7 Fission fragments

The scission event is arguably one of the least understood processes in fission, although some experimental information on scission configurations have just became available Ramos et al. (2020). In a mean-field picture it marks the transition from the final state of the elongated fissioning nucleus to the initial state of the two separated nascent fission fragments. As discussed later in section 7.5, there are good reasons to think that the nascent fragments are entangled at, or immediately after, scission but it is not clear whether this entanglement persists to the stage of the fission fragments or quantum decoherence takes place. Furthermore, the characteristics of these fragments such as their charge, mass, energy, angular momentum, parity, level density, etc., are crucial ingredients in determining the properties of the neutron and $\gamma$ spectra, as well as the $\beta$-decay chains, see section 7.7.

Before microscopic time-dependent descriptions of fission dynamics became available, the scission event was most often treated with ad hoc assumptions, ignoring any role for dynamics. At one extreme, scission is assumed to transform the system into a statistical ensemble of two nuclei having their surfaces separated. In the framework of DFT theory, various criteria were introduced to define scission based on the HFB solution for the fissioning nucleus. The simplest ones define a threshold value for the density between the two pre-fragments or the expectation value of the neck moment. The main problem with such schemes, however, is that both the intrinsic energy of each pre-fragment and their relative interaction energy are extremely poorly described: before the separation of the pre-fragments, both the nuclear and Coulomb interaction energies are vastly overestimated because of the large overlap between the pre-fragments; when the primary fragments are well separated, the minimisation principle underpinning the HFB equation leads to the two fragments in their ground state. Such dramatic simplifications can be mitigated by performing unitary transformations on the total wave function of the fissioning system which, while leaving the whole system invariant, can be designed to minimise the interaction (or equivalently, maximise the localisation) of the pre-fragments Younes and Gogny (2011). In spite of the development of such techniques, it is clear that explicitly non-adiabatic, time-dependent methods provide a much better handle on scission – at least when it comes to defining the initial conditions of the nascent fragments.

Three main methods are used to determine the yields of different fission fragments. Scission-point models assume a statistical distribution of probability among a set of scission configurations of the nucleus Fong (1953); Wilkins et al. (1976), see Lemaître et al. (2019) and Paşca et al. (2019) for recent realisations. These models require the definition of an ensemble of scission configurations that can either be determined by constrained mean-field calculations or from an analytical parameterisation of the shapes of the di-nuclear system. Each of these nuclear configurations is then populated according to a Boltzmann distribution, with the temperature defined in accordance with the initial energy of the system. Since it is computationally effective, this method is used in systematic studies or to investigate the evolution of yields with the excitation energy of the compound nucleus. However, the choice of the ensemble of scission configurations remains arbitrary and may influence the resulting yields. Moreover, an explicit use of temperature for a non-adiabatic and time-dependent process is not really well justified.

To some degree, also the total fragment kinetic energy may be estimated. In particular, some models have sought to predict those quantities exclusively on the basis of the scission configurations Lemaître et al. (2015), but their predictive power has been limited due to the importance of the collective path taken prior to scission. Indeed, experience with a diffusive transport model Randrup et al. (2011) has shown that not only the scission hypersurface but the global topography of the PES may have a qualitative influence on the outcome.

To avoid the assumption of statistical equilibrium at scission, one possibility is to describe the evolution of the compound nucleus from some initial state at lower deformation up to the configurations close to scission. In this approach, one defines an equation of motion for a few collective coordinates associated with a parameterised shape of the nuclear surface. Assuming that these collective degrees of freedom interact with a thermal bath of intrinsic excitations, leads to the Langevin equations in the deformation space; see section 3.5.

These transport equations have been solved in multi-dimensional spaces both in MM and hybrid-DFT frameworks either directly Miyamoto et al. (2019); Sierk (2017); Ishizuka et al. (2017); Usang et al. (2019); Sadhukhan et al. (2016, 2017); Matheson et al. (2019) or in the strongly damped (Smoluchowski) limit Randrup and Möller (2011); Randrup et al. (2011); Ward et al. (2017). The result is the probability of populating different nuclear configurations close to scission and it is then straightforward to determine the resulting mass asymmetry.

Most often these treatments have concentrated on the mass number, assuming that the proton-to-neutron ratio remains constant, but recent progress has been made Möller et al. (2014); Möller and Ichikawa (2015); Möller and Schmitt (2017); Sadhukhan et al. (2016, 2017); Matheson et al. (2019) towards including also the isospin degree of freedom, thus making it possible to determine both the mass and the charge yields. The transport framework takes into account the dissipative effects of the collective dynamics and may even account for the emission of neutrons in the course of the evolution Eslamizadeh and Raanaei (2018). One general limitation is that the Langevin treatment is restricted to the classically allowed region of the collective space, so it cannot treat tunnelling.

As mentioned in section 3.4.1, an alternative approach to computing fission fragment yields within a quantal description is the TDGCM Regnier et al. (2019); Zhao et al. (2019); Younes et al. (2019); Verriere and Regnier (2020). Here, a major difficulty is the determination of a proper manifold of states which usually consists of an ensemble of quadrupole and octupole constrained HFB solutions. While this description of quantum collective dynamics can treat tunnelling, it fails to include diabatic aspects of the dynamics close to scission. Let us also mention recent TDHF Simenel and Umar (2014); Goddard et al. (2015, 2016), TDHF+BCS Scamps et al. (2015); Scamps and Simenel (2018) and TDHFB Bulgac, Jin, Roche, Schunck and Stetcu (2019) studies of fission. In these cases, the initial configurations for the time-dependent calculations are generated by constrained calculations at some elongation beyond the outer turning point. These methods are well suited to investigate the role of shell effects at scission Scamps and Simenel (2018, 2019), and thus provide valuable guidance to more phenomenological models like the scission-point models discussed earlier in this section. As mentioned earlier in section 3.3, time-dependent theories will be challenged to reproduce the tails of the yield distribution, due to non-Newtonian Langevin trajectories, unless a mechanism equivalent to the random force of the Langevin equation is included Aritomo et al. (2014); Sadhukhan et al. (2017). Moreover, the present formulation does not allow for the treatment of quantum tunnelling.

All these state-of-the-art methods have their own strengths and weaknesses. Yet, they all rely on determining the probabilities to populate a set of scission configurations. A common feature in all these approaches is that the fragment yields are computed at scission, where the two nascent fragments still interact through the nuclear force, see section 7.1. As a result, estimates of particle number with projection methods, for example, become extremely sensitive to the condition that define scission configurations. Other observables such as energy sharing between nascent fragments, may not be relevant at this stage of the fission process. Methods should be developed to determine the yields of observables further away from scission.

Current methods have been mostly focused on the yields associated with the mass, charge, and sometimes TKE of the fragments. To go beyond this simple picture, the challenge is to extend the space of configurations in the fission channel in order to be able to make quantitative predictions of correlated yields for these three observables, and eventually additional ones. The new observables of interest are typically the angular momentum and parity of the nascent fragments.

Finally, the prediction of fission yields is essential for a correct description of $r$-process nucleosynthesis and superheavy elements. It would, therefore, be important to carry out systematic large-scale calculations of fission yields in regions of the nuclear chart far from the valley of stability. While such large scale calculations present a serious challenge for computationally intensive models of fission dynamics, some recent progress in this direction has been reported in Giuliani et al. (2018); Lemaître et al. (2018); Giuliani et al. (2019); Rodríguez-Guzmán et al. (2020).

The estimation of $Y(Z,A)$ is usually based on the assumption that the probability density of the mass and charge of the fragments associated with a Bogoliubov wave function is Gaussian. However, in order to describe the odd-even staggering seen in charge distributions more refined methods are required.

In the MM DDD approach, the odd-even effect in fission yields can be attributed to the pair-breaking effect Mirea (2014). In order to assign a particle number to a pre-fragment in the vicinity of the scission configuration, a condition has been introduced in Mirea (2011) based on the position of the neck. A similar approach to particle number identification was proposed in the DFT approach of Younes and Gogny (2011) using the unitary transformations on the total wave function aiming at maximising the localisation of pre-fragments.

Even earlier, a method has been proposed in Simenel (2010) to estimate the exact probability distribution of mass and charge in a nascent fragment created in microscopic models by introducing the particle-number projection for fragments. This method has been applied to determine the transfer probabilities of nucleons during collision reactions and then generalised to superfluid system Scamps and Lacroix (2013) with the use of the Pfaffian method Robledo (2009), and applied to fission Scamps et al. (2015) in TDHF+BCS, thus showing that the odd-even effects can be described with the mean-field dynamics. As discussed in section 3.12.1, these distributions are affected by the lack of one-body fluctuations and correlations (e.g., between mass and charge distributions). As shown in Simenel (2011); Williams et al. (2018); Godbey et al. (2020), the latter can be recovered to some extend for symmetric systems using the TDRPA Balian and Vénéroni (1984).

It will be interesting to couple this approach with configuration-mixing methods and semi-classical descriptions of the fission process. One should also go beyond the approximation of identical occupation of time-reversed canonical HFB states assumed in Verriere et al. (2019) to see whether the proper blocking of one-quasiparticle states in odd-$A$ nascent fragments, associated with breaking of time reversal symmetry, is important for the description of odd-even staggering of fission yields.

Most of the energy released in fission appears in the form of TKE of the fission fragments. Hence, a direct inverse correlation exists between TKE and their total excitation energy available for prompt neutron and gamma emission. Moreover, the distribution of TKE directly influences the prompt neutron multiplicity distribution, which has been measured in a few cases and is important in transport simulations of selected classes of integral experiments.

Once the nascent fragments are separated at scission, the Coulomb repulsion is transformed into kinetic energy. As indicated in section 3.8, however, different models predict different values for the collective kinetic energy at scission. It is typically a few MeV in TDDFT and ranging from zero to 20 MeV in various transport treatments. From a theoretical point of view a tolerance of 20 MeV, representing about 10-15% relative uncertainty, might be deemed acceptable. However, a change of TKE by that much would significantly change the multiplicity of evaporated neutrons (by about two) and it is therefore an important challenge to fission theory to improve on the calculation of TKE.

The available total excitation energy in fission fragments can be calculated from the energy balance in a fission event, knowing the masses of the fissioning system and of the fission fragments, once those are determined via a chosen theoretical model, or extracted from systematics of experimental data. For any model that does not fully separate the fission fragments, the extraction of the energy sharing will be subject to large uncertainties, as energy can flow from one pre-fragment to the other through the neck, and in close proximity the nascent fragments exchange energy via Coulomb interactions. Moreover, the nascent fragments are generally distorted relative to their equilibrium shapes and the associated distortion energy will be converted to additional primary fragment excitation energy, thus affecting the resulting energy sharing.

Guidance on excitation energy partitioning is necessary for simulating neutron and photon emissions when the total excitation energy in the fissioning system increases (as in the case of fission induced by fast neutrons). The only indirect observable related to the excitation-energy sharing is the average number of neutrons per fission event as a function of mass, but the data beyond thermal neutron-induced fission and spontaneous fission reactions are scarce. The results of average neutron multiplicity measurements as a function of mass for significantly different excitation energies in the fissioning systems, suggest that with increasing energy in the fissioning (actinide) system, most of the extra energy is deposited in the heavy fragments Müller et al. (1984); Naqvi et al. (1986).

It has to be emphasised that the process of energy sharing poses an interesting and nontrivial question: does the energy sharing occur in the condition of thermal equilibrium that has developed between nascent fragments? In this case, the details of the process of neck formation and subsequent fracture would be of secondary importance, and only the density of states associated with each of the nascent fragments would play a role. On the other hand, if equilibrium is not reached during the saddle-to-scission evolution the details of the splitting process will be crucial. This issue is still not resolved. Namely, the TDDFT method, which has recently been used to parameterise the energy dependence of the excitation-energy sharing, predicts neutron multiplicities as a function of mass in agreement with experimental observations Bulgac, Jin, Roche, Schunck and Stetcu (2019); Bulgac et al. (2020). At the same time, an approach that models the excitation-energy sharing statistically on the basis of the microscopic level densities within a Brownian shape evolution framework, was also able to reproduce the experimental trend Albertsson et al. (2020).

Spontaneous fission of even-even nuclei is a process by which a $0^{+}$ quantum system decays into two excited nuclei which eventually, after prompt neutron and photon emission, turn into two product nuclei in their ground states, moving apart with opposite momenta. In this sense, the process is analogous to the emission of two electrons from a singlet state, with the additional complication that in fission, neutrons and gamma rays are emitted at or beyond the scission point.

The fission process conserves quantum numbers and, therefore, those that characterise the initial state, such as particle number Bulgac and Jin (2017) and angular momentum, must be shared among all particles and quanta in the exit channel. For example, neglecting neutron emission, the final state would be a superposition of states of the two fragments with numbers of protons and neutrons in one fragment complementing those in the other fragment, so that they add up to the number of protons and neutrons of the initial fissioning nucleus. The particle numbers of the fragments are therefore entangled, and a measurement in one fragment collapses the information about the particle numbers in the other fragment.

The same is true for the measurement of $\gamma$ rays, which are characteristic of a given nucleus and thus uniquely define the other fragment. Summing up the angular momenta of gamma rays emitted from one primary fragment collapses the information about the angular momentum of the other fragment in the exit channel. In the same way the angular-momentum polarisations of the two fragments are also entangled. The real question is whether these effects are ultimately important for experiment? Can they be observed at all? When and how does decoherence of this entanglement occur? Fission fragments may represent a unique opportunity to explore quantum entanglement of mesoscopic systems, that is, they can be the closest realisation of the Schrödinger cat phenomenon Dobaczewski (2019).

An essential ingredient in the microscopic description of fission is the PES, constructed in an intrinsic frame and including pairing correlations in the BCS or HFB approximations, see section 3.2.1. Both of these break symmetries of the underlying Hamiltonian, and their restoration yields additional contributions to the PES; see discussion in section 6.1.

The restoration of particle number has a relatively small effect on the PESs in actinide nuclei, which are far from closed shells Bernard et al. (2019). However, it may have a significant impact on the ATDHFB or GCM inertias that determine the collective Hamiltonians; see section 3.10. In particular, extensive studies are needed in two specific areas. First, one should consistently calculate the PES for symmetry-restored wave functions in the case of particle number projection, possibly by including variation after projection to determine the intrinsic states. Second, and perhaps most important, is the development of a consistent theory of collective inertias for the symmetry-restored wave functions. Recent developments in the description and manipulation of particle number projected HFB states (called Antisymmetrized Germinal Power in the quantum chemistry literature) Dukelsky et al. (2019) could potentially lead to a particle-number-projected ATDHFB theory.

The angular momenta of fragments is an important element that determines neutron yields and other decay properties Wilhelmy et al. (1972). Given a mean-field description of the nucleus and the knowledge of its quasiparticle excitation energies, current theoretical tools can be used to calculate the angular momentum content of the fragments. There are two components to the angular momentum of the newly formed fragments: non-collective and collective.

The non-collective angular momentum is carried by the quasiparticles in the pre-scission configuration that are transferred to the post-scission nascent fragments under diabatic conditions. They will end up in one or the other nascent fragment, depending on the evolution of the corresponding orbitals with elongation; see Bertsch, Younes and Robledo (2019) for an example of this transition. Their angular momentum is conserved, allowing one to estimate its contribution to that of the nascent fragment. The collective angular momentum arises because of the deformation of the compound nucleus. This component can be calculated by well-known projection techniques used to compute rotational bands in deformed nuclei. The only difference in the case of the fissioning system is that scission also affects the angular momentum of the system with respect to the orientation of the fission axis. The collective contribution to the angular momentum of the nascent fragments about the fission axis vanishes. As a result, the distribution of gamma radiation in the subsequent cascade will be anisotropic with respect to this axis Bertsch, Kawano and Robledo (2019). In fact, this anisotropy has been observed in spontaneous fission, and a systematic measurement would provide an invaluable test of our overall understanding of the dynamics at the scission point.

There can be additional angular momentum generated as the pre-fragments separate, due to higher multipole components of the Coulomb field between them, see Bertsch (2019). It is, in fact, straightforward to calculate the effect of the electric quadrupole field on the post-scission nascent fragments, given their deformations and their initial separations. It would therefore be useful to have this information available when reporting fission calculations going through the scission point.

Nascent fragments emerge with significant excitation energies and then primary fragments cool down via various decay modes resulting in particle emission (neutrons and photons from prompt and delayed emission as well as electrons and antineutrinos from $\beta$ decays). In current phenomenological approaches, the neutron emission proceeds after the nascent fragments have fully accelerated becoming primary fragments, whereupon they are treated as compound nuclei that de-excite via particle emissions.

In order to carry out simulations of those decay chains, it is necessary to know the initial states of the primary fragments, in particular their initial excitation energy, angular momentum, and parity. On the other hand, experimental information regarding the fission fragments can only be obtained after neutron emission. Hence, few experimental data can inform phenomenological models, and the microscopic models can play an important role in providing the necessary input for a large range of reactions. The angular momentum of the emerging fission fragments is an important quantity that sets the competition between the neutron and $\gamma$ emission, and it has an important influence on a variety of photon observables, from prompt fission $\gamma$ multiplicity to the prompt fission spectrum and correlations between emitted photons. To a lesser degree, it can also influence the delayed neutron and $\gamma$ properties.

To reduce uncertainties, the angular momentum properties of the fission fragments should be investigated in a framework that allows the total separation of the nascent fragments. It has been demonstrated in TDDFT that scission is followed by a relaxation period in which the nascent fragments transition to a deformation of primary fragments that is close to the ground-state deformation Bulgac, Jin, Roche, Schunck and Stetcu (2019), thereby increasing the energy available for emission.

Both $\gamma$ and $\beta$ decays of the primary fragments can be described using the QRPA, thus defining a consistent framework both for the entrance and exit channels, see section 5. Because $\beta$-decay half lives are long compared to those of $\gamma$ decays, they generally can be assumed to occur from the fragment ground state, thus making a finite-temperature description of $\beta$ decay unnecessary. But it is important to take into account that the $\beta$ decays may generally populate excited states in the daughter fragment, which would then undergo their own emission chain before a subsequent $\beta$ decay could occur. Consequently, $\gamma$ decay should be investigated for each primary fragment as a prompt phenomenon (in principle in competition with but usually after neutron evaporation). $\beta$ decay to the resulting ground-state fission products should also be investigated, including forbidden transitions of particular importance for neutrino studies.