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# Impact of pear-shaped fission fragments on mass-asymmetric fission in actinides

## Abstract

Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass splits1. 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 132Sn, which contains 50 protons and 82 neutrons. However, a systematic study of fission2 has shown that heavy fission fragments have atomic numbers distributed around Z = 52 to Z = 56, indicating that the strong shell effects in 132Sn 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 puzzle3. 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 144Ba (Z = 56)4,5, one of very few nuclei with shell-stabilized octupole deformation6. Using a quantum many-body model of superfluid fission dynamics7, 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 lead8.

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## 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

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.

Correspondence to Guillaume Scamps.

## Extended data figures and tables

1. ### 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) β3 ≈ 0.23–0.27 and β2 ≈ 0.15–0.27. These fragments are much more deformed than those produced by symmetric fission of 258Fm (see Extended Data Fig. 2), where symmetric Sn fragments are formed with β3 ≈ 0.11 at scission. We note that the light fragments also have octupole deformation with β3 ≈ 0.3–0.4 and quadrupole deformation with β2 ≈ 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).

2. ### 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 258Fm, the heavy fragment (with Z ≈ 55) starts with a strong octupole deformation (corresponding to deformation parameter β3 ≈ 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 246Cm fission (black solid line) also exhibits a substantial, yet smaller, deformation (β3 ≈ 0.19 at t = 0). By contrast, symmetric fission of 258Fm produces Sn fragments with a much smaller octupole moment (corresponding to β3 ≈ 0.11 at t = 0) that oscillates around Q30 = 0 (red solid line). These results are compatible with the calculated octupole deformation energy plotted in Fig. 3b, which shows that 138,140Xe (Z = 54) and 144Ba (Z = 56) are less resistant to octupole deformation than 134Te (Z = 52) and 132Sn (Z = 50).

3. ### Extended Data Fig. 3 Identification of the heavy pre-fragment in asymmetric fission of 258Fm.

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 258Fm (red lines, corresponding to calculation 8 in Extended Data Table 1) is found to form a 144Ba 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 258Fm (top half of each panel) and of the octupole-constrained 144Ba (bottom half). The use of the deformation of 144Ba 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 144Ba pre-fragment at times t = 0, 1.875, 3.75 and 4.65 zs (scission occurs at 7.3 zs) are β3 ≈ 0.14, 0.39, 0.39 and 0.42, respectively. Such strong octupole deformations could not be reached in the doubly magic 132Sn nucleus without a high deformation-energy cost (25 MeV for β3 ≈ 0.39), thus hindering the formation of this fragment. The fact that the densities and localization functions of deformed 144Ba match the heavy pre-fragment so well provides a clear signature of the influence of this pre-fragment before and at scission.

4. ### 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 230Th, 234U, 236U and 240Pu. In all four systems, the heavy fragment is identified as 144Ba with a constrained octupole deformation corresponding to β3 ≈ 0.28, 0.28, 0.27 and 0.44, respectively. The matching between deformed 144Ba 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.

5. ### 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 144Ba region than in the 132Sn region, we calculated the total energy of the system using a simple scission-point model40 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 β3 = 0.35 and the quadrupole deformation of the light fragment to be β2 = 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 e2Z1Z2/D with D = 17 fm, where Z1, Z2 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(NZ) 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 132Sn 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 240Pu and 246Cm 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 Zheavy ≈ 54 and overcome the effect of the spherical, doubly magic 132Sn.

6. ### 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 data29,41 (dots) for 240Pu, 250Cf and 234U. 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 234U (lowest red cross at Z = 46 in the middle panel). This calculation describes qualitatively the decrease of TKE for symmetric fission.

7. ### Extended Data Fig. 7 Effect of functional and pairing interaction on octupole deformation.

a, b, Deformation energy for 132Sn (a) and 144Ba (b) with different functionals and with pairing interaction strength V0 (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 144Ba. 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 β3 = 0.165, which is very close to the experimental value4 $${\beta }_{3}=0.1{7}_{-0.06}^{+0.04}$$.

## Supplementary information

1. ### 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 240Pu 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.

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### DOI

https://doi.org/10.1038/s41586-018-0780-0