## Abstract

Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass splits^{1}. Without quantum shell effects, which can give extra binding energy to their mass-asymmetric shapes, these nuclei would fission symmetrically. The strongest shell effects appear in spherical nuclei, such as the spherical ‘doubly magic’ (that is, both its atomic and neutron numbers are ‘magic’ numbers) nucleus ^{132}Sn, which contains 50 protons and 82 neutrons. However, a systematic study of fission^{2} has shown that heavy fission fragments have atomic numbers distributed around *Z* = 52 to *Z* = 56, indicating that the strong shell effects in ^{132}Sn are not the only factor affecting actinide fission. Reconciling the strong spherical shell effects at *Z* = 50 with the different *Z* values of fission fragments observed in nature has been a longstanding puzzle^{3}. Here we show that the final mass asymmetry of the fragments is also determined by the extra stability provided by octupole (pear-shaped) deformations, which have been recently confirmed experimentally around ^{144}Ba (*Z* = 56)^{4,5}, one of very few nuclei with shell-stabilized octupole deformation^{6}. Using a quantum many-body model of superfluid fission dynamics^{7}, we find that heavy fission fragments are produced predominantly with 52 to 56 protons, which is associated with substantial octupole deformation acquired on the way to fission. These octupole shapes, which favour asymmetric fission, are induced by deformed shells at *Z* = 52 and *Z* = 56. By contrast, spherical magic nuclei are very resistant to octupole deformation, which hinders their production as fission fragments. These findings may explain surprising observations of asymmetric fission in nuclei lighter than lead^{8}.

## Access options

Subscribe to Journal

Get full journal access for 1 year

$199.00

only $3.90 per issue

All prices are NET prices.

VAT will be added later in the checkout.

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

## Additional information

**Publisher’s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- 1.
Andreyev, A. N., Nishio, K. & Schmidt, K.-H. Nuclear fission: a review of experimental advances and phenomenology.

*Rep. Prog. Phys*.**81**, 016301 (2018). - 2.
Schmidt, K.-H. et al. Relativistic radioactive beams: a new access to nuclear-fission studies.

*Nucl. Phys. A***665**, 221–267 (2000). - 3.
Schmidt, K.-H. & Jurado, B. Review on the progress in nuclear fission – experimental methods and theoretical descriptions.

*Rep. Prog. Phys*.**81**, 106301 (2018). - 4.
Bucher, B. et al. Direct evidence of octupole deformation in neutron-rich

^{144}Ba.*Phys. Rev. Lett*.**116**, 112503 (2016). - 5.
Bucher, B. et al. Direct evidence for octupole deformation in

^{146}Ba and the origin of large*E*1 moment variations in reflection-asymmetric nuclei.*Phys. Rev. Lett*.**118**, 152504 (2017). - 6.
Gaffney, L. P. et al. Studies of pear-shaped nuclei using accelerated radioactive beams.

*Nature***497**, 199–204 (2013). - 7.
Scamps, G., Simenel, C. & Lacroix, D. Superfluid dynamics of

^{258}Fm fission.*Phys. Rev. C***92**, 011602 (2015). - 8.
Andreyev, A. N. et al. New type of asymmetric fission in proton-rich nuclei.

*Phys. Rev. Lett*.**105**, 252502 (2010). - 9.
Möller, P. & Randrup, J. Calculated fission-fragment yield systematics in the region 74 ≤

*Z*≤ 94 and 90 ≤*N*≤ 150.*Phys. Rev. C***91**, 044316 (2015). - 10.
Sadhukhan, J., Nazarewicz, W. & Schunck, N. Microscopic modeling of mass and charge distributions in the spontaneous fission of

^{240}Pu.*Phys. Rev. C***93**, 011304 (2016). - 11.
Schunck, N. & Robledo, L. M. Microscopic theory of nuclear fission: a review.

*Rep. Prog. Phys*.**79**, 116301 (2016). - 12.
Simenel C. & Umar, A. S. Formation and dynamics of fission fragments.

*Phys. Rev. C***89**, 031601 (2014). - 13.
Goddard, P. M., Stevenson, P. D. & Rios, A. Fission dynamics within time-dependent Hartree–Fock: deformation-induced fission.

*Phys. Rev. C***92**, 054610 (2015). - 14.
Bulgac, A., Magierski, P., Roche, K. J. & Stetcu, I. Induced fission of

^{240}Pu within a real-time microscopic framework.*Phys. Rev. Lett*.**116**, 122504 (2016). - 15.
Tanimura, Y., Lacroix, D. & Ayik, S. Microscopic phase-space exploration modeling of

^{258}Fm spontaneous fission.*Phys. Rev. Lett*.**118**, 152501 (2017). - 16.
Bulgac, A., Jin, S., Roche, K., Schunck, N. & Stetcu, I. Fission dynamics. Preprint at https://arxiv.org/abs/1806.00694 (2018).

- 17.
Möller, P., Madland, D. G., Sierk, A. J. & Iwamoto, A. Nuclear fission modes and fragment mass asymmetries in a five-dimensional deformation space.

*Nature***409**, 785–790 (2001). - 18.
Carjan, N., Ivanyuk, F. A. & Oganessian, Y. T. Pre-scission model predictions of fission fragment mass distributions for super-heavy elements.

*Nucl. Phys. A***968**, 453–464 (2017). - 19.
Becke, A. D. & Edgecombe, K. E. A simple measure of electron localization in atomic and molecular systems.

*J. Chem. Phys*.**92**, 5397–5403 (1990). - 20.
Reinhard, P.-G., Maruhn, J. A., Umar, A. S. & Oberacker, V. E. Localization in light nuclei.

*Phys. Rev. C***83**, 034312 (2011). - 21.
Jerabek, P., Schuetrumpf, B., Schwerdtfeger, P. & Nazarewicz, W. Electron and nucleon localization functions of oganesson: approaching the Thomas–Fermi limit.

*Phys. Rev. Lett*.**120**, 053001 (2018). - 22.
Dasgupta, M., Hinde, D. J., Rowley, N. & Stefanini, A. M. Measuring barriers to fusion.

*Annu. Rev. Nucl. Part. Sci*.**48**, 401–461 (1998). - 23.
Butler, P. & Nazarewicz, W. Intrinsic reflection asymmetry in atomic nuclei.

*Rev. Mod. Phys*.**68**, 349–421 (1996). - 24.
Robledo, L. M. & Bertsch, G. F. Global systematics of octupole excitations in even-even nuclei.

*Phys. Rev. C***84**, 054302 (2011). - 25.
Butler, P. A. Octupole collectivity in nuclei.

*J. Phys. G***43**, 073002 (2016). - 26.
Hulet, E. K. et al. Bimodal symmetric fission observed in the heaviest elements.

*Phys. Rev. Lett*.**56**, 313–316 (1986). - 27.
Unik, J. P., Glendenin, L. E., Flynn, K. F., Gorski, A. & Sjoblom, R. K. Fragment mass and kinetic energy distributions for fissioning systems ranging from mass 230 to 256. In

*Proc. of the Third International Atomic Energy Agency Symposium on the Physics and Chemistry of Fission*, Vol. II 19–45 (IAEA, 1974). - 28.
Böckstiegel, C. et al. Nuclear-fission studies with relativistic secondary beams: Analysis of fission channels.

*Nucl. Phys. A***802**, 12–25 (2008). - 29.
Caamaño, M. et al. Characterization of the scission point from fission-fragment velocities.

*Phys. Rev. C***92**, 034606 (2015). - 30.
Brown, D. A. et al. ENDF/B-VIII.0: the 8th major release of the nuclear reaction data library with cielo-project cross sections, new standards and thermal scattering data.

*Nucl. Data Sheets***148**, 1–142 (2018). - 31.
Kim, K.-H., Otsuka, T. & Bonche, P. Three-dimensional TDHF calculations for reactions of unstable nuclei.

*J. Phys. G Nucl. Phys*.**23**, 1267–1273 (1997). - 32.
Leander, G. A., Nazarewicz, W., Olanders, P., Ragnarssonn, I. & Dudek, J. A new region of intrinsic reflection asymmetry in nuclei around

^{145}Ba?*Phys. Lett. B***152**, 284–290 (1985). - 33.
Scamps, G. & Lacroix, D. Effect of pairing on one- and two-nucleon transfer below the Coulomb barrier: A time-dependent microscopic description.

*Phys. Rev. C***87**, 014605 (2013). - 34.
Bonche, P., Flocard, H. & Heenen, P. H. Solution of the Skyrme HF+BCS equation on a 3D mesh.

*Comput. Phys. Commun*.**171**, 49–62 (2005). - 35.
Scamps, G., Lacroix, D., Bertsch, G. F. & Washiyama, K. Pairing dynamics in particle transport.

*Phys. Rev. C***85**, 034328 (2012). - 36.
Simenel, C. & Umar, A. S. Heavy-ion collisions and fission dynamics with the time-dependent Hartree–Fock theory and its extensions.

*Prog. Part. Nucl. Phys*.**103**, 19–66 (2018). - 37.
Zhang, C. L., Schuetrumpf, B. & Nazarewicz, W. Nucleon localization and fragment formation in nuclear fission.

*Phys. Rev. C***94**, 064323 (2016). - 38.
Sadhukhan, J., Zhang, C., Nazarewicz, W. & Schunck, N. Formation and distribution of fragments in the spontaneous fission of

^{240}Pu.*Phys. Rev. C***96**, 061301 (2017). - 39.
Warda, M., Staszczak, A. & Nazarewicz, W. Fission modes of mercury isotopes.

*Phys. Rev. C***86**, 024601 (2012). - 40.
Wilkins, B. D., Steinberg, E. P. & Chasman, R. R. Scission-point model of nuclear fission based on deformed-shell effects.

*Phys. Rev. C***14**, 1832 (1976). - 41.
Böckstiegel, C.

*Bestimmung der Totalen Kinetischen Energien in der Niederenergiespaltung Neutronenarmer Radioaktiver Isotope*. PhD thesis, TU Darmstadt (1998).

## Acknowledgements

We thank B. Jurado, A. Chatillon and F. Farget for useful discussions at the early stage of this work. We are grateful to D. J. Hinde for continuous support to this project. We thank M. Caamaño for providing references to experimental data. B. Jurado and D. J. Hinde are also thanked for their careful reading of the manuscript. This work has been supported by the Australian Research Council under grant number DP160101254. The calculations were performed in part at the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government, in part using the COMA system at the CCS in the University of Tsukuba, which is supported by the HPCI Systems Research Projects (project hp180041), and using the Oakforest-PACS at the JCAHPC in Tokyo, which is supported in part by the Multidisciplinary Cooperative Research Program in CCS, University of Tsukuba.

## Author information

### Affiliations

### Authors

### Contributions

G.S. and C.S. conceived the project. G.S. performed the numerical simulations. G.S. and C.S. discussed the results. C.S. wrote the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Guillaume Scamps.

## Extended data figures and tables

### Extended Data Fig. 1 Scission configurations.

Isodensity surface and neutron localization just before scission (about 0.1 zs before the neck breaks), used for calculations with different actinides in their asymmetric fission valleys. At scission, all the heavy fragments (left) have octupole deformation parameters (see Methods)

*β*_{3}≈ 0.23–0.27 and*β*_{2}≈ 0.15–0.27. These fragments are much more deformed than those produced by symmetric fission of^{258}Fm (see Extended Data Fig. 2), where symmetric Sn fragments are formed with*β*_{3}≈ 0.11 at scission. We note that the light fragments also have octupole deformation with*β*_{3}≈ 0.3–0.4 and quadrupole deformation with*β*_{2}≈ 0.4–0.8. Such large quadrupole deformation of the light fragment is often found at scission in microscopic calculations (see, for example, figure 4 of ref.^{38}).### Extended Data Fig. 2 Octupole deformation after scission.

Octupole moment (see Methods) in the heavy fragment as a function of time, with a time reference (

*t*= 0) corresponding to the time at which scission occurs in the calculations. In asymmetric fission of^{258}Fm, the heavy fragment (with*Z*≈ 55) starts with a strong octupole deformation (corresponding to deformation parameter*β*_{3}≈ 0.25 at*t*= 0) and remains octupole-deformed, possibly with different orientations (blue dashed and green dotted lines). The fragment with*Z*≈ 52 resulting from^{246}Cm fission (black solid line) also exhibits a substantial, yet smaller, deformation (*β*_{3}≈ 0.19 at*t*= 0). By contrast, symmetric fission of^{258}Fm produces Sn fragments with a much smaller octupole moment (corresponding to*β*_{3}≈ 0.11 at*t*= 0) that oscillates around*Q*_{30}= 0 (red solid line). These results are compatible with the calculated octupole deformation energy plotted in Fig. 3b, which shows that^{138,140}Xe (*Z*= 54) and^{144}Ba (*Z*= 56) are less resistant to octupole deformation than^{134}Te (*Z*= 52) and^{132}Sn (*Z*= 50).### Extended Data Fig. 3 Identification of the heavy pre-fragment in asymmetric fission of

^{258}Fm.**a**, The heavy pre-fragment is identified from its density contour using the technique of ref.^{39}without the assumption of reflection symmetry in the pre-fragment. Proton (left column) and neutron (right column) densities are shown with a difference of 0.01 fm^{−3}between contour lines. The fissioning asymmetric system^{258}Fm (red lines, corresponding to calculation 8 in Extended Data Table 1) is found to form a^{144}Ba pre-fragment with a strong octupole deformation (green lines, obtained from CHF+BCS; see Methods).**b**, Confirmation of the identification of the pre-fragment using the technique of refs^{37,38}with a more general (that is, without assuming reflection symmetry in the pre-fragment) comparison of the proton (left column) and neutron (right column) localization functions of^{258}Fm (top half of each panel) and of the octupole-constrained^{144}Ba (bottom half). The use of the deformation of^{144}Ba as a constraint is chosen to reproduce the nucleon localization function close to the centre of the heavy fragment. The resulting octupole deformations of the^{144}Ba pre-fragment at times*t*= 0, 1.875, 3.75 and 4.65 zs (scission occurs at 7.3 zs) are*β*_{3}≈ 0.14, 0.39, 0.39 and 0.42, respectively. Such strong octupole deformations could not be reached in the doubly magic^{132}Sn nucleus without a high deformation-energy cost (25 MeV for*β*_{3}≈ 0.39), thus hindering the formation of this fragment. The fact that the densities and localization functions of deformed^{144}Ba match the heavy pre-fragment so well provides a clear signature of the influence of this pre-fragment before and at scission.### Extended Data Fig. 4 Identification of the heavy pre-fragment in asymmetric fission of actinides.

**a**,**b**, Same as Extended Data Fig. 3, at configurations around scission for asymmetric fission of^{230}Th,^{234}U,^{236}U and^{240}Pu. In all four systems, the heavy fragment is identified as^{144}Ba with a constrained octupole deformation corresponding to*β*_{3}≈ 0.28, 0.28, 0.27 and 0.44, respectively. The matching between deformed^{144}Ba densities and localization functions with the heavy pre-fragment confirms the strong influence of octupole shell effects associated with*Z*= 56 and*N*= 88 on asymmetric fission.### Extended Data Fig. 5 Effect of octupole deformation of the heavy pre-fragment on total energy at scission.

**a**, To understand why the formation of a fragment is energetically more favourable in the^{144}Ba region than in the^{132}Sn region, we calculated the total energy of the system using a simple scission-point model^{40}for various mass and charge repartitions between the fragments, each system being characterized by the number of protons*Z*and neutrons*N*in one fragment, and with the typical deformations of the fragments observed (in our TDBCS calculations; see Methods) at scission. For simplicity, we only constrain the octupole deformation of the heavy fragment to be*β*_{3}= 0.35 and the quadrupole deformation of the light fragment to be*β*_{2}= 0.6–0.8. The binding energy of each deformed fragment is then computed from CHF+BCS simulations (see Methods) and added to the Coulomb energy between the fragments, which is approximated by the point-like formula*e*^{2}*Z*_{1}*Z*_{2}/*D*with*D*= 17 fm, where*Z*_{1},*Z*_{2}are the atomic numbers of the fragments,*D*is their distance and*e*is the electron charge. (As we are only interested in comparisons between different mass and charge repartitions, the strong nuclear interaction energy between the fragments is neglected because it is not expected to vary much.) The total energy*E*(*N*,*Z*) is then plotted with its minimum value as the reference energy for each system. We note that this is a simple model that does not account for finite-temperature effects, which could potentially dampen shell effects. However, damping of shell effects is expected to occur at higher excitation energies than those involved here. Despite the simplicity of this model, the*Z*and*N*values of the fragments obtained from the TDBCS calculations, shown by red dots, are clearly distributed around the system with minimum energy.**b**, Same as**a**, but without the constraint on the octupole deformation of the heavy fragment (only the quadrupole deformation of the light fragment is constrained). In this case, the formation of^{132}Sn is energetically favoured. This shows that the octupole deformation of the heavy fragment induced in the fission process strongly hinders the impact of spherical shell effects at scission.**c**, Experimental^{240}Pu and^{246}Cm independent fission yield (number of fragments produced after emission of prompt neutrons, but excluding radioactive decay per 100 fission reactions) from ref.^{30}compared to the mean*Z*and*N*values obtained from TDBCS calculations (black crosses). These figures show that taking into account the octupole deformation energy leads to a preference for the fragments to be formed with*Z*_{heavy}≈ 54 and overcome the effect of the spherical, doubly magic^{132}Sn.### Extended Data Fig. 6 Total kinetic energy of the fission fragments.

TKE values obtained from TDBCS calculations (red crosses) are compared with average TKEs from experimental data

^{29,41}(dots) for^{240}Pu,^{250}Cf and^{234}U. As expected from the complexity of many-body dynamics, the results exhibit strong fluctuations (typically a variation of 15–20 MeV between the lowest and highest TKE for each nucleus; that is, of the same order as the experimental fluctuations of TKE). Nevertheless, the TKE values predicted by our TDBCS calculations are essentially distributed around the average experimental TKE, indicating very good agreement between theory and experiment. For consistency, we have calculated the TKE value of a symmetric fission mode of^{234}U (lowest red cross at*Z*= 46 in the middle panel). This calculation describes qualitatively the decrease of TKE for symmetric fission.### Extended Data Fig. 7 Effect of functional and pairing interaction on octupole deformation.

**a**,**b**, Deformation energy for^{132}Sn (**a**) and^{144}Ba (**b**) with different functionals and with pairing interaction strength*V*_{0}(see Methods) varying by ±10%. The Sly4 and Skm* functionals with the centre-of-mass correction and the Sly4d functional without the centre-of-mass correction give similar deformation energy curves. The pairing interaction can slightly change the octupole deformation of the ground state of^{144}Ba. The Sly4d functional with the normal pairing interaction (that is, with the pairing interaction strengths defined in Methods) predicts a ground-state octupole deformation of*β*_{3}= 0.165, which is very close to the experimental value^{4}\({\beta }_{3}=0.1{7}_{-0.06}^{+0.04}\).

## Supplementary information

### Supplementary Video 1: Simulation of asymmetric fission dynamics

The video is in mpg format and it is 14 seconds long. It shows a simulation of fission dynamics for a

^{240}Pu nucleus with an initial quadrupole moment of 45.5b. The evolution of an isodensity surface and of the neutron localisation function are shown as in Fig. 1.

## Rights and permissions

To obtain permission to re-use content from this article visit RightsLink.

## About this article

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.