Nuclear chirality in $^{134}$Pr: surviving chiral geometry and the decisive observables for chiral-band assignments

The nearly degenerate $\Delta I=1$ bands of the same parity in the odd-odd nucleus $^{134}$Pr were the first proposed realization of chiral symmetry breaking in a rotating triaxial nucleus. A 2006 critical analysis by Petrache, together with Doppler-shift lifetime measurements by Tonev et al., showed that the measured branching ratios, lifetimes, and the inferred ratio of the two bands' transition quadrupole moments (2.0(4) over spins $I=14$–$18$) are in clear disagreement with the static chiral picture, and that near-degeneracy of energies alone is insufficient to establish chirality. The problem remains open as of 2026. It asks: whether any remnant of chiral geometry — static, partial, or dynamic — survives in the $^{134}$Pr doublet bands once intrinsic transition moments, branching ratios, and band mixing are accounted for, and which additional measurable quantities would settle the chiral-band assignment. Reviews through 2024 and a 2026 lifetime-methodology paper still list the $^{134}$Pr question as unresolved and the standard chiral fingerprints as unvalidated for real nuclei, while the analogous chiral conundrum in $^{106}$Ag has been resolved, providing an instructive precedent.

Background

Chirality in atomic nuclei is a mode of spontaneous symmetry breaking predicted for triaxial nuclei — nuclei whose charge distribution is deformed with three unequal principal axes (short, intermediate, and long) — rotating about an axis that coincides with none of them. The archetypal system is an odd-odd nucleus with one high-$j$ valence proton particle and one high-$j$ valence neutron hole (in the mass-$130$ region, the configuration $\pi h_{11/2}\,\nu h_{11/2}^{-1}$): the proton angular momentum aligns along the short axis, the neutron-hole angular momentum along the long axis, and the collective rotor angular momentum along the intermediate axis. The three angular momenta then form a system of definite handedness, and its left- and right-handed versions are related by time-reversal (chiral) symmetry. In an intrinsic-frame mean-field description the two handednesses are degenerate minima; restoring the symmetry quantum-mechanically splits them into a pair of near-degenerate $\Delta I=1$ rotational bands of the same parity, called chiral doublet bands. This was predicted by Frauendorf and Meng (1997), and a pair of bands of exactly this kind had already been observed in $^{134}$Pr (Petrache et al., 1996), which thereby became the first chiral candidate; the interpretation was extended to the $N=75$ isotones (nuclei with neutron number $N=75$) $^{130}$Cs, $^{132}$La, $^{134}$Pr, and $^{136}$Pm, with the low-spin behavior attributed to a chiral vibration about the chiral geometry rather than a statically broken symmetry (Starosta et al., 2001).

Beyond energy degeneracy, the static chiral picture makes definite electromagnetic predictions: similar transition strengths within and between the partner bands, characteristic odd-even staggering of intraband $B(M1)/B(E2)$ ratios (where $B(\lambda)$ is the reduced transition probability for radiation of multipolarity $\lambda$), and, in the ideal chiral geometry, selection rules suppressing certain interband transitions (Koike, Starosta, and Hamamoto, 2004). Testing these requires lifetimes, measured by Doppler-shift methods: in the Doppler-shift attenuation method (DSAM) and the recoil distance Doppler-shift method (RDDS), sub-picosecond to picosecond lifetimes are inferred from the Doppler shifts of de-exciting $\gamma$ rays. The first such test on $^{134}$Pr (Tonev et al., 2006) found that the in-band $B(E2)$ values of the two bands differ strongly; a two-band-mixing analysis by Petrache et al. (2006) gave a ratio of 2.0(4) between the transition quadrupole moments at spins $I=14$–$18$, where the bands are almost degenerate, implying considerably different nuclear shapes — so the doublet cannot be static chiral partners. This contradiction, in the very first chiral candidate, became known as the chiral conundrum. Competing non-chiral mechanisms were proposed: shape coexistence of the two bands, mixing of two- and four-quasiparticle configurations in the projected shell model (a shell-model scheme built on angular-momentum-projected deformed quasiparticle states), and $\gamma$-vibrational structure (oscillation of the triaxial asymmetry of the nuclear shape). Self-consistent Skyrme-Hartree-Fock cranking (rotation imposed on a nuclear mean field computed with the Skyrme effective interaction) found tilted-axis chiral solutions only in $^{132}$La among the $N=75$ isotones and none in $^{134}$Pr (Olbratowski et al., 2006), while Tonev et al. left open a dynamical form of intrinsic chirality. Meanwhile adjacent cases were decided both ways: $^{128}$Cs was confirmed by lifetimes as the best established chiral example (Grodner et al., 2006), $^{135}$Nd was shown to evolve from chiral vibration to static chirality with increasing spin (Mukhopadhyay et al., 2007), $^{102}$Rh was found not to possess static chirality (Tonev et al., 2014), and the $^{106}$Ag conundrum was resolved when Doppler-shift lifetimes (Lieder et al., 2014) plus microscopic tilted-axis cranking and time-dependent covariant density functional calculations (relativistic mean-field dynamics) (Zhao et al., 2019; Li et al., 2024) showed its crossing twin bands to have distinct two- and four-quasiparticle configurations. Roughly one hundred candidate chiral doublet bands have been compiled in data tables (Xiong and Wang, 2019) without adjudication, and the identification criteria remain contested: the standard fingerprints were shown to change considerably for real, non-ideal chiral systems (Shirinda and Lawrie, 2012) and to occur in their textbook form only in idealized two-quasiparticle model descriptions (Bark et al., 2024). What has never been settled is the founding case itself — the structure of the $^{134}$Pr doublet bands — and the general question of what evidence decides a chiral-band assignment.

Problem Statement

Determine whether any remnant of chiral geometry — static, partial, or dynamic — survives in the $\Delta I=1$ doublet bands of $^{134}$Pr after accounting for the observed differences in intrinsic transition moments, the measured branching ratios and lifetimes, and the estimated band-mixing strengths. In particular, decide between the chiral interpretation (in a static or dynamic/chiral-vibrational form) and the competing non-chiral mechanisms — shape coexistence, two- versus four-quasiparticle configuration mixing, or $\gamma$-vibrational structure — by simultaneously reproducing the energies, branching ratios, absolute transition strengths, and mixing of the two bands, and state which additional measurable quantities would settle the assignment.

References

1. 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
2. C. M. Petrache, D. Bazzacco, S. Lunardi, C. Rossi Alvarez, G. de Angelis, M. De Poli, D. Bucurescu, C. A. Ur, P. B. Semmes, and R. Wyss, "Rotational bands in the doubly odd nucleus 134Pr", Nuclear Physics A 597, 106 (1996). DOI: 10.1016/0375-9474(95)00416-5, https://doi.org/10.1016/0375-9474(95)00416-5
3. 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
4. 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
5. P. Olbratowski, J. Dobaczewski, and J. Dudek, "Search for the Skyrme-Hartree-Fock solutions for chiral rotation in N=75 isotones", Physical Review C 73, 054308 (2006). DOI: 10.1103/PhysRevC.73.054308, arXiv:nucl-th/0510087, https://arxiv.org/abs/nucl-th/0510087
6. D. Tonev, G. de Angelis, P. Petkov, A. Dewald, S. Brant, S. Frauendorf, D. L. Balabanski, P. Pejovic, D. Bazzacco, P. Bednarczyk, F. Camera, A. Fitzler, A. Gadea, S. Lenzi, S. Lunardi, N. Marginean, O. Möller, D. R. Napoli, A. Paleni, C. M. Petrache, G. Prete, K. O. Zell, Y. H. Zhang, Jing-ye Zhang, Q. Zhong, and D. Curien, "Transition probabilities in 134Pr: A test for chirality in nuclear systems", Physical Review Letters 96, 052501 (2006). DOI: 10.1103/PhysRevLett.96.052501, https://doi.org/10.1103/PhysRevLett.96.052501
7. C. M. Petrache, G. B. Hagemann, I. Hamamoto, and K. Starosta, "Risk of misinterpretation of nearly degenerate pair bands as chiral partners in nuclei", Physical Review Letters 96, 112502 (2006). DOI: 10.1103/PhysRevLett.96.112502, https://doi.org/10.1103/PhysRevLett.96.112502
8. C. M. Petrache, "Chirality in nuclear structure: An experimental view into underlying symmetries", International Journal of Modern Physics E 15, 1897 (2006). DOI: 10.1142/S0218301306005320, https://doi.org/10.1142/S0218301306005320
9. D. Tonev, P. Petkov, D. L. Balabanski, G. de Angelis, A. Gadea, D. R. Napoli, N. Marginean, A. Dewald, P. Pejovic, A. Fitzler, O. Möller, K. O. Zell, S. Brant, S. Frauendorf, D. Bazzacco, S. Lenzi, S. Lunardi, P. Bednarczyk, D. Curien, C. Petrache, Q. Zhong, Y. H. Zhang, and Jing-Ye Zhang, "Lifetime measurements in 134Pr and chirality in nuclei", International Journal of Modern Physics E 15, 1531 (2006). DOI: 10.1142/S0218301306004909, https://doi.org/10.1142/S0218301306004909
10. 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
11. S. Mukhopadhyay, D. Almehed, U. Garg, S. Frauendorf, T. Li, P. V. Madhusudhana Rao, X. Wang, S. S. Ghugre, M. P. Carpenter, S. Gros, A. Hecht, R. V. F. Janssens, F. G. Kondev, T. Lauritsen, D. Seweryniak, and S. Zhu, "From chiral vibration to static chirality in 135Nd", Physical Review Letters 99, 172501 (2007). DOI: 10.1103/PhysRevLett.99.172501, https://doi.org/10.1103/PhysRevLett.99.172501
12. J. Meng and S. Q. Zhang, "Open problems in understanding the nuclear chirality", Journal of Physics G: Nuclear and Particle Physics 37, 064025 (2010). DOI: 10.1088/0954-3899/37/6/064025, arXiv:1002.0907, https://arxiv.org/abs/1002.0907
13. J. Timár, K. Starosta, I. Kuti, D. Sohler, D. B. Fossan, T. Koike, E. S. Paul, A. J. Boston, H. J. Chantler, M. Descovich, R. M. Clark, M. Cromaz, P. Fallon, I. Y. Lee, A. O. Macchiavelli, C. J. Chiara, R. Wadsworth, A. A. Hecht, D. Almehed, and S. Frauendorf, "Medium- and high-spin band structure of the chiral-candidate nucleus 134Pr", Physical Review C 84, 044302 (2011). DOI: 10.1103/PhysRevC.84.044302, https://doi.org/10.1103/PhysRevC.84.044302
14. O. Shirinda and E. A. Lawrie, "Identifying chiral bands in real nuclei", The European Physical Journal A 48, 118 (2012). DOI: 10.1140/epja/i2012-12118-2, https://doi.org/10.1140/epja/i2012-12118-2
15. L. Liu, S. Y. Wang, B. Qi, and C. Liu, "A possible criterion for the static chirality: Very small band interaction strength", International Journal of Modern Physics E 22, 1350060 (2013). DOI: 10.1142/S0218301313500602, https://doi.org/10.1142/S0218301313500602
16. D. Tonev, M. S. Yavahchova, N. Goutev, G. de Angelis, P. Petkov, R. K. Bhowmik, R. P. Singh, S. Muralithar, N. Madhavan, R. Kumar, M. Kumar Raju, J. Kaur, G. Mohanto, A. Singh, N. Kaur, R. Garg, A. Shukla, Ts. K. Marinov, and S. Brant, "Candidates for twin chiral bands in 102Rh", Physical Review Letters 112, 052501 (2014). DOI: 10.1103/PhysRevLett.112.052501, https://doi.org/10.1103/PhysRevLett.112.052501
17. E. O. Lieder, R. M. Lieder, R. A. Bark, Q. B. Chen, S. Q. Zhang, J. Meng, E. A. Lawrie, J. J. Lawrie, S. P. Bvumbi, N. Y. Kheswa, S. S. Ntshangase, T. E. Madiba, P. L. Masiteng, S. M. Mullins, S. Murray, P. Papka, D. G. Roux, O. Shirinda, Z. H. Zhang, P. W. Zhao, Z. P. Li, J. Peng, B. Qi, S. Y. Wang, Z. G. Xiao, and C. Xu, "Resolution of chiral conundrum in 106Ag: Doppler-shift lifetime investigation", Physical Review Letters 112, 202502 (2014). DOI: 10.1103/PhysRevLett.112.202502, https://doi.org/10.1103/PhysRevLett.112.202502
18. J. Meng, Q. B. Chen, and S. Q. Zhang, "Chirality in atomic nuclei: 2013", International Journal of Modern Physics E 23, 1430016 (2014). DOI: 10.1142/S0218301314300161, https://doi.org/10.1142/S0218301314300161
19. K. Starosta and T. Koike, "Nuclear chirality, a model and the data", Physica Scripta 92, 093002 (2017). DOI: 10.1088/1402-4896/aa800e, https://doi.org/10.1088/1402-4896/aa800e
20. M. S. Yavahchova, D. Tonev, N. Goutev, G. de Angelis, R. K. Bhowmik, R. P. Singh, S. Muralithar, N. Madhavan, R. Kumar, M. Kumar Raju, J. Kaur, G. Mahanto, A. Singh, N. Kaur, R. Garg, and A. Sukla, "Examples of dynamic chirality in nuclei", EPJ Web of Conferences 194, 05003 (2018). DOI: 10.1051/epjconf/201819405003, https://doi.org/10.1051/epjconf/201819405003
21. B. W. Xiong and Y. Y. Wang, "Nuclear chiral doublet bands data tables", Atomic Data and Nuclear Data Tables 125, 193 (2019). DOI: 10.1016/j.adt.2018.05.002, arXiv:1804.04437, https://arxiv.org/abs/1804.04437
22. P. W. Zhao, Y. K. Wang, and Q. B. Chen, "Microscopic resolution of the nuclear chiral conundrum with crossing twin bands in 106Ag", Physical Review C 99, 054319 (2019). DOI: 10.1103/PhysRevC.99.054319, arXiv:1905.10544, https://arxiv.org/abs/1905.10544
23. B. F. Lv, C. M. Petrache, Q. B. Chen, J. Meng, A. Astier, E. Dupont, P. Greenlees, H. Badran, T. Calverley, D. M. Cox, T. Grahn, J. Hilton, R. Julin, S. Juutinen, J. Konki, J. Pakarinen, P. Papadakis, J. Partanen, P. Rahkila, P. Ruotsalainen, M. Sandzelius, J. Saren, C. Scholey, J. Sorri, S. Stolze, J. Uusitalo, B. Cederwall, A. Ertoprak, H. Liu, S. Guo, M. L. Liu, J. G. Wang, X. H. Zhou, I. Kuti, J. Timár, A. Tucholski, J. Srebrny, and C. Andreoiu, "Chirality of 135Nd reexamined: Evidence for multiple chiral doublet bands", Physical Review C 100, 024314 (2019). DOI: 10.1103/PhysRevC.100.024314, arXiv:1907.12809, https://arxiv.org/abs/1907.12809
24. 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
25. S. Wang, C. Liu, B. Qi, W. Xu, and H. Zhang, "Experimental studies for nuclear chirality in China", Frontiers of Physics 18, 64601 (2023). DOI: 10.1007/s11467-023-1303-5, https://doi.org/10.1007/s11467-023-1303-5
26. R. A. Bark, E. A. Lawrie, C. Liu, and S. Y. Wang, "Investigations of nuclear chirality at iThemba LABS", Frontiers of Physics 19, 24302 (2024). DOI: 10.1007/s11467-023-1340-0, https://doi.org/10.1007/s11467-023-1340-0
27. B. Li, P. W. Zhao, and J. Meng, "Chiral dynamics in soft triaxial nuclei", Physics Letters B 856, 138877 (2024). DOI: 10.1016/j.physletb.2024.138877, arXiv:2407.08210, https://arxiv.org/abs/2407.08210
28. J. Meng and Y. P. Wang, "The 25th anniversary for nuclear chirality", in "Chirality and Wobbling in Atomic Nuclei" (CRC Press, 2024). DOI: 10.1201/9781032691633-3, arXiv:2405.07805, https://arxiv.org/abs/2405.07805
29. D. Tonev, G. D. Dimitrova, A. Demerdjiev, G. De Gregorio, G. de Angelis, E. Geleva, N. Goutev, N. N. Petrov, I. Pantaleev, and L. Panteleev-Simeonova, "Lifetime measurements — A powerful tool to study nuclear structure", Particles 9, 47 (2026). DOI: 10.3390/particles9020047, https://doi.org/10.3390/particles9020047