Superfluid fission dynamics with microscopic approaches

Recent progresses in the description of the latter stage of nuclear fission are reported. Dynamical effects during the descent of the potential towards scission and in the formation of the fission fragments are studied with the time-dependent Hartree-Fock approach with dynamical pairing correlations at the BCS level. In particular, this approach is used to compute the final kinetic energy of the fission fragments. Comparison with experimental data on the fission of 258Fm are made.


Introduction
Despite important progresses since the discovery of nuclear fission in 1939 [1,2], it remains an important challenge for theorists. Compared with other nuclear dynamical processes such as fusion, the overall time-scale for fission is relatively long. This suggests that the evolution across the potential energy surface is a slow process. However, it is also expected that the systems could encounter rapid shape evolution in the later stages near scission. Theoretical prediction of the fission fragments and their characteristics are often based on the adiabatic approximation [3]. It is assumed that the internal degrees of freedom are equilibrated while the system evolves along the fission path. However, the shape evolution near scission is expected to be non-adiabatic and the approximation may break down in the latter stage of fission [4,5]. Thus, the inclusion of the dynamical effects near scission are crucial. In particular, the dynamics has been shown to generate most of the excitation energy in the fragments [6].
The importance of the dynamical effects is investigated in fission of 258,264 Fm [6,25]. It is shown that dynamics has an important effect on the scission configuration and on the kinetic and excitation energies of the fragments. The vibrational modes of the fragments in the postscission evolution are also analyzed. Quantum shell effects are shown to play a crucial role in the dynamics and formation of the fragments.

Adiabatic calculations
The adiabatic evolution is described in the traditional way by minimizing the HF energy under an external constraint inducing elongation of the system up to the scission point. This constraint can be on the distance R between the centres of mass of the fragments (defined assuming a sharp cut at the neck), or on multipole moments. The static Hartree-Fock equation is solved with BCS pairing correlations using the ev8 code [38]. The SLy4d parametrization [39] of the Skyrme functional [40] is used with a surface pairing interaction [41]. The calculations are performed on a Cartesian grid with mesh size 0.8 fm.
The adiabatic potential has been calculated in Ref. [6] for 264 Fm and in Ref. [25] for 258 Fm. The results are reported on Figs. 1 and 2 for 264 Fm and 258 Fm, respectively. In both cases, the resulting barrier is in good agreement with other calculations [9,10,17,42,43].
An important question is which configuration should we choose as initial state of the time-dependent calculations. A possible choice would be to start with a configuration where the fragments have established their identity. the fragments are thought to be pre-formed well before scission occurs (Younes and Go crucial as it is expected to trigger non-adiabatic effects in the further evolution. Howev the pre-formation of the fragments from the density of the system and one has to look transition. As proposed by the CI (Simenel and Umar 2014), the best way to identify t microscopic structure of the system. This is possible only in fully microscopic approac particular, the distribution of single-particle energies is expected to converge toward th final fragments. An example is shown in Figure 7 where the evolution of the proton sing for the same fission reaction as in Fig. 1 and 5. In this case, the 264 Fm nucleus fission fragments with 50 protons each. We recognise in Fig. 7 the spherical magic gaps assoc numbers. These gaps, and in particular that at Z=50, are present well before scission w Fig. 5). This proves that the fragments are pre-formed well before scission and that n included in the pre-scission dynamics. Figure 7: Evolution of proton single-particle ene function of the distance R between the fragments 132 Sn. The potential energy from the same calcul recognise the nuclear magic numbers in the she The green dashed line indicates the Fermi level.
This novel idea of using the internal structure to i non-adiabatic and adiabatic regimes will be gen systems such as those including pairing cor consistent treatment of both adiabatic and non which has been lacking so far in previous microsc

Time evolution and final properties of the fragments
The main outcome of this project will be to describe the properties of fission and qu include the distribution of their mass and charge, as well as their excitation and kinet momentum. Although TDHF is optimised for the expectation value of one-body obser 1984), nucleon number distributions of the outgoing fragments can be calculated using technique (Simenel, 2010). This is because TDHF is a fully microscopic approach description of the gross features of the dynamics as shown in Fig. 1 and 6, but also give of each single-particle wave function. Recently, the CI significantly improved the theor mass and charge distributions by applying a prescription from Balian and Vénéroni, prescription is also known as the time-dependent RPA (TDRPA). These theoretical tools will be adapted to the fission studies to get a deep insight i fragments. Systematic calculations and comparisons with experimental mass and obtained at the Heavy-Ion Accelerator Facility will provide a fundamental understan shells on the formation of the final fragments. Similar comparison between experiment to understand the role of dissipation in the dynamics, in particular in the repartition of th the fragments.
To describe the non-adiabatic phase of fission, one has to start the time-dependent calculation under one or several external constraints at the transition between adiabat (see previous section). A recent exemple of such a calculation is shown in Fig. 1.
Note that the resulting sequence of shapes around scission presents differences with the ( Fig  The distance between the fragments R is defined in Ref. [6]. (top) Adiabatic potential and isodensities at half the saturation density ρ 0 /2 = 0.08 fm −3 . The energy is defined with respect to the asymptotic final 132 Sn+ 132 Sn state. Single-particle levels are plotted for protons and neutrons in the middle and bottom panels, respectively. Positive and negative parity states are shown in red and blue, respectively. Adapted from [6]. tial energy surface. In a second step, the non-adiabatic descent of the potential towards scission is determined using the time-dependent HF equations with dynamical pairing correlations (TDHF+BCS). The properties of the fragments, in particular their mass, charge, and kinetic energy, are then computed after scission and compared with experimental data from [26]. The mean-field is obtained with the Sly4d [29] Skyrme energy density functional and a constant-G interaction in the pairing channel. The CHF+BCS and TDHF+BCS calculations are obtained with modified versions of the ev8 [30] and tdhf3d [29] codes, respectively, assuming only one plane of symmetry. All calculations are performed on a Cartesian grid of 88⇥19.2⇥19.2 fm 3 with a mesh size 0.8 fm. The time evolution is obtained with a the descent at higher e Let us n sion proces of the pote lations hav along these compact co single part a two-body [18,31]. T phase are r Fig. 2. We tant role in fragments i calculation while in th place and lead to th is likely to fragment m It is also in quire di↵er ⇠ 2 zs, ⇠ modes, res due to a co slopes and to depend The final is another distinguish batic appro scission con energy surf vantage of to include This may be difficult to determine from the total density. However, it is possible to see where fragments are preformed from the single-particle levels [6]. In the case of 264 Fm, the proton and neutron single-particle energies are plotted in the middle and bottom panels of Fig. 1, respectively. Many levels cross at short distance (R < 10 Fm). We also observe sudden jumps in the single-particle energies due to changes of macroscopic shapes. At larger distances, we observe the appearance of energy gaps associated with the spherical magic numbers. As the 132 Sn fragments are doubly magic, there is no crossing of the Fermi surface after the magic gaps at Z = 50 and N = 82 are both formed. This occurs at R ∼ 10.5 fm.
The symmetric fission mode of 258 Fm is also driven by spherical shell effects [44]. Indeed, the fragments have a proton magic number Z = 50. The evolution of proton and neutron single-particle levels are shown in Fig. 3 for the symmetric mode. As in the 264 Fm case, we observe the formation of Z = 50 and N = 82 magic gaps after some global elongation of the fissioning system (quantified by the quadrupole moment) has been reached. However, in the case of neutrons, the levels near the Fermi surface (represented with a dashed line) are only partially filled.
The case of 258 Fm asymmetric fission is more complicated as the fragments are not magic and therefore can be deformed. Indeed, the associated single-particle energies plotted in Fig. 4 show no energy gap neither for protons nor for neutrons. Therefore, it is not always straightforward to identify the pre-formation of the fragments simply by looking at the formation of spherical magic gaps.

TDHF+BCS calculations
In general, it is necessary to consider a range of initial conditions in order to investigate the dynamical evolution along the fission path with time-dependent calculations. Starting with a more compact configuration is desirable in order to capture most of the dynamical effects in the evolution along the fission path. However, the TDHF+BCS approach accounts only for part of the correlations and it is likely that some correlations will be missing for the most compact configurations. The computational time is also a practical limitation. However, starting with a configuration too close to scission, we would miss some of the dynamical effects. In particular, the final observables (mass and charge distributions, total kinetic energy...) may exhibit a dependence with the initial configuration when the latter is too close to scission [25,26]. A compromise has to be found by investigating several initial conditions. As an example, we found that the final properties of the fragments in 258 Fm asymmetric fission were essentially independent on the initial condition if the latter was chosen to be in the range 210 b< Q 20 < 270 b (see Fig. 2). The sensitivity to the initial condition (in TDHF calculations without dynamical pairing) has also recently been investigated in details by Goddard and collaborators in ref. [26].
The importance of dynamical pairing correlations in the evolution was demonstrated in Ref. [25] at the BCS level. The time-dependent BCS equations, giving the evolution of the single-particle occupation numbers, were solved simultaneously with the TDHF equation for the evolution of the single-particle wave-functions. An example of the resulting dynamics of the occupation numbers is shown in Fig. 5 for the symmetric fission mode of 258 Fm. An important rearrangement of the neutron occupation numbers is observed during the evolution. As a re- sult, the time variation of the occupation numbers induces an evolution of the pairing energy [25]. Accounting for dynamical pairing is also crucial for fission to occur. This is illustrated in Fig. 6 which shows the evolution of the quadrupole moment as a proxy for the total elongation of the system. In the frozen occupation approximation (FOA), i.e., assuming that the single-particle numbers are those of the initial configuration, the system is not always able to fission. In addition, trajectories leading to fission in the FOA exhibit a significant dependence of the final observables with the choice of the initial condition [26].
An interesting feature of the dynamical calculations is the possibility to predict the total kinetic energy. As discussed above, the latter is essentially independent of the initial condition if it is not too close to scission and if pairing correlations are included dynamically. The total kinetic energy of the fragments is computed from the sum of their Coulomb repulsion and kinetic energy. In particular, these quantities are well defined after scission as the nuclear interaction between the fragments vanishes due to its short range nature. Indeed, we see in Fig. 6 that the total energy is constant after scission. It is equal to the final total kinetic energy (TKE) as the Coulomb energy is transformed into kinetic energy at large distances. Figure 7 shows a comparison of the predicted fragment mass (top) and TKE (bottom) [25] with experimental distributions [44] in 258    data present a tail in the mass distribution interpreted as an asymmetric mode in competition with the dominant symmetric mode [44]. The calculations along the asymmetric valley predict masses in agreement with this tail. The calculated TKE for both modes also agree with the two main contributions in the experimental distribution, namely a peak at high TKE and a tail at low energy. According to the calculations, the high TKE mode is associated with the  symmetric compact fission which is sensitive to the spherical shell effects in the 132 Sn region, while the asymmetric mode leads to a much lower TKE. This interpretation agrees with the measured correlations between masses and TKE of the fragments [44]. Finally, the time-dependent calculations can also be used to investigate what forms of excitation energy are present in the fragments. The excitation energy is expected to be shared between deformation, collective vibration and rotation, and non-collective excitations [45]. The TDHF approach has been widely used to study giant resonances [46][47][48][49][50][51][52][53][54][55][56][57]. More recently, it has also been used to investigate low-lying collective vibrations [56,58,59], in particular to study their effects on fusion [58][59][60][61].
As an illustration of application to study vibration in fission fragments, the quadrupole and octupole moments computed in the fragments of 264 Fm symmetric fission are shown in Fig. 8 [6]. A comparison with RPA calculations of vibrational modes built on the 132 Sn ground state shows that the low-lying 2 + and 3 − are both excited in the fission fragments. However, the high energy modes, i.e., the giant quadrupole resonance (GQR) and the high-energy octupole resonance (HEOR), are not populated. Similar effects are observed in fusion where the dynamics is dominated by low-lying collective excitations [62].

Conclusions
The dynamics of the fission process in 258,264 Fm has been investigated near scission at the mean-field level with the time-dependent Hartree-Fock approach including dynamical pairing correlations. It is shown that the spherical magic gaps in compact symmetric modes leading to tin fragments are formed well before scission, indicating a pre-formation of the fragments. Accounting for the timedependence of the pairing correlations, leading to an evolution of the occupation numbers, is crucial to allow the system to fission and to reduce the dependence with the initial condition. The results are used to interpret the experimental mass and kinetic energy distributions. It is shown that at least part of the internal excitation energy is stored into low-lying collective vibrations, while, as in fusion, the dynamics is less coupled to high energy modes.
Quantum fluctuations beyond the independent particle/quasi-particle picture need to be incorporated in the future in order to reproduce the experimental distributions. The mass and charge distributions in the final fragments of a TDHF evolution can be calculated with a particle number projection technique [63]. However, these fluctuations are underestimated at the mean-field level [64]. A possible approach is to incorporate beyond mean-field fluctuations with the TDRPA [65] which has already been applied to heavy-ion collisions [66][67][68]. Applications to fission (neglecting pairing correlations) are promising [25]. However, numerical solution of the TDRPA including pairing still has to be done.