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 massasymmetric 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 (pearshaped) deformations, which have been recently confirmed experimentally around ^{144}Ba (Z = 56)^{4,5}, one of very few nuclei with shellstabilized octupole deformation^{6}. Using a quantum manybody 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}.
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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 OakforestPACS at the JCAHPC in Tokyo, which is supported in part by the Multidisciplinary Cooperative Research Program in CCS, University of Tsukuba.
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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.
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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 octupoledeformed, 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 prefragment in asymmetric fission of ^{258}Fm.
a, The heavy prefragment is identified from its density contour using the technique of ref. ^{39} without the assumption of reflection symmetry in the prefragment. 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 prefragment with a strong octupole deformation (green lines, obtained from CHF+BCS; see Methods). b, Confirmation of the identification of the prefragment using the technique of refs ^{37,38} with a more general (that is, without assuming reflection symmetry in the prefragment) comparison of the proton (left column) and neutron (right column) localization functions of ^{258}Fm (top half of each panel) and of the octupoleconstrained ^{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 prefragment 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 deformationenergy 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 prefragment so well provides a clear signature of the influence of this prefragment before and at scission.
Extended Data Fig. 4 Identification of the heavy prefragment 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 prefragment 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 prefragment 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 scissionpoint 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 pointlike 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 finitetemperature 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 manybody 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 centreofmass correction and the Sly4d functional without the centreofmass 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 groundstate 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.
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Scamps, G., Simenel, C. Impact of pearshaped fission fragments on massasymmetric fission in actinides. Nature 564, 382–385 (2018). https://doi.org/10.1038/s4158601807800
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DOI: https://doi.org/10.1038/s4158601807800
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