Static (spectroscopic) quadrupole moments (SQMs) were proposed in 2020 as a new observable for nuclear chirality: particle-rotor-model calculations predict strongly spin-dependent differences of the SQM between the two partner bands when the band pair is a chiral vibration, near-constant equal SQMs when it is static chirality, and high-spin analytic formulas that would allow the triaxiality parameter γ\gamma to be extracted from measured SQMs. The prediction was established for a single particle-hole configuration and one parameter set, and no experimental SQM measurement of any chiral doublet band has been performed since. The problem is to validate (or refute) the SQM criterion: measure SQMs of candidate chiral doublet bands, establish theoretically whether the signatures and analytic formulas survive other configurations, deformations, moments of inertia, mass regions, and triaxiality ranges, and quantify the robustness of extracting γ\gamma from measured SQMs.

Background

A rapidly rotating nucleus with a triaxially deformed (three unequal principal axes) shape, odd proton and odd neutron in high-jj (high angular momentum) orbitals can break chiral symmetry: the proton angular momentum aligns along the short axis, the neutron angular momentum along the long axis, and the collective rotation along the intermediate axis, so the three angular momenta form either a left- or a right-handed system. The two mirror images are degenerate, and quantum-mechanical tunneling between them restores the symmetry, splitting each state into a pair of nearly degenerate ΔI=1\Delta I=1 bands of the same parity, called chiral doublet bands. This was predicted by Frauendorf and Meng (1997) and the first candidates were reported by Starosta et al. (2001); more than 50 candidate doublets are now known in the mass regions A≈80A\approx80, 100, 130, and 190. A key distinction, introduced by Koike, Starosta, and Hamamoto (2004), is between chiral vibration, in which the three angular momenta oscillate about a planar configuration at low spin, and static chirality, in which they settle into an aplanar (genuinely three-dimensional) geometry above some critical spin. Distinguishing the two regimes observationally is difficult: the usual criteria based on similar intra-band and inter-band reduced magnetic-dipole (M1) and electric-quadrupole (E2) transition strengths are suggestive but not conclusive, and the first g-factor (gyromagnetic ratio, the ratio of magnetic moment to angular momentum) measurement in a chiral band, on the Iπ=9+I^\pi=9^+ isomeric bandhead of 128^{128}Cs by Grodner et al. (2018, 2022), indicated a nearly planar geometry at low spin, sharpening the debate.

The static or spectroscopic quadrupole moment Q(I)Q(I) of a state with spin II is its electric quadrupole moment measured in the laboratory frame; it depends both on the intrinsic charge deformation of the nucleus and on how the angular momentum is oriented with respect to the triaxial shape. Chen, Kaiser, Meißner, and Meng (Phys. Lett. B 807, 135568 (2020)) investigated the SQMs of chiral doublet bands for the first time, using the particle-rotor model (PRM), in which a valence proton and a valence neutron are coupled to a triaxial rigid rotor and the Hamiltonian is diagonalized in a strong-coupling basis. For the classic particle-hole configuration π(1h11/2)1⊗ν(1h11/2)−1\pi(1h_{11/2})^{1}\otimes\nu(1h_{11/2})^{-1} (a proton particle and a neutron hole in the 1h11/21h_{11/2} shell, typical of the A≈130A\approx130 region) with quadrupole deformation β=0.23\beta=0.23, coupling constants Cp=0.32C_p=0.32 and Cn=−0.32C_n=-0.32 MeV, irrotational-flow moments of inertia Jk=J0sin⁡2(γ−2kπ/3)\mathcal{J}_k=\mathcal{J}_0\sin^2(\gamma-2k\pi/3) with J0=30 ℏ2/MeV\mathcal{J}_0=30\ \hbar^2/\mathrm{MeV}, intrinsic quadrupole moment Q0′=3.5Q_0'=3.5 eb, and triaxiality parameter 260∘≤γ≤270∘260^\circ\leq\gamma\leq270^\circ (in this convention γ=270∘\gamma=270^\circ is maximal triaxiality, where the short- and long-axis moments of inertia coincide), they found: (i) in the chiral-vibration regime the SQMs of the two partner bands differ strongly and vary with spin II; (ii) in the static-chirality regime the SQMs of both bands become nearly equal and almost constant with spin (vanishing identically at γ=270∘\gamma=270^\circ); and (iii) at high spin the SQMs of both bands are well approximated by analytic formulas, e.g. Q(I)=3I2−I(I+1)(I+1)(2I+3)Q0′cos⁡γQ(I)=\frac{3I^2-I(I+1)}{(I+1)(2I+3)}Q_0'\cos\gamma for the yrast (lower) band, which can be inverted to extract γ\gamma (over 240∘≤γ≤300∘240^\circ\leq\gamma\leq300^\circ) from measured Q(I)Q(I) for both the yrast and yrare (upper) bands, assuming a common Q0′Q_0'. This supplies a potential new criterion for assigning the chirality mode of the many observed candidate doublets, and a possible independent determination of the triaxial deformation. A later refinement by Hu and Chen (Phys. Rev. C 109, L021302 (2024)) studied the SQMs of multiple chiral doublet bands and found that Q(I)Q(I) exhibits a kink — a sharp change in its spin dependence — at specific spin values, located at the spin where the rotation crosses over from planar to aplanar geometry; the presence of this kink was proposed as evidence for the onset of chiral rotation, refining the SQM criterion. The feasibility of bandhead SQM measurements for triaxial rotor bands is demonstrated in the neighboring wobbling case: Laskar et al. (Phys. Rev. C 101, 034315 (2020)) measured both the g factor and the SQM (∣Q∣=1.71±0.34|Q|=1.71\pm0.34 eb) of the 11/2−11/2^- isomeric bandhead of the wobbling band in 133^{133}La, and the companion PRM study (Chen, Frauendorf, Kaiser, Meißner, Meng, Phys. Lett. B 807, 135596 (2020)) reproduced them. For chiral doublet bands, however, the SQM predictions remain purely theoretical, because the partner-band states are short-lived, high-spin excitations populated in heavy-ion fusion-evaporation reactions, for which standard in-beam γ\gamma-ray spectroscopy cannot measure spectroscopic quadrupole moments.

Problem Statement

Establish static quadrupole moments as a validated observable for nuclear chiral doublet bands, as proposed by Chen, Kaiser, Meißner, and Meng (2020), by answering three linked questions. (1) Generality: do the predicted SQM signatures — strongly spin-dependent and band-different Q(I)Q(I) in the chiral-vibration regime, near-constant and equal Q(I)Q(I) in the static-chirality regime (with the kink behavior at the planar-to-chiral transition found in 2024), and the high-spin analytic formulas for Q(I)Q(I) proportional to Q0′cos⁡γQ_0'\cos\gamma — hold quantitatively beyond the single studied case π(1h11/2)1⊗ν(1h11/2)−1\pi(1h_{11/2})^{1}\otimes\nu(1h_{11/2})^{-1} at β=0.23\beta=0.23 with irrotational moments of inertia and 260∘≤γ≤270∘260^\circ\leq\gamma\leq270^\circ, i.e. for other particle-hole and particle-particle configurations, other deformations β\beta, other moments of inertia, other mass regions (A≈80A\approx80, 100, 190), and broader ranges of γ\gamma? (2) Experimental test: measure the static quadrupole moments Q(I)Q(I) — or at least the bandhead SQMs of isomeric states — for both partner bands of a candidate chiral doublet band, and test whether the measured pattern matches the model predictions, thereby checking the SQM criterion for distinguishing chiral vibration from static chirality. (3) Inversion robustness: assess the practical extraction of the triaxiality parameter γ\gamma from measured Q(I)Q(I) via the inversion formulas, including the sensitivity to the assumed common intrinsic quadrupole moment Q0′Q_0' of the two bands and the accuracy of the high-spin approximation at the spins actually accessible to experiment. An answer may resolve any subset decisively, but full resolution of the source question requires confrontation with measured SQM data for at least one chiral doublet band. The dimensions enumerated under (1) describe the scope of the generality question — the directions in which the signatures must be probed — rather than a conjunctive checklist that must be exhausted for (1) to count as resolved: a decisive answer to (1) quantitatively establishes the fate (survival or failure) of the SQM signatures for at least one case beyond the studied configuration and states explicitly which of the enumerated dimensions were tested and which remain open.

References

  1. Q. B. Chen, N. Kaiser, Ulf-G. Meißner, and J. Meng, 'Static quadrupole moments of nuclear chiral doublet bands', Physics Letters B 807, 135568 (2020). DOI: 10.1016/j.physletb.2020.135568; https://doi.org/10.1016/j.physletb.2020.135568 (arXiv:2005.03865)
  2. B. Hu and Q. B. Chen, 'Static quadrupole moment as a criterion to distinguish chiral modes', Physical Review C 109, L021302 (2024). DOI: 10.1103/PhysRevC.109.L021302; https://doi.org/10.1103/PhysRevC.109.L021302
  3. Q. B. Chen, 'g factor of chiral doublets with π(1h11/2)^1⊗ν(1h11/2)^−1 configuration', Physical Review C 109, 024308 (2024). DOI: 10.1103/PhysRevC.109.024308; https://doi.org/10.1103/PhysRevC.109.024308
  4. Y. Wu and Q. B. Chen, 'g factor of chiral doublets with π(1h11/2)^1⊗ν(1h11/2)^−1 configuration. II. Pairing effect', Physical Review C 112, 034321 (2025). DOI: 10.1103/g9kl-2xph; https://doi.org/10.1103/g9kl-2xph
  5. Q. B. Chen and S. Frauendorf, 'Entanglement in two-quasiparticle-triaxial-rotor systems: Chirality, wobbling, and the Pauli effect', Physical Review C 111, 064326 (2025). DOI: 10.1103/s6zp-h83d; https://doi.org/10.1103/s6zp-h83d (arXiv:2504.05608)
  6. Z. C. Wan and Q. B. Chen, 'Effect of pairing on entanglement in nuclear chirality: Neutron pairing in the particle-hole configuration', Physical Review C 113, 054302 (2026). DOI: 10.1103/pmz1-r873; https://doi.org/10.1103/pmz1-r873
  7. Q. B. Chen, S. Frauendorf, N. Kaiser, Ulf-G. Meißner, and J. Meng, 'g-factor and static quadrupole moment for the wobbling mode in 133La', Physics Letters B 807, 135596 (2020). DOI: 10.1016/j.physletb.2020.135596; https://doi.org/10.1016/j.physletb.2020.135596 (arXiv:2006.00259)
  8. Md. S. R. Laskar, R. Palit, S. N. Mishra, N. Shimizu, Y. Utsuno, E. Ideguchi, U. Garg, S. Biswas, F. S. Babra, R. Gala, C. S. Palshetkar, and Z. Naik, 'Structure of the 11/2− isomeric state in 133La', Physical Review C 101, 034315 (2020). DOI: 10.1103/PhysRevC.101.034315; https://doi.org/10.1103/PhysRevC.101.034315
  9. E. Grodner, J. Srebrny, Ch. Droste, L. Próchniak, S. G. Rohoziński, M. Kowalczyk, M. Ionescu-Bujor, C. A. Ur, K. Starosta, T. Ahn, M. Kisieliński, T. Marchlewski, S. Aydin, F. Recchia, G. Georgiev, R. Lozeva, E. Fiori, M. Zielińska, Q. B. Chen, S. Q. Zhang, L. F. Yu, P. W. Zhao, and J. Meng, 'First Measurement of the g Factor in the Chiral Band: The Case of the 128Cs Isomeric State', Physical Review Letters 120, 022502 (2018). DOI: 10.1103/PhysRevLett.120.022502; https://doi.org/10.1103/PhysRevLett.120.022502
  10. E. Grodner, M. Kowalczyk, M. Kisieliński, J. Srebrny, L. Próchniak, Ch. Droste, S. G. Rohoziński, Q. B. Chen, M. Ionescu-Bujor, C. A. Ur, F. Recchia, J. Meng, S. Q. Zhang, P. W. Zhao, G. Georgiev, R. Lozeva, E. Fiori, S. Aydin, and A. Nałęcz-Jawecki, 'Examination of nuclear chirality with a magnetic moment measurement of the I=9 isomeric state in 128Cs', Physical Review C 106, 014318 (2022). DOI: 10.1103/PhysRevC.106.014318; https://doi.org/10.1103/PhysRevC.106.014318
  11. G. Georgiev, D. L. Balabanski, A. E. Stuchbery, and H. Ueno, 'Nuclear electromagnetic moments by spin-precession methods' (2026). arXiv:2607.00900; https://arxiv.org/abs/2607.00900
  12. A. Krakó, D. Sohler, J. Timár, I. Kuti, Q. B. Chen, S. Q. Zhang, J. Meng, K. Starosta, T. Koike, E. S. Paul, D. B. Fossan, and C. Vaman, 'Multiple chiral doublet bands in 104Rh', Physics Letters B 855, 138850 (2024). DOI: 10.1016/j.physletb.2024.138850; https://doi.org/10.1016/j.physletb.2024.138850
  13. S. Frauendorf and J. Meng, 'Tilted rotation of triaxial nuclei', Nuclear Physics A 617, 131 (1997). DOI: 10.1016/S0375-9474(97)00004-3; https://doi.org/10.1016/S0375-9474(97)00004-3
  14. K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R. LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R. Cooper, R. Krücken, J. R. Novak, N. V. Zamfir, K. E. Zyromski, D. J. Hartley, D. L. Balabanski, Jing-ye Zhang, S. Frauendorf, and V. I. Dimitrov, 'Chiral Doublet Structures in Odd-Odd N=75 Isotones: Chiral Vibrations', Physical Review Letters 86, 971 (2001). DOI: 10.1103/PhysRevLett.86.971; https://doi.org/10.1103/PhysRevLett.86.971
  15. T. Koike, K. Starosta, and I. Hamamoto, 'Chiral Bands, Dynamical Spontaneous Symmetry Breaking, and the Selection Rule for Electromagnetic Transitions in the Chiral Geometry', Physical Review Letters 93, 172502 (2004). DOI: 10.1103/PhysRevLett.93.172502; https://doi.org/10.1103/PhysRevLett.93.172502
  16. E. Grodner, J. Srebrny, A. A. Pasternak, I. Zalewska, T. Morek, Ch. Droste, J. Mierzejewski, M. Kowalczyk, J. Kownacki, M. Kisieliński, S. G. Rohoziński, T. Koike, K. Starosta, A. Kordyasz, P. J. Napiorkowski, M. Wolińska-Cichocka, E. Ruchowska, W. Płóciennik, and J. Perkowski, '128Cs as the Best Example Revealing Chiral Symmetry Breaking', Physical Review Letters 97, 172501 (2006). DOI: 10.1103/PhysRevLett.97.172501; https://doi.org/10.1103/PhysRevLett.97.172501
  17. L. Mu, H. Jia, B. Qi, and J. G. Deng, 'Possible new chiral system with two high-j valence protons', Physical Review C 111, 054321 (2025). DOI: 10.1103/PhysRevC.111.054321; https://doi.org/10.1103/PhysRevC.111.054321