pair; (ii) spin-independent energy staggering parameter $S(I)=[E(I+1)-E(I-1)]/2$ (the mid-point energy between band states two spin units apart; for an ideal chiral doublet it varies smoothly with spin $I$, showing no odd-even staggering); (iii) similar spin alignments $i_x$ (the component of the angular momentum aligned with the rotation axis, extracted from the level energies and spins); (iv) similar intra-band reduced electromagnetic transition probabilities $B(M1)$ (magnetic dipole) and $B(E2)$ (electric quadrupole) in the two bands, together with the characteristic odd-even staggering of $B(M1)$; (v) weak inter-band $E2$ transitions; and (vi) small interaction strength between the partners. The transition probabilities are obtained from level lifetimes and branching ratios, so points (iv)–(v) require dedicated lifetime measurements — nowadays mostly with the Doppler Shift Attenuation Method (DSAM), in which $\gamma$ rays from recoiling nuclei slowing down in a backing are line-shape analyzed with Monte-Carlo stopping and side-feeding simulations. By 2019 a compilation counted 59 candidate chiral doublet bands in 47 nuclei across the $A\sim80$, $100$, $130$, and $190$ mass regions, but measured $B(M1)$ and $B(E2)$ values existed for only a handful; lifetime data had in fact refuted or weakened several prominent claims (e.g. $^{103,104}$Rh, $^{106}$Ag, $^{134}$Pr, $^{102}$Rh), establishing that energy degeneracy alone is not a chiral certificate.

Within this landscape, doubly-odd $^{194}$Tl ($Z=81$, $N=113$) became the flagship case. Its negative-parity band pair is based on the $\pi h_{9/2}\otimes\nu i_{13/2}^{-1}$ configuration at low spin and on a four-quasiparticle $\pi h_{9/2}\otimes\nu i_{13/2}^{-3}$ structure above a backbend (a sudden irregularity of the moment of inertia caused by quasiparticle alignment, here at spin $I\approx16$–$19\hbar$). Spectroscopy showed the excitation-energy difference of the pair falling from about $463\ \mathrm{keV}$ at $I=11\hbar$ to about $127\ \mathrm{keV}$ at $I=18\hbar$ and improving further at higher spin, with matching alignments. DSAM lifetime measurements for four bands of $^{194}$Tl, reported by Masiteng et al. in a conference proceeding (J. Phys. Conf. Ser. 724, 012028) and in the journal version (Eur. Phys. J. A 52, 28), then showed that the intra-band $B(M1)$ and $B(E2)$ values of the two partner bands track each other closely, making this pair — remarkably, the four-quasiparticle one above the backbend — 'a prime candidate for best chiral symmetry pair', with the suggestion that a third band may join the chiral scenario. Later reviews (e.g. Bark et al., Front. Phys. 19, 24302 (2024)) still call $^{194}$Tl the best example of chiral degeneracy observed to date.

What the $^{194}$Tl work did not settle, and what this problem targets, is the systematics behind that quality gap: most candidate nuclei show only partial similarity between partners (close energies but differing alignments or transition strengths), several degenerate-looking pairs have been reinterpreted as planar rotation, and it is not understood which structural ingredients (core triaxial stiffness, number and type of quasiparticles, pairing, tunneling between left- and right-handed minima) produce genuinely robust, near-perfect chiral degeneracy, nor whether the persistence of chirality across a quasiparticle alignment seen in $^{194}$Tl is a generic phenomenon or an exception.

Problem Statement

The problem, as motivated by the $^{194}$Tl lifetime study and still open, is to determine what makes nuclear chiral symmetry robust: (1) identify which nuclear-structure ingredients — among them triaxial core stiffness, quasiparticle configuration and number, pairing, and the tunneling between left- and right-handed minima — distinguish nuclei like $^{194}$Tl, whose partner bands are near-degenerate simultaneously in excitation energy, spin alignment, and intra-band $B(M1)$ and $B(E2)$ strengths, from the majority of chiral candidates that show only approximate partner behavior, and express the finding, whether it attributes the spread to a single controlling ingredient or to their interplay, as a quantitative, predictive account of the observed spread of chiral-degeneracy quality across the $A\sim80$, $100$, $130$, and $190$ regions; (2) establish whether the persistence of chirality through quasiparticle alignment (the backbend at $I\approx16$–$19\hbar$ in $^{194}$Tl, where the configuration changes from $\pi h_{9/2}\otimes\nu i_{13/2}^{-1}$ to $\pi h_{9/2}\otimes\nu i_{13/2}^{-3}$) is a generic feature of chiral systems or an exception, by testing multi-quasiparticle degenerate bands in other nuclei for genuinely aplanar (chiral) rotation rather than planar alternatives; and (3) determine, through dedicated lifetime measurements and the fingerprints they yield (near-degenerate energies, spin-independent $S(I)$, equal alignments, similar intra-band $B(M1)$/$B(E2)$ with odd-even $B(M1)$ staggering, weak inter-band $E2$), how far beyond $^{194}$Tl complete chiral-pair status extends, thereby settling the degree to which lifetime data can discriminate truly robust chiral symmetry from merely close energy degeneracy. These three strands form one scientific objective: a validated understanding of when and why nuclear chirality is realized almost perfectly. A full answer to strand (1) subsumes the extension question of strand (3), and strand (2) is separately required, so the objective is met by strands (1) and (2) jointly rather than by any single strand alone.

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