Microscopic origin of shell-correction attenuation near scission and the $N=152$ and $Z=108$ anomalies in spontaneous-fission half-lives of heavy and superheavy nuclei

Macroscopic–microscopic fission models such as the generalized liquid drop model (GLDM) reproduce measured spontaneous-fission half-lives of nuclei from uranium to flerovium to within about two orders of magnitude, but they attenuate the microscopic shell correction with deformation so that it acts only near the ground state — a phenomenological choice justified only qualitatively by the argument that the nuclear proximity interaction between nascent fragments destroys shell structure in necked shapes. The same calculations show unexplained deviations of $10^{3}$–$10^{5}$ concentrated near the $N=152$ sub-shell closure and for $^{264}$Hs ($Z=108$). The open problem is a quantitative microscopic account of shell and pairing corrections at large deformation and near scission — their modification by the proximity interaction and the deformation dependence of the $N=152$ and $Z=108$ sub-shell gaps — that explains these anomalies and places predictions for the still-unmeasured spontaneous-fission half-lives of $Z=114$–$120$ nuclei on a microscopic footing.

Background

Spontaneous fission (SF) is a decay mode of heavy nuclei in which the nucleus quantum-tunnels through a fission barrier into two comparable fragments. In the WKB (Wentzel–Kramers–Brillouin) approximation the half-life is $T_{1/2}\approx\ln 2/(\nu_{0}P)$, where $\nu_{0}$ is the assault frequency (the rate at which the nucleus attempts to cross the barrier; $10^{20}\,\mathrm{s^{-1}}$ is used in the model below) and $P$ is the tunneling probability through the barrier, $P=\exp[-(2/\hbar)\int\sqrt{2B(r)(E(r)-E_{0})}\,dr]$, with $B(r)$ the collective inertia along the deformation coordinate. Because $P$ is exponential in this action integral, an order-of-magnitude-accurate half-life requires sub-MeV control of the barrier and inertia along the whole fission path.

Two frameworks supply the barrier. In the macroscopic–microscopic method, a liquid-drop-type macroscopic deformation energy is corrected by a shell correction extracted from single-particle levels via the Strutinsky procedure — a smoothing prescription that isolates the oscillatory, shell-structure part of the total energy (Strutinsky, 1967). In fully microscopic approaches the barrier comes from self-consistent Hartree–Fock–Bogoliubov (HFB) energy density functional (EDF) calculations, which describe the nucleus through effective mean fields and pairing correlations between nucleons; their time-dependent extension (TD-DFT) can follow fission dynamics explicitly.

The generalized liquid drop model (GLDM) used by Bao, Zhang, Royer and Li (2013) extends the liquid-drop energy to strongly deformed, quasimolecular shapes — two nearly spherical fragments joined by a neck — by adding a proximity energy, the universal short-range attraction between gently curved nuclear surfaces (the proximity force of Błocki, Randrup, Świątecki and Tsang), which strongly lowers the energy of creviced shapes. Their systematic SF study combined the GLDM barrier with a phenomenological pairing correction and with droplet-model shell corrections multiplied by a deformation-dependent attenuation factor, so that by construction the shell correction acts only near the spherical ground state and not at the saddle. The stated justification is qualitative: single-particle calculations cited in that paper indicate that for two separated spheroids the valence nucleon orbitals are strongly perturbed by the nuclear proximity potential, so applying unattenuated shell corrections to creviced shapes overestimates the true shell effects.

The calculation reproduced most measured SF half-lives within a factor of $10^{2}$, but showed deviations of $10^{3}$–$10^{5}$ concentrated near the neutron number $N=152$ (e.g. $^{248}$Cm, $^{250}$Cf, $^{252}$No, $^{256}$Rf, $^{258}$Rf) and for $^{264}$Hs — attributed to deformed sub-shell closures, shell gaps at nucleon numbers that stiffen the shell structure locally at large prolate deformation without being spherical magic numbers. A comparison with the Yukawa-plus-exponential macroscopic energy and unattenuated Woods–Saxon shell corrections of Möller, Nix and Świątecki (1989) showed differences of up to six orders of magnitude in SF half-lives, ascribed mainly to the different shell-correction treatment at deformation; modern macroscopic–microscopic compilations such as the fission barriers of Möller et al. (2009) likewise apply the full Strutinsky shell correction at all deformations, without an attenuation factor. The paper also predicted SF half-lives for then-unknown even–even isotopes of $Z=114$–$120$, nuclei in the region of the predicted island of stability near the next spherical shell closures, and motivated the quasimolecular approach by the then-current assertion that HFB-type microscopic studies could not describe the evolution toward two separated fragments, the formation of a deep neck, the proximity energy, or a well-defined scission point.

What remains unresolved is the microscopic content of these choices: whether and how shell and pairing corrections are really attenuated at large deformation and near scission, with which functional form and scale; and why the half-lives near $N=152$ and $Z=108$ deviate so strongly.

Problem Statement

Provide a quantitative, microscopic account of shell and pairing corrections in fission barriers of heavy and superheavy nuclei at large deformation and near scission that:

(i) derives — rather than imposes phenomenologically — the deformation dependence of these corrections, including the extent to which the nuclear proximity interaction between the nascent fragments attenuates shell effects, with a definite functional form and scale; and

(ii) explains within that account the anomalously large deviations (factors of $10^{3}$–$10^{5}$) between measured spontaneous-fission half-lives and macroscopic–microscopic generalized-liquid-drop-model (GLDM) predictions near the $N=152$ sub-shell closure and at $Z=108$, i.e. the deformation dependence of these sub-shell gaps as it enters fission lifetimes.

Such an account would place predictions for the still-unmeasured spontaneous-fission half-lives of the even–even nuclei of $Z=114$–$120$ on a microscopic footing.

References

1. Xiaojun Bao, Hongfei Zhang, G. Royer, Junqing Li, 'Spontaneous fission half-lives of heavy and superheavy nuclei within a generalized liquid drop model', Nuclear Physics A 906, 1–13 (2013), DOI: 10.1016/j.nuclphysa.2013.03.002, arXiv:1302.7117, https://doi.org/10.1016/j.nuclphysa.2013.03.002
2. V. M. Strutinsky, 'Shell effects in nuclear masses and deformation energies', Nuclear Physics A 95, 420–442 (1967), DOI: 10.1016/0375-9474(67)90510-6, https://doi.org/10.1016/0375-9474(67)90510-6
3. J. Błocki, J. Randrup, W. J. Świątecki, C. F. Tsang, 'Proximity forces', Annals of Physics 105, 427–462 (1977), DOI: 10.1016/0003-4916(77)90249-4, https://doi.org/10.1016/0003-4916(77)90249-4
4. P. Möller, J. R. Nix, W. J. Świątecki, 'New developments in the calculation of heavy-element fission barriers', Nuclear Physics A 492, 349–387 (1989), DOI: 10.1016/0375-9474(89)90403-X, https://doi.org/10.1016/0375-9474(89)90403-X
5. Peter Möller, Arnold J. Sierk, Takatoshi Ichikawa, Akira Iwamoto, Ragnar Bengtsson, Henrik Uhrenholt, Sven Åberg, 'Heavy-element fission barriers', Physical Review C 79, 064304 (2009), DOI: 10.1103/PhysRevC.79.064304, https://doi.org/10.1103/PhysRevC.79.064304
6. N. Schunck, L. M. Robledo, 'Microscopic theory of nuclear fission: a review', Reports on Progress in Physics 79, 116301 (2016), DOI: 10.1088/0034-4885/79/11/116301, https://doi.org/10.1088/0034-4885/79/11/116301
7. Aurel Bulgac, Shi Jin, Kenneth J. Roche, Nicolas Schunck, Ionel Stetcu, 'Fission dynamics of 240Pu from saddle to scission and beyond', Physical Review C 100, 034615 (2019), DOI: 10.1103/PhysRevC.100.034615, arXiv:1806.00694, https://doi.org/10.1103/PhysRevC.100.034615
8. S. E. Agbemava, A. V. Afanasjev, D. Ray, P. Ring, 'Assessing theoretical uncertainties in fission barriers of superheavy nuclei', Physical Review C 95, 054324 (2017), DOI: 10.1103/PhysRevC.95.054324, arXiv:1704.06334, https://doi.org/10.1103/PhysRevC.95.054324
9. Adrián Sánchez-Fernández, Silvia Bara, Wouter Ryssens, Stéphane Goriely, 'Accurate spontaneous fission half-lives from a microscopic large-scale nuclear structure model', Physics Letters B 874, 140287 (2026), DOI: 10.1016/j.physletb.2026.140287, arXiv:2508.16240, https://doi.org/10.1016/j.physletb.2026.140287
10. R. Rodríguez-Guzmán, A. Rakhmankulov, L. M. Robledo, R. N. Bernard, 'Impact of perturbative tensor interactions on the spontaneous fission half-lives of superheavy nuclei', Journal of Physics G 53, 075109 (2026), DOI: 10.1088/1361-6471/ae8c5a, arXiv:2608.12199, https://doi.org/10.1088/1361-6471/ae8c5a
11. K. Pomorski, A. Augustyn, T. Cap, Y. J. Chen, M. Kowal, M. Warda, Z. G. Xiao, 'Survival of Pairing Correlations and Shell Effects at Scission in Finite-Temperature Nuclear Fission: Implications for Odd-Even Staggering', arXiv:2604.24174 (2026), https://arxiv.org/abs/2604.24174
12. F. A. Ivanyuk, S. Chiba, Y. Aritomo, 'The Scission-Point Configuration within the Two-Center Shell Model Shape Parameterization', Physical Review C 90, 054607 (2014), DOI: 10.1103/PhysRevC.90.054607, arXiv:1410.7591, https://doi.org/10.1103/PhysRevC.90.054607
13. E. Minaya Ramirez, D. Ackermann, K. Blaum, M. Block, C. Droese, Ch. E. Düllmann, M. Dworschak, M. Eibach, S. Eliseev, E. Haettner, F. Herfurth, F. P. Heßberger, S. Hofmann, J. Ketelaer, G. Marx, M. Mazzocco, D. Nesterenko, Yu. N. Novikov, W. R. Plaß, D. Rodríguez, C. Scheidenberger, L. Schweikhard, P. G. Thirolf, C. Weber, 'Direct Mapping of Nuclear Shell Effects in the Heaviest Elements', Science 337, 1207–1210 (2012), DOI: 10.1126/science.1225636, arXiv:1406.6477, https://doi.org/10.1126/science.1225636
14. J. Khuyagbaatar, 'Spontaneous fission half-lives of the heaviest nuclei: Semi-empirical predictions', Nuclear Physics A 1002, 121958 (2020), DOI: 10.1016/j.nuclphysa.2020.121958, https://doi.org/10.1016/j.nuclphysa.2020.121958
15. J. Khuyagbaatar, P. Mosat, J. Ballof, R. A. Cantemir, Ch. E. Düllmann, K. Hermainski, F. P. Heßberger, E. Jäger, B. Kindler, J. Krier, N. Kurz, S. Löchner, B. Lommel, B. Schausten, Y. Wei, P. Wieczorek, A. Yakushev, 'Stepping into the Sea of Instability: The New Sub-μs Superheavy Nucleus 252Rf', Physical Review Letters 134, 022501 (2025), DOI: 10.1103/PhysRevLett.134.022501, arXiv:2501.08955, https://doi.org/10.1103/PhysRevLett.134.022501
16. Yi Xie, Ning Wang, Zhongzhou Ren, 'Spontaneous fission half-lives for heavy and super-heavy nuclei from phenomenological models', arXiv:2512.21453 (2025), https://arxiv.org/abs/2512.21453
17. K. Pradeep Kumar, K. P. Santhosh, 'Synthesis of superheavy elements', Pramana – Journal of Physics 97, 47 (2023), DOI: 10.1007/s12043-023-02517-0, https://doi.org/10.1007/s12043-023-02517-0