Shell effects in fission and quasi-fission reactions

Quantum shell effects are responsible for asymmetric fission. They are also expected to affect the formation of fission fragments in quasi-fission reactions occurring in heavy-ion collisions. Systematic time-dependent Hartree-Fock simulations of 40–56Ca+176Yb collisions show that quasi-fission fragment properties share strong similarities with fragments formed in fission of the compound nuclei. This is an indication that similar shell effects are responsible for the final asymmtery in both mechanisms.


Introduction
Unlike fission that happens when a compound nucleus splits into two fragments, quasi-fission occurs in heavy collisions after significant mass-transfer between the fragments but without formation of an equilibrated compound system [1] (see [2] for a recent experimental review on quasifission).In both cases, however, the formation of fission fragments could be affected by quantum shell effects.In fission, shell effects are responsible for the formation of asymmetric fragments, while in quasi-fission, they are expected to stop the mass equilibration process.
Recent static and time-dependent mean-field studies of 226 Th fission modes and of 50 Ca+ 176 Yb reaction at an energy of 13% above the Coulomb barrier indicated that similar shell effects were expected both in asymmetric fission of the 226 Th compound nucleus and in quasi-fission [22].We first summarise results from this work before presenting new results for 40−56 Ca+ 176 Yb systems.

Fission modes of 226 Th
Experimental fission studies of 226 Th showed that it has two main fission modes, one asymmetric and one symmetric, both with similar yields [3][4][5].These observations are qualitatively well reproduced by the potential energy surface (PES) of Fig. 2, confirming the existence of two fission valleys.The PES is obtained from Q 20 and Q 30 constrained Skyrme Hartree-Fock (HF) calculations with BCS pairing correlations using the SkyAx code of [23].The SLy4d parametrisation [24]    is used.The mass asymmetric fission path (solid line) leads to Z H 54 protons in the final heavy fragments.The system may return to symmetric shapes (Q 30 = 0) for little additional cost in energy (dashed line), leading to symmetric elongated fragments.These results are in good agreement with theoretical predictions using other EDF (see, e.g., [25][26][27]), as well as with experimental observations indicating similar yields for both modes at low excitation energy [4; 5].The time-dependent Hartree-Fock (TDHF) approach has been extensively used to investigate quasifission mechanisms in the past decade [17; 28-36] (see [37][38][39][40] for recent reviews of TDHF applications to heavy-ion reactions).The approach has no free parameters except for the Skyrme functional whose parameters are usually not determined from fits to reaction calculations.Note that, although uncertainties in the parameters of the functional are expected to induce uncertainties in reaction observables [41], they are neglected in the present work.To account for the impact of the orientation of the prolately deformed 176 Yb (β 2 ≈ 0.33) on reaction mechanism [42], several orientations have been considered.An example of quasi-fission trajectory is shown in Fig. 3.The long time τ 23 zs is characteristic of the slow mass equilibration [28], and it is typical of quasi-fission reactions [1; 43].become constant for contact times greater than 13 zs.This stop of the mass equilibration process is interpreted as an influence of shell effects in the fragments.In particular, the heavy fragment is formed with Z H 54 protons, indicating a possible influence of the Z = 52 and 56 octupole deformed shell gaps (see Fig. 1) that were already invoked to interpret asymmetric fission in actinides [6].  22Th fission fragments (circles) from Ref. [4] and TDHF predictions of quasi-fission fragments TKE (triangles).From [22].
The total kinetic energy (TKE) of the final fragments can be obtained in TDHF by summing Coulomb and kinetic energies after scission [44] (see also [45; 46] for similar calculations of TKE in 258 Fm fission modes).The TKE resulting from the TDHF calculations of 50 Ca+ 176 Yb quasi-fission reactions are plotted in Fig. 5 and compared with experimental data from low-energy 226 Th fission [4].Similar TKE are observed for the asymmetric fission mode at Z H 54, which is an indication that the systems have similar shapes at scission in both mechanisms.This is further supported by the similar shapes near scission as shown in top-left and top right panels of Fig. 2.
As quantum shell effects are responsible for the valleys in the PES, one would expect that, if fission and quasi-fission are sensitive to the same shell effects, quasi-fission trajectories should probe the fission valleys.Naturally, PES are determined with conditions that do not necessary apply to quasi-fission, such as no orbital angular momentum, no excitation energy, and axial symmetry.Nevertheless, we see in Fig. 2 that a quasi-fission trajectory in the Q 20 − Q 30 plane ends up following the asymmetric fission path, which is an indication that it probes the same fission valley. 40−56 Ca+ 176 Yb collisions Systematic calculations with different neutron and/or proton numbers in the system allow to determine which of proton or neutron shell effects are the main driver to the formation of asymmetric fragments [3; 4].TDHF calculations similar to those reported in [22] have been performed with different calcium isotopes, thus varying the total number of neutrons in the compound system.Figure 6 shows the number of protons and neutrons found once mass equilibration has been stopped due to shell effects.The number of protons remains constant with Z H 54 while the number of neutrons increases with the total number of neutrons in the system.This is an indication that the final mass asymmetry is due to proton shell effects in the fragments.This observation is similar to what was obtained in fission experimental studies [3; 4], further confirming the similarities between both mechanisms in terms of the influence of quantum shell effects in the formation of final fragments.

Conclusions
TDHF simulations of 40−56 Ca+ 176 Yb collisions show that the mass equilibration that is characteristic of quasi-fission reactions stops when the heavy fragment reaches Z H 54 protons.This number of protons is similar to that of the heavy fragments formed in actinide asymmetric fission.In addition, a comparison of TKE in 50 Ca+ 176 Yb quasi-fission fragments (from TDHF) and 226 Th fission fragments (from experiment) indicates that the systems have similar shapes at scission.We conclude that, in these systems, similar shell effects are responsible for the final repartition of nucleons between the fragments in both fission and quasi-fission.The interpretation that octupole deformed shell effects at Z = 52 and 56 contribute to fixing this final asymmetry in fission then also applies to quasi-fission in these systems.

Q 20 Figure 1 .
Figure 1.Evolution of neutron (top) and proton (bottom) single-particle energies as a function of the quadrupole (Q 20 ; lower scale) and octupole (Q 30 ; upper scale) moments in 144 Ba.Adapted from [6].

30 Figure 2 .
Figure 2. Potential energy surface of 226 Th.The red solid line shows the fission path obtained by leaving the octupole moment free.The dashed line shows a possible path towards the symmetric valley.A density distribution (top-left) for the system along the asymmetric fission path is shown for near-scission deformation indicated by the grey arrow.An isodensity surface (top-right) at half the saturation density (ρ 0 /2 = 0.08 fm −3 ) is shown for a similar near-scission configuration (represented by star symbols in Fig. 3) obtained from a TDHF quasi-fission trajectory in 50 Ca+ 176 Yb.The blue dashed line shows a trajectory in the Q 20 − Q 30 plane from a new TDHF calculation at L = 60h.Adapted from [22].

Figure 3 .
Figure 3. Example of TDHF calculation of 50 20 Ca+ 176 70 Yb at E c.m. = 172 MeV with an orbital angular momentum L = 82h leading to quasi-fission fragments 87 35 Br+ 139 55 Cs.The surfaces represent the initial (blue) and final (red) isodensities at half the saturation density ρ 0 /2 = 0.08 fm −3 .The lines represent the evolution of the centres of masses of the fragments.The star symbols indicate the position on the trajectory used to represent the isodensity in Fig. 2 (top-right).The x and y scales correspond to the full numerical box.From [22].

Figure 4 .
Figure 4. (a) Proton (circles) and (b) neutron (squares) numbers in the heavy (open symbols) and light (filled symbols) fragments as a function of contact time.The dashed lines represent possible asymptotic values at Z H 54, Z L 36, N H 84, and N L 52.

Figure 6 .
Figure 6.TDHF calculations of the average number of protons and neutrons in the heavy fragments following40−56 Ca+ 176 Yb quasi-fission reactions with contact times exceeding 20 zs, as a function of the mass number of the compound system.